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 zeroand 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
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 defaultthat is, the probability of default between time t and time t + 1we apply Bayes' theorem to obtain
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.
Now, instead of considering a one-year time interval,
let's consider an arbitrary time interval
We can express this as an average rate of default by dividing by
the time interval
Consider an example. Let
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
where
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:
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).
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t.
Generalizing [
is the first derivative of s with respect to t.
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