"Business, Sauder School of"@en . "DSpace"@en . "UBCV"@en . "Kusy, Martin"@en . "2010-03-05T21:47:45Z"@en . "1978"@en . "Doctor of Philosophy - PhD"@en . "University of British Columbia"@en . "The inherent uncertainty of a bank's cash flows, cost of funds and return on investment, along with the increased variability of economic conditions during the past decade, have emphasized the need for greater efficiency in the management of a bank's assets and liabilities. A consequence has been an increased number of studies on how to structure a bank's assets and liabilities so that an \"optimal\" trade-off exists between risk, return and liquidity. Except for the Bradley and Crane (BC) model, the solution techniques proposed in the literature are computationally tractable only if uncertainty is ignored. Unfortunately, the BC model is not operationally appealing due to severe computational limitations, and a number of undesirable formulation features (such as the restricted feasible region for first period decisions). Given these deficiencies in the literature, the purposes of this dissertation are to develop an asset and liability management model (ALM) that is computationally tractable for large realistic problems and to demonstrate that this model is superior to existing models.\r\nThe ALM model developed in this dissertation is a stochastic linear program with simple recourse (SLPR). This model incorporates the following essential features of asset and liability management: 1) the stochastic nature of the problem (by utilizing a set of random cash flows (deposits) with a given discrete probability distribution),\r\n2) simultaneous consideration of assets and liabilities, 3) transactions costs, and 4) multi-periodicity.\r\nThe ALM model was applied to Vancouver City Savings Credit Union's asset and liability management for a five year planning period in order to demonstrate the effort necessary to implement the model. Computational tractability for this large problem was maintained by using Wets' algorithm for solving SLPR. A simulation was run on a real (uncertain) environment to compare the decision making effectiveness of the solutions generated by the SLPR and stochastic dynamic programming (SDP) models.\r\nThe findings of this dissertation are: 1) the ALM model is superior to an equivalent deterministic model, 2) the solution of the ALM model is sensitive to the asymmetry of the probability distributions of the cash flows, 3) the effort required for the implementation of the ALM model is comparable to that of an equivalent deterministic model, 4) the SLPR formulation is computationally superior to the SDP formulation utilized by Bradley and Crane, and 5) the simulation indicates that the SLPR formulation results in a better initial period decision than the SDP formulation (this is due to the restrictions imposed by the SDP formulation of maintaining feasibility for all possible forecasted economic scenarios for the first period decision)."@en . "https://circle.library.ubc.ca/rest/handle/2429/21551?expand=metadata"@en . "BANK ASSET AND LIABILITY MANAGEMENT b y M a r t i n K u s y B . C o m m . , S i r G e o r g e W i l l i a m s U n i v e r s i t y , 1 9 6 9 M . B . A . S U n i v e r s i t y o f W i n d s o r , 1 9 7 0 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S FOR T H E D E G R E E OF T H E F A C U L T Y O F G R A D U A T E S T U D I E S F a c u l t y o f C o m m e r c e a n d B u s i n e s s A d m i n i s t r a t i o n We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d T H E U N I V E R S I T Y OF B R I T I S H C O L U M B I A M a y 1 9 7 8 - . M a r t i n K u s y , 1 9 7 8 DOCTOR OF P H I L O S O P H Y i n I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may b e g r a n t e d b y t h e H e a d o f my D e p a r t m e n t o r b y h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t b e a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . M a r t i n K u s y D e p a r t m e n t o f C o m m e r c e a n d B u s i n e s s A d m i n i s t r a t i o n T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook Place Vancouver, Canada V6T 1W5 D a t e M a y 1 6 , 1 9 7 8 ABSTRACT T h e i n h e r e n t u n c e r t a i n t y o f a b a n k ' s c a s h f l o w s , c o s t o f f u n d s a n d r e t u r n o n i n v e s t m e n t , a l o n g w i t h t h e i n c r e a s e d v a r i a b i l i t y o f e c o n o -m i c c o n d i t i o n s d u r i n g t h e p a s t d e c a d e , h a v e e m p h a s i z e d t h e n e e d f o r g r e a t e r e f f i c i e n c y i n t h e m a n a g e m e n t o f a b a n k ' s a s s e t s a n d l i a b i l i t i e s . A c o n s e q u e n c e h a s b e e n a n i n c r e a s e d n u m b e r o f s t u d i e s o n how t o s t r u c t u r e a b a n k ' s a s s e t s a n d l i a b i l i t i e s s o t h a t a n \" o p t i m a l \" t r a d e - o f f e x i s t s b e t w e e n r i s k , r e t u r n a n d l i q u i d i t y . E x c e p t f o r t h e B r a d l e y a n d C r a n e ( B C ) m o d e l , t h e s o l u t i o n t e c h n i q u e s p r o p o s e d i n t h e l i t e r a t u r e a r e c o m p u t a t i o n a l l y t r a c t a b l e o n l y i f u n c e r t a i n t y i s i g n o r e d . U n f o r t u n a t e l y , t h e BC m o d e l i s n o t o p e r a t i o n a l l y a p p e a l i n g d u e t o s e v e r e c o m p u t a t i o n a l l i m i t a t i o n s , a n d a n u m b e r o f u n d e s i r a b l e f o r m u l a t i o n f e a t u r e s ( s u c h a s t h e r e s t r i c t e d f e a s i b l e r e g i o n f o r f i r s t p e r i o d d e c i s i o n s ) . G i v e n t h e s e d e f i c i e n c i e s i n t h e l i t e r a t u r e , t h e p u r p o s e s o f t h i s d i s s e r t a t i o n a r e t o d e v e l o p a n a s s e t a n d l i a b i l i t y m a n a g e m e n t m o d e l ( A L M ) t h a t i s c o m p u -t a t i o n a l l y t r a c t a b l e f o r l a r g e r e a l i s t i c p r o b l e m s a n d t o d e m o n s t r a t e t h a t t h i s m o d e l i s s u p e r i o r t o e x i s t i n g m o d e l s . T h e A L M m o d e l d e v e l o p e d i n t h i s d i s s e r t a t i o n i s a s t o c h a s t i c l i n e a r p r o g r a m w i t h s i m p l e r e c o u r s e ( S L P R ) . T h i s m o d e l i n c o r p o r a t e s t h e f o l l o w i n g e s s e n t i a l f e a t u r e s o f a s s e t a n d l i a b i l i t y m a n a g e m e n t : 1 ) t h e s t o c h a s t i c n a t u r e o f t h e p r o b l e m ( b y u t i l i z i n g a s e t o f r a n d o m c a s h f l o w s ( d e p o s i t s ) w i t h a g i v e n d i s c r e t e p r o b a b i l i t y d i s t r i b u t i o n ) , - i -2 ) s i m u l t a n e o u s c o n s i d e r a t i o n o f a s s e t s a n d l i a b i l i t i e s , 3 ) t r a n s a c t i o n s c o s t s , a n d 4 ) m u l t i - p e r i o d i c i t y . T h e A L M m o d e l w a s a p p l i e d t o V a n c o u v e r C i t y S a v i n g s C r e d i t U n i o n ' s a s s e t a n d l i a b i l i t y m a n a g e m e n t f o r a f i v e y e a r p l a n n i n g p e r i o d i n o r d e r t o d e m o n s t r a t e t h e e f f o r t n e c e s s a r y t o i m p l e m e n t t h e m o d e l . C o m p u t a t i o n a l t r a c t a b i l i t y f o r t h i s l a r g e p r o b l e m w a s m a i n t a i n e d b y u s i n g W e t s ' a l g o r i t h m f o r s o l v i n g S L P R . A s i m u l a t i o n w a s r u n o n a r e a l ( u n c e r t a i n ) e n v i r o n m e n t t o c o m p a r e t h e d e c i s i o n m a k i n g e f f e c t i v e n e s s o f t h e s o l u t i o n s g e n e r a t e d b y t h e S L P R a n d s t o c h a s t i c d y n a m i c p r o g r a m m i n g ( S D P ) m o d e l s . T h e f i n d i n g s o f t h i s d i s s e r t a t i o n a r e : 1 ) t h e A L M m o d e l i s s u p e r i o r t o a n e q u i v a l e n t d e t e r m i n i s t i c m o d e l , 2 ) t h e s o l u t i o n o f t h e A L M m o d e l i s s e n s i t i v e t o t h e a s y m m e t r y o f t h e p r o b a b i l i t y d i s t r i b u t i o n s o f t h e c a s h f l o w s , 3 ) t h e e f f o r t r e q u i r e d f o r t h e i m p l e m e n t a t i o n o f t h e A L M m o d e l i s c o m p a r a b l e t o t h a t o f a n e q u i v a l e n t d e t e r m i n i s t i c m o d e l , 4 ) t h e S L P R f o r m u l a t i o n i s c o m p u t a t i o n a l l y s u p e r i o r t o t h e S D P f o r m u l a -t i o n u t i l i z e d b y B r a d l e y a n d C r a n e , a n d 5 ) t h e s i m u l a t i o n i n d i c a t e s t h a t t h e S L P R f o r m u l a t i o n r e s u l t s i n a b e t t e r i n i t i a l p e r i o d d e c i s i o n t h a n t h e S D P f o r m u l a t i o n ( t h i s i s d u e t o t h e r e s t r i c t i o n s i m p o s e d b y t h e S D P f o r m u l a t i o n o f m a i n t a i n i n g f e a s i b i l i t y f o r a l l p o s s i b l e f o r e c a s t e d e c o n o m i c s c e n a r i o s f o r t h e f i r s t p e r i o d d e c i s i o n ) . - i i -T A B L E OF C O N T E N T S P a g e C h a p t e r 1 - I n t r o d u c t i o n 1 . 1 O v e r v i e w o f D i s s e r t a t i o n . 1 1 . 2 D e f i n i t i o n s 4 1 . 3 T h e o r y o f F i n a n c i a l I n t e r m e d i a t i o n 5 1 . 4 A p p r o p r i a t e C r i t e r i o n f o r A s s e t a n d L i a b i l i t y M a n a g e m e n t 8 1 . 5 E s s e n t i a l F e a t u r e s o f a n A s s e t a n d L i a b i l i t y M a n a g e m e n t M o d e l T h a t M a x i m i z e s E x p e c t e d N e t R e t u r n s 12 1 . 6 I m p o r t a n c e o f A s s e t a n d L i a b i l i t y M a n a g e m e n t 1 3 1 . 7 O r g a n i z a t i o n o f t h e D i s s e r t a t i o n , 1 4 C h a p t e r 2 - R e v i e w o f L i t e r a t u r e 2 . 1 I n t r o d u c t i o n 1 5 2 . 2 D e t e r m i n i s t i c M o d e l s 16 2 . 3 S t o c h a s t i c M o d e l s 2 2 C h a p t e r 3 - F o r m a l D e s c r i p t i o n o f t h e A s s e t a n d L i a b i l i t y M a n a g e m e n t ( A L M ) M o d e l 3 . 1 I n t r o d u c t i o n 3 2 3 . 2 F o r m u l a t i o n o f t h e A L M M o d e l 3 4 3 . 3 U s e o f t h e A L M M o d e l 4 7 3 . 4 A p p e n d i x O n e 5 0 3 . 5 A p p e n d i x Two 5 8 C h a p t e r 4 - I m p l e m e n t a t i o n o f t h e A L M M o d e l 4 . 1 I n t r o d u c t i o n 9 3 4 . 2 M o d e l D e t a i l s 9 7 4 . 3 R e s u l t s o f t h e V a n c o u v e r C i t y S a v i n g C r e d i t U n i o n A p p l i c a t i o n 1 0 8 4 . 4 A p p e n d i x O n e 1 1 3 - i i i -P a g e C h a p t e r 5 - A C o m p a r i s o n o f S t o c h a s t i c Dynattri c P r o g r a m m i n g a n d S t o c h a s t i c - L i n e a r P r o g r a m m i n g w i t h S i m p l e R e c o u r s e M o d e l s a s D e c i s i o n T o o l s 5 . 1 I n t r o d u c t i o n . . . . . . . . 1 6 8 5 . 2 S c e n a r i o f o r t h e S i m u l a t i o n 1 7 4 5 . 3 F o r m u l a t i o n o f t h e S t o c h a s t i c D y n a m i c P r o g r a m m i n g M o d e l 1 7 6 5 . 4 F o r m u l a t i o n o f t h e S L P R M o d e l 181 5 ; 5 R e s u l t s o f t h e S i m u l a t i o n 1 8 3 i 5 . 6 A p p e n d i x O n e 1 8 6 C h a p t e r 6 - S u m m a r y , M a j o r F i n d i n g s a n d D i r e c t i o n s f o r F u r t h e r R e s e a r c h 6 . 1 I n t r o d u c t i o n 2 1 4 6 . 2 S u m m a r y 2 1 4 6 . 3 M a j o r F i n d i n g s 2 1 5 6 . 4 D i r e c t i o n s f o r F u r t h e r R e s e a r c h 2 1 7 B i b l i o g r a p h y 2 > 8 . - i v -ACKNOWLEDGEMENTS I w o u l d l i k e t o t h a n k t h e F a c u l t y o f C o m m e r c e f o r p r o v i d i n g t h e s u p p o r t t o f a c i l i t a t e t h e c o m p l e t i o n o f my d i s s e r t a t i o n . I n p a r t i c u l a r , I am i n d e b t e d t o my s u p e r v i s o r , D r . W . T . Z i e m b a , f o r h i s e n c o u r a g e m e n t , a c u m e n a n d a v a i l a b i l i t y , a n d a l s o f o r h i s g u i d a n c e i n a l l a s p e c t s o f my a c a d e m i c d e v e l o p m e n t . I am g r a t e f u l t o D r . R . W . W h i t e , f o r e n c o u r a g i n g me t o s t u d y b a n k a s s e t a n d l i a b i l i t y m a n a g e m e n t . I am a l s o g r a t e f u l t o a l l m e m b e r s o f my d i s s e r t a t i o n c o m m i t t e e , D r . W . T . Z i e m b a , D r . R . W . W h i t e , D r . C . S a r n d a l , D r . L . G . M i t t e n a n d D r . W . E . D i e w e r t , f o r p r o v i d i n g c o n s t r u c t i v e c r i t i c i s m a n d a d v i c e . I w o u l d a l s o l i k e t o t h a n k D r . R . J - B . W e t s f o r b o t h d e v e l o p i n g a n d p r o v i d i n g a c c e s s t o a n a l g o r i t h m f o r s o l v i n g s t o c h a s t i c l i n e a r p r o g r a m s w i t h s i m p l e r e c o u r s e . A l s o , I w o u l d l i k e t o t h a n k V a n c o u v e r C i t y S a v i n g s C r e d i t U n i o n ( i n p a r t i c u l a r M r . H o o k a n d t h e l a t e M r . B e n t l e y ) f o r p r o v i d i n g me w i t h d a t a f o r t h e i m p l e m e n t a t i o n o f t h e A L M m o d e l . I w o u l d l i k e t o t h a n k M s . C h a n , F o n g , H a l l e r a n d M i l l e r f o r t h e e x c e l l e n t t y p i n g o f t h i s d i s s e r t a t i o n . I w o u l d l i k e t o t h a n k a l l my f e l l o w g r a d u a t e s t u d e n t s f o r p r o v i d i n g a c o n d u c i v e a c a d e m i c e n v i r o n m e n t . I n p a r t i c u l a r , I w o u l d l i k e t o t h a n k A . A m e r s h i , V . V . B a b a , D . K i r a a n d S . L a r s s o n . F i n a l l y , I w o u l d l i k e t o t h a n k t w o s p e c i a l f r i e n d s J e r r y K a i l b e r g a n d L a w r e n c e K r y z a n o w s k i f o r t h e i r e n c o u r a g e m e n t , a d v i c e a n d m o r a l s u p p o r t . - v -C h a p t e r 1 INTRODUCTION 1 . 1 O v e r v i e w - o f D i s s e r t a t i o n T h e i n h e r e n t u n c e r t a i n t y o f a b a n k ' s c a s h f l o w s , c o s t o f f u n d s a n d r e t u r n o n i n v e s t m e n t s , a l o n g w i t h t h e u n s e t t l e d e c o n o m i c c o n d i t i o n s o f t h e p a s t d e c a d e , h a v e e m p h a s i z e d t h e n e e d f o r a g r e a t e r e f f i c i e n c y i n t h e m a n a g e m e n t o f a b a n k ' s a s s e t s a n d l i a b i l i t i e s . A c o n s e q u e n c e h a s b e e n a n i n c r e a s e d n u m b e r o f s t u d i e s o n how t o s t r u c t u r e a b a n k ' s a s s e t s a n d l i a b i l i t i e s s o t h a t a n \" o p t i m a l \" t r a d e - o f f e x i s t s b e t w e e n r i s k , r e t u r n a n d l i q u i d i t y [ 7 , 1 1 , 2 0 , 7 0 ] . T h e s e s t u d i e s f o c u s s e d o n t h e d e t e r m i n a t i o n o f - t h e u s e o f f u n d s g i v e n e i t h e r d e t e r m i n i s t i c o r s t o c h a s t i c e c o n o m i c s c e n a r i o s . F a c t o r s t h a t m u s t b e c o n s i d e r e d i n t h e s e d e c i s i o n s i n c l u d e : t h e b a l a n c i n g o f a n t i c i p a t e d s o u r c e s a n d u s e s o f f u n d s t o m e e t l i q u i d i t y a n d c a p i t a l a d e q u a c y c o n s t r a i n t s w h i l e c o n c u r r e n t l y m a x i m i z i n g p r o f i t a b i l i t y [ 1 1 , 2 0 ] , a l l o c a t i n g f u n d s a m o n g a s s e t s b a s e d o n c l a s s i f i c a t i o n , m a t u r i t i e s a n d r a t e s o f r e t u r n [ 5 , 6 ] , a n d a d j u s t i n g a b a n k ' s f i n a n c i a l s t r u c t u r e i n t e r m s o f l i q u i d i t y , c a p i t a l a d e q u a c y a n d l e v e r a g e [ 1 1 , 2 0 ] . C u r r e n t r e s e a r c h h a s s t r e s s e d t w o a p p r o a c h e s . T h e f i r s t a p p r o a c h , b a s e d o n M a r k o w i t z ' s t h e o r y o f p o r t f o l i o s e l e c t i o n , a s s u m e s t h a t r e t u r n s a r e n o r m a l l y d i s t r i b u t e d a n d t h a t b a n k m a n a g e r s a r e r i s k - a v e r s e u t i l i t y 1 2 o f w e a l t h m a x i m i z e r s [ 5 9 , 7 0 ] . I n s u c h a w o r l d , t h e v a l u e o f a n a s s e t d e p e n d s n o t o n l y o n t h e e x p e c t a t i o n a n d v a r i a n c e o f i t s r e t u r n b u t a l s o o n t h e c o v a r i a n c e o f i t s r e t u r n w i t h t h e r e t u r n s o f a l l o t h e r e x i s t i n g a n d p o t e n -t i a l i n v e s t m e n t s . T h e s e c o n d a p p r o a c h a s s u m e s t h a t a b a n k s e e k s t o m a x i m i z e i t s f u t u r e s t r e a m o f p r o f i t s s u b j e c t t o p o r t f o l i o m i x c o n s t r a i n t s [ 1 1 , 2 0 ] . T h e s o l u t i o n t e c h n i q u e s a d v a n c e d b y t h e p r o p o n e n t s o f b o t h a p p r o a c h e s a r e e i t h e r c o m p u t a t i o n a l l y t r a c t a b l e i f t h e y d o n o t c a p t u r e t h e e s s e n t i a l f e a t u r e s o f t h e a s s e t a n d l i a b i l i t y m a n a g e m e n t p r o b l e m [ 1 0 , 1 9 ] o r c o m p u t a t i o n a l l y i n t r a c t a b l e i f t h e y a t t e m p t t o c a p t u r e t h e e s s e n t i a l f e a t u r e s o f t h e a s s e t a n d l i a b i l i t y m a n a g e m e n t p r o b l e m [ 5 , 2 8 , 3 6 ] . A s a n e x a m p l e o f t h e f i r s t s h o r t c o m i n g , C o h e n a n d H a m m e r ' s [ 2 0 ] l i n e a r p r o g r a m m i n g a s s e t m a n a g e m e n t m o d e l i s c o m p u t a t i o n a l l y f e a s i b l e f o r l a r g e p r o b l e m s b u t i s n e i t h e r s t o c h a s t i c i n n a t u r e n o r d o e s i t c o n s i d e r t h e e f f e c t o f t h e c h o i c e ^ o f a s s e t i n s t r u m e n t s o n t h e t o t a l p o r t f o l i o o f t h e b a n k ' s a s s e t s a n d l i a b i l i t i e s . A s a n e x a m p l e o f t h e s e c o n d s h o r t c o m i n g , W o l f ' s [ 9 8 ] s e q u e n t i a l d e c i s i o n t h e o r e t i c m o d e l i n c l u d e s m a n y o f t h e f e a t u r e s i n h e r e n t i n t h e a s s e t a n d l i a b i l i t y p r o b l e m , b u t i s c o m p u t a t i o n a l l y i n f e a s i b l e f o r r e a l i s t i c p r o b l e m s t h a t a r e s t o c h a s t i c i n n a t u r e . T h e s e t w o a p p r o a c h e s s h o w t h a t t h e d i l e m m a e n c o u n t e r e d i n d e v e l o p i n g a n a s s e t a n d l i a b i l i t y m o d e l i s t h e t r a d e -o f f b e t w e e n c o m p u t a t i o n a l t r a c t a b i l i t y a n d r e a l i s m . T h e B r a d l e y a n d C r a n e ( B - C ) m o d e l a t t e m p t s i n a s e r i o u s m a n n e r t o c o p e w i t h t h e a b o v e d i l e m m a [ 5 , 6 , 7 ] . E s s e n t i a l l y , t h e i r m o d e l i s a d e c i s i o n t r e e s t h a t i s f o r m u l a t e d a s a l i n e a r p r o g r a m . B - C h a v e d e v e l o p e d a d e c o m p o s i t i o n a l g o r i t h m t h a t t a k e s a d v a n t a g e o f t h e s p e c i a l s t r u c t u r e o f t h e i r f o r m u l a t i o n . T h e i r m o d e l h a s a n u m b e r o f a p p e a l i n g f e a t u r e s : 3 i t i s d y n a m i c i n n a t u r e ; i t i n c o r p o r a t e s t h e u n c e r t a i n t y o f c a s h f l o w s a n d i n t e r e s t r a t e s ; a n d i t i s c o m p u t a t i o n a l l y t r a c t a b l e f o r p r o b l e m s o f l i m i t e d s i z e . H o w e v e r , t h e r e a r e t h r e e m a j o r s h o r t c o m i n g s o f t h e B - C m o d e l . F i r s t , t h e t y p e s o f d i s t r i b u t i o n f u n c t i o n s t h a t m a y b e u s e d i n t h e i r m o d e l a r e e x t r e m e l y c r u d e - t w o o r t h r e e p o i n t d i s t r i b u t i o n s w i t h c a s h f l o w s a n d i n t e r e s t r a t e s b e i n g h i g h l y c o r r e l a t e d . S e c o n d l y , t h e i r m o d e l i s u n a b l e t o h a n d l e e i t h e r a l a r g e n u m b e r o f d i f f e r e n t f i n a n c i a l i n s t r u m e n t s o r a p l a n n i n g h o r i z o n w i t h m o r e t h a n t h r e e t i m e p e r i o d s w i t h o u t t a x i n g c o m p u t e r c a p a c i t y . F i n a l l y , t h e i r m o d e l i s f o r m u l a t e d s u c h t h a t t h e i n v e s t m e n t d e c i s i o n m a d e n o w , h a s t o s a t i s f y a l l p o s s i b l e f u t u r e e c o n o m i c s c e n a r i o s . T h a t i s , t h e d e c i s i o n w i l l b e o v e r l y i n f l u e n c e d b y t h e w o r s t p o s s i b l e s c e n a r i o . G i v e n t h e s e d e f i c i e n c e s i n t h e l i t e r a t u r e , t h e p r i m a r y p u r p o s e o f t h i s d i s s e r t a t i o n i s t o d e v e l o p a n a s s e t a n d l i a b i l i t y m o d e l ( A L M ) t h a t i s c o m p u t a t i o n a l l y t r a c t a b l e f o r l a r g e r e a l i s t i c p r o b l e m s . I n t h e r e m a i n d e r o f t h i s t h e s i s , t h e f o l l o w i n g p r i n c i p l e a r e a s o f r e s e a r c h w i l l b e d i s c u s s e d i n t u r n : t h e r e a s o n s f o r t h e e x i s t e n c e o f f i n a n c i a l i n t e r m e d i a r i e s , t h e r e a s o n s f o r t h e u s e o f t h e n e t p r e s e n t v a l u e a p p r o a c h a s o p p o s e d t o t h e e x p e c t e d u t i l i t y a p p r o a c h a s a r a t i o n a l e f o r b a n k m a n a g e m e n t , a c r i t i c a l s u r v e y o f t h e n e t p r e s e n t v a l u e m o d e l s c u r r e n t l y i n t h e f i n a n c i a l l i t e r a t u r e , t h e p r e s e n t a t i o n o f a s t o c h a s t i c l i n e a r p r o g r a m w i t h s i m p l e r e c o u r s e m o d e l t o s o l v e t h e a s s e t a n d l i a b i l i t y m a n a g e m e n t p r o b l e m , t h e a p p l i c a t i o n o f t h e p r o p o s e d m o d e l t o a l o c a l f i n a n c i a l i n s t i t u -t i o n ( V a n c o u v e r C i t y S a v i n g s C r e d i t U n i o n ) , a n d t h e d e m o n s t r a t i o n o f t h e ' s u p e r i o r i t y ' o f t h e p r o p o s e d m o d e l t o e x i s t i n g m o d e l s u s i n g a s i m u l a t i o n o f e c o n o m i c s c e n a r i o s . T h e r e m a i n d e r o f t h i s c h a p t e r w i l l c o n s i s t o f t h e d e f i n i t i o n s o f f i n a n c i a l t e r m s u s e d i n t h i s d i s s e r t a t i o n , t h e e c o n o m i c r a t i o n a l e f o r t h e 4 e x i s t e n c e o f f i n a n c i a l i n t e r m e d i a r i e s , t h e f e a t u r e s t h a t a n a s s e t a n d l i a b i l i t y m a n a g e m e n t m o d e l m u s t h a v e , a n d t h e j u s t i f i c a t i o n f o r u s i n g t h e n e t p r e s e n t v a l u e a p p r o a c h i n p r e f e r e n c e t o t h e e x p e c t e d v a l u e a p p r o a c h . 1 . 2 D e f i n i t i o n s T h e liquidity o f a f i n a n c i a l a s s e t w i l l b e d e f i n e d i n t e r m s o f m a r k e t a b i l i t y a n d c a p i t a l c e r t a i n t y . A c c o r d i n g t o V a n H o m e [ 8 3 , p . 7 ] . . . l i q u i d i t y h a s t w o d i m e n s i o n s : ( J ) t h e l e n g t h o f t i m e a n d t r a n s a c t i o n c o s t r e q u i r e d t o c o n v e r t t h e a s s e t i n t o m o n e y , (.2) t h e c e r t a i n t y o f t h e p r i c e r e a l i z e d . . . . T h e t w o f a c t o r s a r e i n t e r r e l a t e d . I f a n a s s e t m u s t b e c o n v e r t e d i n t o m o n e y i n a v e r y s h o r t p e r i o d o f t i m e , t h e r e may b e m o r e u n c e r t a i n t y a s t o t h e p r i c e r e a l i z e d t h a n i f t h e r e w e r e a r e a s o n a b l e t i m e p e r i o d i n w h i c h t o s e l l t h e a s s e t . Financial Intermediaries w i l l b e d e f i n e d a s e n t i t i e s i n v o l v e d i n t h e b u s i n e s s o f h o l d i n g a n d d e a l i n g i n f i n a n c i a l i n s t r u m e n t s ( w h i c h c a n b e e x p r e s s e d i n t e r m s o f m o n e y ) . T h e y i s s u e f i n a n c i a l i n s t r u m e n t s ( i n d i r e c t s e c u r i t i e s ) i n o r d e r t o p u r c h a s e t h e f i n a n c i a l i n s t r u m e n t s o f o t h e r s ( p r i m a r y s e c u r i t i e s ) . F i n a n c i a l i n t e r m e d i a r i e s i n c l u d e s u c h i n s t i t u t i o n s a s c h a r t e r e d b a n k s , c r e d i t u n i o n s a n d l i f e i n s u r a n c e c o m p a n i e s . S i n c e t h i s d i s s e r t a t i o n i s c o n c e r n e d w i t h b a n k s a n d c r e d i t u n i o n s , t h i s i s t h e s e n s e i n w h i c h : t h e t e r m f i n a n c i a l i n t e r m e d i a r y w i l l b e \" u s e d . Portfolio risk i s t h e r i s k a s s o c i a t e d w i t h t h e r a t e o f r e t u r n e a r n e d b y a b a n k . Fund risk i s t h e r i s k a s s o c i a t e d w i t h t h e a b i l i t y o f t h e b a n k t o m e e t i t s c o m m i t m e n t s . Risk independence o c c u r s w h e n t h e f o l l o w -i n g t w o c o n d i t i o n s a r e s a t i s f i e d : (.1) t h e a g g r e g a t e v a l u e o f m u t u a l l y e x c l u s i v e i n v e s t m e n t p r o p o s a l s i s e q u a l t o t h e s u m o f t h e v a l u e s o f t h e p r o p o s a l s c o n s i d e r e d s e p a r a t e l y ( n o s y n e r g i s m ) , a n d ( 2 ) t h e f i n a n c i a l 5 i n s t r u m e n t s u n d e r c o n s i d e r a t i o n a r e p h y s i c a l l y i n d e p e n d e n t [ 6 1 ] . Risk interdependence i s s a i d t o o c c u r i f e i t h e r o f t h e n e c e s s a r y c o n d i t i o n s f o r r i s k i n d e p e n d e n c e a r e n o t m e t A financial market i s a n y m e c h a n i s m o r i n s t i t u t i o n u s e d t o b r i n g t o g e t h e r b u y e r s a n d s e l l e r s o f f i n a n c i a l i n s t r u m e n t s . A -perfect financial market s a t i s f i e s t h e f o l l o w i n g c o n d i t i o n s : (.1) m a n y b u y e r s , s e l l e r s a n d i s s u e r s , a l l o f whom a r e p r i c e t a k e r s , {2\ t r a n s a c t i o n c o s t s ( i n c l u d i n g v o l u m e d i s c o u n t s , p o o l i n g o f i n d e p e n d e n t r i s k s o f d e f a u l t a n d i m p u t e d c o s t s f o r i n c o n v e n i e n c e r e s u l t i n g f r o m i n d i v i s i b i l i t y ) a n d t a x e s d o n o t e x i s t , a n d ( 3 ) a l l i n v e s t o r s h a v e a c c e s s t o a l l r e l e v a n t i n f o r m a t i o n a t n o c o s t [ 5 2 ] . A n imperfect financial market d o e s n o t m e e t o n e o r m o r e o f t h e a b o v e c o n d i t i o n s . 1 . 3 T h e o r y o f F i n a n c i a l I n t e r m e d i a t i o n I n o r d e r t o d e v e l o p t h e o b j e c t i v e s a n d b e h a v i o u r o f a f i n a n c i a l i n t e r m e d i a r y , i t i s e s s e n t i a l t o d i s c u s s t h e r o l e o f f i n a n c i a l i n t e r -m e d i a r i e s i n t h e e c o n o m y . U n f o r t u n a t e l y , t h e t h e o r e t i c a l r a t i o n a l e f o r t h e e x i s t e n c e o f f i n a n c i a l i n t e r m e d i a r i e s h a s n o t b e e n r e s o l v e d i n t h e e c o n o m i c l i t e r a t u r e . ^ F o r e x a m p l e , f i n a n c i a l i n t e r m e d i a r i e s h a v e n o t b e e n i n c o r -p o r a t e d i n t o a g e n e r a l e q u i l i b r i u m m o d e l . E c o n o m i c r a t i o n a l e f o r t h e e x i s t e n c e o f f i n a n c i a l i n t e r m e d i a r i e s i s p r e s e n t e d n e x t . T h e u n d e r l y i n g f r a m e w o r k f o r s u c h a t h e o r y i s H i r s h l e i f e r ' s i n t e r -p r e t a t i o n o f F i s h e r ' s t h e o r y o f t h e i n v e s t m e n t d e c i s i o n [ 4 5 , 4 6 , 4 7 ] . T h e t h e o r y a s s u m e s : C o n t r i b u t i o n s h a v e b e e n m a d e t o s u c h a t h e o r y . S e e f o r e x a m p l e [ 9 , 3 0 , 7 0 , 8 0 ] . 6 1) p e r f e c t m a r k e t s , 2\ c e r t a i n t y , 3 ) n o b o r r o w i n g o r l e n d i n g w i t h t h e a u c t i o n e e r , h) t w o t i m e p e r i o d s , t h e p r e s e n t (.0). a n d t h e f u t u r e ( . 1 ) , 5 ) J i n d i v i d u a l s , w h e r e j < 0 0 6 ) U j (CQj, ) w h e r e U . ' i s t h e j t h i n d i v i d u a l ' s u t i l i t y f u n c t i o n a n d C . i s t h e c o n s u m p t i o n o f i n d i v i d u a l i U i n p e r i o d i \ t h e s e a r e t h e o b j e c t i v e s o f c h o i c e f o r t h e i n d i v i d u a l i n v e s t o r ) , a n d 7 ) e a c h i n d i v i d u a l a t t e m p t s t o m a x i m i z e h i s u t i l i t y f u n c t i o n s u b j e c t t o h i s o p p o r t u n i t y s e t , w h i c h c o n -s i s t s o f h i s i n i t i a l e n d o w m e n t ( . Y o , Y i ) , f i n a n c i a l o p p o r t u n i t i e s ( f i n a n c i a l a s s e t s ) a n d p r o d u c t i v e o p p o r t u n i t i e s ( r e a l a s s e t s ) . F i n a n c i a l o p p o r t u n i t i e s a l o n g t h e m a r k e t l i n e p e r m i t a n i n d i v i d u a l t o t r a n s f o r m h i s i n i t i a l e n d o w m e n t i n t o a l t e r n a t i v e ( C 0 , C i ) c o m b i n a t i o n s . B y i n v e s t i n g o r b o r r o w i n g w i t h o t h e r i n d i v i d u a l s , a n i n d i v i d u a l c a n a t t a i n t h e o p t i m a l p o i n t , P * , o n t h e p r o d u c t i o n p o s s i b i l i t y l o c u s , P P 1 , w h i c h i s t a n g e n t t o t h e h i g h e s t m a r k e t l i n e , N N 1 ( s e e F i g u r e 1 ) . T h e i n d i v i d u a l a t t a i n s t h e o p t i m a l p o i n t a s f o l l o w s . F i r s t t h e i n d i v i d u a l m o v e s t o P * f r o m h i s i n i t i a l e n d o w m e n t ( Y 0 , Y i ) . T h e n h e b o r r o w s o r l e n d s t o a t t a i n h i s u t i l i t y o p t i m u m ( C * , C * ) - I n t h e p a r t i c u l a r c a s e i l l u s t r a t e d i n F i g u r e 1 , t h e i n d i v i d u a l f i r s t i n v e s t s ( Y 0 - P o ) a n d t h e n b o r r o w s ( C 0 - P 0 ) t o r e p l e n i s h c u r r e n t c o n s u m p t i o n . A n i n d i v i d u a l i s d e f i n e d a s a s u r p l u s ( d e f i c i t ) u n i t w h e n ( G * - P 0 ) i s n e g a t i v e ( p o s i t i v e ) . T h e e x i s t e n c e o f s u r p l u s a n d d e f i c i t u n i t s i s a n e c e s s a r y c o n d i t i o n f o r b o t h d i r e c t a n d i n d i r e c t f i n a n c i n g . H o w e v e r , i t i s F i g u r e ! n o t a s u f f i c i e n t c o n d i t i o n f o r t h e e x i s t e n c e o f f i n a n c i a l i n t e r m e d i a r i e s s i n c e a l l f i n a n c i a l t r a n s f e r s c a n t a k e p l a c e d i r e c t l y . E x t e n s i o n s o f t h i s m o d e l b y A r r o w \u00C2\u00A3 1 J a n d H i r s h l e i f e r [ 4 5 , 4 6 ] t o i n c o r p o r a t e u n c e r t a i n t y , s t i l l d o n o t j u s t i f y t h e e x i s t e n c e o f f i n a n c i a l i n t e r m e d i a r i e s . T h e r e f o r e , t h e e x i s t e n c e o f u n c e r t a i n t y , p e r s e , d o e s n o t j u s t i f y t h e e c o n o m i c e x i s t e n c e o f f i n a n c i a l i n t e r m e d i a r i e s . R e l a x i n g t h e a s s u m p t i o n o f p e r f e c t c a p i t a l m a r k e t s d o e s s u g g e s t t w o p o t e n t i a l r e a s o n s w h y f i n a n c i a l i n t e r m e d i a r i e s m a y i n t e r p o s e t h e m s e l v e s b e t w e e n u l t i m a t e b o r r o w e r s a n d l e n d e r s . T h e s e a r e c o s t e c o n o m i e s i n t r a d i n g 8 ( e s p e c i a l l y i n a w o r l d o f n e g o t i a t e d c o m m i s s i o n r a t e s ) , c o s t e c o n o m i e s i n t h e g a t h e r i n g a n d p r o c e s s i n g o f i n f o r m a t i o n , a n d t h e b e n e f i t s o f p o r t -f o l i o d i v e r s i f i c a t i o n ( o f m i n o r i m p o r t a n c e f o r p u r e f i n a n c i a l i n s t i t u t i o n s ) . T h e p r e m i s e b e h i n d t h e s e c o n d r e a s o n i s t h a t s u r p l u s u n i t s m a y n o t b u y p r i m a r y s e c u r i t i e s b e c a u s e t h e y c a n n o t e c o n o m i c a l l y e v a l u a t e t h e b o r r o w e r ' s c r e d i t s t a n d i n g . F u r t h e r m o r e , e c o n o m i e s o f s c a l e i n g a t h e r i n g a n d p r o c e s s i n g i n f o r m a t i o n m a y e n a b l e t h e f i n a n c i a l i n t e r m e d i a r y t o d e v e l o p s u p e r i o r i n f o r m a t i o n a l e x p e r t i s e i n a s u b s e t o f p r i m a r y s e c u r i t i e s . T h i s c o u l d e n a b l e t h e f i n a n c i a l i n t e r m e d i a r y t o f o r m u l a t e m o r e a c c u r a t e p r o b -a b i l i t y d i s t r i b u t i o n s o f p o t e n t i a l o u t c o m e s f o r t h e s a m e d o l l a r e x p e n d i t u r e t h a n s u r p l u s u n i t s . A s a r e s u l t t h e r e q u i r e d r i s k p r e m i u m f o r a f i n a n c i a l i n t e r m e d i a r y m a y b e s m a l l e r t h a n t h a t r e q u i r e d b y a s u r p l u s u n i t . I n s u m m a r y , t h e n e c e s s a r y c o n d i t i o n s f o r t h e e x i s t e n c e o f f i n a n c i a l i n t e r m e d i a r i e s a r e t h e e x i s t e n c e o f s u r p l u s a n d d e f i c i t u n i t s , a n d t h e i m p e r f e c t i o n o f c a p i t a l m a r k e t s . W h e t h e r t h e s e c o n d i t i o n s a r e a l s o s u f f i c i e n t i s a t p r e s e n t u n r e s o l v e d . A l s o , a t t h i s p o i n t i n t i m e , i t i s u n r e s o l v e d w h e t h e r o r n o t t h e a s s e t s o f f i n a n c i a l i n t e r m e d i a r i e s a r e r i s k i n d e p e n d e n t . I f f i n a n c i a l i n t e r m e d i a r i e s c a n n o t c r e a t e a s s e t s t h a t i n v e s t o r s c a n n o t d u p l i c a t e o n t h e i r o w n a c c o u n t , s u c h a s p e r f e c t i n f l a t i o n h e d g e s , t h e n t h e a s s u m p t i o n o f r i s k i n d e p e n d e n c e i s r e a s o n a b l e . 1 . 4 A p p r o p r i a t e C r i t e r i o n f o r A s s e t a n d L i a b i l i t y M a n a g e m e n t A p r i n c i p l e c o n s t r a i n t o n t h e m a n a g e m e n t o f b a n k f u n d s i s t h e n e e d t o m e e t d e p o s i t w i t h d r a w a l c l a i m s o n r e q u e s t . T o i l l u s t r a t e t h e n a t u r e o f t h e r e s u l t i n g p r o b l e m , c o n s i d e r t h e f o l l o w i n g s i m p l i f i e d e x a m p l e f r o m T o b i n [ 7 9 ] : 9 1) c e r t a i n t y , 2) t r a n s a c t i o n c o s t s , a n d 3 ) t w o a s s e t s , o n e o f w h i c h i s i l l i q u i d a n d c a n n o t b e l i q u i d a t e d f o r t w o p e r i o d s ( i n f i n i t e t r a n s a c t i o n s c o s t s u p t o t h e e n d o f t h e s e c o n d p e r i o d ) , a n d t h e s e c o n d w h i c h c a n b e l i q u i d a t e d a t t h e e n d o f t h e f i r s t p e r i o d . T h e r e t u r n s o n t h e a s s e t s a r e r j . a n d r 2 , r e s p e c t i v e l y , w h e r e r i > r 2 . A l t h o u g h t h e a l l o c a t i o n o f i n i t i a l f u n d s b e t w e e n t h e t w o a s s e t s i s t r i v i a l , t h e e x a m p l e d o e s e m p h a s i z e t h a t t o m a k e r e a l i s t i c d e c i s i o n s t h e p l a n n i n g h o r i z o n c a n n o t b e i n f i n i t e s i m a l l y s h o r t . T h i s m u l t i - p e r i o d n a t u r e o f a s s e t a n d l i a b i l i t y m a n a g e m e n t r e s u l t s f r o m t h e e x i s t e n c e o f a s s e t s w i t h d i f f e r e n t d e g r e e s o f l i q u i d i t y , m a t u r i t y a n d y i e l d . T h e e s s e n c e o f t h e l i q u i d i t y m a n a g e m e n t p r o b l e m i s t o a l l o c a t e r e s o u r c e s i n a l e a s t c o s t ( t r a n s a c t i o n s c o s t s a n d o p p o r t u n i t y c o s t s ) m a n n e r . T h e e s s e n c e o f t h e a s s e t a n d l i a b i l i t y m a n a g e m e n t p r o b l e m i s n o t j u s t t o m a n a g e c a s h b u t t o c o n s i d e r a l l a s s e t s a n d l i a b i l i t i e s s i m u l t a n e o u s l y . R e l a x i n g t h e a s s u m p t i o n o f c e r t a i n t y r e s u l t s i n t w o a d d i t i o n a l p r o b l e m s - u n c e r t a i n f u t u r e m a r k e t r a t e s a n d u n c e r t a i n t i m i n g a n d v o l u m e o f c a s h f l o w s . I n v i e w i n g a b a n k a s a n o n g o i n g e n t i t y , i t i s o f t e n a r g u e d t h a t a b a n k m u s t s a t i s f y d e m a n d r e q u e s t s o r f a c e t h e p r o s p e c t o f l o s i n g c u s t o m e r s [ 2 7 ] . I n a n t i c i p a t i o n o f t h e s e u n c e r t a i n ( e . g . l o a n ) r e q u e s t s a b a n k m u s t b e p r e p a r e d t o p r o v i d e l a r g e a m o u n t s o f f u n d s o n r e l a t i v e l y s h o r t n o t i c e . I n t h e c u r r e n t f i n a n c i a l a n d e c o n o m i c l i t e r a t u r e , t w o c r i t e r i a a r e u s e d i n m o d e l l i n g a s s e t a n d l i a b i l i t y m a n a g e m e n t . T h e f i r s t c r i t e r i o n i s d e v e l o p e d f r o m t h e M a r k o w i t z m e a n - v a r i a n c e f r a m e w o r k . I t a s s u m e s t h a t a n i n t e r m e d i a r y p o s s e s s e s a u t i l i t y f u n c t i o n . , w h i c h t h e i n t e r m e d i a r y 1 0 a t t e m p t s t o m a x i m i z e . T h e s e c o n d c r i t e r i o n i s t o m a x i m i z e t h e n e t p r e s e n t v a l u e o f r e t u r n s s u b j e c t t o a n u m b e r o f c o n s t r a i n t s . P y l e ' s p a p e r [ 7 0 ] i s a n e x a m p l e o f t h e u s e o f t h e f i r s t c r i t e r i o n . I t i s t h e m o s t g e n e r a l o f s u c h a p p l i c a t i o n s s i n c e i t c o n s i d e r s a s s e t s a n d l i a b i l i t i e s s i m u l t a n e o u s l y . H o w e v e r , a f u n d a m e n t a l q u e s t i o n a r i s e s - -I s i s p o s s i b l e t o o p e r a t i o n a l i z e p o r t f o l i o t h e o r y f o r c o r p o r a t i o n s ? I n p a r t i c u l a r , w h a t u t i l i t y f u n c t i o n i s a p p r o p r i a t e f o r a c o r p o r a t i o n ? F u r t h e r m o r e , t h e f i r s t c r i t e r i o n l e a d s t o a s t a t i c o n e - p e r i o d m o d e l . T h i s i m p l i e s t h a t t h e i n t e r m e d i a r y c a n s e l e c t t h e a m o u n t o f a s s e t s a n d l i a b i l i t i e s t o b e m a i n t a i n e d o v e r t h e p e r i o d a n d b e c e r t a i n o f h a v i n g t h o s e a m o u n t s a t a l l t i m e s d u r i n g t h e p e r i o d . O r , s t a t e d s o m e w h a t d i f -f e r e n t l y , P y l e ' s p a p e r i g n o r e s f u n d r i s k e x c e p t t o t h e e x t e n t t h a t i t i s r e f l e c t e d i n p o r t f o l i o . r i s k . A n a d v e r s e s y n c h r o n i z a t i o n o f c a s h f l o w s c o u l d r e s u l t i n t r a d i n g c o s t s t h a t m o r e t h a n o f f s e t m a r k e t r e t u r n s a n d i n t h e e x t r e m e c a s e s o l v e n c y . O t h e r p r o b l e m s s u c h a s t h e t r a n s a c t i o n c o s t i n c u r r e d 2 i n t h e s a l e o f a s e c u r i t y p r i o r t o m a t u r i t y a r e i g n o r e d . T h u s t h e o n e -p e r i o d n a t u r e o f t h e m o d e l p r e c l u d e s t h e a b i l i t y o f t h e b a n k t o e x e r c i s e a n y m a t c h i n g ( s y n c h r o n i z a t i o n ) o f m a t u r i t i e s o f a s s e t s a n d l i a b i l i t i e s a n d m a k e s i t d i f f i c u l t t o i n s e r t a d e q u a t e t e r m i n a l c o n d i t i o n s . A n e x a m p l e o f t h e u s e o f t h e s e c o n d c r i t e r i o n , m a x i m i z i n g n e t p r e s e n t r e t u r n , t o m o d e l t h e a s s e t a n d l i a b i l i t y m a n a g e m e n t p r o b l e m i s g i v e n b y C h a m b e r s a n d C h a r n e s Q l ] . A l t h o u g h a r e v i e w o f s u c h l i t e r a t u r e w i l l b e d o n e i n C h a p t e r 2 , a f e w g e n e r a l s t a t e m e n t s a r e i n o r d e r h e r e . F i r s t , m o d e l s u s i n g t h e s e c o n d c r i t e r i o n a s s u m e r i s k i n d e p e n d e n c e C h e n , J e n a n d Z i o n t s h a v e i n c l u d e d t r a n s a c t i o n s c o s t s s i m i l a r m o d e l [ 18] . i n a 11 ( d e f i n e d e a r l i e r ) a s o p p o s e d t o t h e M a r k o w i t z t y p e m o d e l s w h i c h t r e a t r i s k d e p e n d e n c e . S e c o n d , t h e t y p e s o f m o d e l s r a n g e f r o m l i n e a r f o r m u l a t i o n s , w h i c h c a n s o l v e r e l a t i v e l y l a r g e p r o b l e m s , t o s t o c h a s t i c d y n a m i c f o r m u l a t i o n s w h i c h c a n s o l v e l i m i t e d s i z e d p r o b l e m s b e c a u s e o f c o m p u t a t i o n a l i n t r a c t -a b i l i t y . H o w e v e r , n e i t h e r t y p e o f m o d e l , i s g e n e r a l l y c o n s i d e r e d o r a c c e p t e d a s b e i n g t h e 1 b e s t 1 . T h u s t h e q u e s t i o n r e m a i n s : w h i c h o f t h e t w o c r i t e r i a r e s u l t s i n t h e m o d e l w h i c h i s m o s t s u i t a b l e f o r s o l v i n g t h e a s s e t a n d l i a b i l i t y m a n a g e m e n t p r o b l e m ? M y e r s [ 6 1 ] a t t e m p t s t o r e s o l v e t h e c o n t r o v e r s y . M y e r s h a s s h o w n t h a t : 1 ) r i s k i n d e p e n d e n c e i s a n e c e s s a r y c o n d i -t i o n f o r t h e e x i s t e n c e o f s e c u r i t y m a r k e t e q u i l i b r i u m , 2 ) i f s e c u r i t y m a r k e t e q u i l i b r i u m e x i s t s , t h e n t h i s i m p l i e s t h e r i s k i n d e p e n d e n c e o f s e c u r i t i e s , a n d 3 ) i f r i s k i n d e p e n d e n c e o f i n v e s t m e n t o p p o r t u n i t i e s e x i s t s t h e n t h e m a x i m i z a t i o n o f t h e e x p e c t e d n e t p r e s e n t v a l u e i s t h e a p p r o p r i a t e o b j e c t i v e c r i t e r i o n . I n t h e c a s e o f f i n a n c i a l i n s t i t u t i o n s i t i s o b s e r v e d t h a t : 1 ) a s t a t e o f e q u i l i b r i u m e x i s t s f o r t h e s e c u r i t i e s w h i c h a r e h e l d b y f i n a n c i a l i n s t i t u t i o n s , a n d 2 ) s e c u r i t i e s p u r c h a s e d d o n o t h a v e a s y n e r g e t i c e f f e c t ( i m p l y i n g t h e r i s k i n d e p e n d e n c e o f s e c u r i t i e s ) . T h e r e f o r e , t h e i m p l i c a t i o n f r o m ( 1 ) a n d ( 2 ) i s t h a t t h e a p p r o p r i a t e o b j e c t i v e f u n c t i o n f o r a f i n a n c i a l i n s t i t u t i o n i s t h e m a x i m i z a t i o n o f t h e e x p e c t e d n e t p r e s e n t v a l u e . A f u r t h e r t e s t o f t h e t w o a p p r o a c h e s i s t h e i r a p p l i c a b i l i t y a s a n o r m a t i v e t o o l t o t h e a c t u a l p r o b l e m s o l v i n g . T h e r i s k d e p e n d e n t ( M a r k o w i t z ) a p p r o a c h d o e s n o t l e n d i t s e l f t o s o l v i n g l a r g e ( d e c i s i o n v a r i a b l e s ) m u l t i -p e r i o d p r o b l e m s . On t h e o t h e r h a n d , t h e r i s k i n d e p e n d e n t a p p r o a c h , g i v e n c e r t a i n a s s u m p t i o n s , c a n s o l v e p r o b l e m s o f a m o r e r e a l i s t i c s i z e . I f t h e 1 2 a s s u m p t i o n s ( u n d e r l y i n g r i s k i n d e p e n d e n c e } c a n b e r e l a x e d a n d i f i t c a n b e s h o w n i n a n o p e r a t i o n a l s e n s e t h a t t h e r i s k i n d e p e n d e n t a p p r o a c h y i e l d s a s g o o d o r b e t t e r s o l u t i o n s t h a n t h e r i s k d e p e n d e n t a p p r o a c h , t h e n t h i s w o u l d i m p l y t h a t u s i n g t h e m a x i m i z a t i o n o f n e t p r e s e n t v a l u e i s a s u p e r i o r m o d e l l i n g a p p r o a c h . 1 . 5 E s s e n t i a l F e a t u r e s o f a n A s s e t a n d L i a b i l i t y M a n a g e m e n t M o d e l T h a t M a x i m i z e s E x p e c t e d N e t R e t u r n s T h e a s s e t a n d l i a b i l i t y m a n a g e m e n t p r o b l e m i s a n a l y z e d i n t h i s d i s s e r t a t i o n u s i n g a c o n s t r a i n e d o p t i m i z a t i o n m o d e l w h i c h m a z i m i z e s t h e e x p e c t e d n e t r e t u r n s . A g e n e r a l d i s c u s s i o n o f - t h e r e l e v a n t c o n s t r a i n t s f o l 1 o w s . T h e m a j o r c o n s t r a i n t o n t h e m a n a g e m e n t o f a b a n k ' s f u n d s i s t h e c a p a c i t y t o m e e t w i t h d r a w a l c l a i m s o n d e m a n d . S i n c e t h i s c a p a c i t y m u s t b e m a i n t a i n e d a c r o s s t i m e , t h e f o l l o w i n g f i v e f e a t u r e s s h o u l d b e i n c o r -p o r a t e d i n t h e i d e a l o p t i m i z a t i o n m o d e l . 1) m u l t i - p e r i o d i c i t y - i n o r d e r t o i n c o r p o r a t e : a ) t h e c h a n g i n g y i e l d s p r e a d s a c r o s s t i m e , b ) t h e t r a n s a c t i o n c o s t s a s s o c i a t e d w i t h s e l l i n g a s s e t s p r i o r t o m a t u r i t y , a n d c ) t h e s y n c h r o n i z a t i o n o f c a s h f l o w s a c r o s s t i m e b y m a t c h i n g m a t u r i t y o f a s s e t s w i t h e x p e c t e d c a s h o u t f l o w s . 2 ) s i m u l t a n e o u s c o n s i d e r a t i o n o f a s s e t s a n d l i a b i l i t i e s - i n o r d e r t o s a t i s f y b a s i c a c c o u n t -i n g p r i n c i p l e s a n d m o r e i m p o r t a n t l y t o m a t c h t h e l i q u i d i t y q u a l i t i e s o f a s s e t s w i t h t h o s e o f t h e l i a b i l i t i e s . 1 3 3 ) t r a n s a c t i o n c o s t s - i n o r d e r t o i n c o r p o r a t e : a ) b r o k e r a g e f e e s , a n d b ) o t h e r e x p e n s e s i n c u r r e d i n b u y i n g a n d s e l 1 i n g s e c u r i t i e s . h) u n c e r t a i n t y o f c a s h f l o w s - i n o r d e r t o i n c o r -p o r a t e t h e u n c e r t a i n t y i n h e r e n t i n t h e d e p o s i t e r s ' w i t h d r a w a l c l a i m s a n d d e p o s i t s . (.The m o d e l m u s t e n s u r e t h a t t h e s t r u c t u r e o f t h e a s s e t p o r t f o l i o i s s u c h t h a t t h e c a p a c i t y t o m e e t t h e s e c l a i m s i s m a i n t a i n e d b y t h e b a n k . ) 5 ) u n c e r t a i n t y o f m a r k e t r a t e s - i n o r d e r t o i n c o r -p o r a t e f l u c t u a t i n g i n t e r e s t r a t e s i n t o t h e d e c i s i o n -m a k i n g p r o c e s s s o a s t o a v o i d l e n d i n g a n d b o r r o w i n g d e c i s i o n s w h i c h m a y u l t i m a t e l y b e d e t r i m e n t a l t o t h e f i n a n c i a l w e l l - b e i n g o f t h e b a n k . ( F o r e x a m p l e , i f t h e b a n k l e n d s ( b o r r o w s ) l o n g w h e n t h e i n t e r e s t r a t e s a r e r e l a t i v e l y l o w . ( h i g h . ) ) . W h i l e t h e r e a r e o t h e r c o n s t r a i n t s t h a t c a n b e i n c o r p o r a t e d i n t o t h e o p t i m i z a t i o n m o d e l , t h e s e c o n s t r a i n t s a r e n o t u n i v e r s a l . 1 . 6 I m p o r t a n c e o f A s s e t a n d L i a b i l i t y M a n a g e m e n t A s w a s s t a t e d p r e v i o u s l y , t h e r e h a v e b e e n m a n y a t t e m p t s t o m o d e l a s s e t a n d l i a b i l i t y m a n a g e m e n t . T h e c o n c l u s i o n d e r i v e d f r o m t h i s l i t e r a t u r e i s t h a t m u c h w o r k r e m a i n s t o b e d o n e o n t h e p r o b l e m s i n c e n o t o n e o f t h e m o d e l s p r o p o s e d , t h u s f a r , i n c o r p o r a t e s a l l t h e e s s e n t i a l f e a t u r e s o f t h e r e a l w o r l d p r o b l e m w h i l e m a i n t a i n i n g c o m p u t a t i o n a l l y t r a c t a b i 1 i t y . T h e a s s e t a n d l i a b i l i t y m a n a g e m e n t ( A L M ) m o d e l d e v e l o p e d i n t h i s d i s s e r t a t i o n i s a n a t t e m p t t o r e c t i f y t h e a b o v e d e f i c i e n c i e s . T h e A L M m o d e l i n c l u d e s : 1 4 1) t h e s t o c h a s t i c n a t u r e o f t h e p r o b l e m - b y i n c o r p o r a t i n g a s e t o f r a n d o m c a s h f l o w s ( d e p o s i t s ) w i t h a g i v e n d i s c r e t e d i s t r i b u t i o n , 2) s i m u l t a n e o u s c o n s i d e r a t i o n o f a s s e t s a n d l i a b i l i t i e s , 3 ) t r a n s a c t i o n c o s t s , a n d k) m u 1 t i - p e r i o d i c i t y . T h e s e f e a t u r e s w i l l b e i n c o r p o r a t e d i n t o t h e m o d e l w h i l e m a i n t a i n -i n g c o m p u t a t i o n a l t r a c t a b i l i t y f o r l a r g e p r o b l e m s . 