A Bank Asset and Liability Management Model

NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR

A BANK ASSET AND LIABILITY MANAGEMENT MODEL M.I. Kusy*

W.T. Ziemba**

December 1983 CP-83-59

*

Concordia University, Montreal, Quebec, Canada.

**International Institute for Applied Systems Analysis, Laxenburg, Austria.

and

University of British Columbia, Vanouver, B.C., Canada

C o Z Z d b o r a t i v e P a p e r s report work which has not been performed solely at the International Institute for Applied Systems Analysis and which has received only limited review. Views or opinions expressed herein do not necessarily represent those of the Institute, its National Member Organizations, or other organi- zations supporting the work.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS A-2361 Laxenburg, Austria

PREFACE

The area of asset managemeht is rich in potential applications of stochastic programming techniques. This article develops

a multiperiod stochastic programming model for bank asset and liability management, it shows that the results are far superior to those of a deterministic version of such a model. The algorithm used to solve the stochastic problem is part of the soft ware

packages for stochastic optimization problems under development by the Adaptation and Optimization Task at IIASA.

Roger Wets

ABSTRACT

The u n c e r t a i n t y o f a b a n k ' s c a s h f l o w s , c o s t o f f u n d s and r e t u r n on i n v e s t - ments due to i n h e r e n t f a c t o r s and v a r i a b l e economic c o n d i t i o n s h a s emphasized t h e need f o r g r e a t e r e f f i c i e n c y i n t h e management o f a s s e t and l i a b i l i t i e s . A primary g o a l i s t o d e t e r m i n e an o p t i m a l t r a d e o f f between r i s k , r e t u r n , and

l i q u i d i t y . I n t h i s paper we d e v e l o p a m u l t i p e r i o d s t o c h a s t i c l i n e a r programming model ( A m ) t h a t i n c l u d e s t h e e s s e n t i a l i n s t i t u t i o n a l , l e g a l , f i n a n c i a l , and bank r e l a t e d p o l i c y c o n s i d e r a t i o n s , a l o n g with t h e i r u n c e r t a i n a s p e c t s , y e t i s c o m p u t a t i o n a l l y t r a c t a b l e f o r r e a l i s t i c s i z e d problems. A v e r s i o n o f t h e model was developed f o r t h e Vancouver C i t y Savings C r e d i t Union f o r a f i v e year plan- n i n g p e r i o d . The r e s u l t s i n d i c a t e t h a t ALN i s t h e o r e t i c a l l y and o p e r a t i o n a l l y s u p e r i o r to a c o r r e s p o n d i n g d e t e r m i n i s t i c l i n e a r prgramming model and t h e e f f o r t r e q u i r e d f o r t h e implementation o f ALN and t h e c o m p u t a t i o n a l c o s t s a r e compar- a b l e to t h o s e o f t h e d e t e r m i n i s t i c model. W r e o v e r , t h e q u a l i t a t i v e and quant- i t a t i v e c h a r a c t e r i s t i c s o f t h e s o l u t i o n s a r e s e n s i t i v e to t h e s t o c h a s t i c

e l e m e n t s o f t h e model such as t h e asymmetry o f t h e c a s h flow d i s t r i b u t i o n s . ALN was a l s o compared with t h e s t o c h a s t i c d e c i s i o n t r e e (SDT) model developed by B r a d l e y and Crane. ALN i s more c o m p u t a t i o n a l l y t r a c t a b l e on r e a l i s t i c s i z e d problems t h a n SDT and s i m u l a t i o n r e s u l t s i n d i c a t e t h a t A M g e n e r a t e s s u p e r i o r p o l i c i e s .

Without i m p l i c a t i n g them w e would l i k e to thank J. B i r g e , W. ~ S h l e r , G.

Gassmann, J . G . K a l l b e r g , C.E. S a r n d a l , and R.W. White f o r h e l p f u l d i s c u s s i o n s and Messrs. B e n t l y and Hook of the Vancouver C i t y S a v i n g s C r e d i t Union f o r p r o v i d i n g d a t a used i n t h i s s t u d y . T h i s r e s e a r c h was s u p p o r t e d by t h e I n t e r n a t i o n a l I n s t i t u t e f o r Applied Sys terns A n a l y s i s , A u s t r i a , t h e Canada C o u n c i l , and t h e N a t u r a l S c i e n c e s and E n g i n e e r i n g R e s e a r c h C o u n c i l of Canada.

1

.

INTRODUCTION

The i n h e r e n t u n c e r t a i n t y o f a b a n k ' s c a s h f l o w s , c o s t o f f u n d s , and r e t u r n o n i n v e s t m e n t s h a s emphasized t h e need f o r g r e a t e r e f f i c i e n c y i n t h e management o f i t s a s s e t s and l i a b i l i t i e s . T h i s h a s led t o a number o f s t u d i e s concerned w i t h how one should s t r u c t u r e a b a n k ' s a s s e t s and l i a b i l i t i e s so t h a t t h e r e a r e o p t i m a l t r a d e o f f s among r i s k , r e t u r n , and l i q u i d i t y . These s t u d i e s f o c u s on t h e d e t e r m i n a t i o n o f t h e use o f f u n d s f o r d e t e r m i n i s t i c and s t o c h a s t i c economic

s c e n a r i o s . F a c t o r s t h a t must b e c o n s i d e r e d i n t h e s e d e c i s i o n s i n c l u d e : b a l a n c - i n g of a n t i c i p a t e d s o u r c e s and u s e s o f funds t o meet l i q u i d i t y and c a p i t a l adequacy c o n s t r a i n t s w h i l e c o n c u r r e n t l y maximizing p r o f i t a b i l i t y [Chambers and Charnes ( 1 961 )

,

Qhen and Hammer ( 1967 )

1 ,

a l l o c a t i n g funds among a s s e t s b a s e d on r i s k and l i q u i d i t y c l a s s i f i c a t i o n , m a t u r i t y and r a t e o f r e t u r n [Bradley and Crane (1972, 1973, 1976)

1 ,

and a d j u s t i n g a b a n k ' s f i n a n c i a l s t r u c t u r e i n t e r m s o f l i q u i d i t y , c a p i t a l adequacy and l e v e r a g e [Chambers and Charnes (1961 1, Cohen and Hammer ( 1 9 6 7 ) J .

C u r r e n t r e s e a r c h h a s s t r e s s e d t w approaches. The f i r s t approach, b a s e d o n Markowitz's ( 1 9 5 9 ) t h e o r y o f p o r t f o l i o s e l e c t i o n , assumes t h a t r e t u r n s a r e

normally d i s t r i b u t e d and bank managers u t i l i z e r i s k - a v e r s e u t i l i t y f u n c t i o n s . The v a l u e o f an a s s e t t h e n depends n o t o n l y on t h e e x p e c t a t i o n and v a r i a n c e o f i t s r e t u r n b u t a l s o on t h e c o v a r i a n c e o f i t s r e t u r n w i t h t h e r e t u r n s o f a l l o t h e r e x i s t i n g and p o t e n t i a l i n v e s t m e n t s . The second approach assumes t h a t a bank s e e k s t o maximize i t s f u t u r e s t r e m o f p r o f i t s ( o r e x p e c t e d p r o f i t s ) sub-

j e c t t o p o r t f o l i o mix c o n s t r a i n t s .

The most g e n e r a l example o f t h e u s e o f t h e f i r s t approach i s P y l e ( 1971

,

where a s t a t i c model i s developed in which t h e f i n a n c i a l i n t e r m e d i a r y ( b a n k ) can s e l e c t t h e a s s e t and l i a b i l i t y l e v e l s t o b e m a i n t a i n e d t h r o u g h o u t t h e p e r i o d . P y l e ' s a n a l y s i s d e m o n s t r a t e s t h e need f o r f i n a n c i a l i n t e r m e d i a r i e s . He o n l y

c o n s i d e r s t h e r i s k o f t h e p o r t f o l i o and m t o t h e r p o s s i b l e u n c e r t a i n t i e s .

Trading a c t i v i t y , matching a s s e t s and l i a b i l i t i e s , t r a n s a c t i o n s c o s t s , e t c . , a r e o m i t t e d from t h e model. It i s p o s s i b l e to d e v e l o p dynamic models u s i n g con-

s t r u c t s along t h e s e l i n e s , s e e , e.g., K a l l b e r g and Ziemba ( 1 9 8 1 ) . However, g i v e n t h e s e v e r e c o m p u t a t i o n a l d i f f i c u l t i e s due to t h e l e v e l o f c o m p l e x i t y o f a l g o r i t h m s f o r t h e s e problems, it i s not a t p r e s e n t p o s s i b l e t o d e v e l o p u s e f u l o p e r a t i o n a l models f o r l a r g e o r g a n i z a t i o n s such a s b a n k s .

