The Bühlmann Premium as a Linear Predictor

What the Bühlmann Premium Predicts

A risk’s observed loss experience is informative about its underlying expected loss, but it is not the same as that expected loss. Even a risk with a stable underlying level of risk can have unusually high or low observed losses because of random variation. The Bühlmann premium balances those two sources of information: the risk’s own experience and the collective mean for the portfolio.

The premium is a linear predictor of the risk’s expected loss. In this setting, “linear” means an affine function of observed experience: a constant plus a coefficient times the experience average. The coefficient is chosen to minimize the expected squared prediction error across the collective model. It is not generally the risk’s exact conditional expected loss given its observations; it is the best predictor within the specified affine class.

Definition: In the classical Bühlmann model, let $\Theta$ represent a risk’s unobserved risk characteristic. Conditional on $\Theta$, the period losses $X_1,\ldots,X_n$ are independent and identically distributed, with conditional mean $m(\Theta)=E[X_j\mid\Theta]$ and conditional variance $\operatorname{Var}(X_j\mid\Theta)$. The target for the Bühlmann premium is $m(\Theta)$, the risk’s expected loss per period.

The model assumes that the risks in the collective are described by the same distribution of $\Theta$, that the relevant means and variances exist, and that observations have equal exposure. The conditional independence assumption means that, given $\Theta$, one period’s loss does not provide additional information about another period’s random fluctuation. This tutorial treats equal-exposure observations; unequal exposures require the Bühlmann-Straub formulation.

Structural parameters: The collective mean is $\mu=E[m(\Theta)]=E[X_j]$. The expected process variance is $\mathrm{EPV}=E[\operatorname{Var}(X_j\mid\Theta)]$, and the variance of hypothetical means is $\mathrm{VHM}=\operatorname{Var}(m(\Theta))$. Assume $\mathrm{VHM}>0$ when using the conventional credibility factor.

The EPV measures average variation within a risk over time, conditional on that risk’s characteristics. The VHM measures variation between the expected losses of different risks. Large EPV makes a short history less reliable; large VHM makes risk-specific experience more useful because risks differ more substantially in their expected losses.

The Affine Predictor and Its Minimum-Error Condition

Write the observed average for one risk as $\bar X=(X_1+\cdots+X_n)/n$. Consider every predictor of the form $a+b\bar X$, where $a$ and $b$ are constants chosen for the collective, not for an individual risk. Its mean squared prediction error for the target $m(\Theta)$ is

E\left[(m(\Theta)-a-b\bar X)^2\right].

For the minimum-error affine predictor, the prediction residual must have zero expected product with each term used to construct the predictor: the constant 1 and the centered observation $\bar X-E[\bar X]$. These conditions produce the intercept and slope of the best affine predictor.

Center the target and the experience.
Both $m(\Theta)$ and $\bar X$ have collective mean $\mu$, so use the centered quantities $m(\Theta)-\mu$ and $\bar X-\mu$.

Choose the slope.
The least-squares slope is the covariance of the centered target and experience divided by the variance of the centered experience, provided $\operatorname{Var}(\bar X)>0$.

Restore the collective mean.
Because the target and experience have the same mean, the predictor can be written as the collective mean plus the slope times the deviation of experience from that mean.

The classical Bühlmann premium is conventionally written as a blend of the collective mean and the observed average:

P = Z\bar X+(1-Z)\mu =\mu+Z(\bar X-\mu), \qquad Z=\frac{n}{n+K}, \qquad K=\frac{\mathrm{EPV}}{\mathrm{VHM}}.

Here $P$ is the premium per period and $Z$ is the credibility assigned to the risk’s experience. The structural parameters determine how much weight the model assigns to experience; the observed losses determine where that experience lies relative to the collective mean. The detailed derivation of the structural expression for $Z$ is a separate step. For applying the linear-predictor form, it is essential to recognize both the blend and the target it predicts.

Why the Blend Is the Minimum-Error Affine Predictor

A short least-squares argument explains the shape of the result. Let $M=m(\Theta)$, let $W=\bar X$, and consider a predictor $\mu+b(W-\mu)$. Its error is $M-\mu-b(W-\mu)$. Expanding its expected square gives

E[(M-\mu-b(W-\mu))^2] = \operatorname{Var}(M)-2b\operatorname{Cov}(M,W) +b^2\operatorname{Var}(W).

