"Business, Sauder School of"@en . "DSpace"@en . "UBCV"@en . "Woodruff, Charles E."@en . "2011-04-08T19:15:43Z"@en . "1971"@en . "Master of Science in Business - MScB"@en . "University of British Columbia"@en . "Asset management in a commercial banking environment may be viewed as a process of resource allocation. The resources of a bank consist of the funds made available to the bank by its depositors, creditors and shareholders. These funds may be allocated among a variety of earning and non-earning assets. In carrying out this allocation, management must recognize and reconcile the various and often conflicting objectives and requirements of its depositors, its shareholders and the public agencies which regulate its activities. Given these multiple requirements and objectives the senior management of the bank must determine an allocation plan that will provide maximum safety and adequate liquidity for its depositors and an acceptable return for its shareholders. This thesis attempts to demonstrate how bank management can use a mathematical programming model to assist them in formulating short, intermediate and long range plans for the bank's asset management activities. It is shown that such a model is capable of dealing with the multiple goals which represent the bank's objectives and obligations and that the model can be formulated to incorporate the complex interdependencies which exist among the various asset management activities. The solution to the model will represent the most satisfactory course of action available to the bank in terms of its organizational goals."@en . "https://circle.library.ubc.ca/rest/handle/2429/33451?expand=metadata"@en . "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. \u00C2\u00AB4 -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,\u00E2\u0080\u00A2loans 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 . ^ \u00C2\u00BB b . i = l j = i ID D \u00E2\u0080\u0094 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 \u00E2\u0080\u00A2 - - \u00E2\u0080\u00A2 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 \u00C2\u00B1 = 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 \u00E2\u0080\u00A2^ 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 \u00C2\u00A3 = 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! + \u00E2\u0080\u00A29 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)\u00C2\u00A3 = .65(V t - v\u00C2\u00A3) (7) E(V)\u00C2\u00A3 = (.105 - .065) (V* - V m) (8) E(V)\u00C2\u00A3 = (.200 - .105) (V1^ - vf) (9) where E(V)\u00C2\u00A3, 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-\u00E2\u0080\u00A2 47D^ + .36D^ + B t - C*' - .995A^ - .96A^\u00C2\u00B1 - (E(V)\u00C2\u00A3/ .065)<0 .47D^ + .36D^ + Bt - C f c - .995A^, - .96A^ - .90A t - (E ( V ) t 1 z 11 21 32 m / .40) < 0 \u00E2\u0080\u00A2 47D^ + .36D-2 + B t - C t - .995A^\u00C2\u00B1 - .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\u00E2\u0080\u009E + ,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 \u00E2\u0080\u0094 < 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 + \u00E2\u0080\u00A2 -S - K 2 ( D J + D* + B f c + E ) - ( Y 2 t ) + - 0 = 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\u00C2\u00A3 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 \u00E2\u0080\u00A2through 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 \u00C2\u00B1 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 \u00C2\u00B0 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 \u00C2\u00B0 L : . \u00E2\u0080\u00A2< A 2 1 \u00E2\u0080\u0094 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 \u00C2\u00B0 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 \u00C2\u00B0 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 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 T 34 2 4 + P t 24 t 1 / \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 T + P t 14 t 1 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 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\u00C2\u00B0..A\u00C2\u00B0. + t l f - D L1 + t ( f - l ) L i _ i = l n=l n i n i 32 2 1 - r D\u00C2\u00B0 - r B\u00C2\u00B0 - F\u00C2\u00B0 S 2. fc> 1 \" r32 L32 \" r21 L21 + ^ f - 1 \u00C2\u00BB 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 \u00E2\u0080\u009E i=4 n=-3 -i -i r\u00C2\u00B0 - I 2 + 21 S r 1 . . ? 1 . r^-.P 2. - r s D 2 - r B 2 \u00E2\u0080\u00A23 t - 2 t-2 ^ ^ t-1 t-1 ^ <\u00C2\u00A3-3 rT. . P . + Z\u00E2\u0080\u0094 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 \u00C2\u00B0 +ZI A\u00C2\u00B0. + f i * + f L 1 + (D 1 - D\u00C2\u00B0 | iT~i i= 3 2 2 1 i i ) + ( D2 \" \u00C2\u00B02 ) + i = l n=l n \u00C2\u00B1 J B\u00C2\u00B0) + ( E 1 - E\u00C2\u00B0) --58-2 I l j-~f l -? 2 1 ? 1 = IC1 + ZI A 1 \u00C2\u00B1 + 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> + + 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\u00C2\u00B0 ) 1,000,000 - maturing i n 3 years (A\u00C2\u00B02) 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\u00C2\u00B0^) 2,000,000 - maturing i n 3 years (A\u00C2\u00B04) 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\u00C2\u00B0) 5,200,000 - Time Deposits (D\u00C2\u00B0) 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\u00C2\u00B0) 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 \u00C2\u00B1. \u00C2\u00B1. 