Introduction
A reinsurance layer does not pay the full amount of every claim that exceeds its attachment. It pays only the portion of each claim within the layer, subject to the layer limit. Pricing that layer therefore requires the expected payment per claim, not simply the expected claim severity or the probability that a claim reaches the layer. Multiplying the expected payment per claim by expected claim frequency gives the layer’s expected loss cost, or pure premium, before expenses, commissions, risk loads, and profit provisions.
This tutorial develops that calculation for compound frequency-severity models. The focus is the expected value: it can be calculated from the mean claim count without specifying a Poisson model, provided the frequency and severity assumptions are stated. The methods are aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.
The severity distribution should describe individual claim amounts before applying this layer. The frequency \(\lambda\) should use the same exposure unit as the requested premium: for example, claims per policy-year must be multiplied by dollars per claim to produce dollars per policy-year. The pure premium is an expected loss cost, not a quoted reinsurance price with other loads added.
Expected Payment Within a Layer
Write the severity survival function as \(S_X(x)=P(X>x)\). For a nonnegative severity, the expected layer payment can be found by integrating the probability that the layer pays at each point between its attachment and its upper boundary.
This identity holds for discrete as well as continuous severity distributions. In a discrete model, a probability mass at a layer boundary is handled correctly by the integral; the value of the survival function at an individual endpoint does not change the integral.
A short derivation explains why the formula works. For a fixed claim amount \(X\), the layer payment equals the length of the part of the interval from 0 to \(u\) for which \(X>d+y\). In indicator notation, \(Y^L=\int_0^u \mathbf{1}_{\{X>d+y\}}\,dy\). Taking expectations and interchanging the expectation and integral gives the probability integral above. Changing variables from \(y\) to \(x=d+y\) gives the survival-function form. The argument also shows why the integral stops at \(d+u\): losses above the layer’s upper boundary do not increase its payment.
The second expression is another way to see the layer as the difference between two excess-loss payments: the payment above \(d\), less the payment above \(d+u\). Either form is useful if the relevant survival probabilities or excess-loss expectations are available.
From Expected Payment to Pure Premium
Let \(N\) be the number of claims in one exposure unit, and let \(X_1,\ldots,X_N\) be their severities. Suppose \(E[N]=\lambda\), each claim has the same severity distribution, and claim severities are independent of the count. The aggregate layer recovery is \(\sum_{j=1}^N \min((X_j-d)_+,u)\). Conditional on \(N=n\), its expected value is \(nE[Y^L]\). Averaging over the claim count therefore gives:
This result requires a finite expected claim count and finite expected layer payment. The layer payment is bounded by \(u\), so its expectation is finite whenever \(u\) is finite. A Poisson frequency model is not necessary for this expected-value calculation; any frequency model with the stated mean works under the independence and common-severity assumptions. If claim severity distribution differs by class, calculate each class separately and add the expected recoveries.
Record attachment \(d\), limit \(u\), and upper boundary \(d+u\). Check whether the stated limit is the amount of layer capacity, not the upper boundary.
Use the severity model to evaluate \(\int_d^{d+u}S_X(x)\,dx\), or calculate the layer payment for each severity outcome in a discrete model.
Multiply the per-claim expected payment by expected claims per matching exposure unit. If working with a portfolio, use portfolio frequency and then divide by exposures only if a per-exposure result is requested.
The result should be dollars per exposure unit or dollars for the specified portfolio period. It is a pure loss cost, not a fully loaded reinsurance premium.
Worked Frequency-Severity Calculations
Worked Example: Uniform Severity in an Excess Layer
A hypothetical book has expected frequency 0.15 claims per exposure-year. Claim severity is uniform from $0 to $120,000. A reinsurance layer pays $40,000 excess of $20,000. Find the expected layer payment per claim and the pure premium per exposure-year.
Method. The attachment is \(d=\$20,000\), the limit is \(u=\$40,000\), and the layer’s upper boundary is \(d+u=\$60,000\). For \(X\) uniform on \([0,120,000]\), the survival function within the layer is \(S_X(x)=1-x/120,000\).
At the attachment, \(S_X(20,000)=100,000/120,000=5/6\). At the upper boundary, \(S_X(60,000)=60,000/120,000=1/2\). The survival function is linear, so its average value over this interval is the average of its endpoint values: \((5/6+1/2)/2=2/3\).
The expected payment per claim is therefore the layer width multiplied by this average survival probability: \(\$40,000(2/3)=\$26,666.6667\), or $26,666.67 per claim to the nearest cent.
Multiply by frequency to obtain the pure premium: \(0.15(\$26,666.6667)=\$4,000.00\) per exposure-year. The result is less than the layer limit, as it must be. The calculation uses expected frequency, not the probability of at least one claim.
Worked Example: Exponential Severity and a Limited Layer
For a hypothetical portfolio, claim severity is exponential with mean \(\theta=\$80,000\), and expected frequency is 0.08 claims per policy-year. A layer pays $30,000 excess of $20,000. Calculate expected payment per claim and pure premium per policy-year. Assume claim count and severities are independent.
