Deriving the Bühlmann-Straub Premium

Why Unequal Exposures Change the Calculation

When a risk has several periods of experience, those periods need not contribute equally to the credibility estimate. A period with twice the exposure generally has half the conditional variance of a period with one exposure unit, so it should carry twice the weight in the experience estimate. The Bühlmann-Straub model makes this principle precise: it derives the premium that minimizes mean squared prediction error among linear predictors.

The result combines the collective mean with an exposure-weighted experience mean. The credibility assigned to experience depends on the total historical exposure and the ratio of within-risk variation to between-risk variation. This derivation is aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.

Model and Structural Parameters

Definition: Let $\Theta$ denote a risk’s latent parameter, and let $Y_j$ be the observed loss rate per exposure unit in period $j$, whose known positive exposure is $w_j$. The Bühlmann-Straub assumptions are $$ E[Y_j\mid\Theta]=\Theta,\qquad \operatorname{Var}(Y_j\mid\Theta)=\frac{s^2(\Theta)}{w_j}, $$ with observations conditionally independent given $\Theta$. Define the collective mean $m=E[\Theta]$, the EPV $v=E[s^2(\Theta)]$, and the VHM $a=\operatorname{Var}(\Theta)$. Assume the relevant second moments are finite.

The EPV, $v$, measures average variation within a risk after conditioning on its latent parameter. The VHM, $a$, measures variation between risks in their latent means. Conditional independence and the variance assumption imply that a larger exposure reduces the observation’s conditional variance. The model does not require normally distributed observations: the linear-prediction derivation below uses only the stated means, variances, and covariances.

For positive $a$ and $v$, define the credibility constant $K=v/a$, measured in the same exposure units as the $w_j$. If $a=0$, there is no between-risk variation to learn about, so the collective mean is the appropriate predictor. The usual ratio $v/a$ is not defined in that case.

Deriving the Best Linear Predictor

Suppose experience is available from $n$ periods. The target is the loss rate $Y_0$ in a future period. Its exposure may be different, but under the model its conditional mean given $\Theta$ is still $\Theta$. Consider a linear predictor with an intercept:

\widehat{Y}_0=c+\sum_{j=1}^{n}b_jY_j.

Choose $c$ and the $b_j$ to minimize $E[(Y_0-\widehat{Y}_0)^2]$. The intercept makes the prediction error have mean zero at the minimum, so the problem can be written using centered observations $Y_j-m$. By the law of total covariance and conditional independence,

\operatorname{Cov}(Y_j,Y_k)= \begin{cases} a+v/w_j,&j=k,\\ a,&j\ne k, \end{cases} \qquad \operatorname{Cov}(Y_0,Y_j)=a.

The diagonal contains both between-risk variation and the period’s within-risk variation. Distinct periods share the same latent risk mean, which accounts for their covariance $a$. The future period’s own conditional variance affects the size of prediction error, but it does not change its covariance with historical observations.

For a least-squares linear predictor, the prediction error is uncorrelated with each centered observation. Thus the coefficients satisfy

\sum_{k=1}^{n} b_k\operatorname{Cov}(Y_k,Y_j) =\operatorname{Cov}(Y_0,Y_j)=a,\qquad j=1,\ldots,n.

Let $t=\sum_{k=1}^{n}b_k$ and $W=\sum_{k=1}^{n}w_k$. Substituting the covariance expressions into the equation for period $j$ gives

at+\frac{v}{w_j}b_j=a \quad\Longrightarrow\quad b_j=\frac{a}{v}w_j(1-t).

Sum this last expression over periods. Since $K=v/a$, the resulting equation and its solution are

t=\frac{W}{K}(1-t) \quad\Longrightarrow\quad t=\frac{W}{W+K}.

