Normal Losses With Risk-Specific Means

Why Risk-Specific Means Matter

Two risks can have different expected annual losses even when their observations vary around those expectations according to the same process. Bühlmann credibility balances a risk’s own experience against the portfolio mean: more stable and persistent differences between risks give individual experience more weight, while greater random year-to-year variation reduces that weight.

This tutorial applies the equal-exposure Bühlmann model to conditionally normal observations. The model makes the role of each variance explicit, but normality is not needed for the basic Bühlmann linear premium. The essential quantities are the conditional mean, the conditional variance, and how much the conditional means vary across risks.

Definition: Let $\Theta$ represent a risk’s unknown risk-specific mean. Conditional on $\Theta$, suppose the observations $X_1,\ldots,X_n$ are independent and identically distributed as $N(\Theta,\sigma^2)$. Across risks, assume $E[\Theta]=\mu$ and $\operatorname{Var}(\Theta)=\tau^2$. The collective mean is $\mu$; the expected process variance (EPV) is $\sigma^2=E[\operatorname{Var}(X_j\mid\Theta)]$; and the variance of hypothetical means (VHM) is $\tau^2=\operatorname{Var}(E[X_j\mid\Theta])$.

The model treats the risk-specific mean as fixed for a given risk across the observation periods, while individual observations fluctuate around it. The conditional normal distribution has mean $\Theta$ and variance $\sigma^2$, common to all risks. Conditional independence means that, once $\Theta$ is fixed, one period’s deviation gives no information about another period’s deviation. The observations have equal exposure; unequal exposures require the Bühlmann-Straub formulation rather than simply counting periods.

For annual loss applications, $X_j$ denotes the loss in period $j$, and all observations must use consistent dollar and time units. There is no interest or mortality assumption in this single-period loss model: the quantities are not discounted and are not contingent on survival.

Credibility Weight and Premium

For equal exposures, the experience statistic is the arithmetic mean $\bar X=(X_1+\cdots+X_n)/n$. The Bühlmann credibility premium is a weighted average of this risk’s observed mean and the collective mean. Define the credibility constant $K$ as EPV divided by VHM.

K=\frac{\sigma^2}{\tau^2},\qquad Z_n=\frac{n}{n+K} =\frac{n\tau^2}{n\tau^2+\sigma^2},\qquad \widehat{P}=Z_n\bar X+(1-Z_n)\mu.

The weight $Z_n$ lies between zero and one when the variances are positive. It increases with the number of observations and with between-risk variation, and decreases as process variation grows. Thus, $K$ measures how many equal-exposure periods are needed for experience to become influential relative to the collective mean.

Because the observations are conditionally normal, their average satisfies $\bar X\mid\Theta\sim N(\Theta,\sigma^2/n)$. This explains why more observations provide a more precise indication of the risk’s mean: the conditional variance of the average falls from $\sigma^2$ to $\sigma^2/n$.

Why the Credibility Formula Holds

A short derivation shows how the variance components produce the credibility weight. Write each observation as $X_j=\Theta+\varepsilon_j$, where $E[\varepsilon_j\mid\Theta]=0$, $\operatorname{Var}(\varepsilon_j\mid\Theta)=\sigma^2$, and the errors are conditionally independent. The sample mean is then $\bar X=\Theta+\bar\varepsilon$.

Find the mean and variance of the experience mean.
Since the conditional errors have mean zero, $E[\bar X]=\mu$. The risk means contribute variance $\tau^2$, while the average of $n$ independent errors contributes $\sigma^2/n$. Therefore, $\operatorname{Var}(\bar X)=\tau^2+\sigma^2/n$.

Find the covariance with the risk mean.
Because $\bar X=\Theta+\bar\varepsilon$ and the error has conditional mean zero, $\operatorname{Cov}(\Theta,\bar X)=\operatorname{Var}(\Theta)=\tau^2$.

