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# Formula for the Weighted Average Life of a Mortgage

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}