Poisson Claim Counts Under Bühlmann Credibility

Introduction

A Poisson model represents claim counts when claims arrive at a rate, but a portfolio may contain risks with different underlying rates. The observed variation in counts then has two sources: random fluctuation around each risk’s rate and genuine differences in rates between risks. Bühlmann credibility separates these sources to determine how much weight to give an individual risk’s experience.

For conditionally Poisson counts, the structural parameters can be derived directly from the first two moments of the risk-specific rate. The key is to calculate the expected conditional variance and the variance of the conditional mean separately; these quantities are not interchangeable with the unconditional variance of a count.

Definition: Let $\Theta$ be a nonnegative, risk-specific claim rate with finite mean $m=E[\Theta]$ and variance $\tau^2=\operatorname{Var}(\Theta)$. For period $j$ with fixed exposure $w_j>0$, suppose the claim count satisfies $N_j\mid\Theta\sim\operatorname{Poisson}(w_j\Theta)$. Counts in different periods are conditionally independent given $\Theta$. The rate observation is $X_j=N_j/w_j$, with $E[X_j\mid\Theta]=\Theta$ and $\operatorname{Var}(X_j\mid\Theta)=\Theta/w_j$.

Here, exposure is measured in consistent units, such as policy-years. The rate observation $X_j$ is a count per unit of exposure. For a unit-exposure period, $w_j=1$, so the observation is just the count. The model does not require a particular distribution for $\Theta$, but its first two moments must exist. The rate must also be nonnegative, as required for a Poisson parameter.

Collective Mean, EPV, and VHM

The collective mean is the expected claim count per unit exposure across the portfolio. The expected process variance (EPV) measures the average within-risk variation in a rate observation. The variance of hypothetical means (VHM) measures variation between risks’ underlying expected rates.

Theorem: Under the conditionally Poisson model, the collective mean is $m$, the EPV for unit-exposure rate observations is $m$, and the VHM is $\tau^2$. Thus the Bühlmann credibility constant is $K=m/\tau^2$, provided $\tau^2>0$.

For $X=N/w$, the conditional expectation is $\Theta$. Taking expectations gives $E[X]=E[\Theta]=m$, which is the collective mean. For the EPV, average the conditional variance over the rate distribution:

\operatorname{EPV}=E[\operatorname{Var}(X\mid\Theta)] =E\left[\frac{\Theta}{w}\right] =\frac{m}{w}.

For unit exposure, this becomes $m$. The conditional mean of $X$ is $\Theta$, so its variance across risks is $ \operatorname{Var}(E[X\mid\Theta])=\operatorname{Var}(\Theta)=\tau^2$, the VHM.

The unconditional variance of a unit-exposure count is the sum of the EPV and VHM. This identity follows from the law of total variance:

\operatorname{Var}(N) =E[\operatorname{Var}(N\mid\Theta)] +\operatorname{Var}(E[N\mid\Theta]) =m+\tau^2.

This total variance is useful for describing the portfolio’s count distribution, but the credibility constant uses the EPV and VHM separately. Mixing them up changes the credibility weight.

Why the Bühlmann Weight Has This Form

For $n$ conditionally independent unit-exposure counts, let $\bar N$ be their average. A linear credibility premium combines $\bar N$ with the collective mean. The weight on $\bar N$ is the slope of the best linear predictor of $\Theta$ from $\bar N$. The following calculation shows why this weight increases with the amount of experience and decreases when process variation is large relative to rate heterogeneity.

Find the covariance with the risk rate.
Since $E[\bar N\mid\Theta]=\Theta$, the covariance is $\operatorname{Cov}(\Theta,\bar N)=\tau^2$.

Find the variance of the experience average.
Conditional independence gives $\operatorname{Var}(\bar N\mid\Theta)=\Theta/n$. The law of total variance then gives $\operatorname{Var}(\bar N)=\tau^2+m/n$.

Form the linear-predictor slope.
The slope is $\operatorname{Cov}(\Theta,\bar N)/\operatorname{Var}(\bar N)=\tau^2/(\tau^2+m/n)=n/(n+K)$, where $K=m/\tau^2$.

