Equal Exposures: Bühlmann and Bühlmann-Straub

Why Equal Exposures Connect the Two Credibility Models

The Bühlmann-Straub model allows each observation to carry a different amount of exposure. When every observation has the same exposure, its weighted experience mean becomes an ordinary arithmetic mean. The credibility factor also takes the Bühlmann form, provided the Bühlmann calculation uses the correct variance for an exposure-adjusted rate. Keeping track of that variance is essential: the credibility constants in the two formulas are not automatically the same numerical quantity.

This connection is useful both for calculation and for checking results. A balanced portfolio can be priced by either formula, but the observation and variance units must match. The examples below use hypothetical portfolios and state their assumptions explicitly.

Definition: In the Bühlmann-Straub model, let $\Theta$ denote a risk's underlying parameter. Conditional on $\Theta$, observations $X_j$ are independent, with conditional mean $\mu(\Theta)$ and conditional variance $\sigma^2(\Theta)/w_j$, where $w_j>0$ is the known exposure for observation $j$. Define the hypothetical mean $m=E[\mu(\Theta)]$, the expected process variance $v=E[\sigma^2(\Theta)]$, and the variance of hypothetical means $a=\operatorname{Var}(\mu(\Theta))$. The model assumes these quantities are finite and $a>0$ for a nontrivial credibility factor.

Here $X_j$ is an observed loss rate per exposure unit, rather than the total loss for the period. The total exposure is $W=\sum_{j=1}^n w_j$, and the exposure-weighted experience mean is $\bar X_w=\sum_{j=1}^n w_jX_j/W$. Under the Bühlmann-Straub model, the credibility premium for the rate is a weighted average of this experience mean and $m$.

Z_{\mathrm{BS}}=\frac{W}{W+K},\qquad K=\frac{v}{a},\qquad \widehat{\mu}_{\mathrm{BS}}=Z_{\mathrm{BS}}\bar X_w+(1-Z_{\mathrm{BS}})m.

The quantity $K$ is measured in exposure units: $v/a$ has the units of the exposure scale used in the model. If the rate premium is for a future period with exposure $w^*$, the corresponding premium for total loss is $w^*\widehat{\mu}_{\mathrm{BS}}$. The formula prices a rate first; multiplying by future exposure is a separate step.

Deriving the Equal-Exposure Reduction

Suppose all $n$ observations have the same exposure $w$. Then $W=nw$, and each observation receives the same weight in the experience mean:

\bar X_w=\frac{\sum_{j=1}^n wX_j}{nw} =\frac{1}{n}\sum_{j=1}^n X_j=\bar X.

To express this as a Bühlmann model, treat the $X_j$'s as $n$ exchangeable observations of an exposure-adjusted rate. Conditional on $\Theta$, each has variance $\sigma^2(\Theta)/w$. Thus the expected process variance for one such rate observation is $v/w$, not $v$. The Bühlmann credibility constant for these observations is therefore $K_B=(v/w)/a$.

Result: When all $n$ observations have exposure $w$, the Bühlmann-Straub premium for a rate equals the Bühlmann premium applied to the $n$ exposure-adjusted rates, using expected process variance $v/w$: $$ K_B=\frac{v}{wa}=\frac{K}{w},\qquad Z_B=\frac{n}{n+K_B} =\frac{n}{n+K/w} =\frac{nw}{nw+K} =Z_{\mathrm{BS}}. $$ Since the experience means also agree, both premiums are identical.

The variance adjustment has a direct explanation. Conditional independence gives the weighted mean conditional variance $\sum_j w_j^2\sigma^2(\Theta)/w_j\,/\,W^2=\sigma^2(\Theta)/W$. Averaging over $\Theta$ gives $v/W$. Variation in the risk's underlying mean contributes $a$, so the variance of the experience mean is $a+v/W$, while its covariance with $\mu(\Theta)$ is $a$. The linear credibility coefficient is consequently

\frac{\operatorname{Cov}(\mu(\Theta),\bar X_w)} {\operatorname{Var}(\bar X_w)} =\frac{a}{a+v/W} =\frac{W}{W+v/a}.

This justifies the Bühlmann-Straub factor and shows why total exposure, rather than merely the number of records, controls credibility. With equal $w$, writing $W=nw$ and using $K_B=v/(wa)$ gives the Bühlmann coefficient above. This argument presumes conditional independence and the stated variance scaling; it does not establish the reduction for dependent observations or unequal exposures treated as if they were equal.

Worked Example: Converting Equal Exposure Weights

A hypothetical portfolio has four independent rate observations, each with exposure $w=2$. Let $m=\$100$ per exposure unit, $v=3600$, and $a=300$, with conditional variances scaling as $1/w$. The observed rates are $\$90,\$110,\$120,\$140$. Find the credibility premium rate by both methods.

Method: compare the Bühlmann-Straub and equivalent Bühlmann formulas. First, the total exposure is $W=4(2)=8$, and $K=v/a=3600/300=12$. Thus $Z_{\mathrm{BS}}=8/(8+12)=0.4$.

The weighted experience mean is the ordinary mean because all exposures are equal: $\bar X=(90+110+120+140)/4=460/4=\$115$. Therefore, $\widehat{\mu}_{\mathrm{BS}}=0.4(115)+0.6(100)=46+60=\$106$ per exposure unit.

For Bühlmann applied to the four rates, the process variance per rate observation is $v/w=3600/2=1800$. Hence $K_B=1800/300=6$ and $Z_B=4/(4+6)=0.4$. Its premium is $0.4(115)+0.6(100)=\$106$, agreeing exactly. As a check, if the future exposure were $3$, the corresponding total-loss premium would be $3(106)=\$318$.