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\u00E2\u0080\u009E + Y\u00E2\u0080\u009E + 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 \u00E2\u0080\u00A2, 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 $\u00C2\u00AB0\u00C2\u00B0 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\u00C2\u00B0-S 1-\u00E2\u0080\u00A2e +b, \u00E2\u0080\u00A2 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 \u00E2\u0080\u00A2> 1 2 Balance on C a p i t a l : E z - E -S -\u00E2\u0080\u00A2 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 -\u00E2\u0080\u00A2e 2+b 3 = 0 .675 Unb. 6 ,000,000 \u00C2\u00BBi 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 \u00E2\u0080\u00A2< 30 ,000 0 0 Unb. Row 41 18 Deposit Supply @ t=l:D^-D\u00C2\u00B0 = 800 ,000 .928 733, 279 896,171 s 2 *13 19 :Dl - 4,440 ,000 .078 Unb. 5 ,240,000 E\u00C2\u00B0 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 \u00E2\u0080\u00A2\u00E2\u0080\u00A2 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\u00C2\u00B0 = R\u00C2\u00B0A\u00C2\u00B0 - I\u00C2\u00B0D\u00C2\u00B0 - F\u00C2\u00B0 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\u00C2\u00B0 - I AD\u00C2\u00B0 - 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\u00C2\u00B0 = (NP a - NP b) + (NP b - NP\u00C2\u00B0) 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\u00C2\u00B0 = R a ( A a - A\u00C2\u00B0) + I A(D\u00C2\u00B0 - DA) + A\u00C2\u00B0(R a - R\u00C2\u00B0) + D\u00C2\u00B0(I\u00C2\u00B0 - 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\u00C2\u00B0) + i a(D\u00C2\u00B0 - 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\u00C2\u00B0 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\u00C2\u00B0(R a - R\u00C2\u00B0) + D\u00C2\u00B0(I\u00C2\u00B0 - 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 .\u00E2\u0080\u00A2 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\u00C2\u00B0 = (NP a - NP P) + (NPP - NP\u00C2\u00B0) 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\u00C2\u00B0), 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\u00C2\u00B0(I\u00C2\u00B0 - 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\u00C2\u00A3 - 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 \u00C2\u00A3 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\u00C2\u00AB b l \"1 6,000 ,000 - i n f . i n f . Unb. Unb. \u00C2\u00B0l 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. \u00C2\u00BBL 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 \u00C2\u00BB 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 -\u00E2\u0080\u00A2 * i 3 1 4 , 0 8 1 - - 2 . 7 8 . 2 6 2 Row 2 0 b 2 \u00C2\u00BB 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 -\u00C2\u00BBl 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 -\u00C2\u00B0i 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\u00C2\u00BBS \u00E2\u0080\u00A2 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 \u00C2\u00B0 = 1,080,000 1.027 Unb. 2,160,000 C\u00C2\u00B0 2 I n i t i a l Balance - A\u00C2\u00B0 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 -\u00C2\u00BB'\u00C2\u00BB < = 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 *\u00C2\u00B0\u00C2\u00BB 6 I n i t i a l Balance -\u00C2\u00BB\u00C2\u00B034 = 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 - \u00C2\u00B0 1 4 = 2,000,000 .990 Unb. 3,380 ,000 9 I n i t i a l Balance - D\u00C2\u00B0 1 - 4 = 5,200,000 -.099 5,108,760 5,304,726 Y23 \u00C2\u00BBi, 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\u00C2\u00B0 - = 0 0 Unb. 1,040 ,000 B\u00C2\u00B0 12 I n i t i a l Balance - E\u00C2\u00B0 = 800 ,000 .209 787,895 805,808 A 1 13 S2 13 S e l l i n g C onstraint: *\u00C2\u00B021 \" 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, \u00E2\u0080\u00A2< 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\u00C2\u00B0 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\u00C2\u00B0 - 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 \u00E2\u0080\u00A2 S \u00E2\u0080\u00A2 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\u00C2\u00A3 - 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 \u00C2\u00B0l 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\u00C2\u00B0 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\u00E2\u0080\u009E 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 \u00E2\u0080\u00A2 = 0 Goal to Minimize Y33 = 0 Goal to Minimize *41 = 0 Goal to Minimize *\u00C2\u00AB + Y43 = 0 Goal to Minimize = 0 Goal to Minimize Y + 51 + Y52 = 0 Goal to Minimize =\u00E2\u0080\u00A2 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 \u00E2\u0080\u00A2 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 "@en . "Thesis/Dissertation"@en . "10.14288/1.0101660"@en . "eng"@en . "Business Administration"@en . "Vancouver : University of British Columbia Library"@en . "University of British Columbia"@en . "For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en . "Graduate"@en . "A goal programming approach to bank asset management"@en . "Text"@en . "http://hdl.handle.net/2429/33451"@en .