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http://clevelandfed.org/research/workpaper Best available copy Working Paper 8714 DEPOSIT INSURANCE AND THE COST OF CAPITAL by W i l l i a m P. Osterberg and James B. Thomson W i l l i a m P. Osterberg and James B. Thomson a r e economists a t t h e Federal Reserve Bank o f Cleveland. Working papers o f t h e Federal Reserve Bank o f Cleveland are p r e l i m i nary m a t e r i a l s c i r c u l a t e d t o s t i m u l a t e d i s c u s s i o n and c r i t i c a l comment. The views s t a t e d h e r e i n a r e t h o s e of t h e a u t h o r s and n o t n e c e s s a r i l y t h o s e o f t h e Federal Reserve Bank o f C l e v e l a n d o r o f t h e Board o f Governors of t h e Federal Reserve System. December 1987 http://clevelandfed.org/research/workpaper Best available copy ABSTRACT U s i n g a s i n g l e - p e r i o d model, t h e impacts o f d e p o s i t insurance and forbearance on t h e c o s t s and v a l u e o f u n i n s u r e d d e p o s i t s and e q u i t y c a p i t a l a r e shown under t h r e e regimes: w i t h o n l y u n i n s u r e d d e p o s i t s , w i t h b o t h i n s u r e d and u n i n s u r e d d e p o s i t s , and w i t h b o t h i n s u r e d and uninsured d e p o s i t s w i t h forbearance t o u n i n s u r e d d e p o s i t o r s i n some s t a t e s o f the w o r l d . Underpricing of d e p o s i t insurance and forbearance p o l i c i e s t o uninsured d e p o s i t o r s i n c r e a s e t h e v a l u e o f t h e uninsured d e p o s i t s . Under c e r t a i n c o n d i t i o n s , u n d e r p r i c e d d e p o s i t insurance increases t h e v a l u e o f e q u i t y . T h i s paper was presented a t t h e Federal Reserve System Committee Meeting on B a n k i n g and F i n a n c i a l S t r u c t u r e i n Miami, F l a . , i n November 1987. http://clevelandfed.org/research/workpaper Best available copy DEPOSIT INSURANCE AND THE COST OF CAPITAL I. Introduction Research i n t h e a r e a o f d e p o s i t i n s u r a n c e and c a p i t a l r e g u l a t i o n has focused on f o u r b a s i c a r e a s : d e p o s i t i n s u r a n c e p r i c i n g (Ronn and Verma C19861, Marcus and Shaked C19841, and Pennacchi C19871); c a p i t a l r e g u l a t i o n and p o r t f o l i o c h o i c e (Koehn and Santomero C19801, Santomero and Watson 119771, and Chen and Lam C19851); i m p l i c i t d e p o s i t g u a r a n t e e s and c a p i t a l f o r b e a r a n c e (Kane 119861, P y l e C1 9861, P e n a t i and P r o t o p a p a d a k i s C1 9861, and Thomson C1 987a, 1 9 8 7 b l ) ; and d e p o s i t i n s u r a n c e and bank r e g u l a t i o n (Kareken and Wallace l19781, B u s e r , Chen, and Kane C19811, and Benston, E i s e n b e i s , H o r v i t z , Kane, and Kaufman C19861). T h i s s t u d y examines t h e i m p a c t s o f d e p o s i t i n s u r a n c e g u a r a n t e e s and f o r b e a r a n c e s on t h e c o s t o f c a p i t a l f o r banks. The models used i n t h e paper a r e t h e s i n g l e - p e r i o d and m u l t i p e r i o d c a p i t a l a s s e t p r i c i n g models (CAPM) s i m i l a r t o t h e ones used by Chen ( 1 9 7 8 ) . Within t h i s framework we show t h e r e l a t i v e v a l u e s o f d e p o s i t o r and s t o c k h o l d e r p o s i t i o n s i n t h e bank under t h r e e d i f f e r e n t d e p o s i t - i n s u r a n c e r e g i m e s . t h e f i r s t regime, we assume t h a t no d e p o s i t s a r e i n s u r e d . Under Under t h e second r e g i m e , we a l l o w f o r e x p l i c i t d e p o s i t guarantees b u t no i m p l i c i t g u a r a n t e e s . F i n a l l y , under t h e t h i r d regime, we have b o t h i n s u r e d and u n i n s u r e d d e p o s i t s and a l l o w f o r t h e g u a r a n t e e o f t h e u n i n s u r e d d e p o s i t s i n some of t h e f a i l u r e states. F u r t h e r m o r e , we examine t h e e f f e c t s o f m i s p r i c e d d e p o s i t