UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Bank asset and liability management Kusy, Martin 1978

You don't seem to have a PDF reader installed, try download the pdf

Item Metadata

Download

Media
[if-you-see-this-DO-NOT-CLICK]
UBC_1978_A1 K88.pdf [ 10.12MB ]
[if-you-see-this-DO-NOT-CLICK]
Metadata
JSON: 1.0094648.json
JSON-LD: 1.0094648+ld.json
RDF/XML (Pretty): 1.0094648.xml
RDF/JSON: 1.0094648+rdf.json
Turtle: 1.0094648+rdf-turtle.txt
N-Triples: 1.0094648+rdf-ntriples.txt
Original Record: 1.0094648 +original-record.json
Full Text
1.0094648.txt
Citation
1.0094648.ris

Full Text

BANK ASSET AND LIABILITY MANAGEMENT  by  Martin B.  Comm.,  S i r George  M.B.A.  A THESIS THE  S  Kusy  Williams  University  SUBMITTED  University,  of Windsor,  IN  REQUIREMENTS  PARTIAL FOR  DOCTOR OF  THE  1969  1970  FULFILMENT DEGREE  OF  OF  PHILOSOPHY in  THE Faculty  of  FACULTY Commerce  OF  GRADUATE  STUDIES  and Business  Administration  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 to the required standard  THE  UNIVERSITY  OF  May Martin  B R I T I S H COLUMBIA  1978  - .  Kusy,  1978  In  presenting  an  advanced  the  Library  I  further  for  degree shall  agree  scholarly  by  his  of  this  written  this  thesis  in  at  University  the  make  that  it  purposes  for  freely  permission may  representatives. thesis  partial  of  of  Columbia,  British  available for  by  the  is understood  financial  gain  for  extensive  be g r a n t e d  It  fulfilment  shall  Head  be  of  of  University  Commerce of  British  2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5  Date  this  or  allowed  without  May  16,  1978  and  Business  Columbia  Administration  Kusy  for that  study. thesis  Department  permission.  Department  agree  and  of my  I  copying  Martin  The  requirements  reference copying  that  not  the  or  publication my  ABSTRACT  The  inherent  and  return  mic  conditions  greater  on i n v e s t m e n t ,  consequence  a  bank's  (BC)  during  efficiency  A  between  uncertainty  risk,  model,  with  the past  tractable  the  not operationally  limitations, the  to  and a number  restricted  deficiencies develop  tationally that  this  feasible  an a s s e t  model  program  ALM model  following  1)  the stochastic flows  for  large  features  nature  (deposits)  realistic existing  with  of  asset  exists  and  model  Crane are  computational  features  decisions). this  structure  Unfortunately,  severe  (such Given  dissertation  (ALM) and to  -  This  model  and l i a b i l i t y (by  discrete  i  l i a b i l i t i e s .  trade-off  dissertation  (SLPR).  -  of  econo-  that  is  as these  are compu-  demonstrate  models.  the problem  a given  due t o  funds  for  o n how t o  ignored.  problems  in this  of  is  of of  i n the l i t e r a t u r e  period  management  and  the Bradley  formulation  first  recourse  studies  for  proposed  the purposes  developed  simple  essential  to  of  appealing  for  cost  the need  assets  an " o p t i m a l "  undesirable  region  is superior  with  a bank's  uncertainty  and l i a b i l i t y  tractable  the  cash  of  i f  in the l i t e r a t u r e ,  The linear  only  flows,  emphasized  Except  techniques  computationally is  of  cash  increased variability  have  so t h a t  and l i q u i d i t y .  the solution  BC m o d e l  the  an i n c r e a s e d number  and l i a b i l i t i e s  return  a bank's  decade,  i n t h e management  has been  assets  along  of  utilizing probability  is a  stochastic  incorporates management: a set of  random  distribution),  2)  simultaneous  costs,  and 4)  in  ALM model  asset  order  to  demonstrate  the  solutions  (SDP)  model  to  findings  Vancouver  SLPR.  for a five  large A  compare  City  transactions  to  Savings  year  problem  simulation  Credit  planning  implement  the  period  model.  was m a i n t a i n e d  was r u n on a  the d e c i s i o n making  by t h e SLPR and s t o c h a s t i c  ALM  model  4)  t h e SLPR  flows,  to  3)  formulation by B r a d l e y  the  SLPR  formulation  SDP  formulation of  (this  to  by  using  real  effectiveness  dynamic  are:  deterministic  model,  the asymmetry  of  that  required  of  1)  of  programming  2)  t h e ALM model  the solution  the probability for  an e q u i v a l e n t  and C r a n e ,  the simulation  results  in a better  i s due t o  the r e s t r i c t i o n s  feasibility  for the f i r s t  initial  i i  period imposed  for a l l possible  period  -  to  -  decision).  is the  distributions  deterministic  superior  a n d 5)  of  the implementation  i s computationally  maintaining  scenarios  dissertation  the e f f o r t  i s comparable  formulation  this  an e q u i v a l e n t  utilized  economic  of  is sensitive  the cash  tion  3)  models.  superior  of  to  generated  The  ALM  and l i a b i l i t i e s ,  necessary  for this  for solving  environment  to  management  the e f f o r t  tractability  algorithm  (uncertain)  assets  was a p p l i e d  and l i a b i l i t y  Computational Wets'  of  multi-periodicity.  The Union's  consideration  of  the  model,  t h e SDP  formula-  indicates  decision  than  b y t h e SDP forecasted  that the  TABLE  OF  CONTENTS  Page Chapter  1  -  1.1  Introduction Overview  of  Dissertation  1.2  Definitions  1.3  Theory  1.4  Appropriate  1.5  of  Chapter  Management  Essential  Features  5  f o r A s s e t and 8  of  That  an A s s e t  and L i a b i l i t y  Maximizes  Expected  Returns  12  1.7  Organization  Review  of  of Asset of  and L i a b i l i t y  Management  the Dissertation  13 ,  14  Literature  Introduction  15  2.2  D e t e r m i n i s t i c Models  16  2.3  S t o c h a s t i c Models  22  3 3.1  Chapter  Model  Importance  2.1  Chapter  Criterion  1.6  2 -  Intermediation  Liability  Net  1 4  Financial  Management  .  Formal D e s c r i p t i o n o f t h e A s s e t Management (ALM) Model  and L i a b i l i t y  Introduction  32  3.2  Formulation  3.3  Use o f  of  t h e ALM Model  34  3.4  Appendix  One  50  3.5  Appendix  Two  58  t h e ALM M o d e l  4 -  Implementation  4.1  Introduction  93  4.2  Model  97  4.3  Results Union  4.4  of  47  t h e ALM Model  Details of  the Vancouver  City  Application  Appendix  Saving  Credit 108  One  113  -  i i i -  Page  Chapter  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 Programming w i t h Simple Recourse Models  Chapter  as  Decision  5.1  Introduction  .  5.2  Scenario  the  5.3  Formulation  of  Programming  Model  5.4  Formulation  of  5;5  Results  5.6  Appendix  6  -  Summary, Further  for  of  .  Tools .  .  .  168  Simulation  174  the  .  .  Stochastic  Dynamic 176  the  the  .  SLPR Model  181  Simulation  183  i  One  Major  186  Findings  and  Directions  for  Research  6.1  Introduction  214  6.2  Summary  214  6.3  Major  215  6.4  Directions  Findings for  Further  Research  Bibliography  217  2>8.  -  iv  -  ACKNOWLEDGEMENTS  I would the  support  to  particular,  I  like  to  thank  facilitate  the completion  am i n d e b t e d acumen  all  aspects  my a c a d e m i c  for  encouraging  also  grateful  Ziemba,  Dr.  Diewert,  providing  programs City for  with  Savings  excellent  providing thank  for  to  A.  their  C.  me w i t h like of  to  this  like  conducive Amershi,  to  Dr.  to thank  Dr.  R.J-B.  I would  Ms. Chan,  two s p e c i a l  encouragement,  Wets  to  Dr.  in  R.W.  management. Dr.  White, I am  W.T.  and Dr.  f o r both  stochastic  like  W.E.  to  thank  developing  linear Vancouver  Hook a n d t h e l a t e  Fong,  of  Mr.  Bentley)  t h e ALM m o d e l .  Haller  and M i l l e r  for the  dissertation. thank  a l l my f e l l o w  academic environment.  V.V.  for his  and a d v i c e .  for the implementation thank  In  f o r h i s guidance  Mitten  for solving  Also,  providing  Ziemba,  committee,  L.G.  ( i n p a r t i c u l a r Mr.  data  W.T.  am g r a t e f u l  criticism  recourse. Union  Dr.  and l i a b i l i t y  Sarndal,  constructive like  I  for  my d i s s e r t a t i o n .  o f my d i s s e r t a t i o n  simple  would  thank  asset  to an a l g o r i t h m  would  a  bank  access  typing I  like  Dr.  also  Credit  providing I  to  White,  for providing I would  and  study  Commerce  and also  development.  t o a l l members  R.W.  of  and a v a i l a b i l i t y ,  me t o  of  t o my s u p e r v i s o r ,  encouragement, of  the Faculty  Baba,  D.  friends advice  Kira  and S.  Jerry  v  -  students  In p a r t i c u l a r , Larsson.  Kail berg  and moral  -  graduate  I would  Finally,  and Lawrence  support.  for  I  like would  Kryzanowski  Chapter  1  INTRODUCTION  1.1  Overview- o f The  and of  return  inherent  decade,  management  increased  of  number  liabilities liquidity  must  of  assets  based  cash  flows,  the unsettled  t h e need  trade-off  cost  economic  for a greater  and l i a b i l i t i e s .  a bank's  exists  of  funds  conditions  efficiency  A consequence  in  has been  an  a s s e t s and  between  funds  on t h e d e t e r m i n a t i o n  or stochastic  i n these  of  a bank's  decisions  t o meet  maximizing  financial  and leverage  risk,  return  and  economic  include:  liquidity  profitability  structure  of-the  scenarios.  [11,20], and r a t e s of  that  of  anticipated  adequacy  constraints  allocating of  funds  Factors  the balancing  and c a p i t a l  i n terms  use of  return  liquidity,  funds [5,6],  among and  capital  [11,20].  research  on M a r k o w i t z ' s  normally  focussed  on c l a s s i f i c a t i o n , m a t u r i t i e s  Current based  a bank's  o n how t o s t r u c t u r e  an " o p t i m a l "  deterministic  concurrently  adequacy  assets  studies  studies  and uses  adjusting  with  emphasized  a bank's  be c o n s i d e r e d  while  along  of  [7,11,20,70].  either  sources  are  have  so t h a t  These given  uncertainty  on i n v e s t m e n t s ,  the past  the  Dissertation  theory  distributed  has s t r e s s e d of  two a p p r o a c h e s .  portfolio  and t h a t  bank  1  selection,  managers  The f i r s t  assumes  that  are risk-averse  approach, returns  utility  2  of  wealth  not  only  maximizers  on t h e e x p e c t a t i o n  covariance tial  of  i t s return  future  second  stream  of  The approaches essential  features  first  model  that  with  decision a of  the value  of  an a s s e t  depends  i t s return  but also  on t h e  a l l other  existing  and  of  t h e above  decomposition their  advanced  they  poten-  model  that  they  to capture  [20]  feasible  linear  consider of  [11,20]. of  problem  problems  the effect  the features  inherent  and Crane  dilemma  i s formulated that  Their  and  ( B - C ) model  [5,6,7].  model  for  two a p p r o a c h e s  and l i a b i l i t y  asset  the choice  model  [98]  attempts  Essentially,  i n the  asset  realistic show  that  i s the  in a serious  their  program.  advantage  has a number  of of  ^  sequential  the trade-  realism.  as a l i n e a r  takes  infeasible  of  a s s e t s and  i n c l u d e s many o f  tractability  features  but i s  of  Wolf's  an a s s e t  or  As an example  the bank's  These  the  [10,19]  the essential  shortcoming,  in nature.  both  programming  for large  i t s  do n o t c a p t u r e  [5,28,36].  but i s computationally  algorithm  i f  management  problem  portfolio  in developing  formulation.  by t h e p r o p o n e n t s  attempt  i t  to maximize  mix constraints  tractable  nor does  seeks  the second  are stochastic  trees  portfolio  and Hammer's  of  a bank  and l i a b i l i t y  on t h e t o t a l  Bradley  that  management  Cohen  computational  The cope  i f  i n nature  problem,  encountered  between  to  i s computationally  theoretic  problems  the asset  As an example  liability  dilemma  of  instruments  decision  assumes  techniques  and l i a b i l i t y  liabilities.  off  of  the returns  subject  intractable  stochastic  asset  and  with  computationally  shortcoming,  management neither  profits  are either  the asset  of  such a w o r l d ,  and variance  approach  solution  computationally  the  In  investments. The  of  [59,70].  model  appealing  to  i s a  B-C have  the special  manner  developed  structure features:  3  it  is  dynamic  interest size.  rates;  the  interest  of  rates a  horizon  with  more  Finally,their has  to  decision  Given  dissertation  is  computationally In  of  research  financial approach bank in  as  simple  future  influenced  deficiences  the  economic  the  the  City the  in  model  this in  survey  to  of  the  Savings proposed  the for  the  utility the  the  proposed  Credit model  Union), to  model  flows  is  and  unable  or  a  to  planning  capacity.  That  made  i s ,  the  scenario. primary model  purpose (ALM)  that  problems.  of  for  of  a  and  existing  and  limited  decision  l i a b i l i t y  to  principle the  the as  present  and  cash  computer  following  model  of  their  model  the  approach  asset  in  with  taxing  use  net  flows  model.  instruments  reasons  presentation  solve  used  possible  and  the  B-C  scenarios.  realistic  thesis,  of  be  literature,  asset  turn:  the  investment  worst  cash  problems  their  without the  of  of  financial  economic  the  reasons  the  Secondly,  the  an  may  for  distributions  that  large  expected  literature,  application  of  of  discussed  to  by  develop  a critical  recourse  (Vancouver  existence  net a  present  value  models  l i a b i l i t y  the  local  for currently  linear  program  management  financial  demonstration  models  of value  rationale  stochastic  a  areas  using  a  institu-  of  the  simulation  scenarios.  The financial  possible  remainder  opposed  'superiority' of  all  for  that  periods such  to  tractable  different  time  tractable  be  point  correlated. of  uncertainty  shortcomings  formulated  is  the  functions  three  three  intermediaries,  financial  problem, tion  will  management,  the  with  the  major  is  these  this  or  number  overly  of  three  highly  than  satisfy be  incorporates  computationally  two  large  model  will  is are  -  being  either  it  distribution  crude  handle  now,  it  there  types  extremely  nature; and  However,  First, are  in  remainder  terms  used  in  of  this  this  chapter  will  dissertation,  consist  the  of  economic  the  definitions  rationale  for  of the  4  existence  of  liability  management  net  present  1.2  financial  value  intermediaries,  model  must  approach  in  have,  the and  preference  features the  to  that  an  asset  justification  the  expected  for  and  using  value  the  approach.  Definitions  liquidity  The marketability  and  of  capital  a  financial  certainty.  asset  will  According  be  to  defined  Van  in  Home  terms  [83,  of  p.  7]  . . . l i q u i d i t y has two d i m e n s i o n s : (J) the length of t i m e and t r a n s a c t i o n c o s t r e q u i r e d to c o n v e r t the a s s e t i n t o money, (.2) t h e c e r t a i n t y o f t h e p r i c e realized. . . . The two f a c t o r s a r e i n t e r r e l a t e d . If an a s s e t m u s t be c o n v e r t e d i n t o money i n a v e r y short 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 the p r i c e r e a l i z e d than i f there were a reasonable time p e r i o d in which to s e l l the a s s e t .  Financial the  business  expressed  in  securities) (primary as  of  holding  terms in  Intermediaries  of  order  banks,  dissertation  is  in  term  which:the  by  the  bank  ing  two  to  financial  a to  risk  Fund risk  bank. meet  is  its  conditions  l i f e  banks  and  is  risk the  investment  proposals  proposals  considered  separately  instruments include  credit  is  (.1) equal  (no  to  unions,  synergism),  can  in be  (indirect others  institutions  this  Since  is  the  this  sense  be"used. the  rate  value  sum o f and  the  occurs  aggregate the  of  such  associated with  the  (which  companies.  associated with risk  involved  instruments  insurance  will  entities  instruments  Risk independence  satisfied:  exclusive  as  financial  financial  and  commitments.  are  financial  intermediary the  defined  intermediaries  unions with  be  issue  the  Financial  concerned  in  They  purchase  credit  Portfolio earned  dealing  money).  securities).  chartered  and  will  the  (2)  of  ability when  of  of  the  follow-  mutually  values  the  return  of  the  financial  5  instruments  under  interdependence risk  consideration  is  independence A  together  market  volume  satisfies all  of  and  (3)  of  of  resulting  imperfect  is  if  either  have  any  of  independent the  mechanism  financial  Risk  [61].  necessary  independent from  conditions  institution  many  risks  of  to  all  market  buyers,  relevant  does  not  for  and  taxes  one  or  bring  financial and  (including imputed  do  not  information  meet  to  sellers  costs  default and  used  -perfect  A  transaction  indivisibility)  access  financial  (.1)  {2\  takers,  or  instruments.  conditions:  price  pooling  investors  physically  met  following  whom a r e  inconvenience  An  not  occur  sellers  the  discounts,  all  to  market  and  for  [52].  are  financial  buyers  issuers,  said  are  at  more  costs  exist, no  of  cost  the  above  conditions.  1.3  Theory  of  In  Financial  order  intermediary,  it  to  is  in  the  existence  of  financial  porated  into  existence  of The  pretation theory  of  For a  develop  essential  mediaries  literature.^  Intermediation  economy.  discuss  intermediaries financial  equilibrium  financial  framework  theory  of  the  and  the  has  behaviour  role  of  theoretical  not  been  model. is  for  presented a  have  financial inter-  in  not  the  been  rationale  for  for  the  economic incorthe  next.  theory  investment  a  rationale  resolved  Economic  such  of  financial  the  intermediaries  intermediaries  underlying Fisher's  to  objectives  Unfortunately,  example,  general  the  is  decision  Hirshleifer's [45,46,47].  interThe  assumes:  Contributions [9,30,70,80].  have  been  made  to  such  a  theory.  See  for  example  6  1)  perfect  markets,  certainty,  2\ 3)  no  borrowing  h)  two  5)  J  6)  U j (CQj,  time  individuals,  in  )  each  sists  his  lends  -  (C0 An  is  negative  necessary  P*,  to  consumption the  1,  (positive).  condition  the  for  is  both  future  (.1),  utility  individual of  choice  i for  his set,  (.Yo,  assets)  utility which  Yi),  and  con-  financial  productive  assets).  along  the  into  NN1  optimum  as  an  (see  endowment  first  (C0,  individual  combinations. can  PP1,  attain  which  is  1). as  follows.  (Y0,Yi). In  an  Ci)  locus,  Figure point  permit  individual  possibility  (C*,C*)-  current  line  alternative  optimal  initial  the  invests  Then  First  the  he  borrows  particular  (Y0  -  Po)  case  and  then  consumption. a  existence direct  market  individuals,  the  defined  The  maximize  opportunity  individual  replenish  individual  to  endowment  line,  utility  of  objectives  production  his  the  and  his  other  attains  from  his  to  the  market  P*  Figure P0)  with  on  highest  attain  borrows  to  auctioneer,  00  the  endowment  borrowing  moves  in  are  opportunities  individual  illustrated  the  initial  initial  point,  to  is  (real  or  or  individual's  <  opportunities  investing  individual  jth  j  (financial  By  The  and  opportunities  his  the  (.0).  attempts  subject  transform  to  present  investor),  function  to  tangent  is  individual  of  the  U.'  \these  individual  Financial  optimal  C . U  i  with  where  where  and  period  the  the  lending  periods,the  function  7)  or  and  surplus of  (deficit)  surplus  indirect  and  unit  deficit  financing.  when units  (G* is  However,  -  P0)  a it  is  Figure!  not  a  since  sufficient a l l  condition  financial  transfers  Extensions [45, of  46]  to  financial  does  not  between  this  the  reasons  ultimate  existence take  model  by  uncertainty,  economic  the  the can  intermediaries.  justify  potential  of  incorporate  Relaxing two  for  why  borrowers  Arrow s t i l l  existence of  financial and  place  Therefore,  assumption  of  do  of  and  not  justify  existence  capital  intermediaries  lenders.  Hirshleifer  financial  perfect  intermediaries  directly. £1J  the  financial  These  are  of  the  existence  uncertainty,  per  se,  intermediaries. markets  may cost  does  interpose economies  suggest themselves in  trading  8  (especially in  the  folio  in  gathering  buy  superior could  credit  ability  surplus  may  imperfection  are  of  present  of  the  is  Appropriate  A to  meet  this  result  necessary  markets.  surplus  may  evaluate  the  financial  intermediary  of  formulate for  required  surplus  the  risk  required  by  a  these  more same  gathering  dollar  develop  for  prob-  a  financial  unit.  units,  conditions  This  expenditure  existence  deficit  to  accurate  surplus the  in  securities.  premium  for  and  scale  primary  port-  units  of  to  Whether  that  of  institutions).  economies  conditions  of  financial  economically  subset  that  pure  economies  benefits  cannot  outcomes  the  than  deposit  point  as  create perfect  are  of  and  financial the  also  sufficient  in  time, are  assets  it  risk  that  inflation  is  unresolved  whether  independent.  investors  hedges,  If  cannot  then  the  or  not  the  financial  duplicate  assumption  on  their  of  risk  reasonable.  Criterion  principle  resulting  a  a  the  is  the  intermediary  intermediaries  cannot  such  independence  1.4  at  financial  account,  in  and  cost  unresolved.  intermediaries own  enable  existence  capital  Also, assets  may  potential  the  the  they  for  rates),  reason  Furthermore,  smaller  summary,  intermediaries  at  As  be  second  expertise  of  information,  because  financial  commission  importance  the  standing.  units.  In  minor  behind  distributions  intermediary  is  (of  information  the  negotiated  processing of  informational  enable  of  securities  processing  than  and  premise  primary  borrower's and  world  diversification The  not  a  Asset  constraint  withdrawal  problem,  for  on  claims  consider  on  the  and  the  Liability  management  request. following  To  Management  of  bank  funds  illustrate  the  s i m p l i f i e d example  is  the  nature from  need of  Tobin  [79]:  9  Although the  1)  certa i nty,  2)  transaction  3)  two 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 and c a n n o t be l i q u i d a t e d f o r two p e r i o d s ( i n f i n i t e transactions c o s t s up 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 be l i q u i d a t e d a t t h e end o f the first period. The r e t u r n s on t h e a s s e t s a r e rj. and r 2 , r e s p e c t i v e l y , where r i > r 2 .  the  example  horizon  a l l o c a t i o n of does  cannot This  from and  the  resources The  be  in  a of  cash  of  least the  of  cash  that  a  -  flows.  to  customers  [27].  bank  be  must  short  to  the  to  asset  different  assets of  market a  and  bank  demand  provide  two  assets  is  d e c i s i o n s the  trivial,  planning  l i a b i l i t y management  degrees  management costs  and  l i a b i l i t y management  anticipation  prepared  of  liquidity  all  future  viewing  In  with  assumption  satisfy  realistic  (transactions and  the  short.  nature  the  between  and  as  an  and  requests of  these  large  or  problem  uncertain entity,  face  the  uncertain  amounts  of  is  opportunity  results  ongoing  liquidity,  problem  liabilities  certainty  rates  of  is  maturity allocate  costs) not  manner.  just  to  simultaneously. in  two  timing it  is  prospect (e.g.  funds  to  results  additional and often of  loan)  on  volume argued  losing requests  a  relatively  notice. In  are  make  consider  In  must  of  cost  uncertain  bank  that  assets  asset  Relaxing problems  funds  infinitesimally  essence  but  and  initial  multi-period  The  essence  manage  emphasize  existence  yield.  costs,  used  the  current  in modelling  is  developed  from  an  intermediary  the  financial  asset  and  Markowitz  possesses a  and  economic  literature,  l i a b i l i t y management. mean-variance  utility  function.,  The  framework. which  the  two first  It  criteria criterion  assumes  intermediary  that  10  attempts  to  value  returns  of  maximize.  subject  Pyle's It  is  the  most  liabilities Is  is  paper  and  to  what  This  utility  to  at  ferently,  Pyle's  reflected  in  result  trading  extreme  is  paper  solvency.  is  An  that Other  the  the  use  since  can  risk  than  problems  of  it  the  net  for to  present  except  to  a  such as  the  and  arises--  static the be  In  the  one-period  amount  certain  stated  market  assets  corporation?  of of  somewhat  extent  synchronization  offset  criterion.  corporations? a  and  Or,  first  question  select  period  the  considers  for  leads  period.  adverse  more  the  appropriate  over  fund  maximize  theory  criterion  during  to  fundamental  intermediary  ignores  costs  a  portfolio  be m a i n t a i n e d times  of  applications  first the  is  constraints.  example  function  the  of  However,  portfolio .risk.  case  an  such  that  all  criterion  number  operationalize  implies  amounts  in  a  simultaneously.  liabilities  those  to  of  Furthermore, model.  second  [70]  general  possible  particular,  The  of  that  cash  returns  having difit  is  flows  and  transaction  assets  in  cost  could  the incurred  2 in  the  sale  period any  of  nature  matching  makes  it  given will  be  First,  similar  of  the  models  model  to  example to  Chambers done  prior  in  insert of  model and  using  Chen, Jen m o d e l [ 18] .  the  of  use  the  the  and  2,  maturity  a  the  are  ignored.  ability  maturities  adequate  of  terminal  of  the  assets  Thus bank  and  the  second  asset  and  l i a b i l i t y management  Ql]. few  second  Zionts  Although general  criterion  have  criterion,  a  review  statements assume  included  to  the  one-  exercise  liabilities  and  conditions.  of  Charnes  Chapter  to  precludes  (synchronization)  return,  by  security  difficult An  present  a  risk  of  are  maximizing problem  such in  net is  literature  order  here.  independence  transactions  costs  in  a  11  (defined  earlier)  dependence.  as  opposed  Second,  the  types  which  can  solve  relatively  which  can  solve  limited  ability. as  However,  being  the  the  model  management Myers tion  has for  market  then  shown  is  most  that:  and  3)  the  (implying  risk  the and  institution A  period certain  if  [61]  from  model,  is  of  which  linear  stochastic  because  treat  risk  formulations,  dynamic  formulations  computational  generally  which for  risk  of  security  of  to  independence  then  market  this  intract-  considered  the  expected  of  financial  two  the  or  is  criteria  asset  resolve  of  net  and  the  accepted  the  risk  investment present  results  l i a b i l i t y  controversy.  a necessary  equilibrium,  implies  independence  of  the  solving  attempts  1)  risk  case  and  approach  to  models  condi-  2)  if  security  independence  of  opportunities  value  is  the  exists  appropriate  criterion.  institutions,  normative  of  type  range  problems,  suitable  equilibrium exists  (1)  models  remains:  Myers  existence  In of  Markowitz  problems  type  question  maximization  objective  from  sized  equilibrium exists,  the  state  the  problem?  securities,  of  large  neither  which  the  the  best1.  1  Thus in  to  (2) is  2)  is  the  does  to not  problems.  the  that  the  of  test  the  of  the  actual  lend  of  the  two  problem  itself  to  do  not  is  is  solving.  The  On  the  other  hand,  the  assumptions,  can  solve  problems  large risk of  present  their risk  by  1)  effect  the  implication  for  a  financial  value.  dependent  independent  variables)  size.  a  (Markowitz) multi-  approach,  realistic  a  financial  a p p l i c a b i l i t y as  (decision  a more  that:  synergetic  function  net  approaches  solving  a  held  Therefore,  objective  expected  observed are  have  securities).  appropriate  maximization  it  s e c u r i t i e s which  s e c u r i t i e s purchased  independence  further  tool  for  institutions  given If  the  12  assumptions shown good would  in or  an  risk  operational  better  imply  superior  1.5  (underlying  sense  solutions  that  using  modelling  than  the  Features  of  Maximizes  Expected  Net  asset  and  dissertation  using  expected  returns.  net  that the  the  can  risk  risk  be  independent  dependent  maximization  of  relaxed  net  and  if  approach  approach,  present  then  value  is  it  can  be  yields  as  this a  approach.  Essential  The  independence}  an A s s e t  Liability  Management  Model  That  Returns  liability  a constrained A  and  general  management  problem  optimization  model  discussion of-the  is  analyzed  which  in  mazimizes  relevant  this the  constraints  fol1ows. The capacity  to  meet  be m a i n t a i n e d porated  in  major  the  1)  2)  constraint  withdrawal  across ideal  time,  on  the  c l a i m s on the  changing  demand.  following  optimization  multi-periodicity  management  -  five  of  a  Since  bank's this  features  funds  capacity  should  model.  in o r d e r  yield  to  spreads  incorporate:  a)  the  across  time,  b)  the 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 s e l l i n g a s s e t s p r i o r to m a t u r i t y ,  c)  the s y n c h r o n i z a t i o n of cash flows a c r o s s t i m e by 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 expected cash outflows.  with and  simultaneous c o n s i d e r a t i o n of a s s e t s and l i a b i l i t i e s - in order to s a t i s f y b a s i c accounting p r i n c i p l e s and more i m p o r t a n t l y to match the l i q u i d i t y q u a l i t i e s of of the liabilities.  assets  with  those  be  is  the  must  incor-  13  3)  transaction  5)  b)  other expenses incurred sel1ing securi ties.  uncertainty making  process  are  relatively  1.6  Importance  stated  remains  proposed,  thus  far,  dissertation model  and  in  incorporate:  buying  and  is  includes:  an  be  of  the  Liability  to  into  be  incor-  the and  decisionborrowing  detrimental  bank.  (For  to  example,  when  the  interest  that  can  be  the if rates  are  not  there  have  been  the  problem  all  the  essential  since  this not  features  to  model  literature one of  of the  the  (ALM)  above  model  developed  models  real  deficiencies.  The  in ALM  asset  is  world  tractabi1ity.  management the  many a t t e m p t s  from  on  rectify  into  universal.  conclusion derived  done  incorporated  Management  computationally  to  order  lending  long  constraints  liability  attempt  avoid  constraints  incorporates  and  in  rates  low .(high.)).  The  to  -  ultimately  previously,  maintaining asset  to  (borrows)  other  management.  work  as  may  these  Asset  much  The  are  model,  of  was  lends  rates  interest  well-being  bank  there  so  which  the  optimization  while  to  and  market  fluctuating  the  liability  of  porate  While  problem  fees,  order  brokerage  financial  that  in  a)  decisions  and  -  u n c e r t a i n t y of cash flows - in o r d e r to incorporate the u n c e r t a i n t y i n h e r e n t in the d e p o s i t e r s ' withdrawal c l a i m s and d e p o s i t s . (.The m o d e l must ensure that the s t r u c t u r e of the a s s e t portfolio i s s u c h t h a t t h e c a p a c i t y t o meet t h e s e c l a i m s i s m a i n t a i n e d by t h e bank.)  h)  As  costs  this  14  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 - by i n c o r p o r a t i n g a s e t o f random c a s h f l o w s ( d e p o s i t s ) with a given discrete distribution,  2)  simultaneous  consideration  3)  transaction  costs,  These  features  computational  1-7  In  the  a net  ALM model  (SLPR).  of  Chapter  4 presents largest  used  to  tion  i s compared  solve  t h e SLPR  maker.  to  be i n c o r p o r a t e d  return  the results  to  into  t h e model  while  maintain-  problems.  on t h e a s s e t  " criterion  of  is also  an e q u i v a l e n t results  t h e SLPR  3 gives  linear  program  a brief  an a p p l i c a t i o n Computer  summary  of  stochastic  dynamic  operational  by a s i m u l a t i o n and s t o c h a s t i c  t h e c o n c l u s i o n s and p o s s i b l e  of  of  about 5,  relevant of to  the  one  of  algorithm  to  this  formuladetermine  decision-  (economic  formulations. of  recourse  SLPR.  for a  data  the  sto-  t h e SLPR  solutions  extensions  3,  simple  formulation  t h e same  dynamic  with  t h e model  In C h a p t e r  management  In C h a p t e r  development  information  presented.  in better  and l i a b i l i t y  i s reviewed.  and t h e t h e o r e t i c a l  unions.  i s accomplished  f o r both  large  as a s t o c h a s t i c  techniques  credit  for  Chapter  the problem  chapter,  presented.  liabilities,  and  the l i t e r a t u r e  present  approach  This  scenarios) final  2,  The a p p e n d i x programming  if  and  Dissertation  i s developed  chastic  Canada's  the  Chapter  using  will  tractability  Organization  models  assets  mu1ti-periodicity.  k)  ing  of  In  the  research  are  Chapter  2  REVIEW OF LITERATURE  2.1  Introduction  Before models  dealing  results  of  a  recent  study  number  of  tion  (where  [42],  use  that  net  discounted in  models  this using  first  linear  models  by  is  returns returns of  the  taken  in  optimal  utilized  Hester  is  an  by or  was  as  function  category  programming,  liability c o n s i s t s of assume  to  fund  method  of  and  brief  Hester  analyze  presented the  only  are  management  managing  of  a  of  validity Their  either  in  of  a bank's  two  the  Pierce,  management.  in  Chapter  maximization  the  objective asset  a  a  main portfolio.  the  maximiza-  variable  discussed in models  fall  particular  realizations  of  1  and  function  the  expected  function  this into  for  net a management  chapter. two  These all  empirical  for  and'1iabi1ity  d e t e r m i n i s t i c models.  15  summary and  the  P i e r c e was  appropriate  this  objective  order.  bank  Pierce, the  a  analytical  dominant).  arguments and  normative  maximization  Therefore,  and  in  Hester the  the  management,  is  dissertation.  Asset The  result  returns  liability  c r o s s - s e c t i o n a l data  there  discounted  obtained  and  a d i s c u s s i o n of  s e l e c t i o n models  function  a  to  positive  discounted  As  bank  on  is  net  evidence  asset  study  objective of  with  portfolio  conclusion The  proceeding  broad  models random  categories use  events,  16  and  are  models  computationally have  industry  been  models  very the  accepted  second  that  modest  stochastic  are  success  models  constrained theoretic  approach,  are  next  the  needed the 2)  and  by  liability  nature.  to  At  the  these  banking  use  of  best  management  these  computational  achieve  dynamic  linear  two  the  following  programming, under  models  models  have  achieved to  tractability.  techniques:  3)  are  d i f f i c u l t i e s or  computational  programming  sections  1)  sequential  uncertainty,  The  chance-  decision  and  5)  dynamic  to  1961  discussed solve  linear  programming  (LP)  trade-offs  between  review  problem  and  structure  the  the  readily  available  ment  other the  was  the  deterministic  to  and  stochastic  Chambers  optimize deemed  bank  to  be  and  Charnes  portfolios. acceptable  number  of  constraints.  a  linear  large  deterministic  models  summary  format  scale was of  important  because  they  problems.  variables,  in  a  by  are  of  solve  CC m o d e l ,  section  numbers  large  problem  this  portfolio  work  to  large  in  real  seminal model  of  will  Models  programming  cation  asset  inherent  to  linear  of  of  due  4)  models  used  The  to  tool  Furthermore,  techniques.  The  the  normative  in  include  Deterministic  and  useful  problems.  programming.  