1 - 7 O r g a n i z a t i o n o f t h e D i s s e r t a t i o n I n C h a p t e r 2 , t h e l i t e r a t u r e o n t h e a s s e t a n d l i a b i l i t y m a n a g e m e n t m o d e l s u s i n g a n e t p r e s e n t r e t u r n \" c r i t e r i o n i s r e v i e w e d . I n C h a p t e r 3 , t h e t h e A L M m o d e l i s d e v e l o p e d a s a s t o c h a s t i c l i n e a r p r o g r a m w i t h s i m p l e r e c o u r s e ( S L P R ) . T h e a p p e n d i x t o C h a p t e r 3 g i v e s a b r i e f s u m m a r y o f r e l e v a n t s t o -c h a s t i c p r o g r a m m i n g t e c h n i q u e s a n d t h e t h e o r e t i c a l d e v e l o p m e n t o f S L P R . C h a p t e r 4 p r e s e n t s t h e r e s u l t s o f a n a p p l i c a t i o n o f t h e m o d e l t o o n e o f C a n a d a ' s l a r g e s t c r e d i t u n i o n s . C o m p u t e r i n f o r m a t i o n a b o u t t h e a l g o r i t h m u s e d t o s o l v e t h e p r o b l e m i s a l s o p r e s e n t e d . I n C h a p t e r 5 , t h e S L P R f o r m u l a -t i o n i s c o m p a r e d t o a n e q u i v a l e n t s t o c h a s t i c d y n a m i c f o r m u l a t i o n t o d e t e r m i n e i f t h e S L P R a p p r o a c h r e s u l t s i n b e t t e r o p e r a t i o n a l s o l u t i o n s f o r a d e c i s i o n -m a k e r . T h i s i s a c c o m p l i s h e d b y a s i m u l a t i o n o f t h e s a m e d a t a ( e c o n o m i c s c e n a r i o s ) f o r b o t h t h e S L P R a n d s t o c h a s t i c d y n a m i c f o r m u l a t i o n s . I n t h e f i n a l c h a p t e r , t h e c o n c l u s i o n s a n d p o s s i b l e e x t e n s i o n s o f t h i s r e s e a r c h a r e p r e s e n t e d . C h a p t e r 2 REVIEW OF LITERATURE 2 . 1 I n t r o d u c t i o n B e f o r e p r o c e e d i n g t o a d i s c u s s i o n o f t h e n o r m a t i v e a n a l y t i c a l m o d e l s d e a l i n g w i t h a s s e t a n d l i a b i l i t y m a n a g e m e n t , a b r i e f s u m m a r y o f t h e r e s u l t s o f a s t u d y o n p o s i t i v e m o d e l s i s i n o r d e r . H e s t e r a n d P i e r c e , i n a r e c e n t s t u d y [ 4 2 ] , u s e c r o s s - s e c t i o n a l d a t a t o a n a l y z e t h e v a l i d i t y o f a n u m b e r o f p o r t f o l i o s e l e c t i o n m o d e l s i n b a n k f u n d m a n a g e m e n t . T h e i r m a i n c o n c l u s i o n i s t h a t t h e r e i s a n o p t i m a l m e t h o d o f m a n a g i n g a b a n k ' s p o r t f o l i o . T h e o b j e c t i v e f u n c t i o n u t i l i z e d b y H e s t e r a n d P i e r c e w a s e i t h e r t h e m a x i m i z a -t i o n o f n e t d i s c o u n t e d r e t u r n s o r t h e m a x i m i z a t i o n o f a t w o v a r i a b l e f u n c t i o n ( w h e r e n e t d i s c o u n t e d r e t u r n s w a s d o m i n a n t ) . A s a r e s u l t o f t h e a r g u m e n t s p r e s e n t e d i n C h a p t e r 1 a n d t h e e m p i r i c a l e v i d e n c e o b t a i n e d b y H e s t e r a n d P i e r c e , t h e m a x i m i z a t i o n o f e x p e c t e d n e t d i s c o u n t e d r e t u r n s i s t a k e n a s t h e a p p r o p r i a t e o b j e c t i v e f u n c t i o n f o r a b a n k i n t h i s d i s s e r t a t i o n . T h e r e f o r e , o n l y t h e a s s e t a n d ' 1 i a b i 1 i t y m a n a g e m e n t m o d e l s u s i n g t h i s o b j e c t i v e f u n c t i o n a r e d i s c u s s e d i n t h i s c h a p t e r . A s s e t a n d l i a b i l i t y m a n a g e m e n t m o d e l s f a l l i n t o t w o b r o a d c a t e g o r i e s T h e f i r s t c a t e g o r y c o n s i s t s o f d e t e r m i n i s t i c m o d e l s . T h e s e m o d e l s u s e l i n e a r p r o g r a m m i n g , a s s u m e p a r t i c u l a r r e a l i z a t i o n s f o r a l l r a n d o m e v e n t s , 1 5 16 a n d a r e c o m p u t a t i o n a l l y t r a c t a b l e f o r l a r g e p r o b l e m s . F u r t h e r m o r e , t h e s e m o d e l s h a v e b e e n a c c e p t e d a s a u s e f u l n o r m a t i v e t o o l b y t h e b a n k i n g i n d u s t r y [ 2 0 ] . T h e s e c o n d c a t e g o r y o f a s s e t a n d l i a b i l i t y m a n a g e m e n t m o d e l s a r e t h e m o d e l s t h a t a r e s t o c h a s t i c i n n a t u r e . A t b e s t t h e s e m o d e l s h a v e a c h i e v e d v e r y m o d e s t s u c c e s s d u e t o t h e i n h e r e n t c o m p u t a t i o n a l d i f f i c u l t i e s o r t o t h e o v e r s i m p l i f i c a t i o n s n e e d e d t o a c h i e v e c o m p u t a t i o n a l t r a c t a b i l i t y . T h e s t o c h a s t i c m o d e l s i n c l u d e t h e u s e o f t h e f o l l o w i n g t e c h n i q u e s : 1 ) c h a n c e -c o n s t r a i n e d p r o g r a m m i n g , 2 ) d y n a m i c p r o g r a m m i n g , 3 ) s e q u e n t i a l d e c i s i o n t h e o r e t i c a p p r o a c h , 4 ) l i n e a r p r o g r a m m i n g u n d e r u n c e r t a i n t y , a n d 5 ) d y n a m i c l i n e a r p r o g r a m m i n g . T h e n e x t t w o s e c t i o n s w i l l r e v i e w t h e d e t e r m i n i s t i c a n d s t o c h a s t i c m o d e l l i n g t e c h n i q u e s . 2 . 2 D e t e r m i n i s t i c M o d e l s T h e m o d e l s d i s c u s s e d i n t h i s s e c t i o n a r e i m p o r t a n t b e c a u s e t h e y c a n a n d a r e u s e d t o s o l v e r e a l p o r t f o l i o p r o b l e m s . T h e 1 9 6 1 s e m i n a l w o r k b y C h a m b e r s a n d C h a r n e s ( C C ) [ 1 1 ] p r o d u c e d a l i n e a r p r o g r a m m i n g m o d e l t o o p t i m i z e b a n k p o r t f o l i o s . T h e u t i l i z a t i o n o f l i n e a r p r o g r a m m i n g ( L P ) w a s d e e m e d t o b e a c c e p t a b l e g i v e n t h e n u m b e r o f t r a d e - o f f s b e t w e e n l a r g e n u m b e r s o f v a r i a b l e s , t h e i n t e r t e m p o r a l n a t u r e o f t h e p r o b l e m a n d t h e l a r g e n u m b e r o f c o n s t r a i n t s . F u r t h e r m o r e , i t w a s f e a s i b l e t o s t r u c t u r e t h e p r o b l e m i n a l i n e a r f o r m a t a n d e f f i c i e n t a l g o r i t h m s w e r e r e a d i l y a v a i l a b l e t o s o l v e l a r g e s c a l e p r o b l e m s . S i n c e t h e s u b s e q u e n t d e v e l o p -m e n t o f o t h e r d e t e r m i n i s t i c m o d e l s w a s e i t h e r a s l i g h t e x t e n s i o n o r a n a p p l i -c a t i o n o f t h e CC m o d e l , a s u m m a r y o f t h e i r a r t i c l e i s e s s e n t i a l . 17 T h e C C m o d e l m a x i m i z e s n e t d i s c o u n t e d r e t u r n s , s u b j e c t t o b u d g e t ( s o u r c e s a n d u s e s o f f u n d s ) a n d l i q u i d i t y c o n s t r a i n t s . T h e s e c o n s t r a i n t s a r e d e v e l o p e d f r o m t h e c a p i t a l a d e q u a c y f o r m u l a a s p u t f o r t h b y t h e A m e r i c a n F e d e r a l R e s e r v e B o a r d ( F R B ) [ 2 7 ] . T h e s e c o n s t r a i n t s a r e p r e s e n t e d b e l o w ( i n a s o m e w h a t d i f f e r e n t f o r m f r o m t h o s e o r i g i n a l l y u s e d b y C C ) b e c a u s e t h e y a r e u t i l i z e d l a t e r i n t h e A L M m o d e l A b a n k ' s l i q u i d i t y u n d e r ' n o r m a l ' e c o n o m i c c o n d i t i o n s i s s a i d t o b e a d e q u a t e w h e n M a r k e t V a l u e ^ T o t a l , c \u00E2\u0080\u00A2 . , c \u00E2\u0080\u00A2, x / l X o f A s s e t s > L i a b i l i t i e s \" + S u r P l u s > ( ] ) T h e v a l u e o f t h e t o t a l l i a b i l i t i e s i s d e f i n e d a s t h e i r b o o k v a l u e a n d t h e m a r k e t v a l u e o f a s s e t s i s d e f i n e d a s t h e i r l i q u i d a t i o n v a l u e a s g i v e n b y : A t = ^ z i = l X u ( 2 ) w h e r e , A^ _ i s t h e m a r k e t v a l u e o f a s s e t s , i n p e r i o d t , $ ^ i s a p a r a m e t e r c o n -t a i n e d i n t h e c a p i t a l a d e q u a c y f o r m u l a u s e d t o m e a s u r e t h e s h r i n k a g e i n t h e v a l u e o f a s s e t i f r o m b o o k v a l u e i f t h e a s s e t h a s t o b e l i q u i d a t e d q u i c k l y , a n d X - t i s t h e b o o k v a l u e o f a s s e t i h e l d i n p e r i o d t . I f o n l y a s i n g l e b a n k w e r e l i q u i d a t e d , t h e n e q u a t i o n ( 1 ) w o u l d e n s u r e n o l o s s o f p r i n c i p a l t o d e p o s i t o r s . H o w e v e r , i n t h e e v e n t o f s e v e r e r e c e s s i o n ( o r f i n a n c i a l d i s i n t e r m e d i a t i o n ) , o t h e r f i n a n c i a l i n t e r m e d i a r i e s a r e a l s o l i k e l y t o b e i n f i n a n c i a l d i s t r e s s . C o n s e q u e n t l y , a c c o r d i n g t o t h e F R B , t h e d i s c o u n t r e q u i r e d t o l i q u i d a t e a s s e t s i s e x p e c t e d t o b e g r e a t e r ( e x c e p t f o r c a s h a n d t r e a s u r y b i l l s ) t h a n t h e v a l u e u s e d t o c o m p u t e A ^ [ 2 7 ] . 1 8 T h i s a d d i t i o n a l l o s s i s a . f u n c t i o n o f b o t h , t h e a n t i c i p a t e d d e p o s i t w i t h d r a w a l s a n d t h e a s s e t s t r u c t u r e . T h e f u n c t i o n a l r e l a t i o n s h i p s , f o r t h e a d d i t i o n a l l o s s , i s d e f i n e d b y i n e q u a l i t i e s ( 4 ) , w h i c h a r e d e v e l o p e d a s f o l l o w s . T h e d o l l a r v a l u e o f t h e e x p e c t e d d e p o s i t w i t h d r a w a l u n d e r a d v e r s e e c o n o m i c c o n d i t i o n s i s m \u00E2\u0080\u00A2 1 = 1 w h e r e i s t h e a n t i c i p a t e d w i t h d r a w a l o f l i a b i l i t i e s i n p e r i o d t , i s t h e p a r a m e t e r c o n t a i n e d i n t h e c a p i t a l a d e q u a c y f o r m u l a t o m e a s u r e t h e c o n -t r a c t i o n o f l i a b i l i t y i u n d e r a d v e r s e e c o n o m i c c o n d i t i o n s , a n d i s t h e b o o k v a l u e o f l i a b i l i t y i i n p e r i o d t . I n o r d e r t o d e t e r m i n e how t h e a s s e t s t r u c t u r e a f f e c t s l i q u i d a t i o n u n d e r s e v e r e f i n a n c i a l d i s i n t e r m e d i a t i o n , t h e a s s e t s a r e c l a s s i f i e d a s p e r t h e F R B ' s c a p i t a l a d e q u a c y f o r m u l a [ 2 7 ] a s f o l l o w s : 1 ) \" P r i m a r y a n d S e c o n d a r y R e s e r v e s \" ( K i ) w h i c h i n c l u d e s c a s h ( k i ) , t r e a s u r y b i l l s ( k 2 ) , a n d g o v e r n -m e n t b o n d s o f l e s s t h a n f i v e y e a r s m a t u r i t y ( k 3 ) ; 2 ) \" M i n i m u m R i s k A s s e t s \" ( K 2 ) w h i c h i n c l u d e g o v e r n m e n t b o n d s w i t h m o r e t h a n f i v e y e a r s m a t u r i t y ( k 4 ) , m u n i c i p a l b o n d s ( k 5 ) ; 3 ) \" I n t e r m e d i a t e A s s e t s \" ( K 3 ) w h i c h i n c l u d e s m o r t g a g e l o a n s ( k 6 ) ; a n d 4 ) \" P o r t f o l i o A s s e t s \" (Ki+) w h i c h c o n s i s t p r i m a r i l y o f p e r s o n a l l o a n s ( k 7 ) . B a s e d o n t h i s a s s e t c l a s s i f i c a t i o n , l i q u i d i t y r e s e r v e s P . a r e c o n -s t r u c t e d w i t h t h e p r o p e r t y t h a t t h e y i n c r e a s e a s t h e a s s e t s b e c o m e m o r e i l l i q u i d a n d / o r t h e l i a b i l i t i e s b e c o m e m o r e l i q u i d . T h e e x t r a l i q u i d i t y r e q u i r e d a s r e s e r v e s f o r p o s s i b l e a d v e r s e e c o n o m i c c o n d i t i o n s i s d e t e r m i n e d a s f o l l o w s : 1 9 k e K i U . . . L K . a k k , i = l , 2 , 3 , ( 4 ) w h e r e a k a r e t h e p a r a m e t e r s c o n t a i n e d i n t h e c a p i t a l a d e q u a c y f o r m u l a u s e d \u00E2\u0080\u00A2to m e a s u r e t h e s h r i n k a g e i n . t h e v a l u e o f a s s e t k f r o m b o o k v a l u e i f t h e a s s e t h a s t o b e l i q u i d a t e d q u i c k l y u n d e r a d v e r s e e c o n o m i c c o n d i t i o n s , a n d q . i s a m e a s u r e o f t h e r e s e r v e s r e q u i r e d b y t h e b a n k f o r e x c e s s l i q u i d i t y o f l i a b i l i t i e s o v e r a s s e t s i n Kx U \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 U K . . F i n a l l y , t h e F R B ' s c a p i t a l a d e q u a c y f o r m u l a i s I 3 , x . < N e t W o r t h - P i - P 2 - P 3 i = l 1 ( 5 ) S i n c e d e c i s i o n - m a k e r s , o p e r a t i o n a l l y , m a n a g e a s s e t s a n d l i a b i l i t i e s ( a n d n o t n e t w o r t h ) , a n i n t u i t i v e l y m o r e a p p e a l i n g m a n n e r o f s t a t i n g ( 5 ) i s K l - B , X . > P i + P 2 + P 3 + T o t a l r i g h t \u00E2\u0080\u00A2 h a n d s i d e o f - s u r p l u s - e q u i t y -b a l a n c e s h e e t ( 6 ) T h u s t h e g e n e r a l f o r m u l a t i o n o f t h e C C - t y p e m o d e l m a y b e s t a t e d a s m a x f c . x X j ^ O u = l J J s . t . }h S t J X J = S t P . . > q . . i t - H i t T a x Li V T x . e K 1 u \u00C2\u00AB - - u K i x i J ^ x . > y p . . + u 2 0 j , 9 t k j X j = b t k w h e r e i = 1 , 2 , 3 ; t = l , k = l , \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 , K ; c . i s t h e n e t p r e s e n t r e t u r n o n a s s e t j ; J i s t h e a m o u n t o f t h e j t h f i n a n c i a l i n s t r u m e n t ; s ^ i s t h e p e r d o l l a r f l o w o f f u n d s f o r f i n a n c i a l i n s t r u m e n t j i n p e r i o d t : S t i s t h e ^ e x t e r n a l f l o w o f f u n d s i n p e r i o d t , T i s t h e - n u m b e r o f t i m e p e r i o d s ; a ^ i s t h e i n i t i a l , p o l i c y ; - o r t e r m i n a l t e c h n o l o g i c a l c o e f f i c i e n t t y p e k i n p e r i o d t f o r i n s t r u m e n t j ; b ^ i s t h e r e s o u r c e a v a i l a b l e f o r p o l i c y o r t e r m i n a l c o n s t r a i n t t y p e k i n p e r i o d t ; a n d K i s t h e n u m b e r o f i n i t i a l , p o l i c y , a n d t e r m i n a - 1 c o n s t r a i n t s . T h e i n p u t n e c e s s a r y f o r t h e a b o v e f o r m u l a t i o n i s a n i n i t i a l p o r t -f o l i o a n d a n e c o n o m i c s c e n a r i o o f t h e f u t u r e . T h e l i n e a r p r o g r a m y i e l d s a n o p t i m a l c o u r s e o f a c t i o n t h a t i s p r e s e n t e d b y a s e r i e s o f ( T ) b a l a n c e s h e e t s . H o w e v e r , o n l y t h e i m m e d i a t e c h a n g e s i n t h e p o r t f o l i o a r e o f c o n s e q u e n c e t o t h e d e c i s i o n - m a k e r s i n c e m o r e i n f o r m a t i o n w i l l b e a v a i l a b l e a t t h e n e x t d e c i s i o n p o i n t . B e f o r e i m p l e m e n t i n g t h e p o r t f o l i o g e n e r a t e d b y t h e s o l u t i o n , t h e l i n e a r p r o g r a m m i n g t e c h n i q u e c a n b e u s e d t o t e s t t h e s e n s i t i v i t y o f t h e o p t i m a l s o l u t i o n t o p o l i c y c h a n g e s a n d c h a n g e s i n t h e e x p e c t a t i o n s o f t h e e c o n o m i c e n v i r o n m e n t . T h e d u a l v a r i a b l e s a n d t h e r e d u c e d c o s t s h a v e i m p o r t a n t e c o n o m i c i m p l i c a t i o n s f o r a s s e t a n d l i a b i l i t y m a n a g e m e n t . T h e b i n d i n g c o n s t r a i n t s c a n b e i d e n t i f i e d t h r o u g h t h e ( n o n z e r o ) d u a l v a r i a b l e s . T h e v a l u e o f t h e d u a l i s i n t e r p r e t e d a s t h e i n c r e m e n t a l i m p r o v e m e n t ( w o r s e n i n g ) o f t h e o b j e c -t i v e f u n c t i o n b y , r e l e a s i n g o n e u n i t o f r e s o u r c e . T h i s i s o f p r a c t i c a l i m p o r t a n c e s i n c e c o s t s c a n b e a t t a c h e d t o t h e p r o c u r e m e n t o f a d d i t i o n a l r e s o u r c e s . F o r i n s t a n c e , i n t h e i r c a s e s t u d y , C o h e n a n d H a m m e r [ 2 0 J f o u n d t h e d u a l s o f t h e c a p i t a l a d e q u a c y c o n s t r a i n t s t o b e h i g h , w h i c h s u g g e s t e d 21 t h a t a d d i t i o n a l c a p i t a l ( a s s u m i n g t h e m a r g i n a l c o s t , o f p r o c u r i n g f u n d s i s l e s s t h a n t h e d u a l ) , w o u l d r e s u l t i n g r e a t e r p r o f i t a b i l i t y f o r t h e f i r m . I n t h e s a m e m a n n e r , t h e o p p o r t u n i t y c o s t s o f b a n k p o l i c i e s c a n a l s o b e d e t e r m i n e d f r o m t h e d u a l v a r i a b l e s . R e d u c e d c o s t s a l s o p r o v i d e u s e f u l i n f o r m a t i o n t o t h e d e c i s i o n -m a k e r . I f a p o r t f o l i o m a n a g e r w a n t e d t o p u r c h a s e a n a s s e t , n o t c u r r e n t l y i n t h e o p t i m a l s o l u t i o n , t h e n t h e r e d u c e d c o s t i s i n t e r p r e t e d a s t h e p e r u n i t p r o f i t a b i l i t y f o r g o n e i n d i v e r t i n g f u n d s f r o m t h e o p t i m a l p o r t f o l i o t o t h i s n e w p o r t f o l i o . T h e r e a r e t w o o t h e r i m p o r t a n t u s e s o f t h e l i n e a r p r o g r a m m i n g m o d e l . T h e f i r s t i s t o o b s e r v e t h e e f f e c t s o f a n y p o l i c y c h a n g e b y t h e b a n k o n t h e o p t i m a l p o r t f o l i o . T h i s i s a c c o m p l i s h e d b y i n s e r t i n g a d d i t i o n a l c o n -s t r a i n t s i n t h e l i n e a r p r o g r a m m i n g f o r m u l a t i o n . T h e s e c o n d i s t o o b s e r v e t h e e f f e c t s o f a n y c h a n g e s i n t h e b a n k ' s e x p e c t a t i o n s o f t h e e c o n o m i c e n v i r o n -m e n t o n t h e o p t i m a l p o r t f o l i o . T h i s i s a c c o m p l i s h e d b y c h a n g i n g t h e r e t u r n s a n d c o s t s o f f i n a n c i a l i n s t r u m e n t s a n d t h e r e s o u r c e s a v a i l a b l e . T h u s t h e c h a n g i n g e x p e c t a t i o n s a r e r e f l e c t e d i n t h e n e w o p t i m a l s o l u t i o n g e n e r a t e d . D e s p i t e t h e f a c t t h a t t h e l i t e r a t u r e c o n t a i n s m a n y e x a m p l e s o f s u c c e s s f u l a p p l i c a t i o n s [ 2 0 , 5 0 , 5 3 ] o f t h e C C m o d e l , c r i t i c i s m c o n t i n u e s t o b e l e v e l l e d a t t h e m o d e l ; ( s e e f o r e x a m p l e [ 7 , 2 1 , 3 4 ] ) . T h e m a j o r s o u r c e o f t h e d i s e n c h a n t m e n t i s t h e o m i s s i o n o f u n c e r t a i n t y i n t h e m o d e l . P r o b - . a b i l i t y d i s t r i b u t i o n s c a n b e o b t a i n e d f o r d i f f e r e n t e c o n o m i c s c e n a r i o s a n d a l i n e a r p r o g r a m m i n g f o r m u l a t i o n c a n b e a p p l i e d t o e a c h s c e n a r i o i n o r d e r t o g e n -e r a t e o p t i m a l s o l u t i o n s . H o w e v e r , . t h i s w i l l n o t g e n e r a t e a n o p t i m a l s o l u t i o n t o t h e t o t a l p r o b l e m b u t r a t h e r a c t a s a d e t e r m i n i s t i c s i m u l a t i o n t o o b s e r v e p o r t f o l i o b e h a v i o u r u n d e r v a r i o u s e c o n o m i c c o n d i t i o n s . A n o t h e r c r i t i c i s m 2 2 o f t h e m o d e l i s i t s u s e o f t h e F e d e r a l R e s e r v e B o a r d ' s c a p i t a l a d e q u a c y f o r m u l a w h i c h i s l i k e l y t o b e t o o c o n s e r v a t i v e a n d c o u l d l e a d t o p o r t f o l i o s t h a t a r e t o o ' s a f e ' a n d t h u s n o t p r o f i t a b l e e n o u g h . 2 . 3 S t o c h a s t i c M o d e l s T h e m a j o r w e a k n e s s o f t h e CC m o d e l l i n g a p p r o a c h t o a s s e t a n d l i a b i l i t y m a n a g e m e n t , a s p e r c e i v e d b y m o s t r e s e a r c h e r s , i s t h e i n a b i l i t y o f t h e m o d e l t o c o p e w i t h t h e i n h e r e n t u n c e r t a i n t y i n t h e p r o b l e m . T h e m a j o r i t y o f t h e m o d e l s d i s c u s s e d i n t h i s s e c t i o n a r e c o n c e r n e d w i t h t h e a b o v e s h o r t -c o m i n g a n d t h u s a r e m o r e o c c u p i e d w i t h t e c h n i q u e t h a n w i t h c a p t u r i n g t h e e s s e n c e o f a r e a l i s t i c a s s e t a n d l i a b i l i t y m a n a g e m e n t . O n e o f t h e i n i t i a l a t t e m p t s t o i n c o r p o r a t e u n c e r t a i n t y (Util ized c h a n c e - c o n s t r a i n e d p r o g r a m m i n g . C h a r n e s a n d T h o r e \u00C2\u00A3 1 6 ] a n d C h a r n e s a n d L i t t l e c h i l d [ 1 5 ] w e r e t h e p i o n e e r s i n t h i s a r e a . T h e c a p i t a l a d e q u a c y f o r m u l a w a s r e p l a c e d b y c h a n c e - c o n s t r a i n t s o n m e e t i n g w i t h d r a w a l c l a i m s . F u t u r e d e p o s i t s a n d l o a n r e p a y m e n t s w e r e e x p r e s s e d a s j o i n t n o r m a l l y d i s t r i b u t e d r a n d o m v a r i a b l e s . T h o u g h b o t h p a p e r s e n a b l e d t h e d e c i s i o n - m a k e r t o e x p l i c i t l y i n c o r p o r a t e u n c e r t a i n t y i n a m a n n e r t h a t w a s c o m p u t a t i o n a l l y t r a c t a b l e , t h e c h a n c e - c o n s t r a i n e d p r o c e d u r e d o e s n o t h a v e t h e f a c i l i t y t o h a n d l e a d i f f e r e n t i a l p e n a l t y f o r e i t h e r v a r y i n g m a g n i t u d e s o f c o n s t r a i n t v i o l a t i o n s o r d i f f e r e n t t y p e s o f c o n s t r a i n t s . A l s o i n a m u l t i - p e r i o d m o d e l , t h e r e a r e c o n c e p t u a l d i f f i c u l t i e s , a s y e t u n r e s o l v e d i n t h e l i t e r a t u r e d e a l i n g w i t h t h e t r e a t m e n t o f i n f e a s i b i l i t y i n p e r i o d s 2 , \u00C2\u00AB \u00C2\u00AB ' , n [ 3 2 ] . ^ I n o t h e r ^ S e e t h e a p p e n d i x a t t h e e n d o f C h a p t e r 3 f o r a d d i t i o n a l c l a r i f i c a t i o n . 2 3 w o r d s , t h e p r i n c i p a l w e a k n e s s o f c h a n c e - c o n s t r a i n e d p r o g r a m m i n g i s t h a t t h e e c o n o m i c c o n s e q u e n c e s o f v i o l a t i n g a c o n s t r a i n t a r e c o n s i d e r e d o n l y i n d i r e c t l y . A s e c o n d a p p r o a c h w a s d y n a m i c p r o g r a m m i n g . T h i s t e c h n i q u e s o l v e s t h e p r o b l e m o f a s s e t a n d l i a b i l i t y m a n a g e m e n t o n l y f o r v e r y l i m i t e d n u m b e r s o f f i n a n c i a l i n s t r u m e n t s . E p p e n a n d F a m a \u00C2\u00A3 3 4 , 3 5 , 3 6 ] m o d e l l e d t w o a n d t h r e e a s s e t p r o b l e m s , a n d t h e i r w o r k w a s e x t e n d e d b y D a e l l e n b a c h a n d A r c h e r [ 2 8 ] t o i n c l u d e o n e l i a b i l i t y . T h e v i r t u e s o f t h e s e m o d e l s a r e t h a t t h e y a r e d y n a m i c a n d t h a t t h e y t a k e i n t o a c c o u n t t h e i n h e r e n t u n c e r t a i n t y o f t h e p r o b l e m . H o w e v e r , a l b e i t t h e s e a r e u s e f u l t o o l s i n p r a c t i c e , t h e i r a p p l i c - -a b i l i t y i s l i m i t e d b y t h e s m a l l n u m b e r o f f i n a n c i a l i n s t r u m e n t s t h a t c a n b e a n a l y z e d s i m u l t a n e o u s l y . A t h i r d a l t e r n a t i v e , p r o p o s e d b y W o l f [ 9 8 ] f o r a p p r o a c h i n g t h e p r o b l e m , i s a s e q u e n t i a l d e c i s i o n t h e o r e t i c a p p r o a c h . T h e e s s e n t i a l n o t i o n o f h i s m o d e l i s t o e m p l o y s e q u e n t i a l d e c i s i o n a n a l y s i s t o f i n d a n o p t i m a l s o l u t i o n t h r o u g h t h e u s e o f i m p l i c i t e n u m e r a t i o n . T h e f l a w w i t h t h i s t e c h -n i q u e i s t h a t i t d o e s n o t f i n d a n e x p l i c i t o p t i m a l s o l u t i o n t o p r o b l e m s w i t h a t i m e h o r i z o n b e y o n d o n e p e r i o d , b e c a u s e i t . w o u l d b e n e c e s s a r y t o e n u m e r a t e a l l p o s s i b l e p o r t f o l i o s t r a t e g i e s f o r p e r i o d s p r e c e d i n g t h e p r e s e n t d e c i s i o n p o i n t i n o r d e r t o g u a r a n t e e o p t i m a l i t y . I n a n e f f o r t t o e x p l a i n a w a y t h i s d r a w b a c k , W o l f m a k e s t h e d u b i o u s a s s e r t i o n t h a t t h e s o l u t i o n t o a o n e p e r i o d m o d e l w o u l d b e e q u i v a l e n t t o a s o l u t i o n p r o v i d e d b y s o l v i n g a n n p e r i o d m o d e l . ( H e t h u s a v o i d s t h e p r o b l e m o f s y n c h r o n i z i n g t h e m a t u r i t i e s o f a s s e t s a n d 1 i a b i l i t i e s . ) A f o u r t h a p p r o a c h , s u g g e s t e d b y C o h e n a n d T h o r e [ 2 1 ] a n d C r a n e [ 2 4 ] , i s s t o c h a s t i c l i n e a r p r o g r a m m i n g w i t h s i m p l e r e c o u r s e [ S L P R ] . A l t h o u g h r e l a t i v e l y e f f i c i e n t s o l u t i o n a l g o r i t h m s e x i s t e d f o r s o l v i n g S L P R s [ 9 1 , 9 2 ] . 2 4 b o t h m o d e l s w e r e s o l v e d b y u s i n g ' e x t e n s i v e r e p r e s e n t a t i o n . ' ' T h i s t e c h n i q u e e x p l i c i t l y c h a r a c t e r i z e s e a c h r e a l i z a t i o n o f t h e r a n d o m v a r i a b l e s i n t h e m o d e l f o r m u l a t i o n b y a c o n s t r a i n t . S o l a r g e ( . r e a l i s t i c ) p r o b l e m s w e r e c o m p u t a t i o n a l l y i n f e a s i b l e . T h i s h a n d i c a p p e d t h e m o d e l l e r s g r e a t l y , i n f a c t C o h e n a n d T h o r e v i e w e d t h e i r m o d e l m o r e a s a t o o l f o r s e n s i t i v i t y a n a l y s i s ( i n t h e a g g r e g a t e ) r a t h e r t h a n a n o r m a t i v e d e c i s i o n t o o l . T h u s t h e c o m p u t a -t i o n a l i n t r a c t a b i l i t y a n d t h e p e r c e p t i o n s o f t h e f o r m u l a t i o n p r e c l u d e d c o n -s i d e r a t i o n o f p r o b l e m s o t h e r t h a n t h o s e w h i c h w e r e l i m i t e d b o t h i n t e r m s o f t i m e p e r i o d s ( C o h e n a n d T h o r e u s e d o n e a n d C r a n e u s e d t w o ) , a n d i n t h e n u m b e r o f v a r i a b l e s a n d r e a l i z a t i o n s . T h e r e w a s a n a t t e m p t t o a p p l y t h i s f o r m u l a t i o n b y B o o t h [ 4 ] . He l i m i t e d b o t h t h e n u m b e r o f p o s s i b l e r e a l i z a -t i o n s a n d t h e n u m b e r o f v a r i a b l e s c o n s i d e r e d i n o r d e r t o i n c o r p o r a t e t w o t i m e p e r i o d s . I n t h i s d i s s e r t a t i o n a c o m p r e h e n s i v e a n d s y s t e m a t i c a s s e t a n d l i a b i l i t y m a n a g e m e n t m o d e l w i l l b e d e v e l o p e d u s i n g S L P R a n d t h e m o s t e f f i c i e n t a l g o r i t h m [ 9 5 ] a v a i l a b l e w i l l b e u s e d t o s o l v e i t . T h e f i n a l s t o c h a s t i c a p p r o a c h w a s p r o p o s e d b y B r a d l e y a n d C r a n e ( B - C ) [ 5 , 6 , 7 ] . T h i s m o d e l h a s m a n y o f t h e d e s i r a b l e f e a t u r e s e s s e n t i a l t o a b a n k p o r t f o l i o m o d e l . T h e m o d e l h a s a s i t s c o n c e p t u a l o r i g i n s t h e W o l f [ 9 8 ] f o r m u l a t i o n . R e c a l l t h a t W o l f ' s m o d e l b e c a m e c o m p u t a t i o n a l l y i n t r a c t a b l e a s t h e n u m b e r o f t i m e p e r i o d s i n c r e a s e d . I n o r d e r t o o v e r c o m e t h i s s h o r t c o m i n g , t h e y r e f o r m u l a t e d t h e a s s e t a n d l i a b i l i t y p r o b l e m a n d d e v e l o p e d a g e n e r a l l i n e a r p r o g r a m m i n g d e c o m p o s i t i o n a l g o r i t h m t h a t a l l e v i a t e s t h e c o m p u t a t i o n a l d i f f i c u l t i e s . S e e a p p e n d i x a t t h e e n d o f C h a p t e r 3 f o r f u r t h e r c o m m e n t s . 2 5 T h e B - C m o d e l d e p e n d s u p o n t h e d e v e l o p m e n t o f e c o n o m i c s c e n a r i o s . T h e s e e c o n o m i c s c e n a r i o s a r e c o n s i d e r e d t o i n c l u d e t h e s e t o f a l l p o s s i b l e o u t c o m e s . T h e e c o n o m i c s c e n a r i o s c a n b e t h o u g h t o f a s a t r e e d i a g r a m w h e r e e a c h e l e m e n t ( e c o n o m i c c o n d i t i o n s ) i n e a c h p a t h h a s a s e t o f c a s h f l o w s ( v a r y i n g a m o u n t s o f d e p o s i t s ) a n d a s e t o f i n t e r e s t r a t e s . T h e p r o b l e m i s t h e n f o r m u l a t e d a s a l i n e a r p r o g r a m . T h e o b j e c t i v e f u n c t i o n i s t h e m a x i m i z a -t i o n o f t h e e x p e c t e d t e r m i n a l w e a l t h o f t h e f i r m . T h e r e a r e f o u r t y p e s o f c o n s t r a i n t s : 1 ) c a s h f l o w c o n s t r a i n t s , w h i c h d o n o t a l l o w t h e f i r m t o p u r c h a s e m o r e a s s e t s t h a n i t h a s f u n d s a v a i l a b l e ; 2 ) i n v e n t o r y b a l a n c i n g c o n s t r a i n t s , w h i c h e n s u r e t h a t t h e f i r m c a n n o t s e l l a n d / o r h o l d m o r e o f a n a s s e t a t t h e e n d o f a p e r i o d t h a n i t h e l d a t t h e b e g i n n i n g o f a p e r i o d ; 3 ) c a p i t a l l o s s c o n s t r a i n t s , w h i c h d o n o t a l l o w t h e n e t r e a l i z e d c a p i t a l l o s s e s i n a p e r i o d t o e x c e e d s o m e p r e - s p e c i f i e d u p p e r b o u n d ; a n d 4 ) c l a s s c o m p o s i t i o n c o n s t r a i n t s , w h i c h l i m i t t h e h o l d i n g o f a p a r t i c u l a r a s s e t . T h e i r f o r m u l a t i o n i s m a x K r N - l I p ( e N ) I < I e N e E N k = l m=0 h k (e>;) + mn N s . t . C a s h F l o w s y N ( e N } + V N N ( e N } b N ( e N ) K K D I t # e n ) - I k = l \" \u00E2\u0080\u00A2 k = l n-2 m=0 K n - 1 I I k = l rn=0 1 + g ( e ) 3 m , n ^ n ' s ( e ) = f ( e ) m , n v n n v n ' 2 6 2 ) I n v e n t o r y B a l a n c e - b k , ( e J + s k , ( e ) + h k , ( e ) = 0 , n - 1 v n - r n - l , n v n y n - l , n v n ho,o ( eo } = ho> 3 ) C a p i t a l L o s s e s K n - 1 . , - I J g ( e ) s ( e ) < L ( e ) , / , L r s s m , n v n ' m , n v n - n v n ' k = l m=0 ' 4 ) C a t e g o r y L i m i t s k e K 1 m=0 5 ) N o n n e g a t i v i t y b k ( e ) > 0 , s k ( e ) > 0 , h k ( e ) > 0 , m = l , \u00C2\u00BB \u00C2\u00BB \u00C2\u00AB , n - l , m , n n ' - m , n x n ' - m , n v n - ' ' w h e r e e n e E ^ ; n = l , \u00C2\u00AB \u00C2\u00BB \u00C2\u00AB , N ; k = l , \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 , ! < ; e n i s a s e t o f e c o n o m i c c o n d i t i o n f r o m p e r i o d 1 t o n h a v i n g p r o b a b i l i t y p ( e n ) ; E^ i s t h e s e t o f p o s s i b l e e c o n o m i c c o n d i t i o n s f r o m p e r i o d 1 t o n ; K i s t h e n u m b e r o f a s s e t s ; N i s t h e n u m b e r o f t i m e p e r i o d s ; y m ( e n ) i s t h e i n c o m e y i e l d p e r d o l l a r o f p u r c h a s e p r i c e ( p e r i o d m) o f a s s e t k ( c o n d i t i o n a l o n e ) ; v . . ( e . , ) i s t h e e x p e c t e d t e r m i n a l v a l u e m m,IM N r p e r d o l l a r o f p u r c h a s e p r i c e ( p e r i o d m) o f a s s e t k a n d h e l d a t h o r i z o n ( p e r i o d N ) c o n d i t i o n a l o n e ^ ; D n ( e n ) i s t h e d o l l a r a m o u n t o f a s s e t k p u r c h a s e d i n 2 7 p e r i o d n c o n d i t i o n a l o n e ; h ( e ) i s t h e d o l l a r a m o u n t o f a s s e t k p u r -II 111 j 11 I I c h a s e d i n p e r i o d m.. .and s t i l l h e l d i n p e r i o d n c o n d i t i o n a l o n e ; s k ( e ) n m , n v n i s t h e d o l l a r a m o u n t o f a s s e t k p u r c h a s e d i n p e r i o d m a n d s o l d i n p e r i o d n , c o n d i t i o n a l o n e ; g k ( e ) i s t h e c a p i t a l g a i n ( l o s s ) p e r d o l l a r o f p u r c h a s e II III 9 II II p r i c e ( p e r i o d m) o f a s s e t k s o l d i n p e r i o d n ; f n ( e ) i s t h e i n c r e m e n t a l i n c r e a s e ( d e c r e a s e ) o f f u n d s a v a i l a b l e f o r p e r i o d n ; L n ( e n ) i s t h e d o l l a r a m o u n t o f m a x i m u m a l l o w a b l e n e t r e a l i z e d c a p i t a l l o s s e s i n p e r i o d n ; a n d ,.\u00E2\u0080\u00A2 C ^ e ^ ) i s t h e u p p e r ( l o w e r ) b o u n d i n d o l l a r s o n t h e a m o u n t o f f u n d s i n v e s t e d i n a s s e t t y p e i i n p e r i o d n . T h e B - C f o r m u l a t i o n i s d y n a m i c i n n a t u r e . T h e f i r s t d e c i s i o n k k k ( i m m e d i a t e r e v i s i o n , h 0 1 ( e i ) , b ( e j , s 0 i ( e i ) ) h a s a s i t s f e a s i b l e s e t t h e i n t e r s e c t i o n o f a l l p o s s i b l e r e a l i z a t i o n s ( t h a t i s t h e c u r r e n t s o l u t i o n m u s t b e f e a s i b l e f o r t h e s e t E ^ ) . T h i s d e c i s i o n i s c o n d i t i o n a l o n t h e r e a l i -z a t i o n o f e c o n o m i c e v e n t s i n t h e f i r s t p e r i o d . T h e f e a s i b l e s e t f o r t h e s e c o n d d e c i s i o n i s t h e i n t e r s e c t i o n o f a l l p o s s i b l e r e a l i z a t i o n s f r o m t h e s e c o n d d e c i s i o n p o i n t t o t h e h o r i z o n o f t h e m o d e l . I n o t h e r w o r d s , t h e f i n a l s o l u t i o n g e n e r a t e d h a s d e c i s i o n s a t e a c h p o i n t i n t i m e c o n d i t i o n a l o n t h e s t a t e s o f n a t u r e t h a t h a v e o c c u r r e d u p t o t h e c u r r e n t d e c i s i o n p o i n t . T h e r e a r e a n u m b e r o f a d v a n t a g e o u s f e a t u r e s t o t h i s m o d e l i n c l u d i n g i t s d y n a m i c n a t u r e a n d c o m p u t a t i o n a l t r a c t a b i l i t y H o w e v e r , t h e B - C f o r m u l a -t i o n h a s a n u m b e r o f f e a t u r e s t h a t d e t r a c t f r o m i t s p r a c t i c a b i l i t y . T h e c a p i t a l l o s s a n d c a t e g o r y l i m i t c o n s t r a i n t s h a v e a s u p p e r ( o r l o w e r ) b o u n d s a m o u n t s ( r e s o u r c e s ) g e n e r a t e d a r b i t r a r i l y b y p o r t f o l i o m a n a g e r s r a t h e r t h a n t h r o u g h a s y s t e m a t i c p r o c e d u r e . F o r e x a m p l e , n o c o n s i d e r a t i o n i s g i v e n t o t h e p o r t f o l i o m i x i n t h e d e v e l o p m e n t o f b o u n d s ( e x c e p t i n t h e s e n s e t h a t u p p e r ( o r l o w e r ) b o u n d s a r e p l a c e d o n a s s e t c a t e g o r i e s ) . A t s o m e p o i n t i n 2 8 t i m e , t h i s m a y i m p l y t h a t t h e b a n k h a s i n v e s t e d a d i s p r o p o r t i o n a t e a m o u n t o f i t s a v a i l a b l e f u n d s i n l o n g - t e r m b o n d s w h e n c o m p a r e d t o t h e a m o u n t o f s h o r t - t e r m l i a b i l i t i e s h e l d . A l s o t h e f o r m u l a t i o n d o e s n o t u t i l i z e e i t h e r t h e F e d e r a l R e s e r v e B o a r d ' s r e c o m m e n d e d c a p i t a l a d e q u a c y f o r m u l a o r a n y o t h e r s t a t i s t i c a l l y g e n e r a t e d s y s t e m a t i c p r o c e d u r e i n t h e d e v e l o p m e n t o f b o u n d s f o r t h e c o n s t r a i n t s . S i n c e t h e c a p i t a l l o s s a n d c a t e g o r y l i m i t c o n s t r a i n t s a c t u a l l y d e t e r m i n e t h e c o m p o s i t i o n o f t h e s o l u t i o n , t h e a r b i t r a r y n a t u r e o f t h e c h o i c e m a y u n s y s t e m a t i c a l l y b i a s t h e s o l u t i o n . A n o t h e r s h o r t c o m i n g o f t h e m o d e l i s t h a t t h e s o l u t i o n t o t h e i m m e d i a t e r e v i s i o n p r o b l e m h a s t o s a t i s f y a l l f u t u r e e c o n o m i c e v e n t s . I n t h e B - C f o r m u l a t i o n s o m e c o n s t r a i n t s m a y h a v e a v e r y s m a l l p r o b a b i l i t y o f o c c u r r i n g ; t h e s e c o n s t r a i n t s m a y t u r n o u t t o b e b i n d i n g a n d t h i s w o u l d u n d u l y c o n s t r a i n t h e p r o b l e m . 1 T h u s t h e n e t e f f e c t i s t o r e s t r i c t t h e f e a s i b l e r e g i o n f o r t h e i m m e d i a t e r e v i s i o n p r o b l e m t o t h e m o s t p e s s i m i s t i c p o s s i b l e s e t o f e c o n o m i c e v e n t s . T h e s e t w o s h o r t c o m i n g s m a y b e c o r r e c t e d b y r e p l a c i n g t h e c a p i t a l l o s s a n d c a t e g o r y l i m i t c o n s t r a i n t s w i t h o t h e r s t h a t a r e s y s t e m a t i c i n n a t u r e . H o w e v e r , i t m a y n o t b e p o s s i b l e t o c o r r e c t a n o t h e r s h o r t c o m i n g o f t h e B - C f o r m u l a t i o n - c o m p u t a t i o n a l i n t r a c t a b i l i t y f o r l a r g e p r o b l e m s . B - C s t a t e [ 7 ; p . 1 1 2 ] U n f o r t u n a t e l y , t a k i n g u n c e r t a i n t y e x p l i c i t l y i n t o a c c o u n t w i l l m a k e a n a s s e t a n d l i a b i l i t y m a n a g e m e n t m o d e l f o r t h e e n t i r e b a n k c o m p u t a t i o n a l l y i n t r a c t a b l e , u n l e s s i t i s a n e x t r e m e l y a g g r e g a t e d m o d e l . T h e c o m p l e x i t i e s o f t h e g e n e r a l d y n a m i c b a l a n c e s h e e t m a n a g e m e n t p r o b l e m a r e s u c h t h a t t h e n u m b e r o f c o n s t r a i n t s a n d v a r i a b l e s n e e d e d t o a c c u r a t e l y m o d e l t h e e n v i r o n m e n t w o u l d b e v e r y l a r g e . S e e C h a p t e r 5 f o r e v i d e n c e o f t h i s u n d u e c o n s t r a i n t . 2 9 I n a n e f f o r t t o g a i n c o m p u t a t i o n a l t r a c t a b i l i t y t h e y c o n s i d e r o n l y b o n d s i n t h e i r m o d e l . H o w e v e r , e v e n l i m i t i n g t h e i r m o d e l t o t h e s e a s s e t s , t h e B - C m o d e l s t i l l h a s c o m p u t a t i o n a l p r o h l e m s . C o n s i d e r t h e f o l l o w i n g f o u r a s s e t a n d l i a b i l i t y m a n a g e m e n t p r o b l e m s : 1 ) e i g h t a s s e t s , t h r e e t i m e p e r i o d s a n d t h r e e p o s s i b l e r e a l i z a t i o n s p e r p e r i o d ; 2 ) t h i r t y a s s e t s , f i v e c l a s s e s o f a s s e t s , t h r e e t i m e p e r i o d s a n d t h r e e p o s s i b l e r e a l i z a -t i o n s p e r p e r i o d ; 3 ) t h i r t y a s s e t s , f i v e c l a s s e s o f a s s e t s , t h r e e t i m e p e r i o d s a n d f i v e p o s s i b l e r e a l i z a t i o n s p e r p e r i o d ; a n d 4 ) t h i r t y a s s e t s , f i v e c l a s s e s o f a s s e t s , f i v e t i m e p e r i o d s a n d f i v e p o s s i b l e r e a l i z a t i o n s p e r p e r i o d . T h e n u m b e r o f c o n s t r a i n t s a n d d e c i s i o n v a r i a b l e s n e c e s s a r y t o s o l v e e a c h p r o b l e m u s i n g t h e B - C f o r m u l a t i o n i s : 3 1 9 c o n s t r a i n t s a n d 6 5 6 v a r i a b l e s f o r ( 1 ) ; 1 1 4 1 c o n s t r a i n t s a n d 2 4 6 0 v a r i a b l e s f o r C 2 ) ; 2 8 2 7 c o n s t r a i n t s a n d 6 1 2 0 v a r i a b l e s f o r ( 3 ) ; a n d 1 1 6 , 8 2 7 c o n s t r a i n t s a n d 2 4 6 , 1 2 0 v a r i a b l e s f o r ( 4 ) . T h e s e n u m b e r s w e r e c a l c u l a t e d i n t h e f o l l o w i n g m a n n e r : t h e n u m b e r o f v a r i a b l e s i s ( 4 + 5D + 7 D 2 + \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 + ( 2 n + l ) D n _ 1 ) , t h e n u m b e r o f c o n s t r a i n t s i s e q u a l t o t h e s u m o f t h e c a s h f l o w c o n s t r a i n t s (1 + D + D + \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 + D ) t h e c a p i t a l l o s s c o n s t r a i n t s (.1 + D + + \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 + D n _ 1 ) , t h e c a t e g o r y l i m i t c o n s t r a i n t s ( I ) ( l + D + D 2 + \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 + D n _ 1 ) , t h e i n v e n t o r y b a l a n c e c o n s t r a i n t s ( K ) ( L + 2D + 3 D 2 + \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 + n D n _ 1 ) , a n d t h e i n i t i a l c o n d i t i o n s K . D i s t h e n u m b e r o f p o s s i b l e r e a l i z a t i o n s p e r p e r i o d , n i s t h e n u m b e r o f t i m e p e r i o d s , I i s t h e n u m b e r o f a s s e t c l a s s e s , a n d K i s t h e n u m b e r o f a s s e t s . B r a d l e y a n d C r a n e s t a t e t h a t ( 1 ) h a s a r u n n i n g t i m e o f 6 8 s e c o n d s o n a n I B M 3 6 0 / 6 5 [ 5 ] . A m o d e l , o f t h e s a m e s i z e a s ( 1 ) , w o u l d n o t b e o f m u c h b e n e f i t t o a d e c i s i o n - m a k e r i n t h e s e l e c t i o n o f a n a c t u a l p o r t f o l i o , o n e r e a s o n b e i n g t h a t t h e a g g r e g a t i o n o f i n v e s t m e n t o p p o r t u n i t i e s i n t o e i g h t c a t e g o r i e s w o u l d n o t a l l o w f o r t h e s e l e c t i o n o f t h e b e s t o p p o r t u n i t i e s 3 0 w i t h i n a g r o u p o f a s s e t s . A n o t h e r p o t e n t i a l p r o b l e m i s t h e i n a d e q u a t e n u m b e r o f t i m e p e r i o d s t o m a t c h m a t u r i t i e s ( o f a s s e t s a n d l i a b i l i t i e s ) . I n f a c t t h e e x c l u s i o n o f l i a b i l i t i e s m a k e s a n y m a t c h i n g o f a s s e t s a n d l i a b i l i t i e s i m p o s s i b l e . A n o t h e r s h o r t c o m i n g i s t h e l i m i t e d n u m b e r o f p o s s i b l e r e a l i z a t i o n s . D e s p i t e t h e f a c t t h a t u n c e r t a i n t y i s i n c o r p o r a t e d i n t h e m o d e l , t h e p r o b a b i l i t y d i s t r i b u t i o n o f e c o n o m i c e v e n t s i s c r u d e a n d t h u s d o e s n o t a l l o w t h e m o d e l t o e x p l o i t m u c h o f t h e i n h e r e n t u n c e r t a i n t y o f t h e m o d e l . T h e s e d e f i c i e n c i e s w o u l d b e d i m i n i s h e d i f a m o d e l o f t h e s i z e o f p r o b l e m f o u r w e r e u t i l i z e d . H o w e v e r , t h e s i z e o f ( 4 ) ( 1 1 6 , 8 2 7 x 2 4 6 , 1 2 0 ) w o u l d m a k e i t s u s e d i f f i c u l t a s a d e c i s i o n - m a k i n g t o o l . T h e B - C f o r m u l a t i o n c a n b e d e c o m p o s e d [ 5 ] ; b u t e v e n b y d e c o m p o s i n g ( 4 ) t h e b a s i s o f t h e m a s t e r h a s d i m e n s i o n 5 4 6 7 . W h e n t h i s i s c o m p a r e d t o ( 1 ) . . w i t h r a ' m a s t e r b a s i s o f d i m e n s i o n 3 9 a n d n e e d i n g 6 8 s e c o n d s o f r u n n i n g t i m e , t h e c o m p u t a t i o n a l d i f f i c u l t i e s o f s o l v i n g ( 4 ) a r e e v i d e n t . A l s o ( 1 ) h a s i n t h e o r d e r o f 2 2 0 0 n o n z e r o e l e m e n t s a n d ( 4 ) h a s i n t h e o r d e r o f 8 5 0 , 0 0 0 n o n z e r o e l e m e n t s . T h e c o m p u t a t i o n a l a n d d a t a h a n d l i n g d i f f i c u l t i e s o f ( 2 ) a n d ( 3 ) m a y b e l e s s s t r i k i n g b u t n e v e r t h e l e s s , t h e y r e m a i n f o r m i d a b l e . A l t h o u g h i n i t i a l l y t h e B - C f o r m u l a t i o n m a y a p p e a r t o b e a s o u n d a p p r o a c h t o a s s e t a n d l i a b i l i t y m a n a g e m e n t , c o m p u t a t i o n a l t r a c t a b i l i t y a n d p r o b l e m f o r m u l a t i o n s e e m t o p o s s e s s u n d e s i r a b l e f e a t u r e s . I n C h a p t e r 5 c l o s e r a n a l y s i s w i l l b e m a d e o f t h e i n h e r e n t p r o b l e m s i n t h e B - C f o r m u l a t i o n . I n c o n c l u s i o n , t h e s t o c h a s t i c m o d e l s p r e s e n t e d t h u s f a r i n t h e l i t e r a t u r e a r e n o t s a t i s f a c t o r y f o r d e c i s i o n - m a k i n g p u r p o s e s . T h e p r o p o s e d m o d e l s l a c k c o m p u t a t i o n a l t r a c t a b i l i t y f o r l a r g e p r o b l e m s a n d i n a d d i t i o n i n c e r t a i n c a s e s t h e f o r m u l a t i o n h a s b e e n d e v e l o p e d t o f i t t h e t e c h n i q u e r a t h e r t h a n r e f l e c t t h e a c t u a l a s s e t a n d l i a b i l i t y m a n a g e m e n t p r o b l f a c e d b y t h e b a n k . C h a p t e r 3 FORMAL DESCRIPTION OF THE ASSET AND LIABILITY MANAGEMENT (ALM) MODEL 3 . 1 I n t r o d u c t i o n T h e s i z e a n d s t r u c t u r e o f a n a s s e t p o r t f o l i o t h a t a b a n k c a n a c q u i r e i s c o n s t r a i n e d b y t h e u n c e r t a i n t y o f i t s c a s h f l o w s . I n p a r t i c u l a r , d e p o s i t w i t h d r a w a l s m u s t b e s a t i s f i e d o n d e m a n d . I t i s u s u a l l y d i s a d v a n -t a g o u s a n d s o m e t i m e s i m p o s s i b l e f o r b a n k s t o l i q u i d a t e e a r n i n g a s s e t s i n o r d e r t o m e e t u n e x p e c t e d s h o r t a g e s i n c a s h r e q u i r e m e n t s . T h u s a b a n k m u s t h o l d a s u f f i c i e n t p o r t i o n o f i t s a s s e t s i n c a s h a n d l i q u i d a s s e t s t o m e e t u n a n t i c i p a t e d c a s h d r a i n s a s t h e y a r i s e . I t i s i n t h i s e n v i r o n m e n t t h a t a b a n k m u s t f u n c t i o n . T h i s i n v o l v e s a t r a d e - o f f b e t w e e n t h e o p p o r t u n i t y c o s t o f h o l d i n g l o w e r y i e l d i n g l i q u i d a s s e t s a n d t h e p o t e n t i a l l o s s i n c u r r e d i n s e l l i n g a s s e t s p r i o r t o m a t u r i t y . T o i n s u r e a n ' o p t i m a l ' s y n c h r o n i z a t i o n o f t h e m a t u r i t i e s o f a s s e t s a n d l i a b i l i t i e s a n d a n ' o p t i m a l ' r e t u r n o n i n v e s t m e n t , c e r t a i n f e a t u r e s m u s t b e i n c l u d e d i n a n a s s e t a n d l i a b i l i t y m a n a g e m e n t m o d e l . T h e f i r s t f e a t u r e i s m u l t i - p e r i o d i c i t y i n o r d e r t o c a p t u r e t h e s h i f t i n g y i e l d s p r e a d s a c r o s s t i m e , t o i n c o r p o r a t e t h e t r a n s a c t i o n , c o s t s a s s o c i a t e d w i t h c a l l i n g a n d s e l l i n g a s s e t s p r i o r t o m a t u r i t y a n d t o i n c o r p o r a t e a s m o o t h i n g o f n e t c a s h f l o w s a c r o s s t i m e b y m a t c h i n g t h e m a t u r i t y o f a s s e t s w i t h a n t i c i p a t e d 3 2 3 3 c a s h o u t f l o w s . T h e s e c o n d f e a t u r e i s e n v i r o n m e n t a l u n c e r t a i n t y i n o r d e r t o i n c l u d e t h e u n c e r t a i n t y o f c a s h f l o w s ( d e p o s i t s ) a n d i n t e r e s t r a t e s . T h e t h i r d f e a t u r e i s t h e s i m u l t a n e o u s c o n s i d e r a t i o n o f a s s e t s a n d l i a b i l i t i e s i n o r d e r t o s a t i s f y b a s i c a c c o u n t i n g p r i n c i p l e s a n d t o m a t c h t h e l i q u i d i t y q u a l i t i e s o f t h e a s s e t s w i t h t h o s e o f t h e l i a b i l i t i e s . T h e f i n a l f e a t u r e i s t h a t t r a n s a c t i o n c o s t s s h o u l d b e i n c l u d e d i n o r d e r t o i n c o r p o r a t e b r o k e r a g e f e e s a n d o t h e r e x p e n s e s a s s o c i a t e d w i t h t h e p u r c h a s e a n d s a l e o f f i n a n c i a l i n s t r u m e n t s . T h e a p p r o a c h t a k e n i n t h i s d i s s e r t a t i o n t o t h e a s s e t a n d l i a b i l i t y m a n a g e m e n t p r o b l e m h a s a s i t s m o t i v a t i o n t h e C h a m b e r s a n d C h a r n e s f o r m u l a t i o n [ 1 1 ] . H o w e v e r , t h e r e a r e c e r t a i n i n h e r e n t w e a k n e s s e s i n t h e i r f o r m u l a t i o n . F o r e x a m p l e , t h e e x c l u s i o n o f u n c e r t a i n t y [ 7 , 1 5 , 2 1 , 2 8 ] , t h e e x c l u s i o n o f l i a b i l i t i e s a s d e c i s i o n v a r i a b l e s [ 7 , 2 0 ] , t h e u s e o f c o n s e r v a t i v e l i q u i d i t y c o n s t r a i n t s ( a s p r e s c r i b e d b y t h e F e d e r a l R e s e r v e B o a r d [ 2 7 ] ) [ 2 0 ] , t h e a v a i l a b i l i t y o f f u n d s f o r i n v e s t m e n t p u r p o s e s o n l y a t t h e e n d o f a p e r i o d [ 2 0 ] , t h e h o l d i n g o f i n v e s t m e n t s t o m a t u r i t y a n d t h e o m i s s i o n o f d i f f e r e n -t i a t i n g b e t w e e n t h e c o s t s o f v a r i o u s t y p e s o f d e p o s i t s , h a v e b e e n w e l l d o c u m e n t e d i n t h e l i t e r a t u r e . N e v e r t h e l e s s , t h e m o d e l h a s s e r v e d a s a s t a r t i n g p o i n t f o r m a n y a p p l i c a t i o n s t o a c t u a l p r o b l e m s . H o w e v e r , a s t h e l i t e r a t u r e s u r v e y i n C h a p t e r 2 d e m o n s t r a t e d , n o e x i s t i n g m o d e l h a n d l e s t h e s e p r o b l e m s w e l 1 . T h e i n c o r p o r a t i o n o f u n c e r t a i n t y i n a n e f f i c i e n t m a n n e r w a s t h e m a i n d i f f i c u l t y o f r e s e a r c h e r s a t t e m p t i n g t o e x t e n d t h e CC m o d e l . M a n y s t o c h a s t i c o p t i m i z a t i o n m e t h o d s w e r e u n s u c c e s s f u l l y u s e d t o a p p r o a c h t h e p r o b l e m . 