Limited Expected Loss At A Stated Limit

What Limited Expected Loss Measures

A policy limit caps the amount paid for an individual claim. To price the capped coverage, the relevant average is the expected payment after applying that cap to each possible claim severity. This quantity is called the limited expected loss at the stated limit.

Let $X$ be the nonnegative severity of one claim, measured on a specified loss basis, and let $L\geq 0$ be the per-claim limit. The claim payment under this limit is $\min(X,L)$. This tutorial focuses on the expected value of that payment, calculated from the severity distribution. It does not add claim frequency, expenses, profit provisions, or other pricing components.

Definition: The limited expected loss at limit $L$ is $E[\min(X,L)]$, where $X$ is the severity of one claim on the stated loss basis. The limit is applied separately to each claim. The definition assumes $X\geq 0$ and that the expectation exists.

The severity basis matters. For example, $X$ might be ground-up claim severity before a deductible, or it might represent a different explicitly defined payment basis. A calculation is meaningful only when the stated distribution of $X$ matches the loss basis being limited. Unless specified otherwise, the examples below use hypothetical per-claim severities in thousands of dollars and no deductible.

Calculation From a Severity Distribution

For a discrete severity distribution, list the possible values of $X$, apply the limit to each value, and weight the resulting payments by their probabilities. For a continuous distribution with density $f_X$, separate severities at or below the limit from those above it. Severity at or below $L$ is paid in its actual amount; severity above $L$ is paid at $L$.

E[\min(X,L)] = \sum_j \min(x_j,L)P(X=x_j)

E[\min(X,L)] = \int_0^L x f_X(x)\,dx + L\,P(X>L)

The discrete expression applies when the distribution has probability masses at values $x_j$. The density expression applies to the continuous part of a nonnegative severity distribution; if there are atoms, their contributions must also be included. The boundary convention at $X=L$ does not change the result: whether that probability is grouped with the lower or upper part, the payment is $L$.

Why the Calculation Splits at the Limit

The formula follows directly from the payment definition. On the event $X\leq L$, the payment $\min(X,L)$ equals $X$. On the event $X>L$, it equals $L$. These two events partition all possible severities, so the expectation is the sum of the expected payment on each part.

\begin{aligned} E[\min(X,L)] &=E[\min(X,L)\mathbf{1}_{\{X\leq L\}}] +E[\min(X,L)\mathbf{1}_{\{X>L\}}]\\ &=E[X\mathbf{1}_{\{X\leq L\}}] +L\,P(X>L). \end{aligned}

For a continuous distribution with density, the first term is $\int_0^L x f_X(x)\,dx$. If there is probability mass at zero or at another value below $L$, that mass contributes its severity times its probability to the first term. The formula is not a claim that all severity models have a density; the discrete weighted-payment calculation remains valid for discrete distributions.

Worked Example: A Discrete Severity Distribution

A hypothetical insurer is evaluating a per-claim limit of $L=\$50{,}000$. Claim severity $X$, in thousands of dollars, has this distribution: $X=0$ with probability $0.10$, $X=20$ with probability $0.30$, $X=60$ with probability $0.40$, and $X=120$ with probability $0.20$. Calculate the limited expected loss.

Method. Apply the limit to each possible severity before taking the probability-weighted average. In thousands of dollars, the limited payments are $0$, $20$, $50$, and $50$, respectively.

The probability-weighted contributions are $0(0.10)=0$, $20(0.30)=6$, $50(0.40)=20$, and $50(0.20)=10$, all in thousands of dollars.

E[\min(X,50)] =0+6+20+10 =36

Thus, the limited expected loss is $\$36{,}000$ per claim. The probabilities sum to $0.10+0.30+0.40+0.20=1.00$, and the result is between zero and the limit of $\$50{,}000$, as required.

Common mistake avoided: The uncapped mean is $0(0.10)+20(0.30)+60(0.40)+120(0.20)=54$ thousand dollars. Capping that mean gives $\min(54,50)=50$ thousand dollars, which is not the limited expected loss. The limit must be applied to each severity first; the correct answer is $\$36{,}000$.

Worked Example: A Uniform Continuous Severity

For a hypothetical portfolio, suppose an individual claim severity $X$, in thousands of dollars, is uniformly distributed from 0 to 100. A per-claim limit of $L=40$ thousand dollars applies. Calculate the limited expected loss using the density and the probability above the limit.

Method. The uniform density is $f_X(x)=1/100$ for $0\leq x\leq100$. The probability above 40 is the interval length from 40 to 100 divided by the total interval length: $P(X>40)=(100-40)/100=0.60$.