The intercept makes the predictor equal to the collective mean when experience equals the collective mean. Therefore the Bühlmann premium is $Z\bar N+(1-Z)m$, where $Z=n/(n+K)$. If $\tau^2=0$, all risks have the same rate, so past experience does not identify risk-level differences and this formula is interpreted by its limiting value $Z=0$. If $m=0$, nonnegativity implies $\Theta=0$ with probability one; all counts are zero under the model.

Unequal Exposures

When exposure varies by period, raw counts should not be averaged as if they had equal exposure. For fixed exposures $w_1,\ldots,w_r$, define total exposure $W=\sum_{j=1}^r w_j$ and the exposure-weighted rate $\bar X_w=\sum_j w_jX_j/W$. Equivalently, $\bar X_w=\sum_jN_j/W$: total claims divided by total exposure.

Conditional on $\Theta$, this weighted rate has mean $\Theta$ and variance $\Theta/W$. Repeating the linear-predictor calculation with $\bar X_w$ gives $\operatorname{Cov}(\Theta,\bar X_w)=\tau^2$ and $\operatorname{Var}(\bar X_w)=\tau^2+m/W$. Its credibility weight is therefore $W/(W+K)$. This is the Bühlmann-Straub form for this Poisson rate model.

Result: With total exposure $W$, rate experience $\bar X_w$, collective rate $m$, and $K=m/\tau^2$, the credibility premium per unit exposure is $$ P=Z\bar X_w+(1-Z)m,\qquad Z=\frac{W}{W+K}. $$ The corresponding expected count for a future exposure $w_*$ is $w_*P$, if the future period follows the same rate model.

Worked Applications

Worked Example: Equal-Exposure Counts With a Two-Point Rate

A hypothetical book has a latent rate $\Theta$ equal to $0.5$ with probability $0.5$ and $1.5$ with probability $0.5$, measured in claims per unit exposure. For one risk, three successive periods each have unit exposure, and the observed counts are $0,2,2$. Calculate the Bühlmann premium per unit exposure. There is no interest discounting or mortality assumption; counts are conditionally independent Poisson observations given $\Theta$.

Method and rate moments. First calculate the collective rate: $m=0.5(0.5)+0.5(1.5)=1$. The rate variance is $\tau^2=0.5(0.5-1)^2+0.5(1.5-1)^2=0.25$.

Structural parameters. For a unit-exposure Poisson count, the EPV is $m=1$, and the VHM is $\tau^2=0.25$. Thus $K=m/\tau^2=1/0.25=4$.

Experience and credibility. The average observed count is $\bar N=(0+2+2)/3=4/3$. With $n=3$, $Z=3/(3+4)=3/7$.

Premium. The premium is $P=(3/7)(4/3)+(4/7)(1)=4/7+4/7=8/7$, or approximately $1.1429$ claims per unit exposure.

Check and trap avoided. The weights sum to one, and the premium lies between the observed average $4/3$ and collective rate $1$. The unconditional variance of a count is $1+0.25=1.25$, but using $1.25$ as the EPV would incorrectly combine within-risk and between-risk variation.

Worked Example: Unequal Exposures and a Portfolio Rate

A hypothetical risk has exposures $2,1,3$ units in three periods and claim counts $1,1,4$, respectively. Across the collective, the latent rate has mean $m=0.8$ and variance $\tau^2=0.16$. Calculate the credibility premium per unit exposure. Assume conditionally independent Poisson counts with means $w_j\Theta$; there is no interest discounting or mortality assumption.

Method. Use the exposure-weighted Bühlmann-Straub rate. Total exposure is $W=2+1+3=6$, and total claims are $1+1+4=6$. Therefore $\bar X_w=6/6=1$ claim per exposure unit.

Structural parameters. The unit-exposure EPV is $m=0.8$, the VHM is $0.16$, and $K=0.8/0.16=5$.

Credibility weight and premium. The weight is $Z=6/(6+5)=6/11$. Hence $P=(6/11)(1)+(5/11)(0.8)=6/11+4/11=10/11$, or approximately $0.9091$ claims per unit exposure.

Check and trap avoided. The premium is between the observed rate of $1$ and collective rate of $0.8$. Using the unweighted average of the three period rates, $(1/2+1+4/3)/3$, would give too little regard to the period with exposure $3$; the correct experience rate is total claims divided by total exposure.

