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Capital standard, forbearance and deposit insurance pricing under GARCH
Journal of Banking & Finance 23 (1999) 1691±1706 www.elsevier.com/locate/econbase
Capital standard, forbearance and deposit insurance pricing under GARCH q Jin-Chuan Duan a
a,1
, Min-Teh Yu
b,*
Department of Finance, Hong Kong University of Science and Technology, Hong Kong, People's Republic of China b Yuan Ze University and National Central University, Chung-Li 320, Taiwan Received 03 March 1998; accepted 10 February 1999
Abstract We propose a multiperiod deposit insurance pricing model that simultaneously incorporates the capital standard and the possibility of forbearance. The model employs the recently developed GARCH option pricing technique in determining the deposit insurance value. Our model oers two distinctive advantages. First, it explicitly considers the implications of the strict enforcement on capital standard as stipulated in FDIC Improvement Act of 1991. Second, the use of the GARCH model allows us to capture many robust features exhibited by ®nancial asset returns. By the GARCH option pricing theory, the value of a contingent claim is a function of the asset risk premium. This unique feature is found to be prominent in determining the bank's deposit insurance value. We also examine the eects of capital forbearance and moral hazard behavior in this multiperiod deposit insurance setting. Ó 1999 Elsevier Science B.V. All rights reserved. JEL classi®cation: G20; G28 Keywords: Deposit insurance; Capital standard; GARCH; Option pricing
q Earlier versions of this paper were presented at the Seminar of European Group of Risk and Insurance Economists, the PACAP 1996 Annual Conference and the Eastern Economic Association 1997 Annual Meeting. * Corresponding author. Tel.: +886-3-425-2961; fax: +886-3-426-7275. E-mail addresses: [email protected]. (J.-C. Duan), [email protected] (M.-T. Yu) 1 Tel.: +852-2358-7671; fax: +852-2358-1749.
0378-4266/99/$ - see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 4 2 6 6 ( 9 9 ) 0 0 0 2 2 - 9
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1. Introduction Since Merton (1977), deposit insurance has typically been modeled in the literature as a put option. The existing analyses were mostly conducted within the Black and Scholes (1973) option pricing framework, even though dierent model speci®cations were employed. 2 This stream of deposit insurance pricing research has two typical shortcomings. First, the capital standard is not explicitly or properly re¯ected in the models. In a single period deposit insurance pricing model, the capital standard has only a very limited meaning. At most, it determines the initial capital position of a bank. Since the bank is assumed to be liquidated at the end of the coverage period, the closing condition is solely based on the solvency condition, not on the capital standard. This omission of a capital standard is critical, particularly considering the mandatory interventions by the regulatory authority as stipulated in the FDIC Improvement Act (FDICIA) of 1991. Second, the Black±Scholes option pricing framework has poor empirical performance. A large body of empirical studies has conclusively demonstrated that ®nancial asset returns exhibit many robust features such as fat-tailed return distributions, volatility clustering and leverage eects (see for example, the review article by Bollerslev et al., 1992). Because these empirical regularities are incompatible with the Black±Scholes model, the use of such a modeling framework for deposit insurance analysis is at best questionable. In empirical analyses of exchange-traded option data, the Black±Scholes model has also been found to be de®cient in explaining the cross-sectional variation of option prices. When the implied volatilities of the options with the same maturity are plotted against their exercise prices, the curve is typically convex (for example, Rubinstein, 1994), which contradicts the constant volatility assumption of the Black±Scholes model. The Black±Scholes model also fails in the maturity dimension. Heynen et al. (1994) and Xu and Taylor (1994) have shown that the Black±Scholes implied volatility exhibits a term structure phenomenon. In order to address the ®rst shortcoming, our deposit insurance model incorporates (1) the capital standard, (2) the possibility of capital forbearance by regulators and (3) the potential moral hazard behavior of insured banks in a multiperiod setting. Our model setting allows us to examine the interplay between the capital standard and the timing of insolvency caused by delayed resolution. In our model, the insuring agent provides deposit insurance and charges a premium rate based on a multiperiod coverage horizon. Although the insured bank is audited periodically, it may not be closed immediately if found
2
Merton (1978) and Pennacchi (1987a) considered a stochastic audit; Ronn and Verma (1986) incorporated capital forbearance; McCulloch (1985), Pennacchi (1987b) and Duan et al. (1995) considered stochastic interest rates.
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insolvent. The capital standard set by the regulatory authority is typically higher than the solvency requirement, as a preventive measure. The capital standard may or may not be strictly followed. Capital forbearance occurs when a bank not only fails to meet the capital standard but also is allowed to continue its operations. In the literature, capital forbearance has long been recognized as an important determinant of deposit insurance liabilities (see for example, Kane, 1987 and Nagarajan and Sealey, 1995). In order to deal with the second shortcoming, an alternative derivative pricing model is called for. Among many alternatives, one approach to modeling the asset price dynamic is to use, say, the jump-diusion model of Merton (1976). Although such a model can also lead to fat-tailed return distributions, it does not adequately describe the other data features such as volatility clustering and the leverage eect. Because of the lack of volatility reversion, the option pricing model based on the jump-diusion process cannot produce the phenomenon of the term structure of implied volatilities, which is well documented for exchange-traded options. Such a consideration is important because the actual deposit insurance levy under a typical deposit insurance system is never based on market value. The plausibility of a modeling approach must therefore be assessed in other settings. Our belief is that a good modeling approach needs to meet two criteria. First, it must be an adequate empirical speci®cation for the asset value process. Second, the derived option pricing model must be consistent with the price data of the exchange-traded options. These considerations motivate us to adopt the empirically motivated GARCH option pricing technique developed by Duan (1995). Our choice of this pricing framework is supported by the empirical evidence reported by Heynen et al. (1994), Amin and Ng (1994), Duan (1996) and Heston and Nandi (1997). In these studies the GARCH option pricing model has been found to signi®cantly outperform its Black±Scholes counterpart. Apart from the empirical performance, the GARCH option pricing model exhibits a unique feature in terms of the role played by the asset risk premium. It is well known that the Black±Scholes option price is not a function of the asset risk premium. In contrast, the GARCH option price is a direct function of the asset risk premium and should thus aect the price of the derivative contract. Our numerical analyses later show that the asset risk premium is indeed critical for determining the deposit insurance value. 2. Multiperiod deposit insurance under GARCH Consider a discrete-time set-up in which time is indexed as 0, 1, 2 and so on to some large integer time T. The continuously compounded bank asset return from time t ÿ 1 to t is denoted by Rt ; that is, Rt ln A t=A t ÿ 1, where A t is the time-t value of the bank assets. The initial face value of total
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interest-bearing bank deposits is denoted by D. Since the deposits are insured, the interest rate applicable is assumed to be the risk-free rate of return r. Audits of the bank are conducted periodically at time ti ; i 1; 2; . . . , an increasing sequence of integer time points. We assume that the bank asset value is subject to reset at the time of audit. In the event of a failure resolution, the insuring agent typically arranges for a reorganization of the failing institution and then continues to provide deposit insurance coverage. This adjustment resets the assets of the defaulting bank to the level required under the capital standard. After the adjustment, the newly reorganized bank continues to operate with deposit insurance. The historical failure resolution experience using either purchase-and-assumption or government-assisted-merger methods in the United States supports this set-up. 3 From this perspective, the deposit insurance contract is automatically renewed to cover a new period, and it can thus be viewed as a stream of single-period put options with occasional asset value resets. The bank asset value is subject to another type of reset. Since the equity holders of pro®table banks may consider withdrawing excessive capital, a ceiling is placed on the bank's asset value. Speci®cally, the asset value adjustment mechanism at the auditing time ti is 8 < qu Derti if A ti P qu Derti ; 1 A ti A ti if qu Derti > A ti P qDerti ; : q Derti otherwise; l Pt where A ti A tiÿ1 tiiÿ1 1 Rt . The parameters ql and qu 1 6 ql < qu set the upper and lower bounds for the asset value. The parameter ql re¯ects the capital standard set by the regulatory authority, whereas the parameter qu , a threshold level of asset-to-debt ratio, determines the extent to which the pro®table bank equity holders are willing to leave the capital with the bank before paying themselves cash dividends. The parameter q 0 < q 6 ql is used to model capital forbearance. Our asset reset mechanism largely follows that of Duan and Yu (1994) except that they did not consider the role of the capital standard. The capital standard, based on the Basle Accord, calls for the combined tier 1 and 2 capital in the amount exceeding 8% of the risk-adjusted asset value. This capital standard can be translated into ql 1:087. When the forbearance parameter q is less than one, the bank, if insolvent, will not be forced to face an immediate intervention from the insuring agent as long as it remains within the capital forbearance range. A bank in ®nancial distress is able to function
3
According to Bartholomew (1991) and the FDIC Annual Reports, 85.4 percent of thrifts failures during the period from 1980 to 1990 and 70 percent of bank failures during the period from 1945 to 1990 were resolved through these forms of reorganization.
