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 $d \leq b$.

Suppose nonnegative losses with a finite positive mean inflate by factor $1 + r$, where $r>0$. The inflated loss is $X^* = (1+r)X$. With a fixed ordinary deductible $d\geq0$ and no payment limit:

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 ordinary deductible, inflation cannot increase the LER. Equality can occur, including when all losses stay below the deductible.
• 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 ordinary deductible + positive inflation: replace $d$ with $d/(1+r)$ in the LEV; the LER cannot increase.
• Indexed deductible: the LER stays constant, but dollar payments still grow at rate $r$.
• Leveraged inflation: with a positive fixed ordinary deductible and positive original expected payment, unlimited payments grow faster than rate $r$.

Exam shortcut

When you see positive inflation and a fixed ordinary deductible, compute $d_{\text{eff}} = d/(1+r)$. Use the pre-inflation distribution's LEV formula at that value. The new LER cannot exceed the original. A percentage cost increase requires positive original expected payment. For a strict LER decrease, check that the deductible is positive and that some losses exceed its reduced real value. Otherwise equality is possible.

The full lesson (about 1,614 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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