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UBC Theses and Dissertations

A goal programming approach to bank asset management Woodruff, Charles E. 1971

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A GOAL PROGRAMMING APPROACH TO BANK ASSET MANAGEMENT by CHARLES E. WOODRUFF B.A. Sc., U n i v e r s i t y of B r i t i s h Columbia, 1966 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF BUSINESS ADMINISTRATION i n the Fa c u l t y of Commerce and Business A d m i n i s t r a t i o n We accept t h i s t h e s i s as conforming t o the requ i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 19 71 In p re sen t ing t h i s t he s i s in p a r t i a l f u l f i l m e n t o f the requirements fo r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e for reference and s tudy. I f u r t he r agree tha t pe rmiss ion for e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n o f t h i s t h e s i s fo r f i n a n c i a l ga in s h a l l not be a l lowed wi thout my w r i t t e n p e r m i s s i o n . Department of The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8. Canada Date J i ABSTRACT Asset management i n a commercial banking environment may be viewed as a process of resource a l l o c a t i o n . The resources of a bank c o n s i s t of the funds made a v a i l a b l e t o the bank by i t s d e p o s i t o r s , c r e d i t o r s and shareholders. These funds may be a l l o c a t e d among a v a r i e t y of earning and non-earning a s s e t s . In c a r r y i n g out t h i s a l l o c a t i o n , management must recognize and r e c o n c i l e the various and often c o n f l i c t i n g o b j e c t i v e s and requirements of i t s d e p o s i t o r s , i t s shareholders and the p u b l i c agencies which r e g u l a t e i t s a c t i v i t i e s . Given these m u l t i p l e requirements and o b j e c t i v e s the sen i o r management of the bank must determine an a l l o c a t i o n plan t h a t w i l l provide maximum saf e t y and adequate l i q u i d i t y f o r i t s depositors and an acceptable r e t u r n f o r i t s shareholders. This t h e s i s attempts t o demonstrate how bank management can use a mathematical programming model to a s s i s t them i n for m u l a t i n g s h o r t , intermediate and long range plans f o r the bank's asset management a c t i v i t i e s . I t i s shown t h a t such a model i s capable of d e a l i n g w i t h the m u l t i p l e goals which represent the bank's o b j e c t i v e s and o b l i g a t i o n s and tha t the model can be formulated t o incor p o r a t e the complex interdepend-encies which e x i s t among the various asset management a c t i v i t i e s . The s o l u t i o n to the model w i l l represent the most s a t i s f a c t o r y course o f . a c t i o n a v a i l a b l e t o the bank i n terms of i t s o r g a n i z a t i o n a l goals. i i TABLE OF CONTENTS CHAPTER - PAGE NO. I INTRODUCTION 1 The Planning Function 2 Management Science i n Banking 3 I I ASSET MANAGEMENT 8 The S t r u c t u r e of Bank Assets 8 The S t r u c t u r e of L i a b i l i t i e s and C a p i t a l 11 The Asset Management Problem 12 T r a d i t i o n a l Approaches to the Asset Management Problem .. . 16 Lin e a r Programming and Asset Management 2 2 I I I PROGRAMMING TO GOALS 2 8 I n t r o d u c t i o n to Goal Programming 2 8 Formulation of a Goal Programming Problem 29 Goal Programming and Banking 33 IV A MULTI-PERIOD ASSET MANAGEMENT MODEL USING GOAL PROGRAMMING 34 Formulation of the Goals 36 Formulation of the Co n s t r a i n t s 46 Objective Function 5 8 Concluding Comments 59 V APPLICATION OF THE GOAL PROGRAMMING MODEL TO A HYPOTHETICAL BANK 60 Computing a S o l u t i o n 63 The Optimal S o l u t i o n 64 VI ANALYSIS OF THE OPTIMAL SOLUTION 6 8 T e s t i n g the R e l i a b i l i t y of the S o l u t i o n 70 I n t e r p r e t a t i o n of the Dual S o l u t i o n 75 i i i CHAPTER PAGE NO. V I I I DISCUSSIONS AND CONCLUSIONS 88 I m p l i c a t i o n s o f Model Development 88 I m p l i c a t i o n s o f Model Implementation 91 O r g a n i z a t i o n a l I m p l i c a t i o n s : P l a n n i n g and C o n t r o l 92 Conclusi o n s 10 4 BIBLIOGRAPHY , 106 APPENDIX E x h i b i t I M u l t i p l e Goal Programming 109 E x h i b i t I I D e f i n i t i o n of Symbols 113 E x h i b i t I I I Input Data 118 E x h i b i t IV Optimal S o l u t i o n and Cost Ranging R e s u l t s 124 E x h i b i t V Dual S o l u t i o n and Right Hand Side Ranging R e s u l t s 129 CHAPTER I INTRODUCTION This t h e s i s examines the a p p l i c a t i o n of a mathematical programming technique known as goal programming to the resource a l l o c a t i o n problem faced by commercial banks. The d i s c u s s i o n w i l l o u t l i n e the dimensions of the asset planning problem and i d e n t i f y the d e f i c i e n c i e s inherent i n the t r a d i t i o n a l approaches to s o l v i n g the problem. To i l l u s t r a t e the goal programming approach to asset management, a simple model of a h y p o t h e t i c a l bank w i l l be presented. Data w i l l be input to the model and a s o l u t i o n w i l l be determined. The s o l u t i o n w i l l be used to develop numerical examples which i n d i c a t e some of the ways i n which the model can a s s i s t the bank's management i n a n a l y z i n g a number of t y p i c a l banking problems. A l a t e r p a r t of t h i s t h e s i s w i l l d eal w i t h the i m p l i c a t i o n s of developing and implementing t h i s type of model, and p o i n t out the p o s s i b i l i t i e s f o r using the model as a b a s i s f o r a r e s p o n s i b i l i t y accounting system. I t i s assumed th a t the reader has some f a m i l i a r i t y w i t h l i n e a r programming theory and terminology, however, no e x p l i c i t knowledge of goal programming i s assumed and the theory u n d e r l y i n g the technique w i l l be f u l l y e xplained. Without exception, p u b l i c a t i o n s d e a l i n g w i t h the a p p l i c a t i o n of a n a l y t i c a l techniques i n banking have d e a l t w i t h American banking s i t u a t i o n s . In order to r e l a t e t h i s t h e s i s t o the work which has been done i n t h i s area, i t was necessary t o develop the t o p i c w i t h i n the context of the American banking environment. -2-The Planning Function The success of a commercial bank, l i k e t hat of any other business, depends l a r g e l y on the a b i l i t y of i t s management to formulate e f f e c t i v e plans and insure t h e i r implementation. One of the most important elements of planning i s the s e t t i n g of d e s i r a b l e o b j e c t i v e s or goals toward whose attainment the a c t i v i t i e s of the bank w i l l be d i r e c t e d . A normative d e s c r i p t i o n of commercial bank investment behavior might conceive a banker to be an i n v e s t o r who attempts to a l l o c a t e a v a i l a b l e resources i n such as way as to maximize h i s shareholder's wealth, subject t o the l e g a l requirements imposed by the c e n t r a l banking a u t h o r i t y and t o the degree of r i s k exposure t h a t the shareholders are p s y c h o l o g i c a l l y w i l l i n g to t o l e r a t e . 1 This d e s c r i p t i o n i m p l i e s that the choosing of investments f o r a bank i s s i m i l a r t o the s e l e c t i o n of products to be produced by a manufacturing f i r m . The manufacturing f i r m seeks to determine the mix of products which w i l l i n s ure t h a t the goals of the f i r m are most completely s a t i s f i e d , w hile the bank seeks the mix of assets most l i k e l y to r e s u l t i n the attainment of i t s s t a t e d goals. Manufacturing firms have come to r e a l i z e the usefulness of l i n e a r programming and other mathematical models as planning t o o l s . In a t y p i c a l case a manufacturer, faced by the c o n s t r a i n t s of production c a p a c i t y and market demand, attempts to f i n d some optimal mix of products. This optimal mix w i l l then be used as Donald R. Hodgman, Commercial Bank Loan and Investment P o l i c y , Champaign, I l l i n o i s : U n i v e r s i t y of I l l i n o i s , 196 3, p.113. -3-a guide f o r planning f u t u r e o p e r a t i o n s . A l i n e a r programming model permits the high degree of f l e x i b i l i t y which i s d e s i r e -able i n planning. When management i s confronted by changing c o n d i t i o n s or new problems i t w i l l be easy f o r them t o adapt to the new c o n d i t i o n s by a l t e r i n g the c o n s t r a i n t s or the o b j e c t i v e f u n c t i o n expressed by the l i n e a r programming model. Further more l i n e a r programming allows the manufacturing f i r m to q u i c k l y determine the probable e f f e c t of a c t i v i t i e s which deviate from the optimal p l a n . Such a model does not replace the d e c i s i o n making f u n c t i o n of management, but r a t h e r gives management a meaningful framework on which to base d e c i s i o n s . Management Science i n Banking The banking community has c e r t a i n l y lagged the manufacturing s e c t o r i n the a p p l i c a t i o n of mathematical models to planning o problems. There are s e v e r a l important reasons why t h i s i s so. (1) F i n a n c i a l C o n s t r a i n t s Even had the banks been aware of the p o t e n t i a l of a n a l y t i c a l techniques i t appears t h a t only a minute percentage of the t o t a l number of banks i n the United States were large enough to a f f o r d the i n i t i a l o u tlays r e q u i r e d t o finance not only the research and development of the techniques themselves, but a l s o the c o s t l y data r e t r i e v a l and cost accounting systems which would be p r e r e q u i s i t e to t h e i r implementation. Kalman J . Cohen and F r e d r i c k S. Hammer, ed., "Operations Research: A new approach to bank d e c i s i o n making," A n a l y t i c a l Methods i n Banking, Richard D. I r w i n , Inc., Homewood, I l l i n o i s , 1966, pp. 7-9 (2) The Banking Environment The c l i c h e t h a t " n e c e s s i t y i s the mother of i n n o v a t i o n " may a p t l y be a p p l i e d to commercial banking's recent i n t e r e s t i n management sc i e n c e . From the Second World War through the mid 1950's the commercial banks enjoyed growing deposits and p r o f i t s . Only since the l a t e 1950's have the banks been confronted by t i g h t e n i n g money, p e r i o d i c r e c e s s i o n s , and i n c r e a s i n g l y s t i f f competition from other f i n a n c i a l i n s t i t u t i o n s . (3) Personnel C o n s t r a i n t s Banks lacked the t e c h n i c a l l y t r a i n e d people who were able to understand and appreciate the a p p l i c a t i o n s of mathematical techniques t o management problems. I t i s only r e c e n t l y t h a t engineers and mathematicians have j o i n e d bank s t a f f s to analyze i n d u s t r i a l loans and design systems f o r the e x i s t i n g computers whose use had been r e s t r i c t e d t o data processing a p p l i c a t i o n s . (4) The Nature of Banking Problems F i n a l l y , i t i s j u s t i f i a b l e to conclude that the problems faced by bank management are more complex than the c l a s s e s of i n d u s t r i a l management problems to which a n a l y t i c a l techniques were f i r s t a p p l i e d . The normative d e s c r i p t i o n of banking behaviour given above was used to i l l u s t r a t e the s i m i l a r i t i e s between the processes of asset s e l e c t i o n by a bank and the s e l e c t i o n of a product mix by a manufacturing f i r m . I t i s -5-now necessary to discuss the fundamental d i f f e r e n c e s between the two processes. The e a r l y i n d u s t r i a l a p p l i c a t i o n s of a n a l y t i c a l methods d e a l t w i t h the a l l o c a t i o n of p h y s i c a l resources which were both t a n g i b l e and to a larg e degree c o n t r o l l a b l e . The problems faced by bank management centre around fund flows which are ephemeral i n nature. The bank's resources c o n s i s t of an investment fund which i s equal to some f r a c t i o n of the t o t a l of a l l i n v e s t a b l e resources a v a i l a b l e to the e n t i r e banking system. This t o t a l i s determined e s s e n t i a l l y by the p o l i c i e s of the c e n t r a l monetary a u t h o r i t i e s and i s th e r e f o r e u l t i m a t e l y r e l a t e d to the pace of economic a c t i v i t y . The banker was described as an i n v e s t o r who compares a l t e r n a t i v e s i n terms of t h e i r e f f e c t i v e rates of r e t u r n and t h e i r r i s k c h a r a c t e r i s t i c s . He then s e l e c t s t h a t mix of investments which r e s u l t s i n some optimal t r a d e - o f f between y i e l d and r i s k per d o l l a r i n v e s t e d . Such a d e s c r i p t i o n ignores the su b t l e r e l a t i o n s h i p s w i t h which a banker must contend. In the short run, changes i n market f a c t o r s w i l l not a l t e r a manufacturer's production c a p a c i t y . However, the le n d i n g c a p a c i t y of a bank depends to a considerable extent upon the l e v e l of i t s demand deposits which i n turn depend on the bank's loan p o l i c y . The preceding d i s c u s s i o n would i n d i c a t e t h a t the c h a r a c t e r i s t i c s of the environment, w i t h i n which the banker operates, impose upon him a set of goals which are much more -6-complex than those which confront a manufacturer. In banking, the goal of p r o f i t maximization must be subordinated to the goal of maintaining l i q u i d i t y which i s adequate to meet d e p o s i t o r demands f o r cash and c r e d i t . The manufacturing f i r m a l s o has l i q u i d i t y requirements but these are r e l a t i v e l y easy to c a l c u l a t e . A manufacturer knows approximately what h i s b i l l s w i l l be and f u r t h e r more he knows when payment i s due. In c o n t r a s t the banker must be ready to pay on demand, the people to whom the bank owes money. The c o m p l e x i t i e s of bank management, r a t h e r than m i l i t a t i n g against the use of a n a l y t i c a l methods, make the a p p l i c a t i o n of such techniques as l i n e a r programming even more d e s i r a b l e . F i n a l l y , the econometric research, conducted by government, business and the academic community, may soon develop means of i d e n t i f y i n g and measuring some of the complex forces a f f e c t i n g the economic v a r i a b l e s which are c r u c i a l i n banking problems."'' The o b j e c t i v e of a goal programming problem i s to minimize the undesirable d e v i a t i o n s from predefined goals. A model of t h i s k i n d i s appropriate f o r planning since i t w i l l provide an e f f e c t i v e and f l e x i b l e means of expressing the i n t e r r e l a t i o n s which e x i s t among the m u l t i p l e goals of bank management. In developing plans f o r the f u t u r e , bank management w i l l e s t a b l i s h goals or o b j e c t i v e s . These goals w i l l r e f l e c t management's d e s i r e to achieve adequate p r o f i t a b i l i t y w h i l e maintaining s u f f i c i e n t l i q u i d i t y . a n d a balanced p o r t f o l i o . o f Lawrence R. K l e i n , An I n t r o d u c t i o n to Econometrics, P r e n t i c e -H a l l , Englewood C l i f f s , New Jersey, 1962, pp. 236 - 271. -7-loans and investments. The goal programming model which incorporates these c o n s i d e r a t i o n s only promises to do i n a more e f f i c i e n t way the same t h i n g bank management i s already doing. CHAPTER I I ASSET MANAGEMENT Though r o u t i n e i n nature, asset management i s a problem of such importance t h a t i t must be d e a l t w i t h on a day-to-day b a s i s by the bank's top management. Much of the recent l i t e r a t u r e i n the f i e l d of bank management i s devoted t o attempts t o develop a systematic approach t o asset management d e c i s i o n making. This chapter w i l l i n clude a d i s c u s s i o n of the methods c u r r e n t l y i n use as w e l l as those which have been proposed. However, i n order to develop a frame of reference fo r f u r t h e r a n a l y s i s , the f o l l o w i n g s e c t i o n w i l l d eal w i t h the general aspects of the problem. The S t r u c t u r e of Bank Assets Bank management i s faced w i t h two c o n f l i c t i n g requirements: making a p r o f i t , and guaranteeing the s a f e t y of the bank. Every p o s s i b l e mix of assets represents some t r a d e - o f f between earnings and r i s k . When d e c i d i n g among a l t e r n a t i v e uses of funds i t i s necessary to keep i n mind the impact of t h i s d e c i s i o n on the o v e r a l l t r a d e - o f f described by the e x i s t i n g asset mix. Decisions concerning the s e l e c t i o n of bank assets are i n f l u e n c e d or constrained not only by the competitive and economic environment, but a l s o by government r e g u l a t i o n s , bank examiners g u i d e l i n e s , i n t e r n a l p o l i c y and the tendency f o r an i n d i v i d u a l bank to conform t o the general p r a c t i c e s of the banking community. These f a c t o r s combine t o y i e l d the general s t r u c t u r e of banking - 9 -a s s e t s . Since i n d i v i d u a l assets l i e along continuums which describe t h e i r r e l a t i v e r i s k , l i q u i d i t y and earnings c h a r a c t e r i s t i c s , no s i n g l e method of c l a s s i f y i n g bank assets i s s a t i s f a c t o r y i n a l l respects. The groupings used i n t h i s d i s c u s s i o n w i l l c l a s s i f y assets i n terms of t r a d i t i o n a l p r i o r i t i e s which have been e s t a b l i s h e d f o r the use of bank funds. (1) Primary Reserves Since the banks l i a b i l i t i e s are payable i n cash, any use of cash r e s u l t s i n an increase i n r i s k . Holding cash provides the bank w i t h the l i q u i d i t y r e q u i r e d to meet deposit withdrawals and support cheque c l e a r i n g operations and other c r e d i t o b l i g a t i o n s among banks. On top of these cash requirements which may be considered the minimum, are the l e g a l requirements to maintain cash reserves w i t h the c e n t r a l banking a u t h o r i t y . Since cash holdings must not f a l l below the l e g a l reserve requirement f o r any length of time, only cash or h i g h l y l i q u i d asset holdings i n excess of t h i s requirement can be used to meet cash demands. (2) Secondary Reserves The purpose of the bank's secondary reserves i s to provide f o r a n t i c i p a t e d cash needs as w e l l as to insure the bank's s a f e t y by p r o v i d i n g f o r unforseen contingencies. The a n i t i c i p a t e d needs are derived from f o r e c a s t changes i n deposit l e v e l s or loan demand over the short t o intermediate run. The reserve f o r unforseen cash demands provide f o r deposit s h i f t s which r e s u l t from i r r e g u l a r f l u c t u a t i o n s i n the l e v e l of economic a c t i v i t y and l o c a l -10-depressions. The investments which make up the secondary reserves c o n s i s t mainly of bonds issued by F e d e r a l , s t a t e and mun i c i p a l governments, or t h e i r agencies. The i n d i v i d u a l bonds vary as t o maturity date, i n t e r e s t r e t u r n , and l i q u i d i t y . The o v e r a l l composition of the investment p o r t f o l i o r e f l e c t s a n t i c i p a t e d l i q u i d i t y needs and s u f f i c i e n t funds must be i n a form which i s h i g h l y l i q u i d and can be converted to cash without l o s s . (3) Loan P o r t f o l i o T r a d i t i o n a l l y the c h i e f source of a commercial bank's income i s i t s c r e d i t o p e rations. This occurs because the banks, i n comparison w i t h other f i n a n c i a l i n s t i t u t i o n s , are s t r a t e g i c a l l y s u i t e d f o r l e n d i n g . Since the c e n t r a l f u n c t i o n of a commercial bank i s to meet the c r e d i t needs of i t s community, the loan account i s u s u a l l y the l a r g e s t of the asset c a t e g o r i e s . The loans w i t h i n t h i s account range from demand loans, which may be converted to cash on short n o t i c e to the borrower, to term loans of negotiated m a t u r i t y , t o mortgage loans which committ the bank to long term l e n d i n g . Since i n d i v i d u a l loans vary as to the c h a r a c t e r i s t i c s of the borrower, and the purposes f o r which they are made, i t would be expected th a t the c o s t , to the bank, of le n d i n g would a l s o vary from loan to loan. Because of these f a c t o r s the loan p o r t f o l i o e x h i b i t s a wide v a r i e t y of y i e l d s and r i s k s . - l i -l t i s w i t h i n t h i s category of assets t h a t the h i g h e s t i n t e r e s t returns are a v a i l a b l e . However, the bank's r e t u r n on loans may not always be commensurate wi t h the r i s k i n v o l v e d i n l e n d i n g . I t should be noted t h a t i f a bank refuses t o take m a t e r i a l r i s k s i n i t s c r e d i t operations i t must face the p o s s i b i l i t y of l o s i n g deposits to l e s s c onservative competitors. (4) Investment f o r Earnings I f the bank has funds at i t s d i s p o s a l a f t e r s a t i f y i n g i t s l i q u i d i t y needs and customer demand f o r l o a n s , i t should a l l o c a t e these funds t o open market investments f o r earning. I t should be emphasized t h a t because such investments are long term i n nature, l i q u i d a t i o n of these investments before t h e i r m a t u r i t y c a r r i e s a s u b s t a n t i a l r i s k of l o s s . For t h i s reason they should not be considered a source of l i q u i d i t y f o r meeting demands f o r cash or c r e d i t . The S t r u c t u r e of L i a b i l i t i e s and C a p i t a l The types of deposits h e l d by a bank vary as to the nature of the d e p o s i t o r , the v e l o c i t y or turnover r a t e , processing and i n t e r e s t c o s t s , and the l e g a l reserve requirements which they impose on the bank. I t i s the p r i v i l e g e of depositors to draw down t h e i r accounts when they r e q u i r e funds, or to l e t funds accumulate when they have no immediate use f o r them. Though short run f l u c t u a t i o n s i n i n d i v i d u a l accounts may be o f f s e t t i n g , the bank must a n t i c i p a t e and be prepared f o r general movements i n deposit l e v e l s over longer periods of time. -12-C e r t i f i c a t e s of deposit may be the most v o l a t i l e of deposit types. The large depositors who h o l d these c e r t i f i c a t e s , u s u a l l y governments and c o r p o r a t i o n s , are very s e n s i t i v e to the r e l a t i o n between the y i e l d a v a i l a b l e from these deposits and the y i e l d a v a i l a b l e from open market instruments. In periods of high i n t e r e s t r a t e s , the l e g i s l a t e d c e i l i n g on deposit i n t e r e s t r ates may cause the banks t o experience a s u b s t a n t i a l d r a i n i n t h e i r c e r t i f i c a t e of deposit accounts. The costs and r i s k s a s s o c i a t e d w i t h short term promissary notes and long term c a p i t a l debentures d i f f e r s i g n i f i c a n t l y from those a s s o c i a t e d w i t h deposits and e q u i t y c a p i t a l . Borrowing from the Federal Reserve i s considered a l a s t r e s o r t i n money management. This source of funds i s not intended f o r r e g u l a r use but only i n case on unexpected developments.1 A bank may increase the s i z e of i t s c a p i t a l account by i s s u i n g common shares or by r e t a i n i n g earnings. The bank i s i n essence a h i g h l y levered f i r m and i t w i l l be shown i n a l a t e r s e c t i o n t h a t small changes i n the s i z e of the c a p i t a l accounts can cause s i g n i f i c a n t changes i n earnings. The Asset Management Problem Decisions i n the area of asset management focus on the bank's balance sheet. Asset management must determine the s i z e of the bank's t o t a l assets and t h e r e f o r e i t s t o t a l l i a b i l i t i e s . The composition of a s s e t , l i a b i l i t y , and c a p i t a l accounts must Roland I . Robinson, The Management of Bank Funds, Second E d i t i o n , McGraw-Hill, New York, lyf c ^ , p. «4 -13-a l s o be determined. The