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`normally' under such circumstances because the insuring agent guarantees the performance of its deposit liabilities. Failure to mark-to-market immediately the bank's assets and liabilities is one of the interesting features of the regulated deposit-taking industry. An insured institution faces a failure resolution only when its asset value falls below qDerti . Although the parameter q alters the condition for triggering an asset value adjustment, the adjustment will, if taking place, fully restore the asset value to the level dictated by the capital standard. It is clear that when capital forbearance occurs, it amounts to a breach of the capital standard. The scenario that 1 6 q < ql , also a breach of the capital standard, should not be considered capital forbearance because the bank still remains solvent. Under the FDICIA, the deposit insuring agent is required to be vigilant in implementing a tighter capital standard, which implies an earlier closure of any troubled bank even if it is technically solvent. If the capital standard is strictly enforced, then q ql . In the traditional single-period setting, the decision for early closure or capital forbearance is actually irrelevant because depository institutions are assumed to be liquidated at the end of the period anyway. The typical adjustment made to the deposit insurance payo in the single-period setting is somewhat arti®cial and inconsistent with reality. A multiperiod deposit insurance coverage exposes the insuring agent to a stream of put option-like liabilities. The random liability or the put option payo at time ti , denoted by P ti , can be described by 0 if A ti P min q; 1 Derti ; 2 P ti rti De ÿ A ti otherwise: We use min q; 1 to re¯ect the fact that even if q > 1, the cash liability facing the insuring agent in the event of settlement is unaltered. The likelihood of incurring cash payments due to the bank's future insolvency is nevertheless reduced through the asset reset mechanism. At the termination point of this multiperiod coverage, q must be, by de®nition, set to one, regardless of its original value. The last liability can, therefore, be written as P tn MaxDertn ÿ A tn ; 0, a familiar expression for the put option payo. In order to assess the present value of this liability stream, we must characterize the nature of the bank's asset return dynamic. We use a version of the GARCH model, known as the non-linear asymmetric GARCH (NGARCH) model in Engle and Ng (1993), to describe the bank's asset return dynamic: 1 Rt1 r krt1 ÿ r2t1 rt1 et1 ; 2 2 r2t1 b0 b1 r2t b2 r2t et ÿ h ;
3 4
where et1 is, conditional on the time-t information, a standard normal random variable with respect to the physical probability measure governing the actual
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return system. To ensure that conditional volatility stays positive, b0 > 0; b1 P 0 and b2 P 0. If covariance stationarity of the return dynamic is desired, one needs to impose b1 b2 1 h2 < 1. The variable r is the risk-free rate of interest, k the unit risk premium embedded in the bank's asset return, and h a parameter that can be used to re¯ect the potential asymmetric volatility response to return innovations. Notice that if Rt is an i.i.d. sequence of normal Pt random variables, then tiiÿ1 1 Rt will be a normal random variable, and the model is back to the Black±Scholes assumption of lognormality. The NGARCH model becomes the standard linear GARCH model of Bollerslev (1986) when h 0. This simple generalization allows the model to capture asymmetric volatility response, if there is any, to the positive and negative asset return innovations. If the underlying asset is an equity, h is expected to be positive to yield a negative correlation between the return and volatility innovations. Such a negative correlation for equities is known as the leverage eect, referring to the fact that equity is itself a call option subject to the ®rm's risk-sharing arrangement between equity holders and bondholders (see Black, 1976; Christie, 1982). For bank assets, however, there is no a priori reason for a negative correlation between the return and volatility. Our choice of the NGARCH over the standard linear GARCH model is a simple way to obtain more generality without real costs. Alternatively, we may justify this choice by referring to the nature of the bank's asset portfolio. Since the bank's asset portfolio mainly consists of loans subject to credit risks, its return is very likely to exhibit asymmetric response to return innovations just like the equity does. It should be noted that our deposit insurance pricing model is not restricted to the NGARCH speci®cation. Since the GARCH option pricing theory of Duan (1995; to be described below) is applicable to any form of the GARCH model, only a simple substitution is needed if one prefers a dierent GARCH speci®cation. The GARCH option pricing theory developed in Duan (1995) implies that, after locally neutralizing risk, the asset return dynamic for the purpose of pricing contingent claims becomes: 1 Rt1 r ÿ r2t1 rt1 nt1 ; 2
5
r2t1 b0 b1 r2t b2 r2t nt ÿ h ÿ k2 ;
6
where nt1 et1 k is, conditional on the time t information, a standard normal random variable with respect to the risk-neutralized probability measure. The system jointly de®ned by Eq. (5) and Eq. (6) can be used to value the multiperiod deposit insurance coverage. As discussed in Duan (1995), the risk2 ÿ1 neutralized stationary variance equals b0 f1 ÿ b1 ÿ b2 1 h k g , which is always greater than the stationary variance under the physical probability ÿ1 measure, b0 1 ÿ b1 ÿ b2 1 h2 , provided that h and k share the same sign.
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This result has one immediate theoretical implication for long-term options. Since the GARCH price of a long-term option is eectively governed by the risk-neutralized stationary volatility, a higher value relative to the Black± Scholes price is expected. For long-term deposit insurance coverage, the risk premium parameter k is thus expected to play an important role. For shortand intermediate-term deposit insurance coverage, the dierence arising from applying the two pricing frameworks is mainly caused by the two speci®c GARCH parameters ± b1 and b2 . Together, they govern the dynamic evolution of the volatility process. Let dn denote the fairly priced premium rate per period of an n-period (auditing period) deposit insurance coverage. The fairly priced premium rate is a risk-based rate that equates the present value of the whole stream of deposit insurance liabilities with the present value of the total insurance levies at this premium rate. The fairly priced premium rate is, of course, a theoretical entity. The rate-setting practice of the deposit insurance agents around the world can hardly be regarded as setting a fair premium rate. 4 The fairly priced premium rate nevertheless serves as a convenient measure for the intrinsic value of the deposit insurance coverage. The fairly priced premium rate per period in an n-period coverage horizon can be computed as follows: n 1 X eÿrti E0 P ti ; 7 dn nD i1 where E0 denotes expectation taken at time 0 with respect to the dynamic speci®ed in Eq. (5) and (6). Moral hazard behavior in our model can be viewed as changing the magnitude of b0 . An increase in b0 will have an eect similar to a step function in the context of the standard autoregressive (AR) time-series model. The change causes the volatility of the system to rise gradually to a higher stationary level. The speed of adjustment is governed by b1 b2 1 h2 because it, in essence, serves as the AR(1) coecient in the conditional variance process. This point becomes clear if we express the NGARCH conditional variance equation in a dierent way: r2t1 b0 b1 b2 1 h2 r2t b2 r2t et ÿ h2 ÿ 1 h2 :
8
Notice that the last term in the above equation has a zero mean. The typical empirical estimate of b1 b2 1 h2 for a ®nancial time series is very close to one (see for example, Engle and Ng, 1993). This implies that the shock to volatility will take some time to be fully realized. The adjustment for the 4
Although deposit insurance agencies in most countries continue to levy a ®xed premium rate, the FDIC of the U.S. began to assess a risk-adjusted premium rate based on capital adequacy and supervisory reports in January, 1993.
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risk-neutralized system is expected to be slower because the relevant coecient in this case is b1 b2 1 h k2 , which is even closer to one if k and h share the same sign. The speed of adjustment is relevant because deposit insurance amounts to a stream of coverages spanning a large maturity spectrum. We assume that risk-taking behavior (or moral hazard behavior) is governed by the outcomes of the bank's asset value, which is classi®ed into three categories. If the asset value is greater than the level required by the capital standard, the bank functions normally and its portfolio risk characteristics remain unchanged, i.e., b0 ti b0 tiÿ1 . If the bank's asset value breaches the capital standard but is tolerated by the regulatory authority, then the moral hazard behavior occurs; that is, the bank starts to take on more risk in its asset portfolio. One simple way of modeling this eect is to force an increase in b0 . Speci®cally, this action increases the stationary standard deviation of its asset portfolio by 100x%, i.e., b0 ti 1 x2 b0 tiÿ1 . Once the troubled bank breaks the threshold level, we assume that the situation becomes intolerable and the insuring agent steps in to reorganize the bank. As a result of this reorganization, the bank's original risk level is restored, i.e., b0 ti b0 0. This adjustment process can be formally described as follows: 8 if A ti P ql Derti ; < b0 tiÿ1 2 9 b0 ti 1 x b0 tiÿ1 if q Derti > A ti P qDerti ; l : otherwise; b0 0 where b0 is indexed by time to re¯ect its time-varying nature. 5 The analytic framework is now fully speci®ed. We will next carry out a numerical analysis to study the model's implications. 3. Numerical analysis In this section, we study our GARCH-based deposit insurance pricing model by numerically assessing the deposit insurance values under dierent scenarios. We use a more ecient Monte Carlo simulation method recently developed by Duan and Simonato (1998) to compute the deposit insurance value. This simulation method, known as the empirical martingale simulation, imposes the theoretical martingale property of asset prices on the simulated sample paths so that every simulated sample maintains the required martingale 5 In our model, risk-taking is only permitted at the time of an audit. The insured bank may choose to increase the leverage and portfolio risk prior to the time of the audit. In such a case, the value of deposit insurance coverage is expected to increase because such an action aects earlier the bank's asset risk characteristics. Technically, this type of action can be incorporated into our model by setting up a predetermined adjustment mechanism similar to Eq. (9) but the adjustment is allowed to take place at every time point instead of only at auditing times.
