Exam Workshop: Unequal Exposure, Prediction, and Interpretation

Why Unequal Exposure Changes the Calculation

In a credibility problem, each risk’s observed experience provides information about its underlying expected rate. When risks have different amounts of exposure, an observation based on more exposure should generally carry more weight in the experience summary. The Bühlmann-Straub model makes this principle explicit: it uses an exposure-weighted experience mean and lets total exposure determine the credibility assigned to that mean.

The premium produced by the model is a predicted rate. To predict an expected aggregate amount for a future period, multiply that rate by the future exposure. Keeping those two quantities separate is essential, especially when the future exposure differs from the historical exposure.

Definition: For risk $i$, let $Y_{ij}$ be the observed rate in period $j$, based on positive exposure $w_{ij}$. In the Bühlmann-Straub model, conditional on the risk’s unknown expected rate $\Theta_i$, the observations are independent and satisfy $$ E[Y_{ij}\mid\Theta_i]=\Theta_i,\qquad \operatorname{Var}(Y_{ij}\mid\Theta_i)=\frac{\sigma^2}{w_{ij}}. $$ Across risks, assume $E[\Theta_i]=\mu$ and $\operatorname{Var}(\Theta_i)=\tau^2$. Here $\mu$ is the collective mean, $\sigma^2$ is the expected process variance for unit exposure, and $\tau^2$ is the variance of hypothetical means.

This model applies to rate observations whose conditional variance decreases in inverse proportion to exposure. The exposures must be measured on a consistent basis, and the observations must have the stated conditional mean and variance structure. The formula should not be used mechanically if, for example, the recorded values are aggregate amounts rather than rates, or if the exposure scale does not match the variance assumption.

Weighted Experience Mean and Credibility Premium

For one risk, its total historical exposure is $w_i=\sum_j w_{ij}$. Its exposure-weighted experience mean is the sum of exposure times rate, divided by total exposure. Define $k=\sigma^2/\tau^2$ when $\tau^2>0$. The Bühlmann-Straub credibility factor and premium rate are

\bar{Y}_i=\frac{\sum_j w_{ij}Y_{ij}}{w_i}, \qquad Z_i=\frac{w_i}{w_i+k}, \qquad \widehat{\Theta}_i=Z_i\bar{Y}_i+(1-Z_i)\mu.

The estimated rate is a weighted average of the risk’s experience mean and the collective mean. Greater total exposure increases $Z_i$; a larger process variance relative to the variance of hypothetical means increases $k$ and reduces credibility for a given exposure. If $\tau^2=0$, all risks have the same expected rate under the model, so the experience does not shift the premium away from $\mu$.

Why the Credibility Weight Has This Form

A short derivation shows why total exposure, rather than the number of observations alone, determines the weight. Conditional independence gives the variance of the weighted experience mean around the risk’s expected rate:

\operatorname{Var}(\bar{Y}_i\mid\Theta_i) =\operatorname{Var}\left(\frac{\sum_j w_{ij}Y_{ij}}{w_i}\middle|\Theta_i\right) =\frac{1}{w_i^2}\sum_j w_{ij}^2\frac{\sigma^2}{w_{ij}} =\frac{\sigma^2}{w_i}.

Now consider a linear estimator $a\bar{Y}_i+(1-a)\mu$. Its mean squared error for estimating $\Theta_i$ is the sum of the variance of the experience noise, weighted by $a^2$, and the squared shrinkage error, weighted by $(1-a)^2$:

\operatorname{MSE}(a) =a^2\frac{\sigma^2}{w_i}+(1-a)^2\tau^2.

Differentiating and setting the derivative equal to zero gives

\frac{d}{da}\operatorname{MSE}(a) =2a\frac{\sigma^2}{w_i}-2(1-a)\tau^2=0, \qquad a=\frac{\tau^2}{\tau^2+\sigma^2/w_i} =\frac{w_i}{w_i+\sigma^2/\tau^2}.

