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1 The Barrier Option in Deposit Insurance with Bankruptcy Cost Dar Yeh Hwang 2006.12.14 Slide 2 2 I. INTRODUCTION (1/12) Deposit insurance (or DI) is an important instrument defense financial crisis stabilize financial system Purposes of DI prevent run on a bank ensure depositor Slide 3 3 Default events of U.S. banks in 1980 Almost exhaust insurance fund in 1988 FDIC has to take the bank closure risk and bankruptcy cost (hereafter, BC) I. INTRODUCTION (2/12) Slide 4 4 Pricing of DI OPM model Merton (1977) : put option Applied and spanned OPM Merton (1978), Marcus & Shaked (1984), Ronn & Verma (1986), Pyle (1986), Pennacchi (1987a, b), Thomson(1987) Allen& Saunders (1993), Duan & Yu (1999) I. INTRODUCTION (3/12) Slide 5 5 Allen and Saunders (1993) forbearance granted by the FDIC FDIC s regulatory closure policy bank s self-closure policy Callable perpetual American put option bankruptcy cost?? I. INTRODUCTION (4/12) Slide 6 6 Dreyfus, Saunders and Allen (1994) Claim that it may be costly to transfer of the bank s assets or to make early liquidation if the banks is closed I. INTRODUCTION (5/12) Slide 7 7 I. INTRODUCTION (6/12) Warner (1977) bankruptcy cost exists Williamson (1988) also pointed out the bankruptcy cost problems of the corporate special assets. The special assets like intangible assets, such as brand, R&D, charter value, advertisement etc. the intangible assets are short of liquidity when the corporate fall into the bankruptcy Slide 8 8 I. INTRODUCTION (7/12) Cummins et. al. (1995) the franchise value or charter value hold by firm s owner die out Gendreau and Prince (1986), Berger, Kashyap and Scalise's (1995), Rajan (1996), Bordo, Rocko, and Redish (1996) Bank s bankruptcy cost exists Slide 9 9 I. INTRODUCTION (8/12) bankruptcy cost direct bankruptcy costs indirect bankruptcy costs Slide 10 10 I. INTRODUCTION (9/12) direct bankruptcy costs Warner (1977) observed 11 railroad firms found the direct bankruptcy cost is the 4% of the market value on the one year before the bankruptcy time Altman (1984) 4.3% Slide 11 11 I. INTRODUCTION (10/12) direct bankruptcy costs Weiss (1990) observed 37 bankruptcy firms of New York from November, 1979 to December, 1986 3.1% Franks and Torous (1994) 4.5% Betker (1997) 3.51% Branch (2002) 3.1 ~4.3 Gendreau and Prince (1986) : 6% (bank) Slide 12 12 I. INTRODUCTION (11/12) indirect bankruptcy costs Altman (1984) 4.5% for retail business 10.5% for industry Andrade and Kaplan (1988) indirect bankruptcy lie in between 10 ~17 including 31 high financial leverage firms Rajan (1996) 4.2% for bank Slide 13 13 I. INTRODUCTION (12/12) bankruptcy cost direct bankruptcy costs about 3% ~4.5% of the firm market value indirect bankruptcy costs about 4.2% ~17% of the firm market value Bank s bankruptcy costs about 10.2% Bordo, Rocko, and Redish (1996): higher than 40% in Canada between 1880 and 1925 Slide 14 14 II. SOME GENERAL CONSIDERATIONS (1/7) We assume the value of bank assets follow a logarithmic diffusion process and assets are normalized by deposit (denote, a ) Asset/deposit ratio line Slide 15 15 II. SOME GENERAL CONSIDERATIONS (2/7) Without bankruptcy cost the cost of insurer is max(0, 1- a ) noncallable perpetual American put:p( a,;1) Slide 16 16 II. SOME GENERAL CONSIDERATIONS (3/7) Solution where x is denoted the maximum asset value for which it is optimal for the bank to prematurely exercise its deposit insurance put. Then x =/(1+) and Slide 17 17 II. SOME GENERAL CONSIDERATIONS (4/7) Slide 18 18 II. SOME GENERAL CONSIDERATIONS (5/7) The value of the call provision The value of the callable perpetual American put option (p-c) Slide 19 19 II. SOME GENERAL CONSIDERATIONS (6/7) Risk based closure rules Reasonable?? Or Unreasonable ?? = 0.97+0.05 (Reasonable!) Slide 20 20 II. SOME GENERAL CONSIDERATIONS (7/7) = 0.97+0.5 Unreasonable!! Slide 21 21 III. The DI Pricing Model with BC (1/14) Assume The BC is 1-k x on the self-closure point