S i n c e o u r i n t e r e s t i s i n o p e r a t i o n a l models we c o n c e n t r a t e on t h e second approach which h a s t h e o r e t i c a l and e m p i r i c a l s u p p o r t . Myers (1968) a t t e m p t e d to determine which c r i t e r i a i s most s u i t a b l e f o r t h e a s s e t and l i a b i l i t y management problem by showing t h a t : a n e c e s s a r y c o n d i t i o n f o r t h e e x i s t e n c e o f s e c u r i t y market e q u i l i b r i m i s r i s k independence; s e c u r i t y market e q u i l i b r i m i m p l i e s

r i s k independence o f s e c u r i t i e s ; and r i s k independence o f i n v e s t m e n t opportun- i t i e s i m p l i e s t h a t t h e maximization o f t h e e x p e c t e d n e t p r e s e n t v a l u e i s t h e a p p r o p r i a t e o b j e c t i v e c r i t e r i o n .

Thus, i f , a s i s w i d e l y b e l i e v e d , a s t a t e o f e q u i l i b r i u n e x i s t s f o r t h e s e c u r i t i e s which a r e h e l d by f i n a n c i a l i n s t i t u t i o n s , and s e c u r i t i e s purchased do not have s y n e r g e t i c e f f e c t ( i m p l y i n g t h e r i s k independence o f s e c u r i t i e s ) t h e n t h e a p p r o p r i a t e o b j e c t i v e f u n c t i o n s f o r a f i n a n c i a l i n s t i t u t i o n i s t h e maximiz- a t i o n o f t h e e x p e c t e d n e t p r e s e n t v a l u e (ENPV). In a major e m p i r i c a l

s t u d y Hester and P i e r c e ( 1 9 7 5 ) used c r o s s - s e c t i o n a l d a t a to a n a l y z e t h e v a l i d i t y o f a number o f p o r t f o l i o s e l e c t i o n models i n bank f u n d management. They

concluded t h a t b a n k s can b e w e l l managed u s i n g models a s a d e c i s i o n a i d and t h a t t h e b e s t o b j e c t i v e f u n c t i o n s a r e e i t h e r ENPV o r t h e maximization o f a two v a r i - a b l e f u n c t i o n where ENPV i s dominant.

A s s e t and l i a b i l i t y management models u s i n g an ENPV c r i t e r i a f a l l i n t m b r o a d c a t e g o r i e s : d e t e r m i n i s t i c and s t o c h a s t i c . The d e t e r m i n i s t i c models use

l i n e a r programming, assume p a r t i c u l a r r e a l i z a t i o n s f o r all random e v e n t s , and a r e computationally t r a c t a b l e f o r l a r g e problems. These models have been accepted a s a u s e f u l normative t o o l by t h e banking i n d u s t r y [Cohen and Hammer

( 1967 ) 1

.

S t o c h a s t i c models on t h e o t h e r hand have achieved v e r y modest s u c c e s s . T h i s i s due to t h e i n h e r e n t computational d i f f i c u l t i e s , t h e over- s i m p l i f i c a t i o n s needed to achieve computational t r a c t a b i l i t y , and t h e p r a c t i - t i o n e r s ' u n f a m i l i a r i t y with t h e i r p o t e n t i a l . The s t o c h a s t i c models included t h e use of t h e following t e c h n i q u e s : chance-constrained programming; dynamic pro- gramming; s e q u e n t i a l d e c i s i o n t h e o r e t i c approach ; and l i n e a r programming under u n c e r t a i n t y .

E s s e n t i a l l y a l l of t h e d e t e r m i n i s t i c models and many of t h e s t o c h a s t i c models follow t h e approach of Chambers and Charnes' ( 1961) l i n e a r programming model. They maximize n e t discounted r e t u r n s s u b j e c t to budget and l i q u i d i t y c o n s t r a i n t s using t h e F R B ' s c a p i t a l adequacy formulas, s e e S e c t i o n 3 below. The l i t e r a t u r e c o n t a i n s s e v e r a l examples of s u c c e s s f u l a p p l i c a t i o n s o f t h i s model

[Cohen and H a m m e r ( 1967 )

,

Komar ( 1971 )

,

and Lifson and B l a c h a n ( 1973 ) ]

.

However c r i t i c i s m c o n t i n u e s t o b e l e v e l e d l a r g e l y because of t h e omission of u n c e r t a i n t y in t h e model [Bradley and Crane ( 1 976 )

,

Cohen and Thore ( 1 970 )

,

and Eppen and Fama (1968)l. P r o b a b i l i t y d i s t r i b u t i o n s can b e o b t a i n e d f o r d i f f e r e n t economic s c e n a r i o s and a l i n e a r programming f o r m u l a t i o n can b e a p p l i e d to each s c e n a r i o to determine optimal m l u t i o n s . Ibwever

,

t h i s w i l l n o t g e n e r a t e an optimal s o l u t i o n to t h e t o t a l problan b u t r a t h e r a c t a s a d e t e r m i n i s t i c

s i m u l a t i o n to observe p o r t f o l i o behavior under v a r i o u s economic c o n d i t i o n s . One must use c a r e i n d e f i n i n g such models a s it may happen t h a t no s c e n a r i o l e a d s t o an o p t i m a l m l u t i o n , s e e Birge ( 1982 )

.

Charnes and Kirby ( 19651, Charnes and L i t t l e c h i l d ( 1968 1, Charnes and Thore (1966), and o t h e r s developed chance-constrained models i n which f u t u r e d e p o s i t s

and loan repayments were expressed as j o i n t normally d i s t r i b u t e d random vari- ables and t h e c a p i t a l adequacy formula was replaced by chance-constraints on meeting withdrawal claims. These approaches lead to a computationally f e a s i b l e

scheme for r e a l i s t i c s i t u a t i o n s , see e .g., Charnes, Gallegos and Yao ( 1 982)

.

However, the chance-constrained procedure does not have the f a c i l i t y t o handle a d i f f e r e n t i a l penalty for e i t h e r varying magnitudes of c o n s t r a i n t v i o l a t i o n s o r d i f f e r e n t types of c o n s t r a i n t s . bbreover, i n multi-period models t h e r e are con- ceptual d i f f i c u l t i e s , as yet unresolved in t h e l i t e r a t u r e dealing with t h e treatment of i n f e a s i b i l i t y i n periods 2,...,n, s e e , e.g., Eisner, Kaplan, and Soden (1971).

The second approach i s dynamic programming. Eppen and Fama ( 1968, 1969, 1971) modelled t m and three a s s e t problems, and t h e i r work was extended by Daellenbach and Archer (1969) to include one l i a b i l i t y . For a survey of t h i s l i t e r a t u r e see Ziemba and Vickson (1 975). The v i r t u e s of t h e s e models are t h a t they a r e dynamic and take i n t o account the inherent uncertainty of the problem.

However, given t h e m a l l number of f i n a n c i a l instruments t h a t can be analyzed simultaneously, they a r e of limited use i n p r a c t i c e . See Daellenbach (1974) f o r estimates of possible gain using t h e s e models. For a recent survey of r e l a t e d a p p l i c a t i o n s i n banking see Cbhen, Maier and Van Der Weide (1981 1 .

The t h i r d a l t e r n a t i v e , proposed by Wolf (1969) i s a sequential decision t h e o r e t i c approach which employs sequential decision analysis to find an optimal

solution through the use of i m p l i c i t enumeration. The d i f f i c u l t y with t h i s technique i s t h a t it does not find an e x p l i c i t optimal s l u t i o n to problems with a time horizon beyond one period, because it i s necessary to enumerate a l l pos- s i b l e p o r t f o l i o s t r a t e g i e s f o r periods preceding the present decision p i n t i n order t o guarantee optimality. In ^an e f f o r t to explain away t h i s dra-&ack, Wolf makes t h e dubious a s s e r t i o n t h a t the s l u t i o n to a one period model m u l d be

e q u i v a l e n t to a s o l u t i o n provided by s o l v i n g an n p e r i o d model. T h i s among o t h e r t h i n g s i g n o r e s t h e problem of synchronizing t h e m a t u r i t i e s of a s s e t s and l i a b i l i t i e s . Bradley and Crane (1972, 1973, 1976) have developed a s t o c h a s t i c d e c i s i o n t r e e model t h a t has many of t h e d e s i r a b l e f e a t u r e s e s s e n t i a l to an o p e r a t i o n a l bank p o r t f i l i o model. Their model i s c o n c e p t u a l l y s i m i l a r to Wolf's model; t o overcome computational d i f f i c u l t i e s t h e y r e f o r m u l a t e d t h e a s s e t and

l i a b i l i t y problem and developed a g e n e r a l l i n e a r programming decomposition a l g o r i t t m t h a t minimizes t h e computational d i f f i c u l t i e s . T h i s model i s d i s - cussed i n S e c t i o n 5.

