Aggregate Deductible and Limit Calculation Workshop

Translate the Contract Before Calculating

An aggregate coverage calculation starts with the contract wording, not with an expected loss formula. The deductible may apply to the total loss over a policy period, and a limit may cap covered loss before coinsurance or cap the insurer’s payment after coinsurance. Those arrangements can produce different payments for the same aggregate loss.

Let $X$ be the aggregate loss over the period. In this tutorial, unless a problem says otherwise, $d$ is an aggregate deductible, $u$ is the maximum covered amount after the deductible but before coinsurance, and $c$ is the insurer’s coinsurance share, with $0\leq c\leq1$. The insurer pays the share $c$; the insured bears the remaining share.

Definition: For a contract with aggregate deductible $d$, covered-amount limit $u$, and insurer coinsurance share $c$, the insurer’s payment is $$ Y=c\min\bigl((X-d)_+,u\bigr), \qquad (z)_+=\max(z,0). $$ The deductible is applied to aggregate loss first, the limit is applied to the amount remaining, and coinsurance is then applied to that limited amount.

This function is zero when $X\leq d$, increases at rate $c$ while $d<X<d+u$, and is capped at $cu$ when $X\geq d+u$. A different contract may cap the insurer’s payment after coinsurance. If $L$ is that payment limit, the function is instead $Y=\min(c(X-d)_+,L)$. Always identify what the stated limit caps.

Calculating Expected Payment from a Discrete Distribution

For a discrete aggregate loss, list each possible loss, its probability, and the resulting payment. Then weight each payment by its probability. For the contract defined above, a loss value $x$ produces payment $c\min((x-d)_+,u)$. This direct method makes the order of contract operations visible and is easy to check.

Worked Example: A Deductible, a Covered-Amount Limit, and Coinsurance

A hypothetical annual aggregate loss $X$, measured in thousands of dollars, equals 0, 8, or 24 with probabilities 0.25, 0.25, and 0.50, respectively. The contract has a \$10,000 aggregate deductible, a \$16,000 limit on covered amount before coinsurance, and a 50% insurer share. Find the expected insurer payment in dollars.

Method. Apply the deductible, then the covered-amount limit, then coinsurance. The possible payments in thousands are:

If $X=0$, the excess over the deductible is $(0-10)_+=0$, so payment is $0.50\min(0,16)=0$.

If $X=8$, the excess is $(8-10)_+=0$, so payment is $0.50\min(0,16)=0$.

If $X=24$, the excess is $(24-10)_+=14$. The limit does not bind because $14<16$, so payment is $0.50\min(14,16)=7$, or \$7,000.

The expected payment in thousands is

$$ E[Y]=0.25(0)+0.25(0)+0.50(7)=3.50. $$

Therefore, the expected insurer payment is \$3,500.00. The calculation avoids treating the deductible as a claim-by-claim deductible or applying a limit that does not bind. In particular, the 50% share applies to the \$14,000 excess, giving \$7,000 in the loss scenario $X=24$.

A useful check is that every individual payment must be nonnegative and no greater than the maximum payment $cu$. In this example, the maximum is $0.50(16)=8$ thousand, and each listed payment is between 0 and 8 thousand.

Limits and Coinsurance: Keep the Order Explicit

The phrase “limit” is not enough by itself to determine a payment function. Under the convention used in the first example, the limit caps the covered amount before coinsurance. If instead the contract says the insurer’s payment is limited to $L$, the payment is capped after the coinsurance share is calculated. For example, with excess loss of \$20,000, a 50% insurer share, and a \$7,000 insurer-payment limit, the payment is $\min(0.50(20{,}000),7{,}000)=\$7{,}000$. A \$7,000 covered-amount limit before coinsurance would instead produce a payment of $0.50(7{,}000)=\$3{,}500$.

Worked Example: A Limit That Binds After the Deductible

A hypothetical aggregate loss, in thousands of dollars, takes values 0, 5, 10, 15, and 25 with respective probabilities 0.10, 0.20, 0.30, 0.25, and 0.15. The aggregate deductible is 6, the covered-amount limit before coinsurance is 10, and the insurer pays 80% of covered amounts. Calculate the expected payment in dollars.

Method. Calculate the excess above 6, cap that excess at 10, and multiply by 0.80. The loss and payment pairs, in thousands, are:

At $X=0$, the excess is 0 and the payment is 0.

At $X=5$, the excess is 0 and the payment is 0.

At $X=10$, the excess is $10-6=4$, giving payment $0.80(4)=3.2$.

At $X=15$, the excess is $15-6=9$, below the limit, giving payment $0.80(9)=7.2$.

At $X=25$, the excess is $25-6=19$, capped at 10, giving payment $0.80(10)=8$.

Weight each payment by its probability:

$$ E[Y]=0.10(0)+0.20(0)+0.30(3.2)+0.25(7.2)+0.15(8) =0.96+1.80+1.20=3.96. $$

The expected payment is therefore \$3,960.00. The largest possible payment is $0.80(10)=8$ thousand, and the calculated expectation is below that cap.

