LER for Uniform Severities

Why the Uniform Model Makes the Deductible Effect Visible

A deductible removes some ground-up loss from the insurer’s payment. The loss elimination ratio (LER) measures the expected amount removed as a proportion of the expected ground-up loss. With a uniform severity, the density is constant across its support, so the effects of a deductible can be calculated directly and the role of the support endpoints is especially clear.

The key calculation is the limited expected value, the expected value of the smaller of the loss and the deductible. Once that amount is found, dividing by the mean severity gives the LER. This tutorial considers a single ground-up loss with a standard deductible and no policy limit, coinsurance, interest, or mortality component.

Definition: Let $X$ be a nonnegative ground-up loss with $E[X]>0$, and let $d\geq 0$ be an ordinary deductible. The limited expected value at $d$ is $E[\min(X,d)]$. The loss elimination ratio is $$ \operatorname{LER}(d)=\frac{E[\min(X,d)]}{E[X]}. $$ The insurer’s payment per loss is $(X-d)_+=\max(X-d,0)$. Since $X=\min(X,d)+(X-d)_+$, its expected payment is $E[X]-E[\min(X,d)]$.

The numerator represents the portion of each loss retained by the insured, capped at the deductible. A loss below $d$ contributes its full amount to that numerator; a loss above $d$ contributes $d$. The denominator is the average ground-up loss, not the average insurer payment. Thus, with a deductible and no other coverage modification, the expected insurer payment as a proportion of mean ground-up loss is $1-\operatorname{LER}(d)$.

Limited Expected Value for a Uniform Severity

Assume $X\sim\operatorname{Uniform}(a,b)$, where $0\leq a<b$. Its density is $1/(b-a)$ on $[a,b]$, and its mean is $E[X]=(a+b)/2$. The deductible’s position relative to $a$ and $b$ determines the limited expected value.

Result: For $X\sim\operatorname{Uniform}(a,b)$ and $d\geq 0$, $$ E[\min(X,d)]= \begin{cases} d, & 0\leq d\leq a,\\[4pt] \dfrac{2bd-d^2-a^2}{2(b-a)}, & a<d<b,\\[8pt] \dfrac{a+b}{2}, & d\geq b. \end{cases} $$ Therefore, divide the applicable expression by $(a+b)/2$ to obtain the LER.

The three cases have simple interpretations. If $d\leq a$, every possible loss is at least the deductible, so the amount eliminated from each loss is exactly $d$. If $d\geq b$, every possible loss is at or below the deductible, so the entire loss is eliminated from the insurer’s payment. Only when $a<d<b$ do some losses fall below the deductible and others exceed it.

Deriving the Within-Support Formula

For $a<d<b$, losses from $a$ through $d$ contribute their actual value to $\min(X,d)$, while losses above $d$ contribute $d$. Integrating these contributions against the uniform density gives

\begin{aligned} E[\min(X,d)] &=\int_a^d \frac{x}{b-a}\,dx+\int_d^b\frac{d}{b-a}\,dx\\ &=\frac{d^2-a^2}{2(b-a)}+\frac{d(b-d)}{b-a}\\ &=\frac{2bd-d^2-a^2}{2(b-a)}. \end{aligned}

The first integral accounts for losses that do not exceed the deductible; the second accounts for losses that do. This split is why simply multiplying the deductible by a probability is not the correct general calculation: losses below the deductible contribute varying amounts, not all $d$.

For the commonly used special case $X\sim\operatorname{Uniform}(0,b)$, the within-support formula simplifies to

E[\min(X,d)]=d-\frac{d^2}{2b},\qquad \operatorname{LER}(d)=\frac{2d}{b}-\left(\frac{d}{b}\right)^2,\qquad 0\leq d<b.

At $d=0$, the LER is zero. At $d=b$, the expression gives one, matching the above-support case. The endpoints therefore agree with the limiting values of the piecewise formula.

Identify the support and mean.
Write $a$, $b$, and $E[X]=(a+b)/2$ from the stated severity model.

Locate the deductible.
Determine whether $d\leq a$, $a<d<b$, or $d\geq b$. Use the corresponding limited expected value.