g u a r a n t e e s i n r e g i m e s two and t h r e e . The c o s t s o f e q u i t y c a p i t a l and u n i n s u r e d d e p o s i t s a r e e x p l o r e d u n d e r e a c h http://clevelandfed.org/research/workpaper Best available copy regime. Under t h e n o- i n s u r a n c e r e g i m e , o u r r e s u l t s a r e i d e n t i c a l t o C h e n ' s . I f t h e d e p o s i t guarantees a r e c o r r e c t l y p r i c e d , t h e n u n d e r regime two t h e c o s t s o f u n i n s u r e d d e p o s i t s and e q u i t y c a p i t a l a r e i n v a r i a n t t o t h e d e p o s i t mix. T h i s r e s u l t does n o t h o l d up i f t h e i n s u r a n c e i s m i s p r i c e d . Under regime three, t h e c o s t o f uninsured d e p o s i t s i s a f u n c t i o n o f the forbearance p o l i c y o f the deposit guarantor. I n t h i s case, t h e c o s t s o f d e b t and e q u i t y c a p i t a l a r e n o t i n v a r i a n t t o t h e d e p o s i t mix even when t h e e x p l i c i t g u a r a n t e e i s priced correctly. I n s e c t i o n 11, we p r e s e n t t h e s i n g l e - p e r i o d r e s u l t s f o r each r e g i m e . In s e c t i o n 111, we compare t h e v a l u e and c o s t o f u n i n s u r e d d e p o s i t s and e q u i t y a c r o s s t h e t h r e e regimes t o see t h e e f f e c t s o f d e p o s i t i n s u r a n c e m i s p r i c i n g and f o r b e a r a n c e p o l i c y . t h e t h i r d regime. 11. I n s e c t i o n I V , we p r e s e n t t h e m u l t i p e r i o d v e r s i o n for F i n a l l y , i n s e c t i o n V , we p r o v i d e o u r c o n c l u s i o n s . The Cost o f C a p i t a l f o r Banks i n S t a t i c Models To d e t e r m i n e t h e e f f e c t s o f m i s p r i c e d d e p o s i t i n s u r a n c e and FDIC f o r b e a r a n c e p o l i c y on t h e c o s t o f d e b t ( d e p o s i t ) c a p i t a l and e q u i t y c a p i t a l f o r banks, we u t i l i z e t h e s i n g l e - p e r i o d CAPM v a l u a t i o n e q u a t i o n used b y Chen (1978) and d e r i v e d b y Sharpe (19641, L i n t e r (19651, and Mossin ( 1 9 6 6 ) . assumptions u n d e r l y i n g t h i s model a r e : The key ( 1 ) a f i x e d , r i s k - f r e e r a t e of i n t e r e s t , ( 2 ) p e r f e c t l y c o m p e t i t i v e c a p i t a l markets, (3) homogeneous e x p e c t a t i o n s w i t h r e s p e c t t o t h e d i s t r i b u t i o n s and e x p e c t e d y i e l d s on r i s k y a s s e t s , and ( 4 ) r i s k - a v e r s e i n v e s t o r s who seek t o maximize t h e e x p e c t e d u t i l i t y o f terminal wealth. The f o l l o w i n g n o t a t i o n i s used i n t h i s s e c t i o n : http://clevelandfed.org/research/workpaper Best available copy B = Total promised payment when there is no deposit insurance K = Total promised payment when some deposits are insured BI = Total promised payment to insured depositors B, = Total promised payment to uninsured depositors z = Total promised payment to the FDIC P = Deposit insurance premium per dollar of insured deposits YI, I = Value of the end-of-period cash flows to insured (= pBi> depositors Yb I , = Value of the end-of-period cash flows to uninsured depositors Ye = Value of the end-of-period cash flows to stockholders Y,,,, = Value of the end-of-period cash flows to the FDIC Vb I = Value of insured deposits Vb u = Value of uninsured deposits v = Value of bank equity V,, , , = Value of the FDIC claim VF = Value of the bank e (= E(Rbl> = Expected rate of return on insured deposits E(Rbu> = Expected rate of return on uninsured deposits E(Re> = Expected rate of return on bank equity r = Risk-free rate of return F(X> = Cumulative distribution function for X CEQ(X> = Certainty-equivalent of X, equal to E(X> - X COV(X,Rm) X = The market risk premium Rm = Return on the market Bi+B,+z> http://clevelandfed.org/research/workpaper Best available copy We assume t h a t a l l d e p o s i t s a r e d i s c o u n t i n s t r u m e n t s , so