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property. Ten thousand sample paths are used in every Monte Carlo calculation, and common random numbers are used to facilitate the comparison across dierent cases. 6 The basic unit of time is assumed to be one business day and auditing takes place once a year. 7 We use the following set of parameter values: b0 7:254 10ÿ7 ; b1 0:931771; b2 0:037339; h 0:688025 and k 0:027647. This set of parameter values is estimated from the daily S&P 500 index series from January 2, 1990 to June 28, 1996, totaling 1643 observations. These parameter values yield the stationarity volatility of 0:1176152 (annualized based on 252 days) under the physical probability measure. The calculation of the stationary volatility (daily) is based on the formula: r2 b0 1 ÿ b1 ÿ b2 1 h2 ÿ1 . The parameters qu and ql are set to 1:15 and 1:087, respectively. Recall that the parameter value for ql is consistent with the 8% capital standard established under the Basle Accord. To study the impact of a looser capital standard, we also consider the case that ql 1:05. Three values of the asset-to-debt ratio 1:08; 1:10 and 1:12 are examined. These assetto-debt ratios fall inside the range established by qu and ql . The risk-free rate of interest is assumed to be 6% per annum. As discussed in the preceding section, the stationary volatility under the physical probability measure is smaller than the one under the locally riskneutralized probability measure, provided that the unit risk premium, k, of the bank's assets shares the same sign as the asymmetry parameter, h. The value of k should be an important factor, particularly for longer-term coverage, in separating the GARCH deposit insurance pricing model from its Black± Scholes counterpart. We consider, in addition to the estimated value, a different value for k. We use k 0:1 to examine the impact of a higher asset risk premium. Note that using a dierent value for k does not change r2 , which is the stationary variance under the physical probability measure. In other words, banks with dierent expected returns but the same risk level will get the same deposit insurance value under the Black±Scholes model assumption but not in the GARCH setting. The fairly priced premium rates corresponding to dierent coverage horizons are presented in Table 1. The values in this table are based on the assumption that the capital standard is strictly enforced by the regulatory
6 We use Monte Carlo simulation to compute the deposit insurance value because the closedform solution is not available. Although Monte Carlo simulation is usually a less ecient computational algorithm, extensive computing time can still be avoided if the model is needed for repeated business usage. Hanke (1997) has shown how to approximate the GARCH option pricing model by ®rst training a neural network. In a similar vein, we can train a neural network to approximate the GARCH deposit insurance pricing model. 7 The assumption that auditing is performed once a year is typical in the deposit insurance literature (see for example, Ronn and Verma, 1986; Duan et al., 1995).
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Table 1 Comparison of the fairly priced deposit insurance premium rates (in basis points) under dierent model assumptions when the capital standard is strictly enforced, i.e., q ql a ql 1:087 BS
ql 1:05 GARCH
BS
k 0:0276
k 0:1
164.69 141.00 134.87 129.90 127.44
194.62 173.62 168.85 164.41 162.56
249.63 248.23 248.87 248.17 248.55
A=D 1:11 dT 1 dT 3 dT 5 dT 10 dT 20
126.22 125.78 125.70 125.31 125.15
158.98 159.38 160.27 160.12 160.42
A=D 1:13 dT 1 dT 3 dT 5 dT 10 dT 20
95.82 113.33 118.17 121.54 123.26
129.47 147.24 152.94 156.46 158.58
A=D 1:09 dT 1 dT 3 dT 5 dT 10 dT 20
GARCH k 0:0276
k 0:1
164.69 171.19 172.91 173.89 174.06
194.62 199.67 201.58 202.17 202.51
249.63 275.93 283.40 287.94 290.66
211.07 232.94 239.67 243.57 246.25
126.22 153.24 161.89 168.37 170.30
158.99 182.91 191.31 197.02 199.94
211.07 258.33 272.68 282.58 287.98
178.70 219.84 231.77 239.62 244.27
95.82 138.14 152.59 163.71 168.96
129.47 168.55 182.43 192.57 197.71
178.70 243.15 263.38 277.92 285.66
a
This table is based on the following parameter values: qu 1:15; b0 7:254 10ÿ7 ; b1 0:931771; b2 0:037339 and h 0:688025. BS denotes the premium rate using the Black±Scholes pricing framework. The two scenarios under the GARCH speci®cation correspond to two dierent values for k.
authority. The values under the Black±Scholes assumption are computed using the stationary variance r2 . Under the GARCH setting, we report two sets of values corresponding to two dierent values for k. Since the capital standard is strictly observed, there exists no capital forbearance or any moral hazard behavior. The coverage horizons are 1, 3, 5, 10 and 20 years, respectively. As expected, the fairly priced premium rate increases with the leverage ratio. An increase in the coverage horizon causes the premium to rise or fall depending on the initial leverage position. This result is true for both the Black± Scholes and GARCH pricing approaches. When the initial leverage is high (A=D is low), an increase in the coverage horizon reduces the fairly priced premium rates. The reverse is true when the leverage is low. A tighter capital standard is more likely to force the insured bank to inject more equity capital. A longer run deposit insurance coverage has the eect of lowering the fairly priced premium rate. If the capital standard is low relative to the current assetto-debt ratio, the fairly priced premium tends to increase with the coverage horizon. This is re¯ected in the results under ql 1:05. The premium rate
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under the GARCH speci®cation is always higher than its Black±Scholes counterpart. A higher k simply yields a higher fairly priced premium rate. This result conforms to a theoretical prediction of our model discussed earlier. The local risk-neutralization transformation increases the stationary volatility of the underlying asset with respect to the locally-neutralized probability measure if the unit risk premium, k, shares a same sign with the asymmetry parameter h. The GARCH deposit insurance pricing model therefore adds an important, and previously unexplored, dimension to the task of deposit insurance pricing. If the capital standard is not strictly enforced, the insured bank eectively faces a looser capital requirement. Failure to enforce a higher capital standard is not entirely the same as setting a lower capital standard, because capital forbearance is likely to encourage the risk-taking behavior on the part of an insured bank under ®nancial distress. In Tables 2, we consider the eects of capital forbearance with the parameter q equal to 0:97. The risk-taking intensity parameter x is assumed to be 0:2. The results in Table 2 are hardly surprising. The value of deposit insurance increases to re¯ect a potential higher Table 2 Comparison of the fairly priced deposit insurance premium rates (in basis points) under dierent model assumptions when the capital forbearance is present, i.e., q < 1 a ql 1:087 BS
a
ql 1:05 GARCH
BS
k 0:0276
k 0:1
164.69 198.36 211.06 228.21 239.46
194.62 217.81 233.50 252.29 265.80
249.63 291.39 312.46 335.74 351.47
A=D 1:11 dT 1 dT 3 dT 5 dT 10 dT 20
126.22 179.41 199.73 222.57 236.69
158.99 199.72 222.11 246.49 262.85
A=D 1:13 dT 1 dT 3 dT 5 dT 10 dT 20
95.82 163.05 189.57 217.48 234.09
129.47 184.55 212.08 241.14 260.21
A=D 1:09 dT 1 dT 3 dT 5 dT 10 dT 20
GARCH k 0:0276
k 0:1
164.69 203.62 218.05 233.52 243.30
194.62 223.41 238.74 254.67 265.70
249.63 298.78 319.69 339.67 353.25
211.07 272.13 300.08 328.98 348.01
126.22 182.75 205.09 226.90 240.01
158.99 204.47 226.29 248.36 262.59
211.07 278.78 306.91 332.82 349.82
178.70 255.51 289.64 323.51 345.31
95.82 165.06 193.65 220.81 237.05
129.47 188.33 215.54 242.62 259.56
178.70 261.76 295.96 327.25 346.98
This table is based on the following parameter values: qu 1:15; q 0:97; x 0:2; b0 7:254 10ÿ7 ; b1 0:931771; b2 0:037339 and h 0:688025. BS denotes the premium rate using the Black±Scholes pricing framework. The two scenarios under the GARCH speci®cation correspond to two dierent values for k.