s i z e of the bank's primary reserves are determined by f a c t o r s which are l a r g e l y beyond the c o n t r o l of asset management. Such f a c t o r s i n c l u d e the l e g a l reserve requirement and the necessary minimum l i q u i d i t y b u f f e r . I t was p r e v i o u s l y mentioned t h a t the banks investments vary as t o i n t e r e s t r a t e , m a t u r i t y , l i q u i d i t y , m a r k e t a b i l i t y , and r i s k of d e f a u l t . For a given investment p o r t f o l i o the net e f f e c t of these f a c t o r s i s represented by the p o r t f o l i o s market y i e l d . Therefore i t becomes a f u n c t i o n of asset management to determine the t r a d e - o f f which i s acceptable between y i e l d and the other f a c t o r s . In a s i m i l a r manner, determining the appropriate mix of loans and the s i z e and s t r u c t u r e of l i a b i l i t y and c a p i t a l accounts are a l s o important elements of asset management. To t h i s p o i n t asset management has been viewed i n terms of i t s r o l e i n e s t a b l i s h i n g a s u i t a b l e s i z e and composition f o r the loan and investment p o r t f o l i o s , and f o r the l i a b i l i t y and c a p i t a l accounts, i n order to achieve an optimum balance sheet. However, i t would be a mistake to assume t h a t d e c i s i o n s concerning a p a r t i c u l a r account, say the loan p o r t f o l i o , can be made independently of d e c i s i o n s concerning the r e s t of the balance sheet. The problem of asset management i s f u r t h e r complicated by the f a c t t h a t the impact of a d e c i s i o n concerning one p a r t of the balance sheet may a f f e c t the status of a l l other a s s e t s , l i a b i l i t y , and c a p i t a l accounts, as w e l l as a f f e c t i n g the -14-r i s k - y i e l d t r a d e - o f f embodied by the e x i s t i n g asset s t r u c t u r e . I t i s o f t e n p o s s i b l e t o increase the y i e l d from the asset 1 2 mix by reducing i t s l i q u i d i t y . ' However, attempts to maximize y i e l d i n the short run would r e q u i r e lengthening the average maturity of the investment p o r t f o l i o and thereby increase the r i s k of forced l i q u i d a t i o n of investments i n order t o meet loan demand and deposit withdrawals. Conversely, the maintenance of excess l i q u i d i t y may cause the bank to experience a s u b s t a n t i a l opportunity cost. This cost r e s u l t s from earnings forgone and from the l o s s of customers to competitors. To f i n d the optimal balance between earnings and l i q u i d i t y i t i s necessary t o consider the i n t e r a c t i o n s which e x i s t between asset, l i a b i l i t y , and c a p i t a l accounts. Furthermore, current d e c i s i o n s concerning the t r a d e - o f f between y i e l d and l i q u i d i t y are r e f l e c t e d not only i n the current time p e r i o d , but a l s o throughout s e v e r a l f u t u r e time periods. Since current d e c i s i o n s a f f e c t f u t u r e o p p o r t u n i t i e s , i t i s necessary t h a t management considers the impact of t h e i r a c t i o n s over time, up to a s u f f i c i e n t l y d i s t a n t planning h o r i z o n . Present d e c i s i o n s must be based on f o r e c a s t s of f u t u r e loan demand, i n t e r e s t r a t e s , and deposit l e v e l s . C onsideration must a l s o be given to p o s s i b l e changes i n the l e g a l and economic environment. The degree of u n c e r t a i n t y inherent i n such f o r e c a s t s i s no excuse f o r i g n o r i n g the f u t u r e . To operate e f f e c t i v e l y , d e c i s i o n makers r e q u i r e not only r e l i a b l e f o r e c a s t s , but a l s o r e l e v a n t i n f o r m a t i o n concerning 1See J.R. H i c k s , Value and C a p i t a l , Second E d i t i o n , Oxford U n i v e r s i t y Press, London, 1946, pp. 144 - 152. 2 See a l s o J.W. Conrad, The Behaviour of I n t e r e s t Rates, N a t i o n a l Bureau of Economic Research, New York, 1966, pp. 71 - 105. -15-the costs and y i e l d s a s s o c i a t e d w i t h the various types of assets and d e p o s i t s . I t i s recognized that i n f o r m a t i o n p r o v i d i n g marginal cost and revenue measurements i s s u p e r i o r to information which provides measurements of average or standard costs and revenues. Unfortunately i t i s the l a t t e r type of data t h a t i s u s u a l l y a v a i l a b l e from the accounting system, and i n the banking s i t u a t i o n there i s a high p r o b a b i l i t y t h a t a d e c i s i o n based on an average e v a l u a t i o n w i l l d i f f e r from the d e c i s i o n which would have been made i f marginal evaluations had been p o s s i b l e . This occurs because an e v a l u a t i o n based on average costs or y i e l d s tends t o ignore the i n t e r r a c t i o n s which e x i s t among the bank's asset and l i a b i l i t y accounts. For example, long term borrowing by the bank not only provides a d i r e c t r e t u r n from i n v e s t i n g a d d i t i o n a l funds i n earning a s s e t s , but a l s o provides an i n d i r e c t r e t u r n by p e r m i t t i n g an asset mix i n which a l a r g e r p r o p o r t i o n of the bank's funds are inv e s t e d i n higher y i e l d i n g , l e s s l i q u i d a s s e t s . The preceding d i s c u s s i o n of asset management has been conceptual i n nature. I t i s now necessary to discuss s p e c i f i c s t r a t e g i e s and p o l i c y c o n s i d e r a t i o n s w i t h i n t h i s conceptual framework and develop an a n a l y t i c a l approach to the problem. To be v a l i d the approach must give some expression to the complex l i q u i d i t y - y i e l d t r a d e - o f f s which e x i s t . I t must a l s o recognize the nature of the i n t e r r a c t i o n s among the various balance sheet accounts and the ways i n which present d e c i s i o n s e f f e c t the bank's future p o s i t i o n . F i n a l l y , i t must be able to provide the inf o r m a t i o n r e q u i r e d i n order to use marginal a n a l y s i s as the b a s i s f o r making d e c i s i o n s . -16-T r a d i t i o n a l Approaches to the Asset Management Problem The t r a d i t i o n a l approaches t o asset management are s t r a t e g i c i n nature, and are based on h a b i t u a l procedures and h e u r i s t i c judgement c r i t e r i a . In g e n e r a l , i t i s recognized that good banking s t r a t e g y c a l l s f o r the subordination of investment p o l i c y to loan p o l i c y . The bank's competitive advantage f o r lending i s not the sole reason f o r t h i s s t r a t e g y . Bankers a l s o recognize t h a t the l e v e l of demand deposits determine the banks capacity to lend and th e r e f o r e i t s earning c a p a c i t y . Consequently, i t i s v i t a l to a t t r a c t , s a t i s f y , and r e t a i n deposit customers. This can only be done i f the bank provides i t s e l f w i t h adequate l i q u i d i t y i n i t s investment p o r t f o l i o t o meet deposit customer's demands f o r l o a n s . 2 One type of feedback which e x i s t s between loans and demand deposits i s i l l u s t r a t e d by Hodgman.^ He demonstrates the d i f f e r e n c e between lending t o depositors and non d e p o s i t o r s . He p o i n t s out t h a t since a loan to a dep o s i t o r c o n s t i t u t e s a smal l e r d r a i n on a bank's lending c a p a c i t y than a loan to a non d e p o s i t o r , the y i e l d on the two types of loans i s not comparable. This e x p l a i n s the w i l l i n g n e s s of banks, i n c e r t a i n p e r i o d s , to make loans to depositors at c o n t r a c t r a t e s which are Donald R. Hodgman, Commercial Bank Loan and Investment P o l i c y , Champaign, I l l i n o i s : U n i v e r s i t y of I l l i n o i s , 196 3, p. 97. 3 I b i d . , pp. 113 - 12 3. -17-l e s s than the market rate of i n t e r e s t obtainable on a l t e r n a t i v e investment instruments. Granting such loans makes a v a i l a b l e both present and f u t u r e d e p o s i t s . These i n turn generate future earnings which may more than o f f s e t the opportunity c o s t of the i n i t i a l loans at c o n t r a c t r a t e s . (1) Balance Sheet Ratios In order to allow f o r the i n t e r a c t i o n s which e x i s t among the various accounts, bank management makes use of a number of r a t i o s which serve as primary c r i t e r i a i n determining the s i z e and composition of the loan and investment p o r t f o l i o s . Such r a t i o s i n c l u d e : the r a t i o of loans to deposits,•loans to c a p i t a l , r i s k assets to c a p i t a l , and bonds to t o t a l a s s e t s . According to Roland Robinson; "Banks s c r u t i n i z e one anothers f i n a n c i a l statements i n great d e t a i l and tend to j u s t i f y t h e i r p o l i c i e s on the b a s i s of s i m i l a r i t y to p o l i c i e s 4 followed by other banks". As would be expected, t h i s tendency to conform leads to a general concensus among bankers as to what values are d e s i r a b l e f o r key balance sheet r a t i o s . I f one bank departs too g r e a t l y from these s u b j e c t i v e standards, i t i s l i k e l y to f e e l pressure from the t r e a s u r e r s of l a r g e corporate d e p o s i t o r s , investment s e r v i c e s , and supervisory agencies, as w e l l as i n c u r r i n g the d i s a p p r o v a l of other banks. Roland I . Robinson, The Management of Bank Funds, Second E d i t i o n , McGraw-Hill, New York, 1962, pp. 45 - 46. -18-The value of such r a t i o s , as o p e r a t i o n a l t o o l s , i s h i g h l y questionable. Other than the f a c t that they are de r i v e d by concensus, v i r t u a l l y nothing i s known about them. I t i s not known what values of various balance sheet r a t i o s are c r i t i c a l , and why. I t has not been determined how they should vary w i t h the composition of the loan or investment p o r t f o l i o , or how t h e i r values should r e f l e c t such e x t e r n a l phenomena as i n t e r e s t r a t e s , money market c o n d i t i o n s , and future expectations. Furthermore, some measures, such as the r i s k assets to c a p i t a l r a t i o , assume a simple dichotomy between r i s k and non-risk a s s e t s , when i n a c t u a l i t y the r i s k c h a r a c t e r i s t i c s of assets should be measured along a continuum. (2) Examiner's C r i t e r i a In order to r e f l e c t the i n t e r a c t i o n s among balance sheet accounts i n a more comprehensive manner, s e v e r a l r e g u l a t o r y agencies have devised grading systems.^ The most s o p h i s t i c a t e d of these i s the t e s t of bank s a f e t y and c a p i t a l adequacy, designed by the examiners of the Board of Governors of the Federal Reserve System. The c r i t e r i o n , embodied i n t h i s t e s t , r e f l e c t the examiner's judgement 5 I b i d . , pp. 165 - 172. 6 I b i d . , pp. 173 - 190. -19-as t o what a l l o c a t i o n s - o f assets are reasonably safe given the u n c e r t a i n t i e s which the bank faces. In e f f e c t the c r i t e r i a e s t a b l i s h a leverage r e s t r i c t i o n f o r any given asset and deposit mix. As assets become l e s s current or l i a b i l i t i e s more c u r r e n t , the need f o r p o t e n t i a l l i q u i d i t y increases and the leverage requirement becomes more r e s t r i c t i v e i n terms of the amount of c a p i t a l r e q u i r e d to support the asset mix. C a p i t a l i s r e q u i r e d against the r i s k t h a t the bank may be forced to l i q u i d a t e c e r t a i n a s s e t s , as w e l l as against the r i s k of d e f a u l t by borrowers. Losses from the former event occur when some assets must be s o l d f o r l e s s than t h e i r book value. The amount of c a p i t a l r e q u i r e d t o support a given asset, i n the event of such p o s s i b i l i t i e s , i s determined by the examiner's estimate of the asset's r i s k i n e s s . For a given asset mix, the c a p i t a l requirement a l s o increases as the bank's l i a b i l i t y s t r u c t u r e becomes more c u r r e n t . To determine a bank's score on the t e s t , the examiners compute the r a t i o of the bank's e x i s t i n g net worth t o the amount of c a p i t a l r e q u i r e d as c a l c u l a t e d by the c r i t e r i a . The c a p i t a l requirement, determined by the examiner's c r i t e r i a , does not c o n s t i t u t e a l e g a l requirement which must be met by the bank. The c r i t e r i a are the product of the examiner's many years of experience i n the f i e l d of banking. They r e f l e c t a high degree of conservatism and i t i s h i g h l y improbable t h a t a bank which s a t i s f i e s them w i l l become f i n a n c i a l l y i n s o l v e n t . -20-I t i s the "mutatis mutandis" approach to measurement th a t makes the examiner's c r i t e r i a s i g n i f i c a n t l y s u p e r i o r to the t r a d i t i o n a l r u l e s of thumb described i n the previous s e c t i o n . The c r i t e r i a provide a method of d e a l i n g w i t h u n c e r t a i n t y , but i t i s impossible to q u a n t i f y the degree of r i s k aversion which they represent. (3) Asset A l l o c a t i o n Asset a l l o c a t i o n c o n s t i t u t e s a systematic approach to asset management which guides d e c i s i o n s p r i m a r i l y on the b a s i s of l i q u i d i t y c o n s i d e r a t i o n s . I t has a l s o been claimed that asset a l l o c a t i o n provides the b a s i s f o r both p r o f i t 7 planning and r e s p o n s i b i l i t y accounting. The technique of asset a l l o c a t i o n represents an attempt t o recognize the d i f f e r i n g l i q u i d i t y requirements a s s o c i a t e d w i t h the various sources of funds. The amount of funds a l l o c a t e d t o each of the various asset categories i s determined by the sources from which the funds were obtained. The r u l e which governs t h i s a l l o c a t i o n procedure r e s t s on the assumption that the v e l o c i t y or turnover rate of the source of funds d i c t a t e s the type and maturity of the asset i n which the funds should be inv e s t e d . This i m p l i e s t h a t v o l a t i l e sources of funds such as demand deposits be a l l o c a t e d to cash Fred G. Delong, " L i q u i d i t y Requirements and Employment of Funds", A n a l y t i c a l Methods i n Banking, ed. Caiman J . Cohen and F r e d r i c k S. Hammer, Richard D. I r w i n , Inc., Homewood, I l l i n o i s , 1966, pp. 39 - 45. -21-and short term loans and investments, while s t a b l e funds,, such as savings deposits and c a p i t a l be i n v e s t e d i n longer term assets. The short-comings inherent i n the asset a l l o c a t i o n technique severely undermine i t s usefulness as a management o technique. F i r s t , the premise, t h a t a v a i l a b l e funds be used t o support assets appropriate to the turnover r a t e of these funds, ignores the f a c t t h a t a s u b s t a n t i a l amount of the t o t a l volume of demand deposit funds may be used to support long term, high y i e l d , a ssets. The reason f o r t h i s i s t h a t , i n the short run, deposits and withdrawals are roughly equal and o f t e n no l i q u i d a t i o n of assets w i l l be r e q u i r e d . A l s o , since the l e v e l of demand deposits i s d i r e c t l y r e l a t e d to the l e v e l of business a c t i v i t y , asset a l l o c a t i o n tends t o increase the bank's l i q u i d i t y , f o r the purpose of meeting deposit withdrawals, at the same time t h a t loan demand i s r i s i n g . This i s contrary to b a s i c banking s t r a t e g y , which d i c t a t e s , t h a t during periods of i n c r e a s i n g business a c t i v i t y , the l i q u i d i t y requirement f o r loans takes p r i o r i t y over the l i q u i d i t y requirement f o r withdrawals. Asset a l l o c a t i o n r e s t s on the p r i n c i p l e t h a t the uses f o r funds depend on the sources, but i t ignores the converse to the p r i n c i p l e by assuming t h a t the sources are determined independently of the uses. The approach f a i l s to recognize t h a t loan p o l i c y can have a s i g n i f i c a n t e f f e c t on deposit Cohen and Hammer, i b i d . , pp. 45 - 53. -22-l e v e l s . I f a bank f a i l s to keep pace w i t h loan demand i t w i l l l ose deposits to competitors. I t i s recognized t h a t a systematic approach to asset management may provide the b a s i s f o r a system of accounting f o r planning and c o n t r o l . However, as was pointed out e a r l i e r , an accounting system i n order to be an e f f e c t i v e a i d t o d e c i s i o n making, must provide marginal cost and y i e l d data. Since asset a l l o c a t i o n s u p p l i e s only i n f o r m a t i o n on average costs and y i e l d s , i t does not meet t h i s requirement. For a l l these reasons, i t must be concluded t h a t the asset a l l o c a t i o n technique has se r i o u s d e f i c i e n c i e s . L i n e a r Programming and Asset Management As a prelude to the development of a goal programming model f o r asset management, i t i s u s e f u l at t h i s p o i n t to review the past developments upon which the model i s based. The e s s e n t i a l problem of asset management i s one of resource a l l o c a t i o n and i t has been demonstrated t h a t t h i s resource a l l o c a t i o n problem can be formulated as a l i n e a r programming model. (1) Simple One P e r i o d Models Waterman and Gee^, and Teichroew-*-0 have developed l i n e a r ^Robert G. Waterman and Robert E. Gee, "A New Tool For Bank Management: A Mathematical Model i n Banking", A n a l y t i c a l Methods  i n Banking, ed. Kalman J . Cohen and F r e d r i c k S. Hammer, Richard D. I r w i n , Inc., Homewood, I l l i n o i s , 1966, pp. 55 - 62. Daniel Teichroew, An I n t r o d u c t i o n to Management Science, John Wiley and Sons, Inc., New York, 196 4, pp. 49 2 - 49 7. -23-programming models which a l l o c a t e a banks resources among various assets i n such a way as to maximize gross earnings f o r one per i o d . The c o n s t r a i n t s on the a l l o c a t i o n represent l e g a l requirements and p o l i c y c o n s i d e r a t i o n s . The p o l i c y c onsiderations are expressed i n the form of t r a d i t i o n a l balance sheet r a t i o s and other r u l e s of thumb. The problem which they have defined can be described as f o l l o w s : n _ Maximize 3 = ^ c.x. 1=1 ^ D subject to the f o l l o w i n g c o n s t r a i n t s : m n a. .x . ^ » b . i = l j = i ID D — i where x . ^ 0 f o r j = l , 2, ..n D Each of the v a r i a b l e s , x_., represent a f i n a n c i a l asset account, while the c^ represents the net y i e l d a v a i l a b l e from the j t h asset. The balance i n the deposit and c a p i t a l accounts are known, and the ob j e c t i s to a l l o c a t e these funds among the various asset accounts i n such a way as t o maximize p r o f i t without m n _ v i o l a t i n g the c o n s t r a i n t s , defined by the term, ~> ~ 2_Za. . x. i = l j 1 3 3 b^, where the a^ _. represent the c o e f f i c i e n t s of the c o n s t r a i n t matrix and the b^ the c o e f f i c i e n t of the r i g h t hand side column vect o r . The c o n s t r a i n t s attempt to insure t h a t the bank's asset s t r u c t u r e i s s u f f i c i e n t l y d i v e r s i f i e d and t h a t s u f f i c i e n t funds are h e l d i n cash or near cash investments i n order to provide f o r p o t e n t i a l l i q u i d i t y requirements. -24-Th i s type of approach to the asset management problem has s e v e r a l advantages. These advantages w i l l be discussed at length i n chapter V. I t i s more re l e v a n t at t h i s time to elaborate on the inadequacies of t h i s type of model which undermine i t s usefulness i n a c t u a l s i t u a t i o n s . H F i r s t , the model takes i n t o account only one time p e r i o d and t h e r e f o r e i t ignores the impact of present d e c i s i o n s on the bank's future p o s i t i o n . This emphasis on short run earnings may lead to sub-optimal d e c i s i o n s over the long run. The approach has many of the weaknesses inherent i n the asset a l l o c a t i o n technique because i t ignores the p o t e n t i a l l i q u i d i t y of any asset category, and the time p a t t e r n of l i q u i d i t y sources r e s u l t i n g from the maturity s t r u c t u r e of the asset p o r t f o l i o . The model a l s o considers only one side of the balance sheet to be v a r i a b l e and ther e f o r e ignores the r e l a t i o n s h i p between loans and deposits as w e l l as i g n o r i n g managements freedom t o a l t e r the s i z e and composition of i t s l i a b i l i t y and c a p i t a l accounts by i s s u i n g c e r t i f i c a t e s of d e p o s i t s , r e t a i n i n g earnings, or i s s u i n g common stock. The i m p l i c a t i o n s of a l t e r i n g the e x i s t i n g asset mix are also ignored. I f the asset mix generated by the model d i f f e r s s i g n i f i c a n t l y from the e x i s t i n g asset mix, the bank w i l l o b v i o u s l y experience c a p i t a l gains or losses i f i t i s re q u i r e d t o decrease i t s holdings of long term bonds. xKalman J . Cohen and F r e d r i c k S. Hammer, ed., A n a l y t i c a l Methods  i n Banking, Richard D. I r w i n , Inc., Homewood, I l l i n o i s , 1966, pp. 62 - 65 -25-Furthermore, the bank may not be able to c o n t r a c t i t s balances i n c e r t a i n loan accounts without a l i e n a t i n g customers who are a l s o d e p o s i t o r s . The c o n s t r a i n t s expressed incorporated i n the model are d e r i v e d mainly from a r b i t r a r i l y determined balance sheet r a t i o s whose shortcomings have already been noted. In t h i s case the use of such r a t i o s e s t a b l i s h e s l i q u i d i t y r e s t r i c t i o n s on a v a r i a b l e t o v a r i a b l e b a s i s which may r e s u l t i n an excessive l i q u i d i t y accumulation f o r the bank as a whole. Though the model e x h i b i t s too many d e f i c i e n c i e s to be considered an o p e r a t i o n a l t o o l f o r asset management i t w i l l be valuable to compare i t to more s o p h i s t i c a t e d models. 2. M u l t i p e r i o d Models Chambers and C h a r n e s ^ have developed a m u l t i p e r i o d l i n e a r programming model f o r asset management. Their model has few of the d e f i c i e n c i e s inherent i n the model p r e v i o u s l y discussed. I t i s designed to take a simultaneous view of the banks asset, l i a b i l i t y , and c a p i t a l s t r u c t u r e , and insures that the s o l u t i o n w i l l c o n s i s t of balance sheet p o s i t i o n s which conform to the c r i t e r i a set down by the Federal Reserve Board of Governors. I f management's views concerning l i q u i d i t y requirements and bank s a f e t y are compatible w i t h the examiners c r i t e r i a , the s o l u t i o n to the X ZD. Chambers and A. Charnes, "Inter-temporal A n a l y s i s and the O p t i m i z a t i o n of Bank P o r t f o l i o s " , A n a l y t i c a l Methods i n Banking, ed. K.J. Cohen and F.S. Hammer, Richard D. I r w i n , Inc., Homewood, I l l i n o i s , 1966, pp. 67 - 86. -26-model does represent an optimal t r a d e o f f between y i e l d and l i q u i d i t y . Since the model spans s e v e r a l time periods present d e c i s i o n s w i l l not c o n f l i c t w i t h long term o b j e c t i v e s . One extremely valuable by-product of the l i n e a r programming s o l u t i o n are the dual v a r i a b l e s or shadow p r i c e s . These shadow p r i c e s , which w i l l be described i n a l a t e r chapter, make i t p o s s i b l e to apply marginal a n a l y s i s to a v a r i e t y of managerial d e c i s i o n s . The work done i n t h i s area by Cohen and Hammer1^ i s perhaps the most important c o n t r i b u t i o n to the s t a t e of the a r t . The model which they have developed i s c u r r e n t l y i n use at Bankers Trust Company, a large New York bank. They have taken the b a s i c model developed by Chambers and Charnes model and have extended i t to i n c l u d e a multitude of other r e l a t i o n s which are of extreme importance i n banking. For example, t h e i r model includes the very important r e l a t i o n s h i p which e x i s t s between loans and d e p o s i t s . The o b j e c t i v e f u n c t i o n u t i l i z e d by Chambers and Charnes maximizes the simple sum of net earnings over the e n t i r e planning h o r i z o n . In co n t r a s t the o b j e c t i v e f u n c t i o n of the Cohen and Hammer model attempts t o maximize the present value of the future earnings stream. The model which w i l l be developed i n Chapter IV c o n s t i t u t e s an a p p l i c a t i o n of goal programming to a s i m p l i f i e d v e r s i o n of the Cohen and Hammer model. An Kalman J . Cohen and F r e d r i c k S. Hammer, "Linear Programming and Optimal Bank Asset Management D e c i s i o n s " , The J o u r n a l of Finance, XXI (December, 1966), pp. 649 - 674. -27-attempt w i l l a l s o be made to develop an o b j e c t i v e f u n c t i o n which not only measures the time value of money, but a l s o represents the investment o b j e c t i v e s of the banks share-holders . CHAPTER I I I PROGRAMMING TO GOALS In s i t u a t i o n s such as those faced by bank management, i t i s extremely d i f f i c u l t to determine e x a c t l y what c o n s t i t u t e s an optimum a l l o c a t i o n of resources. I t may be more reasonable, given the u n c e r t a i n t y inherent i n the banking s i t u a t i o n , to think i n terms of resource a l l o c a t i o n s which are s a t i s f a c t o r y w i t h respect to the goals of bank management. In other words, a bank manager might be described as a s a t i s f i c e r r a t h e r than an o p t i m i z e r . 