Forecasting Default with the KMV-...
Forecasting Default with the KMV-Merton Model Sreedhar T Bharath and Tyler Shumway��� University of Michigan December 17, 2004 Abstract We examine the accuracy and contribution of the default forecasting model based on Merton���s (1974) bond pricing model and developed by the KMV corporation. Comparing the KMV- Merton model to a similar but much simpler alternative, we find that it performs slightly worse as a predictor in hazard models and in out of sample forecasts. Moreover, several other forecasting variables are also important predictors, and fitted hazard model values outperform KMV-Merton default probabilities out of sample. Implied default probabilities from credit default swaps and corporate bond yield spreads are only weakly correlated with KMV-Merton default probabilities after adjusting for agency ratings, bond characteristics, and our alternative predictor. We conclude that the KMV-Merton model does not produce a su���cient statistic for the probability of default, and it appears to be possible to construct such a su���cient statistic without solving the simultaneous nonlinear equations required by the KMV-Merton model. We include the SAS code we use to calculate KMV-Merton default probabilities in an ap- pendix. ���Department of Finance, University of Michigan Business School, 701 Tappan Street, Ann Arbor, MI 48109. Bharath can be reached at 734-763-0485 or email@example.com. Shumway is visiting Stanford GSB for 2004-05, and can be reached at 650-725-9265 or firstname.lastname@example.org. We thank seminar participants at Michigan and Boston College. We also thank Darrell Du���e, Wayne Ferson, Kyle Lundstedt, Ken Singleton, and Jorge Sobehart for their comments.
Due to the advent of innovative corporate debt products and credit derivatives, academics and practitioners have recently shown renewed interest in models that forecast corporate defaults. One innovative forecasting model which has been widely applied in both practice and academic research1 is a particular application of Merton���s model (Merton, 1974) that was developed by the KMV corporation, which we refer to as the KMV-Merton model2. This paper assesses the accuracy and the contribution of the KMV-Merton model. The KMV-Merton model applies the framework of Merton (1974), in which the equity of the firm is a call option on the underlying value of the firm with a strike price equal to the face value of the firm���s debt. The model recognizes that neither the underlying value of the firm nor its volatility are directly observable. Under the model���s assumptions both can be inferred from the value of equity, the volatility of equity and several other observable variables by solving two nonlinear simultaneous equations. After inferring these values, the model specifies that the probability of default is the normal cumulative density function of a z-score depending on the firm���s underlying value, the firm���s volatility and the face value of the firm���s debt. The KMV-Merton model is a clever application of classic finance theory, but how well it performs in forecasting depends on how realistic its assumptions are. The model is a somewhat stylized structural model that requires a number of assumptions. Among other things, the model assumes that the underlying value of each firm follows geometric Browninan motion and that each firm has issued just one zero-coupon bond. If the model���s strong assumptions are violated, it should be possible to construct a reduced form model with more accuracy. We examine two hypotheses in this paper. First, we ask whether the probability of default implied by the Merton model is a su���cient statistic for forecasting bankruptcy. If the Merton model is literally true, it should be impossible to improve on the model���s implied probability for forecasting. If it is possible to construct a reduced form model with better predictive properties, we can conclude that the KMV-Merton probability (��KMV) is not a su���cient statistic for forecasting default. 1The model is discussed in Du���e and Singleton (2003) and Saunders and Allen (2002). It is applied by Vassalou and Xing (2003), among others. 2While others refer to this model simply as a Merton model, we prefer to call it the KMV-Merton model because (1) deriving the KMV-Merton default probability from observed equity data is a nontrivial extension of the ideas in the classic Merton model and (2) the proprietors of KMV developed this clever extension of the Merton model and we believe they deserve some credit for its development. We do not intend to imply that we are using exactly the same algorithm that Moody���s KMV uses to calculate distance to default. Differences between our method and that of Moody���s KMV are discussed in Section I B and in Table 2. 1
Our second hypothesis is that the Merton model is an important quantity to consider when predicting default. We hypothesize that the information in ��KMV cannot be completely replaced by a reasonable set of simple variables, or that a su���cient statistic for default probabiltity cannot neglect ��KMV . We actually separate the KMV-Merton technique into two potentially important components: the functional form for default probability implied by the Merton model and the solution of two simultaneous nonlinear equations required by the model. It is possible that one of these components is important while the other is not. We test these two hypotheses in five ways. First, we incorporate ��KMV into a hazard model that forecasts defaults from 1980 through 2003. With the hazard model, we compare ��KMV to a naive alternative (��Naive) which is much simpler to calculate, but retains some of the functional form of ��KMV . We also compare it to several other default forecasting variables. Second, we compare the short term, out of sample forecasting ability of ��KMV to that of ��Naive. Third, we examine the forecasting ability of several alternative predictors, each of which calculates KMV- Merton probabilities in a slightly different way. Fourth, we examine the ability of KMV-Merton probabilities to explain the probability of default implied by credit default swaps, and fifth we regress corporate bond yield