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asset risk when the risk-taking behavior is tolerated by the regulatory authority. The relationship between the values based on the Black±Scholes and GARCH approaches remains the same as in Table 1. Since the deposit insurance values corresponding to T 1 are never aected by capital forbearance or moral hazard, they should be the same as those in Table 1. We conduct more numerical analyses to examine how our results depend on the parameter values. The results from these analyses are summarized in Table 3. Speci®cally, we use a higher and a lower value than each of the parameter values in the baseline parameter set to compute the deposit insurance premium rates. The baseline parameter values are given in Table 3. Since we have already computed premium rates for three dierent asset-to-debt ratios in Tables 1 and 2, we will use only the asset-to-debt ratio of 1:09 for our sensitivity analysis. The focus is thus on the changes caused by other parameters. The baseline estimates of the premium rate are taken from column six of the top panel in Table 2. We group our model parameters into three categories: GARCH, capital structure and forbearance. The GARCH parameters: b0 ; b1 ; b2 ; h and k are all positively related to the stationary volatility under the risk-neutralized probability measure, which is a point discussed earlier in Section 2. A higher (lower) value of any GARCH parameter will obviously increase (decrease) the stationary volatility and hence the premium rates. A more informative comparison can be obtained if we vary the parameter value for b1 ; b2 and h while controlling the stationary variance under the physical probability measure, i.e., b0 1 ÿ b1 ÿ b2 1 h2 ÿ1 . In other words, we attempt to ®x the overall risk level of the physical system while the nature of the dynamic evolution is altered. Speci®cally, we make a compensatory adjustment to b0 so that b0 1 ÿ b1 ÿ b2 1 h2 ÿ1 remains ®xed whenever the value of b1 ; b2 or h is altered. We need not make the compensatory adjustment when k is altered because this parameter aects only the risk level of the risk-neutralized system but not the physical system. The corresponding premium rates for these analyses are summarized in the top panel of Table 3. For parameter b0 , it is clear that its increase (decrease) should cause the fairly priced premium rate to increase (decrease) because of a rise (decline) in the overall level of the asset risk. In terms of parameter k, the results are expected as well. An increase (decrease) in its value causes the gap between the riskneutralized stationary volatility and the physical stationary volatility to widen (narrow). Since the put option value is governed by the risk-neutralized stationary volatility, an increase (decrease) in the fairly priced premium rate naturally follows. For the remaining three parameters: b1 ; b2 and h, our comparisons are made by ®xing the level of the physical stationary volatility. The change in their values therefore alters only the dynamic evolution of the conditional volatility but not its level. An increase (decrease) in b1 ; b2 or h, for example, causes a slower (faster) speed of variance reversion to its stationary level, which was accounted for earlier in Eq. (8). As the results in Table 3
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Table 3 Sensitivity analysis of parameter values on deposit insurance premium rates (in basis points)
a
GARCH parameters Baseline
b0 3:627
b0 10:88
b1 0:90
b1 0:94
b2 0:03
b2 0:04
dT 1 dT 3 dT 5 dT 10 dT 20
194.62 223.41 238.74 254.67 265.70
90.73 97.48 99.32 99.82 100.00
284.75 289.01 290.98 291.51 292.04
177.59 181.98 183.46 184.07 184.04
207.77 222.02 228.20 232.11 235.16
181.91 186.21 187.78 188.37 188.42
201.56 209.00 211.61 212.57 213.36
Baseline
h 0:34
h 0:85
k 0:0
k 0:1
dT 1 dT 3 dT 5 dT 10 dT 20
194.62 223.41 238.74 254.67 265.70
174.71 179.57 181.25 181.98 182.03
215.53 236.52 246.75 254.92 261.03
178.85 180.26 181.32 181.31 181.30
249.63 275.93 283.40 287.94 290.66
Capital structure parameters Baseline qu 1:1
qu 1:2
ql 1:02
ql 1:087
dT 1 dT 3 dT 5 dT 10 dT 20
194.62 189.39 185.92 181.38 178.83
194.62 223.06 233.10 240.23 243.52
194.62 173.62 168.85 164.41 162.56
x 0:3
q 0:95
q 1:0
194.62 229.57 249.55 271.03 285.82
194.62 225.70 244.70 264.56 279.59
194.62 214.85 224.38 233.83 240.06
194.62 223.41 238.74 254.67 265.70
194.62 222.06 230.26 235.53 238.74
Forbearance parameters Baseline x 0:1 dT 1 dT 3 dT 5 dT 10 dT 20
194.62 223.41 238.74 254.67 265.70
194.62 217.36 228.27 238.02 244.96
a Both higher and lower values relative to the baseline parameter value are used to compute the premium rates. Note that b0 is stated in terms of 10ÿ7 . Baseline values are computed using the following parameter values: A=D 1:09; b0 7:254; b1 0:931771; b2 0:037339; r 0:06; h 0:688025; k 0:027647; qu 1:15; ql 1:05; q 0:97; x 0:2. Note also that when b1 , b2 and h are varied, we make a compensatory adjustment to b0 so that the stationary variance under the physical probability measure, i.e., b0 1 ÿ b1 ÿ b2 1 h2 ÿ1 , is ®xed.
indicate, the eect of increasing (decreasing) the value for any one of the three parameters is to increase (decrease) the fairly priced premium rate. The capital structure parameter, ql , sets the lower bound of the asset-to-debt ratio. If the capital standard is strictly enforced, a higher (lower) value will decrease (increase) the cost of deposit insurance and hence the fairly priced premium rates. The other capital structure parameter, qu , sets the upper bound of the asset-to-debt ratio. A higher (lower) upper bound implies that more capital is retained within the bank, and it thus decreases (increases) the cost of
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deposit insurance. The corresponding premium rates can be found in the middle panel of Table 3. Finally with regard to the forbearance parameters, either a higher (lower) value of moral hazard intensity, x, or a greater (smaller) degree of forbearance, q, gives rise to a higher (lower) premium rate. The corresponding premium rate estimates for alternative forbearance parameter values are reported in the bottom panel of Table 3. 4. Conclusion Our model explicitly considers the eects of the capital standard, the possibility of forbearance and the potential moral hazard behavior in a multiperiod framework. Given that the FDICIA of 1991 requires the regulatory authority to be vigilant in enforcing a tighter capital standard, our model provides a platform for conducting many policy analyses. Our deposit insurance pricing model also takes advantage of the recently developed GARCH option pricing technique. This pricing approach is attractive for at least two reasons. First, the premise of the model is based on a powerful econometric speci®cation that has now been proved to be very useful in modeling ®nancial time series. Second, the GARCH option pricing model can explain the systematic biases exhibited by the Black±Scholes option pricing model when applied to exchange-traded options. These two factors motivate us to apply the GARCH option pricing technique to deposit insurance pricing in a multiperiod framework. Theoretically, the GARCH option price is a function of risk premium embedded in the underlying assets, which implies that the option price must be a function of the expected return on the underlying assets. This contrasts with the standard conclusion of the Black±Scholes option pricing model. This feature is found to be signi®cant in determining deposit insurance values. This previously unexplored result is important because it suggests that both the asset portfolio's risk level and expected return play critical roles in determining the value of a deposit insurance guarantee. The use of the GARCH framework adds many interesting dimensions to the problem of deposit insurance pricing. It also provides a possibility for more realistic empirical analyses of the current deposit insurance pricing practices. A common belief in the literature is that deposit insurance is subsidized. Appropriate assessment of the extent of the subsidy requires an adequate model. The earlier empirical ®ndings were largely derived from the deposit insurance models that are based on the Black±Scholes framework. Since the Black± Scholes model is known to perform poorly for exchange- traded options, the deposit insurance pricing model based on the same premise is, by a logical inference, unlikely to be adequate. Our proposed deposit model thus oers a potentially rewarding tool for the future analysis of this important public policy issue.
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In the past, researchers faced empirical diculties in implementing the deposit insurance model that is based on the Black±Scholes framework. The diculty arises from the fact that the bank's asset value is not directly observable and hence its volatility cannot be easily estimated. The serious attempt to address the problem was ®rst proposed by Ronn and Verma (1986) and later improved by Duan (1994). Their methods rely on inferring the implied asset value time series from the observed data series of equity prices. The same diculty applies to our deposit insurance pricing model under the GARCH framework. It is thus expected that the same approach should work for our deposit insurance pricing model. Such an empirical analysis will be a fruitful undertaking in future research. Acknowledgements Duan acknowledges the support received as Senior Wei Lun Fellow at the Hong Kong University of Science and Technology. Yu acknowledges the support from the NSC of Taiwan. Part of the paper was completed during Yu's visit to the Drexel University. References Amin, K., Ng, V., 1994. A comparison of predictable volatility models using option data, Working paper. Lehman Brothers, New York. Bartholomew, P., 1991. The cost of forbearance during the thrifts crisis, US Congressional Budget Oces Sta Memorandum, Washington, DC. Black, F., 1976. Studies of stock price volatility changes, Proceedings of 1976 meetings of the Business and Economic Statistics Section, American Statistical Association, pp. 177±181. Black, F., Scholes, M., 1973. The pricing of options and corporate liabilities. Journal of Political Economy 81, 637±659. Bollerslev, T., 1986. Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307±327. Bollerslev, T., Chou, R., Kroner, K., 1992. ARCH modeling in ®nance: A review of the theory and empirical evidence. Journal of Econometrics 52, 5±59. Christie, A., 1982. The stochastic behavior of common stock variances: Value, leverage and interest rate eects. Journal of Financial Economics 10, 407±432. Duan, J.-C., 1994. Maximum likelihood estimation using price data of the derivative contract. Mathematical Finance 4, 155±167. Duan, J.-C., 1995. The GARCH option pricing model. Mathematical Finance 5, 13±32. Duan, J.-C., 1996. Cracking the smile. RISK 9, 55±59. Duan, J.-C., Moreau, A., Sealey, C.W., 1995. Deposit insurance and bank interest rate risk: Pricing and regulatory implications. Journal of Banking and Finance 19, 1091±1108. Duan, J.-C., Simonato, J.-G., 1998. Empirical martingale simulation for asset prices. Management Science 44, 1218±1233. Duan, J.-C., Yu, M.-T., 1994. Forbearance and pricing deposit insurance in a multiperiod framework. Journal of Risk and Insurance 61, 575±591.