Therefore each observation’s coefficient is $b_j=w_j/(W+K)$. The intercept is $m(1-t)$, because the centered predictor has coefficients summing to $t$. The resulting predictor has the familiar premium form:

Theorem: Under the Bühlmann-Straub assumptions, the minimum-mean-squared-error linear predictor of a future loss rate is $$ \widehat{Y}_0=(1-Z)m+Z\overline{Y}_w,\qquad Z=\frac{W}{W+K},\qquad \overline{Y}_w=\frac{\sum_{j=1}^{n}w_jY_j}{W},\qquad W=\sum_{j=1}^{n}w_j,\quad K=\frac{v}{a}. $$ Equivalently, $\widehat{Y}_0=m(1-Z)+\sum_{j=1}^{n}\frac{w_j}{W+K}Y_j$.

The derivation explains both parts of the formula. Exposure determines the relative contribution of each observation, while $W/(W+K)$ determines the total weight assigned to all experience. When $W$ increases, $Z$ increases toward 1. When within-risk variation is large relative to between-risk variation, $K$ is large and $Z$ is smaller.

Worked Applications

Worked Example: Credibility With Three Unequal Periods

A hypothetical risk has collective mean $m=\$100$ per exposure unit, EPV $v=900$ dollars squared per exposure unit, and VHM $a=100$ dollars squared. Its observed loss rates are $\$120$, $\$90$, and $\$110$ per exposure unit, with exposures $1$, $2$, and $3$. Find the Bühlmann-Straub premium rate for a future exposure unit.

Method. Calculate $K$, total exposure, and the weighted experience mean, then combine the collective and experience rates. This avoids treating periods with unequal exposure as equally informative.

First, $K=v/a=900/100=9$ exposure units. Total historical exposure is $W=1+2+3=6$, so

Z=\frac{6}{6+9}=\frac{6}{15}=0.4.

The weighted experience amount is $1(120)+2(90)+3(110)=120+180+330=630$. Divide by $W=6$ to obtain

\overline{Y}_w=\frac{630}{6}=\$105.

The premium rate is $(1-0.4)(\$100)+0.4(\$105)=\$60+\$42=\$102$ per exposure unit. As a coefficient check, the three observation weights are $1/15$, $2/15$, and $3/15$; they sum to $6/15=0.4$, and their contributions are $\$8+\$12+\$22=\$42$. The result is $\$60+\$42=\$102$, as required.

Worked Example: Predicting an Aggregate for a Future Block

A hypothetical book has collective mean $\$250$ per 100 policy-years, EPV $1{,}600$ dollars squared per exposure unit, and VHM $400$ dollars squared. Historical rates are $\$220$, $\$300$, and $\$260$ per exposure unit for exposures $0.5$, $1.5$, and $2.0$, respectively. Find the expected aggregate premium for a future block with exposure $2.5$.

Method. First predict the future rate using historical total exposure; then multiply that rate by the future exposure to obtain the aggregate. The future exposure is not added to the historical exposure in $Z$.

The credibility constant is $K=1{,}600/400=4$, and historical total exposure is $W=0.5+1.5+2.0=4$. Thus $Z=4/(4+4)=0.5$. The weighted historical amount is

$$ 0.5(220)+1.5(300)+2.0(260)=110+450+520=1{,}080. $$

The weighted mean rate is $1{,}080/4=\$270$ per exposure unit. The predicted future rate is $0.5(\$250)+0.5(\$270)=\$260$. For future exposure $2.5$, the predicted aggregate premium is $2.5(\$260)=\$650$. The units check: a rate per exposure unit multiplied by 2.5 exposure units gives a total amount of $\$650$.

Worked Example: Checking the Minimization Directly

For a hypothetical risk, let $m=4$, $a=1$, and $v=4$. Two historical rates are $Y_1=2$ and $Y_2=5$, with exposures $w_1=1$ and $w_2=3$. Find the predictor and verify its mean squared prediction error for a future rate with exposure 1.

Method. Use the covariance equations from the derivation to obtain the coefficients, then calculate the prediction error variance. This confirms that the formula minimizes prediction error rather than merely producing a weighted average.

