How Contract Features Transform Claim Severity
A claim’s loss amount and the payment arising from it are not necessarily the same. A policy limit caps the amount considered for payment, coinsurance assigns only a proportion of that amount to the insurer, and a reinsurance layer transfers a specified band of loss to another party. The calculations are manageable once the contract is written as a payment function of the loss.
Let \(X\geq 0\) be the loss on one claim. The payment per loss, \(Y^L\), is the amount payable for that claim, including zero when the claim produces no payment under the contract. The distribution of \(Y^L\) follows from applying the payment function to \(X\). When a claim may be only partly covered, it is important not to treat the original loss distribution as the payment distribution.
Payment Distributions for Limited and Coinsured Claims
For a nonnegative loss with distribution function \(F_X\), the limited, coinsured payment has a direct distributional description. If \(c>0\), then below the maximum payment \(cu\), the event that the payment is no greater than \(y\) is the event that the loss is no greater than \(y/c\). Thus, for \(0\leq y<cu\), $$ F_{Y^L}(y)=P(Y^L\leq y)=F_X(y/c). $$ For \(y<0\), the CDF is zero; for \(y\geq cu\), it is one. If \(X\) has a continuous distribution, all losses at or above \(u\) produce the maximum payment, so the probability of that payment is \(P(X\geq u)=S_X(u)\), where \(S_X(x)=1-F_X(x)\). This endpoint probability is part of the payment distribution, not a reason to keep using the uncapped severity distribution.
The expectation can be calculated without first finding a density for the payment. For any nonnegative random variable \(Z\), its expectation equals the area under its survival function: \(E[Z]=\int_0^\infty P(Z>y)\,dy\). Applying this identity to a cap gives a useful formula.
To see why the cap formula holds, write \(\min(X,u)=\int_0^u \mathbf{1}_{\{X>t\}}\,dt\), apart from endpoint values that do not change the integral. Taking expectations and interchanging expectation and integration gives $$ E[\min(X,u)] =\int_0^u E[\mathbf{1}_{\{X>t\}}]\,dt =\int_0^u P(X>t)\,dt =\int_0^u S_X(t)\,dt. $$ Coinsurance multiplies each claim payment by \(c\), so it multiplies the expectation by the same factor.
Worked Example: A Coinsured Policy With a Per-Claim Limit
A hypothetical policy pays 80% of each loss, subject to a covered-loss limit of $6,000 applied before coinsurance. Suppose \(X\sim\operatorname{Uniform}(0,10000)\). Find the payment distribution, the expected payment, and the probability that the payment exceeds $3,200.
Method. First write the payment map, then transform the loss CDF and use the survival integral for the mean. This avoids applying the limit to the wrong amount.
The payment is \(Y^L=0.8\min(X,6000)\), so its maximum is \(0.8(6000)=4800\). For \(0\leq y<4800\), $$ F_{Y^L}(y)=F_X(y/0.8)=\frac{y/0.8}{10000}=\frac{y}{8000}. $$ The complete CDF is zero for \(y<0\), \(y/8000\) for \(0\leq y<4800\), and one for \(y\geq4800\). The probability of the maximum payment is \(P(X\geq6000)=1-6000/10000=0.4\).
For a uniform loss on \([0,10000]\), \(S_X(t)=1-t/10000\) on that interval. Therefore, $$ E[Y^L]=0.8\int_0^{6000}\left(1-\frac{t}{10000}\right)dt =0.8\left[6000-\frac{6000^2}{20000}\right] =0.8(6000-1800) =3360. $$ The expected payment is $3,360.00.
The payment exceeds $3,200 exactly when \(0.8\min(X,6000)>3200\), which requires \(X>4000\). Thus, $$ P(Y^L>3200)=1-\frac{4000}{10000}=0.6. $$ As a check, the CDF at $3,200 is \(3200/8000=0.4\), so its complement is also 0.6. A common error is to cap the insurer’s payment at $6,000, yielding a maximum of $6,000; under the stated contract the cap is on loss before coinsurance, so the maximum payment is $4,800.
Coinsurance and the Order of Operations
A limit and coinsurance can be applied in different orders, so the contract wording must determine the payment map. If the limit applies to loss before coinsurance, the payment is \(c\min(X,u)\), with maximum \(cu\). If the insurer’s payment itself is limited to \(u\), the payment is \(\min(cX,u)\). The two maps agree below the point where the cap binds, but they need not have the same maximum or expected payment.
