Posterior Premiums for Risks in Shared Classes

Why Shared Class Experience Matters

An insured risk may have limited claims experience of its own, while other risks in the same class provide relevant information about the class’s underlying claim frequency. A shared-parameter model uses both sources: peer observations update the common class frequency, and the target’s own observations update that same parameter. The target’s future exposure and risk relativity then determine its expected future claim count.

The calculation is sensitive to how exposure is measured. A risk with exposure $e$ and known relativity $r$ contributes $er$ effective exposure units to the baseline frequency estimate. Historical claims and exposures from every observed risk must be pooled consistently, including those of the target when the requested premium is conditional on its experience.

Model and assumptions: Let $\theta$ be the unknown baseline claim frequency shared by a class, with prior $\theta\sim\operatorname{Gamma}(\alpha,\beta)$, using the shape-rate parameterization: the prior mean is $\alpha/\beta$. For each observed risk $j$, let $N_j$ be its claim count, $e_j$ its exposure, and $r_j$ its known frequency relativity. Conditional on $\theta$, assume $N_j\sim\operatorname{Poisson}(\theta e_jr_j)$, with the observed counts conditionally independent. Exposures and relativities are known, and the same baseline parameter applies to peers and target. For pricing, assume future claim severity is independent of frequency and has known mean $m$. These formulas do not apply unchanged if the class risks have separate unknown frequency parameters, relativities are estimated from the same data, or frequency and severity are dependent.

Pooling Observations to Update the Class Frequency

Suppose there are $J$ observed risks in the class. Define the pooled claim count $N=\sum_{j=1}^J N_j$ and pooled effective exposure $E=\sum_{j=1}^J e_jr_j$. The posterior distribution depends on the observations through these two pooled quantities. In particular, a claim from a risk with a larger relativity does not cause a different shape increment, but that risk’s exposure contributes more to the posterior rate.

\theta\mid\{N_j,e_j,r_j\}_{j=1}^J \sim \operatorname{Gamma}\left(\alpha+\sum_{j=1}^J N_j,\ \beta+\sum_{j=1}^J e_jr_j\right).

The first posterior parameter is prior shape plus total claims; the second is prior rate plus total effective exposure. The posterior mean baseline frequency is therefore $\left(\alpha+\sum_j N_j\right)/\left(\beta+\sum_j e_jr_j\right)$. The prior rate $\beta$ is measured in the same effective-exposure units as the observed exposures.

Why the Gamma Update Works

For a Poisson count with mean $\theta e_jr_j$, its likelihood as a function of $\theta$ is proportional to $\theta^{N_j}\exp(-\theta e_jr_j)$; factors that do not involve $\theta$ can be omitted when finding the posterior. Conditional independence means the likelihoods multiply. Multiplying them by the Gamma prior kernel gives the posterior kernel directly:

\begin{aligned} L(\theta) &\propto \prod_{j=1}^J \theta^{N_j}\exp(-\theta e_jr_j)\\ &=\theta^{\sum_jN_j}\exp\left(-\theta\sum_je_jr_j\right),\\ \pi(\theta)L(\theta) &\propto \theta^{\alpha-1}\exp(-\beta\theta) \theta^{\sum_jN_j}\exp\left(-\theta\sum_je_jr_j\right)\\ &= \theta^{\alpha+\sum_jN_j-1} \exp\left[-\theta\left(\beta+\sum_je_jr_j\right)\right]. \end{aligned}

This is the kernel of a Gamma distribution with shape $\alpha+\sum_jN_j$ and rate $\beta+\sum_je_jr_j$, establishing the update. The derivation also explains the pooling rule: counts add in the power of $\theta$, and effective exposures add in its exponential rate.