Thus the minimum-MSE linear weight is $Z_i=w_i/(w_i+k)$. This justification uses the model’s variance structure and the linear credibility approach; it does not mean that a particular realized future rate must equal the premium.

Worked Calculations

Worked Example: Two Risks With Different Historical and Future Exposures

A hypothetical portfolio has collective mean rate $\mu=\$100$, process variance parameter $\sigma^2=300$, and variance of hypothetical means $\tau^2=100$. Risk A has rates of $\$80$ at exposure 1 and $\$120$ at exposure 2. Risk B has a rate of $\$130$ at exposure 1. Predict each risk’s premium rate and expected aggregate amount for next period, when A has exposure 2 and B has exposure 1. Then find the combined expected aggregate. Treat the rates and exposure units consistently.

Method and structural parameter. First calculate $k=\sigma^2/\tau^2=300/100=3$. For each risk, weight observed rates by their exposures.

Risk A’s experience mean. Its total exposure is $w_A=1+2=3$, and its exposure-weighted total is $1(80)+2(120)=80+240=320$. Therefore, $\bar{Y}_A=320/3=\$106.6667$ before final rounding. Its credibility is $Z_A=3/(3+3)=1/2$.

Risk A’s premium rate. The premium is $(1/2)(320/3)+(1/2)(100)=160/3+50=310/3=\$103.3333\ldots$. Rounded to the nearest cent, the rate is $\$103.33$. For future exposure 2, the expected aggregate is $2(310/3)=620/3=\$206.6667\ldots$, or $\$206.67$.

Risk B’s premium rate. Its total exposure is $w_B=1$, and its experience mean is $\bar{Y}_B=\$130$. Thus $Z_B=1/(1+3)=1/4$, and its premium rate is $(1/4)(130)+(3/4)(100)=32.50+75=\$107.50$. At future exposure 1, its expected aggregate is $\$107.50$.

Portfolio prediction and check. Add the two future expected aggregates: $620/3+107.50=620/3+215/2=1{,}885/6=\$314.1667\ldots$, or $\$314.17$. The result is also $2(\$103.3333\ldots)+1(\$107.50)$, confirming that each premium rate was multiplied by the correct future exposure. Common mistake avoided: treating A’s premium rate as $\$100$ or multiplying its rate by historical exposure instead of the stated future exposure.

Worked Example: A Multi-Period Experience With a Different Collective Mean

A hypothetical risk has observed rates of $\$40$ at exposure 1, $\$70$ at exposure 3, and $\$60$ at exposure 2. The collective mean is $\mu=\$50$, with $\sigma^2=200$ and $\tau^2=50$. Find its Bühlmann-Straub premium rate and its expected aggregate for a future exposure of 5.

Method. Compute the exposure-weighted mean and total exposure, then apply the credibility factor. First, $k=200/50=4$. The total exposure is $w=1+3+2=6$, and the weighted rate total is $1(40)+3(70)+2(60)=40+210+120=370$. Hence $\bar{Y}=370/6=\$61.6667$ before rounding.

The credibility factor is $Z=6/(6+4)=0.6$. The premium rate is $0.6(370/6)+0.4(50)=37+20=\$57.00$. The expected aggregate for future exposure 5 is $5(57)=\$285.00$.

Check and interpretation. Since $\$57.00$ lies between the experience mean of about $\$61.67$ and the collective mean of $\$50$, the shrinkage is in the expected direction. The aggregate prediction is five times the premium rate. Common mistake avoided: averaging the three rates equally would give $(40+70+60)/3=\$56.67$, which gives the low-exposure observation the same weight as the exposure-3 observation.

