Asian Review of Financial Research Vol.23 No.1 pp.27-54
A New Approach for Pricing Deposit Insurance
Key Words : Financial Systematic Risk,Deposit Insurance Premia,General Equilibrium Model,Asset Beta Coefficient,Logit Analysis
A financial crisis is often triggered by negative external factors in the credit market. In case of the recent global financial crisis, the trigger was the sudden collapse of Lehman Brothers. Hence, it can be a reasonable notion to reflect the cost of externality on a fair deposit insurance premium in order to absorb consequential market distortion. This is akin to correcting the market distortion by taxing air pollution. Pedersen and Roubini (2008) is the first paper to address this issue by proposing the idea as a viable way to restore financial stability. In this paper, we reframe Merton's (1974) credit risk model based on a general equilibrium context and price the systematic risk for deposit insurance which is considered to be the cost of the negative externality. For that purpose, we make use of the general equilibrium models such as Cox, Ingersoll and Ross (1985), Ahn and Thompson (1988) and Bates (1991). The model assumes that the return dynamics on the firm value process follows a geometric Brownian motion. This kind of stochastic dynamics is made possible by an economy's random technological change over time. Park, Ahn and Kim (2009) has derived the risk-neutral dynamics for the firm value process using the Ahn and Thompson's model (1988). The risk-neutral dynamics allows us to price a risky debt in a general equilibrium setting. Applying Ito's lemma to the equilibrium relationship between the firm (asset) value and its debt value, we derive an equilibrium return dynamics on the debt. From this equilibrium return dynamics, we obtain an equilibrium relationship between the asset risk premium and the debt risk premium for a firm. Particularly, we show that the systematic risk premium depends upon the following variables: 1) the relative riskiness of debt (deposit) to asset, 2) asset beta, and 3) the investor's relative risk aversion. The relative riskiness of debt to asset is largely associated with the bank's financial risk represented by the leverage (debt to asset) ratio. The asset beta is determined by the asset return covariance with the economy's total wealth. According to Merton (1977), the fair deposit insurance premium per dollar is defined as a present value of the expected default loss (per dollar deposited) under Martingale (Q) measure. Deposit insurance premium is, henceforth, computed as a sum of the present value of the expected loss to depositors from their bank default and its systematic risk premium; the risk-neutral default probability under Q measure accounts for the risk premium beyond its expected default loss. The expected default loss is, in turn, estimated using the classical logit model. In order to estimate the debt premium, we impute the firm asset return (unobservable) from its stock return (observable) by employing the iterative procedure used by Vassalou and Xing (2004). This procedure is similar to the KMV's procedure outlined in Crosbie (1999). The procedure is based on the Merton's notion (1974) that a firm's stock is, as a matter of fact, a call option on the firm's asset with the book value of the firm's liability being the strike price. We first estimate the stock return volatility from the daily stock prices and use it as an initial value for the asset return volatility. Using the Merton's relationship, we compute daily asset values for each trading days. We then compute the asset return volatility for the next iteration. The procedure is repeated until it converges. A striking difference of our model from the traditional option based model such as Merton (1977) and Marcus and Shaked (1984) is that our model allows us to estimate the expected default loss and the systematic risk premium explicitly and separately. Our model also differs markedly from that of Cooper and Davydenko (2004),which derives a relationship between the equity risk premium and the debt risk premium, in that ours is, in nature, a general equilibrium model while theirs is not. An empirical analysis is done for the Korean commercial banks and the mutual savings banks and is compared to the results obtained by the prior studies such as Marcus and Shaked (1984), and Duffie, Jarrow, Purnanandam and Yang (2003). Our model generally produces a higher premium than what one would obtain from the Marcus and Shaked's (1984), which tends to underestimate it. We, furthermore, find that the expected default losses are higher for the mutual savings banks than the commercial banks, but the systematic risk premia are the opposite. We think that the commercial banks tend to be more procyclical as evidenced by their higher beta coefficients. The systematic risk premia varied over the business cycle, and reached the peak in the year of 2008. They are negatively related to the degree of financial stability. We recommend that the deposit insurance premia should be charged at the rates reflecting the differences in the systematic risk premia.