The f o u r t h approach i s s t o c h a s t i c l i n e a r programming with simple r e c o u r s e (SLPSR) which i s a l s o c a l l e d l i n e a r programming under u n c e r t a i n t y (LPUU). This t e c h n i q u e e x p l i c i t l y c h a r a c t e r i z e s each r e a l i z a t i o n of t h e r a n d m v a r i a b l e s by a c o n s t r a i n t and l e a d s t o l a r g e problems i n r e a l i s t i c s i t u a t i o n s . !Chis handi- capped m o d e l l e r s g r e a t l y ; i n f a c t Cohen and Thore ( 1970) viewed t h e i r model more a s a t o o l f o r s e n s i t i v i t y a n a l y s i s ( i n t h e a g g r e g a t e ) r a t h e r than a normative d e c i s i o n t o o l . The computational i n t r a c t a b i l i t y and t h e p e r c e p t i o n s of t h e f o r m u l a t i o n precluded c o n s i d e r a t i o n of problems o t h e r than t h o s e which were l i m i t e d b o t h i n terms of time p e r i o d s (Cohen and Thore used one and Crane (1971) use t w o ) and in t h e number of v a r i a b l e s and r e a l i z a t i o n s . Booth (1972) a p p l i e d t h i s f o r m u l a t i o n b y l i m i t i n g t h e number of p o s s i b l e r e a l i z a t i o n s and t h e number o f v a r i a b l e s c o n s i d e r e d in o r d e r to i n c o r p o r a t e t w o t i m e p e r i o d s . Although r e l a t i v e l y e f f i c i e n t s o l u t i o n a l g o r i t h m s e x i s t e d f o r s o l v i n g SLPSR's [Wets

( 1966)l

,

t h e s e models were solved by u s i n g " e x t e n s i v e r e p r e s e n t a t i o n " .

With t h e p o s s i b l e e x c e p t i o n of t h e Bradley-Crane model none of t h e above mentioned models g i v e s an adequate t r e a t m e n t of t h e e s s e n t i a l f e a t u r e s n e c e s s a r y

f o r dn adequate o p e r a t i o n a l bank a s s e t and l i a b i l i t y management model t h a t i s c o m p u t a t i o n a l l y t r a c t a b l e

.

^An i d e a l o p e r a t i o n a l model should c o n t a i n t h e

following f e a t u r e s :

1. m u l t i - p e r i o d i c i t y t h a t i n c o r p o r a t e s : changing y i e l d s p r e a d s a c r o s s t i m e , t r a n s a c t i o n c o s t s a s s o c i a t e d with s e l l i n g a s s e t s p r i o r to matur- i t y , and t h e s y n c h r o n i z a t i o n of cash flows a c r o s s time b y matching m a t u r i t y of a s s e t s with expected cash o u t f l o w s ;

2. simultaneous c o n s i d e r a t i o n s of a s s e t s and l i a b i l i t i e s to s a t i s f y b a s i c accounting p r i n c i p l e s and match t h e l i q u i d i t y of a s s e t s and l i a b i l - i t i e s ;

3. t r a n s a c t i o n c o s t s t h a t i n c o r p o r a t e brokerage f e e s , and o t h e r expenses i n c u r r e d i n buying and s e l l i n g s e c u r i t i e s ;

4. u n c e r t a i n t y of c a s h flows t h a t i n c o r p o r a t e s t h e u n c e r t a i n t y i n h e r e n t i n t h e d e p o s i t e r s ' withdrawal claims and d e p o s i t s (The model must e n s u r e t h a t t h e s t r u c t u r e of t h e a s s e t p o r t f o l i o i s such t h a t t h e c a p a c i t y t o meet t h e s e c l a i m s i s maintained by t h e b a n k ) ;

5. t h e i n c o r p o r a t i o n of u n c e r t a i n i n t e r e s t r a t e s i n t o t h e decision-making p r o c e s s to avoid l e n d i n g and borrowing d e c i s i o n s which may u l t i m a t e l y b e d e t r i m e n t a l to t h e f i n a n c i a l well-being of t h e bank; and

6. l e g a l and p o l i c y c o n s t r a i n t s a p p r o p r i a t e to t h e b a n k ' s o p e r a t i n g environment.

In t h i s paper we develop an SLPSR model t h a t e s s e n t i a l l y c a p t u r e s t h e s e f e a t u r e s of a s s e t and l i a b i l i t y management while m a i n t a i n i n g computational

f e a s i b i l i t y

.

Some background concerning SLPSR models and t h e s o l u t i o n a l g o r i t h m used appear i n S e c t i o n 2. The model ADl i s d e s c r i b e d and formulated i n S e c t i o n 3. In S e c t i o n 4 we apply A M to t h e o p e r a t i o n s o f t h e Vancouver C i t y Savings C r e d i t Union. S e c t i o n 5 p r o v i d e s a comparison of ADl and Bradley and C r a n e ' s Model. F i n a l remarks and c o n c l u s i o n s appear i n S e c t i o n 6.

2 . STOCHASTIC LINEAR PROGRAMS WITH SIMPLE RECOURSE The b a s i c (SLPSR) model i s

min Z ( x ) : c ' x

+

^E

[

min ( q + ' y + + q"y')]

x s

,+,,-Lo

s.t. Ax = b

hc

+

1y+

-

^1y- =

5

n + - + -

where c , x E R

,

^y

,

^y

,

^q

,

^{q E R ~ Z , A} i s rnLxn, T i s mz x n, I i s a mz-

d i m e n s i o n a l i d e n t i t y m a t r i x and

5

i s a mz-dimensional random v a r i a b l e d i s t r i - b u t e d i n d e p e n d e n t l y o f x on t h e p r o b a b i l i t y s p a c e (8,3;~). The SLPSR model is t h e t w s t a g e p r o c e s s : choose a d e c i s i o n v e c t o r x , o b s e r v e t h e random v e c t o r

5

t h e n t a k e t h e c o r r e c t i v e a c t i o n ( y + , y - ) . The model is s a i d to have s i m p l e r e c o u r s e b e c a u s e t h e second s t a g e m i n i m i z a t i o n i s f i c t i t i o u s s i n c e ( y+, y-) are e f f e c t i v e l y unique f u n c t i o n s o f ( x , E ) .

Beale ( 1955) and D a n t z i g ( 1955) i n d e p e n d e n t l y proposed t h e SLPSR model a s a s p e c i a l c a s e o f t h e g e n e r a l l i n e a r r e c o u r s e model where 1y+-1y- i s r e p l a c e d b y Wy f o r a g e n e r a l m a t r i x W. D e t a i l e d p r e s e n t a t i o n s o f t h e t h e o r y o f this

model a p p e a r i n K a l l ( 1976 1, P a r i k h ( 1968)

,

and Ziemba ( 1974 1. Assuming Ax = b

,

x

2

0 h a s a s o l u t i o n x0 and q+

+

q-

2

^{0, ( 1 ) h a s} an o p t i m a l s o l u t i o n and i s a s e p a r a b l e convex program. I f

5

i s a b s o l u t e l y c o n t i n u o u s t h e n Z i s

d i f f e r e n t i a b l e and ( 1 ) may b e s o l v e d u s i n g m o d i f i c a t i o n s o f s t a n d a r d f e a s i b l e d i r e c t i o n a l g o r i t h m s , see, e.g., W e t s ( 1 9 6 6 ) and Ziemba ( 1 9 7 4 ) . I f

5

h a s a f i n i t e d i s t r i b u t i o n t h e n Z i s p i e c e w i s e l i n e a r and ( 1 ) i s e q u i v a l e n t to a l a r g e l i n e a r program. W e t s ( 1 9 7 4 ) n o t e d t h a t t h e d e t e r m i n i s t i c e q u i v a l e n t l i n e a r program can b e w r i t t e n i n t h e form

- - - + -

where i = l,...,mz, A = 2 , - - - , k i , dil

= E i l l

d i A -

ciA _Ei,A-l,

q i - q i + q i t

5 <. . ^.<c

a r e t h e p o s s i b l e v a l u e s o f each

c

t h e i t h component of

5,

with

il i k i i

s- 1 p r o b a b i l i t i e s f il, , f _{i k i} and Fis = P r ( c i

< cis)

⁼ _{f i l -}

A= 1

It i s p o s s i b l e t o develop an algorithm u s i n g g e n e r a l i z e d upper bounding c o n s t r u c t s t h a t w i l l s o l v e ( 2 ) i n a number of p i v o t s t h a t i s o f t h e same o r d e r of magnitude a s t h e number of p i v o t s r e q u i r e d to s o l v e t h e mean l i n e a r program- ming approximation problem, i .e

. ^,

^{( 1 ) with}

⁵

r e p l a c i n g

?.