1 C o m p u t a t i o n a l t r a c t a b i l i t y w a s t h e o b s t a c l e t h a t c o u l d n o t b e o v e r c o m e . V o r a r e v i e w o f s t o c h a s t i c p r o g r a m m i n g s e e t h e a p p e n d i x a t t h e e n d o f t h i s c h a p t e r . 3 4 A s w a s a l r e a d y n o t e d , t h e W e t s a l g o r i t h m [ 9 5 ] ( s e e a p p e n d i x ) w i l l b e u s e d a s t h e s o l u t i o n t e c h n i q u e t o s o l v e t h e a s s e t a n d l i a b i l i t y m a n a g e -m e n t m o d e l ( A L M ) p r e s e n t e d i n t h e n e x t s e c t i o n . T h e A L M m o d e l w i l l a l l o w a b a n k t o a d d r e s s t h e q u e s t i o n o f u n c e r t a i n t y o f i t s c a s h f l o w s i n a s y s t e m a t i c m a n n e r . A l l o f t h e f e a t u r e s n e c e s s a r y f o r a c o m p r e h e n s i v e a s s e t a n d l i a b i l i t y m a n a g e m e n t m o d e l e n u m e r a t e d e a r l i e r w i l l b e i n c o r p o r a t e d i n t h e m o d e l . T e c h n i q u e s t o o v e r c o m e t h e o t h e r s h o r t c o m i n g s o f t h e CC m o d e l w i l l a l s o b e i n c l u d e d i n t h e A L M m o d e l . 3 . 2 F o r m u l a t i o n o f T h e A L M M o d e l T h e a s s e t a n d l i a b i l i t y m a n a g e m e n t - ( A L M ) m o d e l i s a n i n t e r t e m p o r a l d e c i s i o n - m a k i n g o p t i m i z a t i o n t o o l t o d e t e r m i n e a p o r t f o l i o o f a s s e t s a n d l i a b i l i t i e s o f a b a n k , g i v e n d e t e r m i n i s t i c r a t e s o f r e t u r n s a n d c o s t s ( i n t e r e s t r a t e s ) , a n d r a n d o m c a s h f l o w s ( d e p o s i t s ) . A l t h o u g h t h e A L M p r o b l e m i s e s s e n t i a l l y a c o n t i n u o u s d e c i s i o n p r o b l e m a s p o r t f o l i o s a r e c o n s t a n t l y b e i n g r e v i s e d o v e r t i m e . , t h e c o m p u t a t i o n s a n d a n a l y s i s i n v o l v e d w i t h a c o n t i n u o u s t i m e p r o c e s s a r e i n f e a s i b l e f o r a n o r m a t i v e t o o l . T h e r e -f o r e , t h e A L M m o d e l i s d e v e l o p e d a s a m u l t i - p e r i o d d e c i s i o n p r o b l e m i n w h i c h p o r t f o l i o s a r e d e t e r m i n e d a t c o n s e c u t i v e d i s c r e t e p o i n t s i n t i m e ( f o r e x a m p l e , t h e e n d o f e a c h a c c o u n t i n g p e r i o d ) . T h e A L M m o d e l i s d e v e l o p e d i n a m a t h e m a t i c a l p r o g r a m m i n g f r a m e -w o r k . T h e g e n e r a l f o r m u l a t i o n c a n b e s t a t e d a s : 1 . O b j e c t i v e f u n c t i o n m a x i m i z e t h e n e t p r e s e n t p r o f i t s o f a b a n k m i n u s t h e e x p e c t e d p e n a l t y c o s t s f o r i n f e a s i b i 1 i t y . 3 5 2 . C o n s t r a i n t s a . l e g a l , w h i c h a r e a f u n c t i o n o f t h e b a n k 1 s j u r i s d i c t i o n , b . b u d g e t , w h i c h a r e t h e i n i t i a l c o n d i -t i o n s a n d t h e s o u r c e s a n d u s e s o f f u n d s , c . l i q u i d i t y a n d l e v e r a g e , t o s a t i s f y d e p o s i t w i t h d r a w a l s o n d e m a n d , ( t h e F R B 1 s c a p i t a l a d e q u a c y f o r m u l a f o r m t h e b a s i s o f t h e s e c o n s t r a i n t s ) , d . p o l i c y a n d t e r m i n a t i o n , w h i c h c o n s i s t o f c o n s t r a i n t s u n i q u e t o t h e b a n k a n d c o n d i -t i o n s t o e n s u r e t h e b a n k ' s c o n t i n u i n g e x i s t e n c e a f t e r t h e t e r m i n a t i o n o f t h e m o d e l , a n d e . d e p o s i t f l o w s . C o n s t r a i n t s ( a ) a n d ( b ) a r e d e t e r m i n i s t i c , ( c ) c o n s i s t s o f b o t h d e t e r m i n i s t i c a n d s t o c h a s t i c c o n s t r a i n t s , ( d ) c a n c o n s i s t o f e i t h e r d e t e r m i n i s t i c o r s t o c h a s t i c c o n s t r a i n t s , a n d ( e ) c o n t a i n s o n l y s t o c h a s t i c c o n s t r a i n t s . C h a m b e r s a n d C h a r n e s [ 1 1 ] a n d C o h e n a n d H a m m e r [ 2 0 ] h a v e j u s t i f i e d t h e u s e o f l i n e a r f u n c t i o n s t o m o d e l a b a n k ' s a s s e t a n d l i a b i l i t y m a n a g e -m e n t p r o b l e m . T h u s f r o m t h e p o i n t o f v i e w o f l i n e a r i t y , t h e a p p r o p r i a t e -n e s s o f u s i n g L P U l P i s e s t a b l i s h e d . T h e u n c e r t a i n t y a s p e c t o f L P U U i s j u s t i f i e d w i t h t h e f o l l o w i n g a r g u m e n t . I n t h e b a n k i n g b u s i n e s s , c o n s t r a i n t v i o l a t i o n s d o n o t i m p l y t h a t t h e i n t e r m e d i a r y i s p u t i n t o r e c e i v e r s h i p . R a t h e r t h e b a n k i s a l l o w e d t o r e s t r u c t u r e i t s p o r t f o l i o o f a s s e t s t o r e g a i n f e a s i b i l i t y a t s o m e c o s t ( p e n a l t i e s ) . T h e A L M p r o b l e m f i t s w e l l a s a s t o -c h a s t i c l i n e a r p r o g r a m w i t h s i m p l e r e c o u r s e m o d e l . A s w a s s t a t e d p r e v i o u s l y , t h e f o r m u l a t i o n i s a m u l t i p e r i o d m o d e l . H o w e v e r , t h e m o d e l i s a z e r o o r d e r d e c i s i o n r u l e m o d e l i n t h a t d e c i s i o n s L i n e a r p r o g r a m m i n g u n d e r u n c e r t a i n t y . 3 6 f o r p e r i o d l , \u00C2\u00BB \u00C2\u00BB * , n a r e m a d e a s a n i n s t a n t r e v i s i o n i n p e r i o d 1 i n s u c h a w a y t h a t t o t a l p r o f i t s m i n u s e x p e c t e d p e n a l t y c o s t s i n p e r i o d l , \u00C2\u00AB \u00C2\u00BB \u00C2\u00BB , n i s m a x i m i z e d . I t s h o u l d b e n o t e d t h a t t h e d e c i s i o n - m a k e r i s e s s e n t i a l l y i n t e r e s t e d i n t h e i m m e d i a t e r e v i s i o n o f t h e b a n k ' s a s s e t s a n d l i a b i l i t i e s . T h e A L M m o d e l i n c o r p o r a t e s i m m e d i a t e r e v i s i o n b y s e t t i n g t i m e s 0 a n d 1 a n a r b i t r a r i l y s m a l l t i m e p e r i o d a p a r t . T h e p o i n t 0 , i n t i m e , r e f e r s t o t h e b a n k ' s i n i t i a l p o s i t i o n a n d t h e p o i n t 1 r e f e r s t o t h e b a n k ' s p o s i t i o n i m m e d i a t e l y a f t e r r u n n i n g t h e m o d e l . I n p r a c t i c e t h e m o d e l s h o u l d b e ' r o l l e d o v e r ' c o n t i n u o u s l y . A l s o t o p a r t i a l l y o v e r c o m e t h e d r a w b a c k s o f a s t a t i c m o d e l , t h e d e c i s i o n v a r i a b l e s a r e d e f i n e d i n s u c h a m a n n e r t h a t a s e c u r i t y c a n b e p u r c h a s e d i n o n e t i m e p e r i o d a n d s o l d i n o n e o r m o r e s u b s e q u e n t p e r i o d s . I n a d d i t i o n , t h e r e c o u r s e a s p e c t o f t h e m o d e l g i v e s i t a d y n a m i c f l a v o u r . T h e m o d e l b e i n g t w o - s t a g e , m e a n s t h a t i n i t i a l l y t h e d e c i s i o n v a r i a b l e s a r e c h o s e n . N e x t t h e s t o c h a s t i c v a r i a b l e s a r e o b s e r v e d . T h i s d e t e r m i n e s t h e r e c o u r s e v a r i a b l e s ( i n o r d e r t o r e c o v e r f e a s i b i l i t y ) a n d t h e i r c o r r e s p o n d i n g p e n a l t i e s . T h e p e n a l t y i s a f u n c t i o n o f b o t h t h e c o n -s t r a i n t v i o l a t e d a n d t h e m a g n i t u d e o f v i o l a t i o n . T h e r e c o u r s e c o s t h a s t h e e f f e c t o f r e s t r a i n i n g ' a g g r e s s i v e ' c h o i c e s o f d e c i s i o n v a r i a b l e s i f t h e c o s t s i n v o l v e d w i t h r e g a i n i n g f e a s i b i l i t y o u t w e i g h t h e b e n e f i t s . T h u s , t h e ' r o l l i n g o v e r ' o f t h e A L M m o d e l , d e f i n i n g t h e v a r i a b l e s s o a s t o g i v e t h e m f l e x i b i l i t y a n d t h e r e c o u r s e a s p e c t o f S L P R , a r e t h e d y n a m i c f e a t u r e s o f t h e A L M m o d e l . T h e A L M m o d e l c a n n o w b e p r e s e n t e d ( s e e p a g e s 3 9 , 4 0 , 4 1 , 4 2 ) . 3 7 N o t a t i o n f o r A L M M o d e l a s s e t k p u r c h a s e d i n p e r i o d i s o l d i n p e r i o d j ; k = l , . . . , K ; i = 0 , . . . , n - l ; j = i + l , . . . , n , i n i t i a l h o l d i n g s o f s e c u r i t y k , s e c u r i t y p u r c h a s e d i n p e r i o d i a n d t o b e h e l d b e y o n d t h e h o r i z o n o f t h e m o d e l , n e w d e p o s i t s o f t y p e d i n p e r i o d i ; d = l , * \u00C2\u00AB ' , D , i n i t i a l h o l d i n g s o f d e p o s i t t y p e d , f u n d s b o r r o w e d i n p e r i o d i , s h o r t a g e i n p e r i o d j o f s t o c h a s t i c c o n s t r a i n t t y p e s , s u r p l u s i n p e r i o d j o f s t o c h a s t i c c o n s t r a i n t t y p e s , p r o p o r t i o n a l p e n a l t y c o s t a s s o c i a t e d w i t h y * , J s p r o p o r t i o n a l p e n a l t y c o s t a s s o c i a t e d w i t h y~ , J s p a r a m e t e r f o r s h r i n k a g e , u n d e r n o r m a l e c o n o m i c c o n d i t i o n s , i n : p e r i o d j o f a s s e t t y p e k p u r c h a s e d i n p e r i o d i , p a r a m e t e r f o r s h r i n k a g e , u n d e r s e v e r e e c o n o m i c c o n d i t i o n s , i n p e r i o d j o f a s s e t t y p e k p u r c h a s e d i n p e r i o d i , p r o p o r t i o n a l t r a n s a c t i o n c o s t o n a s s e t k , w h i c h i s e i t h e r p u r c h a s e d o r s o l d i n p e r i o d i , r e t u r n o n a s s e t k p u r c h a s e d i n p e r i o d i , t a x r a t e o n c a p i t a l g a i n s ( l o s s e s ) i n p e r i o d j , m a r g i n a l t a x r a t e o n i n c o m e i n p e r i o d j , p r o p o r t i o n a l c a p i t a l g a i n ( l o s s ) o f s e c u r i t y k p u r c h a s e d i n p e r i o d i a n d s o l d i n p e r i o d j , 3 8 Y d - t h e a n t i c i p a t e d f r a c t i o n o f d e p o s i t s o f t y p e d w i t h d r a w n u n d e r a d v e r s e e c o n o m i c c o n d i t i o n s , c . j - r a t e p a i d o n d e p o s i t s o f t y p e d , p . - d i s c o u n t r a t e f r o m p e r i o d i t o p e r i o d o , K 1 - s e t o f c u r r e n t a s s e t s a s s p e c i f i e d b y t h e B r i t i s h C o l u m b i a C r e d i t U n i o n A c t , K i - s e t o f p r i m a r y a n d s e c o n d a r y a s s e t s a s d e f i n e d i n t h e C a p i t a l A d e q u a c y F o r m u l a , . K 2 - s e t o f m i n i m u m r i s k a s s e t s a s d e f i n e d i n t h e C a p i t a l A d e q u a c y F o r m u l a , K 3 - s e t o f i n t e r m e d i a t e r i s k a s s e t s a s d e f i n e d i n t h e C a p i t a l A d e q u a c y F o r m u l a , q n - - p e n a l t y r a t e f o r t h e p o t e n t i a l w i t h d r a w a l o f f u n d s , w h i c h a r e n o t c o v e r e d b y a s s e t s i n K i u \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 u , P . - l i q u i d i t y r e s e r v e s f o r t h e p o t e n t i a l w i t h d r a w a l o f f u n d s n o t c o v e r e d b y a s s e t s i n . . . u K i 5 k . - m i t h m o r t g a g e , a n d 5 . - d i s c r e t e r a n d o m v a r i a b l e i n p e r i o d j o f s t o c h a s t i c c o n s t r a i n t J S t y p e s w h e r e s e S . 3 9 T h e A L M M o d e l M a x x , y , b K I k = l . L x o j r j , r o t ' - ^ + z o V , - T J ^ j=2 0 J ^ \u00C2\u00A3=2 + x 01 z 0 1 t l - T . ) n - 1 n i = l j = i + l J ^ \u00C2\u00A3 = i + l 1 - t , K \u00C2\u00A3 1 J 1 - T . J J \u00C2\u00A3 ' r \u00C2\u00A3 \u00E2\u0080\u00A2 J Z * l r k ( , . T ) P : 1 = T 1 \u00C2\u00A3 = 1+1 D d = l 1 -^ d 1 \" Y , j - l d c . p . + I . 5 y c p . + I I y d. j = l i = l j = i 1 -Y d ) i - y. j - i C . p . b Q e g P ] - ^ b . C j b p . - E?- m i n \u00C2\u00A3 \u00C2\u00A3 [ y V j = l s e S + + -[ P j s y j s p j s y j s J S u b j e c t t o : ( a ) L e g a l c o n s t r a i n t s I I I x k 0 + x k L L L -1 0 l o k e K 1 i=0 ^ \u00C2\u00A3=2 - . 1 D I d = l yo + i - yr J d yo , y i l + bo + b i > o, j = 1 , I I k e K 1 i=0' \u00C2\u00A3 = j + l \u00C2\u00A3 l \u00C2\u00BB I x\ + X v . , 1 \" - . 1 D r j-1 d d = l l-i=0 1 1 2 i - y. , j - i - l + b > 0 , j = 2 , - \u00C2\u00AB \u00C2\u00AB , n , ( b ) B u d g e t c o n s t r a i n t s ( i ) I n i t i a l h o l d i n g s 4 0 j = l * n - ; + x n = x !L> k = 1 , * ' * , K , U j 0 \u00C2\u00B0 \u00C2\u00B0 0 0 5 5 5 y j j = y d , d = 1 , - - - , D J0 Joo ( i i ) S o u r c e s a n d u s e s K I k = l 1=2 1 * 1 1 + t * 0 1 1 + z 01 1 - T n 1 + z 01 D + I d = l - d d-r Y d y 0 y l b , = 0 , J = 1 , K r n , , 1 + t k j - 1 I i = 0 i At*A. \u00C2\u00A3 = j 1 1 + k + X . . 1 J 1 + z r 1 - T . J 1 + z * . h D I d = l I y i i = 0 1 - Y . j - i - 2 I d ) 2 1 - Id d d 2 Y d ] d d d _ y j - 1 C j - U C j - 1 2 + b J - l 1 + i u b , = 0 , j = 2 , ( c ) L i q u i d i t y c o n s t r a i n t s ( i ) I I k e K j . 1=0 v k k k k ) x . . a . . + x . a.. + b . + J D I \ d = l J y i i = 0 1 1 \" Yr j - i - 1 I d ) 2 q l a - H j > ( i i ) . - I I k e K i U K z i = 0 \u00C2\u00A3 = j + l k k k k x . . a . . + x . a.. + b . + J D -d = l J ; 1 d I y i i = 0 1 \" Y , j - i - 1 1 -y \u00E2\u0080\u00A2 <2j P 2 j ( i i i ) - I I k e K i U K 2 U K 3 i = 0 v k k x k k + b d + D I Y D d = l a i = 0 1 > j - i - 1 r Y ^ y \u00E2\u0080\u00A2 Id + i j . 2 j 2 1 - q 3 j P 3 j ( i v ) K I k = l j I i = 0 n 1 J J V + 1 - s . . . ' j s y \u00E2\u0080\u00A2 > P . . + P 0 . + P , . + b . + l j 2 j 3 j 3 D I d = l J\u00E2\u0080\u0094 j - i - 1 d T \" Y d . 1 2 y j i = 0 4 2 ( d ) P o l i c y c o n s t r a i n t s J \u00E2\u0080\u00A21 I \=0 n k , k ,~ I x . m l + x . m 1 \u00C2\u00A3 = j + l U 1 0 0 i = 0 n L v m2 . \u00C2\u00A3 = j + l I J C + y . - y \" < 5 . \u00C2\u00BB j = i , j s j s - J S ( e ) D e p o s i t F l o w s d + K] d y - + J i = 0 v d 1 \" Y , J - i + y . - y . = - \u00C2\u00A3 . , j s J j s \ j s ' j = l , \u00E2\u0080\u0094 9 n ; d = l , \u00E2\u0080\u0094 , D . 4 3 T h e o b j e c t i v e f u n c t i o n c o n s i s t s o f f i v e e x p r e s s i o n s . T h e f i r s t e x p r e s s i o n i s t h e d i s c o u n t e d r e t u r n s ( n e t o f t a x e s ) a n d c a p i t a l g a i n s a n d l o s s e s ( n e t o f t a x e s ) o n a s s e t s . T h e s e c o n d e x p r e s s i o n r e f e r s t o t h e n e t d i s -c o u n t e d c o s t s o f d e p o s i t s . T h e t h i r d a n d f o u r t h e x p r e s s i o n s r e f e r t o t h e c o s t o f d i r e c t b o r r o w i n g e i t h e r f r o m o t h e r b a n k s o r f r o m a c e n t r a l b a n k . T h e f i n a l e x p r e s s i o n i s t h e s u m o f t h e e x p e c t e d p e n a l t i e s f o r v i o l a t i n g t h e s t o c h a s t i c c o n s t r a i n t s . T h e r e a r e n o d i s c o u n t f a c t o r s i n c o r p o r a t e d i n t o t h e c o n s t r a i n t s s i n c e e a c h c o n s t r a i n t r e f e r s t o c o n d i t i o n s i n o n l y o n e p e r i o d . T h e A L M m o d e l t r e a t s t h e f i r s t t w o t y p e s o f c o n s t r a i n t s , l e g a l a n d b u d g e t , s t r i c t l y a s d e t e r m i n i s t i c . T h e l e g a l c o n s t r a i n t s h o w n s t a t e s t h a t t h e c u r r e n t a s s e t s c a n n o t b e l e s s t h a n 10% o f t h e t o t a l l i a b i l i t i e s l e s s r e s e r v e s , s u r p l u s a n d e q u i t y . 1 T h e l e g a l c o n s t r a i n t s a r e , o f c o u r s e , p e c u l i a r t o t h e l o c a l e o f t h e i n s t i t u t i o n b e i n g s t u d i e d . T h e b u d g e t c o n s t r a i n t s i n c l u d e t h e i n i t i a l c o n d i t i o n s a n d a s t a t e m e n t o f t h e a c c o u n t i n g i d e n t i t y - u s e s o f f u n d s a r e e q u a l t o s o u r c e s o f f u n d s . T h e l i q u i d i t y c o n s t r a i n t s , a s d e v e l o p e d i n C h a p t e r 2 , f o l l o w f r o m t h e F e d e r a l R e s e r v e B o a r d ' s c a p i t a l a d e q u a c y f o r m u l a . T h e r e q u i r e -m e n t t h a t t h e m a r k e t v a l u e o f a b a n k ' s a s s e t s i s a d e q u a t e t o m e e t d e p o s i t o r s ' w i t h d r a w a l c l a i m s d u r i n g a d v e r s e e c o n o m i c c o n d i t i o n s i s t h e p r i n c i p a l c o n s t r a i n t i n t h e c a p i t a l a d e q u a c y f o r m u l a . I n o r d e r t o d e v e l o p t h i s c o n s t r a i n t , l i q u i d i t y r e s e r v e s ( f o r a d v e r s e e c o n o m i c c o n d i t i o n s ) a r e f i r s t d e f i n e d . T h e f i r s t t h r e e l i q u i d i t y c o n s t r a i n t s i n t h e A L M m o d e l a r e ^ A s d e f i n e d b y t h e B r i t i s h C o l u m b i a C r e d i t U n i o n A c t [ 8 ] . 2 T h e s a m e n o t a t i o n i s u s e d a s i n C h a p t e r 2 , S e c t i o n 2 . 2 . 4 4 I \u00C2\u00AB k k k e K i U - . - u K . , 1 = 1 , 2 , 3 . T h e p r i n c i p a l c o n s t r a i n t o f t h e c a p i t a l a d e q u a c y f o r m u l a i s 1 = 1 1 - 3 , x . > y p . + 1 - i = i 1 t o t a l r i g h t h a n d s i d e o f b a l a n c e - s u r p l u s - e q u i t y -s h e e t T h e c o n s t r a i n t s t a t e s t h a t t h e m a r k e t v a l u e o f t h e b a n k ' s a s s e t s s h o u l d b e e q u a l o r g r e a t e r t h a n t h e l i q u i d i t y r e s e r v e s f o r d i s i n t e r m e d i a t i o n u n d e r s e v e r e e c o n o m i c c o n d i t i o n s p l u s a l l l i a b i l i t i e s . T h i s c o n s t r a i n t i s i n f a c t t h e l a s t l i q u i d i t y c o n s t r a i n t i n t h e A L M m o d e l . A l t h o u g h t h i s c o n -s t r a i n t i s n o t s t o c h a s t i c i n n a t u r e , a b a n k p o r t f o l i o m a n a g e r m a y v i o l a t e i t b e c a u s e t h e c a p i t a l a d e q u a c y f o r m u l a a s s e t f o r t h b y t h e F R B i s a s u g g e s t e d g u i d e l i n e f o r ' s o u n d ' b a n k m a n a g e m e n t r a t h e r t h a n a s t r i c t 3 r e g u l a t i o n . T h e p e n a l t y f o r a v i o l a t i o n o f t h i s c o n s t r a i n t i s \u00C2\u00A3 q . i = l ( a s p r e s c r i b e d b y t h e F R B ) . T h i s ' p s u e d o - s t o c h a s t i c ' t r e a t m e n t o f t h e F R B ' s r e g u l a t i o n a l l o w s t h e c o n s t r a i n t t o b e v i o l a t e d w h e n t h e b e n e f i t s o f v i o l a t i o n e x c e e d t h e c o s t s . I n t h i s m a n n e r , t h e c r i t i c i s m , l e v e l l e d a t m o d e l l e r s u s i n g F R B ' s c o n s e r v a t i v e c o n s t r a i n t s , c a n b e r e s o l v e d i n a s y s t e m a t i c m a n n e r . T h e f o u r t h s e t o f c o n s t r a i n t s i s a l s o p s u e d o - s t o c h a s t i c . T h e s e c o n s t r a i n t s a r e i n t r o d u c e d t o c a p t u r e t h e i n t e r n a l o p e r a t i o n a l p o l i c y o f t h e i n s t i t u t i o n m o d e l l e d . I n r e a l i t y m i n o r c o n s t r a i n t v i o l a t i o n s o f b a n k p o l i c i e s a r e u s u a l l y t o l e r a b l e w h i l e m o r e s e v e r e v i o l a t i o n s a r e i n c r e a s i n g l y l e s s t o l e r a b l e . T h e i n t r o d u c t i o n o f a p i e c e - w i s e l i n e a r c o n v e x p e n a l t y f u n c t i o n ( v i a a d d i t i o n a l c o n s t r a i n t s ) c a n c a p t u r e t h e d e p e n d e n c y b e t w e e n t h e p e n a l t y c o s t s a n d t h e e x t e n t o f t h e p o l i c y v i o l a t i o n s , 4 5 T h i s i s a c c o m p l i s h e d b y t h e a d d i t i o n o f s u p p l e m e n t a r y c o n s t r a i n t s t o r e f l e c t t h e i n c r e a s e d s e r i o u s n e s s i n t h e m a g n i t u d e o f c o n s t r a i n t v i o l a t i o n s . T h e f i n a l s e t o f c o n s t r a i n t s , d e p o s i t f l o w s , a r e s t o c h a s t i c . S i n c e d e p o s i t f l o w s a r e c o n t i n u a l l y t u r n e d o v e r a n d b e a r v a r i o u s r a t e s o f i n t e r e s t ( t e r m d e p o s i t s ) , t h e m o d e l h a s t o r e f l e c t t h e a c t u a l ( a n d n o t n e t ) f l o w s d u r i n g a n a c c o u n t i n g p e r i o d . T h i s p r o p e r t y o f t h e p r o b l e m w a s - i n c o r p o r a t e d i n t h e m o d e l b y h a v i n g a p r o p o r t i o n a l o u t f l o w ( y ) ^ o f ' o l d f u n d s ' d u r i n g e a c h p e r i o d . T h e t h r e e t y p e s o f l i a b i l i t y e x p r e s s i o n s i n t h e A L M f o r m u l a t i o n w i l l n o w b e d e v e l o p e d . F i r s t , c o n s i d e r t h e d e p o s i t f l o w c o n s t r a i n t s . T h e s e c o n s t r a i n t s r e p r e s e n t t h e t o t a l a m o u n t o f n e w d e p o s i t s i n t h e j t h p e r i o d . T h e t o t a l a m o u n t o f n e w d e p o s i t s o f t y p e d g e n e r a t e d i n p e r i o d j i s y = BS - I y 1 - Y J J i = o ^ J o r d + V d f , D C d y . + ) y . 1 - Y J = B S . . J i = 0 H d J J w h e r e y^ i s t h e t o t a l a m o u n t o f n e w t y p e d d e p o s i t s j , Y D i s t h e a n n u a l r a t e o f w i t h d r a w a l o f t y p e d d e p o s i t s , a n d BS'? i s t h e d i s c r e t e r a n d o m v a r i a b l e r e p r e s e n t i n g b a l a n c e s h e e t f i g u r e o f t y p e d d e p o s i t s a t t h e e n d o f t h e j t h p e r i o d . T h e s e c o n d t y p e o f l i a b i l i t y e x p r e s s i o n r e p r e s e n t s t h e t o t a l a m o u n t o f d e p o s i t s o u t s t a n d i n g d u r i n g a p e r i o d . S i n c e t h e m o d e l i s d i s c r e t e , a n a p p r o x i m a t i o n t o t h e c o n t i n u o u s f l o w s i s m a d e b y a s s u m i n g t h a t h a l f o f a ^ S t a t i s t i c a l l y c a l c u l a t e d b y t h e F R B a n d c o r r o b o r a t e d f o r u s e i n B r i t i s h C o l umb i a i n [ 2 5 ] . 4 6 p e r i o d ' s n e t f l o w s a r r i v e a t t h e b e g i n n i n g o f t h e p e r i o d a n d t h e o t h e r h a l f a r r i v e a t t h e b e g i n n i n g o f t h e n e x t p e r i o d . T h u s d u r i n g t h e f i r s t p e r i o d , t h e f u n d s a v a i l a b l e a r e e q u a l t o d x y0 + 1 Yd y d 2 f o r p e r i o d j o r j - l i = 0 j - i - 1 1 - Y J - 1 d d y \u00E2\u0080\u00A2 y \u00E2\u0080\u00A2 I y i i = 0 1 -Y d l j - i - i y \" + \u00E2\u0080\u0094 1 n. T h e a b o v e e x p r e s s i o n i s u s e d i n t h e o b j e c t i v e f u n c t i o n , l e g a l c o n s t r a i n t a n d l i q u i d i t y c o n s t r a i n t s . T h e t h i r d l i a b i l i t y e x p r e s s i o n i s t h e i n c r e m e n t a l i n c r e a s e ( d e c r e a s e ) o f d e p o s i t s f r o m o n e p e r i o d t o t h e n e x t . T h i s i n c r e m e n t a l d i f f e r e n c e i s u s e d i n t h e s o u r c e s a n d u s e s c o n s t r a i n t . F o r p e r i o d j t h e i n c r e m e n t a l d i f f e r e n c e i s 4 7 3 . 3 U s e o f t h e A L M M o d e l B e f o r e i m p l e m e n t i n g t h e A L M m o d e l , v a r i o u s i n p u t s t o t h e m o d e l h a v e t o b e d e t e r m i n e d . T h e d a t a r e q u i r e d b y t h e m o d e l i n c l u d e : 1) t h e i d e n t i f i c a t i o n o f t h e a s s e t s i n w h i c h t h e b a n k c a n p o t e n t i a l l y i n v e s t ( o r a t l e a s t a r e p r e s e n t a t i v e g r o u p o f a s s e t s ) , 2 ) t h e p o i n t e s t i m a t e s o f t h e r e t u r n s o n t h e s e a s s e t s , 3 ) t h e p o i n t e s t i m a t e s o f c a p i t a l g a i n s ( l o s s e s ) a s a f u n c t i o n o f t h e t i m e t h e b a n k h o l d s t h e a s s e t s , 4 ) t h e i d e n t i f i c a t i o n o f t h e l i a b i l i t i e s w h i c h t h e b a n k c a n p o t e n t i a l l y s e l l , 5 ) t h e p o i n t e s t i m a t e s o f t h e c o s t s o f t h e s e l i a b i l i t i e s , 6 ) t h e r a t e a t w h i c h d e p o s i t s a r e w i t h d r a w n , 7) a n e s t i m a t e d w e i g h t e d c o s t o f f u n d s t o d e t e r m i n e t h e d i s c o u n t r a t e , 8 ) ,. t h e p e r t i n e n t l e g a l c o n s t r a i n t s , 9 ) t h e p a r a m e t e r s u s e d i n t h e d e v e l o p m e n t o f t h e l i q u i d i t y c o n s t r a i n t s , 1 0 ) t h e p o l i c y c o n s t r a i n t s u s e d b y t h e b a n k , 11) t h e e s t i m a t e s o f t h e m a r g i n a l d i s t r i b u t i o n s o f t h e s t o c h a s t i c r e s o u r c e s , a n d 1 2 ) t h e u n i t p e n a l t i e s i n c u r r e d f o r h a v i n g a s h o r t a g e o r a s u r p l u s i n t h e s t o c h a s t i c c o n s t r a i n t s . R e m a r k s , a r e i n o r d e r , a b o u t t h e c h a r a c t e r i s t i c s o f c e r t a i n o f t h e a b o v e i n p u t s . S i n c e t h e S L P R m o d e l h a s a s e p a r a b l e o b j e c t i v e o n l y t h e m a r g i n a l d i s t r i b u t i o n s o f t h e c o m p o n e n t s o f t h e r e s o u r c e v e c t o r a r e n e e d e d t o f i n d t h e o p t i m a l s o l u t i o n . T h i s c h a r a c t e r i s t i c o f S L P R i s m o s t i m p o r t a n t s i n c e i n m o s t p r o b l e m s t r i e c o r r e l a t i o n s o f t h e c o m p o n e n t s o f t h e r a n d o m v e c t o r w o u l d h a v e t o b e i n c o r p o r a t e d i n t h e s o l u t i o n t e c h n i q u e 1 ^ See trie appendix a t the end of the chapter fur a~ f u r t h e r aY 'scussTbTr 4 8 T h e s h o r t a g e ( y + ) a n d s u r p l u s ( y ~ ) v a r i a b l e s h a v e v e r y s p e c i f i c m e a n i n g s i n t h e A L M f o r m u l a t i o n . C o n s i d e r a r e a l i z a t i o n o f t h e r a n d o m d e p o s i t C d s - I f d + V d J i = 0 j s t h e n t h i s w o u l d i m p l y t h a t y + > 0 a n d y ~ = 0 , i f p + + p~ > 0 . y ~ w o u l d b e i n t e r p r e t e d a s t h e a m o u n t o f f u n d s t h a t c o u l d h a v e b e e n u s e d f o r i n v e s t m e n t p u r p o s e s i n t h e A L M . S i n c e t h e c o s t o f d e p o s i t s i s u s u a l l y l o w e r t h a n t h e r e t u r n s - o n a s s e t s , t h e b a n k w o u l d w a n t t o u t i l i z e a l l a v a i l a b l e f u n d s . A p e n a l t y p + > 0 f o r t h e o p p o r t u n i t y c o s t c a n b e d e t e r m i n e d b y a s s u m i n g t h a t t h e f u n d s n o t u s e d c a n b e i n v e s t e d i n - e a r n i n g a s s e t s . T h e y + d o l l a r s w o u l d b e a v a i l a b l e a t s o m e r a t e c a n d c o u l d t h e n b e i n v e s t e d i n s o m e a s s e t a t a r a t e r . T h e p e n a l t y , p + , w o u l d b e e q u a l t o ( r - c ) d i s -c o u n t e d t o p o i n t 0 p l u s t h e n e t d i s c o u n t e d r e t u r n s o n y + ( r - c ) t o t h e h o r i z o n o f t h e m o d e l ( t h a t i s t h e p r o f i t s t h a t c o u l d . h a v e b e e n g e n e r a t e d ) . On t h e o t h e r h a n d , i f J i = 0 Y d J - i d , ^ j s t h e n t h i s w o u l d i m p l y t h a t y > 0 a n d y + = 0 , t h a t i s t h a t a s u r p l u s w o u l d o c c u r . I n t h i s c a s e , t h e b a n k w o u l d h a v e t o d i v e s t i t s e l f o f s o m e e a r n i n g a s s e t s . T h e c o s t , p ~ , o f t h i s a c t i o n w o u l d b e e q u a l t o ( r - c ) d i s c o u n t e d t o p o i n t 0 p l u s t h e n e t d i s c o u n t e d r e t u r n s o n y \" ( r - c ) t o t h e h o r i z o n o f t h e m o d e l ( t h a t i s , t h e p r o f i t s t h a t w o u l d h a v e b e e n g e n e r a t e d w i t h u n a v a i l -a b l e f u n d s ) . 4 9 O n e p o i n t t o n o t i c e i s t h a t i n t h e a b o v e c a s e , b o t h p a n d p \" a r e g r e a t e r t h a n 0 . I n o t h e r w o r d s , p r o f i t i s f o r f e i t e d i f e i t h e r n o t e n o u g h o r t o o m u c h i s i n v e s t e d . A k e y i s s u e i s w h a t r a n d c s h o u l d b e u s e d t o d e t e r m i n e t h e p e n a l t i e s . T h i s p o i n t w i l l b e a d d r e s s e d i n C h a p t e r 4 , w h e r e a c a s e s t u d y o f t h e A L M f o r m u l a t i o n w i l l b e p r e s e n t e d . T h e F e d e r a l R e s e r v e B o a r d ' s p a r a m e t e r s u s e d i n i t s c a p i t a l a d e q u a c y e x a m i n a t i o n s o f a b a n k ' s p o r t f o l i o o f a s s e t s a n d l i a b i l i t i e s a r e w e l l k n o w n , s e e f o r e x a m p l e [ 2 7 ] . B e f o r e u t i l i z i n g t h e i r f i g u r e s i n t h e A L M f o r m u l a t i o n , e s t i m a t e s o f t h e s e p a r a m e t e r s s h o u l d b e m a d e i n o r d e r t o t e s t t h e i r a p p l i c a b i l i t y t o t h e p r o b l e m a t h a n d . When u s i n g t h e W e t s ' a l g o r i t h m t o s o l v e a n a c t u a l p r o b l e m , t w o a d d i t i o n a l n u m b e r s a r e r e q u i r e d f o r e a c h s t o c h a s t i c c o n s t r a i n t , a n d 3 ^ . T h e r e a l i z a t i o n s a r e o r d e r e d < 5 ^ < * * ' < ^ i k - \" ^ e a a n c ' 1 3 a r e c h o s e n s u c h t h a t a.. < E^-j a n d 3.. > 5 ^ \u00E2\u0080\u00A2 A l s o t h e a a n d 3 a r e c h o s e n s o t h a t o p t i m a l i t y i s i n t h i s r e g i o n . T h e W e t s ' a l g o r i t h m s o l v e s a s p e c i a l t y p e o f l i n e a r p r o g r a m . S o a t a n y p o i n t i n t i m e t h e d u a l s g e n e r a t e d c o r r e s p o n d t o t h e d u a l s o f a d e t e r m i n i s t i c l i n e a r p r o g r a m . T h e d u a l o f a s t o c h a s t i c c o n s t r a i n t , a t o p t i m a l i t y , m a y b e n e g a t i v e . T h i s i m p l i e s t h a t a n i n c r e a s e i n t h e v a l u e o f t h e r i g h t h a n d s i d e w o u l d r e s u l t i n a d e c r e a s e i n t h e v a l u e o f t h e s o l u -t i o n ( b e n e f i t ) . H o w e v e r , t h e r e a s o n t h a t t h e s t o c h a s t i c r e s o u r c e c o m p o n e n t d o e s n o t i n c r e a s e i s t h a t t h e m a r g i n a l p e n a l t y c o s t e x c e e d s t h e b e n e f i t s a c c r u e d f r o m t h e i n c r e a s e . 5 0 3 . 4 A p p e n d i x 1 A m o s t i m p o r t a n t a s p e c t o f a s s e t a n d l i a b i l i t y m a n a g e m e n t o p t i m i -z a t i o n m o d e l s i s t h e i n c l u s i o n o f u n c e r t a i n t y . T h e r e h a v e b e e n a n u m b e r o f t e c h n i q u e s d e v e l o p e d t o s o l v e s t o c h a s t i c o p t i m i z a t i o n m o d e l s . T h i s a p p e n d i x s e r v e s t o h i g h l i g h t t h e m a j o r s t o c h a s t i c l i n e a r p r o g r a m m i n g s o l u -t i o n a p p r o a c h e s ( s t o c h a s t i c w i t h r e s p e c t t o r i g h t h a n d s i d e s ) . A l s o , s i n c e t h e A L M f o r m u l a t i o n p r e s e n t e d i n t h i s d i s s e r t a t i o n u s e s t h e s t o c h a s t i c l i n e a r p r o g r a m m i n g w i t h s i m p l e r e c o u r s e ( S L P R ) a p p r o a c h , p a r t i c u l a r e m p h a s i s i s p l a c e d o n S L P R , i t s c h a r a c t e r i s t i c s a n d t h e W e t s ' a l g o r i t h m u s e d t o s o l v e i t . L e t ( f l , F , u ) b e a p r o b a b i l i t y s p a c e . L e t 5 b e a r a n d o m v a r i a b l e d e f i n e d f r o m Jl t o a f i n i t e s u b s e t o f R m . T h e d i s t r i b u t i o n f u n c t i o n o f \u00C2\u00A3 i s d e n o t e d b y F , F ( z ) = u ( { w : KM < z } ) = P U < z } w h e r e z e R m a n d P i s t h e i n d u c e d m e a s u r e o n R m . T h e m o d e l t o b e c o n s i d e r e d i n t h i s a p p e n d i x a s s u m e s 5 i s a d i s c r e t e r a n d o m v a r i a b l e w i t h p o s s i b l e r e a l i z a t i o n s < <\u00C2\u00BB. A g e n e r a l a b s t r a c t f o r m u l a t i o n i s { m i n E [ f ( x , \u00C2\u00A3 ) ] | g ( A x - \u00C2\u00A3 ) > 0 } w h e r e A x > 0 ^ i s m x n , x e R n i s t h e d e c i s i o n v a r i a b l e , f : R n x & -*\u00E2\u0080\u00A2 R a n d g : R m + R k . F o u r b a s i c a p p r o a c h e s h a v e b e e n s u g g e s t e d i n t h e m a t h e m a t i c a l l i t e r a t u r e t o s o l v e t h i s m o d e l . 1 . T h e F a t F o r m u l a t i o n [ 5 5 ] . T h i s p r o c e d u r e i s t o s o l v e m i n c ' x x > 0 s . t . A x = i = l , \u00C2\u00AB \u00C2\u00AB \u00C2\u00AB , L , 51 t h e f e a s i b l e s e t i s L K = x : x > 0 , x e n { X : A x = g 1 } - - , i = l J w h e r e K i s t h e i n t e r s e c t i o n o f t h e f e a s i b l e s e t s f o r e a c h r e a l i z a t i o n ( t h a t i s t h e s a f e s t r e g i o n ) . O n e a d v a n t a g e o f t h e f a t f o r m u l a t i o n i s t h a t i t i s d e t e r m i n i s t i c . A ; m a j o r d i s a d v a n t a g e i s t h a t K m a y b e e m p t y o r u n d u l y . r e s t r i c t e d . T h e f a t f o r m u l a t i o n l e a d s t o : v e r y l a r g e d e t e r m i n i s t i c e a u i v - ' a l e n t s . S i n c e t h e B r a d l e y i a n d C r a n e [ 5 , - 6 , 7 ] . m o d e l ' s s o l u t i o n t o - t h e i m m e d i a t e r e v i s i o n p r o b l e m h a s t o s a t i s f y t h e c o n s t r a i n t s f o r e v e r y p o s s i b l e r e a l i z a t i o n , i t i s a f a t f o r m u l a t i o n . 2 . C h a n c e - C o n s t r a i n e d P r o g r a m m i n g \u00C2\u00A3 1 2 , 1 3 1 . T h i s t e c h n i q u e d e f i n e s x t o b e f e a s i b l e i f i t s a t i s f i e s t h e c o n s t r a i n t s , A x \u00C2\u00A7 w i t h a c e r t a i n p r e s p e c i f i e d p r o b a b i l i t y . T h e r e a r e s e v e r a l t y p e s o f c h a n c e - c o n s t r a i n t s w h i c h m a y b e f o r m u l a t e d . T h e f o l l o w -i n g d i s c u s s i o n f o c u s e s o n t w o t y p e s o f c o n s t r a i n t s : 1 ) m a r g i n a l c h a n c e - c o n s t r a i n t s : P { A . . x > \u00C2\u00A3..}\u00E2\u0080\u00A2. > a. , i = l , \u00C2\u00AB \u00C2\u00AB \u00C2\u00AB , m , w h e r e A i # i s t h e i t h r o w o f A , ^ i s t h e i t h c o m p o n e n t o f \u00C2\u00A3 , a n d a . e [ 0 , 1 ] a r e g i v e n , a n d 2 ) j o i n t c h a n c e - c o n s t r a i n t s : m > a , P f x : x \u00C2\u00A3 n { x : A . x ^ \u00C2\u00A3 . } \u00E2\u0080\u00A2 i = l n \" \u00E2\u0080\u00A2 w h e r e a e [ 0 , 1 ] . 5 2 T h e d e t e r m i n i s t i c e q u i v a l e n t o f t h e m a r g i n a l j o i n t c o n s t r a i n t s p i A i . x > ? i > > a i i s A . x > F f o r i = l , \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 , m T - a i ' w h e r e F i s i n f { y : y e Y . w h e r e Y . = { y : F . ( y ) > a.}}, t h e s m a l l e s t a ^ - f r a c t i l e o f F.. ( w h e r e F.. d e n o t e s t h e m a r g i n a l d i s t r i b u t i o n o f ) \u00E2\u0080\u00A2 S e e [ 6 7 ] f o r a d i s c u s s i o n o f d e t e r m i n i s t i c e q u i v a l e n t s f o r j o i n t c h a n c e -c o n s t r a i n t s . T h e s h o r t c o m i n g s o f t h i s a p p r o a c h i n c l u d e t h e d i f f i c u l t y i n s p e c i f y i n g t h e p r o b a b i l i t i e s a . a n d a i n a s y s t e m a t i c m e t h o d . S e c o n d l y , t h e r e i s n o d i f f e r e n t i a l p e n a l t y f o r : 1 ) s m a l l v e r s u s l a r g e i n f r a c t i o n s o f c o n s t r a i n t s , o r 2 ) t h e t y p e o f c o n s t r a i n t v i o l a t e d . A l s o t h e t r e a t m e n t o f a m u l t i s t a g e p r o b l e m h a s n o t b e e n a d e q u a t e l y c o n c e p t u a l i z e d . T h e p r o b l e m o f h a n d l i n g c o n s t r a i n t v i o l a t i o n s f r o m p e r i o d n t o p e r i o d n + 1 , h a s n o t b e e n r e s o l v e d . E i s n e r , K a p l a n a n d S o d e n [ 3 2 ] h a v e d i s c u s s e d t h r e e a l t e r n a -t i v e a p p r o a c h e s t o t h i s p r o b l e m : 1 ) t o t a l c h a n c e - c o n s t r a i n e d , 2 ) s a f e t y -f i r s t , a n d 3 ) c o n d i t i o n a l - g o . a n d h a v e p r o v i d e d s o m e p r e l i m i n a r y r e s u l t s o n t h i s d i f f i c u l t p r o b l e m . 3 . S t o c h a s t i c L i n e a r P r o g r a m m i n g [ 7 8 ] . G e n e r a l l y s t a t e d , t h i s t e c h n i q u e s t u d i e s t h e d i s t r i b u t i o n o f t h e o b j e c t i v e f u n c t i o n , b y s o l v i n g a l i n e a r p r o g r a m f o r e a c h r e a l i z a t i o n o f t h e r e s o u r c e v e c t o r . F o r e a c h i = l , . . . , L t h e f o l l o w i n g l i n e a r p r o g r a m i s s o l v e d . 5 3 z . = m i n c ' x x s . t . A x = \u00C2\u00A3 x > 0 T h i s s e q u e n c e o f l i n e a r p r o g r a m s g e n e r a t e s t h e d i s t r i b u t i o n o f z b a s e d o n t h e d i s t r i b u t i o n o f \u00C2\u00A3 . C l e a r l y t h i s a p p r o a c h h a s l i t t l e a p p l i c a t i o n a s a n o r m a t i v e t o o l , f o r s t a t i c p r o b l e m s . H o w e v e r , s i n c e r e c o u r s e m o d e l s c o n -t a i n d i s t r i b u t i o n p r o b l e m s a s s u b p r o b l e m s t h e i r u s e i n r e c o u r s e m o d e l s i s q u i t e i m p o r t a n t [ 7 8 ] . 4 . S t o c h a s t i c L i n e a r P r o g r a m m i n g w i t h S i m p l e R e c o u r s e [ 3 , 2 9 , 1 0 3 ] . . . A S L P R c a n b e e x p r e s s e d a s m i n x > 0 - c ' x + E r i n f \u00E2\u0080\u00A2 + - n + - ' : q y + q y ( P I ) s . t . A x T x + i y - I y \" w h e r e c e R n i s t h e c o s t v e c t o r , x e R n i s t h e d e c i s i o n v e c t o r , E, i s a r a n d o m v a r i a b l e d e f i n e d o n R m 2 , b e R m i i s t h e k n o w n r e s o u r c e v e c t o r , y + , y \" e R m 2 a r e t h e r e c o u r s e v a r i a b l e s , q + , q ~ e R m 2 a r e t h e u n i t p e n a l t y 5 4 c o s t s , A i s a '\u00E2\u0080\u00A2m1 x n k n o w n m a t r i x , T i s a m 2 x n k n o w n m a t r i x , a n d I i s a m 2 x m 2 i d e n t i t y m a t r i x . U s u a l l y y + i s t h e ' s h o r t a g e ' v a r i a b l e a n d y \" i s t h e ' s u r p l u s ' v a r i a b l e . T h e o b j e c t i v e i s t o m i n i m i z e t h e s u m o f c ' x a n d t h e e x p e c t e d l i n e a r p e n a l t y c o s t . T h i s p r o b l e m m a y b e v i e w e d a s a t w o - s t a g e p r o b l e m . I n s t a g e o n e a n x i s d e t e r m i n e d . T h e n i n t h e s e c o n d s t a g e t h e r e c o u r s e v a r i -a b l e y + a n d y ~ a r e d e t e r m i n e d i n s u c h a m a n n e r a s t o r e t r i e v e f e a s i b i l i t y w i t h t h e g i v e n T x . T h e r e c o u r s e o r o b l e m i s a l w a y s f e a s i b l e a n d b o u n d e d f r o m b e l o w i f q + + q \" > 0 . P a r i k h [ 6 7 ] s h o w s t h a t i f q t + q . . = 0 , r o w i c a n b e e l i m i n a t e d . C l e a r l y i f q + + q ~ < 0 t h e p r o b l e m i s u n b o u n d e d . I n t h i s d i s s e r t a t i o n , ? ; i s r e s t r i c t e d t o b e d i s c r e t e . T h i s a l l o w s ( P I ) t o b e e x p r e s s e d a s a l i n e a r p r o g r a m . ( P 2 ) m m f c ' x + I - P x , ( y , y }>0 +' + i - ' - i q y + q y s . t . A x = b T x + I y + 1 - I y \" 1 = K1 i = l , \u00E2\u0080\u0094 . L w h e r e P . = P (5 - C'1), i = l L . i r T h i s f o r m , ( P 2 ) , i s k n o w n a s t h e ' e x t e n s i v e r e p r e s e n t a t i o n ' - ' o f t h e d i s c r e t e f o r m o f ( P I ) . C o u h a u l t L 2 3 ] u s e s t h i s f o r m u l a t i o n d i r e c t l y , b u t i t i s c l e a r t h a t f o r r e a s o n a b l y s i z e d p r o b l e m s , t h e n u m b e r o f c o n s t r a i n t s q u i c k l y b e c o m e s u n m a n a g e a b l e . E l - A g i z y [ 3 3 ] u s e s t h i s r e p r e s e n t a t i o n t o r e d u c e ( d e c o m p o s e ) t h e p r o b l e m t o a s m a l l e r s e p a r a b l e c o n v e x p r o g r a m . H o w e v e r , t h e m o s t s a t i s f y i n g t r e a t m e n t o f t h i s p r o b l e m i s m ' v e n b y W e t s . [ 9 5 ] . I n t h i s p a n e r , h e m o d i f i e s t e c h n i q u e s s i m i l a r t o t h o s e d e v e l o p e d f o r 5 5 g e n e r a l i z e d u p p e r b o u n d i n g ( n o t a b l y t h e w o r k i n g b a s i s c o n c e p t [ 3 0 ' , 3 1 ] ) t o r e d u c e t h e p r o b l e m t o o n e t h a t i s t r a c t a b l y c o m p a r a b l e t o a l i n e a r p r o g r a m w i t h m i + m 2 r o w s a n d n v a r i a b l e s . C o n s i d e r t h e s e c o n d s t a g e p r o b l e m Q ( X . 5 ) m m y+\u00C2\u00BBy\">o ! +1 + - ' -q y + q y s . t . I y - I y \" = g - x , w h e r e x = T x \u00E2\u0080\u00A2 I t f o l l o w s t h a t m i Q ( x s 5 ) = I E Q . C x , , ? , ) i = i H 1 1 w h e r e Q ^ ^ - ) = y . f \" . - , 0 f i + V + V V ^ i + \" V = ? i \" i = l , . . . , L . H e n c e t h e r e c o u r s e p r o b l e m i s s e p a r a b l e . T h u s o n l y t h e m a r -g i n a l d i s t r i b u t i o n s a r e o f i m p o r t a n c e . T o i l l u s t r a t e t h e W e t s a l g o r i t h m [ 9 5 ] w r i t e ( P 2 ) a s t h e s e p a r a b l e c o n v e x p r o g r a m m 2 m 2 ( P 3 ) m i n c ' x + \u00C2\u00A3 E { Q . ( X i ) > = m i n c ' x + \u00C2\u00A3 Q ( x . ) x > 0 i = l H 1 1 1 i = l 1 1 s . t . A x = b , The-Q.j(x-j) h a v e t h e u s e f u l r e p r e s e n t a t i o n [ 9 5 ] 5 6 W = A^h ' a i } + m i n f J , p i ^ n K Z y u = xn- - v k . + l k . + l r i i y i \u00C2\u00A3 \u00C2\u00A3 = - 1 \u00E2\u0080\u0094 \u00E2\u0080\u0094 JZ.=\u00E2\u0080\u0094 1 yi_i< o. 0 < y i , k . + i a n d 0 _< y u < d u f o r A = o f 9 . . . , k , w h e r e P . & = - q | + q . F u , \u00C2\u00A3 = 0 , , . . ,k- , ( q . = q t + q T ) ( F u = P r t ^ . _< ? u ) ) , P i , - 1 = P i 0 ' P i , k i + 1 = P 1 , k t ' d U = C i , \u00C2\u00A3 + 1 - ? i \u00C2\u00A3 ' d i 0 = ? i l - a i ' d i , k i = 3 . - \u00C2\u00A3 . . a n d a . < E . , < E . 9 < \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 < 4 . , < ' - 3 - , a n d w h e r e k . i s t h e n u m b e r I I 5 K . J 1 I I 1 <_ 1 j K j I 1 o f p o s s i b l e v a l u e s o f E.. a n d i s t h e m e a n o f T h i s r e p r e s e n t a t i o n i s u s e f u l b e c a u s e i t g r e a t l y s i m p l i f i e s t h e s o l u t i o n o f t h e r e c o u r s e p r o b l e m . I n c r e a s e e a c h o f t y p e y. t o i t s u p p e r b o u n d o r u n t i l t h e s u m o f t h e y . 0 r e a c h e s (x ,- - a . ) s i n c e t h e P . 0 1 X* 1 1 IX/ a r e i n c r e a s i n g i n \u00C2\u00A3 . T h e a l g o r i t h m n e e d s t o r e c o r d t h e v a l u e o f o n l y t h e f i r s t y ^ ( i n e a c h r o w ) w h i c h i s n o t a t i t s u p p e r b o u n d , s a y y . ^ . ( S i n c e y . f o r \u00C2\u00A3 < m i s a t i t s u p p e r b o u n d a n d y . f o r \u00C2\u00A3 > m i s z e r o ) . T h i s r e s e m b l e s t h e u s u a l m o d i f i e d s i m p l e x a l g o r i t h m f o r u p p e r b o u n d e d v a r i a b l e s . N o t e t h a t t h e y. d o n o t e x p l i c i t l y a p p e a r i n t h e ( w o r k i n g ) b a s i s , a g a i n t h i s m e a n s t h a t t h e w o r k i n g b a s i s i s o f d i m e n s i o n m x + m 2 , t h e s a m e s i z e a s i f \u00C2\u00A3 w e r e r e p l a c e d b y i t s m e a n , | . T h e p i v o t i n g r u l e s c a n b e t h o u g h t o f a s e x a m i n i n g t h e v a l u e s o f t h e r e d u c e d c o s t s a n d t h e d u a l s ( a d j u s t e d f o r p e n a l t i e s ) a n d t h e s e l e c t -i n g t h e ' l a r g e s t ' m a r g i n a l v a l u e . I f i t i s a r e d u c e d c o s t , t h e n t h e u s u a l s i m p l e x p i v o t i s p e r f o r m e d . On t h e o t h e r h a n d , i f i t i s a d u a l , t h e n t h e r i g h t h a n d s i d e i s b r o u g h t t o i t s u p p e r b o u n d . A p p e n d i x 2 c o n t a i n s t h e f l o w c h a r t , c o m p u t e r c o d e a n d d e t a i l s o f t h e i m p l e m e n t a t i o n o f t h e c o d e . S i n c e c h a n c e - c o n s t r a i n e d p r o g r a m m i n g ( C C P ) h a s b e e n t h e m o s t w i d e l y u s e d s t o c h a s t i c p r o g r a m m i n g t e c h n i q u e i n t h e l i t e r a t u r e , a c o m p a r i s o n 5 7 o f t h e s o l u t i o n s g e n e r a t e d b y C C P a n d S L P R i s i n o r d e r . R e c a l l t h a t a C C P i s o f t h e f o r m . ( C C P 1 ) m i n c ' x x > 0 . . ' s . t . P r I X x > ? . ] > a. , i = l , \" \u00C2\u00AB , m , a n d t h e c e r t a i n t y e q u i v a l e n t o f ( C C P 1 ) i s ( C C P 2 ) c x s . t . A . x > F i = l , \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 , m . i - a. I t h a s b e e n s h o w n [ 6 7 ] t h a t i f x * i s a n o p t i m a l s o l u t i o n t o ( C C P 2 ) a n d n * i s t h e c o r r e s p o n d i n g d u a l s o l u t i o n t o ( C C P 2 ) , t h e n w i t h q i = 1 - V a n d ^ = \u00C2\u00B0 ' x * i s a n o p t i m a l s o l u t i o n t o t h e c o r r e s p o n d i n g S L P R . S u p p o s e x * i s a n o p t i m a l s o l u t i o n t o ( P 2 ) , t h e S L P R f o r m u l a t i o n , a n d t h a t f o r a l l i , F^ i s s t r i c t l y i n c r e a s i n g a t x * t h e n x * s o l v e s ( C C P 2 ) a n d ( C C P 1 ) w i t h a c o s t v e c t o r c + q \" A + * a n d q . - n . a. = _L 1 , i = l , m , q . - q . w h e r e n i s t h e d u a l f r o m t h e S L P R s o l u t i o n . 5 8 3 . 5 A p p e n d i x Two T h i s a p p e n d i x c o n s i s t s o f t h r e e p a r t s : 1 ) a f l o w c h a r t o f t h e a l g o r i t h m t o s o l v e S L P R , 2 ) a F O R T R A N - I V c o d e f o r t h e a l g o r i t h m , a n d 3 ) a u s e r ' s g u i d e f o r t h e c o d e . T h e f i r s t p a g e o f t h e f l o w c h a r t g i v e s a n o v e r a l l v i e w o f t h e a l g o r i t h m . T h e f o l l o w i n g p a g e s e x p l a i n i n d e t a i l e a c h p a r t . T h e c o m p u t e r c o d e w a s o r i g i n a l l y w r i t t e n b y C o l l i n s [ 2 3 ] a n d m o d i f i e d b y K a i l b e r g a n d K u s y [ 4 9 ] . 5 9 B . P h a s e 1 : i n i t i a l i z a t i o n F i n d m i n i m u m r e d u c e d c o s t ( i n c o r p o r a t e s u s u a l s i m p l e x r e d u c e d c o s t ) P h a s e 2 : i n i t i a l i z a t i o n y e s E . F i n d p i v o t r o w , d o s i m p l e x ( r e v i s e d ) p i v o t y e s S t o p i n f e a s i b l e 1 Do r e v i s e d s i m p l e x p i v o t s t i l l c > 0 H . Do p i v o t s ( r e v i s e d a n d u p p e r b o u n d i n g ) t i l l c > 0 ( S t o p ' o p t i m a l ' 6 0 R e a d i n : t o l e r a n c e ( \u00C2\u00A3 ) n , m i , m 2 ( a ( i ) , 5 ( i , l ) , ' , 5 ( i , k ( i ) ) , e ( . i ) ) ( P ( . i , l ) , - - - , P ( i , k ( i ) ) ) ( q + ( i ) , q \" ( i ) ) A a n d T m a t r i x ( h ( l ) , - \" , h ( m i ) ) i = l , - \u00C2\u00AB - , m 2 I n i t i a l i z e : n ( i ) = w ( i , j ) 1 i f h ( i ) > 0 1 i f h ( i ) < 0 ' 0 i f i t j n ( i j i f i = j i = l , \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 j i r i i '+ m 2 = m\ i = l , \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 , m c ( . i ) = 0 Y d ) = 0 h O + n h ) = a . 