1 In t h i s s a t i s f i c i n g r o l e , he seeks to a t t a i n goals t h a t he has set f o r him s e l f or goals which have been s e t f o r him by oth e r s . In f a c t o p t i m i z a t i o n may be viewed as a s p e c i a l form of goal o r i e n t a t i o n , since even when o p t i m i z a t i o n i s p o s s i b l e , there may be s e v e r a l goals or t a r g e t s , whose attainment i s a l s o d e s i r a b l e during the course of o p t i m i z a t i o n . I n t r o d u c t i o n to Goal. Programming The o p t i m i z a t i o n procedure used i n goal programming seeks a s o l u t i o n which comes c l o s e s t to the achievement of one or more 2 goals, subject to c e r t a i n c o n s t r a i n t s . The goals represent management's d e s i r e s o r . t a r g e t s , w h i l e the c o n s t r a i n t s define the environmental c o n d i t i o n s under which management makes d e c i s i o n s . H.A. Simon, "A Behavioural Model of R a t i o n a l Choice", Qu a r t e r l y J o u r n a l of Economics, LXIX (February, 1955), pp. 99 - 118. 2 A. Charnes, N.W. Cooper and Y. I j i r i , "Breakeven Budgeting and Programming to Goals", J o u r n a l of Accounting Research, Spring, 1963, pp. 1 6 - 4 3 . -29-In l i n e a r programming, the o b j e c t i v e f u n c t i o n contains only one goal which i s to be maximized or minimized. I f other goals e x i s t they are t r e a t e d as c o n s t r a i n t s . The o p t i m i z a t i o n procedure s e l e c t s from the set of s o l u t i o n s which s a t i s f y the c o n s t r a i n t s , the s o l u t i o n ( s ) which o b t a i n the maximum or minimum value of the goal i n the o b j e c t i v e f u n c t i o n . In l i n e a r programming, the goals incorporated i n the c o n s t r a i n t s take p r i o r i t y over the goals represented by the o b j e c t i v e f u n c t i o n . Furthermore, a l l g o a l s , which are expressed as c o n s t r a i n t s , are imputed to have equal importance. In c o n t r a s t , goal programming not only allows a l l goals to be incorporated i n the o b j e c t i v e • - - • f u n c t i o n , but a l s o allows management to e s t a b l i s h p r i o r i t i e s among the goals. There i s a l s o another aspect of goal o r i e n t a t i o n . As a means of p r o v i d i n g an i n c e n t i v e , management sometimes sets goals which are not a t t a i n a b l e w i t h i n the l i m i t s of a v a i l a b l e resources. I f such goals are formulated as c o n s t r a i n t s i n a l i n e a r programming problem, i t w i l l be impossible t o a r r i v e at a f e a s i b l e s o l u t i o n . Formulation of a Goal Programming Problem The f i r s t step i n for m u l a t i n g a goal programming problem i s to express the goals as c o n s t r a i n t s , i n a manner i d e n t i c a l to the procedure used i n l i n e a r programming. For example, consider the f o l l o w i n g g o a l . I t i s assumed th a t a manufacturer wishes the s a l e s of two products, x-^  and ^ ' t o D e greater than some t a r g e t s a l e s volume, C. -30-x l + x 2 - C The goal equations are then constructed by i n t r o d u c i n g s l a c k v a r i a b l e s which represent p o s i t i v e and negative d e v i a t i o n s from the goal. E i t h e r the p o s i t i v e d e v i a t i o n or negative d e v i a t i o n (or both d e v i a t i o n s ) w i l l be terms which w i l l enter the o b j e c t i v e f u n c t i o n . The o b j e c t i v e w i l l be to minimize undesirable d e v i a t i o n s , or maximize the d e s i r a b l e d e v i a t i o n s . In the above example, the goal equation would be s t a t e d as f o l l o w s : x i + x 2 ~ y + + y~ = c The s l a c k v a r i a b l e , y + , measures over-attainment of the sales g o a l , while y~ measures the under-attainment. In t h i s case management would wish to minimize y~, and they may or may not wish to maximize y + . In cases where exact goal attainment i s d e s i r a b l e , the o b j e c t i v e w i l l be to minimize both p o s i t i v e and negative d e v i a t i o n s . When i t i s d e s i r a b l e to s a t i s f y s e v e r a l goals, the o b j e c t i v e w i l l be to minimize the sum of undesirable d e v i a t i o n s from p r e s c r i b e d goals. A problem a r i s e s when there e x i s t goals t h a t are i n c o n f l i c t . To dea l w i t h t h i s s i t u a t i o n i t i s necessary t o determine the r e l a t i o n s h i p between goals. The procedure i s e s s e n t i a l l y one of o r d e r i n g and weighting goals according to t h e i r r e l a t i v e importance. This ranking of goals may be s u b j e c t i v e , but i t i s the type of d e c i s i o n which oft e n confronts management. This type of ranking process i s not p o s s i b l e i n l i n e a r programming. I t i s now necessary to discuss the various components of a goal programming o b j e c t i v e f u n c t i o n . -31-N a t u r a l Slack V a r i a b l e s The n a t u r a l slack v a r i a b l e s measure the d e v i a t i o n s from the s t a t e d goals. Let y t denote the p o s i t i v e d e v i a t i o n from th the i g o a l , and l e t y^ measure the negative d e v i a t i o n from t h a t goal. The f o l l o w i n g r e l a t i o n s w i l l h o l d : + - n y., y ± = 0 y t . y T = 0 A subset of the o b j e c t i v e f u n c t i o n , which deals w i t h a s i n g l e g o a l , can be developed according t o the f o l l o w i n g procedure. I t w i l l be assumed that the s o l u t i o n method attempts to minimize the o b j e c t i v e f u n c t i o n . Five types of goals may e x i s t . (1) Exact attainment of the goal i s d e s i r e d . Minimize Z- = y t + yT (2) P o s i t i v e d e v i a t i o n s are s a t i s f a c t o r y . Minimize Z^ = yT (3) Negative d e v i a t i o n s are s a t i s f a c t o r y . Minimize Z^ = y t (4) P o s i t i v e d e v i a t i o n s are to be maximzed and negative d e v i a t i o n s are to be minimized. Minimize Z^ = - y t + yT (5) Negative d e v i a t i o n s are to be maximized and p o s i t i v e d e v i a t i o n s are t o be minimized. Minimize Z. = y t - yT 1. Ordering Goals To rank the i n d i v i d u a l g o a l s , i t i s necessary to assign a c o e f f i c i e n t to each of the n a t u r a l s l a c k v a r i a b l e s i n the -32-o b j e c t i v e f u n c t i o n . This c o e f f i c i e n t w i l l be c a l l e d a "preemptive p r i o r i t y f a c t o r " . One goal should have a higher preemptive p r i o r i t y f a c t o r than another i f i t must be s a t i s f i e d before any attempt i s made to a t t a i n the other. The value of the f a c t o r s , given by Mj, must be s e l e c t e d so that no number n, however l a r g e , can make nM_._^  as la r g e as M.. An order group c o n s i s t s of those s l a c k v a r i a b l e s i n the o b j e c t i v e f u n c t i o n which o b t a i n the same preemptive p r i o r i t y f a c t o r . Weighing Goals Sometimes i t i s necessary to assign a c o e f f i c i e n t t o each of the v a r i a b l e s w i t h i n an order group. These w i l l be c a l l e d " r e g r e t c o e f f i c i e n t s " , and w i l l be denoted by w^ . The v a r i a b l e s w i t h i n an order group are ranked i n order of importance and the v a r i a b l e t h a t represents the most undesirable d e v i a t i o n i s assigned the highest r e g r e t c o e f f i c i e n t . Forming Subgoals In most cases subgoals are formed a f t e r the i n i t i a l s o l u t i o n t o a goal programming problem. They r e s u l t from the decomposition of an e x i s t i n g goal i n t o a number of goals. The d i v i s i o n i s dependent on the s i z e of the d e v i a t i o n from t h a t e x i s t i n g goal i n the optimal s o l u t i o n . Consider two go a l s , y^ and y7 , where the l a t t e r had a higher preemptive p r i o r i t y f a c t o r . However management may consider values of y7 beyond a c e r t a i n amount, A, t o be more undesirable than -33-small values of y ^ + j , . In t h i s case goal yT can be decomposed i n t o subgoals which are ordered i n such a way that extreme values of y^ w i l l not be admitted i n t o the optimal s o l u t i o n . The v a r i a b l e yt_. would denote values of yT greater than A. The subset of the functions d e a l i n g with these goals would be expressed as f o l l o w s : yT - y t . + yT. = A Minimize 8 = M yt .+ M , y ~ + M yT n i ] n _ 1 i+k n-k i An example of a simple goal programming problem i s given i n E x h i b i t 1 of the Appendix. Goal Programming and Banking The goals of bank management tend to be more complex and more nebulous than the goals of i n d u s t r i a l management. Furthermore, with the exception of l e g a l requirements, bankers are confronted w i t h very few c o n s t r a i n t s which can be e x a c t l y defined i n q u a n t i t a t i v e terms. In l i n e a r programming models which deal w i t h asset management, the p o l i c y c o n s i d e r a t i o n s which are i n f a c t goals are expressed as c o n s t r a i n t s . Goal programming not only d i f f e r e n t i a t e s between goals and c o n s t r a i n t s but a l s o recognizes the f a c t t h a t bankers give d i f f e r e n t weights to d i f f e r e n t p o l i c y c o n s i d e r a t i o n s . Furthermore, as has been i l l u s t r a t e d , i t i s p o s s i b l e w i t h goal programming, to give r e c o g n i t i o n to the f a c t that extreme d e v i a t i o n s from the attainment of a minor goal may be as important as small d e v i a t i o n s from the attainment of a major goal. CHAPTER IV A MULTI-PERIOD ASSET MANAGEMENT MODEL USING GOAL PROGRAMMING This chapter w i l l describe the c o n s t r u c t i o n of a mathematical model which represents a s i m p l i f i e d v e r s i o n of a commerical banking e n t e r p r i s e and the environment i n which i t operates. The model i s expressed as a goal programming problem and though i t i s designed p r i m a r i l y as a t o o l f o r long range planning, i t i s a l s o expected to provide a val u a b l e a i d i n day-to-day d e c i s i o n making. The inputs to the model i n c l u d e the bank's i n i t i a l balance sheet and f o r e c a s t s of the fu t u r e l e v e l s of i n t e r e s t r a t e s , c r e d i t demand and deposit supply. The output from the model defines a sequence of balance sheet p o s i t i o n s , one f o r each p e r i o d up to planning the h o r i z o n . Each balance sheet p o s i t i o n represents an asset mix which i s optimal w i t h respect to the goals which have been incorporated w i t h i n the model. The goals s t a t e d i n the o b j e c t i v e f u n c t i o n of the model r e f l e c t management's d e s i r e to maintain adequate l i q u i d i t y , and d i v e r s i f i c a t i o n w h i l e a t t a i n i n g a s a t i s f a c t o r y p r o f i t . S t r u c t u r e of the Model The m u l t i - p e r i o d model w i l l be a p p l i e d to a bank which has an e x i s t i n g asset and l i a b i l i t y s t r u c t u r e . The i n i t i a l balances i n the various balance sheet accounts w i l l d e f i n e the p o s i t i o n of the bank at time, t = 0. These may be considered the average balances f o r p e r i o d zero, which ends at time, t = 1. -35-I t w i l l be assumed t h a t a l l purchases and l i q u i d a t i o n s of assets occur on the f i r s t day of each p e r i o d . I t w i l l a l s o be assumed th a t changes i n the balances of l i a b i l i t y and c a p i t a l accounts become e f f e c t i v e as of the f i r s t day of each p e r i o d . The balance sheet of the bank a f t e r these changes i n ' i t s accounts w i l l represent i t s p o s i t i o n on the second day of the p e r i o d . The time v a r i a b l e , t , w i l l range from 0 to T. The l a s t day of p e r i o d T defines the h o r i z o n date of the model. To s i m p l i f y the model i t w i l l be assumed th a t the bank has only e i g h t asset accounts, re p r e s e n t i n g primary r e s e r v e s , investments, and loans. A l s o no asset w i l l have a maturity of greater than three p e r i o d s . The model i s designed f o r e x p o s i t o r y purposes and the above assumptions do not undermine the g e n e r a l i t y of the approach. The asset accounts w i l l be d i s t r i b u t e d among the four r i s k c ategories used by the examiners of the Federal Reserve System. The symbol, A ^ w i l l be used to represent the account balance on the second day of p e r i o d t , f o r a non-cash asset, maturing i n "n" p e r i o d s , belonging t o r i s k c l a s s , " i " . The r e p r e s e n t a t i v e a s s e t s , h e l d by the bank at time, t , are l i s t e d as f o l l o w s : (1) Primary and Secondary Reserves Reserves and cash i n v a u l t •^ 11 Government s e c u r i t i e s maturing i n one p e r i o d A21 Government s e c u r i t i e s maturing i n two periods (2) Minimum Risk Assets A ^ Government s e c u r i t i e s maturing i n three p e r i o d s . -36-(3) Intermediate Assets A ^ Other s e c u r i t i e s maturing i n one p e r i o d (4) P o r t f o l i o Assets A ^ Loans maturing i n one p e r i o d A 2 4 Loans maturing i n two periods A^4 Loans maturing i n three p e r i o d s . The grouping above i s not s t r i c t l y r e p r e s e n t a t i v e of the examiner's c l a s s i f i c a t i o n but i t w i l l serve f o r the purposes of t h i s d i s c u s s i o n . The l i a b i l i t y and c a p i t a l accounts of the bank at time t are l i s t e d below. Demand deposits D^ Time deposits Bfc C e r t i f i c a t e s of deposit t E Equity and r e t a i n e d earnings. The remainder of the chapter w i l l be devoted to the c o n s t r u c t i o n of the model. A complete l i s t i n g of the symbols used i s given i n E x h i b i t I I of the Appendix. Formulation of the Goals A goal programming model i s p a r t i c u l a r l y s u i t a b l e f o r the s o l u t i o n of the asset management problem. In very few cases must bank management recognize e x p l i c i t c o n s t r a i n t s . The goal programming technique allows management to e s t a b l i s h p r i o r i t i e s among the s u b j e c t i v e goals and evaluate the consequences of t h e i r p o l i c y g u i d e l i n e s . 1. L i q u i d i t y Goal The c r i t e r i a provided by the examiners of the Fed e r a l -37-Reserve System w i l l be used to c o n s t r u c t a l i q u i d i t y goal which provides f o r the s a f e t y of the bank. The previous d i s c u s s i o n of the c r i t e r i a i n d i c a t e d t h a t a bank which s a t i s f i e d the examiner's c a p i t a l requirement would not f i n d i t s e l f s e r i o u s l y exposed t o the r i s k of losses due t o deposit f l u c t u a t i o n s and i n t e r e s t rate changes. The examiners c l a s s i f i c a t i o n method ranks assets and l i a b i l i t i e s according to the degree to which they are current or non-current. The leverage requirement embodied by the c r i t e r i a tends to r e s t r i c t the upper l i m i t of the r a t i o of t o t a l assets to net worth when the bank reduces the l i q u i d i t y of i t s assets mix or when the bank's l i a b i l i t i e s become more cur r e n t . Since the c a p i t a l requirement i s not mandatory, i t w i l l be considered as a goal r a t h e r than a c o n s t r a i n t . The f o l l o w i n g terms w i l l be used to c o n s t r u c t the l i q u i d i t y goal from the examiner's c r i t e r i a . " ' " Let I*" d e f i n e a weighted index which measure the i l l i q u i d i t y of the assets h e l d by the bank at time, t . n=3 I F C = .005AJ + .04A^ + .04A^L + .06A?,, + .10 A t (1) 11 ^1 32 J-J n = 1 n4 The above r e l a t i o n i s designed to increase the value of I*" as the p r o p o r t i o n of l e s s l i q u i d assets i n the asset mix i n c r e a s e . Howard D. Crosse, Management P o l i c i e s f o r Commercial Banks, P r e n t i c e - H e l l , Englewood C l i f f s , New Jersey, 19 62, pp. 114 -259 . - 3 8 -Let define a weighted index which measures the v o l a t i l i t y of the bank's l i a b i l i t i e s at time, t . Vt = .47D!~ + . 3 6 D | + B*^ t = 1,...,T (2) As the p r o p o r t i o n of demand d e p o s i t s , D^, i n c r e a s e s , the value of v"t i n c r e a s e s . This increase i n d i c a t e s t h a t the v o l a t i l i t y , or p o t e n t i a l magnitude of deposit f l u c t u a t i o n s , has increased. Let E(V) be defined as an i n c r e a s i n g f u n c t i o n of V. E(V) expresses the r e l a t i o n s h i p between the asset and l i a b i l i t y s t r u c t u r e . The f u n c t i o n i s shown g r a p h i c a l l y i n Figure 1. FIGURE 1. >v The leverage r e s t r i c t i o n based on the examiners c r i t e r i a , imposes a c a p i t a l requirement, K^, given by the equation below. K £ = I F C + E ( V ) F C ( 3 ) The shape of the f u n c t i o n , E(V), r e f l e c t s the f a c t t h a t the examiner's are not t r y i n g to encourage banks to reduce t h e i r demand d e p o s i t s , but r a t h e r that they are encouraging banks to maintain the r a t i o of demand deposits to c a p i t a l at a l e v e l which the examiners consider to be safe. -39-The four values of which describe ranges of the f u n c t i o n are defined by the examiners as f o l l o w s : = Cfc + .99 5A1! + •9 6A| (4) Vm = V r + - 9 0 A32 ( 5 ) V i = Vm + ' 8 5 A 1 3 , ( 6 ) The r e l a t i o n s show th a t the c r i t i c a l l e v e l of V t, beyond which E(V) becomes an i n c r e a s i n g f u n c t i o n , i s determined by the s i z e of the bank's primary and secondary reserves. E(V) can be decomposed and expressed i n terms of the f o l l o w i n g v a r i a b l e s : E(V)£ = .65(V t - v£) (7) E(V)£ = (.105 - .065) (V* - V m) (8) E(V)£ = (.200 - .105) (V1^ - vf) (9) where E(V)£, E ( V ) J j , E (V) J > 0 The c a p i t a l requirement i s now given by Equation (11). Kfc = I f c + E ( V ) t + E ( V ) t + E ( V ) f (11) r r m i The f i r s t three c o n s t r a i n t s of the model are obtained by expanding Equations (1) through (11). -40-• 47D^ + .36D^ + B t - C*' - .995A^ - .96A^± - (E(V)£/ .065)<0 .47D^ + .36D^ + Bt - C f c - .995A^, - .96A^ - .90A t - (E ( V ) t 1 z 11 21 32 m / .40) < 0 • 47D^ + .36D-2 + B t - C t - .995A^± - .96A^ - -90Ag - . 85A^ 3 - ( E ( V ) J / .095) _r 0 As was mentioned e a r l i e r , a bank's s c o r e , as c a l c u l a t e d by the examiners, i s taken t o be the r a t i o of the bank's c a p i t a l , E*", to the r e q u i r e d c a p i t a l , K^, at time t . In t h i s case i t w i l l be assumed t h a t the bank's management views the examiners c r i t e r i a t o be o v e r l y c o n s e r v a t i v e and f o r t h i s reason management c o n s i d e r s a sco r e of 0.9 to be a s u i t a b l e l i q u i d i t y g o a l . I t i s now a p p r o p r i a t e t o i n t r o d u c e the n a t u r a l s l a c k v a r i a b l e s , y"l" and yT, , which d e f i n e r e s p e c t i v e l y the i t i t over-attainment and under attainment o f the l i q u i d i t y g o a l i n year t . The s l a c k v a r i a b l e e n t e r i n g the o b j e c t i v e f u n c t i o n w i l l be b r a c k e t e d . The l i q u i t i d y g o a l can now be s t a t e d as f o l l o w s : E t + ( ^ I t ) - *tt = K l K r Sub s t i t u t i n g f o r i n Equation ( i v - e ) , and expanding t h a t e q u a t i o n i n terms of I*" y i e l d s the f i r s t g oal i n c o r p o r a t e d i n the model. -41-. . . . n=3 .005A„ + ,04Ar, + .04A^o + .06A, o + .10 A . 