**Intensity models** (also called **reduced form models**) are a form of default model. Lets start by considering what are known as **mortality models of default**. These are essentially a discrete form of intensity model. Once we have established them, we will take a limit as time intervals go to zero—and out will pop intensity models.

Mortality models derive their name from their similarity to actuarial models of human mortality. Define a survival function *s*(*t*). It might indicate the probability of a human surviving until age *t* or the probability of a bond surviving without default for *t* years. For the rest of this article, we shall use it to denote the latter.

The probability of a bond defaulting in year *t* + 1 is given by

*s*(*t*) – *s*(*t* + 1)

[1]

This is an unconditional probability. It reflects the probability at time 0 (when the bond is issued) of default between time *t* and time *t* + 1. If we want the conditional probability of default—that is, the probability of default between time *t* and time *t* + 1, conditional on survival to time *t*—we apply Bayes’ theorem to obtain

[2]

A survival function can be constructed from historical bond default data. Constructed in this manner, the survival function defines a mortality model of default. Exhibit 1 indicates empirical survival functions by original credit quality. If the numbers were smoothed, it could reasonably be used to specify a mortality model for default.

Exhibit 1

Now, instead of considering a one-year time interval, let’s consider an arbitrary time interval Δ*t*. Generalizing [2], the probability of default between time *t* and time *t* + Δ*t*, conditional on their being no default by time *t*, is

[3]

We can express this as an average rate of default by dividing by the time interval Δ*t*:

[4]

Consider an example. Let *t* be three years. Assume *s*(5) = 0.8921 and *s*(8) = 0.8609. Then the conditional probability of default between years 5 and 8 is

[5]

This is a probability of default over a three year period. By [4], we convert it to an average annual rate of default by dividing by 3. The result is an average rate of .0117 defaults per year over the three-year period.

To obtain an instantaneous rate of default *f* (*t*) at any time *t*, we take the limit as *t* goes to 0 in [4]:

[6]

[7]

[8]

where *s*’(*t*) is the first derivative of *s* with respect to *t*.

The instantaneous rate of default *f* (*t*) is called the **default intensity** or, to borrow a word from insurance, the **hazard rate**. Intensity models work by assuming some functional form for *f* (*t*) and then calibrating that to current interest rate spreads. *f* (*t*) can reflect “real” probabilities to support credit risk management applications. It can reflect risk neutral probabilities to support financial engineering applications. The survival function is recovered from *f* (*t*) by rearranging [8] and integrating:

[9]

From this, probabilities of default can be obtained from [1] or [2] as appropriate.

Note how *f*(*t*) plays a role similar to that of a continuously compounded interest rate in [9]. Use of default intensities tends to simplify mathematics, which is one reason intensity models are popular with financial engineers.

Altman (1989), Asquith, Mullins and Wolff (1989) and Altman and Suggitt (2000) discuss mortality models of default. The first published intensity model appears to be Jarrow and Turnbull (1995). Subsequent research includes Duffie and Huang (1996), Jarrow, Lando and Turnbill (1997) and Duffie and Singleton (1997a, 1997b).

## References

- Altman E. I. (1989). Measuring corporate bond mortality and performance,
*Journal of Finance*, 44 (4), 909-922. - Altman, Edward I., & Suggitt, Heather J. (2000). Default rates in the syndicated bank loan market – a mortality analysis,
*Journal of Banking and Finance*, 24(1-2), 229-253. - Asquith, Paul, David W. Mullins and Eric D. Wolff (1989). Original issue high yield bonds: aging analyses of defaults, exchanges, and calls,
*Journal of Finance*, 44 (4), 923-952. - Duffie, Darrell, and Ming Huang (1996). Swap Rates and Credit Quality,
*Journal of Finance*, 51 (2), 921-49. - Duffie, Darrell and Kenneth Singleton (1997a). Modeling term structures of defaultable bonds,
*Review of Financial Studies*, 12 (4), 687-720. - Duffie, Darrell and Kenneth Singleton (1997b). An Econometric Model of the Term Structure of Interest-Rate Swap Yields,
*Journal of Finance*, 52 (4), 1287-1321. - Jarrow, Robert, Stuart Turnbull (1995). Pricing derivatives on financial securities subject to credit risk’,
*Journal of Finance*, 50 (1), pp. 53-86. - Jarrow, Robert, David Lando and Stuart Turnbull (1997). A Markov model for the term structure of credit spreads,
*Review of Financial Studies*, 10 (2), 481-523.

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