Loss Elimination Ratio and the Effect of Inflation

A health insurer's $2,000 deductible eliminates 45% of expected losses. Three years later, medical costs have inflated 30% but the deductible stays at $2,000. The LER drops to 37%. That 8-point shift means millions more in claims flowing through.

HIGH-FREQUENCY: The LER and its behavior under inflation is one of the most frequently tested concepts in Topic 1.

The loss elimination ratio measures the fraction of expected losses eliminated by a deductible:

$$\text{LER}(d)=\frac{E[X\wedge d]}{E[X]}$$

KEY: The LER ranges from 0 to 1. At $d=0$, LER = 0. As $d\to \infty$, LER approaches 1.

For exponential with mean $\theta$: $\text{LER}(d)=1-e^{-d/\theta }$.

For Pareto with $\alpha >1$: $\text{LER}(d)=1-(\theta /(\theta +d){)}^{\alpha -1}$.

For uniform on $(0,b)$: $\text{LER}(d)=2d/b-d^2/b^2$ for $d\le b$.

Suppose losses inflate by factor $1+r$. The inflated loss is $X^{\ast }=(1+r)X$. With a fixed deductible $d$:

Common mistakes

• Computing inflated LER using the original deductible in the LEV. You must divide $d$ by $1+r$ first. Plugging the original $d$ into the LEV and dividing by $(1+r)E[X]$ gives a different, incorrect answer.
• Claiming LER increases with inflation. With a fixed deductible, inflation always decreases the LER. If your new LER is higher, check your work.
• Forgetting the $(1+r)$ multiplier on insurer payments. After inflation, the insurer pays $(1+r)\cdot E[(X-d/(1+r){)}_{+}]$, not just $E[(X-d/(1+r){)}_{+}]$.

Bottom line

• LER = $E[X\wedge d]/E[X]$: the fraction of expected losses the deductible absorbs.
• Fixed deductible + inflation: replace $d$ with $d/(1+r)$ in the LEV; the LER always drops.
• Indexed deductible: the LER stays constant, but dollar payments still grow at rate $r$.
• Leveraged inflation: with a fixed deductible, insurer costs grow faster than rate $r$.

Exam shortcut

When you see inflation + fixed deductible, immediately compute $d_{\text{eff}}=d/(1+r)$. Use the pre-inflation distribution's LEV formula at $d_{\text{eff}}$. Verify the new LER is strictly less than the original. If the problem asks for percentage cost increase, remember the answer exceeds $r$. "Fixed deductible shrinks" (under inflation, the deductible shrinks in real terms to $d/(1+r)$. "Inflation up, LER down") no exceptions when deductible is fixed.

The full lesson (about 1,579 words, 11 min read) adds 3 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 1c

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