Asian Review of Financial Research Vol.24 No.4 pp.1021-1067
Forecasting Performances of Structural Default Probability Models with a New Iterative Estimation Method
Key Words : Credit Risk,Structural Default Probability Model,Parameters Estimation,Iterative Method,Out-of-Sample Test
We construct a new iterative method to estimate parameters in structural default probability models and compare their forecasting performances using stock return and accounting data of Korea. To adopt the new iterative method, we select four structural default probability models: Longstaff and Schwartz (1995: LS), Leland and Toft (1996: LT), Merton (1974: DD), and the down and out call option model adopted in Brockman and Turtle (2003: DOC). The new method makes it possible to daily estimate the barrier parameter in LT and DOC, which is not possible under the existing methods [see Vassalou and Xing (2004) and Bharath and Shumway (2008)]. When we adopt the new iterative method to LT and DOC and the existing iterative method to LS and DD in order to analyze out-of-sample and accuracy ratio tests, forecasting performances of above calculated default probabilities are more statistically sufficient than those of default probabilities under other estimation methods. Especially, default probabilities in LT using the new iterative method show statistically supported forcasting performances, though those using the other estimation method have no forecasting performance. Moreover, unlike the results in Bharath and Shumway (2008), default probabilities under the new iterative method satisfy properites of sufficient statistics. The new iterative method is significant in three aspects. First, the new iterative method can offer a new approach to solving two puzzles in asset pricing, the equity premium puzzle and the credit spread puzzle, since the new method provides relatively effective default probabilities among other methods in structural default probability models. Vassalou and Xing (2004) construct rank portfolios by DD's default probabilities with the existing iterative method used by KMV and study whether equity returns reflect performances for those portfolios. Goldstein (2009) and Chen et al. (2009b) study the relationship between the credit spread puzzle and the equity premium puzzle with structural default probability models. Second, the new iterative method is a simple alternative to MLE(maximum likelihood estimation) which requires heavy and complex calculation [see Ercsson and Reneby (2005), Chen et al. (2009a), Forte and Lovreta (2009), Wong and Choi (2009), and Chen et al. (2010)]. The new iterative method is, in fact, the only method that can daily provide barriers as well as total firm values. Both methods let the first passage time stochastic process of a firm follow a geometric Brownian motion with a drift rate and a volatility rate in, for instance, DOC and LT. Last, we test whether default probabilities of DOC, LT, LS, and DD are sufficient statistics using the forward induction iterative method. Bharath and Shumway (2008) test sufficient statistics only with DD's default probabilities through the existing forward iterative method of KMV. In order to daily estimate implied barriers and total firm values for DOC using the new iterative method, we follow four steps. First, having fixed a moving window, ordinarily 1 year with 250 business days, we calculate standard deviations of moving averages of equity returns as initial volatility values of a firm under the forward induction. We use the total firm value equal to the book value of the total debt plus the market capitalization as an initial value for a firm on a given day. Second, on that day, we use the Newton-Rahpson method and calculate a implied barrier by taking the derivative of the option value with respect to the barrier. Then using the Newton-Rahpson method again, we calculate a new total firm value by taking the derivative of the option value with respect to the total firm value. Third, we repeat these procedures for the period within the window to get implied barriers and new total firm values. With daily returns of these new total firm values, we compute a new standard deviation and a new mean. Then we update the initial volatility with the new standard deviation for the next iteration, and use the mean as the drift term of the geometric Brownian motion for the total firm value. Finally, we obtain a new volatility following the second and third steps using the new variables generated in the third step. For the period within the window, we repeat the second step until the difference between the existing volatility and the updated volatility converges below the critical level, 10E-4. After we get the converged volatility for the period within the window, we move the window forward by one business day in the forward induction. Next, we follow the first step and repeat the second and third steps until we get a new converged volatility. Hence, we estimate daily total firm values, daily implied barriers, and other parameters to construct the geometric Brownian motion through the full sample period.