Normally when folks calculate an asset’s weighted average life they use the formula $$ WAL = {{\sum_n{n \cdot Prin_n}} \over Par_0} $$ where $$ Par_n = \hbox{Par value at time n} \\[7pt] $$ and $$ Prin_n = \hbox{Principal payment at time n} \\[7pt] $$

In a previous blog post I had presented this formula $$ WAL = {\sum{Par_n} \over Par_0} \\ $$ without any context, other than to leave it to the reader to prove.

A while ago for work I had created a process that required a formula for the weighted average life (WAL) of a mortgage with level payments. I found one online:

$$ W = N D_N - \frac{1}{r} $$

where:

$$ \begin{aligned}
d &= { 1 \over 1 + r} = \hbox{one month discount factor} \\[7pt] r &= {a \over 12} = \hbox{monthly discount} \\[10pt] N &= \hbox{Loan period in months} \\[10pt] D_n &= {1 \over 1-d^n} \end{aligned}
$$

That formula is pretty convenient because it means given a rate and a WAL we can solve pretty easily for how long the loan has to be in order to have a particular WAL without doing a bunch of infinite sums. So where does that fomula come from? Originally I found it here , but I wanted to see if I could derive it.