Limited Expected Value for Deductible Loss Costs

Why Limited Expected Value Matters

A deductible changes the amount an insurer pays on each claim, but it does not change the ground-up severity distribution itself. A useful way to measure the portion of severity retained below a deductible is the limited expected value. Once that quantity is known, the expected insurer payment under an ordinary deductible follows from a simple subtraction. The subtraction is valid because it holds claim by claim, not merely as an approximation.

Let $X$ be a nonnegative ground-up loss and let $d\geq 0$ be an ordinary per-claim deductible. Assume $E[X]$ is finite when calculating expected payments. The insurer payment for one claim is $(X-d)_+=\max(X-d,0)$. This tutorial develops the limited expected value, shows how to compute it, and clarifies a frequent error: subtracting the deductible from losses below it without flooring the result at zero.

Definition: The limited expected value of severity $X$ at deductible $d$ is $$ \operatorname{LEV}(d)=E[\min(X,d)]. $$ It is the expected value of the loss capped at $d$. It is not the expected insurer payment under an ordinary deductible.

Calculating the Limited Expected Value

For a nonnegative severity, the limited loss $\min(X,d)$ never exceeds $d$. Its expectation can be calculated directly from the distribution. For a discrete distribution, multiply each severity below $d$ by its probability and replace each severity at or above $d$ by $d$. For a continuous distribution with density $f_X$, include the density-weighted losses below $d$ and the probability of losses at or above $d$, each capped at $d$.

\operatorname{LEV}(d) = \int_0^d x f_X(x)\,dx + d\,P(X\geq d)

This density expression applies when the distribution has a density. For a discrete severity with possible values $x_j$ and probabilities $p_j$, the corresponding calculation is

\operatorname{LEV}(d) = \sum_j \min(x_j,d)\,p_j.

A general expression, which also accommodates distributions with atoms, is

\operatorname{LEV}(d) = \int_0^d P(X>t)\,dt.

To see why, fix a possible loss value $x\geq0$. The quantity $\min(x,d)$ is the length of the interval from 0 to $d$ over which $x>t$. Equivalently,

\min(x,d)=\int_0^d \mathbf{1}_{\{x>t\}}\,dt.

Taking expectations on both sides gives the general expression: the expectation of the indicator $\mathbf{1}_{\{X>t\}}$ is $P(X>t)$. The possible equality $X=d$ does not create a problem; the single endpoint has zero length in the integral. This representation is also useful when no density exists.

Worked Example: Limited Expected Value for a Uniform Severity

A hypothetical equipment policy has ground-up claim severity $X$ uniformly distributed from \$0 to \$6,000. The contract has an ordinary deductible of \$2,000. Find the limited expected value and the expected insurer payment per claim. State the assumed distribution and parameters.

Method. For $X\sim\operatorname{Uniform}(0,6000)$, the density is $f_X(x)=1/6000$ on $0\leq x\leq6000$, and $E[X]=(0+6000)/2=\$3{,}000$. The deductible is $d=\$2{,}000$.

The probability of a loss at or above the deductible is $P(X\geq2000)=(6000-2000)/6000=2/3$. The part of the limited expectation contributed by losses below the deductible is

\int_0^{2000} x\left(\frac{1}{6000}\right)\,dx = \frac{1}{6000}\left[\frac{x^2}{2}\right]_0^{2000} = \frac{2{,}000{,}000}{6{,}000} = \$333.333\ldots

The capped contribution from losses at or above the deductible is $2000(2/3)=\$1{,}333.333\ldots$. Therefore,

\operatorname{LEV}(2000) = 333.333\ldots+1{,}333.333\ldots = \$1{,}666.666\ldots

The expected insurer payment is $E[X]-\operatorname{LEV}(2000)=3000-1666.666\ldots=\$1{,}333.333\ldots$. Rounding only at the final step to cents, the limited expected value is \$1,666.67 and the expected payment is \$1,333.33. The check is that these two amounts sum to the ground-up mean of \$3,000. The method avoids treating the deductible as a subtraction from every loss, including losses below it.

Why Subtracting LEV Gives the Ordinary-Deductible Payment

The connection between limited expected value and insurer payment comes from a pointwise identity. For every possible loss $X$, split into the two cases $X\leq d$ and $X>d$. If $X\leq d$, then $X-\min(X,d)=X-X=0$, which is the insurer payment. If $X>d$, then $X-\min(X,d)=X-d$, also the insurer payment. Thus the identity holds for every claim outcome:

(X-d)_+=X-\min(X,d).

Taking expectations, using the finite-mean assumption, gives the main result.

Theorem: For a nonnegative ground-up severity $X$ with finite mean and an ordinary per-claim deductible $d\geq0$, $$ E[(X-d)_+]=E[X]-\operatorname{LEV}(d). $$ The result is an expected payment per claim. It does not, by itself, give an annual aggregate amount.

This identity explains why LEV is sometimes described as the expected loss removed by the deductible: for each claim, the ground-up loss is divided into the part $\min(X,d)$ and the insurer-paid part $(X-d)_+$. Their sum is exactly $X$. Taking expectations preserves that sum.

Worked Example: Discrete Severity and the Below-Deductible Trap

For a hypothetical coverage, ground-up severity is \$500 with probability 0.20, \$2,000 with probability 0.50, and \$5,000 with probability 0.30. The ordinary deductible is \$1,500. Calculate the ground-up expected loss, LEV, and expected insurer payment per claim.