Method. For an exponential severity with mean \(\theta\), \(S_X(x)=e^{-x/\theta}\). Integrating over the layer gives \(E[Y^L]=\int_d^{d+u}e^{-x/\theta}\,dx =\theta(e^{-d/\theta}-e^{-(d+u)/\theta})\). Here \(d=\$20,000\), \(u=\$30,000\), and \(d+u=\$50,000\).
Substitute the parameter values: \(e^{-20,000/80,000}=e^{-0.25}=0.7788008\), and \(e^{-50,000/80,000}=e^{-0.625}=0.5352614\). Thus, \(\$80,000(0.7788008-0.5352614)=\$19,483.15\) per claim, rounded to the nearest cent.
The pure premium is \(0.08(\$19,483.1484)=\$1,558.6519\), or $1,558.65 per policy-year to the nearest cent. This is a layer loss cost only; no expenses or risk margin have been added.
Worked Example: Discrete Severity and Portfolio Frequency
A hypothetical portfolio contains 12 identical exposure units. Its annual claim count is Poisson with mean 2.4 claims for the portfolio, and claim severities are independent of the count. Each claim is $10,000 with probability 0.50, $50,000 with probability 0.30, or $100,000 with probability 0.20. A layer pays $40,000 excess of $20,000. Find the expected annual portfolio recovery and the pure premium per exposure-year.
Method. First calculate the payment for each possible claim amount. A $10,000 claim is below the attachment and pays $0. A $50,000 claim is $30,000 above the attachment and pays $30,000, below the $40,000 limit. A $100,000 claim exceeds the layer’s $60,000 upper boundary and pays the full $40,000 limit.
The expected layer payment per claim is \(0.50(\$0)+0.30(\$30,000)+0.20(\$40,000) =\$9,000+\$8,000=\$17,000\). The expected annual portfolio recovery is \(2.4(\$17,000)=\$40,800\).
There are 12 exposure units, so the pure premium per exposure-year is \(\$40,800/12=\$3,400\). The final results are $40,800 per portfolio-year and $3,400 per exposure-year. The Poisson assumption supplies the stated mean count; the expected recovery calculation itself uses only that mean.
Common Mistake: Paying the Full Limit Whenever a Claim Attaches
A frequent error is to multiply the layer limit by the probability that a claim exceeds the attachment. That calculation treats every attaching claim as though it exhausts the layer. In the uniform-severity example, the incorrect calculation would be \(\$40,000P(X>20,000)=\$40,000(5/6)=\$33,333.33\) per claim.
This is wrong because claims between $20,000 and $60,000 pay only the portion above $20,000, while claims above $60,000 pay the $40,000 limit. The correct expected payment accounts for the survival probability across the whole layer: \(\int_{20,000}^{60,000}S_X(x)\,dx=\$26,666.67\) per claim. The incorrect value is too high because it assigns the full layer limit to claims that only penetrate part of the layer.
| Quantity | Meaning | Typical units |
|---|---|---|
| \(E[Y^L]\) | Expected reinsurance payment on one claim | Dollars per claim |
| \(\lambda\) | Expected claim count per exposure unit | Claims per policy-year, for example |
| \(\lambda E[Y^L]\) | Expected layer recovery for one exposure unit | Dollars per exposure-year |
| \(E[N]E[Y^L]\) | Expected recovery for a specified portfolio period | Dollars per portfolio-year |
Another important check is the unit of frequency. If a frequency is given for a whole portfolio, multiplying it by a per-risk pure premium is not valid; use portfolio expected frequency with per-claim payment, or convert the frequency to a per-risk basis. Likewise, if severity is specified conditional on a claim, do not multiply the expected layer payment by a probability of claim again: frequency already accounts for how often claims occur.
Check Your Understanding
For each problem, identify the layer payment per claim before applying frequency. Round monetary answers to the nearest cent where needed.
- Claim severity is uniform from $0 to $100,000, with expected frequency 0.12 claims per exposure-year. A layer pays $25,000 excess of $25,000. Find expected payment per claim and pure premium per exposure-year. (Answer: $15,625.00 per claim; $1,875.00 per exposure-year)
- Claim severity is exponential with mean $50,000, and expected frequency is 0.06 claims per policy-year. A layer pays $30,000 excess of $20,000. Find the pure premium per policy-year. (Answer: $907.32 per policy-year)
- A hypothetical portfolio has annual Poisson claim count with mean 1.5. Each claim is $10,000 with probability 0.40, $30,000 with probability 0.35, or $70,000 with probability 0.25. A layer pays $25,000 excess of $15,000. Find expected annual recovery. (Answer: $17,250.00)
- A hypothetical book has 20 exposure-years in class A and 10 in class B. Class A has frequency 0.03 per exposure-year, exponential severity with mean $60,000, and a $40,000 excess of $20,000 layer. Class B has frequency 0.08 per exposure-year, uniform severity from $0 to $100,000, and a $20,000 excess of $30,000 layer. Find the combined expected recovery and average pure premium per exposure-year across all 30 exposure-years. (Answer: $22,151.47 combined; $738.38 per exposure-year)