6 ( i ) = 0 1 \u00C2\u00A3 ( i ) = 0 K ( i ) = 0 i = l , \u00C2\u00AB - - , m 2 i w ( i ) = - i h ( i ) = | h ( i ) | j z 0 m 1 = 1 1 i = l , \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 , m J C a l l c l m p v t ( c , s , l ) C a l l s m x p v t C c , s , y ) y ( i ) = p C i , k ( . i ) + 1). i = 1 , . . . , m 2 g ( J ) c ( i w ( j ) ) i f i w ( j ) < n ) Y ( i w ( j ) - h ) o t h e r w i s e j = l n ( . j ) = g ( - ) * w ( - , j ) z 0 - 0 6 3 n o C a l l c l m p v t ( c , s , l ) C a l l s m x p v t ( c , s , y ) 6 ( 1 ) = d ( i , D y e s ^ Y d ) - p ( i . D i = l , - \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 , m ^ t o p ' u n b o u n d e ^ ' L o o p f o r j = l \u00C2\u00BB , m a n d i w ( j ) > n : v = i w ( j ) - n J t ( v ) = 1 s u m d ( v , \u00C2\u00AB ) u n t i l , y = I d ( v , i ) > h ( j ) i = l x = p(v,) - p ( v , k ( v ) + 1 ) 6 ( v ) = d ( v , ( j > ) h ( j ) = h ( j ) - y y ( v ) = p ( v , 4 > ) K ( v ) = < H n ( i ) = n ( i ) + w ( j , i ) * x 1 = 1 \u00C2\u00BB - - - , m e n d l o o p . C L M P V T ( c . s . m a u l ^ S t a r t ^ c = -- m i n ( c ( j ) - n ( - ) * A ( - , j ) ) j = 1 , \u00E2\u0080\u0094 , n s = j c o r r e s p o n d i n g t o t h e m i n i m u m y e s R e t u r n 3 n o m i n \" + n U + m J ] , c } j = l , - * * , m 2 u p d a t e s ( i f n e c e s s a r y ) 6 5 A j = l , \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 , m g ( j ) = w ( j , * ) * A ( * , s ) c a l l u p r p v t ( c , s - n , 2 ) n o C a l l c l m p v t ( c , s , 0 ) y e s L o o p f o r i = l , a n d \u00C2\u00A3 ( i ) f 1 , m 2 k k = m i + i c = y ( i ) + I T ( k k ) y e s c - n ( k k ) + p ( i , K ( i ) ) n o g ( \u00C2\u00A3 \u00C2\u00A3 ) = w ( \u00C2\u00A3 \u00C2\u00A3 , k k ) f o r 11 = 1 , . . . , m c a l l u p r p v t ( c , i , l ) g ( \u00C2\u00A3 \u00C2\u00A3 ) = w ( \u00C2\u00A3 \u00C2\u00A3 , k k ) ^ = 1 , . . . , m c a l l u p r p v t ( c , i , 0 ) y e s ^ E n d l o o p ) ^ t o p ' o p t i m a l ^ 6 6 S M X P V T ( c , s , u ) g ( J ) - w ( j , s - n - m a ) \u00E2\u0080\u00A2 . i n gill) = w ( U , * ) * A ( . , s ) C a l l r w p v t ( t , r , y ) i w ( r ) = s c a l l p i v o t ( c , r ) 6 7 RWPVT ( t , r , y ) \u00C2\u00AB*\u00E2\u0080\u00A2 \u00E2\u0080\u00A2> S t a r t \ f t * + 00 t = m i n j = l hM if g(j)>0 h ( i ) _ 6 ( i w ( j ) - n ) i f g ( j ) < 0 n u ; g ( j ) i w ( j ) > n r = j c o r r e s p o n d i n g t o m i n y = 0 y e s no v - 1 / \u00E2\u0080\u0094 ^ I R e t u r n j -P I V O T ( c , r ) 6 8 ^ S t a r t ^ g s = l / g ( r ) . w ( r , j ) = w ( r , j ) * g s j = l , - \" , m h ( r ) = g s * h ( r ) l o o p f o r i = l , ' \u00C2\u00AB ' , m ( i ^ r ) w ( i , j ) = w ( i , j ) - g s * w ( . r , j ) j = l , \" - , m h ( i ) = h ( i ) - g s * h ( r ) n ( i ) = n ( i ) + c * w ( r , i ) e n d l o o p z 0 = z 0 - c * h ( r ) R e t u r n 6 9 U P R P V T ( c , i , k i k ) ^ S t a r t ^ C a l l r w p v t ( t , r , y ) k k = i r i i + i a = 0 F = f a l s e L e t a b e t h e s u m o f d ( i , K ( i ) - j + 1 ) o v e r j = l , \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 , K ( i ) . u n t i l ( i f k i k = 0 ) p ( i , K ( i ) - j + 1 ) < - n ( k k ) ( i f k i k = 1 ) p ( i , K ( i ) - j + 1 ) > - n ( k k ) - o r -t < a , t h e n is = l a s t v a l u e o f j ( ( - j ) i f k i k = 0 ) F = t r u e ( i f n e i t h e r c o n d i t i o n h o l d s g o t o A ) 3 Z h ( j ) = h ( j ) - a * g ( j ) ( j = l , \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 , m ) K ( i ) = K ( i ) + Is c \u00E2\u0080\u0094 ' p ( i , K ( i ) + l ) + n ( k k ) i f k i k = 1 L , p ( i , K ( i ) ) + n ( k k ) i f k i k = 0 Y d ) = p ( i , K ( . i ) + 1 ) = d ( i , K ( i ) + 1 ) t '= t - a p ( i , K ( i ) + 1 ) - n ( k k ) i f k i k = F ,> - n ( k k ) i f k i k = 1 ^ y e s L ^ ^ R e t u r n ^ 7 0 Q C a l l p i v o t ( c , r ) \ y e s K ( 1 ) = K ( i ) - 1 6 ( i ) = d ( i , K ( i ) + 1 ) Y ( i ) = p ( i , K ( i ) + 1 ) h ( r ) = - h ( r ) w ( r , j ) = - w ( r , j ) n ( j ) = n ( j ) + 2 - c - w ( r , j ) - j = l , * - - , m h ( j ) = h ( j ) - w ( j \ i + m i ) * d ( . i , l < ( . i ) + 1 j V V = i w ( r ) i w ( r ) = n + i U i ) = 1 i f i > 0 v = v - n \u00C2\u00A3 ( v ) = 0 i v = m i + v h ( j ) = h ( j ) + w ( j , i v ) * S ( v ) K ( v ) = K ( v ) + 1 6 ( v ) =. d ( v , K ( v ) + 1 ) Y ( v ) = p ( v , K ( v ) + 1 ) j = l ,\"',m 71 o s ? O G B f l M K U Z \" f I N \u00C2\u00BB U T , 0 U T t n j T , T 4 3 \u00C2\u00A3 5 = I N = ' U T , T 4 O E 6 = 0 U T P U T ) C--> C O D F D R O G E R W E T S \".Y H E R M A N C O L L I N S C j L 2 ^ 0 0 I r T C A T I Q N S p.y J . K i L t P E \u00C2\u00B0 G K U S Y r. = \u00C2\u00BB M \u00C2\u00A3 Y T ' l : J M V A L U E S A \u00C2\u00BB F . = - H 2 = 7 3 : M i = 2 2 0 - t l 2 N - - 3 5 0 5 K ( I ) =8 I M P L I C I T P E A L ( A - M . 0 - 7 ) ' I N T E G E R S , R . B M ! , \" . A R T Y 3 \u00C2\u00A3 A L T P ( 7 0 . 1 C ) , T O ( 7 3 , 1 0 ) P E A L P ( 7 C , 1 0 ) , D ( 7 0 , 1 0 ) , A m ; , 2 6 0 l , H ( 1 3 0 > . . C ( Z 6 3 > R E A L W < 1 0 3 , l C G > , G ( l O O ) , D F L T S ( 7 G > , G A M M A < 7 e > , P I ( 1 0 Q > O T \" E N S I C W I W ( i C O ) , K A O D A ( 7 0 ) , L ( 7 0 ) , ' < ( 7 3 > = E A L O c ( 7 Q ) , D M < 7 0 > C O M M O N N . M . H I . M ? , P , D , A , W , C , H , G , D E L T A , G A M M A , \u00C2\u00B0 I , I W , K A P P A , L , E P S , 5 K , O P , O X . Z G , M A R K E R C O M M O N / \" i N T ? / T P , TO ; , ' C = > N 0 7 I C E T H A T IK' T H E D O C U M E N T A T I O N T H A T P ' A N D D U S E 3 - O R I G I N I N D E X I N G , C = > I M T H E F O R T R A N C O D E A L L - S U C H I N D E X E S H A V E B E E N I N C R E M E N T E D B Y 1 . R E A D ( 5 , 9 2 ? ) \u00C2\u00A3 \u00C2\u00B0 S 1 3 ? r O R M A T ( P 1 0 . & ) W R I T E ( 6 , 1 7 & I E P S ; 7 6 P Q P * A T ( 1 H 1 , T l Q , \" T O L E R A N C E I S S E T A T \" . F B . 5 ) , 0 = v R E a D I N N , M 1 . M 2 . K I . R E A D ( 5 , 1 G C ) N , M l , M 2 1 0 0 r O ' M S T ( 7 1 5 ) W R I T E ( 6 , 1 7 \u00C2\u00B0 ) N . M 1 - . M 2 ' 1 3 9 F n ? M A T ( / / / , T 1 0 O F V A R I 4 B L E S = \" , 1 4 , / , T 1 0 , \" * O F D E T E R M I N I S T I C ~ , C \" C G N S T P A I H T S = \" T i p , \"it Q F S T O C H A S T I C C O N S T R A I N T S ^ \" . I fa ) M = M1 M 2 C - > R E A D I N A N D W P I T F O U T T H E X I - V 6'_ U E S ( PO S S I 3 L E V A L U E S ) A N D A L P H A A N D R E TA C = \u00C2\u00BB ( L O W E R A N D U P P E R B O U N O S I I N T O D . W R I T E ( 6 , 1 1 0 ' 1 1 0 F O ? H 4 T l \" l \" , T 2 0 i 3 ' i ( \" * \" ) , / i T 2 0 < \" P O S S I B L E V A L U E S O F R I G H T H A N D S I D E \" , * / . T 2 C - . 3 < < ( \" \u00C2\u00AB \" ) , / ) \u00E2\u0080\u00A2 DO 3 ? L Z = 1 , M 2 \u00E2\u0080\u00A2 P E A D ( 5 , 8 0 0 ) K ( L Z > , D ( L Z , 2 > , P ( L Z . 2 I . T o ( L Z , 1 ) = P ( L 7 , 2 ) T D ( L Z . 1 ) = D ( L Z , 2 ) I - ! K < L Z ) . L E . I I G O T 0 3 7 K P = K f L 7 ) : DO 3 1 L A = 2 , K \u00C2\u00B0 P .EAO C 5 , 8 C 11 D ( L Z , L A + 1 ) , P ( L Z , L A + D T D ( L Z , L A ) = D O . Z , L A + l ) 3 1 T P ( L 7 . L A ) = P ( L 7 , L A + 1 ) .77 = E A O ( 5 , 8 0 2 ) D ( L Z , 1 ) , D ( L Z , K ( L Z ) +2 ) R E A D ( 5 , 1 2 9 ) T P ( L 7 ) , Q M ( L ~ > 4 0 0 F O R M A T d 3 , C 1 D . 2 , F 6 . u ) f>31 = - O R - i A T ( F l D . 2 , F f . . U ) ' P n ? C Q ? I U T ( 2 z \ 0 . 2 ) 1 2 9 F O R M A T ( 2 F 1 0 . ! * ) K I = K ( L Z1 +1 L X - M l * L 2 W P I T h ( 6 , 1 0 2 \u00C2\u00BB L X , ( D ( L 7 , J ) , J = 2 , K I ) 1 0 2 F 0 P M \u00C2\u00A3 T ( T 5 , \" ? 0 W \" , I 3 , T 1 5 , U ( F lif . 2 , 6 X ) , / , 7 2 1 u ( E - i i 4 , 2 , e , y > i 7 2 M ( L 7+ ^ 1 ) = n ( L 7 , 1) _T N P T E T H A T T a g S L \u00C2\u00B0 H A ( I > i O M E T H E R I G H T H A N D T E R M S 1 1 2 \"!=\u00E2\u0080\u00A2 T H E S T O C H A S T I C C O N S T R A O U T T H C L O W E R A N D U P P . E I T E ( 6 , 1 1 2 ' \" 1 S T < / / , T 2 3 , 1.2 ( \" * \" ) , / , T \" \u00C2\u00BB / S \u00C2\u00B0 I A 9 L E S \" , / , T 2 0 . 1 . 2 C ' 1 = 1 . M 2 I N T S P H - J U N O S A N D C A L C U L A T E T H E 0 V A L U E S . 2 0 , \" L O W E ? A N D U P P E R S O U N D S O F R A N D O M \" , DO ^ = > R E K T L x 1,13 n o o < A n ( I ) E ( 6 ? J J ) = A N . 1 0 2 ) L X , 0 ( 1 , i \u00C2\u00BB , D ( I , \" < I U > . = l,\u00C2\u00ABl 0 ( I , J \u00C2\u00AB - 1 I - 0 ( 1 , J ) n W P I T E O U T T H E I N I T I A L P - V A L U E S : C A L C U L A T E T H E A C C U M U L A T E D : > p - V A l . U E S A N D W R I T E T H E M O U T . W = I T E ( 6 , I C S ) I C S ~ O R M AT 2 0 , \" S H O R T A G E P E N A L T Y S U R P L U S P E N A L T Y \" , , / ) L X = M l + I W = T T E ( 6 . 1 G ? \u00C2\u00BB L X . O P ( T ) , O M f DO 3 3 I = 1 , M 2 Kl-K(T)* 1 D ' ( I , i > = - 0 \u00C2\u00B0 ( O A C ~ = 0 . I ) 7 7 = \u00C2\u00BB R Q = Q D < I ) +OM ( I \u00C2\u00BB ' DO 3 \u00C2\u00AB . J = 2 . K T ACC,= A C C + P ( T , J ) \u00C2\u00BB ( T , J ! = - o \u00C2\u00B0 ( I ) + o * a c e C O N T I N U E \u00E2\u0080\u00A2 A O I N ft ' N O H . I S i F O R M A T ( ? F 1 0 . ' \u00C2\u00BB ) DO 1 7 Q J 1 = 1 , M DO 1 7 9 J 2 = i , N A : . ! 1 , J 2 > = 3 . N O T E T H A T T H I S I S AND C 0 D E , W H I C H W 0 ! J L P R E ? T H E ' ' A T ^ I X I N 8 ^ 1 0 . ^ T H E N C N 2 E R 0 E N T R I E S . f I 7 > F O L L O W E D B Y T H E E N T E R A N U L L L I N E . DO 1 B 9 k - I ; S V = I , M -\u00E2\u0080\u00A2 E A 0 ( 5 , 1 , \" \u00C2\u00BB ) I N D , T E M P T H E R D E P A R T U R E F R O M . T H E O R I G I N A L U I R E T-i E U S E R T O E N T E R E A C H ROW QF c O R \" A T 7 W I T H A L A R G E M A T R I X W H I C H \" S O F O R E A C H ROW I N \u00C2\u00B0 U T T H E C O L U M N N O . E N T R Y ( F i C . A ! , W H E N A ROW I S C O M P L E T E I F ( I N D . c O . O ) G O T O 1 9 9 A ( K U S Y , I N D ) = T E MP 7 3 F - 0 T 0 1 8 7 I B ? C O N T I N U E I r < \" l . E C . 3 ) ' \" 0 T O 9 7 3 \u00C2\u00B0 E A D < 5 , 1 C t > ( H ( I ) , 1 = 1 , M l ) \u00C2\u00B0 7 3 C O N T I N U E W R I T E 1 6 , 1 0 6 * 1 0 6 F O ' X J T ( / / , T 2 3 , 30 ( \" * \" ) , / , T 2 0 . \" a - M A T R I X \" , / . T 2 0 . 3 0 ( \u00E2\u0080\u00A2 ' \u00C2\u00BB ' \u00E2\u0080\u00A2 ) , / ) 1 3 3 C Q 3 M A T ( I ? , F 1 3 . < 0 NI.)M9 = N / 1 0 + 1 I F ( M O O ( N , 1 C ) . E O . 0 ) N U M 3 = N U M R - i 0 0 2 21* J = i , N U M f > 1 1 = 1+ 1 C * ( J - l > L 2 = M I N O C 9 + L l , N ) W R I T E ( 6 . ? ? 7 ) ; , ' _ ^ W R I T E ( 6 , 2 2 \u00C2\u00AB ) ( I , I = H , L 2 ) \u00E2\u0080\u00A2 0 0 2 2 1 . < 9=1, M W R I T E ( 6 , 2 2 ? ) \u00C2\u00AB 9 , (A ( K 9 , K \u00C2\u00A3 > , K \u00C2\u00B0 . = L 1 , L 2 ) ZZk C O N T I N U E - 2 2 2 F O R M A T ( / \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 PQW \" , 1 3 , 3 X , 1 0 ( 2 X , ^ 1 C . i , ) ) E 0 R H A T ( / / / 3 Q X , * * * * * * * * * * * * * * * * * * * * * * * 2 2 7 F O R M A T ( / / . T l 0 ) 2 2 8 F O R M A T ( \" C O L U M N S \" , 1 0 1 1 2 ) W = I T E ( & , 1 0 9 ) I C R \" O R M A T f / / , T 2 0 , 70 ( \" * \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 ) , / , T 2 0 , \" I N I T I A L R I G H T H A N D S I D E V E C T O R \" , * T 2 0 , 3 0 ( \u00E2\u0080\u00A2 \" * \" ) ' , / ) 0 0 2 2 1 = 1 , M ; . 2 2 W R I T E f f , , 1 0 2 ) I , H ( I ) 0 = > T N I T I A L I 7 E p I A N D W , 0 0 i t O 0 RL E = 1 . C \u00C2\u00B0 I ( I ) = S I C-N ( D D L E , H ( I ) ) 0 0 U l J = l , \" , , i \u00C2\u00BB l W ( I , J ) = 0 . US W ( I , H = O T ( T ) C W I S 7 E F 0 E X C E P T F O R P I ON T H E D I A G O N A L S 0 = > I N I T I A L l Z E L . K . I W . D E L T A , A N D G A M M A ! C A L C U L A T E H A N D 7 0 . DO 5 0 1 = 1 , M 2 ' G A ^ M A ( I ) = 0 . . D E L T A ( I ) = 1 F 7 0 L ( I ) = 0 5 C K A \u00C2\u00B0 P A ( I > = 0 7 0 = 0 . DO 5 1 1 = 1 , M I w ( I i = - 1 I F ( H ( I ) . G e . G . ) G O T O 5 1 H ! I > = - H ( I > ? 1 Z O = Z O - H ( I > DO 5 2 I = 1 , N 5 2 C ( I ) = 0 . \u00E2\u0080\u00A2J;=>^JUJLU.^LIII-1.^I,IX;. u r n l i i i u j . j L i i i i . x i i C = > P H A S E ' I \u00E2\u0080\u00A2 9 E G I N C O L U M N \u00C2\u00B0 I V O T I N G W I T H M A U = 1 . 2 0 0 C A L L C L M p V T ( C P A R , S , i ) 7 4 T P < C \u00C2\u00BB 4 R . L T . - F . D S ) c o T O 2 0 2 I - ( 7 0 . G \u00C2\u00A3 . - E ^ S ) G O T O 2 0 3 C = > C 3 A \u00C2\u00B0 . G E . 0 A N D 7 Q . t _ T . Q . W R I T E ( 6 , 2 0 7 ) ? 0 7 \" O R M A T ( * \u00E2\u0080\u00A2 I N F E A S I P L E \" ) OAi_L D U M P C = > r . R A R . G E . 0 A N O 7 0 . G E . O . 2 C 3 D O 2 0 U J = 1 , M I F ( I w ( j ) . L T . 5 ) G O T O 2 3 5 2 0 ^ C O N T I N U E T O 3 0 5 2 C 5 W R I T E ( 6 , 2 0 8 ) 2 0 . 8 C 0 R M A T ( / \" D H A S E I D E G E N E R A C Y \" ) C A L L D U M P C = \u00C2\u00BB C A A R < Q . \" 2 3 2 C A L L S M X P V T ( C B 4 R , S , M U ~ > G O T O 2 0 0 C = > O H A S E I I ! R E A D I N C - V E C T O R : 2 0 0 R E 0 D ( 5 , 1 1 9 ) ' ( C ( I > , 1 = 1 , N ) 1 1 9 F O R M A . T ( 3 F 1 0 . V ) W R I T E ( 6 . 8 5 ) -5 5 \" P O R M A T ( / / , T 2 0 , 3 0 ' ( \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 * \" ) , / , T 2 0 , \" C O S T V E C T O R \" , / , T 2 0 t 3 0 ( \" \u00C2\u00BB \" ) , / ) D O 7 Q 9 J 1 = 5 , N 7 C 9 W R I T E ( f c , 7 1 G \u00C2\u00BB J l . C ( J l ) 7 1 0 F O R M A T C . C O S T < \" , 1 3 , \" ) = \" . F l k. k) \u00C2\u00A3 , = > S \u00C2\u00A3 T G A M M A . . ' DO 3 0 1 1 = 1 , M 2 7 0 1 G A M M A ( ] I = P ( I , \c ( I I + 1 ) O = > S E T G , O O 3 C 2 J - - 1 , \" I F ( I W ( J J , L E , N ) G O T O 3 0 3 G ( J ) = G A M M A ( i w ( J ) - N ) G O T O . 3 0 2 3 0 3 G M ) - C < I w ( J ) ) 3 0 2 C O N T I N U E C = \u00C2\u00BB ' S E T P I , 7 0 7 O = 0 . T Q 3 0 a T = 1 . M P I ( I ) = C . D O 3 0 < \u00C2\u00AB J = 1 , M 3 C ' A \u00C2\u00B0 I ( I ) -PI ( D + G ( J ) * W ( J , I ) C = > 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 C - > R- E C- ! C O L U M N P I V O T I N G W I T M M A U = 1 . u \" 0 C A L L C L M P V T ( C ^ A P , S t 1 ) T ^ ~ ( C R A P . G E . - E P S ) G O ~ T O ~ S\"G 0 C A L L S M X F V T ( C B A R , S , M U ) I F ( M l ) . M E . 2 ? G O T O M Q O W R I T E < 6 . i \u00C2\u00BB G ? ) 4 - 0 2 r O R M A T ( / \" U N B O U N D E D \" ) C A L L D U M P \u00E2\u0080\u00A2 , C = > C E < A R > = 0 . ! S E T D E L T A , G A M M A . 5 0 0 D O 5 C 1 1 = 1 , M 2 7 5 H E L T A ( I ) = n ( T i 1 t E 0 1 G A M M A < I ) = o 1 1 , 1 I C = > S E T L . K A P P A . H . G A M M f t , P I . ; 0 0 ' 5 0 2 J = 1 , M _ I P ( I W ( J ) . L E . N ) GO T O 5 0 2 N U = I W ( J \u00C2\u00BB - M L ( M U ) = 1 Y = ] , \u00E2\u0080\u00A2 . P (J T = 1 DO 5 5 3 K K = i , K ! D H ! = K I ( I P ( Y + 0 ( N U , K K > t E P S . G T . H t J ) ! GO T O 5 0 f a 5 0 3 Y = v + |_ ( N U , K t O 5 0 fa Y = P ( N U , P H T ) - \u00C2\u00B0 ( N U , K ( N U ) - H ) D E L T A ( N U ) = D ( N U , P H I ) H ( I) = H < J ) - Y G A M M A ( N U ! = D ( N U , P H I ) . K ftno'A ( N U ) = P h ' T - 1 DO 5 0 5 1 = 1 . M 5 0 5 J I ( I ) = P I f T 1 \u00C2\u00BB y \u00C2\u00BB H ( J , 1 J 5 0 2 C O N T I N U E . 0 = > 3 3 3 7 3 3 3 3 3 2 3 3 2 3 3 7 3 3 3 3 3 3 3 3 3 3 3 3 3 3 7 3 7 3 3 3 3 C = \u00C2\u00BB ^ E G I N C O L U M N P I V O T I N G A G A I N W J T H M A U = 9 . 7 0 G C A L L C L M P V T r C 9 A R , - S , 0 ) I P ( C P A R . G E . - E C S ) GO TO 6 0 0 7 0 1 DO 7 0 3 J - l . M G ( J 1 = 0 . D O 7 0 3 1 = 1 . M 7 0 3 G ( J ) = G ( J ) + W ( J , I ) * A ( I , S ) C A L L U P P P V T ( C E . A R , S - N , 2 ) G O T O 7 0 0 C = > R E G I N U P P E P P O U N D P I V O T I N G . , : , 6 D C D O 6 0 1 1 = 1 . M ? I P ( L ( I > . E C U GO TO 6 0 1 <\u00E2\u0080\u00A2<-\"! *Z ' C = > T E S T 1 . C ? , A R = G A M * A ( I ) + F I ( K K ) T C ( C P . A P . G E . - E P S ) GO TO 6 0 2 . DO 6 0 3 L L = 1 , M 6 0 3 G ( L L ) - - W ( L L . K K I C A L L U P R P V r ( C B A P , I , i ) GO T O 7 0 0 C = > T E S T 2 . 6 0 2 r r i i f i p p A f T ) . E C . J ) GO T O 6 0 1 C 3 A R = P I ( K ' O + P ( 1 , ' K A P P A ( I ! 1 ' \" ~ * \" T P ( C H A D . L E . E P \" ) G O T O 6 0 1 DO 6 0 fa L L = 1 . M 6Gfa G I L D = W i L ! . , * <) C A L L U P P P V T ( C H A R , 1 , 0 ) G O T O 700 ; ; 6 0 1 C O N T I N U E \" 7 C = > W H \u00C2\u00A3 N T H E L O O P T S S A T I S F I E D , W R I T E O U T T H E O P T I M A L S O L U T I O N A N D S T O P ^ . I T f (fc, r 0 \u00C2\u00B0 M - f l T ( \" l \" . T 2 C , 3 Q ( \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 ) , / , T 2 0 , \" / , T 2 0 , 3 0 ( \" * \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 ) , / ) O P T I M A L S O L U T I O N S U 3 R 0 U T I N E S S U B R O U T I N E c I V O T ( C O A R , R > I M P L I C I T p F A L ( A - H . O - Z ) I N T E G E R S . B , O H I t M A o T Y R E A L P l ? C , 1 0 > , n < 7 0 , 1 0 ) . A d C 0 , 2 6 0 ) \u00E2\u0080\u00A2 H \u00C2\u00AB 1 3 0 > , C t 2 6 0 \u00C2\u00BB R E A L W ( 1 0 3 , 1 C 0 ) , G ( 1 0 3 > , C E L T A ( 7 0 ) , 3 A M M A ( 7 0 ) , P I ( 1 0 0 > D I M E N S I O N i n ( 1 0 3 ) , K A \u00C2\u00B0 P A ( 7 0 ) . L ( 7 3 ) , K ( 7 0 ) 3 E C L Q \u00C2\u00B0 ( 7 3 ) . O M ( 7 0 ) C O M M O N N , M , M 1 , M 2 , P , D , A , W , C , H , G , 0 E L T A , G A M H A , \" I , I W , K A P P A , L , E P S , % K , O P , O M , 7.0 . M A R K E R C = > C A L C U ' . A T E P I V O T A L R O W , H ( P ) G S = 1 . 0 / G ( P ) DO 1 0 J = 1 . M - : 1 0 w< = , j ) = w ( o , j ) \u00C2\u00BB r - s H ( R ) = G 3 * H ( P ) C = > \u00C2\u00B0 I V O T O N 0 T H E P R O W S . DO 1 1 1 = 1 , M. I F ( I . E O . P ) GO T O 1 1 G S = G ( I ) DO 1 2 J = 1 , M 1 2 W ( I , J ) = W ( I , J ) - G S * W ( R , J ) H ( I ) = H ( I ) - G S \u00C2\u00BB H ( R ) 1 1 C O N T I N U E C = > C A L C U L A T E \u00C2\u00B0 I AND 7 0 . DO 1 3 J = 1 . M 1 3 P I ('J ) = P I ( J ) + C a AR * W ( R , J ) Z O = Z O - C B A R \u00C2\u00BB H ( R ) 9 7 6 R E T U R N . E N D S U B R O U T I N E R K P V T ( T , P . , M U > I M P L I C I T R E A L ( A - H . O - Z ) I N T E G E R S , \" , p u T , M A R T Y R E A L \u00E2\u0080\u00A2 P ( 7 0 , 1 C ) , D ( 7 C , 1 0 1 , 4 ( 1 1 0 , 2 6 0 1 , H ( 1 0 C > . C ( 2 6Q> R E A L W ( l C C , 1 0 0 ) , G ( i a O > , P E L T A < 7 0 ) , G A M M A ( 7 0 > , P I ( 1 0 0 > D I M E N S I O N IW ( I C G > , K J \u00C2\u00B0 P i C C > , L ( 7 5 I ,< ( 7 0 ) R E A L 0 c ( 7 0 I , OM ( 7 , j | C O M M O N N . M , M 1 . \" 2 , P , D , A , W . C , H , G , D E L T A , G A M M A , P I , I W , K A P R A , L . E P S , 3 K , O P , O M , Z O , M A R K E R T = 1 E 7 G C = > c I N O \" I N ? A T T p H ( J > / G ( J > W H E R E G ( J > > 0 . C = > F I M n M I N P A T I O ( H ( J ) - D E L T A ( I W ( J ) - N ) ) / G ( J ) S T K \u00C2\u00BB 0 ! G ( J ) < 0 . DO 1 1 J = 1 , M I F t G ( J ) . L E . - E \u00C2\u00B0 S ) GO TO 1 0 . I F ( G ( J ) . L T . E P S ) GO T O 1 1 0 = > I F 5 ( J ) > C . R A T T 0 = H < J ) / G ( J ) I F ( R A T I O . G T . T ) G O T O 1 1 T = R A T 1 0 = = J , , , , GO T O 1 1 ?-->I c G U X O . . 1 C . K K = I W ( J ) - N I F ( K K . L F . O GO TO 1 1 P A T T O = ( H ( J ) - D E L T A ( K K ) ) / G ( J ) I F ( R A T I O . G T . T ) GO T O 1 1 T = R A T 1 0 R = J M U = 1 1 1 . C O N T I N U E r.= > I F NO J F O U N D M U = 2 . , , ! I F ( T . G E . 1 E 7 B ) M I J = 2 9 7 F. R E T U R N S U ^ R O U T I K F CIM .PVT ( C 9 4 ? . , S . M A U \u00C2\u00BB I M P L I C I T P r A t ( A - H , Q - Z ) I N T 2 G E P S . , M f l P T V \u00C2\u00B0 E A L P ( 7 0 . 1 0 ) , 0 ( 7 0 , 1 3 ) , A f l J O . 2 6 0 ) , H ( 1 D O ) . C ( 2 6 0 > R E A L W ( I C Q , I C G ) , G ( I O C ) . H E L T A ( 7 C ) , G A M M A ( 7 0 > . P I ( 1 0 0 ) O I M . N . J O ' . ' IW ( 1 0 0 ) , K A P P A ( 7 0 \u00C2\u00BB , L ( 7 0 ) , K ( 7 C ) P E A L Q P ( 7 0 \u00C2\u00BB . O M ( 7 0 I C O M M O N N , M . M l , M 2 , P . 0 , A , W , C , H . G , 0 E L T A , G A M M A < P I , I W , K A P P A , L i E P S j < K , C \u00C2\u00BB , Q M , Z O , M A R K E R C O A P = 1 E 7 0 S = 0 C = > c I N O M T N ( C - P I * A > = C R A R 0 0 I f ? J = 1 , N X = C ( J ) , \u00E2\u0080\u00A2 OO 1 1 1 = 1 , M ' 1 1 X = X - P I ( I > \u00C2\u00BB M I , J ) I F ( X \u00E2\u0080\u00A2 G E . C B A R - E ' S ) GO TO 1 0 C P A R = X S = J 1 0 C O N T I N U E I F ( M A U . E O . O ) R E T U R N C = > F I N D M I N ( C D A R \u00C2\u00AB G A M M A ( * ) + P I ( \u00C2\u00BB * M H ) = C 9 A R 0 0 1 2 1 = 1 . 'M2 X = G 4 M M A ( I t + \u00C2\u00B0 I ( I + M 1 ) I r ( X . G E . C B A P - E ' S ) GO TO 1 2 C R A R = Y . ' S = T + N 1 ? C O N T I N U E P 7 6 R E T U R N E N D 8 0 S U B R O U T I N E l l \u00C2\u00B0 R \u00C2\u00B0 V T ( C 3 A R , I , K I K > I M P L I C I T C \u00C2\u00A3 A L ( A - H . Q - ? ) '.= > T u T S I S V E R S I O N 2 O r U P R \u00C2\u00B0 V T . I N T E G F R S . R . P H T P - A L n I 7 C , 1 3 ) . 0 <7C , 1 0 ) , A ( 10 3 , 2 6 0 > , H<1 3 0 ) , C ( 2 6 0 ) R E V . K ( I C Q . 1 3 0 ) , G ( 1 Q G ) , D E L T A ( 7 C ) . G A M M A ( 7 0 ) i P I ( 1 0 0 ) 0 1 MEM S I O N T W ( 1 0 0 ) . K A D P A ( 7 0 ) . L ( 7 0 ) . K ( 7 0 ) . P F A l n = ( 7 D . OM ( 7 0 ) \u00E2\u0080\u00A2 C O M M O N N , M , M l , M 2 , D , D , A . W , C , H , G , D E L T A , G A M M A , \u00C2\u00B0 I , I ' \u00C2\u00AB I . K A P P A , L . E P S . * K , O P . O M . 7 0 . M A R K E R L O G I C A L P L O G C A L L R W D V T ( T . R , \u00C2\u00AB t J > K K = M 1 * I P L A G = . F A L S E . A L \u00C2\u00B0 ' - I A = Q . I F \u00E2\u0080\u00A2 ( K I K . N E \u00E2\u0080\u00A2 0 > GO T O 2 0 2 , . .., = S U M O F D t l . K A P P A ( I ) - S ) : S = 1 , . . . . L L r = > K I K = 3 P I N O T H E L A R G E S T C,->\u00C2\u00B0 (T , K 4 = P A ( I ( - L L > + R I ( K K ) L S = 1 , \u00C2\u00BB C . K T = K f i p p / _ < _ . OO 1 0 L L ^ l . K T A L \u00C2\u00B0 H A = 0 L D H A + D ( I , K A P = > A ( I ) - L L + 1 ) I F ( P ( I , K A O P A ( I I - L L + 1 ) + D I ( K K ) . L E . E P S .OR. T , L T . A L ' P H A - E P S > G O T O 3 0 L S = L L A S = A L \u00C2\u00B0 H A 1 0 F L A G = . T R U E . G O T O 3 0 C = > K I K = 1 , F I N D T H E L A R G E S T L S = G , 1 , . . . , K ( I ) - K A p P A ( I ) S . T . C = > \u00C2\u00B0 ( I . K A P P A ( I ) + L L X 0 A N D T> = S U M O F D ( I , K A P P A ( I ) - S ) : S = 0 , . . . , L L . 2 0 I r ( K I K . N E . l ) GO T O 4 0 K I = K ( I ) + 1 - K A D O A ( I ) D O 2 1 L L = 1 , K T A L p H A = A L P H A \u00C2\u00BB Q ( I . K A P P A ( I ) + L L ) . I F ( P ( I , K A P P A ( I ) 4 - L L ) + \u00C2\u00B0 I ( K K I . G E . - E P S . O R . T . L T . A L P H A - E P S ) GO T O 3 0 L S ' = L L A S = A L \u00C2\u00B0 H A _ 2 1 F L A G = . T P ' J E . C = > S E E I r S O M E L S - F O U N D . ( I F N O T = I V O T A N D R E T U R N ) \u00E2\u0080\u00A2*D I F ( . N O T . P L A G ) GO TO 4 0 , , C = > S O M E L S F O U N \" I F ( K I K . E O . 3 ) V S = - L S D O 7 1 J = 1 . M H ( J l = H ( J ! - A S * G ( J ! K A P P A ( I ) = K A P \u00C2\u00B0 A ( I ) + L S \u00E2\u0080\u00A2 r c ; K J ' < . E Q . i ) C P A R = _ P _ L L K A _ P \u00C2\u00A3 A J J L i \u00C2\u00B1 i _ ) *:\u00C2\u00B0JJJ 1 ) + P I ( K K > . L T . - E R S \u00C2\u00AB; . O R . K T K \u00E2\u0080\u00A2 F O . 1 . A N D . P ( T , K A P P A ( T ) 4-1 ) + P T ( K K ) . . G X _ - _ . % . C P . T . L T . - E P S ) G O T O 9 ? = > O T H E R W I S E G O P I V O T A N D R E T U R N S i ' \" R O U T I S ' i X P V T ( C B f t R . S . M U ) T M \u00C2\u00B0 L I C T T O F A t ( f l - H , Q - ? ) I N T E G E R S , = . \" H I , M A R T Y R E A L P ( 7 Q , i C > , D ( 7 0 , 1 0 ) . a { 1 0 0 \u00E2\u0080\u00A2 2 6 0 I . H ( 1 0 0 > , C t 2 6 0 I R E M w-< l C C , l C 0 ) , G ( l 3 0 > , O E L T A ( 7 0 > , G A M M a ( 7 0 ) . P T ( 1 0 0 > D I M E N S I O N I W ( 1 0 0 ) , K A P o i ( 7 0 ) , L ( 7 0 ) , < ( 7 0 ) . R E A L O P ( 7 0 ) . O M ( 7 0 1 C O M M O N N . M . M 1 , M g , p , Q , A , W , C , H , G , 3 E L T A , G A M M A , \u00C2\u00B0 I , I W , K A P P A , L ; E P S , ? K , 0 3 . O M . ' O , M A R K E R I c ( S . G T . N ) G O T O 1 0 C = > S < = N . DO 1 1 L L = 1 . M G ( L L ) = 0 DO 1 1 J = 1 . w 1 1 C- ( L L ) =C ( L L ) + W ( L L . J ) * A ( J , S ) GO T O 2 0 C = > S > N . 1 0 K K = ~ - N + M 1 DO 1 2 J = 1 , M 1 2 G ( J ) = - W ( J . K K ) , . , C = > I N E I T H E R C A S E . 2 0 C A L L . P W P V T ( T . R . M I J ) . I F ( M U . E O . 2 ) GO TO 9 9 X W ( R ) = S C A L L P I V O T ( C 9 A = , P ) 9 9 R E T U R N . E N D S U ' i R O U T I NE D R I NT T M ^ L I l ' I T R E A L ( A - H , Q - Z ) I N T E G E R S . R , M f l D T Y \u00C2\u00B0 E A L C ^ I $ ( 7 o I R E A L T \u00C2\u00B0 ( 7 0 \u00C2\u00AB 1 0 ) \u00C2\u00BB T 0 ( 7 0 , 1 0 ) R E A L P I U . 1 0 ) . D ( 7 0 , 1 0 ) , A ( 1 0 0 . 2 6 0 1 , H ( l O Q ) , C ( 2 6 0 ) R E A L W ( 1 C O , 1 0 0 ) , G < 1 0 0 ) . 9 E L T A ( 7 0 ) , G A M M A ( 7 0 ) , P I ( 1 0 0 ) D I M E N S I O N T w g c . Q ) , K A \u00C2\u00B0 P A ( ? Q ) , L ( 7 0 ) , K ( 7 Q ) : . R E A L O P ( 7 0 ) , Y ( 3 5 0 ) , OM ( 7 0 1 C O M M O N N , M , M 1 , M2 , p , o, A , W , C , H \u00C2\u00BB G , D E L T A , G A M M A , P I , I W , K A P P A , L t E P S , * \u00E2\u0080\u00A2 K , 0 \u00C2\u00B0 , 0 \" , 7 0 , M A R K E R C O M M O N / \u00C2\u00B0 R I N T P / T P . T D W R I T E ( 6 , 1 0 0 1 1 0 0 F O R M A T ; / / , T I P . \" B A S I S I N O E Y O U A L V A R I A B L E S \" , / / , 3 0 ( \" - - - \" ) ) 0 0 1 0 J = 1 , M 1 0 W=>ITE ( 6 . 1 0 D I W ( J ) , P I ( J ) 1 0 1 ^ O R M A T ( T l 6 , T 3 , T 2 2 , F l < + . < 4 ) C = > C t L O U L A T F 7 0 . Z S O = 0 . D O ?c J = i \u00C2\u00BB * : : '. 2 5 I c ( I W ( J ) . l . E . N ) Z S 0 = 7 S 0 + C ( I W { J ) ) \u00C2\u00BB H ( J ) C DO 2 1 1 = 1 , M 2 r K I = K A o p A ( T ) C T F ( K T . E O . O ) GO TO 2 1 C DO 2 2 K K = 1 , K I C. 2 2 Z S 0 = 7 S 0 + o ( T , K * ) \u00C2\u00BB 0 ( I , K K ) C 2 1 C O N T I N U E 1 0 8 F O R M A T ( / / / T 2 0 , \" O P T I W A L O B J E C T I V E V A L U E ( W I T H O U T P E N A L T I E S ) = \" , T 6 5 , F l r.= > F I N O T H E Y - V A L U E S F R O M I W I H ; DO 3 1 1 = 1 , N 3i x ( i ) - o . ; , DO 3 2 1 = 1 , M ' 3 2 I F ( I W ( T ) . L E . N ) X ( I W ( I ) \u00C2\u00BB = H ( I ) C = > W R I T E O U T X \" S . W R I T E ( 6 , 3 5 ) 7 - 5 F O R M A T ( / / , T J O . 3 0 ( \" \u00C2\u00BB \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 ) , / , T 2 0 , \" O P T I M A L S O L U T I O N V E C T O R \" , -1 : / , T 2 0 , 3 0 ( \" * \" > , / ) D O 3 3 I = i , t . ' 3 3 H P T T E ( 6 , 1 0 2 > I , Y ( I ) i G 2 F O R M A T ( \" X ( \" , 1 3 . \" ) = \" , F 1 5 . ^ 1 C = > C A L C ' J L A T E T H E A L P H A \" S A N D C H T \" S . W R I T E ( 6 , 7 5 ) 3 6 F O R M A T ; / / , T 2 0 , A 2 ( \" \u00C2\u00BB \" ) , / . T 2 D , \" R I G H T H A N D S I D E F O R S T O C H A S T I C C O * * \" N S T P A I N T S \" , / , T 2 0 , 4 2 ( \" * \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 ) , / ) no 30 1 = 1 , \u00C2\u00AB \u00C2\u00BB A L 3 H A = - ( D ( I . i ) + P T ( M i + T ) j / ( p ( x , K ( I ) 4 - l ) - P ( I , l ) ) C H I = 0 . D O JU J = 1 , K .Ui C M T = C H H - M H - M 1 . J ) \u00C2\u00BB X ( J ) ' C H I S ( I ) = C H I I I = I + M 1 8 3 3 0 W R I T E ( 6 . 1 0 \u00C2\u00B0 ) T I . C H T i._q F _ ? M A T ( T F : , \" = > O W ( \" , 1 3 , \" ) = \" . = v , - 1 4 . 4 ) \u00C2\u00B0 E M = 0 . W R I T E ( 6 , 4 3 ) 4 3 P O ' M A T ( / / , T 2 0 , 3 0 ( \" \u00C2\u00BB \" ) , / , T 2 0 , \" I N O I V I D U A L P E N A L T I E S \" , * . / . T 2 0 , 3 0 , / ) D O 3 * 1 X A = 1 , M _ \u00C2\u00B0 E N 1 = 0 . K I = K ( K A ) DO 3 8 2 K Q = 1 , K I I F ( C H I S < K A ) . L T . T D ( K 4 , K 3 > > P E ^ 1 = D E N 1 + ( T D ( K A . K B ) - C H I S ( K A ) ) * Q P ( K 4 ) \u00E2\u0080\u00A2 * T P ( K A , K B ) 3 8 2 I F - < C H I S ( K f l ) . G T . T 0 < K A , K 3 ) > D E M 1 = P E N 1 + ( C H I S ( K A ) - T D ( K A , K 9 ) ) \u00C2\u00BB Q M ( K A 1 * ; * T P ( K A , K e > P E N = P E N * F E N 1 K B = K A + M 1 W R I T E ( 6 , 3 8 3 ) K B , P E N 1 3 8 3 \" O R M A T f T P , \" \" P E N A L T Y F O R D O W ( \" , I 3 , \" 1 = \" \u00C2\u00BB F 1 5 . 5 ) 3 8 1 C O N T I N U E W R I T E ( 6 . 3 8 4 ) P E N 3 8 4 F O = M A T ( / / / / / / , T 2 0 . \" T O T A L \u00C2\u00B0 E N 4 L T Y = \" , T 6 2 , F 1 8 . 5 ) W R I T E ( 6 , 1 0 - 8 ) 7 S O R E T U R N E N I 8 4 C = > = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = ===3=-=:= = = = = = = = == = = = = == = = = = === = == = = = S i n R O U T T NE D U M P I M P L I C I T J E A L ( A - H . O - 7 ) , I N T E G E R S , C . \u00C2\u00B0 H I , M A R T Y R E A L D ( 7 G , i f : > , D ( 7 C , 1 0 ) , A ( 1 0 0 , 2 6 0 ) , H < 1 3 0 ) , C ( 2 6 0 > R E \" . w < l C C , 1 G 0 > , G ( 1 0 . 0 ) , D E L T A ( 7 0 ! , G A M M A < 7 0 ) , P I ( 1 0 0 ) D I M E N S I O N T W f l O O ) , K A \u00C2\u00B0 P A ( 7 0 ) \u00C2\u00BB L ( 7 0 ) , K ( 7 0 > R E A L 0 \u00C2\u00B0 ( 7 C ) , OM ( 7 0 )' C O M M O N N . M , \" l , M2 ,p, D , A , H , C , . H , C-, D E L T A . G A MM A . P I , I W , K A P P A , L , E P S , S <,OP,OM,70,MARKER W R I T E ( 6 t 1 0 0 > l . ' O F O R M A T ( / / \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 I - 0 M E G A = \u00E2\u0080\u00A2 M= ! P I = \u00E2\u0080\u00A2 W = ? ? / \" \u00E2\u0080\u00A2 \" , \u00C2\u00AB ( \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \" ) ) ^ 0 1 0 J = l , \" , . . 1 0 W R I T E ( o . l O l ) I W ( J ) , h ( J ) I \" I ( J J 1 0 1 F O R M A K \" 1 1 0 , \" ! \" , F l ^ . < \u00E2\u0080\u00A2 , \" ! \" , F l i t . U , ( y \u00C2\u00BB 6 , * * ! \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 , 5 F l < f . M ) W R I T E ( 6 . 1 0 3 1 ( G ( I ) , I = l , M i 1 0 3 F O W M / \" G = \" , ( 6 F 1 5 . 5 1 ) W R I T E ( 6 , 1 0 F ) ( L ( I ) , I = 1 , M 2 > I C S F O = M A T ( \" L = \" . ( 1 2 T 1 0 ) ) , W 9 I T E ( 6 , 1 0 f > ( D E L T A ( I ) , I = 1 , M 2 ) 1 0 6 ' F O R M A T ( \" D E L T A = \" , ( 6 \u00C2\u00A3 1 5 , 5 > > W R I T E ( 6 , 1 0 7 ) ( G A M M A ( I ) , 1 = 1 , M 2 ) 1 0 7 f 0 3 M C T ( \" G A M M A = \" , ( 6 F 1 5 . 5 ) ) W R I T E ( 6 , 1 1 0 ) 7 0 1 1 0 F O R M A T ( \" 7 Q = \" . g 1 5 . 5 i ; . W R I T E ( 6 , 1 1 1 ) ( K A P P A ( I ) , I = 1 , M 2 ) 1 1 1 r O R M A T ( \" K A 0 P A = \" , ( , 1 2 H G ) ) R E T U R N 8 5 P R O G R A M H A P P Y ( T A D E 5 , O U T P U T , T A \u00C2\u00B0 E & = O U T P U T ) I M P L I C I T R E A L ( A - H . O - Z ) T N T E G \" R.. P H T , M A R T Y . R E A L T P ( 7 0 , 1 0 ) , T D ( 7 0 , 1 0 ) R E A L P ( 7 C ' , 1 0 ) , 0 ( 7 u , l J ) , A ( l j 0 . 2 6 0 ) , H ( 1 0 0 ) , C ( 2 6 0 l R E A L W ( 1 0 0 , I C G >' ,G ( 1 0 0 ) , O E L T A ( 7 G > , G A M M A ( 7 0 > , P I ( 1 0 0 > D I M E N S I O N IW ( 1 C J ) , K A P P A < 7 0 ) , L ( 7 0 ) , K ( 7 0 ) R E A L O P ( 7 C I , O M ( 7 0 I 0 i . ; l E N S i g M _ Y ( 1 5 J J L C O M M O N N , M . M i , M 2 , ? , o , A , W , C , H , G , 9 E L T A , G A M M A , P I , I W , K A P ? A , L , E P S , \u00E2\u0080\u00A2f K . Q P , O M , Z O , M A R K E R C O M M O N / P R I N T R / T P , T D C O M M O N / A / y c , \u00C2\u00BB \u00C2\u00BB \u00C2\u00BB , \u00C2\u00BB \u00C2\u00BB \u00C2\u00AB . \u00C2\u00BB \u00C2\u00BB \u00E2\u0080\u00A2 \u00C2\u00BB \u00C2\u00BB G E N E R A T E C A S ' - i F L O W * \u00C2\u00BB \u00E2\u0080\u00A2 * \u00C2\u00BB * * * \u00E2\u0080\u00A2 \u00C2\u00BB \u00C2\u00BB \u00E2\u0080\u00A2 # DO 1 2 3 4 * K = 1 . 2 x i Q 3 = i c o e o o . \u00E2\u0080\u00A2 X l = 3 3 3 3 3 . X 1 9 = 3 3 3 3 3 . X 6 1 = 3 3 3 3 4 . DO 1 2 3 3 K L L = 1 , 3 C A L L K U Z Y ( X 1 0 3 , X 1 , X 1 9 , X 6 1 ) R E W I N O 5 -W R I T E ( 6 , 1 0 2 ) 1 0 2 F O R M A T ( 1 H 1 ) ' W R I T E ( 6 , 1 0 D K K . K L L 1 0 1 F O R M A T ( 1 0 X , 1 h H S I M U L A T I O N R U N , 1 5 , 2 X , 6 H P E R I O D , I 6 ) A X 2 = X ( 2 ) \u00E2\u0080\u00A2 ; _ : i X ( 2 ) = X ( 4 > X ( 4 ) = A X 2 A X 2 0 = X ( 2 0 * A X 2 1 = X ( 2 1 ) A X 2 2 = X ( 2 2 ) . X J 2_0i.= X i _ l 2 J L + X 1 1 3 . L + X J L I JL> : X ( 2 1 ) = X ( 9 ) + X ( l o ) + X ( 1 1 ) X ( 2 2 ) = X ( 8 ) X ( 5 2 ) = X ( 2 3 ) + X ( 2 4 > + X ( 2 5 ) X ( 6 3 ) = A X 2 0 + f l . X 2 1 + A X 2 2 X ( 6 4 ) = X ( 1 9 ) HLHl=LU^ai Y = R A N F ( 0 ) Y Y = 2 G 0 C 0 * Y I Y Y = Y Y - 1 0 0 0 0 R 1 = X < 1 0 3 ) + I Y Y W R I T E ( 6 . 1 0 ( * ) R l _ i l i t E O R\u00C2\u00BBA L < i _ 0 J L j . i ' t n . G A S \u00C2\u00AB _IJL.Ow jR i \u00C2\u00BB = . , F I 6 . . . 4 i _ _ P R 1 = P R ( D U M M Y ) P R 1 = . 5 \u00C2\u00BB o R i + . 0 3 9 7 I F ( I Y Y . G T . C . A N D . ? R l . L E . . i 7 5 ) = R i = P R l + . 0 0 3 I F ( I Y Y , L T . 0 . A N D . P R . l . G E . . 0 5 5 ! P R i = \u00C2\u00B0 = C l - . C 0 5 T 9 1 = P R l + R T 9 ( D U M M Y | T r j _ l = J P J \u00C2\u00BB l J t . 5 T D _ ( J ) U M M X ) . '. : - : A M 1 = P R 1 \u00E2\u0080\u00A2 P M ( D U M M Y ) A L C i = . 7 \u00C2\u00BB ( P R 1 + P L C ( D U M M Y ) ) W R I T E ( 6 , 1 5 0 ) T e i 1 5 0 r O R M A T ( 1 : X , 1 9 H T R E A S U R Y B I L L R AT E = , 2 X , F i 6 . 6 ) W R I T E (_6_,_ 1 51- .>_J.\" 1 i, 1 5 1 F O R M ' A T T I U X ' ; 19~H T E R M O F R O S l f R A T E = i 2 X , [ r l 6 . 6 ) W R I T E < & . 1 5 2 ) A M I 1 5 2 F 0 R M A T ( 1 1 X , 1 3 H M O R T A G E R A T E = , 2 X , F 1 6 . 6 ) W R I T E ( 6 , 1 5 3 ) A L : i 1 5 3 - O R M A T ( 1 0 X , 1 R H L I A B I L I T Y R 4 T E = , 2 X , F l 6 . 6> Y 1 = X ( 2 ) + X ( 2 0 ) + X ( 6 2 I + X ( 3 ) + X ( 2 1 ) + X ( 6 3 > + . C Q 5 \u00C2\u00BB X 1 '* $ + . Q 4 * Y ( 2 2 ) \" + . 0&*x\"\"(64> .\" Z 2 = . u 0 5 * X U ) + . C 4 * X ( 2 2 ) + . D & * x ( 6 4 > A R M = Z 2 I F ( Y i . L T . R l ) X ( 2 ) = X ( 2 ) + R 1 - Y l I F ( Y l . L T . R l ) G O TO 7 9 X 2 = X ( 2 ) - ( Y l - R l > \u00C2\u00BB < . 2 ) I F ( X 2 . GT . . G) Z 2 = Z 2 + ( X ( 2 > - X 2 ) \u00C2\u00BB ( . 0 0 5 ) I F ( X 2 . L E . . O ) 7 2 = Z 2 + ( X ( 2 ) I * ( . 0 0 5 ) X ( 2 ) = X 2 X 2 0 = 0 . 0 I F ( X ( 2 ) . G E . . O ) G O T O 7 7 X 3 = X ( 3 ) + X ( 2 ) \" X 12 ) = 0 . 0 I F ( X 3 . G E . . 0 I Z 2 = Z 2 + ( X ( 3 ) - X 3 ) ' ( . 0 0 5 ) I F ( X 3 . L T . . 0 ) Z 2 = Z 2 + ( X ( 3 ) ) \u00C2\u00BB ( . O 0 5 ) X ( 3 ) = X 3 I F ( X ( 3 ) . G E . . O ) GO T O 7 7 - X 2 0 = - X ( 3 ) _ , X ( 3 ) = b . 0 7 7 C O N T I N U E X 2 3 l = X ( 2 0 ) - X 2 0 - ( Y l - R l ) * ( . i \u00C2\u00BB l I F ( X 2 G 1 . G T . . 0 ) Z 2 = Z 2 + ( X ( 2 G > - X 2 G 1 > , ' < . 0 4 > I F ( X 2 0 1 . L t . . u ) Z 2 = Z 2 + ( X ( 2 Q ) I \u00E2\u0080\u00A2 ( . 0 4 ) X ( 2 0 I = X 2 0 1 X & 2 = 0 . 0 * I F ( X ( 2 0 ) . G E . , . 0 1 GO T O 7 8 X 2 1 = X ( 2 1 ) + X ( 2 0 ) X ( 2 0 ) = 0 . C I F ( X 2 l . G E . . O ) Z 2 = ? 2 + ( X ( 2 1 ) - X 2 l ) \u00C2\u00BB ( . 0 4 ) I F ( X 2 1 . L T . . 0 ) 7 2= 12\u00C2\u00B1 X_( 2 1 ) * ( . 04 ) Y ( 2 1 ) = X 2 1 I F ( X ( 2 1 ) . G E . . 0 > GO TO 7 8 X 6 2 = - X ( 2 1 > X ( 2 1 ) = 0 . 0 . 7 8 C O N T I N U E \u00E2\u0080\u009E J < 6 2 1 = X ( _ 6 2 ) _ - x r , 2 _ - ( Y l - R _ i ) * { . <\u00E2\u0080\u00A2_> _ I F fx b 2 1 r G > ~ . .T) 7 2 = Z 2 + <\"X ( 6 2 ) - X 6 2 1 ) \u00C2\u00BB '(\". 0 6 ) I F ( X 6 2 1 . L T . . 0 ) Z 2 = Z 2 + ( X ( 5 2 ) ) * ( . 0 6 ) X ( 6 2 ) = X 6 2 1 I F ( X ( & 2 S . G E \u00E2\u0080\u00A2 . 0 1 GO T O 7 9 X 5 3 = X ( 6 3 ) + X ( 6 2 ) X ( 6 2 ) = 0 . f : . \" I F \" ( X 6 3 . G E \" . . 0 ) Z 2 = Z 2 + ( X ( 6 3 ) - X 6 3 ) \u00C2\u00BB { . 0 6 ) I F ( X 6 3 . L T . . 0 ) Z 2 = Z 2 + ( X ( 6 3 > ) * ( . 0 6 ) ( 6 3 ) = X 6 3 Jc ( X ( 6 3 > . G \u00C2\u00A3 . . 0 \u00C2\u00BB GO TO 7 9 _ L _ _ _ i _ 2 . . . 7 9 C O N T I N U E 7 _ = T 8 1 * X ( 2 > * T 0 1 * X < 2 0 > + A M 1 * X ( S 2 ) 5 + . 0 5 4 1 * X ( 3 ) + . 0 3 2 7 * X ( 2 i ) + . u 9 9 2 * X ( 6 3 ) Z 3 = Z 1 - Z 2 - R 1 * A L C 1 W 3 I T E ( 6 , 1 0 8 ) K L L . Z 3 101_______A_TLI__>_,. 7 ^ X ( i ) = X ( 2 > + X < 3 ) X ( 1 9 ) =X ( 2 0 > * X ( 2 1 > X ( 6 1 ) =X ( 6 2 ) *\u00E2\u0080\u00A2< ( 6 3 ) x ( 1 3 3 ) = R 1 X 1 0 3 = X ( 1 0 * > x i _ _ l _ ) : X 1 9 = X ( 1 9 ) X 6 1 = X ( 6 1 ) W R T T E ( 6 , 2 C 7 > X l , X 1 9 , X 6 1 , X 1 C 3 2 0 7 F O R M A T ( 1 O X . 2 H X = , < * F 1 6 . 4 > W R I T E ( 6 , 1 6 0 ) i _ 6 _ _ _ _ _ F I ! _ _ _ J _ _ _ l _ ^ ^ W R I T E ( 6 , 1 6 1 ) 1 6 1 F O R M A T ( 1 C X , 2 0 M 5 T A T I 3 T I C S ) S B Z 1 = S B Z 1 + Z 3 S B 7 S = S B Z ? + ( 7 3 ) \u00C2\u00BB * 2 W R I T E ( 6 , 1 6 3 ) 7 1 . 1 6 3 F O R M A T ( 1 C ; X , \u00C2\u00AB GR _____ R ._tf E j l U E S _ _ . X l _ 6 _ \u00C2\u00BB _ 6 J W R I T E ( 6 , 1 6 8 1 A R M 1 6 8 F O R M A T d G X , * C O S T OF S A L E S = * , F 1 5 . 6 ) W R I T E ( 6 , 1 6 5 ) Z 2 1 6 5 F ORM A T ( 1 0 X , * C O S T O F S A L E S A N D F O R C E D S A L E S = * , F l 6 . 6 ) A R M I N = R 1 \u00C2\u00BB A L C 1 w _ J _ _ _ _ _ ^ . : _ _ _ _ A _ _ J . _ 1 6 6 F 0 R M A T ( 1 G V , \u00C2\u00BB C O S T OF F U N 3 S = * , F 1 6 . 6 ) W R I T E ( 6 , 1 6 7 ) S B Z 1 1 6 7 F O R M A T ( 1 0 X , * C U M M U L A T I V E P R O F I T S = \u00C2\u00BB , F 1 5 . ' 6 > W R I T E ( 6 , 1 6 9 ) S 5 2 S 1 6 9 F O R M A T ( 1 3 X , \u00C2\u00BB C U M MU L A T I V E P R O F I T S S Q U A R E D * , F 2 0 . 3 ) . 1 2 3 . 3 _ . 0 N I _ L _ . J _ . 1 2 3 U C O N T I N U E E N D 8 8 T h e p u r p o s e o f t h e c o d e i s t o s o l v e i n - c o r e a s t o c h a s t i c l i n e a r p r o g r a m s w i t h s i m p l e r e c o u r s e . T h e p r o b l e m t o b e s o l v e d i s o f t h e t y p e s u b j e c t t o m m - , x n m 2 + + _ e x . + E { m i n l~ ( p y + p \" y \" j = l J J 0 J f o r j = 1 , \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 , n y + . > 0 f o r i = 1 , \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 , m 2 y ~ > 0 f o r i = l , * * * , m 2 w h e r e : c - i s t h e j t h e l e m e n t o f t h e J x . - i s t h e j t h e l e m e n t o f t h e J a . . - i s t h e ( i , j ) t h e l e m e n t o f 1 J A : \" \u00E2\u0080\u00A2 t . . - i s t h e ( i , j ) t h e l e m e n t o f U T b_j - i s t h e i t h e l e m e n t o f t h e - i s t h e i t h e l e m e n t o f t h e y ~ - i s t h e i t h e l e m e n t o f t h e g i v e n c o s t v e c t o r d e c i s i o n v e c t o r x t h e g i v e n t e c h n o l o g i c a l m a t r i x t h e g i v e n t e c h n o l o g i c a l m a t r i x g i v e n r e s o u r c e v e c t o r b r a n d o m r e s o u r c e v e c t o r , s u r p l u s v e c t o r y ~ 8 9 y i - i s t h e i t h e l e m e n t o f t h e s h o r t a g e v e c t o r y p j - i s t h e i t h e l e m e n t o f t h e p e n a l t y v e c t o r p ( f o r s h o r t a g e ) p.j - i s t h e i t h e l e m e n t o f t h e p e n a l t y v e c t o r p~ ( f o r s u r p l u s ) a l s o d e f i n e - t o b e t h e j t h s m a l l e s t p o s s i b l e r e a l i z a t i o n ( , j = l , \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 , J . ) o f t h e i t h e l e m e n t o f . 1 p | j ) - i s t h e p ^ . . = a n d j = l I1 p| j ) = 1 . f o r i = l , - - - , m 2 9 0 T h e i n p u t d a t a a r e CARD NUMBER C O N T E N T S FORMAT 1 . 2 . 3 . 4 . t o l e r a n c e n , m 1 . m 2 \u00C2\u00BB P i g ( 2 ) ( 2 ) ( F 8 . 5 ) ( 3 1 5 ) ( I 3 , F 1 0 . 2 , F 6 . 4 ) ( F 1 0 . 2 . F 6 . 4 ) 2 + J . 2 + J ! + ! 2 + ^ + 2 2 + J . + 3 2 + J 2 + ( J _ + J _ + 2 + ( J _ + J 2 + -+ J 2 + 4 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 J ) + 2 m 2 r n 2 \u00E2\u0080\u00A2 + J m 2 ) + 2 m 2 + l 2 + ( J i + J 2 + * ) + 2 m 2 + k i 0 1 2 m 1 + m 2 2 + ( J 1 + J 2 + ' \" + J m 2 ) + 2 m 2 + _ k , i = l \u00E2\u0080\u00A2 m i + m 2 m i + m 2 2 + ( J i + J 2 + \u00E2\u0080\u0094 + J )+2m + I k.+lmJS^ m i \u00E2\u0080\u00A2 i = l 1 m_+m2 r ( J i ) n ( J i ) a n 3 i pt> P i J D { 1 ) u 2 , c , 2 , P 2 + p >p K m 2 rri2 j , 0 0 0 0 0 0 . . . 0 0 0 0 0 0 . . . b i , b 2 , b 3 ( F 1 0 . 2 . F 6 . 4 ) ( 2 F 1 0 . 2 ) ( 2 F 1 0 . 4 ) ( I 3 , F 1 0 . 2 , F 6 . 4 ) ( 2 F 1 0 . 4 ) ( 2 F 1 0 . 4 ) ( I 3 . F 1 0 . 4 ) ( 1 1 3 ) ( 1 1 3 ) ( 3 F 1 0 . 4 ) b , 0 , b , , b ( 3 F 1 0 . 4 ) m i - 2 ' I T U - T m i v ' 2 + ( J 1 + J 2 + - - - + J m ) 2 m 2 + I k ^ C n i i / 3 3 + 1 c _ \u00E2\u0080\u009E c 2 . . c 3 ( 3 F 1 0 . 4 ) \u00E2\u0080\u00A2 i = l \u00E2\u0080\u00A2 m i + m 2 2 + ( J 1 + J 2 + - - \u00C2\u00AB + J _ 2 ) + 2 m 2 + ^ k j + D n . / a . + . n / S ] c n _ 2 , c n _ r c n ( 3 F 1 0 . 4 ) i = l i^] i s t h e m i n i m u m o f a l l i n t e g e r s n o t l e s s t h a n ^ y -91 C a r d 1 T h e u s e r m a y p r o v i d e h i s o w n t o l e r a n c e l e v e l , C a r d 2 n - T h e n u m b e r o f v a r i a b l e s ( n o t i n c l u d i n g s u r p l u s a n d s h o r t a g e v a r i a b l e s ) . m_ - T h e n u m b e r o f d e t e r m i n i s t i c c o n s t r a i n t s . m 2 - T h e n u m b e r o f s t o c h a s t i c c o n s t r a i n t s . C a r d 3 ( f i r s t s t o c h a s t i c c o n s t r a i n t ) _ ! - T h e n u m b e r o f r e a l i z a t i o n s o f R H S o f t h e f i r s t s t o c h a s t i c c o n s t r a i n t ( w i t h p o s i t i v e p r o b a b i l i t y ) . - S m a l l e s t p o s s i b l e r e a l i z a t i o n o f RHS o f 1 s t s t o c h a s t i c c o n s t r a i n t . p j ^ - P r o b a b i l i t y o f o c c u r r i n g . C a r d J i + 3 a . - A r e a l n u m b e r e q u a l o r l e s s t h a n ( i . e . a l o w e r b o u n d o n t h e r e a l i z a t i o n s ) . 3 i - A r e a l n u m b e r e q u a l o r g r e a t e r t h a n E,^1^ ( i . e . a n u p p e r b o u n d o n t h e r e a l i z a t i o n s ) . C a r d 2 + J _ + 2 p | - A p e r u n i t p e n a l t y , f o r a s h o r t a g e o n t h e l e f t h a n d s i d e o f t h e f i r s t s t o c h a s t i c c o n s t r a i n t . p~ - A p e r u n i t p e n a l t y , f o r a s u r p l u s o n t h e l e f t h a n d s i d e o f t h e f i r s t s t o c h a s t i c c o n s t r a i n t . C a r d [ 2 + J . + 3 ] t o [ 2 + ( J 1 + J 2 + - \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 + _ + 2 m 2 ] - S e q u e n t i a l l y r e p e a t s p r o c e s s o f c a r d s 3 t o 2 + J i + 2 f o r e a c h c o n s t r a i n t . 9 2 C a r d s 2 + ( J _ + * \" + J ).+2m2+1 - Now s t a r t i n g t o i n p u t t h e t e c h n o l o g i c a l c o e f f i c i e n t s - -r o w s a r e l i s t e d i n o r d e r a n d s e p a r a t e d b y a s t r i n g o f O ' s o r b l a n k s ( a t l e a s t 1 3 ) - - a l l t h e d e t e r m i n i s t i c c o n -s t r a i n t s m u s t p r e c e d e t h e s t o c h a s t i c c o n s t r a i n t s - -a s s u m e k - j - 1 n o n z e r o c o e f f i c i e n t s i n r o w i - - ( u n s p e c i f i e d c o e f f i c i e n t s d e f a u l t t o 0 ) . j - C o l u m n n u m b e r . a - j j - C o e f f i c i e n t f o r f i r s t r o w a n d j t h c o l u m n . m 1 + m 2 C a r d 2 + ( J 1 + J 2 + . \u00C2\u00BB \u00C2\u00AB + J m ) + 2 m 2 + I k.+1 b i - R i g h t h a n d s i d e o f f i r s t d e t e r m i n i s t i c c o n s t r a i n t . b 2 - R i g h t h a n d s i d e o f s e c o n d d e t e r m i n i s t i c c o n s t r a i n t . b 3 - R i g h t h a n d s i d e o f t h i r d d e t e r m i n i s t i c c o n s t r a i n t . m 1 + m 2 C a r d 2 + ( J 1 + J 2 + - + J ) + 2 m 2 + \u00C2\u00A3 k . + [ m _ , 3 ] + 1 m_ \u00E2\u0080\u00A2 . = ] I C i - C o s t c o e f f i c i e n t o f f i r s t v a r i a b l e . c 2 - C o s t c o e f f i c i e n t o f s e c o n d v a r i a b l e . c 3 - C o s t c o e f f i c i e n t o f t h i r d v a r i a b l e . A d d i t i o n a l n o t e s o n i n p u t d a t a a n d r e s t r i c t i o n s . 1 . A l l c o n s t r a i n t s m u s t b e e q u a l i t i e s , s o s l a c k s m u s t b e a d d e d o r s u b t r a c t e d ( s i g n o f RHS i s n o t i m p o r t a n t ) . 2 . T h e n u m b e r o f d e t e r m i n i s t i c c o n s t r a i n t s m i < 1 5 0 + ( 7 0 - m 2 ) . 3 . T h e n u m b e r o f s t o c h a s t i c c o n s t r a i n t s m . < 7 0 . 4 . T h e n u m b e r o f p o s s i b l e r e a l i z a t i o n s f o r E . i s J . < 8 . C h a p t e r 4 IMPLEMENTATION OF THE ALM MODEL 4 . 1 I n t r o d u c t i o n T h i s c h a p t e r i s c o n c e r n e d w i t h r e s u l t s o f a n a p p l i c a t i o n o f t h e A L M m o d e l t o t h e a s s e t a n d l i a b i l i t y p o r t f o l i o p r o b l e m o f V a n c o u v e r C i t y S a v i n g s C r e d i t U n i o n ( V C S ) . I n a d d i t i o n t o t h e s e r e s u l t s , s o m e o f t h e p r o c e d u r a l a s p e c t s o f i m p l e m e n t i n g t h e m o d e l f o r t h i s a n d r e l a t e d p r o b l e m s a r e d i s c u s s e d . T h i s t h e s i s , i n f a c t , w a s p r o m p t e d b y t h e r e a l l i f e p r o b l e m c o n t i n u o u s l y f a c i n g t h i s p a r t i c u l a r c r e d i t u n i o n -a l i q u i d i t y p r o b l e m . S o m e o f t h e s a l i e n t c h a r a c t e r i s t i c s o f V C S d u r i n g t h e f i v e y e a r p l a n n i n g p e r i o d s t u d i e d , 1 9 7 0 t o 1 9 7 4 , a r e : 1 ) t h e f i r m ' s a s s e t s g r e w a t a c o m p o u n d r a t e o f 57% f r o m $ 2 6 m i l l i o n t o $ 1 6 0 m i l l i o n , a n d 2 ) t h e f i r m a d o p t e d a n a g g r e s s i v e p o l i c y o f i n v e s t i n g i n h i g h y i e l d i n g a s s e t s ( p r e d o m i n a n t l y m o r t g a g e s ) . I n 1 9 7 4 , V C S r e a l i z e d t h a t t h e c o m -b i n a t i o n o f t h e i r a g g r e s s i v e i n v e s t m e n t p o l i c y a n d c h a n g i n g m a r k e t c o n d i t i o n s w a s c r e a t i n g s e r i o u s l i q u i d i t y p r o b l e m s . I n v e s t o r s w e r e t r a d i n g t h e i r l o w y i e l d i n g t e r m d e p o s i t s f o r h i g h e r y i e l d i n g d e p o s i t s . A t t h e s a m e t i m e t h e o u t s t a n d i n g m o r t g a g e l o a n s o f V C S w e r e s t i l l e a r n i n g r e t u r n s o n t h e b a s i s o f t h e l o w e r i n t e r e s t r a t e s t r u c t u r e . I t w a s a t t h i s m o m e n t t h a t t h e f i r s t v e r s i o n o f t h i s s t u d y w a s i n i t i a t e d . 