This is a quadratic function of $b$. When $\operatorname{Var}(W)>0$, its minimum occurs where its derivative is zero:

-2\operatorname{Cov}(M,W)+2b\operatorname{Var}(W)=0, \qquad b=\frac{\operatorname{Cov}(M,W)}{\operatorname{Var}(W)}.

The intercept is centered at $\mu$, since both $M$ and $W$ have mean $\mu$. Thus the best affine predictor has the form $\mu+b(W-\mu)$, which is precisely a credibility blend when $b$ is denoted by $Z$. The condition says that, at the optimum, the prediction error is uncorrelated with the centered experience used to make the prediction. It justifies the linear-predictor structure; it does not say that the affine predictor must equal the exact conditional expectation for every possible experience value.

Worked Applications

Worked Example: Blending Three Years of Loss Experience

A hypothetical insurer models annual losses for equal-exposure risks using the classical Bühlmann assumptions. The collective mean annual loss is $\mu=\$500$, $\mathrm{EPV}=900$ dollars squared, and $\mathrm{VHM}=300$ dollars squared. One risk has annual losses of $\$420$, $\$540$, and $\$600$. Calculate its Bühlmann premium per year. Assume the parameters are known and the three observations are conditionally independent given the risk characteristic.

Method. Calculate the credibility factor from the structural parameters, find the risk’s arithmetic experience average, and blend it with the collective mean.

First calculate the credibility constant and factor:

K=\frac{900}{300}=3, \qquad Z=\frac{n}{n+K}=\frac{3}{3+3}=\frac{1}{2}.

The observed average annual loss is

\bar X=\frac{420+540+600}{3} =\frac{1560}{3} =\$520.

Therefore,

P=\frac{1}{2}(520)+\frac{1}{2}(500) =260+250 =\$510.00.

The premium is $\$10$ above the collective mean because the risk’s experience average is $\$20$ above that mean and receives one-half credibility. Check: the premium lies between $\$500$ and $\$520$, as a blend with $0\leq Z\leq1$ must. The common mistake avoided is charging the full $\$520$ experience average without allowing for its random variation.

Worked Example: A Small History With High Process Variation

A hypothetical portfolio has a collective mean annual loss of $\$800$, $\mathrm{EPV}=1{,}600$ dollars squared, and $\mathrm{VHM}=400$ dollars squared. A risk’s two annual losses are $\$700$ and $\$1{,}000$, with equal exposure in both years. Calculate the Bühlmann premium per year, rounding only the final answer to the nearest cent.

Method. Use the equal-exposure classical model. The history contains $n=2$ annual observations; it is not a two-year total to be compared directly with an annual collective mean.

Calculate the credibility constant and factor:

K=\frac{1600}{400}=4, \qquad Z=\frac{2}{2+4}=\frac{1}{3}.

The average annual loss is

\bar X=\frac{700+1000}{2} =\frac{1700}{2} =\$850.

The experience exceeds the collective mean by $\$850-\$800=\$50$. One-third of that difference is $\$50/3=\$16.666\ldots$, so

P=800+\frac{1}{3}(850-800) =800+\frac{50}{3} =\$816.67.

Check: $\$816.67$ is between the collective mean and the experience average, and it is closer to the collective mean because $Z=1/3$. The common mistake avoided is using the two-year total $\$1{,}700$ as though it were an annual loss average; the premium and collective mean here are both per year.

Worked Example: Separating the Risk Mean From Next Year’s Loss

For a hypothetical equal-exposure portfolio, $\mu=\$120$, $\mathrm{EPV}=900$ dollars squared, and $\mathrm{VHM}=100$ dollars squared. A risk has three annual losses of $\$80$, $\$140$, and $\$160$. Calculate its Bühlmann premium per year. Then calculate the mean squared prediction error for (a) the risk’s expected annual loss and (b) its next annual observed loss. Assume the next loss is conditionally independent of the three observed losses given the risk characteristic.

Method. First predict $m(\Theta)$ with the Bühlmann blend. For the next observed loss, add the process variance to the prediction error for $m(\Theta)$, because the next loss includes new random variation around that risk mean.

The credibility constant and factor are

K=\frac{900}{100}=9, \qquad Z=\frac{3}{3+9}=\frac{1}{4}.

The experience average is

\bar X=\frac{80+140+160}{3} =\frac{380}{3} =\$126.666\ldots.

Thus the premium is

P=120+\frac{1}{4}\left(\frac{380}{3}-120\right) =120+\frac{1}{4}\left(\frac{20}{3}\right) =\frac{365}{3} =\$121.67.