The BC is 1-k on the regulatory closure point To distinguish p(a,;1), we assume p bc (a,;1) is noncallable perpetual American put with BC. Slide 22 22 III. The DI Pricing Model with BC (2/14) p bc (a,;1) satisfies the following ordinary differential equation: Slide 23 23 III. The DI Pricing Model with BC (3/14) Solution Similarly, where x is denoted the maximum asset value for which it is optimal for the bank to prematurely exercise its deposit insurance put. Then x =/[(1+) k x ] and Slide 24 24 III. The DI Pricing Model with BC (4/14) The value of the noncallable perpetual American put with BC Relationship between p bc and k x Slide 25 25 III. The DI Pricing Model with BC (5/14) Slide 26 26 III. The DI Pricing Model with BC (6/14) The call provision with BC will satisfies the following ODE: subject to the following boundary conditions: Slide 27 27 III. The DI Pricing Model with BC (7/14) Solution The value of the callable perpetual American put option (p bc -c bc ) Slide 28 28 III. The DI Pricing Model with BC (8/14) Slide 29 29 III. The DI Pricing Model with BC (9/14) Slide 30 30 III. The DI Pricing Model with BC (10/14) Slide 31 31 III. The DI Pricing Model with BC (11/14) Slide 32 32 III. The DI Pricing Model with BC (12/14) Slide 33 33 III. The DI Pricing Model with BC (13/14) Risk based closure rules => OUT Risk based bankruptcy cost => IN Reasonable?? Or Unreasonable ?? k = 1-0.05 (Reasonable!) Slide 34 34 III. The DI Pricing Model with BC (14/14) k = 1-0.25 Reasonable! Slide 35 35 III.5. Conclusion Using the insights of bankruptcy cost, we model the deposit insurance with forbearance as a callable put option. This paper solved the unreasonable results, risk based closure rules, in Allen and Saunders (1993) by using risk based bankruptcy cost . Slide 36 36 IV.1. The BO in DI without BC (1/9) Allen and Saunders (1993) forbearance granted by the FDIC FDIC s regulatory closure policy bank s self-closure policy => Callable perpetual American put option Slide 37 37 IV.1. The BO in DI without BC (2/9) Forbearance? Kane (1986) deposit insurer have enforced to forbear the assured bank s closure point Allen and Saunders (1993) Forbearance is granted whenever the FDIC fails to enforce its known regulatory closure point. Slide 38 38 IV.1. The BO in DI without BC (3/9) self-closure policy? Allen and Saunders (1993) even if the option expires in the money, bank shareholders may choose not to exercise the put since exercise implies voluntary bank closure self-closure point > regulatory closure point self-closure point < regulatory closure point Slide 39 39 IV.1. The BO in DI without BC (4/9) Brockman and Turtle (2003) They take the equity and debt claims as a barrier option. regulatory closure point = lower barrier So pricing the deposit insurance could be viewed as a barrier option (hereafter, BO) Slide 40 40 IV.1. The BO in DI without BC (5/9) We assume the value of bank assets follow a logarithmic diffusion process and assets are normalized by deposit (denote, a t ) down-and-out put (hereafter, DOP) option DOP = where is a regulatory closure point Slide 41 41 IV.1. The BO in DI without BC (6/9) Where is self-closure point? < a T 50 IV.2. Application of the BO to Pricing DI with BC (6/11) self-closure region > regulatory closure point ??? We don t need it! Owing to the above reasons and considering the BC Slide 51 51 IV.2. Application of the BO to Pricing DI with BC (7/11) Close form solution and Slide 52 52 IV.2. Application of the BO to Pricing DI with BC (8/11) Slide 53 53 IV.2. Application of the BO to Pricing DI with BC (9/11) Slide 54 54 IV.2. Application of the BO to Pricing DI with BC (10/11) Slide 55 55 IV.2. Application of the BO to Pricing DI with BC (11/11) Slide 56 56 V. CONCLUSION Our model can improve the unreasonable points in some aspects on barrier option applied on pricing deposit insurance directly. The bankruptcy cost is a key factor for pricing deposit insurance in the real world and should not be ignored.