9 3 9 4 T h e a p p r o a c h f i r s t t a k e n w a s t o c o n s t r u c t a f i v e y e a r l i n e a r p r o g r a m m i n g p l a n n i n g m o d e l b a s e d u p o n t h e C h a m b e r s a n d C h a r n e s f o r m u -l a t i o n [ 1 1 ] . T h e o b j e c t i v e o f t h e V C S f o r m u l a t i o n w a s t o m a x i m i z e n e t d i s c o u n t e d r e t u r n s w h i c h a r e g i v e n b y t h e t o t a l d i s c o u n t e d r e t u r n s m i n u s t h e t o t a l d i s c o u n t e d c o s t s . T h e r e w e r e f o u r t y p e s o f c o n s t r a i n t s : 1 ) t h e l e g a l c o n s t r a i n t s a s p r e s c r i b e d b y t h e C r e d i t U n i o n s A c t o f B r i t i s h C o l u m b i a [ 8 ] , 2 ) t h e l i q u i d i t y c o n s t r a i n t s w h i c h a r e s i m i l a r t o i n e q u a l i t i e s ( 4 ) a n d ( 5 ) i n c h a p t e r 2 , 3 ) t h e b u d g e t c o n s t r a i n t s t h a t i n c l u d e t h e i n i t i a l c o n d i t i o n s a n d a s t a t e m e n t o f t h e a c c o u n t i n g i d e n t i t y - t h e u s e s o f f u n d s a r e e q u a l t o t h e s o u r c e s o f f u n d s , a n d 4 ) t h e p o l i c y c o n s t r a i n t s w h i c h i n c l u d e t h e i n t e r n a l o p e r a t i n g p o l i c i e s o f V C S a n d t h e t e r m i n a l c o n d i t i o n s t o i n s u r e t h a t t h e s t r u c t u r e o f t h e f i n a l p o r t f o l i o o f a s s e t s a n d l i a b i l i t i e s m a i n t a i n s c o n t i n u i t y o f o p e r a t i o n s . T h e b a s i c s h o r t c o m i n g o f t h e a b o v e f o r m u l a t i o n i s t h a t i t d o e s n o t i n c o r p o r a t e t h e i n h e r e n t u n c e r t a i n t y o f u n k n o w n c a s h f l o w s a n d i n t e r e s t r a t e s . A s a n i n i t i a l a t t e m p t t o o v e r c o m e t h i s d r a w b a c k , a d e c i s i o n t h e o r e t i c a p p r o a c h w a s t a k e n . T h e p r o c e d u r e w a s t o f i r s t m a k e p o i n t e s t i m a t e s o f f u t u r e i n t e r e s t r a t e s t r u c t u r e s a n d p o t e n t i a l g r o w t h r a t e s o f V C S ' s a s s e t s . T h e n , f o r e a c h p o s s i b l e c o m b i n a t i o n o f i n t e r e s t r a t e s t r u c t u r e a n d g r o w t h r a t e , t h e l i n e a r p r o g r a m w a s e x e c u t e d . T h i s w o u l d t h e n y i e l d a s e t o f s o l u t i o n s . T h e s t e p s n e c e s s a r y t o f i n d t h e ' b e s t ' s o l u t i o n c a n b e s u m m a r i z e d : 1 ) f i n d i n g t h e o p t i m a l s o l u t i o n v e c t o r f o r e a c h s t a t e o f n a t u r e , 2 ) c o m p u t i n g t h e r e s u l t i n g n e t 9 5 p r e s e n t v a l u e s f o r e a c h o p t i m a l s o l u t i o n f r o m s t e p 1 ) f o r e a c h o f t h e r e m a i n i n g s t a t e s o f n a t u r e , ( T h i s c a n b e a c c o m p l i s h e d b y f o r c i n g t h e s o l u t i o n t o b e t h e s a m e a s i n s t e p 1 ) f o r e a c h s t a t e o f n a t u r e . I f i n f e a s i b i l i t y i s r e a c h e d t h e n t h e d e b t c o n s t r a i n t s a r e r e l a x e d t o a t t a i n f e a s i b i l i t y ) , , 3 ) u s i n g s u b j e c t i v e p r o b a b i l i t i e s f o r t h e l i k e l i -h o o d o f e a c h s t a t e o f n a t u r e , a n e x p e c t e d v a l u e f o r e a c h d e c i s i o n i n s t e p 1 ) w a s c o m p u t e d , a n d 4 ) f r o m s t e p 3 ) s e l e c t i n g t h a t a c t i o n w h i c h h a s t h e h i g h e s t e x p e c t e d n e t p r e s e n t v a l u e . M a t h e m a t i c a l l y , s t e p s 3 ) a n d 4 ) a r e r e p r e s e n t e d b y m n E ( N P V k ) - ^ ^ N P V i j k P ( e , . . ) a n d c h o o s i n g k * s u c h t h a t E ( N P V k J > E ( N P V k ) f o r a l l k . W h e r e e ^ - i s t h e s t a t e o f n a t u r e w i t h t h e i t h i n t e r e s t r a t e s t r u c t u r e a n d t h e j t h g r o w t h r a t e a n d i = 1 , . . . , m a n d j = l , . . . , n ; P ( e . - ) i s t h e p r o b a b i l i t y o f e . . o c c u r r i n g ; N P V . . . i s t h e n e t p r e s e n t v a l u e o f c h o o s i n g s t r a t e g y k w h e i t e g y . n 6 - o c c u r s f o r k = 1 , . . . , ( n x m ) ; a n d k * i s t h e o p t i m a l s t r a -A l t e r n a t i v e c r i t e r i a s u c h a s m i n i m a x c a n b e u s e d t o f i n d a ' b e s t ' s o l u t i o n i n s t e a d o f s t e p s 3 ) a n d 4 ) . H o w e v e r , i n g e n e r a l , t h i s a p p r o a c h i s n o t v e r y a p p e a l i n g f o r t h e f o l l o w i n g r e a s o n s : 1 ) t h e k * c h o s e n i s i n n o w a y o p t i m a l , f o r t h e r e m a y e x i s t a s o l u t i o n k w h i c h i s n o t o p t i m a l f o r a n y p a r t i c u l a r s t a t e o f n a t u r e , b u t h a s a h i g h e r e x p e c t e d n e t p r e s e n t v a l u e ; a n d 2 ) t h e m o d e l d o e s n o t i n c o r p o r a t e a n y m e a n s o f 9 6 e v a l u a t i n g t h e e c o n o m i c c o n s e q u e n c e s o f i n f e a s i b i l i t y , f o r e x a m p l e t h e r e m a y b e a p a r t i c u l a r l y d i s a s t r o u s r e a l i z a t i o n t h a t r e s u l t s i n i n s o l v e n c y . A s d e s c r i b e d i n C h a p t e r 2 , a n u m b e r o f o t h e r a p p r o a c h e s h a v e b e e n p r o p o s e d t o m o d e l t h e p r o b l e m , b u t , a t b e s t t h e s e m o d e l s h a v e o n l y l i m i t e d a p p l i c a b i l i t y . H e n c e a d i f f e r e n t a p p r o a c h t o t h e p r o b l e m w a s n e c e s s a r y . T h i s l e d t o t h e A L M f o r m u l a t i o n . T h e A L M m o d e l d o e s i n c o r -p o r a t e u n c e r t a i n t y w h i l e m a i n t a i n i n g c o m p u t a t i o n a l t r a c t a b i l i t y f o r l a r g e p r o b l e m s . A s a l r e a d y s t a t e d t h e p u r p o s e o f t h i s c h a p t e r i s t o d e m o n -s t r a t e t h e a p p l i c a b i l i t y o f t h e A L M m o d e l . S p e c i f i c a l l y , t h e r e a r e t h r e e m a j o r d o m a i n s w h e r e i t c a n b e d e m o n s t r a t e d : 1 ) t h e u s e f u l n e s s o f t h e m o d e l i n t e r m s o f t h e r e s u l t s t o b e u s e d b y m a n a g e m e n t , i n c r e a s e d p r o f i t a b i l i t y a n d s u p e r i o r i t y o f t h e e q u i v a l e n t d e t e r m i n i s t i c p r o b l e m , 2 ) t h e e a s e o f a p p l i c a t i o n , a n d 3 ) t h a t t h i s m o d e l d o e s i n f a c t h a v e t h e f e a t u r e s a t t r i b u t e d t o i t , b e f o r e c o m p a r i n g i t t o t h e ' b e s t ' a l t e r -n a t i v e s o l u t i o n t e c h n i q u e - t h e B r a d l e y a n d C r a n e m o d e l . H e n c e t h e r e m a i n d e r o f t h i s c h a p t e r i s c o n c e r n e d w i t h t h e i m p l e m e n t a t i o n a n d r e s u l t s o f t h e A L M m o d e l t o V C S f o r t h e p l a n n i n g p e r i o d 1 9 7 0 - 7 4 . 9 7 4 . 2 M o d e l D e t a i l s T h e a i m o f t h i s s e c t i o n i s t o d e s c r i b e t h e i n p u t n e c e s s a r y f o r t h e i m p l e m e n t a t i o n o f t h e A L M m o d e l t o V a n c o u v e r C i t y S a v i n g s C r e d i t U n i o n . I t w i l l i n c l u d e t h e m e t h o d o f d a t a c o l l e c t i o n , t h e c h o i c e o f d e c i s i o n v a r i a b l e s , a n d t h e a c t u a l c o n s t r a i n t s a n d o b j e c t i v e f u n c t i o n u s e d i n t h e a p p l i c a t i o n . T h e p u r p o s e i s t o i n d i c a t e t h e e f f o r t r e q u i r e d t o i m p l e m e n t t h e A L M m o d e l r a t h e r t h a n t o d e m o n s t r a t e t h e v e r y d i f f i -c u l t p r o b l e m s o f e s t i m a t i o n . T h e a c t u a l d a t a u s e d a r e g i v e n i n A p p e n d i x 1 a t t h e e n d o f t h i s c h p a t e r . S i n c e p r e s e n t i n g t h e d a t a i n m a t r i x f o r m w o u l d b e r a t h e r c u m b e r s o m e ( t h e m a t r i x i s 9 2 b y 2 5 7 ) , t h e y a r e p r e s e n t e d i n i n p u t f o r m ( a s d e s c r i b e d i n C h a p t e r 3 , A p p e n d i x 2 ) . T h e A L M m o d e l b e i n g a S L P R m o d e l i m p l i e s t h a t t h e r e a r e f i r s t a n d s e c o n d s t a g e d e c i s i o n v a r i a b l e s . T h e f i r s t s t a g e v a r i a b l e s a r e k d d i v i d e d i n t o a s s e t s , X . ^ , a n d l i a b i l i t i e s , a n d ( a s d e f i n e d i n C h a p t e r 3 ) . E l e v e n t y p e s o f a s s e t s a r e c o n s i d e r e d i n t h i s a p p l i c a t i o n . T h e y a r e : 1 ) c a s h , 2 ) B r i t i s h C o l u m b i a C r e d i t U n i o n s h a r e s , 3 ) f e d e r a l g o v e r n m e n t b o n d s m a t u r i n g i n i y e a r s ( i = 1 4 ) , 4 ) f e d e r a l g o v e r n m e n t b o n d s m a t u r i n g i n f i v e t o t e n y e a r s , 5 ) p r o -v i n c i a l g o v e r n m e n t b o n d s m a t u r i n g i n m o r e t h a n t e n y e a r s , 6 ) f i r s t a n d s e c o n d m o r t g a g e s w i t h a t h r e e y e a r t e r m , a n d 7 ) p e r s o n a l l o a n s . S i x t y p e s o f l i a b i l i t i e s a r e c o n s i d e r e d . T h e y i n c l u d e : 1 ) d e m a n d d e p o s i t s , 2 ) s h a r e c a p i t a l o f V C S , 3 ) b o r r o w i n g f r o m b a n k s , a n d 4 ) t e r m d e p o s i t s m a t u r i n g i n i y e a r s ( i = 1 , 3 , 5 ) . S p e c i f i c a l l y , i f a 9 8 f o u r y e a r f e d e r a l g o v e r n m e n t b o n d i s p u r c h a s e d a t t h e b e g i n n i n g o f t h e 6 6 t h i r d t i m e p e r i o d , t h i s w i l l g e n e r a t e d e c i s i o n v a r i a b l e s X g ^ , X - g a n d 6 6 6 X - ^ , w h e r e X ^ a n d X g ^ a r e t h e p o r t i o n s o f t h e i n i t i a l i n v e s t m e n t t o b e s o l d i n p e r i o d s f o u r a n d f i v e , r e s p e c t i v e l y a n d i s t h e p o r t i o n t o b e h e l d a t t h e h o r i z o n o f t h e m o d e l . T h e e l e v e n t y p e s o f a s s e t s g e n e r a t e 1 3 6 v a r i a b l e s ( i n c l u d i n g t h e i n i t i a l p o s i t i o n s ) a n d t h e s i x t y p e s o f l i a b i l i t i e s g e n e r a t e a n a d d i t i o n a l 3 6 v a r i a b l e s ( i n c l u d i n g t h e i n i t i a l p o s i t i o n s ) f o r t h e f i v e y e a r p l a n n i n g p e r i o d . T h e c h o i c e o f t h e s e a s s e t s a n d l i a b i l i t i e s w a s b a s e d o n V C S ' s h i s t o r i c a l p o r t -f o l i o s o f a s s e t s a n d l i a b i l i t i e s [ 8 5 ] . T h e r e a s o n f o r s u c h a c h o i c e w a s t o m a i n t a i n a b a s i s o f c o m p a r i s o n b e t w e e n t h e a c t u a l p o r t f o l i o s a n d t h e p o r t f o l i o s y i e l d e d b y t h e A L M f o r m u l a t i o n . B e f o r e d e t a i l i n g t h e s p e c i f i c s o f t h e m o d e l , i t i s i m p o r t a n t t o n o t e t h a t a l t h o u g h t h e c a s h f l o w s a r e c o n t i n u o u s o v e r t i m e , t h e m o d e l a s s u m e s t h a t a l l t r a n s a c t i o n s o c c u r a t t h e b e g i n n i n g o f p e r i o d s . C a s h f l o w s d u r i n g a n y p e r i o d a r e t r e a t e d a s s u m i n g t h a t h a l f t h e c a s h f l o w s o c c u r a t t h e b e g i n n i n g o f t h e p r e s e n t p e r i o d a n d t h e o t h e r h a l f a t t h e b e g i n n i n g o f t h e n e x t p e r i o d . T h e c o n s t r a i n t s w i l l n o w b e d e s c r i b e d . 9 9 a . L e g a l C o n s t r a i n t s T h e s o u r c e f o r t h e l e g a l c o n s t r a i n t s i s t h e C r e d i t U n i o n A c t o f B r i t i s h C o l u m b i a [ 8 J . T h i s a c t p l a c e s t h r e e o p e r a t i o n a l r e s t r i c -t i o n s o n t h e c o m p o s i t i o n o f t h e p o r t f o l i o o f a s s e t s a n d l i a b i l i t i e s . T h e f i r s t c o n s t r a i n t i s t h a t c r e d i t u n i o n s m a i n t a i n a t l e a s t 10% o f t h e t o t a l a s s e t s , \u00C2\u00A3 X . . , i n h i g h l y l i q u i d a s s e t s , Y X . . , t h a t i s i e l 1 1 i e I L 1 I X > .1 I X i e I L 1 i e l 1 T h e s e c o n d r e q u i r e m e n t i s t h a t c r e d i t u n i o n s m a i n t a i n a t l e a s t 1% o f t h e i r t o t a l d e b t , Y . . , i n c a s h a n d t e r m d e p o s i t s , X , t a n d Xr,^, r e s p e c t i v e l y , X l t + X 2 t ^ 0 1 J n Y i t \" 1 e u T h e f i n a l c o n s t r a i n t r e s t r i c t s t h e c r e d i t u n i o n ' s b o r r o w i n g , I Y . . t o o n e h a l f o f t h e t o t a l l i a b i l i t i e s , b _ B D Z I Y b t i ' 5 I Y i f b e B D t i e D i r S i n c e t h e p l a n n i n g h o r i z o n i s f o r f i v e p e r i o d s , t h e l e g a l r e q u i r e m e n t s a c c o u n t f o r f i f t e e n c o n s t r a i n t s i n t h e f o r m u l a t i o n . 1 0 0 b . B u d g e t C o n s t r a i n t s O f t h e t w e n t y - t w o c o n s t r a i n t s i n t h i s s e t , t h e f i r s t s e v e n -t e e n e s t a b l i s h t h e i n i t i a l p o s i t i o n s o f t h e e l e v e n t y p e s o f a s s e t s a n d s i x t y p e s o f l i a b i l i t i e s , w h i l e t h e o t h e r f i v e c o n s t r a i n t s r e q u i r e t h e s o u r c e s o f f u n d s t o b e e q u a l t o t h e u s e s o f f u n d s i n e a c h p e r i o d . T h e s e c o n s t r a i n t s w e r e c o n s t r u c t e d d i r e c t l y f r o m t h e b u d g e t c o n s t r a i n t s i n t h e A L M f o r m u l a t i o n , a s i n C h a p t e r 3 . T h e w a y i n w h i c h t h e a c t u a l n u m b e r s u t i l i z e d i n t h e s e e q u a t i o n s w e r e d e t e r m i n e d , w i l l b e p a r t o f t h e d i s c u s s i o n o n t h e o b j e c t i v e f u n c t i o n . c . L i q u i d i t y C o n s t r a i n t s T h e f u n c t i o n o f t h e l i q u i d i t y c o n s t r a i n t s i s t o e n s u r e t h a t t h e f i r m h a s s u f f i c i e n t c a p i t a l r e s e r v e s t o m e e t s e v e r e w i t h d r a w a l c l a i m s u n d e r a d v e r s e e c o n o m i c c o n d i t i o n s . T h e c o n s t r a i n t s f o l l o w f r o m t h e F e d e r a l R e s e r v e B o a r d ' s c a p i t a l a d e q u a c y f o r m u l a [ 2 7 ] . T h e a p p l i -c a t i o n o f t h e F R B ' s c a p i t a l a d e q u a c y f o r m u l a t o B r i t i s h C o l u m b i a ' s c r e d i t u n i o n s i s j u s t i f i e d i n a s t u d y p u b l i s h e d b y t h e C r e d i t U n i o n R e s e r v e B o a r d [ 2 5 ] . T h e f i r s t t h r e e c o n s t r a i n t s e s t a b l i s h c a p i t a l r e s e r v e s b a s e d u p o n t h e s t r u c t u r e o f t h e p o r t f o l i o o f a s s e t s a n d l i a b i l i t i e s . ^ P . > q . ( W - I a . k ) i = 1 , 2 , 3 ( 1 ) 1 1 k e ^ u . ^ u K . K T h e s a m e n o t a t i o n i s u s e d a s i n C h a p t e r 3 , S e c t i o n 2 . 101 w h e r e W i s t h e d o l l a r v a l u e o f t h e e x p e c t e d w i t h d r a w a l . c l a i m s u n d e r m a d v e r s e e c o n o m i c c o n d i t i o n s , W = \u00C2\u00A3 y^y-, w h e r e y . m e a s u r e s t h e c o n -i = l 1 1 1 t r a c t i o n o f l i a b i l i t y y . u n d e r a d v e r s e e c o n o m i c c o n d i t i o n s . T h e y.'s u s e d w e r e . 4 7 f o r d e m a n d d e p o s i t s , . 3 6 f o r t e r m d e p o s i t s a n d 1 . 0 f o r b o r r o w i n g . T h e p a r a m e t e r s a r e j u s t i f i e d i n [ 2 5 _ . T h e i n ( 1 ) i s a p a r a m e t e r t h a t m e a s u r e s t h e r e a l i z a b l e p o r t i o n i n t h e v a l u e o f a s s e t k i f t h e a s s e t i s t o b e l i q u i d a t e d q u i c k l y u n d e r a d v e r s e e c o n o m i c c o n d i -t i o n s . T h e q . m e a s u r e s t h e r e s e r v e s r e q u i r e d f o r p o t e n t i a l w i t h d r a w a l c l a i m s t h a t e x c e e d t h e r e a l i z a b l e p o r t i o n o f t h e a s s e t s c o n t a i n e d i n K-| u . . . L b K . j . P . i s t h e r e q u i r e d r e s e r v e n e c e s s a r y t o m e e t t h e e x c e s s w i t h d r a w a l c l a i m s . F i n a l l y , t h e p r i n c i p a l c o n s t r a i n t i n t h e c a p i t a l a d e q u a c y f o r m u l a c a n b e s t a t e d a s , K 3 [ t o t a l r i g h t h a n d - e q u i t y - s u r p l u ^ I 0 - 8 , - ) x - >_ I P . + \u00E2\u0080\u00A2 j s i d e o f b a l a n c e i = l i = l s h e e t w h e r e g . i s a p a r a m e t e r t o m e a s u r e t h e s h r i n k a g e o f a s s e t i , w h e n t h e a s s e t i s t o b e l i q u i d a t e d q u i c k l y . T h e a c t u a l n u m b e r s u s e d f o r o ^ , q ^ , a n d 3 . a r e t h e s a m e a s t h o s e p r e s c r i b e d b y t h e F R B [ 2 7 ] . S i n c e t h e p u r p o s e h e r e i s n o t t o d e v e l o p a n o p e r a t i o n a l m o d e l f o r V C S , b u t r a t h e r t o d e m o n s t r a t e t h e a p p l i c a b i l i t y o f t h e A L M m o d e l , t h e n u m b e r s u s e d f o r t h e a b o v e p a r a m e t e r s p r o v i d e a n a d e q u a t e p r o x y . H o w e v e r , i n t h e d e v e l -o p m e n t o f a n o p e r a t i o n a l m o d e l i t w o u l d b e n e c e s s a r y t o e s t i m a t e t h e p a r a m e t e r s . 1 0 2 S i n c e t h e s e c o n s t r a i n t s h a v e t o h o l d f o r a l l f i v e p e r i o d s , i t i s i m p l i e d t h a t t h e r e a r e t w e n t y l i q u i d i t y c o n s t r a i n t s . d . P o l i c y C o n s t r a i n t s T w o t y p e s o f p o l i c y c o n s t r a i n t s a r e i n c l u d e d : 1 ) p e r s o n a l l o a n s m a d e i n p e r i o d t ( x t L ) s h o u l d b e e q u a l t o o r l e s s t h a n . 2 o f t h e f i r s t m o r t g a g e l o a n s m a d e i n p e r i o d t U t m ) , X t L <_ . 2 X t m , a n d 2 ) s e c o n d m o r t g a g e s m a d e i n p e r i o d t ( X t ) s h o u l d b e e q u a l o r l e s s t h a n . 1 2 5 o f f i r s t m o r t g a g e s , X ^ s _< . 1 2 5 X t m . T h e r a t i o n a l e f o r s u c h i n v e s t m e n t p o l i c i e s i s t h a t t h e r e t u r n s o n t h e f i r s t m o r t g a g e s a r e l e s s r i s k y ( s m a l l e r d e v i a t i o n s ) , c o m p a r e d t o s e c o n d m o r t g a g e s o r p e r s o n a l l o a n s e v e n t h o u g h t h e l a t t e r m a y y i e l d a b e t t e r r e t u r n . T h i s i s c o n s i s t e n t w i t h m a n a g e m e n t ' s p r e f e r e n c e m a y b e v i o l a t e d w i t h o u t a n y l e g a l i m p l i c a t i o n s . T h e s e f e a t u r e s a r e r e a d i l y i n c o r p o r a t e d b y t r e a t i n g t h e c o n s t r a i n t s a s t h o u g h t h e y w e r e s t o c h a s t i c . S i n c e t h e o b j e c t i v e h e r e i s m e r e l y t o d e m o n s t r a t e t h e a p p l i c a b i l i t y o f t h e A L M m o d e l , i n t h i s a p p l i c a t i o n ( p + , p ~ ) i s ( 0 , 1 . 0 ) , w h i c h s u g g e s t s t h a t t h e c o n s t r a i n t s w i l l n o t b e v i o l a t e d . F o r t h e f i v e p e r i o d s , t h e a b o v e t w o p o l i c y c o n d i t i o n s g e n e r a t e t e n c o n s t r a i n t s . e . D e p o s i t F l o w s T h e v a r i a b l e y^ r e p r e s e n t s t h e n e w d e p o s i t s o f t y p e d ( d = 1 , . . . , 5 ) g e n e r a t e d i n p e r i o d j ( j = 1 , . . . , 5 ) a n d i s a d i s c r e t e 1 0 3 r a n d o m v a r i a b l e r e p r e s e n t i n g t h e b a l a n c e s h e e t f i g u r e o f d e p o s i t t y p e d a t t h e e n d o f t h e j t h p e r i o d . T h e a m o u n t o f y ! ? g e n e r a t e d i s e s t a b l i s h e d \u00E2\u0080\u00A2J i n t h e d e p o s i t f l o w c o n s t r a i n t a s f o l l o w s : T h e y ' s u s e d w e r e 1 . 0 f o r d e m a n d d e p o s i t s a n d . 3 6 f o r t e r m d e p o s i t s . T h e y ' s a r e i n c l u d e d t o r e f l e c t t h e a c t u a l ( a n d n o t n e t ) f l o w o f d e p o s i t f u n d s . T h e d i s t r i b u t i o n o f w a s e s t i m a t e d b y u s i n g t h e a c t u a l b a l a n c e j d s h e e t f i g u r e s o f V C S f o r 1 9 7 0 - 7 4 a s t h e m o d e o f t h e d i s t r i b u t i o n a n d c o n s t r u c t i n g a d i s t r i b u t i o n o f v a l u e s a r o u n d t h i s m o d e . ^ . ^ h a d d i f f e -r e n t p r o b a b i l i t y d i s t r i b u t i o n a s s u m p t i o n s f o r v a r y i n g r u n s o f t h e V C S a p p l i c a t i o n . T h e f i r s t d i s t r i b u t i o n u s e d i s s h o w n i n A p p e n d i x 1 . T h e p e n a l t i e s f o r s h o r t a g e s a s s o c i a t e d w i t h t h e s e c o n s t r a i n t s a r e : 1 ) . f o r d e m a n d d e p o s i t s a n d s h a r e c a p i t a l , p + i s t h e t o t a l d i s -c o u n t e d r e t u r n s o n a o n e y e a r t e r m d e p o s i t m i n u s t h e d i s c o u n t e d c o s t o f t h e f u n d s c a l c u l a t e d t o t h e h o r i z o n o f t h e m o d e l , 2 ) f o r t e r m d e p o s i t s m a t u r i n g i n o n e o r t h r e e y e a r s , p + i s t h e t o t a l d i s c o u n t e d r e t u r n s o n a f i v e y e a r t e r m d e p o s i t m i n u s t h e d i s c o u n t e d c o s t o f t h e f u n d s c a l c u l a t e d t o t h e h o r i z o n o f t h e m o d e l , a n d 3 ) f o r t e r m d e p o s i t s m a t u r i n g i n f i v e y e a r s , p + i s t h e t o t a l d i s c o u n t e d r e t u r n s o n a t e n y e a r p r o v i n c i a l g o v e r n m e n t b o n d m i n u s t h e d i s c o u n t e d c o s t o f t h e f u n d s c a l c u l a t e d t o t h e h o r i z o n o f t h e m o d e l . T h e p e n a l t i e s p \" , f o r s u r p l u s e s a s s o c i a t e d w i t h 1 0 4 t h e d e p o s i t f l o w c o n s t r a i n t s a r e t h e t o t a l d i s c o u n t e d r e t u r n s o n f i r s t m o r t g a g e s m i n u s t h e d i s c o u n t e d c o s t s o f f u n d s c a l c u l a t e d t o t h e h o r i z o n o f t h e m o d e l . T h e p + a n d p ~ a t t e m p t t o r e f l e c t a c o n s e r v a t i v e s t r a t e g y , o n t h e p a r t o f m a n a g e m e n t , a s t o w h a t p o l i c y d e c i s i o n s t o m a k e : 1 ) w i t h t h e s u r p l u s f u n d s a v a i l a b l e w h e n r e a l i z e d s o u r c e s e x c e e d u s e s , a n d 2 ) w i t h t h e s h o r t a g e o f f u n d s w h e n u s e s e x c e e d r e a l i z e d s o u r c e s , r e s p e c -t i v e l y . f . O b j e c t i v e F u n c t i o n T h e o b j e c t i v e i s t o m a x i m i z e t h e e x p e c t e d t o t a l d i s c o u n t e d r e v e n u e s m i n u s t h e e x p e c t e d t o t a l d i s c o u n t e d c o s t s a n d m i n u s t h e e x p e c t e d p e n a l t y c o s t s . T h e d a t a w e r e g a t h e r e d f r o m a n u m b e r o f s o u r c e s . T h e s o u r c e f o r t h e r e t u r n s o n t h e f e d e r a l a n d p r o v i n c i a l g o v e r n m e n t b o n d s w a s [ 1 0 ] . T h e s o u r c e f o r t h e r e t u r n s o n B C C U s h a r e s , m o r t g a g e s a n d p e r s o n a l l o a n s a n d t h e c o s t o f t h e t e r m d e p o s i t s , d e m a n d d e p o s i t s a n d s h a r e c a p i t a l w a s [ 8 5 ] . T h e d i s c o u n t r a t e u s e d w a s t h e t i m e v a l u e o f m o n e y . T o o b t a i n i t , t h e r i s k f r e e r a t e ( t h e a v e r a g e y i e l d o n t h r e e m o n t h t r e a s u r y b i l l s ) w a s u s e d . T h e s e r a t e s a r e a s f o l l o w s [ 1 0 ] : 1 9 7 0 1 9 7 1 1 9 7 2 1 9 7 3 1 9 7 4 A v e r a g e y e a r l y . 0 5 9 9 . 0 3 5 6 . 0 3 5 6 . 0 5 4 7 . 0 7 8 2 y i e l d d i s c o u n t f a c t o r 1 . 9 4 3 5 . 9 1 1 . 8 7 9 7 . 8 3 4 1 1 . 0 5 9 9 1 . 0 3 5 6 1 . 0 3 5 6 1 . 0 5 4 7 1 . 0 7 8 2 . 9 4 3 5 = . 9 1 1 = . 8 7 9 7 = . 8 3 4 1 = . 7 7 3 6 1 0 5 T h e r e t u r n s o n t h e a s s e t s a r e a s f o l l o w s [ 1 0 , 8 5 ] : R e t u r n s o n A s s e t i n Y e a r 1 9 6 9 1 9 7 0 1 9 7 1 1 9 7 2 1 9 7 3 1 9 7 4 T y p e o f A s s e t 1 y e a r f e d e r a l g o v e r n m e n t b o n d ( f g b ) . 0 7 2 5 . 0 6 2 0 . 0 4 5 0 . 0 5 1 0 . 0 6 1 0 . 0 8 0 0 2 y e a r f g b . 0 7 4 9 . 0 6 5 7 . 0 4 9 0 . 0 5 5 0 . 0 6 5 4 . 0 8 0 3 3 y e a r f g b . 0 7 5 8 . 0 6 8 4 . 0 5 2 5 . 0 5 9 0 . 0 6 8 0 . 0 8 0 7 4 y e a r f g b . 0 7 6 7 . 0 7 1 0 . 0 5 5 5 . 0 6 2 6 . 0 6 9 8 . 0 8 1 0 5 y e a r f g b . 0 7 7 6 . 0 7 5 8 . 0 6 1 5 . 0 6 7 4 . 0 7 1 7 . 0 8 2 7 1 0 y e a r p r o v i n -c i a l g o v e r n m e n t b o n d . 0 8 4 0 . 0 9 0 4 . 0 8 0 3 . 0 8 1 3 . 0 8 3 6 . 0 9 9 1 f i r s t m o r t g a g e . 0 9 3 8 . 1 0 4 0 . 0 9 4 3 . 0 9 2 1 . 0 9 5 9 . 1 1 2 4 s e c o n d m o r t g a g e . 1 0 5 0 . 1 2 2 0 . 1 1 0 8 . 1 0 8 3 . 1 1 2 3 . 1 3 2 1 p e r s o n a l l o a n s . 1 0 4 0 . 1 1 7 0 . 1 0 7 5 . 1 0 5 0 . 1 0 7 5 . 1 2 7 5 B . C . C . U . s h a r e s . 0 6 0 0 . 0 6 0 0 . 0 6 0 0 . 0 6 0 0 . 0 7 0 0 . 0 7 0 0 I f o n e w e r e t o p u r c h a s e a f i v e y e a r f e d e r a l g o v e r n m e n t b o n d i n 1 9 7 0 , t h e d e c i s i o n v a r i a b l e s X ^ , X ^ , X ^ , X ^ , a n d X ^ w o u l d b e g e n e -r a t e d . T h e r e t u r n s w o u l d b e c a l c u l a t e d a s f o l l o w s : 1 0 6 D e c i s i o n V a r i a b l e X . . R e t u r n r . . 13 J x j 2 ( . 0 7 5 8 ) ( . 9 4 3 5 ) = . 0 7 2 0 x j 3 ( . 0 7 5 8 ) ( . 9 4 3 5 + . 9 1 1 0 ) = . 1 4 1 0 X ^ 4 ( . 0 7 5 8 ) ( . 9 4 3 5 + . 9 1 1 0 + . 8 7 9 7 ) = . 2 0 7 0 X J 5 ( . 0 7 5 8 ) ( . 9 4 3 5 + . 9 1 1 0 + . 8 7 9 7 + . 8 3 4 1 ) = . 2 7 0 0 X , 7 ( . 0 7 5 8 ) ( . 9 4 3 5 + . 9 1 1 0 + . 8 7 9 7 + . 8 3 4 1 + . 7 7 3 6 ) = . 3 2 9 0 T h e r e t u r n i s t h e i n t e r e s t e a r n e d e v e r y y e a r d i s c o u n t e d b a c k t o t h e b e g i n n i n g o f t h e p l a n n i n g h o r i z o n . T h e r e t u r n s o n a l l a s s e t s w e r e d e t e r m i n e d s i m i l a r l y . T h e c o s t s o f t h e l i a b i l i t i e s a r e a s f o l l o w s [ 8 5 ] : T y p e o f L i a b i l i t y C o s t o f L i a b i l i t y i n Y e a r 1 9 6 9 1 9 7 0 1 9 7 1 1 9 7 2 1 9 7 3 1 9 7 4 1 y e a r t e r m . 0 7 1 2 . 0 7 8 0 . 0 7 2 0 . 0 6 8 0 . 0 7 8 0 . 0 9 9 0 d e p o s i t 3 y e a r t e r m . 0 7 1 2 . 0 8 2 0 . 0 7 6 0 . 0 6 9 0 . 0 8 2 0 . 0 9 8 0 d e p o s i t 5 y e a r t e r m . 0 7 8 5 . 0 8 5 0 . 0 8 0 0 . 0 8 0 0 . 0 8 5 0 . 0 9 7 5 d e p o s i t d e m a n d d e p o s i t . 0 4 0 0 . 0 4 6 0 . 0 4 1 0 . 0 4 2 0 . 0 5 6 0 . 0 7 7 0 s h a r e c a p i t a l . 0 5 0 0 . 0 5 0 . 0 5 0 . 0 5 5 . 0 5 7 5 . 0 8 0 0 T h e c o s t o f a f i v e y e a r t e r m d e p o s i t ( y - j ) s o l d d u r i n g 1 9 7 0 w o u l d b e d e t e r m i n e d a s f o l l o w s : 1 0 7 Y e a r i C o s t I n c u r r e d i n Y e a r i 1 9 7 0 ( . 5 ) ( . 0 8 5 0 ) ( . 9 4 3 5 ) = . 0 4 0 1 1 9 7 1 ( . 8 2 ) ( . 0 8 5 0 ) ( . 9 1 1 0 ) = . 0 6 3 5 1 9 7 2 ( . 8 2 ) ( . 6 4 ) ( . 0 8 5 0 ) ( . 8 7 9 7 ) = . 0 3 9 2 1 9 7 3 ( . 8 2 ) ( . 6 4 ) 2 ( . 0 8 5 0 ) ( . 8 3 4 1 ) = . 0 2 3 8 1 9 7 4 ( . 8 2 ) ( . 6 4 ) 3 ( . 0 8 5 0 ) ( . 7 7 3 6 ) = . 0 1 4 1 T h e t o t a l d i s c o u n t e d c o s t o f y-j i s . 1 8 0 7 . I n a n a c t u a l i m p l e m e n t a t i o n f u r t h e r r e f i n e m e n t s w o u l d b e r e q u i r e d . F i r s t l y , a l t h o u g h t h e t i m e v a l u e o f m o n e y w a s u t i l i z e d a s t h e d i s c o u n t r a t e , a r i s k a d j u s t e d d i s c o u n t r a t e s h o u l d b e u s e d t o r e f l e c t v a r y i n g d e g r e e s o f r i s k i n e s s o f i n v e s t m e n t s , s e e f o r e x a m p l e [ 8 4 ] . S e c o n d l y , l i a b i l i t y m a n a g e -m e n t , f o r e x a m p l e \" c o n t r o l l i n g \" t h e d e p o s i t ~ f l o w s , c a n n o t b e d i r e c t l y i n c l u d e d i n t h e f r a m e w o r k o f t h e A L M m o d e l . I n c o n t r a s t t o t h e l i n e a r t r e a t m e n t o f a s s e t m a n a g e m e n t , l i a b i l i t y m a n a g e m e n t m i g h t t y p i c a l l y i n v o l v e a l t e r n a t i v e i n t e r e s t r a t e s t r u c t u r e s i n d u c i n g a l t e r n a t i v e d i s t r i b u t i o n s o f d e p o s i t f l o w s . F i n a l l y , m o s t a s s e t a n d l i a b i l i t y m a n a g e m e n t m o d e l s d o n o t i n c l u d e a n y s y s t e m a t i c a p p r o a c h t o l i q u i d i t y c o n s t r a i n t s , b u t r a t h e r m a k e u s e o f t h e j u d g e m e n t o f b a n k m a n a g e r s i n p r e s c r i b i n g m a x i m u m l e v e l s f o r e i t h e r c a p i t a l l o s s e s o r l i m i t s o n t h e a m o u n t o f i n v e s t m e n t i n a s s e t s . I t i s d e s i r a b l e t o m a i n t a i n s o m e l e v e l o f c o n s i s t e n c y i n m a t c h i n g t h e l i q u i -d i t y c h a r a c t e r i s t i c s o f a s s e t s a n d l i a b i l i t i e s a c r o s s b a n k s . T h u s a l t h o u g h t h e p a r a m e t e r s u t i l i z e d b y t h e F e d e r a l R e s e r v e B o a r d m a y n o t b e v a l i d , t h e g e n e r a l s t r u c t u r e o f t h e s e c o n s t r a i n t s a r e , a n d t h e s e w o u l d h a v e t o b e m o r e c a r e f u l l y e s t i m a t e d . 1 0 8 4 . 3 R e s u l t s o f T h e V a n c o u v e r C i t y S a v i n g s C r e d i t U n i o n A p p l i c a t i o n I t w i l l b e r e c a l l e d t h a t t h e r e a r e t w o p u r p o s e s o f t h i s a p p l i -c a t i o n : 1 ) t o d e m o n s t r a t e t h e a p p l i c a b i l i t y o f t h e A L M m o d e l , a n d 2 ) t o t e s t t h e s e n s i t i v i t y o f t h e s o l u t i o n g e n e r a t e d . T o a c c o m p l i s h t h e s e g o a l s , t h e m o d e l w a s r u n i n i t i a l l y w i t h t h e d a t a g i v e n i n A p p e n d i x I ( b a s i c m o d e l ) , s e c o n d l y b y r e p l a c i n g a l l r a n d o m v a r i a b l e s w i t h t h e i r e x p e c t e d v a l u e s ( d e t e r m i n i s t i c e q u i v a l e n t ) , a n d f i n a l l y w i t h v a r i a n t s o f t h e b a s i c m o d e l ( w i t h r e s p e c t b o t h t o . t h e p e n a l t y c o s t s a n d t h e p r o b a b i l i t y d i s t r i b u t i o n s ) . B e f o r e p r e s e n t i n g t h e d e t a i l e d r e s u l t s o f t h e a p p l i c a t i o n , s e v e r a l g e n e r a l s t a t e m e n t s c a n b e m a d e c o n c e r n i n g t h e A L M m o d e l i n p a r t i c u l a r a n d S L P R m o d e l s i n g e n e r a l : 1 ) t h e i n i t i a l p o r t f o l i o h e l d b y V C S v i o l a t e s t h e l i q u i d i t y c o n s t r a i n t s ( a s i t u a t i o n w h i c h w a s k n o w n t o m a n a g e m e n t ) , 2 ) t h e s t o c h a s t i c m o d e l s y i e l d e d s o l u t i o n s t h a t w e r e s u p e r i o r t o t h e d e t e r m i n i s t i c e q u i v a l e n t , ^ a n d 3 ) t h e n a t u r e o f t h e p r o b a b i l i t y d i s t r i b u t i o n s a n d t h e p e n a l t y c o s t s m a r k e d l y a f f e c t s t h e o p t i m a l s o l u t i o n . T h e b a s i c m o d e l h a s s y m m e t r i c t h r e e p o i n t p r o b a b i l i t y d i s t r i -b u t i o n s ( . 2 , . 6 , . 2 ) f o r a l l t h e d e p o s i t f l o w c o n s t r a i n t s a n d d e g e n e -r a t e p r o b a b i l i t y d i s t r i b u t i o n s f o r t h e l i q u i d i t y a n d p o l i c y c o n s t r a i n t s . T h e p e n a l t i e s f o r a l l s t o c h a s t i c c o n s t r a i n t s a r e a s y m m e t r i c . T h e o p t i m a l M a d a n s k y i n [ 5 6 ] h a s s h o w n t h a t t h e ' d e t e r m i n i s t i c e q u i v a l e n t ' p r o v i d e s a l o w e r b o u n d o n t h e o p t i m a l v a l u e o f a S L P R . 1 0 9 v a l u e o f t h e b a s i c m o d e l i s $ 2 , 5 2 0 , 3 1 6 . 0 1 ( $ 8 , 2 8 8 , 9 4 1 . 5 3 i n e x p e c t e d p r o f i t s m i n u s $ 5 , 7 6 8 , 6 2 5 . 5 2 i n e x p e c t e d p e n a l t i e s ) . T h e d e t e r m i n i s t i c e q u i v a l e n t m o d e l h a s a n o p t i m a l v a l u e o f $ 2 , 2 7 8 , 1 8 7 ( $ 8 , 5 6 5 , 0 6 8 i n e x p e c t e d p r o f i t s m i n u s $ 6 , 2 8 6 , 8 8 5 i n e x p e c t e d p e n a l t i e s ) . T h u s t h e b o u n d p r o v i d e d b y t h e d e t e r m i n i s t i c e q u i v a l e n t i s 1 0 . 6 % b e l o w t h e o p t i m a l v a l u e o f t h e b a s i c m o d e l . T h e s t r u c t u r e o f t h e t w o p o r t f o l i o s i s s i m i l a r i n t h e i n i t i a l p e r i o d . H o w e v e r , t h e i n v e s t m e n t p a t t e r n s d i f f e r b e y o n d t h e f i r s t t i m e p e r i o d . T h e b a s i c m o d e l d i d n o t i n v e s t a s h e a v i l y i n t h e l e s s l i q u i d a s s e t s ( n a m e l y m o r t g a g e s ) . T h e d e t e r m i n i s t i c e q u i v a l e n t m o d e l a l s o h a d a n e m p t y f e a s i b l e s e t w h e n i t w a s f i r s t r u n . T h e r e a s o n w a s t h a t V C S ' s i n i t i a l p o r t f o l i o d i d n o t s a t i s f y t h e l i q u i d i t y c o n s t r a i n t s . I n o r d e r t o s e c u r e f e a s i -b i l i t y , v a r i a b l e s w e r e a d d e d t o e a c h o f t h e l i q u i d i t y c o n s t r a i n t s . T h e o b j e c t i v e c o e f f i c i e n t s o f t h e s e v a r i a b l e s w e r e t h e s a m e a s t h e p e n a l t i e s a s s o c i a t e d w i t h v i o l a t i n g t h e s t o c h a s t i c l i q u i d i t y c o n s t r a i n t s i n t h e b a s i c m o d e l . A s a f u r t h e r i n s i g h t i n t o t h e o p e r a t i o n s o f V C S , t h e p e n a l t i e s c o u l d b e s e t a r b i t r a r i l y h i g h s o t h a t t h e m o d e l w o u l d v i o l a t e t h e l i q u i d i t y c o n s t r a i n t s o n l y t o a t t a i n f e a s i b i l i t y . T h e a m o u n t b y w h i c h t h e c o n s t r a i n t s a r e v i o l a t e d w i l l b e t h e a m o u n t o f l i q u i d r e s e r v e s t h a t t h e f i r m n e e d s t o m e e t t h e F R B ' s l i q u i d i t y r e q u i r e m e n t s . V a r i a n t s o f t h e b a s i c m o d e l w e r e r u n i n o r d e r t o a s c e r t a i n t h e e f f e c t s o f d i f f e r e n t p r o b a b i l i t y d i s t r i b u t i o n s , v a r i o u s p e n a l t y c o s t a n d p a r a m e t e r c h a n g e s . T h e i n i t i a l c h a n g e i n s t i t u t e d i n t h e b a s i c m o d e l 1 1 0 w a s t h e a l t e r a t i o n o f t h e f i r s t l e g a l c o n s t r a i n t f r o m t h e r e q u i r e m e n t t h a t c u r r e n t a s s e t s b e e q u a l t o o r g r e a t e r t h a n 10% o f t h e l i a b i l i t i e s t o e q u a l o r g r e a t e r t h a n 1% o f t h e l i a b i l i t i e s . T h e e f f e c t o f t h i s c h a n g e w a s t o i n c r e a s e t h e o p t i m a l v a l u e t o $ 2 , 9 0 6 , 7 7 3 . 5 3 ( $ 8 , 6 5 7 , 6 1 9 . 2 4 i n e x p e c t e d p r o f i t s m i n u s $ 5 , 7 5 0 , 8 4 5 . 7 1 i n e x p e c t e d p e n a l t i e s ) . F o r t h e i n i t i a l t w o p e r i o d s , t h e i n v e s t m e n t p a t t e r n d e v i a t e d s u b s t a n t i a l l y f r o m t h a t o f t h e b a s i c m o d e l i n t h a t m o r e o f t h e i n c r e m e n t a l f u n d s w e r e a l l o c a t e d t o l o n g e r t e r m a s s e t s . A f t e r t h e f i r s t t w o p e r i o d s t h e r e d i d n o t s e e m t o b e a n y g e n e r a l i z e d b e h a v i o r i n t h e i n v e s t m e n t p a t t e r n s o f t h e t w o m o d e l s . H o w e v e r , t h e r e w a s a l a r g e r t o t a l a m o u n t i n v e s t e d i n l o n g e r t e r m a s s e t s i n t h e m o d i f i e d f o r m u l a t i o n . T h e b a s i c m o d e l w a s t h e n f u r t h e r a l t e r e d t o i n c l u d e a c h a n g e i n t h e p r o b a b i l i t y d i s t r i b u t i o n ( . 0 5 , . 5 0 , . 4 5 ) o f t h e c a s h f l o w s . T h e i n c r e a s e i n t h e o p t i m a l v a l u e w a s m u c h m o r e d r a m a t i c i n t h i s c a s e . T h e o p t i m a l v a l u e j u m p e d t o $ 3 , 2 5 6 , 5 0 0 . 6 5 ( $ 8 , 8 7 2 , 9 1 1 . 5 3 i n e x p e c t e d p r o f i t s m i n u s $ 5 , 6 6 1 , 4 1 0 . 8 0 i n e x p e c t e d p e n a l t i e s ) . T h e e x p e c t e d n e t p r o f i t r o s e w i t h r e s p e c t t o b o t h t h e b a s i c m o d e l a n d t h e m o d e l w i t h t h e p a r a -m e t e r c h a n g e w h i l e a t t h e s a m e t i m e t h e e x p e c t e d p e n a l t y c o s t s d e c r e a s e d w i t h r e s p e c t t o b o t h m o d e l s . T h i s i s e x p l a i n e d b y : 1 ) a l l t h e v i o -l a t i o n s o f s t o c h a s t i c c o n s t r a i n t s a r e n o w i n f e a s i b l e o n l y w i t h a p r o b a -b i l i t y . 0 5 i n s t e a d o f . 2 ( t h a t i s t h e p e n a l t i e s d e c r e a s e d ) a n d 2 ) c o n -s t r a i n t s w h i c h w e r e n o t p r e v i o u s l y v i o l a t e d b e c a u s e o f e x c e s s i v e p e n a l t i e s a r e now v i o l a t e d b y 15% ( i m p l y i n g m o r e p r o f i t s ) . T h i s d e m o n -i n s t r a t e s t h e n e e d f o r a c c u r a t e e s t i m a t e s a r o u n d t h e v a l u e s o n t h e l e f t h a n d s i d e o f a s t o c h a s t i c c o n s t r a i n t . A l t h o u g h i t i s n o t p o s s i b l e t o m a k e d e f i n i t i v e g e n e r a l i z a -t i o n s f r o m t h e r u n s o f t h e A L M m o d e l d e s c r i b e d a b o v e , s o m e g e n e r a l c o n c l u s i o n s m a y b e i n f e r r e d . F i r s t , t h e a s y m m e t r y o f t h e p r o b a b i l i t y d i s t r i b u t i o n s m a y h a v e a s u b s t a n t i a l e f f e c t o n t h e o p t i m a l s o l u t i o n s a n d v a l u e s . A l s o a n i m p o r t a n t p o i n t t o n o t e i s t h e s e n s i t i v i t y o f t h e e s t i m a t e o f t h e p r o b a b i l i t y d i s t r i b u t i o n a r o u n d t h e v a l u e o n t h e l e f t h a n d s i d e o f t h e s t o c h a s t i c c o n s t r a i n t s . S e c o n d , t h e s o l u t i o n s a r e s e n s i t i v e t o c h a n g i n g p e n a l t y c o s t s . T h i r d , t h e v a r i o u s s t o c h a s t i c m o d e l s h a v e s u b s t a n t i a l l y d i f f e r e n t s o l u t i o n s t h a n t h e e q u i v a l e n t d e t e r m i n i s t i c m o d e l s . T h i s i n d i c a t e s t h a t r e l i a n c e o n t h e d e t e r m i n i s t i c m o d e l s a s n o r m a t i v e t o o l s c a n l e a d t o e r r o n e o u s s o l u t i o n s . F o u r t h , t h e i m p l e m e n t a t i o n o f t h i s m o d e l i s n o t m o r e d i f f i c u l t t h a n t h e i m p l e m e n -t a t i o n o f a s i m i l a r d e t e r m i n i s t i c m o d e l . F i n a l l y , t h e c o m p u t a t i o n s n e c e s s a r y t o s o l v e t h e A L M f o r m u l a t i o n a r e o f t h e s a m e o r d e r a s t h e c o m p u t a t i o n s n e c e s s a r y f o r a n e q u i v a l e n t d e t e r m i n i s t i c p r o b l e m . A l l t h e r u n s w e r e c o m p u t e d o n t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a ' s I B M 3 7 0 / 1 6 8 . I t w i l l \" b e r e c a l l e d , t h a t t h e A L M f o r m u l a t i o n i s 9 2 b y 2 5 7 w i t h 4 0 s t o c h a s t i c c o n s t r a i n t s . U s i n g t h e S L P R c o d e , t h e s o l u t i o n o f t h e A L M m o d e l t o o k a b o u t 3 7 s e c o n d s o f C P U t i m e . T o s o l v e a n e q u i v a l e n t s i z e d d e t e r m i n i s t i c p r o b l e m u s i n g t h e S L P R c o d e t o o k a b o u t 3 0 s e c o n d s o f C P U t i m e . T h e s a m e d e t e r m i n i s t i c l i n e a r p r o g r a m w a s a l s o s o l v e d o n 1 1 2 a s t a n d a r d L . P . c o d e , UBC L I P w h i c h t o o k 1 7 s e c o n d s o f C P U t i m e . T h e s e a n d a n u m b e r o f o t h e r r u n s i n d i c a t e t h a t C P U t i m e s f o r s t o c h a s t i c m o d e l s a r e g e n e r a l l y a b o u t d o u b l e t h a t o f e q u i v a l e n t d e t e r m i n i s t i c m o d e l s . A p p e n d i x O n e T h e f o l l o w i n g i s t h e i n p u t d a t a f o r t h e b a s i c m o d e l . 1 1 4 l . 0 0 0 0o o i p 0025700052000/jn 7 _J 1 0 .0 f . l\ - 1 0 o o o o 0 o ' . 1 o o o 0 n 0 0 . 5 .2 0. 1 0 .0 I . 7 - 1 0 0 0 0 0 0 0 1 o o 0 o 0 0 o . 8 ... ,? o. q 1 0 . 0 f . 10 -1 o ooooo o'. l noooooo . ...11 . 2 \u00E2\u0080\u009E_.,J1 _\u00E2\u0080\u009E . . 1? 1 0,0 1. . 13 -1 00 0 ( .\u00C2\u00BB00 o . . t o o o o o o o . l ' l . .. ...2 0 . 15 1 0 . 0 1 . 16 - 1 0 0 0 0 0 0 o . J. 0 0 0 0 0 o o . 17 2 . 0 IB 1 0 . 0 t '. - 1 0 0 0 0 0 . 1 0 0 0 0 0 '\u00E2\u0080\u009E 2.0 0 . 1 . 21 1 n . 0 1. 22 - 1 0 0 0 0 0 . \ <> 0 0 0 0 . ..23 - o . __ ...1..\u00E2\u0080\u0094 2'\ 1 0.0 1 . 25 - 1 0 0 il 0 0 . . 1 0 0 0 0 0 . 26 0 . 1 . -! I i . t 1 o.o .1 28 - 1 0 i.) () o o . t 0 0 0 o 0 . . 29 . 0 . 1 30 i o .o r . 31 - 1 0 0 0 0 0 . 1 0() 0 o 0 . 32 0 . 1. \u00E2\u0080\u00A27 7 .J J 1 O.O 1 . 3'l - 1 0 0 0 0 0 . 1 0 0 0 0 0 . 35 Q - - l . . 36 i o . o r . 7 T _> l -1 0 00 0 0 . 1 0 0 0 0 0 . 3B 0.. . 1 .. 39 l o . 0 1 . 4 0 - 1 0 0 0 0 0 . 1. 0 0 0 0 o . ..4 1 . 0 . . 1 ... \u00E2\u0080\u00A2\u00E2\u0080\u00A2'12 1 0 . 0 1 . 4 3 -1 0 0 0 0 0 . 1 fi o 0 o 0 . /.Ml 0..... 1 . 4 5 1 0 . 1 1 1 . 4 6 - 1 0 0 0 0 0 . 1 0 0 0 0 0 . 4 7 . P . . . . . 1 . ...... flfl 3 6 0 0 (\u00E2\u0080\u00A2 o 0 0 . # ;> 4 9 7 3626 0 0 . .6 5 0 B 0 0 0 0 0 0 . . 2 1 1 5 51 5 0 0 0 0 0 0 . 9 0 0 0 0 0 0 . 52 0 . 0 . . 5 3 . . 3...17\u00C2\u00BBCOG<\0. ... 5'l 1 8 8 o 0 8 0 0 . .6 55 ? 0 0 0 0 0 0 0 . . 2 56 16000 0 00 . 2 1 0 o ') 0 00 . 57 0 .0 . 0 y 9 1 58 3 2 6 0 0 0 O O O . .2 .59 _ ?7(. 1 '37 0 0 ...... 6 6 0 2900 0 0 00 , .2 61 25000000 . .300 0(10 0 0 . 62 O.'.O . 07 ? 6 3 3 60000 o o o . .2 6 0 6 7433 :. 1 6 0 I 1 0 0 0 () 0 0 . . 2 1 6 1 8 0.0.0.0 0 0 . . . . 1 2 0 O.Q.O 0.0.. . 1 6 2 . 0 . 1 2 7 0 1 6 3 3 n o o o o o c , 2 \u00E2\u0080\u00A2 1 6 2 0 1 . 1 8 7 2 1 1 . 188 . . . . 2 2 1 1 8 9 2 3 1 . 1 9 0 2 5 1 . 1 9 . 1 . . ._26._ j . J .9 2 2 7 l . 1 9 3 2 8 1 . 19/1 2 9 1 . 3 5 1 . 1 9 6 3 6 1 . 1 9 ? _ 3 . L . ..1 1 9 8 3 8 r . _ 3 9 2 0 0 _ 0 _ 1 1 8 2 0 1 4 1 1 . 2 0 2 4 2 1 , 2 0 3 4 3. . 1 . 2 0 4 4 4 1 . 2 0 5 4 5 1 . 2 0 6 4 6 3 . 2 0 7 4 7 1 . 2 0 H 4 \u00C2\u00AB 1 . 2 O^ 4 9 . ] . . 2 I 0 5 0 1 . 21 1 5 1 1. 2 1 2 5 2 1 . 2 1 3 5 3 1 . 2 14 5 4 1 . 2 1 5 5 5 . J . 2 1 6 5 6 1 . 2 1 7 1 5 2 1 2 1 8 1 5 3 1 2 1 9 1 5 \u00C2\u00AB M OS 2 2 0 1 5 9 O f l ? 2 2 1 . 1 6 Q . 0 8 2 - i *\u00E2\u0080\u00A2} C i- <- 1 6 1 0 7 6 5 1 1 7 C C -J 1 6 2 0 7 6 5 2 2 4 1 6 3 n 5 2 2 5 1 6 4 0 5 2 2 6 1 6 5 OS 1 6 6 -jr. f 0 5 2 2 8 1 6 7 * 0 5 2 2 9 188 \u00E2\u0080\u00A2 2 3 0 0 0 0 0 0 0 0 0 0 2 3 1 \" p * . 2 3 2 1 1 i . 2 3 3 1 2 i . 2 3 4 1 5 i . 2 3 5 1 6 i . 2 3 6 17 i . \u00E2\u0080\u00A2> 7 1 . > , 2 0 3 . 2 3 8 2 1 1 . 2 3 9 22 1 . 2 4 0 2 3 i . 2 4 1 2 6 1 '. 2 4 2 2 7 1 . 2 4 3 2P 1 . 2 4 4 2 9 1 . 2 -:i 5 3 9 1 , 2 :'l 6 ~ 4 r . . . 2 4 7 4 2 1 . 2 4 8 4 4 1 . 2 4 9 a 5 1 . 2 5 0 4 6 1 '. 0-0 0 o o 0 o 0 0 0 0 0 0 0 0 1 1 1 9 2 5 ] / J 8 1 , 2 5 ? 4 9 1 . 2 5 3 5 0 1 . 2 5 '1 5 1 1 . 2 5 5 5 3 1 . 2 5 6 5 4 1 . . 2 . 5 7 _ 5 5 .. 1. _ 2 5 B 5 6 1 . 2 5 9 6 2 1 . 2 6 0 6 3 1 . 2 6 ] 6 / 1 1 . 2 6 2 6 5 1 . 2 6 3 . . . . 6 6. . . . 1 . . _ _ . 2 6 ' l 6 7 1 . 2 6 5 6 8 1 . 2 6 6 6 9 1 . 2 6 7 7 0 1 . 2 6 B 7 1 1 . 2 6 9 _ 7 2 1 . . 2 7 0 7 3 1 . 2 7 1 7 4 1 . \u00E2\u0080\u00A2> T 1 C 1 .- 7 5 1 . 2 7 3 7 6 1 . 2 7 4 7 7 1 . 2 7 5 . . . . 7 . 8 . . . 1 . _ . . 2 7 6 7 9 J . 2 7 7 8 0 1 . 2 7 8 , . 8 1 1 . 2 7 9 1 5 ' J - . 1 2 B 0 1 5 9 - . 0 5 2 5 2 9 1 1 6 0 - . o r ^ 2 5 2 8 2 1 6 1 - . 0 / ( 0 5 2 8 3 1 6 2 - . 0 a 0 5 2 8 1 6 6 , 0 7 6 5 . 2 8 8 1 6 7 - . 0 7 6 5 2 8 9 1 6 8 - . O S 21 o 1 6 9 - . 0 5 2 C M . 1 7 0 - . 0 S \u00E2\u0080\u00A2 > \u00C2\u00AB ? i . . ' >-1 7 1 - . 0 5 2 9 3 . _ 1 7 . 2 . - . 0 _ 2 9 < l 1 8 9 - 1 . 2 9 5 oooo o () 0 o o o oooooo 0 0 0 o 0 o o 2 9 6 1 2 1 . 2 9 7 1 6 1 . 2 9 8 1 7 1 . 2 9 9 2 1 _ 1 . 3 0 0 2 2 1 . 3 0 . ? 2 7 3 0 3 2 H 3 0 0 2 9 . 3 0 5 4 ? _ 3 0 6 \u00C2\u00AB V 3 0 7 \u00C2\u00AB 6 3 0 8 4 \" 3 0 \u00C2\u00AB 5 0 3 1 0 5 1 3 1 1 5 . : ) . . 3 1 2 5 5 3 1 3 5 6 3 i \" 6 6 3 1 5 6 8 3 J o 6 9 3 1 7 7 1 . . . . 3 1 8 7 2 5 1 9 7 3 3 2 0 7 5 3 2 1 7 6 3 2 7 8 H 3 2 8 8 9 3 2 9 9 0 3 3 5 _ _ 9 6 3 3 6 9 7 3 3 7 9 H 5 3 8 9 9 5 3 9 1 0 0 3 -:i 0 1 0 1 3'\u00E2\u0080\u00A2]..._ _ 1 0 2 77 7\u00C2\u00B0 8 0 81 87 91 9 2 9.3 9U 9 5 103 155 159 160 161 1 h 2 1 6 'J 165 166 3 s 2 353. 3 5 /I 3 5 5 3 5 6 3 5 ? 3 5 a 3 6 0 3 6 1. 3 6 , ? Ji 6 3 3 6 4 3 6 5 3 6 6 3 6 7 3 6 B 3 6 9 3 7 0 3 7 3 3 7 4 3 7 S 3 7 6 377 _.. 3 7 8 3 7 \" 3 8 0 3 8 j 3 8 ? 3 8 3 3 8 4 3 8 5 3 * 6 3 8 7 3 8 8 3 3 9 _ . . 3 9 0 3 9 ) . 3 9 ? 3 \" 3 3 n 4 395 3 9 6 3 9 7 3 9 8 3 9 9 4 0 0 1 6 7 - . 0 4 0 5 1 6 8 - . 0 5 1 6 9 \u00E2\u0080\u00A2a . O R ? 1 7 0 \u00E2\u0080\u00A2a . 0 8 ? 1 7 1 - . 0 7 6 5 1 7 ? - . 0 7 6 5 1 7 3 - . '0 5 1 7 4 - . 0 5 1 7 5 - . 0 5 1 7 6 - . 0 5 1 7 7 - . ( J 5 1 9 0 - 1 . 0 0 0 0 o 0 o 0 0 17 _ 2 2 2 3 2 8 2 9 . n 6 5 0 . . . 5 1 . . 5 5 5 6 6 9 7 2 7 3 7 6 .. 7 7 8 0 81 1 9? . . . 9 / 1 . . . 9 6 9 7 99 1 0 0 1 0 2 . 1 0 3 . . . 1 0 9 1 1 0 1 1 1 1 ! 2 1 1 3 . 1 1 4 . . 1 1 5 1 1 6 1 1 7 1 1 8 1 1 9 '1 0 1... 120. .J... .._ . '1 0 2 121 1 . 4 0 3 156 -.1 4 0 '.1 16 0. - . 0 2 15 .'I 0 5 161 - '. 0 1 1 4 4 0 6 162 - . 0 1 1 4 4 0 7. 1 6 4,_ . - . 0 ? C 5 4 06 1 6 5 - . \" 3 36 4 0 9 1 6 6 - . 0 2 1 5 4 J 0 167 - . 0 2 15 4 1 ! 169 - . 0 5 2 5 4 1? 170 - . \" 5 25 413 171_ . .0 HXt'i-... 4 I 4 172 - . 0 4 0 5 4 15 173 - . 0 5 \u00E2\u0080\u00A2'11 6 174 - , 0 \u00C2\u00AB 2 417 175 - . 0 82 4 18 176 - . 1 ) 7 65 4 i 9 _ 177.. .-..0.7 6 fL 420 178 - . 0 5 4 2 1 179 - . O S 4;:.? 180 - . O S 4 2 3 181 - . \u00C2\u00B0 5 4 ? 4 1 8 2 - . 0 5 -'125 191 .-.} .. 4 ? 6 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 \" 0 0 0 0 427 2 3 1 . 4 28 29 1 . 429 51 1 . 4 Z0 5 6 1 . \u00C2\u00AB 3 J 73 A . 4 32 77 l . 4 33 81 l . 4 34 94 1 . 4 35 97 1 . 4 36 100 1 . 437 1 0 3 1 . 4 38 1 1 3 1 . /1 39 1 15 1 . 4 4 0 117 1 . 4 4 J 119 1 . 4 4 ? 121 1 . 4 4 3 _ 126 J . 4 4 4 127 1 . 4 4 5 .128 1 . 4 4 6 129 1 . 4 4 7 130 1 . 4 4 8 131 1 . 44 9 _ ; 132 L. 4 5 0\" ~ 133 1 . 1 2 3 '151 1 5 7 - . 1 '152 . . . 160 - . 0 1 3 0 'I 5 3 161 - . 0 06 0 5 5 .'17 0 1 8 2 - . 0 7 6 5 '171 183 - . O S '17 2 16/) - ' . 0 _ 0 73 1.8 5..._-..0.s '17'I 186 - . 0 5 ' i 75 1 67 - . O S '176 192 -1 , '17 7 O O O O O o . 0 0 9 0 0 0 0 0 000000 0 5 \u00E2\u0080\u00A2'178 2 1 . ' !79_ ___ 35... 1. _ _ _ _. ' 1 8 0 152 - . 0 1, '181 153 - . 0 1 0 8 2 158 - . 0 05 '183 159 - . 0 0 8 2 '18 i.\ 16 0 - ' . 0 0 8 2 0 8 5. _ . 161 - .007.65 n8 6 16 2 - . 0 0 7 6 5 '187 163 - ' . 0 0 5 \u00E2\u0080\u00A2'18 8 1 6 /I - . 0 0 5 0 8 9 165 - . 0 05 ' 1 9 0 1 66 - ' , 0 o 5 '10 1 _ 1 6 7 . 0 0 5_ ' 1 9 2 \" 1 9 3 - 1 . '1^3 ooooooo o o o o o o o o o o o o o o o 6 ' 19 'I 6 2 1. \u00E2\u0080\u00A2'19 5 \u00E2\u0080\u00A2 15 '4 - . 0 1 ' 1 9 6 1 5 9 - . 0 0 5 2 5 ' 1 9 7 160 - . 0 0 5 2 5 _ ' 1 9 8 16 1 - . 0 0/105 ' I ' ) 9 162 - . 000 OS 500 163 - ' . 005 1 2 4 5 01 164 - , 0 0 8 2 5 0-? 165 - . 0 08? .5 0 3 ...16 6 - . 0 0 7 6 5 . . . 5 04 167 - . 0 0 7 6 5 5 0 5 168 - . 0 0 5 506 169 - . 0 05 . 5 07 170 - . 0 0 5 5 0 8 171 - . 0 0 5 5 09... 17.2 - . 0 05 510 19.g -1 . 5 1 1 0 0 0 0 0 0 0 0 0 n 0i 512 67 r . 513 155 - . 0 1. 5 1 'I 159 - . 00 2 0 5 515.. ._. 16 0. .'.0 0 33 . ( 3 - -5 1 6 161 - . 00 2 1 5 517 16? - . 0 0 2 1 5 5 1 C 164 - . 0 0 5 2 5 5 1 \u00C2\u00B0 165 - . 0 0 5 2 5 52 0 166 - . 0 0 4 0 5 521 ... 1 6 7 o o / i a.5. . 522 1 6 8> - . 0 0 5 523 169 - . 0 0 8 2 524 . 170. - . 0 0 \" 2 .. . 525 171 - . 0 0 7 6 5 526 17? - . 0 0 7 6 5 527... 173 _-...Q.05 5 ?a t 74 - . 0 0 5 5 2 \u00C2\u00B0 175 - , 0 r, <:\u00E2\u0080\u00A2, 530 176 ..-.005 531 177 - . 0 0 5 532 195 -1 . 533... ...0.0 0 0.0 0.!) 0.0.0.0. 5 3 \" 1 09 1 . 535 156 - . 0 1 536 160 . - . \" 0 2 1 5 r > * 16 1 - . 0 0 I 1 Si 2 Si 0 0 * - t!L\ T I S . Si I 2 o o * - OlS. Si 1201. * - \ li- t , 9 ' . 9 i v: o o * - on 0 9 S. S 0 _ 0 0 ' - 6 9 t 19 Si ti 11 0 0 ' - 19\ 99S. 1)110 0 \" - 991 Vi 9 Si Si T 2 o 0 ' - S.9 t l ? 9 S i 9 0 D 0 * - 29 t 9 0 0 u ' - T 9 l 2 9 Si \i \u00C2\u00A3 1 0 0 * - 091 1 0 ' - z _ t 0 9 Si \u00E2\u0080\u00A2 I 92 I o ; / i t i 0 0 0 0 0 U 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 b i S - - \u00E2\u0080\u00A2 V - - . ' - r i -fe. _ i - i Si o o * - 2 9 T 9 S i S i Si 0 0 * - I _ l S Si Si _ 0 0 * - 0 8 T i / S . S i Si 0 0 * - o / . J i ' j i S i Si 0 0 \ - \u00E2\u0080\u00A2 * L \ C S J S \u00C2\u00BB \" ' ' \" \" \" ~ S i 9 7 . 0 6 * ' - \u00E2\u0080\u00A2~L1T~ 9 2 L 126 60 1 1 5 \" 1. 602 159 - . 2 6 2 5 60 3 160 - . 2 6 25 6 0 \" 161 - . 2 0 2 5 6 05 162_.-,2 0 25__ 606 163 - . 2 5 6 0 7 1 6 a - . 4 i 6 0 8 . 1 6 5 - . ''i 1 609 \66 - . 5 8 2 5 6 ) 0 167 - . 3 8 ? 5 6 1 1 \u00E2\u0080\u009E l 6 4 5 1 6 6 - '. 1 0 7 5 64 6 167 - . 1 0 75 64 7. 1 6 9_. - . 2 6 2 5 .... . . ......... ' \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 1 7 0 - . 2 6 2 5 6 49 171 - . 2 0 25 650 172 - . 2 H 2 5 1 2 7 65 1 173 25 6 5 ? 174 \" 1 653... L75 .4.1 6 5 4 176 mm 3825 655 177 -' 3 6 25 656 178 25 657 1 70 25 6 5 0 1 8 0 25 0 5 \u00C2\u00B0 181.. 2 5 6 6 0 18? 25 661 20 1 1 i 6 6 ? 0 0 0 0 0 oooooo 66 3 157 1 . 6 6 / ! 160 P\" 0 6 9 '\u00E2\u0080\u00A265 . . 1.6 1. W .9 3 0? .. 6 6 6 1 6 ? mm 0 3 0 ? 66 7 16^ - . 1 0 7 5 h 6 8 1 6 6 mm 0 5 6.9 h 6 ? 167 0 5 6 9 67 0 1 6 9 mm 1 025 671 170 166 67? 171 1 0 7 5 673 17? 1 0 7 5 6 7/1 . 174 mm 2625 675 175 2 6 2 5 676 176 - t 2 0 25 67 7 1 77 2 0 ? 5 6 7 6 178 - \ 25 67 c ' 179 4 1 6 0 180 \u00C2\u00AB 1 681 1 8 1 3825 68? 1 8 ? 3825 683. . 183 M 25 6 8 /I 184 25 6 8 5 185 2 5 6 8 6 186 \u00E2\u0080\u00A2>* ? S 6 fl 7 167 - . 25 6 8 8 2 0 ? 1 . 6 8 9 . _ . 0 0.OP.0 0 o_-...o\u00E2\u0080\u009Eo 6 0 0 1 1 . 69 1 ? 1 . 69? oooooo 0 oo 693 3 1 . 6 9 /I 0 0 0 0 0 0 0 0 0 0 6 95 4 6 9 fc s 1 . 697 oooooo oooo 69 8 6 1 . 6 9 1 7 1 . 7 0 0 ft 1 . 0 0 00 0 0 00 00.0 00 0 0... 1.4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 8 1 2 8 .7 01 0 0 0 0 0 0 0 0 0 0 0 0 0 Q 0 0 0 0 0 0 1 9 7 0 2 9 1 . 7 03 1\" J . 7 o . 1 1 1 . 7 0? 1? 1. 7 0 6 0 0 0 0 0 0 0 o 0 0 0 0 0 0 0 0 0 0 0 0 2 0 707.. 1 1 ! . . _ 708 \u00E2\u0080\u00A214 1. 709 15 1. 7 1 0 1 6 J . 711 17 r . 71?. 0 0 0 oo ooo oono 00000000ooo 717. 18...1 .. 7 1 \" 19 1. 715 20 1. 7 1 6 .. . 21 \ . 717 22 J . 718 23 1. 7 . i ' J .D0 0 0 o o 0 0 0 0 i o o 0 0 0 0 0 0 0 0 22 720 2 4 1 . 721 25 1 , 7 22 2 6 1 . 723 CI J . 7 2 \" 2 8 1 . 7.2 5. . 2? 1 . 726 0 0 0 0 0 0 0 o 0 0 0 o 0 o o o 0 0 0 0 0 0 0 727 30 1 . 728 . . . 