Worked Example: Rate Moments From a Gamma Distribution

Suppose a hypothetical portfolio has risk rates $\Theta\sim\operatorname{Gamma}(\text{shape}=2,\text{scale}=0.5)$, so $E[\Theta]=1$ and $\operatorname{Var}(\Theta)=0.5$. A risk has total exposure $W=4$ and made 5 claims over that experience. Find its credibility premium per unit exposure. Conditional counts are Poisson with mean equal to exposure times rate; periods are conditionally independent. No interest discounting or mortality assumption applies.

Method and moments. For the stated shape-scale parameterization, the mean is $2(0.5)=1$, and the variance is $2(0.5)^2=0.5$. Thus the EPV for a unit-exposure count is $1$, the VHM is $0.5$, and $K=1/0.5=2$.

Experience rate. Total claims divided by total exposure gives $\bar X_w=5/4=1.25$.

Premium. The credibility weight is $Z=4/(4+2)=2/3$. Therefore $P=(2/3)(1.25)+(1/3)(1)=5/6+1/3=7/6$, or approximately $1.1667$ claims per unit exposure.

Check and trap avoided. The premium lies between the observed rate $1.25$ and collective rate $1$. The Gamma assumption is used here to obtain the rate moments; once those moments are known, the Bühlmann calculation requires no further details of the rate distribution.

Common Exam Traps and Interpretation

A useful way to organize a calculation is to identify the rate moments first, then calculate the experience rate and exposure, and only then apply credibility. The method is aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.

Using total variance as EPV. The unconditional variance $m+\tau^2$ includes both Poisson process variation and rate heterogeneity. The unit-exposure EPV is $m$; the VHM is $\tau^2$.
Forgetting the exposure scale. A count over exposure $w$ has conditional mean and variance $w\Theta$. To use rate credibility formulas, divide by exposure and use the total exposure in the credibility weight.
Taking an unweighted average of unequal-exposure rates. The correct observed rate is total claims divided by total exposure, not the arithmetic average of period-specific rates.
Reversing EPV and VHM. $K=m/\tau^2$. High process variation increases $K$ and reduces credibility for a fixed exposure; high between-risk variation decreases $K$ and increases credibility.
Applying the weight to the wrong quantity. $Z$ multiplies the risk’s observed rate, while $1-Z$ multiplies the collective rate. For a future exposure $w_*$, multiply the resulting rate premium by $w_*$ only if an expected count for that future exposure is requested.

Key takeaway: For conditionally Poisson counts, the risk-rate mean $m$ supplies the unit-exposure EPV, and the risk-rate variance $\tau^2$ is the VHM. With total exposure $W$, credibility is $W/(W+m/\tau^2)$, applied to total claims divided by total exposure.

Check Your Understanding

For each problem, use the stated Poisson rate model and report a premium per unit exposure unless the question asks otherwise.

A latent rate is $0.4$ or $1.2$, each with probability $0.5$. A risk has four unit-exposure periods with counts $0,1,1,2$. Find $m$, $\tau^2$, $K$, and the credibility premium. (Answer: $m=0.8$, $\tau^2=0.16$, $K=5$, $P=8/9\approx0.8889$.)
A collective has rate mean $1.5$ and rate variance $0.75$. A risk has total exposure $3$ and 6 claims. Find its experience rate, credibility weight, and premium. (Answer: $\bar X_w=2$, $K=2$, $Z=3/5$, $P=1.8$.)
Rates follow a Gamma distribution with shape $3$ and scale $0.4$. A risk has total exposure $5$ and 8 claims. Calculate the EPV, VHM, credibility weight, and premium. (Answer: EPV $=1.2$, VHM $=0.48$, $K=2.5$, $Z=2/3$, $P=22/15\approx1.4667$.)
A candidate uses $m+\tau^2$ as the EPV when $m=2$ and $\tau^2=1$, with four unit-exposure counts averaging $3$. State the correct EPV, VHM, and credibility premium, and identify the error. (Answer: EPV $=2$, VHM $=1$, $K=2$, $Z=2/3$, $P=8/3\approx2.6667$; $m+\tau^2=3$ is the unconditional variance, not the EPV.)
A rate model has $m=0.6$ and $\tau^2=0.09$. A risk’s three periods have exposures $1,2,3$ and counts $0,1,3$. Calculate total exposure, weighted experience rate, credibility weight, and premium. (Answer: $W=6$, $\bar X_w=4/6=2/3$, $K=20/3$, $Z=9/19$, $P=12/19\approx0.6316$.)

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