Default models are a category of models that assess the likelihood of default by an obligor. They differ from credit scoring models in two ways:

• Credit scoring is usually (but not always) applied to smaller credits—individuals or small businesses. Default models are applied more to larger credits—corporation or sovereigns.
• Credit scoring models are largely statistical, regressing instances of default against various risk indicators, such as a obligor's income, home renter/owner status, etc. Default models directly model the default process, and are typically calibrated to market variables, such as the obligor's stock price or the credit spread on its bonds.

Default models find many applications within financial institutions. They are used

• to support or supplant credit analysis;
• to calculate utilization of counterparty credit risk limits;
• to extend standard financial engineering techniques to value credit derivatives or other credit sensitive instruments.

Default models may be integrated with some sort of correlation model to facilitate modelling the credit risk of portfolios with exposures to multiple obligors. Such extensions of default models—called portfolio credit risk models—can be used

• to calculate utilization of industry, country or portfolio credit risk limits;
• to price collateralized debt obligations (CDOs) or other securitizations;
• to support capital calculations.

Consider a time horizon staring at the current time 0 and ending at some future time t. A one year horizon is typical, but financial institutions usually consider credit risk over several horizons. Let L represent the financial loss, if any, due to default on a particular obligation—a bond, loan, derivative instrument, etc.—over the horizon. L is a random variable. Its expected value E(L) is a metric of the credit risk of the obligation. It can be calculated as the product

E(L) = Pr(default) EAD LGD        [1]

where

• Pr(default) is the probability of default on the obligation during the horizon—what is called the default probability.
• EAD is exposure at default—the credit exposure on the obligation at the time of default. In [1], this is treated as a known constant.
• LGD is loss given default—the fraction of EAD that will not be recovered following default. EAD is simply 1 minus the recovery rate. In [1], it too is treated as a known constant.

The essential purpose of a default model is to calculate the default probability. However, sophisticated models may do more than this. For example, models might treat EAD and LGD as random, and substitute their expectations into [1]. Treating both in this manner requires an assumption that they are independent. Such an assumption is difficult to justify, but it may be made to simplify models.

A simple default model can be constructed by calibrating credit ratings to historical frequencies of migrations between ratings. Exhibit 1 indicates a ratings transition matrix. constructed by Standard & Poor's. It indicates one-year ratings migration probabilities based upon bond rating data from the period 1981-2000.

One-Year Ratings Transition Matrix Exhibit 1

original rating probability of migrating to rating by year end (%) AAA AA A BBB BB B CCC Default AAA 93.66 5.83 0.40 0.08 0.03 0.00 0.00 0.00 AA 0.66 91.72 6.94 0.49 0.06 0.09 0.02 0.01 A 0.07 2.25 91.76 5.19 0.49 0.20 0.01 0.04 BBB 0.03 0.25 4.83 89.26 4.44 0.81 0.16 0.22 BB 0.03 0.07 0.44 6.67 83.31 7.47 1.05 0.98 B 0.00 0.10 0.33 0.46 5.77 84.19 3.87 5.30 CCC 0.16 0.00 0.31 0.93 2.00 10.74 63.96 21.94 Default 0.00 0.00 0.00 0.00 0.00 0.00 0.00 100.00

One-year ratings migration probabilities based upon bond rating data from 1981-2000. Data is adjusted for rating withdrawals. Numbers in each row should sum to 100%. Due to round-off error, they may not do so exactly. Source: Standard & Poor's.

For example, based upon the matrix, a BBB-rated bond has a 4.44% probability of being downgraded to a BB-rating by the end of one year. The matrix is based upon raw data, so it exhibits statistical anomalies. A CCC-rated bond is given a 0.16% probability of being upgraded to AAA, but a B-rated bond has a 0.00% probability of such an upgrade. If it were used to model defaults, the numbers in the matrix might be smoothed.

To use a ratings transition matrix as a default model, we simply take the default probabilities indicated in the last column and ascribe them to bonds of the corresponding credit ratings. For example, with this approach, we would ascribe an A-rated bond a 0.04% probability of default within one year.

If we want two-year default probabilities, we simply multiply the matrix by itself once (i.e. employ matrix multiplication as defined in linear algebra) to obtain a two-year ratings transition matrix. The last column of that matrix will provide the desired default probabilities. For three-year default probabilities, we multiply the matrix by itself three times, etc. Exhibit 2 indicates a five-year ratings transition matrix obtained by multiplying the one-year matrix of Exhibit 1 by itself five times.

Five-Year Ratings Transition Matrix Exhibit 2

original rating probability of rating after five years (percent) AAA AA A BBB BB B CCC Default AAA 72.39 21.69 4.74 0.86 0.20 0.08 0.01 0.02 AA 2.49 66.45 25.05 4.45 0.75 0.51 0.09 0.18 A 0.39 8.19 68.22 18.05 3.19 1.32 0.18 0.50 BBB 0.16 1.72 16.80 60.61 13.16 4.68 0.79 2.08 BB 0.13 0.53 3.81 19.50 44.77 19.84 3.09 8.34 B 0.06 0.42 1.62 4.15 15.18 46.97 6.54 25.15 CCC 0.34 0.20 1.21 3.05 6.33 18.10 12.36 58.51 Default 0.00 0.00 0.00 0.00 0.00 0.00 0.00 100.00

Five-year ratings migration probabilities obtained by multiplying the matrix of Exhibit 1 by itself five times.

