The BARRA Brainteaser for Winter 1998

While flying to Pebble Beach to attend the BARRA Fixed Income Seminar, a trader at MBS "" US engaged in a conversation about his work with the passenger in the adjacent seat. After learning about various pass-throughs and CMOs, the trader's companion—who until now has known nothing about mortgages but who loves puzzles—posed the following question:

Suppose that in a pool of N (N £ 360) thirty-year mortgages at least one mortgage is prepaid in the very first month, and, subsequently, the number of mortgages prepaying never decreases from one month to the next. Suppose further that all prepayment scenarios are equally likely to occur. For example, if the pool contained only four mortgages, the following scenarios would each occur with 20% probability:

Month 1	Month 2	Month 3	Month 4
1	1	1	1
1	1	2
1	3
2	2
4

For a given scenario, let X be the number of months during which only a single loan prepaid, and let Y be the number of distinct prepayment levels. Which is greater: the expected value of X, or the expected value of Y? Prove it!