modelling  can  large  stochastic  programming,  The  2.2  a  category  oversimplifications  linear  as  for  [20]. The  the  tractable  the  and  their  a  The  given  [11]  the  Since slight  article  number  intertemporal  efficient  is  produced  utilization  Furthermore,  problems. either  (CC)  of  was  algorithms  the  of  nature it  essential.  or  of  feasible were  subsequent  extension  a  an  developappli-  17  The (sources are  a  they  of  from  Reserve  somewhat are  model  uses  developed  Federal (in  and  CC  the  (FRB)  later  value  market  in  liquidity  of  the of  where,  A^_  is  tained  in  the  and  X-t  total  no  recession are FRB,  also the  (except  i  from  book  only  loss (or  originally  ALM  model 'normal'  These  put  those  defined  cash  =  forth  are  to  budget  constraints by  the  American  presented  used  economic  of  book  of  single  be  in  as  defined their  assets,  if  the i  were  by  CC)  below  because  conditions  to  treasury  as  in  used  is  said  to  period  in  has  t,  to  value as  and  the  given  by:  $^  other  is  the  shrinkage  equation in  the  financial  expected  value  parameter  liquidated  Consequently, is  a  in  conthe  quickly,  t.  then  However,  the  (])  value  be  period  assets  than  / l X  >  book  to measure  distress.  bills)  their  x  l u s  (2)  liquidated,  liquidate  P  u  asset  held  •,  c S u r  liquidation  depositors.  financial  , +  X  asset  bank  •.  c  disintermediation),  required and  is  formula  value  p r i n c i p a l to  to  , "  ^ i=l  adequacy  financial  discount  tz  value  value  a  of  likely  for  as  from  ^ Total > Liabilities  is  market  capital  the If  ensure  formula  constraints  under  subject  constraints.  These  liabilities  assets  the  asset is  liquidity  returns,  when  value  of  discounted  adequacy  the  A  value  net  [27].  form  Market Value of Assets  The  and  capital  different  bank's  be a d e q u a t e  funds)  Board  utilized A  maximizes  used  to  (1)  event  would of  severe  intermediaries according to  be  to  the  greater  compute  A^  [27].  18  This withdrawals the as  additional  and  additional follows.  adverse  the  asset  loss,  The  is  is  a.function  structure.  defined  dollar  economic  loss  value  conditions  by  of  The  of  both,the  functional  inequalities  the  expected  anticipated  deposit  relationships,  (4),  which  deposit  are  for  developed  withdrawal  under  is  m• 1=1  where the  is  the  parameter  traction book  of  value  the  of  (Ki)  bonds  of  (K2)  which  include  loans loans  withdrawal the  under i  in  period  economic  the  asset  disintermediation,  than  five  government 3)  [27]  cash years  in  formula  period to  conditions,  t,  is  measure  the  and  is  conthe  t.  how  formula  liabilities  adequacy  adverse  includes  (k5);  of  capital  determine  adequacy  less  bonds  (k6);  to  which  ment  municipal  i  financial  capital  Reserves"  in  l i a b i l i t y order  severe  FRB's  contained  l i a b i l i t y  In under  anticipated  the  as  (ki),  with  "Intermediate  assets  follows: treasury  maturity  bonds  structure  (k3);  affects  are 1)  classified "Primary  bills 2)  (k2),  "Minimum  more  than  five  Assets"  (K3)  which  and  4)  "Portfolio  Assets"  (Ki+)  which  Based  on  this  classification, liquidity  liquidation  consist  as  and  per Secondary  and  govern-  Risk  Assets"  years  maturity  includes primarily  (k4),  mortgage of  personal  (k7).  structed  with  illiquid  and/or The  economic  the  asset  property  the  extra  conditions  that  liabilities liquidity is  they  increase  become  required  determined  as  more as  as  the  assets  P.  become  are more  liquid.  reserves  follows:  reserves  for  possible  adverse  con-  19  a  k  k  ,  i=l,2,3,  (4)  keKiU...LK.  where  a  k  are  •to m e a s u r e  the  the  asset  has  q.  is  a measure  of  liabilities  adequacy  to  parameters  shrinkage  be  of  the  over  not  net  value  quickly  reserves  assets  in  K  of  asset  under  x  U  •••  3,x.  capital k from  adverse by  the  U K..  adequacy book  economic bank  for  Finally,  formula  value  if  the  conditions, excess  the  used  and  liquidity  FRB's  capital  < Net  Worth  X.  the  an  -P2  operationally,  intuitively  > Pi  +  P2  general  +  P3  +  -P3  (5)  h  }  q.. i t  H  appealing  of  S  the  tJXJ  T x . e K 1 u « -Li- u K  x.  >  y  S  model  t  x V  i  p..  +  a T i  x J  u  of  surplus  CC-type  =  assets  manner  c .x J J  u=l  s.t.  manage  Total right •hand s i d e o f balance sheet  f  Xj^O  > -  more  formulation  max  P.. it  -Pi  1  K  Thus  the  required  decision-makers,  worth),  l-B,  in  is  I i= l  (and  in. the  liquidated  formula  Since  contained  ^  and  liabilities  stating  -  (5)  equity-  may  be  stated  is  (6)  as  20  j ,  where  i=1,2,3;  9  t = l , k = l , •  tkj  j  X  • •,K;  =  c.  b  is  tk  the  net  present  return  on  asset  j ;  J  is  the  amount  funds  for  financial  in  period  terminal is  the  and  K  t,  T  is  resource the  and  point.  linear  optimal  period  time  t:  k  or  the  above  of  the  future.  presented in  the  information  technique  can  changes  the be  and  is  t  for  instrument  to  changes  of  are  an  in  of  or  b ^  period  the  the  port-  yields  an  balance  sheets.  consequence  to  at  the  by  next  the  solution,  sensitivity  expectations  of  of  dual for  asset  identified  dual  is  tive  function  variables  through  interpreted  since  resources.  For of  as  the  the  costs  can  instance, capital  the  liability the  by,releasing  importance  duals  and  and  reduced  costs  management.  (nonzero)  dual  The  improvement  one  resource.  be in  of  attached their  adequacy  case  to  the  study,  constraints  important  binding  variables.  incremental unit  have  The  the the  constraints  (worsening) This  economic  value of  of  the  the  objec-  is  of  practical  procurement  of  additional  Cohen to  be  and high,  Hammer [ 2 0 J which  of  funds  j ;  initial  (T)  generated  test in  of  policy;-  k  program  available  portfolio used  is  linear  series  be  type  flow  constraints.  portfolio will  flow  i n i t i a l ,  formulation  a  dollar  the  constraint  The  per  is  period  by  the  the^external  termina-1  for  changes  is  a ^  in  and  s ^  t  terminal  policy,  is  S  periods;  type  implementing  policy  instrument;  environment.  implications  the  that  s i n c e more  to  in  policy  scenario  immediate  programming  The  be  for  necessary  Before  solution  economic  can  the  of  i n i t i a l ,  action  decision-maker  decision the  of  j  coefficient  of  economic  only  financial  instrument  available  input  course  However, the  an  jth  the-number  number  The  optimal  the  technological  is  folio  of  found  suggested  t;  21  that  additional  less  than  In  capital  the dual),  t h e same m a n n e r ,  determined  from  in  new  first  the  optimal  straints the  and  portfolio.  of  expectations  the disenchantment  programming optimal  portfolio  f o r the  bank  can also  policies  purchase cost funds  to  the  an a s s e t ,  is  is  firm. be  decision-  not  interpreted  from  This  that  [20,  of  currently  as the per  the optimal  the l i n e a r  any p o l i c y  portfolio  i s accomplished  and the r e s o u r c e s  50,53]  t h e CC m o d e l ,  (see f o r example  c a n be o b t a i n e d  for different  However,.this  various  will  solution  environreturns the  generated.  many e x a m p l e s  not generate  Prob-.  scenarios  scenario  and a  in order  an o p t i m a l  simulation Another  to  source  in the model. economic  of  continues  The major  conditions.  con-  the economic  Thus  to each  on  observe  available.  a c t as a d e t e r m i n i s t i c economic  i s to  criticism  uncertainty  c a n be a p p l i e d  additional  the  [7,21,34]).  model.  by t h e bank  by c h a n g i n g  contains  of  under  of  i n t h e new o p t i m a l  of  programming  The s e c o n d  expectations  the l i t e r a t u r e  but rather  change  by i n s e r t i n g  i s the omission  solutions.  behaviour  of  i n the bank's  formulation  problem  uses  formulation.  are reflected  the model;  distributions  the total  funds  profitability  information  i s accomplished  instruments  applications at  important  programming  the f a c t  of  to  to  in diverting  This  financial  levelled  erate  wanted  portfolio.  be  linear  useful  the effects  any changes  Despite  ability  of  the reduced  a r e two o t h e r  to observe  of  successful  then  forgone  on t h e o p t i m a l  changing  costs  provide  manager  in the linear  costs  in greater  procuring  portfolio.  is  effects  ment  also  solution,  There The  result  cost, of  variables.  costs  profitability  this  the marginal  the opportunity  a portfolio  the optimal  unit to  If  would  the dual  Reduced maker.  (assuming  to  to  gen-  solution observe  criticism  22  of  the  model  formula  is  which too  its  is  that  are  2.3  Stochastic  use  of  the  likely  to  be  'safe'  The  and  major  the  to  the  cope  models  coming  and  essence  a  with  are  of  [15]  replaced  deposits random  more  the  chance-constrained  was  and  by  with  the  most  could  lead  and  are  attempts  adequacy to  portfolios  were  both  in  penalty  treatment  the  of  types  for of  area.  appendix  at  as  a manner  the  end  The  joint the  and  inability The  of  majority  above  short-  capturing  the  not  and  unresolved  was  in  3  for  formula  Future  distributed to  computationally  have  in  and  adequacy  claims.  normally  2 , « « ' , n  Chapter  Charnes  capital  the  magnitudes  Also  (Utilized  uncertainty  decision-maker  that  varying  periods  of  the  with  withdrawal  does  constraints.  infeasibility in  than  Thore £ 1 6 ]  enabled  d i f f i c u l t i e s , as y e t  the  concerned with  and  this  either  is  asset  problem.  incorporate  expressed  papers  the  to  management.  to  Charnes in  in  technique  liability  pioneers  approach  researchers,  uncertainty  section  uncertainty  different  conceptual  ^See  modelling  chance-constrained procedure  differential  clari f icat ion.  asset  Though  the  are  this  CC by  inherent  repayments  tractable,  there  and  capital  enough.  c h a n c e - c o n s t r a i n t s on m e e t i n g  loan  or  the  occupied with  the  incorporate  violations  profitable  programming.  were  variables.  a  of  initial  explicitly  handle  Board's  conservative  perceived  the  realistic  One  Littlechild  as  discussed in  thus  of  not  weakness  management,  of  too  Reserve  Models  liability model  thus  Federal  of  facility  constraint  a multi-period the  to  model,  literature  [32].^  In  additional  other  dealing  23  words,  the  economic  principal  weakness  consequences  of  of  chance-constrained  violating  a  constraint  are  programming  is  that  considered  only  the  indirectly. A the of  problem  of  financial  asset to  second  dynamic  ability analyzed  problem,  The  the  for  technique  very  of  account  by  limited  Daellenbach  these  the  useful of  models  inherent tools  in  are  two  numbers  and  and  Archer  that  they  uncertainty practice,  financial  solves  of  [28] are the  their  instruments  three  applic--  that  can  approaching  the  be  simultaneously. third  is  a  alternative,  sequential is  horizon  to  it  to  Wolf  would  be  the  the  to  problem  a of  [98]  for  approach.  The  essential  to  enumeration.  flaw  explicit  for  periods In  assertion  solution  The  optimal  it.would  optimality.  dubious  Wolf  decision analysis  because  strategies  equivalent  avoids  an  period,  guarantee  makes  implicit  find  by  theoretic  sequential of  not  portfolio  order  drawback,  use  one  proposed  decision  employ  beyond  possible  thus  into  number  does  (He  virtues  small  that  model  extended  by  nique  in  was  only  This  Fama £ 3 4 , 3 5 , 3 6 ] m o d e l l e d  are  the  point  and  these  through  all  take  programming.  management  albeit  solution  time  Eppen  liability. they  dynamic  liability  work  limited  model  is  was  their  that  A  his  and  However, is  and  instruments.  one  and  problem.  of  asset  problems,  include  approach  solution  be  an  effort  that  the  synchronizing  the  by  the  to  an  with to  necessary  preceding  provided  find  optimal this  tech-  problems to  explain to  with  a  enumerate  present  solution  notion  decision  away a one  this period  solving  an  n period  maturities  of  assets  model.  and  1iabilities.) A is  fourth  stochastic  relatively  approach,  linear  efficient  suggested  programming solution  with  by  Cohen  simple  algorithms  and  Thore  recourse  existed  for  [21]  [SLPR]. solving  and  Crane  [24],  Although SLPRs  [91,92].  24  both  models  explicitly model  were  solved  by  characterizes  formulation  by  using  each  'extensive  realization  a constraint.  computationally  infeasible.  This  Cohen  viewed  model  (in  and  the  tional  Thore  aggregate)  time  number  of  time  problems  periods of  and  (Cohen  variables  formulation tions  rather  by  the  number  this  management  algorithm  [95]  more  [4]. of  as  perceptions than  Thore  those used  He  a  limited  variables  the  for  decision the  which  There  Crane was  the  considered  greatly,  Thus  limited used  an  number in  order  con-  terms in  apply  possible  to  computa-  in and  to  fact  analysis  precluded  two),  of  in  the  both  attempt  the  were  sensitivity  tool.  technique  in  problems  formulation  were  and  both  variables  modellers  tool  of  one  random  (.realistic)  handicapped  realizations.  (B-C)  [5,  6,  model  available  The  final 7 ].  This  bank  portfolio  Wolf  [98]  formulation.  intractable  as  the  shortcoming,  developed  a  computational  See  will  will  model.  they  developed  used  has  Recall  model that  time  was  has  Wolf's  periods the  programming  and  the this  realiza-  incorporate  systematic  using  solve  many o f  reformulated  linear  to  approach  The  of  be  be  model  number  general  a comprehensive  stochastic  a  the  and  the  large  a normative  dissertation  liability  this  other  and  Booth  the  of  This  two  periods. In  to  than  i n t r a c t a b i l i t y and  sideration of  their  So  representation.' '  SLPR and  as  by  desirable its  most  Bradley  became  increased. and  In  efficient  Crane  essential  origins  the  computationally order  liability  decomposition  and  features  conceptual  model  asset  the  and  i t .  proposed  the  asset  to  overcome  problem  algorithm  that  and alleviates  difficulties.  appendix  at  the  end  of  Chapter  3  for  further  comments.  25  The These  economic  outcomes. each  B-C model scenarios  The economic  element  (varying  (economic  amounts  then  formulated  tion  of  of  1)  purchase  assets  more  constraints,  3)  at  losses  loss  flow than  to exceed  constraints,  e  N  e E  of  the firm i t  which some  which  path  rates.  the f i r m .  available; cannot  held  at  K  2)  sell  and/or  the beginning  upper  the holding  of  r  bound;  h k (e>;) mn N  N  N  ( e  N  }  +  V  NN(eN}  bN(eN)  s.t. Flows  K D  I k=l  t # e "  n  )  •  a  K  n-2  I k=l  m=0  K  n-1  I  I  k = l rn=0  1  + g (e ) 3 m,n^ n'  s  m,nv  (e  n  )  = f  n  (v e  ) n'  more  +  to  of  an  period; capital  and 4)  N-l  of  balancing  hold of  is  maximiza-  types  the firm  a particular  m=0  k=l  flows  i s the  the net realized  pre-specified  where  The p r o b l e m  inventory  I< I  p(eN)  y  Cash  cash  are four  do n o t a l l o w  possible  diagram  function There  scenarios.  a l l  as a tree  has a s e t o f  do n o t a l l o w  limit  of  interest  which  economic  the set of  The o b j e c t i v e  has funds  than  include  of  is  I  max  wealth  that  constraints,  formulation  and a s e t o f  constraints,  a period  to  i n each  program.  i t  ensure  in a period  composition Their  which  the development  c a n be t h o u g h t  conditions)  terminal  cash  the end of  capital  scenarios  as a l i n e a r  the expected  upon  are considered  deposits)  constraints:  asset  depends  class  asset.  26  2)  Inventory  Balance  -  b k , (v e J n-1 n - r  + s  h  3)  Capital  n-1  - / I, Category  .  ,  Limits  keK1  m=0  Nonnegativity  bk (e ) > 0 , m,n n' -  where  e  period  n  e E^;  1  to  conditions time m)  o>  = h  J g (e ) s ( e ) < L v( e ) , L s r s m,nv n' m,nv n n n' m=0 '  k=l  5)  o,o ( e o }  hk , (e ) = 0 , n-l,nv n  Losses K  4)  k , (e ) + n-l,nv ny  of  asset  from  k  k (e ) > 0 , m,nx n' -  n=l,«»«,N;  n having  periods;  s  y  m  (  e  n  )  k=l,•••,!<;  probability  period  1  is  h  to  n;  the  (conditional  dollar  of  purchase  income  N)  conditional  on e ^ ;  price D  n  (  e  n  );  yield v  (period )  is  n  the  m per  e  p(en);  K is  on e  k (e ) > 0 , m,nv n -  m=l,»»«,n-l, ' '  i s a set of E^  is  number  the of  set of assets;  per d o l l a r  ..(e.,) m,IM  N  m)  of  the d o l l a r  economic  of  condition  possible N is  purchase  i s the expected  from  economic  t h e number price  terminal  of  (period value  r  asset  k and h e l d  amount  of  asset  at  horizon  k purchased  (period in  27  period  n  conditional  on  e  ;  h  chased  in  period  m...and  (e  still  )  11  II  held  in  111 j  II  is  the  period  dollar n  amount  of  asset  conditional  on  e  ;  s  n is  the  dollar  conditional  amount  on  e  ;  gk III  II  price  (period  increase  m)  of  of  maximum  C^e^)  is  the  in  asset  type The  (e 9  II  of  )  in  B-C  k purchased  is  the  k sold  funds  net  (lower) period  intersection must  be  of  all  feasible  h  of  second  decision  second  decision point  solution states  economic  nature There  its  dynamic  tion  the  generated  of  has  is  nature  of  loss  amounts  (resources)  through  a  the  portfolio  upper  (or  mix  lower)  E^).  in  the  the  in  dynamic  ( e j ,  )  period  on  of  first  per  is  n;  in  period  dollar  the  of  n,  purchase  incremental  Ln(en)  losses the  in  amount  generated  (that  all  of  The  up  the  detract  on  is  is  the  period of  dollar n;  and  funds  first  ,.•  invested  decision  in  by  time  of  bounds  asset  its  no  other  as  set  on  set  the  for  this  reali-  the  from  words,  decision to  the  solution  conditional  However,  the  the on  final the  point. model  the  including  B-C  formula-  practicability.  The  upper  bounds  portfolio  example,  current  realizations  features  have  feasible  feasible  current  from  its  conditional  In  tractability  constraints  placed  the  model.  advantageous  development  is  The  point  to  as  possible  the  each  For  has  decision  arbitrarily  procedure.  nature.  period.  of  at  that  limit  are  in  s0i(ei))  This  occurred  features  bounds  sold  ) n  v  k  computational  the  fn(e  dollars  horizon  number  category  systematic  m and  (loss)  capital  realizations  decisions  and  capital  b  set  have  a  in  n;  for  realized  intersection  has  are  and  (ei),  to  that  a number  0 1  events  is  gain  period  k  the  zation  period  (e  k  m, n  n.  possible  for  in  bound  formulation  revision,  capital  available  k (immediate  in  pur-  II  allowable  upper i  asset  asset  (decrease)  amount  of  k  (or  lower)  managers  consideration  (except  in  categories).  the At  rather is  given  sense some  than to  that  point  in  28  time, of  this  its  may  available  short-term the  funds  Reserve  for  the  constraints nature  in  of  held.  Board's  actually  the  immediate B-C  unduly  these  some  the for  possible  set  economic  of  r e p l a c i n g the are  the  in  However, i t  B-C  the  B-C  state  ;  p.  the  when  has  to  capital  capital  model  satisfy may  turn Thus  all have  out the  to  of  revision These  two  loss  category  and  that  utilize  solution,  the  small  very  binding is to  the  shortcomings  the  events.  this  restrict most may  constraints  be  In of  would the  pessimistic corrected  with  others  -  not  be  possible  computational  to  correct  another  intractability  for  shortcoming  large  problems.  112]  Chapter  arbitrary  probability  and to  to  Unfortunately, taking uncertainty e x p l i c i t l y into account w i l l make an 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 model f o r the e n t i r e bank 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 an extremely aggregated model. The c o m p l e x i t i e s o f the 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 management p r o b l e m a r e s u c h t h a t t h e number o f c o n s t r a i n t s and v a r i a b l e s needed t o a c c u r a t e l y model t h e e n v i r o n m e n t w o u l d be v e r y l a r g e .  See  of  limit the  solution  a  limit  any  solution.  economic  problem  or  category  of  either  development  future  effect  amount  amount  formula  the  the  the  be  net  events. and  is  in  the  not  adequacy  loss  bias  to  does  procedure  composition  constraints may  compared  formulation  the  the  a disproportionate  nature.  may  formulation [7  of  immediate  capital  systematic  the  problem.1  region  bonds  systematic Since  constraints  constrain  invested  unsystematically  problem  formulation  has  recommended  shortcoming  feasible  of  Also  determine  c h o i c e may  revision  occurring;  that  bank  long-term  constraints.  Another  by  the  s t a t i s t i c a l l y generated  bounds  the  that  liabilities  Federal  other  imply  5 for  evidence  of  this  undue  constraint.  29  In only  bonds  assets,  tions  and f i v e  classes  period. each  problem  6120  These variables  is  equal  I  is  of  Bradley on  an IBM 360/65  much one  benefit reason  eight  the cash (.1  +  ••• + nD  realizations asset  A model,  categories  that would  of  assets, and 4)  and f i v e  possible  for  n _ 1  n _ 1  ),  ),  t h e same  to  manner:  ),  of  has a running as of  investment  f o r the s e l e c t i o n  (1),  of +  for  constraints )  limit  constraints  K. of  and  ••• + D  balance  D i s the time  periods,  assets. time would  an a c t u a l  of  the best  68 seconds  n o t be o f portfolio,  opportunities of  solve  number  the category  n i s t h e number  size  the  + D + D  n _ 1  per  and 656 v a r i a b l e s  conditions  in the selection of  assets,  realizations  t h e number  (1  realiza-  time  variables  a n d K i s t h e number  of  three  and 246,120  the inventory  (1)  possible  constraints  ••• + D  per period,  thirty  C 2 ) ; 2827  constraints +  2)  necessary  319 c o n s t r a i n t s  (2n+l)D  that  eight  thirty  variables  and t h e i n i t i a l  the aggregation not allow  classes  per period;  + D +  state  to a decision-maker  being  flow  classes,  and Crane [5].  ••• +  ),  assets,  i n the following  ••• + D  n _ 1  1)  constraints  calculated +  problems:  and three  variables  and 116,827  were  the  and d e c i s i o n  and 2460  these  Consider  periods  i s :  to  prohlems.  time  five  consider  model  per period;  periods  constraints  + D + D2  possible  i s t h e number  time  their  they  realizations  realizations  constraints  (I)(l  three  assets,  ( 4 + 5D + 7 D 2  + 2D + 3 D 2 + of  possible  five  (3);  limiting  management  t h e B-C f o r m u l a t i o n  for  loss  constraints  number  of  t o t h e sum o f  capital  (K)(L  assets,  numbers  of  thirty  possible  even  tractability  has computational  assets,  constraints  variables  (4).  still  of  3)  using  1141  However,  and three  The number  (1);  the  of  computational  and l i a b i l i t y  classes  per period;  to gain  model.  asset  periods  five  periods  for  four  time  assets,  five  in their  t h e B-C model  following three  an e f f o r t  into  opportunities  30  within  a group  of  number  of  periods  In  fact  time  the  exclusion  liabilities possible in  the  thus of  does  size  not  246,120)  would  formulation of  the  can  master  master  liabilities  basis  be  has of  Another  were its  dimension  may  be  less  computational  striking  Although approach problem closer  to  asset  In literature models in  and  lack  certain  be made  conclusion,  are  not  the  the  B-C  of  68  economic of  the  are  has  in  formulation  has  compared  (4)  they  B-C  basis  time,  (1)  of  the  has  in  the  nonzero  (2)  and  (3)  formidable.  may  appear  computational  problems  in  presented  to  In  a  sound  the  Chapter B-C  thus  purposes.  problems to  be  tractability  features.  developed  The  850,000  difficulties  large  x  (1)..withra'  of  decision-making  the  the  running  order  remain  inherent  been  to  and  (116,827  (4)  the  undesirable  for  of  of  tool.  Also  formulation  tractability  of  crude  uncertainty  a model  evident.  handling  is  inherent  size  of  incorporated  events  if  and  number  is  decomposing  is  s t o c h a s t i c models for  the  the  assets  limited  seconds  management,  possess  satisfactory  computational cases  the  liability  seem t o  will  data  of  uncertainty  by  this  (4)  (4)  the  inadequate liabilities).  a decision-making  needing  nevertheless,  initially  formulation analysis  but  is  the  and  diminished  even  When  and  and  matching  much  as  solving  elements  any  However,  but  5467.  of  assets  of  be  is  (of  that  exploit  [5];  39 and  nonzero  The  fact  difficult  dimension  order  elements.  to  decomposed  difficulties  2200  the  utilized.  use  problem  shortcoming  distribution  model  computational of  makes  d e f i c i e n c i e s would  four make  potential  maturities  Despite  the  These  problem  match  probability  allow  model.  of  of  realizations. the  Another  to  impossible.  model,  the  assets.  fit  and the  and 5  formulation.  far The in  in  the  proposed addition  technique  rather faced  than by  the  reflect bank.  the  actual  asset  and  l i a b i l i t y  management  probl  Chapter  3  FORMAL DESCRIPTION OF THE ASSET AND LIABILITY MANAGEMENT (ALM) MODEL  3.1  Introduction  The  size  and s t r u c t u r e  acquire  i s constrained  deposit  withdrawals  tagous order hold  t o meet  unanticipated a  bank  cost in  must  of  liabilities  i n cash It  liquid  assets  prior  across  and l i a b i l i t y  disadvanassets  in  Thus  a bank  must  and l i q u i d  assets  to  environment  between  the  meet that  opportunity  and the p o t e n t i a l  to maturity  loss  matching  certain  model.  associated  incorporate  the maturity  32  the maturities  the shifting  costs  and t o  of  on i n v e s t m e n t ,  management  to capture  the transaction,  by  particular,  incurred  maturity.  in order  time  In  earning  is in this  a trade-off  a bank c a n  i s usually  requirements.  return  incorporate  flows  arise.  It  liquidate  'optimal'  multi-periodicity  cash  to  and an  i n an a s s e t  assets  flows.  synchronization  is  selling  i t s cash  'optimal'  included  to  to  that  an  be  time,  yielding  prior  insure  i t s assets  involves  portfolio  on demand.  i n cash  as they  This  lower  assets  of  of  f o r banks  shortages  drains  function.  To and  impossible  portion  cash  holding  selling  be s a t i s f i e d  unexpected  a sufficient  an a s s e t  by t h e u n c e r t a i n t y  must  and sometimes  of  of  yield  spreads  with  calling  assets  with  assets  features  The f i r s t  a smoothing  of  of  must  feature across and net  anticipated  33  cash  outflows.  include third in  the  to  qualities is  that  fees  second  uncertainty  feature  order  The  is  the  satisfy  of  the  and  other  of  cash  assets  costs  those  should  environmental (deposits)  consideration  accounting  with  expenses  is  flows  simultaneous basic  transaction  feature  and of  the  to  the  in  order  purchase  in  order  rates.  and  to  The  liabilities  match  the  liquidity  The  final  feature  liabilities.  included  associated with  interest  assets  p r i n c i p l e s and  of  be  uncertainty  to  incorporate  and  sale of  the  asset  brokerage  financial  instruments. The management [11]. For  approach  problem  However, example,  liabilities  as  constraints  (as  availability [20],  the  tiating  these  of  the  are  this  certain  for  problems  for  of  uncertainty  the  investments of  many in  [7,  20],  Federal  to  various  Chambers  [7,  15,  the  maturity  to  and  use  of  actual  2 demonstrated,  their  at  the  formulation  formulation. exclusion  liquidity  [20],  the  of  a  omission  of  differen-  model  have has  been  served However,  existing  model  period  well as  a  as  the  handles  wel1. incorporation  difficulty  of  of  researchers  stochastic  optimization  problem.1  Computational  methods  uncertainty attempting were  in to  an  efficient  extend  the  the  obstacle  manner  CC m o d e l .  u n s u c c e s s f u l l y used  t r a c t a b i l i t y was  to  that  was  Many  approach could  the  not  the be  overcome. V o r end  of  this  a  review  chapter.  of  s t o c h a s t i c programming  see  of  end  problems.  no  the  [27])  deposits, the  liability  conservative  Board  the  and  Charnes  28],  of  only and  in  21,  Reserve  purposes  types  to  weaknesses  Nevertheless,  applications  Chapter  the  inherent  investment  costs  dissertation  motivation  literature.  survey  The main  funds  the  point  literature  its  p r e s c r i b e d by  between  starting  in  decision variables  of  in  as  exclusion  holding  documented  has  there  the  taken  the  appendix  at  the  34  As be  used  ment a  as  the  model  bank  was  already  solution  (ALM)  to  the  manner.  and  liability  the  model.  will  also  3.2  Formulation  be  All  of  management  included  The  in  to in  The  ALM  asset  and  liability  liabilities  costs  ALM  (interest  problem  constantly with  a  (for  work.  essentially  being  revised  continuous  fore, the which  is  ALM  model  portfolios  model  its  cash  of  management-(ALM)  optimization a  bank,  and  given  random  time.,  flows  decision  determined  at  consecutive  ALM  model  developed  formulation  Objective  infeasible  each accounting is  will  manage-  will  flows  allow  in  a  comprehensive be  can  be  in  model  asset  incorporated of  is  determine  the  CC  in  model  for  an a  rates  (deposits). problem  computations  a multi-period  The  are  the  to  as  of  a  deterministic  cash  a continuous  over  tool  developed  end  for  shortcomings  is  the  1.  ALM  process  are  general  The  time  example,  The  of  rates),  liability  will  Model  of and  appendix)  and  earlier  other  (see  model.  of  decision-making  and  the  [95]  asset  necessary  enumerated  ALM  intertemporal assets  uncertainty  overcome the  the  section.  features  model  algorithm  solve  next  of  the  Wets  to  the  question  Techniques  the  technique  presented  address  systematic  noted,  and  as  portfolio of  Although  the  portfolios  are  analysis  a normative  points  involved  tool.  d e c i s i o n problem  discrete  returns  in  Therein time  period). a mathematical  stated  as:  function  maximize the net present p r o f i t s of a bank minus the e x p e c t e d p e n a l t y c o s t s for infeasi b i1i ty.  programming  frame-  35  2.  Constra i nts  Constraints and  (a)  stochastic  stochastic  a.  legal, bank1s  b.  budget, which t i o n s and the  c.  l i q u i d i t y and l e v e r a g e , to 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 on demand, ( t h e F R B 1 s c a p i t a l adequacy formula form the b a s i s of these c o n s t r a i n t s ) ,  d.  p o l i c y and 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 the bank and c o n d i t i o n s to ensure the bank's continuing e x i s t e n c e a f t e r the termination of the model, and  e.  deposit  and  use  constraints,  of  linear  ment  problem.  ness  of  and  LPUlP  with  the  do  Rather  bank  is  at  some  feasibility chastic  linear As  However,  not  the  imply  Linear  (e)  contains  to  the  cost  is  and  model  a  the  of  only Cohen  The  of  In  order  under  ALM  deterministic  deterministic  or  constraints.  and  [20]  Hammer and  put  have  l i a b i l i t y the  aspect  banking  of  manage-  LPUU  business,  into  justified  appropriateis  constraint  receivership.  portfolio  of  assets  problem  fits  well  to as  regain a  sto-  model.  formulation  decision  both  stochastic  is  its  recourse  of  linearity,  the  The  the  either  uncertainty  restructure  previously,  consists  asset  intermediary  simple  programming  of  view  (penalties).  a zero  (c)  bank's  argument.  to  with  the  initial condiand uses of funds,  consist  [11]  point  that  allowed  stated  model  can  established.  program  was  (d)  Charnes  is  of  flows.  following  violations the  and  from  are the sources  deterministic,  functions  Thus  using  justified  are  constraints,  Chambers the  (b)  which are a function j u r i sd i c t ion ,  rule  uncertainty.  is  model  a multiperiod in  that  model.  decisions  36  for  period  way  that  l , » » * , n  total  maximized.  in  arbitrarily  be  after  the  purchased In  in  flavour.  The  variables  are chosen.  determines their  costs the  of  involved 'rolling  them of  violated  effect  over'  magnitude  of  recourse  to  in  that  outweigh  defining of  should  the  SLPR,  of  be  a  static  a security  gives  dynamic  i t  the  a  decision  are observed.  This  feasibility) of  both  recourse  decision the  'rolled  periods.  initially  of  the  subsequent  is a function  choices  to  an  position  a manner t h a t  The  is  0 and 1  refers  bank's  recover  a  liabilities.  times  t h e model  variables to  and  time,  the  such  essentially  assets  t h e model  violation.  aspect  is  in  l,«»»,n  the drawbacks  such  of  1  in period  one o r more  order  feasibility  t h e ALM m o d e l ,  and the  in  penalty  'aggressive'  regaining of  in  means  The  0,  overcome  the stochastic  penalties.  period  setting  practice  aspect  (in  by  refers  and s o l d  variables  and the  with  1 In  two-stage,  Next  restraining  flexibility  t h e ALM  being  recourse  corresponding  straint the  the  point  recourse  bank's  point  are defined  period  the  the  The  partially  variables  one t i m e  model  to  costs  revision  apart.  in  decision-maker  of  the model.  Also  addition,  the  immediate  and t h e  running  decision  that  revision  penalty  revision  period  position  instant  expected  be n o t e d  time  continuously.  model,  minus  incorporates  initial  as an  immediate  small  immediately over'  should the  The ALM model  bank's  profits  It  interested  a r e made  the  cost  variables  if  give  features  model. The  ALM m o d e l  c a n now be p r e s e n t e d  (see pages  39,  40,  the  Thus,  so as to  dynamic  con-  has  variables  benefits.  are the  and  41,  42).  can  37  Notation  asset k purchased in i=0, . . . , n - l ; j=i+l,  initial  holdings  of  s e c u r i t y purchased of the model,  new  deposits  initial  funds  of  borrowed  shortage  surplus  in  in  in  of  in  proportional  i  d  in  sold  j  j  penalty  of  of  i  and  period  deposit  Model  in  period  to  be  j ;  k=l,  . . . ,  K;  k,  period  period  period  period  period . . . ,n,  ALM  security  type  holdings  for  i ;  type  held  beyond  the  horizon  d = l , * « ' , D ,  d,  i ,  stochastic  stochastic  cost  constraint  constraint  associated  with  y*  type  type  s,  s,  ,  J s proportional  penalty  cost  associated with  y~ , J s  parameter f o r s h r i n k a g e , under j of asset type k purchased in  normal period  economic i ,  conditions,  parameter f o r shrinkage, under j of asset type k purchased in  severe period  economic i ,  conditions,  proportional transaction or sold in period i ,  return  tax  on  rate  marginal  asset  on  tax  cost  k purchased  capital  rate  on  proportional capital i and s o l d i n p e r i o d  gains  asset  period  (losses)  income  gain j ,  in  on  in  in  period  (loss)  of  k,  which  is  either  in:period  in  period  purchased  i ,  period  j ,  j ,  security  k purchased  in  period  38  Yd  -  the a n t i c i p a t e d f r a c t i o n of d e p o s i t s adverse economic c o n d i t i o n s ,  c.j  -  rate  p.  -  discount  K1  -  set of current Union Act,  Ki  -  s e t o f p r i m a r y and s e c o n d a r y Adequacy Formula, .  K2  -  s e t o f minimum Formula,  K3  -  set of intermediate Formula,  qn-  -  penalty covered  P.  -  liquidity  reserves  by  in  k  .  5. J  S  paid  assets  on  deposits  rate  from  assets  risk  . . .  type  period  as  i  risk  type  to  period  u  by  assets  as  assets  the K  under  as  defined  as  o,  the  in  defined  potential  British  defined  potential withdrawal Ki u ••• u ,  for  d withdrawn  d,  specified  assets  rate f o r the by a s s e t s i n  mortgage,  of  of  the  the  the  of  Adequacy  Capital  funds,  withdrawal  Credit  Capital  Capital  in  of  in  Columbia  which  funds  Adequacy  are  not  i 5  -  mith  and  -  d i s c r e t e random v a r i a b l e type s where s e S.  in  period  j  of  stochastic  constraint  not  covered  39  The  ALM  Model  K  I  Max x,y,b  .L j=2  k=l  n-1  x  r o j^£=2 j, ot'-^ r  +  z  0 J  ,  T  1-t, K  ^£=i+l  J  £  1J  Z  *  l r  (,.  k  )  T  bQ  .5 y  c  p.  eg  -  P ]  ^  b  .  I  I  i=l  j=i  +  C j  b  p.  -  Y  1  y d.  E?-  -  min [ y V  Subject  tl-T.)  1  -  i  -  j - l  ^d  1  "  d c.  Y,  p.  j - i  d)  C .  y.  +  £  £  j=l  seS  p .  +  [Pjs  y  -  j s  j s  p  jsJ  y  to:  Legal  constraints  I  L  0 1  £'r£  d=l  j=l  I  L  D x  k  0  +  x  -1 0  L  i=0 ^£=2  keK1  z  JJ  :  P  £=1+1  1  + I  I  01  1-T.  D • J  (a)  x  n  i = l j=i+l  1=T  +  oV - J^  -  k  I  .1  y  o  +  i  y  -  r  yJ  d  o , il y  +  d=l  lo  o  b  +  j  I keK1  I  I  i=0' £ = jv +. , l  D  x\£ 1"  + X  -  l »  b  > 0,  l  -i=0  j  =  1  2  2 , - « « , n ,  i  -  y.  > o,  i  =  , j - i - l  d  1  d=l  +  rj-1  .1  b  1,  40  (b)  Budget  constraints  (i)  Initial  holdings  *n-; Uj  j =l  +  x  n = 0°°  yjj  x  !L>  = y  0  J  (ii)  Sources  and  k  1 , * ' * ,K 5 5  =  00  d  oo  ,  d =  +  z  , 5  1,---,D  J  uses  K  I  1  1=2  k=l  1  *  +  t  -  D  d  Yd  + I  01  1  -  1  Tn  +  z  01  d-r  0  y  1  *01  1  l  y  b,  =  0,  J  =  1  ,  d=l  r  K  n  ,  ,  j-1 1  I  +tk  i=0  + +  D  I  I  k  X.  1J  1  .  1  zr  -  T. J  1  -  Id) 2  Y.  d d]  d C j-1  _  y  +  1  z*.  h d  j-i-2 yi  At*A. 1  1  i=0  d=l  Y  +  i  £=j  1  d  - Id 2  d  j-1  C  2  j - U +  b  J-l  1+  iu  b,  =  0  ,  j  =  2,  (c)  Liquidity  constraints  I  (i)  I  keKj.  v  k  )  k  j-i-1  \ J  I d=l  y  i=0  -  J;  "  +  b. J  Id) 2  Yr  k  I  d y  i  k  k  x..a..  i=0  1  I  d=l  a..  +  qla-  k  + x.  a..  +  b. J  £=j + l  j-i-1 1  "  +  y • 1  Y,  -  P  <2j  i=0  -  (iii)  v  I  I  keKiUK2UK3  d=l  K  j  k=l  i=0  I  (iv)  D  a  i=0  +  P0 . + 2j  P,. 3j  j-i-1  IY d^  r  1JJ  +  b.  3  +  I  d=l  V  +  J—  i=0  b  d  +  iyj .• 2  1 -  - s...  1  +  q  3 j  'js  j-i-1 "  Y  d.  2j  k  2j  n  I  k  x  1  D P. . lj  k  +  >  Y  k  i=0  D  I  Hj>  1  keKiUKz  D  1  i  - I  .  k  + x.  1=0  D  (ii)  k  x..a..  P  3j  y• >  y 1  2  dT j  42  (d)  Policy  constraints  n  J •1  I  \=0  I  Deposit  d yJ  Kv  ]  +  +  i=0  x.  £=j+l  +  (e)  k  ,  k +  m l  x.  U  y. js  -  +  y . js  ,~  v i=0  1 0 0  y" js  < -  5.  L m2  n  m 1  »  £=j + l  .  IJC  j=i,  JS  Flows  dd  J-i 1  "  Y,  -  J  y . j s  =-£. , \js'  j =l , —  9  n;  d=l, —  ,D.  43  The expression losses  objective  is  (net  the  of  counted  costs  cost  direct  of  The  final  the  stochastic  each  model  treats  and  the  locale  the  initial  of  are  no  constraint the  cannot  funds  be  from  the  ment  that  less  The  to  expressions. and  expected  or  from  Reserve  market  in  capital  the  liquidity The  to  and  the  refer  net  to  a central  penalties  10%  a  of  the  constraints  of  dis-  the  bank.  for  violating  of  as  bank's  the  constraints  period.  legal  states  are,  of  The  the  and  that  The  the  course,  strictly  current reserves,  peculiar  notation  to  constraints  accounting  identity  -  in  adequacy  assets  formula. adequate  conditions  formula. adverse  liquidity  is  Chapter  In  is  order  economic  include uses  Columbia  is  as  Credit  follow  The to  the  to  require-  meet  develop  in  Union  depositors'  principal this  conditions)  constraints  British  2,  the  Act  are  ALM  used  in  Chapter  2,  Section  model  [8].  2 same  ALM  budget,  less  budget  developed  economic  (for  three  one  liabilities  studied.  capital  a  only  into  funds.  adequacy  the  shown  total  statement  of  in  constraints,  being  reserves  by  incorporated  constraint  adverse  first  defined  of  Board's  value  constraint  The  first  gains  refers  expressions  banks  The  capital  expression  fourth  other  the  constraints,  during  ^As  taxes)  conditions  types  sources  claims  defined.  of  and  factors  legal  and  withdrawal  first  of  to  institution  liquidity  Federal  constraint,  from  sum  legal than  1  equal  the  the  five  second  third  either  two  The  the  The  The  refers  conditions  are  The  discount  first  equity. of  assets.  (net  of  constraints.  deterministic.  surplus  is  consists  returns  deposits.  expression  since  assets  on  borrowing  There  as  discounted  taxes) of  function  2.2.  are  44  I  «  ,  k  k  1=1,2,3.  keKiU-.-uK.  The  principal  constraint  of  the  total 1  -  x.  3,  1  1=1  The should under in  be  fact  straint it  the  -  i=i  p.  or  last  stochastic  because  the  capital  guideline  regulation.  (as  The  prescribed  regulation  for  a  This  costs.  in  surplus  a  In  value  of  reserves  is  -  equity-  the  bank's  bank as  ALM  model.  portfolio set  forth  management of  manager  be  this  conservative  violated  manner,  the  is  be  con-  a  strict 3 £ i=l  q.  of  the  benefits  criticism, can  a  is  violate  is  treatment  when  this  may  FRB  than  constraint  the  constraints,  the  rather  this  constraint  Although  by  assets  disintermediation  This  'psuedo-stochastic' to  for  liabilities. the  violation  constraint  FRB's  all  bank  the  systematic  of  levelled  resolved  FRB's  in  at  a  manner. The  fourth  constraints institution  policies  increasingly  are less  penalty  dependency  'sound'  penalty  the  plus  formula  allows  using  convex  adequacy  for  -  market  liquidity  nature,  FRB).  modellers  bank  in  formula  hand  balance  the  constraint  the  exceed  the  the  by  violation  These  that  conditions  not  of  adequacy  sheet  than  liquidity  right  side  states  greater  economic  +  1  is  suggested  of  y  constraint  equal  severe  >  capital  set  are  constraints  introduced  modelled. usually  In  the  to  (via  The  also  capture  the  minor  while  more  introduction  additional  penalty  is  reality  tolerable  tolerable.  function  between  of  costs  and  psuedo-stochastic. internal constraint severe of  a  constraints) the  operational  extent  violations  violations  piece-wise  the  of  are  linear  can capture of  policy  policy  the violations,  45  This  is  accomplished  reflect  the  increased  The deposit (term an  flows  accounting by  now  be  constraints The  are  total  the  set  of  the  a  model  to  represent of  of  new  bear  reflect  the  actual  (y)^  amount  = BS  -  of  the of  type  I  y  1  i=o  J  (and  problem  was  of  funds'  'old  in  deposit  new d  are  various  expressions  consider  total  J  the  -  the  deposits  violations.  stochastic. rates not  of  net)  flows  ALM  during  in  each  the period.  formulation  constraints.  in  in  during  Since  interest  -incorporated  flow  generated  to  constraint  flows,  and  of  constraints  of  over  liability  deposits  y  deposit  outflow  First, the  supplementary  the magnitude  property  proportional  developed.  in  turned  has  This  types  of  constraints,  continually  three  amount  addition  seriousness  period.  having The  will  final  deposits),  model  by  the  jth  period  j  These  period. is  Y  J  ^  or  d + y. + J  y^  where rate  of  the  the  of  representing  jth  amount type  deposits  of  new  -  sheet  YJ d  D C d BS. J  = J  type  d deposits,  balance  second  type  outstanding  approximation  to  the  C o l umb i a  in  of  d  and  figure  liability  during  continuous  ^Statistically Br i t i sh  df, y. 1 H  ) i=0  .  deposits BS'? of  is  j ,  the  type  Y  D  is  the  discrete  d deposits  annual random  at  the  end  period. The  of  total  withdrawal  variable of  is  V  [25].  a  expression  period. flows  calculated  by  is  Since made  the  FRB  represents  the  by  model  assuming  and  is  the  total  discrete,  that  corroborated  half  for  of  use  amount an a  in  46  period's half  net  arrive  period,  flows at  the  arrive  the  funds  at  beginning available  y0  the of  the  are  d  beginning next  equal  x  1  +  of  the  period  period.  Thus  and  during  the  other  the  first  function,  legal  to  Yd y  d 2  for  period  j j-i-1  J-1  1  j - l  -  d y •  Y  d y •  i=0  or  I  i=0  The constraint the This For  above  and  incremental period  j  the  i  1  -  increase  j - i - i  dl  y"  + —  is  used  in  the  objective  constraints.  The  (decrease)  of  deposits  in  the  difference incremental  n.  1  expression  liquidity  incremental  Y y  is  used  difference  is  third  liability from  sources  one  and  expression  period  uses  to  is  the  constraint.  next.  47  3.3  Use o f  t h e ALM  Before have  to  Model  implementing  be d e t e r m i n e d .  The d a t a  3)  the point a function  4)  the i d e n t i f i c a t i o n of the l i a b i l i t i e s bank can p o t e n t i a l l y sell,  5)  the point  6)  the rate  7)  an e s t i m a t e d  estimates  returns  estimates of capital o f t h e t i m e t h e bank  estimates at  which  of  weighted  di scount  gains holds  the costs  deposits cost  are of  on t h e s e  of  a  Remarks, above  legal  which  the  liabilities,  funds  to  determine  constraints,  constraints  estimates  the unit  assets,  rate,  the parameters used in the development liquidity constraints,  the  model  withdrawn,  9)  the policy  the  (losses) as the assets,  these  the pertinent  or  of  used  by  the marginal  the  of  the  bank,  distributions  of  the  resources, and  penalties  surplus  in  incurred  for  the stochastic  are in order,  about  having  a  shortage  constraints.  the c h a r a c t e r i s t i c s  of  certain  of  inputs. Since  distributions optimal  of  t h e SLPR model  to  has a s e p a r a b l e  t h e components  solution.  problems  have  the  8 ),.  12)  would  of  to  include:  the point  stochastic  most  by t h e model  2)  11)  in  inputs  the i d e n t i f i c a t i o n of the assets in which the bank c a n p o t e n t i a l l y invest (or a t l e a s t a r e p r e s e n t a t i v e group of a s s e t s ) ,  10)  the  required  various  1)  the  the  t h e ALM m o d e l ,  trie  This  of  be i n c o r p o r a t e d  ^ See trie appendix  the resource  characteristic  correlations  objective  of  of  vector  SLPR  the components  in the solution  only  a r e needed  i s most of  the  marginal to  important  t h e random  find since  vector  technique1  a t the end of the chapter  fur a~ f u r t h e r aY'scussTbTr  48  The meanings deposit  in  shortage  the  Cds-  ALM  (y+)  and  formulation.  V  +  would  interpreted  as  purposes  the  the  in  assuming  y  dollars  some  asset  that  amount  ALM.  y  a  the  A  penalty  that  the  funds  would  be  at  a  to  point  horizon  of  the On  r.  the  other  cost  realization  p+  > 0 for  not  used at  penalty, net  is  hand,  want  have  to  the be  would  imply  that  this  case,  the  bank  would  of  this  action  0 plus  the  net  discounted  (that  i s ,  the  profits  able  funds).  p~,  > 0 and y  to  model  cost,  y  The  the  random  been  used  for  usually  returns that  investment  lower  than  all cost  can  in-earning  could be  be  then  equal  to  on y + ( r  could.have  be  determined  assets.  The  be  invested  in  (r  -  -  c)  been  c) to  disthe  generated).  if  assets. point  would  y~  is  c and would  profits  i=0  J  In  the  > 0.  utilize  J-i  occur.  specific  of  p~  invested  rate p+,  +  opportunity  discounted  the  p+  deposits  can  some  = 0,if could  of  would  the  (that  that  bank  The  0 plus  model  the  funds  Y  this  very  js  > 0 and y~  available  rate  counted  then  have  d  +  of  Since  assets,  funds.  by +  imply the  returns-on  available  Consider  variables  i=0  J  this  (y~)  If  d  then  surplus  d  +  ,  ^js  = 0,  that  to  divest  have  would  returns  that  d  would  be  that  itself  equal  on y " ( r have  is  -  been  a  surplus  of  some  to  (r  -  c)  c)  to  the  generated  would  earning  discounted horizon with  of  unavail-  49  One are or  greater too  a  than  much  determine case  point  is  the  study  well ALM  of  their  see  for  numbers  such  The any  point  optimality,  a  of  right  is  in  may  be  hand  (benefit).  does  not  increase from  the  this  time  the  negative. would  However, that  increase.  < 5 ^  <  > 5 ^  •  solves  duals  program.  side  and  c  if  in  and  either  should  addressed  p  be  p"  not  used  Chapter  enough to  4,  where  presented.  of  used  in  assets  utilizing  at  its  and  liabilities  their  should  capital  figures  be made  are  in  in  the  order  to  hand.  to  each  both  solve  an  stochastic **'  actual  constraint,  < ^ik-"  Also  the  problem,  a  ^  e  and  a  and  a n c  3 are  two  '  13  a  r  3^.  e  chosen  so  region.  algorithm  is  for  3..  r  parameters  algorithm  case,  forfeited  parameters  Before  ordered  in  be be  problem  required  and  will will  these  the  is  portfolio  [27].  above  i s what  Board's  bank's  the  profit  point  Wets'  a.. < E^-j  linear  tion  accrued  of  are  Wets'  deterministic  the  Reserve  in  issue  formulation  the  are  that  optimality  at  key  example  using  that  words,  This  estimates  realizations  that  A  a p p l i c a b i l i t y to  additional  chosen  ALM  Federal  When  of  the  is  other  invested.  formulation,  The  In  examinations  known,  test  So  0.  notice  penalties.  The adequacy  to  a  special  generated  The This  dual  of  implies  result  in  the  reason  the  marginal  a  that  type  of  correspond a  linear to  stochastic  that  an  decrease the  penalty  the  the  stochastic cost  duals  of  constraint,  increase in  program.  in  value  the of  exceeds  the  at value  the  resource  a  solu-  component  benefits  50  3.4  Appendix  1  A most zation of  models  techniques  appendix tion the  i s the  to  approaches  on S L P R ,  defined  denoted  simple  z  e Rm  discrete general  uncertainty.  with  respect  in this  subset  of  m x  model  Four to  basic  solve  to  the induced  approach,  sides).  uses  number This solu-  Also,  since  the stochastic  particular algorithm  Let  a  programming  emphasis used  5 be a random  The d i s t r i b u t i o n  with  is  to  linear is  solve  variable  function  approaches  have  been  on  <  of  £  Rm. appendix  [f(x,£)]|g(Ax  variable, suggested  This  The F a t  procedure  min x>0 s.t.  i s to  f:  =  5 is  a  < <». -  £)  > 0}  R n x & -*• R a n d g :  i n the mathematical  Formulation  [55] .  solve  c'x  Ax  assumes  where  A A  ^  model. 1.  z}  realizations  {min E  the decision  = P U  in this  possible is  z})  measure  be c o n s i d e r e d  formulation  e Rn  this  hand  been  models.  linear  right  space.  <  KM  = u({w:  variable  abstract  x  have  optimi-  F,  random  n,  to  Rm.  management  optimization  dissertation  (SLPR)  x>0 is  There  stochastic  be a p r o b a b i l i t y  a finite  and P i s  The  and l i a b i l i t y  stochastic  recourse  F(z)  where  asset  i t s c h a r a c t e r i s t i c s and the Wets'  Jl t o  by  of  the major  presented  (fl,F,u)  from  solve  (stochastic  with  Let  to  highlight  ALM f o r m u l a t i o n  placed  aspect  inclusion of  developed  serves  programming  is  important  i = l , « « « , L ,  Rm +  Rk.  literature  i t .  51  the  feasible  set  is  K  =  x:  x  > 0,  x  L n  e  {X:  Ax  =  g1}--  i=l  where is  K is  the  the  intersection  safest  region).  deterministic. restricted. alents.  Since  immediate possible  fat  the  revision  This constraints, types  ing  discussion  1)  marginal  to  it  has  is  a  with  x  to  chance-constraints two  for  be  large  empty  [0,1]  given,  the  i  t  h  row  constraints  types  Programming  £12,  for  it  unduly  be  if  feasible  it  A,  ^  probability.  which  formulated.  of  may  be  eaui v-'  every  the  There The  are  follow-  constraints:  is  i=l,«««,m,  the  i  A.  x  t  h  component  of  chance-constraints:  x  £  n i=l  [0,1].  .  to-the  satisfies  prespecified  > £..}•. > a. ,  of  is  131.  m  a e  or  (that  that  solution  and  Pfx:  where  is  deterministic  7].model's  the  realization  formulation  K may  to:very  each  formulation.  a certain  on  fat  that  [5,-6,  Chance-Constrained  defines  the  satisfy  fat  sets  chance-constraints:  is  joint  leads  problem  focuses  where A i #  2)  of is  Crane  P{A..x  are  advantage  Bradleyiand  §  of  feasible  disadvantage  technique  Ax  the  formulation  realization,  2.  several  One  A;major  The  of  ,  J  {x:  n  "  •  ^  £.}•  > a ,  £,  and  a.  e  52  The  deterministic  equivalent  p  i  i .  A  of  > ?i>  x  the  >  a  marginal  joint  constraints  i  is A. T  where  F  is  a^-fractile See  [67]  inf{y: of  for  F..  y  e Y.  (where  where F..  x  Y.  > F  =  denotes  a d i s c u s s i o n of  for  - a {y:  the  i = l,•••,m  '  i  F.(y)  > a.}},  marginal  deterministic  the  smallest  distribution  equivalents  for  of  ) •  joint  chance-  constraints. The specifying there  is  shortcomings  the  no  probabilities  differential  constraints,  or  a multistage  problem  of  handling  of  2)  the  a.  has  not  of  been  resolved.  Eisner,  Kaplan  tive  approaches  to  problem:  on  and  this  3)  conditional-go.  difficult  the  solved.  from  small  1)  and  have  large  Also  infractions  the  n to  period  have  discussed  treatment The  n+1,  some  of  not  three  preliminary  of  problem  has  chance-constrained,  provided  in  Secondly,  conceptualized.  [32]  total  difficulty  method.  versus  violated.  period  Soden  the  systematic  adequately  and  Stochastic  Generally  stated,  function,  by  resource  1)  include  2)  alternasafety-  results  problem. 3.  objective  a  constraint  been  first,  a in  for:  violations  this  approach  and  penalty  type  constraint  this  vector.  this  solving For  Linear  each  Programming  technique  a  linear  i=l,  ...,L  studies  program the  for  [78].  the  distribution  each  following  realization linear  of  the of  program  is  53  z.  = min  c'x  x  s.t.  Ax  =  x  This on a  sequence  the  linear  distribution  normative  tain  of  tool,for  distribution  quite  of  important  4.  programs £.  generates  Clearly  static  problems  0  the  distribution  approach However,  subproblems  has  l i t t l e  since  their  use  of  z  application  recourse  in  based  models  recourse  as  con-  models  is  [78].  Stochastic  Linear  Recourse  A  this  problems. as  >  £  SLPR can  be  expressed  mi n - c ' x x>0  +  Programming  with  Simple  [3,29,103]...  as  Er  inf • +  -  q  y  +  +  q  - ' : y  n  (PI)  s.t.  where  c e Rn  is  random  variable  y+,  e Rm2  y"  are  the  cost  defined the  Ax Tx  +  iy  vector,  x  e Rn  is  the  decision  b e Rmi  is  the  known  on  Rm2,  recourse  -  variables,  Iy"  q  +  ,  q~  e Rm2  vector,  resource are  the  E, i s  a  vector, unit  penalty  54  costs, m2  x  A  is  m2  a  '•m 1  identity  linear In  the  y  given  an  Tx.  if  q  +  x  (PI)  to  be  is  +  is  is  The  recourse  +  this  q"  discrete  but  it  is  this  x  n  known  in  q  as  a  in  the  such  linear  fc'x  ,y  s.t.  Ax  sum o f  Then  matrix,  is  always  as  as  a  and  I  is  a  stage  restricted  to  the  expected  and  qt  is  problem.  the  retrieve  if  problem  'surplus'  two-stage  to  that  the  and  feasible  shows  < 0 the  is  c'x  second  a manner  [67]  + q~  +  the  and y "  viewed  Parikh  if  variable  be  dissertation,?; is  x,(y  recourse  feasibility  bounded  + q..  vari-  = 0,  with  from row  i  can  unbounded.  be  discrete.  This  allows  program.  +'  I-P  +  q  +i  -'  y  - i  + q  y  —  .L  }>0  +  (5 - C' ), 1  form,  form  clear  becomes  the  Iy+1  -  Iy"1  i=l  =  b  =  K  i =l ,  1  L.  (P2),  is  (PI). for  known  the  as  Couhault reasonably  unmanageable.  most  paner,  of  that  (decompose)  However,  m2  r  the  reduce  a  may  oroblem  > 0.  expressed  This  quickly  is  minimize  determined.  Clearly  = P  to  problem  mm  P. i  T  'shortage'  This  Tx  In  the  determined  (P2)  where  matrix,  are  eliminated. In  y  cost.  and y~  +  below be  one  known  objective  penalty  stage  able  The  n  matrix.  Usually variable.  x  he m o d i f i e s  L23]  to  a  'extensive uses  sized  El-Agizy  problem  satisfying  the  problems,  [33]  smaller  treatment  techniques  of  this  uses  formulation the  this  separable  this  problem  similar to  representation'-'of  those  number  directly, of  constraints  representation convex  to  program.  i s m'ven developed  by  Wets.  for  [95].  55  generalized to  reduce  program  upper  the  with  bounding  problem mi  + m2  to  one  rows  Consider  (notably  the  that  and  n  the  is  working  tractably  basis  concept  comparable  to  [30',31]) a  linear  variables.  second  stage  problem  !+ + -' q y + q y 1  mm  Q(X.5)  y »y">o +  s.t.  where  x  Iy  = Tx•  It  -  Iy"  = g  follows  -  ,  x  that  mi  Q(xs5) =  E  I i=i  where  i = l , . . . , L . ginal  =  Q^^-)  Hence  y  the  distributions To  .f".-,0  recourse  are  of  illustrate  separable  convex  (P3)  min c ' x +  f i  V  +  Q.Cx,,?,) H  +  problem  1  V is  V ^ i  1  +  have  =  i  ?  Thus  £ i=l  Wets  algorithm  [95]  write  (P2)  Ax  the  m2  {Q.( H  1  1  X i  )>  = min c ' x +  1  = b ,  useful  the  importance. the  E  "  only  program  x>0  The-Q.j(x-j)  V  separable.  m2  s.t.  "  representation  [95]  £  Q  i=l  1  (x.) 1  as  the  mar-  56  r  W  A^h  =  '  i  a  }  +  m  i y  yi_i< o. 0 < y i, k .+i where  P  P .  i,-1  3.  =  -  P  = - q |+  i 0 '  £. .  I  of  &  i , k  P  i +  =  1  possible  P  solution bound  ,  p  £=-1  i £  _<  a n d0  1,kt'  U  d  < E.9  i ^ n K Z —  y  =  d  f  u  o  < • • • < 4.  ?  increasing  ,  <'-3-,  first y.  y ^  ( i n each  i s useful  problem. 0  Note  that  t h e y.  this  means  £ were  that  because  The of  t h e reduced  ing  right flow  the working  pivot  hand  costs  chart,  used  marginal  computer  code  k.  d  i ,  k  ) ) ,  =  i  i s t h e number  simplifies  y.  a . ) since  toi t s theP.  1  0  IX/  the value  bound,  of only the  sayy . ^ .  f o r £ > m i s zero). f o r upper  mx + m  2  (Since This  bounded  i n t h e (working)  i s of dimension  c a n be thought  value.  (adjusted If  variables.  basis,  again  , t h e same  t o i t s upper andd e t a i l s  programming  o f as examining f o r penalties)  i t i s a reduced  On t h e o t h e r  chance-constrained  stochastic  i '  u  size as  | .  and t h e duals  i s brought  Since widely  rules  i s performed.  side  basis  a  o f type  (x,- -  to record  algorithm appear  i l -  i t greatly  1  andy .  simplex  by i t s mean,  pivoting  the 'largest'  simplex  modified  bound  ?  = P r t ^ . _< ?  u  a n dwhere  each  reaches  needs  do n o t e x p l i c i t l y  replaced  A=of9...,k,  r  1  i s n o t a t i t s upper  f o r £ < m i s a t i t s upper t h e usual  if  The a l g o r i t h m  v  of  Increase  t h e sum o f t h e y .  row) which  resembles  xn- -  I  i s t h e mean  representation  i n £.  =  i £ 'd i 0 =  1 j Kj  and  of the recourse  -  1 X*  are  u  y  — JZ.=— 1  <  u  i,£+1  C  1 <_  o f E..  or until  f J  n  k.+l i  £ = 0 , , . . ,k- , ( q . = q t + q T ) ( F  II  values This  upper  1  ,  u  and a . < E.,  I5K.J  the  q . F  k.+l i  hand,  bound.  cost,  technique  and the s e l e c t then  i f i t i s a dual, Appendix  t h e usual then t h e  2 contains the  of the implementation programming  the values  of the code.  ( C C P ) h a s been  in the literature,  t h e most a comparison  57  of CCP  the solutions i s of  the  generated  the certainty  PrIXx  equivalent  (CCP2)  c  It a n d n*  has been i s the  i s an o p t i m a l  and  that  for a l l  and  (CCP1)  with  x*  i ,  A.x i  [67]  i  =  to  1  a  a cost  is  is  i=l,•••,m.  that  i f  dual  solution  - V  a  n  ^  i s an o p t i m a l  strictly  to  (CCP2),  solution then  solution  to  (P2),  t h e SLPR  x*  x*  then  q.  -  a. = _L from  t h e SLPR  * -  n. 1 q.  solution.  ,  i = l ,  formulation,  solves  + q"A +  with  SLPR.  increasing at  q.  to  °'  =  vector  and  i s the dual  d  x*  the corresponding  c  n  that  i= l , " « , m ,  > F a.  i s an o p t i m a l  F^  > a. ,  (CCP1)  corresponding  solution  Suppose  > ?.]  of  shown  q  where  Recall  x  s.t.  x*  in order.  min c ' x x>0 ..'  s.t.  (CCP2)  is  form.  (CCP1)  and  by CCP a n d SLPR  m,  (CCP2)  58  3.5  Appendix  Two  This algorithm user's view The  of  to  solve  guide the  for  SLPR,  the  and  code Kusy  consists 2)  code.  algorithm.  computer  K a i l berg  appendix  was  The  a  of  three  FORTRAN-IV  The  first  following  [49].  code  page pages  originally written  parts:  by  of  1)  a flowchart  for  the  the  flowchart  explain Collins  algorithm,  in  [23]  and  gives  detail and  of  each  the 3)  an  a  overall  part.  modified  by  59  B. Phase 1: initialization  F i n d minimum r e d u c e d cost (incorporates usual simplex reduced cost)  E. yes  yes  do  Find pivot row, simplex (revised) pivot  Stop infeasible  Phase 2: initialization  Do r e v i s e d simplex pivots t i l l c > 0  H. Do  pivots (revised bounding) t i l l  (  Stop 'optimal'  and c >  upper 0  1  60  Read  in:  tolerance n,  mi,  (£)  m2 ,5(i,k(i)),  (a(i),5(i,l),'  e(.i))  (P(.i,l),---,P(i,k(i))) (q+(i),  q"(i))  A and T  matrix  i=l,-«-,m  2  (h(l),-",h(mi))  Initialize:  n(i)  1  if  h(i)  >  0  1  if  h(i)  <  0  i = l , • • • j i r i i '+  =  '0  if  i  t  n(ij  c(.i)  =  iw(i)  i=j  0 = 0 0 0  a.  i=l,«--,m  1  = - i =  if  0  Y d ) = hO+nh) 6(i) = £(i) = K(i) =  h(i)  j i=l,•••,m  w(i,j)  |h(i)|j m  z0 1=1  1  J  2  i=l, • ••,m  m2  = m\  Call clmpvt (c,s,l)  Call smxpvt Cc,s,y)  y(i)  = pCi,k(.i)+  c(iw(j))  i =1,  1).  if  iw(j)<  n  g(J) Y(iw(j)-h)  n(.j)  =  z0  0  -  g(-)*w(-,j)  otherwise  . . . ,  ) j=l  m2  63  Call clmpvt (c,s,l)  no  6(1)  Call smxpvt  yes  ^  (c,s,y)  Loop ^top  'unbounde^'  for  j=l» and i w ( j )  ,m > n:  v = iw(j) Jt(v)  sum d ( v , « )  n  = 1  y  until,  =  d(v,i)  I  >  h(j)  i= l x  = p(v,<j>)  -  6(v) h(j)  =  K(v) = n(i)  =  1)  -  y  p(v,4>) =  <H  + w(j,i)*x end  +  d(v,(j>)  = h(j)  y(v)  n(i)  p(v,k(v)  loop.  = d ( i , D  Y d ) - p ( i . D i = l , - • • ,m  1 = 1 » - - - ,m  CLMPVT  ^  c  =  (c.s.maul  Start  -- m i n  (c(j)  ^  -  n(-)*A(-,j))  j=1, — , n s  = j  corresponding t h e minimum  to  Return  yes  no  min  "  + nU+mJ],  j=l,-**,m update  s  2  (if  necessary)  c}  3  65  Call clmpvt (c,s,0)  A j=l,  • • •,  m  g(j)=w(j,*)*A(*,s) call  uprpvt  Loop no  yes  and  for £(i)  i=l,  f  ,  m2  1  (c,s-n,2)  kk c  = mi  +  = y(i)  i +  IT(kk)  c  yes  +  -  n(kk) p(i,K(i))  no  g(££)  =  11 =  for call  =  call  1,  . . . ,  uprpvt  g(££) ^  w(££,kk)  = 1,  m  ( c , i , l )  w(££,kk) . . . ,  m  uprpvt  (c,i,0) yes ^End  ^ t o p  'optimal^  loop)  66  SMXPVT(c,s,u)  g(J)  -w(j,s-n-ma) • .in  gill)  = w(U,*)*A(.  Call  rwpvt  (t,r,y)  call  iw(r) = s pivot (c,r)  ,s)  67  RWPVT  (t,r,y)  «*•  •>  Start  \f t  h t  =  M  * + 00  if  g(j)>0  min j=l  (i) n u ; h  r  = j  y  =  _  6(iw(j) g(j)  corresponding to  n)  if  g(j) < 0 iw(j) > n  min  0  yes  no /  —  I  Return  ^ j-  v - 1  68  PIVOT  ^  Start  gs w(r,j)  =  w(i,j)  ^  l/g(r).  = w(r,j)*gs h(r)  loop  (c,r)  for  = w(i,j)  =  gs*h(r)  i = l , ' « ' , m -  (i^r)  gs*w(.r,j)  h(i)  = h(i)  -  gs*h(r)  n(i)  = n(i)  +  c*w(r,i) end  z0  j = l , - " , m  = z0  loop -  Return  c*h(r)  j = l , " - , m  69  UPRPVT  ^  Start  Call  rwpvt  kk  Let  a  be  the  (c,i,kik)  ^  (t,r,y)  = irii  +  i  a  =  0  F  =  false  sum o f  d(i,K(i)  j=l,•••,K(i).  -  j+1)  over  until  (if  kik  = 0)  p(i,K(i)  -  j+1)  <  -n(kk)  (if  kik  =  p(i,K(i)  -  j+1)  >  -n(kk)  ((-j)  if  1)  -  or t  then  is  =  last  value F  (if  =  -  <  a,  of  j  kik  =  0)  true  neither condition  holds  go  to  A)  3Z h(j)  = h(j)  -  a*g(j)  K(i)  = K(i)  +  Is  'p(i,K(i)+l) c  L  +  (j=l,•••,m)  n(kk)  if  kik  = 1  if  kik  =  kik  =  —  ,p(i,K(i)) Y d )  + =  p(i,K(.i) +  1)  =  d(i,K(i) +  1)  t  p(i,K(i)  +  n(kk)  1)  '= t  -  0  a  -  n(kk)  ,> -  n(kk)  if if  kik  = 1  F  ^yesL^^  Return  ^  70  Q Call pivot (c,r)  yes  \ K(1)  = K(i)  6(i)  = d(i,K(i)  +  1)  Y(i)  = p(i,K(i)  +  1)  h(r)  = -h(r)  w(r,j)  =  -  1  -w(r,j)  n(j)  = n(j)  +  h(j)  = h(j)  -  2-c-w(r,j ) w(j\i+mi)*d(.i,l<(.i) + 1 j  V  V =  iw(r) U i )  v  = v  £(v) iv  -  iw(r)  = n + = 1  i  if  i  > 0  n  = 0  = mi +  v  h(j)  = h(j)  +  K(v)  = K(v)  + 1  6(v)  =. d ( v , K ( v )  +  1)  (v)  = p(v,K(v)  +  1)  Y  -j = l , * - - , m  w(j,iv)*S(v)  j=l  ,"',m  71  os?OGBflM KUZ" f IN » U T , 0 U T t n j T , T 4 3 £ 5 = I N = ' U T , T 4 O E 6 = 0 U T P U T ) C--> CODFD R O G E R W E T S ".Y H E R M A N C O L L I N S CjL2 ^ 0 0 I r T C A T I Q N S p.y J . K i L t P E ° G KUSY r. = » M £ Y T ' l : J M V A L U E S A»F.=H2=73: M i=2 2 0 - t l 2 N--350 5 K ( I ) =8 IMPLICIT PEAL (A-M.0-7) ' INTEGER S , R . B M ! , ".ARTY 3 £AL T P ( 7 0 . 1 C ) ,TO ( 7 3 , 1 0 ) PEAL P ( 7 C , 1 0 ) , D ( 7 0 , 1 0 ) , A m ; , 2 60l,H(130>..C(Z63> REAL W<103,lCG>,G(lOO),DFLTS(7G>,GAMMA<7e>,PI(10Q> O T " E N S I C W I W ( i C O ) , K A O D A ( 7 0 ) ,L (70) ,'<(73> = EAL Oc (7Q),DM<70> COMMON N . M . H I . M ? , P , D , 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 , K,OP,OX.ZG,MARKER COMMON / " i N T ? / T P , TO ; , ' C=>N07ICE THAT 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 USE 3 - O R I G I N INDEXING, C=>IM THE FORTRAN CODE A L L -SUCH I N D E X E S HAVE BEEN INCREMENTED BY 1. READ(5,9 2?) £ ° S r 13? O R M A T ( P 1 0 . &) WRITE ( 6 , 17&I EPS ;76 P Q P * A T ( 1H1 , T l Q , " T O L E R A N C E I S S E T AT " . F B . 5 ) , 0 = vRE a D I N N , M 1 . M 2 . KI . READ (5,1GC)N,Ml,M2 10 0 r O ' M S T (715) WRITE ( 6 , 1 7 ° ) N.M1-.M2 ' 139 F n ? M A T ( / / / , T 1 0 O F VARI4BLES= " , 1 4 , / , T 1 0 , " * OF D E T E R M I N I S T I C ~ , 5  "CGNSTPAIHTS= " 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 M2 C - > RE 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 ( P O 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=»(LOWER AND U P P E R BOUNOSI INTO D. WRITE(6,110' C  110  F O ? H 4 T l " l " , T 2 0i3'i ("*") *  •  31 .77 400 f>31 '  ,/iT20<"POSSIBLE  /.T2C-.3<<("«") ,/) • DO 3 ? LZ=1,M2 PEAD(5,800) K(LZ>,D(LZ,2>,P(LZ.2I. To(LZ,1)=P (L7,2) TD(LZ.1)=D (LZ,2) I-!K<LZ) .LE.IIGOT037 KP=K f L7) DO 3 1 L A = 2 , K ° P . E A O C 5 , 8 C 11 D ( L Z , TD(LZ,LA)=DO.Z,LA TP(L7.LA)=P(L7,LA = E A O ( 5 , 8 0 2 ) D ( LZ ,  LA+1),P(LZ,LA + D +l ) +1) 1 ) , D ( L Z , K ( L Z ) +2 )  READ ( 5 , 1 2 9 ) T P (L7) , Q M ( L ~ > F O R M A T d 3 , C 1 D . 2 , F 6 . u) =-OR-iAT(FlD.2,Ff.. U)  P n ?  C Q ? I U  12 9  F O R M AT ( 2 F 1 0 . ! * ) K I = K ( L Z1 +1  T ( 2 \ 0. 2 ) z  L X- M l * L 2 102  WPITh(6,102» L X , ( D ( L 7 , J ) , J = 2 , K I ) F0PM£T(T5,"?0W " , I 3 , T 1 5 , U ( F lif . 2 , 6 X ) , / ,  VALUES  OF  RIGHT  :  HAND  SIDE",  72 1 7  2  u (E-ii4,2,e,y> i M ( L 7+ ^ 1 )  = n ( L 7 , 1) iOME THE NPTE THAT Tag SL°HA(I> "!=• T H E S T O C H A S T I C C O N S T R A I NTS  _T  OUT  AND  UPP.E  " 1 S T < / / , T 2 3 , 1.2 ( " * " )  ,/,T  ITE 112  "  TH  (6,  LOWER  C  112'  »/S°I A 9 L E S " , / , T 2 0  P  RIGHT  H-JUNOS  HAND  AND  .1.2C'20,"LOWE?  TERMS  CALCULATE  AND  UPPER  THE  0  SOUNDS  VALUES.  OF  RANDOM",  1=1.M2  DO KT  (I)  L x E (6 . 1 0 2 ) L X , 0 ( 1 , i » , D ( I , " < I U>. 1,13 l,«l ? J= no J ) = 0 ( I , J «-1 I - 0 (1 , J ) o < W P I T E OUT THE INITIAL P-VALUES: CALCULATE AN n ^ = >RE A n ALSO WRITE O U T - 0 - ° L U S AND :>p-VAl.UES AND WRITE THEM OUT.  THE  ACCUMULATED  Q-MINU3.  W= I T E ( 6 , I C S ) ICS  ~ O R M AT </ / , T 2 0 , 3 3  ("*")  , / .T 2 0 , " P R O B A B I L I T I E S  / . T ? c , 3 0 ( " » " ) DO  30  EVENTS",  *i  LX=M1+1 .7 D  RANDOM  I = 1 , M ?  ( I )  •<!-«  OF  , /)  WRITE ( 6 , 1 0 2 )  L X,  !  D  11, J i , J= 2,<I>  WPTTE ( 6 , i f . ) 10 A  r  O R M A T ( / / , T 2 C » 3 3  •  ("*")  / , T2 0 , 3 3  ,/,T 20,"SHORTAGE  PENALTY  SURPLUS  PENALTY",  ( " * " ) , /)  I - l.M?.  t*Z  LX=Ml+I LX.OP(T),OMf I)  W = T T E ( 6 . 1 G ? » DO  33  I=1,M2  Kl-K(T)* 1 D '(I , i > = - 0 °  (O  AC~=0. Q = Q D < I ) +OM ( I » ' DO  3«.  J=2.KT  ACC,= A C C + P ( T , J ) » ( T , J ! = - o °  77  (I)+o* ace  CONTINUE  = » R •AO  IN  ft  'NO  H.  F O R M A T ( ? F 1 0 . ' » )  DO  17Q  J1=1,M  DO  179  J 2 = i , N  A :.!1 , J 2 >  =3.  NOTE  THAT  C0DE , THE  ''AT^IX  THE  NCN2ER0  fI7>  ISi  1 B 9  A  AND T H E R RE?UIRE  IS  W0!JLP IN  8^10.^ ENTRIES.  FOLLOWED  ENTER DO  THIS  WHICH  NULL  BY  THE  LINE.  k I;SV=I,M -  -• E A 0 ( 5 , 1 , " » )  IND,  TEMP  IF (IND.cO.O) GOTO 1 9 9 A ( K U S Y , I N D ) = T E MP  c  DEPARTURE T-i E  OR"AT 7  SO  FOR  ENTRY  USER WITH  EACH  FROM.THE TO  A ROW  ( F i C . A ! ,  ENTER  LARGE  MATRIX  I N ° U T WHEN  ORIGINAL EACH  A  ROW  QF  WHICH"  THE  COLUMN  ROW  IS  NO.  COMPLETE  73 F-0T0 187 CONTINUE  IB?  Ir<"l  .EC .3)  ' " 0 TO  ° E A D <5 , 1 C t >( H ( I ) , °7 3 10 6 13 3  973 1=1,Ml)  CONTINUE WRITE 1 6 , 1 0 6* F O ' X J T ( / / , T 2 3 , 30 ( " * " ) , / , T 2 0 . " / . T 2 0 . 3 0 (•'»'•) , /) C  a  WRITE ( 6 , 2 2 « ) ( I , I =H , L 2 ) 0 0 2 2 1 . < 9=1, M ZZk  WRITE ( 6 , 2 2 ? ) CONTINUE  222  F O R M A T  (/••  '  _ ^  •  « 9 , (A ( K 9 , K £ > , K ° . = L 1 , L 2 )  PQW  " ,  E0RHAT(///3QX, 227  MATRIX",  Q3MAT(I?,F13.<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 > L2 = MINO C9+ L l , N ) WRITE ( 6 . ? ? 7 ) ; ,  -  -  13,  3X  , 10  ( 2X,^  1C  .i,)  )  * * * * * * * * * * * * * * * * * * * * * * *  FORMAT(//.Tl0)  228  FORMAT(" COLUMNS",10112) W=ITE(&,109) I C R " O R M A T f / / , T 2 0 , 70 ( " * • • ) , / , T 2 0 , " I N I T I A L * T2 0 , 3 0 ( • " * " ) ' , / ) 0 0 2 2 1= 1 , M WRITE ff,, 102)  22  W(I,J)=0 W(I, H  US  C  W IS DO  = OT  '  50  (I)  C  ,  ,  (T) EXCEPT  FOR  PI  L.K.IW.DELTA,  ON AND  THE  DIAGONALS  GAMMA!  CALCULATE  1=1,M2  GA^MA ( I ) = 0 . DELTA(I)=  5  .  1F70  L (I) = 0 K A ° P A (I > = 0 70=0. DO  51  1=1,M  I w (I i = -1 IF  ( H ( I ) . G e . G . )  H!I> ?1 52  GOTO  51  = -H(I>  ZO=ZO-H(I> DO  52  I=1,N  C(I)=0.  •J;=>^JUJLU.^LIII-1.^I,IX;. u r n l i i i u j . j L i i i i . x i i C = >PHASE' 200  I •  CALL  SIDE  .  7EF0  0=>INITIALlZE  HAND .  ;  I ,H  p 0=>TNITIALI7E I AND W , 0 0 itO 0 RL E = 1 . C ° I ( I ) = S I C-N ( D D L E , H ( I ) ) 00 U l J = l , "  i»l  RIGHT  9 EG I N CLM  p  COLUMN  ° I V O T I N G  VT(CPAR,S,i)  WITH  MAU = 1 .  H  AND  70.  VECTOR" ,  74 T P < C » 4 R . L T . - F . D S ) I-  (70 . G £ . - E ^ S )  C = >C3 A ° . GE . 0 WRITE ? 0 7  AND  TO 2 0 2  Q . t _ T . Q .  7  ( 6 ,207)  " O R M A T OAi_L  (*•  I N F E ASI P L E " )  D U M P  C =>r.RAR.GE . 0  ANO  2 C 3  J =  7 0 . G E . O .  DO  2 0 U  I F  ( I w ( j ) . L T . 5 )  2 0 ^  c o  GOTO 2 0 3  1 , M GO  TO  2 3 5  C O N T I N U E TO  2C5  W R I T E  20.8  C  3 0 5 ( 6 , 2 0 8 )  0R MA T  C A L L C= » C  A  ( / "  D  H A S E  I  D E G E N E R A C Y " )  DUMP  A R < Q .  "232  C A L L GO  C=>OHASE 2 0 0  S M X P V T ( C B 4 R , S , M U ~ >  TO  R E 0 D  1 1 9  READ  I N  ( 5 , 1 1 9 )  C - V E C T O R :  ' ( C (I > , 1 = 1 , N )  F O R M A . T ( 3 F 1 0 . V ) W R I T E  55  2 0 0  I I !  ( 6 .85)-  "PORMAT ( / / , T 2 0 , 30'(••*") / , T 2 0 t DO  7Q9  W R I T E ( f c , 7 1 G »  7 1 0  F O R M A T C .  . F l k.  k) .  3 0 1  1 = 1 , M 2  G, OO  3 C 2  I F  ( I W ( J J , L E , N )  J - - 1 , "  G ( J ) =GAMMA GO 3 0 3  <", 1 3 , " ) = "  G A M M A ( ] I = P ( I , \c ( I I + 1 )  O= > S E T  GO  TO  3 0 3  (iw ( J ) - N )  TO . 3 0 2  G M ) - C < I w ( J ) )  30 2  C O N T I N U E  C =»'SET  P I , 7 0 7O=0 TQ  .  3 0 a  T = 1 . M  P I ( I ) = C . DO 3C'A  C - > R- E C- ! u"0  30<«  J =1 , M  ° I ( I ) -PI  C = > 2 2 2 2 2  ( D + G  C O L U M N C L  M  T ^ ~ ( C R A P C A L L I F r  . G E . - EPS )  MAU=1  .  G O ~ T O ~ S"G 0  GO  TO  MQO  < 6 . i»G?) ( / "  U N B O U N D E D " )  DUMP  C=>CE<AR> = 0 . ! DO  W I T M 1 )  S M X F V T ( C B A R , S , M U )  ORMA T  CALL 5 0 0  P I V O T I N G  P V T ( C ^ A P , S t  (Ml) . M E . 2 ?  WRITE 4-02  ( J ) * W ( J , I )  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  CALL  V E C T O R " ,  J l . C ( J l ) COST  GAMMA. DO  70 1  COST  , / )  J 1 = 5 , N  7C9 £, => S £ T  , / , T 2 0 , "  30 ( " » " )  5 C 1  S E T  • D E L T A , G A M M A .  1 = 1 , M 2  ,  '  75 H E L T A ( I ) = n (Ti 1 t E01  GAMMA <I)=o 1 1 , 1 I  C=>SET  L . K A P P A . H . G A MMft , P I . 00 '5 0 2 IP  J = 1 ,  ;  M _  ( I W ( J ) . L E . N )  GO  TO 502  NU = I W ( J » - M L(MU)=1 Y = ] , P (J T = 1 DO D  553  5 0  .  K K = i , K !  H ! = KI(  IP 503  •  (Y+0(NU,KK>  t E P S . G T . H t J )  !  GO  TO  50fa  Y = v + |_ ( N U , K t O fa  Y = P ( N U , P H T ) - ° ( 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 ! = K  ftno'A  DO  D  ( N U , P H I ) .  ( N U) =Ph'T - 1  5 0 5 1 = 1 . M  505  J  I ( I ) = P I  5 02  CONTINUE  f T 1 » y » H ( J , 1 J .  0=>3337333332332337333333333333337373333 C = » ^ E G I N 7  0G  7 01  COLUMN  CALL  PIVOTING  AGAIN  WJTH  MAU=9.  C L M P V T r C 9 A R , - S , 0)  I P  ( C P A R . G E . - E  DO  7 03  S )  C  GO  TO 6 0 0  J - l . M  G (J1 = 0 . DO 703  70 3  CALL GO  U P P P V T  TO  C=>REGIN 6DC  1=1 . M  G(J) = G ( J ) + W ( J , I ) * 70 0  UPPEP  POUND  DO 6 0 1 IP  A ( I , S )  ( C E . A R , S - N , 2 ) PIVOTING.  ,  :  ,  1=1.M?  (L (I> . E C U  <•<-"! *Z  GO  C=>TEST1.  TO  601  '  C?,AR=GAM*A ( I ) + F I ( K K )  603  TC  (CP.AP . G E . - E P S )  DO  60 3  G ( L L ) - - W CALL GO  TO  GO  TO  602  .  LL=1,M (LL.KKI  UPRPVr  ( C B A P , I , i )  7 00  C=>TEST2. 60 2  r r  ii f i p p A f T) . E C . J )  GO  TO  C3AR = P I ( K ' O +P ( 1 , ' K A P P A ( I ! 1  6Gfa  TP  (CHA  DO  6 0 fa L L = 1 . M  . LE.EP")  GO  '  "  ~*  "  TO 601  G I L D = W i L ! . , * <) CALL GO  601  D  601  TO  UPPPVT  700  (CHAR,1,0)  ;  ;  C O N T I N U E  C=>WH£N  T H E LOOP  " TS  S A T I S F I E D ,  WRITE  O U T T H E O P T I M A L  S O L U T I O N  AND STOP  7  r  ^ . I T f (fc, 0 ° M - f l T ( " l " . T 2 C , 3 Q (•••••),/, T20, " / , T2 0 , 3 0 ("*••)  ,/)  OPTIMAL  SOLUTION  SU3R0UTINES c SUBROUTINE IVOT(COAR,R> p IMPLICIT FAL (A-H.O-Z) INTEGER S.B,OHItMAoTY  REAL P l ? C , 1 0 > , n<70, 1 0 ) . A d C 0 , 260) • H « 1 3 0 > ,Ct260» REAL W(103,1C0),G(103>,CELTA(70),3AMMA(70),PI(100> DIMENSION i n ( 1 0 3 ) , K A ° P A ( 7 0 ) . L (73) ,K(70) 3 ECL Q°(73).OM(70) COMMON 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 , % K , O P , O M , 7.0 . M A R K E R C=>CALCU'.ATE PIVOTAL ROW, H(P) GS = 1 . 0 / G ( P ) DO  10  10  J=1.M  -  w< = , j ) = w ( o , j ) » r - s H(R)=G3*H(P)  C=>°IVOT  ON  0THEP  ROWS.  DO 1 1 1 = 1 , M. IF ( I . E O . P ) GO T O 11 GS = G ( I ) DO 1 2 J= 1 , M W ( I , J ) =W ( I , J ) - G S * W ( R , J ) H ( I ) = H ( I ) - G S » H ( R )  12 11  CONTINUE  C=>CALCULATE DO 13  13  ° I  AND  70.  J=1.M  P I ('J ) = P I ( J ) + C a AR * W ( R , J ) ZO=ZO-CBAR»H(R) 976  RETURN END  .  :  L,EPS,  S U B R O U T I N E R K P V T ( T , P., MU> IMPLICIT REAL (A-H.O-Z) INTEGER S,",puT,MARTY REAL• 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> REAL W ( l C C , 1 0 0 ) , G ( i a O > , P E L T A < 70 ) , 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 ° P i C C > , L ( 7 5 I ,< ( 7 0 ) REAL 0 c ( 7 0 I , OM ( 7 , j | COMMON 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,OP,OM,ZO,MARKER  T = 1E 7 G C=> c I N O " I N ? A T T p H ( J > / G ( J > WHERE 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 ) DO 1 1 J=1,M IF tG ( J ) . L E . - E ° S ) GO TO 1 0 I F (G ( J ) . L T . E P S ) GO T O 11 0=>IF 5(J)>C. R AT T 0=H < J ) / G ( J ) IF(RATIO .GT.T) GO TO 11 T = R A T 10 ==J  ,  ,  GO TO 11 G U X O . . . KK=IW(J)-N IF ( K K . L F . O GO T O 11 PATTO=(H(J)-DELTA(K K))/G(J) IF (RATIO.GT.T) GO T O 11 T = RA T 1 0 R= J MU=1 11. CONTINUE r.= > I F NO J F O U N D M U = 2 . IF (T.GE.1E7B) MIJ=2 9 7 F. R E T U R N  ST  K » 0 !  G(J)<0.  .  ,  ,  ?-->I c 1C  ,  ,  !  S U ^ R O U T I K F CIM.PVT ( C 9 4 ? . , S . M A U » IMPLICIT PrAt ( A - H , Q-Z) IN T 2 G E P S . , MflPTV °EAL P(70.10),0(70,13),AflJO.260),H(1DO).C(260> REAL W( I C Q , I C G ) , G ( I O C ) . H E L T A(7C) ,GAMMA(70 > . PI (100) O I M . N . J O ' . ' IW ( 1 0 0 ) , K A P P A ( 7 0 » , L ( 7 0 ) , K ( 7 C ) PEAL Q P ( 7 0 » . O M (70 I COMMON 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 » ,QM,ZO,MARKER COAP=1E70 S=0 C = > c I N O MTN(C - P I * A >=CRAR 0 0 If? J=1,N X=C ( J ) , OO 1 1 1=1, M ' 11 X = X - P I ( I > » M I , J ) IF (X•GE .CBAR-E'S) GO TO 10 CPAR=X S=J 10 CONTINUE IF (MAU.EO.O) RETURN C=>FIND M I N ( C D A R « GAMMA(*)+PI(»*MH ) 0 0 1 2 1 = 1 . 'M2 X=G4MMA(It + ° I ( I + M 1 ) Ir (X.GE .CBAP-E'S) GO TO 12 CRAR= Y S=T + N 1? CONTINUE P76 RETURN END  •  =  C9AR  .  '  80  SUBROUTINE IMPLICIT  C  ll°R°VT £AL  (C3AR,I,KIK> ( A - H . Q - ? )  '.= > T u T S IS VERSION 2 O INTEGFR S.R.PHT  UPR°VT.  r  P - A L  n I7C,  REV.  K (ICQ.130) , G ( 1 Q G ) , D E L T A ( 7 C )  1  3 ) . 0 <7C , 1 0 ) , A ( 10 3 , 2 6 0 > , H < 1 3 0 ) , C ( 2 6 0 )  0 1 MEM S I ON T W ( 1 0 0 ) . K A PFAl n = ( 7 D . OM ( 7 0 ) COMMON *  N , M , M l , M 2 ,  D  P A ( 7 0 ) . L  D  .GAMMA(70)  iPI(100)  (70) . K ( 70) .  C , H, G , D E L T A , G A M M A , ° I  , D , A.W,  ,I'«I.KAPPA,L  •  .EPS.  K ,OP .OM.70.MARKER LOGICAL CALL  PLOG  RWDVT(T.R,«tJ>  KK=M1*I PL A G = . F A L S E . AL°'-IA = Q. IF  • ( K IK . N E • 0 >  r =>KIK =3  C,->°  PINO  THE  (T , K 4 = P A ( I KT=Kfipp/_ OO  10  (  +  »  RI ( K K )  2 , . ..,<A°°«(H  C.  AND  T>=  SUM  OF  S . T .  D t l . K A P P A ( I ) - S ) :  S = 1 , . . . . L L  L L ^ l . K T  AL°HA =0L IF  - LL > < _.  TO 20 LS=1,  GO  LARGEST  D  H A + D ( I , K A P = > A ( I ) - L L + 1)  ( P (I , K A O P A (II - L L + 1 ) +  D  .OR.  I (KK) . L E . EPS  T , L T . A L ' P H A - E P S > GOT O  30  LS = LL AS =A L °H A 1 0  FLAG=.TRUE.  GO TO 30 C =>KIK =1, F I N D  THE  LARGEST  C => ° (I .KAPPA (I ) + L L X 0 20  AND  I r ( K I K . N E . l ) GO TO K I = K ( I ) + 1 - K A D O A ( I )  . . . , K ( I ) - K A p P A OF  (I)+LL)  C=>SEE  I  •*D  IF  r  SOME  LS-FOUND.  ( . N O T . PL AG)  7 1  S= 0 , . . . , L L .  .OR.  T . L T . AL P H A - E P S )  _  GO  C=>SOME LS FOUN" IF ( K I K . E O . 3 ) DO  S . T .  .  (P ( I , KAPPA (I) 4 - L L ) + ° I ( K K I . G E . - E P S  LS'=LL AS= A L ° H A FLAG=.TP'JE.  21  (I)  D (I , KAPP A ( I ) - S) :  40  DO 21 LL=1,KT A L p H A = AL P H A » Q ( I . K A P P A IF  LS=G,1, T> = S U M  (IF TO  NOT  = IVOT  40  AND  RETURN)  ,  ,  VS=-LS  J =1. M  H ( J l = H ( J ! - A S * G ( J! KAPPA r c IF7  ( I ) = K A P ° A (I)+LS  ; K J'< . E Q . i ) ( K I K . E O . 0 )  GAMMA ( I ) =  D  •  *:°JJJ<S±.  C P A R = _ P _ L L K A_P£ AJ J L i ± i _ ) C P A R =P ( I , K A P P A ! I ) 1 + P I ( K K )  ( I . K A ° ° A ( I )  •  D  D E L T A ( I ) = D ( I . K A P P A ( I ) + 1 ) T = T - C'S TF  (K-IK.FO.Q  . A N D . P ( I , K 4 P P A ( I ) - > 1 ) + P I ( K K >  «;  . O R .  %  . C P . T . L T . - E P S )  =>OTHERWISE  G O  K T K • FO . 1 P I V O T  AND  . L T . - E R S  . A N D . P ( T , K A P P A ( T ) 4-1 ) + P T ( K K ) GO TO 9 ? RETURN  . . G X _ - _ .  GO  TO  30  Si'"ROUT I  S'iXPVT(CBftR.S.MU)  T M ° L I C T T  OFAt  INTEGER  S,=."HI,MARTY  (fl-H,Q-?)  REAL  P  R E M  w-< l C C , l C 0 ) , G ( l 3 0 > , O E L T A ( 7  ( 7 Q , i C > , D ( 7 0 , 1 0 ) . a { 1 0 0 • 2 6 0 I . H ( 1 0 0 > , C t 2 6 0 I  DIMENSION  IW(100),KAPoi(7  0>,GAMMa(70).PT(100>  0 ) , L ( 7 0 ) , < ( 7 0 ) .  REAL  OP (70) .OM  COMMON  N . M . M 1 , M g , p , Q , A , W , C , H , G , 3 E L T A , G A M M A , ° I , I W, KAPPA,L  ?  K , 0 I  c  C= > S < = N . DO  3  .OM.'O,MARKER  (S.GT.N) 11  (701  GO  TO  10  L L = 1 . M  G(LL)=0 DO 11  11  J = 1 .  w  C- ( L L ) =C ( L L ) GO  TO  + W (LL  . J) *A  (J,S)  20  C=>S>N. 10  KK=~-N+M1 DO  12  J=1,M  12  G (J) = - W ( J . K K )  C=>IN  EITHER  20  CALL. IF  ,  .  ,  CASE.  PWPVT ( T . R . M I J )  (MU.EO.2)  GO  TO  . 99  X W(R)=S CALL 99  RETURN END  PIVOT(C9A=,P) .  ;  E P S ,  S U ' i R O U T I NE  D  R I NT  TM^LIl'IT  REAL  INTEGER  S . R , M f l D T Y  (A-H,Q-Z)  7o  ° E AL  C^ I $ (  REAL  T ° (7 0 « 1 0 ) »  REAL  P I U . 1 0  REAL  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 )  DIMENSION REAL  I 0 (7 0 ,10 )  T  ) . D (7 0 , 1 0 ) , A ( 1 0 0 . 2 6 0 1 , H ( l O Q )  Twgc.Q)  , K A ° P A ( ? Q ) , L  , C (260) ,PI(100)  (70) , K ( 7 Q )  :  .  O P ( 7 0 ) , Y ( 3 5 0 ) , OM ( 7 0 1  COMMON  *  100  N , M , M 1 , M2 , 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 t E P S , • K,0°,0",70,MARKER COMMON / ° R I N T P / TP.TD WRITE(6,1001 FORMAT;//,TIP."BASIS 00  10  10  W=>ITE ( 6 . 1 0 D  101  OUAL  V A R I A B L E S " , / / , 30 ( " - - - " ) )  I W ( J ) , P I ( J )  ^ O R M A T ( T l 6 , T 3 , T22,Fl<+.<4)  C=>CtLOULATF ZSO=0 25 C r C C C. C  INOEY  J=1,M  70.  .  DO  ?c  I  (IW(J)  c  DO  J = i » *  21  Z S 0 = 7 S 0 + C ( I W {J)  :  '.  ) » H ( J)  1=1,M2  KI =KAopA  22  :  .l.E.N) (T)  TF  (KT.EO.O)  DO  22  GO  TO  21  KK=1,KI  Z S 0 = 7 S 0 + o ( T , K * ) » 0 ( I , K K )  21  CONTINUE  108  FORMAT(///T20,"OPTIWAL  r.= > F I N O  3i  THE  Y-VALUES  DO  31  1=1,N  DO  32  1=1,M  x(i)-o. ' 3 2  I F  FROM  IW  I  VALUE(WITHOUT  PE N A L T I E S ) = " , T 6 5 , F l  H;  ;  ( I W ( T ) . L E . N )  C=>WRITE  OBJECTIVE  , X(IW  ( I ) » = H ( I )  O U T X " S .  W R I T E ( 6 , 3 5 ) 7-5  -1  F O R M A T ( / / , T J O . 3 0 : DO  33  33  HPTTE  iG2  FORMAT  C=>CALC'JLATE  ("»••)  , / , T 2 0 , " O P T I M A L  S O L U T I O N  VECTOR",  / , T 2 0 , 3 0 ( " * " > , / )  I=i,t.' (6,102> ( "  I , Y ( I )  X ( " , 1 3 . " ) = " , F 1 5 . ^ 1  T H E  A L P H A " S  A N D  C H T " S .  W R I T E ( 6 , 7 5 ) 36  F O R M A T ; / / , T 2 0 , A 2 ( " » " ) , / . T 2 D , " R I G H T *  " N S T P A I N T S " , / ,  no 30 1 = 1 , « » A L  3  HAND  S I D E  T 2 0, 4 2 ( " * • • ) , / )  H A = - ( D ( I . i ) + P T ( M i  + T ) j / ( p  (  x , K ( I ) 4 - l ) - P ( I , l ) )  C H I = 0 . DO .Ui  JU J = 1 , K  C M T = C H H - M H - M 1 . J ) » X ( J ) C H I S ( I ) = C H I I I = I + M 1  '  FOR  S T O C H A S T I C  CO*  83 30  W R I T E ( 6 . 1 0 ° )  T I . C H T  F _ ? M A T ( T F : , "=>OW ( " , 1 3 , " ) =" . = v , - 1 4 . 4 )  i._q  ° E M = 0 . W R I T E ( 6 , 4 3 ) 43  P O ' M A T ( / / , T 2 0 *  . DO  3*1  , 3 0( " » " ) , / , T 2 0 , " I N O I V I D U A L / . T 2 0 , 3 0  ,  /  PENALTIES"  ,  )  XA=1,M_  ° E N 1 = 0 . K I = K ( K A ) DO  38 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 )  • 38 2  * T P ( K A , K B ) 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 ) ) » Q M ( K A 1  ;  * T P ( K A , K e >  PEN= P E N * F E N 1 K B = K A + M 1 W R I T E ( 6 , 3 8 3 ) 383 381  "ORMATfTP,""  FOR  D  OW(" , I 3 , " 1  = " » F 1 5 . 5 )  CONTINUE WRITE  3 8 4  KB,PEN1 PENALTY  ( 6 . 384)  PEN  FO=M A T ( / / / / / / , T 2 0 . " T O T A L WRITE  ( 6 , 10-8)  RETURN EN I  7SO  ° E N 4 L T Y = " , T 6 2 , F 1 8 . 5 )  84 C = > = = = = = = = = = = = = = == = = = = = === = = = = = = = = = = =  S i n R O U T T NE DUMP J IMPLICIT E AL  ===3=-=:= = = = = = = = == = = = = == = = = = === = == = = =  (A-H.O-7)  ,  INTEGER S , C . ° H I , M A R T Y D REAL (7G,if:>,D(7C,10),A(10 0,260),H<130),C(260> RE". w <l C C , 1G0 > , G ( 1 0 . 0 ) , D E L T A ( 7 0 ! , GAMMA < 7 0 ) , P I (1 0 0 ) DIMENSION TWflOO) , K A ° P A ( 7 0 ) » L (70) , K ( 7 0> REAL 0 ° ( 7 C ) , OM ( 7 0 )' COMMON  N . M , " l , M2 , p ,  S  W R I T E ( 6 t 100> FORMAT (//••  l.'O  ? ? 10 101 103 ICS 106 107  •  M=  /"•",«(••  !  PI=  •  ,  FORMAK" 110 WRITE ( 6 . 1031 F O W M / " G=", WRITE(6,10F)  , " ! " , Fl^.<•,"!", Flit.U, ( G ( I ) , I = l , M i (6 F 1 5 . 5 1 ) (L(I),I=1,M2>  I  .  .  " I ( J J ( y » 6 , * * ! • • • , 5Fl<f. M )  FO=MAT( " L = " . (1 2 T 1 0 ) ) W9ITE(6,10f> (DELTA(I) ,I = 1,M2) '  W=  ")) I W ( J ) , h ( J )  ,  FORM A T ( " D E L T A = " , ( 6 £ 1 5 , 5 > > WRITE ( 6 , 107) (GAMMA ( I ) ,1 = 1 , M 2 ) f  0  3MCT(  FORMAT WRITE  111  I-0MEGA=  ^ 0 10 J = l , " WRITE(o.lOl)  "  GAMMA=",(6F15.5))  WRITE ( 6 , 110) 110  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 ,  <,OP,OM,70,MARKER  r  ORMAT  RETURN  (  "  (6,111) (  "  70  7Q=".  g  1 5 . 5 i  ;  ( K A P P A ( I ) , I = 1 , M 2 )  KA0PA=", ( , 12HG  ) )  .  85 PROGRAM  H A P P Y ( T A D E 5 , O U T P U T , T A ° E & = OUT PUT )  IMPLICIT  REAL  TNT E G "  (A-H.O-Z)  R.. P H T , M A R T Y T P(70,10)  REAL  P ( 7 C ' , 1 0 ) , 0 ( 7 u , l J ) , A ( l j 0. 2 6 0 ) , H ( 1 0 0 ) , C ( 260l  REAL  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  DIMENSION REAL  ,TD  .  REAL  (70 , 10)  IW ( 1 C J ) , K A P P A < 7 0 ) , L  (70)  ,K  (100>  (70)  OP(7CI,OM(70 I  0i.;lENSigM_Y(15JJL COMMON 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 ,  •f  c  , » » » ,  K . Q P  ,OM,ZO,MARKER  COMMON  /PRINTR/  COMMON  /A/  » » « . » » • » » DO  1234  TP,TD  y GENERATE  CAS'-i  FLOW  * » • * » * * * • » » • #  *K=1.2  xiQ3=icoeoo. •  Xl=33333. X19=33333. X61=33334. DO  1233  CALL  KL L =1, 3  KUZY(X103,X1,X19,X61)  R E W I N O  5  -  W R I T E ( 6 , 1 0 2 ) 102  FORMAT(1H1) '  101  W R I T E ( 6 , 1 0 D  KK.KLL  F O R M A T ( 1 0 X , 1 h H S I M U L A T I O N AX2 = X (2 )  R U N , 1 5 , 2 X , 6 H P E R I O D , I  •  ;  6)  _  i  :  X(2)=X(4> X (4) = AX2 AX20=X(20* AX21=X(21) AX22=X(22) . X J 2_0i.= X i _ l 2 J L + X 1 1 3 . L + X J L I JL>  :  X(21)=X(9)+X(lo)+X(11) X(22)=X(8) X ( 5 2 ) =X(  23)+  X(24>+X(25)  X(63)=AX20+fl.X21+AX22 X(64)=X(19)  HLHl=LU^ai Y=RA N F ( 0) YY=2G0C0*Y IYY=YY-10000 R1=X  < 103)+IYY  W R I T E ( 6 . 10(*) _  ilit  R l  EOR»AL<i_0JLj.i'tn.GAS«_IJL.OwjRi»=., F I 6 . . . 4 i _ _ PR1=PR(DUMMY) P R 1 = . 5 » o R i + . 0 3 97 I F ( I Y Y . G T . C . A N D . ? R l . L E . . i 7 5 )  =Ri=PRl+.003  IF(I  PRi =°=Cl-. C05  Y Y , L T . 0 . AND. PR.l . G E . . 0 5 5 !  T91=PRl+RT9(DUMMY| Trj_l=JPJ»lJt.5TD_(J)UMMX) . AM1=PR1•PM(DUMMY) ALCi= . 7 » (PR1 + P L C ( D U M M Y ) )  '.  :  -  :  WRITE(6,150) 150  r  Tei  ORMAT(1 : X,19HTREASURY  BILL  R AT E = , 2 X , F i 6 . 6 )  i,  W R I T E (_6_,_ 1 5 1 - . > _ J . " 1 151  F O R M ' A T T I U X ' ; 19~H WRITE<&.152)  152  TERM  153  RATE=i2X,  MORTAGE  RATE=,2X,F16.  LIABILITY  + . Q 4 * Y ( 2 2 ) " + . 0&*x""(64>  +.CQ5»X  ."  Z 2 = . u 0 5 * X U ) + . C 4 * X ( 2 2 ) + . D & * x ( 6 4 > ARM=Z2 I F ( Y i . L T . R l )  X  I F ( Y l . L T . R l )  GO  (2)=X(2)+R1-Yl TO  ( Y l - R l > » < . 2  79 )  I F ( X 2 . GT . . G)  Z2=Z2+ ( X ( 2 > - X 2 ) »  I F ( X 2 . L E . . O )  72=Z2+(X(2)I  X(2)  (. 005)  * (.0 05)  = X2  X20 = 0 . 0 I F ( X ( 2 ) . G E . . O )  G O T O  77  X3 = 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 ) ) » ( . O 0 5 ) X(3)  = X3  I F ( X ( 3 ) . G E . . O )  GO  TO  77  -X20=-X ( 3 )  _  ,  X ( 3 )= b. 0 77  CONTINUE X23l =X(2 0)-X20  - ( Y l - R l ) * ( . i » l  I F ( X 2 G 1 . G T . . 0 ) Z 2 =  Z2+(X(2G>-X2G1>,'<.04>  I F ( X 2 0 1 . L t . . u ) Z 2 = Z2 + ( X ( 2 Q )  I •( . 0 4 )  X (20 I =X2 0 1 X&2=0.0 IF(X(  *  20) .GE.,. 0 1  X21=X(21  GO  TO  78  ) +X(20)  X (20)=0 . C I F ( X 2 l . G E . . O ) Z 2 = ? 2 + ( X ( 2 1 ) - X 2 l ) » ( . 0 4 ) I F ( X 2 1 . L T . . 0 ) 7  2= 12± X_( 2 1 ) * ( . 0 4 )  Y(21)=X21 IF ( X ( 21)  .GE.. 0>  GO  TO  78  X62=-X(21> X(21)=0. 0 . 78  CONTINUE „ J < 6 2 1 = X ( _ 6 2 ) _ - x r , 2 _ - ( Y l - R _ i ) * { . <•_> _ I F fx b 2 1 r G > ~ . . T ) IF(X6  7 2 = Z 2 + <"X ( 6 2 ) - X 6 2 1 ) » '(". 0 6 )  2 1 . L T . . 0 ) Z 2 = Z 2 + ( X ( 5 2 ) ) * ( . 0 6 )  X (62)=X621 IF ( X (&2 S . G E • . 0 1  GO  TO  79  X53=X(63)+X(62) X ( 6 2 ) =0 . f  :  . " I F " ( X 6 3 . GE". . 0 ) Z 2 = Z 2 + (X ( 6 3 ) - X 6 3 ) » { . 0 6 ) IF(X6  6)  R 4 T E = , 2 X , F l 6 . 6>  Y1 = X ( 2) + X (20 ) + X ( 6 2 I + X ( 3) + X ( 2 1 ) + X ( 6 3 >  X2=X(2)-  l6.6)  A L : i  -ORMAT(10X,1RH $  [ r  AMI  F0RMAT(11X,13H WRITE ( 6 , 1 5 3 )  OFROSlf  3 . L T . . 0 ) Z 2 = Z 2 + ( X ( 6 3 > ) * ( . 0 6 )  1 '*  (6 3 ) = X 6 3 c (X ( 6 3 > . G £ . . 0 »  J  GO  _L___i_2. 79  TO  79  .  .  CONTINUE 7_=T81*X(2>*T01*X<20>+AM1*X(S2) 5 +.0 5 4 1 * X ( 3 ) + . 0 3 2 7 * X ( 2 i ) + . u 9 9 2 * X ( 6 3 ) Z3=Z1-Z2-R1*ALC1 W3ITE(6 , 108) KLL.Z3  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 ) *•< ( 6 3 ) x(13 3 ) = R 1 X10 3 = X (1 0 * >  xi__l_)  :  X19=X(19) X61 = X ( 6 1 ) WRTTE(6,2C7> Xl,X19,X61,X1C3 207 FORMAT (1 OX. 2HX = , <*F16. 4 > W R I T E ( 6 , 1 6 0) i_6_____FI!___J___l_^^ WRITE ( 6 , 1 6 1 ) 161 FORMAT(1CX,20M5 T A T I 3 T I C S ) SBZ1=SBZ1+Z3 SB7S=SBZ?+ ( 7 3 ) » * 2 WRITE(6,163) 71. 163 FO RMA T ( 1 C ; X , « G R _____ R ._tf E j l U E S __.Xl_6_»_6J WRITE(6,1681 ARM 168 FORMATdGX,* C O S T OF SALES = * , F 1 5 . 6 ) WRITE(6,165) Z2 165 F ORM A T ( 1 0 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 l 6 . 6) ARMIN=R1»ALC1 166 167 169  w_J_____^.:____ A _ _ J . _ F0RMAT(1GV,» C O S T OF F U N 3 S = * , F 1 6 . 6 ) WRITE(6,167) SBZ1 FORMAT (1 0X , * C U M M UL AT I V E P R O F I T S = »,F15.'6> WRITE(6,169) S52S F O R M A T ( 1 3 X , »  .123.3_.0NI_L_.J_ 123U CONTINUE END  C U M MU L A T I V E  PROFITS .  SQUARED  * , F 2 0 . 3 )  88  The programs  with  purpose  of  simple  recourse.  n mm-, _ j=l x  subject  the code  i s to  solve  The problem  to  m2 e x . J J  + E  {min + _ y-y •  <H  =  be s o l v e d  +  i  1  i s of  the  linear type  +  (p y  l~ i  in-core a stochastic  +  1  p " y " )} 1  1  to  T j=i  a. .x. 1 J  +  n  Y j =l  t . -x.  x.  bi  for  i  =  1,•••,m_  l•  for  i  =  1,•••,m  for  j  =  1, • • • , n  for  i  =  1,•••,m  2  for  i  =  l , * * * , m  2  J  + y.  -  y.  =  > 0  2  J  y+.>  y~  where:  0  > 0  -  i s  the j t h  element  of  the given  x. J  -  i s the j t h  element  of  the decision  a..  -  c  cost  vector  J  i s the  (i,j)th  A  1 J  the  (i,j)th  x  of the  given  technological  matrix  of the  given  technological  matrix  •  t.. U  -  b_j  -  i s the  i t h element  of  the given  -  i s the  i t h element  of  t h e random  -  i s the  i t h element  of  the surplus  y~  is  element : "  vector  element  T  resource resource vector  vector vector y~  b ,  89  y  also  -  is  the  ith  element  of  the  pj  -  is  the  ith  element  of  the  penalty  vector  p  (for  shortage)  p.j  -  is  the  ith  element  of  the  penalty  vector  p~  (for  surplus)  i  shortage  vector  y  define  -  p|  j )  -  to of  be 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 the i t h element of .  is  the  and  I1 j =l  p ^ . .  =  p| j )  =  1.  for  i  =  l , - - - , m  2  (,j=l,• •  •,J.) 1  90  The  input  data  are  CARD  NUMBER  CONTENTS  1.  tolerance  (F8.5)  2.  n,m1.m2  (315)  3.  (I3,F10.2,F6.4)  »Pi  4.  g  (2)  (2)  2+J.  r  (Ji)  n  (F10.2.F6.4)  (Ji)  (F10.2.F6.4)  2+J!+!  a n  3i  (2F10.2)  2+^+2  pt>  Pi  (2F10.4)  2+J.+3  Ju 2 , c , 2  2+J  ••J  2+(J_+J2+-  rn 2  )+2m2  p  +  >p  m2 j ,  •+Jm2)+2m2+l  + k i m1+m2 2+(J1+J2+'"+Jm2)+2m2+ _ k, +  (I3,F10.2,F6.4)  1 )  (2F10.4)  K  2  { , PD2  +J2+4  2+(J_+J_+  2 + ( J i + J  FORMAT  * ) + 2 m  2  (2F10.4) rri2  (I3.F10.4)  000000...  (113)  000000...  (113)  0 1 2  i =l • mi+m2 bi,  b2,  (3F10.4)  b3  mi+m2  2+(J i + J 2 + — + J  )+2m + I  k.+lmJS^  b  1 i=l m_+m 2 2+(J1+J2+---+Jm)2m2+ I k^Cnii/33+1 • i=l • mi+m2 2+(J1+J2+--«+J_2)+2m2+ ^ kj+Dn./a.+.n/S] i=l m i  i^]  •  i s t h e minimum o f  a l l integers  not  less  ,  0  ,b  mi-2'  c _ „ c  2  c  , c  n  _  2  , , b  ITU-T  . . c  n  t h a n ^y-  mi  (3F10.4) '  (3F10.4)  3  _  v  r  c  n  (3F10.4)  91  Card  1 The  Card  Card  may p r o v i d e  n  -  The number o f v a r i a b l e s shortage variables).  m_  -  The number  m2  -  The number o f  3 (first  p j ^  Ji  stochastic  of  (not  deterministic stochastic  including  surplus  and  constraints.  constraints.  constraint)  -  The number o f r e a l i z a t i o n s constraint (with positive  -  Smallest possible constraint.  -  Probability  -  A real (i.e.  3i  -  A real (i.e.  2 + J_  o f RHS o f t h e f i r s t probability).  realization  of  of  RHS o f  1st  stochastic  stochastic  occurring.  number e q u a l a lower  or less  bound  on t h e  number equal an upper  than realizations).  or greater  bound  on t h e  than  E,^ ^ 1  realizations).  + 2  p|  -  A per unit of  p~  -  [2+J.+3]  to  -  penalty,  the f i r s t  A per unit of  Card  level,  + 3 a.  Card  h i s own t o l e r a n c e  2  _!  Card  user  penalty,  the f i r s t  Sequentially  hand  side  on t h e  left  hand  side  constraint.  +2m2]  repeats  constraint.  on t h e l e f t  constraint.  for a surplus  stochastic  [2+(J1+J2+-••+_  each  f o r a shortage  stochastic  process  of  cards  3 to 2+Ji+2  for  92  Cards  2+(J_+*"+J  j  ).+2m 2 +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 - rows a r e l i s t e d i n o r d e r and s e p a r a t e d by 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 13) - - 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 must precede 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 - (unspecified coefficients default to 0).  -  Column  a-jj  -  number.  Coefficient  for first  row and j t h  column.  m1+m2  Card  2+(J1+J2+.»«+J  )+2m2+  m  I  k.+1  bi  -  Right  hand  side  of  first  deterministic  b2  -  Right  hand  side  of  second  b3  -  Right  hand  side  of  third  constraint.  deterministic deterministic  constraint. constraint.  m1+m2 Card  2+(J1+J2+  -  +J  )+2m2+ m_  Additional  £ .  •  =  k.+[m_,  3]+1  I  ]  Ci  -  Cost  coefficient  of  first  c  2  -  Cost  coefficient  of  second  c3  -  Cost  coefficient  of  third  notes 1.  A l l  on i n p u t  data  constraints  added  variable. variable. variable.  and r e s t r i c t i o n s . must  or subtracted  be e q u a l i t i e s , (sign  of  2.  The number  of  deterministic  3.  The number  of  stochastic  4.  The number  of  possible  so s l a c k s  RHS i s n o t constraints  constraints  realizations  m.  must  be  important). mi < 150 + <  f o r E.  (70  70. is J.  < 8.  -  m2).  Chapter  4  IMPLEMENTATION OF THE ALM MODEL  4.1  Introduction  This of  t h e ALM model  Vancouver some  the  City  to  real  l i f e  problem  grew  a t a compound the firm  trading  of  at  their  of  (VCS).  an  This  thesis,  facing  application problem  In a d d i t i o n  implementing  continuously  studied,  to  t h e model in fact,  this  of  these  results,  for this  and  was p r o m p t e d  particular  credit  of  1970  aggressive  that  policy  term  of  are:  deposits mortgage the lower version  93  1)  by  union  i n high that  and changing  problems. for higher  assets  yielding  this  rate study  and  yielding the  were deposits.  still  structure. was  com-  market  Investors  o f VCS w e r e  interest of  five  to $160 m i l l i o n ,  VCS r e a l i z e d  loans  the  the f i r m ' s  investing  policy  liquidity  the f i r s t  of  1974,  investment  serious  on t h e b a s i s  1974,  $26 m i l l i o n  In  the outstanding  moment  to  57% f r o m  mortgages).  low y i e l d i n g  time  c h a r a c t e r i s t i c s o f VCS d u r i n g  an a g g r e s s i v e  was c r e a t i n g  returns this  Union  aspects  rate  adopted  their  t h e same  earning  period  (predominantly  conditions  was  Credit  the s a l i e n t  planning  bination  of  problem.  year  assets  results  and l i a b i l i t y p o r t f o l i o  are discussed.  Some o f  At  the asset  Savings  problems  liquidity  2)  i s concerned with  of the procedural  related  a  chapter  It  initiated.  -  94  The programming lation  the  total  1)  the  4)  the VCS  final  which  -  the  the  policy and  are  initial of  funds  terminal  portfolio  of  by  There  were  which  and  was  to  by  types the  of  3)  to  the the  insure  liabilities  net  returns  Unions  which  minus  of  sources  of  similar  constraints  the  of  Act  are  budget  a statement  formu-  constraints:  Credit  the  linear  maximize  discounted  constraints  and  to  total  four  include  conditions  assets  formulation  2,  equal  year  Charnes  liquidity  are  five  and  the  chapter  a  Chambers  prescribed  in  construct  the  VCS  conditions  constraints  the  upon  to  given  the  (5)  was  the  as  2) and  uses  of  costs.  [8], (4)  taken  based  constraints  Columbia  identity  model  discounted  include  first  objective  returns  inequalities  that  of  The  legal  British to  planning  [11].  discounted  approach  accounting  funds,  and  internal  operating  policies  that  structure  of  the  maintains  continuity  the  of  operations. The does  not  basic  incorporate  interest  rates.  decision  theoretic  point  estimates  rates  of  rate  VCS's  structure  would  shortcoming  then  As  of  the an  and  yield  a  approach future  the  attempt  was  for  each  set  solutions.  solution  can  be  vector  for  state  the  summarized: of  nature,  formulation  The  2)  this  procedure  possible  and  was  that  cash  was  to  first  potential of  to  the  optimal  computing  the  resulting  a make  interest  find  finding  and  growth  executed.  necessary  it  flows  drawback,  combination  program steps  is  unknown  structures  linear  1)  of  overcome  The  rate  rate,  of  to  taken.  interest  Then,  above  uncertainty  growth  'best'  each  inherent  initial  assets.  of  This the  solution net  95  present  values  remaining solution  states to  of  optimal  nature,  be t h e same  infeasibility attain  f o r each  state  of  each  step  1)  was computed,  has  the highest  and  4)  then  3)  hood  of  (This  are represented  from  k*  the  state  nature  of  such  -  that with  and i  = 1 , . . .  of  occurring;  NPV...  6 -  occurs  of  of  value  3)  the If  to  for the  f o r each  l i k e l i -  decision  selecting that  value.  the  nature.  are relaxed  probabilities  NPV  E(NPVkJ  i j k  action  Mathematically,  in which  steps  3)  P(e,..)  > E(NPVk)  the i t h interest  rate  whhee ni k w  n ^  ^  growth e . .  step  net present  m  choosing  f o r each  by  E(NPVk)  and  1)  state  constraints  an expected  and 4)  step  f o r each  subjective  nature,  expected  1)  the debt  using  from  can be a c c o m p l i s h e d by f o r c i n g  as i n step  i s reached  feasibility),,  solution  ,m a n d j  f o r a l l k.  rate  = l , . . . , n ;  P(e.-)  i s the net present  for k = 1 , . . . , ( n  x m);  structure  value  and k*  of  Where  e^- i s  and the j t h  i s the  probability  choosing  strategy  i s the optimal  stra-  tegy. Alternative 'best'  solution  approach chosen  criteria  instead  i s not very  of  steps  appealing  i s i n no way o p t i m a l ,  3)  as minimax and 4 ) .  optimal  f o r any p a r t i c u l a r  net  present  value;  c a n be u s e d  However,  for the following  f o r there  not  a n d 2)  such  state  t h e model  may e x i s t  of  nature,  does  to  find  in general,  reasons: a solution  1)  this  the  k which  b u t has a h i g h e r  not incorporate  a  k* is  expected  a n y means  of  96  evaluating there  may  the be  economic  consequences  a particularly  of  disastrous  infeasibility, realization  for  that  example  results  in  insolvency. As been  described  proposed  limited  This  Chapter  the  problem,  Hence  to  the  while  2,  a  a  number  but,  at  different  ALM  of  other  approaches  best  these  models  approach  formulation.  maintaining  The  to ALM  computational  the  have  have  problem  model  does  only was  incor-  tractability  for  problems. As  strate  the  led  uncertainty  large  three  model  applicability.  necessary. porate  to  in  the  stated  the  purpose  a p p l i c a b i l i t y of  the  ALM  major  model  already  domains  in  terms  profitability 2)  the  the  features  native  ease  of  of  of  of  superiority  to  technique  ALM  can  chapter model  be  results  application,  this  the  i t  the  attributed  solution  remainder results  and  where  of and  i t , is  to  the  of  this  model.  demonstrated:  to  be  used  by  the  equivalent  3)  that  before  this  and  for  the  1)  the  to  demon-  there  i t  Crane the  planning  to  does the  model.  in  problem, fact  'best'  have alter-  Hence  implementation period  of  increased  deterministic model  are  usefulness  management,  comparing  Bradley  is  Specifically,  concerned with VCS  chapter  the and  1970-74.  97  4.2  Model  Details  The for  the  It  decision  to  this  implementation  Union.  used  aim of  will  include  variables,  in  the  of  and  the  ALM  constraints  The  model  end of  this  be  rather  cumbersome  and  estimation.  The  chpater. (the  (as  described in  The  ALM  model  second stage  assets,  Chapter  3).  Eleven 1)  k X.^,  and  types  cash,  of  2)  to  data  Chapter  3,  Appendix  a SLPR model The  by  first d  l i a b i l i t i e s ,  bonds  maturing  in  i  4)  federal  government  bonds  maturing  in  five  bonds  and  second mortgages  Six  types  deposits, 4)  term  of 2)  with  liabilities share  deposits  maturing a  three  are  capital  maturing  in  in  more  year  that  stage  and  to  than  term,  in  (i  ten  7)  They  of  borrowing  i  3)  years  (i  this  form  presented  are  are in  shares, 4), 5)  6)  1)  pro-  first  personal  from  f i r s t  application.  years,  include:  = 1,3,5).  are  defined  years,  considered. VCS,  matrix  = 1  ten  and  Appendix  variables  Union  years  in  there  (as  Credit  government  they  d i f f i -  2).  considered  B r i t i s h Columbia  required  very  in  of  function  given  data  Credit  choice  effort  the  257),  implies  federal  government  the  the  92  Savings  the  used are  is  necessary  objective  demonstrate  presenting  are  City  indicate  3)  vincial  and  matrix  assets  input  collection,  to  actual  decision variables.  into  are:  is  than  Since  form  divided  They  rather  being  data  purpose  the  Vancouver  actual  the  input  to  the  1  in  model  describe  of  problems  would  the  ALM  to  method  cult at  of  is  the  application.  implement  section  loans.  demand  banks,  and  Specifically, i f  a  98  four  year  third  federal  time  government  period,  6  this  6  X-^  ,  where  be  sold  to  be  in  X^  at  and  the  are  the  four  and  five,  horizon  136  types  liabilities  the of  initial these  variables  assets  of  was  to  maintain  and  the  and  assets a  model Cash  flows  the  occur  at  described.  the  all any  the  beginning  of  detailing  during  of  an  the  the  beginning  variables  the  five  the cash  was  beginning the  year  of  6  6  Xg^,  X-g  the  and  next  are of  The  period.  the  types  of  the  the  six  (including  period.  reason  portion assets  and  variables  to  The  choice  historical  port-  for  choice  such  actual  a  portfolios  formulation.  are occur  treated the  is  VCS's  between  investment  positions)  36  on  s p e c i f i c s of flows  and  planning  based  ALM  initial  eleven  initial  [85].  transactions period  The  additional  the  the  respectively  comparison by  of  model.  liabilities  yielded  that  the  liabilities  although  assumes  flows at  that  for  basis  portfolios  of  generate  and  Before note  at  decision  portions  (including  positions)  folios  to  purchased  generate  Xg^  generate of  will  is  6  periods  held  bond  model,  continuous at  the  it  over  is  period  that and  constraints  important  time,  beginning  assuming  present The  the  half the will  of  the periods.  the  cash  other now  be  half  99  a. The of  British  tions  on  source  for  Columbia  the  constraint  the  total  assets,  is  £  I  The  and  1%  Xr,^,  of  their  unions  highly  operational  assets  maintain  liquid  >  I  .1  assets,  and at  Union  Act  restric-  liabilities. least  Y  X.., L  10%  of  that  is  1  X  iel  1  requirement  total  Credit  ieI  is  debt,  that  Y..,  l t  +  X  2 t ^  0  J  1  in  credit cash  unions  and  term  maintain deposits,  at X,  t  The  final  one  half  constraint of  the  bD tt  i  n  Y  it"  eu  1  to  of  the  respectively,  X  I Y.. b_B D  three  portfolio  credit  in  1  second  places  is  1 1  X L  constraints  act  the  that  X..,  iel  ieI  of  Constraints  legal  This  composition  first  least  the  [8J.  The  Legal  restricts  total  the  credit  union's  borrowing,  liabilities,  Z  I  Y  beB  Since requirements  the  account  '  5  planning for  I  Y  ieD  i  ir f  horizon  fifteen  is  for  constraints  five in  periods, the  the  legal  formulation.  100  b. Of teen six  establish types  sources These in  of  of  twenty-two  the  funds  ALM  numbers  initial  liabilities, to  constraints  the  the  the  Budget  be  in  d i s c u s s i o n on  while  as  these  the  positions  in  the  firm  claims the  has  sufficient  under  Federal  adverse Reserve  cation  of  credit  unions  Reserve  upon  the  the  is  The  first  structure  same  > q.(W  the  -  I  used  The  of  each  assets  and  require  the  period.  budget  in which will  constraints the  be  of  in  is  severe  formula to  actual  part  of  British by  the  i  and  3,  that  follow The  from  appli-  Columbia's Credit  Union  reserves  based  liabilities.^  = 1,2,3  (1)  K  Chapter  ensure  withdrawal  [27].  establish capital assets  to  constraints  published  a.k)  as  way  meet  formula  constraints portfolio  to  adequacy  a study  k e ^ u . ^ u K .  is  the  constraints  conditions.  adequacy  the  from  in  determined,  reserves  capital  of  types  seven-  Constraints  liquidity  capital  three  were  first  constraints  funds  The  the  function.  Board's  in  five of  3.  set,  eleven  directly  Chapter  capital  economic  1  notation  uses  Liquidity  justified  [25].  1  The  FRB's  Board  P.  of  this  the  the  objective  function  of  other  equations  c. The  to  in  the  constructed  formulation,  utilized  constraints  equal  were  Constraints  Section  2.  101  where  W is  the  dollar  value  of  the  expected  withdrawal. claims  under  m adverse  economic  traction  of  used were  liability  .47  borrowing.  if  the  for  The  parameter  that  asset  tions.  The  claims  that  measures  q.  to  exceed P.  can  be  g.  is  a  asset  is  to  be  purpose to the  are  of  realizable required  the  the  con-  economic  conditions.  The  y.'s  for  deposits  and  1.0  for  in  (1)  is  term  in  [25_.  portion under  required  portion reserve  of  The  in  the  value  adverse for  the  of  assets  necessary  to  asset  economic  potential  k  condi-  withdrawal  contained  meet  a  the  in  excess  is  parameters.  constraint  >_  3 I P. i=l  not the  to  the  capital  to  [total r i g h t hand + •jside of balance sheet  measure  the  liquidated quickly. same a s  in  adequacy  as,  - 8,-)x-  those  The  develop  provide  operational  shrinkage actual  p r e s c r i b e d by an  model  an i t  the  operational  a p p l i c a b i l i t y of  parameters an  principal  parameter  demonstrate  opment  reserves  measures  1  justified  realizable  the  the  stated  the  here  above  are  .36  y.  claims.  where  3.  adverse  where  liquidated quickly  the  is  £ y^y-, i=l 1 1  deposits,  the  measures  K I 0 i=l  and  be  W =  under  demand  Finally, formula  y.  parameters  is  K-| u . . . L b K . j . withdrawal  conditions,  the  ALM  adequate would  be  -  of  equity  asset  numbers FRB  model,  proxy.  for the  to  surplu^  when for  VCS,  in  q^,  the  but  numbers  the  o^,  Since  However,  necessary  i ,  used  [27].  model  -  rather  used the  estimate  for  develthe  102  is  implied  Since  these  constraints  that  there  are  twenty  d. Two  types  loans  made  first  mortgage  mortgages first  in  period  made  in  mortgages, The  on  loans  the  f i r s t  return.  the  ALM  that above  or This  without  the  by  two  any  _<  in  policy  for are  is  (d  = 1,...,5)  five  periods,  it  1)  personal  constraints.  period  t  are  equal U  should  to  ),  t m  be  included:  X  or  less  <_ . 2 X  t L  equal  or  such  investment  than  t m  ,  less  risky  policies  (smaller  .2  and  of  2)  than  even  though  consistent  with  management's  implications.  the is  constraints merely  to  application  will  not  conditions  y^ in  be  violated. ten  Deposit  represents period  j  latter  though  is  that  the second  .125  they the  (0,1.0),  For  the  the  of  returns  compared  may  yield  preference  features  demonstrate  (p+,p~)  generate  the  These as  is  deviations),  loans  here  generated  be  less  legal  variable  all  Constraints  should  )  for  .125Xtm.  e. The  hold  constraints  (Xt  personal  this  constraints  in t  treating  objective  model,  the  made  mortgages  better  Since  (xtL)  rationale  mortgages  incorporated  t  to  liquidity  Policy  policy  period X^s  second  violated  of  have  are were  may  five  a be  readily stochastic.  applicability which  to  of  suggests  periods,  the  constraints.  Flows the  (j  =  new  deposits  1,...,5)  and  of  type is  d a  discrete  103  random  variable  at  end  the  of  representing  the  jth  the  period.  balance  The  sheet  amount  of  figure  of  deposit  y!? g e n e r a t e d  is  type  d  established  •J  in  the  deposit  flow  constraint  The  y's  used were  The  y's  are  included  The  distribution  funds.  1.0  for to  as  demand  reflect of  follows:  deposits  and  the  actual  was  estimated  .36  (and  for  not  by  term  net)  using  deposits.  flow  the  of  deposit  actual  balance  jd sheet  figures  constructing rent  of  probability  are:  1).  counted  maturing five to  years,  penalties  for  demand on  a  calculated  year  the  first  The  in  one  term  horizon p+  is  or  bond  horizon  the  of  of  for  term  to  around  the  years,  is  of is  the the  total  discounted  model,  and  3)  The  for  returns  discounted penalties  cost p",  term  2)  for  of  the  a  ten  of  the  year  diffe-  the  VCS  1.  constraints total  discounted  deposits  on  for  the  discounted  cost  of  these  and  had  Appendix  is  the  ^.^  runs  in  p+  model,  the  the  shown  minus  minus  discounted  mode.  varying  capital,  deposit  p+  distribution  associated with  share  horizon  the  this  for  used  shortages  year  mode o f  assumptions  one  total  model.  values  and  the  the  distribution  three  minus  as  deposits  deposit  the  government of  1970-74  distribution  The  returns  funds  for  a distribution  application.  the  VCS  discost  term  deposits  returns  funds maturing  of  on  a  calculated in  five  provincial  funds  calculated  to  the  surpluses  associated  with  104  the  deposit  mortgages  flow  minus  constraints  the discounted  of  funds  a n d p~ a t t e m p t  to  reflect  the model.  The  on  the part  management,  the  surplus  2)  with  funds  p  +  as  available  the shortage  discounted  costs  of  of  are the total  of  to what  when  funds  policy  realized  when  uses  returns  c a l c u l a t e d to  sources  exceed  horizon strategy,  make: uses,  realized  f i r s t  the  a conservative  d e c i s i o n s to  exceed  on  1)  with  and  sources,  respec-  tively.  f. The revenues penalty source was  minus  for  personal  The s o u r c e loans  capital  it,  the r i s k  was  used.  discount  These  yearly  factor  the expected  discounted  gathered  on t h e for  federal  the returns of  from  costs  total  and minus  a number  of  and p r o v i n c i a l  deposits,  the  expected  sources.  The  government  o n BCCU s h a r e s ,  the term  discounted  mortgages  demand  bonds and  deposits  and  [85].  discount free  total  were  and the cost was  The  Average yield  The d a t a  the returns  Function  i s to maximize  the expected  costs.  [10].  share  objective  Objective  rate rates  rate  used was t h e t i m e  (the average  yield  a r e as f o l l o w s  value  on t h r e e  of  money.  month  To  obtain  treasury  bills)  [10]:  1970  1971  1972  1973  1974  .0599  .0356  .0356  .0547  .0782  1  .9435  .911  .8797  .8341  1.0547  1.0782  1.0599 .9435  1.0356 =  .911  1.0356 =  .8797  =  .8341  =  .7736  105  The  returns  on  the  assets  are  as  follows  [10,85]:  Returns  Type  of  on  Asset in  Year  1969  1970  1971  1972  1973  1974  .0725  .0620  .0450  .0510  .0610  .0800  Asset  1 year federal government bond (fgb) 2 year  fgb  .0749  .0657  .0490  .0550  .0654  .0803  3 year  fgb  .0758  .0684  .0525  .0590  .0680  .0807  4 year  fgb  .0767  .0710  .0555  .0626  .0698  .0810  5 year  fgb  .0776  .0758  .0615  .0674  .0717  .0827  10 y e a r provincial government bond  .0840  .0904  .0803  .0813  .0836  .0991  first  .0938  .1040  .0943  .0921  .0959  .1124  .1050  .1220  .1108  .1083  .1123  .1321  mortgage  second  mortgage  personal  loans  .1040  .1170  .1075  .1050  .1075  .1275  B.C.C.U.  shares  .0600  .0600  .0600  .0600  .0700  .0700  federal  government  If 1970, rated.  the  one  were  decision  The  returns  to  purchase  variables would  be  X^,  a  five  X^,  calculated  year  X^, as  X^,  and  follows:  X ^  would  be  bond  in  gene-  106  Decision Variable X.. 13  Return  . J  xj2  (.0758)  (.9435)  xj3  (.0758)  (.9435  +  .9110)  X^4  (.0758)  (.9435  +  .9110  +  .8797)  XJ5  (.0758)  (.9435  +  .9110  +  .8797  +  .8341)  X, 7  (.0758)  (.9435  +  .9110  +  .8797  +  .8341  The to  r.  the  were  return  beginning  determined The  Type  of  the  the  .0720  interest  planning  =  earned  horizon.  .1410 =  every The  .2070  year  returns  = +  .2700 .7736)  discounted on  all  =  .3290  back  assets  similarly.  costs  of  the  liabilities  are  as  follows  [85]:  of  Liability  1 year  is  =  Cost  term  of  Liability  in  Year  1969  1970  1971  1972  1973  1974  .0712  .0780  .0720  .0680  .0780  .0990  .0712  .0820  .0760  .0690  .0820  .0980  .0785  .0850  .0800  .0800  .0850  .0975  .0400  .0460  .0410  .0420  .0560  .0770  .0500  .050  .050  .055  .0575  .0800  deposit 3 year  term  deposit 5 year  term  deposit demand share  deposit capital  The be  determined  cost as  of  a  five  follows:  year  term  deposit  (y-j)  sold  during  1970  would  107  Year  i  Cost  Incurred  1970  (.5)  (.0850)  1971  (.82)  (.0850)  1972  (.82)  (.64)  1973  (.82)  (.64)  2  (.0850)  (.8341)  =  .0238  1974  (.82)  (.64)  3  (.0850)  (.7736)  =  .0141  The  In Firstly, a  risk  adjusted of  included  the time  investments,  of asset  alternative  interest  any s y s t e m a t i c  is  desirable  dity the  losses  of  carefully  =  o f money  .0392  .1807.  further  refinements  was u t i l i z e d  be u s e d  of  liability  structures asset  bank  to  reflect  [84].  to  some assets  estimated.  might  of  to  Reserve  Board  a r e , and these  the  linear involve  models  levels  make for  in assets. the Thus  It  liquialthough  may n o t b e v a l i d , have  of  do n o t  but rather  would  of  manage-  distributions  banks.  rate,  directly  i n matching  across  degrees  typically  investment  consistency  and l i a b i l i t i e s  constraints  be  i n p r e s c r i b i n g maximum of  required.  liability  management  constraints,  on t h e amount  level  cannot  contrast  and l i a b i l i t y  by t h e F e d e r a l  these  In  be  varying  Secondly,  management  liquidity  would  as the d i s c o u n t  inducing alternative  managers  or limits  utilized  structure  (.8797)  t h e ALM m o d e l .  most  to maintain  parameters  .0635  o f y-j i s  should  approach  characteristics of  general more  Finally,  the judgement capital  of  rate  include  either  value  management,  flows.  of  cost  =  "controlling" the deposit~flows,  deposit  use  = .0401  see f o r example  i n the framework  treatment  (.0850)  rate  i  (.9110)  implementation  discount  f o r example  (.9435)  discounted  an a c t u a l  although  riskiness ment,  total  i n Year  t o be  the  108  4.3  Results  of  It cation: 2)  to  these I  to  expected  demonstrate  t h e model  model), values  the basic  general  by r e p l a c i n g  (with  both  Application  of  with  a l l  the data  random  To  this  variables  penalty  appli-  and  accomplish  given  and f i n a l l y  to.the  of  t h e ALM m o d e l ,  generated.  equivalent),  respect  Union  a r e two p u r p o s e s  solution  presenting  the detailed  in general:  to  probability  the l i q u i d i t y 2)  distributions  results  of  concerning 1)  constraints  the s t o c h a s t i c  the deterministic  in with  with  costs  Appendix their  variants  and the  equivalent,^  and the penalty  application,  t h e ALM model  the i n i t i a l (a  models  the  situation  portfolio which  solutions  and 3)  the nature  markedly  held  was  yielded  costs  in  known  that  were  of  the  affects  the  solution. The  The  the  and SLPR models  management),  rate  of  c a n be made  to  butions  there  statements  VCS v i o l a t e s  optimal  that  Credit  distributions).  by  superior  Savings  the a p p l i c a b i l i t y  (deterministic  Before  particular  City  was r u n i n i t i a l l y  secondly  model  probability  several  be r e c a l l e d  the s e n s i t i v i t y  goals,  (basic  of  will  1) test  The Vancouver  (.2,  b a s i c model .6,  probability penalties  .2)  has symmetric  for a l l the deposit  distributions  for  for a l l stochastic  three flow  point  constraints  the l i q u i d i t y  constraints  probability and  and p o l i c y  are asymmetric.  Madansky i n [ 5 6 ] h a s shown t h a t t h e ' d e t e r m i n i s t i c a l o w e r bound on t h e o p t i m a l v a l u e o f a SLPR.  distridegene-  constraints. The  equivalent'  optimal  provides  109  value  of  profits  the minus  equivalent expected bound  model  similar  differ  by of  the the  did  not  bility,  i t  was  first  variables  with  model.  penalties  As  could  liquidity the  were  violating a be  needs  and  of  of  different  parameter  constraints. each  the  variables  stochastic  high  so  attain  violated  meet  the  will  FRB's  b a s i c model  probability The  an  VCS's order  to  the  that  the  were  run  change  not  invest  empty  as  of  the  VCS,  model  would  The  as  feasible portfolio  constraints  amount o f  liquidity  patterns  feasi-  constraints.  feasibility. be  portfolios  secure  same  the  the  the  initial  operations  distributions,  initial  had  liquidity  arbitrarily  did  liquidity  were  the  the  changes.  of  In  into  are  to  that  two  investment  also  in  Thus  below  the  mortgages).  reason  to  of  (namely  was  only  10.6%  b a s i c model  The  the  ($8,565,068  The  insight  constraints  Variants effects  set  the  model  these  further  structure  equivalent  to  is  expected  deterministic  penalties).  equivalent  However,  in  The  $2,278,187  expected  The  assets  added of  ($8,288,941.53  penalties).  of  period.  liquidity  constraints firm  time  run.  coefficients  associated  in  period.  liquid  the  value  basic model.  deterministic  satisfy  objective  the  less  expected  deterministic  first  the  in  $6,286,885  the  when  that  minus  beyond in  $2,520,316.01  optimal  initial  set  which  an  the  The  the  is  in  heavily  basic  has  profits  value  model  $5,768,625.52  provided  optimal is  basic  penalties in  the  the violate  amount  liquid  The  by  reserves  requirements. in  order  various  instituted  in  to  ascertain  penalty the  the  cost  basic  model  110  was  the  that to  current  equal  change in  alteration  or  was  to  two the  allocated  to  longer  seem t o  the  two  longer  be  term  the  optimal minus rose  value  with  change respect  lations bility  of .05  straints penalties  in  to  while  both  After  was  a  (.05,  the same  of  the in  models.  to  .45)  the  are  now  not  previously  violated  by  15%  in  include  a  change  cash f l o w s .  the  expected  the  expected  explained  penalties  were  invested  expected  the  which  of  in  and  model  1)  infeasible only  violated (implying  decreased)  because more  of  The  profits  profit the  costs all  The  case.  net  with  penalty by:  did  patterns  ($8,872,911.53 The  were  there  this  instead  is  funds  in  now  (that  For  substantially  periods  altered  are  .2  penalties).  amount  of  this  ($8,657,619.24  investment  stochastic constraints of  of  dramatic  b a s i c model  is  liabilities  effect  two  total  penalties).  This  the  incremental  f i r s t  larger  .50,  requirement  deviated  the  the  much more  time  The  pattern  further  then  of  expected  was  was  10%  the  $2,906,773.53  formulation.  expected  the  in  more  $3,256,500.65  both  at  that  to  modified  value  to  value  behavior  distribution  jumped  to  in  than  from  l i a b i l i t i e s .  investment  there  the  optimal  respect  the  assets.  However, in  greater  optimal  the  term  or  constraint  $5,750,845.71  b a s i c model  the  of  generalized  assets  in  the  minus  $5,661,410.80  meter with  any  probability  increase  1%  legal  to  b a s i c model  models.  The in  than  periods,  of  first  equal  increase  that  not  the  be  profits  initial  from  assets greater  expected  the  of  para-  decreased  the  with  a  and  vioproba2)  con-  excessive  profits).  This  demon-  in strates hand  the need  side  of  for accurate  a stochastic  Although tions  from  the runs  conclusions  values.  estimate hand  of  of  may h a v e Also  i s not possible  t o make  t h e ALM model  described  an i m p o r t a n t  sensitive  to  changing  deterministic  as normative  of  necessary  runs  168. 40 ALM  It  to  sized of  of  this  solve  model  is  around  that  some  of  the  than  the  reliance  than  Finally,  the  t h e same  on t h e U n i v e r s i t y  of  constraints. about  deterministic  problem  T h e same  t h e ALM f o r m u l a t i o n  Using  37 s e c o n d s  British  of  using  t h e SLPR  code,  CPU t i m e . t h e SLPR  deterministic  linear  To  code  program  are  deterministic Fourth,  the  the  implemen-  computations order  as t h e  problem.  All IBM 3 7 0 /  i s 92 by 257  the solution  took  left  equivalent  Columbia's  solve  the  stochastic  solutions.  computed  that  of  on the  on t h e  difficult  are of  solutions  the solutions  the various  to erroneous  model.  general probability  the value  deterministic  took  left  generaliza-  the s e n s i t i v i t y  Second,  solutions  t h e ALM f o r m u l a t i o n  w i l l " be r e c a l l e d ,  CPU t i m e .  note  i s n o t more  deterministic  above,  f o r an e q u i v a l e n t  were  on t h e  on t h e o p t i m a l  Third,  indicates  can lead  definitive  necessary  stochastic model  to  costs.  different  This  tools  a similar  computations the  penalty  models.  implementation tation  point  constraints.  the stochastic  models  effect  distribution  substantially  the values  t h e asymmetry  the probability  of  have  First,  a substantial  side  models  around  constraint.  may b e i n f e r r e d .  distributions and  i t  estimates  an  about  of  with the  equivalent 30  was a l s o  seconds solved  on  112  a  standard  and  a  L.P.  number  models models.  are  of  code,  UBC  other  runs  generally  about  LIP  which  took  17  indicate  that  CPU  double  that  of  seconds times  of for  equivalent  CPU  time.  stochastic deterministic  These  Appendix  The  One  following  is  the  input  data  for  the  basic  model.  114  . 0 0 0 0o o i 0025700052000/jn 1 0 .0 f. - 1 0 o o o o 0 o'. 1 o o o 0 n 0 0 . .2 0. 1 0.0 I . - 1 00 00 0 0 0 1 o o 0 o 0 0 o. ... ,? o. 1 0.0 f. -1 o o o o o o o'. l n o o o o o o .  l p 7  _J l\ 5  7 8 q  10 ...11 1? 13 l'l  15 16 17 IB 2.0  21 22 ..23 2'\  25 26 -! I  28t . 29 30 31 32 •7 7 .J J 3'l 35 36 i.  7  T  _ >l 3B 39 40 ..4 1 ••'12 4 3 /.Ml  5 46 4 7 flfl 49 50 4  .2  . ..  „_.,J1  _„ . .  1 0,0 1. . -1 00 0 (.»00 o . . t o o o o o o o . ...2 0. 1 0.0 1 . - 1 0 0 0 0 0 0 o . J. 0 0 0 0 0 o o . 2  .0  1 0.0 t '. - 1 00 000. 1 0 0 0 0 0 '„ 0. 1 . 1 n .0 1. - 1 0 00 0 0 . \< > 000 0 . - o . __ ...1..— 1 0.0 1 . - 1 0 0 il 0 0 . .10 0 0 0 0 . 0. 1 . 1 o.o .1 - 1 0 i.) () o o . t00 0o 0 . .0. 1 i o .o r. - 100000 . 1 0() 0 o 0 . 0. 1. 1 O.O 1 . - 100000. 100000 . Q - l . . i  o.o  r.  -1 0 00 0 0 . 1 00000. 0.. . 1 .. l o. 0 1 . - 100000 . 1. 0 0 0 0 o . . 1 ... . 0. 1 0.0 1 . -1 0 0 0 0 0 . 1 fi o 0 o 0 . 0..... 1. 1 0.11 1 . - 1 0 0 000 . 100000. . P.. ...1. ...... 3 6 0 0 (• o 0 0 . ;> 7 3626 0 0 . .6 B00 0 00 0 . .2 #  115  51 52 .53. . 5'l 55 56 57 58 .59 _ 60 61 62 63 60 .65 66 67 68 70 .71 -r 2 T T  7'! 75 76 ...7.7...... 78 79 no fll  82 83 8<l  85 86 87 88 89 00 91 9? 93 9/|  . ' ' 5 ... 96 97 <7 0  i  0o  5 0 0 00 0 0. 9000000 . 0.0 . 3...17»COG<\0. ... 1 8 8 o 0 8 0 0 . .6 ? 00 0 00 0 0. . 2 1 6 0 0 0 0 00 . 2 1 0 o ') 0 00 . 0.0 .0 9 1 3 26000OOO. .2 ?7(. 1 '37 0 0 ...... 6 2 9 0 0 0 0 00 , .2 2 5 0 0 0 0 0 0 . .300 0(10 0 0 . O.'.O .07? 3 6 0 0 0 0 o o o . .2 6 7 4 3 3 <J 0 0 . .6 _7_. 0 00.0.0.0 .. . 2 _ _ 5 8 0 0 0 0 0 0 . 7'": 0 0 0 0 0 0 , 0.0 . 0 '\ 6 y  3.. 8 ' 5 0 0 0 O O Q .  ...?.. .  9 3 9 6 80 9 6 . . 6 1 0 0 00 0 0 0 0 ..2 8.0 M i L O J I Q.^_L1.0 0.0.0 o.CIO. 0.0 . 02 81 3 2000000 . . ? 2635007 . .6 3oooooo. . 1800000 . 35 0 o n 0 0 . o.o. .1;?75 3 2 0 0 0 0 0 0 . .2 ?87372<l. .6 350 0 0 0 0 . .2 1800000 . 3 7 0 0 0 fi 0 . 0.0 .0001. 3 70 0 0 O O O . .2 8 155152. .6 9000000. .2 65000 0 0 . 9_000 0 0 , 0.0 .07? 3 1 20 0 00 0 0. . ? 1 3078 1 7 6 , . 6 i a o o o oo o . .2 1 1 0 0 0 () 0 0 . 1 , 0 0 0 0 0 0 . 0.0 .0 ah 3 1 2 0 0 0 0 0 0 .'. 2 1 32<»9_0«. . 6 15 0 0 0 o o 0 . ,2 1 1 0 0 0 0 0 0 . 16 0 0 0 0 0 0 . 0.0 .028 1 3 25 000 00 . '. ? 3070900. .6 350000 0. .2 • r  116 0 1 0 2 oz on OS" 06 0 7 06 09  20.00.0_o_o {4.0noQon , O.o" " " . 1 27 3 3 6M0OO0f:# .2 7 4 3 2 0 () 0 . ,6 8 000000 . .2 6 5 0 0 00 0 . 0 , 0 . 3  10  0.0 3  .6 .2  20000000. ?H0 0000 0 0.0. .0^6 3 22000000.'. 2 2 4.0.1) 0 0 0 0 . . . . . . 6 2 6000 000. .2  IB 19. 20 21 22  2 0 0 0 0 () 0 0 ,. 2 8 0 0 0 0.0 .0281 3 75000 0 0 . . ? 3641735'. ,6  T  C J 2n 25 26  000  qsooopo. ,2. 700 0 0 0 0 . 10 0 0 0 0 0 0 .0. .2546 3.. 1 6O.)00n.O .  ~»  28 29  17623159. 1 9 0 0 0 0 00 . 1.5 0.0.0 0 . 0 . 0 . .  30 3J. 3?.  .6 .2 2.01\0.0.0.0 0  .0  .19?6 3 25OOOOOO..2 2722833H. .6  34 35 36  3 000000 0 .  .2  24 0 0 0 0 00 . 3100 00 0 0 .0 . 1 2.7 9 3 2900 0000 .'. 2 3279/1637. .6 36 0 0 0 0 0 0 . .2 28000000. 37000000 . 0 . 067 7  7 "7 -J .  38  39 •'•! 0  '1 1 4 2 4 3. 4n 45 46 4 7 48 4 9.. 5 0  .2  .6 .2 j. 6 0 0 0 0 0 0  .072 2 2 0 0 00 n o . . ?  ?4Q 0 00 00 . ? 6 0 0 0 0 0().  15 16 17  -1  1 3 O O 0 0 0 0 .  1 4 0 0 0 (1 0 0 . 15500000. 1 2 0 0 0 i) 0 0 .  11 12 13 in  85^0000 . .. . 0 9 . ' L i  _.  _.3_ 2 7 . 0 0 0 0 0 0 . . 2 ?9.'J0674?. .6 33000 0 0 0 . 2 6 0 0 0 0 0 (). ,0 3  ._  3 .'1 0 0 0 0 0 0 .030/1  6 7OO00O.  7 45695 7 8 200000.  6 .2  .2  .  6 5 0 0 0 IJ 0 .  151  .0  i  3  5:. 1 5 '-i 155.. 156 157 158 159  .6 .2  7500000. .0  i. 0 0 0 0 0 0 0 .  .0  . 1 270 3  166  17'\  865125 4. .9.5.0.0 0 . 0 . 0 .  n  ...  _o__.  . 0 6 7.7 3 1 1 0 0 0 0 o n .'. ? 12386S.1. .6 ... .1 3 5 0 0 0 0 0 . , . , . 2 1 0000000 . Mi 00 0 0 0 0 , . 03 on .0 ?. 1 . .  175 176  3 5 7  1 . 1 . 1 .  177 178  8 1 0  1 .  J 1.1 . .1 12 i  17" 1 8 0 181 18?  1/4  15 16  183 18<l  17  19_ A  185 1  8  t>  187  188 189  190 19.1 J.9 2 193 19/1  19 6 19?_ 19 8  _  200  __.  ooo oo c , 2 • 1 2 2 5 '16 1 0 . . . . . 13500000. .2 1 0 0 0 o OOO. 1 41 0 o 0 0 0 0 .  163 1 6 <i 165  16^ 170 171 172 173.  .25'.!6 7 800000 . . ?  . 1 926 . ?_ 3 o 00000o , ,>:. 10 3 5 5 2 / 0 . I 1 0 0 0 () 0 0 . . 2 8 0.0.0.0 0 0 . . . . 1 2 0 O.Q.O 0.0...  16 0 161 16 2  1 6 7. 168  8 . 0 (in.io,  ....  . i. i. i. i. .  20  1 .  21  1. 1  22 23 25  1 . 1 . ..._26._ j . 2 7 28 29 35  l . 1 .  1 . 1 . 36 1 . 3 . L . ..1 38 r. 39 _0_  .  .__  118 20 1 20 2 203  20 4 205 2 06 207 20H  2 O^ 2I0 21 1 212 213 2 14 215 216 217 218 219  - i *•} C i- <-  117 C C -J  224 225 226  1 .  4«  1 . 4 9. ] . . 5 0 1 . 51 1.  5 2 53 54  1 .  1 .  1 .  55.J 56  .  1 . 1  234 235 236  1  2. 3>8 , 2 39 2 40 2 4 1 2 42  1 OS Ofl?  M  082 076 5  16 1 162 163 164  0 765 5  n  0 5 OS  -jr. f0 *  188  2 3 1 " 2 32 233  250  3.  4 7  167  229 230  2 48 2 49  1 . 1 .  4 5 4 6  165 166  228  24 3 244 2 -:i 5 2 :'l 6 24 7  44  152 153 15« 159 .16Q.  220 221  •> 7  1 1 . 42 1 , 4 3. . 1 . 4  5 0 5  •  000000 p  *.  1 1i . 12 i . 15 i . 16 i . 17 i . 20 3. 21  1 .  22  1 . i .  23 26  1 '.  27  1 .  2P  1 . 1 .  29 39  ~ 4  1 , ...  r  42 44  1 . 1 .  a5  1 . 1 '.  4 6  000 0-0 0  oo0o00000000  1  119 /J8  1  ,  2 5  ?  4 9  1  .  2 5  2 5 ]  3  5 0  1  .  2 5 '1  51  1  .  2 5 5  5 3  1 .  2 5 6 .2.5  7 _  2 5  B  5 4  1  5 5  .. 1.  5  6  . _  1 .  2 5 9  6 2  1  2 6  6 3 6/1  1 . 1 .  6 5  1  0  2 6 ] 2 6 2 2 6  3....  .  .  6. . . . 1 . .  6  __.  2 6 ' l  6 7  1  .  2 6 5  6 8  1  .  2 6 6  6 9  1  .  2 6  7  70  1  .  2 6  B  71  1  .  7 2  1..  2 6 9  _  27  0  73  2 7 1 •> T 1 C  1 .-  2 7 3 27  4  27  5...  7  27  8  .  4  1  .  7  5  1  .  7  6  1 . 1 .  77  . 7 . 8 . .. 1  2 7 6 27  1  7  ,  . _  79  J  8 0  1 .  .  81  1 .  2 7 9  15'J  - .  2 B 0  1 5 9  - . 0 5 2 5  2 9 1  1 6 0  - .o  161  - . 0 / ( 0 5  2 8  2  ..  .  1  r  ^ 2 5  1 6 2  -  2 8 <l  16  - . O S  2 8 5  164  -  2 8 6 • v u ->  1  -  2 8 8  167  - . 0  2 8 9  1 6 8  - . O S  21 o  1 6 9  -  .  2CM.  170  -  . 0 S  •>« ? i . . ' >-  171  - . 0 5  2 9 3 . _  17.2.  -  29<l  1 8 9  -1  2 9 5  o o o o o () 0 o o o o o o o o o 0 0 0 o 0 o o  2 8 3  3  6 5  . 0 a 0 5  . 0 8 2 . 0 8 2 , 0 7  1 6 6  . 0_ .  12  2 9 7  16  1  .  17  1  .  21_  1  .  2 2  1  .  8  2 9 9 3 0  0  6 5  0 5  2 9 6  2 9  7  6 5 .  1 .  30.?  2 7  3 0 3  2 H  3 0 0  2 9  .3 0 5  4?_  3 0 6  « V  3 0 7  «6  3 0 8  4"  3 0 «  5  3 1 0  5 1  3 1 1  0  5.:)..  3 1 2  5 5  3 1 3  5 6  3  i "  6 6  3 1 5 3 J  6 8  o  6 9  317  71....  3 1 8  7  5 1 9  7  3  3 2 0  7  5  3 2 1  2  7 6  77 7° 80 81 87  3 2 7  8 H  3 2 8  8 9  3 2 9  9 0  91 92 9.3  9U 95  3 3 5  __  9 6  3 3 6  9 7  3 3 7  9 H  5 3 8  9 9  539  1 0 0  3 -:i 0  3'•]..._  10 1  _ 1 02 103 155 159 160 161 1 1  h  2  6 'J  165 166  3  167 1 68 169  s2  353.  170  3 5 /I 355 35 6 35? 35  175 176 177  ( J  22 2 3 28 29 .  n6 5 0 ...51.. 55 5 6 6 9  7 2  _..  73 7 6 ..  37 8  77  37"  80 81  395 39 6 39 7 39 8 399 40 0  . 08?  . 0  17 _  37 3 37 4 37S 376  38 0 38 j 38? 383 3 84 385 3*6 387 388 3 39 39 0 39). 39? 3 "3 3n4  05 .OR?  190 0 000o0 o  367 36 B 36 9 37 0  377  •a  173 174  36 0 3 6 1. 36,? Ji 6 3 36 4 365 36 6  •a  04 0 5  - . 0 765 - . 0 7 65 - . '0 5 - . 05 - . 05 - 5 -. 5 -1 .  171 17?  a  -. -.  1  9?  ...9/1... 9 6 9 7  99 100  1 02 _ . . . 1 0 3... 1 09 1 10 111 1 ! 2 113 .114.. 115 1 16 117 118 1 19  00  -'125  1 2 0 . .J... .._ 121 1 . 156 -.1 16 0. - . 0 2 15 161 - '. 0 1 1 4 162 - . 0 1 1 4 1 6 4,_. - . 0 ? C 5 1 6 5 - . " 3 36 1 66 -.0215 167 - . 0 2 15 169 - . 0 5 2 5 170 - . " 5 25 171_ . .0 HXt'i-... 172 - . 0 4 0 5 173 - . 0 5 174 - , 0 « 2 175 - . 0 82 176 - . 1 ) 7 65 177...-..0.7 6 f L 178 - . 0 5 179 - . O S 180 - . O S 181 -.°5 1 82 -.05 191 .-.} ..  4? 6  0 0000 0 0 0  '1 0 1... '1 0 2 403 4 0 '.1 .'I 0 5 406 4 0 7. 4 06 4 4 4 4  09 J 0  1 ! 1? 413 4I 4 4 15 •'11 6 417 4 18 4 i 9_ 420 421 4;:.? 423 4? 4  427 4 28 429 4 Z0 «3J 4 32 4 33 4 34 4 35 4 36 437 4 38 /1 39 44 0 44 J 44 ? 443 _ 444 445 446 447 448 4 4 9 _; 4 5 0" ~  23 29 51 56 73 77 81 94  97  1. 1 .  1. 1. A. l . l . 1 . 1 .  100 1 . 103 1 .  1 3 1. 15 1 . 117 1 . 119 1 . 121 1 . 126 J . 127 1 . .128 1 . 129 1 . 130 1 . 131 1 . 132 L. 133 1 . 1 1  .  0 00 0 0 0 0 0 0 0 " 0 0 0 0  123  '151 '152 'I 5 3 0 5 <l  157  - . 1 . . . 160 - . 0 1 3 0 161 - . 0 06 1 62 - , 0 06  .155. 'I56  L6.5_-.0.215 1 6 6 - . 0 1 1 i\ 1 6 7 - . 0 1 1 i\  0 5 7 .'!5B '159 •'16 0 ft 6.1 '162 '163  169 - . 0 2 0 5 1 70 - . 0 3 3 6 171 -.0215 172 _ - . " 2 1 5 17/J - ' . 0 5 2 5 175 - . 0 5 2 5  'I6'l  -'16 5 '166 'I 6 7 _ '16 B .'16° .'17 0 '171 '17 2 0 73 '17'I  'i75 '176 '17 7 •'178 '!79_ '18 0 '181 0 8 2 '183 '18 i.\ 0 8 5. n8 6 '187  _ .  89 0 '10 1 '19 2 '1^3  0  '19  ' 1 9 'I  5•  '19 6  '19 7 '19 8 'I')9  500  :  -.0/405  177 - . 0 0 0 5 17B - . 0 5 . i J o ft 2 1 80 -.082 181 -.07/>5 1 82 - .0765 183 - . O S 16/) - ' . 0 _ 1.8 5..._-..0.s 186 - . 0 5 1 67 - . O S 192 -1 , OOOOOo. 2 1.  0 0 9 0 0 0 0 0  ___ 35... 1. _ 152 - . 0 1, 153 - . 0 1 158 - . 0 05 159 - . 0 0 8 2 16 0 - ' . 0 0 8 2 _ . 161 - . 0 0 7 . 6 5 16 2 - . 0 0 7 6 5 163  •'18 8  •'19  176  _  000000 0 5 _  _  - ' . 0 0 5  1 6 /I - . 0 0 5 165 - . 0 05 1 66 - ' , 0 o 5 _ 167 . 0 0 5_ " 193-1 . ooooooo ooooooooooooooo 6 6 2 1. 15'4 - . 0 1 1 59 - .0 0525 160 - . 0 0 5 2 5 _ 16 1  - . 00/105  162 - . 000 OS 163 - ' . 0 0 5  _.  124  164 - , 0 0 8 2 5 01 165 - . 0 08? 5 0-? ...16 6 - . 0 0 7 6 5 . . . .5 0 3 167 - . 0 0 7 6 5 5 04 168 - . 0 0 5 505 169 - . 0 05 . 506 5 07 170 - . 0 0 5 171 - . 0 0 5 508 5 09... 17.2 - . 0 05 19.g -1 . 510 0 0 0 0 0 0 0 0 0 0i 51 1 67 r . 512 155 - . 0 1. 513 159 - . 00 2 0 5 5 1 'I 5 1 5 . .._. 16 0. .'.0 0 3 3 . ( 3 - 161 - . 00 2 1 5 516 16? - . 0 0 2 1 5 517 164 - . 0 0 5 2 5 51C 165 - . 0 0 5 2 5 51° 166 - . 0 0 4 0 5 52 0 o o / i a.5. . 167 521 ... 1 6 8>- . 0 0 5 522 169 - . 0 0 8 2 523 524 . 170. - . 0 0 " 2 .. . 525 171 - . 0 0 7 6 5 17? - . 0 0 7 6 5 526 173 _-...Q.05 527... 5?a t 74 - . 0 0 5 175 - , 0 r, <:•, 52° 176 . . - . 0 0 5 530 177 - . 0 0 5 531 195 -1 . 532 ...0.0 0 0.0 0.!) 0.0.0.0. 533... 1 09 1 . 53" 156 - . 0 1 535 536 160 . - . " 0 2 1 5 * 16 1 - . 0 0 I 1 <i 16? - . 0 0 1 1 4 538 1.6 4.........0 0 2.0 5. 5 39 165 - . 0 0 3 3 * 5«P 166 - . 0 0 ? J 5 54 1 54 2. . 167 - . 0 0 2 1 5 169 - . 0 052 5 5 43 544 170 - . 0 0 5 2 5 111........0 0.4 05.... 5J 5_. 546 172 - . 0 0 a 0 5 54? 173 - . 0 0 5 54 8 . . 1 7 4 .. - . 008?. 175 - . 0 o 8 ? 5 '19 176 - . 0 0 76 5 55 0 n  r  >  1  1) 0 () 0 0 0 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 " " 1 8 b "\ _9t 991 S.9 I _ _ " - t?9 t  \I  S!»;•'(!•;•'-  i9I 29 191 09 I "6  0 09 " 6 0 Si 96Si  Lb'.;  9b!i Si 6 I'  1  0000C 00 00 U  "  :  *' I ' I '_-T " 0 00 00 00000 00 • 1 - £61 Si 0 0 * _9I Si 0 0 " - 9tf T _ 0 0 " - S,9l siu o " - r/« I Si 0 0 * S 9 Z. 0 0 ' i y I _ 9 0 o * - Out 2 G o 0 * •• o L \  ~  Siu o  Si 0  w  -  0" hi 0 f/ u 0 * Si 2 Si U 0 " Si 2 Si 0 0 * Si I 2 o o * Si 1201. * ty  0  i v: o o * 0 _00 ' ti 11 0 0 ' 1)110 0 " Si T 2 o 0 ' 9 0D0 * 900u ' \i £ 1 0 0 * 9 S  1  ti  0  0000U 0 000000 00 -  " ' 92 L  '  -  •  2  6 Si  T6S1  ST  - _ I  0  Si  1/ ( i Si  6 9 Si  yB  Si  9tf Si S.9G  1/ 9s.  £9S 2uSi  tssi 0 9 Si oiS, yis. I I S ,  9L\  9 IS, bis; t/lSi £i!i  SiAI  2IS>  "vi i' ill t!L\  \ lion  6  9t  19\  991 S.9 t 29 t T9l 091 0 ' - z_t 92 I • I 0000000 V-  Si o o * - 2 9 T Si 0 0 * - I _ l _00* - 08 T Si 0 0 * - o / . J Si 0 0 \ - • * L \ " " " ~ S i 9 7 . 0 6 * ' - •~L1T~  TIS.  OlS. t,9'.  0 9 S. 19 Si 99S. Vi  9 Si  l?9Si  2  9 Si  0 9 Si o ; / i 9biS -.'-rife. _ i - i 9SiSi S  Si Si  i/S.Si i'jiSi CSJS»  126 60 1 602 60 3 6 0" 6 05 606 607 608. 609 6)0 61 1 612 613 614 615 i-16  1 5 " 1. 159 - . 2 6 2 5 160 - . 2 6 25 161 - . 2 0 2 5 162_.-,2 0 25__ 163 - . 2 5 1 6a - . 4 i 1 6 5 - . ''i 1 \66 - . 5 8 2 5 167 - . 3 8 ? 5 „ l <?!<_:%.? 5" 169 - . 2 5 170 - , 2 5 17 1 - . 2 5 172 - ' . 2 5 19 9  _ "  '  1'.  6) 7__; 0 0.0.0 0.0 0... 0.0.0.0"0.0.0.00.00000 t 2 6J 8 155 1. 61 9 159 - . 1025 62 0 16 0 - . 1 6 8 0 621 161 - . 1 0 7 5 622 162 - . 1 0 7 5 623. 164..-.2625... 6?4 165 - . 2 6 2 5 625 166 - . 2 0 2 5 626 167 - . 2 0 2 5 627 168 - . 2 5 6 28 169 - . 4 1 6 29 _ .170. - . " 1 63 0 171 - . 3 8 2 5 631 172 - . 3 0 25 632 173 - . 2 5 633 174 - . 2 5 634 175 - . 2 5 6 35.. .176__r.'.25 636 177 - . 2 5 637 20 0 1 . 6 38 .. 0 0 0 0 0 0 0 0 (I 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .1.3 63" 156 1. 640 16 0 - . 1 0 7 5 6 '! 1 ...161 . . - . 0 5 6 9 . 642 162 - . 0 5 6 9 643 164 - . t o r n 6 4 4 16 5 - . 1 (,i> 645 1 6 6 - '. 1 0 7 5 64 6 167 - . 1 0 75 64 7. 1 6 9_. - . 2 6 2 5 .... . . ......... ' • • 1 7 0 -.2625 6 49 171 - . 2 0 25 650 172 - . 2 H 2 5  127  65 1 65 ?  653... 65 4 655 656 657 6 50 0 5°  66 0  661 66? 66 3 6 6/! '•65 . . 666  66 7 h68  h6  ?  67 0 671 67? 673 6 7/1  675 676 67 7 6 76 67 ' 6 0 681 68? 683. . 6 8 /I 68 5 68 6 6 fl 7 688 6 89._. 6 0 0 69 1 69? 693 6 9 /I 6 95 6 9 fc 697 69 8 c  691  700  173 25 174 "1 .4.1 L75 176 mm 3 8 2 5 177 -' 3 6 25 178 25 1 70 25 180 25 181.. 25 18? 25 20 1 1 i 0 0 0 0 0 oooooo 0 0 0 0 0 0 0 0 00.0 0 0 0 0... 157 1 . 160 P" 0 6 9 1.6 1. W .9 3 0? .. 1 6 ? mm 0 3 0 ? 16^ - 1 0 7 5 1 6 6 mm 0 5 6.9 167 0569 1 6 9 mm 1 025 166 170 171 1075 17? 1075 . 174 mm 2 6 2 5 175 2625 176 - t 2 0 25 1 77 20? 5 \ 25 178 4 1 179 180 « 1 18 1 3825 18? 3825 183 M 25 184 25 25 185 186 •>* ? S 167 25  .  -  -.  20?  1 .  0 0.OP.0 0 o_-...o„o 1 1. ? 1. oo oooooo 0 3 1 . 000 0000  00 0  4  s 1 . o o o o o o 0o o0 o0o0 0 0 6 1. 7 1 . ft 1 .  0  00000000  1  8  1.4  128  000 0 0 0 0 0 0 0 0 0 0 Q 0 00 0 0 0 1 9 .7 01 702 9 1 . 7 03 1" J . 7o . 1 1 1 . 7 0? 1? 1 . 7 06 00 0 00 00 o 0 0 00 0 0 00 0 0 0 0 20 707.. 11 !.. _ 708 •14 1. 709 15 1 . 7 10 16 J . 711 17 r . 71?. 0 0 0 oo ooo oono 0 0 0 0 0 0 0 0 o o o 717. 18...1 .. 71" 19 1. 20 1 . 715 . 21 \ . 7 1 6 .. 717 22 J . 718 23 1. .D0 0 0oo 0 0 0 0 i o o 0 0 0 0 0 0 0 0 22 7.i' 720 24 1 . 25 1 , 721 26 1 . 7 22 CI J . 723 28 1 . 72" 7.2 5. . 2? 1 . 726 00 0 0 0 0 0o 000o0ooo0000000 30 1 . 727 728 . . . 0 0 0 0 o 000 o o 00 00 0 0000 0 2 4 7 29 31 1 . 730 00 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 25 731 32. 1.. _ 732 33 1. -> f T 34 1 . J .> 71 34 0 0000 000 (10 0 0 0 0 0 0 0 o o o o o o 735 158 l . 7 36 oooooo oooooon 0 0 0 0 0 u 0 0 0 2 7.37 __ . 1.59._l.......... 0 0 0 (.) 0 0 00 0 0 n o 0 0 0 0 0 0 0 0 28 738 1ft0 1 . 7 3 00 000 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 2 9 7 '1 0 7 41 161 1. 00 0 00 0 0 0 o0 0 0 0 0 0 0 0 0 0 0 3 0 74? ...16 2.. 1 . 74 3 744 (i OOOOO 0 0 n 0 n o 0 0 0 0 0 0 0 0 0 . 31 152 1 . 7 45 0000 000 00 0 0 0 0 0 0 0 0 0 0 .3 2 7 46 1 -1 . 7 47 4 -.99/5 748 7 4 9....... ... . 6 . r . 9 9 / 5 9 -.997 5 75 0 J  q  751 7 5.? 753 7 5'i 7 5 5.. 75 6 757 758 . 75? 76 0 .7.6 i ... 76? 767, 7 61\ 765 766 767  13 - . 9 9 7 5 18 - . 9 9 75 24 -.9075 35 3 6 . •*• 9 •'- •1.0025 37 38 1.0 0 25 30 1 , 0 02 5 1 . 0 0 25 " 1 1 . 0 1; ? 5 4 2 _ .l....«.\0.25_. . 0 3 1.0o?s 1.0 025 a 5 1.0 025 . n 6 1.0 025 '17 UP.  1 . 0 025 1 . 00?5  766  no  1.('0 25  7 6'-) 77 0  50  J.0025  771 7 7?  773_ 77'i 775 7 76 777 778 77°.. 7H0 78 1 78? 78 3 7 8 '1. 785. 7 86* 787 78 8 78" 700 79.1.. 79? 79 3  51 5?  1.0 025 1.0 025 53 1.0 0 2 5 5.4... J . . 0.0 2 5. 55 1 . 0 0 2 5 56 J.0 025 57 58 59  *  6.0 .J . 61 153 -1 . 158 • 5. 159 . 1«  16 0 .18 16JL , 2 3 5 16?  ,2 3 5  163 - . 5 1 6 4 . .-'.5 165 -.5 166 5 167 ? . .•. . •5 000 0 0 0 0000  79/1 7C?5  2 3 5  -1 . - J .06  7"6  7  - ) . 072'J  -1.0725  700  P. - . 0 7 '19 1 0 -1.0733 1 1 -.0758  800  12  79 7 79 8  - . 07 5 8  14 15 16 17 19 20 21 22 23 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41  801 802 803 8 0 ••'! 805 806 807 808 8p9 810 811 812 813 814 815 816 817 818 819 82 0 821 822 823 82« 825 826 827 828 829 83 0 831 8 32 8 33 83<! 835 8 36 837 838 839 84 0 841 842 84 3 84" 8«5 84 6  a?.  43 a a  45 «6 47  48 49 50 51 52 53 5-'J  8-17  848 81\ 9 850  '  55 56 57 58 59 60 61 62 63 6a 65  -1.0742 -'.0767 -'.0767 -,0767 -1.0751 - ' . 077.6  * . 0 776 -'.0776 -.0776 -1 . 0 8 1 5 -'.084 1 »v0 8U 1 ~.08ai -.08/4 1  -'.0938 -.105 -1 . 10<l -1.10 a . ._ -1 1oa -1. -1.06 -1 . 0 5 9 5 -1.0632 -.0657 -1 . 0 6 5 9 <». 0 68 4 - .0684 -1 . 0 6 8 5 -.071 -'.071 -.071 -1 . 0 7 3 3 -".0758 -.0758 - . 0 758 - 0758 -1 . 0 8 7 5 -.09 -'.09 -.09 -.090 -,104 -.122 -1.117 - ' l 17 -.117 .  _  _  a  r  r.  i. 1.0025 1 . 0 02.5  _  131  851 852 853  66  85-4  69  855 856 857  70  858 859 66 0 861 862  73 74 75  67 68  71 72  76 77 78  R63  79 80 81 82 83  66 4 865 866  ....'J..iJ..u  867 866  1.0025 1.0025 1.0025 CO025 1.0025 1 . 0 0 25 1.0025 T.0025 1.0025 U0Q25 T . 0 025 i'.0025 1,OO?* T.0025 1 . 0 025 1 . 0 0.25  84  t;  85  r.  871  86  if  — i——  ..Q../....I-.—  —  A-.V-.i-.  153 154  873 87-'* 875 876 877 8 73  .J.J..X....V.  879 860 8 81  -  -  - -  -  -  - .-  „5"*56  159 160 16JL 162 163  .3567 ,3596 ,391 0 ,3979 ,039 ™',?79 -.2775 -.242 -„24 . - . 5  -  - -  -- -  887  170  888  -'.5 171 172 - . 5 o oooo nooo o 0 o o o o o .Oil.Q O.a.0.0. Q ^ . 3 « a «1,0749 -1.0733 H I ? - ' 0756 15 - 1 . 0 7 4 2  889 890 J*l....£,Ju 891 892 tt f 394 895 7  16 L7 20 21  -'.0767 -*0I67 -1.0751 -.0776  399  ??  -',077*  90 0  23  -'.0776  . . i8 J - .9 . t . _6 i»  8'97  898  ™.  --  - — -  -  835 886  88 4  ..  -1.  165 166 167 168 169  882 883  . .  —  1 a 082 1.082  158  164  • iJ. ! - -A  .  1. 1.  870  —W l . / ,  _ _  -  ...  -.-  .  -  _  -  901 , J9J0L2 903 904 90S 906 907 908 909 910 9 U  _  26 - 1 . 0 8 1 5 27 - '„ 0 8 4 28 - \ 0 8 4 29 - . 0 8 4 3 0 -'.09-58 31 - . 1 0 5 3 3 -1 . 1 0 4 34_. - . 1 0 4 39 -1 . 0 6 5 7 41 -1 . 0 6 5 9 up  44 45 - .. . J U i 48 49 5n 918 51 919 53 54 920 55 921 922 56 923 _5JL 58 924 60 925 • 61 92.6 62 927 928 63 929 64 65 93 0 66 931 67 932 68 933 69 934 935 70 71 936 72 937 73 938 74 939 75 94 0 76 94 i 9a2 77 76 94 3 94 4 79 80 94 5 81 94 6 94? 82 83 9 48 94 9 84 85 95 0  912 913 .94-4 915 916  -;0AR4  -1.0687 -.071 -.071 -1 . 0 7 3 3 -,0758 -.0758  -.0758 -1.0875 -.09 -.09 -.09 - 1 04 -.122 -1,117 -1117 - 1 a -1.06 -1 . 0 4 5 -1,0465 -.049 -1.05 -.0525 -'.0525 -1 . 0 5 3 -.0555 -'.0555 -.0555 -1 . 0 5 9 -.0615 -.0615 -.0615 -1 . 0 7 7 5 - o8 -.08 -.08 -.0943 . - . 1 108 -1.1075 -.1075  „  t  9  _  - -  _._  _  951 952 95 3 954 9'55 956 957 958 9_5_9 96 0 961 9_62 963 964 965 966 967 968 969 970 971 972 973 _7__ 975 976 97 7 978 979 <l_LQ_ 981 982 983 984 985 9.8.6 987 988 —9A9 990 991 9.92... 993 994 995 996 997 99.8 999 1000  86 - . 1 0 7 5 87 1. 88 1 . 89 1 . 0 0 2 5 90 1 , 0 0 2 5 91 1 . 0 0 2 5 92 1.0025 93 T . 0 0 2 5 94 l ' . 0 0 ? 5 95 1 . 0 025 96 1 , 0 0 2 5 97 1 . 0 0 2 5 98 1.0025 99 1.0025 100 l f 0 Q 2 5 101 1.0025 102 1.0025 103..J/.AQ.25 1 0 4 1'. 105 f. 1 0 6 1". 1 0 7 l'p 108 1. 154 . 1 . 0 6 5 155 -1. 159 . 3 5 9 4 160 . 2 3 0 1  ,  161 . 2 0 7 2 162 , 2 1 0 9 163 . 5 39 _ 164 .3624 165 . 3 6 4 9 L66 ,39.3 : 167 ,3979 168 , 0 3 6 16.9 -.,..2.8.2 _ _ _ 170 - ' . 2 8 171 -.2445 172 - . 2 4 . 173 - . 5 174 - . 5 1JJ5_-J,5 _ 176 - . 5 177 - . 5 aooonoQn noonnnnnonnnnnn 35 12 - 1 . 0 7 5 8 16 -1.0742 ..._L7......-;...0.16! _ 21 -1.0751 22 - . 0 7 6  OiJjQJ  23  - . 0 7 6  $002 1003  27 28  - X.0815 -.084  lOOq 1005 J006 1007 1008 $009 iOl.0 1011 1012 1013 1014 1015 1.01.6  29 »'.Q84 30 - 1 . 0 9 3 8 31 -1 . 1 0 5 34 - 1 . 1 0 4 42 -1,0684 45 - 1 . 0 6 8 5 M . - .071 49 -1.0733 50 -.0758 51 -.0758 54 -1.0875 55 - . 0 9 56__~lj)9.  1017 1018  57 58  -1.104 -1.122  JJLLB.  61  .1,117  1020 1021 1022. 1023 1024 1025 1026 1027 l02 8 1029 1030 1031 1032 1033 1034_ 1035 1036 LQ3_7 1038 1039 1040 104 1 1042 1043 1044 1045 1Q4 6 10 4 7 10 4 8 1 Q49 1050  66 68 69 71 72 73 75 76 77 79 80 81 82 83 __85 86 87 68 89 90 91 92 93 94 95 96 97 98 99 100 101  _  -1,049 -1.05 -.0525 -1.053 -.0555 -'.0555 -1.059 -.0615 - ' . 0 6_L5 _ - 1 . 0 775 -.08 -.06 -.0943 -'.1108 -1 . 1 0 7 5 -.1075 -1. -1.06 -1.051 -1.0525 -.055 -1.0565 -.059 -',059 -1.060 1 -'.0626 -',0 626 - 1 ,0ft4 9 - . 067 4 -.0674 -1.0786  105} 1052 1053 1054 1055 1056 1057 1058_ 1059 1060 1061 1062 1063  102 103 104 105 106 107 108 LQ.9 110 111 112 113 114  -.0813 „-.08.13 f'.6<?2l -.1083 -1,105 »'.105 -.105 1_ r. 1.0025 1.0025 1.0025 T.0025  i._6._  1JL5__L_J_0J_  1065 1066 1067 1068 |069 107 0 1071 1072  116 1 . 0 0 2 5 117 1.0025 118 r . 0 0 25 119 1 . 0 0 25 120 T . 0 0 2 5 _21_li0_025 122 1 . 123 1 .  I<m  1 2 4  1074 1075 1076 1077 1078 iOZS 1080 1081 1082 1083 1034 1085 1086 1087 L088 1089 1090 1091 1092 1093 109 4 1095 1096 1097  i  .  125 1 . 155 1 . 0 6 L5„..._!l__ 159 . 2 2 0 2 160 . 1 4 7 3 1_LL___U  162 . 1 1 2 8 164 . 3 6 3 165 . 2 3 3 5 1 6 6 ,20~76 167 . 2 1 2 9 166 .536 169 . 3 5 7 5 170 . 3 6 0 8 lll__.».39.1.fe 172 , 4 0 1 7 173 , 0 3 4 174 - ' . 2 3 5 5 175 - . 2 8 176 - . 2 4 4 1X7. - . . 2 . 3 7 5 178 - ' , 5 179 - . 5 ,  180 -,'.5  1098 1099  181 - . 5 182 - . 5  11.0 i l _  0.0. C O 0.0  0..0.i)..O..0.Q.Q..Q.a.Q...aO..Q..0..O  3.6  1101 U02 -il-0.3 1104 1105 1106 1107 1108 •1109 1110 1111 1 1 1 2. 1113 1114 ']) |5 1116 1117 1118 1119 1120 JUUJ 1122 1 123 I . 1.2.4. 1125 1126  JJJU 1128 1129 .113.0 1131 H32 1 133 1134 1135 II.36 1137 1138 1139 1140 1141 114 2 1143 1144 1145  17 -1.0767 22 -1 , 0 7 5 1 23 -'.0776 26 - 1 , 0 8 1 5 29 - . 0 8 4 _JL6_-i^_QXl 50 - 1 . 0 7 3 3 51 -.0758 5-5, -1 . 0 8 7 5 56 - . 0 9 69 -1.0525 _ - ...7.2.. . . . - L . .05.3 73 - ' . 0 5 2 5 76 - 1 . 0 5 9 77 - , 0 6 1 5 80 "1.0775 81 -'.08 £.2 ...-1 . 0 9 4 3 83 - 1 . 1 1 0 8 86 - 1 , 1 0 7 5 SJ, -1 . 0 5 5 . 93 - 1 , 0 5 6 5 94 - ' . 0 5 9 96...-l^Q601. 97 - . 0 6 2 6 99 - 1 . 0 6 4 9 100 - , 0 6 I 4 _ 102 - 1 . 0 7 8 8 103 - . 0 8 1 3 \<ia - ' . 0 9 2 1 105 - ' . 1 0 8 3 1 0 7 -1 . 1 0 5 108 - ' . 1 0 5 109 - 1 . 110 -1,07 L U _ - l i O M 112 -1.0629 113 -.0654 \\g - 1 . Q 6 5 5 115 - . 0 6 8 116 -1.0673 117 -'.0696 118 -1.0692 119 - ' . 0 7 1 7 120 - 1 . 0 8 I t  1146  121  - . 0 6 3 6  1147  122  - . 0 9 5 9  1.1.4.8. 1149 1150  1.2.I.--..L1.2.3„ 124 -1.1075 125 - . 1 0 7 5  .1151  1?*> 1 .  1152 1153 1154 1155 1156 jig? 1158 1159 1160 1161 1162  127 1. 128 1.0025 JJ5JLJUM25. 130 1.0025 131 1.0025 1 3 ? l'.Q025 133 1.0025 13/J 1. _ 1 3 5 1... 136 f. 156 1.076  I1.6JS 1164 1165 1166 1167 1168 116 9 1170 1171  ,  1172  1173 1174 .1115 1176 1177 1178 1179 1180 ;UL8J 1182 1183 1184 1185 1186 1187 1188 1169 119 0 1192 1193 1194 1 195 1196 1197 1198 1 199 ~1200  157 160 161 16.2 164 165 U L 6  167 169  ..