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# THE EXPONENTIAL DISTRIBUTION AND TIME TO DEFAULT

The exponential distribution is a continuous probability distribution. It describes the waiting time between events in a Poisson process, i.e. a process in which events occur continuously and independently at a constant average rate, the intensity or hazard rate, X. The probability density function (PDF) of an exponential distribution has the form, for x > 0:

The distribution is supported on the interval [0, oo). The probability of no event over a time interval At is the exponential function. Hence the probability of no event up to t is the summation, or, at the limit, the integral of Xexp(-Xt) between 0 and t:

This cumulative distribution function (CDF) is the probability of no event until t. It is given by the integral of this exponential density function and is equal to:

This formula results from the using the Poisson probability of observing k events and setting k - 0. The probability of a single occurrence3 is the complement to 1: 1 - Xexp(-Xt).

The standard deviation of the exponential distribution is llX, equal to the mean. Consider as rare event a default with intensity X. For example, for 1 year, the intensity is the default probability 1% for one year. Let t be the time to default t. The CDF of the exponential function is the probability of default not occurring between dates 0 and t, or survival probability from 0 to t. The CDF of an exponential distribution is a survival function. It represents the probability that a default does not occur before horizon 1 year, here exp(-Xt). For example, if X -5% and the horizon is 1 year, the survival probability is exp(-5%) — 95.123%. The average of the default time is 1/5% — 20 years, or 1 in 20 years.

Note that the survival time should be, in discrete time, exactly 95% and the discrete probability of default is 5% in the first year. The difference with the values comes from continuous intensity to default. The intensity to default matching exactly the 5% discrete default probability is such that cxp(-Xt) = 95%, with t = 1 year, or X = -ln(95%) = +ln(5%) = 5.129%. In other words, the discrete probability of default in 1 year of 5% is equivalent to a continuous intensity of default of 5.129%.

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