0 0 0 0 o 0 0 0 o o 0 0 0 0 0 0 0 0 0 0 2 4 7 29 31 1 . 730 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 25 731 32. 1.. _ 732 33 1. -> f T 1 J .> 34 1 . 7 34 0 0 0 0 0 0 0 0 (10 0 0 0 0 0 0 0 o ooooo 735 158 l . 7 36 oooooo oooooon 0 0 0 0 0 u 0 0 0 2 7.37 _ _ . 1.59._l.......... 738 0 0 0 (.) 0 0 0 0 0 0 n o 0 0 0 0 0 0 0 0 2 8 7 3 q 1 ft 0 1 . 7 '1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 9 7 4 1 1 6 1 1 . 7 4 ? 0 0 0 0 0 0 0 0 o0 0 0 0 0 0 0 0 0 0 0 3 0 74 3 ...16 2.. 1. 7 4 4 (i OOOOO 0 0 n 0 n o 0 0 0 0 0 0 000. 31 7 45 152 1 . 7 46 0000 000 00 0000000000 .3 2 7 47 1 -1 . 748 4 - . 9 9 / 5 7 4 9....... ... . 6 . r . 9 9 / 5 75 0 9 - . 9 9 7 5 7 5 1 1 3 - . 9 9 7 5 7 5 . ? 1 8 - . 9 9 7 5 7 5 3 2 4 - . 9 0 7 5 7 5 ' i 3 5 7 5 5.. 3 6 . \u00E2\u0080\u00A2*\u00E2\u0080\u00A2 9 \u00E2\u0080\u00A2'- \u00E2\u0080\u00A2-7 5 6 3 7 1 . 0 0 2 5 7 5 7 3 8 1 . 0 0 2 5 7 5 8 . 30 1 , 0 0 2 5 7 5 ? 1 . 0 0 2 5 7 6 0 \" 1 1 . 0 1; ? 5 .7.6 i ... 4 2 _ . l . . . . \u00C2\u00AB . \ 0 . 2 5 _ . . 7 6 ? 0 3 1 . 0 o ? s 7 6 7 , 1 . 0 0 2 5 7 61\ a 5 1 . 0 0 2 5 . 7 6 5 n 6 1 . 0 0 2 5 7 6 6 '17 1 . 0 0 2 5 7 6 7 UP. 1 . 0 0 ? 5 766 no 1 . ( ' 0 2 5 7 6'-) 5 0 J . 0 0 2 5 77 0 51 1 . 0 0 2 5 7 7 1 5? 1 . 0 0 2 5 7 7 ? 5 3 1.0 0 2 5 773_ 5.4.. . J . . 0.0 2 5. 7 7 ' i 55 1 . 0 0 2 5 7 7 5 5 6 J . 0 0 2 5 7 7 6 5 7 7 7 7 5 8 7 7 8 5 9 * 7 7 \u00C2\u00B0 . . 6.0 . J . 7 H 0 6 1 7 8 1 1 5 3 -1 . 7 8 ? 1 5 8 \u00E2\u0080\u00A2 5. 7 8 3 1 5 9 . 1 \u00C2\u00AB 7 8 '1. 1 6 0 . 1 8 7 8 5 . 16JL ,2 3 5 7 8 6 * 1 6 ? , 2 3 5 7 8 7 1 6 3 - . 5 7 8 8 1 6 4 . .-'.5 7 8 \" 1 6 5 - . 5 7 0 0 1 6 6 5 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 79.1.. 1 6 7 ? . . . . 5 7 9 ? 000 0 0 0 0 0 0 0 7 9 3 2 - 1 . 79/1 3 - J . 0 6 7 C ? 5 5 - 1 . 0 7 2 5 7 \" 6 7 - ) . 0 7 2 ' J 7 9 7 P. - . 0 7 '19 7 9 8 1 0 - 1 . 0 7 3 3 7 0 0 1 1 - . 0 7 5 8 8 0 0 1 2 - . 0 7 5 8 801 14 - 1 . 0 7 4 2 802 15 - ' . 0 767 803 16 - ' . 0 767 8 0 \u00E2\u0080\u00A2'! 17 - , 0 7 6 7 805 19 - 1 . 0 7 5 1 - ' . 077.6 806 20 807 21 * . 0 776 808 22 - ' . 0 776 8p9 23 - . 0 7 7 6 810 25 -1 . 0 8 1 5 811 26 - ' . 084 1 812 27 \u00C2\u00BB v 0 8 U 1 813 814 28 29 ~ . 0 8 a i - . 0 8 / 4 1 815 30 - ' . 0 938 816 31 - . 1 0 5 817 32 -1 . 10 1 . 1 1 5 2 1 2 7 1 . 1 1 5 3 1 2 8 1 . 0 0 2 5 1 1 5 4 J J 5 J L J U M 2 5 . 1 1 5 5 1 3 0 1 . 0 0 2 5 1 1 5 6 1 3 1 1 . 0 0 2 5 j i g ? 1 3 ? l ' . Q 0 2 5 . . - . 1 1 5 8 1 3 3 1 . 0 0 2 5 1 1 5 9 1 3 / J 1 . 1 1 6 0 _ 1 3 5 1... . 1 1 6 1 1 3 6 f . 1 1 6 2 1 5 6 1 . 0 7 6 I1.6JS , 1 5 7 ' I t 1 1 6 4 1 6 0 , 0 9 4 3 1 1 6 5 1 6 1 , 0 6 0 1 1 1 6 6 16.2 . 0 6 1 1 6 7 1 6 4 , 2 2 1 6 1 1 6 8 1 6 5 , 1 4 9 4 1 1 6 9 U L 6 . 1 1 3 1 1 7 0 1 6 7 , 1 1 3 4 1 1 7 1 1 6 9 , 3 5 9 9 1 1 7 2 11SL-\u00C2\u00BB23 S \u00C2\u00A3 t c _ f . T t s s t L H 1 4 0 1 2 - 1 . 1 4 0 2 3 - I t ] ' 4 0 3 5 - . 9 9 5 1 4 0 4 7 - . 9 6 1 4 0 5 8 - ' > 6 1 4 0 6 1 0 - . ; . i 6 , , 1 4 0 7 1 1 - , 9 b 1 4 0 8 1 2 - . 9 6 1 4 0 9 1 4 - ' . 9 f t 1 4 ] 0 1 5 - . 9 6 1 4 1 1 1 6 - . 9 6 1 4 1 2 . J U L - ' . 9 6 1 4 1 3 1 9 - . 9 1 4 1 4 2 0 - . 9 1 4 J 5 2 1 - 1 9 1 4 1 6 2 2 \" . 9 1 4 1 7 2 3 -. 8 9 1 6 0 , 1 8 8 9 1 .4.9.0. l M _ . . . . a . l i 0 6 1 4 9 1 1 6 2 , 1 9 0 6 1 4 9 2 1 6 3 . 1 8 1 A 9 . 3 1 M , ? 9 J j 2 , ] ' 4 9 4 1 6 5 , 2 9 5 2 1 4 9 5 1 6 6 . 3 5 9 6 . 1 . 4 9 6 1 6 7 . 3 5 9 6 1 4 9 7 1 6 8 , j 8 1 4 9 8 1 6 9 , 1 8 170 T 1 a 1 5 0 0 1 7 1 , 2 3 5 1501 1 7 2 . 2 3 5 15.Q2.... 2D 9 1. 1 5 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 4 1 5 0 4 1 2 - ' . 9 9 5 1 5 Q 5 1 6 - \u00E2\u0080\u009E 9 f e 1 5 0 6 1 7 - . 9 6 1 5 0 7 2 1 - . 9 I 5 J L A 22_J=.\u00C2\u00BB9 1 5 0 9 2 3 - . 9 1 5 1 0 4 2 - ' . 9 9 5 t 5 M 4 5 - 1 9 6 1 5 1 2 4 6 - . 9 6 1513 4 9 - . 9 15.1.4 5.0. - . . . 9 1 5 1 5 5 1 - ' . 9 1 5 1 6 6 6 - . 9 9 5 1 5 1 7 6 8 - ' ^ 9 . 6 1 5 1 8 6 9 - . 9 6 1 5 1 9 7 1 - ' . 9 6 1 5 2 0 . _ 7 2 - . 9 6 1 5 2 1 7 3 - . 9 6 1 5 2 2 7 5 - . 9 _ I 5 _ 2 J 7 6 - ^ g 1 5 2 4 7 7 - . 9 1 5 2 5 8 7 - 1 , . . .15.2.6 _ 8 8 - l . . \u00C2\u00AB _ 1 5 2 7 8 9 - . 9 9 5 1 5 2 8 9 0 - . 9 6 1 5 2 9 9 1 - . 9 6 1 5 3 0 9 2 - . 9 6 1 5 3 1 9 3 - . 9 6 . 1 . 5 . 1 2 . . . 9 J L \u00E2\u0080\u009E - \u00C2\u00BB ? 6 . . . 1 5 3 3 9 5 - . 9 6 1 5 3 4 9 6 - . 9 6 1 5 3 5 9 7 - . 9 6 1 5 3 6 9 8 - . 9 1 5 3 7 9 9 - . 9 1 5 3 8 . . 1 0 . 0 . . . . . . 9 _ __ 1 5 3 9 1 4 4 - 2 5 , 1 5 4 0 1 5 5 T . J 5 - ' U 1 5 9 f Q 7 3 7 1 5 4 2 1 6 0 , 1 2 0 9 1 5 4 3 1 6 1 , 1 0 1 1 5 4 4 1 6 2 , i 0 t 1 5 4 5 1 6 4 . 1 8 8 9 J 5 4 6 1 6 5 , 1 8 8 9 J 5 4 7 1 6 6 . 1 9 Q 6 : _ 1 5 4 8 1 6 7 . 1 9 0 6 1 5 4 9 1 6 8 , 1 8 1 5 5 0 1 6 9 , 2 9 5 2 1 5 5 1 1 7 0 , 2 9 5 2 ^ 5 5 2 1 7 1 , 3 5 9 6 1 5 5 3 1 7 2 . 3 5 9 6 1 5 5 4 1 7 3 , 1 A 1 5 5 5 1 7 4 , 1 3 1 5 5 6 1 7 5 . 1 8 1 5 5 7 1 7 6 , 2 3 5 1 5 5 8 1 7 7 , 2 3 5 1 5 5 9 2 1 0 1 ' . 1 5 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 5 1 5 6 1 1 7 - ' . 9 9 5 1 5 6 2 2 2 _ _ . _ 1 5 6 3 2 3 - . 9 1 5 6 4 4 6 - . 9 9 5 1 5 6 5 5 0 - ' . 9 1 5 6 6 5 1 - , 9 1 5 6 7 6 9 1 - . 9 9 5 15_68_ . 7 _ _ J = _ 1 6 _ 1 5 6 9 7 3 - . 9 6 1 5 7 0 7 6 - . 9 1 5 7 1 7 7 - ' . 9 1 5 7 2 9 1 - . 9 9 5 1 5 7 3 9 3 - . 9 6 157.4 _ _ . 9 4 _ - ' e 9 6 1 5 7 5 9 b - . 9 6 1 5 7 6 9 7 - . 9 6 1 . 5 7 7 9 9 - . 9 1 5 7 8 1 0 0 - . 9 1 5 7 9 1 0 9 - 1 , 1 1 0 1 5 8 1 1 1 1 - . 9 9 5 1 5 8 2 1 1 2 - . 9 6 X5AZ U 3 _ _ J l i i _ 1 5 8 4 1 1 4 - . 9 6 1 5 8 5 1 1 5 - ' . 9 6 . 1 5 . 8 . 6 1 1 6 . - . . 9 . 6 . 1 5 8 7 1 1 7 - . 9 6 1 5 8 8 1 1 8 - . 9 X \u00C2\u00A3 L S _ L L 9 - ' . 9 1 5 9 0 1 4 5 - 2 5 . 1 5 9 1 1 5 6 f . . 1 5 3 . 2 .1..&...Q . 0 . 7 7 4 1 5 9 3 1 6 1 . 0 5 3 5 1 5 9 4 1 6 2 . 0 5 3 5 15-9.5. 1 6 4 , 0 7 3 7 1 5 9 6 1 6 5 , 1 2 0 9 1 5 9 7 1 6 6 , 1 0 1 . 1 5 . 9 5 . . . . 1 6 . 7 . . M L 1 5 9 9 1 6 9 , 1 6 8 9 1 6 0 0 1 7 0 , 1 8 8 9 1 6 0 1 1 7 1 , 1 9 0 f t 1 6 0 2 1 7 2 , 1 9 0 6 1 6 0 3 1 7 3 , 1 8 lh.M l l f l _ , 2 3 5 2 _ _ \u00E2\u0080\u009E _. . 1 6 0 5 1 7 5 , 2 9 5 2 \u00C2\u00A3 6 0 6 1 7 6 , 3 5 9 6 1 . 6 0 7 1 7 7 . 3 5 9 6 1 . 6 0 8 1 7 8 , 1 8 1 6 0 9 1 7 9 , 1 8 I M S I 8 j j _ a a . 1611 1 9 1 . 2 3 5 1 6 1 2 1 8 2 , 2 3 5 1613 211 1. 1 6 1 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 6 1 6 1 5 2 3 - . 9 14.16. 5 1 \u00E2\u0080\u009E - A 9 1 6 1 7 7 3 - ' . 9 9 5 1 6 1 8 7 7 - . 9 1 6 j 9 9 4 - ' , 9 9 5 1 6 2 0 9 7 i . 9 6 1 6 2 1 1 0 0 - . 9 1 6 . 2 . 2 1 1 3 - * . 9 9 5 1 6 2 3 1 1 5 - ' . 9 6 1 6 2 4 1 1 7 - . 9 6 lh25 1 1 9 - . 9 1 6 2 6 1 2 6 - 1 . \ 6 2 7 1 2 7 - 1 , . 1 . 6 2 8 1 2 J L _ T U 9 \u00C2\u00A3 5 -1 6 2 9 1 2 9 - ' . 9 6 1 6 3 0 1 3 0 - . 9 6 1 6 3 1 1 3 1 - ' . 9 6 1 6 3 2 1 3 2 - . 9 1 6 3 3 1 4 6 - 2 5 , 1 6 3 4 . 1 5 7 )..,.._ _ 1 6 3 5 1 6 0 , 0 4 9 5 1 6 3 6 1 6 1 . 0 2 8 4 JLhZl^^AiiZ . 0 2 6 4 1 6 3 8 1 6 5 , 0 7 7 4 1 6 3 9 1 6 6 , 0 5 3 5 1 6 4 0 , 1 6 ? , 0 5 3 5 1 6 4 1 1 6 9 , 0 7 3 7 1 6 4 2 1 7 0 , 1 2 0 9 _ 1 M 1 _ L I l _ _ a l i 1 6 4 4 1 7 2 , 1 0 1 1 6 4 5 1 7 4 , 1 8 8 9 1 6 . 4 6 1.7.5 , 1 . 8 8 . 9 1 6 4 7 1 7 6 . 1 9 0 6 1 6 4 8 1 7 7 . 1 9 0 6 1 6 4 9 1 7 6 . 1 6 1 6 5 0 1 7 9 , 2 9 5 2 1 6 5 1 1 8 0 , 2 9 5 2 : I . 6 5 l 1 8 1 _ u 3 _ L ? _ _ _ _ _ _. 1 6 5 3 1 . 8 2 , 3 5 9 6 1 6 5 4 1 8 3 as 1 6 5 5 1 8 4 . 1 8 1 6 5 6 1 8 5 tie 1 6 5 7 1 8 6 . 2 3 5 1 6 5 8 1 8 7 __j_35_ _ _ 1 6 5 \" 2 1 2 1 . 1 6 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 7 1 6 6 1 2 - 1 . 1 6 6 2 3 - 1 . 1 6 6 3 5 - . 9 9 5 . . . 1 6 6 4 7 - ' , 9 6 1 6 6 5 8 1 6 6 6 1 0 - ' . 9 6 1 6 6 7 1 1 - . 9 6 1 6 6 8 1 2 - . 9 6 1 6 6 9 1 4 - . 9 6 1 6 7 0 1 5 - ' . 9 6 1 6 7 1 1 6 - . 9 6 1 6 7 2 1 7 - ' . 9 6 1613 1 9 - . 9 1 6 7 4 2 0 - . 9 1 6 7 5 2 1 - . 9 1 6 7 6 2 2 V ? _ _ _ _ 1 6 7 7 2 3 1 6 7 8 2 5 - ' . 8 5 1 6 7 9 2 6 - 1 8 5 1 6 8 0 2 7 - \" , 8 5 1 6 8 1 2 8 - . 8 5 1 6 8 2 2 9 1 6 8 3 3 0 - . 8 2 5 1 6 8 4 3 1 - . 8 1 6 8 5 3 5 - 1 . 1 6 8 6 3 6 - 1 * 1 6 8 7 3 7 - . 9 9 5 1 6 8 8 . 3 8 - ' . 9 6 1 6 8 9 39~ - . 9 6 1 6 9 0 ^ 0 - . 9 6 1 6 9 1 4 1 - ' . 9 6 1 6 9 2 4 2 - . 9 6 i ' 6 9 3 4 3 - , 9 6 1 6 9 4 4 4 - . 9 6 1 6 9 5 4 5 - . 9 6 1 6 9 6 4 6 - . 9 6 1 6 9 7 4_7 - T 9 1 6 9 8 4 8 - q 0 ' 1 6 9 9 4 9 - . 9 1 7 0 0 . 5.0... \u00C2\u00BB _ 9 1 7 0 1 5 1 - . 9 1 7 0 2 5 2 - . 8 5 i i _ L 5 3 - ' . a s 1 7 0 4 5 4 - . 8 5 1 7 0 5 5 5 - ' . 8 5 1 7 0 6 5 6 , - l i _ 5 1 7 0 7 5 7 - . 8 2 5 1 7 0 8 58 - . 8 1 7 0 9 1 4 7 - 1 0 . 5 2 6 1 7 1 0 1 5 2 1 * . 1 7 1 1 1 5 3 r . \u00E2\u0080\u00A2 1 7 1 2 1 5 8 \u00E2\u0080\u009E,____ 1 7 1 3 1 5 9 . 2 9 5 2 1 7 1 4 1 6 0 . 2 9 5 2 1 7 - 1 5 1 6 1 , 3 5 9 6 1 7 1 6 1 6 2 , 3 5 9 6 1 7 1 7 1 6 3 , 1 8 1 7 1 3 1 6 4 . a . 9 _ - - _ 1 7 1 9 1 6 5 , 1 8 1 7 2 0 1 6 6 , 2 3 5 1 7 2 3 1 6 7 . 2 3 5 1 7 2 2 2 1 3 1 . 1 7 2 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 8 1 7 2 \u00C2\u00AB _ 8 _ - _ 9 9 5 1 7 2 5 1 1 - . 9 6 1 7 2 6 1 2 - ' . 9 6 1 1 2 7 1 5 - . 9 6 1 7 2 8 1 6 - . 9 6 1 7 2 9 1 7 - . 9 6 1 7 3 0 2 0 . _ - , 9 1 7 3 1 2 1 - . 9 1 7 3 2 2 2 - ' . 9 \u00C2\u00B1112 2 3 - . 9 1 7 3 4 2 6 - ' . 8 5 1 7 3 5 2 7 - ' . 8 5 1 1 1 6 . 2 8 - ' . 8 5 : 1 7 3 7 2 9 - . 8 5 1 7 3 8 3 0 - ' . 8 2 5 _ J . _ _ 3 1 - . 8 1 7 4 0 3 9 - . 9 9 5 1 1 7 4 1 4 1 - . 9 6 . 1 1 4 2 _L2_____?_ . 1 7 4 3 4 4 - . 9 6 1 7 4 4 4 5 - . 9 6 1 7 4 5 4 6 - . 9 6 1 7 4 6 4 8 - . 9 1 7 4 7 \u00C2\u00AB9 - . 9 i ' 7 4 . 8 . 5 0 . . . - . 9 _ 1 7 4 9 5 1 - . 9 1 7 5 0 5 3 - . 8 5 1 7 5 1 5 4 - ' . 8 5 1 7 5 2 5 5 - . 8 5 1 7 5 3 5 6 - ' . 8 5 1 7 3 4 5 7 \u00E2\u0084\u00A2 , 8 2 5 1 7 5 5 5 8 - . 8 1 7 5 6 6 2 - 1 . 1 7 5 7 6 3 \u00E2\u0080\u00A2 h , , 1 7 5 8 6 4 \u00E2\u0080\u00A2 * . 9 9 5 1 7 5 9 6 5 - . 9 6 1 7 6 Q 6 6 . - . 9 6 1 7 6 1 6 7 - . 9 6 1 7 6 2 6 8 - . 9 6 1 7 6 3 6 9 - . 9 6 1 7 6 4 7 0 - . 9 6 1 7 6 5 7 1 - . 9 6 1 1 6 6 _ 7 2 - ' . 9 6 1 7 6 7 7 3 - ' . 9 6 1 7 6 8 7 4 - . 9 1 7 6 9 7 5 - ' . 9 1 7 7 0 7 6 - . 9 1 7 7 1 7 7 \u00C2\u00AB , 9 1 7 7 2 7 8 - . 8 5 1 7 7 3 7 9 \" . 8 5 1 7 7 4 8 0 - . 8 5 1 7 7 5 8 1 - 1 8 5 1 7 7 6 3 2 - ' . 8 2 5 1 7 7 7 8 3 - . 8 1 7 7 8 1 4 8 \" 1 0 . 5 2 6 1 7 7 9 1 5 4 1 . 1 7 8 0 1 5 9 . 1 8 8 9 . 1 1 8 1 _ L M _ . 1. 8 8 9 1 7 8 2 1 6 1 \u00C2\u00AB 1 ^ 0 6 1 7 8 3 1 6 2 . 1 9 0 6 1 1 8 4 _.. 1 6 . 3 . _ . i _ a _ _ _ . _ 1 7 8 5 1 6 4 , 2 9 5 2 1 7 8 6 1 6 5 , 2 9 5 2 1 7 8 7 1 6 6 . 3 5 9 6 1 7 8 8 1 6 7 , 3 5 9 6 1 7 8 9 1 6 8 , 1 8 1 7 9 0 1 6 9 . . . a * _ 1 7 9 1 1 7 0 , 1 3 1 7 9 2 1 7 1 . 2 3 5 l 7 _ 9 3 1 7 2 . 2 3 5 1 7 9 4 2 1 4 1 . 1 7 9 5 0 0 0 0 0 0000 0 0 0 0 0 0 0 0 0 0 0 4 9 1 7 9 6 1 2 - . 9 9 5 1 7 9 7 1 6 - . 9 6 1 7 9 8 1 7 - . 9 6 1 7 9 9 2 1 - ' . 9 1 8 0 0 2 2 - . 9 1 8 0 1 . I 8 0 . ? _ 1 8 0 3 1 8 0 4 1 8 0 5 . 1 8 0 6 i \" 8 0 7 j 8 0 8 . 1 8 0 \" ? 1 8 1 0 1 8 J , 1 -1 8 1 2 1 8 1 3 1 . 8 . 1 1 . 1 8 1 5 1 8 1 6 1 8 1 8 1 8 1 9 18.2.0... 1 8 2 1 1 8 2 2 l a ' 1 5 2 7 - , 9 - . 8 5 CO 2 9 3 0 - . 8 5 - . 8 5 - . 8 2 5 3 1 - . 8 4 2 , 9 9 5 4 5 - . 9 6 4 6 4 9 5 0 > 9 6 5 1 - . 9 5 4 - . 8 5 5 5 \u00E2\u0080\u009E - * 8 5 . D O 5 7 5 8 - . 8 5 - . 8 2 5 6 6 - . 9 9 5 6 8 - . 9 6 . 6 . 9 . . . . . . 9 . 6 7 1 - . 9 6 7 2 - , , 9 6 7 3 - . 9 6 -1 8 2 4 1 8 2 5 . . . 1 8 2 6 . . 1 8 2 7 1 8 2 8 1 . 8 . 2 9 1 8 3 0 1 8 3 1 1 8 3 2 1 8 3 3 1 8 3 4 l M 5 _ 1 8 3 6 1 8 3 7 1 8 3 8 . 1 8 3 9 1 8 4 0 1 3 4 2 1 8 4 3 ...1.81.4... 1 8 4 5 1 8 4 6 l 8 4 _ 7 _ 1 8 4 8 1 8 4 9 18.5.0... 7 5 - . 9 7 6 - . 9 ...7.7 - . 9 . 7 9 - . 8 5 8 0 - . 8 5 . . f l l - . 8 5 8 2 - . 8 2 5 8 3 - . 8 . 8 . . 7 . . . . . - L . - 1 . - ' , , 9 9 5 - r 9 6 - . 9 6 8 8 8 9 _ 9 J L 9 1 9 2 . 9 3 . , 9 6 - . 9 6 9 4 9 5 _ 9 A . - p 9 6 - . 9 6 - ' . 9 6 9 7 9 8 9 9 - . 9 6 - . 9 1 0 0 1 0 1 1 0 2 - . 9 - . 8 5 - ' . 8 5 1 0 3 . 1 0 4 1 0 5 - . 8 5 - . 8 2 5 m B 1 8 5 1 1 4 9 - 1 0 . 5 2 6 1 8 5 2 1 5 5 1 . 1 8 5 3 1 5 9 . 0 7 3 7 1 8 5 4 1 6 0 , 1 2 0 9 1 8 5 5 1 6 1 , 1 0 1 \u00C2\u00B1$Sh 1 6 2 . 1 0 , 1 1 8 5 7 1 6 4 . , 1 8 8 9 1 8 5 8 1 6 5 , 1 8 8 9 . 1 6 5 9 L 6 6 , 1 9 0 6 I 8 6 0 1 6 7 , 1 9 0 6 1 6 6 1 1 6 8 , 1 8 lM2\u00E2\u0080\u0094-\u00E2\u0080\u0094\u00C2\u00B1&5-*2252 1 8 6 3 1 7 0 , 2 9 5 2 1 8 6 4 1 7 1 , 3 5 9 6 1 8 6 5 1 7 2 f 3 5 9 6 1 8 6 6 1 7 3 , 1 6 1 6 6 7 1 7 4 . 1 6 Iftfcfi iJ-5_*lJB 1 8 6 9 1 7 6 , 2 3 5 1 8 7 0 1 7 7 . 2 3 5 1 6 7 1 2 U S _ X 1 8 7 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 1 8 7 3 1 7 - . 9 9 5 . 1 8 . 7 4 2 2 - . 9 , 1 8 7 5 2 3 - . 9 1 8 7 6 2 8 - ' . 8 5 1 8 7 ? _ _ 2 9 _ - . 8 5 1 8 7 8 4 6 ^ . 9 9 5 1 8 7 9 5 0 - . 9 1 8 8 0 _ g j ^ 9 1 8 8 1 5 5 - ' , 8 5 1 8 8 2 5 6 - . 8 5 J L 8 6 3 6 9 - ' . 9 9 5 1 8 8 4 7 2 - e 9 6 1 8 8 5 7 3 - ' . 9 6 1 8 8 6 7 6 - . 9 1 8 8 7 7 7 - . 9 1 8 8 8 8 0 - . 8 5 I M 2 8 1 - . 6 5 1 8 9 0 8 ? - ' . 8 2 5 1 8 9 1 8 3 - ' . 8 1 8 9 2 ? 1 - , 9 9 5 1 8 9 3 9 3 - . 9 6 1 8 9 4 94 - . 9 6 1 8 9 5 9 6 - . 9 6 1 8 9 6 9 7 - . 9 6 i ' 8 9 ? 99 - . 9 1 8 9 8 1 0 0 - . 9 1 3 9 9 1 0 2 - , 8 5 \" ~ 1 9 0 0 1 0 3 - . 8 5 1 9 0 2 1 0 5 - . 8 1 9 0 3 1 0 9 - 1 . 1 9 0 4 1 1 0 - } , 1 9 0 5 i l l - . 9 9 5 1 9 0 6 1 1 2 - ' . 9 6 1 9 Q 7 1 1 3 - . 9 6 1 9 0 8 1 1 4 - . 9 6 1 9 p 9 1 1 5 - ' . 9 6 \u00E2\u0080\u009E 9 1 0 L 1 6 _ - l 9 6 _ 1 9 1 1 1 1 7 - . 9 6 1 9 1 2 1 1 8 - . 9 1 9 1 3 1 1 9 - ' . 9 . 1 9 1 4 1 2 0 - . 8 5 1 9 1 5 1 2 1 - . 8 5 1 9 1 6 1 2 2 - . 8 2 5 _ 1 9 1 7 1 2 3 - . 8 1 9 1 8 1 5 0 - 1 0 . 5 2 6 1 9 j q 1 5 6 r . 1 9 2 0 1 6 0 t 0 7 7 a 1 9 2 1 1 6 1 , 0 5 3 5 1 9 2 2 1 6 2 . 0 5 3 5 _ 1 9 2 3 1 6 4 , 0 7 3 7 1 9 2 4 1 6 5 , 1 2 0 9 1 9 2 5 1 6 6 . 1 0 1 1 9 2 6 1 6 7 , 1 0 1 1 9 2 7 1 6 9 , 1 8 8 9 I 9 _ 2 J 3 _ 7 ___ J _ _ 8 _ 1 9 2 9 1 7 1 , 1 9 0 6 1 9 3 0 1 7 2 , 1 9 0 6 1931 L 7 _ L _ * J _ 8 1 9 3 2 1 7 4 , 2 9 5 2 1 9 3 3 1 7 5 . 2 9 5 2 1 9 3 . 4 1 X 6 \u00E2\u0080\u009E . . , . . 1 5 9 . 6 1 9 3 5 1 7 7 , 3 5 9 6 1 9 3 6 1 7 8 , 1.8 d3M^~XIl^d3 1 9 3 8 1 8 0 , 1 8 1 9 3 9 1 8 1 , 2 3 5 - 1 8 . 2 , 2 3 5 ; - _ 1 9 / 4 1 2 1 6 1 ' . | 9 4 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 1 4 _ _ 3 : 2 3 - ^ 9 : 1 9 4 4 2 9 - . 8 5 1 9 4 5 5 1 - . 9 .1 .9i! . .6 5 . 6 -... .8.5...._ 1 9 4 7 7 3 - , 9 9 5 1 9 4 8 7 7 - . 9 I _ _ L 2 , _ _ _ - _ U _ S : 1 9 5 0 9 4 - . 9 9 5 1 9 5 1 9 7 - ' , 9 6 1 9 5 2 - , 9 1 9 5 3 1 0 3 - . 8 5 1 9 5 4 1 0 4 - . 8 2 5 ( 9 5 5 1 0 5 \" ' . 8 1 9 5 6 1 1 3 - ' . 9 9 5 1 9 5 7 1 1 5 - ' . 9 6 [ 9 5 8 1 1 7 f . 9 6 . _ -. 1 9 5 9 1 1 9 - . 9 I 9 6 0 1 2 1 - p j . 8 5 1 9 6 1 1 2 ? - a 8 ? 5 1 9 6 2 1 2 3 - . 3 1 9 6 3 1 2 6 \" I s 19i>4_ . . . . 1 2 7 . . . - 1 \u00C2\u00BB i ' 9 6 5 1 2 8 - . 9 9 5 1 9 6 6 1 2 9 - ' . 9 6 ' ( 9 6 ? 1 3 0 - , 9 6 1 9 6 8 1 3 1 - ' . 9 6 1 9 6 9 1 3 2 1 9 7 0 1 3 3 - ' . 8 5 1 9 7 1 1 3 4 - . 8 2 5 1 9 7 2 1 3 5 - ' . 8 J 9 7 \" * , 1 5 1 - 1 0 ' , 5 ? 6 1 9 7 4 1 5 7 r . 1 9 7 5 1 6 0 . 0 4 9 5 1 9 7 6 1 6 1 . . . 0 2 8 4 1 9 7 7 1 6 2 . 0 2 8 4 1 9 7 8 1 6 5 . 0 7 7 4 1 9 7 9 1 6 6 , ' 1 5 3 5 1 9 8 0 1 6 7 , 0 5 3 5 1 9 8 1 1 6 9 , 0 7 3 7 1 9 8 2 1 7 0 J L 2 Q 9 . . . 1 9 8 3 1 7 1 , 1 0 1 1 9 8 4 1 7 2 . 1 0 1 1 9 8 5 1 7 4 , 1 8 8 9 1 9 8 6 1 7 5 . 1 3 8 9 1 9 8 7 1 7 6 , 1 9 0 6 i ' 9 8 8 . L 7 J _ \u00E2\u0080\u009E \u00E2\u0080\u009E . 1 . 9 Q i > . . . . . ._ _ _ _ _ 1 9 8 9 1 7 8 , 1 8 1 9 9 0 1 7 9 , 2 9 5 2 1 9 9 1 1 8 0 f ? 9 5 ? 1 9 9 2 1 8 1 , 3 5 9 6 1 9 9 3 1 8 2 , 3 5 9 6 \u00E2\u0080\u00A2 1 9 9 4 1 8 3 . . . . 1 8 . . . . . _ \u00E2\u0080\u009E 1 * 9 9 5 1 8 4 , 1 8 1 9 9 6 1 8 5 . 1 8 1 9 9 7 1 8 6 . . . 2 3 5 . . . . . . 1 9 9 8 1 8 7 , 2 3 5 1 9 9 9 2 1 7 1 . . 2 . 0 0 0 0 0.0.0 0. , 0 0 . 0 . 0 0 0 0 0 . 0 . 0 0 0 0 0.0 5 2 2 0 0 1 2 1 . 2 0 0 2 2 0 0 3 3 5 1 . . 9 9 5 2 0 0 4 7 , 9 6 2 0 0 5 2 0 0 6 8 1 0 . 9 6 , 9 9 6 \u00E2\u0080\u009E , _ 2 0 0 7 1 1 , 9 6 2 0 0 8 1 2 , 9 6 2 0 0 9 it . 9 6 2 0 1 0 2 0 J l 2 0 ^ 2 1 5 1 6 17. , 9 6 , 9 6 . . 9 6 _ . . . 2 0 1 3 1 9 , 9 6 2 0 1 4 2 0 . 9 6 2 0 ^ 5 2 1 . 9 6 2 0 1 6 2 2 , 9 6 2 0 1 7 2 3 , 9 6 2018 2 5 . . * 9 _ L _ ... _ _ 2 0 1 9 2 0 2 0 2 0 2 1 2 6 2 7 2 8 . 9 < l , 9 4 . 9 4 2 0 2 2 2 9 . 9 4 2 0 2 3 3 0 , 9 3 2 0 2 4 3 1 , 9 2 _ . 2 0 2 5 3 2 , 8 8 2 0 2 6 3 3 , 8 8 2 0 2 7 3 4 . 8 8 2 0 2 8 3 5 1 . 2 0 2 9 3 6 1 . 2 0 3 0 3 7 , 9 9 5 2 0 3 1 3 8 . 9 6 2 0 3 2 3 9 , 9 6 2 0 3 3 4 0 . 9 6 2 0 3 4 4 1 , 9 6 2 0 3 5 4 2 , 9 6 2 0 3 6 4 3 . 9 6 2 0 3 7 4 4 , 9 6 2 0 3 8 4 5 , 9 6 2 0 3 9 4 6 , 9 6 2 0 4 0 4 7 . 9 6 2 0 4 1 4 8 . 9 6 2 0 4 2 4 9 , 9 6 2 0 4 3 5 0 , 9 6 2 0 4 4 5 1 . 9 6 2 0 4 5 5 2 , 9 4 2 0 4 6 5 3 . 9 4 2 0 4 7 5 4 2 Q 4 8 5 5 . 9 4 \u00E2\u0080\u00A2 _ . .... , .... 2 0 4 9 5 6 . 9 4 2 0 5 0 5 7 , 9 3 2 0 5 1 5 8 P 9 2 2 0 5 2 5 9 , 8 8 2 0 5 3 6 0 , 8 8 2 0 5 4 6 1 a.8 8 ...... ... . . . _. 2 0 5 5 1 3 7 - 1 , 2 0 5 6 . 1 4 2 - 1 , . . . 2 0 5 7 . i \u00C2\u00AB 7 - 1 . 2 0 5 8 1 5 2 - 1 , 2 0 5 9 1 5 3 - 1 , . 2 0 . 6 . 0 . _ . 1 5 8 . - \" , 5 2 0 6 1 1 5 9 - . 8 2 2 0 6 2 1 6 0 - ' . 8 2 2 0 6 3 1 6 1 2 0 6 4 1 6 2 - . 7 6 5 2 0 6 5 1 6 3 - . 5 2 0 6 6 .. 1 6 4 . - . 5 2 0 6 7 1 6 5 - . 5 2 0 6 8 1 6 6 - . 5 2 0 6 9 1 6 7 - . 5 2 0 7 0 2 1 8 - 1 , 0 0 0 2 0 7 1 2 0 7 2 o o o o o o O O O O O O 0 0 0 0 0 0 0 0 0 5 3 8 9 9 5 2 0 7 3 1 1 , 9 6 2 0 7 4 1 2 . 9 6 2 0 7 5 1 * 2 0 7 6 1 6 . 9 6 2 0 7 7 1 7 , 9 6 . 2 . 0 . 7 8 2.Q.. . 9 6 _ 2 0 7 9 2 1 . 9 6 2 0 8 0 2 2 . 9 6 2 0 8 1 _ _ 2 J 3 _ , 9 6 2 0 8 2 2 6 . 9 4 2 0 8 3 2 7 . 9 4 . 2 . 0 . 8 4 2 8 . . . . .9 .4 2 0 8 5 2 9 . 9 f l 2 0 8 6 3 0 , 9 3 2 0 8 7 3 1 . 9 2 2 0 8 8 3 3 . 8 8 2 0 8 9 3 4 . 8 8 2 . 0 9 0 . . 3 9 , 9 9 5 2 0 9 1 4 1 , 9 6 2 0 9 2 4 2 . 9 6 2 0 9 3 4 4 . 9 6 2 0 9 4 4 5 . 9 6 2 0 9 5 4 6 . 9 6 2 0 9 6 4 8 . 9 6 2 0 9 7 4 9 . 9 6 2 0 9 8 5 0 , 9 6 2 0 9 9 5 1 _ . 9 6 2 1 0 0 5 3 , 9 4 2 1 0 1 5 4 . 9 4 ? _ i Q 2 5 5 . 9 4 2 1 0 3 - 5 6 , 9 4 2 1 0 4 5 7 , 9 3 / l i _ 5 5 8 , 9 2 1 0 6 6 0 , 8 8 2 1 0 7 6 1 , 8 8 \u00C2\u00A3 1 0 8 & _ _ J _ _ _ _ 2 1 0 9 6 3 1 . 2 1 1 0 6 4 , 9 9 5 2 1 1 1 6 5 . 9 6 2 1 1 2 6 6 , 9 6 2 1 1 3 6 7 . 9 6 2 1 1 4 6 8 . _ _ 9 . 6 _ . . . 2 1 1 5 6 9 , 9 6 2 1 1 6 7 0 , 9 6 21A2 _ 7 J _ _ . . 9 _ _ _ 2 1 1 8 7 2 , 9 6 2 1 1 9 7 3 , 9 6 2 1 2 0 7.4 , 9 6 2 1 2 1 7 5 , 9 6 2 1 2 2 7 6 , 9 6 2 1 2 3 7 7 \u00E2\u0080\u009E 9 6 2 1 2 4 7 8 , 9 4 2 1 2 5 7 9 . 9 4 2 1 2 . 6 . . 8.0.. ._.. ,94.. . 2 1 2 7 8 1 , 9 4 2 1 2 8 8 2 , 9 3 2 1 2 9 8 3 . 9 2 2 1 3 0 8 4 , 8 8 2 1 3 1 8 5 , 8 8 .2.1 .3.2. _ 8 . 6 . . . _ . , . M . . . 2 1 3 3 1 3 8 - 1 2 1 3 4 1 4 3 - 1 2123 1 4 8 - i _ 2 1 3 6 1 5 4 - i , 2 1 3 7 1 5 9 - . 5 2 4 8 2 1 3 . 8 1 . 6 0 . . _ . - _ . 5 . 2 . 4 . f i . 2 1 3 9 1 6 1 - . 4 0 5 5 2 1 4 0 1 6 2 - ' . 4 0 5 5 2 J _ L L _ _ i _ J _ _ _ _ 5 _ 2 1 4 2 1 6 4 - . 8 2 2 1 4 3 1 6 5 - . 8 2 2 1 4 4 1 6 6 - ' . 7 6 5 2 1 4 5 1 6 7 - . 7 6 5 2 1 4 6 1 6 8 - . 5 ?JJU 1A?___!L_5 2 1 < ! 8 1 7 0 - . 5 2 1 4 9 1 7 1 - . 5 5 1 ^ ( 1 1 7 2 \u00C2\u00AB _ 5 \ 1 5 7 2 * 5 1 2 1 9 - 1 , 2 1 5 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 4 2 1 5 3 1 2 - 9 9 5 2 1 5 4 2 1 5 5 2 1 5 6 1 6 , 9 6 1 7 , 9 6 2 1 . 9 6 .. 2 1 5 7 2 1 5 8 2 1 5 9 2 2 , 9 6 2 3 , 9 6 2 7 - 9 4 2 1 6 0 2 1 6 1 2 1 6 2 2 8 , 9 4 2 9 , 9 4 3 0 , 9 3 2 1 6 3 2 1 6 4 2 1 6 5 3 1 , 9 2 3 4 , 8 8 4 2 . 9 9 5 2 1 6 6 2 1 6 7 2 1 6 8 4 5 , 9 6 4 6 , 9 6 4 9 n 9 6 -II*. \u00E2\u0080\u009E\u00E2\u0080\u0094 2 1 6 9 2 1 7 0 2 1 7 1 5 0 , 9 6 5 1 , 9 6 5 4 . 9 4 2 1 7 2 2 1 7 3 2 1 7 4 5 5 , 9 4 5 6 , 9 4 5 7 , 9 3 I 2 1 7 5 2 1 7 6 2 f 7 7 5 8 , 9 2 6 1 . 6 8 6 6 r 9 9 S 2 1 7 8 2 1 7 9 2 1 8 0 6 8 , 9 6 6 9 , 9 6 7 1 . 9 6 - -C i . U A l 2 1 8 1 2 1 3 2 ? ! 6 7 2 , 9 6 7 3 , 9 6 7=; . 9 6 2 1 8 4 2 1 8 5 2 1 8 6 2 1 8 7 2 1 8 8 7 6 . 9 6 7 7 . 9 6 7 9 . . 9 4 8 0 . 9 4 8 1 , 9 4 6 ? \u00E2\u0080\u009E Q 3 \u00E2\u0080\u0094 \u00E2\u0080\u0094 - -2 1 9 0 2 1 9 1 ? 1 9 2 8 3 , 9 2 8 5 , 6 8 8 6 . 8 8 r..X i A\u00E2\u0080\u0094 -2 1 9 3 2 1 9 4 ? I 2 2 9 6 \" O S S r / 2 2 5 6 6 * 9*7 I7t722 6 2 \u00C2\u00A3 t / 2 2 17 6 * 9 2 2(7 2 2 9 6 * \u00C2\u00A3 2 t r ; 2 2 9 6 * 2 2 017 2 2 5 6 6 * I T b \u00C2\u00A3 2 2 S \u00C2\u00A3 o o o o o o o o o o o o o o o 0 0 0 0 0 & \u00C2\u00A3 2 2 - 1 - 0 2 2 i . i . c c s ' - LL\ 9 \u00C2\u00A3 2 2 s \" > 9 LI S \u00C2\u00A3 2 2 - \" ' 17 t l \u00C2\u00BB O s ' - \u00C2\u00A3 \u00C2\u00A3 2 2 s ' - 1L\ 2 \u00C2\u00A3 2 2 ZL\ S 9 Z ' - \L\ 0 \u00C2\u00A3 2 2 Ql\ 6 2 2 2 ~ 6 9 T 8 2 7 2 2 s S 8 9 T 1 2 2 2 s s o (7 \u00E2\u0080\u00A2 - \u00C2\u00A3 9 T 9 2 2 2 9 9 1 & 2 2 2 8 r / 2 s \" \u00C2\u00AB \u00C2\u00BB S 9 T 1 7 2 2 2 8 r ; 2 s \" - 179 J \u00C2\u00A3 2 2 2 T 9 ~ T ' - - \u00C2\u00AB - ^ , - , . \u00E2\u0084\u00A2 -6 r / T 2 * ' - T 9 T T 2 2 2 feS\u00C2\u00A3\u00C2\u00A3'\u00C2\u00BB 0 9 1 0 2 2 2 S T 7 U 2 \" - b i t 6 1 2 2 * t - S S I S t ' 2 2 * t - 6 f71 i t 2 2 M - r/i7\"T ' 9 T 2 2 \" * T - b \u00C2\u00A3 T & T 2 2 8 9 * 8 0 1 I 7 T 2 2 y y * \u00C2\u00A3 0 T \u00C2\u00A3 < 2 2 9 8 * 9 0 1 2 T 2 2 2 6 * S O T t t 2 2 D O T t,b a \u00C2\u00A3 0 1 b ' 6 \" 2 2 2 0 T 8 0 2 2 t 7 6 - T O T Z 0 2 2 9 6 * 0 0 T 9 0 2 2 9 6 * 6 6 & 0 2 2 9 b ' 9 6 \" 9 6 * 1 6 \u00C2\u00A3 0 2 2 9 6 * 9 6 2 0 2 2 9 b - b b 1 0 2 2 2 2 5 1 7 3 , 9 6 2252 7 6 a.9 6 _ _ 2 2 5 3 7 7 , 9 6 2 2 5 4 8 0 , 9 4 2 2 5 5 8 1 . 9 4 2 2 5 6 8 2 , 9 3 2 2 5 7 2 2 5 8 8 3 8 6 , 9 2 _ 8 8 _ _ _ 2 2 5 9 9 1 , 9 9 5 2 2 6 0 9 3 , 9 6 2 2 6 1 9 4 . 9 6 2 2 6 2 9 6 . 9 6 2 2 6 3 9 7 , 9 6 2 2 6 4 9 9 . 9 6 2 2 6 5 1 0 0 , 9 6 2 2 6 6 2 2 6 7 1 0 2 1 0 3 , 9 4 . 9 4 2 2 6 8 1 0 4 . 9 3 2 2 6 9 1 0 5 , 9 2 2 2 7 0 1 0 7 t f l 8 2 2 7 1 1 0 8 , 8 8 2 2 7 2 1 0 9 \ \ 2 2 7 3 1 1 0 r . 2 2 7 4 1 1 1 , 9 9 5 2 2 7 5 1 1 2 , 9 6 2 2 7 6 1 1 3 , 9 6 2 2 7 7 2 2 7 8 1 1 4 1 1 5 , 9 6 , 9 6 2 2 7 9 1 1 6 . 9 6 2 2 8 0 1 1 7 , 9 6 2 2 8 1 1 1 8 , 9 6 2 2 8 2 1 1 9 . 9 6 2 2 8 3 1 2 0 . 9 4 2 2 8 4 1 2 1 , 9 4 2 2 8 5 1 2 2 . 9 3 2 2 8 6 1 2 3 , 9 2 2 2 8 7 1 2 4 . 8 8 2 2 . 3 8 1 2 5 , 8 8 _ 2 2 8 9 1 4 0 \" 1 \u00C2\u00BB 2 2 9 0 1 4 5 - 1 , 2 2 9 1 1 5 0 \u00E2\u0080\u00A2 1 . . 2 2 9 2 1 5 6 - 1 . 2 2 9 3 1 6 0 - ' . 2 1 5 2 2 9 4 1 6 1 - . 1 1 3 9 2 2 9 5 1 6 2 - . 1 1 3 9 2 2 9 6 1 6 4 - , 2 0 4 8 2 2 9 7 1.6.5 - . 3 3 5 9 ..... - - '\u00E2\u0080\u00A2- -2 2 9 8 2 2 9 9 1 6 6 1 6 7 - . 2 1 4 9 - . 2 1 4 9 2 3 0 0 . 1 6 9 - . 5 2 . 4 8 2 3 0 1 1 7 0 - . 5 2 4 8 2 3 0 2 1 7 1 - . 4 0 5 5 1 7 ? - > 0 5 5 2 3 0 4 1 7 3 - . 5 2 3 0 5 1 7 4 - . 8 2 2 3 0 6 - . 1 1 3 . . . ...-.J.2 - . - . -2 3 0 7 1 7 6 - ' . 7 6 5 2 3 0 8 1 7 7 - \" . 7 6 5 2 3 0 9 1 7 8 - . 5 2 3 1 0 1 7 9 - . 5 2 3 1 1 1 8 0 \"r 5 2 3 1 2 1 8 . 1 . . - ' 5 2 3 1 3 1 8 2 r - . 5 2 3 1 4 2 2 1 - I t o o o o o o o n o n o n n n n o n o n o 5 6 2 3 1 6 2 3 , 9 6 2 3 j 7 2 9 , 9 4 . 2 3 1 8 _ _ . . . 5 . 1 \u00E2\u0080\u009E . . 9 6 2 3 1 9 5 6 , 9 4 2 3 2 0 7 3 , 9 9 5 2 3 2 1 7 7 2 3 2 2 8 1 . 9 4 , 2 3 2 3 9/1 . 9 9 5 2 3 2 4 . . . . . . . . . . .9.1... . 9 6 2 3 2 5 1 0 0 . 9 6 2 3 2 6 1 0 3 . 9 4 2 ^ 2 7 1 0 4 T 9 3 2 3 2 8 1 0 5 , 9 2 2 3 2 9 1 0 8 , 8 8 . 2 3 1 0 . . ... . 1 1 3 . . 9 9 5 2 3 3 1 1 1 5 . 9 6 2 3 3 2 1 1 7 , 9 6 2 3 3 3 1 1 9 f 9 6 2 3 3 4 1 2 1 . . 9 4 2 3 3 5 1 2 2 . 9 3 2 3 3 6 1 2 3 . 9 2 2 3 3 7 1 2 5 , 8 8 2 3 3 8 1 2 6 1 . 2 3 3 9 1 2 7 r . 2 3 4 0 1 2 8 , 9 9 5 2 3 ^ 1 1 2 9 , 9 6 2 3 4 2 .. Q I L . 9 6 2 3 4 3 1 3 1 , 9 6 2 3 4 4 1 3 2 . 9 6 2 3 4 5 1 3 3 . 9 4 2 3 4 6 1 3 4 . 9 3 2 3 4 7 1 3 5 . 9 2 2 1 4 . 8 1.3.6 . . , . 8 8 _ . 2 3 4 9 1 4 1 ~ i . 2 3 5 0 1 4 6 - l . 3 i s o o o o o c o c o o o o o o o o o 0 0 t 7 2 _ _ _ tTbTU \u00E2\u0080\u00A2 I 9 0 1 8 6 E 2 \u00E2\u0080\u00A2 X \u00E2\u0080\u009E.. \u00E2\u0080\u009E.___\u00C2\u00A3 i O l i 6 \u00C2\u00A3 2 9 0 1 9 6 E 2 r \u00C2\u00BB t ? o i S 6 E Z u z s o o o o c o c o o o o o o o o o t76_c_ r / S _ \u00C2\u00A3 f c E _ \u00E2\u0080\u00A2 I 9 8 2 6 \u00C2\u00A3 Z u S 9 T 6 E Z ' ~ ~ ~ ~ ~ ~ \u00E2\u0080\u00A2 ( r? e 0 6 E 2 1 \u00C2\u00B0 * 2 9 6 8 E 2 v i s o o o o o o o c o o o o o o o o o B B E 2 -\u00E2\u0080\u00A2 I _ _ 2 i B E 2 *l 1 9 9 B E 2 \u00E2\u0080\u00A2 'I 0 9 S 8 E 2 \" ~ ' \" - ~ \" * l 6 S i > 8 \u00C2\u00A3 2 z S i _ E 6 E 2 \u00E2\u0080\u00A2 T P \u00C2\u00A3 Z 8 E Z \u00E2\u0080\u00A2 I \u00C2\u00A3 \u00C2\u00A3 T B E 2 \" I ? \u00C2\u00A3 0 8 E 2 2 \" ' - 0 \u00C2\u00A3 6 i \u00C2\u00A3 2 _ S O d O O O O O O O O O O O O O 0 0 0 0 0 8 I E 2 * I - 2 2 2 i i \u00C2\u00A3 2 s i 8 l 9 i \u00C2\u00A3 2 c - 9 8 1 S i \u00C2\u00A3 2 S 8 1 l 7 _ E 2 1731 E I E 2 \" \u00E2\u0080\u0094 ~ \u00C2\u00A3 _ T 2 I E 2 S 9 _ \" - 2 8 1 U E 2 S 9 i ' - I 8 1 0 _ E 2 oet b 9 \u00C2\u00A3 2 2 B \" > 6 i l B 9 E 2 s > an _ 9 \u00C2\u00A3 Z 5 S O f 7 > i i l 9 9 \u00C2\u00A3 _ & _ 0 t ? e - 9 i l S 9 E 2 8 t 7 2 _ \u00C2\u00B0 - a_i I / 9 E 2 8 1 7 _ S > & _ \u00C2\u00BB E 9 E Z 2 i l 2 9 E 2 6 t > l 2 - U l T 9 E 2 \u00C2\u00A3 > _ \u00C2\u00A3 _ \u00E2\u0080\u00A2 - O i l 0 9 E 2 8 1 7 0 Z > 6 9 1 6 S E 2 6 _ T X \" i 9 1 8 S E 2 fail 1 ' \"9\"9\" I i & t e S T 2 \" - S 9 l 9 S E 2 1 7 0 9 0 \" - 2 9 1 S S E 2 \" T 9 T ~ t > _ \u00C2\u00A3 2 9 i \u00C2\u00A3 l \" - 0 9 1 E S E 2 * f - i S l 2 S E Z I b E C 9 9 o o o o o c u o o o u o o c o \" o o o o o o d s V 2 \"i \u00C2\u00A3 \u00C2\u00A3 2 617*72 . \u00E2\u0080\u00A2 I J7_9I 8t7t?2 t / ^ ' b & I I W 2 \" 19 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9J7572 ' \ 2 \u00C2\u00A3 2 _ S r / ! 7 2 ' 1 \u00C2\u00A3 8 1 \" l M ? t ; 2 9 9 O O O O O O O O O O O O O O O O O O O O O \u00C2\u00A31717 2 T U 2 2 l 7 t / 2 T _ _ n TTFTTlT 5 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 O O O O O O G Ol7t7 2 M 0 \u00C2\u00A3 2 6 \u00C2\u00A3 t 7 2 \u00E2\u0080\u00A2 \u00E2\u0080\u0094 _ _ _ * ' t E T T \" \" \u00C2\u00A5 \u00C2\u00A3 \u00C2\u00BB ? 2 t7 9 O O O O O O O O O O O O O O O 0 0 0 0 0 i \u00C2\u00A3 t ? 2 'I 6 2 2 9 Z t 7 2 \" T ~ F 9 r - b T t T 2 ~ \u00C2\u00A3 9 O O O O O O O O O O O O O O O 0 0 0 0 0 t 7 \u00C2\u00A3 r / 2 M 8 2 2 \u00C2\u00A3\u00C2\u00A3t? 2 : X \u00C2\u00A3 9 1 - 2 I \u00C2\u00A5 2 2 9 O O O O O O O O O O O O O O O O O O O O O T \u00C2\u00A3 t 7 2 ' I LZZ 0 \u00C2\u00A3 r ; 2 \" I S \u00C2\u00A3 J \" & 2 T f 2 -S 2 T - - fr\u00C2\u00A3I W 2 t ? 2 T 9 O O O O O O O O O O O O O O O 0 0 0 0 \u00C2\u00A3 2 \u00C2\u00BB 2 ~ ~ \u00E2\u0080\u00A2 H 9 2 2 \u00E2\u0080\u00A2 9 2 1 7 2 * l \u00C2\u00A3 2 1 S 2 r / 2 S 2 f - 2 2 1 i ? 2 ) ? 2 o T ~ r n r o T w u o o o o o u c u \u00E2\u0080\u0094 i r o w o zzwr \u00C2\u00B0 t S 2 2 2 2 t 7 2 ' \ S O I T 2 t ? 2 \" ~ ~ \u00E2\u0080\u0094 ~ ! 5 7 T ' ' = - \u00C2\u00A5 i n 0 2 1 / 2 6 5 O O O O O O O O O O O O O O O 0 0 0 0 0 6 T t 7 2 M t 7 2 2 8 l t ? 2 __ , - n ~ r \u00C2\u00A7 Z T F S \" 5 2 1 * \u00C2\u00BB 2 8 9117 2 8 5 O O O O O O O O O O O O O O O 0 0 0 0 _ _ s f l ? 2 - r j - ^ - \u00E2\u0080\u0094 j j ^ g \" * I 8 5 \u00C2\u00A3 I t / 2 S 2 T ' > I S 2 T 1 7 2 H T 2 \" S 2 T > . 0 \u00C2\u00A3 . O t t ? 2 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 bOtiZ - T i s r - ~ 8 0 ) 7 2 \u00E2\u0080\u00A2 | 9 \u00C2\u00A3 I i 0 t ? 2 - - - - - I t ? \u00C2\u00A3 I 90-172 a i s \" o o o o o o o o o o o d o o o o o 50172 * J 9 5 2 - 17 0*7 2 ' 1 S 2 t \u00C2\u00A3 0 t ? 2 ~ * \ 172 X \" 2 0 1 ? 2 1 % 2 2 1 T 0 U 2 LI 0 0 o o o o o o o o o o'o 0 0 o o o o o o 0 0 b 2 * 5 2 ( 7 2 6 6 t ? 2 \" \u00E2\u0080\u00A2 \" t S 9 1 8 6 t 7 2 b 9 * 0 8 1 1 6 1 7 2 9 6 0 ( 7 * iiL'i 9 6 1 7 2 - 2 9 2 ' O i l a 6 1 ? 2 8 1 9 1 * \u00C2\u00A3 9 1 (76*72 J 7 i 0 l * 0 9 1 \u00C2\u00A3 6 t 7 2 \" 9 1 O O O O O O O O O O C o O O O 0 0 0 0 0 2 6 t ? 2 * I l t ? 2 T 6 1 7 2 \" I O B I 0 6 1 7 2 6 8 1 7 2 9 6 0 1 7 * O i l 8 8 1 / 2 2 9 2 * S 9 l I 8 1 ; 2 - \" \" \" 8 1 9 1 * 0 9 1 9 8 ( 7 2 5/_ O O O O O O O O O O O O O O O 0 0 0 0 0 \u00C2\u00A3 8 1 7 2 M 0172 (78172 - ' \u00E2\u0080\u00A2 \u00E2\u0080\u0094 'Mr\u00E2\u0080\u0094T* ; \" - r \" t b i l Q l 7 Z h 9 * O i l 2 8 i ; 2 9 6 0 ( 7 * \u00C2\u00A3 9 l 1 8 1 7 2 \" - _ 2 9 2 * 0 9 1 \" 0 8 l ? 2 fr?I O O O O O O O O O O O O O O O 0 0 0 0 6 1 1 / 2 \" I 6 \u00C2\u00A3 2 8 I 1 ? 2 \" r u n I I 1 7 2 \" 9 1 1 7 2 9 6 0 1 7 * 0 9 1 S I 1 7 2 ~ ~TT O O O O O W O O 0 0 u 0 0 0 0 0 0 ~ D 0 0 0 l ? I t ? 2 \" 1 8 \u00C2\u00A3 2 \u00C2\u00A3 1 1 7 2 * 1 S 9 I 2 I i ; 2 t ? 9 ' 0 9 1 1 1 ( 7 2 2 1 O O O O O O O O O O O O O O O 0 0 0 0 0 0 i t 7 2 ' I i \u00C2\u00A3 2 b 9 ( ? 2 _ . * 1 (781 a 7 ? (72 f ? 9 * bLX 1 9 1 7 2 9 6 Q t 7 * fell 9 9 1 7 2 U O O O O O O O O O O O O O O O 0 0 0 0 0 b 9 t 7 2 ' 1 9 \u00C2\u00A3 2 (7 9 * 7 2 \u00C2\u00B0 1 6 i l \u00C2\u00A3 9 1 7 2 t ? 9 * ( 7 i l 2 9 1 7 2 9 6 0 ( 7 * 6 9 1 1 9 1 7 2 0 1 O O O O O O O O O O O O O O O 0 0 0 0 0 9 1 7 2 \" I b \u00C2\u00A3 2 6bt7<-' * 1 (7I-T 8 S 1 7 2 ( 7 9 * 6 9 1 I S 1 7 2 9 6 0 ( 7 ' t ? 9 1 9 S 1 7 2 6 9 O O O O O O O O O O O O O O O O O O O O O \u00C2\u00A3 \u00C2\u00A3 1 ? 2 'I I 7 \u00C2\u00A3 2 17 S 17 2 \" 1 6 9 1 \u00C2\u00A3 b t 7 2 ( 7 9 * (791 2 S 1 7 2 9 6 0 ( 7 * 6 S 1 1 S 1 7 2 \u00C2\u00A3 9 L 2 5 0 1 1 6 1 . 5 3 2 5 0 2 1 6 6 1 . 2 5 0 3 2 4 3 r . . 2 5 0 4 _ _ .. 0 0 . 0 . 0 - 0 . M . M M . Q M M . Q O A . . . 7 8 2 5 0 5 1 6 1 , 2 8 0 9 2 5 0 6 1 6 6 , 5 3 . . . 2 5 0 7 171 i . 2 5 0 8 2 4 4 1 . 2 5 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 9 2 5 } 0 1 6 1 , 1 4 8 9 2 5 1 1 1 6 6 , 2 8 0 9 2 5 1 2 2 5 1 3 1 7 1 , 5 3 1 7 6 1 . 2 5 1 \u00C2\u00AB 2 4 5 1 , 2 5 1 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 2 5 l 6 1 6 1 . 0 7 9 2 5 1 7 2 5 1 8 1 6 6 , 1 4 8 9 1 7 1 , 2 8 0 9 2 5 1 9 1 7 6 . 5 3 2 5 2 0 1 8 1 1 , 2 5 2 1 2 4 6 f . 2 5 2 2 0 0 0 0 0 o o o o o o o o o o o o o o o 8 1 2 5 2 3 1 6 1 . 0 4 1 8 2 5 2 4 1 6 6 , 0 7 9 2 5 2 5 1 7 1 . 1 4 8 9 2 5 2 6 2 5 2 7 1 7 6 . 2 8 0 9 1 8 1 , 5 3 2 5 . 2 8 . _ .1 .8 6 J _ . . _ .. 2 5 2 9 2 4 7 1 , 2 5 3 0 0 0 0 0 O O O O O O O O O O O O O O O 8 2 2 5 3 1 1 6 2 . 5 3 2 5 3 2 1 6 7 1 . 2 5 3 3 2 4 8 1 . 2 5 3 \u00C2\u00AB 0 0 0 0 0 O O O O O O O O O O O O O O O 8 3 2 5 3 5 1 6 2 , 2 8 0 9 2 5 3 6 1 6 7 . 5 3 2 5 3 7 1 7 2 1 . _ . 2 5 3 8 2 5 3 9 2 5 A O 2 4 9 1 . 0 0 0 0 0 O O O O O O O O O O O O O O O 8 4 1 6 . 2 . 1 . 4 8 9 2 5 4 1 1 6 7 . 2 8 0 9 2 5 4 2 1 7 2 . 5 3 7 5 4 3 1 7 7 r . 2 5 4 4 2 5 0 f . 2 5 4 5 0 0 0 0 0 0 O O O O O O O O O O O O O O O 8 5 2 5 4 6 1 6 2 . 0 7 9 2 5 4 7 1 6 7 . 1 4 8 9 2 5 4 8 1 7 2 . 2 8 0 9 1 7 7 . 5 3 2 5 5 0 1 3 2 1 . 1 6 5 2 5 5 1 2 5 1 1 . d f l n n n n n n o o o o o o 0 0 0 0 0 0 0 8 6 C 2 5 5 3 1 6 2 , 0 4 1 8 2 5 5 4 1 6 7 . 0 7 9 T \u00C2\u00AB : ; 5 5 17? . 1 4 8 9 t _ . 1 11 2 5 5 6 1 7 7 , 2 8 0 9 2 5 5 7 1 8 ? , 5 3 ? 5 5 R 1 8 7 1 ' . / _ _1 J A J 2 5 5 9 , JL J O- *-\u00E2\u0080\u0094 \u00E2\u0080\u0094 2 5 2 1 . 2 5 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 7 \u00E2\u0080\u00A2 7> ^ A 1 0 , 0 . 0 r. i >) 1 2 5 6 2 \u00E2\u0080\u0094 , 0 . 0 . 0 2 5 6 3 ,0 ,o . 0 7 5 6 4 . 0 . o_ _ . . . . .D . . ..fZ..-t..&2..zK 2 5 6 5 , 0 . 0 . 0 2 5 6 6 2 1 0 7 3 1 . 5 1 0 0 0 0 . 2 0 0 0 0 0 0 . 7 5 6 7 . 0 T n _...,o f >'yJ 1 2 5 6 8 \u00E2\u0080\u0094 \u00C2\u00A7\u00E2\u0080\u00A2 \" \u00E2\u0080\u0094 ,0 . 0 1 8 1 4 1 4 1 4 . 2 5 6 9 1 9 0 8 9 9 6 , 3 1 2 4 8 1 3 . 4 1 0 8 6 0 0 . 2 5 7 0 4 8 1 L 8 l , 3 8 0 5 2 4 ? . 6 J X . 6 3 6 8 . . . . 3 . _ . 2 6 3 1 , 0 8 4 . 0 3 4 . 1 0 8 2 6 3 2 . , 1 4 4 . 0 6 5 . 0 7 3 2 6 3 3 . 0 7 2 . 1 6 9 . 1 8 1 2 6 3 4 . 0 8 2 . 0 9 5 , 0 6 4 2 6 3 5 . 1 3 5 . 1 5 . 0 7 9 2 6 3 6 . 0 9 3 , 0 5 8 , 1 0 6 _ 2 6 3 7 , 1 2 2 . 0 7 8 , 0 8 6 2 6 3 8 , 0 6 3 , 0 8 6 . 0 8 9 2 6 3 9 . 0 6 9 . 0 7 1 . 0 3 8 2 6 4 0 . 0 4 , 0 3 8 . 0 3 2 6 4 1 , 0 3 1 . 0 . 0 2 6 4 2 . 0 . , .0. _ o 2 6 4 3 , 0 , o , o 2 6 4 4 , 0 , 0 . 0 2 6 4 5 . 0 . 0 . 0 2 6 4 6 , 0 . 0 , 0 2 6 4 7 . 0 , 0 . 0 2 6 4 8 . 0 . 0 . ...0. 2 6 4 9 . 0 . 0 . 0 2 6 5 0 , 0 . 0 , 0 1 6 7 2 6 5 1 \u00C2\u00ABJ> J 3 J l 2 6 5 2 , 0 . 0 f 0 2 6 5 3 , 0 . 0 , 0 2 6 5 \u00C2\u00AB +Q. *...!) 2 6 5 5 , 0 , 0 , 0 2 6 5 6 , 0 . 0 , 0 . 2 . 6 5 J L ..Q 1_Q j Q . 2 6 5 8 8 0 , 0 . 0 2 6 5 9 , 0 , 0 , 0 2 6 6 f! , 0 - f...O - \u00E2\u0080\u009E-0. 2 6 6 1 , 0 , 0 , 0 2 6 6 2 , 0 , 0 , 0 - 2 6 6 3 \u00C2\u00BBJ3 ^ 0 ^ 0 . 2 6 6 4 \u00E2\u0080\u009E 0 , 0 C h a p t e r 5 A COMPARISON OF STOCHASTIC DYNAMIC PROGRAMMING AND STOCHASTIC LINEAR PROGRAMMING WITH SIMPLE RECOURSE MODELS AS DECISION TOOLS 5 . 1 I n t r o d u c t i o n T h e a s s e t a n d l i a b i l i t y m a n a g e m e n t p r o b l e m i s a c o n t i n u o u s d e c i s i o n p r o b l e m i n w h i c h a c t i o n s c a n b e t a k e n a t a n y p o i n t i n t i m e . A t a n y d e c i s i o n p o i n t , t h e b a n k h a s a p o r t f o l i o o f a s s e t s a n d l i a b i l i t i e s o n h a n d . B a s e d o n f o r e c a s t s o f f u t u r e i n t e r e s t r a t e s a n d c a s h f l o w s , t h e b a n k m u s t d e c i d e w h i c h a s s e t s t o h o l d i n i t s p o r t f o l i o , w h i c h a s s e t s t o s e l l f r o m i t s p o r t f o l i o a n d w h i c h a s s e t s t o b u y f o r i t s p o r t f o l i o . T h e s e d e c i s i o n s a r e m a d e s u b j e c t t o s u c h c o n s t r a i n t s a s c a s h f l o w s a n d l i q u i d i t y . T h i s d e c i s i o n - m a k i n g p r o c e s s i s p e r f o r m e d r e p e a t e d l y . T h e p r o c e s s i s d y n a m i c i n t h a t t h e o p t i m a l s o l u t i o n t o t h e i m m e d i a t e p r o b l e m i s d e p e n d e n t o n t h e a c t i o n s t h a t w i l l b e t a k e n a t e a c h f u t u r e d e c i s i o n p o i n t ( a n d d e p e n d e n t o n t h e r e a l i z a t i o n s o f t h e r a n d o m v a r i a b l e s ) . T h e b e s t a p p r o a c h t o s o l v i n g t h e a s s e t a n d l i a b i l i t y m a n a g e m e n t p r o b l e m w o u l d p r o b a b l y b e a c o n t i n u o u s t i m e s t o c h a s t i c d y n a m i c o p t i m i z a t i o n f o r m u l a t i o n . D e c i s i o n s c o u l d b e m a d e a t e a c h p o i n t i n t i m e c o n d i t i o n a l o n t h e e n t i r e h i s t o r y o f e v e n t s u p t o a n d i n c l u d i n g t h e p r e s e n t . S u c h a t e c h n i q u e w o u l d b e a b l e t o u t i l i z e a g r e a t e r a m o u n t o f i n f o r m a t i o n t h a n c a n b e s u m m a r i z e d i n a d i s c r e t e - t i m e m o d e l . H o w e v e r , c o m p u t a t i o n a l t r a c t a b i l i t y i n h i b i t s t h e d e v e l o p m e n t o f a n o p e r a t i o n a l m o d e l o f t h e c o n t i n u o u s t y p e . On t h e o t h e r h a n d , s o m e o p e r a -t i o n a l m o d e l s h a v e b e e n d e v e l o p e d w h e r e t h e t i m e p a r a m e t e r a n d t h e p r o b a b i l i t y d i s t r i b u t i o n h a v e b e e n d i s c r e t i z e d t o a p p r o x i m a t e t h e m o r e g e n e r a l c a s e . T h e B r a d l e y a n d C r a n e m o d e l [ 5 , 6 , 7 ] d e s c r i b e d i n C h a o t e r 2 i s o n e e x a m p l e . 1 6 8 1 6 9 T h e B - C m o d e l c a p t u r e s m a n y o f t h e e s s e n t i a l f e a t u r e s o f t h e a s s e t a n d l i a b i l i t y m a n a g e m e n t p r o b l e m a n d a c h i e v e s a c e r t a i n a m o u n t o f c o m p u t a t i o n a l t r a c t a b i 1 i t y . H o w e v e r , a s w a s s h o w n i n C h a p t e r 2 , t h e B - C m o d e l e n c o u n t e r s s i g n i f i c a n t c o m p u t a t i o n a l d i f f i c u l t y a s t h e s i z e o f t h e m o d e l i s e n l a r g e d . A n o t h e r s h o r t c o m i n g o f t h e B - C m o d e l i s t h e i n h e r e n t p r o b l e m i n t h e f o r m u l a t i o n . T o a t t a i n a n o p e r a t i o n a l m o d e l , B - C d i s c r e t i z e a c o n t i n u o u s p r o b -a b i l i t y d i s t r i b u t i o n . S o r a t h e r t h a n o b t a i n i n g o p t i m a l i t y o n t h e e n t i r e s u p p o r t o f t h e r a n d o m v a r i a b l e , o n e g e n e r a t e s o p t i m a l i t y o n l y f o r a f i n i t e s e t o f p o i n t s i n t h e s u p p o r t . T h a t i s , t h e d e c i s i o n v a r i a b l e m u s t s a t i s f y t h e c o n s t r a i n t s e t c o n s t r u c t e d f r o m t h e r e p r e s e n t a t i v e p o i n t s . F o r r e a l i -z a t i o n s d i f f e r e n t f r o m t h e r e p r e s e n t a t i v e p o i n t s , t h e B - C m o d e l d o e s n o t c o n s i d e r t h e p r o f i t f o r e g o n e o r t h e c o s t i n c u r r e d i n a d j u s t i n g t h e p o r t -f o l i o . A l s o s i n c e t h e f i r s t p e r i o d f e a s i b i l i t y i s a c h i e v e d b y t h e s i m u l -t a n e o u s s a t i s f a c t i o n o f t h e c o n s t r a i n t s g e n e r a t e d f r o m t h e d i s c r e t e n u m b e r o f s c e n a r i o s , t h e f o r m u l a t i o n i s t y p i c a l o f t h e f a t f o r m u l a t i o n ^ f o r t h e d i s c r e t e n u m b e r o f r e a l i z a t i o n s o f t h e r a n d o m v a r i a b l e . F o r e x a m p l e , c o n s i d e r a t w o - p e r i o d p r o b l e m i n w h i c h a n i n v e s t o r f a c e s t h e f o l l o w i n g s i t u a t i o n : 1 ) h e h a s $ 1 0 0 i n p e r i o d o n e t o i n v e s t i n e i t h e r a s s e t X n w i t h a r e t u r n r n = . 1 a n d m a t u r i n g a f t e r o n e p e r i o d o r X i 2 w i t h a r e t u r n r i 2 = . 2 p e r p e r i o d a n d m a t u r i n g a f t e r t w o p e r i o d s ; 2 ) i n . t h e s e c o n d p e r i o d t h e i n v e s t o r e i t h e r r e c e i v e s a n a d d i t i o n a l $ 5 0 t o i n v e s t w i t h p r o b a b i l i t y . 9 o r h e m u s t r e t u r n $ 5 0 w i t h p r o b a b i l i t y . 1 ; 3 ) i n . t h e s e c o n d p e r i o d , t h e i n v e s t o r a l s o h a s t h e o p p o r t u n i t y t o i n v e s t i n a o n e - p e r i o d a s s e t x 2 i w i t h a r e t u r n r 2 i = . 1 o r t o s e l l a p a r t o f h i s h o l d i n g s ^ S e e A p p e n d i x 1 i n C h a p t e r 3 -1 7 0 i n x i 2 a t a 2 0 % d i s c o u n t ; a n d 4 ) t h e i n v e s t o r s t i p u l a t e s t h a t h i s r e a l i z e d c a p i t a l l o s s e s c a n n o t e x c e e d 10% o f t h e o u t s t a n d i n g e x o g e n o u s f u n d s \" i n a n y p e r i o d . T h e l i n e a r p r o g r a m m i n g f o r m u l a t i o n o f t h e a b o v e p r o b l e m i n t h e B - C - f r a m e w o r k i s g i v e n i n T a b l e 5 . 1 . T h e o p t i m a l s o l u t i o n t o t h i s p r o b l e m i s baUai) = 8 0 . 0 0 h i 2 U 2 i ) = 8 8 . 8 9 s j 2 ( \u00C2\u00A3 2 i ) = 1 1 . 1 1 S 1 2 U 2 . O = 0 a n d t h e o p t i m a l v a l u e i s $ 4 2 . 8 7 . T h e f i n a l c o n s t r a i n t i n t h e m o d e l , ( w h i c h i s a b o u n d o n r e a l i z e d c a p i t a l l o s s e s ) , i s b i n d i n g . I f t h e r i g h t h a n d s i d e w a s i n c r e a s e d t o 7 . 5 ( m a x i m a l l o s s o f 1 5 % ) , t h e n t h e o p t i m a l s o l u t i o n w o u l d b e t o p u r c h a s e $ 1 0 0 o f a s s e t x i 2 ( a n d s e l l $ 3 7 . 5 0 o f x i 2 a t t h e e n d o f p e r i o d 1 , i f t h e i n v e s t o r h a s t o r e p a y $ 5 0 , ) . T h e o p t i m a l v a l u e t o t h i s p r o b l e m i s $ 4 4 . 1 1 . T h u s , b e c a u s e o f t h e f a t f o r m u l a t i o n u s e d b y t h e B - C m o d e l , w i t h r e s p e c t t o t h e r e p r e s e n t a t i v e p o i n t s i n t h e s u p p o r t o f t h e r a n d o m v a r i a b l e , d e c i s i o n f l e x i b i l i t y i s r e d u c e d . T h i s i s a r e s u l t o f c o n s i d e r i n g c o n s t r a i n t s c o r r e -s p o n d i n g t o r e a l i z a t i o n s w i t h a s m a l l p r o b a b i l i t y o f o c c u r r e n c e . F o r e x a m p l e , i t i s c l e a r f r o m t h e a b o v e p r o b l e m t h a t i n t h e B - C f o r m u l a t i o n a n u m b e r o f c o n s t r a i n t s ( s u c h a s t h e c a p i t a l l o s s c o n s t r a i n t s ) a r e n o t w e i g h t e d w i t h r e s p e c t t o t h e i r p r o f i t a b i l i t y . = 1 1 . 1 1 = 8 8 . 8 9 h i U 2 2 ) = 0 h f U i 2 ) = 6 3 . 8 9 S 1 2 U 2 2 ) = 1 1 . 1 1 s\z{lzz) = 2 5 . 0 0 T a b l e 5 . 1 b i U i ) b?Ui) b 2 U 2 i ) h ! 2 ( \u00C2\u00A3 2 i ) s l 2 ( A 2 i ) s ? 2 U 2 i ) b 2 ( \u00C2\u00A3 2 2 ) hJ 2 U 2 2 ) s\2(l2Z) S i 2 ( \u00C2\u00A3 2 2 ) O b j e c t i v e F u n c t i o n . 2 ( . 9 ) ( . l ) ( . 9 ) ( . 2 ) ( . 9 ) ( - . 2 ) ( : ! ) ( . ! ) ( . ! ) ( . 2 ) ( \u00E2\u0080\u00A2 ! ) ( - . 2 ) C a s h F l o w i l l A 2 i \u00C2\u00A3 2 2 - . 2 \u00E2\u0080\u00A2 1 . 1 - 1 . 0 - . 2 \u00E2\u0080\u00A2 1 . 1 = 1 0 0 = 5 0 \u00E2\u0080\u00A21 = - 5 0 I n v e n t o r y B a l a n c e lZi 1 111 1 - 1 - 1 - 1 - 1 = 0 = 0 = 0 = 0 C a p i t a l L o s s i2\ In < 1 5 < 5 1 7 2 On t h e o t h e r h a n d , c o n s i d e r t h e s i m p l e r e c o u r s e f o r m u l a t i o n d e s c r i b e d i n t h i s d i s s e r t a t i o n . . I n t h e r e c o u r s e f o r m u l a t i o n , t h e r i g h t h a n d s i d e s a r e n o t b i n d i n g o n t h e d e c i s i o n v a r i a b l e s . R e c o u r s e i s a l l o w e d a n d t h e p e n a l t i e s f o r t h e r e c o u r s e d e c i s i o n c o m p e n s a t e f o r d e c i s i o n i n f e a s i -b i l i t y . T h u s t h e r e i s m o r e f i r s t p e r i o d d e c i s i o n f l e x i b i l i t y i n t h e r e c o u r s e m o d e l t h a n i n a n y f a t f o r m u l a t i o n . T h e u s e o f r e p r e s e n t a t i v e s a m p l e p o i n t s a n d t h e f a c t t h a t t h e f i r s t p e r i o d d e c i s i o n i s c o n s t r a i n e d b y a l l f u t u r e e c o n o m i c e v e n t s m a y c a s t d o u b t o n t h e v a l u e o f t h e B - C f o r m u l a t i o n . I n t h i s c h a p t e r , i n o r d e r t o u p h o l d t h e a b o v e c o n t e n t i o n s , a s i m u l a t i o n w i l l b e p e r f o r m e d o n a s m a l l a s s e t m a n a o e m e n t p r o b l e m . T h e r e f o r e , c o u p l e d w i t h t h e . . c o m p u t a t i o n a l s u p e r i o r i t y o f t h e r e c o u r s e m o d e l , t h e A L M m o d e l p r e s e n t e d i n t h i s d i s s e r t a t i o n s h o u l d t h e n b e c o n s i d e r e d b e t t e r o p e r a t i o n a l l y t h a n t h e B - C m o d e l . T h e s i m u l a t i o n u t i l i z e s t w o a s s e t . a n d l i a b i l i t y m a n a g e -m e n t m o d e l s . I n o n e m o d e l , a s t o c h a s t i c d y n a m i c p r o g r a m m i n g f o r m u l a -t i o n w i l l b e u s e d - t h e s a m e a p p r o a c h B - C u s e . I n t h e s e c o n d m o d e l , a s t o c h a s t i c l i n e a r p r o g r a m w i t h s i m p l e r e c o u r s e f o r m u l a t i o n w i l l b e u s e d . T h e s i m u l a t i o n c a n b e f l o w c h a r t e d a s i n E x h i b i t 5 . 1 . T h i s p r o c e s s i s r e p e a t e d f o r b o t h f o r m u l a t i o n s . A f t e r e a c h r e c o n c i l i a t i o n t h e p r o f i t ( l o s s ) f o r t h e p e r i o d i s g e n e r a t e d a n d s t o r e d . T o e v a l u a t e t h e s i m u l a t i o n a s t a t i s t i c a l c o m p a r i s o n w i l l b e m a d e o f t h e p r o f i t s g e n e r a t e d b y t h e t w o a p p r o a c h e s . E x h i b i t 5 . 1 S T A R T S e t T = 1 S e t 1 = 1 A I n i t i a l i z e M o d e l G e n e r a t e F i r s t P e r i o d S o l u t i o n O b t a i n ( r a n d o m l y ) a n E c o n o m i c S c e n a r i o R e c o n c i l e G e n e r a t e d P o r t f o l i o a n d R a n d o m E v e n t s , G e n e r a t i n g a New I n i t i a l P o s i t i o n C a l c u l a t e P r o f i t s f o r P e r i o d 1 = 1 + 1 No T = T + 1 R e t r i e v e I n i t i a l P o r t f o l i o 1 7 4 5 . 2 S c e n a r i o f o r t h e S i m u l a t i o n T h e q u e s t i o n a d d r e s s e d i n t h i s s i m u l a t i o n i s W h i c h f o r m u l a t i o n t e c h n i q u e , S L P R o r S D P , c a n b e s a i d t o b e b e t t e r i n s o m e o p e r a t i o n a l s e n s e ? T h i s q u e s t i o n m u s t b e a n s w e r e d f r o m t w o p o i n t s o f v i e w . F i r s t , w h i c h i s b e t t e r f r o m a c o m p u t a t i o n a l s t a n d p o i n t ? A n d s e c o n d , w h i c h t e c h n i q u e i s b e t t e r f r o m a n o r m a t i v e s t a n d p o i n t ? When o n e c o n s i d e r s t h e d i f f e r e n c e i n t h e s i z e o f t h e m o d e l s f o r s i m i l a r p r o b l e m s , ^ t h e a n s w e r t o t h e f i r s t q u e s t i o n i s s e l f -e v i d e n t . T h e a n s w e r t o t h e s e c o n d q u e s t i o n i s n o t a s c l e a r . A l t h o u g h t h e o r e t i c a l l y S D P s h o u l d p r o v i d e a b e t t e r n o r m a t i v e s o l u t i o n , t h e r e s t r i c -t i o n s i n h e r e n t i n t h e f o r m u l a t i o n - f e w r e p r e s e n t a t i v e s a m p l e p o i n t s a n d t h e r e s t r i c t i v e c o n s t r a i n t s e t - m a y r e d u c e t h e e f f e c t i v e n e s s o f S D P . I t i s t h i s q u e s t i o n o f n o r m a t i v e e f f e c t i v e n e s s t h a t t h e s i m u l a t i o n a d d r e s s e s . T o a n s w e r t h e a b o v e q u e s t i o n , a n a s s e t a n d l i a b i l i t y m a n a g e m e n t s c e n a r i o i s c r e a t e d i n w h i c h t h e p r o b l e m w i l l b e s o l v e d b y t h e t w o t e c h n i q u e s . E s s e n t i a l l y t h e p r o b l e m i s t o d e t e r m i n e t h e o p t i m a l p o r t f o l i o o f a s s e t s a n d l i a b i l i t i e s , g i v e n r a n d o m f u t u r e r a t e s o f r e t u r n , c o s t s a n d c a s h f l o w s . T o m a i n t a i n c o m p u t a t i o n a l f e a s i b i l i t y f o r t h e S D P a p p r o a c h , o n l y 3 p l a n n i n g p e r i o d s , 3 a s s e t s a n d 1 l i a b i l i t y a r e c o n s i d e r e d . T h e a s s e t s c o n s i d e r e d a r e : 1 ) a o n e p e r i o d t r e a s u r y b i l l , 2 ) a t e r m d e p o s i t m a t u r i n g b e y o n d t h e h o r i z o n o f t h e m o d e l , a n d 3 ) a l o n g - t e r m m o r t g a g e . T h e l i a b i l i t y u s e d i s a d e m a n d d e p o s i t . T h e r e t u r n s a n d c o s t s o f t h e s e f i n a n c i a l i n s t r u m e n t s w e r e g e n e r a t e d f r o m 2 6 c o n s e c u t i v e o b s e r v a t i o n s u s i n g d a t a f r o m C e n t r a l M o r t g a g e a n d H o u s i n g C o r p o r a t i o n [ 1 0 ] . T o g e t a S e e c h a p t e r s 2 a n d 4 f o r c o m p a r i s o n s . 1 7 5 r e a s o n a b l e c o r r e l a t i o n o f i n t e r e s t r a t e s , t h e r e t u r n s a n d c o s t s w e r e m a d e a f u n c t i o n o f t h e p r i m e r a t e . T h e d i s t r i b u t i o n o f t h e p r i m e r a t e ( R ) i s r P r ( R = r ) . 0 6 : 6 / 2 6 . 0 6 5 3 / 2 6 . 0 6 7 5 1 / 2 6 . 0 7 5 2 / 2 6 . 0 7 7 5 1 / 2 6 . 0 8 2 / 2 6 . 0 8 5 4 / 2 6 . 0 9 2 / 2 6 . 0 9 5 2 / 2 6 . 1 1 2 / 2 6 . 1 1 5 1 / 2 6 D i s t r i b u t i o n s w e r e t h e n d e r i v e d f o r t h e d i f f e r e n c e b e t w e e n t h e p r i m e r a t e a n d t h e ' r a t e o f r e t u r n o f e a c h o f t h e f o u r f i n a n c i a l i n s t r u -m e n t s . T h e s e d i s t r i b u t i o n s a r e m P ( M _ m ) d P ( D < d ) t P ( T < t ) i P ( L _ \u00C2\u00A3 ) . 0 0 3 7 0 . 0 - . 0 1 0 4 0 . 0 - . 0 3 8 8 0 . 0 - . 0 2 7 5 0 . 0 . 0 0 8 8 0 . 2 - . 0 0 7 2 0 . 2 - . 0 3 0 6 0 . 2 - . 0 2 5 0 . 2 . 0 1 9 8 0 . 4 2 + . 0 0 0 8 0 . 4 4 - . 0 2 5 3 0 . 5 - . 0 2 2 5 0 . 3 1 . 0 2 3 5 0 . 6 2 + . 0 0 4 0 . 5 - . 0 2 2 5 0 . 7 7 - 0 . 2 0 . 9 2 . 0 2 9 7 0 . 8 1 + . 0 1 1 8 0 . 7 8 - . 0 1 7 4 0 . 8 1 - . 0 1 7 5 1 . 0 0 . 0 3 3 8 1 . 0 0 + . 0 1 9 5 1 . 0 0 - . 0 0 5 1 1 . 0 0 H e r e t h e r a n d o m v a r i a b l e s M , D , T a n d L a r e d e f i n e d t o b e t h e d i f f e r e n c e b e t w e e n t h e p r i m e r a t e a n d e a c h o f t h e f o l l o w i n g : t h e m o r t g a g e r a t e , t e r m d e p o s i t r a t e , t r e a s u r y b i l l r a t e a n d t h e l i a b i l i t y r a t e , r e s p e c t i v e l y . 1 7 6 A t t h e i n i t i a l p o i n t , t h e i n v e s t o r h a s $ 1 0 0 , 0 0 0 i n d e m a n d d e p o s i t s w h i c h i s e q u a l l y i n v e s t e d i n t h e t h r e e t y p e s o f a s s e t s . T h e d e m a n d d e p o s i t s w i l l b e a s s u m e d t o i n c r e a s e ( d e c r e a s e ) f r o m o n e p e r i o d t o t h e n e x t u n i -f o r m l y i n t h e i n t e r v a l [ - 2 0 , 0 0 0 , 2 0 , 0 0 0 ] . I f t h e d e m a n d d e p o s i t s d e c r e a s e s o t h a t a s s e t s h a v e t o b e l i q u i d a t e d , t h e n t h e F R B ' s p a r a m e t e r s f o r q u i c k l i q u i d a t i o n a r e u s e d . T h e d i s c o u n t s f o r t r e a s u r y b i l l s i s . 