t h a t t h e t o t a l promised payment t o each c l a s s o f d e p o s i t o r i n c l u d e s both p r i n c i p a l repayment and i n t e r e s t . Below we p r e s e n t t h e expected end- of- period payments t o deposi t o r s , s t o c k h o l d e r s , and t h e FDIC under t h r e e d i f f e r e n t d e p o s i t - i n s u r a n c e regimes. Under a l l t h r e e regimes, bankruptcy costs and t a x e s a r e assumed t o be z e r o . The first regime r e p l i c a t e s Chen's r e s u l t s f o r t h e t h e c o s t o f d e b t ( d e p o s i t ) c a p i t a l and e q u i t y c a p i t a l when t h e r e i s no d e p o s i t insurance. regime, banks i s s u e b o t h i n s u r e d and uninsured d e p o s i t s . I n t h e second However, t h e FDIC does extend forbearances t o any c l a s s o f c r e d i t o r s o r s t o c k h o l d e r s . We b u i l d i n FDIC forbearance p o l i c y i n t h e t h i r d regime. A. Regime I: No D e p o s i t Insurance Banks i s s u e u n i n s u r e d d e p o s i t s o n l y , and t h e end- of- period cash f l o w s a c c r u i n g t o t h e d e p o s i t o r s , Yb, a r e : The v a l u e o f t h e d e p o s i t s , V,, i s t h e r i s k - a d j u s t e d discounted v a l u e of Y,. The c o s t o f debt ( d e p o s i t ) c a p i t a l , o r t h e r e q u i r e d r a t e o f r e t u r n on t h e d e p o s i t s , i s E(R,) = E(Yb)/V,. http://clevelandfed.org/research/workpaper Best available copy As i n Chen, the c o s t o f debt ( d e p o s i t ) c a p i t a l i s a f u n c t i o n o f t h e r i s k - f r e e i n t e r e s t r a t e , t h e f i r m ' s systematic r i s k (as measured by XCOV(X,R,)>, and t h e p r o b a b i l i t y o f bankruptcy ( F ( B > > . For t h e s t o c k h o l d e r s , t h e expected end- of- period cash f l o w s , Y e , e q u a l s : Ye = I X - B 0 and f r o m t h e s i n g l e - p e r i o d CAPM v a l u a t i o n equation, t h e value o f s t o c k h o l d e r e q u i t y , V,, equals: As i n t h e case o f debt c a p i t a l , t h e c o s t o f e q u i t y c a p i t a l , o r t h e r e q u i r e d r a t e o f r e t u r n on e q u i t y , i s E ( R e ) = Ye/Ve. As i n Chen, t h e c o s t o f e q u i t y c a p i t a l f o r a bank i s a f u n c t i o n of i t s s y s t e m a t i c r i s k , t h e l e v e l o f t o t a l promised payment, t h e probabi 1 it y of b a n k r u p t c y , and t h e r i s k - f r e e i n t e r e s t r a t e . The v a l u e o f t h e f i r m i s t h e sum o f t h e v a l u e o f a l l claims a g a i n s t t h e firm, that is, Vf = Vb + Ve. http://clevelandfed.org/research/workpaper Best available copy B. Regime 11: I n s u r e d and Uninsured Deposits, But No FDIC Forbearances I n t h i s regime, we a l l o w f o r t h e e x p l i c i t insurance o f some of t h e b a n k ' s deposit l i a b i l i t i e s . We assume t h a t t h e FDIC charges t h e bank a premium o f p on each d o l l a r o f i n s u r e d d e p o s i t s and t h a t t h e premium i s c o l l e c t e d a t t h e end o f t h e p e r i o d . The t o t a l l i a b i l i t y c l a i m a g a i n s t t h e bank, K, i s t h e sum o f t h e end- of- period promised payments t o t h e i n s u r e d d e p o s i t o r s , B , ; t h e u n i n s u r e d d e p o s i t o r s , B,; and t h e FDIC, z = pB,. The e f f e c t o f d e p o s i t insurance on t h e value and c o s t o f c a p i t a l depends on whether t h e guarantees a r e u n d e r p r i c e d , f a i r l y p r i c e d , o r o v e r p r i c e d . ' However, i n t h i s paper we g e n e r a l l y assume t h a t t h e FDIC u n d e r p r i c e s i t s guarantees and, t h e r e f o r e , t h a t K < D. The end- of- period cash flows f o r t h e i n s u r e d d e p o s i t s , Y,,,, promised payment t o i n s u r e d d e p o s i t o r s , B , , i n e v e r y s t a t e . value o f the insured deposits i s V b , on t h e i n s u r e d d e p o s i t s i s E(R,,) t o t h e bank i s r + = = r. equals t h e Therefore, t h