The l i n e a r program

( 2 has t h e same number of working b a s i s elements, (ml+mz) a s t h e mean problem.

Wets ( 1 974, 1983a) has developed an a l g o r i t h t h a t has been coded by C o l l i n s

( 1975 )

,

K a l l b e r g and Kusy ( 1976). The code was w r i t t e n to s o l v e problems with up t o 70 s t o c h a s t i c c o n s t r a i n t s , 220 t o t a l c o n s t r a i n t s and 8 r e a l i z a t i o n s p e r random element. The code can b e expanded to s o l v e much l a r g e r problems. The development o f more s o p h i s t i c a t e d codes to handle l a r g e r problems i s c u r r e n t l y b e i n g undertaken a t IIASA. See Wets (1983b) f o r e x t e n s i o n o f h i s a l g o r i t h m to t h e convex c a s e .

The f o r m u l a t i o n ( 1 ) i s e s s e n t i a l l y s t a t i c while t h e a s s e t and l i a b i l i t y management problem i s dynamic. We u t i l i z e t h e model ( 2 ) and i t s e f f i c i e n t

computational scheme while a t t h e same t i m e r e t a i n i n g a s many of t h e dynamic a s p e c t s o f t h e model a s p o s s i b l e . To do t h i s we u t i l i z e t h e approximation d e s c r i b e d below. The g e n e r a l n-stage SLPSR problem i s

n ' n

+

min [ c x + E

{

^min

En lEn-1

, .. .,

^{1 n+ n- [qn+' yn*}

xn >0

- ⁵

^{Y , Y -}

^>

⁰

Anxn=bn

The approximation p r o c e d u r e a g g r e g r a t e s x 2 ,

. . ^{. ,}

xn i n w i t h x1 and

c2, . . . ^{, En}

1 1 n

w i t h

5 .

Thus i n ( 1 ) one chooses xE(x

,...,

^{x 1 '} i n s t a g e one, o b s e r v e s

1 n

5

( 5 , .. . ^{, 5 '}

a t t h e end of s t a g e one and t h e s e t o g e t h e r d e t e r m i n e

( y + I y ~ ) E [ ( y l + , y l - ) I . . . I (yn+,yn')l i n s t a g e two. T h i s approach y i e l d s a f e a s i b l e procedure f o r t h e t r u e dynamic model ( 3 ) t h a t is c o m p u t a t i o n a l l y f e a s i b l e f o r

l a r g e problems and i n c o r p o r a t e s p a r t i a l dynamic a s p e c t s s i n c e p e n a l t y c o s t s f o r p e r i o d s 2,.

. .

, n a r e c o n s i d e r e d i n t h e c h o i c e of x l , .

.

^.,xn. The d e c i s i o n

maker is p r i m a r i l y i n t e r e s t e d i n t h e immediate r e v i s i o n of t h e b a n k ' s a s s e t s and l i a b i l i t i e s . The ALM model i n c o r p o r a t e s immediate r e v i s i o n by s e t t i n g times 0 and 1 an a r b i t r a r i l y s m a l l time p e r i o d a p a r t . P o i n t 0 r e f e r s to t h e b a n k ' s i n i t i a l p o s i t i o n and p o i n t 1 r e f e r s to t h e b a n k ' s p o s i t i o n immediately a f t e r r u n n i n g t h e model. I n p r a c t i c e t h e model is r o l l e d o v e r c o n t i n u o u s l y . Also to p a r t i a l l y overcome t h e drawbacks of a s t a t i c s o l u t i o n t e c h n i q u e t h e d e c i s i o n v a r i a b l e s a r e d e f i n e d s o t h a t a s e c u r i t y can be purchased i n one time p e r i o d and s o l d i n one o r more s u b s e q u e n t p e r i o d s .

The r e c o u r s e a s p e c t of t h e model g i v e s it a dynamic f l a v o u r . The model is two-stage: i n i t i a l l y t h e d e c i s i o n v a r i a b l e s a r e chosen, n e x t t h e s t o c h a s t i c v a r i a b l e s a r e observed and t h i s d e t e r m i n e s t h e r e c o u r s e v a r i a b l e s ( i n o r d e r to r e c o v e r f e a s i b i l i t y ) and t h e i r c o r r e s p o n d i n g p e n a l t i e s . The p e n a l t y is a

f u n c t i o n o f b o t h t h e c o n s t r a i n t v i o l a t e d and t h e magnitude of v i o l a t i o n . The r e c o u r s e c o s t h a s t h e e f f e c t o f r e s t r a i n i n g "aggressive" c h o i c e s o f d e c i s i o n v a r i a b l e s i f t h e c o s t s involved with r e g a i n i n g f e a s i b i l i t y outweigh t h e bene- f i t s . Thus, t h e r o l l i n g over o f t h e ALM model, d e f i n i n g t h e v a r i a b l e s t o g i v e them f l e x i b i l i t y and t h e r e c o u r s e a s p e c t o f SLPSR, a r e t h e dynamic f e a t u r e s o f t h e ALM model.

3. FORMULATION OF THE AIM MODEL

The a s s e t and l i a b i l i t y management ( U ) m d e l i s an i n t e r t e m p o r a l

decision-making o p t i m i z a t i o n t o o l t o determine a b a n k ' s p o r t f o l i o o f a s s e t s and l i a b i l i t i e s given d e t e r m i n i s t i c r a t e s o f r e t u r n s and c o s t ( i n t e r e s t r a t e s )

,

and random cash flows ( d e p o s i t s ) . Although t h e a s s e t and l i a b i l i t y management problem i s a continuous d e c i s i o n problem a s p o r t f o l i o s a r e c o n s t a n t l y b e i n g r e v i s e d over t i m e , t h e computations and a n a l y s i s involved with a continuous t i m e p r o c e s s a r e i n f e a s i b l e f o r a normative t o o l . T h e r e f o r e , t h e A W model i s devel- oped a s a m u l t i - p e r i o d d e c i s i o n problem in which p o r t f o l i o s a r e determined a t d i s c r e t e p o i n t s i n time (e.g

. ^,

t h e end of each accounting

pried) .

The ALM model has t h e following f e a t u r e s : 1. & j e c t i v e f u n c t i o n :

maximize t h e n e t p r e s e n t v a l u e o f bank p r o f i t s minus t h e expected p e n a l t y c o s t s f o r i n f e a s i b i l i t y .

2. C o n s t r a i n t s :

a . l e g a l , b e i n g a f u n c t i o n o f t h e b a n k ' s j u r i s d i c t i o n ,

b . b u d g e t : i n i t i a l c o n d i t i o n s and t h e s o u r c e s and u s e s of f u n d s , c . l i q u i d i t y and l e v e r a g e , t o s a t i s f y d e p o s i t withdrawals on demand,

( t h e FRB's c a p i t a l adequacy formula i s t h e b a s i s o f t h e s e c o n s t r a i n t s )

,

d. p o l i c y and t e r m i n a t i o n : c o n s t r a i n t s unique to t h e bank and c o n d i t i o n s to e n s u r e t h e b a n k ' s c o n t i n u i n g e x i s t e n c e a f t e r t h e t e r m i n a t i o n o f t h e model, and

e . d e p o s i t f l o w s .

C o n s t r a i n t s ( a ) and ( b ) a r e d e t e r m i n i s t i c , ( c ) c o n s i s t s o f b o t h d e t e r m i n i s t i c and s t o c h a s t i c c o n s t r a i n t s , ( d ) c a n c o n s i s t o f e i t h e r d e t e r m i n i s t i c o r s t o c h - a s t i c c o n s t r a i n t s

,

^{and (} e ) c o n t a i n s on1 y s t o c h a s t i c c o n s t r a i n t s

.

