Newsletter #166 Home Newsletter Contributors
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The BARRA Brainteaser
for Spting 1998
Solution to the Winter
1998 Brainteaser
Brainteaser from Winter 1998
By Eugene Reznik 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 scenarios shown in the table above right would each occur with 20% probability:
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!
| Month 1 | Month 2 | Month 3 | Month 4 |
|
|
|||
| 1 | 1 | 1 | 1 |
| 1 | 1 | 2 | |
| 1 | 3 | ||
| 2 | 2 | ||
| 4 | |||
Solution
Let’s consider the example shown in the table above, where N= 4. The values of X and Y for each prepayment scenario are shown in TABLE 1 below.
Clearly, for N= 4, 
Let’s
try to establish this relationship for the general case.
For a pool of N mortgages, let 
and
let SN be the number of ways
in which the loans might be prepaid. The following relationship holds,
as shown by the example in TABLE 2:
If we define SO to be 1, we can write XN as a sum:
To better understand YN, consider an SNxN table constructed as follows: If the jth prepayment scenario has at least one month during which exactly k loans will prepay, we will put a check mark in the cell (j,k) of the table. For example, for N= 4, see TABLE 3.
By definition, YN is the total number of check marks in the table. The number of check marks in column k is the number of ways a pool with N-k loans can be prepaid. Therefore:
and this completes the proof.
TABLE 1
Prepayment scenarios for a pool of 4 loans
|
|
Month | |||||
|
Scenario
No. |
Month
1 |
Month
2 |
Month
3 |
Month
4 |
X
|
Y
|
|
|
||||||
|
1
|
1
|
1
|
1
|
1
|
4
|
1
|
|
2
|
1
|
1
|
2
|
2
|
2
|
|
|
3
|
1
|
3
|
1
|
2
|
||
|
4
|
2
|
2
|
0
|
1
|
||
|
5
|
4
|
0
|
1
|
|||
 |
|
|||||
TABLE 2
Prepayment scenarios for a pool of 5 loans
|
|
Month | ||||
|
Scenario
No. |
Month
1 |
Month
2 |
Month
3 |
Month
4 |
Month
5 |
|
|
|||||
|
1
|
1*
|
1
 |
1
|
1
|
1
|
|
2
|
1
|
1
|
1
|
2
|
|
|
3
|
1
|
1
|
3
|
||
|
4
|
1
|
2
|
2
|
||
|
5 (=S4)
|
1
|
4
|
|||
|
6
|
2
|
3
|
|||
|
7
|
5
|
||||
* Italic figures indicate all the added months during which a single loan prepays. Their number equals S4.
Bold
figures indicate the prepayment scenarios of a four-loan pool.
TABLE 3
Distinct prepayment levels for a pool of 4 loans
|
|
Month | |||
|
Scenario
No. |
Month 1
|
Month 2
|
Month 3
|
Month 4
|
|
|
||||
|
1
|
 |
|||
|
2
|
|
|
||
|
3
|
|
|
||
|
4
|
|
|||
|
5
|
|
|||