For example, with \(c=0.8\) and \(u=6000\), the first contract caps payment at \(4800\). The second caps it at \(6000\), and that cap begins to bind only when \(X\geq7500\), since \(0.8X=6000\) at \(X=7500\). Before integrating or calculating probabilities, identify whether the stated limit is a limit on covered loss or on the insurer’s payment.
Layered Payments and Reinsurance
A layer covers losses between an attachment point \(a\) and an upper attachment point \(b\), where \(0\leq a<b\). Its width is \(b-a\). For a layer with no coinsurance, payment is zero below \(a\), increases with the loss between \(a\) and \(b\), and is capped at the layer width above \(b\). If the layer pays fraction \(c\), its payment per loss is $$ Y^L=c\min\big((X-a)_+,b-a\big), \qquad (z)_+=\max(z,0). $$ Below the maximum payment \(c(b-a)\), its CDF is \(F_X(a+y/c)\) for \(0\leq y<c(b-a)\). The expected payment follows by integrating the survival function over the layer: $$ E[Y^L]=c\int_a^b S_X(t)\,dt. $$ The lower boundary matters: losses up to \(a\) generate no layer payment. The upper boundary matters too: losses above \(b\) do not increase that layer’s payment.
In reinsurance, a ceded layer is the payment made by the reinsurer to the insurer under the treaty. The insurer’s retained loss is the original loss less that ceded amount, if the insurer is responsible for the full original claim. A layer therefore describes a band of loss transferred; it does not automatically mean that every dollar above the attachment point is ceded.
Worked Example: An Excess Layer With Coinsurance
A hypothetical contract pays 90% of a layer attaching at $2,000 and ending at $6,000. Claim severity is exponential with mean $5,000, so \(S_X(t)=e^{-t/5000}\) for \(t\geq0\). Find the payment distribution and expected payment, and calculate the probability that the payment exceeds $1,800.
Method. Use the layer payment map and transform the severity CDF. This avoids treating the layer as unlimited excess above its attachment.
The layer width is \(6000-2000=4000\), and its maximum payment is \(0.9(4000)=3600\). For \(0\leq y<3600\), a payment no greater than \(y\) means the loss is no greater than \(2000+y/0.9\). Therefore, $$ F_{Y^L}(y)=1-\exp\left(-\frac{2000+y/0.9}{5000}\right). $$ The CDF is zero for \(y<0\) and one for \(y\geq3600\). At zero payment, the CDF is \(1-e^{-0.4}\), corresponding to losses at or below the attachment. At the cap, losses at or above $6,000 produce the maximum payment.
For the mean, the layer survival integral gives $$ E[Y^L]=0.9\int_{2000}^{6000}e^{-t/5000}\,dt =4500\left(e^{-0.4}-e^{-1.2}\right). $$ Using \(e^{-0.4}=0.670320\) and \(e^{-1.2}=0.301194\), the mean is \(4500(0.369126)=1661.067\), or $1,661.07.
A payment above $1,800 requires \(0.9(X-2000)>1800\), with the layer cap not yet exceeded at that threshold. This is equivalent to \(X>4000\), so $$ P(Y^L>1800)=e^{-4000/5000}=e^{-0.8}=0.4493. $$ The answer is consistent with the payment map: losses up to $2,000 pay zero, and losses of $4,000 produce exactly $1,800, not a payment greater than $1,800.
Separating Retained and Ceded Amounts
For a treaty that cedes the layer from \(a\) to \(b\), define the ceded payment \(C=\min((X-a)_+,b-a)\). If the insurer otherwise pays the full loss, its retained amount is \(N=X-C\), so \(X=N+C\) for every claim. This identity provides an important check on both claim-level calculations and expectations: $$ E[X]=E[N]+E[C]. $$ The ceded amount is capped at \(b-a\); for losses above \(b\), the insurer again bears increases in loss unless another treaty provision applies.
