In this article we survey alternative models of default. Default models can be broken down into two broad categories as shown in Table 1 below. Structural models model default in terms of a firm's assets and liabilities. Reduced-form models introduce default as an exogenous process unrelated to the firm's assets and liabilities.
CREDIT RATING MODELS
The traditional actuarial approach to measuring credit risk is based on credit ratings. These models specify transition probabilities between ratings for a fixed period of time. The ratings are typically, though not necessarily, assigned by rating agencies such as Standard & Poor's or Moody's Investor Services.
Jarrow, Lando and Turnbull proposed an arbitrage-free model of credit risk that uses a discrete time finite-state Markov chain to model the probability of moving from one rating to another. Transition matrices that give the probability of rating changes are typically calibrated from historical data (Brand, Rubbia and Bahar, or Carty). As an example, consider the case of three ratings A, B and C (default) along with the transition matrix below.
The entry 0.2 means that the probability of an A rated firm becoming a B rated firm over the period is 20%. Note that the last row C corresponds to default and that once a firm defaults it is assumed not to recover. This is reflected in the zero transition probabilities to non-default states from the default state in the last row of the matrix.
Given a one-period transition matrix, the two period transition matrix T2 can be calculated by simple matrix multiplication
Similarly, the m-period transition matrix is simply Tm.
The main advantage of a model in this class is the availability of historical data for calibration. It should be noted, however, that since transitions between extreme ratings are rare, estimates of these transitions are less accurate. The main disadvantages include the variation of credit quality within classes of credit, questions regarding timeliness of ratings data and that this approach does not relate default probabilities to market prices. The transition matrix only contains information on historical default frequencies and that alone is not sufficient for pricing.
STRUCTURAL MODELS
Structural models relate default probabilities to the value of the underlying credit's assets and liabilities. Black and Scholes and Merton first observed that the value of a firm's assets is the sum of its equity and debt and that a firm can be viewed as going into default when the value of its assets falls below the value of its debt. In this way, a risky bond can be viewed as a contingent claim on the firm's assets. Merton modeled the market value of a firm as the lognormal diffusion process:
where * is the instantaneous return on the firm's assets, * is the volatility parameter and C is the net dollar payout from the firm. It is easy to see that the payoff of risky debt is equal to the payoff on risk-free debt minus a put on the firm's value.
where BT is the price of a risk-free bond of maturity T and P(V,B,T) is the value of put option on the firm's assets with strike B and expiration T.
This model has been developed and expanded to cater for stochastic interest rates, more complex capital structures and more complex recovery processes most recently by Longstaff and Schwartz, Saa-Requejo and Santa-Clara, and Leland and Toft.
This approach relates default to a firm's capital structure and asset value. While in principal it is attractive to model different seniority classes, in practice this is very difficult to implement and calibrate. Moreover, using equity as a proxy for asset value introduces additional problems.
One additional issue with this type of model is that default does not occur as a surprise. In reality, credit deterioration and default often occur as a surprise, especially in an emerging markets context.
Finally, models of this class imply zero short-term credit spreads, which is contradicted by data.
SPREAD MODELS
Using this approach, a model of the spread process is typically overlaid on a process for the risk-free rate with some assumed spread level implying default.
Longstaff and Schwartz proposed a model for the short rate r and credit spread s as
This approach falls into the category of reduced form models.
DEFAULT INTENSITY MODELS
Default intensity models characterize default as a Poisson arrival process with stochastic intensity ht. Duffie and Singleton proposed a model:
where h is the arrival intensity of a Poisson process. Default occurs with the first jump in the Poisson process.
The value of a defaultable zero-coupon bond is then given by:
where rt is the risk-free spot rate process and Lt is the default loss process.
REFERENCES
* Jarrow, R., Lando, D., and Turnbull, S., 1994, "A Markov
Model for the Term Structure of Credit Spreads,"
working paper, Cornell University.
* Brand, L., Rabbia, J., and Bahar, R., 1997 "Rating
Performance 1996: Stability and Transition,"
Standard & Poor's.
* Carty, L., 1997, "Moody's Rating Migration and Credit
Quality Correlation, 1920-1996," Moody's.
* Black, F., and Scholes, M., 1973, "The Pricing of
Options and Corporate Liabilities," Journal of Political
Economy 81, 637-654.
* Merton, R.C., 1994, "On the Pricing of Corporate Debt:
The Risk Structure of Interest Rates," The Journal of
Finance 29, 449-470.
* Longstaff, F.A, and Schwartz, E.S., 1994, "A Simple
Approach to Valuing Risky Fixed and Floating Rate
Debt," working paper, UCLA.
* Saa-Requejo, J., Santa-Cara, P., 1997, "Bond Pricing with
Default Risk," working paper, UCLA.
* Leland, H.E. and Toft, K.B., 1996, "Optimal Capital
Structure, Endogenous Bankruptcy, and the Term
Structure of Credit Spreads," Journal of Finance 51, 987-1019.
* Duffie, D., and Singleton, K., 1994, "Modeling Term
Structures of Defaultable Bonds," working paper,
Stanford University.
This weeks' Learning Curve was written by Rohan Douglas, a director in the emerging markets research group at Salomon Smith Barneyin New York.