-  .  .  'It ,0943 ,0601 .06 ,2216 ,1494 .113 ,1134 ,3599  11SL-»23<IS.  _  _  171 .2133 172 .2139 il3__*5!4 174 ,3518 175 ,3608 _A1(\ .AQ.24 _ 177 ,4036 1 7 8 ,0 3 9 lJL2_=i2ig 180 -.2775 181 ='.237 _ 182 - . 2 3 6 2 _ 183 - . 5 184 - . 5 185 - p 5 186 - . 5 187 - . 5 0000_00 0 000000000000000 3 5 7 8 10_ 11 12 14 15  - 1 , -'.995 -.96 -'.96 - .96 _ -.96 -.96 -.9 6 -.96  _  37  1201 1202 1203 120'! 1205 1206 1207 1208 1209 1210 i 2 U 1212 4213 1214. 1215 1216  16 17 35 36 37 38 39 40  .  «t 42 43 44 45 46 137 152  153 „1  12JJL 1218 i219 122.0 1221 1222 .1.2.23 122'! 1225 122.6 1227 1220 I.22S 1230 1231 .1.2.3.2 1233 1234 12Z5 1236 1237 .12.3.8. 1239 1240  158 .18 159 .2952 160 ,29.5.2... 161 .3596 162 .3596. -1.63 .18 164 .18 165 ,18 1.66 . 2 3 5 167 „?.35 20 3 1 . ooooo ooonnnnnnnooono 8 -.995 11 - . 9 6 .12 - . 9 6 _ 15 - 9 9 6 16 - . 9 6 L7._r.jL6 39 - . 9 9 5 ^* t - ' . 9 6 4? » . 9 6 44 - , 9 6 A5 - ' . 9 6 46 - f 96 62 -1 6 3 - 1 . _ .64 .-'_99.5... 65 - ' . 9 6 66 * , 9 6  J21U  1242 1243 12«4.... 1245 1246  ±2Jil  124 8 1249 1250  - f 9 6 -.96 -1 . -is -.995 -.96 -.96 -.96 -',96 -.96 -'.96. -.96 -'.96 ..-..96...... _ -15,385 1'.  6_L___9_ft_  _  68 69 70  - . 96 - . 9 6 -.96  38  - . 9 6  1256  71 72 73 1.38 154 159  1257 1258 1 259  160 161 1 62  .1889 ,1906 .1906  126 0 1261 1262 1263  163 164  ,18 ,29 ,29 .35 .35 .1.8  4251 1252 I?5_ _25« "1255  126« 1265 1266 1267 i'268 1269 1270  1271 1272 1273  165 166 167 168 169  -*.96 -'.96 -15  3  385  r.  . 1889  _.. ...  .19  170 _ .. .... . L7l_ , 2 3 5 .2 35 172 2 04 i " . ooooon 00no00000 000000 12 - . 9 9 5 16 - . 9 6 - , 9 6 - . 9 9 5  66 68 69 71  -  1282 1283  72 73  - . 9 6 - . 9 6  1284 1285 1286  87 88 89  -1  1287 1288 1289  90 91  129 0 1291  93 94  - . 9 6  1292  95  -.96... _ «.9fe  1277 1278 1279 128 0 1281  9 2  129 3 129 4 1295  96 97 139  1296 1297 129 8 1299  155 159  1300  ...  52 52 96 96  17 42 45 4 6  127 « 1275 127 6  .  .  39  ..... - -  -.96 - . 9 6 . . . .  9 9 9 9  9 5 6 6 6  ,  .  -1. - . 9 9 5 - . 9 6 - . 9 6 - . 9 6 - . 9 6  - . 9 6 -15.385  r.  160  .0737 , 1209.........  161 162  .101 .101  _  _  13J2J 1302 1303 J 3 OA. 1305 1306 1307 1308 1309 1310 1311 1312 1313 1314 1315 JJJJ6 1317 1318 1319 1320 1321 •322 1323 1324  iM_JLlft«?_  165 , 1 8 8 9 166 , 1 9 0 6 167 .1906 168 ,18 169 ,2952 170 .2952 171 ,3596 172 ,3596 17_3_al8 174 .18 175 ,18 176 .235 177 ,235 205 1. 0000 0 OOOQOOOO0 0 0 0 0 0 0 17 - . 9 9 5 " 46 - . 9 9 5 69 - . 9 9 5 72 - . 9 6 73 - ' . 9 6 ?1_JV?95_ _ _ 93 - . 9 6 94 - . 9 6 96 - . 9 6 1326 97 - . 9 6 1327 109 -1. JLI2.8 __J1Q °il_*_ 1329 111 - . 9 9 5 1330 112 - ' . 9 6 A311 U 3 " . 9 6 1332 114 - . 9 6 1333 115 - . 9 6 J.3.3.4 I.i.6....-.;.9.6 _ _ 1335 117 - . 9 6 1336 140 -15.385 1337 156 1. 1338 160 ,0774 1339 161 ,0535 1 3 4 Q. _ ...16.2 , 0 5 3 5 1341 164 ,0737 134 2 165 ,1209 .43iG X6_6_*4.QX-_ 1344 167 ,101 1345 169 . 1 8 8 9 1346 1.70 . 1 . 8 8.9. 13-47 171 ,1906 J348 172 ,1906 -LZMJi .173 . 1 8 135 0 174 .2952  2  0000 0000 00 0 00C 0  17  oooooo * t / ft? i B l  8b£T  98T  _6£T  £81  96£t  9 1 ' •J79T GT" £ 9 1 281 96_£*  S6ET  TftT db'fo^'*"" O B T b_T _S62* 9t* 9_I  26£T I6EI  b£2  4  -  e i ' -  -  "  "  96S£*  III 9061* 6991* 6891 * TOT0 TOT* '  "  " • "  '  '  u6t?T" 66£T  9 LI SLX uL'i ZL\ \L\  602P~ Oil _£Z0*  69T  t7__0' T/820  S9T 291 T9T  S/6ET £6£T  0 6ET '69ET B8ET _8£l 98£T S8£T i/BET £6£T _8£T~  test "6_rr  naso * _6l;0 * T »  96* 96" 96 Sb6" _  M -  LH  I" T£T 0£T 621 821 9 2 r  9 6 ' -  SU  0 0 0 0 0 0 0 0 0 0 0 00 0 0 •I  —  1ST  _I1  STbT*-  9_£'T SZ.ET"  t 17  9fa°-  9b"» S 6 6 * Tr;  09T  BiET U E T  Lb 1^6  U . 00000 902 2 g T  cAET U E T l i l t 69ET 8 9 ET T w r 99ET S9ET ~ TP)ET _9£T 29ET T T E T 09 £ f 6SE"f  "SSET  S£2*  181  •_SET  9T* 81*  091  BT* 9fcS£* " - 9 V _ £ ' 2S62'  9_t LL\ 911 __T  9SET bbE" i>S£t  611  c_f.T  tsst  1401  2  -1  1402  3  ]'403  5  140 4 1405  7 8  -It - . 9 9 5 - . 96  1406  10  -.;.i6  11  - , 9 b  1409  12 14  - . 9 6 -'.9ft  14]0  15  - . 9 6  1411  16  - . 9 6  JUL 19  -'.96 - . 9  14 J 5  20 21  - . 9 -19  1416  22  " . 9  1417  23  14J8 1419  35  1420 1421  37 38 39  1407 1408  1412  .  1413 1414  -'>6  .-1.0  - . 9 6 - . 9 6  1425  42  ..-.^.96 - . 9 6  1426  43 44  - . 9 6  1427 i"428  45  - . 9 6  1429  46  1430  47  - . 9 6 -'.9  1431 1432  48 49  - . 9 - . 9  ..  1433  50  -'.9  51  1435  142  - . 9 - 2 5 .  1436  15?  1 .  14  153 158 159  1.  144 0 1 « 4 1  160  ,2952  161  ,3596  1442  162  ,,3596  1443 144 4  163 164  ,19  1445  165  .18 .18  166  .235 ,235  1 4 4 8_  167 206  1 4 4 9  0 00 0  1438 1439  i  4 4 6  144  7  14 5 0  8  _  _  -'.96  1434  37  _  -1. - , 9 9 5 - . 9 6  40  1423 14 2 4  , ,  -.<?  36  1422  .  .  _  ,18 ,?952  1'  __  . „ 000000000000000  - . 9 9 5  _ _ 4 3  1452  12  -  1455 1456 1457  17 20 21  - f 9 6 - . 9 - . 9  1453 1A5JI  1453 1459 ...JL4M  p  9 6  15 -,96 16 -^9.6—  _  22 - . 9 23 - . 9 3.9...-.,.995...,  1461  41  - . 9 6  1462  42  -',96  J J L A 6 3  44  -  B  9 6  1464  45  1465 1466  46 - ' . 9 6 4.0 . . - . . 9 .  1467 1468  49 50  -iAh9. 1470 1471 1472  - . 9 6  - . 9 - . 9  5JL_-J^9_ 62 - 1 . 63 64  - 1 , «.,.9<i5..  1476  65 66 67 68  - . 9 6 -'.96 -JL6__ - . 9 6  1477  69  - . 9 6  1473 1474  i(LZfi  7JL_i»9_6 _  1479  71  -'.96  1480  72  -'.96  J.JL8J  .73  - . 9 6  1482  7 4  1483  75  - . 9  1.4 8.4  7.6.  - . 9  - , 9  1485  77  1486  143  - 2 5 .  - . 9  1487  154  l  I486  159  .1889 ,1889  r  |>89  160  1.4.9.0.  lM_....a.li06  1491  162  1492  163  .18  1A9.3  1 M  ,?9Jj2,  ,1906  ]'494  165  ,2952  1495  166  .3596  .1.496 149 7 1498  167 168 169  1500  171  .3596 , j 8 ,18 1 a ,235  170  T  1501  172  15.Q2....  2D 9  1503 1504 15Q5 1506 1507  0000 12 16 17 21  . 2 3 5  1. 0  0000 00 0 0 0 0 0 0000 -'.995 -„9fe - . 9 6 - . 9  22_J=.»9  I5JLA 1509 1510  23 42  - . 9 -'.995 -196  t5 M  45  1512  46 - . 9 6  1513 15.1.4  49  5.0. - . . . 9  1515 1516 1517 1518 1519 1520.  51 66 68 69 71 72  _  1521 1522 _I5_2J 1524 1525 ...15.2.6 1527 1528  73 75 76 77 _  - . 9 - ' . 9 - . 9 9 5 -'^9.6 - . 96 -'.96 - . 9 6 -  . 9 6 . 9 ^ g . 9  87 - 1 , 8 8 -l..« 89 - . 9 9 5 90 - . 9 6  1529  91  1530  92 - . 9 6  1531 .1.5.12... 1533 1534 1535 1536 1537 1538.. 1539  44  _  - . 9 6  93 - . 9 6 9JL„-»?6.. 95 - . 9 6 96 - . 9 6 97 - . 98 - . 99 - . 10.0...... 144 - 2  1540 J5-'U 1542 1543 1544 1545 J546 J547  155 159 160 161 162 164 165 166  1548 1549 1550  167 168 169  9 6 9 9 9 5 ,  T. Q 737 , 1 2 0 9 ,101 , i 0 t . 1 8 8 9 , 1 8 8 9 . 19Q6  .  _  __  f  . 1 9 0 6 ,18 ,2952  :  _  1551  170  ,2952  ^552 1553  171 172  ,3596 .3596  1554 1555 1556  173 174 175  ,1A ,13 .18  1557 1558 1559 1560  176 177 210 0000  1561  17  1562 1563 1564 1565 1566  22__._ 23 - . 9 46 - . 9 9 5 50 - ' . 9 51 - , 9  1567  69  15_68_ 1569 1570 1571  .7__J=_16_ 73 - . 9 6 76 - . 9 77 - ' . 9  1572 1573 157.4 1575  _ _ .  1576 1.577  91 93 9 4 9b  , 2 3 5 , 2 3 5 1'. 0000 -'.995  1-.995  - . 9 9 5 - . 9 6 _-'e96 - . 9 6  97 - . 9 6 99 - . 9  1578 1579  100 109 110  - . 9 - 1 ,  1581  111  - . 9 9 5  1582  112  - . 9 6  X5AZ 1584  U 3 _ _ J l i i _ 114 - . 9 6  1585 .15.8.6 1587 1588 X£LS_  1 15 - ' . 9 6 116. - ..9.6 . 117 - . 9 6 118 - . 9 LL9 - ' . 9  159 0  145 - 2 5 .  1591  156  .153.2 1593 1594 15-9.5.  .1..&...Q 161 162 164  .0.774 .0535 .0535 , 0 737  1596  165  , 1 2 0 9  1597 . 1 5 . 9 5 ... 1599  166 .16.7 169  ,101 . . M L , 1 6 8 9  170  , 1 8 8 9  1600  f.  000000000000000  45  1601  171  ,190ft  1602  172  ,1906  1603  173  ,18  lh.M 1605  1.607  l l f l _ , 2 3 5 2 _ _ 175 ,2952 176 ,3596 177 .3596  1.608 1609  178 179  IMS 1611  I8jj_aa.  1612  182  £606  191  . 2 3 5 , 2 3 5  211  1614  00000  1. 000000000000000  1615  23  14.16.  5 1 „ -  1617  73  1618  77  - . 9  16j 9  94  -',995  1620  97  i . 9 6  1621  100  - . 9  16.2.2  113  -*.995  1623  115  -'.96  1624  117  - . 9 6  lh25 1626  119  - . 9  126  - 1 .  \627  127  - 1 ,  .1.628 1629  12JL_TU9£5129 - ' . 9 6  - . 9 A  9  -'.995  1630  130  - . 9 6  1631  131  -'.96  1632  132  - . 9  1633  146  - 2 5 ,  1634.  157  )..,.._  1635  160  1636  161  ,0495 .0284 .0264  1638  165  ,0774  1639  166  ,0535  164 0  ,16?  1641  169  ,0737  170  ,1209  1642 _ 1 M 1 1644  _. .  , 18 ,18  1613  JLhZl^^AiiZ  „  ,0535  _ L I l _ _ a l i 172 ,10 1  1645  174  ,1889  16.4 6  1.7.5  ,1.88.9  1647  176  .1906  1648  177  .1906  16 4 9  176  .16  1650  179  ,2952  _  46  1651  180 181  :I.65l  ,2952 _u3_L?__ ,3596  1653  1.82  1654 1655  183 184  as .18  1656  185  1657 1658  186  tie  165"  212  187  _  1661 1662  3  1663  5  -1. - . 9 9 5  ...1664  7  -',96  1665 1666  8 10  1667  11  -'.96 - . 9 6  1668  12  - . 9 6  1669  14  - . 9 6  1670  15  -'.96  1671  16  - . 9 6  1672  17 19  -'.96 - . 9  1674  20  1675  21  - . 9 - . 9  1676  22  V ?  1677  23 25  -'.85  26  -185  1680  27  -",85  1681  - . 8 5  1682  28 29  1683  30  168 4  31  - . 8 2 5 - . 8  1685  35  -1  1686  36  -1* - . 9 9 5  1613  1678 1679  _  .  37  1688. 1689  38 39~  1690 1691  ^0 41  1692  42  - . 9 6  i'693 1694  43 4 4  - , 9 6  1695 1696  45 4 6  - . 9 6  1697  4_7  - . 9 6 - T9  1698  48  -  49 .  _.  _  _  _  _  1687  1699  _  .235 __j_35_  1. 00000 000000000000000 2 -1 .  1660  1700  _  -'.96 - . 9 6 - . 9 6 -'.96  - . 9 6  q 0 '  - . 9 5.0... » _ 9  47  _  1701  51  - . 9  1702  52 53  -.85 -'.as  i i _ L 1704  54  - . 8 5  1705 1706  55 56,  -'.85 -li_5  1707 1708 17 0 9  58 14 7  1710  152  1711 •1712 1713 1714 17-15  153 158 159 160 161  r. „,____ .2952 .2952 ,3596  1716  162  ,3596  1717  163  ,18  1713 1719 1720 172 3 1722  164. 165 166 167 213  a.9_ ,18 ,235 .235 1.  1723 1 7 2 « _  57  - . 8 2 5  -.8 - 1 0 . 5 2 6 1*.  000000  -  -  _  000000000000000  48  8_-_995  1725 1726 1127 1728 1729 1730 1731 1732  11 - . 9 6 12 -'.96 15 - . 9 6 16 - . 9 6 17 - . 9 6 20._-,9 21 - . 9 22 -'.9  ±112 1734  23 26  - . 9 -'.85  1735  27  - ' . 8 5  1116. 1737 1738  28 29 30  -'.85 - . 8 5 -'.825  _J.__ 1740 11741  31 39 41  - . 8 - . 9 9 5 - . 9 6  .1142 1743 1744  _L2_____?_ 44 - . 9 6 45 - . 9 6  1745 1746  46 48  - . 9 6 - . 9  -.9  1747  «9  i'74.8. 1749  50...-.9 51 - . 9  1750  53  - . 8 5  :  .  _  1751  54  -'.85  1752  55  - . 8 5  1753  56 57  -'.85 ™, 825  58  - . 8  62  -1.  63 64  • h •*.995 - . 9 6  1734 1755 1756 1757 1758 1759 176Q  65 66 . - . 9 6  1761 1762 1763  67 68 69  - . 9 6 - . 9 6 - . 9 6  1764  70 71 72  - . 9 6  1767 1768 1769  73 74  -'.96  1770  76  1771 1772  77 78 79  1765 116 6  _  75  1773 1774  80 81  1775 1776 1777 1778 1779 178  0  .1181 1782 1783 1184 1785 1786 1787 1788 1789  _..  - . 8 5 " . 8 5 - . 8 5 -185  1. .1889  - . 8 "10  .526  . 1. 8 8 9 «  1^06  162 .1906 1 6 . 3 ._ . i _ a _ _ 164 ,2952 165 166  ,2952 .3596  167 168  ,3596  169  1792  171 172 214  1795 1796  - . 9 « , 9  154  179 0 1791 l7_93 1794  - . 9 -'.9  -'.825  LM_ 161  170  _ . _  ,18 ...a* ,13  _  .235 .235 1.  0 0 0 00 12  ,  - . 9 6 -'.96  32 83 148 159 _  ,  0000  - . 9 9 5 - . 9 6  1797 1798 1799  16 17 21  - . 9 6 -'.9  1800  22  - . 9  00000000000  4 9  1801. I80.?_ 1803 1804 1805. 1806 i"807 j 808. 180"? 1810 18J,11812 1813 1.8.11. 1815 1816 1818 1819 18.2.0... 1821 1822  - , 9 - . 8 5  27 CO  29 30 31  - . 8 5 - . 8 5 - . 8 2 5  42 45 46 49 50 51 54 5 5 „ DO 57 58 66 68  - . 8 ,995 - . 9 6 >96  - . 9 - . 8 5 - * 8 5 . - . 8 5 - . 8 25 - . 9 9 5 - . 9 6  .6.9 . . . . . . 9 . 6 71 72  - . 9 6 -,,96  l a ' 1 5 1824  73 75  - . 9 6 - . 9  1825  76  ...1826.. 1827 1828 1.8.29 1830 1831 1832 1833 1834  ...7.7 79  - . 9 . - . 8 5  80  - . 8 5  l M 5 _ 1836  ..fll  82 8 3  91 92  1838. 1839  .93. 94  184 0  95  l84_7_ 1848 18 4 9 18.5.0...  - . 8 5 - . 8 2 5 - . 8  .8..7.....-L. 88 - 1 . 89 -',,995 _9JL - r 9 6  1837  1342 1843 ...1.81.4... 184 5 1846  - . 9  - . 9 6 ,96 - . 9 6  - p 9 6 - . 9 6 _9A. - ' . 9 6 9 7 - . 9 6 98 99 - . 9  100 101 102  - . 9 - . 8 5 -'.85  103  - . 8 5  .104  - . 8 2 5 m B  105  1851  149  - 1 0 . 5 2 6  1852 1853  155 159  1. .0737  1854  160  ,1209  1855  161  ,101  ±$Sh  162 .10,1  1857 1858 .1659  164. 165 L66  I860  167  ,1906  1661  168  ,18  ,1889 ,1889 ,1906  lM2—-—±&5-*2252 1863 1864 1865  170 171 172  1866 1667  173 174  Iftfcfi  ,2952 ,3596 f 3 5 9 6 ,16 .16  iJ-5_*lJB  1869  176  ,235  1870  177  .235  1671 1872 1873 .18.74  2 U S _ X 00000 000000000000000 17 - . 9 9 5 22 - . 9 ,  1875 1876 187? 1878 1879 1880  23  - . 9  28 - ' . 8 5 _ _ 2 9 _ - . 8 5 46 50 gj  ^ . 9 9 5 - . 9 ^ 9  1881 1882 JL863  55 56 69  -',85 - . 8 5 -'.995  1884  72  -  1885  73  -'.96  1886  76  - . 9  1887  77  - . 9  1888  80  - . 8 5  81  - . 6 5  I M 2  _  9 6  e  1890  8?  -'.825  1891  83  -'.8  189 2  ?1  - , 9 9 5  1893  93  - . 9 6  1894  94  - . 9 6  1895  96  - . 9 6  1896  97  - . 9 6  i'89?  99  - . 9  1898  100  - . 9  1399  102  -  1900  103  - . 8 5  ,  8  5  "  ~  50  1902 1903 1904 19 05 1906  105 109 110 i l l 112  - . 8 - 1 . -} , - . 9 9 5 -'.96  19Q7 1908 19p9 „910  113 - . 9 114 - . 9 115 - ' . 9 L 1 6 _ - l 9  1911 1912 1913 1914 1915  117 118 119 120 121  -  1916 1917 1918 19jq 1920 1921  122 123 150 156 160 161  - . 8 2 5 - . 8 -10.526 r . t 0 7 7 a ,0535  1922 1923 1924 1925 1926 1927  162 164 165 166 167 169  .0535 ,0737 ,1209 .101 ,101 , 1 8 8 9  6 6 6 6 _  . 9 6 . 9 ' . 9 . 8 5 . 8 5  .  _  _  _7___J__8_  I9_2J3 1929 1930  171 172  1931 1932 1933 193.4  L7_L_*J_8 174 ,2952 175 .2952 1X6„..,..159.6  1935 1936  177 178  1938 1939  180 181 18.2 216 0000 23  d3M^~XIl^d3 19/41 |94 2 4__3  :  1944 1945  , 1 9 0 6 , 1 9 0 6  ,3596 , 1.8 ,18 , 2 3 5 , 2 3 5 ; 1'. 00 0000 000 0000 0000 - ^ 9 :  29 - . 8 5 51 -.9  .1.9i!..6  5 . 6 -....8.5...._  1947 1948 I__L2 1950  73 - , 9 9 5 77 - . 9 ___-_U_S 94 - . 9 9 5  ,  :  _ 51  1951 1952  97  1953 1954 (955  103 104 1 05  1956  113 115  -',96 - , 9 - . 8 5 - . 8 2 5 "'.8 -'.995 -'.96  1957 [958 1959  117 119  I960 196 1  121 12?  f . 9 6 - . 9 -pj.85 - a 8 ? 5  1962 1963 19i>4_  123  - . 3  126  "Is -1 » - . 9 9 5 -'.96  i'965 1966 '(96?  . ...127... 128 129 130  1968 1969 197 0  131 132 133 134  1971 1972 J97"*,  135 151  -'.85 - . 8 2 5 -'.8 -10',5?6  157 160 161  1977 1978 1979  162 165 1 66  1980 1981  167 169  19 8 2 1983 1984  170 171  ,0737 JL2Q9 ,101  1985  172 174  .101 ,1889  1986  175  .1389  1987  176  ,1906  i'988 1989 199 0 1991 199 2  r. .0495 ...0284 .0284 .0774 ,'1535 ,0535  L7J_„„.1.9Qi>. 178 179 180 181  ....  ._ _ _  _  _  f ? 9 5 ? ,3596  1*995 199 6 1997 1998 1999  217  .2.00 0  ...  ,18 ,2952  ,3596 182 1 8 3 . . ..18..... 184 ,18 185 .18 186 ...235 ,235 187  1993 •1994  _  - , 96 -'.96  1974 1975 1976  .  .  1. 0 0.0.0 0.,  _ „  0 0 . 0 . 0 0 0 0 0 . 0 . 0 0 0 0 0.0  .  52  .....  -.  2001 2002 2003 2004 2005 2006 2007 2008 2009  2 3 5  7 8 10 11 12  it  2010  15  20  16  Jl  20^2 2013 2014 20^5 2016 2017  2018 2019 2020 2021 2022 2023 2024 2025 2026 2027 2028 2029 2030  1. 1. .995 ,96 .96 ,996 , . , ,  9 9 9 9  21  6 ,96  32 33 34 35 36  ,9 ,8 ,8 .8  20 34  41  ,96  4 2 43 44  ,96  45 46  ,96  47 4 8 49  .96 .96 ,96  2043 20 44  50  2045  52  ,96 .96 ,94  20 46  53 54 55  .94  56  .94  57  ,93  2047 2Q4 8 2049 2050  _  ...  _  _  ,96 .96  .96 ,96 ,96  51  ..  1. 1 . ,995 .96  2035 2036  2 04 1 20 4 2  .  2 8 8 8  37 38 39 40  204 0  _  ,93  2031 2032 2033  2037 2038 2039  _  .96 .96  ,96 22 ,96 23 25 ..*9_L .9<l 26 ,94 27 28 .94 29 .94 30 31  ,  6 6 6 6  17. . . 9 19 20  „  ,96  .94  • _  . .... , ....  .  _  59  P 9 2 ,88  2054  60 61  a.8 8  2055  137  58  2051 2052 2053  .  2056  142  ,88 - 1 , - 1 ,  . i « 7  2058 2059  152 - 1 , 153 - 1 , .158. -",5 159 - . 8 2  _  2061 2062 2063 2064 2065 2066 2067 2068 2 069  ..  2070 2071 20 72  2075 2076 .2.0.78 2079 2081 2082 2083 .2.0.84  -'.82  162  - . 7 6 5  163 164. 165  - . 5 - . 5 - . 5  166 167 218  - . 5 - . 5 - 1 , 0 0 0  12  .96  1* 16  .96  .96 22 __2J3_ ,96 .94 26 .94 27 2 8 .....9.4 29 .9fl , 93 30 31 .92 .88 33  2085 2086 2087 2088 2089 2.09 0.  160 161  ,96 17 2.Q.. . 9 6 .96 21  2077  2080  .  34 39  .88  2091  41  ,995 ,96  2092  4 2 44  .96 .96  45  .96  46 48  .96 .96 .96  2093 2094 2095 2096 2097 2098 2099 2100  ...  oooooo OOOOOO0 0 0 0 0 0 0 0 0 8 995 ,96 11  2073 2074  49 50  ...  -1.  2057  .20.6.0.  ...... ...  ,96 51_ . 9 6 ,94 53  _  53  _.  2101  54  ?_iQ2  55  .94 .94  2103  -56  ,9 4  2104  57  ,93  /li_5 106  58 60  ,92 ,88  2107  61  ,88  £108  &__J____  2109  63  1.  2110  64  ,995  2111  65  .96  2112  66  ,96  2113  67  .96  2114  68.__9.6_...  2115  69  ,96  2116  70  ,96  _7J__..9___  21A2 2118  72  ,96  2119 2120  73 7.4  ,96 ,96  2121 2122 2 123  75 76 77  ,96 ,96 „96  2124 2125 212.6..  78 ,94 79 .94 8.0..._..,94...  2127 2128  81 82  ,94 ,93  2129 2130  83 84  .92 ,88  85  ,88  2131 .2.1.3.2. 2133  _  8.6..._.,.M... 138 -1  2134  143  2123  2136  148 154  2137 213.8  159 - . 5 2 4 8 1.60.._.-_.5.2.4.fi.  2139  161  - . 4  2140  162  -'.4055  2J_LL_ 2142  _i_J____5_ 164 - . 8 2  2143 2144  165 166  -1 - i _  -  i  ,  055  - . 8 2 -'.765  2145  167  - . 7 6 5  2146  168  - . 5  ?JJU 21<!8 2149  1A?___!L_5 170 - . 5 171 - . 5 172 « _5  51^(1  \  157  2*51 2152  219 -1, 00000 000000000000000 12 16  -995 ,96  17 21  ,96 .96  2157 2158 2159  22  ,96  23 27  ,96 -94  2160  28  ,94  2161  29  ,94  2162  30 31 34  ,93 ,92 ,88  42 45 46 49  .995 ,96 ,96 n 9 6  50 51 54  ,96 ,96  .94  55  ,94  56  ,94  57 58 61 66  ,93 ,92 .68 r 9 9 S  68  ,96  2153 2154 2155 2156  2163 2164 2165 2166 2167 2168 -II*.  „—  2169 2170 2 171 2172 2173 2 1 7I 4 2175 2176 2 f 77 2178 2179 2 18 0  C  69 71  i.UAl  ..  ,96 .96  72  ,96  73 7=;  .96  76  .96  2186  77 79.  .96 .9 4  2187 2188  80 81  6?  .94 ,94 „Q3  2190 2191 ? 19 2  83 85 86  ,92 ,68 .88  87 88 89 90  f. 1, ,,995 .96  2181 2132  ?2 1!864 2185  r..X  i A—  2193 2194 ? I y<J5  -  x - ——  2196 2197 2198 2199 2200  3  91 92  54  -  -  ,96  ,96 „96  93 ,96 9 4 , 9 6  —  —  -  -  96* 1/6* !7b*  2.',  0 522 617 2 2 8r?22  69 95  <U22 TS OS 9*7  9I>22 Sr/22  17 6 *  62 92  £t/22 2(7 2 2  96* 96*  £2 22  9b* 96" 566*  I7t722  tr;22 017 2 2 b£22  566* S£  ooooooooooooooo  -  "  IT 00000 -1022 LL\ s ' s"> 9 LI  &£22 i. i . c c 9£22 S£22 17 t l » O ££22 2£22  '  s ' s ' -  1L\  ZL\ \L\ Ql\ ~69T 89T s S s s o (7 • - £ 9 T  0£22 6222 8 2722  S 9 Z ' -  8r/2s"«» 8r;2s"-  1222 9222 &222 17222  991 S9T  £222 ' - -«-^,-,.™T222 0222  179 J T9~T 6r/T2*'T9T feS££'» ST7U2"* t * t M * T89* yy* 98* 26*  t,b a t769  6  *  96* 9b' 96*  091 bit SSI 6 f71  6122 St'22 i t 22  r/i7"T b£T  '  801 £0T 901 SOT  9T22" &T22 I7T22 £<22 2T22 tt 22  DOT £01 20T  b'6"22 8022  TOT  Z022  00 T  9022  66 96  &022 "  16  £022  96*  96  2022  9b-  bb  1022  225 1  2252 2253 2254 2255 2256 2257 2258 2259 2260 2261 2262 2263 2264 2265 2266 2267 2268 2269  73 76  ,96  77 80 81  ,96 ,94 .94  82 83 86  ,93 ,92 _88  91 93 94  ,995 ,96 .96  96  .96 ,96 .96  97 99 100 102 103 104 105  a.9 6  ,92 fl8 ,88  2273  110  r.  2274 2275 2276  111 112 113 114  ,995 ,96 ,96  t  \ \  115 116  ,96 ,96 .96  117 118 119  ,96 ,96 .96 .94  2285  120 121 122  2286  123  2287  124  ,92 .88  22.3 8 2289  125 140  229 0 2291  145  2292  156  2293 229 4  160 161  2295 2296  162 164  229 7  1.6.5  - . 1 1 3 9 - , 2 0 48 - . 3 3 5 9  2298 2299  166  - . 2 1 4 9  2279 228 0 2281 2282 2283 2284  2300  150  167 .169  __  .94 .93  2271 2272  2277 2278  _  _  ,96 ,94  107 108 109  227 0  _  ,94 .93  , 8 8  _  "1 » - 1 , •1.. - 1 . -'.215 - . 1 1 3 9  - . 2 1 4 9 - . 5 2.48  .....  -  -  '•- -  2301 2302 2304 2305 2306 2307 2308 2309  - . 5 2 4 8  170 171  .  -  2310 2311 2312  17?  - . 4 0 5 5 - > 0 5 5  173 174  - . 8 2  - . 5  1 1 3 . . ....-.J.2 176  -'.765  177 178 179  -".765 - . 5  23j 7 .23 1 8 2319 232 0 2321 2322 2323 2324 2325 2326 2^27  r  .2310. 2331  o o n o n o n n n n o n o n o  23 29  ,96  73 77 81 9/1  ,9 95 .94,  .995 ...... .....9.1... . 9 6 .96 100 .94 103 1 04 T 9 3  2328 2329  ,92 ,88  105 108  .113. . 9 9 5 1 15 .96 ,96 117 119 f 9 6  ....  2332 2333 2334 2335 2336  121.  .94  122 123  .93 .92  2337 2338 2339  125  ,88  126 127  1. r.  23 4 0  128  23^1  129  ,995 ,96 .96  234 2 2343 2344 2345  ..  Q I L  131  .  - . 5 - I t  ,94 _ ...5.1„. . 9 6 ,94 56  _  -  "r 5 - ' 5  ooooo  2316  .  - . 5  180 18.1.. 182 221  2313 2314  -  ,96 .96 .9 4  2346  132 133 134  2347  135  214.8 234 9 235 0  1.3.6  .92 ..,.88  141  ~ i .  146  - l .  .93 _  .  56  -  3  i s  ooooococooooooooo 901  t8T6bT U E2  • X „.___£  i O l  i6£2  9 01  96E2  r»  t?oi  oooococooooooooo  S6EZ t76_c_  ___  •I „.. u  ' ~  zs  ~ ~ ~ v  ~  ~  r/S_  £fcE_  •I u  9 8  26£Z  S9  T6EZ  • (  r? e  06E2  29 1 ° * ooooooocooooooooo  i s  ~ '  "  - ~  i BE 2  19  •'I  09  9BE2 S8E2  * l  6S  i>8£2  z S  i _  E6E2  •T  P£  Z8EZ  • I  ££ ?£  0 8E2  0£  6 i £ 2  "  2"'-  OdOOOOOOOOOOOOO 00000 * I - 222 s c -  "  —  BBE2-  __2  " I _S  68E2  •I  *l "  00t72  ~ S 9 _ " S 9 i ' -  TBE2  8 I E 2 i i £ 2  i 8 l  9 i £ 2  9 81  Si£2  S81  l7_E2  1731  EIE2  £_T 281  2IE2 U E 2  I 81  0_E2  oet  b9£2  2B">  6 i l  B9E2  s > 5SOf7> &_0t?e-  an i i l  _9£Z 99£_  9 i l  S9E2  8t72_°-  6t>l2  a_i  -  £>_£_•8170Z> 6_T X " fail 1  2 i l U l O i l  0 9E2  691  6SE2  i91  8SE2  ' "9"9" I S9l ST2"-  17090"9 i £ l " * f -  I/9E2 E9EZ 29E2  &_»  817_S>  291 " T 9 T ~  T9E2  i & t e 9SE2 SSE2 t>_£2  091  ESE2  i S l  2SEZ IbEC  9 9 o o o o o c u o o o u o o c o "  ooooo  "i • I  . 000000000000000  19  '\ OOOOOOOOOOOOOOO  0000000 00000000  •  — t7 9  £9  _  _  *  '  8  I W 2 " 9J7572 _  1  Sr/!72  "  lM?t;2 £1717 2  T  2l7t/2 TTFTTlT  U 2 _ _ n  OOOOOOG  t  E  Ol7t7 2  0£2 T  OOOOOOOOOOOOOOO  OOOOOOOOOOOOOOO  V2 617*72 8t7t?2  2 £ 2 £  M _  s  OOOOOO T  59  d  t / ^ ' b & I 000 0 0 '1  99  o  ££2 J7_9I  6£t72  T  ""¥£»? 2  00000  i£t?2  'I 622 " T ~ F 9 r 00000  9Zt72  M  ££t?2  :  bTtT2~ t7£r/2  822  X £ 9 1 - 2 I ¥ 2  29  OOOOOOOOOOOOOOO  OOOOOO  T£t72  ' I  LZZ  0£r;2  "I  S£J"  &2Tf2-  S 2 T - fr£I OOOOOOOOOOOOOOO 0000  T9 ~~  •  H  9  W2t?2 £2»2  2 2 • 9 * l £ 2 1  2  1  7 2 S2r/2  S 2 f 221 o T ~ r n r o T w u ooooouc u — i r o w o  "  ~  ~  ~  —  65  °t  S22  ' \  SO I  ! 5 7 T ' ' = - ¥ i n 0 OOOOOOOOOOOOOOO 0000 0  __  T2t?2 2  • | -  /  - ---  I  9117 2 _ _ s f l ? 2 —jj^g"  -  85 IS  £It/2 2T172 HT2" .  ~  '1 * \ 1 %  ~  -  9 £ I t?£I  80)72 i0t?2 90-172  50172  952S2t 172 X 221  Ott?2  bOtiZ  i s "ooooooooooodooooo * J  2 6Tt72 8lt?2 ZTFS"  S 2 T > . 0 £ 000000000000000000 - T i s r  3  1  28  OOOOOOOOOOOOOOO 0 0 0 0 - r j - ^ *I S2T'>  a  22t72  M t722 -n~r§  ,  521 * » 85  i?2)?2 zzwr  17 0*7 2 "  £0t?2 201?2 T0U2  LI  0 0 o o o o o o o o o o'o 0 0 "  •  " t b9*  2  "91  9  2  O i l £91 091  a61?2 (76*72  00000 * I lt?2  26t?2 T6172  " I 9 6017* 292* -  "  "  " 5/_  8 1 9 1 * OOOOOOOOOOOOOOO  OBI  06172  O i l  68172 881/2  S 9 l  I81;2  0172  " t b i l h 9 * O i l -  960(7*  £ 9 l  2 92* OOOOOOOOOOOOOOO "I  091 0000 6£2  _  fr?I  " r u n 96017*  ~  ~TT  000~D000 "1 8 £ 2 *1 S9I t?9' 0 9 1 OOOOOOOOOOOOOOO 00000  .  OOOOOOOOOOOOOOO  01  69  £9L  (78172 - '  • — 'Mr—T*  ; " - r  Q l 7 Z 28i;2 18172  "  08l?2 611/2 8I1?2 II172" 91172 SI172 l?It?2 £1172 2Ii;2 11(72 0it72 b9(?2  (781 bLX fell  a 7 ? (72 19172  00000 '1 9 £ 2  b9t72 (7 9 * 7 2  ° 1 6 i l t?9* ( 7 i l  £9172  * 1 f?9* 96Qt7* U  98(72 £8172  i £ 2  'I _  091  OOOOOWOO00u00 0  21  £6t72  091 00000 M  "  86t72 16172 96172  iiL'i  ' 8191* J7i0l*  OOOOOOOO OOCoOOO  66t?2  S91 081  960(7* -  00b2  oooooo * 5 2(72  960(7* OOOOOOOOOOOOOOO " I * 1 (79*  691 0000 b£2 (7I-T 691  9 6 0 ( 7 ' t?91 OOOOOOOOOOOOOOO OOOOOO  99172  29172 19172 09172 6bt7<-' 8S172 IS172 9S172 ££1?2  'I "1 (79*  I7£2  17 S 17 2  691  960(7*  6S1  £bt72 2S172 1S172  (791  2501  161  . 5 3  2502 2503 . 2 5 0 4 _ _ .. 2505 2506 2507 2508 2509 25}0 2511  166 243  1.  2512  251« 2515 25l6 2517 2518 2 5 19 2520 2521 2522 2523 2524 2525 2526 2527  2530 2531  0 0.0.0- 0 . M . M M . Q M M . Q O A . . . 7 8 161 ,2809 166  171  , 5 3 . i .  244 1. 0000 000000000000000 161 , 1 4 8 9 166 ,2809  _  .1.8 6 J _ . . _ 24 7 1, 0 0 0 0 OOOOOOOOOOOOOOO 162  167 1. 248 1 . 0000 0  2535 2536  162  2538 2539 2 5 AO 25 41 254 2 7543 254 4 2545  . 5 3 1.  249  1 .  00000 16.2 167 172 177  255 0  132  82  OOOOOOOOOOOOOOO  83  _  .  OOOOOOOOOOOOOOO  84  .1.4 8 9 .2809 . 5 3 r .  25 0 f. 0 00000  2547 254 8  ..  ,2809  167 172  162 167 172 17 7  254 6  79  . 5 3  2532 2533 253«  2537  ..  171 , 5 3 176 1. 245 1 , 00000 0 0 0 0 0 0 0 0 0 0 000 0 0 8 0 161 .079 166 , 1 4 8 9 171 ,2809 176 . 5 3 181 1, 246 f . 00000 ooooooooooooooo 81 161 . 0 4 1 8 166 , 0 7 9 171 . 1 4 8 9 176 .2809 181 , 5 3  2513  25.28. 2529  r .  OOOOOOOOOOOOOOO  . 0 79 . 1 4 8 9 .2809 . 53 1.  85  165  251 1. dflnnnn  255 1 C  n n o o o o o o 0 0 00 0 00  2553  162  ,0418  2554 T«:;55 t _ . 1 11 2556 2557 ?55R / _ _1 J A J  167  .079 .1489  17?  ,  2559 2560 i >)  1  2562 2563 7564 ..fZ..-t..&2..zK 2565 2566 7567 f >'yJ 1  ,2809  18?  ,53  187  1'.  JL  J  O-  *-—  252 1 . 00000 0  • 7> ^ A 1 r.  177  —  000000000000000 87 .0 ,0 —  ,0  .0  ,0  ,o . o_  .0 ,0 210731  .  .0  2568 2569 257 0 .<U--J..l..-\.  —  §• " —  ,0 1908996, 481 L 8 l , 6864025.  2571 2572 2573  - 1 0 3 4 5 4 6 . - 3 3 7 8 4 9 3 . ,0  2574 2575 7576  „_  2577 2578 ? 57<3 fry J i •  .0 SL..T.0 .0 .0  2580  .0  2581 258?  -',071 -,072  ..feL.--.-.™..^  2583 2584 2585 2586 258 7 ? 58 8 2589  86  .0 .0 _.....D.  T n .0 3124813. 38 0 5 2 4 ? . 3960949 .  _...,o 1814  14  6JX.6368... .0 - 2 5 2 0 8 9 9 . .0  n n .0  .0 .0  ,0 _ _ 0 _ .. • o  ...0 .0  ..  ,9  -.207  ,0 -.21 -',073 -.277 -.079 » , 3  -.072 - . 2 74 - ; i 4 4  -.102 .0  -.337  .0 - . 2 3  -'.287 -'.284  - . 0 9 8  -:.  -.127  259 3  -.067 - , 2 5 3  -'.132  -'.194  -.072 -.27  -...1.4.1 - . 3 2 9  -.167  - . 2 4 6 -..2,84  2 5 9 i7 f-t  1  2598 2599 2600  -.321 -_33:4  __-l39__  . ...  - . 0 5 7  - . 0 65  - . 0 8 5  .  .. - . 2 5 6 -'.193  -'.058  -.207  _  2i2  2592 259 4 2595 2596  -  .0  ,0 - . 0 6 ?  2590 2591  -  -'.fl.57  -_!!___ - .  -.156 -.36.5  14.  4108600.  -1657966,  r o -'.068 -.139  .  .0 2000000.  .0 510 0 0 0 .  _=__.!? ? -.187  -.312  -.11 .0  -.217 - . 0 5 4 6  -.041  -".0 45  - , 0 8 8  __  .  _  ..  -.048 -',051 -.189  2601 2602 2603 2604 2605 2606 2607 2603 2609 2610 2611  -.161 -.143 -.-,248 -.193 -".053 -.094 -'.147 -'.156 - » t * 8 __ . -.202 -',092 .0  2 H 2  2613 2614 2615 2616 2617 261 a 2619 2620 2621 2622 2623 2624 2625 2626 2627 2628 2629 2630 2631 2632 . 2633 2634 2635 2636 2637 2638 2639 264 0 2641 2642 2643 264 4 2645 2646 264 7 2648 2649 2650  -',0 55  -.109 -.06 -.134 -.09 -.054 -.062 -.077 -.098 • '  ,o .0 .0 ,0 ,0 .059 ,084 ,144 .072 .082 .135 .093 ,122 ,063 .069 .04 ,031 .0 ,0 ,0 . 0 ,0 .0 .0 .0 ,0  -.094 -.099 -'.056 *.209 -.21 -'.291 -.282 - . 0 45 -'.05? -.055 -'.059 -;.oi.2 -.229 -,18 -.058 -.105 -'.058 -_1J5 -.154 -.173 -'.06? -.063 -'.087 »3 ,0  .0 .0 ,0 .075 .053 .034 .065 .169 .095 .15 ,058 .078 ,086 .071 ,038 .0 .,.0. ,o ,0 .0 .0 ,0 .0. .0 .0  -.138 -.146 -.11 -:.073  ...  -.272 -,098  ....  ,0 -.048 -.101 -.107 -.116 .........1.3.9 _ -.269 -.261 -.049 -.057 -.112 -..Q.7 _ -.18  .0 _o ,o .0 .0  ,0 .0  ...0. .0  ,0  .  _  -.062 -.064 -.102 .0. .0 .0 .0 ,0 , 0 77 ...«.„*> 3 . .108 .073 .181 ,064 .079 ,106 ,086 .089 .038 .03  ; :  . ......  _  .  _  167  2651 2652  «J> ,0  J3 .0  f  2653  ,0  .0  ,0  265«  +Q.  *...!)  2655 2656 .2.65JL  ,0 ,0 ..Q  ,0 .0 1_Q  ,0 ,0 jQ.  0 ,0 ,0  ,0 ,0 f...O  .0 ,0 „-0.  2658 2659 2 6 6 f!  8  -  -  J l 0  2661  ,0  ,0  ,0  2662 -2663 2664  ,0 »J3 „0  ,0 ^0 ,0  ,0 ^0.  Chapter  5  A COMPARISON OF STOCHASTIC DYNAMIC PROGRAMMING AND STOCHASTIC LINEAR PROGRAMMING WITH SIMPLE RECOURSE MODELS AS DECISION TOOLS 5.1  Introduction The  decision any  asset  problem  and l i a b i l i t y  i n which  decision point,  hand.  Based  on f o r e c a s t s  must  decide  sell  from  i t s portfolio  decisions This in  actions the  which  a r e made  the optimal that  will  realizations The  problem  be t a k e n of  approach  Decisions  on  history  the entire  technique be  would  summarized  rates  to  to  to  at  models  distribution Bradley  have have  and Crane  future  of  been been model  in time.  flows,  which  repeatedly.  flows  assets  The  problem  and  to These  liquidity.  process  i s dependent  point  on  the  its portfolio.  as cash  decision  At  and l i a b i l i t i e s  and cash  buy f o r  immediate  continuous  is  dynamic  on t h e  (and dependent  on  variables). solving  be made  events to  to  constraints  the  assets  its portfolio,  assets  each  to  could  of  be a b l e  model  in  i s performed  the asset time  utilize  stochastic  a t each  a greater  point  amount  management  dynamic  i n time  optimization  conditional  the present. of  information  Such  a  than  can  model. tractability  the continuous developed discretized [5,6,7]  and l i a b i l i t y  up t o a n d i n c l u d i n g  in a discrete-time  operational  tional  hold  such  However,computational an  interest  is a  any p o i n t  future  be a c o n t i n u o u s  formulation.  at  of  t h e random  probably  c a n be t a k e n  and which  process  problem  has a p o r t f o l i o  assets  solution  best  would  of  subject  decision-making  that  actions  t h e bank  bank  management  where  type.  inhibits  On t h e o t h e r  the time  to approximate  described 168  the development  parameter t h e more  i n Chaoter  hand,some and the general  2 i s one  of opera-  probability case.  example.  The  169  The asset  and  B-C  liability  computational  encounters  model  is  management  the To  an  distribution.  support  of  the  points  the  constraint  zations  the  in  the  set  the  number  in  either  period  or  X i  $50 3) a  to  in .the  with  one-period  ^See  than one  or  first  of  the  B-C  discretize  obtaining  generates  the  the  amount the  B-C  of  the  size the  of  inherent  is  optimality  of a  situation:  a return second  r  i 2  x  2  i  Appendix  the  with  1  in  he  random  $100  .2  period  .9  or  return  Chapter  =  investor he m u s t also r  2  3-  i  =  for  from  fat  the  a  finite satisfy  For  model  achieved  by  reali-  does  not  the  port-  the  simul-  discrete  formulation^  in  in  period  .1  and  which  number  for  return the or  to  an  one  the  after  after  receives $50 w i t h  an  a  one two  additional  probability  opportunity sell  investor  to  maturing  and m a t u r i n g  either  has .1  B-C  entire  must  adjusting  problem  =  per  only  prob-  variable.  r n  the  in  is  the  return  investor a  of  has  the  points.  the  generated  a  period  probability  points,  two-period  1)  with  the  on  decision variable  incurred  typical  a continuous  optimality  feasibility  constraints  consider  X n  cost  period  realizations  period,  asset  model,  representative  foregone  asset  in. the  second  the  formulation  with  invest  is  of  2,  the  model  representative  following  2)  B-C  the  the  of  invest  periods;  the  from  example,  2  of  as  constructed  satisfaction  faces  difficulty  the  taneous  the  rather  variable,  from  since  For  Chapter  i s ,  Also  discrete  in  That  profit  scenarios,  shown  features  a certain  support.  folio.  of  achieves  as was  shortcoming  operational So  random  different  consider  and  essential  formulation.  ability  of  problem  the  computational  Another  attain  set  many o f  However,  significant  enlarged. in  captures  tractabi1ity.  model  problem  model  part  to of  invest his  .1; in  holdings  170  in  x i  at  2  capital  a 20% d i s c o u n t ;  losses  cannot  and  4)  exceed  the  10%  of  investor the  stipulates  outstanding  that  his  exogenous  realized  funds" i n  any  period. The  linear  B-C- f r a m e w o r k  is  programming  given  in  Table  formulation 5.1.  The  of  the  optimal  above  problem  solution  to  in  this  the  problem  is  baUai)  =  88.89  sj  =  11.11  2  i)  =  88.89  hiU22) = 0  =  (£  11.11  80.00  hi2U2i) 2  =  hfUi2)  S12U2.O = 0  and  the  optimal The  capital  loss  of  x  asset to  final  losses),  (maximal  has  value  i 2  repay  because  of  of  (and  15%),  then  sell  $37.50  fat  flexibility  is  reduced.  sponding  to  realizations  example,  it  is  with  clear  constraints  respect  to  the  their  in  from  the  the  x  S12U22)  =  11.11  s\z{lzz)  =  25.00  at  i 2  used  is  with  a  as  a  by  the  this  of  capital  bound  is  that loss  to 1,  if  in  constraints)  $100  investor  Thus, to  constraints  B-C  7.5  the  decision  occurrence. the  to  the  respect  variable,  of  realized  purchase  $44.11.  with  considering  on  increased  be  period  model,  probability  problem  profitability.  would  random  of  a was  problem  B-C  the  result  above  end  is  side  solution  the  of  small  hand  the  to  support  the  model,(which right  value  This  (such  the  optimal  of  formulation  points  of  If  optimal  representative  number  in  binding.  The  63.89  $42.87.  constraint  is  $50,).  the  is  =  corre-  For  formulation are  not  a  weighted  Table  biUi) Objective Function  Cash Flow  b?Ui) .2  b2U2i) (.9)(.l)  h !  2  ( £  2  i )  sl (A2i) 2  (.9)(.2)  5.1  s?2U2i) (.9)(-.2)  b 2 (£ 2 2 ) (:!)(.!)  hJ 2 U 2 2 )  s\2(l2Z)  (.!)(.2)  S i  2  ( £  2  )  (•!)(-.2)  ill  = 100 -.2  A2i  £  2  •1.1  -1.0  = 50 -.2  2 2  •1.1  •1  = -50  Inventory  lZi  Balance  1  = 0  -1 -1  111  =  0  = 0  1 -1  -1  =  0  <  15  Capital Loss  i\ 2  In  < 5  172  On  the  described  in  this  sides  not  the  are  penalties  bility.  Thus  recourse  model  cast to  period doubt  uphold  small  asset  this  the  B-C  recourse  than  in  of  any  In  is  value  first  formulation,  B-C  contentions, problem. of  Recourse  compensate  period  formulation  for  the  is  right  allowed  decision  decision flexibility  hand and  infeasi-  in  the  formulation.  the  should  recourse  recourse  decision  sample  constrained  superiority  simple  decision variables.  fat  of  the  the  representative  dissertation  by  points  all  simulation  Therefore,  the then  future  formulation.  a  recourse be  and  this be  coupled  model,  considered  fact  economic  In  will  the  ALM  better  the  events  may  chapter,  performed  with  the  that  the  .  in  on  order  a  .  model  presented  operationally  than  model.  models.  tion  will  be  stochastic  simulation In  one  used  -  linear  simulation This  reconciliation evaluate  profits  the  i s more  manaoement  ment  To  the  above  The  The  for  the  computational in  on  decision  the  consider  binding  use  on  hand,  dissertation..  there  The first  other  the  can  model, the  be  is  the  asset.and  approach  as  repeated (loss)  for  for  use.  recourse in  Exhibit both  the  a  statistical  two  approaches.  l i a b i l i t y  dynamic  B-C  simple  flowcharted  profit  by  two  stochastic  with  simulation  generated  a  same  program  process the  utilizes  manage-  programming In  the  second model,  formulation  will  be  a  used.  5.1.  formulations.  period  formula-  is  comparison  After  generated will  be  and made  each stored. of  the  Exhibit  5.1  START  Set  T  =  1  Set  1  =  1  Initialize Model  Generate Period  First  Solution  Obtain (randomly) an Economic Scenario  A  Reconcile Generated P o r t f o l i o Random 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 Position  Calculate Profits for Period  1 = 1  +  1  T  +  1  No  Retrieve Initial Portfolio  = T  and  174  5.2  Scenario  for  The  the  question  t e c h n i q u e , SLPR o r This  question  from  a  models  must  for  is  The  SDP  restrictive  To is  to  the  the  of  answer created  this  to  from  When o n e  in  some  two  points  of  view.  second,  answer  second  formulation -  to  better -  few  may  reduce  the  question, the  is  first  an  problem  First,  not  the  in  as  is  the  sample  effectiveness  a  the  self-  restric-  of  asset  and  liability by  from  of  points  simulation  solved  size  the  the  be  better  Although  that  will  is  better  is  clear.  solution,  sense?  which  question  representative  effectiveness  in which  operational  difference  the  formulation  technique  normative  normative above  which  the  question  a  i s Which  better  considers  the  set  simulation  be  And  provide  constraint  question  scenario  said  answered  should  in  in  standpoint?  answer  inherent  this  be  s i m i l a r problems,^  theoretically  the  be  standpoint?  evident.  tions  addressed  SDP, c a n  computational  normative  Simulation  and  SDP.  It  addresses. management  the  two  techniques. Essentially of  assets  and  the  problem  liabilities,  given  is  to  random  cash  flows.  To  maintain  computational  only  3 planning  periods,  3 assets  The term  deposit  mortgage. of  these  using  assets maturing  The  from  See  beyond  liability  financial  data  considered  used  is  instruments  Central  chapters  4  future  1  liability  1)  a demand  and  for  a one of  of  for are  from  and  approach,  treasury and  3)  b i l l , a  returns  [10].  