Choose the best linear predictor.
A linear predictor of the form $\mu+b(\bar X-\mu)$ has minimum mean squared prediction error when $b=\operatorname{Cov}(\Theta,\bar X)/\operatorname{Var}(\bar X)$. Substitution gives $b=\tau^2/(\tau^2+\sigma^2/n)=n\tau^2/(n\tau^2+\sigma^2)=Z_n$. Thus the predictor is $Z_n\bar X+(1-Z_n)\mu$.

This derivation establishes the Bühlmann premium as the best linear predictor of the risk-specific mean under squared-error loss. It does not, by itself, assert that the premium is the exact conditional expectation $E[\Theta\mid X_1,\ldots,X_n]$. If the risk means are also normally distributed, $\Theta\sim N(\mu,\tau^2)$, normal conjugacy makes this linear predictor the exact conditional mean. Without that additional assumption, the Bühlmann formula remains the best linear predictor, but the exact conditional mean need not be linear.

Worked Example: Three Annual Loss Observations

A hypothetical insurer models a risk’s annual losses as conditionally independent $N(\Theta,90000)$ observations, in dollars squared, where $\Theta\sim N(500,40000)$. The collective mean is $\$500$, the VHM is $\$40{,}000$, and the EPV is $\$90{,}000$. The risk’s three observed annual losses are $\$420$, $\$540$, and $\$480$. Calculate the Bühlmann premium. No discounting or mortality is involved.

Method and assumptions. Use the equal-exposure Bühlmann formula. Conditional on the risk mean, each annual loss is normal with process variance $\sigma^2=90{,}000$; the risk means have variance $\tau^2=40{,}000$. The normal distribution for $\Theta$ is specified here, so the Bühlmann result also equals the exact conditional mean.

Step 1: Calculate the credibility constant.

K=\frac{90{,}000}{40{,}000}=2.25.

Step 2: Calculate the risk’s observed mean. The total is $420+540+480=1{,}440$, so $\bar X=1{,}440/3=\$480$.

Step 3: Calculate the weight.

Z_3=\frac{3}{3+2.25}=\frac{3}{5.25}=\frac{4}{7}=0.571428\ldots.

Step 4: Calculate the premium.

\widehat{P}=\frac{4}{7}(480)+\frac{3}{7}(500) =\frac{1{,}920+1{,}500}{7} =\frac{3{,}420}{7} =\$488.57.

Rounded to the nearest cent, the premium is $\$488.57$. It lies between the risk’s observed mean of $\$480$ and the collective mean of $\$500$, as it must for a weight between zero and one. The method avoids treating the noisy observed average as fully credible.

Worked Example: Process Variance Given as a Standard Deviation

For a hypothetical portfolio, a risk’s annual loss conditional on its mean is $N(\Theta,3600)$. The risk means have mean $\mu=\$1{,}000$ and variance $\tau^2=1{,}600$. The annual process standard deviation is therefore $\$60$, and the standard deviation of risk means is $\$40$. The risk has five annual losses: $\$940$, $\$1{,}030$, $\$1{,}100$, $\$970$, and $\$1{,}060$. Find the Bühlmann premium. Assume equal exposures, conditional independence, and no discounting or mortality.

Method and assumptions. The stated conditional distribution is $N(\Theta,3600)$; thus the EPV is the variance $3{,}600$, not the standard deviation $60$. The VHM is $1{,}600$.

Step 1: Calculate the observed mean. The sum is $940+1{,}030+1{,}100+970+1{,}060=5{,}100$, so $\bar X=5{,}100/5=\$1{,}020$.

Step 2: Calculate $K$ and the weight.

K=\frac{3{,}600}{1{,}600}=2.25,\qquad Z_5=\frac{5}{5+2.25}=\frac{20}{29}.

Step 3: Blend the experience with the collective mean.

\widehat{P}=\frac{20}{29}(1{,}020)+\frac{9}{29}(1{,}000) =\frac{20{,}400+9{,}000}{29} =\frac{29{,}400}{29} =\$1{,}013.79.

The premium, rounded to the nearest cent, is $\$1{,}013.79$. The portfolio mean is lower than the risk’s observed mean, so credibility pulls the result down. The method avoids squaring the already-squared process variance or using a standard deviation in the variance ratio.