Calculate the ratio and check it.
Divide by $E[X]$. The eliminated amount should not exceed the mean loss, and the expected insurer payment should equal $E[X]-E[\min(X,d)]$.

Worked Calculations

Worked Example: A Deductible Within a Zero-Based Support

A hypothetical coverage has ground-up severity $X\sim\operatorname{Uniform}(0,1200)$, with amounts measured in dollars. The ordinary deductible is $d=\$300$, with no policy limit. Calculate the limited expected value, LER, and expected insurer payment. No interest or mortality assumptions apply to this single-loss severity calculation.

Step 1: Find the mean. Here $a=0$, $b=1200$, and $0<300<1200$, so the deductible is within the support. The mean ground-up loss is

E[X]=\frac{0+1200}{2}=\$600.

Step 2: Find the limited expected value. Apply the within-support formula:

E[\min(X,300)] =\frac{2(1200)(300)-300^2-0^2}{2(1200-0)} =\frac{720000-90000}{2400} =\frac{630000}{2400} =\$262.50.

Step 3: Calculate the LER and insurer payment.

\operatorname{LER}(300)=\frac{262.50}{600}=0.4375=43.75\%, \qquad E[(X-300)_+]=600-262.50=\$337.50.

Check. The two expected amounts sum to the mean loss: $\$262.50+\$337.50=\$600$. The result says that 43.75% of expected ground-up loss is eliminated by the deductible. The method is to calculate the limited expected value before dividing; this avoids mistaking the probability of being below the deductible for the fraction of loss eliminated.

Worked Example: A Deductible Inside a Positive Support

A hypothetical equipment-loss severity is $X\sim\operatorname{Uniform}(200,1000)$ dollars. The policy has an ordinary deductible $d=\$600$, and no other coverage modification. Find the LER and expected insurer payment. This is a severity calculation, so no interest or mortality assumptions are needed.

Step 1: Determine the case and mean. Since $200<600<1000$, use the within-support formula. The mean loss is

E[X]=\frac{200+1000}{2}=\$600.

Step 2: Calculate the limited expected value.

\begin{aligned} E[\min(X,600)] &=\frac{2(1000)(600)-600^2-200^2}{2(1000-200)}\\ &=\frac{1200000-360000-40000}{1600}\\ &=\frac{800000}{1600} =\$500. \end{aligned}

Step 3: Calculate the LER and insurer payment.

\operatorname{LER}(600)=\frac{500}{600}=\frac{5}{6}=0.8333\ldots, \qquad E[(X-600)_+]=600-500=\$100.

The LER is $83.33\%$ when expressed as a percentage to two decimal places. As a check, the proportion of losses above the deductible is $(1000-600)/(1000-200)=1/2$, and their average excess above $600$ is $\$200$. Thus the expected insurer payment is $(1/2)(\$200)=\$100$, consistent with the subtraction calculation. The method avoids treating the deductible itself as the eliminated amount for every loss: losses below $\$600$ contribute only their actual amounts.

Worked Example: Comparing an Interior Deductible with One Above the Support

A hypothetical repair-cost severity is $X\sim\operatorname{Uniform}(100,500)$ dollars. Compare an ordinary deductible of $\$350$ with one of $\$550$, calculating the LER and expected insurer payment for each. There is no policy limit; interest and mortality are not part of this single-loss model.

Step 1: Find the common mean.

E[X]=\frac{100+500}{2}=\$300.

Step 2: Evaluate the $\$350$ deductible. Because $100<350<500$, use the within-support formula:

\begin{aligned} E[\min(X,350)] &=\frac{2(500)(350)-350^2-100^2}{2(500-100)}\\ &=\frac{350000-122500-10000}{800}\\ &=\frac{217500}{800} =\$271.875. \end{aligned}

The expected eliminated amount is $\$271.875$, the expected insurer payment is $\$300-\$271.875=\$28.125$, and

\operatorname{LER}(350)=\frac{271.875}{300}=0.90625=90.625\%.

Step 3: Evaluate the $\$550$ deductible. Since $550\geq500$, every possible loss is below the deductible. Therefore,

E[\min(X,550)]=E[X]=\$300,\qquad \operatorname{LER}(550)=\frac{300}{300}=1,\qquad E[(X-550)_+]=\$0.