2)  a  long-term and  consecutive  Corporation  comparisons.  SDP  costs  considered.  The 26  portfolio  return, the  model,  deposit.  Housing  optimal  period  the  generated  the  rates  feasibility  horizon  were  Mortgage  2 and  and  are:  the  determine  To  costs  observations get  a  175  reasonable function  of  correlation  of  interest  the prime  rate.  rates,  The d i s t r i b u t i o n  r  the  prime  ments.  m  rate  These  3/26  .0675  1/26  .075  2/26  .0775  1/26  .08  2/26  .085  4/26  .09  2/26  .095  2/26  .11  2/26  .115  1/26  and t h e ' r a t e distributions  P(M_m)  d  then of  the prime  rate  were (R)  derived  return  is  of  f o r the difference each  of  the four  between  financial  instru-  t  P(T<t)  i  P(L_£)  .0037  0.0  -.0104  0.0  -.0388  0.0  -.0275  0.0  .0088  0.2  -.0072  0.2  -.0306  0.2  -.025  0.2  .0198  0.42  +.0008  0.44  -.0253  0.5  -.0225  0.31  .0235  0.62  +.004  0.5  -.0225  0.77  -0.2  0.92  .0297  0.81  +.0118  0.78  -.0174  0.81  -.0175  1.00  .0338  1.00  +.0195  1.00  -.0051  1.00  Here  t h e random  variables  between  the prime  rate,  deposit  rate,  term  a  are  P(D<d)  difference  made  :6/26  .065  were  of  and c o s t s  Pr(R=r)  .06  Distributions  the returns  rate  treasury  M,  D,  T and L a r e defined  and each bill  rate  of  the following:  and the l i a b i l i t y  to  be t h e  the rate,  mortgage respectively.  176  At which will  is be  formly so  the  equally assumed  in  that  the  deposits  4%, The  the  have  and  4)  that  investor  the  outstanding  The  class  than  $50,000  $60,000  in  total  period  in  in  highest  the  discounts  model  is  to  are  dynamic  1)  cash  flows,  of  program,  over  the  tive  distribution  four  financial  it  conditions.  is  period for  be  of 2)  The  the  in  periods  1  restrain  investments  in  terminal so  assets, of  necessary  As  the  constraints that  all  the  normal  expected  cash  The  demand  deposits  the  next  uni-  deposits  is  .5%,  decrease  for  quick  for  term  and  3)  more 4%  class  include  1  horizon  having and  2,  simply  of The  3%  of  period^.  a discount not  com-.,  assume  than  in  from  periods  discounts.  the  more and  on  the  invested  model.  objective  of  the  returns.  Dynamic  posed first  This  will and  already  Programming  in  to  flows  the  of  are  constraints  constraints  investor in  The  losses,  loss  2,  funds  to  type.  losses  and  asset  held  problem  instruments.  any  and  Stochastic  the  deposits  parameters  B-C  capital  deposits  horizon.  demand  to  bills  capital  net  the  in  demand  FRB's  treasury  constraints  the  the  realize  net  period  the  to  the  assets.  one  want  portfolio  formulate  three  will  one-half  $100,000  6%.  used  The  maximize  for  of  If  then  set  not  yielding  Formulation  To  3.  current  These  5.3  does  demand  in  assets the  discounts  terminal  from  20,000].  The  has  types  (decrease)  mortgages  composition  three  liquidated,  include:  position  the  investor  be  constraint  investor  the  to  for  the  [-20,000,  used.  and  in  increase  interval  are  point,  invested to  assets  liquidation  on  initial  Section  establish include the  rate  stated,  5.2 an  Model  as  a  stochastic  economic  obtaining  a  of  of  the  return use  of  scenario  representaeach  of  stochastic  the  177  dynamic  programming  tions,  otherwise  of  simulation  the  variables  end  of  the  [-20,000, the  the tion  1.0  x  the  be  The  this  although  symmetrically.  100000 ( w . p . ' l )  will  limited are  will  probability  for  the  of  the  random  the  to  mean  $100,000. in  the  and the  smaller  approximation  is  reasonable  similarly.  third  Thus  the  decision cash  the  in  with  underlying  is  the  used  $110,000  variance  For  At  interval  distribution  of  purposes  two.  lie  .5  distribu-  So,  currently  representative  the  probability  realizations  difference  with  of  unwieldly.  be  distribution  However,the  constructed  hecome  deposits  point  $90,000  approximations  possible  period  demand  two  be  Using  105).  of  incremental  maintained  distribution will  time the  will  .5.  is  number  each  20,000].  probability  versus  the  period  crude  computations  before,  formulation  tribution  the  during As  implies  (.1.33 as  point  flows  have  x it the the  dis-  105 divides distribudistribution  178  Using for  a  be  the  prime  particular median rate  estimate  in  are:  1)  R plus  simulation  1)  mortgage  rate  P(M  actual  (R)  =  .25. P(.M  < m)  term  are:  rate  deposit  The =  .375,  1)  rate  .0827 (w.p.  1)  median the  of  the  mortgages  rates  .875, and  first  4) the  2)  period  mortgage  m,where  four  of  the  (assume  .is R p l u s  < m) =  of  distributions  (w.p.  above,  the  return  .0992  2)  plus  period  < m)  as  instrument  of  m,where  m,where The  rate  second  P(M  R plus  approach  financial  the  the  m,where  same  prime  and  minus  3)  the  of  rates  CM).  The  of  two  in  the  third  m,where  P(M  < m)  return  < m)  used  in  the  point  and  P(.M  to  between  .75  m,where  return  taken  =  return  of  is  difference  PCM < m)  R plus  R plus  rate),  rate  R^ period =  = the  .625, .125.  179  3)  treasury  bill  rate  .0541 (w.p.  and  4)  nonchequing  1)  rate  .0577 (w.p.  For used  as  the  treasury bills  purposes  demand  bill  impinge  on  the  demonstrate  This  ad  one  simulation,  rate  (This  usefulness  that  the  deposit  rate.  a priori.)  of  1)  would hoc of  the  have  of  of  simulation  technique  the  nonchequing  nonchequing  precluded  derivation the  solution  since  70.%  may  the  investment demand  because be  rate  the  rate  dominates in  the  treasury  deposit  rate  objective  operationally  was  does  is  better  not  to than  another. The fined  in  variables the  term  decision  Chapter  2.  completely deposits  and  variables  Since define  for  treasury all  mortgages  the bills  potential mature  B-C  model  will  mature a f t e r investment  beyond  the  be one  the  same a s  period,  opportunities.  horizon  of  the  de-  eighteen Since  model,  180  42  variables  of  these  two  flows  the  in  three.  in  problem.  the  flow uses  require demand 15  to  of  funds  deposits 35  the  limit in  the  the  loss  are  each  equal  to  capital period  one  the  amount  of  demand  the  funds  and  the  composition  139  of its  the  each  portfolio  occurrence.  the  to  89  four  add  size  the  period  each  of  deposit  1  to  oppor-  7  are  scenario; 8  the  namely to  14  outstanding  three. the  (inventory  financial  Constraints  assets  as  pre-  balancing)  instruments  consist  and  scenario.  constraint  available  total  in  one,  for  for  the  1  investment.  another  of  period  B-C  28  places Also  slack  an  the  capital  variables  formulation  is  upper  89  to  the  con-  variables.  objective  variable  36  in  period  Constraints  3% o f  each  deposits  investment  economic  funds.  4%  invested  constraints  straints  from  than  economic  flow  each  be  and  the  in  demand  demand  Constraints  of  less  in  four  define  sources  two,  of  each  potentially  Therefore,the  The  and  each  in  formulation. with  to  funds  deposit  on  the  the  flows  the  constraints. under  opportunities  define  and  variables  Constraints of  to  deposit  two,  period  losses  in  holdings  110  of  investment  demand  period  a l l ,  types  all  necessary  the  in  for  transactions  The bound  four  problem.  initial  records  In  requirements  realized  scribed of  are  variables  flows  period  There  describe  position,  deposit  the  to  The  initial  demand  tunities  cash  required  categories.  include: the  are  is  over the  is the  to  maximize  horizon  product  of  the  expected  of  the  model.  the  net  return  value That  and  of  i s ,  the  the the  net  returns  coefficient  probability  of  181  5.4  Formulatioh  The Also  manner  i s fewer  i n which  financial  penalties  buying flows  variables,  for  4)  will  of  of  this  being  number  of  the r e s u l t  of  i n t h e SLPR m o d e l .  interest  i s used  function.  i n a manner  by  In  are defined  of  Also,  The d e c i s i o n  for  contrast the the  incorporating  similar  opportunities  conthe  f o r each  realizations.  the constraints  deposits  formulation.  to  the  treasury  variables  variables b i l l s ,  term  by s i x , e l e v e n ,  eleven  respectively.  of  i n t h e SLPR model  the i n i t i a l the asset,  periods,  3)  holding  2)  three  three  requirements,  of  the three  constraints  of  which  assets  being  the flow  of  constraints  in total  of which  five  composition  an a s s e t  constraints  four  demand  to  1)  to 5)  three  the  equate  f o r each  of  future the  the  deposits.  are stochastic.  three In  describe three  Adding  as  there  nine  the  the  capital  stochastic  short,  and one f o r  cash  three  is deterministic  and 6)  constraints  of:  with  constraints  period)  stochastic,  describe  f o r the class  of  and one l i a b i l i t y ,  one ( t h e f i r s t  and t h e o t h e r s  are comprised  constraints 4)  which  variables  as the B-C  but the total  the possible  be d e f i n e d  position  straints  rate  than  a n d demand  composition  above,  type,  incorporated  investment  balance  f o r the three  initial loss  to  information  i n the objective  constraints  and s e l l i n g  assets  rather  The  mortgages  constraints  is  for violations  t h e SLM m o d e l .  four  t h e same  t h e mean  recourse  The  25  only  t h e SLPR model  and  t h e same  i n the B-C model;  uncertainty  allows for  deposits,  to  than  Model  uses  are of  instruments  SLPR model  in  SLPR model  the B-C m o d e l ,  for  t h e SLPR  the constraints  straints  to  of  1) conare  slack  deterministic  182  capital  loss  constraint,  the  SLPR  formulation  has  25 c o n s t r a i n t s  and  42  variables. The are  representative  model. many of  However,  points of  model,  chosen any  highest  •5[(1  is  n  the  of  the  the  the  than  ability  in  constraints  the is  B-C  net  their  portfolio  is  + r  is  the  return  +  .5[(1  period;  on  term  rm  t  is  penalty  )  4  "  the  deposits;  -  n  a is  1]  -  median  and  constraints  used  the  r^  The  to  penalty  to  the  priori,  calculated  as  [(1  +  the  rrf ) 4 "  on  the  number  for  viola-  and be  of  50%  term  1]  mortgages;  median  the  potentially  -  n  B-C  handle  horizon  to  return  is  in  algorithm  50% m o r t g a g e s  considered,  -1]  n  deposit  difficulties,  return  c o n s i s t i n g of  )4~  Wets'  model.  portfolio  +  median  the  a  This  = 1,2.3  distribution  of  the  portfolio.  m  demand  computational  yielding  r  stochastic  uniform  creating  larger  by  of  from  without  these  since  sides  points  is  generated  deposits,  where  of  hand  because  realizations  tions  the  right  cost  of  r  t  demand  deposits. The  right  are  the  for  violations  policy as  representative  in of  penalties the  net  second  of  constraints  those  penalty  hand  the 4.1%  points  these  rather  for  constraint  stage  in is  stochastic the a  B-C  paragraph.  In  or  loss  formulation.  loss  physically  capital  percentage legally  this  constraints  The  penalty  because  restrictive  particular  these  are  constraints  formulation,  a  used.  objective  for  the  used  than  preceding is  of  constraints  The  return  sides  the  variables.  is  to  maximize  violations.  first  stage  the The  net  returns  coefficient  variables  and  the  minus of  the  each  penalty  expected  variable  for  the  is  183  5.5  to  Results  of  the  In  most  normative  financial  determine  what  portfolio  changes  multi-perodicity shifting is  to  determine  superior  first In  Again, able  initial  time  models  However,  be i m p l e m e n t e d  determines  which  is  immediately.  The  i s to the  compensate  purpose  immediately.  technique  (SDP  at  model  so as to  incremental  the beginning security  of  of  one aggregates each  cash of  period  flows  of  The  for  the  the  model  simulation  o r SLPR)  -  any p o i n t  facing  yields  are given  period.  in a period,  consider  random  are aggregated  the current  holdings  the chapter.  folio.  Both period.  funds  spent  amount  portfolio  period, this  A  (.45  then  to of  rates  so t h a t  In  both  a l l  the  of  flows  return.  one-half  the simulation  is  for  however,  decisions  is  formulations  and the cash  i s then  avail-  the  the next  cash  the r e a l i z a t i o n  t h e sum o f  the  must  flows  same  period  (known)  If  and.'.l  exceed  returns  flow,  from  the  treasury  bills).  bills. are  the assets  of  present  in treasury returns  the  then  during  of  for  t h e amount  cash  the  port-  solution  spending  t h e random  at  an i n i t i a l  an o p t i m a l  be d i v e s t e d  invested of  with  t h e random  deposits  is  i n an appendix  generated.  exceeds  .45 term  amount  starts  determine  spending  t h e random  the incremental  are  models  period  the excess  if  the process  flow  the f i r s t  reconcil1iation, revenues  cash  mortgages,  hand,  for  Essentially  random  during  equal  the other  flowchart  t h e SDP a n d S L P R  first  The  be e f f e c t e d  the objective  random.  end  On  time.  models,  d e c i s i o n s may be made  the s t a r t  A detailed  an  should  financial  across to  planning  periodosolution.  reality,  at  the  at  are  section  a discrete  b e made  scenarios  the changes  in this  using  c h a r a c t e r i s t i c of  economic  analyzed  to  Simulation  the  first  After  determined.  held  since  184  the  start  of  at  the start  of  demand  SDP  of  The r e c o n c i l e d  which  is  whole  (random)  The c o s t s  and the d i s c o u n t  then  used  a n d SLPR m o d e l s .  This  and the  the period.  deposits  maturity. folio  the period  process  portfolio to  This  i s repeated  on t h e a s s e t s  a r e t h e sum o f  for selling serves  generate cycle  returns  (random)  securities  prior  a s t h e new i n i t i a l  t h e new s o l u t i o n s  i s repeated  fifty  the  bought  times  for  for a total  for a  total  of  to  port-  both  of  cost  the  eight  four  times.  hundred  iterations. The used  to  test  simulation  two h y p o t h e s e s .  H  is  used  to  superior  test  to  SDP.  of  0  :  y  for  first  of  (2)  the standard  (3)  the correlation sample,  SDP  or  "  y  SLPR  -  for  ° '  i s tested  the i n i t i a l  the paired  period  profit  used  of  between  to  -251.37 150.43/750  test  f o r SLPR  =  t h e 50 c y c l e s the f i r s t  differences  t h e SDP a n d S L P R  statistic:  _  n  the paired  is  profits  the paired  diffe-  f o r SLPR a n d  hypothesis  ($251.37 i n favour  the paired  the s i g n i f i c a n c e of test  by e x a m i n i n g  run of  differences  deviation  the following  are  hypothesis,  not the i n i t i a l  information  t h e mean  using  y  formulations  SDP.  the p r o f i t s  The s p e c i f i c  large  =  f o r t h e SDP a n d S L P R  The f i r s t  hypothesis  (1)  the  d  whether  that  The rences  results  of  SLPR),  ($150.43), (0.958).  differences  i s :  and Given  is  tested  185  Since the a  null  the test  hypothesis  statistically  statistic  is rejected.  significant  is significant Thus,  better  t h e SLPR  initial  at  the 0.001  formulation  solution  than  level,  yields  t h e SDP  formulation. The  second  H  is  used  that  to  for  test  t h e mean  SDP.  :  y  SDP  y  or  hypothesis  profits  t h e mean  2)  the standard  3)  the c o r r e l a t i o n  y  SLPR  >  0  n o t t h e mean  of  the eight  the paired deviation  the sample using  of  is tested  between  size,  profit  f o r SLPR  the following  Since  this  is rejected.  significant  better  To simulation  test  using  by e x a m i n i n g  runs  used  of  to  is superior  differences of  this  ($297.26  the paired  to  in favour  of  differences  f o r SLPR a n d  hypothesis  t h e SDP a n d S L P R mean  test  cycles  differences  i s :  of  SLPR),  ($308.74),  profits  the paired  and  (.785).  Again,  differences  is  statistic:  =-6.81.  is significant  at  Thus,  formulation  solution  the paired  the f i f t y  test  the significance  - 2 9 7 - 2 l 308.74//50  thesis  "  information  1)  tested  =  whether  The s p e c i f i c  given  d  SDP. This  of  0  hypothesis,  t h e SLPR than  the s t a b i l i t y SLPR was r u n .  the  t h e SDP of  .001  level,  the null  yields  a  hypo-  statistically  formulation.  the above  The r e s u l t s  summary of  this  statistics, simulation  a  are  second  186  analyzed  as above:  cycles,  a n d 2)  cycles.  The  1)  t h e mean  $4672.23 runs  a test  have  t h e mean  and $482.15 t h e same  0-  the standard  respectively).  the  the 8 runs  SLPR  of  fifty the  hypothesis  runs  fifty i s :  ($4645.85  deviations  for  The h y p o t h e s i s  and  t h e two  that  both  mean,  ^SLPR  standard  the pooled  deviation  variance.  the test  i s no r e a s o n The  test  a s i m i l a r manner  used  statistic  i s again  =  that for  statistic  statistic  is  the  is  .291.  i s not s i g n i f i c a n t t h e mean  the second  i s not  at  the  .1  level,  stable.  hypothesis  is  established  and i s  -  4720.15  86.84  Since  f o r the test  The t e s t  statistic  to believe  4783.13  there  the f i r s t  test  of  first.  Since  in  for  and second  a n d 2)  solution  profits  the f i r s t  for  4672.23 - 4645.85 90.53  there  initial  profits  The of  the  to  respectively),  tested  root  of  of  necessary  H  is  a test  information  ($421.11  samples  1)  the test no r e a s o n  =  statistic to believe  .73.  i s not s i g n i f i c a n t at that  t h e mean  i s not  the  .1  stable.  level,  187  A CDC 6 4 0 0 w a s u s e d total was  CPU t i m e .240 hours  explains periods  to  perform  the above  the 400 i t e r a t i o n s  a n d f o r t h e SDP f o r m u l a t i o n s  why o n l y  a limited  and r e a l i z a t i o n s  highlights  to perform  number  were  used  t h e gap i n t r a c t a b i l i t y  of  for  computations.  t h e SLPR  was 6 . 3 8 5  financial  between  formulation  hours.  instruments,  i n the simulation. t h e SLPR  It  The  This time  also  a n d SDP  further  techniques  188  5.6  Appendix  1  This performed  appendix  i n Chapter  5,  consists a n d 2)  of:  1)  a flowchart  a computer  code  for  for  the  executing  simulation the  simu-  lation. First, is the  t h e amount initial  purchasing  as  generated X(3)  invested  portfolio;  to  3)  by  (X(21),  is  in  period  in  the  2)  X(2)  is  mortgages)  t h e amount  of  1;  XI03  initial  bills  the optimal X(63))  and 5)  variables  in treasury  new t r e a s u r y  (term deposits, X(64))  the following  (X(20), (term  solution  (term  the  B-C  of  still  in period  is  held bills  t h e amount  (SLPR)  of  period 1;  4)  deposits,  demand  X61)  mortgages)  mortgages)  initial  (term  X1(X19,  i s t h e amount  deposits,  to  1)  deposits,  x(62))  the amount  treasury  portfolio.  bills  are defined:  in  allocated in period  formulation; treasury X(4)  (X(22),  mortgages)  deposits  bills  sold  outstanding  1  189  ^  Start  Set  T  ^  =  1  V Initialize  portfolio  XI  =  33333  X19  =  33333  X61  =  33334  X103  =  100000  Set  I  = 1  Formulate (B-C  Call  or  Kuzy  XI03)  to  period  problem SLPR)  (XI,  X19,  generate  X61, first  solution  Generate: 1)  exogenous  cash  flow  1YY  and  Rl  = XI03  +  set  2)  prime  rate  3)  treasury  4)  term  5)  mortgage  6)  demand  e [TIOOOO, 1YY  PR1  bill  deposit  rate rate  rate  deposit  TBl(PRl) TD(PRI)  AMI(PRl) rate  ALCI(PRl)  10000]  190  .005*X(4)  12  =  YI  = X(2)  ARM ==  +  04*X(22)  + X(3)  -n X ( 2 0 )  +  .06*X(64)  + X(21)  + X(62)  + X(63)  +  Z2  Z2  \  X2  =  X(2)  Z2  = 12  +  Z2  = Z2  + X(2)*.005  X(2) X20  = =  -  .2*(Y1  (X(2)  -  -  Rl)  X2)*.005  if  X2  >  0  if  X2  <  0  ©  X2 0.0  yes  X3  = X(3)  X(2)  =  +  X(2)  0.0  Z2  = Z2  +  (X(3)  Z2  = Z2  +  X(3)*.005  X(3)  =  -  X3)*.005  X3  (?)  if  X3  > 0  if  X3 <  0  (!)  191  yes  ©  X201  = X(20)  Z2  -  Z2  +  (X(20)  Z2  = Z2  +  ( X ( 2 0 ) ) * 04  X(20)  =  X62  0.0  =  -  X20 -  (YI  -  - X201)* .04  Rl)*.4 if  X201  > 0  if  X201  _ 0  X201  yes  © X21  = X(21)  X(20)  =  = Z2  +  Z2  = Z2  +  =  X(20)  0.0  Z2  X(21)  +  (X(21)  -  X21)*.04  (X(21))*.04  X21  0  if  X21  > 0  if  X21  < 0  192  yes  ©  = - X(21)  X62  X(21)  =  0.0  \!  X621  = X(62)  Z2  = Z2  +  Z2  = Z2  +  X(62)  =  -  X62 -  (Yl  •-  - X621)*  (X(62)  Rl)*.4 06  ( X ( 6 2 ) ) * 06  if  X621  > 0  if  X621  <: 0  X621  yes  © X63 = X ( 6 3 ) X(62)  -  +  X(62)  0.0  Z2  =. Z 2  +  (X(63)  Z2  = Z2  +  (X(63))*.06  X(63)  -  X63)*.06  = X63  <5  if  X63 > 0  if  X63 < 0  193  yes  c  1  Zl  = TB1*X(2) +  + T D 1 * X ( 2 0 ) + AMI * X ( 6 2 )  .0541*X(3)  Z3 = Z l XI  STOP problem infeasiblel  -  = X(2)  Z2 +  -  +  .0827*X(21) +  R1*ALCI  X(3)  X I 9 == X ( 2 0 )  +  X 6 1 == X ( 6 2 )  + X(63)  X103  -  .0992*X(63)  X(21)  Rl  no  no  -.194,' P R O 3 ? . AM  HApoY(TAPE5,0UTP'IT,TAoE6 = 0UT°UT>  IMPLICIT  - t AL  I N T E G E~_ _ S .  c  (A-H,0-Z)  . J > H . I , M AR T Y  ,  R £ U " TP(7b,Tc"i ,"T6 {7 3 , T : I  REAL 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 SEAL W(lCC.10G),G(iOQ)tDELTA(7Q>,GAMMA(7 j ) , P I ( l 0 0 ) DIMENSION IW ( 1 C J ) , K A P ° A ( 7 0 > , L ( 7 0 ) , K ( 7 0 > REAL 0 " ( 70 I , O M ( 7 0 ) OI1£NSION_J«<150) 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  $  K,QP  COMMON / P R I N T R / TP,TD COMMON / A / x Q * » » » » * » * » » * » » » GENERATE CASH DO_  , E  D  S .  .OM,ZO,MARKER  1 2 3U_xK  PLOW  * • * » * » * « * » • * *  = 1_,_2  XI 0 3=15 0 COO. Xl=33333. X19=33333. X61=33334. DO 1 2 3 3 K L l = l , 3 C ALL KUZY(X1J3.X1.X19..»_X_6.1 J  10 2 101  REWIND 5 WRITE(6,102) FORMAT(1H11 WRIT E (6 . 101) ' « < KLL FORMAT(10Y,1<«HSIMULATION A X 2 - X (2 ) X ( 2 ) = X (t* 1 X !4I =AX2 AX20=X(2C> AX2i=XC21) AX22=X(22)  R U N , 1 5 , 2 X , 6 H P E R I 0 0 , I 6> :  :  X(23)=X<12)+X(13)+X(14) x <2 1 ) = x ( q i * x t i u ) + x < i i > X (22 ) =X { P» X ( 5 2 I =X t 2 3 )  *X  X (o3> =AX2G +  ( 2 4 I +x ( 2 5 ) ftX2i+ftX22  X ( 6 4 > ="/ ( I P ) x ( i o 3 ) - x (?q> Y=RANF(CI YY=20G6C*Y TYY= YY -1GC0G Ri=X(l03)+IYY W R I T E ( 6 , 19 4) 104  P i  '  FORMAT<.10X,1UHCASH  FLQW(R1) = , - 1 6 . 4 )  _..  ° R 1 = P R { DUMMY) PR1=.5*PRl+.0397 I F ( I Y Y . G T . c . A N D . a . R i , L E . .~.7 5 ) I F U Y V . L T . G . A N O .  P R i . G E . . 0 85)  3  R i = PR 1 + . u 0 5 PR1=?R1-.CQ5  T 9 1 = P R l + RT 8 ( D U M M Y ) X O i s . P § < _ t . ~ L P i D U M M YI A ' l l = P R i + =M ( D U M M Y ) A L C 1 = . 7 * ( P R l +RLC  (DUMMY))  ;  i  195 W R I T E ( 6 , 150) TR1 ORMAT(1 ; r .19HTREASURY BILL * ATE-t2X,F16 WRITE 16, 151) TQl J FORMAT ( l & X . i q n TERM DEPOSIT R A T £ = , 2 X , ^ 1 W » I T E ( & , i 5 2 ) AMI FORMAT(llX,i3H M O R T A G E R i T E = . 2X , F l 6 . WRITE(6.153) AL-1 c OR^AT(lCX,iqH LIABILITv R ATE = , 2 X . F l 6  150  r  151 152 153  .6) 6 . 6 > 6) . 6)  Y 1 = X ( 2 ) + X ( 2 3 J J - X ( 6 2 L + _ t _» L21- ' tAS 6 3 ) + . 0 0 5 » X ? + . 0 4 * X ( 2 2 ) + . o " 6 * X~( 6 <•) Z2=. G 0 5 * X (li) + . * X ( 2 2 ) + . j & * X < 5<t» ARM=Z2 I F ( Y l . L T . P l ) X (2)=X(2)+R1-Y1 " I F ( Y l . L T . R l ) G O TO 7 9 X 2 = X t 2 ) - ( Y l - R l ) » (_._2J IF(X2.GT ,.C) Z 2 = Z 2 + ( X " ( 2 ) * - X 2 > » ( . 0 0 51 IFIX2.LE..0) 11=12* ( X ( 2 ) ) * ( . 3 0 5 ) X(2)=X2 X2Q=G.C IF(X(21.GE..0) 3 0 TO 77 X3=X(3>»X(?) X (2)=0. 0 I F C X 3 . G E . . O Z2=Z2+(X ( 3 ) - X 3 ) » (. 003) IF(X3.LT..0)Z2=Z2+(X(3))*(.035) X(3> = X3 IF(X(3) .GE ..D) GO T G 77 X20 = - x ( 3 ) . X (3 ) = 0 . 0 77 CONTINUE X231 = X < 2 0 > - X 2 0 - ( Y 1 - R 1 )*(.<•! I F ( X 2 J 1 . G T . . 0 ) 7 2 = Z2+<X(2G>-X?G1><'(.0<4) I F ( X 2 0 1 . IZ . . J ) 7.2 = Z 2 + ( X ( 2 0 ) ) * ( . 0<*> ^ X ( 20 ) =X2 01 ; X52=0.0 I F ( X ( 2 0 ) . G E . . 0 ) GO TO 7 f l X2l=X(21 ) +X(20) X(2G)=0.C I F I X 2 1 . G E . . 0 > Z 2 = 7 2 + (X ( 2 1 ) - X 2 l ) * ( . 0 < « > IF(X21.LT..0)Z2=Z2+X(21)'l(.0'4V x'(?i)"=X21 IF(X(21I .GE..Q) GO TO 78 X62=-X(21) X ( 21 > = 0 . 0 78  CONTINUE X 6 2 1 f X(6?J_-X62-f Yl-Rl)_*(_.4_)_ 'TF(xV2l.~GT7*.TiT2 =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 = Z2 + (X ( 6 2 ) > * ( . 0 6 ) X(62)=X62i IF<X(62 > , GE..0>  GO  TO  79  X 5 3 = X f 6 3 ) «-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 ) ) » ( . 0 6 t  \  _ "  196 x(63)=X63 I " ( X ( 6 3 » . G E . . 0 ) GO TO 7 9 STOP 1G0 79 CONTINUE 7l = T 3 1 * X ( 2 l * r r ) l » X ( 2 G > + A " U » X ( i ; 5 ) 1 +. ' J 5 i t i * X < 3 > + .C-32 7 * X ( 2 1 ) + . u 9 9 2 * X ( 6 3 > Z 3 = Z l - Z 2 - f i l » A L C l WRITE (6,10ft) KLL.Z3 108 FQRMAT(1CX,17HFR0FIT 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 <63 > X(10 3) = R1 X l a J - X l l O 7 )  20 7  X1 = X ( 1 ) X19=X(19) X61=X(61> WRTTE(6,2G7) X1,X19,X61,X103 FORMAT ( 1 0 V , 2 H X =, ivF16.it) WRITE(6,160)  160 161  POPM/\T  (1  f)Y  '1  168  F O R M A T ( 1 C'X WRITE(6,168) F O R M A T ( 1 0 X , »  165  WRITE(6,165) FORMATdOX,*  163  c u » * * * » * * » * * » * * * * * * * * * * » * * * * * * * * * * » * * * * * * * *  WRITE(6,161) FORMAT!1 OX,20HS T A SSZ1=SBZ1+Z3 SB7S=S8ZS+ ( 7 3 ) * » 2 WRITE(6,163) 71  T  I  3  GROSS  T  I  C S  >  REVENUES  = » , F 1 6 . 6 )  ARM COST Z2 COST  OF  O"  SALES  SALES  AND  =  * , F 1 6 . 6 )  FORCED  S A L E S = * , F 1 5 . 6)  A R M I N =R 1 * A L C 1 166  WRITE(6,166) FORM A T ( 1 CY , *  ARMIN COST S9Z1  167  WRITE(6,167) FORMATdOX,* WPITE(6.169) FORMATdOX-,*  S9ZS CUMMiJLATIVE  169 1233 1231*  CONTINUE CONTINUE END  OF  FUN3S=  CUM M U L A T I V E  PROFITS PROFITS .  » , F 1 6 . 6 ) = SQUARED  ••.F16.6) * , F 2 0 . 3)  ,  ' * ' * * * >  FUNCTION  PP(O)  Y=RANF<0> IFj_v . L £ . „ ? . 2 A 1 L_E3 06 . D i F ( V . G T . . 2 3 l . f t N 0 . V . L E . . 3 - 7 ) R=.0 65 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 ° - 5 . A NO . Y . I E . 2 ) PR = . 0 7 5 I F ( v . G T . . ^62.A N O . Y . L £ . .5) » R = .0 775 I F ( Y . G T . . 5 . A N O . Y . L £ . . 5 77> °R= .08 I F ( Y . GT . IF(Y.GT". I F ( Y . GT . I C ( Y . G T . WRITE(6, 101  . 5 7 7 .AND. .731^ AN O . . 8 0 8 . A NQ . . 88 5 . A N D . 101) PR  Y . L £ . ._?1<J P ? = . Q B 5 Y . L £ . ".8-J8)°? =. 0 9 Y . |_ £ . . 8.3 5 ) ° R = . 0 9 5 Y . L £ . . 9 6 2 > ° * = . l l  F O R M A T ( 1 t X . l l K P . R I ME RETURN ENO  R A TE= . 2 X , F l 6 . 6 )  198 FUNCTION "TR(OI Y=RAMF(0 ) lr---'J-KJ 4 2 = - . 0 306 13=-.0253 A4 = - . 0 2 2 5 A5 = - . 0 1 7 4 A6=-.0C5 1 a  l " c ! Y . C-f . IF(Y.GT. I F f Y . G T . . IF < Y . G T . RT>3=T0 _RE_T_U°_N END"  .  _  . 7 . AND . Y . L E . . 5 > T 3 = A 2 * ( ( Y - . 2) / . 3 ) .5.AND.Y.IE..77> T 6 = A 3 + ( ( Y - . 5 ) / . 7 7 . A N 0 . Y . L E . . 3 1 ) T3=A4f((Y-.77) . S i . A N D . Y . L E . 1 . > TB = A 5 * ( ( Y - . 8 1 ) /  » (A3-A2) 2 7 ) * ( A 4 - A 3 I /.G41*(A5-A4) , 1 9 ) » ( A6-A51  FUNCTION Y=RANFID)  RTOfO)  A l = - . 01 0 4 A2=-.0072 A 3= . 0 0 0 9 A4=.00't A5=.G118 A6=.0195 T F ( Y . L £ . . 2 1 T D = A i M Y / . 2 ) » (  A 2^5  i F ( Y . G T . . 2 . A , N 0 . Y . L £ . . 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 » T 0 = A 3 + < < Y - . 44) / . j 6 >*'( A 4 - A3 > I F ( Y . GT • . 5 . A N 0 . Y . L E . . 7 R ) TD=A4+(<Y-.5>/.23)*'<A5-A4> IF(Y.GT. . 7 3 . A N 0 . Y . L E . i . ) Tn=A5+((Y-.78)/.22)*<A6-A5l 3 TO=TD RETURN _ ^ ENO  -.200 FUNCTION  R"(Q)  Y=RANF(0> al = . u03 7  .  ,  A2=.C088 A3=.D19? A4=.  C235  A5=.02P7 66=.0338 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 ) I F ( Y . G T . . 4 2 . A N D . Y . L E . . 6 2 ) I F ( Y . GT . . 6 2 . A N D . Y . I E .  AM = A 2 + ( ( Y - . 2 ) / ( . 2 2 ) AV,=  . S l > A M = A 4+ (.( Y - . 6 2 )  I P < Y . G T . . 8 1 . A N D . Y . L E . 1 . )  )*(A3-A2)  A 3 « - ( ( Y - . 4 2 ) / . 2 ) * ( A 4 - A 3 ) . 1 9 ) * < A 5 - A4>  A M = A 5 M ( Y - . 8 1 ) / . i 9 > * < A6- A5)  R M= A M RETURN. END  _  •201. FUNCTION  RLCfOI  Y=RANF(D>  I E J * * J U L r . . J t t £ l ?.L?.£=....0.2.r.5. I r ! Y . G T . . IF(Y.GT . I c ( Y . G T . I r ( v . G T . RETURN  i l 5 . A N 0 . Y . L E . . 1 9 2 > R . C = - . G 2 ; . 1 9 2 . A N D . v . L E . . 3 0 H ) R LC = - . Q 2 2 5 . 3 0 3 . A N D . v . L E . . 9 2 3) RLC=-.02 . 9 2 3 . A N D . Y . |_E . 1 . ) RI_:=-.Q175  202 KUZY;xio3,xi,xi9,x6i)  S U B R O U T I N E  MOTIF ICATIOKS  r. = >  3Y  C  J .  KALL3ERG  V  1.  <U3Y  H?~a.2±JtI*ZSMZ}&.._L_?50:  £ l 5 > : i i ^ ^ l . J ^ L U J i _ 4 . S l =  K m  .  I M P L I C I T  REAL  INTEGER  ( A - H . O - Z )  S , R , P H I , MARTY  REAL  T  REAL  P (7 0 , 1 0 ) , 0 (7 0 , 10) . A ( U 0 , 2 60 1 . H ( 1 0 0 ) , C ( 2 6 0 I  :_=S4L_  D  (70.10)  ,TD(7 3 ,10 I  _ W U . 0 0 , i 0 . 0 J _ . _ _ ( l 00 ) . v O E L J A J _ 7 0 . L , GAMM4_(iu_)_,.P.I U . O 0 J  DIMENSION 'REAL  IW ( 1 00 ) , K A ? P A ( 7 0 ) , _  0 ° i 701  DIMENSION COMMON  ( 7 0 ) , K ( 70 )  ,OM(70)  X(15Q>  / A /  COMMON  X  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  .JO.,.MARKER COMMON  /PPINTR/  C=>NOTICE  THAT  C => I N  FOPT°SN  THE  TN  READ(5,9321  T P , TD  THE  DOCUMENTATIOM  COPE  ALL  THAT  SUCH - I N D E X E S  FORM4T(F10.6)  176  F Q R M A T < 1 M l , T l 0 , " T O L Eg.a_NC.E_ I S  SET  C =>REA D IN N,M1,M2, KI . READ (5,100)N,Ml,M2 100 FORMAT (315) 17 139 O R M A T ( / / / , T 1 0 , " ' < OF V A R I A RLE S = C "CONST<»AINTS= " , I 4 , / , T10, "# _r.li±i12 C=>READ  IN  C = >( L O W E R  AND AND  P  AND  HAVE  0  USE  SEEN  0-ORIGIN  INCREMENTED  INDEXING. B Y  1.  .  E ° S  932  110  : _ .. '  :  WRITE IJP^ER  OUT  THE  BOUNDS)  AT  " . 7 9 .51  " , 1 4 , / , T 1 0 , " # OF D E T E R M I N I S T I C :•, OF S T O C H A S T I C C O N S T R A I NT S = " , I 4 ) -  XI-VALUES<POSSIBL£ INTO  VALUES!  AND  ALPHA  A N D  BETA  O.  FORMA T ( " 1 " , T 2 G , 3 4 < • ' * " > , / , T 2 0 , " P O S S I B L E  VALUES  OF  RIGHT  HAND  SIDE",  / , T 2 0 , 3 4 ( " • " ) , / ) DO  32  L7=l,M2  _ E A D 13,  3 00)  K LLZ±jO_  ( L Z _ , 2__._P_ I Z , 2 J  ,  TP(LZ,11=P(LZ,2) TO(LZ,l)=D(LZ,2> I F ( K ( L 7 )  .LE.l)G0TO37  ..  KP=K(LZ) DO  31  LA=2.KP  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 A4-1) 31 37 S00  TP(LZ,LA)=P(LZ,LA+1) READ(5,8G2>  O 1 L Z , 11 , O ( L Z , K  READ(5,1291  QP ( L Z ) , Q M  (LZI+2)  ! LZ)  F O R M A T ( I 3 , F 1 0 . 2 , F 6 . 4 )  MA^lSSMJJIM.O.rl&j.ftl 802  F0RMAT(2F1G.2>  129  - O R M A T < 2  r  i C 4 )  KI=K(LZ)+1 LX=M1+LZ 102 •_ 1 32 C  ^ 0 S M A T ( T 5 , " R 0 W " , 13 , T 1 5 , 4 < F l 4 . 2 , 6X } , / , M _ f _ J _ . l *M.U H(LZ+M1)= D(LZ,1) NOTE THAT THE AL ° HA(I) BECOME THE RIGHT  : HAND  TERMS  •_  203' C  F O =  C = > M R I T E 1 1 2  T H E  S T O C H A S T I C  O U T  T H E  C O N S T R A I N T S  LOWER  A N O  U P P E °  FORMAT.L/V..129...<£.<"*_"' 1  V A R TA T - E S "  1 1  00  35  KI  SOUNDS  A N O  C A L C U L A T E  T H E  0  V A L U E S .  ._/>l20,:'_LpWE.R_ANDJJ5.?E^_^oJNJJs_^.3AfJDOMJI,  » / » T 2 0 » ' » 2  ( " * " ) »  / )  I = 1 , M 2  - K ( I ) + 1  L X = M 1 * I DO  35  J = 1 . « I  PJ.l_»jJJ «pji <-4 * _ I . L - _ D <_U J.1  31  C = > R E A D  I N  A N D  WRITE  C = >P - V A L U E S " A N D 103  O U T  W R I T E  T H E  THEM  F O R M A T </ / . T 2 0 , 3 0('*•*") /, DO  3 0  ALSO  , / , T 2 0t"  T2 0 , 3 0 < "*"•)  P  P - V A L U E S ! W R I T E  C A L C U L A T E  O U T  Q - P L U S  R OBA3 1 L I T I E S  OF  T H E  A N D  A C C U M U L A T E D  Q - M I N U S .  PANOOM  E V E N T S " ,  . / )  1 = 1 , M?  K I =K t I ) *  _  . I N I T I A L  O U T .  1  LJf^iTi 30  —  C O N T I N U E 10A  F O R M A T ! / / , T 2 0 , 3 3( " * " ) , / , T 2 C , " S H O R T A G E /. DO  A 2  T2C . 3 3 ( " • " >  S U R P L U S  P E N A L T Y " ,  1 = 1 , M 2  M^li+i A2  P E N A L T Y  , / )  ,  :  :  C O N T I N U E DO  Kl D  3 3  ( 1 , 1 >  ACC=C Q.= Q  3*.  1 = 1 , M 2  = < ( I ) + 1 = - Q P( I ) .  ( I ) + Q M ( I )  D  DO  3<t  ACC  = ACC  I  J - 2 , K I + ?  ( I , J >  P ( I , J l= - O P(T >+0* 33  C = > R  t  1 0 1  A D  1 7 9  I N  A  A N D H .  F O R M A T ( 3 F l C t ) O  1 7 9  J 1 = 1 , M  DO  1 7 9  J 2 = 1 . N  N  C  . A ( J l , NJ O2 T) E = T0 H. A 'T  T H I S  C  C O D E .  C  T H E _MAT R I X  C  T H E  C  ( 1 3 )  WHICH  DO  A  ANOTHER  D E P A R T U R E  R E Q U I R E  T H E  3 F 1 0 • ] * _ F 0 RM A T .  E N T R I E S .  NULL  B Y  T H E  L I N E .  1 6 9 KUSY=i,M INO.TEMP  U N D . E O . O )  GOTO. •139  I N  F O L L O W E D  R E A D ( 5 , 1 3 « ) IF  I S  WOULD  NONZERO  E N T E R  r. 197  ACC  C O N T I N U E  GOTO  189  1S7  CONTINUE I F ( M l . E Q . O )  GO  R E A O ( 5 , 1 0 1 ) ( H ( I )  r r ^ C O U j J H U E H (19 > = X1 0 3 H(16)=X1  TO  973 , 1 = 1 , M l )  S O  - O R  ENTRY  FROM  U S E R W I T H  E A C H  TO A  T H E  E N T E R  U R G E WHEN  ROW OF  H & TR I X . . . . W H I C _ H _  ROW I N P U T  ( F 1 0 . A ) .  O R I G I N A L E A C H  A  T H E  C O L U M N  ROW I S  N O .  C O M P L E T E  H ( 1 7 ) =  X 1 9  H (1 H > = X 6 1 13b  F O R M A T ( / / , T ? 3 , 7 . ( " * " ) , / » J _ 2 . G . , _ _ / , T 2 0 , 3 0  188  FORMAT  ("»••)  &  -  K A T R I  X_ ,  , / >  <I 3 . F 1 - ] . 4 )  N U M R = N / 1 C + l I "  (MOO (N , 1 2I . £ 0 . 0) NUH3=NUM«?-1  2 2 2  F O R M A T ! / "  2 2Z  ^O^JM^T I / / / 3 0 X ? * * * * * * * * * * * *  *  v - i ^ * * * * * * * . * * * ^ * . * * . * * *  <77TTIO)  ~~"_"_7~ F ' O R M T T 2 2 8  ROW " , I 3 , 3 X , 1 0 ( 2 X , - 1 0 . A ) )  ^ORM AT ( "  C O L U M N S " , 1 0 1 1 2 )  W R I T E ( 6 , 9 0 1 ) 9 0 1  ^ O R M A T ( 1 H I ) W R I T E ( 6 , 1 0 9 )  1 0 9  F Q R M A T ( / / , T ? Q , *' OO  22  3Q ( _ _ _ L _ _ _ , A . X 2 J „ J ^  / , T 2 0 , 3 0 ("*••> 2 2  , /)  1 = 1 , M-  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 . OO  4 0  I = 1 , M  O R L E _ L i ._5 I  D  OO 41  4 1  J = i . M  W ( I , J ) = 0 .  4 0 C  •  ( I ) = S I C- N ( O 3 L E , H ( I ) )  W ( I , I ) = R I ( I ) W  I S  Z E R O  E X C E P T  C = > I N I T I A L I Z E DO  5 0  L , K , I W  ^OR  ? I  ON  T H E  D I A G O N A L S  . D E L T A , __ANO__GAM M A !  C A L C U L A T E  H  AND- Z Q .  1 = 1 , M 2  G A M M A ( I ) = G . D E L T A ( I ) = 1 E 7 0 L 50  t i l = 0  KA ° P A  ( I ) = 0  Z O = 0 . DO  ,  5 1  ,  1 = 1 , M  I W ( I ) = - T IF  ( H ( I ) . G E . O . I  GOTO  5 1  H i I ) = - H ( I ) 5 1  Z O = Z O - H ( I ) DO  5 2  5 2  1 =  :  .  :  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 2 0 0 .  It  B E G I N  C O L U M N  3  1 V O T I N G  WITH  MAU = 1 .  C A L L C L M P V T C C B A R , S , 11 I F ( C B A R . L T . - E P S ) G O TO 2 0 2 I F j T  "C => C B A R . G E ,  0  . G E ._-Eo.S»_2QTP_20 0  AND"  3  Z O . L T . G .  W R I T E ( 6 , 2 S 7 ) 2 0 7  FORMAT C A U L  ( "  I N F E A SI B L E " )  DUMP  C => C B A R . G E . 0  ANO  * Z O . G E . 0 .  20 3  DJ)__0j__J=___  2 0 4  I F ( I W ( J ) . L T . 0 C O N T I N U E  )  GO  TO 2 0 5  .  205 208  " 0 TO 3 G ? WRITE(6,2D8I c ORHAT (/" °MASE  I  OEGENEgACY")  CALL' 0UMD C= > C B 5 R < 0 . 202 CALL SMX°VT(CBAR,S,MU> GO T O 2 0 0 ~ =>?HASE 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 ) _ 119 FORMAT!3r10.U) 35 FORMAT<//,T2 0 , 3 0 ( " t , / , T 2 0 , " * / , T 2 G , 3 0 ( " * • • ) , /> DO 7 0 9 J1=1,N 709 CONTINUE 710 FORMAT!" COST ( " , 1 3 . " ) = " . ^ 1 C=>SET GAMMA. DO 3 0 1 1=1,M? 301 GAMMA(I)=P (I,K(I)+1) C=>SET G. DO 3 0 2 J = 1 , M IF ( I W J J ) . L E . N) GO T O 3 0 3 G ( J ) = G A M * A (IW(J) -N)  A.A)  GO T O 3 0 2 303 n [ J ) = C ( I « ! J D 302 CONTINUE C=>SET FI,ZO ZO=0 . [ DO 3 0 A I = 1 , M PI(I)=0 . DO 3 0 A J=1,M ?-G'u P I ( I ) = P I ( I i +G ( J ) » W { J , I ) C = > 2 2 2 2 2 2 2 2 2 2 2 22 2 2 2 2 22 2 2 2 22 2 222 2 2 2 2 2 2 2 2 2 C=>3EGIN COLUMN FIVOTING WITH MAU=1. AGO CALL C L K P V T ( " C B A R , S , 1) IF (C9A<= . G E . - E F 3 ) G O T O 5 0 0 CALL S M X P V T I C 8 A R . S , MU) I F ( M U . N E . 2 ) GO TO A 0 0 WRITE(6,A02> A02 FQ?MAT ! / " UNBOUNDED"' . ^ CALL DUMP . C=>CRAR>=0.! SET DELTA,GAMMA. 5G0 DO 5 0 1 1 = 1 , M ? DELTA(I)=D!I,1) 501 GAMMA ( 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 IF <IW(Jt . L E . N ) NU=IW!J)-N L <NU)=1 Y=0. P H I = 1_ KI = K (NU1 «•! DO 5 0 3 K K = 1 , K I  " GO  ;  TO  50 2  206 P H I = K K I  50J3.  ( Y + O ( N U . K K ) + E P S . G T . H < J ) }  R  GO  TO  5 C (•  Y =V+n ( N U . . K J O I t =~3 < N U ,'°  *5GA  H I ) -  ,  _  ( N'J , K V N U ) + 1 1  C  0 E L T . f i ( M l ) ).