Chambers and Charnes (1961) and Cohen and Hammer (1967) have j u s t i f i e d t h e u s e o f l i n e a r f u n c t i o n s t o model a b a n k ' s a s s e t and l i a b i l i t y management

problem. Thus from t h e p o i n t o f view o f l i n e a r i t y , t h e a p p r o p r i a t e n e s s of SLPSR i s e s t a b l i s h e d . The r e c o u r s e a s p e c t i s j u s t i f i e d w i t h t h e f o l l o w i n g argument.

In t h e b a n k i n g b u s i n e s s , c o n s t r a i n t v i o l a t i o n s do n o t imply t h a t t h e i n t e r - mediary i s p u t i n t o r e c e i v e r s h i p . Rather t h e bank i s allowed to r e s t r u c t u r e i t s p o r t f o l i o o f a s s e t s to r e g a i n f e a s i b i l i t y a t some c o s t ( p e n a l t i e s ) . With t h e i n h e r e n t u n c e r t a i n t i e s t h e a s s e t and l i a b i l i t y management problem i s =ll modeled a s a s t o c h a s t i c l i n e a r program w i t h s i m p l e r e c o u r s e .

3 1 N o t a t i o n f o r t h e A M Model

x

-

amount o f a s s e t k purchased in p e r i o d i s o l d i n p e r i o d j ; k = l I . . . , K ; i j

i=O,...,n-1; j = i + l , . . . , n

x

-

i n i t i a l h o l d i n g s o f s e c u r i t y k 0 0

x

-

amount o f s e c u r i t y k purchased i n p e r i o d i to b e h e l d beyond t h e h o r i z o n

i m

o f t h e model

y:

-

new d e p o s i t s o f t y p e d i n p e r i o d i; d=l

, . . . ^,

^D

-

i n i t i a l h o l d i n g s o f d e p o s i t t y p e d Yo

bi

-

f u n d s borrowed i n p e r i o d i

+

-

s h o r t a g e i n p e r i o d j i n s t o c h a s t i c c o n s t r a i n t s Y j s

Y

is ^-

s u r p l u s i n period j i n stochasic c o n s t r a i n t s

+

-

proportional penalty c o s t associated with y+

P j s j s

p; s

-

proportional penalty c o s t associated with y- j s

-

parameter f o r shrinkage, under normal economic c o n d i t i o n s , i n period j of

p i j

a s s e t type k purchased i n period i

a

-

parameter f o r shrinkage, under severe economic c o n d i t i o n s , i n period j of i j

a s s e t type k purchased i n period i

-

proportional t r a n s a c t i o n c o s t on a s s e t k, which i s e i t h e r purchased o r

ti sold in period i

r

-

r e t u r n on a s s e t k purchased i n period i i

T

-

t a x r a t e on c a p i t a l gains ( l o s s e s ) i n period j j

T j

-

marginal t a x r a t e on income in period j

z

-

proportional c a p i t a l gain ( l o s s ) on s e c u r i t y k purchased i n period i and i j

sold i n period j

yd

-

t h e a n t i c i p a t e d f r a c t i o n of d e p o s i t s of type d withdrawn under adverse economic c o n d i t i o n s

c

-

r a t e paid on d e p o s i t s of type d i

'i

-

discount r a t e from period i to period 0

-

s e t of p o s s i b l e c u r r e n t a s s e t s a s s p e c i f i e d by t h e B r i t i s h Columbia Credit Union A c t

K 1

-

s e t of primary and secondary a s s e t s a s defined i n t h e c a p i t a l adequacy formula ( c a f

K2

-

s e t of minimum r i s k a s s e t s a s defined i n t h e caf K3

-

s e t of intermediate r i s k a s s e t s a s defined i n t h e caf

q i

-

penalty r a t e f o r the p o t e n t i a l withdrawal of funds, i n period i, which a r e not covered by a s s e t s i n K1 U

...

_K3

'i

-

l i q u i d i t y r e s e r v e s f o r t h e p o t e n t i a l withdrawal of f u n d s , i n p e r i o d i , n o t covered by a s s e t s i n K ₁ U

. . .

_K3

k m i

-

m - t h mortgage i

-

d i s c r e t e random v a r i a b l e i n p e r i o d j f o r s t o c h a s t i c c o n s t r a i n t t y p e s , 'Is s e S where S i s t h e set o f s t o c h a s t i c c o n t r a i n t s .

3.2 The AlN Model

1

k n-1 n

I

d i s c o u n t e d r e t u r n s and

+

^X

01 zOl i = l j = i + l r ; ( l + r R ) P R c a p i t a l g a i n s ( n e t o f

R=i+ 1 t a x e s ) on a s s e t s

i= 1 n

n e t d i s c o u n t e d c o s t o f d e p o s i t s

(demand and t i m e )

c o s t o f d i r e c t borrowing from o t h e r banks and a c e n t r a l bank

e x p e c t e d p e n a l t y c o s t s f o r c o n s t r a i n t v i o l a t i o n s

S u b j e c t to:

( a ) Legal c o n s t r a i n t s

( b ) Budget c o n s t r a i n t s i . I n i t i a l h o l d i n g s

i i . Sources and u s e s

( c ) L i q u i d i t y c o n s t r a i n t s

iii.

- C E _[ !

^{xi, a:,}

⁺

^x:matj

ksK1 UK UK i = O e = j + l 2 3

n

i v .

- 1

⁽¹

-

6 . . ) x i , ^k_{1 3} ^k

+

(1

-

_{1 3}

( d ) P o l i c y c o n s t r a i n t s

+ Y ; ~

-

^{Y ; ~}

^{5 E j S}

^{j = 1 ,}

...,

^{n , SES}

( e ) D e p o s i t f l o w s

d j-1

d j-i

+ - _-

y . * Y i ( l - Y d ) + Y j s - Y j s - E j s 3

i = o

j = 1 , .

. .

, n ; d = 1 , .

.

^{.D, SES}

( f ) Nonnegativi t y

k d

X i j ' b i t yi, y?s.~;s

2

0 f o r a11 i I j I k I d

There a r e no d i s c o u n t f a c t o r s i n c o r p o r a t e d i n t o t h e c o n s t r a i n t s s i n c e e a c h c o n s t r a i n t r e f e r s t o c o n d i t i o n s i n o n l y one p e r i o d . The ALM model t r e a t s t h e f i r s t two t y p e s of c o n s t r a i n t s , l e g a l and budget, as d e t e r m i n i s t i c . The l e g a l c o n s t r a i n t s t a t e s t h a t t h e c u r r e n t a s s e t s c a n n o t be l e s s t h a n 10% of t h e t o t a l l i a b i l i t i e s l e s s r e s e r v e s , s u r p l u s and e q u i t y [as d e f i n e d by t h e B r i t i s h

Columbia C r e d i t Union Act ( B r i t i s h Columbia Government, 1 9 7 3 ) l . The l e g a l con- s t r a i n t s a r e , of c o u r s e , p e c u l i a r t o t h e l o c a l e of t h e i n s t i t u t i o n b e i n g

s t u d i e d . The budget c o n s t r a i n t s i n c l u d e t h e i n i t i a l c o n d i t i o n s and t h e a c c o u n t i n g i d e n ti ty--uses and s o u r c e s of funds a r e e q u a l .

The l i q u i d i t y c o n s t r a i n t s f o l l o w from t h e F e d e r a l Reserve Board's c a p i t a l adequacy formula ( c a f ) . The r e q u i r e m e n t t h a t t h e market v a l u e of a b a n k ' s

a s s e t s is a d e q u a t e to meet d e p o s i t o r

'

s withdrawal c l a i m s d u r i n g a d v e r s e economic c o n d i t i o n s is t h e p r i n c i p a l c o n s t r a i n t i n t h e c a f . To d e v e l o p t h i s c o n s t r a i n t , l i q u i d i t y r e s e r v e s ( f o r a d v e r s e economic c o n d i t i o n s ) a r e f i r s t d e f i n e d . The f i r s t t h r e e l i q u i d i t y c o n s t r a i n t s a r e

-

The p r i n c i p a l c o n s t r a i n t of t h e caf i s

K 3 t o t a l r i g h t hand

1

^{( 1-Bi)xi}

¹ 1

^{P. 1 +} s i d e of b a l a n c e - s u r p l u s - e q u i t y

.

i = l i =l s h e e t

Thus t h e market v a l u e of t h e b a n k ' s a s s e t s s h o u l d be n o t less t h a n t h e l i q u i d i t y r e s e r v e s f o r d i s i n t e r m e d i a t i o n under s e v e r e economic c o n d i t i o n s p l u s l i a b i l i t i e s . T h i s c o n s t r a i n t i s t h e f i n a l l i q u i d i t y c o n s t r a i n t i n ALM.