5 % , f o r t e r m d e p o s i t s 4 % , a n d f o r m o r t g a g e s 6 % . T h e c o n s t r a i n t s e t u s e d w i l l b e o f t h e B - C t y p e . T h e c o n s t r a i n t s o n t h e i n v e s t o r i n c l u d e : 1 ) c a s h f l o w s , 2 ) c a p i t a l l o s s e s , 3 ) c l a s s c o m - . , p o s i t i o n a n d 4 ) t e r m i n a l c o n d i t i o n s . T h e c a p i t a l l o s s c o n s t r a i n t s a s s u m e t h a t t h e i n v e s t o r d o e s n o t w a n t t o r e a l i z e n e t l o s s e s o f m o r e t h a n 3 % o f t h e o u t s t a n d i n g d e m a n d d e p o s i t s i n p e r i o d s 1 a n d 2 , a n d 4% i n p e r i o d ^ . T h e c l a s s c o m p o s i t i o n c o n s t r a i n t s r e s t r a i n t h e i n v e s t o r f r o m h a v i n g m o r e t h a n $ 5 0 , 0 0 0 i n t o t a l i n v e s t m e n t s i n a n y a s s e t i n p e r i o d s 1 a n d 2 , a n d $ 6 0 , 0 0 0 i n p e r i o d 3 . T h e t e r m i n a l c o n s t r a i n t s i n c l u d e a d i s c o u n t o n t h e a s s e t s i n t h e c u r r e n t p o r t f o l i o s o t h a t a l l f u n d s a r e n o t s i m p l y i n v e s t e d i n t h e h i g h e s t y i e l d i n g a s s e t s , a n d h e l d t o t h e h o r i z o n o f t h e m o d e l . T h e s e d i s c o u n t s a r e o n e - h a l f o f t h e n o r m a l d i s c o u n t s . T h e o b j e c t i v e o f t h e m o d e l i s t o m a x i m i z e t h e n e t e x p e c t e d r e t u r n s . 5 . 3 F o r m u l a t i o n o f t h e S t o c h a s t i c D y n a m i c P r o g r a m m i n g M o d e l T o f o r m u l a t e t h e p r o b l e m p o s e d i n S e c t i o n 5 . 2 a s a s t o c h a s t i c d y n a m i c p r o g r a m , i t i s n e c e s s a r y t o f i r s t e s t a b l i s h a n e c o n o m i c s c e n a r i o o v e r t h e t h r e e p e r i o d h o r i z o n . T h i s w i l l i n c l u d e o b t a i n i n g a r e p r e s e n t a -t i v e d i s t r i b u t i o n f o r t h e c a s h f l o w s a n d t h e r a t e o f r e t u r n o f e a c h o f t h e f o u r f i n a n c i a l i n s t r u m e n t s . A s a l r e a d y s t a t e d , t h e u s e o f s t o c h a s t i c 1 7 7 d y n a m i c p r o g r a m m i n g i m p l i e s c r u d e a p p r o x i m a t i o n s o f p r o b a b i l i t y d i s t r i b u -t i o n s , o t h e r w i s e t h e c o m p u t a t i o n s h e c o m e u n w i e l d l y . S o , f o r t h e p u r p o s e s o f t h e s i m u l a t i o n t h e n u m b e r o f p o s s i b l e r e a l i z a t i o n s o f t h e r a n d o m v a r i a b l e s d u r i n g e a c h t i m e p e r i o d w i l l b e l i m i t e d t o t w o . A s b e f o r e , t h e d e m a n d d e p o s i t s a r e c u r r e n t l y $ 1 0 0 , 0 0 0 . A t t h e e n d o f t h e p e r i o d t h e i n c r e m e n t a l d i f f e r e n c e w i l l l i e i n t h e i n t e r v a l [ - 2 0 , 0 0 0 , 2 0 , 0 0 0 ] . T h e t w o p o i n t r e p r e s e n t a t i v e d i s t r i b u t i o n u s e d i n t h e f o r m u l a t i o n w i l l b e $ 9 0 , 0 0 0 w i t h p r o b a b i l i t y . 5 a n d $ 1 1 0 , 0 0 0 w i t h p r o b a b i l i t y . 5 . U s i n g t h i s d i s t r i b u t i o n t h e m e a n o f t h e u n d e r l y i n g d i s -t r i b u t i o n i s m a i n t a i n e d a l t h o u g h t h e v a r i a n c e i s s m a l l e r ( . 1 . 3 3 x 1 0 5 v e r s u s 1 . 0 x 1 0 5 ) . H o w e v e r , t h e a p p r o x i m a t i o n i s r e a s o n a b l e a s i t d i v i d e s t h e d i s t r i b u t i o n s y m m e t r i c a l l y . F o r t h e t h i r d d e c i s i o n p o i n t t h e d i s t r i b u -t i o n w i l l b e c o n s t r u c t e d s i m i l a r l y . T h u s t h e c a s h f l o w s h a v e t h e d i s t r i b u t i o n 1 0 0 0 0 0 ( w . p . ' l ) 1 7 8 U s i n g t h e s a m e a p p r o a c h a s a b o v e , t h e f i r s t p e r i o d r a t e o f r e t u r n f o r a p a r t i c u l a r f i n a n c i a l i n s t r u m e n t ( a s s u m e m o r t g a g e r a t e ) , i s t a k e n t o b e t h e m e d i a n p r i m e r a t e ( R ) p l u s t h e m e d i a n o f t h e d i f f e r e n c e b e t w e e n t h e p r i m e r a t e a n d t h e r a t e o f r e t u r n o f t h e m o r t g a g e s C M ) . T h e t w o p o i n t e s t i m a t e i n t h e s e c o n d p e r i o d . i s R p l u s m , w h e r e PCM < m) = . 7 5 a n d R^ m i n u s m , w h e r e P ( M < m) = . 2 5 . T h e f o u r r a t e s o f r e t u r n i n t h e t h i r d p e r i o d a r e : 1 ) R p l u s m , w h e r e P(.M < m) = . 8 7 5 , 2 ) R p l u s m , w h e r e P ( M < m) = . 6 2 5 , 3 ) R p l u s m , w h e r e P ( M < m) = . 3 7 5 , a n d 4 ) R p l u s m , w h e r e P(.M < m) = . 1 2 5 . T h e a c t u a l d i s t r i b u t i o n s o f t h e r a t e s o f r e t u r n u s e d i n t h e s i m u l a t i o n a r e : 1 ) m o r t g a g e r a t e . 0 9 9 2 ( w . p . 1 ) 2 ) t e r m d e p o s i t r a t e . 0 8 2 7 ( w . p . 1 ) 1 7 9 3 ) t r e a s u r y b i l l r a t e . 0 5 4 1 ( w . p . 1 ) a n d 4 ) n o n c h e q u i n g r a t e . 0 5 7 7 ( w . p . 1 ) F o r p u r p o s e s o f t h e s i m u l a t i o n , 70.% o f t h e n o n c h e q u i n g r a t e w a s u s e d a s t h e d e m a n d d e p o s i t r a t e s i n c e t h e n o n c h e q u i n g r a t e d o m i n a t e s t h e t r e a s u r y b i l l r a t e . ( T h i s w o u l d h a v e p r e c l u d e d i n v e s t m e n t i n t r e a s u r y b i l l s a p r i o r i . ) T h i s a d h o c d e r i v a t i o n o f t h e d e m a n d d e p o s i t r a t e d o e s n o t i m p i n g e o n t h e u s e f u l n e s s o f t h e s i m u l a t i o n b e c a u s e t h e o b j e c t i v e i s t o d e m o n s t r a t e t h a t o n e s o l u t i o n t e c h n i q u e m a y b e o p e r a t i o n a l l y b e t t e r t h a n a n o t h e r . T h e d e c i s i o n v a r i a b l e s f o r t h e B - C m o d e l w i l l b e t h e s a m e a s d e -f i n e d i n C h a p t e r 2 . S i n c e t r e a s u r y b i l l s m a t u r e a f t e r o n e p e r i o d , e i g h t e e n v a r i a b l e s c o m p l e t e l y d e f i n e a l l p o t e n t i a l i n v e s t m e n t o p p o r t u n i t i e s . S i n c e t h e t e r m d e p o s i t s a n d m o r t g a g e s m a t u r e b e y o n d t h e h o r i z o n o f t h e m o d e l , 1 8 0 4 2 v a r i a b l e s a r e r e q u i r e d t o d e s c r i b e a l l i n v e s t m e n t o p p o r t u n i t i e s i n e a c h o f t h e s e c a t e g o r i e s . T h e v a r i a b l e s n e c e s s a r y t o d e f i n e t h e d e m a n d d e p o s i t s i n c l u d e : t h e i n i t i a l p o s i t i o n , t h e d e m a n d d e p o s i t f l o w s i n p e r i o d o n e , t h e t w o d e m a n d d e p o s i t f l o w s i n p e r i o d t w o , a n d t h e f o u r d e m a n d d e p o s i t f l o w s i n p e r i o d t h r e e . I n a l l , 1 1 0 v a r i a b l e s d e f i n e t h e i n v e s t m e n t o p p o r -t u n i t i e s i n t h e p r o b l e m . T h e r e a r e f o u r t y p e s o f c o n s t r a i n t s . C o n s t r a i n t s 1 t o 7 a r e t h e c a s h f l o w r e q u i r e m e n t s f o r e a c h p e r i o d u n d e r e a c h e c o n o m i c s c e n a r i o ; n a m e l y t h e u s e s o f f u n d s a r e e q u a l t o t h e s o u r c e s o f f u n d s . C o n s t r a i n t s 8 t o 1 4 r e q u i r e r e a l i z e d c a p i t a l l o s s e s t o b e l e s s t h a n 3 % o f t h e o u t s t a n d i n g d e m a n d d e p o s i t s i n p e r i o d o n e a n d t w o , a n d 4% i n p e r i o d t h r e e . C o n s t r a i n t s 1 5 t o 3 5 l i m i t t h e a m o u n t o f f u n d s i n v e s t e d i n e a c h o f t h e a s s e t s a s p r e -s c r i b e d i n t h e p r o b l e m . C o n s t r a i n t s 3 6 t o 8 9 ( i n v e n t o r y b a l a n c i n g ) c o n s i s t o f t h e i n i t i a l h o l d i n g s o f e a c h o f t h e f o u r f i n a n c i a l i n s t r u m e n t s a n d r e c o r d s t h e t r a n s a c t i o n s i n e a c h e c o n o m i c s c e n a r i o . T h e d e m a n d d e p o s i t f l o w c o n s t r a i n t f o r p e r i o d 1 p l a c e s a n u p p e r b o u n d o n t h e f u n d s p o t e n t i a l l y a v a i l a b l e f o r i n v e s t m e n t . A l s o t h e c a p i t a l l o s s a n d t h e c o m p o s i t i o n c o n s t r a i n t s a d d a n o t h e r 2 8 s l a c k v a r i a b l e s t o t h e f o r m u l a t i o n . T h e r e f o r e , t h e t o t a l s i z e o f t h e B - C f o r m u l a t i o n i s 8 9 c o n -s t r a i n t s w i t h 1 3 9 v a r i a b l e s . T h e o b j e c t i v e i s t o m a x i m i z e t h e e x p e c t e d v a l u e o f t h e n e t r e t u r n s f r o m t h e p o r t f o l i o o v e r t h e h o r i z o n o f t h e m o d e l . T h a t i s , t h e c o e f f i c i e n t o f e a c h v a r i a b l e i s t h e p r o d u c t o f t h e n e t r e t u r n a n d t h e p r o b a b i l i t y o f i t s o c c u r r e n c e . 181 5 . 4 F o r m u l a t i o h o f t h e S L P R M o d e l T h e S L P R m o d e l u s e s t h e s a m e i n f o r m a t i o n a s t h e B - C f o r m u l a t i o n . A l s o t h e c o n s t r a i n t s a r e o f t h e s a m e t y p e , b u t t h e t o t a l n u m b e r o f c o n -s t r a i n t s i s f e w e r t h a n i n t h e B - C m o d e l ; t h i s b e i n g t h e r e s u l t o f t h e m a n n e r i n w h i c h u n c e r t a i n t y i s i n c o r p o r a t e d i n t h e S L P R m o d e l . I n c o n t r a s t t o t h e B - C m o d e l , o n l y t h e m e a n r a t e o f i n t e r e s t i s u s e d f o r e a c h o f t h e f i n a n c i a l i n s t r u m e n t s r a t h e r t h a n t h e p o s s i b l e r e a l i z a t i o n s . A l s o , t h e S L P R m o d e l a l l o w s f o r v i o l a t i o n s o f t h e c o n s t r a i n t s b y i n c o r p o r a t i n g p e n a l t i e s f o r r e c o u r s e i n t h e o b j e c t i v e f u n c t i o n . T h e d e c i s i o n v a r i a b l e s f o r t h e S L P R m o d e l w i l l b e d e f i n e d i n a m a n n e r s i m i l a r t o t h e v a r i a b l e s i n t h e S L M m o d e l . T h e i n v e s t m e n t o p p o r t u n i t i e s f o r t r e a s u r y b i l l s , t e r m d e p o s i t s , m o r t g a g e s a n d d e m a n d d e p o s i t s a r e d e f i n e d b y s i x , e l e v e n , e l e v e n a n d f o u r v a r i a b l e s , r e s p e c t i v e l y . T h e c o n s t r a i n t s i n t h e S L P R m o d e l a r e c o m p r i s e d o f : 1 ) t h r e e c o n s t r a i n t s t o b a l a n c e t h e i n i t i a l h o l d i n g o f a n a s s e t w i t h t h e f u t u r e b u y i n g a n d s e l l i n g o f t h e a s s e t , 2 ) t h r e e c o n s t r a i n t s t o e q u a t e t h e c a s h f l o w s f o r t h e t h r e e p e r i o d s , 3 ) t h r e e c o n s t r a i n t s f o r e a c h o f t h e t h r e e a s s e t s f o r c o m p o s i t i o n r e q u i r e m e n t s , 4 ) f o u r c o n s t r a i n t s t o d e s c r i b e t h e i n i t i a l p o s i t i o n o f t h e t h r e e a s s e t s a n d o n e l i a b i l i t y , 5 ) t h r e e c a p i t a l l o s s c o n s t r a i n t s o f w h i c h o n e ( t h e f i r s t p e r i o d ) i s d e t e r m i n i s t i c a s 1 ) t o 4 ) a b o v e , a n d t h e o t h e r s b e i n g s t o c h a s t i c , a n d 6 ) t h r e e s t o c h a s t i c c o n -s t r a i n t s w h i c h d e s c r i b e t h e f l o w o f d e m a n d d e p o s i t s . I n s h o r t , t h e r e a r e 2 5 c o n s t r a i n t s i n t o t a l o f w h i c h f i v e a r e s t o c h a s t i c . A d d i n g n i n e s l a c k v a r i a b l e s f o r t h e c l a s s c o m p o s i t i o n c o n s t r a i n t s a n d o n e f o r t h e d e t e r m i n i s t i c 1 8 2 c a p i t a l l o s s c o n s t r a i n t , t h e S L P R f o r m u l a t i o n h a s 2 5 c o n s t r a i n t s a n d 4 2 v a r i a b l e s . T h e r i g h t h a n d s i d e s o f t h e s t o c h a s t i c d e m a n d d e p o s i t c o n s t r a i n t s a r e r e p r e s e n t a t i v e p o i n t s f r o m t h e u n i f o r m d i s t r i b u t i o n u s e d i n t h e B - C m o d e l . H o w e v e r , b e c a u s e o f t h e a b i l i t y o f t h e W e t s ' a l g o r i t h m t o h a n d l e m a n y r e a l i z a t i o n s w i t h o u t c r e a t i n g c o m p u t a t i o n a l d i f f i c u l t i e s , t h e n u m b e r o f p o i n t s c h o s e n i s l a r g e r t h a n i n t h e B - C m o d e l . T h e p e n a l t y f o r v i o l a -t i o n s o f a n y o f t h e s e c o n s t r a i n t s i s t h e n e t r e t u r n t o t h e h o r i z o n o f t h e m o d e l , g e n e r a t e d b y a p o r t f o l i o c o n s i s t i n g o f 5 0 % m o r t g a g e s a n d 5 0 % t e r m d e p o s i t s , s i n c e t h e i r p o r t f o l i o i s c o n s i d e r e d , a p r i o r i , t o b e p o t e n t i a l l y t h e h i g h e s t y i e l d i n g p o r t f o l i o . T h i s p e n a l t y i s c a l c u l a t e d a s \u00E2\u0080\u00A2 5 [ ( 1 + r m ) 4 ~ n - 1 ] + . 5 [ ( 1 + r t ) 4 \" n - 1 ] - [ ( 1 + r r f ) 4 \" n - 1 ] w h e r e n = 1 , 2 . 3 i s t h e p e r i o d ; r m i s t h e m e d i a n r e t u r n o n m o r t g a g e s ; r t i s t h e m e d i a n r e t u r n o n t e r m d e p o s i t s ; a n d r ^ i s t h e m e d i a n c o s t o f d e m a n d d e p o s i t s . T h e r i g h t h a n d s i d e s o f t h e s t o c h a s t i c c a p i t a l l o s s c o n s t r a i n t s a r e t h e r e p r e s e n t a t i v e p o i n t s u s e d i n t h e B - C f o r m u l a t i o n . T h e p e n a l t y f o r v i o l a t i o n s o f t h e s e c o n s t r a i n t s i s a l o s s p e r c e n t a g e b e c a u s e t h e s e a r e p o l i c y c o n s t r a i n t s r a t h e r t h a n p h y s i c a l l y o r l e g a l l y r e s t r i c t i v e c o n s t r a i n t s a s t h o s e i n t h e p r e c e d i n g p a r a g r a p h . I n t h i s p a r t i c u l a r f o r m u l a t i o n , a p e n a l t y o f 4 . 1 % i s u s e d . T h e o b j e c t i v e i s t o m a x i m i z e t h e n e t r e t u r n s m i n u s t h e e x p e c t e d p e n a l t i e s f o r c o n s t r a i n t v i o l a t i o n s . T h e c o e f f i c i e n t o f e a c h v a r i a b l e i s t h e n e t r e t u r n f o r t h e f i r s t s t a g e v a r i a b l e s a n d t h e p e n a l t y f o r t h e s e c o n d s t a g e v a r i a b l e s . 1 8 3 5 . 5 R e s u l t s o f t h e S i m u l a t i o n I n m o s t n o r m a t i v e f i n a n c i a l p l a n n i n g m o d e l s , t h e o b j e c t i v e i s t o d e t e r m i n e w h a t p o r t f o l i o c h a n g e s s h o u l d b e e f f e c t e d i m m e d i a t e l y . T h e m u l t i - p e r o d i c i t y c h a r a c t e r i s t i c o f f i n a n c i a l m o d e l s i s t o c o m p e n s a t e f o r t h e s h i f t i n g e c o n o m i c s c e n a r i o s a c r o s s t i m e . H o w e v e r , t h e p u r p o s e o f t h e m o d e l i s t o d e t e r m i n e t h e c h a n g e s t o b e i m p l e m e n t e d i m m e d i a t e l y . T h e s i m u l a t i o n a n a l y z e d i n t h i s s e c t i o n d e t e r m i n e s w h i c h t e c h n i q u e ( S D P o r S L P R ) y i e l d s t h e s u p e r i o r f i r s t p e r i o d o s o l u t i o n . I n r e a l i t y , d e c i s i o n s m a y b e m a d e a t a n y p o i n t i n a p e r i o d , h o w e v e r , u s i n g a d i s c r e t e t i m e m o d e l o n e a g g r e g a t e s s o a s t o c o n s i d e r a l l d e c i s i o n s t o b e m a d e a t t h e s t a r t o f e a c h p e r i o d - f a c i n g r a n d o m r a t e s o f r e t u r n . A g a i n , t h e i n c r e m e n t a l c a s h f l o w s a r e a g g r e g a t e d s o t h a t o n e - h a l f i s a v a i l -a b l e a t t h e b e g i n n i n g o f t h e c u r r e n t p e r i o d . I n b o t h f o r m u l a t i o n s t h e s a m e i n i t i a l s e c u r i t y h o l d i n g s a r e g i v e n a n d t h e c a s h f l o w s f o r t h e n e x t p e r i o d a r e r a n d o m . A d e t a i l e d f l o w c h a r t f o r t h e s i m u l a t i o n i s i n a n a p p e n d i x a t t h e e n d o f t h e c h a p t e r . E s s e n t i a l l y t h e p r o c e s s s t a r t s w i t h a n i n i t i a l p o r t -f o l i o . B o t h t h e S D P a n d S L P R m o d e l s d e t e r m i n e a n o p t i m a l s o l u t i o n f o r t h e f i r s t p e r i o d . A r a n d o m c a s h f l o w i s t h e n g e n e r a t e d . I f t h e a m o u n t o f f u n d s s p e n t d u r i n g t h e f i r s t p e r i o d e x c e e d s t h e r a n d o m c a s h f l o w , t h e n a n a m o u n t e q u a l t o t h e e x c e s s s p e n d i n g m u s t b e d i v e s t e d f r o m t h e p r e s e n t p o r t f o l i o ( . 4 5 o f m o r t g a g e s , . 4 5 t e r m d e p o s i t s a n d . ' . l t r e a s u r y b i l l s ) . On t h e o t h e r h a n d , i f t h e r a n d o m c a s h f l o w s e x c e e d s p e n d i n g d u r i n g t h e f i r s t p e r i o d , t h e n t h e i n c r e m e n t a l a m o u n t i s i n v e s t e d i n t r e a s u r y b i l l s . A f t e r t h i s r e c o n c i l 1 i a t i o n , t h e r e a l i z a t i o n o f t h e r a n d o m r e t u r n s a r e d e t e r m i n e d . T h e r e v e n u e s a r e t h e s u m o f t h e ( k n o w n ) r e t u r n s o f t h e a s s e t s h e l d s i n c e 1 8 4 t h e s t a r t o f t h e p e r i o d a n d t h e ( r a n d o m ) r e t u r n s o n t h e a s s e t s b o u g h t a t t h e s t a r t o f t h e p e r i o d . T h e c o s t s a r e t h e s u m o f t h e ( r a n d o m ) c o s t o f d e m a n d d e p o s i t s a n d t h e d i s c o u n t f o r s e l l i n g s e c u r i t i e s p r i o r t o m a t u r i t y . T h e r e c o n c i l e d p o r t f o l i o s e r v e s a s t h e n e w i n i t i a l p o r t -f o l i o w h i c h i s t h e n u s e d t o g e n e r a t e t h e n e w s o l u t i o n s f o r b o t h t h e S D P a n d S L P R m o d e l s . T h i s c y c l e i s r e p e a t e d f o r a t o t a l o f e i g h t t i m e s . T h i s w h o l e p r o c e s s i s r e p e a t e d f i f t y t i m e s f o r a t o t a l o f f o u r h u n d r e d i t e r a t i o n s . T h e s i m u l a t i o n r e s u l t s f o r t h e S D P a n d S L P R f o r m u l a t i o n s a r e u s e d t o t e s t t w o h y p o t h e s e s . T h e f i r s t h y p o t h e s i s , H 0 : y d = y S D P \" y S L P R - \u00C2\u00B0 ' i s u s e d t o t e s t w h e t h e r o r n o t t h e i n i t i a l p e r i o d p r o f i t f o r S L P R i s s u p e r i o r t o t h a t f o r S D P . T h e f i r s t h y p o t h e s i s i s t e s t e d b y e x a m i n i n g t h e p a i r e d d i f f e -r e n c e s o f t h e p r o f i t s f o r t h e i n i t i a l r u n o f t h e 5 0 c y c l e s f o r S L P R a n d S D P . T h e s p e c i f i c i n f o r m a t i o n u s e d t o t e s t t h e f i r s t h y p o t h e s i s i s : ( 1 ) t h e m e a n o f t h e p a i r e d d i f f e r e n c e s ( $ 2 5 1 . 3 7 i n f a v o u r o f S L P R ) , ( 2 ) t h e s t a n d a r d d e v i a t i o n o f t h e p a i r e d d i f f e r e n c e s ( $ 1 5 0 . 4 3 ) , a n d ( 3 ) t h e c o r r e l a t i o n b e t w e e n t h e S D P a n d S L P R p r o f i t s ( 0 . 9 5 8 ) . G i v e n t h e l a r g e s a m p l e , t h e s i g n i f i c a n c e o f t h e p a i r e d d i f f e r e n c e s i s t e s t e d u s i n g t h e f o l l o w i n g t e s t s t a t i s t i c : - 2 5 1 . 3 7 = _ n 1 5 0 . 4 3 / 7 5 0 1 8 5 S i n c e t h e t e s t s t a t i s t i c i s s i g n i f i c a n t a t t h e 0 . 0 0 1 l e v e l , t h e n u l l h y p o t h e s i s i s r e j e c t e d . T h u s , t h e S L P R f o r m u l a t i o n y i e l d s a s t a t i s t i c a l l y s i g n i f i c a n t b e t t e r i n i t i a l s o l u t i o n t h a n t h e S D P f o r m u l a t i o n . T h e s e c o n d h y p o t h e s i s , H 0 : y d = y S D P \" y S L P R > 0 i s u s e d t o t e s t w h e t h e r o r n o t t h e m e a n p r o f i t f o r S L P R i s s u p e r i o r t o t h a t f o r S D P . T h i s h y p o t h e s i s i s t e s t e d b y e x a m i n i n g t h e p a i r e d d i f f e r e n c e s o f t h e m e a n p r o f i t s o f t h e e i g h t r u n s o f t h e f i f t y c y c l e s f o r S L P R a n d S D P . T h e s p e c i f i c i n f o r m a t i o n u s e d t o t e s t t h i s h y p o t h e s i s i s : 1 ) t h e m e a n o f t h e p a i r e d d i f f e r e n c e s ( $ 2 9 7 . 2 6 i n f a v o u r o f S L P R ) , 2 ) t h e s t a n d a r d d e v i a t i o n o f t h e p a i r e d d i f f e r e n c e s ( $ 3 0 8 . 7 4 ) , a n d 3 ) t h e c o r r e l a t i o n b e t w e e n t h e S D P a n d S L P R m e a n p r o f i t s ( . 7 8 5 ) . A g a i n , g i v e n t h e s a m p l e s i z e , t h e s i g n i f i c a n c e o f t h e p a i r e d d i f f e r e n c e s i s t e s t e d u s i n g t h e f o l l o w i n g t e s t s t a t i s t i c : - 2 9 7 - 2 l = - 6 . 8 1 . 3 0 8 . 7 4 / / 5 0 S i n c e t h i s i s s i g n i f i c a n t a t t h e . 0 0 1 l e v e l , t h e n u l l h y p o -t h e s i s i s r e j e c t e d . T h u s , t h e S L P R f o r m u l a t i o n y i e l d s a s t a t i s t i c a l l y s i g n i f i c a n t b e t t e r s o l u t i o n t h a n t h e S D P f o r m u l a t i o n . T o t e s t t h e s t a b i l i t y o f t h e a b o v e s u m m a r y s t a t i s t i c s , a s e c o n d s i m u l a t i o n u s i n g S L P R w a s r u n . T h e r e s u l t s o f t h i s s i m u l a t i o n a r e 1 8 6 a n a l y z e d a s a b o v e : 1 ) a t e s t o f t h e i n i t i a l s o l u t i o n o f t h e f i f t y c y c l e s , a n d 2 ) a t e s t o f t h e m e a n p r o f i t s f o r t h e 8 r u n s o f t h e f i f t y c y c l e s . T h e i n f o r m a t i o n n e c e s s a r y t o t e s t t h e f i r s t h y p o t h e s i s i s : 1 ) t h e m e a n p r o f i t s f o r t h e f i r s t a n d s e c o n d S L P R r u n s ( $ 4 6 4 5 . 8 5 a n d $ 4 6 7 2 . 2 3 r e s p e c t i v e l y ) , a n d 2 ) t h e s t a n d a r d d e v i a t i o n s f o r t h e t w o r u n s ( $ 4 2 1 . 1 1 a n d $ 4 8 2 . 1 5 r e s p e c t i v e l y ) . T h e h y p o t h e s i s t h a t b o t h s a m p l e s h a v e t h e s a m e m e a n , H 0 - ^ S L P R i s t e s t e d f i r s t . T h e s t a n d a r d d e v i a t i o n u s e d f o r t h e t e s t s t a t i s t i c i s t h e r o o t o f t h e p o o l e d v a r i a n c e . T h e t e s t s t a t i s t i c i s 4 6 7 2 . 2 3 - 4 6 4 5 . 8 5 9 0 . 5 3 = . 2 9 1 . S i n c e t h e t e s t s t a t i s t i c i s n o t s i g n i f i c a n t a t t h e .1 l e v e l , t h e r e i s n o r e a s o n t o b e l i e v e t h a t t h e m e a n i s n o t s t a b l e . T h e t e s t s t a t i s t i c f o r t h e s e c o n d h y p o t h e s i s i s e s t a b l i s h e d i n a s i m i l a r m a n n e r a n d i s 4 7 8 3 . 1 3 - 4 7 2 0 . 1 5 8 6 . 8 4 = . 7 3 . S i n c e t h e t e s t s t a t i s t i c i s n o t s i g n i f i c a n t a t t h e .1 l e v e l , t h e r e i s a g a i n n o r e a s o n t o b e l i e v e t h a t t h e m e a n i s n o t s t a b l e . 1 8 7 A CDC 6 4 0 0 w a s u s e d t o p e r f o r m t h e a b o v e c o m p u t a t i o n s . T h e t o t a l C P U t i m e t o p e r f o r m t h e 4 0 0 i t e r a t i o n s f o r t h e S L P R f o r m u l a t i o n w a s . 2 4 0 h o u r s a n d f o r t h e S D P f o r m u l a t i o n s w a s 6 . 3 8 5 h o u r s . T h i s e x p l a i n s w h y o n l y a l i m i t e d n u m b e r o f f i n a n c i a l i n s t r u m e n t s , t i m e p e r i o d s a n d r e a l i z a t i o n s w e r e u s e d i n t h e s i m u l a t i o n . I t a l s o f u r t h e r h i g h l i g h t s t h e g a p i n t r a c t a b i l i t y b e t w e e n t h e S L P R a n d S D P t e c h n i q u e s 1 8 8 5 . 6 A p p e n d i x 1 T h i s a p p e n d i x c o n s i s t s o f : 1 ) a f l o w c h a r t f o r t h e s i m u l a t i o n p e r f o r m e d i n C h a p t e r 5 , a n d 2 ) a c o m p u t e r c o d e f o r e x e c u t i n g t h e s i m u -l a t i o n . F i r s t , t h e f o l l o w i n g v a r i a b l e s a r e d e f i n e d : 1 ) X 1 ( X 1 9 , X 6 1 ) i s t h e a m o u n t i n v e s t e d i n t r e a s u r y b i l l s ( t e r m d e p o s i t s , m o r t g a g e s ) i n t h e i n i t i a l p o r t f o l i o ; 2 ) X ( 2 ) ( X ( 2 0 ) , x ( 6 2 ) ) i s t h e a m o u n t a l l o c a t e d t o p u r c h a s i n g n e w t r e a s u r y b i l l s ( t e r m d e p o s i t s , m o r t g a g e s ) i n p e r i o d 1 a s g e n e r a t e d b y t h e o p t i m a l s o l u t i o n t o t h e B - C ( S L P R ) f o r m u l a t i o n ; 3 ) X ( 3 ) ( X ( 2 1 ) , X ( 6 3 ) ) i s t h e a m o u n t o f i n i t i a l p e r i o d t r e a s u r y b i l l s ( t e r m d e p o s i t s , m o r t g a g e s ) s t i l l h e l d i n p e r i o d 1 ; 4 ) X ( 4 ) ( X ( 2 2 ) , X ( 6 4 ) ) i s t h e a m o u n t o f t r e a s u r y b i l l s ( t e r m d e p o s i t s , m o r t g a g e s ) s o l d i n p e r i o d 1 ; a n d 5 ) X I 0 3 i s t h e a m o u n t o f d e m a n d d e p o s i t s o u t s t a n d i n g i n t h e i n i t i a l p o r t f o l i o . 1 8 9 ^ S t a r t ^ S e t T = 1 V I n i t i a l i z e p o r t f o l i o X I = 3 3 3 3 3 X 1 9 = 3 3 3 3 3 X 6 1 = 3 3 3 3 4 X 1 0 3 = 1 0 0 0 0 0 S e t I = 1 F o r m u l a t e p r o b l e m ( B - C o r S L P R ) C a l l K u z y ( X I , X 1 9 , X 6 1 , X I 0 3 ) t o g e n e r a t e f i r s t p e r i o d s o l u t i o n G e n e r a t e : 1 ) e x o g e n o u s c a s h f l o w 1YY e [ T I O O O O , 1 0 0 0 0 ] a n d s e t R l = X I 0 3 + 1YY 2 ) p r i m e r a t e PR1 3 ) t r e a s u r y b i l l r a t e T B l ( P R l ) 4 ) t e r m d e p o s i t r a t e T D ( P R I ) 5 ) m o r t g a g e r a t e A M I ( P R l ) 6 ) d e m a n d d e p o s i t r a t e A L C I ( P R l ) 1 9 0 12 = YI = ARM = . 0 0 5 * X ( 4 ) + X ( 2 ) + X ( 3 ) -= Z 2 0 4 * X ( 2 2 ) + . 0 6 * X ( 6 4 ) n X ( 2 0 ) + X ( 2 1 ) + X ( 6 2 ) + X ( 6 3 ) + Z 2 \ X 2 = X ( 2 ) - . 2 * ( Y 1 - R l ) Z 2 = 12 + ( X ( 2 ) - X 2 ) * . 0 0 5 i f X 2 > 0 Z 2 = Z 2 + X ( 2 ) * . 0 0 5 i f X 2 < 0 X ( 2 ) = X 2 X 2 0 = 0 . 0 (?) y e s X 3 = X ( 3 ) + X ( 2 ) X ( 2 ) = 0 . 0 Z 2 = Z 2 + ( X ( 3 ) - X 3 ) * . 0 0 5 i f X 3 > 0 Z 2 = Z 2 + X ( 3 ) * . 0 0 5 i f X 3 < 0 X ( 3 ) = X 3 \u00C2\u00A9 (!) 191 y e s \u00C2\u00A9 X 2 0 1 = X ( 2 0 ) - X 2 0 - ( Y I - R l ) * . 4 Z 2 - Z 2 + ( X ( 2 0 ) - X 2 0 1 ) * . 0 4 i f X 2 0 1 > 0 Z 2 = Z 2 + ( X ( 2 0 ) ) * 0 4 i f X 2 0 1 _ 0 X ( 2 0 ) = X 2 0 1 X 6 2 = 0 . 0 y e s X21 = X ( 2 1 ) + X ( 2 0 ) X ( 2 0 ) = 0 . 0 Z 2 = Z 2 + ( X ( 2 1 ) - X 2 1 ) * . 0 4 i f X21 > 0 Z 2 = Z 2 + ( X ( 2 1 ) ) * . 0 4 i f X21 < 0 X ( 2 1 ) = X 2 1 \u00C2\u00A9 0 1 9 2 X 6 2 = - X ( 2 1 ) X ( 2 1 ) = 0 . 0 \ ! y e s \u00C2\u00A9 X 6 2 1 = X ( 6 2 ) - X 6 2 - ( Y l \u00E2\u0080\u00A2 - R l ) * . 4 Z 2 = Z 2 + ( X ( 6 2 ) - X 6 2 1 ) * 0 6 i f X 6 2 1 > 0 Z 2 = Z 2 + ( X ( 6 2 ) ) * 0 6 i f X 6 2 1 <: 0 X ( 6 2 ) = X 6 2 1 y e s \u00C2\u00A9 X 6 3 = X ( 6 3 ) + X ( 6 2 ) X ( 6 2 ) - 0 . 0 Z 2 =. Z 2 + ( X ( 6 3 ) - X 6 3 ) * . 0 6 Z 2 = Z 2 + ( X ( 6 3 ) ) * . 0 6 X ( 6 3 ) = X 6 3 i f X 6 3 > 0 i f X 6 3 < 0 <5 1 9 3 c S T O P 1 p r o b l e m i n f e a s i b l e l y e s Z l = T B 1 * X ( 2 ) + T D 1 * X ( 2 0 ) + AMI * X ( 6 2 ) + . 0 5 4 1 * X ( 3 ) + . 0 8 2 7 * X ( 2 1 ) + . 0 9 9 2 * X ( 6 3 ) Z 3 = Z l - Z 2 - R 1 * A L C I X I = X ( 2 ) + X ( 3 ) X I 9 = = X ( 2 0 ) + X ( 2 1 ) X 6 1 = = X ( 6 2 ) + X ( 6 3 ) X 1 0 3 - R l n o n o - . 1 9 4 , ' P R O 3 ? . AM H A p o Y ( T A P E 5 , 0 U T P ' I T , T A o E 6 = 0 U T \u00C2\u00B0 U T > I M P L I C I T - t A L ( A - H , 0 - Z ) I N T E G E~_ _ S . c . J > H . I , M AR T Y , R \u00C2\u00A3 U \" TP(7b,Tc\"i ,\"T6 {7 3 , T : I R E A L P ( 7 C t l 3 ) . n ( 7 o . i : ) , f i ( 1 0 0 i 2 6 0 ) . H ( 1 0 0 ) t C ( 2 6 0 l S E A L W ( l C C . 1 0 G ) , G ( i O Q ) t D E L T A ( 7 Q > , G A M M A ( 7 j ) , P I ( l 0 0 ) D I M E N S I O N IW ( 1 C J ) , K A P \u00C2\u00B0 A ( 7 0 > , L ( 7 0 ) , K ( 7 0 > R E A L 0 \" ( 7 0 I , O M ( 7 0 ) O I 1 \u00C2\u00A3 N S I O N _ J \u00C2\u00AB < 1 5 0 ) C O M M O N \"u . M . M 1 , M2 , ' p , O T A , W \" , C , H \ G ,~DE L T A , G AM M A , P I , I W , K A P P A , L , E D S . $ K,QP . O M , Z O , M A R K E R C O M M O N / P R I N T R / T P , T D C O M M O N / A / x Q * \u00C2\u00BB \u00C2\u00BB \u00C2\u00BB \u00C2\u00BB * \u00C2\u00BB * \u00C2\u00BB \u00C2\u00BB * \u00C2\u00BB \u00C2\u00BB \u00C2\u00BB G E N E R A T E C A S H P L O W * \u00E2\u0080\u00A2 * \u00C2\u00BB * \u00C2\u00BB * \u00C2\u00AB * \u00C2\u00BB \u00E2\u0080\u00A2 * * DO_ 1 2 3 U _ x K = 1_,_2 X I 0 3 = 1 5 0 C O O . X l = 3 3 3 3 3 . X 1 9 = 3 3 3 3 3 . X 6 1 = 3 3 3 3 4 . DO 1 2 3 3 K L l = l , 3 C A L L K U Z Y ( X 1 J 3 . X 1 . X 1 9 . . \u00C2\u00BB _ X _ 6 . 1 J R E W I N D 5 W R I T E ( 6 , 1 0 2 ) 1 0 2 F O R M A T ( 1 H 1 1 W R I T E ( 6 . 1 0 1 ) ' \u00C2\u00AB < K L L 1 0 1 F O R M A T ( 1 0 Y , 1 < \u00C2\u00AB H S I M U L A T I O N R U N , 1 5 , 2 X , 6 H P E R I 0 0 , I 6> A X 2 - X ( 2 ) : : X ( 2 ) = X (t* 1 X ! 4 I = A X 2 A X 2 0 = X ( 2 C > A X 2 i = X C 2 1 ) A X 2 2 = X ( 2 2 ) X ( 2 3 ) = X < 1 2 ) + X ( 1 3 ) + X ( 1 4 ) x < 2 1 ) = x ( q i * x t i u ) + x < i i > X ( 2 2 ) = X { P \u00C2\u00BB X ( 5 2 I =X t 2 3 ) *X ( 2 4 I +x ( 2 5 ) X ( o 3 > = A X 2 G + ftX2i+ftX22 X ( 6 4 > =\"/ ( I P ) x ( i o 3 ) - x ( ? q > Y = R A N F ( C I Y Y = 2 0 G 6 C * Y T Y Y = Y Y - 1 G C 0 G R i = X ( l 0 3 ) + I Y Y W R I T E ( 6 , 1 9 4 ) P i ' 1 0 4 F O R M A T < . 1 0 X , 1 U H C A S H F L Q W ( R 1 ) = , - 1 6 . 4 ) _ . . \u00C2\u00B0 R 1 = P R { D U M M Y ) P R 1 = . 5 * P R l + . 0 3 9 7 I F ( I Y Y . G T . c . A N D . a . R i , L E . . ~. 7 5 ) 3 R i = P R 1 + . u 0 5 I F U Y V . L T . G . A N O . P R i . G E . . 0 8 5 ) P R 1 = ? R 1 - . C Q 5 T 9 1 = P R l + RT 8 ( D U M M Y ) X O i s . P \u00C2\u00A7 < _ t . ~ L P i D U M M YI ; i A ' l l = P R i + =M ( D U M M Y ) A L C 1 = . 7 * ( P R l + R L C ( D U M M Y ) ) 195 W R I T E ( 6 , 1 5 0 ) T R 1 1 5 0 r O R M A T ( 1 ; r . 1 9 H T R E A S U R Y B I L L * A T E - t 2 X , F 1 6 . 6 ) W R I T E 1 6 , 1 5 1 ) T Q l J 1 5 1 F O R M A T ( l & X . i q n T E R M D E P O S I T R A T \u00C2\u00A3 = , 2 X , ^ 1 6 . 6 > W \u00C2\u00BB I T E ( & , i 5 2 ) A M I 1 5 2 F O R M A T ( l l X , i 3 H M O R T A G E R i T E = . 2 X , F l 6 . 6 ) W R I T E ( 6 . 1 5 3 ) A L - 1 1 5 3 c O R ^ A T ( l C X , i q H L I A B I L I T v R A T E = , 2 X . F l 6 . 6 ) Y 1 = X ( 2 ) + X ( 2 3 J J - X ( 6 2 L + _ t _\u00C2\u00BB L21- ' tAS 6 3 ) + . 0 0 5 \u00C2\u00BB X ( ^ ) ? + . 0 4 * X ( 2 2 ) + . o \" 6 * X~( 6 <\u00E2\u0080\u00A2) Z 2 = . G 0 5 * X ( l i ) + . * X ( 2 2 ) + . j & * X < 5 \u00C2\u00BB ( . 0 0 51 I F I X 2 . L E . . 0 ) 11=12* ( X ( 2 ) ) * ( . 3 0 5 ) X ( 2 ) = X 2 X 2 Q = G . C I F ( X ( 2 1 . G E . . 0 ) 3 0 T O 7 7 X 3 = X ( 3 > \u00C2\u00BB X ( ? ) X ( 2 ) = 0 . 0 I F C X 3 . G E . . O Z 2 = Z 2 + ( X ( 3 ) - X 3 ) \u00C2\u00BB ( . 0 0 3 ) I F ( X 3 . L T . . 0 ) Z 2 = Z 2 + ( X ( 3 ) ) * ( . 0 3 5 ) X ( 3 > = X 3 I F ( X ( 3 ) . G E . . D ) G O T G 7 7 X 2 0 = - x ( 3 ) . . : X ( 3 ) = 0 . 0 7 7 C O N T I N U E X 2 3 1 = X < 2 0 > - X 2 0 - ( Y 1 - R 1 ) * ( . < \u00E2\u0080\u00A2 ! I F ( X 2 J 1 . G T . . 0 ) 7 2 = Z 2 + < X ( 2 G > - X ? G 1 > < ' ( . 0 < 4 ) I F ( X 2 0 1 . IZ . . J ) 7.2 = Z 2 + ( X ( 2 0 ) ) * ( . 0<*> ^ X ( 2 0 ) = X 2 0 1 ; X 5 2 = 0 . 0 I F ( X ( 2 0 ) . G E . . 0 ) GO TO 7 f l X 2 l = X ( 2 1 ) + X ( 2 0 ) X ( 2 G ) = 0 . C I F I X 2 1 . G E . . 0 > Z 2 = 7 2 + ( X ( 2 1 ) - X 2 l ) * ( . 0 < \u00C2\u00AB > I F ( X 2 1 . L T . . 0 ) Z 2 = Z 2 + X ( 2 1 ) ' l ( . 0 ' 4 V x ' ( ? i ) \" = X 2 1 I F ( X ( 2 1 I . G E . . Q ) GO TO 7 8 X 6 2 = - X ( 2 1 ) X ( 2 1 > = 0 . 0 7 8 C O N T I N U E X 6 2 1 f X ( 6 ? J _ - X 6 2 - f Y l - R l ) _ * ( _ . 4 _ ) _ _ _ ' T F ( x V 2 l . ~ G T 7 * . T i T 2 = Z ^ + T x ( 6 2 ) - X 5 2 l f * ( 7 j 6 ) ~ \" ~ ~ - - - - -I F C X 6 2 1 . I T . . 0 ) Z 2 = Z 2 + ( X ( 6 2 ) > * ( . 0 6 ) X ( 6 2 ) = X 6 2 i I F < X ( 6 2 > , G E . . 0 > GO TO 7 9 X 5 3 = X f 6 3 ) \u00C2\u00AB - X ( 6 2 > X 1 6 2 > =0 . r . _ . \ _ I F ( X 6 3 . G E . ' . O ) Z 2 = Z 2 + ( X ( 6 3 ) - X 6 3 > * ( . 0 6 ) ' \" I F ( X 6 3 . L T . . 0 ) Z 2 = Z 2 + ( X ( 6 3 ) ) \u00C2\u00BB ( . 0 6 t 1 9 6 x ( 6 3 ) = X 6 3 I \" ( X ( 6 3 \u00C2\u00BB . G E . . 0 ) GO TO 7 9 S T O P 1 G 0 7 9 C O N T I N U E 7 l = T 3 1 * X ( 2 l * r r ) l \u00C2\u00BB X ( 2 G > + A \" U \u00C2\u00BB X ( i ; 5 ) 1 + . ' J 5 i t i * X < 3 > + . C - 3 2 7 * X ( 2 1 ) + . u 9 9 2 * X ( 6 3 > Z 3 = Z l - Z 2 - f i l \u00C2\u00BB A L C l W R I T E ( 6 , 1 0 f t ) K L L . Z 3 1 0 8 F Q R M A T ( 1 C X , 1 7 H F R 0 F I T F O R ? E R I O P , 2 X , 1 1 0 , 1 M = , F 1 5 . I*) X < 1 ) = X ! 2 ) + X ( 7 ) X ( 1 9 ) =X ( 2 0 ) +X ( 2 1 ) X ( 6 1 ) = X ( 6 2 ) + X < 6 3 > X ( 1 0 3 ) = R 1 X l a J - X l l O 7 ) X 1 = X ( 1 ) X 1 9 = X ( 1 9 ) X 6 1 = X ( 6 1 > W R T T E ( 6 , 2 G 7 ) X 1 , X 1 9 , X 6 1 , X 1 0 3 2 0 7 F O R M A T ( 1 0 V , 2 H X = , i v F 1 6 . i t ) W R I T E ( 6 , 1 6 0 ) 1 6 0 P O P M / \ T ( 1 f ) Y '1 c u \u00C2\u00BB * * * \u00C2\u00BB * * \u00C2\u00BB * * \u00C2\u00BB * * * * * * * * * * \u00C2\u00BB * * * * * * * * * * \u00C2\u00BB * * * * * * * * , ' * ' * * * > W R I T E ( 6 , 1 6 1 ) 1 6 1 F O R M A T ! 1 O X , 2 0 H S T A T I 3 T I C S > S S Z 1 = S B Z 1 + Z 3 S B 7 S = S 8 Z S + ( 7 3 ) * \u00C2\u00BB 2 W R I T E ( 6 , 1 6 3 ) 7 1 1 6 3 F O R M A T ( 1 C'X G R O S S R E V E N U E S = \u00C2\u00BB , F 1 6 . 6 ) W R I T E ( 6 , 1 6 8 ) A R M 1 6 8 F O R M A T ( 1 0 X , \u00C2\u00BB C O S T O\" S A L E S = * , F 1 6 . 6 ) W R I T E ( 6 , 1 6 5 ) Z 2 1 6 5 F O R M A T d O X , * C O S T OF S A L E S A N D F O R C E D S A L E S = * , F 1 5 . A R M I N = R 1 * A L C 1 W R I T E ( 6 , 1 6 6 ) A R M I N 6 ) 1 6 6 F O R M A T ( 1 C Y , * C O S T O F F U N 3 S = \u00C2\u00BB , F 1 6 . 6 ) W R I T E ( 6 , 1 6 7 ) S 9 Z 1 1 6 7 F O R M A T d O X , * C U M M U L A T I V E P R O F I T S = \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 . F 1 6 . 6 ) W P I T E ( 6 . 1 6 9 ) S 9 Z S 1 6 9 F O R M A T d O X - , * C U M M i J L A T I V E P R O F I T S S Q U A R E D * , F 2 0 . 3 ) 1 2 3 3 C O N T I N U E . 1 2 3 1 * C O N T I N U E E N D F U N C T I O N P P ( O ) Y = R A N F < 0 > I F j _ v . L \u00C2\u00A3 . \u00E2\u0080\u009E ? . 2 A 1 L _ E 3 0 6 . i F ( V . G T . . 2 3 l . f t N 0 . V . L E . . 3 - 7 ) D R = . 0 6 5 I F ( Y . GT . . 7 i . 7 . A N D . Y . I E . . 3 3 5 ) =R = . 0 6 7 5 I F ( Y . GT . . 3 \u00C2\u00B0 - 5 . A NO . Y . I E . 2 ) PR = . 0 7 5 I F ( v . G T . . ^ 6 2 . A N O . Y . L \u00C2\u00A3 . . 5 ) \u00C2\u00BB R = . 0 7 7 5 I F ( Y . G T . . 5 . A N O . Y . L \u00C2\u00A3 . . 5 77> \u00C2\u00B0 R = . 0 8 I F ( Y . GT . . 5 7 7 . A N D . Y . L \u00C2\u00A3 . . _ ? 1 < J P ? = . Q B 5 I F ( Y . G T \" . . 7 3 1 ^ A N O . Y . L \u00C2\u00A3 . \" . 8 - J 8 ) \u00C2\u00B0 ? = . 0 9 I F ( Y . GT . . 8 0 8 . A NQ . Y . |_ \u00C2\u00A3 . . 8.3 5 ) \u00C2\u00B0 R = . 0 9 5 I C ( Y . G T . . 8 8 5 . A N D . Y . L \u00C2\u00A3 . . 9 6 2 > \u00C2\u00B0 * = . l l W R I T E ( 6 , 1 0 1 ) P R 1 0 1 F O R M A T ( 1 t X . l l K P . R I ME R A T E = . 2 X , F l 6 . 6 ) R E T U R N E N O 198 F U N C T I O N \" T R ( O I Y = R A M F ( 0 ) a l r - - - ' J - K J . _ 4 2 = - . 0 3 0 6 1 3 = - . 0 2 5 3 A 4 = - . 0 2 2 5 A 5 = - . 0 1 7 4 A 6 = - . 0 C 5 1 l \" c ! Y . C-f . . 7 . A N D . Y . L E . . 5 > T 3 = A 2 * ( ( Y - . 2 ) / . 3 ) \u00C2\u00BB ( A 3 - A 2 ) I F ( Y . G T . . 5 . A N D . Y . I E . . 7 7 > T 6 = A 3 + ( ( Y - . 5 ) / . 2 7 ) * ( A 4 - A 3 I I F f Y . G T . . 7 7 . A N 0 . Y . L E . . 3 1 ) T 3 = A 4 f ( ( Y - . 7 7 ) / . G 4 1 * ( A 5 - A 4 ) I F < Y . G T . . S i . A N D . Y . L E . 1 . > T B = A 5 * ( ( Y - . 8 1 ) / , 1 9 ) \u00C2\u00BB ( A 6 - A 5 1 R T > 3 = T 0 _RE_T_U\u00C2\u00B0_N E N D \" F U N C T I O N R T O f O ) Y = R A N F I D ) A l = - . 0 1 0 4 A 2 = - . 0 0 7 2 A 3= . 0 0 0 9 A 4 = . 0 0 ' t A 5 = . G 1 1 8 A 6 = . 0 1 9 5 T F ( Y . L \u00C2\u00A3 . . 2 1 T D = A i M Y / . 2 ) \u00C2\u00BB ( A 2 ^ 5 i F ( Y . G T . . 2 . A , N 0 . Y . L \u00C2\u00A3 . . 4 i t ) T O = A2 + ( ( Y - . 2 ) / ' ( . 2 4 ) ) * ( A 3 - A 2 ) I r ( Y . G T . . 4 4 . A N n . Y . L E . . 5 \u00C2\u00BB T 0 = A 3 + < < Y - . 4 4 ) / . j 6 > * ' ( A 4 - A 3 > I F ( Y . GT \u00E2\u0080\u00A2 . 5 . A N 0 . Y . L E . . 7 R ) T D = A 4 + ( < Y - . 5 > / . 2 3 ) * ' < A 5 - A 4 > I F ( Y . G T . . 7 3 . A N 0 . Y . L E . i . ) T n = A 5 + ( ( Y - . 7 8 ) / . 2 2 ) * < A 6 - A 5 l 3 T O = T D R E T U R N _ ^ E N O - . 2 0 0 F U N C T I O N R \" ( Q ) Y = R A N F ( 0 > a l = . u 0 3 7 . , A 2 = . C 0 8 8 A 3 = . D 1 9 ? A 4 = . C 2 3 5 A 5 = . 0 2 P 7 6 6 = . 0 3 3 8 I F 1 Y . L E . . 2 ) AM = A l - K Y / . 2 > * ( A * - A p I F ( Y , GT . . 2 . A N D . Y . L E . . 4 2 ) AM = A 2 + ( ( Y - . 2 ) / ( . 2 2 ) ) * ( A 3 - A 2 ) I F ( Y . G T . . 4 2 . A N D . Y . L E . . 6 2 ) AV,= A 3 \u00C2\u00AB - ( ( Y - . 4 2 ) / . 2 ) * ( A 4 - A 3 ) I F ( Y . GT . . 6 2 . A N D . Y . I E . . S l > A M = A 4+ (.( Y - . 6 2 ) . 1 9 ) * < A 5 - A4> I P < Y . G T . . 8 1 . A N D . Y . L E . 1 . ) A M = A 5 M ( Y - . 8 1 ) / . i 9 > * < A 6 - A 5 ) R M = A M R E T U R N . _ E N D \u00E2\u0080\u00A2 2 0 1 . F U N C T I O N R L C f O I Y = R A N F ( D > I E J * * J U L r . . J t t \u00C2\u00A3 l ?.L?.\u00C2\u00A3=....0.2.r.5. I r ! Y . G T . . i l 5 . A N 0 . Y . L E . . 1 9 2 > R . C = - . G 2 ; I F ( Y . G T . . 1 9 2 . A N D . v . L E . . 3 0 H ) R L C = - . Q 2 2 5 I c ( Y . G T . . 3 0 3 . A N D . v . L E . . 9 2 3 ) R L C = - . 0 2 I r ( v . G T . . 9 2 3 . A N D . Y . |_E . 1 . ) R I _ : = - . Q 1 7 5 R E T U R N 2 0 2 S U B R O U T I N E K U Z Y ; x i o 3 , x i , x i 9 , x 6 i ) r. = > M O T I F I C A T I O K S 3 Y J . K A L L 3 E R G V 1 . < U 3 Y \u00C2\u00A3 l 5 > : i i ^ ^ l . J ^ L U J i _ 4 . S l = H?~a.2\u00C2\u00B1JtI*ZSMZ}&.._L_?50: K m . :_ .. C : ' I M P L I C I T R E A L ( A - H . O - Z ) I N T E G E R S , R , P H I , M A R T Y R E A L T D ( 7 0 . 1 0 ) , T D ( 7 3 , 1 0 I R E A L P ( 7 0 , 1 0 ) , 0 ( 7 0 , 1 0 ) . A ( U 0 , 2 6 0 1 . H ( 1 0 0 ) , C ( 2 6 0 I : _ = S 4 L _ _ W U . 0 0 , i 0 . 0 J _ . _ _ ( l 0 0 ) . v O E L J A J _ 7 0 . L , G A M M 4 _ ( i u _ ) _ , . P . I U . O 0 J D I M E N S I O N I W ( 1 0 0 ) , K A ? P A ( 7 0 ) , _ ( 7 0 ) , K ( 7 0 ) ' R E A L 0 \u00C2\u00B0 i 7 0 1 , O M ( 7 0 ) D I M E N S I O N X ( 1 5 Q > C O M M O N / A / X C O M M O N N , M , M 1 , M 2 , P , 0 , 4 , W , C , H , G , D E L T A , G A M M A , P I , I W , K A P P A , L , E P S , 5 . J O . , . M A R K E R C O M M O N / P P I N T R / T P , TD C = > N O T I C E T H A T T N T H E D O C U M E N T A T I O M T H A T P A N D 0 U S E 0 - O R I G I N I N D E X I N G . C = > I N T H E F O P T \u00C2\u00B0 S N C O P E A L L S U C H - I N D E X E S H A V E S E E N I N C R E M E N T E D B Y 1 . . R E A D ( 5 , 9 3 2 1 E \u00C2\u00B0 S 9 3 2 F O R M 4 T ( F 1 0 . 6 ) 1 7 6 F Q R M A T < 1 M l , T l 0 , \" T O L Eg.a_NC.E_ I S S E T A T \" . 7 9 . 5 1 C = > R E A D I N N , M 1 , M 2 , K I . R E A D ( 5 , 1 0 0 ) N , M l , M 2 1 0 0 F O R M A T (315) 1 3 9 1 7 O R M A T ( / / / , T 1 0 , \" ' < O F V A R I A R L E S = \" , 1 4 , / , T 1 0 , \" # OF D E T E R M I N I S T I C : \u00E2\u0080\u00A2 , C \" C O N S T < \u00C2\u00BB A I N T S = \" , I 4 , / , T 1 0 , \" # O F S T O C H A S T I C C O N S T R A I NT S = \" , I 4 ) _ r . l i \u00C2\u00B1 i 1 2 -C = > R E A D I N A N D W R I T E O U T T H E X I - V A L U E S < P O S S I B L \u00C2\u00A3 V A L U E S ! A N D A L P H A A N D B E T A C = > ( L O W E R A N D I J P ^ E R B O U N D S ) I N T O O . 1 1 0 F O R M A T ( \" 1 \" , T 2 G , 3 4 < \u00E2\u0080\u00A2 ' * \" > , / , T 2 0 , \" P O S S I B L E V A L U E S O F R I G H T H A N D S I D E \" , / , T 2 0 , 3 4 ( \" \u00E2\u0080\u00A2 \" ) , / ) DO 3 2 L 7 = l , M 2 _ E A D 13, 3 0 0 ) K LLZ\u00C2\u00B1jO_ ( LZ_, 2__._P_I Z , 2 J , T P ( L Z , 1 1 = P ( L Z , 2 ) T O ( L Z , l ) = D ( L Z , 2 > I F ( K ( L 7 ) . L E . l ) G 0 T O 3 7 . . K P = K ( L Z ) D O 3 1 L A = 2 . K P R E AD_L 5 _ a 0 1 ) D ( L Z , L A + 1 ) , P J . L Z,}. A+.1). T D ( L Z . L A ) = D ( L Z , L A 4 - 1 ) 3 1 T P ( L Z , L A ) = P ( L Z , L A + 1 ) 3 7 R E A D ( 5 , 8 G 2 > O 1 L Z , 11 , O ( L Z , K ( L Z I + 2 ) R E A D ( 5 , 1 2 9 1 Q P ( L Z ) , Q M ! L Z ) S 0 0 F O R M A T ( I 3 , F 1 0 . 2 , F 6 . 4 ) MA^lSSMJJIM.O.rl&j.ftl 8 0 2 F 0 R M A T ( 2 F 1 G . 2 > 1 2 9 - O R M A T < 2 r i C 4 ) K I = K ( L Z ) + 1 L X = M 1 + L Z 1 0 2 ^ 0 S M A T ( T 5 , \" R 0 W \" , 1 3 , T 1 5 , 4 < F l 4 . 2 , 6 X } , / , \u00E2\u0080\u00A2_ 1 M _ f _ J _ . l *M.U : \u00E2\u0080\u00A2_ 3 2 H ( L Z + M 1 ) = D ( L Z , 1 ) C N O T E T H A T T H E A L \u00C2\u00B0 H A ( I ) B E C O M E T H E R I G H T H A N D T E R M S 2 0 3 ' C F O = T H E S T O C H A S T I C C O N S T R A I N T S C = > M R I T E O U T T H E L O W E R A N O U P P E \u00C2\u00B0 S O U N D S A N O C A L C U L A T E T H E 0 V A L U E S . 1 1 2 F O R M A T . L / V . . 1 2 9 . . . < \u00C2\u00A3 . < \" * _ \" ' . _ / > l 2 0 , : ' _ L p W E . R _ A N D J J 5 . ? E ^ _ ^ o J N J J s _ ^ . 3 A f J D O M J I , 1 1 1 V A R T A T - E S \" \u00C2\u00BB / \u00C2\u00BB T 2 0 \u00C2\u00BB ' \u00C2\u00BB 2 ( \" * \" ) \u00C2\u00BB / ) 0 0 3 5 I = 1 , M 2 K I - K ( I ) + 1 L X = M 1 * I D O 3 5 J = 1 . \u00C2\u00AB I 31 PJ.l_\u00C2\u00BBjJJ \u00C2\u00ABpji <-4 * _ I . L - _ D <_U J.1 C = > R E A D I N A N D W R I T E O U T T H E . I N I T I A L P - V A L U E S ! C A L C U L A T E T H E A C C U M U L A T E D C = > P - V A L U E S \" A N D W R I T E T H E M O U T . A L S O W R I T E O U T Q - P L U S A N D Q - M I N U S . 1 0 3 F O R M A T < / / . T 2 0 , 3 0 ( ' * \u00E2\u0080\u00A2 * \" ) , / , T 2 0 t \" P R O B A 3 1 L I T I E S O F P A N O O M E V E N T S \" , / , T 2 0 , 3 0 < \" * \" \u00E2\u0080\u00A2 ) . / ) D O 3 0 1 = 1 , M ? K I = K t I ) * 1 _ LJf^iTi \u00E2\u0080\u0094 3 0 C O N T I N U E 1 0 A F O R M A T ! / / , T 2 0 , 3 3 ( \" * \" ) , / , T 2 C , \" S H O R T A G E P E N A L T Y S U R P L U S P E N A L T Y \" , / . T 2 C . 3 3 ( \" \u00E2\u0080\u00A2 \" > , / ) D O A 2 1 = 1 , M 2 M ^ l i + i , : : A 2 C O N T I N U E D O 3 3 1 = 1 , M 2 Kl = < ( I ) + 1 D ( 1 , 1 > = - Q P ( I ) A C C = C . Q.= Q D ( I ) + Q M ( I ) I D O 3 3 * . P ( I , J l = - O P ( T > + 0 * A C C 3 3 C O N T I N U E C = > R t A D I N A A N D H . 1 0 1 F O R M A T ( 3 F l C t ) N O 1 7 9 J 1 = 1 , M D O 1 7 9 J 2 = 1 . N 1 7 9 A ( J l , J 2 ) = 0 . ' C . N O T E T H A T T H I S I S A N O T H E R D E P A R T U R E F R O M T H E O R I G I N A L C C O D E . W H I C H W O U L D R E Q U I R E T H E U S E R T O E N T E R E A C H R O W O F C T H E _ M A T R I X I N 3 F 1 0 \u00E2\u0080\u00A2 ] * _ F 0 R M A T . W I T H A U R G E H & T R I X . . . . W H I C _ H _ C T H E N O N Z E R O E N T R I E S . S O - O R E A C H R O W I N P U T T H E C O L U M N N O . C ( 1 3 ) F O L L O W E D B Y T H E E N T R Y ( F 1 0 . A ) . W H E N A R O W I S C O M P L E T E r. E N T E R A N U L L L I N E . D O 1 6 9 K U S Y = i , M 1 9 7 R E A D ( 5 , 1 3 \u00C2\u00AB ) I N O . T E M P I F U N D . E O . O ) G O T O 1 8 9 G O T O . 1 S 7 \u00E2\u0080\u00A2 1 3 9 C O N T I N U E I F ( M l . E Q . O ) GO T O 9 7 3 R E A O ( 5 , 1 0 1 ) ( H ( I ) , 1 = 1 , M l ) r r ^ C O U j J H U E H ( 1 9 > = X 1 0 3 H ( 1 6 ) = X 1 H ( 1 7 ) = X 1 9 H ( 1 H > = X 6 1 1 3 b F O R M A T ( / / , T ? 3 , 7 . ( \" * \" ) , / \u00C2\u00BB J _ 2 . G . , _ _ & - K A T R I X _ , . / , T 2 0 , 3 0 ( \" \u00C2\u00BB \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 ) , / > 1 8 8 F O R M A T , / ) O O 2 2 1 = 1 , M-2 2 W R I T E ( 6 , 1 0 2 ) I , H ( I > C = > I N I T I A L I Z E P I A N 9 W . O O 4 0 I = 1 , M O R L E _ L i ._5 \u00E2\u0080\u00A2 D I ( I ) = S I C- N ( O 3 L E , H ( I ) ) O O 4 1 J = i . M 4 1 W ( I , J ) = 0 . 4 0 W ( I , I ) = R I ( I ) C W I S Z E R O E X C E P T ^ O R ? I O N T H E D I A G O N A L S C = > I N I T I A L I Z E L , K , I W . D E L T A , _ _ A N O _ _ G A M M A ! C A L C U L A T E H A N D - Z Q . D O 5 0 1 = 1 , M 2 G A M M A ( I ) = G . D E L T A ( I ) = 1 E 7 0 L t i l = 0 5 0 K A \u00C2\u00B0 P A ( I ) = 0 Z O = 0 . , , D O 5 1 1 = 1 , M I W ( I ) = - T I F ( H ( I ) . G E . O . I G O T O 5 1 H i I ) = - H ( I ) 5 1 Z O = Z O - H ( I ) D O 5 2 1 = : . : 5 2 C ( I ) = 0 . c=> i n i n i i i l i i u r n i l i n m i n n i l m i l C = > P H A S E I t B E G I N C O L U M N 3 1 V O T I N G W I T H M A U = 1 . 2 0 0 C A L L C L M P V T C C B A R , S , 1 1 . I F ( C B A R . L T . - E P S ) G O T O 2 0 2 I F j T 0 . G E . _ - E o . S \u00C2\u00BB _ 2 Q T P _ 2 0 3 \" C = > C B A R . G E , 0 A N D \" Z O . L T . G . W R I T E ( 6 , 2 S 7 ) 2 0 7 F O R M A T ( \" I N F E A S I B L E \" ) C A U L D U M P * C = > C B A R . G E . 0 A N O Z O . G E . 0 . 2 0 3 D J ) _ _ 0 j _ _ J = _ _ _ I F ( I W ( J ) . L T . 0 ) G O T O 2 0 5 2 0 4 C O N T I N U E \" 0 T O 3 G ? 2 0 5 W R I T E ( 6 , 2 D 8 I 2 0 8 c O R H A T ( / \" \u00C2\u00B0 M A S E I O E G E N E g A C Y \" ) C A L L ' 0 U M D C= > C B 5 R < 0 . 2 0 2 C A L L S M X \u00C2\u00B0 V T ( C B A R , S , M U > GO T O 2 0 0 ~ = > ? H A S E I I : R E A D Jt< C - V E C T O R : _3 C_G REA_D_ ( 5 , 1 1 9 ) _(C ( 11 . I = 1 . H ) _ 1 1 9 F O R M A T ! 3 r 1 0 . U ) 3 5 F O R M A T < / / , T 2 0 , 3 0 ( \" t , / , T 2 0 , \" * / , T 2 G , 3 0 ( \" * \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 ) , /> DO 7 0 9 J 1 = 1 , N 7 0 9 C O N T I N U E 7 1 0 F O R M A T ! \" C O S T ( \" , 1 3 . \" ) = \" . ^ 1 A . A ) C = > S E T G A M M A . DO 3 0 1 1 = 1 , M ? 3 0 1 G A M M A ( I ) = P ( I , K ( I ) + 1 ) C = > S E T G . DO 3 0 2 J = 1 , M I F ( I W J J ) . L E . N ) GO T O 3 0 3 G ( J ) = G A M * A ( I W ( J ) - N ) GO T O 3 0 2 3 0 3 n [ J ) = C ( I \u00C2\u00AB ! J D 3 0 2 C O N T I N U E C = > S E T F I , Z O Z O = 0 . [ DO 3 0 A I = 1 , M P I ( I ) = 0 . DO 3 0 A J = 1 , M ?-G'u P I ( I ) = P I ( I i +G ( J ) \u00C2\u00BB W { J , I ) C = > 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 C = > 3 E G I N C O L U M N F I V O T I N G W I T H M A U = 1 . A G O C A L L C L K P V T ( \" C B A R , S , 1 ) I F (C9A<= . G E . - E F 3 ) G O TO 5 0 0 C A L L S M X P V T I C 8 A R . S , M U ) I F ( M U . N E . 2 ) G O T O A 0 0 W R I T E ( 6 , A 0 2 > A 0 2 F Q ? M A T ! / \" U N B O U N D E D \" ' . ^ C A L L D U M P . C = > C R A R > = 0 . ! S E T D E L T A , G A M M A . 5 G 0 DO 5 0 1 1 = 1 , M ? D E L T A ( I ) = D ! I , 1 ) 5 0 1 G A M M A ( I ) = P ( 1 , 1 > _ C = > S E T L , K A P P A , H , G A M M A , P I . DO 5 j 2 J = ' l ,~M \" I F < I W ( J t . L E . N ) GO T O 5 0 2 N U = I W ! J ) - N L < N U ) = 1 Y = 0 . P H I = 1_ ; K I = K ( N U 1 \u00C2\u00AB \u00E2\u0080\u00A2 ! DO 5 0 3 K K = 1 , K I 2 0 6 P H I = K K I R ( Y + O ( N U . K K ) + E P S . G T . H < J ) } G O T O 5 C (\u00E2\u0080\u00A2 5 0 J 3 . Y = V + n ( N U . . K J O , _ * 5 G A I t =~3 < N U ,'\u00C2\u00B0 H I ) - C ( N'J , K V N U ) + 1 1 0 E L T . f i ( M l ) ) . = 0 ( M U , P H I ) H ( J | = H ( J ) - Y G A M M A ( N U ) = P ( N U . = H I ) K A = P A ( N U ) = P H I - 1 0_0_5 G J L . . I = 1 \u00E2\u0080\u009E , M 5 0 5 P I ( I 1 i = P I ( I ) + X * W ( J , 1 > 5 0 2 C O N T I N U E C = > 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 C = > B \u00C2\u00A3 G I N C O L U M N P I V O T I N G A G A I N W I T H M A U = 0 . 7 0 0 . C A L L C L M P V T ( C B A R , S , 0 ) I_F_ I C B A R . G E . - E P S ) __GO_ _T_Q. A O 0 7 0 1 0 0 7 0 3 J = 1 , M G ( J ) = 0 . D O 7 0 3 1 = 1 , M 7 0 3 G I J ) = G I J ) * W ( J , I ) \u00C2\u00BB A ( I . S ) C A L L U P P . P V T ( C 8 A . P , S - N , 2) G O T O 7 0 0 , C = ^ B E G I N U P P E R B C U N O P I V O T I N G . 6 0 0 D O 6 0 1 I = 1 . M 2 I F ( L ( I ) . E O . l ) G O T O 6 0 1 K K = M 1 + ' I C = > T E S T 1 . C B A R = G A M M A ( II+ P I J K K . ) _ , : . _ I F ( C B A R . G E \u00E2\u0080\u00A2 - E P S > G O T O 6 0 2 D C 6 0 3 L L = 1 . M 6 0 3 G ( L L > = - \u00C2\u00AB ( L L , K K > C A L L U P R P V T ( C 9 A R , I , 1 ) G O T O 7 0 0 C = > T E S T 2 . , ; 6 0 2 I p ( K A P P A ( I ) . E O . 