e ( l + r ) - ' B , and t h e r e q u i r e d r e t u r n The c o s t o f i n s u r e d d e p o s i t c a p i t a l p. For t h e uninsured d e p o s i t o r s , t h e end- of- period cash flows, Y d u , depend on t h e promised payment t o the u n i n s u r e d d e p o s i t o r s and on t h e t o t a l l e v e l o f promised payments. http://clevelandfed.org/research/workpaper Best available copy The value of t h e uninsured d e p o s i t s , V,,,, and t h e c o s t o f t h e r e q u i r e d r a t e of r e t u r n on these d e p o s i t s , E ( R b u > , a r e : V b u = (1+r)-'CBuC1-F(K)I + (B,/K)CEQE(X)l, (1b) and A s i n t h e f i r s t regime, t h e c o s t o f d e b t ( u n i n s u r e d d e p o s i t ) c a p i t a l i s a f u n c t i o n o f systematic r i s k , t o t a l promised payments, t h e p r o b a b i l i t y of bankruptcy, and t h e r i s k - f r e e r a t e o f i n t e r e s t . However, t h e c o s t o f d e b t c a p i t a l i s e x p l i c i t l y a f u n c t i o n o f t h e d e p o s i t mix when t h e F D I C guarantees are mispriced. That i s , u n d e r p r i c e d d e p o s i t guarantees lower t h e b a n k r u p t c y t h r e s h o l d , F(K>, and increase t h e p r o p o r t i o n a l c l a i m o f t h e u n i n s u r e d d e p o s i t o r s r e l a t i v e t o t h e i n s u r e d d e p o s i t o r s and t h e FDIC. The degree t o which t h i s e f f e c t operates i s a f u n c t i o n o f t h e FDIC's p r i c i n g e r r o r p e r d o l l a r o f i n s u r e d d e p o s i t s and o f t h e d e p o s i t m i x . As i n regime one, t h e s t o c k h o l d e r s ' expected end - of - period cash f l o w s a r e e a r n i n g s l e s s t o t a l promised payments i n t h e nonbankruptcy s t a t e s , and z e r o i n t h e bankruptcy s t a t e s . However, t h e t o t a l promised payment and t h e p r o b a b i l i t y o f a bankruptcy s t a t e a r e now a f u n c t i o n o f t h e d e p o s i t m i x and o f t h e p r i c i n g o f t h e d e p o s i t guarantees. ye = 1 X - K if 0 I X > K K > X http://clevelandfed.org/research/workpaper Best available copy The v a l u e of e q u i t y and t h e e x p e c t e d r e t u r n t o t h e s t o c k h o l d e r s a r e : (3b) V, = ( 1 + r>-'(CEQ,(X> - KC1-F(K>I), and As i n t h e case o f u n i n s u r e d d e p o s i t s , t h e c o s t of e q u i t y c a p i t a l i s a f u n c t i o n o f t h e d e p o s i t i n s u r a n c e p r i c i n g and t h e d e p o s i t m i x . I n other words, t h e s u b s i d y t h a t a r i s e s when t h e FDIC m i s p r i c e s i t s d e p o s i t g u a r a n t e e s affects t h e c o s t and v a l u e o f d e b t ( u n i n s u r e d d e p o s i t ) c a p i t a l and e q u i t y capital. The FDIC s u b s i d y can be seen when we a g g r e g a t e t h e c l a i m s of t h e d e b t - h o l d e r s ( i n s u r e d and u n i n s u r e d ) and e q u i t y - h o l d e r s i n t h e f i r m . The t o t a l v a l u e o f t h e bank i s : The e n d - o f- p e r i o d cash f l o w s and t h e v a l u e o f t h e FDIC's p o s i t i o n i n t h e bank ( t h e FDIC's s u b s i d y ) a r e : The v a l u e o f t h e FDIC's c l a i m i s a f u n c t i o n o f t h e p r o b a b i l i t y of b a n k r u p t c y , F(K); t h e l e v e l o f promised payments t o i n s u r e d d e p o s i t o r s , B i ; t h e http://clevelandfed.org/research/workpaper Best available copy systematic r i s k o f t h e bank as r e f l e c t e d i n cEQ?