11 21 32 13 n 1 n4 n=l -EVKJ^ + E ( V ^ ) + E ( V ^ ) + E ( V ^ ) - (y~_t) + y+ t = 0 2. D i v e r s i f i c a t i o n Goals D i v e r s i f i c a t i o n r e s t r i c t s the p r o p o r t i o n of funds i n v e s t e d i n c e r t a i n asset types. One purpose f o r doing t h i s i s t o l i m i t the v u l n e r a b i l i t y of the bank t o being fo r c e d to l i q u i d a t e such assets at a l o s s . D i v e r s i f i c a t i o n i s a l s o aimed at a c h i e v i n g a balanced mix of asset m a t u r i t i e s i n order to promote an even flow of funds i n t o the bank. T r a d i t i o n a l l y d i v e r s i f i c a t i o n has been measured by various balance sheet r a t i o s . Though the weakness of such measurements has already been noted, they are s t i l l considered to be important measures of bank s a f e t y by lar g e d e p o s i t o r s , shareholders, investment s e r v i c e s and the banking community i n general. For t h i s reason i t i s necessary t h a t bank management remain s e n s i t i v e to i t s balance sheet r a t i o s and prevent them from d e v i a t i n g too g r e a t l y from t h e i r g e n e r a l l y accepted values. A number of these r a t i o s w i l l be incorporated i n the model as goals. The goals w i l l r e f l e c t management's views concerning the range over which the value of a r a t i o may vary. -42-(1) R a t i o of Loans to T o t a l Assets The acceptable range i n the r a t i o of t o t a l loans t o t o t a l assets l i e s between a maximum value of K2 and a minimum value of K3, so t h a t n=3 t v An4 Tjr  K 3 — < K2 This r e s t r i c t i o n enters the model as the second and t h i r d g o a l s. ( a ) ^ t t t t t + • -S - K 2 ( D J + D* + B f c + E ) - ( Y 2 t ) + - 0 <b) n=3 t t 21 A r - K (D + D. + B r + E^) - Y ^ + (Y,. ) = 0 ^TJi n4 3 1 2 3t 3t (2) Ratio of Long Term Loans to Demand Deposits The value of the r a t i o i s l i m i t e d to K4 and the fou r t h goal i s expressed as f o l l o w s : A34 " K 4 D 1 " ( Y4 t) + Y 4 t = 0 (3) Ratio of Long Term Assets t o C a p i t a l The f i f t h goal i s to l i m i t three p e r i o d loans and three p e r i o d government bonds to l e s s than times the bank's c a p i t a l . A34 + A32 " K 5 E t " ( Y 5 t } + Y 5 t = ° (4) Ratio, of Government S e c u r i t i e s t o T o t a l Assets The s i x t h and l a s t d i v e r s i f i c a t i o n goal i s to -43-maintain the r a t i o of government s e c u r i t i e s to t o t a l assets at a minimum of Kg. A^ - Kg (D? + + Bt + E f c) - Y+ + (Y~ ) = 0 n = l n i b x 2 o t 6 t 3. Growth Goal Management d e s i r e s t h a t the bank achieves a growth r a t e which w i l l r e s u l t i n a volume of t o t a l assets equal t o G d o l l a r s at the h o r i z o n , T. This goal may be expressed as f o l l o w s : i=4 n=3 i = l n=l P r o f i t Goal Ct + 3 T ^ A*. - y^ t + (y- t) = G I t w i l l be assumed t h a t a goal of management i s to increase the wealth of the bank's shareholders over the h o r i z o n . In other words, the h o r i z o n value of the share-holder's e q u i t y must be greater than the current value by an amount s u f f i c i e n t t o y i e l d the shareholders a reasonable r e t u r n on t h e i r investment. The p r i n c i p l e of maximizing the h o r i z o n value of the shareholders e q u i t y w i l l a l s o be a p p l i e d t o d e c i s i o n s concerning d i v i d e n d payments and eq u i t y s u b s c r i p t i o n s . I t w i l l be assumed t h a t the shareholders are i n c u r r i n g a p o s i t i v e opportunity cost as a r e s u l t of owning equity i n the bank. This opportunity cost i s measured by Rt which defines the best r a t e of r e t u r n a v a i l a b l e , i n p e r i o d t , from investments i n equity of the same r i s k c l a s s as the bank. Shareholders w i l l forgo dividends or i n v e s t new -44-e q u i t y i n the bank only i f by doing so they can earn r e t u r n i n excess of Rt. I t w i l l a l s o be assumed t h a t the market value of the bank's e q u i t y i s p r i m a r i l y a f u n c t i o n of the a n i t i c i p a t e d value of the banks earnings f o r the forthcoming p e r i o d , and t h a t the p r i c e - e a r n i n g s m u l t i p l e a p p l i e d to the banks earnings i s i d e n t i c a l to the m u l t i p l e a p p l i e d to determine the value of other equity shares of the same r i s k c l a s s as the banks. Let e^ equal the net the banks a n t i c i p a t e d earnings f o r the p e r i o d commencing at time t , and l e t a be the p r i c e -earnings r a t i o which i s expected to p r e v a i l over the planning h o r i z o n . The market value of the banks equity at the h o r i z o n , d e f i n e d by MVT i s then a f u n c t i o n of the earnings expected f o r the p e r i o d commencing at the h o r i z o n and i s given as f o l l o w s : MVT = a e T Excluding f o r the moment the i n f l u e n c e of dividends or e q u i t y s u b s c r i p t i o n s , i t i s i n the best i n t e r e s t s of shareholders t h a t MV o b t a i n a value which i s gr e a t e r than T the present value of the banks e q u i t y , MV^ by at l e a s t an amount s u f f i c i e n t t o compensate the shareholders f o r the opportunity cost i n c u r r e d by them as a r e s u l t of t h e i r investment i n the banks e q u i t y . This c o n d i t i o n i s defined by the f o l l o w i n g r e l a t i o n i n e q u a l i t y : MV 5- MV (1 + R ) T T — 0 o -45-Let denote the amount of dividends p a i d out at time t , and l e t S t denote the amount of new e q u i t y subscribed by shareholders at time t . Since the shareholders can i n v e s t any amounts b t and Sfc at the opportunity r a t e R t, they w i l l a l s o i n c u r r an opportunity cost i f earnings are r e t a i n e d as a d d i t i o n a l c a p i t a l i s i n v e s t e d . The t o t a l wealth of the shareholders at the h o r i z o n , denoted by WT i s defined as f o l l o w s : t=T t=T - T-t _ +- T-t WT = ae + (1 + R )l C - S t (1 - R ) Let Q T be management's t a r g e t value f o r WT where Q T i s determined by management's t a r g e t f o r the ra t e of r e t u r n on e q u i t y , y e/ s o t h a t : Q T = MV0 (1 + r e ) T The p r o f i t goal can now be s t a t e d by the f o l l o w i n g equation: t=T t=T - - T—+• -t- T - t -ae™ + 2T- b. (1 + R . ) 1 - S (1 + R . ) 1 ^ + (Y p) 1 t = l r t = l ^ 0 <*8> = Q T The goal i s constructed i n such a way t h a t earnings are r e t a i n e d or a d d i t i o n a l e q u i t y i s subscribed only i f doing so w i l l provide a l a r g e r c a p i t a l gain or l a r g e r d i v i d e n d i n l a t e r years. -46-Formulation of the C o n s t r a i n t s The r e l a t i o n s described by the c o n s t r a i n t s can be grouped i n t o three c a t e g o r i e s . The f i r s t group of r e s t r i c t i o n s r e f l e c t the l i m i t a t i o n s on the bank imposed by exogenous f a c t o r s which r e s u l t from the economic, l e g a l and competitive aspects of i t s environment. The second group may be thought of as m a t e r i a l balance equations which express the flow of funds between accounts, across time p e r i o d s , and i n and out of the bank. The t h i r d group of equations expresses the r e l a t i o n between exogenous and endogenous f a c t o r s . 1. Exogenously Determined Factors The f o l l o w i n g c o n s t r a i n t s and equations are der i v e d from c o n s i d e r a t i o n s which are l a r g e l y exogenous to the model. (1) Reserve Requirement and L i q u i d i t y B u f f e r The minimum amount of cash and reserves that the bank must ho l d i s determined by the Federal Reserve. I t w i l l be assumed th a t the law r e q u i r e s the bank to h o l d f i f t e e n percent of i t s demand deposits and f i v e percent of i t s time deposits i n cash or i n reserves. The amount of cash t o be h e l d f o r day-to-day t r a n s a c t i o n s , correspondent bank r e l a t i o n s , and f l o a t c o n s i d e r a t i o n s , i s determined by the i n d i v i d u a l bank. 1 In t h i s model i t w i l l be assumed th a t the bank wished to keep one percent of i t s t o t a l assets i n cash i n a d d i t i o n t o the r e q u i r e d l e g a l reserves. The cash balance c o n s t r a i n t can be expressed as f o l l o w s : Roland I . Robinson, The Management of Bank Funds, Second E d i t i o n , McGraw-Hill, New York, 1962, p.76. C f c> ,15D^ + -05D^ + .01 (D^ + + B f c + E11) t = 1,.. (2) I n i t i a l Balance Sheet The balances i n the bank's asset and l i a b i l i t y accounts at the beginning of the planning p e r i o d must be c l a s s i f i e d as to type and maturity. (3) Forecast Loan Demand The f o r e c a s t demand f o r new loans, of various m a t u r i t i e s , i n each p e r i o d , w i l l be taken as an upper bound c o n s t r a i n t on loans made during the p e r i o d . L a t e r , i t w i l l be shown how the bank's loan p o l i c y a f f e c t s i t s capacity to lend. I t i s now necessary to define a new v a r i a b l e , B ^ i which w i l l define the amount of an asset of r i s k n i c l a s s i , maturing i n n p e r i o d s , acquired by the bank on the f i r s t day of p e r i o d t . For example, the amount of new two p e r i o d loans made on the f i r s t day of p e r i o d 3 three i s given by ^ 2 4 * The c o n s t r a i n t s on loans made are expressed below, where A t. defines the f o r e c a s t demand i n p e r i o d t f o r a n4 c loan maturing i n n p e r i o d s . Pn4 - An4 t = 1,...,T n = 1, 2, 3 I n t e r - P e r i o d R e l a t i o n s and Endogenous E f f e c t s The f o l l o w i n g equations represent a mixture of exogenous and endogenous c o n s i d e r a t i o n s . They provide a means of l i n k i n g the v a r i a b l e s p e r t a i n i n g to d i f f e r e n t time periods and expressing the e f f e c t of i n i t i a l c o n d i t i o n s and present -48-d e c i s i o n s on the future p o s i t i o n of the bank. (1) Feedback Between Loans and Deposits The e f f e c t of the bank's loan p o l i c y on the s i z e of i t s deposits has already been discussed. In order to achieve optimal asset management d e c i s i o n s t h i s r e l a t i o n between loans and deposits must be q u a n t i f i e d and incorporated i n t o the model. Two feedback mechanisms can be i d e n t i f i e d . The f i r s t l i n k s changes i n the l e v e l of the banks deposits to changes i n the bank's market share of t o t a l loan demand. The d e r i v a t i o n of both feedback e f f e c t s w i l l be based on the assumption that borrowers are u s u a l l y a l s o d e p o s i t o r s . This assumption should h o l d even i n the absence of any e x p l i c i t compensating balance requirement. I f t h i s assumption i s v a l i d i t can be shown t h a t i f the bank expands c r e d i t t o an extent which enables i t to e x a c t l y maintain i t s market share of loans, i t w i l l experience a growth i n deposits roughly equal to i t s market share of the t o t a l growth of deposits experienced by a l l the banks i n i t s market. The magnitude of the growth w i l l depend on the average r a t i o of compensating balances to loans, and the l e g a l reserve requirement. The reverse e f f e c t occurs i f the bank con t r a c t s i t s loan volume to an extent t h a t i t maintains or i t continues t o maintain i t s market share of loans. -49-Th e second feedback e f f e c t can be i d e n t i f i e d i f the bank gains or loses share of market as a r e s u l t of i t s c r e d i t operations. When the bank f a i l s to keep pace w i t h competitors i n meeting loan demand i t w i l l lose deposit customers. Conversely i f the bank i s able to pursue a more agressive l e n d i n g p o l i c y than other banks i t w i l l gain depositors at the expense of i t s competitors. In other words a bank's deposit growth due to an increase i n the absolute volume of i t s loans might be o f f s e t by deposit losses i f the bank f a i l s to maintain i t s market share. For the purpose of b u i l d i n g the model i t i s necessary to recognize the existence of i n t e r a c t i o n s between loans and d e p o s i t s , however, the e x p l i c i t d e r i v a t i o n of such loan-deposit feedback mechanisms l i e s beyond the scope of t h i s paper. In expressing the f o l l o w i n g equations i t w i l l be assumed th a t the feedback e f f e c t s are l i n e a r and t h a t the bank has d e r i v e d the c o e f f i c i e n t s which define the r e l a t i o n s . Let define the banks estimated market share of loans i n p e r i o d t . Let p]_ and P2 be the two feedback c o e f f i c i e n t s which r e l a t e current expansions or c o n t r a c t i o n s i n loan volume to changes i n future l e v e l s of demand d e p o s i t s . The v a r i a b l e s d* and d f c d e f i n e , r e s p e c t i v e l y , the amount by which demand deposits are increased or decreased endogenously during p e r i o d t . -50-d - d = p (^ E. A - 21 At X) + p {JT_ A 1 - M t ) t t 1 n = l n4 n = i n4 2 ^ r j n4 d*.d~ = 0 t t To s i m p l i f y the c o n s t r u c t i o n of the model i t w i l l be assumed th a t only demand deposit l e v e l s are d i r e c t l y a f f e c t e d by loan p o l i c y . Deposit Supply The model w i l l i n c o r p o r a t e f o r e c a s t s of fut u r e deposit l e v e l s over the planning h o r i z o n . These f o r e c a s t s consider changes i n deposit l e v e l s which occur due to f a c t o r s which are exogenous to the model and independent of loan p o l i c y . The form of the f o r e c a s t w i l l vary w i t h the deposit type. The amount of time deposits h e l d by the bank at time t , w i l l be equal to the f o r e c a s t l e v e l of time d e p o s i t s . However, the r e l a t i o n expressing the l e v e l of demand deposits w i l l be more complicated i n that i t in c l u d e s the f o r e c a s t change i n the l e v e l of demand deposits w i t h respect to the previous p e r i o d , and a l s o takes i n t o account the endogenous changes i n the s i z e of demand deposits which r e s u l t from loan p o l i c y . The f o r e c a s t supply of c e r t i f i c a t e s of deposits w i l l be taken as an upper bound c o n s t r a i n t on the bank's holdings of such d e p o s i t s . I t w i l l be assumed th a t the bank must a c t i v e l y a t t r a c t t h i s form of deposit by paying a premium i n t e r e s t r a t e . The equations d e f i n i n g the s i z e of the various deposits are given below. -51-Let be the f o r e c a s t exogenous change i n the l e v e l of demand deposits between times, t and t - 1 . The v a r i a b l e s d^ and d£ have already been de f i n e d , however, i t w i l l be assumed t h a t the endogenous changes i n demand deposits w i l l l a g by one p e r i o d the changes i n loan volume which produced them. The l e v e l of a banks demand d e p o s i t s , D^, can be given as f o l l o w s : D l = D l _ 1 + 5 1 + d t - l " d t - l I f i s defined as the f o r e c a s t absolute l e v e l of time deposits h e l d by the bank at time t , the f o l l o w i n g r e l a t i o n o b t a i n s : I f B*" i s defined as the p o t e n t i a l supply of c e r t i f i c a t e s of deposit at time t , the amount of these deposits h e l d by the bank i s given as f o l l o w s : (3) C a p i t a l A v a i l a b i l i t y The balance i n the bank's c a p i t a l account at time t w i l l be equal to the balance i n the previous p e r i o d plus or minus any endogenous or exogenous changes i n c a p i t a l . Endogenous changes r e s u l t from increases or decreases i n r e t a i n e d earnings over the previous p e r i o d . Exogenous changes i n c a p i t a l r e s u l t from the issue of common shares. As was p r e v i o u s l y s t a t e d , the s i z e of d i v i d e n d -52-payments at time t i s given by the v a r i a b l e b.j_. The value o f di v i d e n d s to the share h o l d e r s i s taken to be equal t o the o p p o r t u n i t y r a t e of r e t u r n a v a i l a b l e i n year t f o r investments b e l o n g i n g t o the same r i s k c l a s s as the bank. This r a t e , p r e v i o u s l y d e f i n e d as Rfc, i s used to e v a l u a t e d i v i d e n d payments i n the p r o f i t a b i l i t y g o a l . Given the value o f di v i d e n d s to s h a r e h o l d e r s , i t i s now p o s s i b l e f o r the model t o determine an optimum d i v i d e n d p o l i c y . F i n a n c i n g d e c i s i o n s with regard t o the f l o t a t i o n o f common stock can a l s o be i n c o r p o r a t e d i n t o the model. I f E** i s d e f i n e d to be the maximum amount of new e q u i t y the e x i s t i n g s h a r e h o l d e r s would be w i l l i n g to purchase •through p r i v i l e d g e d s u b s c r i p t i o n , the optimum s i z e o f the s u b s c r i p t i o n , S*-, can be determined by the model. F i n a l l y , given t h a t e^-_^ equals the net income accrued d u r i n g p e r i o d t - 1 , the equation which determines the balance i n the bank's c a p i t a l account on the second day of p e r i o d t may now be expressed: E t = E t - 1 + s t + e _ b t ~ 1 b t ^ S t - l S f c ± E t (4) S e l l i n g C o n s t r a i n t A bank can i n c r e a s e i t s p r o f i t p o t e n t i a l by s p e c u l a t i n g on s e c u r i t y p r i c e s . However, the a c t i v i t i e s of buying and s e l l i n g s e c u r i t i e s ,. on the b a s i s o f f o r e c a s t movements i n i n t e r e s t r a t e s , w i l l be - 5 3 -i n c o r p o r a t e d i n t o the model only to a l i m i t e d extent. The model i s designed f o r planning purposes and the emphasis w i l l be placed on asset management con s i d e r a t i o n s and not on t r a d i n g o p p o r t u n i t i e s . Because of t h i s o r i e n t a t i o n the major i n t e n t of the model i s to determine the optimal s i z e , composition, and maturity s t r u c t u r e , of the bank's balance sheet f o r each p e r i o d up to the h o r i z o n . Since the i n i t i a l balance sheet i s g i v e n , i t i s d e s i r e a b l e t o provide the model w i t h enough f l e x i b i l i t y to permit the s a l e of s e c u r i t i e s from i n i t i a l holdings i f doing so w i l l permit the bank to more r a p i d l y a t t a i n optimal balance sheet p o s i t i o n s i n f u t u r e p e r i o d s . Beyond p e r i o d zero, the model s t r u c t u r e s the asset mix i n such a way t h a t only s e c u r i t i e s of d e s i r e d maturity are purchased, and the need to l i q u i d a t e s e c u r i t i e s before t h e i r maturity i s e l i m i n a t e d . Let denote the amount l i q u i d a t e d , at time t , of an asset of r i s k c l a s s i , h e l d at t - 0 , maturing i n n periods from t=0. The c o n s t r a i n t on the s a l e s of i n i t i a l holdings of t h r e e - p e r i o d government s e c u r i t i e s can be expressed as f o l l o w s : 1 2 ° L + L A O N 3 2 3 2 - 32 For two-period government s e c u r i t i e s the f o l l o w i n g r e l a t i o n h o l d s : 1 ° L : . •< A 2 1 — 32 -54-(5) M a t u r i t y S t r u c t u r e Since s e v e r a l balance sheet categories are disaggregated i n t o m u l t i p l e maturity c l a s s e s , the opening balances i n these accounts must inco r p o r a t e the run-out i n t o s h o r t e r maturity classes and the r u n - i n from longer m a t u r i t i e s . For example, the opening balance at t a l , f o r government s e c u r i t i e s , maturing i n two p e r i o d s , would be equal to the r u n - i n of government s e c u r i t i e s maturing i n three p e r i o d s , h e l d at t - 0 , plus any purchases of two-period government s e c u r i t i e s at t = l . The r e l a t i o n s d e f i n i n g the maturity s t r u c t u r e are given below: Three-Period Government S e c u r i t i e s : Afc = P f c t = 1, ... ,T 32 32 Two-Period Government S e c u r i t i e s ; 1 ° 1 1 A = A - L + P 21 32 32 21 A 2 1 = 421 + P 2 1 t = 2 T One-Period Government S e c u r i t i e s : 1 ° 1 1 A = A - L + P 11 21 21 11 2 1 2 2 A + A - L + P n , 11 21 32 11 t t-1 t A l l = A 2 1 + P l l t = 3 T Other One-Period S e c u r i t i e s ; -55-Loans 34 t 1 i • • • T 34 2 4 + P t 24 t 1 / • • • T + P t 14 t 1 • • • • T 14 (6) Income C a l c u l a t i o n The net income f o r a given p e r i o d w i l l depend on the i n t e r e s t r e c e i v e d and i n t e r e s t p a i d as a r e s u l t of the bank's holdings of various assets and l i a b i l i t i e s d u r i n g the p e r i o d . Earnings w i l l a l s o be a f f e c t e d by c a p i t a l gains or losses a s s o c i a t e d w i t h s e c u r i t y s a l e s , promotional expenses i n c u r r e d to a t t r a c t depositors and borrowers, and a wide v a r i e t y of other c o s t s . Since the purpose of the model i s to determine the optimal r i s k - y i e l d t r a d e - o f f , i t i s necessary to evaluate i n d i v i d u a l investment o p p o r t u n i t i e s and sources of funds i n terms of t h e i r marginal costs and revenues. For t h i s reason the i n t e r e s t rate r e l e v a n t f o r a given asset i s i t s y i e l d net of taxes, marginal p r o c e s s i n g c o s t s , and marginal d e f a u l t allowance. C a p i t a l gains and losses r e s u l t i n g from s e c u r i t y sales should a l s o be expressed on an a f t e r tax b a s i s . The y i e l d s a s s o c i a t e d w i t h i n d i v i d u a l assets vary not only w i t h the type and maturity of the a s s e t , but a l s o r e f l e c t the y i e l d s a s s o c i a t e d w i t h s e v e r a l time p e r i o d s . The f o l l o w i n g r e l a t i o n s w i l l be designed to r e f l e c t t h i s d i v e r s i t y of y i e l d s i n order to measure -56-th e impact of present asset management d e c i s i o n s on future earnings. F i r s t i t i s necessary to define the f o l l o w i n g v a r i a b l e s . Let r 1". d e f i n e the net rate of r e t u r n a f t e r d i r e c t n i expenses and taxes f o r an asset of r i s k c l a s s i , maturing i n n p e r i o d s , purchased at the beginning of pe r i o d t . x Let f ^ denote the r a t i o of purchase p r i c e to s e l l i n g p r i c e f o r a s e c u r i t y of r i s k c l a s s i , h e l d at t=0, s o l d before maturity i n p e r i o d t . Let t be the rate at which the bank's income i s taxed, and l e t r and r, define the a f t e r - t a x r a t e s of i n t e r e s t s b on time deposits and c e r t i f i c a t e s of d e p o s i t s , r e s p e c t i v e l y . F i n a l l y , l e t F t be equal t o the t o t a l amount, a f t e r t a x e s , of a l l u n a l l o c a t e d expenses f o r p e r i o d t - 1 , accrued by the beginning of p e r i o d t . The f o l l o w i n g equations can now be s t a t e d : 1=4 n=3 e o = ^ ^ r°..A°. + t l f - D L1 + t ( f - l ) L i _ i = l n=l n i n i 32 2 1 - r D° - r B° - F° S 2. fc> 1 " r32 L32 " r21 L21 + ^ f - 1 » L 3 2 + i=4 i i i i i l ^ r . P . - r,D - r, B - F I i = l n=l nx n i d 2 b j R. Haydon and J . Wicks, "A Model of Bank Earning A s s e t s " , J o u r n a l of F i n a n c i a l and Q u a n t i t a t i v e A n a l y s i s , V o l . 1, No. 2, June 1966, pp. 99-113. -57-e2 = i=4 z: i = l o ,o r A_ . 3 i 3 i i=4 n=4 i = l n=l „ i=4 n=-3 -i -i r° - I 2 + 21 S r 1 . . ? 1 . r^-.P 2. - r s D 2 - r B 2 •3 t - 2 t-2 ^ ^ t-1 t-1 ^ <£-3 rT. . P . + Z— r .P + 21 2f"_ 3 1 3 1 i = l n=2 n l n i 1=1 n=l r t . . P t . - r Dfc - r , B t - F t| t=3,...,T n i n i s b I (7) Funds A v a i l a b i l i t y C o n s t r a i n t The next group of equations i n the model e s t a b l i s h the necessary c o n d i t i o n that i n any p e r i o d the t o t a l uses of funds must be equal to the t o t a l sources of funds. Funds are used to meet cash and reserve requirements, purchase s e c u r i t i e s , extend c r e d i t , meet deposit withdrawals and pay dividends. Funds are de r i v e d from maturing loans and s e c u r i t i e s , the s a l e of s e c u r i t i e s , a d d i t i o n a l d e p o s i t s , r e t a i n e d earnings and the s u b s c r i p t i o n of a d d i t i o n a l e q u i t y . The b a l a n c i n g equations f o r funds w i l l be expressed i n terms of the opening cash balance f o r p e r i o d t . C 1 = f c ° +ZI A°. + f i * + f L 1 + (D 1 - D° | iT~i i= 3 2 2 1 i i ) + ( D2 " °2 ) + i = l n=l  n ± J B°) + ( E 1 - E°) --58-2 I l j-~f l -? 2 1 ? 1 = IC1 + ZI A 1 ± + f L 2 3 2 + (D 2 - DJ) + (D 2 - D 2) + (B 2 - B 1) + (E 2 ! i=4 n=3 "I - E 1) - P.I i = l n=l n J C * = | c t = 1 + JET 4Z1 + (D* - D*" l> + <D* - D*" 1 ) + i = l • L X 1 1 2 2. t t-1 t t-1 i ^ 4 . £<3 1 (B - B^ A) + (E r - E r X) - 21 SI P | t » 3., . , i = l n=l n i f Objective Function The only v a r i a b l e s to enter the o b j e c t i v e f u n c t i o n are those which measure d e s i r e a b l e or undesireable d e v i a t i o n s from the goals which have been p r e v i o u s l y defined. With the exception of the goal v a r i a b l e which measures the overattainment of the p r o f i t g o a l , the values of a l l the v a r i a b l e s are t o be minimized. The l i q u i d i t y and d i v e r s i f i c a t i o n goals have p r i o r i t y over the p r o f i t g o a l , which i n turn has p r i o r i t y over the growth g o a l . The l i q u i d i t y and d i v e r s i f i c a t i o n goals w i l l be assigned to the same order group and w i l l be assigned a p r i o r i t y c o e f f i c i e n t which w i l l insure that they are s a t i s f i e d before any attempt i s made to s a t i s f y the p r o f i t and growth goals. The o b j e c t i v e f u n c t i o n i s s t a t e d below: Minimize 2 =|M, 21 ( Y ^ + Y ^ + + Y ^ + Y ^ + Y^) 1 t=T + M2 <Y8 " Y 8 > + M 3 Y ? J where: M^>> M^ > > M^ - 5 9 -ConcTuding Comments The model developed i n t h i s chapter i s fundamentally a t o o l f o r long range planning. I t i s designed to generate a sequence of t a r g e t balance sheet p o s i t i o n s which become goals f o r the managers of various departments i n the bank. In i t s present form the model does not lend i t s e l f t o e x p l i c i t l y d e f i n s hort term s h i f t s i n the asset mix. A s u b s t a n t i a l increase i n the complexity of the model would be r e q u i r e d to r e a l i s t i c a l l y approximate the dynamics of the bank's operations over the short term. Furthermore, i t i s p o s s i b l e to question t h e . d e s i r a b i l i t y of developing a more complex model to de a l e x p l i c i t l y w i t h day-to-day d e c i s i o n making. The model which has been developed defines long term g o a l s , but doesn't take away from the bank's management the r e s p o n s i b i l i t y of f i n d i n g the best paths to the attainment of these goals. In the next chapter i t w i l l be shown how a model of t h i s l e v e l of a b s t r a c t i o n can be used t o a i d day-to-day d e c i s i o n making by p r o v i d i n g management w i t h the type of inform a t i o n r e q u i r e d to evaluate these d e c i s i o n s . Furthermore, i t w i l l be shown, th a t the model provides in f o r m a t i o n which enables management to a l t e r the long term goals i n response t o changing c o n d i t i o n s i n the environment. This combination of management d i s c r e t i o n and mathematical programming permits f l e x i b i l i t y i n d e c i s i o n making to a degree which would be impossible to achieve w i t h even the most complex mathematical model. CHAPTER V APPLICATION OF THE GOAL PROGRAMMING MODEL TO A HYPOTHETICAL BANK The model developed i n the previous chapter w i l l now be ap p l i e d to a problem which has been synthesized from an assumed set of inputs which describe the e x i s t i n g asset s t r u c t u r e of a h y p o t h e t i c a l bank and the c h a r a c t e r i s t i c s of the environment i n which the bank w i l l conduct i t s business. The model w i l l cover the operations of a bank over four time p e r i o d s , and to s i m p l i f y the c o n s t r u c t i o n of data, i t w i l l be assumed th a t each p e r i o d spans one year. Given the banks i n i t i a l balance sheet at time, t=0, the model w i l l generate balance sheets which represent the banks p o s i t i o n at the beginning of each p e r i o d . The planning h o r i z o n i s the f o u r t h year and the asset mix at the beginning of the f o u r t h year w i l l determine the earnings f o r the year and hence the market value of the banks common stock at the h o r i z o n . Input Data The balances i n various accounts at the beginning of the planning p e r i o d are given below i n Table 1. The f o r e c a s t values of fu t u r e c r e d i t demand, deposit supply, asset y i e l d s and i n t e r e s t on deposits appear i n E x h i b i t I I I i n the Appendix. E x h i b i t I I in the Appendix provides a glo s s a r y which summarizes a l l the symbols used to -61-TABLE 1 INITIAL BALANCE SHEET POSITION ASSETS Cash & Reserves $ 1,080,000 Government S e c u r i t i e s - maturing i n 1 year ( A ^ ) $1,000,000 - maturing i n 2 years (A° ) 1,000,000 - maturing i n 3 years (A°2) 500,000 - T o t a l 2 ,500 ,000 Other S e c u r i t i e s 1,380,000 Loans , o . - maturing i n 1 year (A.