Method. First calculate the ground-up mean:

E[X] = 0.20(500)+0.50(2000)+0.30(5000) = 100+1000+1500 = \$2{,}600.

For LEV, cap each severity at \$1,500. The three limited amounts are \$500, \$1,500, and \$1,500, so

\operatorname{LEV}(1500) = 0.20(500)+0.50(1500)+0.30(1500) = 100+750+450 = \$1{,}300.

Subtracting gives an expected insurer payment of $2600-1300=\$1{,}300$ per claim. Directly checking the payment function gives payments of \$0, \$500, and \$3,500, respectively, so

0.20(0)+0.50(500)+0.30(3500) = 0+250+1050 = \$1{,}300.

Common mistake. Writing $0.20(500-1500)+0.50(2000-1500)+0.30(5000-1500)=\$1{,}100$ incorrectly assigns a negative \$1,000 payment to the \$500 loss. Under an ordinary deductible, that claim produces zero insurer payment, not a negative payment. The correction is to apply the positive-part function to each claim: $\max(X-1500,0)$. The direct check confirms the correct result is \$1,300.

From Per-Claim Payment to Expected Aggregate Payment

LEV describes the deductible’s effect on one severity. To calculate an expected aggregate amount over a period, a claim count assumption is also needed. If $N$ is the number of claims, each claim has the same severity distribution, and the severities are independent of $N$, then the expected aggregate insurer payment is $E[N]E[(X-d)_+]$. This follows by conditioning on the claim count: given $N=n$, the expected sum of $n$ payments is $nE[(X-d)_+]$, and averaging over $N$ gives the product. Independence between claim severities is not required for this expectation if each claim has the same expected payment conditional on $N$; independence of frequency and severity is the key assumption in the stated product form.

Worked Example: Expected Annual Payment With a Claim Frequency Assumption

A hypothetical portfolio has an expected annual claim count of 2.4. Each claim has severity \$500 with probability 0.60, \$2,000 with probability 0.10, or \$5,000 with probability 0.30. Assume claim severities are independent of the count and have this same distribution for each claim. With a \$1,500 ordinary deductible, find the expected payment per claim and expected annual aggregate payment.

Method. The payment for the three severity outcomes is $\max(X-1500,0)$, giving \$0, \$500, and \$3,500. Therefore the expected payment per claim is

E[(X-1500)_+] = 0.60(0)+0.10(500)+0.30(3500) = 0+50+1050 = \$1{,}100.

As a check through LEV, the ground-up mean is $0.60(500)+0.10(2000)+0.30(5000)=300+200+1500=\$2{,}000$. The LEV is $0.60(500)+0.10(1500)+0.30(1500)=300+150+450=\$900$. The difference is $2000-900=\$1{,}100$, matching the direct payment calculation.

Using the assumed expected annual count, expected annual aggregate payment is $2.4(\$1{,}100)=\$2{,}640$. This multiplication uses expected frequency and expected payment per claim; it does not mean that 2.4 claims occur in any particular year. The final amounts are \$1,100 per claim and \$2,640 per year in expectation.

Interpretation and Exam Traps

LEV is increasing in $d$: raising the cap cannot reduce $\min(X,d)$ for any loss value. At $d=0$, $\operatorname{LEV}(0)=0$. As $d$ increases without bound, $\operatorname{LEV}(d)$ approaches $E[X]$ when the mean is finite. These checks can help catch an incorrectly coded or calculated limited value. In particular, an LEV cannot exceed $E[X]$, because $\min(X,d)\leq X$ for nonnegative losses.

The formula $E[X]-\operatorname{LEV}(d)$ applies to an ordinary deductible, whose payment function is $(X-d)_+$. Do not substitute a different contract payment function without deriving the matching relationship. Also distinguish the expected payment per claim from an expected portfolio total: the latter requires a claim-count assumption. When using a discrete severity table, apply the cap to each outcome before weighting; do not subtract $d$ from all possible losses and allow negative terms to remain.

Key takeaway: Calculate $\operatorname{LEV}(d)=E[\min(X,d)]$ by capping each loss at the deductible. For an ordinary deductible, subtract that limited expected value from ground-up expected severity to obtain expected insurer payment per claim.

Check Your Understanding

For each problem, identify the limited value or ordinary-deductible payment requested. Answers are rounded only where needed.

A hypothetical severity is \$100 with probability 0.25, \$1,000 with probability 0.50, and \$3,000 with probability 0.25. At deductible \$800, find the ground-up mean, LEV, and expected insurer payment. (Answer: ground-up mean \$1,275; LEV \$625; expected payment \$650.)
Let $X\sim\operatorname{Uniform}(0,4000)$, and let $d=\$1,000$. Find the LEV and expected insurer payment. (Answer: LEV \$875; expected payment \$1,125.)
A hypothetical severity is \$1,000 with probability 0.30, \$2,500 with probability 0.40, and \$6,000 with probability 0.30. For a \$2,000 ordinary deductible, calculate LEV and expected insurer payment. (Answer: LEV \$1,700; expected payment \$1,400.)
The expected claim count is 1.8 per year, independent of severity. Each claim is \$1,000 with probability 0.60 or \$3,000 with probability 0.40. Find expected annual insurer payment with a \$1,000 ordinary deductible. (Answer: expected payment per claim \$800; expected annual aggregate payment \$1,440.)
A hypothetical severity is \$500 with probability 0.20, \$1,500 with probability 0.50, and \$4,000 with probability 0.30. At deductible \$2,500, find LEV and expected insurer payment. (Answer: LEV \$1,600; expected payment \$450.)

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