Common mistake avoided: using $v=3600$ as the Bühlmann rate-observation variance would give $K_B=12$ and a different weight. The rate observations each represent exposure $2$, so their process variance is $v/2$.

A Balanced-Portfolio Numerical Check

For a balanced portfolio, computing both routes is a practical way to catch an exposure-unit error. The following example uses a compact set of parameters and shows every part of the calculation.

Worked Example: Matching Premiums for a Balanced Portfolio

A hypothetical insurer observes three independent annual loss rates, each based on exposure $w=2$. Suppose $m=\$95$, $v=240$, and $a=60$, with conditional rate variance $\sigma^2(\Theta)/w$. The three observed rates are $\$90,\$100,\$110$. Calculate the credibility premium rate using both formulas.

Method: use total exposure for Bühlmann-Straub and exposure-adjusted variance for Bühlmann. The total exposure is $W=3(2)=6$, and $K=v/a=240/60=4$. The Bühlmann-Straub factor is $Z_{\mathrm{BS}}=6/(6+4)=0.6$.

The weighted experience mean equals the ordinary mean: $\bar X=(90+100+110)/3=300/3=\$100$. The premium rate is $0.6(100)+0.4(95)=60+38=\$98$.

For the Bühlmann calculation, $v/w=240/2=120$, so $K_B=120/60=2$. Then $Z_B=3/(3+2)=0.6$, and the Bühlmann premium is $0.6(100)+0.4(95)=\$98$. Both methods agree. The result is a rate; for a future exposure of $2$, for example, its corresponding total-loss premium would be $2(98)=\$196$.

Check: the weighted mean, experience weight, and complement are internally consistent: $0.6+0.4=1$, and the premium $\$98$ lies between the experience mean $\$100$ and the hypothetical mean $\$95$.

When the Units Do Not Match

The equal-exposure identity does not mean that one may move between formulas while leaving every parameter unchanged. In the Bühlmann-Straub expression, $K=v/a$ is compared with total exposure $W$. In the equivalent Bühlmann expression, $K_B=v/(wa)$ is compared with the number of rate observations $n$. The conversion is $K_B=K/w$, not $K_B=K$.

Worked Example: Finding and Correcting a Unit Error

A hypothetical portfolio has three rate observations with equal exposure $w=2$, hypothetical mean $m=\$95$, $v=240$, $a=60$, and observed rates $\$90,\$100,\$110$. A calculation uses $K=v/a=4$ directly in the ordinary Bühlmann formula, obtaining $Z=3/(3+4)=3/7$. It then reports a premium of $(3/7)(100)+(4/7)(95)=\$97.14$, rounded to cents. Diagnose the error and calculate the correct rate premium.

Method: convert the Bühlmann-Straub constant to the rate-observation scale. The experience mean is $(90+100+110)/3=\$100$. The three observations have exposure $2$ each, so the expected process variance for each rate observation is $v/w=240/2=120$. Therefore $K_B=120/60=2$, not $4$.

The correct Bühlmann factor is $Z_B=3/(3+2)=3/5=0.6$. The premium rate is $0.6(100)+0.4(95)=60+38=\$98$. The Bühlmann-Straub check gives $W=6$, $K=4$, and $Z_{\mathrm{BS}}=6/(6+4)=0.6$, also producing $\$98$.

The incorrect calculation compares a constant measured in exposure units with a count of rate observations. The correction changes the constant to the matching per-observation scale before using $n/(n+K_B)$.

Exam Checks and Practical Interpretation

For equal exposures, the arithmetic average and exposure-weighted average are the same, but the credibility adjustment still depends on the total exposure. Before calculating, label each quantity: $w$ is exposure per observation, $W=nw$ is total exposure, $v$ is the process variance parameter before exposure scaling, and $v/w$ is the expected process variance of one observed rate.

A useful exam check is to calculate the credibility factor both ways. If the model assumptions hold, the factors and premiums must agree exactly before rounding. If they do not, check whether the experience values are rates or aggregate losses, whether the exposure scale is consistent, and whether the complement $1-Z$ multiplies the hypothetical mean.

If all exposures are equal, the experience mean simplifies to the ordinary mean.
For an equivalent Bühlmann calculation on rates, use $K_B=v/(wa)$.
Apply a premium rate to future exposure only when a total-loss premium is requested.
The reduction does not justify treating unequal exposures as equal; their weighted mean and variance structure must be retained.

Check Your Understanding

For each hypothetical portfolio, use the stated model and report the requested premium. Exposure weights refer to rate observations with conditional variance $\sigma^2(\Theta)/w$.

There are five observations, each with exposure $w=2$. Let $m=\$70$, $v=180$, $a=30$, and the observed rates be $\$60,\$70,\$80,\$90,\$100$. Find the premium rate using either equivalent method. (Answer: $\$76.25$ per exposure unit.)
Three observations each have exposure $w=4$, with $m=\$200$, $v=800$, $a=100$, and rates $\$180,\$210,\$240$. Find the Bühlmann-Straub factor and premium rate, then state the equivalent Bühlmann constant. (Answer: $Z=0.6000$, premium rate $\$206$, and $K_B=2$.)
Four observations each have exposure $w=0.5$, with $m=\$40$, $v=90$, $a=15$, and rates $\$30,\$35,\$45,\$55$. Find the credibility premium rate. (Answer: $\$40.31$ per exposure unit, rounded to cents.)
Two observations each have exposure $w=3$, with $m=\$50$, $v=120$, $a=40$, and rates $\$40,\$80$. Find the premium rate and the total-loss premium for a future exposure of $5$. (Answer: premium rate $\$56.67$ per exposure unit and total-loss premium $\$283.33$, both rounded to cents.)

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