Choose a Model for Loan Default When Prepayment Is a Competing Risk

A lender wants to estimate the probability that a newly issued 36-month loan defaults, to price credit risk. A loan ends in default or in full prepayment (the borrower pays it off early), whichever comes first, or it reaches the end of the term. The risk team's first analysis treats prepayments as censored and reports 1-\text{KM} for default at 36 months: 30.2 percent. Management says it looks too high compared with realized defaults.

Suppose, to examine this, that default has a constant hazard of 0.01 per month and prepayment 0.03 per month, independent, and that there is no other censoring within 36 months.

1. Compute the true probability of default within 36 months, the probability of prepayment, and the probability the loan is still active. Check that the figures the risk team computed are what 1-\text{KM} gives.
2. Explain in words what the 1-\text{KM} number answers, and why it is not the quantity needed for pricing.
3. Which models would you use to (a) estimate the probability of default by month 36, (b) learn how credit score changes the rate of default, (c) predict absolute default risk for a borrower with a given score? Mention the interpretation of each coefficient type.
4. A collaborator notes that the cause-specific hazard ratio of a "new job" variable on default is 0.8 but its effect on the cumulative default probability is smaller. Explain how that happens.