^) 2 ,000 ,000 - maturing i n 2 years (A°^) 2,000,000 - maturing i n 3 years (A°4) 1,040,000 - T o t a l 5 ,040 ,000 T o t a l Assets 10,000,000 LIABILITIES & EQUITY Deposits - Demand Deposits (D°) 5,200,000 - Time Deposits (D°) 4,000 ,000 - T o t a l 9,200,000 C e r t i f i c a t e s of Deposits (B°) T o t a l L i a b i l i t i e s 9,200,000 C a p i t a l Accounts - P a i d i n C a p i t a l 200,000 - Retained Earnings 600,000 - T o t a l C a p i t a l (Et) 800 ,000 T o t a l L i a b i l i t i e s & Equity 10,000,000 -62-define the a c t i v i t i e s , v a r i a b l e s and parameters. In a d d i t i o n to the data given i n the Appendix, i t w i l l be assumed th a t the u n a l l o c a t e d overhead and f i x e d costs are $75,000 per year a f t e r taxes. The values of the two parameters which measure the i n t e r a c t i o n between c r e d i t expansion, market share, and deposit supply w i l l be s e t equal to .10 so t h a t : . n=3 . n=3 t - i n=3 ±. ±. C - d" = .1 ( r A - A ) + .1 ( A r - Mr) v The parameters which define the l i q u i d i t y and d i v e r s i f i c a t i o n goals are shown i n E x h i b i t I I I . As i t s growth g o a l , management de s i r e s t h a t the bank achieve a l e v e l of t o t a l assets amounting to $14,000,000 by time t=3. The current market value of the banks common stock i s set at $2,800,000. Managements goal i s to increase the current value of the shareholders wealth to an amount at the h o r i z o n which provides a compound i n t e r e s t rate of r e t u r n to the shareholders of at l e a s t 15%. This means tha t the shareholders wealth at the h o r i z o n must achieve a value of at l e a s t $4,260,000. The current market rate of r e t u r n f o r common stock of the same r i s k c l a s s i s 10%, and i t i s assumed that the banks common stock commands a p r i c e - e a r n i n g s m u l t i p l e which i s equal t o 15. The p r o f i t goal can now be s t a t e d as f o l l o w s : t= 3 t= 3 15e, + S b. (1 + . 1 0 ) 3 - t - !2E~- S11 (1 + . 1 0 ) 3 _ t + 3 t = l t = l Y~ - = $4,260,000 The o b j e c t i v e f u n c t i o n can be s t a t e d as f o l l o w s : - 6 3 -Minimize = 1000 (Y + Y„ + Y„ + Y; + Y + YZ\ I t 2t 3t 3t 4t 5t + Y 6 t ) + Y 8 " Y 8 + - 0 0 0 1 Y7 The p r i o r i t y c o e f f i c i e n t s i n d i c a t e t h a t management regrets d e v i a t i o n s from the l i q u i d i t y and d i v e r s i f i c a t i o n goals a thousand times more than they r e g r e t d e v i a t i o n s from the p r o f i t g o al. S i m i l a r l y , achievement of the p r o f i t goal i s considered to be ten thousand times more important than the growth g o a l . Computing a S o l u t i o n The goal programming model f o r the h y p o t h e t i c a l bank c o n s i s t s of a matrix w i t h 109 rows, 135 s t r u c t u r a l v e c t o r s , 35 s l a c k v e c t o r s , and a t o t a l of 634 elements. The den s i t y of the matrix i s 3.19 percent. The a l g o r i t h m used to solve the problem i s the l i n e a r programming a l g o r i t h m designed f o r the IBM 70 44 computer at the U n i v e r s i t y of B r i t i s h Columbia Computing Centre, the IBM 70 40/7044 LP System I I (LP40). When s o l v i n g a l i n e a r program wi t h LP40 the user has at h i s d i s p o s a l s e v e r a l o p t i o n a l r o u t i n e s or agenda which are designed to perform a v a r i e t y of f u n c t i o n s . Those of major importance i n t h i s problem are described below: (1) NORMAL provides the algorithm t o solve the optimal s o l u t i o n of both the p r i m a l and dual l i n e a r programming problems. (2) RHS RANGING c a l c u l a t e s the ranges over which the r i g h t hand s i d e values of the rows i n the matrix may vary without changing the optimal b a s i s . -64-(3) COST RANGING - defines the range over which the c o e f f i c i e n t s i n the o b j e c t i v e f u n c t i o n can vary without changing the optimal b a s i s . (4) DO DJ - computes the cost of f o r c i n g i n t o the s o l u t i o n one u n i t of a v a r i a b l e not s e l e c t e d as p a r t of the optimal b a s i s . (5) DO PLP - performs parametric programming on the r i g h t hand side values of s p e c i f i e d rows. (6) DO PCR - performs parametric programming on the o b j e c t i v e f u n c t i o n c o e f f i c i e n t s f o r s p e c i f i e d v e c t o r s . The Optimal S o l u t i o n The optimal s o l u t i o n to the goal programming problems can be expressed d i r e c t l y w i t h two fundamental accounting statements. Table 2 gives the banks i n i t i a l and p r o j e c t e d balance sheet p o s i t i o n s , and Table 3 gives a p r o j e c t i o n of sources and a p p l i c a t i o n s of funds over the planning h o r i z o n . The p r o j e c t e d balance sheet p o s i t i o n s shows the optimal asset s t r u c t u r e at the beginning of the second, t h i r d , and f o u r t h years. The sources and a p p l i c a t i o n s of funds statement i n d i c a t e the flow of funds r e q u i r e d to a r r i v e at the optimal asset s t r u c t u r e . The balance sheet p o s i t i o n at the h o r i z o n permits the bank to earn $452,9 89 i n the year commencing at t=3. The value of the o b j e c t i v e f u n c t i o n i s minimized at -3,183,375. This represents the underattainment of the growth goal by an amount equal to $85 7,4 32 and the overattainment of the p r o f i t goal by an amount equal to $3,184,29 0. There were -65-TABLE 2 PROJECTED BALANCE SHEET POSITIONS Date of Statements ASSETS t=0 t = l t=2 t=3 Cash & Reserves (Cfc) $ 1,080,000 1,237,513 1,420 ,014 1,435,223 Government S e c u r i t i e s - maturing i n 1 y r . (A^ ) 1,000,000 1,000,000 2,319,439 - maturing i n 2 y r s . C A ^ ) 1,000,000 1,026,927 - 3,260,642 - maturing i n 3 y r s . ( A ^ 2 ) 500,000 - 919,002 - - T o t a l 2,500,000 2,026,927 3,238,441 3,260,642 Other S e c u r i t i e s ( A ^ ) 1,380,000 84,311 Loans - maturing i n 1 y r . (A^ ) 2,000,000 4,794,543 2,208,006 4,700,000 - maturing i n 2 y r s . f A ^ ) 2 ,000 ,000 2 ,208,006 4,700 ,000 2 ,341,640 - maturing i n 3 y r s . f A ^ ) 1,040 ,000 1,200 ,000 1,387 ,302 1,405 ,063 - T o t a l 5,040,000 8,202,549 8,295,308 8,446,703 T o t a l Assets $10,000,000 11,551,300 12,953,763 13,142,568 LIABILITIES & EQUITY Deposits Demand Deposits (D^) $ 5,200,000 6,000,000 6,936,510 7,025,316 - Time Deposits (D^) 4,000,000 4,440,000 5,000,000 5,000,000 - T o t a l 9,200,000 10,440,000 11,936,510 12,025,316 C e r t i f i c a t e s of Deposits (Bt) - 100 ,000 100 ,000 T o t a l L i a b i l i t i e s 9,200,000 10,540,000 11,936,510 12,125,316 C a p i t a l Accounts - P a i d i n C a p i t a l 200,000 220,000 220,000 220,000 - Retained Earnings 600 ,000 791,300 797,253 797 ,253 - T o t a l C a p i t a l (Et) 800,000 1,011,300 1,017,253 1,017,253 T o t a l L i a b i l i t i e s & $10,000,000 11,551,300 12,953,763 13,142,568 Equity -66-TABLE 3 PROJECTED STATEMENT OF SOURCES AND APPLICATIONS OF FUNDS Sources of Funds Year 1 Year 2 Year 3 Net Income (e f c) 191,300 314,081 334,714 Maturing A s s e t s Government S e c u r i t i e s 1,000,000 1,000,000 2,319,439 Other S e c u r i t i e s 1,380,000 84,311 L o a n s 2,000,000 4,794,543 2,208,006 Sale of S e c u r i t i e s ( L ^ ) -I n c r e a s i n g Deposits Demand Deposits 800,000 936,510 88,805 Savings Deposits 440,000 560,000 C e r t i f i c a t e s o f Deposits 100,000 (100,000) 100,000 C a p i t a l S u b s c r i p t i o n s 20 ,000 5 ,952 T o t a l Sources 5,931,300 7,595,397 5,050,964 A p p l i c a t i o n s of Funds Increases i n Reserves 157,513 182,501 15,209 Purchases of Government S e c u r i t i e s one year ( P ^ ) - 1,292 ,512 two year ( P ^ ) 526 ,927 - 2 ,341,640 three year ( P | 2 * " 919,002 Purchase of Other S e c u r i t i e s (P^ ^ ) 84,311 Loans one year ( P ^ 4 ) 2 ,794,543 two year ( P ^ ) 1,168,006 3,500 ,000 954 ,338 three year ( P ^ ) 1,200 ,000 1,387,302 1,405 ,063 Dividend Payments (b t) ' 314 ,081 334 ,714 T o t a l A p p l i c a t i o n s 5,931,300 7,595,397 5,050,964 -67-no undesireable d e v i a t i o n s from the other goals. The e n t i r e s o l u t i o n appears i n E x h i b i t s IV and V i n the Appendix. E x h i b i t IV gives the s o l u t i o n to the p r i m a l problem and the r e s u l t s generated by the COST RANGING agendum. E x h i b i t V gives the s o l u t i o n to the dual problem and the r e s u l t s of the RHS RANGING agenda. The s o l u t i o n and i t s a p p l i c a t i o n s w i l l be discussed i n the f o l l o w i n g chapter. CHAPTER VI ANALYSIS OF THE OPTIMAL SOLUTION A r r i v i n g at the optimal s o l u t i o n to a problem of t h i s type i s only the beginning step i n o b t a i n i n g the i n f o r m a t i o n r e q u i r e d about the model or system under study. I t i s u s u a l l y of equal importance to determine what changes i n the optimal s o l u t i o n would occur i f the statement of the problem i s changed. For example, i t i s d e s i r e a b l e to know i f the changes w i l l r e s u l t i n an increase or decrease i n the l e v e l of goal attainment, or i f the changes would r e s u l t i n a change i n the set of a c t i v i t i e s i n c l u d e d i n the optimal b a s i s . F o r t u n a t e l y t h i s i n f o r m a t i o n can be obtained without r e s t a t i n g the problem and s o l v i n g i t again. The NORMAL agendum of LP40 which produces the optimal s o l u t i o n a l s o produces the s o l u t i o n to the dual problem. The s o l u t i o n to the dual problem c o n s i s t s of the "shadow p r i c e s " or marginal values a s s o c i a t e d with each of the goal or c o n s t r a i n t rows i n c l u d e d i n the problem. The shadow p r i c e a s s o c i a t e d w i t h a row defines the amount of change i n the l e v e l of goal attainment which w i l l r e s u l t from a one u n i t increase i n the r i g h t hand side value of t h a t row. The COST RANGING agendum determines the s e n s i t i v i t y of the optimal s o l u t i o n to changes i n the costs or y i e l d s a s s o c i a t e d w i t h the a c t i v i t i e s i n c l u d e d i n the optimal b a s i s . The agendum operates on the o b j e c t i v e row c o e f f i c i e n t s f o r the b a s i c a c t i v i t i e s and defines f o r each c o e f f i c i e n t the range over which the value of the c o e f f i c i e n t w i l l be permitted to vary. An -69-a c t i v i t y i s removed from the optimal b a s i s and replaced by another a c t i v i t y , only i f i t s o b j e c t i v e row c o e f f i c i e n t i s given a new value which exceeds the upper l i m i t of the range o r f a l l s below the lower l i m i t . The s p e c i f i c a c t i v i t y to enter the s o l u t i o n depends on whether the upper or the lower l i m i t i s v i o l a t e d . In t h i s problem the o b j e c t i v e row c o e f f i c i e n t s f o r a c t i v i t i e s , other than goal v a r i a b l e s , are equal to zero. For these a c t i v i t i e s the COST RANGING agendum defines the maximum allowable p o s i t i v e and negative changes i n the cost or y i e l d a s s o c i a t e d w i t h the a c t i v i t y . The RHS RANGING agendum determines the s e n s i t i v i t y of the optimal s o l u t i o n to changes i n the r i g h t hand side values of the rows. The agenda operates on the r i g h t hand side values and defines f o r each row the range over which i t s r i g h t hand si d e value may vary without changing the optimal p r i m a l or dual s o l u t i o n . I f a r i g h t hand side value i s changed so th a t i t s new value exceeds the upper l i m i t of the range or f a l l s below the lower l i m i t , an a c t i v i t y w i l l be forced out of the optimal s o l u t i o n and the value of the shadow p r i c e a s s o c i a t e d w i t h the row w i l l be changed. The a c t i v i t y t o leave the s o l u t i o n depends on whether the upper l i m i t or the lower l i m i t i s v i o l a t e d . The DO DJ agendum defines the amount by which the l e v e l of goal attainment i s reduced when one u n i t of a non b a s i c a c t i v i t y i s fo r c e d i n t o the optimal s o l u t i o n . Conversely, the agendum i n d i c a t e s the increase i n y i e l d or decreases i n -70-cost r e q u i r e d before an a c t i v i t y becomes e l i g i b l e to enter the optimal s o l u t i o n . Examination of the output of the above agendum can o f t e n provide i n f o r m a t i o n which i s as valuable as the optimal s o l u t i o n i t s e l f . The remainder of t h i s chapter w i l l be devoted to i l l u s t r a t i n g the a p p l i c a t i o n of post optimal a n a l y s i s to the problems of t e s t i n g the r e a l i a b i l i t y of the s o l u t i o n , i n t e r p r e t i n g the s o l u t i o n , and u t i l i z i n g the s o l u t i o n i n the d e c i s i o n making process. T e s t i n g the R e l i a b i l i t y of the S o l u t i o n In g e n e r a l , the s o l u t i o n to any problem i s only as r e l i a b l e as the data upon which the s o l u t i o n i s based. Therefore i t i s extremely important t o examine the s e n s i t i v i t y of the s o l u t i o n to the type of e r r o r s i n the data t h a t r e s u l t from inaccurate f o r e c a s t s and measurements. One of the valuable a t t r i b u t e s of the LP40 l i n e a r programming al g o r i t h m i s t h a t the output provides the b a s i s f o r a n a l y z i n g the s e n s i t i v i t y of the s o l u t i o n t o various types of e r r o r s . In some cases i t w i l l be found t h a t a small e r r o r i n the value of a parameter may r e s u l t i n a l a r g e change i n the composition of the optimal s o l u t i o n , w h i l e i n other cases the optimal s o l u t i o n remains i n s e n s i t i v e to l a r g e changes i n the value of an input parameter. I t should be remembered t h a t c e r t a i n e r r o r s , such as underestimating annual o p e r a t i n g costs by 15 percent, may r e s u l t i n an unfavorable variance i n the value of the o b j e c t i v e f u n c t i o n but may not cause a change -71-i n the s t r u c t u r e of the optimal s o l u t i o n . In other words the s o l u t i o n may not be as p r o f i t a b l e but i t i s s t i l l the best a v a i l a b l e . The obvious candidates f o r s e n s i t i v i t y a n a l y s i s are the parameters which.measure the net r e t u r n a v a i l a b l e from the various assets. The rates of r e t u r n , \S •, f o r the various classes of assets \ n i are based on f o r e c a s t s , accounting measurements, and s u b j e c t i v e judgements, a l l of which are sub j e c t to v a r y i n g degrees of e r r o r . The output from the COST RANGING agenda i n d i c a t e s the s e n s i t i v i t y of the s o l u t i o n to e r r o r s i n the estimates of the rates of r e t u r n . A sample of the COST RANGING r e s u l t s f o r s e v e r a l of the v a r i a b l e s d e a l i n g w i t h asset purchases appears below i n Table 4. The COST RANGING r e s u l t s i n d i c a t e t h a t the s o l u t i o n i s r e l a t i v e l y i n s e n s i t i v e t o e s t i m a t i n g e r r o r s of the magnitude which might be expected. The most s e n s i t i v e parameter, could t o l e r a t e an e r r o r of plus or minus 16 percent, without changing the optimal b a s i s . The COST RANGING agendum not only i n d i c a t e s the s e n s i t i v i t y of the s o l u t i o n to changes i n the values of the parameters, but a l s o i n d i c a t e s what w i l l happen when the change i n the value of parameter i n i t i a t e s a change i n the optimal s o l u t i o n . For example, I f the r e t u r n a s s o c i a t e d w i t h P 1 (purchases of two year government s e c u r i t i e s at t=l) f a l l s from 2.4 percent to 2.0 percent the optimal b a s i s w i l l change. The s l a c k v a r i a b l e , Row 20, w i l l enter the s o l u t i o n , which means tha t l e s s than TABLE 4 SENSITIVITY ANALYSIS FOR OPTIMAL ACTIVITIES ^ t Maximum Allowable Change Incoming V e c t o r V a r i a b l e Optimal Value \ n i Increase Decrease Upper L i m i t Lower L i m i t 13 ,1 14 ,1 21 ,1 32 ,1 34 >2 11 >2 24 84,311 2,794,543 526 ,927 0 1,200,000 1,259,512 3,500 ,000 .034 .044 .024 .03 .018 .06 .057 .237 .004 .018 .068 999.773 .004 i n f . .004 .042 .004 i n f . .227 .004 .073 21 2 P14 Row 23 A32 Y + M l Row 20 Row 2 3 Row 20 Unbounded 41 Row 2 3 i - j I Unbounded Row 25 -73-the maximum amount of c e r t i f i c a t e s of deposit w i l l be purchased at t = l . Reference to the RHS RANGING r e s u l t s f o r Row 20 shows ( 21 ( t n e r e t u r n on p 2 i ^ o f $«0° 4 reduces the marginal value of c e r t i f i c a t e s of deposits to zero and no funds would be obtained from t h i s source. In the event t h a t the r e t u r n on increases from 2.4 percent to 2.8 percent, the s l a c k v a r i a b l e , Row 23, w i l l enter the bas i s and l e s s than the maximum amount of c a p i t a l w i l l be subscribed at t = l . This suggests t h a t a d d i t i o n a l funds would be a l l o c a t e d to purchases of two year governments, thereby i n c r e a s i n g the l i q u i d i t y of the asset mix and reducing the amount of c a p i t a l r e q u i r e d t o support the mix. The COST RANGING r e s u l t s only give p a r t of the in f o r m a t i o n r e q u i r e d to measure the s e n s i t i v i t y of the s o l u t i o n to e r r o r s i n e s t i m a t i n g the returns a v a i l a b l e from the various a s s e t s . Other i n f o r m a t i o n i s a v a i l a b l e from the output of the DO.DJ agendum. For example, the a c t i v i t y P ^ (purchase of one year government s e c u r i t i e s at t=l) i s not i n c l u d e d i n the optimal b a s i s . However, i t can be determined from the output of the DO.DJ agendum that the rat e of r e t u r n f o r P ^ must increase by .018 i f the a c t i v i t y i s to be in c l u d e d i n the optimal s o l u t i o n . An examination of the s e n s i t i v i t y of other non optimal a c t i v i t i e s shown i n Table 5 confirms the r e s u l t s of the COST RANGING output i n t h a t the s o l u t i o n i s r e l a t i v e l y i n s e n s i t i v e t o e r r o r s i n e s t i m a t i n g the returns a v a i l a b l e from the non optimal a c t i v i t i e s . t h a t the shadow p r i c e a s s o c i a t e d with a d d i t i o n a l c e r t i f i c a t e s of deposits i s $.00 4. I t may be concluded t h a t a decrease i n -74-TABLE 5 SENSITIVITY ANALYSIS OF NON OPTIMAL ACTIVITIES A c t i v i t y ^ n i ,Increase Required P ^ .014 .018 P 2 1 * 0 2 8 * 0 4 P 2 3 .038 .196 P 2 4 .048 .089 p l l - 0 1 4 * 2 7 P 3 .044 .241 14 P 3 2 .03 . , .029 v. -75-I n t e r p r e t a t i o n of the Dual S o l u t i o n In order t o i n t e r p r e t the optimal p r i m a l s o l u t i o n and gain an understanding of the i n t e r a c t i o n s among the various goals and c o n s t r a i n t s , i t i s necessary to analyze the s o l u t i o n to the dual problem. The goals and c o n s t r a i n t s may impose on the problem opportunity costs which cannot be measured i n the course of f o r m u l a t i n g the problem. These costs are defined by the shadow p r i c e s which make up the optimal dual s o l u t i o n . Once the opportunity cost a s s o c i a t e d w i t h a goal i s i d e n t i f i e d , i t i s p o s s i b l e f o r management to evaluate the t r a d e - o f f between the cost imposed by the goal and the b e n e f i t s of m a i n t a i n i n g i t i n i t s present form. In the case of a c o n s t r a i n t i t i s p o s s i b l e to evaluate the cost of r e l a x i n g the c o n s t r a i n t and compare i t to the r e s u l t i n g r e d u c t i o n i n the opportunity cost imposed on the s o l u t i o n . The f o l l o w i n g a n a l y s i s w i l l be concerned w i t h an examination of the c o n s t r a i n t s and goals which determine the a v a i l a b i l i t y of funds and e f f e c t t h e i r deployment. 1. A v a i l a b i l i t y of Funds As has been p r e v i o u s l y e x p l a i n e d , the composition of the banks l i a b i l i t y and c a p i t a l accounts determines to a l a r g e extent the composition of i t s asset accounts. This s e c t i o n w i l l i l l u s t r a t e the type of i n f o r m a t i o n provided by the dual s o l u t i o n w i t h respect to the bank's sources of funds. -76-Shareholders Funds The balance i n the c a p i t a l account at the beginning of a p e r i o d , i s a f u n c t i o n of the earnings r e a l i z e d over the previous p e r i o d , and the dividends payed and c a p i t a l subscribed at the end of the previous p e r i o d . Dividends are constrained to an amount which i s l e s s than o r equal to the c u r r e n t year's earnings and c a p i t a l s u b s c r i p t i o n s are constr a i n e d to an amount which i s l e s s than o r equal to the estimated a v a i l a b i l i of funds from t h i s source. The shadow p r i c e s and RHS RANGING r e s u l t s f o r the rows d e f i n i n g these c o n s t r a i n t s appear i n Table 6. Since any earnings r e a l i z e d can be p a i d out to shareholders, the minimum value of funds from t h i s source i s equal t o t h e i r v a l u e i n the hands of share-h o l d e r s . This value i s determined by the market rate of r e t u r n a v a i l a b l e t o shareholders who i n v e s t t h e i r dividends i n e q u i t i e s of the same r i s k c l a s s as the bank. This e x t e r n a l r a t e was set at 10 percent and t h e r e f o r e $1.00 i n d i v i d e n d s , i n v e s t e d at t = l , w i l l y i e l d the shareholder $1.21 at t=3. One d o l l a r i n v e s t e d at t=2, w i l l y i e l d $1.10, and $1.00 i n v e s t e d at t=3 i s worth $1.00. The problem has been s t a t e d so that earnings w i l l be r e t a i n e d or c a p i t a l usbscribed only i f the amounts r e t a i n e d or subscribed can earn a rate of r e t u r n which i s i n excess of the e x t e r n a l rate of r e t u r n a v a i l a b l e -77-TABLE 6  SOURCES OF FUNDS  DUAL SOLUTION AND RHS RANGING RESULTS Row RHS Dual Ran eye o f Values Outgoing Vector Row Row Desc r i p t i o n s Type Value Value Minimum Maximum At Min. At Max. Earnings Accounts 42 C a l c u l a t i o n of e Q -75,000 1.237 - 8 8 , 252 -68,990 A 1  A 1 3 s2 45 C a l c u l a t i o n of e^ = -75,000 1.100 Unb. -75,000 A 2  A 2 1 46 C a l c u l a t i o n of e^ = -75,000 1.000 Unb. -75,000 A 2  A 1 3 51 C a l c u l a t i o n of C a p i t a l Accounts -75,000 15.00 Unb. 1 ,312,302 p 34 21 Balance on Capital:E 1-E°-S 1-•e +b, • i 0 1.237 Unb. 1 ,026 ,927 A 1  A 21 22 L i m i t on Dividends ^i - eo 0 -191 , 300 Unb. Row 22 23 L i m i t on New C a p i t a l S 1 < 20 ,000 .027 6 , 747 26,010 S 2 30 •> 1 2 Balance on C a p i t a l : E z - E -S -• e l + b 2 = 0 1.10 Unb. 84 ,311 A 1 34 31 L i m i t on Dividends b 2 ~ e ^ 0 0 0 24,047 s 2 Row 32 L i m i t on New C a p i t a l S 2 30 ,000 5,952 0 Unb. Row 32 39 Balance on C a p i t a l : E -E^-S -•e 2+b 3 = 0 .675 Unb. 6 ,000,000 »i 40 L i m i t on Dividends b-j - e 2 0 .325 -190 , 386 156,250 s 2 Row 41 L i m i t on New C a p i t a l Deposit Accounts S 3 •< 30 ,000 0 0 Unb. Row 41 18 Deposit Supply @ t=l:D^-D° = 800 ,000 .928 733, 279 896,171 s 2 *13 19 :Dl - 4,440 ,000 .078 Unb. 5 ,240,000 E° 20 Demand for C e r t i f i c a t e s : B < 100,000 .004 0 1 ,361,870 B 1 p 2 11 27 Deposit Supply @ t=2:D|-D]- = 400 ,000 .786 313, 519 564,545 s 2 A 1  A 1 3 28 :Dl = 5,000 ,000 - .161 Unb. 5 ,000 ,000 P 1 32 29 Demand f o r C e r t i f i c a t e s : B 120,000 0 0 Unb. Row 29 36 Deposit Supply @ t=3:D:jl-D2 (100,000) .617 Unb. 1 ,068,000 ' J . 37 : Dj = 5 ,000 ,000 .637 Unb. 9 ,794,543 38 . . 3 Demand f o r C e r t i f i c a t e s : B < 100 ,000 .743 0 195,232 B 3 S 2 -78-t o shareholders. Reference t o the optimal s o l u t i o n shows tha t the maximum allowable amount of c a p i t a l was subscribed at t = l . At t=2 the dividend payment exceeded the c a p i t a l s u b s c r i p t i o n with the r e s u l t that the net amount re c e i v e d by shareholders was l e s s than the previous years earnings. At t=3, the maximum allowable d i v i d e n d payment was made. These r e s u l t s can be ex p l a i n e d i n terms of the shadow p r i c e s given i n Table 6. The shadow p r i c e a s s o c i a t e d w i t h an a d d i t i o n a l $1.00 i n earnings at t = l obtained a value of $1,237. This i n d i c a t e s that the marginal r a t e of r e t u r n on r e i n v e s t e d funds i s roughly 11.2 percent. Since each $1.00 i n c a p i t a l s u p p l i e d by shareholders imposes upon them an opportunity cost of $1.21, t h e i r net marginal gain from earnings r e i n v e s t e d and c a p i t a l subscribed i s equal to $.027. This amount i s i n f a c t equal to the shadow p r i c e a s s o c i a t e d with the c o n s t r a i n t , Row 23, which l i m i t s the s i z e of c a p i t a l s u b s c r i p t i o n s at t = l . Reference t o RHS RANGING r e s u l t s f o r Row 2 3 i n d i c a t e s that at l e a s t another $6,010 i n new e q u i t y , i f a v a i l a b l e , could be r e i n v e s t e d at a rate of r e t u r n i n excess of 11.1 percent. I f the amount of new c a p i t a l subscribed f e l l below $6,747, the marginal rate of r e t u r n on c a p i t a l funds would be forced t o in c r e a s e . -79-The shadow p r i c e s a s s o c i a t e d with the a v a i l a b i l i t y of earnings at t - 2 , and t=3 are $1.10 and $1.00, r e s p e c t i v e l y . 'Both of"these shadow p r i c e s r e f l e c t the e x t e r n a l r a t e of r e t u r n which can be earned on each $1.00 i n dividends. At t=2 the shadow p r i c e f o r Row 2 3 which measures the marginal r e t u r n a v a i l a b l e on r e i n v e s t e d earnings a l s o obtains a value of $1.10. This i n d i c a t e s that at the margin the r e t u r n on c a p i t a l employed i n the bank i s equal to the e x t e r n a l r a t e and shareholders are i n d i f f e r e n t as to whether or not they r e c i e v e an a d d i t i o n a l $1.00 i n dividends. At t=3 the shadow p r i c e f o r Row 39 which measures the marginal value of an a d d i t i o n a l $1.00 i n c a p i t a l employed i n the bank obtains a value of $.6 75, which i s l e s s than the value of an a d d i t i o n a l $1.00 i n dividends. The RHS RANGING r e s u l t s f o r Row 40 i n d i c a t e s t h a t shareholders would be b e t t e r o f f i f an a d d i t i o n a l $156,250 i n dividends were p a i d out. Their net gain would be $.325 per $1.00 r e c e i v e d i n dividends. The foregoing a n a l y s i s i n d i c a t e s t h a t i n the absence of c o n s t r a i n t s the balance i n the bank's c a p i t a l account w i l l tend to a r r i v e at an amount where the marginal r e t u r n on an a d d i t i o n a l $1.00 i n c a p i t a l i s equal to the marginal opportunity cost to the shareholders of p r o v i d i n g t h a t a d d i t i o n a l $1.00. This f o l l o w s the F i s h e r i a n a n a l y s i s of investment and i n t e r e s t rates which d i c t a t e s t h a t shareholders w i l l i ncrease t h e i r -80-investment only i f the incremental returns on an a d d i t i o n a l investment o f f s e t s the opportunity cost i n c u r r e d . The c o n s t r a i n t s imposed on d i v i d e n d payments and c a p i t a l s u b s c r i p t i o n s prevent the balance i n the c a p i t a l accounts from reaching i t s e q u i l i b r i u m l e v e l at t = l and t=3. At t=2, e q u i l i b r i u m i s a t t a i n e d as dividends are p a i d out to the p o i n t where the marginal r e t u r n on r e t a i n e d earnings r i s e s to a rate equal to the e x t e r n a l rate of r e t u r n . Any f u r t h e r dividend payments would force up the i n t e r n a l r a t e of r e t u r n . A t = l , a c a p i t a l r a t i o n i n g s i t u a t i o n e x i s t s because the marginal r e t u r n on a d d i t i o n a l c a p i t a l investment exceeds the e x t e r n a l r a t e . An a d d i t i o n a l c a p i t a l investment i n excess of $6,010 would d r i v e down the i n t e r n a l r a t e of r e t u r n . At t=3, the shareholders are e x p e r i e n c i n g d i m i n i s h i n g r e t u r n s . Their c a p i t a l investment has exceeded the e q u i l i b r i u m p o i n t by at l e a s t $156,250. Each a d d i t i o n a l $1.00 r e d u c t i o n i n the bank's c a p i t a l at t=3 w i l l reduce the market value of the bank's common stock by only $.6 75. Given t h a t the market value i s equal to 15 times the bank's earnings f o r the year f o l l o w i n g the h o r i z o n , a $1.00 r e d u c t i o n i n the bank's c a p i t a l account at t=3 w i l l only reduce earnings f o r the year by $.045. Therefore, at t=3, an optimal t r a d e - o f f between dividends and c a p i t a l gains has not been achieved. -81-(2) Depositors Funds The balances i n the deposit accounts at the beginning of each p e r i o d are l a r g e l y determined by f o r e c a s t s . The l e v e l of demand deposits can be i n f l u e n c e d by the banks lending a c t i v i t i e s and i t i s assumed t h a t the bank has some c o n t r o l over the amount of funds r a i s e d through c e r t i f i c a t e s of d e p o s i t . The shadow p r i c e s and RHS RANGING r e s u l t s f o r the rows which determine the a v a i l a b i l i t y of deposits appear i n Table 6. The shadow p r i c e f o r Row 18, the c o n s t r a i n t row governing the a v a i l a b i l i t y of demand deposits at t = l obtains a value of $.9 28. The f o r e c a s t exogenous change i n the l e v e l of demand deposits at t = l i s $800,000 and the range of v a l i d i t y f o r the shadow p r i c e has a minimum value of $733,279 and a maximum value of $896,171. Given a tax ra t e of 50 percent, the r e s u l t s i n d i c a t e t h a t the bank would be w i l l i n g to pay $1,95 6 before taxes f o r each a d d i t i o n a l $1.00 i n demand de p o s i t s . No i n t e r e s t i s p a i d on demand deposits which t o t a l $6,000,000 at t = l . This s i t u a t i o n cannot be i n t e r p r e t e d to mean th a t the bank w i l l pay an i n t e r e s t rate of up to 195.6 percent on an a d d i t i o n a l $96,171 i n d e p o s i t s . However, i t does i n d i c a t e that the bank can increase i t s p r o f i t a b i l i t y by paying up to 3.4 percent i n t e r e s t , before t a x e s , on a l l demand deposits i f doing so -82-increases the l e v e l of demand deposits by $96,171. At t = l , time deposits t o t a l $4,440,000. The value of an a d d i t i o n a l $1.00 i n time d e p o s i t s , as defined by the shadow p r i c e f o r Row 19, i s equal to $.078. At t = l the i n t e r e s t r a t e p a i d on time deposits i s 2 percent, a f t e r t axes, and 4 percent before taxes. The RHS RANGING r e s u l t s i n d i c a t e t h a t the bank can improve i t s p r o f i t s by i n c r e a s i n g the before tax i n t e r e s t r a t e on time deposits to 6.4 percent i f doing so a t t r a c t s an a d d i t i o n a l $800,000 i n d e p o s i t s . C e r t i f i c a t e s of deposit provide $100,000 i n funds at t = l . Each a d d i t i o n a l $1.00 i n c e r t i f i c a t e s of deposit up to a maximum of $1,361,870 w i l l i ncrease the banks p r o f i t by $.00 4 per $1.00 of new d e p o s i t s . At t = l the i n t e r e s t rate p a i d to c e r t i f i c a t e holders i s 4 percent before taxes. An increase i n t h i s rate of l e s s than .37 percent would be j u s t i f i e d i n order to a t t r a c t the a d d i t i o n a l $1,361,870 i n c e r t i f i c a t e s of depos i t . 2. Uses of Funds In only one year was the bank unable to achieve an optimal asset mix because of i n s u f f i c i e n t demand f o r c r e d i t . At t=2 the shadow p r i c e a s s o c i a t e d w i t h the demand f o r two year loans obtained a value of $.073. The RHS RANGING r e s u l t s i n d i c a t e t h a t an a d d i t i o n a l $443,000 i n loans would be s u p p l i e d i f there was s u f f i c i e n t demand. The bank could a f f o r d to cut i n t e r e s t rates on these loans -83-by up to 1.09 percent i f doing so would increase the demand f o r these loans by $443,000. 3. Cost of the Goals The dual s o l u t i o n to the problem provides e x p l i c i t i n f o r m a t i o n about management's various goals and t h e i r impact on one another. Since the r e l a t i o n s h i p s which determine the bank-'s. pxo f i t a b i l i t y do not e x p l i c i t l y deal . w i t h the r i s k i n e s s of various l e v e l s of p r o f i t attainment, i t was necessary to introduce other goals whose attainment took p r i o r i t y over the attainment of the p r o f i t g o a l . These other goals taken together attempt to r e s t r a i n the bank from exposing i t s e l f to undue r i s k s . The l i q u i d i t y and d i v e r s i f i c a t i o n goals tend to r e s t r i c t the l e v e l of p r o f i t attainment i n a d e t e r m i n i s t i c sense but not n e c e s s a r i l y i n an absolute sense. Therefore, any opportunity costs a s s o c i a t e d with the r i s k l i m i t i n g goals cannot be evaluated i n terms of marginal cost versus marginal p r o f i t a b i l i t y . Rather, they must be evaluated i n terms of marginal cost versus marginal u t i l i t y , where u t i l i t y i s considered to be some f u n c t i o n which r e f l e c t s the share-holders and depositors aversion t o r i s k , management's propensity t o f o l l o w other banks, and the views of the Federal Reserve. For example, at t=2, the shadow p r i c e f o r Row 59, which defines the l i q u i d i t y g o a l , obtains a value of $.212. The banks c a p i t a l at t=2 i s $1,017,253. This amount i s 90 percent -84-of the amount defined t o be adequate by the examiners c r i t e r i a . For each $1.00 increase i n the d i f f e r e n c e between adequate c a p i t a l and a c t u a l c a p i t a l , the banks p r o f i t a b i l i t y w i l l increase by $.212. Given t h i s i n f o r m a t i o n , management must weigh the d i s u t i l i t y i n v o l v e d i n reducing the banks degree of c a p i t a l adequacy against the increase i n p r o f i t s which would r e s u l t . 4. Marginal A n a l y s i s f o r D e c i s i o n Making I t has already been pointed out that there e x i s t s p o t e n t i a l l y s e r i o u s d e f i c i e n c i e s i n the type of accounting i n f o r m a t i o n a v a i l a b l e to the bank's d e c i s i o n makers. I t i s not p o s s i b l e f o r accounting measurements to take i n t o c o n s i d e r a t i o n the complex i n t e r a c t i o n s which e x i s t among the banks asset and l i a b i l i t y accounts and because of t h i s , the cost and y i e l d data a v a i l a b l e t o managers may •• not r e f l e c t marginal c o n s i d e r a t i o n s . I t i s h i g h l y p o s s i b l e t h a t the greatest p o t e n t i a l b e n e f i t of the goal programming approach to asset management i s the by-product i n f o r m a t i o n provided by the s o l u t i o n to the dual problem. The dual s o l u t i o n i n d i c a t e s the incremental b e n e f i t s which w i l l be r e a l i z e d from a marginal r e l a x a t i o n of any of the models c o n s t r a i n t s . Therefore, the s o l u t i o n provides the r e l e v a n t i n f o r m a t i o n r e q u i r e d to apply marginal a n a l y s i s to various types of d e c i s i o n s . The i n f o r m a t i o n developed i n the previous s e c t i o n i n d i c a t e d t h a t p r o f i t a b i l i t y was r e s t r i c t e d by the a v a i l a b i l i t y -85-of funds f a r more than i t was r e s t r i c t e d by a lack of p r o f i t a b l e investment o p p o r t u n i t i e s . I t would t h e r e f o r e be reasonable t o assume t h a t management would place a high p r i o r i t y on determining ways of a t t r a c t i n g a d d i t i o n a l d e p o s i t s . The a n a l y s i s i n d i c a t e d t h a t the bank could increase i t s p r o f i t a b i l i t y by paying i n t e r e s t on demand deposits i f doing so would a t t r a c t a s u f f i c i e n t amount of a d d i t i o n a l deposit - f u n d s . However, even i f t h i s course of a c t i o n was l e g a l l y a v a i l a b l e , i t would probably not be f e a s i b l e . I f one bank announced th a t i t planned to pay i n t e r e s t on i t s demand d e p o s i t s , i t s competitors would probably f o l l o w the move i n order t o p r o t e c t t h e i r d e p o s i t s . The net r e s u l t of t h i s a c t i o n would be to reduce the p r o f i t s of a l l the banks. The r e s u l t s could be even more damaging t o the i n i t i a t o r of such an a c t i o n , i f i t s competitors d i d not f o l l o w . The bank would soon f i n d i t s e l f w ith f a r more deposit funds than i t could p r o f i t a b l y i n v e s t and would face the prospect of i n c u r r i n g s u b s t a n t i a l l o s s e s . A l t e r n a t i v e s t r a t e g i e s a v a i l a b l e to the bank might incl u d e more s u b t l e competitive moves such as reducing s e r v i c e charges or o f f e r i n g s p e c i a l s e r v i c e s t o d e p o s i t o r s . In any case, part of the inf o r m a t i o n r e q u i r e d to evaluate " t h e costs and b e n e f i t s of such s t r a t e g i e s i s a v a i l a b l e from the optimal s o l u t i o n . The s o l u t i o n i n d i c a t e s t h a t the marginal revenue which would r e s u l t from a t t r a c t i n g an a d d i t i o n a l $96,000 i n demand amounts to $89,000 a f t e r taxes. -86-Assuming that a d e c i s i o n was made and r e s u l t e d i n a t t r a c t i n g the a d d i t i o n a l $96,000 i n d e p o s i t s , reference to the RHS RANGING r e s u l t s i n d i c a t e s t h a t other a c t i o n i s re q u i r e d to maintain o p t i m a l i t y , namely, the purchase of "other s e c u r i t i e s " at t = l , i s no longer a d e s i r e a b l e a c t i v i t y . The a d d i t i o n a l deposits and the funds d i v e r t e d from "other s e c u r i t i e s " are a v a i l a b l e f o r other investments. However, i t i s necessary t o compute a new s o l u t i o n to the problem i n order to determine t h e i r deployment. This new s o l u t i o n can be r e a d i l y determined by performing parametric programming on the r i g h t hand side value f o r Row 18. This f u n c t i o n i s performed by the DO PLP agenda of LP40. 5. Borrowing Funds Management may wish to consider the p o s s i b i l i t y of borrowing funds from other banks. The shadow p r i c e f o r Row 44, which defines the marginal value of a d d i t i o n a l funds, obtains the value of $1,027. The d e s i r e a b i l i t y of borrowing w i l l depend on the terms of the loan. This shadow p r i c e measures the value of an i n t e r e s t f r e e perpetual loan. I f , f o r example, the loan must be rep a i d i n one year, i t i s necessary to s u b t r a c t from the above shadow p r i c e the amount of $.989 which measures the marginal value of funds at t=2, as define d by Row 47. This c a l c u l a t i o n y i e l d s a net value of $.03 8 f o r each $1.00 r a i s e d from one year borrowings. This amount defines the maximum i n t e r e s t r a t e , a f t e r taxes, t h a t the bank can a f f o r d to pay on a one year loan at t = l . -87-Reference to the RHS RANGING r e s u l t s f o r Row 44 i n d i c a t e s t h a t at l e a s t $85,000 would be borrowed at rates below 3.8 percent. CHAPTER V I I DISCUSSIONS AND CONCLUSIONS The previous chapters have emphasized the t h e o r e t i c a l aspects of a h y p o t h e t i c a l model. I t i s now re l e v a n t to discuss some of the p r a c t i c a l i m p l i c a t i o n s a s s o c i a t e d w i t h the development and implementation of such models and the r e l a t i o n s h i p that e x i s t s between d e c i s i o n models used f o r planning and the accounting systems used f o r c o n t r o l . I m p l i c a t i o n s of Model Development I f c a r r i e d out p r o p e r l y , the process of model b u i l d i n g can by i t s e l f , y i e l d valuable r e s u l t s . 1 At the o u t s e t , i t i s e s s e n t i a l that the u l t i m a t e users of the model be i n v o l v e d w i t h and committed t o the development of the model. I t must be recognized that the o r i e n t a t i o n of the model b u i l d e r w i l l be q u i t e d i f f e r e n t than the o r i e n t a t i o n of the ul t i m a t e users of the model. The focus of the mathematical programming expert, tends to be d i r e c t e d towards s o l v i n g the resource a l l o c a t i o n problem, wh i l e the managers tend t o be concerned w i t h the broader aspects of the banking business. I f these two o r i e n t a t i o n s are s u c c e s s f u l l y combined, a l l the l o g i c , r i g o r and computational power of the l i n e a r programming approach can be brought to bear on the asset management problem without s a c r i f i c i n g the o v e r a l l perspective t h a t i s possessed only by G.B. Dantzig, L i n e a r Programming and Extensions, P r i n c e t o n U n i v e r s i t y Press, New Jersey, 196 3, p. 34. -89-the bank's management . The model b u i l d e r depends on management to provide him w i t h a p i c t u r e of the banking environment and define f o r him the o p e r a t i o n a l goals and p o l i c y c o n siderations which must be i n c l u d e d i n the model. Many of the important r e l a t i o n s which should be i n c l u d e d i n the model must be based on management's ab s t r a c t d e f i n i t i o n s of the r e a l world s i t u a t i o n . These a b s t r a c t d e f i n i t i o n s define t a n g i b l e concepts about p a r t i c u l a r facets of the banking business and must be t r a n s l a t e d by the model b u i l d e r i n t o a set of l i n e a r a l g e b r a i c r e l a t i o n s . Therefore i n the course of b u i l d i n g the model, the i m p l i c i t assumptions under which managers operate, must be described i n e x p l i c i t terms. Once described e x p l i c i t l y , these assumptions provide a b a s i s f o r d i s c u s s i n g various areas of asset management. These d i s c u s s i o n s may improve management's conception of the.asset management problem and lead to a refinement of the assumptions un d e r l y i n g investment d e c i s i o n s . Another i n t r i n s i c b e n e f i t of model b u i l d i n g i s t h a t i t u s u a l l y brings accounting data under close s c r u t i n y . C o n s t r u c t i o n of the model r e q u i r e s a p r e c i s e d e f i n i t i o n of i n f o r m a t i o n requirements. In many cases the type of data a v a i l a b l e from the N.L. Chervany, J.S. Strom and R.F. Boehlky, "An Operations Planning Model f o r the Northwestern N a t i o n a l Bank of Minneapolis", Corporate S i m u l a t i o n Models, ed. A.N. S c h r e i b e r , College of S i m u l a t i o n and Gaming and the I n s t i t u t e of Management Science, Providence, Rhode I s l a n d , 1970, pp. 208 - 263. - 9 0 -accounting system may not be r e l e v a n t t o the model. For example, i t i s extremely important t h a t the net y i e l d s developed f o r the various assets and the net costs a s s o c i a t e d w i t h deposits do not inc l u d e a r b i t r a r y a l l o c a t i o n s of f i x e d costs and overhead. A review of the present b a s i s of accounting measure-ments may lead t o the development of an improved accounting system, one which not only c o i n c i d e s more c l o s e l y w i t h the requirements of the model, but a l s o more adequately s a t i s f i e s management's present i n f o r m a t i o n needs. The development of a planning model requires managers to discuss t h e i r conception of the future environment. These d i s c u s s i o n s may help to determine the adequacy of present f o r e -c a s t i n g techniques and may lead to the design and implementation of new f o r e c a s t i n g models. However, i t should be recognized that the major purpose of foc u s s i n g on the future i s to d i s c e r n impending changes i n the nature of the banking business and i d e n t i f y the op e r a t i n g s t r a t e g i e s which are appropriate i n l i g h t of these changes. In summary, s u c c e s s f u l model development blends the mathematical s k i l l s of the model b u i l d e r w i t h the experience and p e r s p e c t i v e of the bank's management. The development process by i t s e l f can lead to improving management's understanding of the, asset management problem. I t a l s o b r i n g s i n t o the open the d e f i c i e n c i e s of the present i n f o r m a t i o n system, and f i n a l l y i t helps to emphasize the importance of f o c u s s i n g management's a t t e n t i o n on the f u t u r e . -91-I m p l i c a t i o n s of Model Implementation The previous chapter i l l u s t r a t e d the type of informat i o n a v a i l a b l e from a s i n g l e run of a goal programming model. However, i n a c t u a l planning s i t u a t i o n s , i t i s u s u a l l y d e s i r a b l e to t e s t a number of a l t e r n a t i v e s t r a t e g i e s . In the face of u n c e r t a i n t y , management w i l l wish to analyze the consequences of a v a r i e t y of future s i t u a t i o n s . The model provides a f a s t , and e f f i c i e n t method of performing such an a n a l y s i s . Therefore, the model has a s i m u l a t i o n c a p a b i l i t y as w e l l as an o p t i m i z i n g c a p a b i l i t y . Use of the model a l s o provides a common b a s i s f o r planning d i s c u s s i o n s . Cohen and Hammer3 have pointed out tha t from t h e i r experience, group asset management d e c i s i o n s r e f l e c t the opinions of the most f o r c e f u l , r a t h e r than the most i n s i g h t f u l members of the group. Discussions centered on the input t o or output from the model w i l l tend to concentrate on s p e c i f i c w e l l defined items r a t h e r than on broad g e n e r a l i z a t i o n s . The previous chapter i l l u s t r a t e d s e v e r a l i n s t a n c e s , where the shadow p r i c e s given by the dual s o l u t i o n might enable management to a r r i v e a t ' d e c i s i o n s based on q u a n t i t a t i v e measurements ra t h e r than on s p e c u l a t i o n . Another p o t e n t i a l l y powerful a p p l i c a t i o n of model l i e s i n i t s use as a research l a b o r a t o r y . The previous chapter has already demonstrated the way i n which the model can be used to Kalman J . Cohen and F r e d r i c k S. Hammer, "Linear Programming and Optimal Bank Asset Management D e c i s i o n s " , The Jo u r n a l  of Finance, XXI (December, 1966), pp. 649 - 674 - 9 2 -perform s e n s i t i v i t y a n a l y s i s . The r e s u l t s of such an a n a l y s i s q u i c k l y p o i n t s out the probable impact of i n a c c u r a c i e s i n the input data and i n d i c a t e s where greater refinement i n measurement i s r e q u i r e d . The model a l s o provides a l o g i c a l and e f f i c i e n t method of experimenting w i t h a l t e r n a t i v e c a p i t a l adequacy for m u l a t i o n s . For example, the c o e f f i c i e n t s used to measure the v o l a t i l i t y of the banks l i a b i l i t i e s and the l i q u i d i t y of the banks assets under the examiners c r i t e r i a may be modified to b e t t e r r e f l e c t the unique c h a r a c t e r i s t i c s of a p a r t i c u l a r bank. The model a l s o could be used to i n v e s t i g a t e p o t e n t i a l a c q u i s i t i o n s or the opening of new branches. O r g a n i z a t i o n a l I m p l i c a t i o n s : Planning and C o n t r o l The cost of developing and implementing a s o p h i s t i c a t e d d e c i s i o n model f o r planning can be j u s t i f i e d only i f the o r g a n i