A Tail-Integral Formula for Capped Excess

For a continuous aggregate loss model, a tail-integral expression can be more convenient than listing payments at every possible loss. Let $S_X(x)=P(X>x)$ be the survival function of $X$. For nonnegative $X$, a deductible $d\geq0$, a covered-amount limit $u>0$, and insurer share $c$, the same payment function satisfies

E[Y]=c\int_d^{d+u}S_X(t)\,dt.

This expression integrates only over the range where an additional dollar of aggregate loss increases the covered amount: above the deductible and below the point at which the limit is exhausted. It applies when the expectation exists; for bounded $u$, the payment itself is bounded, so its expectation is finite.

Derivation: For any fixed aggregate loss $X$, the length of the part of the interval $[d,d+u]$ lying below $X$ is exactly $\min((X-d)_+,u)$. Equivalently, $$ \min((X-d)_+,u)=\int_d^{d+u}\mathbf{1}_{\{X>t\}}\,dt. $$ Multiplying by $c$ gives the insurer payment. Taking expectations and interchanging expectation and integration is valid because the integrand is nonnegative and bounded over a finite interval. Since $E[\mathbf{1}_{\{X>t\}}]=P(X>t)=S_X(t)$, it follows that $$ E[Y]=c\int_d^{d+u}S_X(t)\,dt. $$

Worked Example: Continuous Aggregate Loss with a Capped Payment

Suppose a hypothetical aggregate loss $X$, in thousands of dollars, is uniformly distributed on $[0,20]$. The contract has an aggregate deductible of 4 thousand, a covered-amount limit of 10 thousand before coinsurance, and an insurer share of 80%. Find the expected payment in dollars.

Method. For $0\leq t\leq20$, the uniform model gives $S_X(t)=(20-t)/20$. The deductible and limit mean the tail integral runs from $d=4$ to $d+u=14$:

E[Y]=0.80\int_4^{14}\frac{20-t}{20}\,dt =\frac{0.80}{20}\left[20t-\frac{t^2}{2}\right]_4^{14}.

At the upper endpoint, $20(14)-14^2/2=280-98=182$. At the lower endpoint, $20(4)-4^2/2=80-8=72$. Thus the integral is $(182-72)/20=110/20=5.5$ thousand. Applying the 80% insurer share gives $0.80(5.5)=4.4$ thousand, so the expected payment is \$4,400.00.

As a check, the covered amount cannot exceed 10 thousand, so the insurer payment cannot exceed $0.80(10)=8$ thousand. The expected payment of 4.4 thousand is within that bound.

Common Mistakes and Checks

One common error is applying the deductible after the limit or after coinsurance. For the contract in the second example, the loss $X=25$ produces a correct payment of $0.80\min(25-6,10)=8$ thousand. The incorrect calculation $0.80\min(25,10)-6=2$ thousand first caps gross loss and then subtracts the deductible; it does not implement the stated contract.

A second error is to treat a covered-amount limit as an insurer-payment limit. With a 50% insurer share and a covered-amount limit of 10, the maximum insurer payment is 5, not 10. A third error is mixing units: if losses and contract terms are in thousands, the expected payment calculated from them is also in thousands. Convert to dollars only after the calculation.

Identify the aggregate variable.
Confirm that $X$ is the total loss over the period to which the deductible applies, not an individual claim loss.

Translate the contract.
Write the payment function and state whether the limit caps covered loss before coinsurance or the insurer’s payment after coinsurance.

Calculate and check.
For a discrete model, weight each possible payment by its probability. For a continuous capped excess, use the tail integral when suitable. Check nonnegativity, the payment cap, and units.

These calculations matter because the expected payment is the pure expected cost of the specified aggregate coverage. Changing the order of the deductible, limit, or coinsurance changes the contract and therefore changes the expected cost. A careful payment function is the most reliable way to prevent that error.

Check Your Understanding

For each problem, write or use the payment function before calculating. Amounts are in thousands of dollars unless stated otherwise.

A discrete aggregate loss $X$ is 0, 4, 9, or 17 with probabilities 0.20, 0.30, 0.30, and 0.20. The deductible is 6, the covered-amount limit before coinsurance is 8, and the insurer share is 75%. Find the expected payment in dollars. (Answer: \$1,875.00)
A hypothetical aggregate loss is 3, 12, or 20 with probabilities 0.25, 0.50, and 0.25. The deductible is 5, the covered-amount limit before coinsurance is 9, and the insurer share is 60%. Find the expected payment in dollars. (Answer: \$3,450.00)
Let $X$, in thousands of dollars, be uniform on $[0,20]$. A contract has deductible 4, covered-amount limit 10 before coinsurance, and insurer share 80%. Find the expected payment. (Answer: \$4,400.00)
A hypothetical aggregate loss is \$30,000. The deductible is \$8,000, coinsurance is 70%, and the stated \$15,000 limit caps covered amount before coinsurance. Find the insurer payment. (Answer: \$10,500.00)
Two hypothetical claims in one policy period are \$4,000 and \$9,000. The contract has one \$5,000 aggregate deductible, no binding limit, and an 80% insurer share. Find the insurer payment for the period. (Answer: \$6,400.00)

What You'll Learn