Although t h i s c o n s t r a i n t is n o t s t o c h a s t i c , a bank p o r t f o l i o manager may v i o l a t e i t because t h e caf s e t f o r t h by t h e E'RB is a s u g g e s t e d g u i d e l i n e f o r sound bank management r a t h e r t h a n a s t r i c t r e g u l a t i o n . The p e n a l t y f o r a v i o l a t i o n s of

3

t h i s c o n s t r a i n t is

1

^q ( a s p r e s c r i b e d by t h e E'RB). T h i s e l a s t i c t r e a t m e n t of i = l i

FRB's r e g u l a t i o n a l l o w s t h e c o n s t r a i n t to be v i o l a t e d when t h e b e n e f i t s of v i o l a t i o n exceed t h e c o s t s . I n t h i s manner, t h e c r i t i c i s m , l e v e l l e d a t

m o d e l l e r s u s i n g FRB's c o n s e r v a t i v e c o n s t r a i n t s , can be r e s o l v e d i n a s y s t e m a t i c manner. See S e c t i o n 4.1.3 f o r more d i s c u s s i o n c o n c e r n i n g t h e s e c o n s t r a i n t s .

The f o u r t h s e t of c o n s t r a i n t s i s a l s o elastic. These c o n s t r a i n t s a r e i n t r o d u c e d to c a p t u r e t h e i n t e r n a l o p e r a t i o n a l p o l i c y of t h e i n s t i t u t i o n

modelled. I n r e a l i t y minor c o n s t r a i n t v i o l a t i o n s of bank p o l i c i e s are u s u a l l y t o l e r a b l e w h i l e more s e v e r e v i o l a t i o n s are i n c r e a s i n g l y less t o l e r a b l e . The i n t r o d u c t i o n of a piece-wise l i n e a r convex p e n a l t y f u n c t i o n ( v i a a d d i t i o n a l con- s t r a i n t s ) can c a p t u r e t h e dependency between t h e p e n a l t y c o s t s and t h e e x t e n t of t h e p o l i c y v i o l a t i o n s . T h i s is accomplished by the a d d i t i o n of s u p p l e m e n t a r y c o n s t r a i n t s to r e f l e c t t h e i n c r e a s e d s e r i o u s n e s s of t h e magnitude of c o n s t r a i n t v i o l a t i o n s .

The f i n a l s e t of c o n s t r a i n t s , d e p o s i t f l o w s , is s t o c h a s t i c . S i n c e d e p o s i t f l o w s a r e c o n t i n u a l l y t u r n e d o v e r and b e a r v a r i o u s r a t e s of i n t e r e s t t h e model has to r e f l e c t t h e g r o s s (and n o t n e t ) flows d u r i n g an a c c o u n t i n g p e r i o d . T h i s p r o p e r t y of t h e problem was i n c o r p o r a t e d i n the model by having a p r o p o r t i o n a l o u t f l o w [ s t a t i s t i c a l l y c a l c u l a t e d by t h e FRB and c o r r o b o r a t e d f o r u s e i n B r i t i s h Columbia i n C r e d i t Union Reserve Board ( 1 9 7 3 ) l of o l d f u n d s d u r i n g e a c h p e r i o d .

The three t y p e s of l i a b i l i t y e x p r e s s i o n s i n the ALM f o r m u l a t i o n are now developed. The d e p o s i t f l a w c o n s t r a i n t s r e p r e s e n t the t o t a l amount of new d e p o s i t s i n t h e j t h p e r i o d . The t o t a l amount of new d e p o s i t s of t y p e d gener- a t e d i n p e r i o d j is

d j - i d

yd _j = BS

- 1

^yi(1

-

^{Y d ) j -i}

j

i=o

d j - i

d j - i

-

Y . +

I

¹

-

^Y

^-

^{BS d}

7

i = o

1 '

where yd is t h e t o t a l amount of new type d d e p o s i t s i n p e r i o d j , yd is t h e

j d

annual r a t e of withdrawal of type d d e p o s i t s , and BS i s t h e d i s c r e t e random j

v a r i a b l e r e p r e s e n t i n g balance s h e e t f i g u r e of type d d e p o s i t s a t t h e end of t h e j th period.

The second type of l i a b i l i t y e x p r e s s i o n r e p r e s e n t s t h e total amount of d e p o s i t s o u t s t a n d i n g d u r i n g a p e r i o d . Since t h e model i s d i s c r e t e , an approx- imation to t h e continuous flow is made by assuming t h a t half of a p e r i o d ' s n e t flows a r r i v e a t t h e beginning of t h e period and t h e o t h e r h a l f a r r i v e a t t h e beginning of t h e n e x t p e r i o d . During t h e f i r s t p e r i o d , t h e funds a v a i l a b l e a r e

and f o r p e r i o d j

This e x p r e s s i o n i s used i n the o b j e c t i v e f u n c t i o n , and t h e l e g a l and l i q u i d i t y c o n s t r a i n t s . The t h i r d l i a b i l i t y e x p r e s s i o n is t h e i n c r e m e n t a l i n c r e a s e

( d e c r e a s e ) of d e p o s i t s from one period to t h e next. This i n c r e m e n t a l d i f f e r e n c e i s used i n the s o u r c e s and u s e s c o n s t r a i n t . For p e r i o d j t h e i n c r e m e n t a l

d i f f e r e n c e is

3.3 Data Required to Implement t h e ALM b d e l

To implement t h e AM model r e q u i r e s t h e following d a t a :

1. t h e i d e n t i f i c a t i o n of t h e a s s e t s i n h i c h t h e bank can p o t e n t i a l l y i n v e s t ( o r a t l e a s t a r e p r e s e n t a t i v e group of a s s e t s ) ;

2. p o i n t e s t i m a t e s o f t h e r e t u r n s on t h e s e a s s e t s ;

3. p o i n t e s t i m a t e s o f c a p i t a l g a i n s ( l o s s e s ) a s a f u n c t i o n of t h e time t h e bank h o l d s t h e a s s e t s ;

4. i d e n t i f i c a t i o n of t h e l i a b i l i t i e s which t h e bank can p o t e n t i a l l y s e l l ; 5. p o i n t e s t i m a t e s o f t h e c o s t s of t h e s e l i a b i l i t i e s ;

6. t h e r a t e a t which d e p o s i t s a r e withdrawn;

7. an e s t i m a t e d weighted c o s t o f funds t o determine t h e d i s c o u n t r a t e ; 8. p e r t i n e n t l e g a l c o n s t r a i n t s ;

9. p a r a m e t e r s used in t h e development of t h e l i q u i d i t y c o n s t r a i n t s ; 10. p o l i c y c o n s t r a i n t s used by t h e bank;

11. e s t i m a t e s o f t h e marginal d i s t r i b u t i o n s of t h e s t o c h a s t i c r e s o u r c e s ; and

12. u n i t p e n a l t i e s i n c u r r e d f o r s h o r t a g e o r s u r p l u s i n t h e s t o c h a s t i c c o n s t r a i n t s .

Remarks :

a. Since t h e AM model h a s a s e p a r a b l e o b j e c t i v e o n l y t h e marginal d i s t r i b u t i o n s of t h e components of t h e r e s o u r c e v e c t o r a r e needed to f i n d t h e o p t i m a l s o l u t i o n .

b. The s h o r t a g e ( y + ) and s u r p l u s ( y - ) v a r i a b l e s have s p e c i f i c meanings in t h e ALM formulation. Consider a r e a l i z a t i o n

gd'

_js o f t h e randcm d e p o s i t

E

_js d

.

I f

j-i d j-i d'

+ 1

^{y i ( l}

-

^{y d )}

_{< Sjs}

y j

iio

then y+

>

0 and y' = 0, assuming p+

+

^p'

>

0; y + would b e i n t e r p r e t e d a s t h e amount o f funds t h a t c o u l d have been used f o r investment purposes i n t h e AIA.

Since t h e c o s t o f d e p o s i t s i s u s u a l l y lower t h a n t h e r e t u r n s on a s s e t s , t h e bank would want to u t i l i z e all a v a i l a b l e funds. A p e n a l t y p+

>

0 f o r t h e

o p p o r t u n i t y c o s t can b e determined by assuming t h a t t h e funds n o t used can b e i n v e s t e d i n e a r n i n g a s s e t s . The y+ d o l l a r s m u l d b e a v a i l a b l e a t some r a t e c and could then b e i n v e s t e d i n some a s s e t a t a r a t e r . The p e n a l t y , p+, would b e equal t o ( r - c ) discounted to p o i n t 0 p l u s t h e n e t d i s c o u n t e d r e t u r n s on y + ( r - c ) to t h e h o r i z o n o f t h e model ( t h a t i s , t h e p r o f i t s t h a t c o u l d have been g e n e r a t e d )

.

