Bühlmann Credibility for Average Claim Costs

Why Credibility Is Applied to Average Claim Costs

A risk’s observed average claim cost can differ from its underlying expected cost because of both persistent differences among risks and random variation in claims. Bühlmann credibility balances those two sources of information: it gives more weight to a risk’s own experience when that experience is sufficiently informative, and more weight to the collective mean when random fluctuation is comparatively large.

Here, an observation is a claim-cost amount on a consistent basis, such as the average paid amount per covered service. The observed average is calculated over comparable units for one risk. This is not automatically the same as a total loss divided by policy years: the numerator and denominator must match the cost measure being modeled. Interest and mortality assumptions do not apply to these examples; each uses a Bühlmann moment model for claim costs.

Definition: Let $\Theta$ represent the unknown risk type of a randomly selected risk. Conditional on $\Theta$, let $X_1,\ldots,X_n$ be independent, identically distributed claim-cost observations with conditional mean $\mu(\Theta)=E[X_j\mid\Theta]$ and conditional variance $\sigma^2(\Theta)=\operatorname{Var}(X_j\mid\Theta)$. The EPV (expected process variance) is $EPV=E[\sigma^2(\Theta)]$. The VHM (variance of hypothetical means) is $VHM=\operatorname{Var}(\mu(\Theta))$. The collective mean is $m=E[\mu(\Theta)]=E[X_j]$. These quantities are assumed finite, and $EPV>0$, $VHM>0$ for the usual credibility formula.

The model assumes that observations for a given risk have the same conditional mean and variance and are conditionally independent given its risk type. Different risks may have different conditional means. The Bühlmann method uses the first two moments; it does not require claim costs to follow a particular named distribution. The risk-specific hypothetical mean $\mu(\Theta)$ is not directly observed, so credibility estimates it from the risk’s experience and the collective mean.

The Bühlmann Estimate for an Average

For $n$ observations on a risk, define its sample average by $\bar X=\frac{1}{n}\sum_{j=1}^{n}X_j$. The Bühlmann parameter $K$ and credibility factor $Z$ are

K=\frac{EPV}{VHM}, \qquad Z=\frac{n}{n+K}.

The credibility estimate of the risk’s hypothetical mean is the weighted average

\widehat{\mu}=Z\bar X+(1-Z)m.

Here, $n$ is the number of comparable observations, not the dollar total of those observations. The formula presumes observations on a consistent basis and equal conditional variance for each observation. If exposure varies materially or observations have different variances, this equal-observation form should not be applied mechanically.

Interpretation: $Z$ is the weight on the individual risk’s observed average; $1-Z$ is the weight on the collective mean. Under the Bühlmann model, $\widehat{\mu}$ estimates the risk’s expected cost per observation. It is also the Bühlmann linear prediction of the next observation’s cost, conditional on the observed experience. It is not a guarantee that the next cost will equal the estimate.

Worked Example: Combining an Observed Average With the Collective Mean

A hypothetical portfolio has $EPV=800$ dollars squared and $VHM=200$ dollars squared for claim costs per covered service. For one risk, five comparable service costs have an observed average of $700. The collective mean is $640. Find the Bühlmann estimate. Use the distribution-free moment model described above; no interest rate or mortality assumption is involved.

Method. Calculate $K$, then $Z$, then combine the two means. The observed average is already supplied, so its five underlying costs need not be reconstructed.

First, $K=EPV/VHM=800/200=4$. With $n=5$,

Z=\frac{5}{5+4}=\frac{5}{9}, \qquad 1-Z=\frac{4}{9}.

The weighted estimate is

\widehat{\mu} =\frac{5}{9}(700)+\frac{4}{9}(640) =\frac{5(700)+4(640)}{9} =\frac{3{,}500+2{,}560}{9} =\frac{6{,}060}{9} =673.333\ldots.

Rounded to the nearest cent, the estimate is $673.33 per covered service. It lies between the risk’s observed average, $700, and the collective mean, $640, as a weighted average should.

Common mistake and correction. The products $5(700)=3{,}500$ and $4(640)=2{,}560$ are correct numerators. The error is to add them and report $3{,}500+2{,}560=6{,}060$ as the estimate, omitting the division by 9. The correct result is $(3{,}500+2{,}560)/9=673.33$ dollars after rounding.

Why the Credibility Factor Has This Form

The factor $Z$ can be justified by finding the best linear predictor of the risk’s hypothetical mean using the observed average. This derivation explains why credibility increases with the number of observations and why the variance components determine its rate of increase.