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Engle, R., Ng, V., 1993. Measuring and testing the impact of news on volatility. Journal of Finance 48, 1749±1778. Hanke, M., 1997. Neural network approximation of option pricing formulas for analytically intractable option pricing models. Journal of Computational Intelligence in Finance 5, 20±27. Heston, S., Nandi, S., 1997. A closed-form GARCH option pricing model, Unpublished manuscript, Washington University. Heynen, R., Kemna, A., Vorst, T., 1994. Analysis of the term structure of implied volatilities. Journal of Financial and Quantitative Analysis 29, 31±56. Kane, E., 1987. Dangers of capital forbearance: The case of the FSLIC and ``Zombie'' S and Ls. Contemporary Policy Issues 5, 77±83. McCulloch, J.H., 1985. Interest-risk sensitive deposit insurance premia. Journal of Banking and Finance 9, 137±156. Merton, R., 1976. Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics 3, 125±144. Merton, R., 1977. An analytic derivation of the cost of deposit insurance and loan guarantee. Journal of Banking and Finance 1, 3±11. Merton, R., 1978. On the cost of deposit insurance when there are surveillance costs. Journal of Business 51, 439±451. Nagarajan, S., Sealey, C.W., 1995. Forbearance, prompt closure, and incentive compatible bank regulation. Journal of Banking and Finance 19, 1109±1130. Pennacchi, G., 1987a. Alternative forms of deposit insurance: Pricing and bank incentive issues. Journal of Banking and Finance 11, 291±312. Pennacchi, G., 1987b. A reexamination of the over or under pricing of deposit insurance. Journal of Money Credit and Banking 19, 340±360. Ronn, E., Verma, A., 1986. Pricing risk-adjusted deposit insurance: An option-based model. Journal of Finance 41, 871±895. Rubinstein, M., 1994. Implied binomial trees. Journal of Finance 49, 771±818. Xu, X., Taylor, S., 1994. The term structure of volatility implied by foreign exchange options. Journal of Financial and Quantitative Analysis 29, 57±74.
Capital standard, forbearance and deposit insurance pricing under GARCH q Jin-Chuan Duan a
a,1
, Min-Teh Yu
b,*
Department of Finance, Hong Kong University of Science and Technology, Hong Kong, People's Republic of China b Yuan Ze University and National Central University, Chung-Li 320, Taiwan Received 03 March 1998; accepted 10 February 1999
Abstract We propose a multiperiod deposit insurance pricing model that simultaneously incorporates the capital standard and the possibility of forbearance. The model employs the recently developed GARCH option pricing technique in determining the deposit insurance value. Our model oers two distinctive advantages. First, it explicitly considers the implications of the strict enforcement on capital standard as stipulated in FDIC Improvement Act of 1991. Second, the use of the GARCH model allows us to capture many robust features exhibited by ®nancial asset returns. By the GARCH option pricing theory, the value of a contingent claim is a function of the asset risk premium. This unique feature is found to be prominent in determining the bank's deposit insurance value. We also examine the eects of capital forbearance and moral hazard behavior in this multiperiod deposit insurance setting. Ó 1999 Elsevier Science B.V. All rights reserved. JEL classi®cation: G20; G28 Keywords: Deposit insurance; Capital standard; GARCH; Option pricing
q Earlier versions of this paper were presented at the Seminar of European Group of Risk and Insurance Economists, the PACAP 1996 Annual Conference and the Eastern Economic Association 1997 Annual Meeting. * Corresponding author. Tel.: +886-3-425-2961; fax: +886-3-426-7275. E-mail addresses: [email protected]. (J.-C. Duan), [email protected] (M.-T. Yu) 1 Tel.: +852-2358-7671; fax: +852-2358-1749.
0378-4266/99/$ - see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 4 2 6 6 ( 9 9 ) 0 0 0 2 2 - 9
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1. Introduction Since Merton (1977), deposit insurance has typically been modeled in the literature as a put option. The existing analyses were mostly conducted within the Black and Scholes (1973) option pricing framework, even though dierent model speci®cations were employed. 2 This stream of deposit insurance pricing research has two typical shortcomings. First, the capital standard is not explicitly or properly re¯ected in the models. In a single period deposit insurance pricing model, the capital standard has only a very limited meaning. At most, it determines the initial capital position of a bank. Since the bank is assumed to be liquidated at the end of the coverage period, the closing condition is solely based on the solvency condition, not on the capital standard. This omission of a capital standard is critical, particularly considering the mandatory interventions by the regulatory authority as stipulated in the FDIC Improvement Act (FDICIA) of 1991. Second, the Black±Scholes option pricing framework has poor empirical performance. A large body of empirical studies has conclusively demonstrated that ®nancial asset returns exhibit many robust features such as fat-tailed return distributions, volatility clustering and leverage eects (see for example, the review article by Bollerslev et al., 1992). Because these empirical regularities are incompatible with the Black±Scholes model, the use of such a modeling framework for deposit insurance analysis is at best questionable. In empirical analyses of exchange-traded option data, the Black±Scholes model has also been found to be de®cient in explaining the cross-sectional variation of option prices. When the implied volatilities of the options with the same maturity are plotted against their exercise prices, the curve is typically convex (for example, Rubinstein, 1994), which contradicts the constant volatility assumption of the Black±Scholes model. The Black±Scholes model also fails in the maturity dimension. Heynen et al. (1994) and Xu and Taylor (1994) have shown that the Black±Scholes implied volatility exhibits a term structure phenomenon. In order to address the ®rst shortcoming, our deposit insurance model incorporates (1) the capital standard, (2) the possibility of capital forbearance by regulators and (3) the potential moral hazard behavior of insured banks in a multiperiod setting. Our model setting allows us to examine the interplay between the capital standard and the timing of insolvency caused by delayed resolution. In our model, the insuring agent provides deposit insurance and charges a premium rate based on a multiperiod coverage horizon. Although the insured bank is audited periodically, it may not be closed immediately if found
2
Merton (1978) and Pennacchi (1987a) considered a stochastic audit; Ronn and Verma (1986) incorporated capital forbearance; McCulloch (1985), Pennacchi (1987b) and Duan et al. (1995) considered stochastic interest rates.
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insolvent. The capital standard set by the regulatory authority is typically higher than the solvency requirement, as a preventive measure. The capital standard may or may not be strictly followed. Capital forbearance occurs when a bank not only fails to meet the capital standard but also is allowed to continue its operations. In the literature, capital forbearance has long been recognized as an important determinant of deposit insurance liabilities (see for example, Kane, 1987 and Nagarajan and Sealey, 1995). In order to deal with the second shortcoming, an alternative derivative pricing model is called for. Among many alternatives, one approach to modeling the asset price dynamic is to use, say, the jump-diusion model of Merton (1976). Although such a model can also lead to fat-tailed return distributions, it does not adequately describe the other data features such as volatility clustering and the leverage eect. Because of the lack of volatility reversion, the option pricing model based on the jump-diusion process cannot produce the phenomenon of the term structure of implied volatilities, which is well documented for exchange-traded options. Such a consideration is important because the actual deposit insurance levy under a typical deposit insurance system is never based on market value. The plausibility of a modeling approach must therefore be assessed in other settings. Our belief is that a good modeling approach needs to meet two criteria. First, it must be an adequate empirical speci®cation for the asset value process. Second, the derived option pricing model must be consistent with the price data of the exchange-traded options. These considerations motivate us to adopt the empirically motivated GARCH option pricing technique developed by Duan (1995). Our choice of this pricing framework is supported by the empirical evidence reported by Heynen et al. (1994), Amin and Ng (1994), Duan (1996) and Heston and Nandi (1997). In these studies the GARCH option pricing model has been found to signi®cantly outperform its Black±Scholes counterpart. Apart from the empirical performance, the GARCH option pricing model exhibits a unique feature in terms of the role played by the asset risk premium. It is well known that the Black±Scholes option price is not a function of the asset risk premium. In contrast, the GARCH option price is a direct function of the asset risk premium and should thus aect the price of the derivative contract. Our numerical analyses later show that the asset risk premium is indeed critical for determining the deposit insurance value. 2. Multiperiod deposit insurance under GARCH Consider a discrete-time set-up in which time is indexed as 0, 1, 2 and so on to some large integer time T. The continuously compounded bank asset return from time t ÿ 1 to t is denoted by Rt ; that is, Rt ln A t=A t ÿ 1, where A t is the time-t value of the bank assets. The initial face value of total
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interest-bearing bank deposits is denoted by D. Since the deposits are insured, the interest rate applicable is assumed to be the risk-free rate of return r. Audits of the bank are conducted periodically at time ti ; i 1; 2; . . . , an increasing sequence of integer time points. We assume that the bank asset value is subject to reset at the time of audit. In the event of a failure resolution, the insuring agent typically arranges for a reorganization of the failing institution and then continues to provide deposit insurance coverage. This adjustment resets the assets of the defaulting bank to the level required under the capital standard. After the adjustment, the newly reorganized bank continues to operate with deposit insurance. The historical failure resolution experience using either purchase-and-assumption or government-assisted-merger methods in the United States supports this set-up. 3 From this perspective, the deposit insurance contract is automatically renewed to cover a new period, and it can thus be viewed as a stream of single-period put options with occasional asset value resets. The bank asset value is subject to another type of reset. Since the equity holders of pro®table banks may consider withdrawing excessive capital, a ceiling is placed on the bank's asset value. Speci®cally, the asset value adjustment mechanism at the auditing time ti is 8 < qu Derti if A ti P qu Derti ; 1 A ti A ti if qu Derti > A ti P qDerti ; : q Derti otherwise; l Pt where A ti A tiÿ1 tiiÿ1 1 Rt . The parameters ql and qu 1 6 ql < qu set the upper and lower bounds for the asset value. The parameter ql re¯ects the capital standard set by the regulatory authority, whereas the parameter qu , a threshold level of asset-to-debt ratio, determines the extent to which the pro®table bank equity holders are willing to leave the capital with the bank before paying themselves cash dividends. The parameter q 0 < q 6 ql is used to model capital forbearance. Our asset reset mechanism largely follows that of Duan and Yu (1994) except that they did not consider the role of the capital standard. The capital standard, based on the Basle Accord, calls for the combined tier 1 and 2 capital in the amount exceeding 8% of the risk-adjusted asset value. This capital standard can be translated into ql 1:087. When the forbearance parameter q is less than one, the bank, if insolvent, will not be forced to face an immediate intervention from the insuring agent as long as it remains within the capital forbearance range. A bank in ®nancial distress is able to function
3
According to Bartholomew (1991) and the FDIC Annual Reports, 85.4 percent of thrifts failures during the period from 1980 to 1990 and 70 percent of bank failures during the period from 1945 to 1990 were resolved through these forms of reorganization.