On t h e o t h e r hand, i f

d j - i d j-i d *

Y j +

1

^{y i ( 1}

-

^{l d )}

i=

o > S j s

t h e n y-

>

0 and y+ = 0, and a s u r p l u s o c c u r s . In t h i s c a s e , t h e bank m u l d have to d i v e s t i t s e l f o f some e a r n i n g a s s e t s . The c o s t , p-, o f t h i s a c t i o n i s

( r - c ) discounted to p i n t 0 p l u s t h e n e t discounted r e t u r n s on y-(r-c) to t h e h o r i z o n of t h e model ( t h a t i s , t h e p r o f i t s t h a t m u l d have been g e n e r a t e d with u n a v a i l a b l e f u n d s )

.

Thus b o t h p+ and p- a r e p o s i t i v e and p r o f i t i s lowered i f e i t h e r t o o

l i t t l e o r t o o much i s i n v e s t e d . The key i s s u e of what r and c should b e used to determine t h e p e n a l t i e s i s now addressed where a c a s e s t u d y using t h e ALM

f o r m u l a t i o n i s p r e s e n t e d .

4. APPLICATION OF AIA TO THE VANCOUVER CITY SAVINGS CREDIT U N I O N

T h i s s e c t i o n i s concerned with an a p p l i c a t i o n o f t h e A M model to t h e a s s e t and l i a b i l i t i y p o r t f o l i o problem of Vancouver C i t y Savings C r e d i t Union (VCS).

T h e r e is a l s o a d i s c u s s i o n o f p r o c e d u r a l a s p e c t s o f implementing t h e model to t h i s and r e l a t e d i n s t i t u t i o n s . T h i s s t u d y was prompted b y t h e VCS's c o n t i n u a l l i q u i d i t y problem and f o c u s e s o n t h e f i v e year p l a n n i n g p e r i o d 1970-1974.

During t h i s p e r i o d t h e f i r m ' s a s s e t s grew a t a compound r a t e of 57%/year from $26 m i l l i o n to $160 m i l l i o n and t h e r e was an a g g r e s s i v e p o l i c y o f i n v e s t i n g in high y i e l d i n g a s s e t s , predominantly mortgages. In 1974, VCS r e a l i z e d t h a t t h e combination o f t h e i r a g g r e s s i v e i n v e s t m e n t p o l i c y and changing market

c o n d i t i o n s was c r e a t i n g s e r i o u s l i q u i d i t y problems. I n v e s t o r s were t r a d i n g low y i e l d t e r m d e p o s i t s f o r h i g h e r y i e l d d e p o s i t s . Meanwhile t h e o u t s t a n d i n g

mortgage l o a n s were s t i l l e a r n i n g r e t u r n s on t h e b a s i s o f t h e lower f i x e d r a t e s . It was a t t h i s moment t h a t t h i s s t u d y was i n i t i a t e d .

4.1 Model D e t a i l s

We now d e s c r i b e t h e i n p u t n e c e s s a r y t o implement t h e ATA model a t VCS. The d i s c u s s i o n h e r e i s on g e n e r a l c o n c e p t s c o n c e r n i n g methods o f d a t a c o l l e c t i o n , c h o i c e o f d e c i s i o n v a r i a b l e s , c o n s t r a i n t s and ab j e c t i v e f u n c t i o n . The a c t u a l d a t a , a 92 x 257 i n p u t m a t r i x , appear i n Kusy ( 1 9 7 8 ) .

T h f i r s t s t a g e v a r i a b l e s a r e a s s e t s ( x k _{i j} ) and l i a b i l i t i e s ( yd and b i ) _i

.

There a r e e l e v e n a s s e t t y p e s : 1. c a s h ;

2. B r i t i s h Columbia C r e d i t Union s h a r e s ;

3-6. f e d e r a l government bonds m a t u r i n g in i = 1,

. . .

^{, 4} y e a r s ; 7. f e d e r a l government bonds m a t u r i n g in f i v e t o t e n y e a r s ; 8. p r o v i n c i a l government bonds m a t u r i n g i n more t h a n t e n years;

9-10. f i r s t and second mortgages w i t h a t h r e e y e a r t e r m , and 11. p e r s o n a l l o a n s .

S i x t y p e s o f l i a b i l i t i e s a r e c o n s i d e r e d : 1. demand d e p o s i t s ;

2. s h a r e c a p i t a l o f VCS;

3. borrowing from banks; and

4-6. term d e p o s i t s maturing in i = 1 , 3 , 5 y e a r s .

These a s s e t and l i a b i l i t y t y p e s g e n e r a t e 132 and 36 v a r i a b l e s ,

r e s p e c t i v e l y , i n c l u d i n g i n i t i a l p o s i t i o n s . For example a four year f e d e r a l government bond purchased a t t h e beginning o f t h e t h i r d time p e r i o d g e n e r a t e s d e c i s i o n v a r i a b l e s x 6

and x

34' X35f where x 6 and x 6 a r e t h e amounts o f t h e

34)' 34 3 5

i n i t i a l investment to b e s o l d in p e r i o d s f o u r and f i v e , r e s p e c t i v e l y , and x 6 i s 34) t h e amount to b e h e l d a t t h e horizon. The c h o i c e o f a s s e t s and l i a b i l i t i e s was based on VCS' s h i s t o r i c a l p o r t f o l i o s ( 1968-1975) so t h a t comparison between

a c t u a l p o r t f o l i o s and A W g e n e r a t e d p o r t f o l i o s could b e e a s i l y made. Although cash flows are c o n t i n u o u s over time t h e model assumes t h a t a l l t r a n s a c t i o n s occur a t t h e beginning of p e r i o d s . Cash flows d u r i n g any p e r i o d are modeled by assuming t h a t h a l f t h e flow o c c u r s a t t h e beginning o f t h e p r e s e n t p e r i o d and t h e o t h e r h a l f a t t h e b e g i n n i n g of t h e n e x t p e r i o d . The model h a s t h e f o l l o w i n g c o n s t r a i n t s .

4.1.1 Legal C o n s t r a i n t s

The source f o r t h e l e g a l c o n s t r a i n t s i s t h e C r e d i t Union Act of B r i t i s h Columbia [ B r i t i s h Columbia Qvernment ⁽ 1973

) I ,

which p l a c e s t h r e e o p e r a t i o n a l r e s t r i c t i o n s on t h e composition o f t h e p o r t f o l i o o f a s s e t s and l i a b i l i t i e s . The

f i r s t c o n s t r a i n t i s t h a t c r e d i t unions maintain a t l e a s t 10% of t h e t o t a l a s s e t s (denoted b y t h e s e t I ) in high l i q u i d a s s e t s (denoted b y t h e s e t I ) :

L

The second requirement i s t h a t c r e d i t unions m a i n t a i n a t l e a s t 1% of t h e i r t o t a l d e b t i n c a s h and t e r n d e p o s i t s :

The f i n a l c o n s t r a i n t r e s t r i c t s t h e c r e d i t u n i o n ' s borrowing from o p p o r t u n i t i e s denoted by t h e set B, t o one h a l f o f t h e t o t a l l i a b i l i t i e s :

Since t h e planning b r i z o n has f i v e p e r i o d s , t h e l e g a l requirements account f o r f i f t e e n c o n s t r a i n t s .