Write a linear predictor.
Consider $m+b(\bar X-m)$, where $b$ is a constant. This form centers the risk’s average on the collective mean; $b$ determines how much of the observed deviation from $m$ is retained.

Calculate the covariance.
Conditional independence gives $\operatorname{Cov}(\mu(\Theta),\bar X)=VHM$. Indeed, the conditional mean of $\bar X$ is $\mu(\Theta)$, so its covariance with $\mu(\Theta)$ is $\operatorname{Var}(\mu(\Theta))$.

Calculate the variance of the observed average.
The law of total variance gives $\operatorname{Var}(\bar X)=VHM+EPV/n$: variation among risk means contributes $VHM$, and conditional sampling variation in an average of $n$ observations contributes $EPV/n$.

Choose the best linear coefficient.
The least-squares coefficient is covariance divided by variance, so $b=VHM/(VHM+EPV/n)=n/(n+EPV/VHM)=n/(n+K)$. Thus $b=Z$ and the predictor is $Z\bar X+(1-Z)m$.

This is a best linear predictor based on the specified moments. In general, it should not be described as the exact conditional expectation $E[\mu(\Theta)\mid X_1,\ldots,X_n]$; that stronger equality requires additional distributional assumptions. The Bühlmann estimate is useful precisely because the moment model can produce a credibility estimate without specifying a full probability distribution for every risk type.

Calculating the Variance Components From a Risk Model

Sometimes EPV and VHM are supplied. In other problems, a model for risk types and conditional costs is supplied, and the actuary must calculate the components. EPV is an average of within-risk variances. VHM is the variance of the conditional means across risks. Keeping those calculations separate prevents a frequent error: treating variability within a risk as though it were variability among risk means.

Worked Example: Finding K From Two Possible Risk Types

A hypothetical service portfolio has two equally likely risk types, A and B. Conditional on type A, a service cost is $600 or $1,000, each with probability $1/2$. Conditional on type B, it is $900 or $1,100, each with probability $1/2$. A particular risk has five observed services averaging $940. Find its credibility estimate. These discrete conditional distributions are the complete claim-cost model; interest and mortality are not relevant.

Method. Calculate each type’s conditional mean and variance, average the variances for EPV, and calculate the variance of the conditional means for VHM.

For type A, the mean is $(600+1{,}000)/2=800$. Its variance is $\{(600-800)^2+(1{,}000-800)^2\}/2=(40{,}000+40{,}000)/2=40{,}000$. For type B, the mean is $(900+1{,}100)/2=1{,}000$, and its variance is $\{(900-1{,}000)^2+(1{,}100-1{,}000)^2\}/2=(10{,}000+10{,}000)/2=10{,}000$.

The two types are equally likely, so

EPV=\frac{1}{2}(40{,}000)+\frac{1}{2}(10{,}000)=25{,}000.

The collective mean is $m=(800+1{,}000)/2=900$. The two conditional means are each 100 away from 900, so

VHM=\frac{1}{2}(800-900)^2+\frac{1}{2}(1{,}000-900)^2=10{,}000, \qquad K=\frac{25{,}000}{10{,}000}=2.5.

With $n=5$, $Z=5/(5+2.5)=5/7.5=2/3$. Therefore,

\widehat{\mu}=\frac{2}{3}(940)+\frac{1}{3}(900) =626.666\ldots+300 =926.666\ldots.

Rounded to the nearest cent, the estimate is $926.67 per service. It exceeds the collective mean because the observed average exceeds it. A common error would be to use the variance of all possible costs as EPV; EPV is instead the probability-weighted average of the conditional variances, here $25,000.

What the Estimate Predicts

The estimate is a risk-level expected cost on the same basis as the observations. For example, if $X_j$ is cost per covered service, then $\widehat{\mu}$ is an estimate in dollars per covered service. It does not directly give a total premium. To obtain an expected total for a specified number of future services, multiply by the number of services only when the same cost basis and exposure assumptions apply. Any additional expense, trend, coverage, or rating adjustments are separate from the credibility calculation.

The next example makes the prediction interpretation explicit. It also checks that the calculation uses the number of observations, rather than the dollar amount of the observed costs, in $Z$.