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`normally' under such circumstances because the insuring agent guarantees the performance of its deposit liabilities. Failure to mark-to-market immediately the bank's assets and liabilities is one of the interesting features of the regulated deposit-taking industry. An insured institution faces a failure resolution only when its asset value falls below qDerti . Although the parameter q alters the condition for triggering an asset value adjustment, the adjustment will, if taking place, fully restore the asset value to the level dictated by the capital standard. It is clear that when capital forbearance occurs, it amounts to a breach of the capital standard. The scenario that 1 6 q < ql , also a breach of the capital standard, should not be considered capital forbearance because the bank still remains solvent. Under the FDICIA, the deposit insuring agent is required to be vigilant in implementing a tighter capital standard, which implies an earlier closure of any troubled bank even if it is technically solvent. If the capital standard is strictly enforced, then q ql . In the traditional single-period setting, the decision for early closure or capital forbearance is actually irrelevant because depository institutions are assumed to be liquidated at the end of the period anyway. The typical adjustment made to the deposit insurance payo in the single-period setting is somewhat arti®cial and inconsistent with reality. A multiperiod deposit insurance coverage exposes the insuring agent to a stream of put option-like liabilities. The random liability or the put option payo at time ti , denoted by P ti , can be described by 0 if A ti P min q; 1 Derti ; 2 P ti rti De ÿ A ti otherwise: We use min q; 1 to re¯ect the fact that even if q > 1, the cash liability facing the insuring agent in the event of settlement is unaltered. The likelihood of incurring cash payments due to the bank's future insolvency is nevertheless reduced through the asset reset mechanism. At the termination point of this multiperiod coverage, q must be, by de®nition, set to one, regardless of its original value. The last liability can, therefore, be written as P tn MaxDertn ÿ A tn ; 0, a familiar expression for the put option payo. In order to assess the present value of this liability stream, we must characterize the nature of the bank's asset return dynamic. We use a version of the GARCH model, known as the non-linear asymmetric GARCH (NGARCH) model in Engle and Ng (1993), to describe the bank's asset return dynamic: 1 Rt1 r krt1 ÿ r2t1 rt1 et1 ; 2 2 r2t1 b0 b1 r2t b2 r2t et ÿ h ;
3 4
where et1 is, conditional on the time-t information, a standard normal random variable with respect to the physical probability measure governing the actual
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return system. To ensure that conditional volatility stays positive, b0 > 0; b1 P 0 and b2 P 0. If covariance stationarity of the return dynamic is desired, one needs to impose b1 b2 1 h2 < 1. The variable r is the risk-free rate of interest, k the unit risk premium embedded in the bank's asset return, and h a parameter that can be used to re¯ect the potential asymmetric volatility response to return innovations. Notice that if Rt is an i.i.d. sequence of normal Pt random variables, then tiiÿ1 1 Rt will be a normal random variable, and the model is back to the Black±Scholes assumption of lognormality. The NGARCH model becomes the standard linear GARCH model of Bollerslev (1986) when h 0. This simple generalization allows the model to capture asymmetric volatility response, if there is any, to the positive and negative asset return innovations. If the underlying asset is an equity, h is expected to be positive to yield a negative correlation between the return and volatility innovations. Such a negative correlation for equities is known as the leverage eect, referring to the fact that equity is itself a call option subject to the ®rm's risk-sharing arrangement between equity holders and bondholders (see Black, 1976; Christie, 1982). For bank assets, however, there is no a priori reason for a negative correlation between the return and volatility. Our choice of the NGARCH over the standard linear GARCH model is a simple way to obtain more generality without real costs. Alternatively, we may justify this choice by referring to the nature of the bank's asset portfolio. Since the bank's asset portfolio mainly consists of loans subject to credit risks, its return is very likely to exhibit asymmetric response to return innovations just like the equity does. It should be noted that our deposit insurance pricing model is not restricted to the NGARCH speci®cation. Since the GARCH option pricing theory of Duan (1995; to be described below) is applicable to any form of the GARCH model, only a simple substitution is needed if one prefers a dierent GARCH speci®cation. The GARCH option pricing theory developed in Duan (1995) implies that, after locally neutralizing risk, the asset return dynamic for the purpose of pricing contingent claims becomes: 1 Rt1 r ÿ r2t1 rt1 nt1 ; 2
5
r2t1 b0 b1 r2t b2 r2t nt ÿ h ÿ k2 ;
6
where nt1 et1 k is, conditional on the time t information, a standard normal random variable with respect to the risk-neutralized probability measure. The system jointly de®ned by Eq. (5) and Eq. (6) can be used to value the multiperiod deposit insurance coverage. As discussed in Duan (1995), the risk2 ÿ1 neutralized stationary variance equals b0 f1 ÿ b1 ÿ b2 1 h k g , which is always greater than the stationary variance under the physical probability ÿ1 measure, b0 1 ÿ b1 ÿ b2 1 h2 , provided that h and k share the same sign.
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This result has one immediate theoretical implication for long-term options. Since the GARCH price of a long-term option is eectively governed by the risk-neutralized stationary volatility, a higher value relative to the Black± Scholes price is expected. For long-term deposit insurance coverage, the risk premium parameter k is thus expected to play an important role. For shortand intermediate-term deposit insurance coverage, the dierence arising from applying the two pricing frameworks is mainly caused by the two speci®c GARCH parameters ± b1 and b2 . Together, they govern the dynamic evolution of the volatility process. Let dn denote the fairly priced premium rate per period of an n-period (auditing period) deposit insurance coverage. The fairly priced premium rate is a risk-based rate that equates the present value of the whole stream of deposit insurance liabilities with the present value of the total insurance levies at this premium rate. The fairly priced premium rate is, of course, a theoretical entity. The rate-setting practice of the deposit insurance agents around the world can hardly be regarded as setting a fair premium rate. 4 The fairly priced premium rate nevertheless serves as a convenient measure for the intrinsic value of the deposit insurance coverage. The fairly priced premium rate per period in an n-period coverage horizon can be computed as follows: n 1 X eÿrti E0 P ti ; 7 dn nD i1 where E0 denotes expectation taken at time 0 with respect to the dynamic speci®ed in Eq. (5) and (6). Moral hazard behavior in our model can be viewed as changing the magnitude of b0 . An increase in b0 will have an eect similar to a step function in the context of the standard autoregressive (AR) time-series model. The change causes the volatility of the system to rise gradually to a higher stationary level. The speed of adjustment is governed by b1 b2 1 h2 because it, in essence, serves as the AR(1) coecient in the conditional variance process. This point becomes clear if we express the NGARCH conditional variance equation in a dierent way: r2t1 b0 b1 b2 1 h2 r2t b2 r2t et ÿ h2 ÿ 1 h2 :
8
Notice that the last term in the above equation has a zero mean. The typical empirical estimate of b1 b2 1 h2 for a ®nancial time series is very close to one (see for example, Engle and Ng, 1993). This implies that the shock to volatility will take some time to be fully realized. The adjustment for the 4
Although deposit insurance agencies in most countries continue to levy a ®xed premium rate, the FDIC of the U.S. began to assess a risk-adjusted premium rate based on capital adequacy and supervisory reports in January, 1993.
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risk-neutralized system is expected to be slower because the relevant coecient in this case is b1 b2 1 h k2 , which is even closer to one if k and h share the same sign. The speed of adjustment is relevant because deposit insurance amounts to a stream of coverages spanning a large maturity spectrum. We assume that risk-taking behavior (or moral hazard behavior) is governed by the outcomes of the bank's asset value, which is classi®ed into three categories. If the asset value is greater than the level required by the capital standard, the bank functions normally and its portfolio risk characteristics remain unchanged, i.e., b0 ti b0 tiÿ1 . If the bank's asset value breaches the capital standard but is tolerated by the regulatory authority, then the moral hazard behavior occurs; that is, the bank starts to take on more risk in its asset portfolio. One simple way of modeling this eect is to force an increase in b0 . Speci®cally, this action increases the stationary standard deviation of its asset portfolio by 100x%, i.e., b0 ti 1 x2 b0 tiÿ1 . Once the troubled bank breaks the threshold level, we assume that the situation becomes intolerable and the insuring agent steps in to reorganize the bank. As a result of this reorganization, the bank's original risk level is restored, i.e., b0 ti b0 0. This adjustment process can be formally described as follows: 8 if A ti P ql Derti ; < b0 tiÿ1 2 9 b0 ti 1 x b0 tiÿ1 if q Derti > A ti P qDerti ; l : otherwise; b0 0 where b0 is indexed by time to re¯ect its time-varying nature. 5 The analytic framework is now fully speci®ed. We will next carry out a numerical analysis to study the model's implications. 3. Numerical analysis In this section, we study our GARCH-based deposit insurance pricing model by numerically assessing the deposit insurance values under dierent scenarios. We use a more ecient Monte Carlo simulation method recently developed by Duan and Simonato (1998) to compute the deposit insurance value. This simulation method, known as the empirical martingale simulation, imposes the theoretical martingale property of asset prices on the simulated sample paths so that every simulated sample maintains the required martingale 5 In our model, risk-taking is only permitted at the time of an audit. The insured bank may choose to increase the leverage and portfolio risk prior to the time of the audit. In such a case, the value of deposit insurance coverage is expected to increase because such an action aects earlier the bank's asset risk characteristics. Technically, this type of action can be incorporated into our model by setting up a predetermined adjustment mechanism similar to Eq. (9) but the adjustment is allowed to take place at every time point instead of only at auditing times.