For the minimum-error linear predictor of the risk mean, the residual variance is $\mathrm{VHM}(1-Z)$. In this example it is

100\left(1-\frac{1}{4}\right)=75 \quad\text{dollars squared}.

A future observed loss has, in addition, expected process variance $\mathrm{EPV}=900$ dollars squared. Conditional independence makes the future process fluctuation uncorrelated with the past-based prediction error, so its mean squared prediction error is $75+900=975$ dollars squared. Check: the prediction error for a future loss exceeds the error for the underlying risk mean by exactly the stated EPV. The common mistake avoided is treating the premium as a prediction with no future process variation.

Interpretation and Exam Traps

The premium responds to experience through the deviation $\bar X-\mu$, not through the experience average in isolation. If $\bar X=\mu$, the premium equals $\mu$ for any credibility factor. If experience is above the collective mean and $Z>0$, the premium rises; if it is below, the premium falls. Under the model, $Z=n/(n+K)$ lies between zero and one for positive $n$ and $K$, so the premium lies between the collective mean and the experience average.

Do not confuse the target with an observation. The Bühlmann premium predicts $m(\Theta)$, not a particular future $X_j$. A prediction error for a future observed loss also includes process variation.
Keep the unit of the premium consistent. If $X_j$ is annual loss per risk, $\mu$, $\bar X$, and $P$ are all annual amounts. Do not combine an annual mean with a multi-year total.
Use the right experience summary. In the equal-exposure model, the observed average is the arithmetic mean of the period losses. Unequal exposures are not handled by simply treating every period as equally informative.
Do not apply credibility to the whole observed average and then add the collective mean. The correct expression is $Z\bar X+(1-Z)\mu$, or equivalently $\mu+Z(\bar X-\mu)$. Adding $Z\bar X$ to $\mu$ counts the collective level incorrectly.
Do not assume exact conditional expectation. The Bühlmann premium is optimal among affine predictors under squared-error loss. The model alone does not make that affine predictor equal to the full conditional expectation for every possible data history.

Key takeaway: The Bühlmann premium is the collective mean adjusted toward a risk’s experience average by the credibility weight. Its defining prediction target is the risk’s expected loss, and its affine form is justified by minimizing mean squared prediction error.

Check Your Understanding

For each problem, use the classical equal-exposure Bühlmann model and treat the stated structural parameters as known. Give monetary answers to the nearest cent where needed.

A hypothetical portfolio has $\mu=\$240$, $\mathrm{EPV}=400$ dollars squared, and $\mathrm{VHM}=100$ dollars squared. One risk’s two annual losses are $\$200$ and $\$260$. Calculate $K$, $Z$, the experience average, and the premium. (Answer: $K=4$, $Z=1/3$, $\bar X=\$230$, $P=\$236.67$.)
A portfolio has $\mu=\$500$, $\mathrm{EPV}=1{,}200$ dollars squared, and $\mathrm{VHM}=300$ dollars squared. A risk has four equal-exposure annual losses of $\$350$, $\$450$, $\$500$, and $\$550$. Find its credibility factor and premium. (Answer: $Z=1/2$, $P=\$481.25$.)
For a portfolio with $\mu=\$1{,}000$, $\mathrm{EPV}=2{,}500$ dollars squared, and $\mathrm{VHM}=500$ dollars squared, a risk has five annual losses of $\$900$, $\$950$, $\$1{,}000$, $\$1{,}100$, and $\$1{,}200$. Calculate the premium per year and the predicted total expected loss over the next three years, assuming the risk mean remains constant. (Answer: $Z=1/2$, $P=\$1{,}015.00$ per year, three-year expected total $=\$3{,}045.00$.)
A hypothetical risk portfolio has $\mu=\$75$, $\mathrm{EPV}=300$ dollars squared, and $\mathrm{VHM}=150$ dollars squared. A risk has two annual losses, $\$60$ and $\$70$. Find the Bühlmann premium, and state whether it predicts the next observed loss exactly. (Answer: $Z=1/2$, $\bar X=\$65$, $P=\$70.00$; no, it predicts the risk’s expected loss, not the next random observed loss.)
A portfolio has $\mu=\$250$, $K=6$, and a risk has four equal-exposure observations with average loss $\$310$. Calculate the credibility factor and premium. (Answer: $Z=4/(4+6)=0.4$, $P=0.4(310)+0.6(250)=\$274.00$.)

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