Deriving Combined Deductible-and-Limit Costs

Introduction

A deductible removes the first part of each covered loss from the insurer’s payment, while a limit caps the amount the insurer pays. When both apply, the expected payment depends on where the limit is measured: it may be an insurer-payment limit above the deductible, or a total limit on the covered loss before the deductible is subtracted. Confusing these interpretations shifts the upper endpoint of the calculation.

The severity survival function provides a direct way to calculate the expected payment. The key is to view the insurer’s payment as a layer of loss amounts: the insurer pays at level $t$ only when the loss exceeds the deductible plus $t$. This tutorial derives that result and applies it to hypothetical contracts. No interest or mortality assumption is needed; losses are measured in fixed dollars.

Definition: Let $X$ be a nonnegative loss, $d\geq 0$ an ordinary deductible, and $S_X(x)=P(X>x)$ the severity survival function. If the insurer’s payment limit is $u\geq 0$ above the deductible, the payment per loss is $$ Y=\min\big((X-d)_+,u\big), \qquad (z)_+=\max(z,0). $$ Thus the insurer pays nothing when $X\leq d$, pays $X-d$ when $d<X<d+u$, and pays $u$ when $X\geq d+u$.

Expected Payment as a Survival Integral

For any nonnegative random variable bounded by $u$, its expectation can be written as the integral of its tail probabilities over payment levels from zero to $u$. Applied to $Y$, a payment greater than $t$ occurs precisely when the loss exceeds $d+t$, for $0\leq t<u$. Therefore,

P(Y>t)=P(X>d+t)=S_X(d+t), \qquad E[Y]=\int_0^u P(Y>t)\,dt =\int_0^u S_X(d+t)\,dt =\int_d^{d+u}S_X(x)\,dx.

This is the expected cost per loss for the combined contract. The integral covers the loss layer from $d$ through $d+u$: the deductible determines the attachment point, and the above-deductible limit determines the layer width. The identity works for continuous or discrete severity distributions; atoms at endpoints do not change the integral.

Result: For an ordinary deductible $d$ and a limit $u$ that caps the insurer’s payment above the deductible, $$ E[Y]=\int_d^{d+u}S_X(x)\,dx. $$ If the payment is not capped, the corresponding deductible-only expectation is $E[(X-d)_+]=\int_d^\infty S_X(x)\,dx$, when that expectation is finite.

The derivation follows from the tail-integral identity. For a fixed payment $Y$, the length of the interval of payment levels $t\geq 0$ for which $Y>t$ equals $Y$. In symbols, $Y=\int_0^u \mathbf{1}_{\{Y>t\}}\,dt$, where $\mathbf{1}$ is one when its condition holds and zero otherwise. Taking expectations gives $E[Y]=\int_0^u P(Y>t)\,dt$. Since $Y>t$ exactly when $X>d+t$, substituting the survival probability and changing variables to $x=d+t$ gives the stated formula.

What Does the Limit Measure?

A contract’s limit wording determines the upper endpoint. If $u$ is the insurer’s maximum payment above the deductible, the insurer pays at most $u$, and the loss threshold at which the cap is reached is $d+u$. If instead $L$ is a total limit on the loss amount before the deductible is applied, with $L\geq d$, the payment is $\min((X-d)_+,L-d)$. The maximum insurer payment is then $L-d$, not $L$.

Limit interpretation	Payment	Expected payment
$u$ above the deductible	$\min((X-d)_+,u)$	$\int_d^{d+u} S_X(x)\,dx$
Total loss limit $L$, where $L\geq d$	$\min((X-d)_+,L-d)$	$\int_d^L S_X(x)\,dx$

If $L\leq d$, the total loss limit does not exceed the deductible, so the insurer’s payment is zero. If contract wording is ambiguous, do not infer the interpretation from the number alone: identify whether the stated limit caps the insurer’s payment or the loss amount before the deductible.

Worked Example: Exponential Severity with a Limit Above the Deductible

A hypothetical equipment policy has an ordinary deductible of $5,000 and an insurer-payment limit of $15,000 above that deductible. Suppose the loss severity is exponential with mean $10,000, so $S_X(x)=e^{-x/10{,}000}$ for $x\geq 0$. Find the expected insurer payment per loss. The loss amounts are fixed dollars; there is no discounting.

Method. The limit is stated above the deductible, so the payment layer runs from $d=\$5{,}000$ to $d+u=\$20{,}000$. For exponential severity, integrate the survival function:

E[Y]=\int_{5{,}000}^{20{,}000}e^{-x/10{,}000}\,dx =10{,}000\left(e^{-5{,}000/10{,}000}-e^{-20{,}000/10{,}000}\right).

The two survival terms are $e^{-0.5}=0.60653066$ and $e^{-2}=0.13533528$. Their difference is $0.47119538$, so

E[Y]=10{,}000(0.47119538)=\$4{,}711.9538\approx \$4{,}711.95.

Check. The deductible-only expected payment is $10{,}000e^{-0.5}=\$6{,}065.31$. Capping the payment can only reduce that amount, and the calculated $4,711.95 is smaller. A common mistake is to use $15,000 as the upper loss endpoint. That would treat the $15,000 as a total loss limit; here it is the insurer-payment cap, so the correct endpoint is $5,000+$15,000=$20,000.

Worked Example: Comparing Two Limit Interpretations

A hypothetical liability contract has a $5,000 deductible, and loss severity is exponential with mean $10,000. Compare (a) a $20,000 total loss limit, which includes the deductible, with (b) a $20,000 insurer-payment limit above the deductible. Find both expected payments per loss. Assume fixed-dollar losses and no discounting.

