How do I use the exponential distribution for default timing?

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AcadiFi · Accepted Answer

The exponential distribution describes waiting times between events in a Poisson process. Its PDF is f(t) = lambda * e^(-lambda*t), CDF F(t) = 1 - e^(-lambda*t), mean 1/lambda, variance 1/lambda^2. Its memoryless property (P(T > s+t | T > s) = P(T > t)) makes it central to reduced-form credit models.

If Larksong Technologies has a constant default hazard of 2% per year, the probability of survival beyond 5 years is e^(-0.02*5) = 90.48%. Expected time to default is 1/0.02 = 50 years. In intensity-based models, the default time tau = -ln(U)/lambda where U is uniform(0,1). The assumption of constant hazard is restrictive; real-world hazard rates vary with macroeconomic conditions, so practitioners extend to piecewise-exponential or Cox processes with stochastic intensity.

The exponential connects to Poisson: if event counts are Poisson(lambda*t), inter-arrival times are Exp(lambda). This duality lets analysts move between counting defaults and timing defaults. Always stress-test the constant-hazard assumption by comparing predicted vs realized default curves over multi-year cohorts. Memorylessness also enables tractable pricing of defaultable bonds: price = risk-free price * E[e^(-integral of hazard)].

How do I use the exponential distribution for default timing?

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