0 ) G O T O 6 0 1 C 3 A R = ? I ( K K 1 + P ( I , K A P P A ( I > ) I F ( C B A R . L E . E P S ) G O T O 6 0 1 D O 6 0 A L L = 1 , M 6 0<\u00E2\u0080\u00A2 G ( L L ) = W ( L L , K K I C A L L U P R P V T ( C B A R . I . 3 ) . G O T O 7 0 0 6 0 1 C O N T I N U E C = > W H E N T H E L O O P I S S A T I S F I E D , W P I T E O U T T H E O P T I M A L S O L U T I O N A N D S T O \u00C2\u00B0 7 0 2 F O R M A T ; \" ! \" , T 2 D , 3 0 ( \" * \" ) , / , T 2 0 , \" O P T I M A L S O L U T I O N \" , / , T 2 0 , 3 0 ( \" * \" ) , / ) C A L L _ ? _ R I f J T R E T U R N \" E N O 2 0 7 C = > S U B R O U T I N E S SLZZS. S U B R O U T I N E P I V O T ( C B 4 R , R ) I M P L I C I T R F A L ( A - H , 0 - Z t I N T E G E R S , P . I , M A R T Y R E A L P ( 7 G , 1 3 > . D < 7 j , 1 0 ) . A ( i : ; J , 2 & G ) , H < l G O ) . C < 2 f c 0 i R E A L W ( 1 0 C , 1 0 0 ) . G ( : 0 0 > ? O E L T A ( 7 0 ) , G A M K A ( 7 0 > . P I ( 1 G G > _ Q I M E N S I G N I N ( l Q j ) , K A P P A t 7 . 0 1 . , . L J L I O J U J L O . 0 . 1 P ^ A L O P ( 7 0 ) , O M ( 7 0 I \u00E2\u0080\u00A2 C O M M O N N , M , M 1 , M2 , P , 0 , A , W , C M . G , 0 E L T A , G A M M A , P I , I W , K A P P A , L . E P S , \u00E2\u0080\u00A2 S K , O P , O M , Z O , M A R K E R C = > C A L C U L A T E P I V O T A L R O W , H ( R > G S = 1 . 0 / G ( P > \u00E2\u0080\u00A2 0 0 1 0 J = 1 . M \u00E2\u0080\u0094 \u00E2\u0080\u0094 1 0 . W ( R , J ) = W ( R , J ) * G S H ( R ) = G S * H ( ? ) C = > P I V 0 T O N O T H E R R O W S . . 0 0 1 1 1 = 1 , M I F ( I . E Q . R ) G O T O 1 1 G S = G ( I ) DO 1 2 J = 1 , M 1 2 W ( I , J t = W ( T , J ) - G S * W ( R , J ) H ( I ) = H ( I i - G S * H ( R ) 1 1 C O N T I N U E C = > C A L C U L A T E P I A N D Z O . DO 1 3 J=_1_,_M 1 3 P I ! J ) = P I ( J ) + C 9 A R * W ( R , J ) Z0.= Z 0 - C 9 A R \" H ( R ) 9 7 6 R E T U R N E N D 2 0 8 (-, = = = = = = = = = === = = = == = = == = ====== = == = = ===== = = = = = = = = = = = = = = === = = = = = = = = == = S U B R O U T I N E RW=>VT ( T , R , MU> I J L P J J f RE.A L. _ A - _ t i ! _ J } I N T E G E R S , R , \u00C2\u00B0 H I , M A R T Y R E A L P ( 7 _ , l G t , D ( 7 G , i O I . A ( 1 0 G . 2 6 G > , H ( l G G > , C ( 2 6 0 ) R E A L W ( l C-j , I C O ) . G d O i i l . O E L T A I / O ) \u00E2\u0080\u00A2 G A M M A ( 7 0 ) \u00C2\u00BB \u00C2\u00B0 I ( 1 G O ) D I M E N S I O N I'-l ( 1 0 0 ) , K A P = A ( 7 0 ) , L ( 7 j ) , < ( 7 0 ) R E A L Q af70>,QM(70) C O M M O N N , M ,_M 1 . M2. ,_P ,_D , A_ W,_jC , H , G , D E L T A , GA M M A , P I , I W. K A P R A . L , E \u00C2\u00B0 S , ? K , O P . O M , 7 0 , M A R K E R T = 1 E 7 C C = > F I N D M I N R A T I O H ( J ) / G ( J ) W H E R E G ( J ) > 0 . 0 = > F I N D M I N R A T I O ( H ( J ) - 0 E L T A ( I W (J> - N ) ) / G < J ) S T K > 0 ? G I J K B . 0 0 1 1 J = 1 , M . I F ( G ( J ) . L E . - E P S ) GO TO 1 0 f F ( G ( J ) . L T . E P S ) GO T O 1 1 C = > I F G < J ) > 0 . R A T I O = H ( J ) / G ( J ) I F ( R A T I O . G T . T ) GO T O 1 1 T = R A T I O _ _ J M U = 0 GO T O 1 1 C = > I F G ( J ) < 0 . 1 0 K K = I W ( J ) - N I F ( K K . L E . O ) GO TO 1 1 _ R A T I Q = ( M ( J ) - Q E L T A IKK) \u00C2\u00B1/0\u00C2\u00B1J ) I F ( R A T I C . G T , T ( GO T O 1 1 T = R A T 1 0 R = J MU = 1 1 1 C O N T I N U E C = > I F MO J F Q U N O MiJ = 2 . I F ( T . G E . 1 E 7 0 ) M U = 2 9 7 6 R E T U R N E N D 2 0 9 S U B R O U T I N E C L M P t f T ( C B A R . S . M A U ) I J _ _ L I C . U _ \u00E2\u0080\u009E F A . L ( A - H , 0-21 , -I N T E G E R S . R , P H I , M A R T Y R E A L P ( 7 0 , 1 0 J . D ( 7 0 , I D ) , A ( 1 3 0 , 2 6 0 ) . H ( l 0 0 ) , C ( 2 6 0 > R E A L W ( 1 0 0 , 1 0 G ) , G ( i j O ) , n E L T A ( 7 C ) , 3 A M M A ( 7 U ) , P I ( l u O ) D I M E N S I O N I W ( 1 C . O ) , < A \u00C2\u00B0 P A ( 7 0 ) . L ( 7 J ) , < ( 7 0 ) R E A L 0 p 1 7 o i , O M ( 7 0 ) C O M M O N N , M , M I , M ; , ? j _ o _ j L . _ n J . j _ i ^ 5 K , O P , O M , Z O . M A R K E R C R A R = 1 E 7 0 S = 0 C = > F T N D M I N ( C - \u00C2\u00B0 I * A ) = C B A R D O 1 3 J = 1 , N X = C ( J J : DO 1 1 1 = 1 , M 1 1 X = X - P I ( I } * A < I , J ) I F < X . G E . C B A R - E P S ) GO T O 1 0 C R A R = X S = J _1_Q C O N T I N U E I F ( M A U . E O . 0 ) R E T U R N C = > F I N D M I N ( C R A P , G A M M A ( \u00E2\u0080\u00A2 ) + P I ( \u00C2\u00BB + M l ) ) = C 3 A R O O 1 2 1 = 1 , M 2 X = G A M M A ( I ) <\u00E2\u0080\u00A2 P K I + M l ) I F ( X . G E . C B A R - E P S ) GO T O 1 2 C 3 A R = X , ; S = I + N 1 2 C O N T I N U E B 7 6 R E T U R N E N D 2 1 0 C = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = S U B R O U T I N E U P R P V T ( C - 4 R . I . K K ) I M P L I C I T P f A L ( A - M , 0 - Z . \u00C2\u00BB C = > T H I S I S V E R S I O N 2 0 F U 0 R \u00C2\u00B0 v f . I N T E G E R S . R , \u00C2\u00B0 H I R E A L P ( 7 0 . 1 0 ) . 0 ( 7 S . 1 0 ) , A ( 1 3 0 , ' 2 o 0 I , H ( 1 0 u t , C ( 2 6 0 ) R E A L W ( 1 0 0 , l C G > , G ( i j 3 > , D E L T A < 7 C > , G A M M A ( 7 C > . P I ( 1 J O ) 0 1 M E N S I O N I W ( 1 0 3 ) , < A \u00C2\u00B0 \u00C2\u00B0 A ( 7 0 I , . ( 7 0 I , K ( 7 0 1 R E _ A L Q D ( 7 0 1 _ . f l M ( 7 G J _ _ C O M M O N N , M , M 1 , M 2 \u00C2\u00B0 7 o , A , W , C , H , G ,~D E L T A , G~A M M A , ? I , I w , K A P P A , L , E P S , 5 K . O P . O M , Z O . M A R K E R L O G I C A L F L A G C A L L R W P V T ( T , R . M U ) K K = M 1 + I \u00E2\u0080\u00A2 P L A G = . F A L S E . . ' A L S H A = 0 . I F ( K I K . N E . O ) G O T O 2 0 C = > K I K = 0 F I N D T H E L A R G E S T L S = l , 2 , . . . , K A p o A ( I ) . S . T . C = > P ( I , K A P P A ( I ) - L L ) + P I < K K > > 0 . A N D T > = S U M O F D ( I , K A P P A < I ) - S ) ; S = 1 , . . . , L L K I = K A P P A ( I > DO 1 0 L L = 1 . K I , A L \u00C2\u00B0 H A = A L D H A + 0 ( I . K A P P A ( D - L L + l ) I F ( P ( I . K A P P A ( I ) - L L + 1 ) + P I ! K K > . L E . E P S . O R . T . L T . A L P H A - E P S ) G O T O 3 0 L S = L L A S = 4 L P H A 1 0 R L A G = . T R U E . G O T O 3 0 C = > K I K = 1 . F I N D T H E L A R G E S T L S = 0 , 1 . . . . . K ( I ) - K A P P A ( I ) S . T . C = > \u00C2\u00B0 ( I , K A P P A ( I ) + L L ) < 0 A N D T > = S U M O F D ( I , K A P P A ( I ) - S ) : S = 0 L L . 2 0 I F ( K l K . H E . l t G O T O 4 0 K I = K ( I ) *\u00E2\u0080\u00A2 1 - K A P P A ( I ! D O 2 1 L L = 1 . K I A L \u00C2\u00B0 H A = A L P H A \u00C2\u00BB p ( I , K A P P A ( I ) + L L ) I F ( P ( I . K A F 3 4 ( I } * L L ) + P I (.< K t . 3 E \u00E2\u0080\u00A2 - E P S . O R . T . L T . 4 L P H A - E P S ) G O T O 3 0 L S = L L A S = A L P H A 2 1 F L A G = . T R U E . C = > S E E I F S O M E L S . F O U N D . ( I F N O T = > I V O T A N D R E T U R N ) ' 3 0 I F ( . N O T . F L A G ) G O T O 4 0 ~ C = > S O M E \" \" L S F O U N D 1 ^ ( K I K . E Q . 0 ) L S = - L S D O 3 1 J = 1 , M 3 1 H ( J ) = H ( J ) - A S * G ( J ) K A \u00C2\u00B0 P A ( I ) = K A P P 4 ( I ) + L S I F _ 0 _ T K \u00E2\u0080\u00A2 ~ 1 * A > _ R a p i ~ \u00C2\u00B0J J ' < * p t i ? < I> 1 1 1 1 _ I < K K J \" \" I F \" ( K I K . ' t O . O ! T. \" A '\u00E2\u0080\u00A2\u00E2\u0080\u00A2-'\u00E2\u0080\u00A2:> ( I , < A = =\u00C2\u00BBi i 1 1 > * P I ( K < ) G A M M A ( I ) = P ( I , K A \u00C2\u00B0 P A ( I ) + 1 ) D E L T A ( I ) = D f I , K A \u00C2\u00B0 P A ( I ) + 1 ) T = T - A S I F ( K I K . E O . O . A N D . P ! I , KA P P A ( I ! + 1 ) + P I ( K .<) . L T . - E \u00C2\u00B0 S __C_R . K I K . E Q . . 1 . A N D . P ( I , K A \u00C2\u00AB > P f t ( I ) \u00E2\u0080\u00A2 ! > \u00C2\u00BB \u00C2\u00B0 I ( K K > . G T \u00E2\u0080\u00A2 \u00C2\u00A3 P S , $ ' \" . O f f . \" T . L T . - E P S ) G O T O 9 9 \" C = > O T H E R W I S E G G P I V O T A N D R E T U R N C = > P I V O T A N D R E T U R N UQ C A L L P I V O T ( C 3 A P , R ) I F ( K I K . N E . C > G O T O U2 ; K A D P A ( f > = K A ? P A ( I > - 1 D E L T A ( I ) = D ( X , K C P P A ( I t + l ) G A ' I H A ( T \u00C2\u00BB =P ( I , K A P \u00C2\u00B0 A ( I ) + 1 ! DO A l J = 1 , M U l W ( R , J > = - W ( R , J ) H ( R ) = - H ( P ) DO A A J = i , M A A \u00C2\u00B0 I (J ) =P-I ( J ) + 2 . * C ? A R \u00C2\u00BB W ( R , J ) DO A 3 J = 1 , M U 3 H ( J ) = H ( J ) - W ( J , I + M 1 ) * D ( I , K A P P A ( I ) + l 4 2 N U = I W ( R \u00C2\u00BB I W ( R ) =N+ I \" I F ( I . G T . O ) L ( I ) = 1 I F ( N U . L E . N ) GO TO 9 9 N U = N U - N L ( N U ) = 0 I F ( M U . E O . O ) GO T O 9 9 I.V = M j , . \u00C2\u00B1 N y DO 5 1 J = i , M 5 1 M ( J > = H ( J ) + W ( J , I V ) \" D E L T A ( N U ) K A P P A ( N U ) = K A P P A ( N U ) + 1 D E L T A ( N U > = n ( N U , < A P P A ( N U ) + l ) G A H M A ( N U ) = P ( N U , < A P P A ( N U ) + 1 ! 9 9 R E T U R N E N D 2 1 2 S U B R O U T I N E S M X P ' V T ( C 3 A R . S , M U ) I H P L I C I T R E A L ( A - H j J W J \u00E2\u0080\u00A2- , I N T E G E R S \u00C2\u00AB R \" , \" P H I , M A R T Y R E A L P ( 7 0 , 1 0 > . D ( 7 G , 1 0 ) , A ( 1 3 Q , 2 6 0 ) , H ( l J O ) , C ( 2 6 0 > R E A L W ( I C G , 1 G 0 > , G ( 1 G O ) , D E L * A ( 7 0 I , G AMM A ( 7 0 ) , P I ( 1 0 0 ) D I M E N S I O N I W ( 1 0 0 ) , K A P P A ( 7 0 ! , L ( 7 0 ) , < ( 7 0 ) R E A L Q P ( 7 0 ) , O M ( 7 0 ) C O M M O N N , M , M l , M 2 , P , O . A , W , C , H , G , D E L T A , G A M M A , P I , I w , K A P P A , L , E P S , j $ \" K , Q P , O M , Z ' 0 , M A R K E R I F ( S . G T . N ) GO T O 1 0 C = > S < = N . DO 1 1 L L = 1 , M G ( L L ) = 0 DO 1 1 J = 1 , M 1 1 G ( L L ) = G ( L L ) + W ( L L , J ) * A ( J , S ) GO T O 2 0 C = > S > N . 1 0 K K = S - N + M 1 DO 1 2 J = 1 , M 1 2 G ( J ) = - W ( J , K K ) ; ; C = > I N E f T H E R C A S E . 2G C A L L R W P V T ( T , R , M U ) I F ( M U . E O . 2 ) GO TO 9 9 I W ( R ) = S C A L L P I V O T ( C 9 A P , R ) 9 9 R E T U R N ; : ; ^ E N D . \" 2 1 3 S U B R O U T I N E P = T M T J M P L . I C I T R E A L ( A - H , _ Q - Z . i _ _. . ... I N T E G E R S , R , P H I . M A R T Y R E A L C H I S ( 7 0 ) R E A L T P ( 7 0 , 1 0 ) , T O ( 7 0 , 1 0 ) R E A L P ( 7 C , 1 0 > , Q ( 7 0 , 1 0 ) , A ( l i ) 0 , 2 6 0 > , H ( 1 0 G > , C ( 2 6 0 ) R E A L W ( 1 0 0 , 1 0 0 ) , G ( 1 0 0 ) , D E L T A ( 7 0 ) , G A M M A ( 7 0 ) , P I ( 1 0 C ) D H S N S I O N IWJX30 )jiCA\u00C2\u00A3\u00C2\u00A3l<-7_0_>_,. . J..73J_,.' R E A L ~QP ( 7 0 ) , X ( 1 5 0 ) , Q M ( 7 0 ) C O M M O N N , M , M i , M ? , P , o , A , W , C , H , G , D E L T A , G A M M A , P I , I W , K A P P A , L , E 3 S , $ K , Q P , Q M , Z O , M A R K E R C O M M O N / P R I N T R / T P , T O C O M M O N / A / X 1 0 0 F O R M A T S / , T I P . \" B A S I S I N D E X D U A L V A R I A 3 L E S \" , / / , 3 0 1 \" \" > > D O 1 0 J = 1 , M 1 0 C O N T I N U E 1 0 1 F 0 R M A T ( T 1 6 . I 3 , T 2 2 , F 1 < , . 4 ! C = > C A L C U L A T E Z O . Z S O = 0 . D O 2 0 J = 1 . M ' Z C I F ( I W ( J ) . L E . N ) Z S 0 = 7 S 0 \u00C2\u00BB - C ( I W ( J l ) * H ( J ) C D O 2 1 1 = 1 , M 2 C K I = K A P P A ( I ) C I F [ K I . E O . 0 I G O T O 2 1 C D O 2 2 K K = 1 , K I C 2 2 Z S O - Z S O + P ( I . K K ) \u00C2\u00BB D ( I , K K ) C 2 1 C O N T I N U E 1 0 3 F O R M A T ( / / / T 2 0 , \" O P T I M A L O B J E C T I V E V A L U E ( W I T H O U T P E N A L T I E S ) = \" , T 6 5 , F 1 $H. A ) C = > F I N D T H E X - V A L U E S F R O M I W li H . D O 3 1 1 = 1 , N 3 1 X ( I ) = Q . : D O 3 2 1 = 1 , M 3 2 I F ( I W ( I ) . L E . N ) X ( I W ( I ) ) = H ( I ) C = > W R I T E O U T X \" S . W R I T E ( 6 , 1 2 3 7 ) 1 2 3 7 F O R M A T ( l H l ) W R I T E ( 6 , 3 5 ) 3 5 F O R M A T ( / / . T 2 0 , 3 0 ( \" \u00C2\u00BB \" ) , / , T 2 0 , \" O P T I M A L S O L U T I O N V E C T O R \" , * / , T 2 0 , 3 0 ( \" * \" > , / ) D O 3 3 1 = 1 , N 3 3 W R I T E ( 6 . 1 0 2 ! I , X ( I ) 1 0 2 F O R M A T ( \" X ( \u00C2\u00BB , I 3 , - ) = \" , F 1 5 . A ) C = > C A L C U L A T E T H E . A L P H A . * ' S _ A _ N O . J P \u00C2\u00AB . I ~ S , 3 6 F O R M A T ( / V , T 2 0 , A 2 ( \" * ' * ) , / \u00C2\u00BB T 2 G , \" R I G H T H A N D S I D E F O R S T O C H A S T I C C O \" , * \" N S T R A I M T S \" , T 2 0 , A 2 ( \" \u00C2\u00BB \" ) , / ) D O 3 0 1 = 1 . M 2 A L \u00C2\u00B0 H A = - ( P ( 1 , 1 I * P I ( M 1 \u00E2\u0080\u00A2 I ) ) / ( P ( I , K ( I ) t i ) - P ( I , i i I C H I = 0 . D O 3 A J = I , N ; .. : \u00E2\u0080\u00A2_ 3 A C H I = C H I + A ( T + M 1 , J ) * X ( J ) C H I S ( I \u00C2\u00BB = C H I 2 1 4 [1=1+ \"u 3 0 C O N T I N U E 1 0 j _ J L O _ M J J ^ P E N = 0 4 3 F O R M A T ( / / , T 2 J . 30 ( \" * \" > , / t T 2 0 \u00C2\u00BB \" I N D I V I D U A L D _ N A L T I Z S \" , * / , T 2 0 . 3 0 ( \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 * \" > . / ) M . Z = M 1 * 1 OO 3 8 1 K A = 1 , M 2 P E N 1 = 0 . '< I = K. ( K A I DO 3 8 7 K R = 1 . K I T F ( C M I S ( K A ) . L T . T D ( K A , K 9 ) > P E M 1 = =>E N 1 + - C H I S (< A ) > * Q P < KA > * ~ * T P ( K A t K S ) 3 3 ' I F ( C H I S I K A ) . G T . T D C K A . KB>> P E N 1 = \u00C2\u00B0 E N 1 * ( C H I S ( K A ) - T O ( K A . K D ) I * Q M < KA I ' * T 3 ( K A , K B ) . _ _ P E N = P E N + \u00C2\u00B0 E N 1 K B = K A + t ' l \u00E2\u0080\u00A2 3 . 3 3 F O R M A K T F , \" P E N A L T Y F O R R 9 W ( \" , 1 3 \u00C2\u00BB \" ) = \" \u00C2\u00BB F 1 5 . 5 > 3\u00C2\u00ABi C O N T I N U E 3 3 4 F O R M A T ( / / / / / / \u00C2\u00AB T 2 C , \" T O T A L P E N ! L T Y = \" , T 6 2 . F 1 8 . 5 > R E T U R N : '. \u00E2\u0080\u0094 E N D . v 2 1 5 ' S U B R O U T I N E D U M P I M P L I C I T P E A L ( A - H . O - 2 1 . . . -I N T E G E R S , R . \u00C2\u00B0 H I , M A R T Y R E A L P ( 7 0 . 1 0 ) , 0 ( 7 0 , 1 3 ) . A ( 1 ] 3 , 2 6 0 ) , H ( I C Q I , C ( 2 6 0 ) R E A L W ( 1 C O , I S O > , G < 1 0 0 ) , \" E L L A < 7 0 ) , G A M M A < 7 0 ) , P I ( 1 O G ) D I M E N S I O N IW ( 1 0 0 ) , K A R P A ( 7 0 ) , L ( 7 0 ) , K ( 7 0 I R E A L 0 \u00C2\u00B0 ( 7 0 ) , O M ( 7 0 ) C O M M O N N , M , M l , * 2 , P , P , A , W. C . H . G . D E L T ft , G A MM A , P I . I K \u00C2\u00AB . . K A P _ _ L \u00C2\u00AB _ _ t _ P J u > % \" . \" K , Q P , Q M , Z D . M A R K E R W R T T E ( 6 , 1 0 G > 1 0 0 F O R M A T ( / / \" T - 0 M E G A= ! H = ! \u00C2\u00B0 I = ! W = % \", S / \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 + \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 . \u00C2\u00AB ( \" _ _ \" ) ) DO 1 0 J = l . M : : 1 0 W R I T E ( 6 , 1 0 1 > I W ( J ) , H ( J ) , \u00C2\u00B0 I ( J > 1 0 1 F O R M A T ( \" \" . 1 1 0 F i u ,u, \u00E2\u0080\u00A2\u00E2\u0080\u00A2>\", F n , i \u00C2\u00BB , ( T 4 6 , \" > . \" , 5 F 1 4 . 4 ) ) W R I T E 1 6 . 1 0 3 ) ( G ( I > , I = 1 , M > 1 0 3 F O R M A T ( / \" G = \" , ( 6 F 1 5 . 5 ) ) W R I T E ( 6 , 1 Q 5 ) ( L ( I > . I = 1 . , M 2 ) 1 0 5 F O R M A T ( \" L = \" , ( 1 2 1 1 0 ) ) W R I T E ( 6 , 1 0 6 ) ( D E L T A ! I ) , 1 = 1 , M 2 ) 1 0 6 F O R M A T ! \" D E L T A = \" , ( 6 E 1 5 . 5 ) ) W R I T E ( 6 . 1 C 7 ) ( G A M M A ( I ) , 1 = 1 , M 2 ) 1 0 7 F O R M A T ( \" G A M M A = \" , ( 6 F 1 5 . 5 ) ) W R I T E ( 6 , 1 1 0 ) 7 0 1 1 0 \u00E2\u0080\u00A2 _F ORM A T ( \" Z O = \" , F 1 5 . 5 J , W R I T E ( 6 . 1 1 1 ) ( K A P P A ( I ) , I = 1 , M 2 ) 1 1 1 F O R M A T ( \" K A P P A = \" , ( 1 2 1 1 0 ) ) R E T U R N E N D 2 1 6 C h a p t e r 6 SUMMARY, MAJOR FINDINGS AND DIRECTIONS FOR FURTHER RESEARCH 6 . 1 I n t r o d u c t i o n I n t h i s c h a p t e r a s u m m a r y o f t h e d i s s e r t a t i o n ( 6 . 2 ) , m a j o r f i n d i n g s o f t h e d i s s e r t a t i o n ( 6 . 3 ) , a n d d i r e c t i o n s f o r f u r t h e r r e s e a r c h ( 6 . 4 ) a r e p r e s e n t e d . 6 . 2 S u m m a r y I n t h e l i t e r a t u r e t h e s t u d y o f a s s e t a n d l i a b i l i t y m a n a g e m e n t o f b a n k s h a s b e e n a p p r o a c h e d f r o m t w o p o i n t s o f v i e w . T h e f i r s t i s b a s e d u p o n a m e a n - v a r i a n c e a p p r o a c h t o p o r t f o l i o s e l e c t i o n . T h e s e c o n d a p p r o a c h i s b a s e d o n a n o b j e c t i v e o f m a x i m i z i n g n e t e x p e c t e d r e t u r n s . I n C h a p t e r 1 , b y u s i n g M y e r s ' c r i t e r i a [ 6 1 ] , i t w a s s h o w n t h a t t h e a p p r o p r i a t e c r i t e r i o n f o r a f i n a n c i a l i n s t i t u t i o n i s t h e m a x i m i z a t i o n o f e x p e c t e d n e t r e t u r n s . I n C h a p t e r 2 , d e t e r m i n i s t i c a n d s t o c h a s t i c m o d e l s w h i c h a s s u m e d t h i s c r i t e r i o n w e r e s u r v e y e d . I n t h e m o s t c o m o r e h e n s i v e o f t h e s e m o d e l s , B r a d l e y a n d C r a n e [ 5 ] a t t e m p t e d t o o v e r c o m e t h e c r u c i a l o b s t a c l e ( t o a s s e t a n d l i a b i l i t y m a n a g e m e n t ) o f i n c o r p o r a t i n g u n c e r t a i n t y w h i l e m a i n t a i n i n g c o m p u t a t i o n a l t r a c t a b i l i t y f o r l a r g e p r o b l e m s . U n f o r -t u n a t e l y , t h e i r f o r m u l a t i o n i s n o t a p p e a l i n g f o r d e c i s i o n - m a k i n g b e c a u s e i t n o t o n l y h a s s e v e r e c o m p u t a t i o n a l l i m i t a t i o n s b u t a l s o p o s s e s s e s s u c h u n d e s i r a b l e f e a t u r e s a s a r b i t r a r y c o n s t r a i n t s o n c a p i t a l l o s s e s , a n a b s e n c e o f p o r t f o l i o m i x c o n s t r a i n t s a n d a n i m m e d i a t e r e v i s i o n t h a t m u s t s a t i s f y a l l p o s s i b l e f o r e c a s t e d e c o n o m i c s c e n a r i o s . 2 1 7 G i v e n t h e s e d e f i c i e n c i e s , t h e p u r p o s e o f t h i s d i s s e r t a t i o n w a s t o p r e s e n t a n a s s e t a n d l i a b i l i t y m a n a g e m e n t m o d e l t h a t i s b o t h c o m p u t a t i o n a l l y t r a c t a b l e a n d r e a l i s t i c f o r l a r g e p r o b l e m s . I n C h a p t e r 3 , t h e A L M f o r m u l a -t i o n w a s d e v e l o p e d a s a n a l t e r n a t i v e a p p r o a c h t o a s s e t a n d l i a b i l i t y m a n a g e -m e n t . T h i s m o d e l i n c o r p o r a t e s t h e i n h e r e n t u n c e r t a i n t y i n a s s e t a n d l i a b i l i t y m a n a g e m e n t , w h i l e m a i n t a i n i n g c o m p u t a t i o n a l t r a c t a b i T i t y . I n C h a p t e r 4 , t h e A L M f o r m u l a t i o n w a s a p p l i e d t o V C S i n o r d e r t o d e m o n s t r a t e t h e e f f o r t n e c e s s a r y t o e x e c u t e t h e m o d e l . T h e r e s u l t s o f t h i s a p p l i c a t i o n i n d i c a t e t h a t : 1 ) t h e A L M m o d e l i s s u p e r i o r t o e q u i v a l e n t d e t e r m i n i s t i c m o d e l s , a n d 2 ) t h e r e s u l t s i m p r o v e a s t h e i n f o r m a t i o n i n c o r p o r a t e d i n t o m o d e l i n c r e a s e s . I n C h a p t e r 5 , b y u s i n g a s i m u l a t i o n t o r e f l e c t a r e a l ( u n c e r t a i n ) e n v i r o n m e n t , t h e f l e x i b i l i t y o f S L P R a n d SDP f o r m u l a t i o n s w a s c o m p a r e d . T h e r e s u l t s o f t h e s i m u l a t i o n i n d i c a t e t h a t t h e S L P R f o r m u l a t i o n l e a d s t o b e t t e r i n i t i a l p e r i o d d e c i s i o n s . T h i s i s d u e t o t h e r e s t r i c t i v e n a t u r e o f h a v i n g f i r s t p e r i o d p o r t f o l i o s , i n t h e S D P f o r m u l a t i o n , f e a s i b l e f o r a l l p o s s i b l e f o r e c a s t e d e c o n o m i c s c e n a r i o s . 6 . 3 M a j o r F i n d i n g s T h e o b j e c t i v e s o f t h i s d i s s e r t a t i o n , f i r s t , t o o b t a i n a c o m p u t a -t i o n a l l y t r a c t a b l e a s s e t a n d l i a b i l i t y m a n a g e m e n t m o d e l , a n d , s e c o n d , t o d e v e l o p a f o r m u l a t i o n t h a t c a p t u r e s t h e e s s e n c e o f t h e a s s e t a n d l i a b i l i t y m a n a g e m e n t p r o b l e m , a r e s u c c e s s f u l l y a c h i e v e d . M o r e s p e c i f i c a l l y , t h e c o m p u t a t i o n a l t r a c t a b i l i t y o f t h e A L M m o d e l w a s f o u n d t o b e s u p e r i o r f o r a n u m b e r o f r e a s o n s . F i r s t , t h e C P U t i m e u s e d t o s o l v e a S L P R w a s a p p r o x i m a t e l y t w i c e t h a t u s e d t o s o l v e 2 1 8 a n e q u i v a l e n t s i z e ; l i n e a r p r o g r a m . S e c o n d , t h e C P U t i m e u s e d f o r 5 0 0 i t e r a t i o n s o f t h e s i m u l a t i o n o f t h e s t o c h a s t i c d y n a m i c f o r m u l a t i o n ( B - C t y p e m o d e l ) w a s m u c h l a r g e r t h a n t h a t n e e d e d f o r 4 0 0 i t e r a t i o n s f o r t h e S L P R f o r m u l a t i o n ( 6 . 3 9 t o 0 . 2 4 h o u r s ) . T h i r d , w h e n a d d i t i o n a l t i m e p e r i o d s a n d / o r r e a l i z a t i o n s a r e a d d e d , t h e g r o w t h i n t h e s i z e o f t h e s t o c h a s t i c d y n a m i c f o r m u l a t i o n i s a p p r o x i m a t e l y e x p o n e n t i a l w h i l e t h e g r o w t h o f t h e S L P R i s a p p r o x i m a t e l y l i n e a r . C l e a r l y t h e s e f a c t s d e m o n s t r a t e t h e s u p e r i o r i t y o f t h e A L M f o r m u l a t i o n i n t e r m s o f c o m p u t a t i o n s w h e n c o m p a r e d t o t h e B - C f o r m u l a t i o n . F o u r t h , t h e A L M f o r m u l a t i o n i s e a s y t o o p e r a t i o n a l i z e a n d i s e q u i v a l e n t t o i m p l e m e n t i n g a l i n e a r p r o g r a m w h i l e p r o v i d i n g s u p e r i o r r e s u l t s . I n f a c t t o a p p l y t h e A L M m o d e l , t h e f o l l o w i n g i n f o r m a t i o n m u s t b e d e t e r m i n e d : 1 ) a n e s t i m a t e o f d e p o s i t f l o w s , 2 ) a n e s t i m a t e o f t h e t e r m s t r u c t u r e o f i n t e r e s t r a t e s , 3 ) a n e s t i m a t e o f w i t h d r a w a l r a t e s o f d e p o s i t s u n d e r v a r i o u s e c o n o m i c c o n d i t i o n s , 4 ) l e g a l c o n s t r a i n t s g o v e r n i n g t h e b e h a v i o u r o f t h e f i n a n c i a l i n s t i t u t i o n , 5 ) p o l i c y c o n s t r a i n t s , 6 ) t h e F e d e r a l R e s e r v e B o a r d ' s r e c o m m e n d e d r e s e r v e s f o r m a i n t a i n i n g a l i q u i d p o s i t i o n , a n d 7 ) t h e i n i t i a l p o s i t i o n o f t h e f i r m . T h i s i s t h e s a m e i n f o r m a t i o n t h a t i s n e c e s s a r y t o i m p l e m e n t a n e q u i v a l e n t d e t e r m i n i s t i c m o d e l . C l e a r l y , t h e s e f o u r p o i n t s i n d i c a t e t h a t c o m p u t a t i o n a l t r a c t a b i l i t y i s n o t a c o n s t r a i n i n g f a c t o r i n t h e A L M m o d e l . W i t h r e g a r d t o p r o b l e m f o r m u l a t i o n , t h e A L M m o d e l , T i k e t h e C h a m b e r s a n d C h a r n e s l i n e a r p r o g r a m m i n g m o d e l , i n c o r p o r a t e s m o s t o f t h e e s s e n t i a l c h a r a c t e r i s t i c s o f t h e a s s e t a n d l i a b i l i t y m a n a g e m e n t p r o b l e m . H o w e v e r , u n l i k e t h e C h a m b e r s a n d C h a r n e s f o r m u l a t i o n , t h e A L M f o r m u l a t i o n o v e r c o m e s t w o i m p o r t a n t d r a w b a c k s : 1 ) t h e i n h e r e n t u n c e r t a i n t y o f t h e p r o b l e m a n d 2 ) t h e c o n s e r v a t i v e n a t u r e o f t h e C h a m b e r s a n d C h a r n e s 2 1 9 f o r m u l a t i o n . T h e f i r s t d r a w b a c k h a s a l r e a d y b e e n d i s c u s s e d . B y u s i n g s t o c h a s t i c l i q u i d i t y c o n s t r a i n t s t h e s e c o n d d r a w b a c k i s d e a l t w i t h e f f e c -t i v e l y . F u r t h e r m o r e , w h e n t h e A L M f o r m u l a t i o n i s c o m p a r e d t o t h e B r a d l e y a n d C r a n e f o r m u l a t i o n , i t h a s b e e n d e m o n s t r a t e d i n a s i m u l a t i o n t h a t t h e A L M m o d e l i s m o r e f l e x i b l e a n d t h u s p r o v i d e s s u p e r i o r r e s u l t s . 6 . 4 P i r e c t i o n s f o r F u r t h e r R e s e a r c h T w o a r e a s f o r f u t u r e r e s e a r c h a r e d i s c u s s e d b e l o w . O n e s h o r t c o m i n g o f t h e A L M m o d e l i s t h a t i t i s n o t a d y n a m i c f o r m u l a t i o n d u e t o t h e l a c k o f e f f i c i e n t a l g o r i t h m s t o s o l v e s u c h p r o b l e m s . A s B r a d l e y a n d C r a n e h a v e s h o w n f o r a d y n a m i c m o d e l , a s t h e n u m b e r o f t i m e p e r i o d s a n d p o s s i b l e r e a l i z a t i o n s o f t h e r a n d o m v a r i a b l e s i n c r e a s e s t h e f o r m u l a t i o n \" b l o w s u p \" i n s i z e . T h e r e f o r e , t h e d e v e l o p m e n t o f a n e f f i c i e n t s t o c h a s t i c d y n a m i c p r o g r a m m i n g a l g o r i t h m w o u l d b e u s e f u l f o r t h i s p r o b l e m a n d f o r o p t i m i z a t i o n p r o b l e m s i n g e n e r a l . A n o t h e r a r e a t h a t h a s h o t b e e n d e a l t w i t h i n t h e d i s s e r t a t i o n , i s t h e p r o b l e m o f f o r e c a s t i n g . D e p o s i t f l o w s , i n t e r e s t r a t e s a n d w i t h d r a w a l r a t e s w e r e t a k e n a s g i v e n . T h e p r o b l e m o f f o r e c a s t i n g h a s b e e n t h e s u b j e c t o f n u m e r o u s r e s e a r c h p r o j e c t s a n d i s v i e w e d a s b e i n g b e y o n d t h e s c o p e o f t h i s d i s s e r t a t i o n . H o w e v e r , i n o r d e r t o p r o p e r l y i m p l e m e n t t h e A L M m o d e l , o n e w o u l d h a v e t o e s t i m a t e t h e d e p o s i t f l o w s , i n t e r e s t r a t e s a n d w i t h d r a w a l r a t e s u s i n g t h e a v a i l a b l e f o r e c a s t i n g t e c h n i q u e s . G i v e n t h e e x i s t i n g s t a t e o f k n o w l e d g e i n t h e a r e a s p e r t a i n i n g t o a s s e t a n d l i a b i l i t y m a n a g e m e n t , t h e A L M m o d e l f o r m u l a t e d i n t h i s d i s s e r t a t i o n a p p e a r s t o b e s u p e r i o r t o e x i s t i n g m o d e l s a s a n o r m a t i v e t o o l . 2 2 0 BIBLIOGRAPHY A r r o w , K . I . , \" T h e R o l e o f S e c u r i t i e s i n t h e O p t i m a l A l l o c a t i o n o f R i s k -B e a r i n g , \" Review of Economic Studies, V o l . 31 ( 1 9 6 4 ) , p p . 9 1 - 9 6 . B a u m o l , W . J . a n d Q u a n d t , R . E . , \" I n v e s t m e n t a n d D i s c o u n t R a t e s U n d e r C a p i t a l R a t i o n i n g : A P r o g r a m m i n g A p p r o a c h , \" The Economic Journal, V o l . 7 5 ( 1 9 6 5 ) , p p . 3 1 7 - 3 2 9 . B e a l e , E . M . L . , \" O n M i n i m i z i n g a C o n v e x F u n c t i o n S u b j e c t t o L i n e a r I n e q u a l i t i e s , \" Journal of the Royal Statistical Society, Series B. V o l . 1 7 ( 1 9 5 5 ) , p p . 1 7 3 - 1 8 4 . B o o t h , G . G . , \" P r o g r a m m i n g B a n k P o r t f o l i o s U n d e r U n c e r t a i n t y : A n E x t e n s i o n , \" Journal of Bank Research, V o l . 2 ( 1 9 7 2 ) , p p . 2 8 - 4 0 . B r a d l e y , S . P . a n d C r a n e , D . B . , \" A D y n a m i c M o d e l f o r B o n d P o r t f o l i o M a n a g e m e n t , \" Management Science, V o l . 1 9 ( . 1 9 7 2 ) , p p . 1 3 9 - 1 5 1 . , \" M a n a g e m e n t o f C o m m e r c i a l B a n k G o v e r n -m e n t S e c u r i t y P o r t f o l i o s : A n O p t i m i z a t i o n A p p r o a c h U n d e r U n c e r -t a i n t y , \" Journal of Bank Research, V o l . 4 . ( 1 9 7 3 ) , p p . 1 8 - 3 0 . , Management of Bank Portfolios, J o h n W i l e y I n c . , New Y o r k , 1 9 7 6 . B r i t i s h C o l u m b i a G o v e r n m e n t , Credit Unions Act of British Columbia, 1 9 7 3 . C a s s , D . a n d S t i g l i t z , J . E . , \" T h e S t r u c t u r e o f I n v e s t o r P r e f e r e n c e s a n d A s s e t R e t u r n s a n d t h e S e p a r a b i l i t y i n P o r t f o l i o A l l o c a t i o n : A C o n t r i b u t i o n t o t h e P u r e T h e o r y o f M u t u a l F u n d s , \" Journal of Economic Theory, V o l . 2 ( 1 9 7 0 ) , p p . 3 3 1 - 3 5 4 . C e n t r a l M o r t g a g e a n d H o u s i n g C o r p o r a t i o n , \" C a n a d i a n H o u s i n g S t a t i s t i c s , 1 9 7 5 . C h a m b e r s , D . a n d C h a r n e s , A . , \" I n t e r - T e m p o r a l A n a l y s i s a n d O p t i m i z a t i o n o f B a n k P o r t f o l i o s , \" Management Science, V o l . 7 ( 1 9 6 1 ) , p p . 3 9 3 -4 1 0 . 221 [ 1 2 ] C h a r n e s , A . , C o o p e r , W . W . a n d S y m o n d s , G . H . , \" C o s t H o r i z o n s a n d C e r -t a i n t y E q u i v a l e n t s : A n A p p r o a c h t o S t o c h a s t i c P r o g r a m m i n g o f H e a t i n g O i l , \" Management Science, V o l . 6 ( . 1 9 5 9 ) , p p . 7 3 - 7 9 . [ 1 3 ] C h a r n e s , A . a n d C o o p e r , W . W . , \" C h a n c e - C o n s t r a i n e d P r o g r a m m i n g , \" Management Science, V o l . 6 ( 1 9 5 9 ) , p p . 7 3 - 7 9 . [ 1 4 ] C h a r n e s , ' A . a n d K i r b y , M . J . L . . , \" A p p l i c a t i o n o f C h a n c e - C o n s t r a i n e d P r o g r a m m i n g t o t h e S o l u t i o n o f t h e S o - C a l l e d ' S a v i n g s a n d L o a n ' A s s o c i a t i o n T y p e o f P r o b l e m , \" R e s e a r c h A n a l y s i s C o r p o r a t i o n , 1 9 6 5 . [ 1 5 ] C h a r n e s , A . a n d L i t t l e c h i l d , S . C . , \" I n t e r t e m p o r a l B a n k A s s e t C h o i c e w i t h S t o c h a s t i c D e p e n d e n c e , \" Systems Research Memorandum No. 188, The Technological Institute, Northwestern University, A p r i l 1 9 6 8 . [ 1 6 ] C h a r n e s , A . a n d T h o r e , S . , \" P l a n n i n g f o r L i q u i d i t y i n F i n a n c i a l I n s t i t u t i o n s : T h e C h a n c e - C o n s t r a i n e d M e t h o d , \" Journal of Finance, V o l . 21 ( 1 9 6 6 ) , p p . 6 4 9 - 6 7 4 . [ 1 7 ] C h e n , A . H . Y . , J e n , F . C . a n d Z i o n t s , S . , \" T h e O p t i m a l P o r t f o l i o R e v i s i o n P o l i c y , \" Journal of Business, V o l . 4 4 ( 1 9 7 1 ) , p p . 5 1 - 6 1 . [ 1 8 ] : , \" P o r t f o l i o M o d e l s w i t h S t o -c h a s t i c C a s h D e m a n d s , \" Management Science, V o l . 1 9 ( 1 9 7 2 ) , p p . 3 1 9 - 3 3 2 . [ 1 9 ] C h e n g , P . I . , \" O p t i m u m B o n d P o r t f o l i o S e l e c t i o n , \" Management Science, V o l . 8 ( 1 9 6 2 ) , p p . 4 9 0 - 4 9 9 . [ 2 0 ] C o h e n , K . J . a n d H a m m e r , F . S . , \" L i n e a r P r o g r a m m i n g a n d O p t i m a l B a n k A s s e t M a n a g e m e n t D e c i s i o n , \" Journal of Finance, V o l . 2 2 ( 1 9 6 7 ) , p p . 1 4 7 - 1 6 7 [ 2 1 ] C o h e n , K . J . a n d T h o r e , S . , \" P r o g r a m m i n g B a n k P o r t f o l i o s U n d e r U n d e r -t a i n t y , \" Journal of Bank Research, V o l . 1 ( 1 9 7 0 ) , p p . 4 2 - 6 1 . [ 2 2 ] C o l l i n s , H . , \" A C o d e f o r S t o c h a s t i c L i n e a r P r o g r a m s w i t h S i m p l e R e c o u r s e , \" D e p t . o f M a t h e m a t i c s , U n i v e r s i t y o f K e n t u c k y , 1 9 7 5 . [ 2 3 ] C o u h a u l t , A . , \" Q u e l q u e s m e ' t h o d e s d e r e s o l u t i o n d ' u n p r o b l e m e d e p r o g r a m m a t i o n s t o c h a s t i q u e l i n e \" a i r e v e n a n t d e l a g e s t i o n d u s t o c k s , \" Cahiers de 1'I.R.I.A., V o l . 9 ( 1 9 7 2 ) , p p . 7 7 - 1 0 0 . 2 2 2 C r a n e , D . B . , \" A S t o c h a s t i c P r o g r a m m i n g M o d e l f o r C o m m e r c i a l B a n k B o n d P o r t f o l i o M a n a g e m e n t , \" Journal of Financial and Quantitative Analysis, V o l . 6 ( / I 9 7 1 ) , p p . 9 5 5 - 9 7 6 . C r e d i t U n i o n R e s e r v e B o a r d , \" A R e p o r t o n t h e A d e q u a c y o f t h e F i n a n c i a l C a p a c i t y o f t h e C r e d i t U n i o n R e s e r v e B o a r d , \" 1 9 7 3 . , \" F i n a n c i a l S t a t i s t i c s , \" o n t h e B r i t i s h C o l u m b i a C r e d i t U n i o n s , 1 9 7 0 - 7 4 . C r o s s e , H . D . a n d H e m p e l , G . H . , Management Policies for Commercial Banks, 2 n d E d i t i o n , P r e n t i c e - H a l l I n c . , E n g l e w o o d C l i f f s , New J e r s e y , 1 9 7 3 . D a e l l e n b a c h , H . a n d A r c h e r , S . A . , \" T h e O p t i m a l B a n k L i q u i d i t y : A M u l t i - P e r i o d S t o c h a s t i c M o d e l , \" Journal of Financial and Quanti-tative Analysis, V o l . 4 ( 1 9 6 9 ) , p p . 3 2 9 - 3 4 3 . D a n t z i g , G . B . , \" L i n e a r P r o g r a m m i n g U n d e r U n c e r t a i n t y , \" Management Science, V o l . 1 ( 1 9 5 5 ) , p p . 1 9 7 - 2 0 6 . , \" U p p e r B o u n d s , S e c o n d a r y C o n s t r a i n t s , a n d B l o c k T r i -a n g u l a r i t y i n L i n e a r P r o g r a m m i n g , \" Econometrica, V o l . 2 3 ( 1 9 5 5 ) , p p . 1 7 4 - 1 8 3 . D a n t z i g , G . B . a n d V a n S l y k e , R . , \" G e n e r a l i z e d U p p e r B o u n d e d T e c h n i q u e s f o r L i n e a r P r o g r a m m i n g I I , \" Journal Comput. Systems Sci., V o l . I ( 1 9 6 7 ) , p p . 2 1 3 - 2 2 6 . E i s n e r , M . J . , K a p l a n , R . S . a n d S o d e n , J . V . , \" A d m i s s i b l e D e c i s i o n R u l e s f o r t h e E - M o d e l o f C h a n c e - C o n s t r a i n e d P r o g r a m m i n g , \" Management Science, V o l . 17 ( 1 9 7 1 ) , p p . 3 3 7 - 3 5 3 . E l - A g i z y , \" T w o S t a g e P r o g r a m m i n g U n d e r U n c e r t a i n t y w i t h D i s c r e t e D i s -t r i b u t i o n F u n c t i o n , \" Operations Research, V o l . 1 5 ( 1 9 6 7 ) , p p . 5 5 - 7 0 . E p p e n , G . D . a n d F a m a , E . F . , \" S o l u t i o n s f o r C a s h B a l a n c e a n d S i m p l e D y n a m i c P o r t f o l i o P r o b l e m s , \" Journal of Business, V o l . 41 ( 1 9 6 8 ) , p p . 9 4 - 1 1 2 . , \" O p t i m a l P o l i c i e s f o r C a s h B a l a n c e a n d S i m p l e D y n a m i c P o r t f o l i o M o d e l s w i t h P r o p o r t i o n a l C o s t s , \" Inter-national Economic Review, V o l . 1 0 ( 1 9 6 9 ) , p p . 1 1 9 - 1 3 3 . 22'3 [ 3 6 ] E p p e n , 6 . D . a n d F a m a , E . F - , \" T h r e e A s s e t C a s h B a l a n c e a n d D y n a m i c P o r t f o l i o P r o b l e m s , \" Management Science, V o l . 17 0 9 7 1 ) , p p . 3 1 1 - 3 1 9 . [ 3 7 ] F a m a , E . F . a n d M i l l e r , M . H . , The Theory of Finance, H o l t R i n e h a r t a n d W i n s t o n I n c . , New Y o r k , 1 9 7 2 . [ 3 8 ] G u r l e y , J . G . a n d S h a w , E . S . , Money in a Theory of Finance, B r o o k i n g s I n s t i t u t e , 1 9 6 0 . [ 3 9 ] H a l e y , C . W . a n d S c h a l l , D . W . , The Theory of Financial Decisions, M c G r a w - H i l l , New Y o r k , 1 9 7 3 . [ 4 0 ] H e m p e l , G . H . , \" B a s i c I n g r e d i e n t s o f C o m m e r c i a l B a n k s ' I n v e s t m e n t P o l i c i e s , \" The Bankers Magazine, V o l . 1 5 5 ( 1 9 7 2 ) , p p . 5 9 - 5 9 . [ 4 1 ] H e s p o s , R . F . , a n d S t r a s s m a n n , P . A . , \" S t o c h a s t i c D e c i s i o n T r e e s f o r t h e A n a l y s i s o f I n v e s t m e n t D e c i s i s o n s , \" Management Science, V o l . 11 ( 1 9 6 5 ) , p p . B - 2 4 4 - B - 2 5 9 . [ 4 2 ] H e s t e r , D . D . a n d P i e r c e , J . L . , Bank Management and Portfolio Behaviour, Y a l e U n i v e r s i t y P r e s s , New H a v e n , 1 9 7 5 . [ 4 3 ] H i l l i e r , F . S . , \" T h e D e r i v a t i o n o f P r o b a b i l i s t i c I n f o r m a t i o n f o r t h e E v a l u a t i o n o f R i s k y I n v e s t m e n t s , \" Management Science, V o l . 9 ( 1 9 6 3 ) , p p . 4 4 3 - 4 5 7 . [ 4 4 ] , \" A B a s i c M o d e l f o r C a p i t a l B u d g e t i n g o f R i s k y I n t e r r e l a t e d P r o j e c t s , \" Engineering Economist, V o l . 2 0 ( 1 9 7 4 ) , p p . 3 7 - 4 9 . [ 4 5 ] H i r s h l e i f e r , J . , \" I n v e s t m e n t D e c i s i o n U n d e r U n c e r t a i n t y : C h o i c e - T h e o -r e t i c A p p r o a c h e s , \" Quarterly Journal of Economics, V o l . 1 9 ( 1 9 6 5 ) , p p . 5 0 9 - 5 3 6 . [ 4 6 ] , \" I n v e s t m e n t D e c i s i o n U n d e r U n c e r t a i n t y : A p p l i c a t i o n o f t h e S t a t e - P r e f e r e n c e A p p r o a c h , \" Quarterly Journal of Economics, V o l . 8 0 ( 1 9 6 6 ) , p p . 2 5 2 - 2 7 7 . [ 4 7 ] , Investment, Interest and Capital, P r e n t i c e - H a l l I n c . , E n g l e w o o d C l i f f s , New J e r s e y , 1 9 7 0 . [ 4 8 ] K a i l , P . , Stochastic Programming, S p r i n g e r - V e r l a g , B e r l i n , 1 9 7 6 . [ 4 9 ] K a l l b e r g , J . G . a n d K u s y , M . I . , \" A S t o c h a s t i c L i n e a r P r o g r a m ' . w i t h S i m p l e R e c o u r s e , \" F a c u l t y o f C o m m e r c e , T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1 9 7 6 . 2 2 4 K o m a r , R . I . , \" D e v e l o p i n g a L i q u i d i t y M a n a g e m e n t M o d e l , \" Journal of Bank Research, V o l . 2 0 9 7 1 ) , p p . 3 8 - 5 2 . L a s d o n , L . S . , Optimisation Theory for Large Systems, M a c M i l l a n , New Y o r k , 1 9 7 0 . L e v y , H . a n d S a r n a t , M . , Investment and Portfolio Analysis, J o h n W i l e y a n d S o n s I n c . , New Y o r k , 1 9 7 2 . L i f s o n , K . A . a n d B l a c k m a n , B . R . , \" S i m u l a t i o n a n d O p t i m i z a t i o n M o d e l s f o r A s s e t D e p l o y m e n t a n d F u n d s S o u r c e s , B a l a n c i n g P r o f i t , L i q u i d i t y a n d G r o w t h , \" Journal of Bank Research, V o l . 4 ( 1 9 7 3 ) , p p . 2 3 9 - 2 5 5 . L u e n b e r g e r , D . W . , Introduction to Linear and Nonlinear Programming., A d d i s o n - W e s l e y , R e a d i n g , M a s s a c h u s e t t s , 1 9 7 3 . M a d a n s k y , A . , \" M e t h o d s o f S o l u t i o n s o f L i n e a r P r o g r a m s U n d e r U n c e r t a i n t y , \" Operations Research, V o l . 1 0 ( 1 9 6 2 ) , p p . 1 6 5 - 1 7 6 . , \" I n e q u a l i t i e s f o r S t o c h a s t i c L i n e a r P r o g r a m m i n g P r o b l e m s , \" Management Science, V o l . 6 ( 1 9 6 0 ) , p p . 1 9 7 - 2 0 4 . M a o , J . C . T . , \" A p p l i c a t i o n o f L i n e a r P r o g r a m m i n g t o S h o r t - t e r m F i n a n c i n g D e c i s i o n , \" Engineering Economist, V o l . 1 3 ( 1 9 6 8 ) , p p . 2 2 1 - 2 4 1 . M a r k o w i t z , H . M . , \" P o r t f o l i o S e l e c t i o n , \" Journal of Finance, V o l . 6 ( 1 9 5 2 ) , p p . 7 7 - 9 1 . , Portfolio Selection, Efficient Diversification of Investments, J o h n W i l e y a n d S o n s I n c . , New Y o r k , 1 9 5 9 . M o s s i n , J . , Theory of Financial Markets, P r e n t i c e - H a l l I n c . , E n g l e w o o d C l i f f s , New J e r s e y , 1 9 7 3 . M y e r s , S . C . , \" P r o c e d u r e s f o r C a p i t a l B u d g e t i n g U n d e r U n c e r t a i n t y , \" Industrial Management Review, V o l . 9 ( 1 9 6 8 ) , p p . 1 - 2 0 . N a s l u n d , B . , \" A M o d e l o f C a p i t a l B u d g e t i n g U n d e r R i s k , \" Journal of Business, V o l . 3 9 ( 1 9 6 6 ) , p p . 2 5 7 - 2 7 1 . \u00E2\u0080\u00A2 N a s l u n d , B . a n d W h i n s t o n , A . , \" A M o d e l o f M u l t i - P e r i o d I n v e s t m e n t U n d e r U n c e r t a i n t y , \" Management Science, V o l . 8 ( 1 9 6 2 ) , p p . 1 8 4 - 2 0 0 . 2 2 5 O r g l e r , Y . E . , \" A n U n e q u a l - P e r i o d M o d e l f o r C a s h . M a n a g e m e n t D e c i s i o n , \" Management Science, V o l . 1 6 ( 1 9 6 9 ) , p p . B - 7 7 - B - 9 2 . : , Cash Management Methods and Models, W a d w o r t h P u b l i s h i n g C o . I n c . , B e l m o n t C a l i f o r n i a , 1 9 7 0 . O r r , D . , Cash Management and the Demand for Money, P r a e g e r P u b l i s h i n g C o . , I n c . , 1 9 7 0 . P a r i k h , S . C , Notes on Stochastic Programming, u n p u b l i s h e d , I . E . O . R . D e p a r t m e n t , U n i v e r s i t y o f C a l i f o r n i a , B e r k e l e y , 1 9 6 8 . P o g u e , G . A . a n d B u s s a r d , R . N . , \" A L i n e a r P r o g r a m m i n g M o d e l f o r S h o r t -T e r m F i n a n c i a l P l a n n i n g U n d e r U n c e r t a i n t y , \" Sloan Management Review, V o l . 1 3 ( 1 9 7 2 ) , p p . 6 9 - 9 8 . P y e , G . , \" S e q u e n t i a l P o l i c i e s f o r B a n k M o n e y M a n a g e m e n t , \" Management Science, V o l . 2 0 ( 1 9 7 3 ) , p p . 3 8 5 - 3 9 5 . P y l e , D . H . , \" O n t h e T h e o r y o f F i n a n c i a l I n t e r m e d i a t i o n , \" Journal of Finance, V o l . 2 6 ( 1 9 7 1 ) , p p . 7 3 7 - 7 4 8 . R o b i c h e k , A . A . a n d M y e r s , S . C . , Optimal Financing Decisions, P r e n t i c e -H a l l I n c . , E n g l e w o o d C l i f f s , New J e r s e y , 1 9 6 5 . R o l l , R . , \" I n v e s t m e n t D i v e r s i f i c a t i o n a n d B o n d M a t u r i t y , \" Journal of Finance, V o l . 2 6 ( 1 9 7 1 ) , p p . 5 1 - 6 6 . S h a r p e , W . F . , Portfolio Theory and Capital Markets, M c G r a w - H i l l , New Y o r k , 1 9 7 0 . S y m o n d s , G . H . , \" C h a n c e - C o n s t r a i n e d E q u i v a l e n t s o f S o m e S t o c h a s t i c P r o -g r a m m i n g P r o b l e m s , \" Operations Research, V o l . 1 9 ( 1 9 6 8 ) , p p . 1 1 5 2 -1 1 5 9 . T e l s e r , L . , \" S a f e t y F i r s t a n d H e d g i n g , \" Review of Economic Studies, V o l . 2 3 ( 1 9 5 5 - 1 9 5 6 ) , p p . 1 - 6 . T h o m s o n , M . R . , \" F o r e c a s t i n g f o r F i n a n c i a l P l a n n i n g , \" Jo-umal of Bank Research, V o l . 4 ( 1 9 7 3 ) , p p . 2 2 5 - 2 3 1 . T h o r e , S . , \" P r o g r a m m i n g B a n k R e s e r v e s U n d e r U n c e r t a i n t y , \" Swedish Journal of Economics, V o l . 7 0 ( 1 9 6 8 ) , p p . 1 2 3 - 1 3 7 . 2 2 6 ' [ 7 8 ] T i n t n e r , G . , \" S t o c h a s t i c L i n e a r P r o g r a m m i n g w i t h A p p l i c a t i o n s t o A g r i -c u l t u r a l E c o n o m i c s , \" Proceedings 2nd Symposium, Linear Programming, E d i t e d b y H . A . A n t o s i e w i c z , 1 9 5 5 . [ 7 9 ] T o b i n , J . , \" T h e o r y o f P o r t f o l i o S e l e c t i o n , \" The Theory of Interest Rates, E d i t e d b y F . H . H a h n a n d R . P . R . B i e c h l i n g , M a c M i l l a n , L o n d o n , 1 9 6 5 , p p . 7 - 9 . [ 8 0 ] T o b i n , J . a n d B r a i n a r d , W . C . , \" F i n a n c i a l I n t e r m e d i a r i e s a n d t h e E f f e c -t i v e n e s s o f M o n e t a r y C o n t r o l s , \" American Economic Review, V o l . 5 3 ( 1 9 6 3 ) , p p . 3 8 3 - 4 0 0 . [ 8 1 ] T u t t l e , D . I . a n d L i t z e n b e r g e r , R . H . , \" L e v e r a g e , D i v e r s i f i c a t i o n a n d C a p i t a l M a r k e t E f f e c t s o n a R i s k - A d j u s t e d C a p i t a l B u d g e t i n g F r a m e -w o r k , \" Journal of Finance, V o l . 2 3 ( 1 9 6 8 ) , p p . 4 2 7 - 4 4 3 . [ 8 2 ] V a n H o m e , J . C . , \" A L i n e a r - P r o g r a m m i n g A p p r o a c h t o E v a l u a t i n g R e s t r i c -t i o n s U n d e r a B o n d I n d e n t u r e o r L o a n A g r e e m e n t , \" Journal of Financial and (Quantitative Analysis, V o l . 1 ( 1 9 6 6 ) , p p . 6 8 - 8 3 . [ 8 3 ] , Function and Analysis of Capital Market Rates, P r e n t i c e - H a l l I n c . , E n g l e w o o d C l i f f s , New J e r s e y , 1 9 7 0 . [ 8 4 ] , Financial Management and Policy, 4 t h E d i t i o n , P r e n t i c e -H a l l I n c . , E n g l e w o o d C l i f f s , New J e r s e y , 1 9 7 7 . [ 8 5 ] V a n c o u v e r C i t y a n d S a v i n g s C r e d i t U n i o n , Financial Statements, 1 9 6 8 -1 9 7 5 . [ 8 6 ] W a g n e r , H . M . , Principles of Operations Research, P r e n t i c e - H a l l I n c . , E n g l e w o o d C l i f f s , New J e r s e y , 1 9 6 9 . [ 8 7 ] W a l k u p , D . W . a n d W e t s , R . J . B . , \" S t o c h a s t i c P r o g r a m s w i t h R e c o u r s e , \" SI AM Journal on Applied Mathematics, V o l . 1 5 ( 1 9 6 7 ) , p p . 1 2 9 9 - 1 3 1 4 . [ 8 8 ] W e i n g a r t n e r , H . M . , Mathematical Programming and the Analysis of Capital Budgeting Problems, P r e n t i c e - H a l l I n c . , E n g l e w o o d C l i f f s , New J e r s e y , 1 9 6 3 . [ 8 9 ] , \" C a p i t a l B u d g e t i n g o f I n t e r r e l a t e d P r o j e c t s : S u r v e y a n d S y n t h e s i s , \" Management Science, V o l . 1 4 ( 1 9 6 6 ) , p p . 4 8 5 - 5 1 6 . [ 9 0 ] W e t s , R . J . B . , \" P r o g r a m m i n g U n d e r U n c e r t a i n t y : T h e E q u i v a l e n t C o n v e x P r o g r a m , \" SI AM Journal on Applied Mathematics, V o l . 1 4 ( 1 9 6 6 ) , p p . 8 9 - 1 0 5 . 2 2 7 [ 9 1 ] W e t s , R . J . B . , \" P r o g r a m m i n g U n d e r U n c e r t a i n t y : T h e S o l u t i o n S e t , \" SIAM Journal on Applied Mathematics, V o l . 1 4 ( 1 9 6 6 ) , p p . 1 1 4 3 - 1 1 5 1 . [ 9 2 ] , \" P r o g r a m m i n g U n d e r U n c e r t a i n t y : T h e C o m p l e t e P r o b l e m , \" Z. Wahrsch. verw. Geb., V o l . 4 ( 1 9 6 6 ) , p p . 3 1 6 - 3 3 9 . [ 9 3 ] , \" C h a r a c t e r i z a t i o n T h e o r e m s f o r S t o c h a s t i c P r o g r a m s , \" Mathematical Programming, V o l . 2 ( 1 9 7 2 ) , p p . 1 6 6 - 1 7 5 . [ 9 4 ] , \" S t o c h a s t i c P r o g r a m s w i t h F i x e d R e c o u r s e : T h e E q u i v a l e n t D e t e r m i n i s t i c P r o g r a m , \" SIAM Review, V o l . 1 6 ( 1 9 7 4 ) , p p . 3 0 9 - 3 3 9 . [ 9 5 ] , \" S o l v i n g S t o c h a s t i c P r o g r a m s w i t h S i m p l e R e c o u r s e , I , \" Mathematical Programming ( f o r t h c o m i n g ) . [ 9 6 ] W i l l i a m s , A . C . , \" O n S t o c h a s t i c L i n e a r P r o g r a m m i n g , \" SIAM Journal Applied Mathematics, V o l . 1 3 ( 1 9 6 5 ) , p p . 9 2 7 - 9 4 0 . [ 9 7 ] , \" A p p r o x i m a t i o n F o r m u l a s f o r S t o c h a s t i c L i n e a r P r o g r a m m i n g , \" SIAM Journal of Applied Mathematics, V o l . 1 4 ( 1 9 6 6 ) , p p . 6 6 9 - 6 7 7 . [ 9 8 ] W o l f , C . R . , \" A M o d e l f o r S e l e c t i n g C o m m e r c i a l B a n k G o v e r n m e n t S e c u r i t y P o r t f o l i o s , \" The Review of Economics and Statistics, V o l . 51 ( 1 9 6 9 ) , p p . 4 0 - 5 2 . [ 9 9 ] Z a n g w i l l , W . I . , Nonlinear Programming: A Unified Approach, P r e n t i c e -H a l l I n c . , E n g l e w o o d C l i f f s , New J e r s e y , 1 9 6 9 . [ 1 0 0 ] Z i e m b a , W . T . , \" A M y o p i c C a p i t a l B u d g e t i n g M o d e l , \" Journal of Financial and Quantitative Analysis, V o l . 6 ( 1 9 6 9 ) , p p . 3 0 5 - 3 2 7 . [ 1 0 1 ] 1 \" T r a n s f o r m i n g S t o c h a s t i c D y n a m i c P r o g r a m s i n t o N o n l i n e a r P r o g r a m s , \" Management Science, V o l . 17 ( 1 9 7 1 ) , p p . 4 5 0 - 4 6 2 . [ 1 0 2 ] , \" S o l v i n g N o n l i n e a r P r o g r a m m i n g P r o b l e m s w i t h S t o c h a s t i c O b j e c t i v e F u n c t i o n s , \" Journal of Financial and Quantitative Analysis, V o l . 7 ( . 1 9 7 2 ) , p p . 1 8 0 9 - 1 8 2 7 . [ 1 0 3 ] , \" S t o c h a s t i c P r o g r a m s w i t h S i m p l e R e c o u r s e , \" p p . 2 1 3 - 2 7 3 , i n P . L . H a m m e r a n d G . Z o u t e n d i j k , e d i t o r s , Mathematical Programming: Theory and Practice, N o r t h H o l l a n d P u b l i s h i n g , A m s t e r d a m , 1 9 7 5 . [ 1 0 4 ] Z i e m b a , W . T . a n d V i c k s o n , R . G . , e d i t o r s , Stochastic Optimization Models in Finance, A c a d e m i c P r e s s I n c . , New Y o r k , 1 9 7 5 . "@en . "Thesis/Dissertation"@en . "10.14288/1.0094648"@en . "eng"@en . "Business Administration"@en . "Vancouver : University of British Columbia Library"@en . "University of British Columbia"@en . "For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en . "Graduate"@en . "Bank asset and liability management"@en . "Text"@en . "http://hdl.handle.net/2429/21551"@en .