# 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:

$$ \begin{aligned} WAL \cdot Par_0 &=& \sum_{n=1}^{N}{Par_n} \\[7pt] WAL &=& { \sum_{n=0}^{N}{Par_n} \over Par_0 } \end{aligned} $$