Default models that base default probabilities on empirical ratings transition matrices are called ratings migration models. CreditMetrics is an example of a commercial portfolio credit risk model that calculates default probabilities with a ratings migration model. CreditMetrics also uses its ratings migration matrices to model the evolution of bonds' credit spreads based upon migrations in their ratings. This allows for the modeling of bond portfolios' market (or mark-to-model) values over time.

Ratings migration models have a number of shortcomings. First, credit ratings reflect overall credit quality, which depends on both probabilities of default as well as likely recovery rates. If two bonds have the same credit rating, but one bond is senior and the other is subordinated, the senior bond is likely to have a higher default probability offsetting its likely higher recovery rate.

Second, ratings migration models are not dynamic. Because they are based upon long-term empirical probabilities of ratings transitions, they are not sensitive to business cycles or other fluctuations in the business environment.

Ratings migration models are just one type of default model. Many different default models have been proposed in the literature or implemented by financial institutions. With few exceptions, those that are not ratings migration models are implementations of:

• asset value models, or
• intensity models.

Both types of models are sophisticated, flexible approaches to credit risk modelling that support a variety of analyses. They can be calibrated to current business conditions (typically using a firm's stock price or bond spreads for this purpose). They can be implemented with "real probabilities" to support credit risk measurement or with risk neutral probabilities to support financial engineering applications.

The structural credit risk model is a model for assessing credit risk, typically of a corporation's debt. It is also sometimes called the Merton model or asset value model. It was proposed in Black and Scholes' (1973) seminal paper on option pricing and a more detailed paper by Merton (1974). Merton anticipated the model in Merton (1970), and he actively supported the work of Black and Scholes, which is why the model is often called the Merton model. A popular implementation of the model is the commercial KMV model. KMV was a boutique software firm that is now owned by Moodys.

The model considers a corporation financed through a single debt and a single equity issue. The debt comprises a zero-coupon bond that matures at time t = t*, at which time it is to pay investors b dollars. The equity pays no dividends.

An unobservable process V describes the firm's value ^tV>=0 at any time t. We ascribe the outstanding debt and equity values ^tF and ^tE, respectively. Accordingly, at any time t

tV= tF+tE	[1]
At time  t*, the firm's debt matures. At that time, either t*V will exceed the bond's maturity value b, or it won't. In the former case, the firm will pay off the bondholders. The remaining value of the firm will belong to the equity holders, so
t*E = t*V– b	[2]
In the latter case, the firm defaults on its debt. The bondholders take ownership of the firm, and the stockholders are left with nothing:
t*E= 0	[3]
Combining the above two results, we obtain a general expression for the value of the firm's stock at the maturity of its debt:
t*E= max( t*V– b, 0)	[4]
Look closely at this formula. It is precisely the payoff of a call option on the firm's value  t*V with strike price b. Based upon this realization, the asset value model treats the firm's equity as a call option on the value of the firm struck at the maturity value b of its debt. By put-call parity, the firm's debt comprises a risk-free bond that guarantees payment of b plus a short put option on the value of the firm struck at b. Accordingly
t*F= b –max(b – t*V, 0)	[5]
The asset value model treats tV just like any underlier. It assumes tV follows a geometric Brownian motion with volatility σ. Further, it makes all the other simplifying assumptions of the Black-Scholes (1973) option pricing formula. Accordingly, the firm's equity can be valued at any time t as
tE= c( tV, b, , r, t* – t )	[6]
where c is the Black-Scholes formula for the value of a call option, and r is the risk-free rate. By [5], we can similarly value the firm's debt as
tF= b – p(tV , b, , r, t* – t )	[7]

where p is the Black-Scholes formula for the value of a put. Note that we discount the payment b at the risk free rate because that payment is risk-free in formula [5]—we have stripped out the credit risk as a put option.

At any time t, the distance to default for a the firm's debt is defined as

( tV– b) / σ

[8]

This is a metric indicating how many standard deviations the equity holders' call option is in-the-money. The smaller the distance to default, the more likely a default is to occur. The probability of default is precisely the probability of the call option expiring out-of-the-money. This is approximately equal to one minus the option's normalized delta (if investors were risk neutral, equality would be exact). See this glossary's article Black-Scholes (1973) option pricing formula for the Black-Scholes formula for delta. To normalize that value, divide the delta by the underlier's value.

Three shortcomings of the asset value model are:

1. Its assumption that the firm's debt financing consists of a one-year zero-coupon bond is, for most firms, an oversimplification..

2. The Black-Scholes (1973) simplifying assumptions are questionable in the context of corporate debt, and

3. The firm's value ^tV is not observable, which makes assigning values to it and its volatility problematic.

Still, the model provides a useful context for considering and modeling credit risk. Practical implementations of the model are used by financial institutions and institutional investors. These extend the model in some manner to facilitate the assigning of values to ^tV and σ. Such techniques generally relate ^tV to the observable market capitalization of the firm.