(X>; t h e r i sk- free r a t e , r ; and t h e insurance premium, z. I f we add (5b) and ( 6 b > , t h e r e s u l t i n g e q u a t i o n i s i d e n t i c a l t o e q u a t i o n (5a) i n t h e no- insurance regime. when VFDIC = I n fact, 0, t h e insurance i s c o r r e c t l y p r i c e d and e q u a t i o n (5b) i s i d e n t i c a l t o e q u a t i o n (5a). C. Regime 111: I n s u r e d and Uninsured Deposits and FDIC Forbearances We extend t h e a n a l y s i s t o i n c l u d e t h e c o n d i t i o n a l guarantee o f u n i n s u r e d d e p o s i t s i n bankruptcy s t a t e s when earnings, X, f a l l between S 1 and S r . We assume t h a t S, and S 2 a r e determined by FDIC p o l i c y and a r e known t o market p a r t i c i p a n t s . For s i m p l i c i t y , we model o n l y one s e t o f bounds f o r FDIC b a i l o u t s , b u t t h e a n a l y s i s h o l d s f o r m u l t i p l e and d i s j o i n t b a i l o u t s t a t e s . The p o s i t i o n o f t h e i n s u r e d d e p o s i t o r s i s n o t a f f e c t e d by FDIC b a i l o u t policies. However, Y d u , Vdu and E ( R d u > a r e a l l f u n c t i o n s o f t h e FDIC bai l o u t pol i c y . X > K K>X>S;? if S 2 > X > S1 S1 > X > O The c o s t o f debt ( u n i n s u r e d d e p o s i t ) c a p i t a l i s now a f u n c t i o n o f t h e p r o b a b i l i t y o f an FDIC b a i l o u t and t h e s i z e o f t h e u n i n s u r e d d e p o s i t o r s ' http://clevelandfed.org/research/workpaper Best available copy bailout. The value o f t h e FDIC b a i l o u t i s e n t i r e l y captured by t h e uninsured d e p o s i t o r s and does n o t a f f e c t t h e p o s i t i o n o f t h e s t o c k h o l d e r s . below, Y e , V,, As seen and E ( R e ) a r e i d e n t i c a l t o those i n t h e p r e v i o u s regime. However, as seen i n e q u a t i o n (Sc), t h e value o f t h e f i r m i s a f u n c t i o n of t h e FDIC's b a i l o u t p o l i c y . The l a s t two terms on the r i g h t s i d e o f (Sc) r e f l e c t t h e n e t v a l u e o f t h e FDIC forbearances t o uninsured d e p o s i t o r s . The end- of- period cash f l o w s t o t h e FDIC and t h e v a l u e o f t h e FDIC's c l a i m on t h e bank now i n c l u d e t h e c o s t o f guaranteeing t h e u n i n s u r e d d e p o s i t s i n t h e bai l o u t states. http://clevelandfed.org/research/workpaper Best available copy The e x p e c t e d c o s t o f p r o v i d i n g f o r b e a r a n c e s t o u n i n s u r e d d e p o s i t o r s i n t h e b a i l o u t s t a t e s i s r e f l e c t e d i n t h e l a s t two terms on t h e r i g h t s i d e o f e q u a t i o n ( 6 ~ ) . As b e f o r e , t h e sum o f ( 5 c ) and ( 6 c ) e q u a l s t h e v a l u e o f t h e f i r m i n 5(a). 111. The E f f e c t s o f M i s p r i c e d I n s u r a n c e and Forbearance on t h e C o s t and Value o f Debt and E q u i t y C a p i t a l As seen i n s e c t i o n 11, m i s p r i c e d d e p o s i t i n s u r a n c e and FDIC f o r b e a r a n c e s a f f e c t t h e c o s t o f c a p i t a l and t h e v a l u e o f d e b t and e q u i t y shares i n b a n k s . The presence o f m i s p r i c e d FDIC g u a r a n t e e s and FDIC f o r b e a r a n c e s has an i m p a c t on t h e v a l u e o f u n i n s u r e d d e p o s i t s . mispriced d e p o s i t guarantees. The v a l u e o f e q u i t y i s a l s o a f f e c t e d b y I n t h i s s e c t i o n , we show t h e d i r e c t i o n of change i n t h e v a l u e o f c a p i t a l f r o m FDIC p o l i c i e s . The v a l u e o f t h e u n i n s u r e d d e p o s i t s ( p e r d o l l a r o f p r o m i s e d payment) i s i n c r e a s e d by t h e presence o f m i s p r i c e d d e p o s i t i n s u r a n c e . To d e m o n s t r a t e t h i s , http://clevelandfed.org/research/workpaper Best available copy we d i v i d e Yb and Ybu i n regimes one and two by I3 and B,,, respectively. By s u b t r a c t i n g Y,,/B f r o m YbU/Bu, we can s p l i t t h e u n i n s u r e d d e p o s i t i n regime two i n t o two i n s t r u m e n t s : one t h a t i s i d e n t i c a l t o t h e uninsured d e p o s i t i n regime one, and a second t h a t has t h e f o l l o w i n g p a y o f f s n e x t p e r i o d . I f t h e value o f AgYbu i s p o s i t i v e ( n e g a t i v e ) , then s t o c h a s t i c dominance r e q u i r e s t h a t t h e v a l u e o f an uni n sured d e p o s i t ( w i t h a p a r v a l u e of one d o l l a r ) i n regime two must be g r e a t e r ( l e s s ) t h a n i t s v a l u e i n regime one. I n equation ( 7 ) , AgVbu i s t h e v a l u e o f t h e income stream t h a t accrues t o uninsured d e p o s i t o r s when t h e FDIC u n d e r p r i c e s i t s d e p o s i t guarantees. The f i r s t term on t h e r i g h t s i d e i s p o s i t i v e , t h e second t e r m i s n e g a t i v e , and t h e t h i r d term i s p o s i t i v e when d e p o s i t guarantees a r e u n d e r p r i c e d . y equals 1 l K minus l I B , which i s p o s i t i v e because K < B. From t h e bankruptcy c o n d i t i o n i n regime one, we know t h a t B[F(B) - F ( K ) I > EE(X> > CEQE(X), and t h a t F(B> - F(K) > (l/B)CEQF(X). Therefore, equation ( 7 ) i s p o s i t i v e , and t h e value o f t h e uninsured d e p o s i t s i n c r e a s e s when d e p o s i t insurance i s u n d e r p r i c e d . AgVb,,Bu can be i n t e r p r e t e d as t h e v a l u e of FDIC s u b s i d i e s a c c r u i n g t o u n i n s u r e d d e p o s i t o r s f r o m u n d e r p r i c e d d e p o s i t guarantees. http://clevelandfed.org/research/workpaper Best available copy The i n t r o d u c t i o n o f FDIC forbe arances when S , < X < S r f u r t h e r increases t h e value o f uninsured d e p o s i t s ( p e r do1 l a r o f promised payment). F o l l o w i n g t h e procedure used i n t h e p r e v i o u s case, we separate an u n i n s u r e d d e p o s i t i n regime t h r e e i n t o two i n s t r u m e n t s : one t h a t i s i d e n t i c a l t o an uninsured d e p o s i t i n regime two, and a second t h a t has the f o l l o w i n g end- of- period p a y o f f s : The v a l u e o f AFYbu is: A s u f f i c i e n t c o n d i t i o n f o r e q u a t i o n ( 8 ) t o be p o s i t i v e i s K 2 Sr, which h o l d s by d e f i n i t i o n . which i m p l i e s t h a t Furthermore, FDIC forbearances r e p r e s e n t a c a l l o p t i o n , AFVd, must be n onnegative. Therefore, e q u a t i o n ( 8 ) i s p o s i t i v e , and t h e e x t e n s i o n o f f o r b e a r a n c e t o u n i n s u r e d d e p o s i t o r s i n some b a n k r u p t c y s t a t e s increases t h e v a l u e o f those d e p o s i t s . For t h e e q u i t y - h o l d e r s , s e c t i o n I 1 shows t h a t FDIC forbearances t o u n i n s u r e d d e p o s i t o r s do n o t a f f e c t t h e v a l u e o f t h e i r shares. On t h e o t h e r hand, u n d e r p r i c e d d e p o s i t insurance does a f f e c t t h e v a l u e o f bank e q u i t y . change i n t h e payments t o e q u i t y - h o l d e r s from regime one t o regime two i s : The http://clevelandfed.org/research/workpaper Best available copy As b e f o r e , we have d i v i d e d t h e e q u i t y c l a i m i n regime two i n t o a c l a i m i d e n t i c a l t o the e q u i t y c l a i m i n regime one and an i n s t r u m e n t whose v a l u e i s : We know t h a t E!(X> > K[F(B)-F(K>I; however, i t i s n o t c l e a r t h a t CEQF(X) > K[F(B)-F(K>I because EF(X) > CEQ!(X). w i l l be p o s i t i v e i f (B-K)[1-F(B)I (B-K>[l-F(B>I - xCOV~(X,R,) - lCov;(X,R,) < E!(X) Equation ( 9 ) 2 0, o r i f - KCF(B)-F(K)I. It is l i k e l y t h a t t h i s c o n d i t i o n holds and t h a t e q u a t i o n ( 9 ) i s p o s i t i v e . Furthermore, i f p a y i n g t o o l i t t l e f o r d e p o s i t i n s u r a n c e lowers a b a n k ' s v a l u e , t h e bank c o u l d always remove t h i s r e d i s t r i b u t i v e e f f e c t by v o l u n t a r i l y i n c r e a s i n g i t s payments t o t h e FDIC. Therefore, e q u a t i o n ( 9 ) must be nonnegative. IV. TheMultiperiodModel We a n a l y z e t h e m u l t i p e r i o d v e r s i o n o f t h e t h i r d regime discussed above: t h e bank i s s u e s both i n s u r e d and u n i n s u r e d d e p o s i t s ; t h e r e i s a f i x e d d e p o s i t i n s u r a n c e premium t h a t may o r may n o t be equal t o t h e " f a i r " premium t h a t reduces t o z e r o t h e v a l u e o f t h e FDIC's c l a