Fast Formulas #5: Quasi-Geometric Series (The Other Trick)

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

Q=q+q^{2}+q^{3}+...+q^{n-1}+q^{n}

If n is large, it would be tedious to calculate the terms of Q.  Fortunately, there is a shortcut, as discussed in Fast Formulas #4:

Q=\dfrac {q-q^{n+1}}{1-q}                                                                                                            (Formula 4.4)

If you multiply each term in the geometric series by its exponent, you get this variation:

S=q+2q^{2}+3q^{3}+...+(n-1)q^{n-1}+nq^{n}

The series S doesn’t have its own name, but it is an example of a quasi-geometric series.  It has applications to the duration of a bond or annuity.

We can find a shortcut formula for S 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 S by q:

qS=q^{2}+2q^{3}+3q^{4}+...+(n-1)q^{n}+nq^{n+1}

Most of these terms nearly match the corresponding terms of S.  For example, S has the term 3q^{3} while qS has the term 2q^{3}.  We need one more q^{3} in the new series.  We also need a q, one more q^{2}, one more q^{4}, and so on.  In short, we need to add Q to the new series.  This gets us a perfect match for all of the terms of S, with an additional term:

qS+Q=q+2q^{2}+3q^{3}+4q^{4}+...+nq^{n}+nq^{n+1}

Subtracting S from this, almost everything cancels out:

qS+Q-S=nq^{n+1} ,

leading to

S=\dfrac {nq^{n+1}-Q} {q-1}.

Applying Formula 4.4 and a little more algebra, we find the shortcut formula for S:

S=\dfrac {q-(n+1)q^{n+1}+nq^{n+2}} {(q-1)^{2} }                                                                                 (Formula 5.1)

We could also find Formula 5.1 by exploiting some basic calculus.  You might have noticed that S is very similar to the derivative of the geometric series Q:

Q'=1+2q+3q^{2}+...+{n}q^{n-1}

Multiplying by q we see that  S=qQ' .  So all we need is a short formula for Q' .  By Formula 4.4, we can find Q'  by taking the derivative of \dfrac {q-q^{n+1}}{1-q} , 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.

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2 Responses to Fast Formulas #5: Quasi-Geometric Series (The Other Trick)

  1. Pingback: Fast Formulas #3: Pool Average Life with CPR Prepayments | The Well-Tempered Spreadsheet

  2. Pingback: Fast Formulas #3: The Derivation | The Well-Tempered Spreadsheet

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