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The Barrier Option in Deposit Insurance with Bankruptcy Cost. Dar Yeh Hwang 2006.12.14. 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.

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i introduction 1 12
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
i introduction 2 12
I. INTRODUCTION (2/12)
  • 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 3 12
I. INTRODUCTION (3/12)
  • 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 4 12
I. INTRODUCTION (4/12)
  • 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 5 12
I. INTRODUCTION (5/12)
  • 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 6 12
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
i introduction 7 12
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
i introduction 8 12
I. INTRODUCTION (8/12)
  • bankruptcy cost
    • direct bankruptcy costs
    • indirect bankruptcy costs
i introduction 9 12
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%
i introduction 10 12
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)
i introduction 11 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
i introduction 12 12
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’sbankruptcy costs
      • about 10.2%
      • Bordo , Rocko, and Redish (1996): higher than 40% in Canada between 1880 and 1925
ii some general considerations 1 7
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
ii some general considerations 2 7
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)
ii some general considerations 3 7
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 =γ/(1+γ) and

ii some general considerations 5 7
II. SOME GENERAL CONSIDERATIONS(5/7)
  • The value of the call provision
  • The value of the callable perpetual American put option (p-c)
ii some general considerations 6 7
II. SOME GENERAL CONSIDERATIONS(6/7)
  • Risk based closure rules
    • Reasonable?? Or Unreasonable ??
  • ā = 0.97+0.05σ (Reasonable!)
ii some general considerations 7 7
II. SOME GENERAL CONSIDERATIONS(7/7)
  • ā = 0.97+0.5σ
  • Unreasonable!!
iii the di pricing model with bc 1 14
III. The DI Pricing Model with BC(1/14)
  • Assume
    • The BC is 1-kx on the self-closure point
    • The BC is 1-kā on the regulatory closure point
    • To distinguish p(a,∞;1), we assume pbc(a,∞;1) is noncallable perpetual American put with BC.
iii the di pricing model with bc 2 14
III. The DI Pricing Model with BC (2/14)
  • pbc(a,∞;1) satisfies the following ordinary differential equation:
iii the di pricing model with bc 3 14
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 =γ/[(1+γ) kx] and
iii the di pricing model with bc 4 14
III. The DI Pricing Model with BC(4/14)
  • The value of the noncallable perpetual American put with BC
  • Relationship between pbc and kx
iii the di pricing model with bc 6 14
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:

iii the di pricing model with bc 7 14
III. The DI Pricing Model with BC(7/14)
  • Solution
  • The value of the callable perpetual American put option (pbc-cbc)
iii the di pricing model with bc 13 14
III. The DI Pricing Model with BC(13/14)
  • Risk based closure rules => OUT
  • Risk based bankruptcy cost => IN
    • Reasonable?? Or Unreasonable ??
  • kā = 1-0.05σ (Reasonable!)
iii 5 conclusion
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”.
iv 1 the bo in di without bc 1 9
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

iv 1 the bo in di without bc 2 9
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.
iv 1 the bo in di without bc 3 9
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
iv 1 the bo in di without bc 4 9
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)
iv 1 the bo in di without bc 5 9
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, at)
  • down-and-out put (hereafter, DOP) option

DOP =

where

ā is a regulatory closure point

iv 1 the bo in di without bc 6 9
IV.1. The BO in DI without BC (6/9)
  • Where is self-closure point?
    • ā < aT <1
  • We call the interval as “self-closure region.”
iv 1 the bo in di without bc 7 9
IV.1. The BO in DI without BC (7/9)
  • Rebate

where τis the first passage time for at to hit the ā

iv 2 application of the bo to pricing di with bc 1 11
IV.2. Application of the BO to Pricing DI with BC(1/11)
  • Buser, Chen, and Kane (1981)
    • The FDIC has to take the bank closure risk and bankruptcy cost
  • Kane (1986)
    • the FDIC which consider supervise cost has enforced to forbear the assured bank’s closure point
iv 2 application of the bo to pricing di with bc 2 11
IV.2. Application of the BO to Pricing DI with BC(2/11)
  • Kaufman (1992)
    • recommended the government and superintendent would not hope to see the circumstance that the bank is easy to close for some society stable.
  • Allen and Saunders (1993)
    • Forbearance is granted whenever the FDIC fails to enforce its known regulatory closure point.
iv 2 application of the bo to pricing di with bc 3 11
IV.2. Application of the BO to Pricing DI with BC(3/11)
  • (1-kā) is the asset regulatory point chosen by deposit insurer
  • (1-ka) is the bankruptcy cost on the self-closure point chosen by bank
  • Then the value of deposit insurance premium

where MDOP means the modify down-and-out put option in the rule of the rebate

iv 2 application of the bo to pricing di with bc 6 11
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
v conclusion
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.
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