(a) Total loss limit. The insurer’s maximum payment is $20,000-$5,000=$15,000. The loss layer ends at $L=\$20{,}000$:

E[Y_a]=\int_{5{,}000}^{20{,}000}e^{-x/10{,}000}\,dx =10{,}000(e^{-0.5}-e^{-2}) =10{,}000(0.60653066-0.13533528) =\$4{,}711.95.

(b) Limit above the deductible. The insurer can pay up to $20,000 in addition to the deductible. The covered layer ends at $d+u=\$25{,}000$:

E[Y_b]=\int_{5{,}000}^{25{,}000}e^{-x/10{,}000}\,dx =10{,}000(e^{-0.5}-e^{-2.5}).

Here $e^{-2.5}=0.08208500$, so $E[Y_b]=10{,}000(0.60653066-0.08208500)=\$5{,}244.4566$, or $5,244.46 to the nearest cent. The above-deductible limit produces the larger expected payment because it covers an additional $5,000 loss layer. Treating both stated limits as total loss limits would miss that difference.

Worked Example: Pareto Severity and the Antiderivative Sign

Suppose a hypothetical property loss has Pareto Type II (Lomax) severity with survival function $S_X(x)=(1+x/\theta)^{-\alpha}$ for $x\geq 0$, where $\theta=\$5{,}000$ and $\alpha=2$. A contract has a $5,000 deductible and a $10,000 insurer-payment limit above the deductible. Calculate the expected payment per loss. The model has finite mean because $\alpha>1$; amounts are in fixed dollars, with no discounting.

The attachment is $5,000 and the upper endpoint is $5,000+$10,000=$15,000. With these parameters, the survival function is $(1+x/5{,}000)^{-2}$. Integrating gives

\begin{aligned} E[Y] &=\int_{5{,}000}^{15{,}000}\left(1+\frac{x}{5{,}000}\right)^{-2}dx\\ &=-5{,}000\left[\left(1+\frac{x}{5{,}000}\right)^{-1}\right]_{5{,}000}^{15{,}000}\\ &=-5{,}000\left(\frac{1}{4}-\frac{1}{2}\right) =\$1{,}250.00. \end{aligned}

The antiderivative has a negative sign because the derivative of $(1+x/5{,}000)^{-1}$ is $-\frac{1}{5{,}000}(1+x/5{,}000)^{-2}$. Equivalently, the evaluation can be written $5{,}000[(1+x/5{,}000)^{-1}]_{15{,}000}^{5{,}000}$. Omitting the negative sign would give a negative expected payment, which contradicts the nonnegative payment definition.

Calculation Checks and Common Traps

The survival integral makes the contract’s structure visible, but several errors can still produce plausible-looking results. Check the endpoint, the interpretation of the stated limit, and the sign of any antiderivative. The following process is useful when setting up a calculation.

Write the payment function.
For an above-deductible limit, use $Y=\min((X-d)_+,u)$. For a total loss limit $L\geq d$, use $Y=\min((X-d)_+,L-d)$.

Set the loss-layer endpoints.
The lower endpoint is $d$. The upper endpoint is $d+u$ for an above-deductible limit and $L$ for a total loss limit.

Integrate the survival function.
Evaluate $\int_d^{d+u}S_X(x)\,dx$ or $\int_d^L S_X(x)\,dx$, as appropriate, and check that the answer is nonnegative and no greater than the payment cap.

A frequent mistake is to subtract the deductible from the limit even when the limit is explicitly stated above the deductible. That produces too short a layer. The opposite error—adding the deductible to a total loss limit—produces too long a layer. Another trap is to evaluate an antiderivative without checking its derivative. For a decreasing power function, the antiderivative may carry a negative sign even though the final definite integral must be positive.

These calculations produce an expected payment per loss, not an expected annual loss cost by themselves. Any subsequent conversion to an annual cost would also require claim frequency and an exposure basis. Keeping the severity payment calculation separate helps avoid mixing contract mechanics with frequency. The survival-integral method and the careful distinction between the two limit conventions are aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Key takeaway: Combined deductible-and-limit cost is the area under the severity survival function across the covered loss layer. For a limit $u$ above deductible $d$, the layer is $[d,d+u]$; for a total loss limit $L$, it is $[d,L]$, provided $L\geq d$.

Check Your Understanding

For each problem, identify the limit convention before calculating. Round dollar answers to the nearest cent.

A hypothetical contract has an ordinary deductible of $4,000 and an $8,000 insurer-payment limit above the deductible. Severity is exponential with mean $8,000. Calculate expected payment per loss. (Answer: $3,067.20)
A hypothetical policy has a $3,000 deductible and a $12,000 total loss limit. Severity is exponential with mean $6,000. Calculate the expected payment per loss. (Answer: $2,827.17)
For a hypothetical severity with $S_X(x)=(1+x/4{,}000)^{-3}$, the deductible is $4,000 and the insurer-payment limit above it is $8,000. Calculate expected payment per loss. (Answer: $375.00)
A hypothetical loss takes values $0, $10,000, and $30,000 with probabilities 0.50, 0.30, and 0.20, respectively. An ordinary deductible is $5,000 and the insurer-payment limit above it is $10,000. Calculate expected payment per loss by evaluating the possible payments. (Answer: $3,500.00)
For exponential severity with mean $10,000 and a $5,000 deductible, calculate the additional expected payment from increasing the insurer-payment limit above the deductible from $10,000 to $20,000. (Answer: $1,410.45)

Limit interpretation	Payment	Expected payment
\(u\) above the deductible	\(\min((X-d)_+,u)\)	\(\int_d^{d+u} S_X(x)\,dx\)
Total loss limit \(L\), where \(L\geq d\)	\(\min((X-d)_+,L-d)\)	\(\int_d^L S_X(x)\,dx\)

What You'll Learn