Expected Life and Competing Hazards

Competing hazards models are statistical models that analyze multiple durations that begin at the same time for a subject. In credit risk, particular for retail products, the competing hazards are the risk of default and the risk of prepayment (early refinance). The expected life of a loan will determine the horizon over which the loan can be expected to generate a revenue (ie interest) stream for the lender. For loan products there is also the contractual life of the loan, which governs the loan's maximum terms and truncates the default and prepayment hazards.

Truncated Weibull distribution

The truncated Weibull distribiution allows for lower and upper limits on the potential distribution of life. For truncated Weibull distribution see: Crenin 2015, the PDF is given by, where :

The cumulative truncated Weibull distribution (CDF) is given by:

Probability density and cumualtive density for the truncated weibull distribution. If k=1, the result is equivalent to the exponential distribution.

Competing hazards

Expected loan life combining default and prepayment risk requires calculating the joint density distribution for the two competing hazards, and considering the probability of the two hazards occuring simultaneously. The logical position is that the hazards are inversely correlated, a customer with higher default risk usually has a lower refinance risk, however given a common strategy among borrowers is to capitalise arrears by refinancing during rising house prices, the correlation should not be assumed to be perfectly negative.

Joint density distribution

References: