Loans as Options: The KMV and Moody’s Models Chapter 4 Loans as Options: The KMV and Moody’s Models

The Link Between Loans and Optionality: Merton (1974) Figure 4.1: Payoff on pure discount bank loan with face value=0B secured by firm asset value. Firm owners repay loan if asset value (upon loan maturity) exceeds 0B (eg., 0A2). Bank receives full principal + interest payment. If asset value < 0B then default. Bank receives assets.

Using Option Valuation Models to Value Loans Figure 4.1 loan payoff = Figure 4.2 payoff to the writer of a put option on a stock. Value of put option on stock = equation (4.1) = f(S, X, r, , ) where S=stock price, X=exercise price, r=risk-free rate, =equity volatility,=time to maturity. Value of default option on risky loan = equation (4.2) = f(A, B, r, A, ) where A=market value of assets, B=face value of debt, r=risk-free rate, A=asset volatility,=time to debt maturity.

Problem with Equation (4.2) A and A are not observable. Model equity as a call option on a firm. (Figure 4.3) Equity valuation = equation (4.3) = E = h(A, A, B, r, ) Need another equation to solve for A and A: E = g(A) Equation (4.4) Can solve for A and A with equations (4.3) and (4.4) to obtain a Distance to Default = (A-B)/ A Figure 4.4

Estimating the Option Value Using the Black-Scholes Model

Merton’s Theoretical PD Assumes assets are normally distributed. Example: Assets=$100m, Debt=$80m, A=$10m Distance to Default = (100-80)/10 = 2 std. dev. There is a 2.5% probability that normally distributed assets increase (fall) by more than 2 standard deviations from mean. So theoretical PD = 2.5%. But, asset values are not normally distributed. Fat tails and skewed distribution (limited upside gain).

KMV’s Empirical EDF Utilize database of historical defaults to calculate empirical PD (called EDF): Fig. 4.5

Accuracy of KMV EDFs Comparison to External Credit Ratings Enron (Figure 4.8) Comdisco (Figure 4.6) USG Corp. (Figure 4.7) Power Curve (Figure 4.9): Deny credit to the bottom 20% of all rankings: Type 1 error on KMV EDF = 16%; Type 1 error on S&P/Moody’s obligor-level ratings=22%; Type 1 error on issue-specific rating=35%.

Monthly EDF™ credit measure Agency Rating

Problems with KMV EDF Not risk-neutral PD: Understates PD since includes an asset expected return > risk-free rate. Use CAPM to remove risk-adjusted rate of return. Derives risk-neutral EDF (denoted QDF). Bohn (2000). Static model – assumes that leverage is unchanged. Mueller (2000) and Collin-Dufresne and Goldstein (2001) model leverage changes. Does not distinguish between different types of debt – seniority, collateral, covenants, convertibility. Leland (1994), Anderson, Sundaresan and Tychon (1996) and Mella-Barral and Perraudin (1997) consider debt renegotiations and other frictions. Suggests that credit spreads should tend to zero as time to maturity approaches zero. Duffie and Lando (2001) incomplete information model. Zhou (2001) jump diffusion model.

Moody’s Public Firm Model Uses non-linear artificial neural network to weight Merton Distance to Default and 8 other key variables: Moody’s credit rating, ROA, firm size, operating liquidity, leverage, stock price volatility, equity growth rate, ROE. Relative importance of the variables changes over time – Fig. 4.10 (a) & (b) Power curve – Figure 4.11: Moody’s empirical EDF has Type 1 error of 20%.

Appendix 4.1 Merton’s Valuation Model B=$100,000, =1 year, =12%, r=5%, leverage ratio (d)=90% Substituting in Merton’s option valuation expression: The current market value of the risky loan is $93,866.18 The required risk premium = 1.33%