z a t i o n can e f f e c t i v e l y implement the plan generated by the model. Therefore, once a plan has been e s t a b l i s h e d the primary concern o f s e n i o r management i s the performance of the bank's op e r a t i n g departments i n c a r r y i n g out the p l a n . In other words, o r g a n i z a t i o n a l emphasis s h i f t s from planning to c o n t r o l . As the goal programming model provided the f o c a l p o i n t f o r the bank's planning a c t i v i t i e s , the bank's accounting system provides a f o c a l p o i n t f o r c o n t r o l a c t i v i t i e s . The f o l l o w i n g d i s c u s s i o n w i l l examine the r e l a t i o n s h i p t h a t e x i s t s between d e c i s i o n models f o r planning and accounting systems f o r c o n t r o l . -93-The d i s c u s s i o n w i l l i n v e s t i g a t e means of developing accounting information which can be used to evaluate planning performance as w e l l as o p e r a t i n g performance. The aim w i l l be to develop accounting feedbacks which provide i n f o r m a t i o n f o r c o n t r o l and at the same time provide improved inputs f o r the planning model i t s e l f . (A) The Planning Sequence In the development of the model, the planning f u n c t i o n has 4 been c h a r a c t e r i z e d as a process of s e t t i n g goals . I t was shown th a t the highest order g o a l s , those s e t by s e n i o r management, could be d i r e c t l y i n c orporated i n t o the model. The model, given these g o a l s , then generates a plan made up of a set of a c t i v i t i e s which i f c a r r i e d out would lead t o the highest degree of goal attainment. The model in f a c t , decomposes the o r g a n i z a t i o n a l goals defined by s e n i o r management i n t o a set of sub-goals defined by the planned a c t i v i t i e s . These sub-goals become goals f o r the operating managers responsible f o r c a r r y i n g out the a c t i v i t i e s . Since the a c t i v i t i e s i n c l u d e d i n the model represent an aggregation of other a c t i v i t i e s , i t may be necessary to f u r t h e r decompose the sub-goals to represent goals f o r lower l e v e l subordinates. The planning process which commenced w i t h the d e f i n i t i o n of the highest order o r g a n i z a t i o n a l goals i s completed once the lowest order sub-goals have been defined. Y u j i I j i r i , Management Goals and Accounting f o r C o n t r o l , North Holland P u b l i s h i n g Co., 1965, pp. 1 - 5 0 . - 9 4 -(B) The C o n t r o l Process The fundamental purpose of a c o n t r o l accounting system i s to monitor operations and p e r i o d i c a l l y provide s e n i o r management wi t h a p i c t u r e of the or g a n i z a t i o n s a c t u a l performance r e l a t i v e to planned performance. This p i c t u r e i s g e n e r a l l y provided by the accounting system i n the form of data feedbacks which measure progress towards va r i o u s types of goals f o r the various parts of the o r g a n i z a t i o n . This feedback when assembled i s organized and expressed i n terms of accounting v a r i a n c e s . The i n d i v i d u a l variances are analyzed and wherever p o s s i b l e the cause of the variance i s noted. In t h i s a n a l y s i s an attempt i s made to separate variances which r e f l e c t managerial performance from variances which were unavoidable or r e s u l t e d from e s t i m a t i n g , f o r e c a s t i n g , or budgeting e r r o r s . Only a f t e r t h i s r e c o n c i l l i a t i o n i s completed does s e n i o r management re c e i v e the feedback which i t needs to take remedial a c t i o n aimed at improving both operating and planning performance. The i n f o r m a t i o n thus provided i s v i t a l , however, the e f f o r t r e q u i r e d t o prepare i t o f t e n means t h a t the response t o events lags the events themselves by considerable amount of time. This delay may represent a s i g n i f i c a n t opportunity cost to the f i r m . For t h i s reason, i t i s u s e f u l to subdivide c o n t r o l feedback i n t o two c l a s s e s . The f i r s t c l a s s i s the t y p i c a l c o n t r o l accounting feedback which i s only p e r i o d i c a l l y a v a i l a b l e . The second c l a s s of feedback i s i n the form of i n d i c a t o r s which though l e s s p r e c i s e , are more r e a d i l y a v a i l a b l e . Both classes of feedback w i l l be examined, commencing w i t h the l a t t e r . - 9 5 -(C) Feedback I n d i c a t o r s The achievement of each of the o r g a n i z a t i o n a l goals i n c o r p o r a t e d i n the planning model i s dependent on the outcome of a l a r g e number of a c t i v i t i e s . However, f o r a given goal i t may be found that c e r t a i n a c t i v i t i e s are more c r i t i c a l than others. I f these a c t i v i t i e s can be i s o l a t e d and monitored, i t may be p o s s i b l e to impute the degree of goal attainment without p r e c i s e l y measuring i t . Therefore these a c t i v i t i e s are s a i d to be i n d i c a t o r s . I j i r i proposes that feedback i n d i c a t o r s can be evaluated i n terms of t h e i r e f f i c i e n c y and e f f e c t i v e n e s s ^ . The e f f i c i e n c y of an i n d i c a t o r i s a f u n c t i o n of the cost and time r e q u i r e d to make i t a v a i l a b l e , whereas the e f f e c t i v e n e s s of an i n d i c a t o r i s a f u n c t i o n of i t s a b i l i t y t o measure the a c t u a l degree of goal attainment. For example, the c a l c u l a t e d net p r o f i t i s a p e r f e c t l y e f f e c t i v e i n d i c a t o r of the degree of p r o f i t goal attainment, but i t i s not always r e a d i l y a v a i l a b l e and i t s c a l c u l a t i o n u s u a l l y r e s u l t s i n considerable accounting c o s t s . Since to a l a r g e extent the p r o f i t a b i l i t y of the bank i s determined by i t s lending o p e r a t i o n s , i t might be f e a s i b l e to use loan volume as an i n d i c a t o r of the degree of p r o f i t goal attainment. Given th a t banks do i n f a c t c o n t i n u a l l y monitor t h e i r asset and l i a b i l i t y s t r u c t u r e , the t o t a l volume of loans outstanding i s r e a d i l y a v a i l a b l e , and may be a v a i l a b l e i n a form which breaks down t o t a l volume i n t o volume by type of loan. To determine the e f f e c t i v e n e s s of loan volume as an i n d i c a t o r , I b i d , pp. 63 - 64. -96-i t i s necessary to estimate the p o s s i b l e range of values f o r p r o f i t t h a t might be obtained f o r a given loan volume. I j i r i has demonstrated t h a t i n s i t u a t i o n s where a f i r m employs a goal programming model f o r plan n i n g , i t may be p o s s i b l e to use parametric programming to estimate the e f f e c t i v e n e s s of an i n d i c a t o r ^ . For example, i f the bank wished to determine the e f f e c t i v e n e s s of t o t a l loan volume as an i n d i c a t o r of the degree of p r o f i t goal attainment, i t would f i r s t measure net p r o f i t f o r a l l a t t a i n a b l e l e v e l s of loan volume. ( i . e . , the range over which loan volume can vary and s t i l l y i e l d a f e a s i b l e s o l u t i o n . ) This r e l a t i o n s h i p can be p l o t t e d to show the maximum a t t a i n a b l e p r o f i t f o r a given loan volume. This parametric programming procedure i s repeated with an o b j e c t i v e f u n c t i o n which attempts to minimize the degree of p r o f i t goal attainment. The r e s u l t i n g p l o t of net p r o f i t versus loan volume i n d i c a t e s the minimum p r o f i t a t t a i n a b l e f o r a given'loan volume. Given the d i f f e r e n c e between the maximum and minimum p r o f i t f o r a given loan volume, management can assess the e f f e c t i v e n e s s of loan volume as an i n d i c a t o r of p r o f i t goal attainment. Once management has synthesized an i n d i c a t o r which represents an acceptable t r a d e - o f f between e f f e c t i v e n e s s and e f f i c i e n c y , i t i s i n a p o s i t i o n to take c o r r e c t i v e a c t i o n s f a r more r a p i d l y than i t would be able to i f i t had to depend s o l e l y on t r a d i t i o n a l c o n t r o l accounting feedback. For example, i f management noted t h a t the bank's loan volume I b i d . pp. 73 - 75 -97-was expanding less r a p i d l y than had been anticipated, i t could quickly determine whether the rate of growth i n loan demand had been overestimated or whether there were d i f f i c u l t i e s i n s p e c i f i c areas. Action can then be taken to improve performance or revise the bank's o v e r a l l strategy i n response to a new set of exogenous" Variables*. (D) Accounting Feedback The worth of a control accounting system depends l a r g e l y on the usefulness of the feedback information i t provides. In general, feedback may be considered u s e f u l i f i t leads to learning or remedial action. I t i s not useful i f the variances reported are uncontrollable or i f t h e i r source cannot be determined. The p o t e n t i a l usefulness of feedback ind i c a t o r s has been demonstrated and the following discussion w i l l examine the usefulness of the feedback more commonly associated with a i control accounting system. J o e l Demski suggests that firms which use a well defined decision model f o r planning and a t r a d i t i o n a l form of accounting system f o r control may not be getting a l l the u s e f u l feedback that i s available . Dem3ki describes an accounting system structured on a planning model and proposed that t h i s system provides control feedbacks and planning inputs which are superior to those available from t r a d i t i o n a l techniques of standard cost variance analysis. 1 J o e l S. Demski, "An Accounting System Structured on a Linear Programming Model", i n Management Planning and Control, Edited by John L e s l i e Livingstone, New York, McGraw-Hill Inc., 1970, pp. 249 - 266. -98-Th e t r a d i t i o n a l c o n t r o l accounting system a p p l i e s variance a n a l y s i s t o two sets of r e s u l t s , the planned or ex ante r e s u l t s and the observed r e s u l t s . T y p i c a l l y , the i n d i v i d u a l variances are reported e i t h e r as p r i c e and e f f i c i e n c y variances or mix and volume var i a n c e s . An accounting system s t r u c t u r e d on a goal programming model would apply variance a n a l y s i s not only to the ex ante and observed r e s u l t s , but a l s o to a t h i r d set of r e s u l t s , the ex post r e s u l t s . The ex post r e s u l t s are computed by r e - s o l v i n g the goal programming model using r e v i s e d data i n p u t s . These r e v i s e d data inputs are based on informat i o n acquired i n the course of implementing the o r i g i n a l p l a n . The d i f f e r e n c e s between the o r i g i n a l and r e v i s e d i n p u t data are the r e s u l t of i n a c c u r a c i e s i n measurements and f o r e c a s t s , so that the ex post r e s u l t s i n d i c a t e what would have been planned had more 'accurate i n f o r m a t i o n been a v a i l a b l e . The ex post r e s u l t s can be compared to both the ex ante and a c t u a l r e s u l t s and a d d i t i o n a l variances w i l l be reported. Therefore, the s t r u c t u r e d accounting system w i l l , i f nothing e l s e , generate more feedback than i s a v a i l a b l e from the t r a d i t i o n a l system. However, the s u p e r i o r i t y of the s t r u c t u r e d system depends s o l e l y on the usefulness of t h i s a d d i t i o n a l feedback. I t i s t h e r e f o r e , necessary to analyze the type of feedback a v a i l a b l e from both systems. The three sets of r e s u l t s associated w i t h a bank's p r o f i t goal can be synthesized as f o l l o w s : Let A be the column ve c t o r r e p r e s e n t i n g the bank's asset mix and l e t R be a row ve c t o r -99-r e p r e s e n t i n g the c o n t r i b u t i o n margins f o r the various a s s e t s . S i m i l a r l y , l e t D, a column v e c t o r , and I, a row v e c t o r , represent r e s p e c t i v e l y the bank's deposit mix and the u n i t costs a s s o c i a t e d w i t h each type of deposit. F i n a l l y , l e t F, a s c a l a r , denote t o t a l f i x e d c o s t s . The three sets of r e s u l t s can now be w r i t t e n : Observed Net P r o f i t : NP° = R°A° - I°D° - F° Ex Ante Net P r o f i t : NP a = R aA a - I AD A - F a Ex Post Net P r o f i t : NP P = R?A? - IPDP - FP (1) Feedback from T r a d i t i o n a l Varhnce A n a l y s i s T r a d i t i o n a l variance a n a l y s i s compares the d i f f e r e n c e s between observed and ex ante net p r o f i t . F i r s t a set of h y b r i d r e s u l t s are constructed. This set of r e s u l t s , denoted by NP*3, represents the observed asset and deposit mix evaluated at ex ante c o n t r i b u t i o n rates and deposit c o s t s , so t h a t : NP b = R aX° - I AD° - F a The t o t a l d i f f e r e n c e between ex ante and observed net p r o f i t can now be w r i t t e n : NP a - NP° = (NP a - NP b) + (NP b - NP°) Ignoring f i x e d c o s t s , t h i s r e l a t i o n s h i p can be expressed to y i e l d the f o l l o w i n g equation: NP a - NP° = R a ( A a - A°) + I A(D° - DA) + A°(R a - R°) + D°(I° - I A) This equation expresses the t o t a l d i f f e r e n c e between ex ante and observed net p r o f i t i n terms of a volume and mix variance and a p r i c e and e f f i c i e n t y v a r i a n c e . The volume and mix va r i a n c e , denoted by V,, i s defined -100-as f o l l o w s : V 1 = . R a ( A a - A°) + i a(D° - D a) This variance compares the c o n t r i b u t i o n and cost of the planned asset and deposit mix to the c o n t r i b u t i o n and cost of the a c t u a l asset mix, evaluated at ex ante r a t e s . The components of vector A a defines the planned s e t of a c t i v i t i e s s e l e c t e d by the goal programming model. Therefore, the d i f f e r e n c e s between A a and A° evaluates the degree of sub-goal attainment. The variance a l s o i n d i c a t e s the extent to which overestimating or underestimating deposit volumes might e f f e c t the degree of sub-goal attainment. The t o t a l p r i c e and e f f i c i e n t y v a r i a n c e , denoted by V"2, i s defined as f o l l o w s : V 2 = A°(R a - R°) + D°(I° - I a ) This variance measures the d i f f e r e n c e s between planned and a c t u a l rates of c o n t r i b u t i o n and cost f o r the observed mix of assets and d e p o s i t s . In order to evaluate performance on the b a s i s of these v a r i a n c e s , i t i s necessary t o i s o l a t e those variances which were unavoidable or r e s u l t e d from e r r o r s i n f o r e c a s t i n g , measurement or the model i t s e l f , from those which r e s u l t e d from sub-standard performance. There are of course, many d i f f i c u l t i e s encountered when c a r r y i n g out such an assessment, however, the primary concern of t h i s d i s c u s s i o n i s not the i n t e r p r e t a t i o n of i n d i v i d u a l v a r i a n c e s , but the e v a l u a t i o n of the types of feedback a v a i l a b l e from the t r a d i t i o n a l c o n t r o l accounting system. -101-Demski proposed that the major weakness of t r a d i t i o n a l standard cost variance a n a l y s i s i s t h a t i t f a i l s to monitor a l l of the input parameters and th e r e f o r e may f a i l to re p o r t the true magnitude of the opportunity costs i n c u r r e d by the f i r m . For example, i f a given asset, A^, i s not i n c l u d e d i n e i t h e r the planned b a s i s , A or the observed b a s i s , A , the t r a d i t i o n a l a n a l y s i s would not r e p o r t any variance r e s u l t i n g i n d i f f e r e n c e s between R a and R?, the ex ante and ex post c o n t r i b u t i o n r a t e s f o r .• I t i s conceivable t h a t e r r o r s i n f o r e c a s t i n g the value of R^ r e s u l t e d i n i t s e x c l u s i o n from the planned and a c t u a l asset mix. I f t h i s , i n f a c t , happened, the bank would have i n c u r r e d an opportunity cost. Furthermore, s i n c e the variance a n a l y s i s d i d not r e q u i r e t h a t R^ be r e c a l c u l a t e d , i t i s p o s s i b l e t h a t t h i s f a i l u r e to monitor R^ m i g h t " r e s u l t i n subsequent planning e r r o r s . In g e n e r a l , the t r a d i t i o n a l a n a l y s i s only monitors the c o e f f i c i e n t s a s s o c i a t e d w i t h the a c t i v i t i e s which appear i n the observed b a s i s . (2) Feedback from Ex Post Variance A n a l y s i s The s t r u c t u r a l accounting system proposed by Demski produces variances based on a comparison of the ex ante, observed and ex post r e s u l t s . The t o t a l d i f f e r e n c e between the ex ante and observed r e s u l t s i s expressed as f o l l o w s : NP a - NP° = (NP a - NP P) + (NPP - NP°) The f i r s t term, the d i f f e r e n c e between ex ante and ex -102-post p r o f i t measures the d i f f e r e n c e between what the bank planned to do and what i t should have planned to do. The s i z e of t h i s d i f f e r e n c e i s an i n d i c a t o r of the firms planning e f f e c t i v e n e s s . Again, i g n o r i n g f i x e d c o s t s , the components of t h i s planning variance may be expressed as f o l l o w s : NP a - NP P = R P ( A a - A P) + I p ( D P - Da) + A a ( R a - R p) + D a ( I p - I a ) The volume and mix v a r i a n c e , given by the f i r s t two terms, compares the ex ante and ex post b a s i s evaluated at ex post r a t e s . Demski r e f e r s to t h i s variance as a b a s i s variance and i m p l i e s that the variance i n d i c a t e s f o r e c a s t i n g performance. However, i n the context of the goal programming model, both A a and A p are determined by the model and are i n f a c t not c o n t r o l l a b l e and therefore t h i s variance i s of questionable usefulness. The second p a r t of the planning v a r i a n c e , the p r i c e and e f f i c i e n c y v a r i a n c e , measures the d i f f e r e n c e between ex ante and ex post rates of c o n t r i b u t i o n and cost. This variance does i n f a c t monitor the bank's a b i l i t y t o f o r e c a s t the values of the parameters used by the planning model, and i s th e r e f o r e u s e f u l feedback. In the t o t a l variance equation the second term, (NP P - NP°), measures the d i f f e r e n c e between the ex post r e s u l t s , determined by the model, and the observed r e s u l t s . I t t h e r e f o r e , measures the d i f f e r e n c e between the p r o f i t the bank could have p o t e n t i a l l y earned and p r o f i t i t a c t u a l l y d i d earn. -103-This d i f f e r e n c e i s equal to the t o t a l opportunity cost i n c u r r e d by the bank. The components of t h i s opportunity cost variance can be expressed as f o l l o w s : N P P _ N P O = R o ( A P _ A o } + jo^o _ d P ) + A o ( R p _ R o } + D°(I° - IP) The volume and mix v a r i a n c e , given by the f i r s t two terms, compares.the ex ante and observed b a s i s evaluated at observed r a t e s . I t compares the a c t u a l asset mix to the best p o s s i b l e mix and i s therefore an absolute i n d i c a t o r of o v e r a l l asset management e f f e c t i v e n e s s . In the event th a t the ex post asset mix d i f f e r s from the planned asset mix t h i s variance i s of dubious value as an i n d i c a t o r of managerial performance. The p r i c e and e f f i c i e n c y v a r i a n c e , given by the l a s t two terms of the opportunity cost v a r i a n c e , measures the d i f f e r e n c e between ex post and observed c o n t r i b u t i o n and deposit rates f o r the observed asset and deposit mix. I t i s t h e r e f o r e , a d i r e c t i n d i c a t o r of the operating performance of managers res p o n s i b l e f o r a c t i v i t i e s which a f f e c t c o n t r i b u t i o n rates and deposit r a t e s . This variance would evaluate the a b i l i t y of asset managers to get the best a v a i l a b l e rate of i n t e r e s t f o r each r i s k c l a s s of a s s e t s . I t has been demonstrated t h a t ex post a n a l y s i s produces variances which represent u s e f u l feedback of a type not a v a i l a b l e from t r a d i t i o n a l variance a n a l y s i s . The ex post a n a l y s i s insures that a l l input parameters are monitored and t h a t the t o t a l o p portunity cost i s reported. However, since the goals set f o r -104-operating managers are based on the planned r e s u l t s and goal attainment i s the c r i t e r i o n against which performance i s to be evaluated, i t can be concluded that the ex post system augments t r a d i t i o n a l variance a n a l y s i s but cannot replace i t . Use of the combined r e s u l t s of both methods "of variance a n a l y s i s can provide s u p e r i o r feedback f o r c o n t r o l as w e l l as improved inpu t data f o r the goal programming model. Conclusions This t h e s i s has attempted to demonstrate the p o t e n t i a l f o r the use of goal programming models as a b a s i s f o r f o r m u l a t i n g the o p e r a t i n g plans of commercial banks. A review of the dynamics of bank asset management focussed on the complex resource a l l o c a t i o n problem faced by the bank's s e n i o r management. I t was noted that the best s o l u t i o n to t h i s problem was one i n which the bank came c l o s e s t to s a t i s f y i n g a number of goals r a t h e r than one which optimized the attainment of a s i n g l e goal such as net p r o f i t . The p l a n n i n g process c a r r i e d out by banks was c h a r a c t e r i z e d as a process of t r a n s l a t i n g a s e t of o r g a n i z a t i o n a l goals i n t o a set of operating sub-goals which defined the resource a l l o c a t i o n a c t i v i t i e s to be c a r r i e d out by the banks operating managers. I t was shown that t h i s process could be c a r r i e d out by a goal programming model. To i l l u s t r a t e the a p p l i c a t i o n of goal programming to asset management, a mathematical model was formulated to represent the a c t i v i t i e s of a h y p o t h e t i c a l banking e n t e r p r i s e . An -105-assumed set of data was input to the model and a s o l u t i o n determined. This s o l u t i o n represented a m u l t i - p e r i o d plan f o r the bank's o p e r a t i o n . A n a l y s i s of the r e s u l t s demonstrated th a t the s o l u t i o n provided not only planning i n f o r m a t i o n , but a l s o i n f o r m a t i o n which could be used by managers i n day-to-day d e c i s i o n making. The i m p l i c a t i o n s of developing and implementing a goal programming model were discussed and i t was noted t h a t such a model could provide managers w i t h a common frame of reference i n f o r mulating planning and operating d e c i s i o n s as w e l l as the means of t e s t i n g the r a m i f i c a t i o n s of various o p e r a t i n g s t r a t e g i e s . The d i s c u s s i o n concluded w i t h an i n v e s t i g a t i o n of r e l a t i o n s h i p t h a t e x i s t s between a goal programming model which provides a b a s i s f o r planning and the accounting system which provides a b a s i s f o r c o n t r o l . I t was shown how the accounting system could provide feedback which would f a c i l i t a t e e f f e c t i v e c o n t r o l as w e l l as feedback which could improve the performance of the planning model i t s e l f . In g e n e r a l , i t may be concluded t h a t the a p p l i c a t i o n of goal programming t o bank asset management represents a p o t e n t i a l l y powerful blend of management i n t u i t i o n and management sc i e n c e . -106-BIBLIOGRAPHY B a i l e , Andrew D. J r . and Douglas, L. Mahrer, "Analysis of a Li n e a r Programming Model f o r Budgeting and F i n a n c i a l P l a n n i n g " , i n Management Planning and C o n t r o l , E d i t e d by John L e s l i e L i v i n g s t o n e , New York, McGraw-Hill I n c . , 1970 pp. 539-554. Baumol, W i l l i a m J . , Economic Theory and Operations A n a l y s i s , Englewood C l i f f s , New Jersey: P r e n t i c e - H a l l , Inc. 1961. 