Check. For the first deductible, $\$271.875+\$28.125=\$300$; for the second, all $\$300$ of mean loss is eliminated. The method changes at the upper support endpoint. Extending the interior formula beyond $b$ is unnecessary and can obscure the fact that the insurer payment is exactly zero once the deductible is at least the maximum possible loss.

Common Exam Trap: Confusing the LER with a Probability

The LER is a ratio of expected loss amounts. It is not the probability that a loss is below the deductible. For example, in the first worked example, $X\sim\operatorname{Uniform}(0,1200)$ and $d=300$. The probability that the loss is at most the deductible is

P(X\leq300)=\frac{300}{1200}=0.25.

A mistaken calculation might therefore report $\operatorname{LER}=0.25$. That is the wrong quantity: it counts how often a loss is below $d$, not how much expected loss is eliminated across all losses. The correct limited expected value is $\$262.50$, giving

\operatorname{LER}(300)=\frac{262.50}{600}=0.4375.

The correction is to average $\min(X,d)$ over the entire severity distribution, then divide by the ground-up mean. Some losses exceed the deductible and contribute the full deductible to the numerator, while losses below it contribute different amounts.

Using the Piecewise Result Reliably

Before substituting numbers, verify that the deductible and support use the same monetary basis and units. Then identify the appropriate case. For a uniform severity, the expected insurer payment can be obtained either by evaluating $E[(X-d)_+]$ directly or by subtracting the limited expected value from $E[X]$. The latter is often the fastest check, and the two expected amounts must add to the ground-up mean.

Deductible position	Limited expected value	Interpretation
$0\leq d\leq a$	$d$	Every possible loss reaches the deductible.
$a<d<b$	$\dfrac{2bd-d^2-a^2}{2(b-a)}$	Some losses are below $d$, and some exceed it.
$d\geq b$	$\dfrac{a+b}{2}$	Every possible loss is at or below the deductible.

This piecewise treatment is specific to a continuous uniform ground-up severity. The formulas depend on its constant density and bounded support; they should not be transferred unchanged to other severity distributions. For a uniform model, however, they provide a direct route from support and deductible to limited expected value, LER, and expected insurer payment.

Key takeaway: For $X\sim\operatorname{Uniform}(a,b)$, first compare the deductible with both support endpoints. Calculate $E[\min(X,d)]$ from the matching case, then divide by $(a+b)/2$. Do not substitute a probability for the expected amount eliminated.

Check Your Understanding

For each hypothetical single-loss model, calculate the requested LER and, where asked, the expected insurer payment. Assume an ordinary deductible and no policy limit or other coverage modification; interest and mortality are not part of these severity calculations.

$X\sim\operatorname{Uniform}(0,800)$ dollars and $d=\$200$. Find the limited expected value and LER. (Answer: limited expected value $\$175.00$; LER $0.4375$.)
$X\sim\operatorname{Uniform}(100,700)$ dollars and $d=\$250$. Find the LER and expected insurer payment. (Answer: LER $0.5781$; expected insurer payment $\$168.75$.)
$X\sim\operatorname{Uniform}(300,900)$ dollars and $d=\$150$. Find the LER and expected insurer payment. (Answer: LER $0.2500$; expected insurer payment $\$450.00$.)
$X\sim\operatorname{Uniform}(250,750)$ dollars and $d=\$900$. Find the limited expected value and LER. (Answer: limited expected value $\$500.00$; LER $1.0000$.)
$X\sim\operatorname{Uniform}(50,650)$ dollars and $d=\$500$. Find the limited expected value, LER, and expected insurer payment. (Answer: limited expected value $\$331.25$; LER $0.9464$; expected insurer payment $\$18.75$.)

Deductible position	Limited expected value	Interpretation
\(0\leq d\leq a\)	\(d\)	Every possible loss reaches the deductible.
\(a<d<b\)	\(\dfrac{2bd-d^2-a^2}{2(b-a)}\)	Some losses are below \(d\), and some exceed it.
\(d\geq b\)	\(\dfrac{a+b}{2}\)	Every possible loss is at or below the deductible.

What You'll Learn