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property. Ten thousand sample paths are used in every Monte Carlo calculation, and common random numbers are used to facilitate the comparison across dierent cases. 6 The basic unit of time is assumed to be one business day and auditing takes place once a year. 7 We use the following set of parameter values: b0 7:254 10ÿ7 ; b1 0:931771; b2 0:037339; h 0:688025 and k 0:027647. This set of parameter values is estimated from the daily S&P 500 index series from January 2, 1990 to June 28, 1996, totaling 1643 observations. These parameter values yield the stationarity volatility of 0:1176152 (annualized based on 252 days) under the physical probability measure. The calculation of the stationary volatility (daily) is based on the formula: r2 b0 1 ÿ b1 ÿ b2 1 h2 ÿ1 . The parameters qu and ql are set to 1:15 and 1:087, respectively. Recall that the parameter value for ql is consistent with the 8% capital standard established under the Basle Accord. To study the impact of a looser capital standard, we also consider the case that ql 1:05. Three values of the asset-to-debt ratio 1:08; 1:10 and 1:12 are examined. These assetto-debt ratios fall inside the range established by qu and ql . The risk-free rate of interest is assumed to be 6% per annum. As discussed in the preceding section, the stationary volatility under the physical probability measure is smaller than the one under the locally riskneutralized probability measure, provided that the unit risk premium, k, of the bank's assets shares the same sign as the asymmetry parameter, h. The value of k should be an important factor, particularly for longer-term coverage, in separating the GARCH deposit insurance pricing model from its Black± Scholes counterpart. We consider, in addition to the estimated value, a different value for k. We use k 0:1 to examine the impact of a higher asset risk premium. Note that using a dierent value for k does not change r2 , which is the stationary variance under the physical probability measure. In other words, banks with dierent expected returns but the same risk level will get the same deposit insurance value under the Black±Scholes model assumption but not in the GARCH setting. The fairly priced premium rates corresponding to dierent coverage horizons are presented in Table 1. The values in this table are based on the assumption that the capital standard is strictly enforced by the regulatory
6 We use Monte Carlo simulation to compute the deposit insurance value because the closedform solution is not available. Although Monte Carlo simulation is usually a less ecient computational algorithm, extensive computing time can still be avoided if the model is needed for repeated business usage. Hanke (1997) has shown how to approximate the GARCH option pricing model by ®rst training a neural network. In a similar vein, we can train a neural network to approximate the GARCH deposit insurance pricing model. 7 The assumption that auditing is performed once a year is typical in the deposit insurance literature (see for example, Ronn and Verma, 1986; Duan et al., 1995).
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Table 1 Comparison of the fairly priced deposit insurance premium rates (in basis points) under dierent model assumptions when the capital standard is strictly enforced, i.e., q ql a ql 1:087 BS
ql 1:05 GARCH
BS
k 0:0276
k 0:1
164.69 141.00 134.87 129.90 127.44
194.62 173.62 168.85 164.41 162.56
249.63 248.23 248.87 248.17 248.55
A=D 1:11 dT 1 dT 3 dT 5 dT 10 dT 20
126.22 125.78 125.70 125.31 125.15
158.98 159.38 160.27 160.12 160.42
A=D 1:13 dT 1 dT 3 dT 5 dT 10 dT 20
95.82 113.33 118.17 121.54 123.26
129.47 147.24 152.94 156.46 158.58
A=D 1:09 dT 1 dT 3 dT 5 dT 10 dT 20
GARCH k 0:0276
k 0:1
164.69 171.19 172.91 173.89 174.06
194.62 199.67 201.58 202.17 202.51
249.63 275.93 283.40 287.94 290.66
211.07 232.94 239.67 243.57 246.25
126.22 153.24 161.89 168.37 170.30
158.99 182.91 191.31 197.02 199.94
211.07 258.33 272.68 282.58 287.98
178.70 219.84 231.77 239.62 244.27
95.82 138.14 152.59 163.71 168.96
129.47 168.55 182.43 192.57 197.71
178.70 243.15 263.38 277.92 285.66
a
This table is based on the following parameter values: qu 1:15; b0 7:254 10ÿ7 ; b1 0:931771; b2 0:037339 and h 0:688025. BS denotes the premium rate using the Black±Scholes pricing framework. The two scenarios under the GARCH speci®cation correspond to two dierent values for k.
authority. The values under the Black±Scholes assumption are computed using the stationary variance r2 . Under the GARCH setting, we report two sets of values corresponding to two dierent values for k. Since the capital standard is strictly observed, there exists no capital forbearance or any moral hazard behavior. The coverage horizons are 1, 3, 5, 10 and 20 years, respectively. As expected, the fairly priced premium rate increases with the leverage ratio. An increase in the coverage horizon causes the premium to rise or fall depending on the initial leverage position. This result is true for both the Black± Scholes and GARCH pricing approaches. When the initial leverage is high (A=D is low), an increase in the coverage horizon reduces the fairly priced premium rates. The reverse is true when the leverage is low. A tighter capital standard is more likely to force the insured bank to inject more equity capital. A longer run deposit insurance coverage has the eect of lowering the fairly priced premium rate. If the capital standard is low relative to the current assetto-debt ratio, the fairly priced premium tends to increase with the coverage horizon. This is re¯ected in the results under ql 1:05. The premium rate
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under the GARCH speci®cation is always higher than its Black±Scholes counterpart. A higher k simply yields a higher fairly priced premium rate. This result conforms to a theoretical prediction of our model discussed earlier. The local risk-neutralization transformation increases the stationary volatility of the underlying asset with respect to the locally-neutralized probability measure if the unit risk premium, k, shares a same sign with the asymmetry parameter h. The GARCH deposit insurance pricing model therefore adds an important, and previously unexplored, dimension to the task of deposit insurance pricing. If the capital standard is not strictly enforced, the insured bank eectively faces a looser capital requirement. Failure to enforce a higher capital standard is not entirely the same as setting a lower capital standard, because capital forbearance is likely to encourage the risk-taking behavior on the part of an insured bank under ®nancial distress. In Tables 2, we consider the eects of capital forbearance with the parameter q equal to 0:97. The risk-taking intensity parameter x is assumed to be 0:2. The results in Table 2 are hardly surprising. The value of deposit insurance increases to re¯ect a potential higher Table 2 Comparison of the fairly priced deposit insurance premium rates (in basis points) under dierent model assumptions when the capital forbearance is present, i.e., q < 1 a ql 1:087 BS
a
ql 1:05 GARCH
BS
k 0:0276
k 0:1
164.69 198.36 211.06 228.21 239.46
194.62 217.81 233.50 252.29 265.80
249.63 291.39 312.46 335.74 351.47
A=D 1:11 dT 1 dT 3 dT 5 dT 10 dT 20
126.22 179.41 199.73 222.57 236.69
158.99 199.72 222.11 246.49 262.85
A=D 1:13 dT 1 dT 3 dT 5 dT 10 dT 20
95.82 163.05 189.57 217.48 234.09
129.47 184.55 212.08 241.14 260.21
A=D 1:09 dT 1 dT 3 dT 5 dT 10 dT 20
GARCH k 0:0276
k 0:1
164.69 203.62 218.05 233.52 243.30
194.62 223.41 238.74 254.67 265.70
249.63 298.78 319.69 339.67 353.25
211.07 272.13 300.08 328.98 348.01
126.22 182.75 205.09 226.90 240.01
158.99 204.47 226.29 248.36 262.59
211.07 278.78 306.91 332.82 349.82
178.70 255.51 289.64 323.51 345.31
95.82 165.06 193.65 220.81 237.05
129.47 188.33 215.54 242.62 259.56
178.70 261.76 295.96 327.25 346.98
This table is based on the following parameter values: qu 1:15; q 0:97; x 0:2; b0 7:254 10ÿ7 ; b1 0:931771; b2 0:037339 and h 0:688025. BS denotes the premium rate using the Black±Scholes pricing framework. The two scenarios under the GARCH speci®cation correspond to two dierent values for k.