From the Posterior Frequency to a Premium

For a target risk with future exposure $e_*$ and relativity $r_*$, let $N_*$ denote its future claim count. Conditional on $\theta$, its expected count is $\theta e_*r_*$. Averaging over the posterior distribution of $\theta$ gives the posterior predictive expected count. If the expected severity per claim is $m$, the pure premium is the expected aggregate claim amount, without expenses, profit provision, or other loadings.

\begin{aligned} E[N_*\mid\text{all observed data}] &=e_*r_*E[\theta\mid\text{all observed data}],\\ \text{Pure premium} &=m e_*r_* \frac{\alpha+\sum_jN_j}{\beta+\sum_je_jr_j}. \end{aligned}

The target’s historical record belongs in the sums when the premium is conditional on both class and target experience. The target’s future exposure and relativity enter separately, as a multiplier of its posterior expected baseline frequency. Historical exposure updates the posterior; future exposure scales the prediction.

Worked Example: Updating With Peer and Target Experience

A hypothetical class has prior baseline frequency $\theta\sim\operatorname{Gamma}(2,3)$, where the second parameter is a rate. Peers report 5 claims over 4 effective exposure units. A target risk reports 1 claim over 2 exposure units at relativity 1. For its next period, the target has exposure 1.5 at relativity 1.2. Assume independent claim severities with mean $500. Find the target’s posterior expected future claim count and pure premium.

Method: pool all historical observations. The prior shape and rate are $\alpha=2$ and $\beta=3$. The observed total claims are $5+1=6$. The observed effective exposure is $4+2(1)=6$ units.

Update the posterior. Its shape is $2+6=8$, and its rate is $3+6=9$. Thus $\theta\mid\text{data}\sim\operatorname{Gamma}(8,9)$, with posterior mean $8/9$.

Predict the target’s claims. Future effective exposure is $1.5(1.2)=1.8$. The expected future claim count is $1.8(8/9)=14.4/9=1.6000$ claims.

Calculate the pure premium. Expected claim amount is $1.6000($500)=$800.00$.

Check and trap avoided. The target’s historical claim and exposure have both been included in the posterior, and its future relativity has been applied once to future exposure. Omitting target history would give posterior mean $(2+5)/(3+4)=1$, leading to an incorrect 1.8 expected claims and $900.00 premium. The correct results are 1.6000 expected claims and $800.00.

Worked Applications

Worked Example: A High Future Relativity

A hypothetical class has prior $\theta\sim\operatorname{Gamma}(1,2)$. Peer risks report 7 claims over 5 effective exposure units. The target reports 2 claims over 1 year at relativity 1. Its next period has exposure 0.75 years at relativity 2. Assume the mean severity is $900 and is independent of claim frequency. Find the posterior mean baseline frequency, expected future claim count, and pure premium.

Pool the historical data. Total observed claims are $7+2=9$. The peer effective exposure is 5, and the target effective exposure is $1(1)=1$, for a total of 6.

Update. The posterior is $\operatorname{Gamma}(1+9,2+6)=\operatorname{Gamma}(10,8)$. Its posterior mean baseline frequency is $10/8=1.25$.

Predict and price. Future effective exposure is $0.75(2)=1.5$. Expected future claims are $1.5(1.25)=1.875$. The pure premium is $1.875($900)=$1,687.50$.

Check and trap avoided. The target’s past relativity is 1, so its historical exposure contributes 1, not 2, effective units. The future relativity of 2 applies only to the future exposure. The results are 1.25 baseline frequency, 1.875 expected claims, and $1,687.50.

Worked Example: Pooling Several Relativities

A hypothetical class has prior $\theta\sim\operatorname{Gamma}(2,5)$. One peer group reports 4 claims over 2 years at relativity 1; another reports 3 claims over 1 year at relativity 2. The target reports 2 claims over 1.5 years at relativity 1.5. For the next period, its exposure is 2 years at relativity 1.1. Assume mean severity $750. Find the posterior mean baseline frequency, expected future count, and pure premium.

Pool claims. Total claims are $4+3+2=9$, so posterior shape is $2+9=11$.

