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)
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2 Comments

Filed under Coverage Modifications, Deductibles

2 responses to “The Loss Elimination Ratio

  1. kriston

    I want to know,what is expected cost per loss and expected cost per payment in ordinary deductible and then why is important to derive the expected cost and variance.

    thx

  2. Vens Lee

    Thanks lot!
    This blog do help me a lot in preparing SOA exams.

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