= 0 ( M U , P H I ) H ( J | = H ( J ) - Y GAMMA KA =  P I ( I 1i=PI  50 2  = H I )  GJL..I = 1„,M  0_0_5 5 0 5  ( N U ) = P ( N U .  P A ( N U ) = P H I - 1 ( I ) + X * W ( J  ,1>  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 £ G I N 700  .  C O L U M N  C A L L  P I V O T I N G  I_F_ I C B A R . G E . - E P S 7 0 1  0 0  A G A I N  WITH  M A U= 0 .  C L M P V T ( C B A R , S , 0 )  7 0 3  )  __GO_ _T_Q. A O 0  J = 1 , M  G ( J ) = 0 . DO 7 0 3  7 0 3  1 = 1 , M  G I J ) = G I J ) * W ( J , I ) » A ( I . S ) C A L L GO  UPP.PVT  TO  ( C 8 A . P , S - N , 2)  7 0 0  ,  C = ^ B E G I N  U P P E R  B C U N O  6 0 0  DO  6 0 1  I F  ( L (I ) . E O . l )  KK =  M1+'I  P I V O T I N G .  I = 1 . M 2 GO  TO  6 0 1  C = > T E S T 1 . C B A R = G A M M A (II+ I F D C 6 6 0 3  0  = - «  C A L L  U P R P V T  TO  ;  ( C B A R . L E . E P S )  DO  6 0 A  6 0 1  TO  GO  TO  6 0 1  (I>) TO  6 0 1  L L = 1 , M  G ( L L ) =W ( L L , GO  GO  ( K K 1 + P ( I , K A P P A  I F  C AL L  U P R P V T  KKI ( C B A R . I . 3 )  .  7 0 0  C O N T I N U E  C=>WHEN 7 0 2  ( C 9 A R , I , 1 )  ,  C 3 A R = ? I  6 0<•  6 0 2  7 0 0  ( K A P P A ( I ) . E O . 0 )  p  . _  :  TO  ( L L , K K >  C = > T E S T 2 . I  GO  3 L L = 1 . M  G ( L L > GO  6 0 2  P I J K K . ) _ ,  ( C B A R . G E • - E PS >  T H E  L O O P  F O R M A T ; " ! " ,  I S  S A T I S F I E D ,  T 2 D , 3 0 ( " * " ) , / ,  W P I T E T 2 0 , "  / , T 2 0 , 3 0 ( " * " ) , / ) C A L L _?_R I fJT R E T U R N " ENO  OUT  T H E  O P T I M A L  O P T I M A L  S O L U T I O N  S O L U T I O N  " ,  AND  S T O °  207  C=>  SUBROUTINES  SLZZS.  SUBROUTINE IMPLICIT  PIVOT  RFAL  INTEGER  (C B 4 R , R) (A-H,0-Zt  S , P.  I,MARTY  REAL  P(7G,13>.D<7j,10).A(i:;J,2&G),H<lGO).C<2fc0i  REAL  W ( 1 0 C , 1 0 0 ) . G ( : 0 0>?OELTA(70),GAMKA(70>  .PI(1GG>  _QIMENSIGN I N ( l Q j ) ,KAPPAt7.01.,.LJLIOJUJLO.0.1 P^AL OP(70) ,OM(70 I •  COMMON S  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 , •  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  ROW, H(R>  GS=1.0/G(P>• 00 10.  10  J=1.M  —  W ( R , J ) = W ( R , J ) * G S H ( R ) = G S * H ( ? )  C=>PIV0T .00  ON  OTHER  11  1 = 1, M  ROWS.  IF ( I . E Q . R) GO TO 11 GS = G (I) DO 1 2 J=1,M W(I,Jt=W ( T , J ) - G S * W ( R , J )  12  H(I)=H(Ii-GS*H(R) 11  CONTINUE  C=>CALCULATE  PI  AND  ZO.  DO 1 3 J=_1_,_M P I ! J ) =PI ( J ) + C 9 A R * W ( R , J )  13  Z0.= Z 0 - C 9 A R " H ( R ) 976  RETURN END  —  208  (-, = = = = = = = = = = = = = = = == = = == = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =  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 - _ti!_J} INTEGER S , R , ° H I , M A R T Y REAL 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 ) REAL W ( l C-j , I C O ) . G d O i i l . O E L T A I / O ) • G A M M A ( 7 0 ) » ° 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 ) REAL 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 ° S , ? K,OP.OM,70,MARKER T=1E7C C=>FIND MIN RATIO H ( J ) / G ( J ) WHERE G ( J ) > 0 . 0=>FIND MIN RATIO ( H ( J ) - 0 E L T A ( I W (J> - N ) ) / G < J ) 00 11 J = 1 , M . IF (G(J) .LE.-EPS) GO TO 1 0 fF(G(J) .LT .EPS) GO T O 1 1 C=>IF G<J)>0. RATIO=H(J)/G(J) IF(RATIO.GT.T) GO T O 1 1 T=RATIO  C=>IF 10 _  11 C =>IF 976  __J MU=0 GO T O 1 1 G(J)<0. KK=IW(J)-N IF (KK.LE.O) GO TO 1 1 R A T I Q = ( M ( J ) - Q E L T A IKK) ±/0±J IF (R A T I C . G T , T ( GO T O 1 1 T=RAT10 R= J MU = 1 CONTINUE MO J F Q U N O MiJ = 2 . IF ( T . G E . 1 E 7 0) MU=2 RETURN END  )  ST  K>0?  G I J K B .  209  SUBROUTINE  CLMPtfT  INTEGER REAL  ,  -  S . R , P H I , M A R T Y P ( 7 0 , 1 0 J . D ( 7 0 , I D ) , A (13 0 , 2 6 0 )  REAL  . H ( l 0 0 ) , C ( 2 60 >  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 )  DIMENSION REAL  0  I W ( 1 C . O ) , < A ° P A ( 7 0 ) . L ( 7 J ) , < ( 7 0 ) 1 7 o i , O M ( 7 0 )  p  N,M,MI,M;,?j_o_jL._nJ.j_i^  COMMON  5  (CBAR.S.MAU)  ( A - H , 0-21  IJ__LI C . U _ „ F A . L  K,OP,OM,ZO.MARKER CRAR=1E7 0 S = 0  C=>FTND  M I N (C - ° I * A ) = CB A R 13  DO  J = 1 , N  X=C (JJ DO 11  11  : 1 =  X= X - P I ( I IF  1,M }*A<I,J)  <X.GE .CBAR-EPS)  GO  TO  10  CRAR=X S =  _1_Q  J  CONTINUE IF  C =>FIND  (MAU.EO.0) MIN (CRAP ,  OO  12  X=G  A MMA ( I )  IF  ( X . G E .CBAR-EPS)  S =I + N CONTINUE B76  RETURN END  )  =  C3AR  1=1,M2  C3AR=X 12  RETURN GA MM A ( • ) + P I ( » + M l )  <•  P K I + M l ) GO  TO  12 ,  ;  210  C=============================================================================== S U B R O U T I N E IMPLICIT C = > T H I S  I S  U P R P V T  PfAL  V E R S I O N  I N T E G E R  ( C - 4 R . I . K K )  ( A - M , 0-Z.» 2  0 F  U  0  R°vf.  S . R , ° H I  REAL  P (70.10  REAL  W ( 1 0 0 , l C G > , G ( i j 3 > , D E L T A < 7 C >  0 1 M E N S I O N  I W ( 1 0 3  RE_AL  Q  COMMON  N , M , M 1 ,  ) , < A ° ° A ( 7 0 I , .  I , H ( 1 0 u t , C (260) , G A M M A ( 7 C > . P I ( 1J O )  ( 7 0I  , K ( 7 0 1  M2  __  ° 7 o , A , W , C , H , G ,~D E L T A , G~A M M A , ? I ,  I w , K A P PA , L , E P S ,  . O M , Z O . M A R K E R  L O G I C A L C A L L  , A (13 0 , ' 2 o 0  (701_.flM (7GJ  D  K . O P  5  ) . 0 (7S.10)  F L A G  R W P V T ( T , R . M U )  KK=M1+I •  P L A G = . F A L S E . A L  S  .  '  H A= 0 .  I F  ( K I K . N E . O )  C = > K I K = 0  F I N D  G O  TO  T H E  L A R G E S T  C => P ( I , K A P P A ( I ) - L L )  +PI<KK>  K I = K A P P A DO  1 0  L S = l , 2 , . . . , K A p o A ( I ) . S . T . >  0.  A N D  T > =  S U M OF  D ( I , K A P P A < I ) - S )  ;  S = 1 , . . . , L L  (I >  L L = 1 . K I  A L ° H A = A L I F  2 0  ,  H A +0 ( I . K A P P A  D  ( D - L L+ l )  ( P ( I . K A P P A ( I ) - L L + 1 ) + P I  !KK>. L E . E P S  . O R .  T . L T . A L P H A - E P S ) G O T O  30  L S = LL A S = 4 L P H A 10  R  L  A G = . T R U E .  GO  TO  C = > K I K = 1 .  3 0 F I N D  C => ° ( I , K A P P A ( I 20  I F  T H E  L A R G E S T  ) +LL)<0  ( K l K . H E . l t  L S = 0 , 1 . . . . . K ( I ) - K A P P A ( I )  A N D G O  T>=SUM  TO  OF  S . T .  D ( I , K A P P A ( I ) - S ) :  S=0  L L .  4 0  K I = K ( I ) *• 1 - K A P P A ( I ! DO  2 1  L L = 1 . K I  A L ° H A = A L P HA»p ( I , IF  (P ( I . K AF  3  K A P P A  ( I ) +L L )  4 ( I } * L L ) + P I (.< K t . 3 E • - E P S  . O R .  T . L T . 4 L PH A - E P S )  GO  TO  3 0  LS = LL AS= A L P H A 21  F L A G = . T R U E .  C =>SEE  I F  3 0  I F  S O M E  ~C = > S O M E " " L S  3 1  LS. F O U N D .  ( . N O T . F L A G )  ( I F  GO  TO  N O T  =>IVOT  A N D R E T U R N )  '  4 0  F O U N D  1^  ( K I K . E Q . 0 )  DO  3 1 J = 1 , M  L S = - L S  H ( J ) = H ( J ) - A S * G ( J ) K A ° P A IF ""IF"  ( I ) = K A P P 4  _0_TK • ~1 * A  ( I ) + L S  >  _  R  a  p  i ~ °J  J ' < *  p t  i?  T. " A '••-'•:>  ( KIK.'tO.O!  GAMMA  ( I ) = P ( I , K A ° P A ( I ) +1)  D E L T A  ( I ) = D f I , K A ° P A ( I )  < I> 1 1 1 1  _I <  K  KJ  ( I , < A = =»i i 1 1 > * P I ( K < )  +1)  T=T-AS IF $  '  ( K I K . E O . O  . A N D .  __C_R .  K I K . EQ..1  .Off."  T . L T . - E P S )  "  C = > O T H E R W I S E  G G  P I V O T  A N D  P ! I , K A P P A ( I ! + 1 ) + P I ( K .<) . L T . - E ° S . A N D. GO  P (I, TO  R E T U R N  9 9  KA«>Pft ( I ) • ! > » ° I ( K K > . GT • £ P S "  ,  C=>PIVOT AND RETURN UQ CALL P I V O T ( C 3 A P , R) IF (KIK.NE.C > G O T O U2 K A D P A (f> =KA?PA ( I > - 1 DELTA(I)=D(X,KCPPA(It+l) G A ' I H A ( T » =P ( I , K A P ° A ( I ) + 1 !  ;  DO A l J=1,M W(R,J> = - W ( R , J ) H (R) = - H (P )  Ul  DO A A J = i , M ° I (J ) = P - I ( J ) + 2 . * C ? A R » W ( R , J )  A A  DO A 3 J=1,M H ( J ) = H ( J ) - W ( J , I + M 1 ) * D ( I , K A P P A ( I ) + l NU=IW(R» IW ( R ) =N+ I "IF (I.GT.O) L(I)=1 IF (NU.LE.N) GO TO 9 9 NU=NU-N L(NU)=0  U3 42  IF  51  99  (MU.EO.O)  GO  TO  99  I.V = M j , . ± N y DO 5 1 J = i , M M ( J > = H ( J ) + W ( J , I V ) " D E L T A (NU ) K APP A (NU)=KAPPA(NU)+1 D E L T A (NU> = n ( N U , < A P P A ( N U ) + l ) GAHMA(NU)=P(NU,<APPA(NU)+1! RETURN END  212  SUBROUTINE  SMXP'VT(C3AR.S,MU)  IHPLICIT REAL (A-H j J W J I N T E G E R S « R","P H I , M A R T Y  •-  ,  REAL P ( 7 0 , 1 0 > . D ( 7 G , 1 0 ) , A ( 1 3 Q , 2 6 0 ) , H ( l J O ) , C ( 260> REAL W ( I C G , 1 G 0 > , G ( 1 G O ) , D E L * A ( 7 0 I , G AMM A ( 7 0 ) ,PI(100) D I M E N S I O N I W ( 1 0 0 ) , K A P P A (70 ! , L (70) ,< (70) REAL QP(70),OM(70) $  COMMON 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 , " K ,QP ,OM,Z'0,MARKER  IF C=>S<=N. DO  (S.GT.N) 11  GO  TO  j  10  LL=1,M  G (L L ) = 0 DO 1 1 J=1,M G(LL)=G(LL)+W(LL,J)*A ( J,S)  11  GO T O 20 C=>S>N. 10 KK=S-N+M1 DO 1 2 J=1,M 12 G ( J ) = - W ( J , K K )  ;  C=>IN  E f T HER  2G  CALL RWPVT(T,R,MU) IF (MU.EO.2) GO TO  99  IW(R)=S CALL PIVOT(C9AP,R) RETURN  ;  99  END  ;  CASE.  .  "  :  ;  ^  213  S U B R O U T I N E J M P L . I C I T I N T E G E R  S ,  (A-H,_Q-Z.i_  R E A L  C H I S ( 7  T P ( 7 0 , 1 0 )  R E A L  P (7C , 1 0 > , Q (7 0 , 1 0 )  R E A L  W ( 1 0 0 , 1 0 0 ) , G ( 1 0  R E A L  0 )  $  ,TO  ( 70  , 1 0 ) , A (li) 0 , 2 6 0 > , H ( 1 0 G  > , C ( 2 6 0 )  0 ) , D E L T A ( 7 0 ) , G A M M A ( 7 0 ) , P I ( 1 0 C )  I W J X 3 0 )jiCA££l<-7_0_>_,.. J..73J_,.'<JL7_0_>  ~QP ( 7 0 )  COMMON  _ . . ...  R , P H I . M A R T Y  R E A L  DHSNSION  100  P=TMT  R E A L  , X( 1 5 0 ) , QM (7 0 )  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 COMMON  / P R I N TR/  COMMON  / A /  T P , T O  X  F O R M A T S / , T I P . " B A S I S DO  10  1 0  INDEX  DUAL  V A R I A 3 L E S " , / / , 3 0 1"  "> >  J = 1 , M  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 .  ZSO = 0 . DO  2 0  I F  ( I W ( J ) . L E . N )  C  DO  2 1  C  K I = K A P P A  C  I F  [ K I . E O . 0 I  C  DO  2 2  ZC  C  2 2  C  2 1  J = 1 . M  ' Z S 0 = 7 S 0 » - C (IW (Jl  ) * H ( J )  1 = 1 , M 2 ( I ) GO  TO  2 1  K K = 1 , K I  Z S O - Z S O + P ( I . K K ) » D ( I , K K ) 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  C = > F I N D  THE  DO 31  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  A)  $H.  X - V A L U E S  3 1  FROM  IW  li  H .  1 = 1 , N  X ( I ) = Q. DO 3 2  3 2  I F  : 1 = 1 , M  ( I W ( I ) . L E . N )  C = > W R I T E  OUT  X ( I W  ( I ) ) = H ( I )  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 ) WRITE  3 5  ( 6 , 3 5 )  F O R M A T ( / / . T 2 0 , 3 0 ( " » " ) * DO  3 3  33  W R I T E  1 0 2  FORMAT  C = > C A L C U L A T E 3 6  , / , T 2 0 , " O P T I M A L  / , T 2 0 , 3 0 ("*"> 1 = 1 ,  V E C T O R " ,  N  ( 6 . 1 0 2 ! ( " T H E  I , X ( I )  X ( » , I 3 , - ) = " , F 1 5 . A ) .ALPHA.*'S_A_NO.JP«.I~S,  F O R M A T( / V , T2 0 , A2 ( " * ' * ) *  S O L U T I O N  , / )  , / » T 2 G , " R I G H T  HAND  S I D E  FOR  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 ( " » " ) , / ) DO  3 0  A L °  H A =- ( P (1,1  1 = 1 . M 2 I * P  I (M1 • I) ) / ( P (I , K ( I )  t i ) - P (I,  i i I  C H I = 0 . DO 3A  3A  J = I , N  C H I = C H I + A C H I S ( I » = C H I  ; ( T + M 1 , J ) * X ( J )  ..  :  •_  214  [1=1+  "u  30 CONTINUE 10j_JLO_MJJ^ P E N = 0 43 F O R M A T ( / / , T 2 J . 30 ( " * " > *  , / t T 2 0 » " I N D I V I D U A L  D  _NALTIZS",  / , T 2 0 . 3 0 (••*"> . /) M.Z=M1*1 OO 3 8 1 K A = 1 , M 2  PEN1=0. '< I = K. ( K A I DO 3 8 7 KR=1.KI TF(CMIS(KA).LT.TD(KA,K9)> *~ 3 3 ' I F ( C H I S I K A ) . G T . T D C K A . KB>> '  P E M 1 = =>E N 1 + <TD * PEN1 =° E N 1 * ( C H *  < K A , K 3 > - C H I S (< A ) > * Q P < KA > T P(K A t KS) I S ( K A ) - T O ( K A . K D ) I * Q M < KA I T 3 (K A ,KB).__  PEN=PEN+°EN1 •3.33 3«i 334  KB=KA+t'l F O R M A K T F , "  PENALTY  FOR  CONTINUE F O R M A T ( / / / / / / « T 2 C , " T O T A L RETURN : END  R 9 W (" , 1 3 » " ) = " » F 1 5 . 5 >  PEN!LTY=",T62.F18.5> '. —  .v 215'  SUBROUTINE  DUMP  IMPLICIT PEAL (A-H.O-21 . . INTEGER S , R . ° H I , M A R T Y REAL 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 ) REAL 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 ) DIMENSION IW ( 1 0 0 ) , K A R P A ( 7 0 ) , L ( 7 0 ) , K ( 7 0 I REAL 0 ° ( 7 0 ) , O M ( 7 0 ) % 100  % S 10 101 103 105 106 107 110 111  •  .  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 « . . K A P _ _ L « _ _ t _ P J u > " . " K , Q P , Q M , ZD . M A R K E R WRTTE(6,10G> FORMAT (//" T - 0 M E G A= ! H= ! ° I = !  ", /••+••.«("__ ")) DO 1 0 J = l . M : WRITE(6,101> I W ( J ) , H ( J ) , ° I ( J > FORMAT (" " . 1 1 0 F i u ,u, ••>", F n , i » , ( T 4 6 , " > . " , 5 F 1 4 . 4 ) ) WRITE16.103) (G(I>,I=1,M> FORM A T ( / " G=", ( 6 F 1 5 . 5 ) ) WRITE(6,1Q5) ( L ( I > . I = 1.,M2 ) FORMA T ( " L = " , ( 1 2 1 1 0 ) ) W R I T E ( 6 , 1 0 6) (DELTA! I),1=1,M2) FORMAT! " DELTA=",(6E15.5)) WRITE ( 6 . 1C7) (GAMMA ( I ) ,1=1,M2) FORMAT ( " G A M M A = " , ( 6 F 1 5 . 5 ) ) WRITE(6,110) 70  _F ORM A T ( " ZO=", F 1 5 . 5J WRITE (6.111) (KAPPA(I),I=1,M2) FORMAT ( " K A P P A = " , ( 12110)) RETURN END  ,  :  -  W=  216  Chapter  6  SUMMARY, MAJOR FINDINGS AND DIRECTIONS FOR FURTHER RESEARCH 6.1  Introduction  In findings (6.4)  of  are  6.2  this  of  banks  the l i t e r a t u r e  has been  and d i r e c t i o n s  a mean-variance  based  Chapter  expected  these  models,  Bradley  while  (to asset  maintaining  tunately,  their  not only  undesirable  In  (6.2),  major  for further  were  and Crane  formulation  of  portfolio  satisfy  a l l possible  criteria  2,  research  attempted  i s not appealing  forecasted  of  [61],  based  approach  i t was shown i s the  maximization  the  uncertainty  for decision-making  on c a p i t a l  Unforbecause  possesses losses,  revision  of  crucial  problems.  but also  models  comorehensive  incorporating  for large  that  and s t o c h a s t i c  t h e most  scenarios.  is  returns.  and an i m m e d i a t e  economic  The second  t o overcome  limitations  constraints  mix constraints  In  management  The f i r s t  deterministic  tractability  as a r b i t r a r y  view.  institution  management)  computational  of  selection.  surveyed.  [5]  and l i a b i l i t y  net expected  Myers'  Chapter  computational  absence  two p o i n t s  maximizing  and l i a b i l i t y  features  asset  for a financial  criterion  has severe  of  to portfolio  by u s i n g  net returns. this  obstacle  1,  of  criterion  assumed  from  approach  on an o b j e c t i v e  appropriate  the study  approached  which  it  the dissertation  presented.  In the  (6.3),  of  Summary  upon is  a summary  the dissertation  In of  chapter  such  an  that  must  217  Given present  an a s s e t  tractable tion  these  and l i a b i l i t y management  and r e a l i s t i c  was d e v e l o p e d  ment.  d e f i c i e n c i e s , the purpose  for  large  This  model  incorporates  management,  while  maintaining  ALM  formulation  was a p p l i e d  necessary  to execute  that:  t h e ALM model  1)  2)  the results  In  Chapter  5,  flexibility  the  simulation  of  to  i s superior  a simulation  S L P R a n d SDP  indicate  that  decisions.  This  period  portfolios,  i n t h e SDP  6.3  Major  economic  develop  time  asset  a formulation problem, More  model  was f o u n d used  to  to  asset  t h e ALM  in asset  to  the  of  demonstrate this  incorporated  into  the r e s t r i c t i v e  nature  feasible  the  indicate models,  and  increases.  (uncertain)  leads  4,  effort  model  was compared.  formulation  Chapter  deterministic  manage-  and l i a b i l i t y  application  a real  formula-  and l i a b i l i t y  In  reflect  formulation,  3,  computationally  tractabiTity.  equivalent  formulations  i s due t o  Chapter  to  environment,  The r e s u l t s  to  better  of  for a l l  of  initial  having  first  possible  scenarios.  objectives  tractable  management  to  i s both  was  Findings The  tionally  results  t h e SLPR  period  forecasted  VCS i n o r d e r The  dissertation  uncertainty  computational  as the i n f o r m a t i o n  by u s i n g  the  approach  this  that  In  the inherent  the model.  improve  model  problems.  a s an a l t e r n a t i v e  of  to  of  this  dissertation,  and l i a b i l i t y management  that  captures  are successfully  the essence  solve  be s u p e r i o r  of  to  model,  obtain  a  computa-  and, second,  the asset  to  and l i a b i l i t y  achieved.  s p e c i f i c a l l y , the computational to  f i r s t ,  f o r a number  a SLPR was a p p r o x i m a t e l y  of  tractability reasons.  twice  that  of  First, used  to  t h e ALM t h e CPU solve  218  an  equivalent  iterations  of  size ; linear  the simulation  type  model)  SLPR  formulation  and/or  SLPR of  was much  larger  (6.39 to  realizations  dynamic  equivalent  to  implementing  In  fact  apply  1)  an e s t i m a t e  economic  conditions,  an e s t i m a t e  reserves  position  the f i r m .  implement indicate the  ALM  legal 5)  that  2)  these  This  of  a liquid  6)  tractability  periods  of  the to  the  superiority  the  operationalize  of  must  be  Clearly,  results.  determined:  under  the behaviour  of  the  of various the  Reserve  a n d 7)  that  and i s  structure  deposits  position,  B-C  superior  the Federal  information  the  stochastic  compared  the term  governing  for  the growth  providing  rates  d e t e r m i n i s t i c model.  computational  the  information  constraints,  i s t h e same  of  (B-C  time  demonstrate  to  while  an e s t i m a t e  constraints  policy  additional  when  i s easy  program  f o r 500  formulation  while  facts  of withdrawal  for maintaining  an e q u i v a l e n t  Board's initial  i s necessary  these  four  i s not a constraining  to  points  factor  in  model. With  Chambers  regard  and Charnes  essential However,  to  problem  linear  characteristics of unlike  overcomes problem  4)  institution,  recommended  exponential  the following  flows,  when  computations  a linear  used  f o r 400 i t e r a t i o n s  i n the size  t h e ALM f o r m u l a t i o n  deposit  rates,  of  of  dynamic  Third,  Clearly  t h e ALM m o d e l ,  interest  needed  the growth  i n terms  Fourth,  3)  that  0.24 hours).  formulation.  t h e CPU t i m e  the stochastic  than  linear.  t h e ALM f o r m u l a t i o n  of  Second,  i s approximately  i s approximately  financial  of  are added,  formulation  to  program.  t h e Chambers  two i m p o r t a n t  and 2)  formulation,  programming the asset  model,  the conservative  1)  nature  incorporates  and l i a b i l i t y  and Charnes  drawbacks:  t h e ALM m o d e l ,  of  inherent  most  the of  the  management  problem.  t h e ALM  formulation  formulation,  the  Tike  uncertainty  t h e Chambers  and  of  Charnes  the  219  formulation. stochastic tively.  The  first  liquidity  drawback  has  already  constraints  the  second  Furthermore,  when  and  Crane  formulation,  ALM  model  is  6.4  Pi r e c t i o n s  One  shortcoming  and  and  formulation  is  for  the  rates of  one  Crane  possible  that  were  taken  research  have  using  to  the  Given and to  However,  available the  liability be  The  projects  estimate  in  in  a  using  with  to  the  simulation  superior  is  that  effecBradley  that  the  results.  dynamic  the  is  to  model,  random  below.  it  algorithms  the  not  such  the  number  as  useful  dealt  with  problems. of  increases  development be  dynamic  solve  variables  would  a  of  for  an  this  time the  efficient problem  general. hot  been  Deposit  flows,  problem  of  is  order  to  deposit  state the  existing  of  in  as  being  properly  flows,  the  interest  forecasting  viewed  forecasting  management, to  a  of  and  the  existing  superior  dealt  compared  discussed  Therefore,  in  has  given.  model  algorithm  forecasting. as  for  size.  Another of  ALM  are  efficient  shown  problems  problem  appears  of  optimization  would  asset  the  programming  dissertation.  rates  of  research  realizations  area  provides  is  By  Research  lack  in  is  demonstrated  thus  discussed.  drawback  formulation  been  future  have  up"  dynamic  numerous  this  the  "blows  stochastic and  to  for  ALM  and  Further  areas  Bradley  periods  for  due  has  flexible  Two  formulation As  more  it  the  been  dissertation,  rates has  and  been  withdrawal  the  subject  beyond  the  scope  implement  the  ALM  interest  of  model,  rates  and  withdrawal  areas  pertaining  techniques. knowledge  ALM  model  models  as  in  the  formulated a normative  in  this  tool.  to  dissertation  220  BIBLIOGRAPHY  Arrow, K . I . , "The R o l e B e a r i n g , " Review Baumol,  W.J.  and  Capital Vol. 75  of  Securities  in  the  of Economic Studies,  Quandt,  Rationing: (1965), pp.  R.E.,  "Investment  A Programming 317-329.  Optimal A l l o c a t i o n V o l . 31 ( 1 9 6 4 ) , p p .  and  Discount  Approach,"  Rates  The Economic  of Risk91-96.  Under  Journal,  B e a l e , E . M . L . , "On 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 Linear I n e q u a l i t i e s , " Journal of the Royal Statistical Society, Series Vol. 17 ( 1 9 5 5 ) , p p . 173-184.  Booth, G.G., "Programming E x t e n s i o n , " Journal B r a d l e y , S . P . and Management,"  Crane,  Bank  Portfolios  of Bank Research, D.B.,  "A  Under Vol.  Uncertainty: 2 (1972), pp.  B.  An 28-40.  Dynamic Model f o r Bond P o r t f o l i o V o l . 19 ( . 1 9 7 2 ) , p p . 139-151.  Management Science,  , "Management o f C o m m e r c i a l Bank G o v e r n ment S e c u r i t y P o r t f o l i o s : An 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 . 18-30.  , Management of Bank Portfolios, Wiley  Inc.,  B r i t i s h Columbia 1973.  Cass,  D.  and  New  York,  Returns  A Contribution  to  Economic Theory, Central Mortgage 1975.  C h a m b e r s , D. o f Bank 410.  Credit Unions Act of British  Government,  Stiglitz,  and A s s e t  and  J.E.,  "The  and  the  the  Pure  Vol.  2  Housing  and C h a r n e s , Portfolios,"  John  1976.  A.,  Structure  of  Investor  Separability  in  Portfolio  Theory  (1970),  of  pp.  Corporation,  Mutual  Funds,"  Columbia,  Preferences Allocation:  Journal of  331-354.  "Canadian  "Inter-Temporal  Management Science,  Housing  Statistics,  A n a l y s i s and Optimization V o l . 7 (1961), pp. 393-  221  [12]  Charnes,  A.,  tainty  Cooper,  Heating  [13]  Charnes,  A.  W.W.  and  Equivalents:  Cooper,  W.W.,  Management Science, [14]  Charnes,' A.  and  Kirby,  Programming Association 1965.  [15]  Charnes, with  A.  and  [17]  "Cost  M.J.L..,  6  Horizons  Stochastic Vol.  6  (1959),  pp.  and  pp.  Programming,"  73-79.  " A p p l i c a t i o n of  Chance-Constrained  S.C.,  Dependence,"  Institute,  "Intertemporal  Bank  Asset  Choice  Systems Research Memorandum No. 188, Northwestern University, April 1968.  Finance,  Chen,  Revision  A.H.Y.,  Jen,  F.C.  Journal  and  Zionts,  of Business,  S.,  Cheng,  P.I.,  Vol.  , Demands,"  "Optimum  8  "The  Vol.  :  c h a s t i c Cash 319-332.  (1962),  Bond  pp.  44  Optimal  (1971),  Portfolio  pp.  51-61.  " P o r t f o l i o Models with StoV o l . 19 ( 1 9 7 2 ) , pp.  Management Science,  Portfolio  Selection,"  Management Science,  490-499.  [20]  Cohen, K.J. a n d Hammer, F . S . , " L i n e a r Programming and Optimal 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 pp. 147-167  [21]  Cohen,  K.J.  and  tainty,"  Collins,  H.,  Couhault,  A.,  Thore,  Journal "A  Recourse,"  [23]  of  73-79.  C h a r n e s , A. and T h o r e , S . , "Planning for Liquidity in Financial Institutions: 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 Vol. 21 ( 1 9 6 6 ) , p p . 649-674.  [18]  [22]  Cer-  Programming  (.1959),  "Chance-Constrained  Vol.  Littlechild,  Stochastic  Policy,"  [19]  G.H., to  to the S o l u t i o n of the S o - C a l l e d ' S a v i n g s and Loan' Type of P r o b l e m , " Research A n a l y s i s C o r p o r a t i o n ,  The Technological [16]  Symonds, Approach  Management Science,  Oil,"  and  An  Code Dept.  S.,  Bank  Portfolios  Vol.  1  for of  "Quelques  programmation stocks,"  "Programming  of Bank Research, Stochastic  Linear  Mathematics,  me'thodes  stochastique  de  Programs  University  of  resolution d'un  line"aire venant  Cahiers de 1'I.R.I.A.,  (1970),  Vol.  9  de  (1972),  Under pp.  with  Bank (1967),  Under-  42-61.  Simple  Kentucky,  probleme la  gestion  pp.  1975.  de du  77-100.  222  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 Model f o r C o m m e r c i a l Bank Bond 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 Union Reserve Board, Capacity of the Credit  Columbia  Crosse,  Credit  , "Financial Unions, 1970-74.  H.D.  and Hempel, 2nd E d i t i o n , Jersey, 1973.  Banks,  Daellenbach,  H.  "A R e p o r t o n t h e A d e q u a c y o f Union Reserve Board," 1973.  Dantzig,  and A r c h e r ,  S.A.,  "The Optimal  Stochastic Model," Analysis, V o l . 4 ( 1 9 6 9 ) ,  G.B.,  Science,  on t h e  Financial  British  G . H . , Management Policies for Commercial 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  Multi-Period  tative  Statistics,"  the  Bank  Liquidity:  Journal of Financial pp.  329-343.  Management  " L i n e a r Programming Under U n c e r t a i n t y , " V o l . 1 (1955), pp. 197-206.  , "Upper Bounds, Secondary a n g u l a r i t y i n Linear Programming," pp. 174-183.  Dantzig, G . B . and Van S l y k e , R . , f o r L i n e a r Programming I I , " (1967), pp. 213-226.  A  and Quanti-  Constraints,  Econometrica,  "Generalized  Upper  and B l o c k T r i V o l . 23 ( 1 9 5 5 ) ,  Bounded  Techniques Vol. I  Journal Comput. Systems Sci.,  E i s n e r , M . J . , Kaplan, R.S. and Soden, J . V . , "Admissible Decision Rules 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 . El-Agizy,  "Two S t a g e  tribution pp.  Eppen,  Programming  Function,"  Under  Operations  Uncertainty  Research,  with  V o l . 15  Discrete  Dis-  (1967),  55-70.  G.D.  and Fama,  Dynamic P o r t f o l i o pp. 94-112.  Simple  national  E.F.,  "Solutions  Problems,"  f o r Cash  Balance  Journal of Business,  and  Simple  V o l . 41  (1968),  , "Optimal P o l i c i e s f o r Cash Balance and P o r t f o l i o Models with Proportional C o s t s , " InterEconomic Review, V o l . 1 0 ( 1 9 6 9 ) , p p . 1 1 9 - 1 3 3 .  Dynamic  22'3  [36]  [37]  Eppen, 6 . D . and Fama, E.F-, Portfolio Problems,"  Fama,  E.F.  and  Winston  Miller,  Inc.,  "Three  Cash  B a l a n c e and Dynamic V o l . 17 0 9 7 1 ) , p p .  The Theory of Finance,  M.H.,  New  Asset  Management Science,  York,  Holt  Rinehart  and  1972.  Money in a Theory of Finance,  [38]  Gurley, J.G. and Shaw, Institute, 1960.  [39]  H a l e y , C.W. and S c h a l l , D.W., The M c G r a w - H i l l , New Y o r k , 1973.  [40]  Hempel, G.H., "Basic P o l i c i e s , " The  E.S.,  Ingredients  Theory of Financial  of  Commercial V o l . 155  Bankers Magazine,  Brookings  Decisions,  Banks' (1972),  Investment pp. 59-59.  [41]  Hespos, R.F., and S t r a s s m a n n , P . A . , " S t o c h a s t i c Decision Trees for 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, Vol. (1965), pp. B-244-B-259.  [42]  Hester,  D.D.  Yale [43]  Hillier,  and  Pierce,  University F.S.,  "The  Evaluation of pp. 443-457.  [44]  ,  "A  [45]  New  Derivation Risky  Basic  Haven, of  Model  for  Management Science,  Capital  Economist,  the 11  Behaviour,  1975.  P r o b a b i l i s t i c Information  Investments,"  Engineering  Projects,"  Bank Management and Portfolio  J.L.,  Press,  311-319.  Budgeting  Vol.  20  of  (1974),  Risky pp.  for  the  Vol.  9  (1963),  Interrelated 37-49.  Hirshleifer, J . , "Investment D e c i s i o n Under U n c e r t a i n t y : Choice-Theor e t i c A p p r o a c h e s , " Quarterly Journal of Economics, V o l . 1 9 (1965), pp. 509-536.  [46]  ,  "Investment  D e c i s i o n Under  of the State-Preference Approach," Vol. 80 ( 1 9 6 6 ) , pp. 252-277.  [47] Englewood  P.,  Quarterly  , Investment, Interest and C l i f f s , New J e r s e y , 1970.  Stochastic  Programming,  Uncertainty:  Application  Journal of Economics,  Capital,  [48]  Kail,  Springer-Verlag,  [49]  Kallberg, J.G. and Kusy, M . I . , "A S t o c h a s t i c L i n e a r R e c o u r s e , " F a c u l t y o f Commerce, The U n i v e r s i t y 1976.  Prentice-Hall  Berlin,  Inc.,  1976.  Program'.with Simple of B r i t i s h Columbia,  224  Komar,  R.I.,  "Developing  Bank Research, Lasdon,  Levy,  a  Liquidity  2  0971),  Optimisation  L.S.,  York,  Vol.  Management  pp.  Theory for Large Systems,  and Portfolio  Lifson, K.A. and B l a c k m a n , B . R . , "Simulation f o r A s s e t Deployment and Funds S o u r c e s , L i q u i d i t y a n d G r o w t h , " Journal of Bank pp. 239-255. Luenberger, D.W., Introduction Addison-Wesley, Reading, Madansky,  Journal of  MacMillan,  New  1970.  a n d S a r n a t , M . , Investment S o n s I n c . , New Y o r k , 1972.  H. and  Model,"  38-52.  A.,  "Methods  Operations ,  of  Research,  Analysis,  John  Wiley  and O p t i m i z a t i o n Models Balancing Profit, Research, V o l . 4 ( 1 9 7 3 ) ,  to Linear and Nonlinear Programming., Massachusetts,  1973.  Solutions of Linear V o l . 10 ( 1 9 6 2 ) , p p .  Programs 165-176.  Under  Uncertainty,"  " 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 Programming V o l . 6 (1960), pp. 197-204.  Problems,"  Management Science, Mao,  J . C . T . , " A p p l i c a t i o n of D e c i s i o n , " Engineering  Markowitz, H.M., pp. 77-91.  Linear  Programming to S h o r t - t e r m Financing V o l . 13 ( 1 9 6 8 ) , p p . 221-241.  Economist,  "Portfolio  Selection,"  Journal  of Finance,  Vol.  , Portfolio Selection, Efficient Diversification 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 . Mossin,  J . ,  Theory of Financial  Cliffs,  Myers,  New  S.C.,  B.,  B.  for  Capital  Management Review, "A  Business, Naslund,  Markets,  and  Model Vol.  39  of  Capital  (1966),  Whinston,  Uncertainty,"  Prentice-Hall  Inc.,  (1952),  of  Englewood  1973.  "Procedures  Industrial Naslund,  Jersey,  6  A.,  Budgeting  pp.  "A  Budgeting Under Uncertainty," V o l . 9 (1968), pp. 1-20.  Under  257-271.  Model  Management Science,  of  Risk,"  Multi-Period  Vol.  Journal of  •  8  (1962),  Investment pp.  Under  184-200.  225  Orgler,  Y.E.,  "An U n e q u a l - P e r i o d Model f o r Cash. Management V o l . 16 ( 1 9 6 9 ) , p p . B - 7 7 - B - 9 2 .  Decision,"  Management Science,  , Cash Management Methods and Models,  :  Orr,  Wadworth  Co.  I n c . , Belmont  D., Co.,  Cash Management and the Demand for Money, P r a e g e r  Publishing  C a l i f o r n i a , 1970.  Publishing  I n c . , 1970.  P a r i k h , S . C , Notes on Stochastic Programming, u n p u b l i s h e d , Department, U n i v e r s i t y of C a l i f o r n i a , Berkeley, 1968.  I.E.O.R.  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 Model 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 . Pye,  G.,  "Sequential Policies V o l . 20 (1973),  Science, Pyle,  D.H.,  "On t h e T h e o r y  Finance, Robichek,  A.A.  Hall  Roll,  "Investment  Finance, Sharpe,  and Myers,  I n c . , Englewood  R.,  Financial pp.  S.C.,  Management,"  Intermediation,"  New J e r s e y ,  Portfolio  Journal of  Decisions,  Prentice-  1965.  D i v e r s i f i c a t i o n a n d Bond pp.  Management  737-748.  Optimal Financing  Cliffs,  V o l . 26 ( 1 9 7 1 ) ,  W.F.,  York,  of  V o l . 26 ( 1 9 7 1 ) ,  f o r Bank Money pp. 385-395.  Maturity,"  Journal of  51-66.  Theory and Capital Markets,  McGraw-Hill,  New  1970.  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 Some S t o c h a s t i c 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 . 1159.  Telser, L., "Safety First Vol. 23 ( 1 9 5 5 - 1 9 5 6 ) ,  Thomson,  M.R.,  Research, Thore,  S.,  "Forecasting Vol. 4  Bank  of Economics,  Review of Economic Studies,  for Financial  (1973),  "Programming  Journal  and Hedging," pp. 1-6.  pp.  Planning,"  Jo-umal of Bank  225-231.  Reserves  V o l . 70  Pro1152-  Under  (1968),  Uncertainty," pp.  123-137.  Swedish  226'  [78]  Tintner, G., " S t o c h a s t i c L i n e a r Programming w i t h A p p l i c a t i o n s to 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 by H.A. Antosiewicz, 1955.  [79]  Tobin,  J . ,  "Theory  of  Portfolio  Selection,"  Edited  by  F.H.  and  Rates, 1965,  pp.  Hahn  R.P.R.  The Theory of Biechling,  Interest  MacMillan,  London,  7-9.  [80]  T o b i n , J . and 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 and t h e Effect 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 . 53 ( 1 9 6 3 ) , p p . 3 8 3 - 4 0 0 .  [81]  T u t t l e , D.I. Capital work,"  [82]  Van  Home, tions  and L i t z e n b e r g e r , M a r k e t E f f e c t s on  Journal  J.C., Under  "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 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 and (Quantitative Analysis, V o l . 1 ( 1 9 6 6 ) , p p . 6 8 - 8 3 .  Financial [83]  , Function Prentice-Hall  [84]  , Hall  Inc.,  [85]  Vancouver C i t y 1975.  [86]  Wagner,  Walkup,  D.W.  Inc.,  and Analysis Englewood  Financial  and  and  C l i f f s ,  Savings  C l i f f s ,  Wets,  of Capital  C l i f f s ,  New  Market Rates,  Jersey,  Management and Policy,  Englewood  Principles  H.M.,  Englewood  [87]  of Finance,  R.H., "Leverage, D i v e r s i f i c a t i o n and a Risk-Adjusted Capital Budgeting FrameV o l . 23 ( 1 9 6 8 ) , pp. 4 2 7 - 4 4 3 .  New  Credit  Union,  of Operations New  Jersey,  R.J.B.,  Jersey,  4th  1970. Edition,  Prentice-  1977.  Financial  Research,  Statements,  1968-  Prentice-Hall  Inc.,  1969.  " S t o c h a s t i c Programs w i t h R e c o u r s e , " V o l . 15 ( 1 9 6 7 ) , p p . 1299-1314.  SI AM Journal on Applied Mathematics, [88]  Weingartner,  Mathematical Programming and the Analysis 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  Jersey,  [90]  ,  Wets,  R.J.B., pp.  "Capital  Synthesis,"  Program,"  f s ,  Capital New  1963.  [89] and  of  H.M.,  Budgeting  "Programming  SI AM Journal  89-105.  Budgeting  of  Management Science, Under  Interrelated Vol.  Uncertainty:  14  The  on Applied Mathematics,  Projects:  (1966),  pp.  Equivalent Vol.  14  Survey 485-516.  Convex (1966),  227  [91]  Wets,  R.J.B.,  "Programming  Under  Uncertainty: Vol.  The S o l u t i o n Set," 14 ( 1 9 6 6 ) , p p . 1143-1151.  Uncertainty:  The  SIAM Journal on Applied Mathematics, [92]  ,  "Programming  Under  Z. Wahrsch. verw. Geb., V o l . [93]  ,  "Characterization  Mathematical [94]  Programming,  ,  "Solving  Mathematical Williams,  A.C.,  Mathematics, [97]  [100]  ,  "Approximation  Wolf,  C.R.,  "A  Zangwill, Hall  Model  Ziemba,  W.I., Inc.,  W.T.,  "A  [101]  1  for  Problem,"  Simple  Recourse,  for Stochastic Linear V o l . 14 ( 1 9 6 6 ) , p p .  Commercial  Nonlinear  Programming:  Englewood  C l i f f s ,  Capital  Analysis,  I,"  SIAM Journal  Bank  New  A Unified Jersey,  Stochastic  Dynamic  Vol.  17  51  (1969),  Prentice-  1969.  Budgeting Model," V o l . 6 (1969), pp.  Management Science,  Security  Vol.  Approach,  Applied  Programming," 669-677.  Government  The Review of Economics and Statistics,  Myopic  Equivalent 309-339.  Journal  Programs  (1971),  of  Financial  305-327.  into  pp.  Nonlinear  450-462.  , " S o l v i n g N o n l i n e a r Programming Problems 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, Vol. 7 (.1972), pp. 1809-1827.  [103]  , P.L.  "Stochastic  Hammer a n d  G.  Ziemba,  W.T.and  in Finance,  Programs  Zoutendijk,  Theory and Practice, [104]  with  Programming," 927-940.  Formulas  Selecting  "Transforming  Programs,"  Programs  of Applied Mathematics,  and Quantitative  [102]  Complete  316-339.  (forthcoming).  "On S t o c h a s t i c L i n e a r V o l . 13 ( 1 9 6 5 ) , p p .  Portfolios," pp. 40-52.  [99]  pp.  Theorems f o r S t o c h a s t i c P r o g r a m s , " V o l . 2 (1972), pp. 166-175.  Stochastic  Programming  SIAM Journal [98]  (1966),  , " S t o c h a s t i c Programs w i t h Fixed Recourse: The 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 .  [95]  [96]  4  Vickson, Academic  with  Simple  editors,  Recourse,"  Mathematical  Publishing,  pp.  213-273,  Programming:  North  Holland  Amsterdam,  R.G., Press  e d i t o r s , Stochastic Optimization I n c . , New Y o r k , 1975.  1975.  Models  in  

Cite

Citation Scheme:

    

Usage Statistics

Country Views Downloads
United States 23 2
India 15 1
Germany 5 0
Serbia 4 0
Norway 4 0
Philippines 4 0
Lebanon 3 0
Japan 3 0
Italy 3 0
Egypt 3 0
Indonesia 3 0
Malaysia 2 0
Bangladesh 2 0
City Views Downloads
Unknown 42 7
Mountain View 5 0
Essen 4 0
Belgrade 4 0
Sunnyvale 4 0
Cairo 3 0
Tokyo 3 0
Mumbai 3 0
Banská Bystrica 2 0
Tempe 2 0
Hyderabad 2 1
Toronto 2 0
Kuala Lumpur 2 0

{[{ mDataHeader[type] }]} {[{ month[type] }]} {[{ tData[type] }]}
Download Stats

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0094648/manifest

Comment

Related Items