In Fast Formulas #3: Pool Average Life with CPR Prepayments, I described Formula 3.1., a compact formula for the average life of a prepaying loan pool, and I wrote that the formula had been found by using “a couple of tricks.” I explained one of the tricks in Fast Formulas #4: Geometric Series (One of the Tricks). This post describes the other trick, which concerns a variation on the geometric series.
A geometric series is a sum such as
If n is large, it would be tedious to calculate the terms of . Fortunately, there is a shortcut, as discussed in Fast Formulas #4:
If you multiply each term in the geometric series by its exponent, you get this variation:
We can find a shortcut formula for with the same strategy we used to find Formula 4.4. We construct a new series that shares most of its terms with the original series, and then cancel out the matching terms on our way to a compact formula.
The first step is to multiply by :
Most of these terms nearly match the corresponding terms of . For example, has the term while has the term . We need one more in the new series. We also need a , one more , one more , and so on. In short, we need to add to the new series. This gets us a perfect match for all of the terms of , with an additional term:
Subtracting from this, almost everything cancels out:
Applying Formula 4.4 and a little more algebra, we find the shortcut formula for :
We could also find Formula 5.1 by exploiting some basic calculus. You might have noticed that is very similar to the derivative of the geometric series :
Multiplying by we see that . So all we need is a short formula for . By Formula 4.4, we can find by taking the derivative of , using a rule for the derivative of the quotient. If you work it out, you will end up with Formula 5.1.
Formula 5.1 is the other trick I used to find Formula 3.1.