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APPENDIX -109-EXHIBIT I MULTIPLE GOAL PROBLEM The f o l l o w i n g m u l t i p l e goal problem may be used t o i l l u s t r a t e the goal programming technique. A production manager faces the problem of a l l o c a t i n g jobs between two production teams. Team number 1 can produce one u n i t per hour, w h i l e team number 2 which i s p r e s e n t l y being t r a i n e d , can produce only one-half a u n i t per hour. The production manager has been given a production t a r g e t of 15 u n i t s per day. In a d d i t i o n , he d e s i r e s t h a t both teams work as clo s e to e i g h t hours as p o s s i b l e . I d l e time should be avoided, but only a f t e r the other goals have been s a t i s f i e d . I f overtime o p e r a t i o n i s necessary t o achieve the production t a r g e t , he w i l l assign i t to the teams i n such a way t h a t any overtime beyond two hours i s avoided f o r team number 1. S o l u t i o n Procedure (1) Determine the equations f o r goals and c o n s t r a i n t s . Let = number of hours worked by team one x 2 = number of hours worked by team two Production goal x x + .5x 2 + y7_ - = 15 Hours worked goals x 1 + y£ - yf = x 2 + Y2 ~ Y~3 = 8 8 -110-(2) Develop the minimizing o b j e c t i v e f u n c t i o n . Production goal Over-attainment i s s a t i s f a c t o r y . Therefore the element f o r that goal i n the o b j e c t i v e f u n c t i o n i s v l -Hours worked goals Exact attainment i s d e s i r e d . Therefore both s l a c k v a r i a b l e s f o r each goal w i l l enter the o b j e c t i v e f u n c t i o n . T o t a l o b j e c t i v e f u n c t i o n Minimize Z = y^ + y~ + Y\ + Y~ + (3) Order the v a r i a b l e s i n the o b j e c t i v e f u n c t i o n . M i n i m i z i n g i d l e time This goal i s subordinate to a l l other goals and w i l l be assigned a preemptive p r i o r i t y f a c t o r equal to M^. This order group contains the v a r i a b l e s , y^ and y~. M i n i m i z i n g overtime This goal takes p r i o r i t y over the goal of minimizing i d l e time but i s subordinate t o the attainment of the production t a r g e t . A preemptive p r i o r i t y f a c t o r equal to M w i l l be assigned to the v a r i a b l e s y* and y + . 2. 2. 3 Production goal The production goal has the h i g h e s t p r i o r i t y and the v a r i a b l e , y , w i l l be assigned a preemptive p r i o r i t y f a c t o r equal to M^. T o t a l o b j e c t i v e f u n c t i o n Minimize Z = M^y^ + M 2 V 2 + M1^2 + M2^3 + M1^3 -111-(4) Solve the problem using the simplex method. Optimal S o l u t i o n x1 = 11 x 2 = 8 y 2 = 3 Since y* i s greater than 3 hours, the problem i s not y e t completed. (5) Form sub-goals where they are appropriate. The problem s t a t e d t h a t any overtime beyond 2 hours must be avoided f o r team 1. Since the s o l u t i o n y i e l d e d a value f o r y* equal t o 3, a sub-goal must be formulated. Yo + y~ - y + = 2 (6) Weigh the elements of order groups which contain p o s i t i v e elements. y+ i s assigned a r e g r e t c o e f f i c i e n t which i s equal t o 1. y* i s assigned a r e g r e t c o e f f i c i e n t which i s equal t o 2. (7) Order the v a r i a b l e s i n the o b j e c t i v e f u n c t i o n . Four l e v e l s of p r i o r i t y have now been e s t a b l i s h e d . The order groups f o r M^ and M 2 remain the same. L i m i t on overtime goal Subordinated only to the production g o a l , i s the goal t h a t any overtime f o r team 1, beyond 2 hours, be minimized. Therefore the v a r i a b l e , Y ^ ' i s assigned a preemptive p r i o r i t y f a c t o r of M^. Production goal The preemptive p r i o r i t y f a c t o r f o r y-^  now becomes . Revised o b j e c t i v e f u n c t i o n Minimize Z = M 4y~ + M 3 y 2 1 + M 2 (y+ + 2y^) + M^y" + y~) -112-Solve the r e v i s e d problem by the simplex method Optimal s o l u t i o n x 1 = 10 y+ = 2 x 2 = 10 y ; = 2 Since both y 2 and y* belong to the same order group and a l l higher order goals are s a t i s f i e d the problem i s completed. -113-EXHIBIT I I DEFINITION OF SYMBOLS Section A: V a r i a b l e s , A c t i v i t i e s and Goals D e s c r i p t i o n  Cash Balance at time t Value of Government S e c u r i t i e s , maturing i n one year, h e l d at time t Value of Government S e c u r i t i e s , maturing i n two years, h e l d at time t Value of Government S e c u r i t i e s , maturing i n three years, h e l d at time t Value of other s e c u r i t i e s maturing i n one year, h e l d at time t Values of loans due i n one year, outstanding at time t Values of loans due i n two yea r s , outstanding at time t Values of loans due i n three years, outstanding at time t Balance of demand deposits h e l d at time t Balance of time deposits h e l d at time t Value of C e r t i f i c a t e s of Deposits outstanding at time t Balance of the c a p i t a l accounts at time t Amount of one year Government S e c u r i t i e s purchased at time t Amount of two year Government S e c u r i t i e s purchased at time t Amount of three year Government S e c u r i t i e s purchased at time t Amount of other one year s e c u r i t i e s purchased at time t Amount of one year loans granted at time t Amount of two year loans granted at time t Amount of three year loans granted at time t -114-IT I I (Continued) D e s c r i p t i o n Sale of i n i t i a l holdings of two year government s e c u r i t i e s at t = 1 Sale of i n i t i a l h oldings of three year government s e c u r i t i e s at time t Endogenous increase i n demand deposits created at time t , which are a v a i l a b l e at time t = t+1 Endogenous decrease i n demand deposits created at time t , and which are a v a i l a b l e at time t = t+1 Net income f o r p e r i o d ending at time t Amounts of dividends payed at time t Amount of new equ i t y subscribed at time t The amount of demand deposits created endogenously during p e r i o d t , which are a v a i l a b l e at time t = t+1 The c o n t r a c t i o n i n demand deposits created endogenously during p e r i o d t , which are a v a i l a b l e at time t = t+1 Amount by which the c a p i t a l adequacy goal i s o v e r a t t a i n e d Amount by which the c a p i t a l adequacy goal i s underattained The amount by which t o t a l loans outstanding at time t exceed the amount s e t by the goal which l i m i t s t o t a l loans to a maximum of percent of t o t a l assets The amount by which t o t a l loans f a l l below K percent of t o t a l assets The amount by which t o t a l loans outstanding f a l l s short of s a t i s f y i n g the goal which c a l l s f o r mai n t a i n i n g t o t a l loans at a l e v e l i n excess of percent of t o t a l assets The amount by which loans outstanding exceeds the K 3 percent of t o t a l assets The amount by which three year loans outstanding at t exceeds the amount set by the goal which l i m i t s these loans to a maximum of percent of demand deposits The amount by which t o t a l loans outstanding f a l l below percent of demand deposits The amount by which the sum of three year loans outstandin and the holdings of three year government s e c u r i t i e s excee the amount set by the goal which l i m i t s the t o t a l of these assets to K 5 times the banks c a p i t a l -115-EXHIBIT I I (Continued) Symbol D e s c r i p t i o n Y The amount by which the sum of three year loans and three year government s e c u r i t i e s f a l l below K times the banks c a p i t a l Y The amount by which holdings of government s e c u r i t i e s f a l l below the amount re q u i r e d to s a t i s f y the goal which c a l l s f o r m a i n t a i n i n g t o t a l government s e c u r i t i e s at l e v e l i n excess of Kg percent of t o t a l assets Yg The amount by which holdings of government s e c u r i t i e s 8 exceed Kg percent of t o t a l assets Y^ The amount by which t o t a l assets at the h o r i z o n exceed the l e v e l defined by the growth goal YTj The amount by which t o t a l assets at the horiz o n f a l l s h ort of s a t i s f y i n g the growth goal Y* The amount by which the increase i n wealth over the p e r i o d exceeds the amount set as a t a r g e t i n the p r o f i t goal Y The amount by which the increase i n the shareholders wealth f a l l s short of s a t i s f y i n g the p r o f i t goals -116-EXHIBIT I I (Continued) SECTION B: Parameters Symbol D e s c r i p t i o n ' K ' Minimum value f o r r a t i o of the bank's l e v e l of eq u i t y to the l e v e l of equity defined as adequate by the Examiner's C r i t e r i a K 2 Maximum value f o r r a t i o of t o t a l loans to t o t a l assets K3 Minimum value f o r r a t i o of t o t a l loans to t o t a l assets K Maximum value f o r r a t i o of long term loans assets to demand deposits Kg Maximum value f o r r a t i o of long term assets t o the bank's e q u i t y Kg Minimum value f o r r a t i o of government s e c u r i t i e s to t o t a l assets G The t a r g e t e d l e v e l of the bank's assets at the end o f the planning h o r i z o n Q The ta r g e t e d increase i n shareholders wealth over the planning p e r i o d - The r e t u r n i n each of n-years r e s u l t i n g from the purchase n i of one d o l l a r of asset at the beginning of p e r i o d t . m  A^^ The i n t e r e s t p a i d on time deposits The i n t e r e s t p a i d on c e r t i f i c a t e s of deposits f t . The r a t i o of s e l l i n g p r i c e t o purchase p r i c e which r e s u l t s n i i f one d o l l a r of asset A^^ i s s o l d before maturity a^_ The p r i c e earnings r a t i o a p p l i e d by the market to the bank's e q u i t y at time t V The r a t e of r e t u r n a v a i l a b l e i n p e r i o d t f o r investments ' t i n eq u i t y of the same r i s k c l a s s as the banks Management t a r g e t f o r the average annual growth r a t e i n the wealth of the shareholders over the planning p e r i o d Mfc Forecast amount of the bank's share of the loans market at time t P The amount by which i t i s assumed demand deposits increase or decrease as a r e s u l t of a one d o l l a r expansion or c o n t r a c t i o n i n loan volume -117-EXHIBIT I I (Continued) Symbol D e s c r i p t i o n P 2 The amount by which i t i s assumed demand deposits expand or c o n t r a c t due t o a one d o l l a r increase or decrease i n market share of loan volume A*" Forecast demand f o r new loans maturing i n n years at time t n4 D^  Forecast of exogenous change i n the l e v e l of demand deposits from time t-1 to t D^  Forecast l e v e l of time deposits at time t B*" Forecast demand f o r c e r t i f i c a t e s of deposits at time t E*~ Forecast of the amount of new equ i t y c a p i t a l a v a i l a b l e from shareholders at time t -118-EXHIBIT I I I ( a ) FORECAST OF LOAN DEMAND AND MARKET SHARE Year Type of Loan Market Share £ 514 524 *34 M t 1 3,000,000 3,000,000 3,000,000 6,000,000 2 3,500,000 3,500,000 3,500,000 6,500,000 3 3,500,000 3,500,000 3,500,000 7,000,000 -119-EXHTBIT 111(b) FORECAST OF AVAILABILITY OF DEPOSITS AND CAPITAL Forecast of Year Forecast Balance Exogenous Change Forecast of Time C e r t i f i c a t e s Demand New C a p i t a l Deposits of Deposit Deposits A v a i l a b l e t D^  B1 5^ E t 1 4,440,000 100,000 800,000 20,000 2 5,000,000 120,000 400,000 30,000 3 5,000,000 100,000 (100,000) 30,000 -120-EXHIBIT III(c) FORECAST OF ASSET YIELDS Year Government Secur i t i e s Other Loans 1 year 2 year 3 year Secur i t i e s 1 year 2 year 3 year t tn X21 ^32 X l 4 \l \l 0 .01 .02 .025 .03 .04 .05 .06 1 .014 .024 .03 .034 .044 .056 .068 2 .018 .028 .032 .038 .048 .06 .072 3 .014 .024 .03 .034 .044 .058 .068 -121-EXHIBIT 111(d) FORECAST OF DEPOSIT INTEREST RATES Year Time Deposits C e r t i f i c a t e s ' of Deposit t ^ 0 .015 .020 1 .020 .025 2 .025 .025 3 .020 .022 -122-EXHIBIT 111(e) FORECAST GAIN OR LOSS ON SALE OF SECURITIES BEFORE MATURITY Ra t i o of Purchase P r i c e to S e l l i n g P r i c e  Year 3 Year Governments 2 Year Governments t f t f t r r21 r32 1 .88 .87 2 .83 -123-EXHIBIT 111(f) PARAMETERS OF LIQUIDITY AND DIVERSIFICATION GOALS Parameter D e s c r i p t i o n Value Degree of C a p i t a l Adequacy 100% K Maximum Value of R a t i o of Loans to .65 T o t a l Assets K 3 Minimum Value of Ratio of Loans t o .55 T o t a l Assets K Maximum Value of R a t i o of 3 Year Loans .20 to Demand Deposits K Maximum Value of Rat i o of 3 Year Assets 5.0 to C a p i t a l Kg Minimum Value of R a t i o of Government .25 S e c u r i t i e s t o T o t a l Assets -124-EXHIBIT IV OPTIMAL , SOLUTION AND COST RANGING 1 RESULTS Optimal S o l u t i o n Cost Ranging Results A c t i v i t y or V a r i a b l e Optimal Value X S i Cost i n Problem Maximum Increase Change Decrease Incoming Limit 1 Vector L i m i t 2 Not i n Basis -,1 L32 Not i n Basis -L2 L32 Not i n Basis -C l 1,237,513 - i n f . .440 Unb. Row 20 4l 1,026 ,927 - -.004 .004 Row 23 Row 20 526,927 .024 - -.004 .004 Row 2 3 Row 20 - i i 1,000 ,000 - -.018 .186 L2~l >11 Not i n Basis .014 -A 1  A32 Not i n Basis -32 0 .03 - -.018 i n f . Unb. A13 84,311 - -.057 .004 *21 Row 2 3 * i 3 84,311 .034 - -.057 .004 Y21 Row 2 3 *34 1,200 ,000 - -999.773 .227 Y + 41 P 1 34 1,200 ,000 .06 8 - -999 .773 .227 Y+ *41 Y41 A 1  A24 2,208,000 - -.072 .083 Row 25 P 2 *14 ^ 4 1,168,000 .056 - -.072 .083 Row 25 P 2 14 -14 4,794,543 - -.237 .042 b l ' *l< 2,794,543 .044 - -.237 .042 *i« b l "1 6,000 ,000 - i n f . i n f . Unb. Unb. °l 4,440,000 - i n f . i n f . Unb. Unb. B 1 100 ,000 - i n f . .004 Unb. Row 20 E 1 1,011,300 - i n f . .027 Unb. Row 23 536,510 - 0 .194 d l b i d l Not i n Basis -e o 191,300 - i n f . 3.098 Unb. b l Not i n Basis -s 1 20 ,000 - i n f . .027 Unb. Row 2 3 v 1 84,466 -.046 .068 Row 52 Row 2 3 ' V 1 m 51,964 - -.037 .111 Row 53 Row 2 3 116,676 - -.031 .363 Row 54 Row 53 -125-EXHIBIT IV (Continued) Optimal Solution Cost Ranging Results  Cost A c t i v i t y Optimal ' t i n Maximum Change Incoming Vector or V a r i a b l e Value \ n i Problem Increase Decrease Limit 1 L i m i t 2 c 2 1 , 4 2 0 , 0 1 4 - - . 2 0 2 1 . 2 0 5 Row 8 2 Row 2 3 0 - - . 0 1 1 i n f . Unb. »L Not i n Basis . 0 2 8 -2 , 3 1 9 , 4 3 9 - - . 0 3 1 . 0 9 6 3 P 1 4 P2 A 3 2 1 , 2 9 2 , 5 1 2 9 1 9 , 0 0 2 . 0 1 8 -- . 0 0 4 - . 0 9 4 . 0 0 4 . 0 1 1 Row P 3 1 4 2 0 Row A 1 3 2 2 3 9 1 9 , 0 0 2 . 0 3 2 - - . 0 9 4 . 0 1 1 P 3 1 4 A 1 3 2 *L 0 - - . 1 9 6 i n f . P 2 1 3 Unb. P 2 1 3 Not i n Basis . 0 3 8 -A 2  A 3 4 1 , 3 8 7 , 3 0 2 - - 2 . 7 7 . 1 5 0 Row 2 5 Y 4 2 A. 1 , 3 8 7 , 3 0 2 . 0 7 2 - - 2 . 7 7 . 1 5 0 Row 2 5 Y 4 2 4 , 7 0 0 , 0 0 0 - - 9 9 9 . 7 7 3 . 0 7 3 + Y 4 1 Row 2 5 3 , 5 0 0 , 0 0 0 . 0 6 - i n f . . 0 7 3 Unb. Row 2 5 2 , 2 0 8 , 0 0 0 - - . 0 7 2 . 1 3 6 Row 2 5 A 3 2 » i . Not i n Basis . 0 4 8 -<>? 6 , 9 3 6 , 5 1 0 - - . 5 5 5 . 1 9 6 Row 2 5 b 2 -1 5 , 0 0 0 , 0 0 0 - i n f . i n f . Unb. Unb. B 2 Not i n Basis -E 2 1 , 0 1 7 , 2 5 3 - - . 2 3 4 . 1 0 6 Row 5 7 A 3 2 <$ 1 8 8 , 8 0 7 - . 9 7 3 d 2 A 1 3 2 *i Not i n Basis -• * i 3 1 4 , 0 8 1 - - 2 . 7 8 . 2 6 2 Row 2 0 b 2 » 2 3 1 4 , 0 8 1 - - 2 . 7 8 Row 2 0 Row 3 1 S 2 5 , 9 5 2 - - . 0 2 7 b l Row 3 1 V 2 r 9 1 , 9 0 4 - - . 2 2 1 . 1 8 1 Row 5 6 V 2 m 2 3 , 4 5 5 - - . 2 1 4 4 . 0 3 2 Row 5 7 V 2 ' 1 5 5 , 7 4 0 - - . 2 1 6 1 . 6 9 7 Row 5 8 C 3 - 1 , 4 3 5 , 2 2 3 - . 7 4 9 1 . 9 1 2 Row 8 3 Row 2 3 A 3  A 2 1 3 , 2 6 0 , 6 4 2 - - . 2 6 2 . 0 1 0 + Y 6 3 A 1 3 2 *l 2 , 3 4 1 , 6 4 0 . 0 2 4 - . 0 7 4 . 0 9 6 Row 2 5 , P 3 1 4 *L 0 - - . 0 1 1 i n f . Unb. Not i n Basis . 0 1 4 --126-EXHIBIT IV(Continued) Optimal S o l u t i o n Cost Ranging Results  Cost A c t i v i t y Optimal . i n Maximum Change Incoming Vector or V a r i a b l e Value X n i Problem Increase Decrease Limit 1 L i m i t 2 A 3 A32 0 - -.029 i n f . *332 Unb. P 3 *32 Not i n Basis .032 -Not i n Basis -Not i n Basis .034 -»l 1,405,063 - -1.93 .268 B 2 Y43 'I. 1,405,063 2,341,640 .068 --1.93 -.074 .268 .096 B 2 Row 25 Y~ 43 '5. 954,338 .058 - -.076 .096 Row 25 A 3  A14 4,700,000 - -.091 .073 P 3 *14 Row 25 >14 Not i n Basis .044 -°i 7,025,316 - -.385 .308 B 2 b 2 D2 3 5,000,000 - i n f . i n f . Unb. Unb. B 3 100,000 - i n f . .743 Unb. Row 32 E 3 1,017,253 - -.234 .106 Row 57 A32 -r 159,810 - 0 1.145 d 3 d 3 Not i n Basis -e2 334,714 -.833 .717 Row 54 Row 20 b 3 334,714 - -.833 .322 Row 54 Row 40 s 3 Not i n Basis -v r 33,962 - 4.074 Row 60 Y + 63 V 3 m 185,580 0 0 Y13 V i 49,652 - 0 0 Row 62 A13 e 3 452,989 - -4.338 1.445 Row 57 Y + *11 Not i n Basis -Y l l Not i n Basis 1000 Y + 12 Not i n Basis -Y12 Not i n Basis 1000 Y + 13 " Not i n Basis -Y~ 13 Not i n Basis 1000 Y + 21 Not i n Basis 1000 Y21 Not i n Basis -Y + 22 Not i n Basis 1000 Y22 124,638 - -.186 .369 + Y62 B2 -127-EXHIBIT IV (Continued) Optimal S o l u t i o n Cost Ranging Results Cost A c t i v i t y Optimal \ n i i n Maximum Change Incoming Vector or V a r i a b l e Value Problem Increase Decrease L i m i t 1 L i m i t 2 Y +  x23 Not i n Basis 1000 Y23 95,966 -.242 3.025 P^ 4 Row 57 Y + 31 1,261,931 -.721 .102 Row 25 Y21 Y31 Not i n Basis 1000 Y + 32 1,170,738 -.262 .179 B 2 + Y32 Not i n Basis 1000 y + 33 1,218,291 -1.179 .244 Row 57 Y33 Not i n Basis 1000 Not i n Basis 1000 Y41 Not i n Basis Y + 42 Not i n Basis 1000 Y42 Not i n Basis Y + 43 Not i n Basis 1000 Y43 Not i n Basis Y + *51 Not i n Basis 1000 Y51 3,856,500 -.227 .005 Y ^ Row 2 3 Y52 Y~ 52 Not i n Basis 1000 2,779 ,957 -.025 .127 A ^ Row 82 Y + 53 Y53 Not i n Basis 1000 3,681,199 -.047 .021 Row 57 A 1 32 < 1 401,615 -.004 .004 Row 23 Row 20 Y61 Not i n Basis 1000 <2 Not i n Basis Y62 Not i n Basis 1000 Y + *63 Not i n Basis Y63 Not i n Basis 1000 + Y 7 Not i n Basis Y 7 - 857,432 .0001 -.743 .029 Row 35 Row 61 y; 3,184,290 -1.0 -1.000 -.013 Y~ B 2 Y8 Not i n Basis 1.0 -12 8-EXHIBIT IV (Continued) Optimal S o l u t i o n Cost Ranging Results A c t i v i t y o r V a r i a b l e Optimal Value Cost i n Problem Maximum Change Increase Decrease Incoming Vector L i m i t 1 Limit2 Row 13 1,000 ,000 - i n f . .186 Unb. Row 14 500,000 - i n f . .216 Unb. Row 15 1,800 ,000 - -.227 999.773 Y41 Row 16 1,831,994 - -.083 .072 *14 Row 25 Row 17 205,457 - -.042 .237 b l P 2 14 Row 22 191,300 - i n f . .027 Unb. b l Row 24 2,112,698 - -.150 2.77 Y42 Row 25 Row 26 3,500,000 - i n f . .088 Unb. *14 Row 29 120,000 - i n f . .131 Unb. B 2 Row 32 24,047 - .027 Row 31 b l Row 33 2,094,937 - -.268 1.927 Y~ 43 B 2 Row 34 2,545,662 - -.096 .076 P 3 14 Row 25 Row 35 3,500,000 - i n f . .091 Unb. Row 41 30,000 - i n f . .325 Unb. S 3 Row 61 4,117,167 - 0 0 A13 Y13 Row 81 0 _ i n f . i n f . Unb. Unb. EXHIBIT V DUAL SOLUTION AND RIGHT HAND SIDE RANGING RESULTS Range of Values Outgoing Vector Row R * H»S • Dual At At Row Row D e s c r i p t i o n Type Value Value Minimum Maximum Minimum Maximum 1 I n i t i a l Balance - C ° = 1,080,000 1.027 Unb. 2,160,000 C° 2 I n i t i a l Balance - A° 21 = 1,000 ,000 1.040 Unb. 2 ,000 ,000 A0  A21 3 I n i t i a l Balance = 1,000 ,000 1.040 757,122 1,097,374 s 2 x23 4 I n i t i a l Balance -»'» < = 500 ,000 1.060 Unb. 1,500,000 A 0  A32 5 I n i t i a l Balance - A 0  A13 = 1,380 ,000 1.064 Unb. 1,880,000 *°» 6 I n i t i a l Balance -»°34 = 1,040 ,000 .904 984 ,951 1,136,000 Y2_3 7 I n i t i a l Balance - A 0  A2 4 = 2,000,000 1.017 Unb. 2 ,000 ,000 8 I n i t i a l Balance - ° 1 4 = 2,000,000 .990 Unb. 3,380 ,000 9 I n i t i a l Balance - D° 1 - 4 = 5,200,000 -.099 5,108,760 5,304,726 Y23 »i, 10 I n i t i a l Balance 4,000 ,000 -1.046 3,902,624 4,304,385 Y2 _3 s2 11:. I n i t i a l Balance - B° - = 0 0 Unb. 1,040 ,000 B° 12 I n i t i a l Balance - E° = 800 ,000 .209 787,895 805,808 A 1 13 S2 13 S e l l i n g C onstraint: *°21 " 4l 0 0 Unb. 1,000 ,000 Row 13 14 S e l l i n g C o n s t r a i n t : 4 - ^  > 0 0 Unb. 500 ,000 Row 14 15 Loan Demand @ t = 1: A* 3,000 ,000 0 " 1,200 ,000 Unb. Row 15 16 Loan Demand @ t = 1: 1 P24 3,000,000 0 1,168,000 Unb. Row 16 17 Loan Demand @ t = 1: 'I, •< 3,000 ,000 0 2,794,543 Unb. Row 17 18 Deposit Supply C 3 t = = 800 ,000 .928 733,279 896,171 s 2 A13 19 Deposit Supply { a t = 4,440,000 .078 Unb. 5,240,000 E° 20 Demand f o r C e r t i f i c a t e s : B < 100 ,000 .004 0 1,361,870 B 1 -ll 21 Balance on C a p i t a l : E 1 - E° - S 1 - e Q + b x = 0 1.237 Unb. 1,026,927 A 2 1 22 L i m i t on Dividends: b l " e0 -< 0 0 -191,300 Unb. Row 22 23 L i m i t on New C a p i t a l : S 1 -c 20 ,000 .027 6,747 26 ,010 A 1 13 Row s 2 24 Loan Demand @ t = 2: P 2  F34 3,500 ,000 0 1,387,302 Unb. 24 25 Loan Demand @ t = 2: p2 F24 3,500 ,000 .073 3,424,410 3,943,040 s 2 EXHIBIT V (Continued) Range o f Values Outgoing Vector Row R r H • S • Dual At At Row Row D e s c r i p t i o n Type Value Value Minimum Maximum Minimum Maximum 26 2 Loan Demand @ t = 2: P ^ 3,500 ,000 0 0 Unb. Row 26 27 2 n Deposit Supply @ t = 2: D.^  - D^ = 400 ,000 .786 313,519 564 ,545 S 2 A 1  A13 28 Deposit Supply @ t = 2: D 2 = 5,000 ,000 -.161 Unb. 5,000,000 .'1. 29 2 Demand f o r C e r t i f i c a t e s @ t = 2: B < 120 ,000 0 0 Unb. Row 29 30 2 1 9 Balance on C a p i t a l : E - E - S - + b2 0 1.1 Unb. 84,311 31 L i m i t on Dividends: E>2 - e^ < 0 0 0 24 ,047 S 2 Row 32 32 L i m i t on New C a p i t a l : 30,000 0 0 Unb. Row 32 33 Loan Demand @ t = 3: P 3 34 < 3,500,000 0 1,405,063 Unb. Row 33 34 Loan Demand @ t = 3: P 24 < 3,500 ,000 0 954,338 Unb. Row 34 35 Load Demand @ t = 3: P 3^ 3,500,000 0 0 Unb. Row 35 36 3 2 Deposit Supply @ t = 3: D£ - D^ = -100,000 .617 Unb. 1,068,000 37 Deposit Supply @ t = 3: D 3 5,000 ,000 .637 Unb. 9 ,794,543 S 2 38 Demand f o r C e r t i f i c a t e s @ t = 3: B 3 100 ,000 .743 0 195,232 B 3 39 Balance on C a p i t a l : E 3 - E 2 - S 3 - e 2 + b 3 0 .675 Unb. 6 ,000 ,000 °l 40 L i m i t on Dividends: - e 2 <- 0 .325 -190 ,386 156 ,250 s 2 Row 61 41 L i m i t on New C a p i t a l : S 3 30 ,000 0 0 Unb. Row 41 42 C a l c u l a t i o n of e^ = -75 ,000 1.237 -88,252 -68,990 S 2 43 C a l c u l a t i o n of d^ and d^ = -600 ,000 .785 Unb. -408,700 e° 44 Balances of Sources and Uses of Funds ( a t = 1 = 0 1.027 Unb. 84,466 V 1 r 45 C a l c u l a t i o n of e 1 = -75,000 1.100 Unb. -75,000 A 2  A 2 1 46 C a l c u l a t i o n of dt and d„ 2 2 -650,000 .617 -838,991 16,354 < S 2 47 Balance of Sources and Uses of Funds @ t = 2 = 0 .989 Unb. 2,319,439 *211 48 C a l c u l a t i o n of = -75,000 1.000 Unb. -75,000 *13 49 C a l c u l a t i o n of d^ and d^ -700,000 0 -859,810 0 50 Balance of Sources and Uses of Funds @ t = 2 = 0 .751 Unb. 1,387,302 A 2 EXHIBIT V (Continued) Row R.H.S. Row Row D e s c r i p t i o n : Type V a l u e 51 C a l c u l a t i o n s of e ? = -75,000 52 C a l c u l a t i o n of = 0 53 C a l c u l a t i o n of v [ = 0 54 C a l c u l a t i o n of v| = 0 55 L i q u i d i t y Goal @ t = 1 = 0 56 C a l c u l a t i o n of V 2 = 0 r 57 C a l c u l a t i o n of V 2 = 0 2 58 C a l c u l a t i o n of V i = 0 59 L i q u i d i t y Goal @ t = 2 = 0 3 60 C a l c u l a t i o n of V_ = 0 61 C a l c u l a t i o n of V 3 = 0 62 C a l c u l a t i o n o f V? = 0 63 L i q u i d i t y Goal at t = 3 = 0 64 65 66 67 68 69 70 71 72 73 74 75 Goal to Minimize *_1 = 0 Goal to Minimize Y +  y22 = 0 Goal to Minimize + Y23 = 0 Goal to Minimize Y31 = 0 Goal to Minimize Y32 • = 0 Goal to Minimize Y33 = 0 Goal to Minimize *41 = 0 Goal to Minimize *« + Y43 = 0 Goal to Minimize = 0 Goal to Minimize Y + 51 + Y52 = 0 Goal to Minimize =• 0 Goal to Minimize + Y5 3 = 0 Range of Values Outgoing Vector Dual Value Minimum Maximum At Minimum At Maximum 15.00 Unb. 1,312,302 .002 -224,690 1,070,306 .001 -365,231 1,686,455 m .003 -153,689 732,097 + Y61 .029 -14,609 69 ,591 + Y61 .014 .008 -408,774 -664,458 101,185 164,474 Row Row 32 32 S 2 2 S .020 -279,604 69,211 Row 32 s 2 .212 Unb. 1,017,253 E 2 0 -2,532,881 522,337 Row 61 V 3 r 0 -4,117,167 Unb. Row 61 0 -1,732,504 522,337 Row 61 V 3 l 0 Unb. 91,904 V 2 r .103 0 Unb. Unb. 23,455 55,740 < 2 V. l 0 Unb. 1,435,223 c 3 0 Unb. 3,260 ,242 21 0 Unb. 2 ,341,640 P 3 21 0 .227 Unb. -75,590 0 443,040 A 1 13 A 3 • 11 Row 32 .146 .269 Unb. -93,290 0 376,879 s 2 A 3 32 Row 32 0 -3,856,500 Unb. Y51 0 0 -2,779,959 Unb. Unb. 1,405,063 Y52 EXHIBIT V (Continued) Row Row D e s c r i p t i o n 76 Goal to Minimize Y 61 77 Goal to Minimize Yg 2 78 Goal to Minimize Y b 3 79 Growth Goal Minimize h 80 P r o f i t Goal 81 Minimum Cash Balance @ t 82 Minimum Cash Balance @ t 83 Minimum Cash Balance @ t -10 4 Balancing Equations Range o f Values Outgoing Vector Row R.H.S. Dual At At Type Value Value Minimum Maximum Minimum Maximum = 0 0 Unb. 1,405,063 = 0 - .188 Unb. 2 ,341,640 = 0 - .272 Unb. 954 ,338 'I, = 14,000,000 - .0001 13,142,468 Unb. Y 7 -1.000 Unb. - .032 Unb. 7,025,316 'l - .196 -60,128 21,316 *13 S 2 - .751 -93,290 376,879 s 2 Row 32 

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