Alternative formulation of Weighted Average Life
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.
I first encountered this formula in a Milliman MG-ALFA model, and it took me a bit to understand how it works. Computationally it is much, much easier to calculate than the traditional way and that sometimes makes it easier to use when calculating closed form solutions to things. I certainly hadn’t seen it before (or at least had completely forgotten that I had) and I think that’s mostly because while it’s computationally convenient, it doesn’t give folks much of a feel for what the weighted average life actually means, whereas the “traditional” one is much clearer about what the intention of the formula is - the average time the asset will be around.
Derivation of the formula
If we let $Prin_n$ be the principal payment at the beginning of time n then:
$$ \begin{aligned} WAL \cdot Par_0 =& \sum_{n=1}^{N}{n \cdot Prin_n} \\[7pt] =& Prin_1 + 2 Prin_2 + 3 Prin_3 + \ldots \\[7pt] \end{aligned} $$ rearranging the terms and recognizing that $\sum_{i=n}^N{Prin_i}$ is just the principal remaining at the beginning of time i, which is $Par_i$: $$ \begin{aligned} WAL= & Prin_1 &+ Prin_2&+ Prin_3 &+ Prin_4 & + \ldots & + Prin_N & \qquad \left( = Par_0 \right) \\[7pt] & & + Prin_2& + Prin_3 &+ Prin_4 & + \ldots & + Prin_N& \qquad \left( = Par_1 \right) \\[7pt] & & & + Prin_3 &+ Prin_4 & + \ldots & + Prin_N & \qquad \left( = Par_2 \right) \\[7pt] & & & &+ Prin_4 & + \ldots & + Prin_N & \qquad \left( = Par_3 \right) \\[7pt] & \ldots \\ & & & & & & + Prin_N & \qquad \left( = Par_{N-1} \right) \\[7pt] \end{aligned} $$
Leaving us with our final formula:
$$ \begin{aligned} WAL \cdot Par_0 &=& \sum_{n=1}^{N}{Par_n} \\[7pt] WAL &=& { \sum_{n=0}^{N}{Par_n} \over Par_0 } \end{aligned} $$