# The Loss Elimination Ratio

If you impose a deductible $d$ on an insurance policy that you’ve written, what fraction of expected losses do you eliminate from your expected liability?  This is measured by the Loss Elimination Ratio $LER(d)$.

$\displaystyle LER(d) = \frac{E\left[X \wedge d\right]}{E\left[X\right]}$

Definitions:

1. Ordinary deductible $d$— The payment made by the writer of the policy is the loss $X$ minus the deductible $d$.  If the loss is less than $d$, then nothing is paid.
2. Franchise deductible $d_f$—  The payment made by the writer of the policy is the complete amount of the loss $X$ if $X$ is greater than $d_f$.
A common type of question considers what happens to LER if an inflation rate $r$ increases the amount of all losses, but the deductible remains unadjusted.  Let $X$ be the loss variable.  Then $Y=(1+r)X$ is the inflation adjusted loss variable.  If losses $Y$ are subject to deductible $d$, then
$\begin{array}{rll} \displaystyle LER_Y(d) &=& \frac{E\left[(1+r)X\wedge d\right]}{E\left[(1+r)X\right]} \\ \\ \displaystyle &=&\frac{(1+r)E\left[X\wedge \frac{d}{1+r}\right]}{(1+r)E\left[X\right]} \\ \\ &=& \frac{E\left[X \wedge \frac{d}{1+r}\right]}{E\left[X\right]}\end{array}$
Memorize:
$\displaystyle E\left[X \wedge d\right] = \int_0^d{x f(x) dx} + d\left(1-F(x)\right)$