Pool effective exposure. The three contributions are $2(1)=2$, $1(2)=2$, and $1.5(1.5)=2.25$. Their sum is $2+2+2.25=6.25$. The posterior rate is $5+6.25=11.25$, giving posterior mean $11/11.25=0.977777\ldots$.

Predict future claims. Future effective exposure is $2(1.1)=2.2$. Expected claims are $2.2(11/11.25)=24.2/11.25=2.151111\ldots$.

Calculate the premium. The pure premium is $(2.2)(11/11.25)($750)=$1,613.333\ldots$, or $1,613.33 to the nearest cent. The posterior mean baseline frequency is approximately 0.9778, and expected future claims are approximately 2.1511.

Check and trap avoided. The second peer group contributes 2 effective exposure units, not 1, because its relativity is 2. The target contributes 2.25 units. The total posterior rate is therefore 11.25, not a rate formed by adding unweighted years or multiplying all exposures by the future relativity.

Common Traps and Scope

A shared class parameter makes peer information relevant to a target, but it does not make every part of the calculation interchangeable. Use historical counts and their matching historical effective exposures to form the posterior. Only after that update should the target’s future exposure and future relativity be applied. If the target’s own historical experience is part of the information specified, include it once—not zero times and not twice.

Ignoring historical relativities. Convert each observed exposure to $e_jr_j$ before adding it to the posterior rate.
Using future relativity on past exposure. Historical effective exposure uses historical relativity; future pricing uses future relativity.
Updating with peers only. If the premium is conditional on target and class data, the target’s observed claims and exposure also update $\theta$.
Confusing baseline frequency with the target’s frequency. The posterior mean estimates $\theta$; the target’s future expected count additionally multiplies by future effective exposure.
Calling the frequency estimate the premium. Convert expected claims to expected dollars using mean severity, under the stated independence and common-severity-mean assumptions.

This shared-class calculation is aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus. Its main discipline is to keep the pooled historical update separate from the target-specific future prediction.

Check Your Understanding

For each hypothetical scenario, use a Gamma prior in shape-rate form, pool all stated historical observations, and calculate the requested posterior predictive quantity.

A class has prior $\operatorname{Gamma}(3,4)$. Peers report 6 claims over 3 effective exposure units. A target reports 1 claim over 2 years at relativity 0.5. Its next period has exposure 1.2 years at relativity 1.5, and mean severity is $400. Find expected claims and pure premium. (Answer: 2.2500 claims; $900.00)
A class has prior $\operatorname{Gamma}(2,6)$. One peer group reports 3 claims over 2 years at relativity 1.5; another reports 4 claims over 4 years at relativity 0.5. The target reports 1 claim over 1 year at relativity 2. Its future exposure is 0.5 years at relativity 2, with mean severity $650. Find the pure premium. (Answer: $500.00)
A class has prior $\operatorname{Gamma}(1,3)$. Peers report 2 claims over 1.5 years at relativity 2. The target reports 3 claims over 2 years at relativity 0.5. Its future exposure is 1.75 years at relativity 1.4, and mean severity is $800. Find expected claims and pure premium. (Answer: 2.1000 claims; $1,680.00)
A class has prior $\operatorname{Gamma}(4,10)$. Two peer groups report 5 claims over 2 years at relativity 1.5 and 2 claims over 3 years at relativity 0.5. The target reports zero claims over 1 year at relativity 2. Its future exposure is 0.8 years at relativity 1.25, and mean severity is $1,200. Find the target’s posterior mean baseline frequency and pure premium. (Answer: $11/16.5=0.6667$; $800.00)
A class has prior $\operatorname{Gamma}(2,5)$. Peer groups report 8 claims over 4 years at relativity 1.5 and 1 claim over 2 years at relativity 0.5. The target reports 2 claims over 1 year at relativity 2. Its future exposure is 1.25 years at relativity 1.2, with mean severity $700. Find the posterior mean baseline frequency, expected claims, and pure premium. (Answer: $13/14=0.9286$; 1.3929 claims; $975.00)

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