Skip to content

Loss Elimination Ratio

Definition

For 0<E[X]<∞0<\mathbb{E}[X]<\infty and ordinary deductible d≥0d\ge0, the loss elimination ratio (LER) is

LER⁡(d)=E[X]−E[(X−d)+]E[X]=E[min⁡(X,d)]E[X].\operatorname{LER}(d)=\frac{\mathbb{E}[X]-\mathbb{E}[(X-d)_+]}{\mathbb{E}[X]} =\frac{\mathbb{E}[\min(X,d)]}{\mathbb{E}[X]}.

Interpretation

The ratio measures the share of expected ground-up loss removed from insurer payments. It concerns expected amounts, rather than the fraction of claims receiving no payment.

Exponential survivor function with deductible 100: the area under the curve left of d is shaded blue (about 63.2) and the area to its right red (about 36.8).

For X∼Exp⁡(0.01)X\sim\operatorname{Exp}(0.01) the whole area under FˉX(t)=e−0.01t\bar F_X(t)=\mathrm{e}^{-0.01t} is E[X]=100\mathbb{E}[X]=100. The blue part over [0,d][0,d] is E[min⁡(X,d)]=100(1−e−1)≈63.2\mathbb{E}[\min(X,d)]=100(1-\mathrm{e}^{-1})\approx63.2, the expected amount that the deductible d=100d=100 removes from insurer payments, while the red tail is πX(d)=E[(X−d)+]=100e−1≈36.8\pi_X(d)=\mathbb{E}[(X-d)_+]=100\mathrm{e}^{-1}\approx36.8, the expected payment that remains. The loss elimination ratio is the blue share of the total area, 63.2/100≈0.63263.2/100\approx0.632, as in the example.

The stop-loss transform and reinsurance gives πX(d)=E[(X−d)+]\pi_X(d)=\mathbb{E}[(X-d)_+], the expected payment remaining above the deductible. Subtracting this from the ground-up mean and dividing by that mean produces the eliminated share.

Key Results

  • 0≤LER⁡(d)≤10\le\operatorname{LER}(d)\le1 and it increases with dd.
  • LER⁡(d)=∫0dFˉX(t) dt/E[X]\operatorname{LER}(d)=\int_0^d\bar F_X(t)\,\mathrm{d}t/\mathbb{E}[X].
  • It is zero at d=0d=0 and tends to one as d→∞d\to\infty.
  • A zero or infinite ground-up mean makes this ratio inappropriate without further conventions.

Examples

For exponential losses at rate 0.010.01 and deductible 100, LER is 1−e−1≈0.63211-\mathrm{e}^{-1}\approx0.6321. The payment probability is e−1≈0.3679\mathrm{e}^{-1}\approx0.3679, a different quantity.

Prerequisites

Sources

Aliases: LER, deductible loss reduction ratio Keywords: LER(d) = E[min(X, d)]/E[X], expected loss eliminated by a deductible, 1 - E[(X - d)+]/E[X], exponential loss elimination ratio Symbols: infinity, integral, pi

Last updated: 22 October 2026 19:04