A Multi-factor Approach for Systematic Default and Recovery Risk

Abstract

Banks face the challenge of forecasting losses and loss distributions in relation to their credit risk exposures. Most banks choose a modular approach in line with the current proposals of the Basel Committee on Banking Supervision (2004), where selected risk parameters such as default probabilities, exposures at default and recoveries given default are modelled independently. However, the assumption of independence is questionable. Previous studies have shown that default probabilities and recovery rates given default are negatively correlated [Carey (1998), Hu and Perraudin (2002), Frye (2003), Altman et al. (2005), or Cantor and Varma (2005)]. A failure to take these dependencies into account will lead to incorrect forecasts of the loss distribution and the derived capital allocation.

This article originally appeared in the September 2005 issue of The Journal of Fixed Income and is reprinted with permission from Institutional Investor, Inc. For more information please visit http://www.iijournals.com.

Appendix: Results of Monte-Carlo Simulations

In order to prove the reliability of our estimation method, a Monte-Carlo simulation was set up which comprises four steps:

• Step 1: Specify model (1) and model (9) with a given set of population parameters w, c, b, μ, and ρ.

• Step 2: Draw a random time series of length T for the defaults and the recoveries of a portfolio with size N from the true model.

• Step 3: Estimate the model parameters given the drawn data by the Maximum-Likelihood method.

• Step 4: Repeat Steps 2 and 3 for several iterations.

We used 1,000 iterations for different parameter constellations and obtained 1,000 parameter estimates which are compared to the true parameters. The portfolio consists of 10,000 obligors. The length of the time series T is set to T = 20 years. We fix the parameters at w = 0.2, μ = 0.5, and b = 0.5 and set the correlations between the systematic factors to 0.8, 0.1, and −0.5. In addition, we analyze three rating grades A, B, and C where the default probabilities and thresholds c in the grades are:

• A: $\pi = 0.005$, i.e., $c = - 2.5758$

• B: $\pi = 0.01$, i.e., $c = - 2.3263$

• C: $\pi = 0.02$, i.e., $c = - 2.0537$

Table 7.10 contains the results from the simulations. The numbers without brackets contain the average of the parameter estimates from 1,000 simulations. The numbers in round (.)-brackets represent the sample standard deviation of the estimates (which serve as an approximation for the unknown standard deviation). The numbers in square [.]-brackets give the average of the estimated standard deviations for each estimate derived by Maximum-Likelihood theory. It can be seen in each constellation that our ML–approach for the joint estimation of the default and recovery process works considerably well: the averages of the estimates are close to the originally specified parameters. Moreover, the estimated standard deviations reflect the limited deviation for individual iterations. The small downward bias results from the asymptotic nature of the ML-estimates and should be tolerable for practical applications.

Table 7.10 Results from Monte-Carlo simulations