Worked Example: Premium With a Small Number of Observations

A hypothetical insurer assumes $\Theta\sim N(800,10000)$ for a risk’s annual mean loss and $X_j\mid\Theta\sim N(\Theta,40000)$, independently across years. The risk has two annual losses, $\$600$ and $\$900$. Calculate the Bühlmann premium, giving the weight as well. All amounts are in dollars; no discounting or mortality is involved.

Method and assumptions. The collective mean is $\mu=\$800$, the VHM is $10{,}000$, and the common process variance is $40{,}000$. Equal exposures allow the equal-period Bühlmann formula.

Step 1: Calculate the observed mean. The total is $600+900=1{,}500$, so $\bar X=1{,}500/2=\$750$.

Step 2: Calculate the credibility constant and weight.

K=\frac{40{,}000}{10{,}000}=4,\qquad Z_2=\frac{2}{2+4}=\frac{1}{3}.

Step 3: Calculate the premium.

\widehat{P}=\frac{1}{3}(750)+\frac{2}{3}(800) =250+\frac{1{,}600}{3} =\$783.33.

The premium rounded to the nearest cent is $\$783.33$. It is closer to the collective mean because two observations provide limited experience relative to the process variance. The calculation avoids applying the weight to the collective mean rather than to the risk’s experience.

Common Exam Trap: Reverse the Variance Ratio

The credibility constant is EPV divided by VHM, not VHM divided by EPV. A tempting but incorrect calculation in the first example would use $K=40{,}000/90{,}000=4/9$, giving $Z=3/(3+4/9)=27/31\approx0.8710$. That would give much more credibility to three observations than the model supports.

The correction is $K=90{,}000/40{,}000=2.25$, so $Z=3/(3+2.25)=4/7\approx0.5714$, and the premium is $\$488.57$. A useful check is direction: increasing process variance should reduce credibility, while increasing variation between risk means should increase it. Reversing the ratio violates both relationships.

Another frequent error is to insert the total $X_1+\cdots+X_n$ in place of $\bar X$. The formula weights the average period loss. Equivalently, one could algebraically rewrite the premium using a sum, but then the coefficient and units must also be transformed; simply substituting the total produces an incorrect result.

Key takeaway: For equal-exposure conditionally normal observations with common process variance, compute $K=\text{EPV}/\text{VHM}$, use $Z_n=n/(n+K)$, and blend the observed average with the collective mean. Check that the weight responds in the correct direction to both variance components.

Check Your Understanding

For each problem, use the equal-exposure Bühlmann model and round money to the nearest cent where needed. Unless stated otherwise, observations are conditionally independent and there is no discounting or mortality.

A risk has $\Theta\sim N(400,300)$ and $X_j\mid\Theta\sim N(\Theta,900)$. Its three losses are $\$350$, $\$450$, and $\$500$. Find the credibility weight and premium. (Answer: $Z=0.5000$; premium $=\$416.67$)
A portfolio has collective mean $\$1{,}200$, VHM $10{,}000$, and process variance $2{,}500$. A risk’s four observations are $\$1{,}100$, $\$1{,}300$, $\$1{,}200$, and $\$1{,}400$. Find the credibility weight and premium. (Answer: $Z=16/17\approx0.9412$; premium $=\$1{,}247.06$)
The credibility constant is $K=2.5$. What is the smallest whole number of equal-exposure observations needed for the credibility weight to be at least $0.75$? (Answer: 8 observations)
A hypothetical risk has collective mean $\$700$, VHM $900$, and process variance $3{,}600$. Its three annual losses are $\$600$, $\$750$, and $\$900$. Calculate the credibility weight and premium. (Answer: $Z=3/7\approx0.4286$; premium $=\$721.43$)
Two risks each have four observations and observed mean $\$90$, with collective mean $\$100$. For Risk A, EPV is $400$ and VHM is $100$; for Risk B, EPV is $400$ and VHM is $400$. Calculate each weight and premium, and identify which risk receives greater credibility. (Answer: Risk A: $Z=0.5000$, premium $\$95.00$; Risk B: $Z=0.8000$, premium $\$92.00$; Risk B receives greater credibility.)

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