Here $K=4/1=4$, $W=1+3=4$, and $Z=4/(4+4)=0.5$. The weighted experience mean is $(1(2)+3(5))/4=17/4=4.25$. Hence the predicted rate is $0.5(4)+0.5(4.25)=4.125$.

The covariance vector between the future rate and the two historical rates is $(1,1)$. The historical covariance matrix has diagonal elements $1+4/1=5$ and $1+4/3=7/3$, with off-diagonal elements 1. The coefficients from the derived formula are $b_1=1/(4+4)=1/8$ and $b_2=3/(4+4)=3/8$. They satisfy the normal equations:

5(1/8)+1(3/8)=1,\qquad 1(1/8)+(7/3)(3/8)=1.

The future-rate variance is $a+v/1=5$, and $b_1+b_2=1/2$. The variance of the linear prediction is

a(b_1+b_2)^2+v\left(\frac{b_1^2}{w_1}+\frac{b_2^2}{w_2}\right) =\frac14+4\left(\frac{1}{64}+\frac{9}{64\cdot3}\right) =\frac14+\frac14=\frac12.

The covariance of the future rate with the prediction is $1(b_1+b_2)=1/2$. Therefore the prediction error variance is $5-2(1/2)+1/2=4.5$. Both normal equations hold, so the prediction error is uncorrelated with the historical observations; this is the least-squares linear predictor.

Interpretation and Common Traps

The premium is a compromise: the collective mean is the stable reference point, while the weighted experience mean reflects information specific to the risk. The model is a linear credibility model, so its predictor is optimal among linear predictors under the stated assumptions. It need not equal the full conditional expectation in a model with arbitrary distributions.

A frequent mistake is to use an ordinary average for experience while correctly calculating $Z$. In the first example, the ordinary average is $(120+90+110)/3=106.67$ to the nearest cent. Combining it with $Z=0.4$ gives $0.6(100)+0.4(106.67)=102.67$, which is not the Bühlmann-Straub premium. The correction is to weight each rate by its exposure: the weighted mean is $\$105$, giving $\$102$.

Do not confuse rate and aggregate. The credibility formula predicts a rate per exposure unit; multiply by the target exposure only when an aggregate amount is requested.
Keep the exposure basis consistent. The exposure units used for $w_j$, $v$, and $K=v/a$ must agree.
Do not count future exposure as historical experience. It scales the predicted aggregate but does not belong in $W$ when pricing the future rate.
Do not swap EPV and VHM. The ratio is $K=v/a$: within-risk variation in the numerator, between-risk variation in the denominator.

Key takeaway: Unequal exposures lead to an exposure-weighted experience mean. The total experience credibility is $Z=W/(W+K)$, where $W$ is historical exposure and $K=\mathrm{EPV}/\mathrm{VHM}$.

Check Your Understanding

For each problem, find the requested Bühlmann-Straub premium using the stated model quantities.

A hypothetical risk has $m=\$80$, $v=36$, and $a=12$. Its rates are $\$70$ and $\$100$ at exposures 1 and 2. Find the predicted rate. (Answer: $\$85.00$)
A portfolio has $m=\$500$, $v=900$, and $a=100$. A risk has rates $\$450$ and $\$550$ at exposures 2 and 4. Find its predicted rate. (Answer: $\$506.67$)
For $m=\$30$, $v=25$, and $a=5$, a risk has rates $\$26$ and $\$34$ at exposures 0.5 and 1.5. Find the predicted rate, to the nearest cent. (Answer: $\$30.57$)
A hypothetical book has $m=\$120$, $v=200$, and $a=50$. Historical rates are $\$100$ and $\$140$ at exposures 1 and 3. Find the predicted aggregate premium for future exposure 6. (Answer: $\$750$)
Let $m=10$, $v=160$, and $a=40$. Historical rates are 8, 12, and 9 at exposures 1, 2, and 5. Find the predicted rate. (Answer: $9.75$)

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