Worked Example: Ceded Layer and Insurer Retention
A hypothetical insurer buys reinsurance that cedes $3,000 excess of $3,000: the reinsurer pays the layer from $3,000 to $5,000, with a maximum payment of $2,000 per claim. Suppose \(X\sim\operatorname{Uniform}(0,10000)\). Find the expected ceded amount and expected retained amount, then calculate both amounts for losses of $4,200 and $6,500.
Method. Apply the layer map to find ceded payment, subtract it from the original loss to find retention, and check the expected values using the identity \(X=N+C\). The common mistake is to cede every dollar above $3,000, ignoring the treaty limit.
For this treaty, $$ C=\min\big((X-3000)_+,2000\big), \qquad N=X-C. $$ Since \(S_X(t)=1-t/10000\) for \(0\leq t\leq10000\), $$ E[C]=\int_{3000}^{5000}\left(1-\frac{t}{10000}\right)dt =2000-\frac{5000^2-3000^2}{20000} =2000-800 =1200. $$ The expected original loss is \(E[X]=10000/2=5000\). Therefore, $$ E[N]=E[X]-E[C]=5000-1200=3800. $$ An independent check is to integrate retention over the loss range: $$ E[N]=\frac{1}{10000}\left(\int_0^{3000}x\,dx+ \int_{3000}^{5000}3000\,dx+ \int_{5000}^{10000}(x-2000)\,dx\right) =450+600+2750 =3800. $$
For \(X=4200\), the ceded amount is \(\min(4200-3000,2000)=1200\), and retention is \(4200-1200=3000\). For \(X=6500\), the layer is exhausted, so the ceded amount is $2,000 and retention is \(6500-2000=4500\). In each case, retained plus ceded equals the original loss. Ceding \(6500-3000=3500\) at the second loss would be incorrect because it exceeds the treaty’s $2,000 limit.
Common Calculation Traps
- Applying coinsurance to the wrong base. Translate the contract wording into a payment map before calculating. A loss limit before coinsurance gives \(c\min(X,u)\); a cap on insurer payment gives \(\min(cX,u)\).
- Ignoring the attachment point. A layer from \(a\) to \(b\) has expected payment \(c\int_a^b S_X(t)\,dt\), not \(c\int_0^b S_X(t)\,dt\).
- Forgetting the layer limit. The layer payment stops increasing once \(X\geq b\). The amount above \(b\) is not ceded by that layer.
- Confusing payment with loss. A probability about \(Y^L\) must be translated through the payment map before using \(F_X\) or \(S_X\).
- Failing to reconcile the treaty. When the insurer otherwise bears the whole claim, retained plus ceded must equal the original loss claim by claim.
Check Your Understanding
For each problem, write the payment map before calculating. Unless stated otherwise, losses are nonnegative.
- A hypothetical policy pays 75% of loss, limited to $4,000 of loss before coinsurance. If \(X\sim\operatorname{Uniform}(0,8000)\), find the maximum payment, \(E[Y^L]\), and \(P(Y^L=3000)\). (Answer: Maximum $3,000; expected payment $2,250.00; probability of maximum payment 0.5000.)
- A hypothetical contract pays a layer attaching at $1,000 and ending at $4,000, with no coinsurance. If \(X\) is exponential with mean $4,000, find the probability of a positive payment and the expected payment. (Answer: \(P(Y^L>0)=e^{-0.25}=0.7788\); expected payment \(4000(e^{-0.25}-e^{-1})=\)$1,643.69.)
- A policy pays 60% of loss, but the insurer’s payment itself is capped at $3,000. If \(X\sim\operatorname{Uniform}(0,10000)\), find the expected payment and the probability the payment reaches the cap. (Answer: Expected payment $2,250.00; probability at the cap 0.5000.)
- A reinsurance treaty cedes the layer from $2,000 to $7,000. If \(X\) is exponential with mean $5,000\), find the expected ceded amount and expected retention when the insurer otherwise bears the full loss. (Answer: Expected ceded amount \(5000(e^{-0.4}-e^{-1.4})=\)$2,118.62; expected retention $2,881.38.)
- A hypothetical coinsured layer pays 80% of losses between $1,000 and $4,000. If \(X\sim\operatorname{Uniform}(0,6000)\), find the maximum payment, probability of a positive payment, and expected payment. (Answer: Maximum payment $2,400; probability of a positive payment \(5/6=0.8333\); expected payment $1,400.00.)