Forbearance and Pricing Deposit Insurance in a Multiperiod Framework

Abstract

Introduction Since Merton (1977) first suggested an analogy between deposit insurance and a put option to value deposit insurance contracts, there has been a tradition of modeling deposit insurance as a one-period European put option (Merton, 1978; McCulloch, 1985; Marcus and Shaked, 1984; Ronn and Venna, 1986; Pennacchi, 1987; and Duan, Moreau, and Sealy, 1992). In this literature, the put option formula is derived under the assumption that, at the time of audit, either deterministic or stochastic, the put is exercised if the insured institution is found to be insolvent. The deposit insuring agent renegotiates the terms for the next period if the insured institution is solvent. Allen and Saunders (1993) depart from this tradition and argue that deposit insurance is best described as a callable put in the sense that deposit insurance is a perpetual put option with the insuring agent holding the right to terminate the put prematurely.(1) In this article, we propose an alternative way of interpreting deposit insurance in a multiperiod framework. The defaulting banks in our model are assumed to have their assets reset to the level of the outstanding deposits plus accrued interests when an insolvency resolution takes place. According to the deposit insurance contract, the amount required to reset assets is indeed the legal liability of the insuring agent. The historical experience of deposit insurance in the United States supports this set-up. The majority of defaulting depository institutions were resolved through either purchase-and-assumption or the government-assisted merger method. Bartholomew (1991) reported data for 1,730 thrifts that were resolved during the period from 1980 through 1990. Of these 1,730 thrifts, 1,478 institutions (or 85.4 percent) were resolved through this form of reorganization. According to Table 125 of the Federal Deposit Insurance Corporation's (FDIC) 1990 Annual Report, 1,813 banks were closed during the period from 1945 through 1990. Among them, 1,261 banks (or 69.6 percent) were resolved through this form of reorganization. Since the majority of defaulting banks after reorganization continue to operate with deposit insurance, these banks can thus be regarded as receiving an at-the-money put option at the point of insolvency resolution. From this perspective, deposit insurance can be viewed as a stream of one-period Merton-type put options with occasional asset value resets. The banks are assumed to pay out cash dividends whenever the asset value exceeds the level required by a threshold debt-to-asset ratio. This threshold debt-to-asset ratio can be regarded as the maximum level of paid-in capital above which the bank equity holders would consider excessive and start to distribute cash dividends. In other words, this threshold level is dictated by the dividend policy of the bank. The deposits in our model are assumed to bear interests with the interest payments being added back to the deposit base. The deposit base is therefore growing at a rate equal to the interest rate. The premium rate levied by the insuring agent is assumed to be constant over a particular coverage horizon. The fixed premium rate coverage horizon can be one year or any number of years. In reality, charging a fixed premium rate over a period of several years is the standard practice of most deposit insuring agencies. A fairly-priced deposit insurance premium rate can be determined by setting equal the present value of premium proceeds and that of puts until the terminal point of the coverage horizon. The stream of one-period Merton-type put options can be priced by the risk-neutral valuation technique. Although a closed-form solution cannot be derived, the present value can be computed using a Monte Carlo simulation method. Our multiperiod deposit insurance model can be compared with that of Merton (1977) through the use of the fairly-priced deposit insurance premium rate. A deposit insurance premium rate determined by Merton's model cannot be fairly priced according to the multiperiod model. …