The expected payment from severities up to the limit is $\int_0^{40}x(1/100)\,dx=(1/100)(40^2/2)=8$ thousand dollars. The payment from severities above the limit contributes $40(0.60)=24$ thousand dollars.

E[\min(X,40)] = 8+24 = 32 \quad\text{thousand dollars}

The limited expected loss is $\$32{,}000$ per claim. As a check, the answer cannot exceed the limit of $\$40{,}000$. The uncapped mean of this uniform distribution is $(0+100)/2=50$ thousand dollars, so the capped expected payment being below 40 thousand dollars is consistent with the distribution having a substantial share of claims below the limit.

Common mistake avoided: Do not use the full uncapped mean as the payment for claims above the limit. Those claims contribute $L$ times their probability, here $40(0.60)$, not $50(0.60)$ or their average uncapped severity.

Worked Example: A Probability Mass at Zero

Consider a hypothetical coverage where a claim has zero severity with probability $0.20$. Conditional on a positive claim, severity $X$ is uniform from 0 to 80 thousand dollars. The per-claim limit is $L=30$ thousand dollars. Find the limited expected loss per claim, including the probability of zero severity.

Method. First calculate the limited expectation conditional on a positive claim. Under the conditional uniform distribution, the density is $1/80$. The probability that this severity exceeds 30 is $(80-30)/80=50/80=0.625$.

For the conditional distribution, the contribution from severities up to 30 is $\int_0^{30}x(1/80)\,dx=30^2/(2\cdot80)=900/160=5.625$ thousand dollars. The contribution above 30 is $30(0.625)=18.75$ thousand dollars. The conditional limited expected loss is therefore $5.625+18.75=24.375$ thousand dollars.

A zero-severity outcome contributes zero. Positive severity occurs with probability $1-0.20=0.80$, so the unconditional limited expected loss is $0.80(24.375)=19.5$ thousand dollars.

E[\min(X,30)] = 0.20(0)+0.80(24.375) = 19.5 \quad\text{thousand dollars}

The answer is $\$19{,}500$ per claim. As a check, the unconditional probability of exceeding the limit is $0.80(0.625)=0.50$, so claims above the limit contribute $30(0.50)=15$ thousand dollars in total. The remaining contribution is $19.5-15=4.5$ thousand dollars from positive claims at or below the limit.

Common mistake avoided: The conditional severity model applies only when a positive claim occurs. Using its conditional capped mean of $\$24{,}375$ as the unconditional answer would omit the 20% probability of zero severity. Weight by the probability of a positive claim to obtain $\$19{,}500$.

Interpretation and Checks

Limited expected loss is an average of capped payments per claim, not necessarily an average per exposure or policy. If a pricing calculation also requires the expected number of claims, the frequency component must be supplied separately; it is not included in $E[\min(X,L)]$. Likewise, a result based on a severity distribution for reported claims should not silently be treated as a result for all exposures if the distribution was conditioned on a claim occurring.

For nonnegative severity and $L\geq0$, the payment always satisfies $0\leq\min(X,L)\leq L$. Therefore:

The limited expected loss must lie between 0 and $L$.
Increasing the limit cannot decrease the payment for any individual severity, so it cannot decrease the limited expected loss.
If all possible severities are at or below $L$, the limited expected loss equals $E[X]$.
If the distribution has probability above $L$, those outcomes each contribute exactly $L$ times their probability, not their uncapped severity.

These checks can reveal common calculation errors, but they do not replace matching the distribution to the stated loss basis. Keep track of whether severities are measured in dollars or thousands of dollars, apply the limit in the same units, and include every probability mass. The calculations here are aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.

Key takeaway: To find limited expected loss at limit $L$, cap each possible claim payment at $L$ and then average using the severity distribution. Do not cap the uncapped mean severity.

Check Your Understanding

For each hypothetical model, calculate the limited expected loss on the stated severity basis. Answers are in thousands of dollars unless stated otherwise.

Severity is 10 with probability 0.20, 30 with probability 0.50, and 70 with probability 0.30. The limit is 40. (Answer: 29 thousand dollars)
Severity is uniformly distributed from 0 to 60, and the limit is 24. (Answer: 19.2 thousand dollars)
Severity is zero with probability 0.25; conditional on a positive severity, it is uniform from 0 to 100. The limit is 40. (Answer: 24 thousand dollars)
Severity is 0, 25, 50, or 100 with respective probabilities 0.10, 0.20, 0.40, and 0.30. The limit is 50. (Answer: 40 thousand dollars)

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