Worked Example: Predicting a Future Average Cost

A hypothetical risk’s four service costs are $900, $700, $1,100, and $900. The model specifies $EPV=40{,}000$ dollars squared and $VHM=7{,}500$ dollars squared, with collective mean $m=$850$. Assume the moment-based Bühlmann model and conditional independence described above. Find the risk estimate and state what it predicts. There is no interest or mortality assumption.

Method. Compute the observed average, $K$, and $Z$, then form the credibility-weighted estimate.

The total observed cost is $900+700+1{,}100+900=3{,}600$, so $\bar X=3{,}600/4=$900$. Next,

K=\frac{40{,}000}{7{,}500}=\frac{16}{3}, \qquad Z=\frac{4}{4+16/3}=\frac{4}{28/3}=\frac{3}{7}.

Thus $1-Z=4/7$, and

\widehat{\mu}=\frac{3}{7}(900)+\frac{4}{7}(850) =\frac{2{,}700+3{,}400}{7} =\frac{6{,}100}{7} =871.428571\ldots.

Rounded to the nearest cent, the estimate is $871.43 per service. Under the model, this is the Bühlmann prediction of the risk’s expected cost per service, and hence the prediction of the next service cost in the linear-credibility sense. It is not a prediction that the realized next cost will be exactly $871.43.

Check. The observed average ($900) is above the collective mean ($850), so the estimate must be between them. The result, $871.43, passes that check.

Exam Checks and Practical Interpretation

For a calculation, identify the observation unit before substituting for $n$. If each observation is one claim amount, $n$ counts claims; if each observation is a per-service average, $n$ counts the comparable service units represented by the model. Do not substitute total dollars for the observation count. Confirm that EPV and VHM are expressed in squared units of the same claim-cost measure, so their ratio $K$ is dimensionless and can be added to $n$.

A quick reasonableness check is also valuable. When $EPV$ and $VHM$ are positive, $0<Z<1$; the estimate must lie between $\bar X$ and $m$. If $n$ increases while the variance components are fixed, $Z$ increases. If $EPV$ increases relative to VHM, $K$ increases and the weight on the observed average decreases. The calculation produces a credibility-weighted estimate, not a regulatory approval, a rate indication after all adjustments, or a certainty about future claims. These distinctions matter when applying the method to ratemaking.

Key takeaway: Bühlmann credibility for average claim costs uses $K=EPV/VHM$, $Z=n/(n+K)$, and $\widehat{\mu}=Z\bar X+(1-Z)m$. The estimate is a moment-based linear prediction on the same per-observation basis as the modeled costs.

Check Your Understanding

For each problem, use the Bühlmann moment model, keep the stated cost basis, and round monetary estimates to the nearest cent where needed.

A hypothetical portfolio has $EPV=900$, $VHM=300$, collective mean $500 per repair, and a risk average of $620 from six repairs. Find $K$, $Z$, and the credibility estimate. (Answer: $K=3$, $Z=2/3$, estimate $=$580.00$ per repair.)
Two equally likely risk types have conditional means $700 and $900. Their conditional variances are $16,000 and $24,000, respectively. Find the collective mean, EPV, VHM, and $K$. (Answer: collective mean $=$800$, $EPV=20{,}000$, $VHM=10{,}000$, and $K=2$.)
A risk has three observed costs totaling $2,700. The collective mean is $800, $EPV=12{,}000$, and $VHM=3{,}000$. Calculate the credibility estimate. (Answer: $\bar X=$900$, $K=4$, $Z=3/7$, estimate $=$842.86$.)
A risk’s observed average is $1,050 based on eight units; the collective mean is $950, $EPV=18{,}000$, and $VHM=6{,}000$. A calculation uses $Z=8/(8+3)$ and then reports $8(1{,}050)+3(950)=$11{,}250$. Identify both errors and give the correct estimate. (Answer: $Z=8/11$, $1-Z=3/11$; the correct estimate is $[8(1{,}050)+3(950)]/11=$1{,}022.73$.)
For a hypothetical discrete model, types A and B have probabilities $1/4$ and $3/4$. Conditional on A, a service cost is $500 or $900 with equal probabilities; conditional on B, it is $700 or $1,100 with equal probabilities. A risk’s four observed costs average $900. Find EPV, VHM, $K$, $Z$, and its estimate. (Answer: conditional means are $700 and $900; each conditional variance is $40{,}000$; $EPV=40{,}000$, $m=$850$, $VHM=7{,}500$, $K=16/3$, $Z=3/7$, estimate $=$871.43$ per service.)

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