Expected Payment Under a Deductible and Policy Limit

Introduction

A deductible and a policy limit both modify the amount paid on a claim, but they act at different points in the loss range. The deductible leaves the initial portion of a loss with the policyholder; a payment limit caps what the insurer pays after the deductible. A useful way to calculate the expected payment is to view it as the expected loss in a layer of the severity distribution.

This layer interpretation leads to a compact formula: expected payment equals the limited expected value at the deductible-plus-limit threshold, less the limited expected value at the deductible. The threshold is $d+u$, not $u$, because a payment of $u$ is reached only after the loss has passed through the deductible.

Definition: Let $X$ be a nonnegative loss, $d\geq 0$ an ordinary deductible, and $u\geq 0$ a limit on the insurer’s payment. The payment per loss is $Y^L=\min((X-d)^+,u)$, where $z^+=\max(z,0)$. For $a\geq0$, the limited expected value of $X$ at threshold $a$ is $\operatorname{LEV}_X(a)=E[\min(X,a)]$.

The payment is zero when $X\leq d$, increases with the loss between $d$ and $d+u$, and equals $u$ when $X\geq d+u$. These definitions use an ordinary deductible and a cap on the payment itself. Assume $X$ is nonnegative; no assumption that $X$ has a continuous distribution is required.

The Limited Expected Value Identity

For any nonnegative loss $X$, the expected payment can be calculated directly from two limited expected values:

Theorem: If $d\geq0$ and $u\geq0$, then $$ E[Y^L]=E[\min((X-d)^+,u)] =\operatorname{LEV}_X(d+u)-\operatorname{LEV}_X(d). $$

The result follows from a pointwise identity. For each possible loss amount $x\geq0$, subtracting $\min(x,d)$ from $\min(x,d+u)$ isolates exactly the part of the loss between the deductible and the deductible-plus-limit threshold.

\min((x-d)^+,u)=\min(x,d+u)-\min(x,d),\qquad x\geq0.

To verify it, consider the three ranges of $x$. If $x\leq d$, both sides are zero. If $d<x<d+u$, the right side is $x-d$, which is also the payment. If $x\geq d+u$, the right side is $(d+u)-d=u$, the capped payment. Taking expectations gives the theorem. Both terms on the right are bounded, so this argument does not require $E[X]$ to be finite.

For a nonnegative severity, the limited expected value also has a survival-function representation:

\operatorname{LEV}_X(a)=\int_0^a S_X(t)\,dt, \qquad E[Y^L]=\int_d^{d+u}S_X(t)\,dt.

Here $S_X(t)=P(X>t)$. The first identity follows by writing $\min(X,a)$ as the area under the indicator that the loss exceeds $t$, from $0$ to $a$, and then taking expectations. Subtracting the integrals at $d+u$ and $d$ gives the second identity. This representation is valid for discrete and mixed distributions as well as continuous ones; individual point masses do not change the integral.

Identify the coverage thresholds.
The payment starts above $d$ and reaches its cap $u$ at a loss of $d+u$.

Evaluate both limited expected values.
Calculate $\operatorname{LEV}_X(d+u)$ and $\operatorname{LEV}_X(d)$ using the severity model or the loss data.

Subtract in the correct order.
Expected payment is the higher-threshold limited expected value minus the lower-threshold one.

Worked Calculations

Worked Example: Exponential Loss Severity

A hypothetical policy covers one claim with an ordinary deductible of $200 and a payment limit of $500. Assume the loss $X$ has an exponential distribution with mean $1,000, so $S_X(x)=e^{-x/1000}$ for $x\geq0$. Calculate the expected payment per loss.

Method. Use the difference of limited expected values. For an exponential loss with mean $\theta$, $$ \operatorname{LEV}_X(a)=\int_0^a e^{-t/\theta}\,dt =\theta(1-e^{-a/\theta}). $$ The model parameter here is $\theta=1000$, and the two thresholds are $d=200$ and $d+u=200+500=700$.

First evaluate the limited expected value at $700: $$ \operatorname{LEV}_X(700)=1000(1-e^{-700/1000}) =1000(1-e^{-0.7}) \approx1000(1-0.4965853) =\$503.4147. $$ Then evaluate it at $200: $$ \operatorname{LEV}_X(200)=1000(1-e^{-200/1000}) =1000(1-e^{-0.2}) \approx1000(1-0.8187308) =\$181.2692. $$ Subtract: $$ E[Y^L]=503.4147-181.2692=\$322.1455. $$

The expected payment is $322.15, rounded to the nearest cent. As a check, the survival-integral form gives $1000(e^{-0.2}-e^{-0.7})=\$322.1455$, the same result. The common mistake avoided here is using $u=500$ as the upper loss threshold: the correct threshold is $d+u=700$.

Worked Example: Discrete Severity With a Payment Cap

For a hypothetical equipment policy, the loss $X$, in dollars, has the following discrete distribution. The contract has a deductible of $100 and a payment limit of $250. Find the expected payment per loss.