i m ; and i n s t a t e s S , t h r o u g h http://clevelandfed.org/research/workpaper Best available copy S 2 , t h e r e i s FDIC forbearance t o u ninsured d e p o s i t o r s . I n a multiperiod c o n t e x t , i f t h e bank cannot meet a l l o f i t s o b l i g a t i o n s f r o m i t s cash f l o w , i t i s a b l e t o issue e q u i t y t o meet t h e claims o f t h e d e p o s i t o r s . I f t h e t o t a l of cash f l o w and e q u i t y i s i n s u f f i c i e n t t o meet a l l c l a i m s , t h e t o t a l a v a i l a b l e funds a r e s p l i t p r o p o r t i o n a t e l y among a l l c l a i m a n t s . We make several assumptions i n a d d i t i o n t o those s t a t e d above f o r t h e s i n g l e - p e r i o d model. F i r s t , e q u i t y i s issued t o h e l p meet promised payments as l o n g as shareholder w e a l t h i s p o s i t i v e . Bankruptcy can be avoided even i f t h e t e r m i n a l value o f t h e bank i s n e g a t i v e , as l o n g as t h e t o t a l o f t h e e q u i t y Second, v a l u e and t h e n e t o p e r a t i n g income a t l e a s t meets promised payments. i n d e r i v i n g the expressions f o r market values and t h e i m p l i e d r e q u i r e d r a t e s o f r e t u r n below, we assume t h a t t h e term s t r u c t u r e o f i n t e r e s t r a t e s i s f l a t . T h i r d , we assume t h a t t h e p r i c e o f r i s k i s c o n s t a n t over t i m e . F o u r t h , we deal w i t h a zero- growth bank and assume t h a t t h e d i s t r i b u t i o n o f n e t o p e r a t i n g income i s constant o v e r t i m e . F i n a l l y , we assume t h a t , ex- ante, t h e b a n k r u p t c y p o i n t i n f u t u r e p e r i o d s equals i t s v a l u e i n t h e f i r s t p e r i o d . A1 1 terms a r e d e f i n e d as above, except t h a t we r e p l a c e B, and B u y t h e t o t a l promised payments due i n s u r e d and uninsured d e p o s i t o r s i n t h e s i n g l e - p e r i o d model, by D , and D,, t h e p r i n c i p a l s due on d e p o s i t s . The and , Id", the o t h e r p o r t i o n s o f t h e payments r e c e i v e d by d e p o s i t o r s a r e I d i n t e r e s t payments due i n s u r e d and uninsured d e p o s i t o r s , r e s p e c t i v e l y . The w e a l t h o f insured d e p o s i t o r s i s equal t o t h e i r i n t e r e s t payments r e c e i v e d o v e r t h e p e r i o d p l u s t h e v a l u e o f t h e i r c l a i m s a t t h e end o f t h e p e r i o d ( V d , discounted a t the r i s k - f r e e r a t e o f i n t e r e s t , r. +,), http://clevelandfed.org/research/workpaper Best available copy + Vdi Y d l = I d i Vdi = [ l / ( l + r ) I ( I d i + I + Vdi +I). By repeated s u b s t i t u t i o n , t h i s leads t o : V d i = (l/r)I,, i E ( R d i ) = r. Uninsured d e p o s i t o r s w i l l r e c e i v e o n l y a p o r t i o n o f t h e t o t a l o f e a r n i n g s and funds r a i s e d through e q u i t y i s s u e unless earnings and e q u i t y funds a r e s u f f i c i e n t t o meet t h e t o t a l o f a l l c l a i m s (K = Id,+Idu+Di+Du+z) o r unless t h e FDIC b a i l s o u t t h e bank. The FDIC i s assumed t o b a i l o u t u n i n s u r e d d e p o s i t o r s i f bank e a r n i n g s f a l l between S 1 and S p . The expected, o r r e q u i r e d , r e t u r n on uninsured d e p o s i t s i s d e r i v e d from t h e c o n d i t i o n t h a t Vdu = E(YdU)/[E(R dU)+F(h)l. http://clevelandfed.org/research/workpaper Best available copy The v a l u e o f t h e u n i n s u r e d d e p o s i t s and t h e r a t e o f r e t u r n r e q u i r e d o n them by d e p o s i t o r s depend on t h e expected value of e q u i t y as w e l l as on t h e f a c t o r s emphasized i n t h e s i n g l e - p e r i o d model. The v a l u e o f t h e d e p o s i t s depends on t h e expected v a l u e o f e q u i t y through t h e d e f a u l t premium, F ( h > , and as i t a f f e c t s t h e t o t a l funds a v a i l a b l e t o