Loss Elimination Ratio

Study guide: Exam 5

Loss Elimination Ratio (LER) is the proportion of ground-up losses removed by a deductible — the share of expected loss that the insured retains rather than transfers.

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

\text{Deductible Relativity} = 1 - \text{LER}(d)

• E[X \wedge d] = E[\min(X,d)] is the limited expected value — the expected loss capped at d. Because every loss contributes at most d, the LER captures both the small claims eliminated entirely and the first d dollars of the large ones.
• The LER is the bridge from a full-coverage rate to a deductible rate: multiply the full-coverage loss cost by (1 - \text{LER}) to get the loss cost net of the deductible. See Deductible Rating.
• LERs are computed from an empirical size-of-loss distribution (where the ground-up loss is known for every claim) or from a fitted continuous distribution. If the data hold only losses net of the deductible, policies with a deductible above the one being priced cannot be used, because the losses they eliminated were never reported.
• The LER falls as losses inflate, because a fixed deductible eliminates a shrinking fraction of a growing loss — the leveraged effect described under Inflation. Deductible relativities must therefore be reviewed as costs rise, or they silently become inadequate.
• The relativity (1-\text{LER}) applied to the whole rate rests on Werner's assumption that all expenses are variable. Per-policy fixed expenses do not shrink with a deductible — where they are material and not charged as a fee, including the fixed-expense loading flattens the relativity — and on a large deductible policy the insurer often handles every claim, including those entirely within the deductible, so the premium must cover those handling costs too.

LER from an Empirical Loss Distribution A book of 1{,}000 claims has this size-of-loss distribution:

Loss size	Claims	Total losses
\$0–500	400	\$100{,}000
\$500–1{,}000	300	\$225{,}000
over \$1{,}000	300	\$1{,}175{,}000

Compute the LER for a \$500 deductible.

Answer Every claim contributes \min(X, 500):

• The 400 claims under \$500 contribute their full amount, \$100{,}000.
• The 600 claims above \$500 each contribute exactly \$500: 600 \times \$500 = \$300{,}000.

\begin{align*}
E[X \wedge 500] \times n &= \$100{,}000 + \$300{,}000 = \$400{,}000 \\[4pt]
\text{Total losses} &= \$1{,}500{,}000 \\[6pt]
\text{LER}(500) &= \frac{\$400{,}000}{\$1{,}500{,}000} = 26.7\%
\end{align*}

A \$500 deductible eliminates 26.7\% of losses, so the loss cost net of the deductible is 73.3\% of full coverage — even though 40\% of claims disappear entirely. Claim counts and loss dollars give very different answers, and the rate depends on the dollars.

Inflation Erodes the Loss Elimination Ratio Losses follow an exponential distribution with mean \theta = \$2{,}000, and the deductible is \$500. Losses then inflate 20\%.

Show what happens to the LER and to the deductible relativity.

Answer For an exponential, E[X \wedge d] = \theta\left(1 - e^{-d/\theta}\right) and E[X] = \theta, so

\text{LER}(d) = 1 - e^{-d/\theta}

\begin{align*}
\text{Before: } \text{LER} &= 1 - e^{-500/2000} \\
&= 1 - e^{-0.25} = 22.1\% \\[6pt]
\text{After } (\theta = 2{,}400): \; \text{LER} &= 1 - e^{-500/2400} \\
&= 1 - e^{-0.2083} = 18.8\%
\end{align*}

The deductible relativity therefore rises from 0.779 to 0.812 — the deductible is worth less to the insured and costs the insurer more.

Two consequences follow. First, an insurer that leaves its deductible relativities untouched through an inflationary period systematically under-charges its higher-deductible policies. Second, the ground-up rate increase understates the increase needed on deductible business: at 20\% inflation, losses excess of the \$500 deductible grow by

\frac{2{,}400 \times 0.812}{2{,}000 \times 0.779} - 1 = +25.1\%

five points more than the ground-up trend.