spreads on ��KMV , ��Naive and other variables. Assessing the KMV-Merton model���s value is of importance for two reasons. Perhaps the most important reason is that many researchers and practitioners are applying the model without know- ing very much about its statistical properties. For example, Vassalou and Xing (2003) use ��KMV to examine whether default risk is priced in equity returns. As a second example, the Basel Commit- tee on Banking Supervision (1999) considers exploiting the KMV-Merton model a viable practice currently employed by numerous banks. To have confidence in both the risk management of the banking sector and the accuracy of academic research, the power of the KMV-Merton model must be examined. A second reason to assess the KMV-Merton model is to test the Merton (1974) model in a new way. If the Merton model is literally true, ��KMV should be the best default predictor available. The Merton model has been rejected previously for failing to fit observed bond yield spreads.3 Comparing the model to reduced form alternatives gives us a fresh perspective about how realistic the model���s assumptions are. Over the past several years, a number of reserchers have examined the contribution of the KMV- 3see Jones et al. (1984). 2
Merton Model. The first authors to examine the model carefully were practitioners employed by either KMV or Moody���s. A couple of years ago, several papers addressing the accuracy of the KMV-Merton model were available on the internet. Some papers, including Stein (2000), Sobehart and Stein (2000) and Sobehart and Keenan (1999) argued that KMV-Merton models can easily be improved upon. Other papers, including Kealhofer and Kurbat (2001), argued that KMV-Merton models capture all of the information in traditional agency ratings and well known accounting variables. Curiously, while some practitioner papers can now be found in print, including Sobehart and Keenan (2002a) and (2002b) and Falkenstein and Boral (2001), it has become very di���cult to find electronic copies of some of the papers cited above since Moody���s acquired KMV in April 2002. Perhaps in response, an academic literature has recently developed that critically assesses the model. Both Hillegeist, Keating, Cram and Lundstedt (2004) and Du and Suo (2004) examine the model���s predictive power in ways that are similar to some of our analyses. Du���e and Wang (2004) show that KMV-Merton probabilities have significant predictive power in a model of default probabilties over time, which can generate a term structure of default probabilities. Campbell, Hilscher and Szilagyi (2004) estimate hazard models that incorporate both ��KMV and other variables for bankruptcy, finding that ��KMV seems to have relatively little forecasting power after conditioning on other variables. While our findings are consistent with the findings of all of these papers, we analyze the performance of ��KMV in several novel ways. In particular, we introduce and assess our naive predictor and we examine the ability of ��KMV to explain credit default swap premia and bond yield spreads. Like all of these researchers, we have no particular interest in finding evidence for or against the KMV-Merton model. Therefore, we hope to help resolve confusion about some of the issues raised in the practitioner literature described above. We find that it is fairly easy to reject hypothesis one, or that ��KMV is not a su���cient statistic for default probability. We also find that after conditioning on ��Naive, it appears to be possible to construct a reduced form model that does not benefit by conditioning on ��KMV (or in which ��KMV is not statistically significant). We therefore conclude that while ��KMV has some predictive power for default, most of the marginal benefit of ��KMV comes from its functional form rather than from the solution of the two nonlinear equations on which it is based. The contribution of ��KMV to a well-specified reduced form model is fairly low. The paper proceeds as follows. The next section details the KMV-Merton model, our naive 3
alternative default probability, and the hazard models that we use to build reduced form models. Section I also lists several ways in which our KMV-Merton model differs from the model that Moodys KMV actually sells. Section II discusses the data that we use for our tests and Section III outlines our results. We conclude in Section IV. I. Default Forecasting Models As discussed above, we examine our hypotheses by examining the statistical and economic signif- icance of the KMV-Merton default probabilities (��KMV) and a simple, naive alternative (��Naive). Before examining the empirical value of these variables, we need to describe them carefully. The KMV-Merton model was developed by the KMV corporation in the late 1980s. It was successfully marketed by KMV until KMV was acquired by Moodys in April 2002. The model is now sold to subscribers by Moody���s KMV. A. The KMV-Merton Model The KMV-Merton default forecasting model produces a probability of default for each firm in the sample at any given point in time. To calculate the probability, the model subtracts the face value of the firm���s debt from from an estimate of the market value of the firm and then divides this difference by an estimate of the volatility of the firm (scaled to reflect the horizon of the forecast). The resulting z-score, which is referred to as the distance to default, is then substituted into a cumulative density function to calculate the probability that the value of the firm will be less than the face value of debt at the forecasting horizon. The market value of the firm is simply the sum of the market values of the firm���s debt and the value of its equity. If both these quantities were readily observable, calculating default probabilities would be simple. While equity values are readily available, reliable data on the market value of firm debt is generally unavailable. The KMV-Merton model estimates the market value of debt by applying the Merton (1974) bond pricing model. The Merton model makes two particularly important assumptions. The first is that the total value of a firm is assumed to follow geometric Brownian motion, dV = ��V dt + ��V V dW (1) 4