4.1.2 Budget C h n s t r a i n t s

There a r e twenty-two budget c o n s t r a i n t s , seventeen e s t a b l i s h t h e i n i t i a l p o s i t i o n s o f t h e e l e v e n a s s e t and s i x l i a b i l i t y t y p e s and f i v e r e q u i r e t h e

s o u r c e s and u s e s of funds t o b e e q u a l i n each p e r i o d . 4.1.3 L i q u i d i t y C h n s t r a i n t s

The l i q u i d i t y c o n s t r a i n t s ensure t h a t t h e f i r n has s u f f i c i e n t c a p i t a l

r e s e r v e s t o meet s e v e r e withdrawal claims under adverse economic c o n d i t i o n s . The c o n s t r a i n t s f o l l o w from t h e Federal &serve Board's c a p i t a l adequacy formula

[Crosse and Hempel (197311. The a p p l i c a t i o n o f t h e FRB's c a f t o B r i t i s h

Columbia's c r e d i t unions i s j u s t i f i e d i n a s t u d y p u b l i s h e d by t h e C r e d i t Union Reserve Board (19731,

The f i r s t t h r e e c o n s t r a i n t s e s t a b l i s h c a p i t a l r e s e r v e s based upon t h e s t r u c t u r e o f t h e p o r t f o l i o o f a s s e t s and l i a b i l i t i e s :

where P i s t h e r e q u i r e d r e s e r v e n e c e s s a r y t o meet t h e excess withdrawal c l a i m s , i

%

measures t h e r e s e r v e s r e q u i r e d f o r p o t e n t i a l withdrawal claims t h a t exceed t h e r e a l i z a b l e p o r t i o n o f t h e a s s e t s c o n t a i n e d in K U...UKi, ak i s a parameter

1

t h a t measures t h e r e a l i z a b l e p o r t i o n o f t h e v a l u e of a s s e t k i f t h e a s s e t i s to

b e l i q u i d a t e d q u i c k l y under adverse economic c o n d i t i o n s , W =

1

^yiyi i s t h e i = l

d o l l a r v a l u e of t h e expected withdrawal c l a i m s under adverse c o n d i t i o n s , where y

.

measures t h e c o n t r a c t i o n of l i a b i l i t y y under adverse economic c o n d i t i o n s .

1 i

The y 's used were 0.47 f o r demand d e p o s i t s , 0.36 f o r term d e p o s i t s and 1.0 f o r borrowing; s e e C r e d i t Union & s e r v e Board ( 1973 ) f o r j u s t i f i c a t i o n .

The a s s e t s a r e c l a s s i f i e d a s per t h e FFtB's caf a s follows:

1. "primary and Secondary Reserves: ( K 1 ) which i n c l u d e s c a s h , t r e a s u r y b i l l s , and government bonds of l e s s t h a n f i v e y e a r s m a t u r i t y ;

2. "Minimum Risk Assets" (K ) which i n c l u d e government bonds with more 2

t h a n f i v e y e a r s m a t u r i t y , and municipal bonds; and

3. " I n t e r m e d i a t e Assets" which i n c l u d e s mortgage and p e r s o n a l l o a n s . F i n a l l y , t h e p r i n c i p a l c o n s t r a i n t in t h e c a f i s

K 3 t o t a l r i g h t - e q u i t y - s u r p l u s

1

¹

- p i

^Xi

1

^{Pi +} hand s i d e

i = l i = l of b a l a n c e

s h e e t

where

pi

i s a parameter t o measure t h e shrinkage of a s s e t i , when t h e a s s e t i s t o b e l i q u i d a t e d q u i c k l y . The a c t u a l n m b e r s used f o r ak, q i , and

pi

a r e t h o s e p r e s c r i b e d by t h e

Fm

[Cross and Hempel ( 1973 ) 1

.

Since t h e purpose here i s n o t t o develop an o p e r a t i o n a l model f o r VCS, b u t r a t h e r t o demonstrate t h e

a p p l i c a b i l i t y o f t h e ALM model, t h e parameter v a l u e s used provide an adequate proxy. I n t h e development of an o p e r a t i o n a l model it w u l d b e n e c e s s a r y t o e s t i m a t e t h e parameters. Since t h e s e c o n s t r a i n t s have to hold f o r a l l f i v e p e r i o d s , t h e r e a r e twenty l i q u i d i t y c o n s t r a i n t s .

4.1.4 P o l i c y m n s t r a i n t s

TvJo t y p e s of p o l i c y c o n s t r a i n t s a r e included:

1. p e r s o n a l l o a n s should not exceed 20% of t h e f i r s t mortgage l o a n s i n any period t , i.e., x

<

0.2 xtm; and

t L

-

2. second mortgages should not exceed 12.5% of f i r s t mortgages, i.e.,

x

<

0.125 x

t P

-

^tm*

The r a t i o n a l e i s t h a t r e t u r n s on f i r s t mortgages a r e l e s s r i s k y t h a n second mortgages o r p e r s o n a l l o a n s and some of t h e l a t t e r a r e d e s i r a b l e (even though t h e y may have lower r e t u r n s ) t o respond to management's p r e f e r e n c e f o r a l e s s r i s k y o v e r a l l p o r t f o l i o . These c o n s t r a i n t s may b e v i o l a t e d without l e g a l

i m p l i c a t i o n s and a r e modelled by t r e a t i n g t h e c o n s t r a i n t s a s s t o c h a s t i c using (P+,P- ) = ( 0 , l )

.

There a r e t e n such c o n s t r a i n t s over t h e f i v e p e r i o d s .

4.1.5 Deposit Flows

The v a r i a b l e yd r e p r e s e n t s t h e new d e p o s i t s o f t y p e d = 1,

. . .

, 5 generated I

i n p e r i o d j = 1,

. . .

^{, 5 and}

⁵

i s a d i s c r e t e random v a r i a b l e r e p r e s e n t i n g t h e j d

b a l a n c e s h e e t of d e p o s i t t y p e d a t t h e end of p e r i o d j. The d e p o s i t flow c o n s t r a i n t s a r e

d ^{j - i} d j - i +

- ^-

+ 1

^{y i ( l}

-

^{y d )} _{+ Y j d}

-

^{Y j d}

^{- Cjd}

where t h e y ' s (1.0 f o r demand d e p o s i t s and 0.36 f o r term d e p o s i t s ) a r e included to r e f l e c t t h e g r o s s flow o f d e p o s i t funds. The d i s t r i b u t i o n of

5

was

j d

e s t i m a t e d using t h e b a l a n c e s h e e t f i g u r e s of VCS f o r 1970-1974; see Kusy (1978) f o r s p e c i f i c e s t i m a t e s .

The p e n a l t i e s f o r s h o r t a g e s a s s o c i a t e d with t h e s e c o n s t r a i n t s a r e : 1. f o r demand d e p o s i t s and s h a r e c a p i t a l , p+ i s t h e t o t a l discounted

r e t u r n on a one year term d e p o s i t minus t h e d i s c o u n t e d c o s t of t h e funds c a l c u l a t e d to t h e horizon of t h e model;

2. f o r t e r m d e p o s i t s maturing i n one o r t h r e e y e a r s , p+ i s t h e t o t a l

d i s c o u n t e d r e t u r n on a f i v e y e a r t e r m d e p o s i t minus t h e d i s c o u n t e d c o s t of t h e funds c a l c u l a t e d to t h e h o r i z o n of t h e model; and

3. f o r t e r m d e p o s i t s maturing i n f i v e y e a r s , p+ is t h e t o t a l d i s c o u n t e d r e t u r n on a t e n y e a r p r o v i n c i a l government bond minus t h e d i s c o u n t e d c o s t of t h e funds c a l c u l a t e d to t h e h o r i z o n of t h e model.

The p e n a l t i e s p', f o r s u r p l u s e s a s s o c i a t e d w i t h t h e d e p o s i t f l o w c o n s t r a i n t s a r e t h e t o t a l d i s c o u n t e d r e t u r n s on f i r s t mortgages minus t h e d i s c o u n t e d c o s t s of funds c a l c u l a t e d to t h e h o r i z o n of t h e model. The p e n a l t y approach a t t e m p t s to model a c o n s e r v a t i v e management s t r a t e g y w i t h s u r p l u s f u n d s when r e a l i z e d s o u r c e s exceed u s e s and when t h e r e a r e s h o r t a g e s .

4.1.6 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 is to maximize t h e e x p e c t e d t o t a l d i s c o u n t e d revenues minus e x p e c t e d total d i s c o u n t e d c o s t s i n c l u d i n g p e n a l t y c o s t s . The s o u r c e f o r d a t a on t h e r e t u r n s on t h e f e d e r a l and p r o v i n c i a l government bonds is t h e C e n t r a l

Mortgage and Housing C o r p o r a t i o n (1 975 )

.

The s o u r c e f o r t h e re t u r n s on WJCU s h a r e s , demand d e p o s i t s and s h a r e c a p i t a l is Vancouver C i t y and Savings C r e d i t Union (1 968-1 975 1 .

The d i s c o u n t r a t e used was t h e time v a l u e of money. The r i s k f r e e r a t e ( t h e average y i e l d on t h r e e month t r e a s u r y b i l l s ) was [ C e n t r a l Mortgage and Housing C o r p o r a t i o n ( 1 975 ) 1

.

^:

Average y e a r l y y i e l d .0599 .0356 .03 56 .0547 .0782

M u l t i p e r i o d d i s c o u n t f a c t o r .9435 .9110 .8797 .8341 .7736