Loss $x$	Probability $P(X=x)$
$0	0.20
$100	0.30
$300	0.30
$600	0.20

Method. The limited-value thresholds are $d=100$ and $d+u=100+250=350$. At $350, the limited loss values for losses $0, $100, $300, and $600 are respectively $0, $100, $300, and $350. Therefore, $$ \operatorname{LEV}_X(350) =0(0.20)+100(0.30)+300(0.30)+350(0.20) =0+30+90+70 =\$190. $$ At $100, the limited loss values are $0, $100, $100, and $100. Thus, $$ \operatorname{LEV}_X(100) =0(0.20)+100(0.30)+100(0.30)+100(0.20) =0+30+30+20 =\$80. $$ The expected payment is $$ E[Y^L]=190-80=\$110. $$

A direct check gives payments of $0, $0, $200, and $250 at the four loss values. Their expected value is $0(0.20)+0(0.30)+200(0.30)+250(0.20)=60+50=\$110$. This check confirms the difference-of-limited-values calculation, including at the deductible boundary.

Worked Example: Applying the Formula to Observed Losses

An actuary uses five hypothetical observed claim losses, in dollars, to estimate the average payment under a deductible of $200 and a payment limit of $500. The observed losses are $0, $150, $400, $900, and $1,200. Treat the observations as equally weighted and calculate the empirical expected payment.

Method. For equally weighted observations $x_1,\ldots,x_n$, estimate $\operatorname{LEV}_X(a)$ by averaging $\min(x_i,a)$. The upper threshold is $d+u=200+500=700$.

At $700, the limited observations are $0, $150, $400, $700, and $700: $$ \widehat{\operatorname{LEV}}_X(700) =\frac{0+150+400+700+700}{5} =\frac{1950}{5} =\$390. $$ At $200, they are $0, $150, $200, $200, and $200: $$ \widehat{\operatorname{LEV}}_X(200) =\frac{0+150+200+200+200}{5} =\frac{750}{5} =\$150. $$ Consequently, the empirical estimate of expected payment is $$ \widehat{E[Y^L]}=390-150=\$240. $$

Check by calculating payments individually: the five payments are $0, $0, $200, $500, and $500. Their average is $(0+0+200+500+500)/5=\$240$. The estimate is therefore $240.00 per observed loss, rounded to the nearest cent. This is an empirical average, not a claim that the five observations determine the exact expected payment under an unknown population model.

Using the Formula Correctly

The order and location of the thresholds are essential. A deductible shifts the beginning of the covered layer to $d$; a payment limit of $u$ makes the layer end at $d+u$. The limited expected value at the higher threshold includes all loss up to that point, while the limited expected value at $d$ removes the portion below the layer.

Worked Example: Spotting a Wrong Upper Threshold

Return to an exponential hypothetical loss with mean $1,000, deductible $d=200$, and payment limit $u=500$. A proposed calculation uses $\operatorname{LEV}_X(500)-\operatorname{LEV}_X(200)$. Identify the error and calculate the correct expected payment.

The proposed calculation uses the limit $u=500$ as though it were the maximum loss at which payment reaches the cap. But a loss must first exceed the $200 deductible, so the payment reaches $500 only at a loss of $200+500=\$700$. The proposed result is $$ \operatorname{LEV}_X(500)-\operatorname{LEV}_X(200) =1000(1-e^{-0.5})-1000(1-e^{-0.2}) \approx393.4693-181.2692 =\$212.2001, $$ which is not the expected payment for this contract.

The correct calculation uses $700: $$ \operatorname{LEV}_X(700)-\operatorname{LEV}_X(200) =503.4147-181.2692 =\$322.1455. $$ The expected payment is therefore $322.15, rounded to the nearest cent. The lower proposed result corresponds to taking a loss layer from $200 to $500, only $300 wide, rather than the intended $500-wide layer from $200 to $700.

A related error is to calculate $E[\min(X,u)]-E[\min(X,d)]$. Even if this expression is numerically defined, its upper threshold is still wrong when $u$ is the payment limit. Another error is to omit the positive part and use $X-d$ below the deductible; that would assign negative payments to losses less than $d$. The limited expected value identity avoids both problems when the contract terms are translated into the correct thresholds.

Key takeaway: For a nonnegative loss and an ordinary deductible $d$ with a payment cap $u$, calculate expected payment as $\operatorname{LEV}_X(d+u)-\operatorname{LEV}_X(d)$. The formula applies to continuous, discrete, and mixed severity distributions.

Check Your Understanding

For each hypothetical situation, find the expected payment or empirical estimate using the limited expected value identity. Round money to the nearest cent where needed.

A loss is exponential with mean $800. A policy has deductible $100 and payment limit $300. Calculate the expected payment using the two limited expected values. (Answer: $\$220.77$)
A discrete loss has values $0, $200, $500, and $800 with respective probabilities 0.10, 0.40, 0.30, and 0.20. For a $150 deductible and $400 payment limit, calculate $\operatorname{LEV}_X(550)$, $\operatorname{LEV}_X(150)$, and the expected payment. (Answer: $\operatorname{LEV}_X(550)=\$340$, $\operatorname{LEV}_X(150)=\$135$, expected payment $=\$205$)
Four equally weighted observed losses are $100, $300, $700, and $1,000. Estimate expected payment with a $250 deductible and a $400 payment limit by calculating both empirical limited expected values. (Answer: $\widehat{\operatorname{LEV}}_X(650)=\$425$, $\widehat{\operatorname{LEV}}_X(250)=\$212.50$, estimated payment $=\$212.50$)
A hypothetical loss is zero with probability 0.25; with probability 0.75 it is exponential with mean $600. A policy has a $100 deductible and a $500 payment limit. Use the mixture structure and the limited expected value identity to find expected payment. (Answer: $0.75[600(e^{-100/600}-e^{-600/600})]=\$215.37$)

What You'll Learn