be s p l i t among c l a i m a n t s . The d e f a u l t premium e n t e r s t h e r i s k - a d j u s t e d r a t e , r + F(h), now used t o d i s c o u n t cash f l o w s . Because i t i s now t h e t o t a l o f n e t o p e r a t i n g income p l u s expect ed funds r a i s e d through e q u i t y i s s u e t h a t w i l l be s p l i t among c l a i m a n t s , t h e v a l u e o f e q u i t y d i r e c t l y e n t e r s t h e expressions f o r t h e value and r e q u i r e d r a t e o f r e t u r n on u n i n s u r e d d e p o s i t s . As usual, t h e e q u i t y - h o l d e r s have a r e s i d u a l c l a i m : The v a l u e and r e q u i r e d r a t e o f r e t u r n on e q u i t y depend on t h e p r o b a b i l i t y of bankruptcy , which i s a f u n c t i o n o f t h e d i s t r i b u t i o n o f n e t o p e r a t i n g income p l u s funds r a i s e d f r o m e q u i t y i s s u e a t t h e end of t h e p e r i o d (Ve , [ I . http://clevelandfed.org/research/workpaper Best available copy We can e x p l o r e t h e determina nts o f t h e f a i r i n s u r a n c e premium by examining t h e determinants o f t h e v a l u e o f t h e FDIC's c l a i m on t h e bank. Once a g a i n , a f a i r premium s e t s t h e v a l u e o f t h e c l a i m t o zero. V . Conclusion Using t h e s i n g l e - p e r i o d and m u l t i p e r i o d c a p i t a l a s s e t p r i c i n g model u t i l i z e d by Chen (1978), we examined the impact o f m i s p r i c e d d e p o s i t guarantees and FDIC forbearances on t h e r a t e s o f r e t u r n and on t h e v a l u e of uninsured d e p o s i t s and e q u i t y c a p i t a l . I n t h e absence o f m i s p r i c e d FDIC d e p o s i t guarantees and forbearances, t h e c o s t of uninsured d e p o s i t s and t h e i r v a l u e does n o t depend on t h e d e p o s i t m i x . However, when we a l l o w f o r t h e i n s u r a n c e http://clevelandfed.org/research/workpaper Best available copy to be mispriced, the required rate of return on the uninsured deposits becomes a function of both the deposit mix, the degree of mispricing, and FDIC forbearance policy. Similar results hold for the return on equity. Furthermore, these policies affect the values of the uninsured deposits and equity. The values of equity and uninsured deposits both increase when the FDIC underprices its insurance. The value of uninsured deposits also increases in response to FDIC forbearances. In addition, we show that the values of FDIC guarantees and forbearances are a function of the insured bank's systematic risk, its probability of bankruptcy, and its deposit mix (when the insurance is mispriced). Finally, we replicate the results of section 11, part C in a multiperiod setting. http://clevelandfed.org/research/workpaper Best available copy Footnotes 1) Fair pricing of deposit insurance guarantees implies that the total value of the firm and the rates of return are the same as in the no-insurance case. The correct value of deposit insurance is a function of the deposit mix and the earnings distribution. However, if correctly priced, the' deposit insurance premium offsets the effects of the shift from "no insured deposits" t o "some insured deposits." http://clevelandfed.org/research/workpaper Best available copy References Benston, George J . , R o b e r t A. E i s e n b e i s , Paul M. H o r v i t z , Edward J . Kane, and George G. Kaufman. P e r s p e c t i v e s on Safe and Sound Banking: P a s t , P r e s e n t , and F u t u r e . Cambridge, MA: MIT Press, 1986. B u s e r , Steven A . , Andrew H. Chen, and Edward J . Kane. " F e d e r a l D e p o s i t I n s u r a n c e , R e g u l a t o r y P o l i c y , and Optimal Bank C a p i t a l , " J o u r n a l o f Finance, v o l . 36 (March 1981>, 51-60. Chen, Andrew H . "Recent Developments i n t h e Cost o f Debt C a p i t a l , " of Finance, v o l . 33 (June 1978), 863-877. Journal , and Chun H. Lam. 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