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asset risk when the risk-taking behavior is tolerated by the regulatory authority. The relationship between the values based on the Black±Scholes and GARCH approaches remains the same as in Table 1. Since the deposit insurance values corresponding to T 1 are never aected by capital forbearance or moral hazard, they should be the same as those in Table 1. We conduct more numerical analyses to examine how our results depend on the parameter values. The results from these analyses are summarized in Table 3. Speci®cally, we use a higher and a lower value than each of the parameter values in the baseline parameter set to compute the deposit insurance premium rates. The baseline parameter values are given in Table 3. Since we have already computed premium rates for three dierent asset-to-debt ratios in Tables 1 and 2, we will use only the asset-to-debt ratio of 1:09 for our sensitivity analysis. The focus is thus on the changes caused by other parameters. The baseline estimates of the premium rate are taken from column six of the top panel in Table 2. We group our model parameters into three categories: GARCH, capital structure and forbearance. The GARCH parameters: b0 ; b1 ; b2 ; h and k are all positively related to the stationary volatility under the risk-neutralized probability measure, which is a point discussed earlier in Section 2. A higher (lower) value of any GARCH parameter will obviously increase (decrease) the stationary volatility and hence the premium rates. A more informative comparison can be obtained if we vary the parameter value for b1 ; b2 and h while controlling the stationary variance under the physical probability measure, i.e., b0 1 ÿ b1 ÿ b2 1 h2 ÿ1 . In other words, we attempt to ®x the overall risk level of the physical system while the nature of the dynamic evolution is altered. Speci®cally, we make a compensatory adjustment to b0 so that b0 1 ÿ b1 ÿ b2 1 h2 ÿ1 remains ®xed whenever the value of b1 ; b2 or h is altered. We need not make the compensatory adjustment when k is altered because this parameter aects only the risk level of the risk-neutralized system but not the physical system. The corresponding premium rates for these analyses are summarized in the top panel of Table 3. For parameter b0 , it is clear that its increase (decrease) should cause the fairly priced premium rate to increase (decrease) because of a rise (decline) in the overall level of the asset risk. In terms of parameter k, the results are expected as well. An increase (decrease) in its value causes the gap between the riskneutralized stationary volatility and the physical stationary volatility to widen (narrow). Since the put option value is governed by the risk-neutralized stationary volatility, an increase (decrease) in the fairly priced premium rate naturally follows. For the remaining three parameters: b1 ; b2 and h, our comparisons are made by ®xing the level of the physical stationary volatility. The change in their values therefore alters only the dynamic evolution of the conditional volatility but not its level. An increase (decrease) in b1 ; b2 or h, for example, causes a slower (faster) speed of variance reversion to its stationary level, which was accounted for earlier in Eq. (8). As the results in Table 3
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Table 3 Sensitivity analysis of parameter values on deposit insurance premium rates (in basis points)
a
GARCH parameters Baseline
b0 3:627
b0 10:88
b1 0:90
b1 0:94
b2 0:03
b2 0:04
dT 1 dT 3 dT 5 dT 10 dT 20
194.62 223.41 238.74 254.67 265.70
90.73 97.48 99.32 99.82 100.00
284.75 289.01 290.98 291.51 292.04
177.59 181.98 183.46 184.07 184.04
207.77 222.02 228.20 232.11 235.16
181.91 186.21 187.78 188.37 188.42
201.56 209.00 211.61 212.57 213.36
Baseline
h 0:34
h 0:85
k 0:0
k 0:1
dT 1 dT 3 dT 5 dT 10 dT 20
194.62 223.41 238.74 254.67 265.70
174.71 179.57 181.25 181.98 182.03
215.53 236.52 246.75 254.92 261.03
178.85 180.26 181.32 181.31 181.30
249.63 275.93 283.40 287.94 290.66
Capital structure parameters Baseline qu 1:1
qu 1:2
ql 1:02
ql 1:087
dT 1 dT 3 dT 5 dT 10 dT 20
194.62 189.39 185.92 181.38 178.83
194.62 223.06 233.10 240.23 243.52
194.62 173.62 168.85 164.41 162.56
x 0:3
q 0:95
q 1:0
194.62 229.57 249.55 271.03 285.82
194.62 225.70 244.70 264.56 279.59
194.62 214.85 224.38 233.83 240.06
194.62 223.41 238.74 254.67 265.70
194.62 222.06 230.26 235.53 238.74
Forbearance parameters Baseline x 0:1 dT 1 dT 3 dT 5 dT 10 dT 20
194.62 223.41 238.74 254.67 265.70
194.62 217.36 228.27 238.02 244.96
a Both higher and lower values relative to the baseline parameter value are used to compute the premium rates. Note that b0 is stated in terms of 10ÿ7 . Baseline values are computed using the following parameter values: A=D 1:09; b0 7:254; b1 0:931771; b2 0:037339; r 0:06; h 0:688025; k 0:027647; qu 1:15; ql 1:05; q 0:97; x 0:2. Note also that when b1 , b2 and h are varied, we make a compensatory adjustment to b0 so that the stationary variance under the physical probability measure, i.e., b0 1 ÿ b1 ÿ b2 1 h2 ÿ1 , is ®xed.
indicate, the eect of increasing (decreasing) the value for any one of the three parameters is to increase (decrease) the fairly priced premium rate. The capital structure parameter, ql , sets the lower bound of the asset-to-debt ratio. If the capital standard is strictly enforced, a higher (lower) value will decrease (increase) the cost of deposit insurance and hence the fairly priced premium rates. The other capital structure parameter, qu , sets the upper bound of the asset-to-debt ratio. A higher (lower) upper bound implies that more capital is retained within the bank, and it thus decreases (increases) the cost of
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deposit insurance. The corresponding premium rates can be found in the middle panel of Table 3. Finally with regard to the forbearance parameters, either a higher (lower) value of moral hazard intensity, x, or a greater (smaller) degree of forbearance, q, gives rise to a higher (lower) premium rate. The corresponding premium rate estimates for alternative forbearance parameter values are reported in the bottom panel of Table 3. 4. Conclusion Our model explicitly considers the eects of the capital standard, the possibility of forbearance and the potential moral hazard behavior in a multiperiod framework. Given that the FDICIA of 1991 requires the regulatory authority to be vigilant in enforcing a tighter capital standard, our model provides a platform for conducting many policy analyses. Our deposit insurance pricing model also takes advantage of the recently developed GARCH option pricing technique. This pricing approach is attractive for at least two reasons. First, the premise of the model is based on a powerful econometric speci®cation that has now been proved to be very useful in modeling ®nancial time series. Second, the GARCH option pricing model can explain the systematic biases exhibited by the Black±Scholes option pricing model when applied to exchange-traded options. These two factors motivate us to apply the GARCH option pricing technique to deposit insurance pricing in a multiperiod framework. Theoretically, the GARCH option price is a function of risk premium embedded in the underlying assets, which implies that the option price must be a function of the expected return on the underlying assets. This contrasts with the standard conclusion of the Black±Scholes option pricing model. This feature is found to be signi®cant in determining deposit insurance values. This previously unexplored result is important because it suggests that both the asset portfolio's risk level and expected return play critical roles in determining the value of a deposit insurance guarantee. The use of the GARCH framework adds many interesting dimensions to the problem of deposit insurance pricing. It also provides a possibility for more realistic empirical analyses of the current deposit insurance pricing practices. A common belief in the literature is that deposit insurance is subsidized. Appropriate assessment of the extent of the subsidy requires an adequate model. The earlier empirical ®ndings were largely derived from the deposit insurance models that are based on the Black±Scholes framework. Since the Black± Scholes model is known to perform poorly for exchange- traded options, the deposit insurance pricing model based on the same premise is, by a logical inference, unlikely to be adequate. Our proposed deposit model thus oers a potentially rewarding tool for the future analysis of this important public policy issue.
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In the past, researchers faced empirical diculties in implementing the deposit insurance model that is based on the Black±Scholes framework. The diculty arises from the fact that the bank's asset value is not directly observable and hence its volatility cannot be easily estimated. The serious attempt to address the problem was ®rst proposed by Ronn and Verma (1986) and later improved by Duan (1994). Their methods rely on inferring the implied asset value time series from the observed data series of equity prices. The same diculty applies to our deposit insurance pricing model under the GARCH framework. It is thus expected that the same approach should work for our deposit insurance pricing model. Such an empirical analysis will be a fruitful undertaking in future research. Acknowledgements Duan acknowledges the support received as Senior Wei Lun Fellow at the Hong Kong University of Science and Technology. Yu acknowledges the support from the NSC of Taiwan. Part of the paper was completed during Yu's visit to the Drexel University. References Amin, K., Ng, V., 1994. A comparison of predictable volatility models using option data, Working paper. Lehman Brothers, New York. Bartholomew, P., 1991. The cost of forbearance during the thrifts crisis, US Congressional Budget Oces Sta Memorandum, Washington, DC. Black, F., 1976. Studies of stock price volatility changes, Proceedings of 1976 meetings of the Business and Economic Statistics Section, American Statistical Association, pp. 177±181. Black, F., Scholes, M., 1973. The pricing of options and corporate liabilities. Journal of Political Economy 81, 637±659. Bollerslev, T., 1986. Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307±327. Bollerslev, T., Chou, R., Kroner, K., 1992. ARCH modeling in ®nance: A review of the theory and empirical evidence. Journal of Econometrics 52, 5±59. Christie, A., 1982. The stochastic behavior of common stock variances: Value, leverage and interest rate eects. Journal of Financial Economics 10, 407±432. Duan, J.-C., 1994. Maximum likelihood estimation using price data of the derivative contract. Mathematical Finance 4, 155±167. Duan, J.-C., 1995. The GARCH option pricing model. Mathematical Finance 5, 13±32. Duan, J.-C., 1996. Cracking the smile. RISK 9, 55±59. Duan, J.-C., Moreau, A., Sealey, C.W., 1995. Deposit insurance and bank interest rate risk: Pricing and regulatory implications. Journal of Banking and Finance 19, 1091±1108. Duan, J.-C., Simonato, J.-G., 1998. Empirical martingale simulation for asset prices. Management Science 44, 1218±1233. Duan, J.-C., Yu, M.-T., 1994. Forbearance and pricing deposit insurance in a multiperiod framework. Journal of Risk and Insurance 61, 575±591.
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