Worked Example: Separate Risk Predictions and a Combined Future Amount

A hypothetical portfolio has $\mu=\$200$, $\sigma^2=800$, and $\tau^2=200$. Risk C has rates $\$180$ at exposure 2 and $\$220$ at exposure 4; its future exposure is 3. Risk D has rates $\$240$ and $\$160$, each at exposure 1; its future exposure is 4. Calculate each premium rate, each future expected aggregate, and the combined expected aggregate.

Method. Here $k=800/200=4$. For C, total exposure is $w_C=2+4=6$. Its weighted total is $2(180)+4(220)=360+880=1{,}240$, so $\bar{Y}_C=1{,}240/6=\$206.6667$. Its credibility is $Z_C=6/(6+4)=0.6$. The premium rate is $0.6(1{,}240/6)+0.4(200)=124+80=\$204.00$. At future exposure 3, the expected aggregate is $3(204)=\$612.00$.

For D, total exposure is $w_D=2$, and the weighted total is $1(240)+1(160)=400$. Thus $\bar{Y}_D=400/2=\$200$, $Z_D=2/(2+4)=1/3$, and the premium rate is $(1/3)(200)+(2/3)(200)=\$200.00$. At future exposure 4, the expected aggregate is $4(200)=\$800.00$.

The combined expected aggregate is $\$612+\$800=\$1{,}412.00$. As a check, both premium rates equal their corresponding predicted rates before exposure scaling; the portfolio total adds the two future-period amounts, not the two historical experience totals. Common mistake avoided: using the same future exposure for both risks or combining their rates without multiplying each by its own future exposure.

Prediction, Units, and Exam Traps

A premium rate is an estimate of expected cost per exposure unit under the model. If the future exposure is $w_i^{\mathrm{new}}$, then the predicted future aggregate amount is

\widehat{M}_i^{\mathrm{new}}=w_i^{\mathrm{new}}\widehat{\Theta}_i.

This is a conditional-mean prediction based on the fitted credibility premium, not a promise about the realized amount. For a portfolio, calculate each risk’s future expected aggregate using its own future exposure and premium rate, then add the results. Do not average premium rates across risks unless the question specifically asks for an average rate and supplies an appropriate weighting basis.

The model also distinguishes exposure from the number of observations. Two records with exposures 1 and 9 contribute total exposure 10, not two equal units of information. The weighted experience mean is $ (1Y_1+9Y_2)/10$, and the credibility factor uses total exposure 10. If an exposure is zero, it contributes neither to the weighted mean nor to total exposure; it must not be treated as a positive-weight observation.

Establish the model inputs.
Identify the collective mean, process variance, variance of hypothetical means, observed rates, and exposures. Confirm that the observations are rates with the stated variance scaling.

Summarize each risk’s experience.
Calculate $w_i=\sum_j w_{ij}$, then $\bar{Y}_i=\sum_j w_{ij}Y_{ij}/w_i$. Retain enough precision for later steps.

Calculate credibility and premium rate.
Find $k=\sigma^2/\tau^2$, $Z_i=w_i/(w_i+k)$, and $\widehat{\Theta}_i=Z_i\bar{Y}_i+(1-Z_i)\mu$.

Predict the requested future amount.
Multiply each premium rate by that risk’s stated future exposure. Add the resulting aggregates only if a combined prediction is requested.

A reliable final check is to ask whether each answer has the right units and falls in a plausible range. A premium rate should lie between the risk’s experience mean and the collective mean when $0\leq Z_i\leq1$. A future aggregate should equal that rate multiplied by future exposure. Keep full precision through the calculation and round money only at the end.

Key takeaway: Unequal exposure enters twice: it weights the risk’s observed rates in $\bar{Y}_i$, and its total determines $Z_i$. The credibility premium is a rate; future exposure converts it into an expected aggregate amount.

Check Your Understanding

For each problem, calculate with the stated exposure weights and distinguish the premium rate from any future aggregate prediction.

A risk has collective mean $\$80$, $k=2$, and one observed rate of $\$100$ at exposure 3. Find its credibility premium rate. (Answer: $\$92.00$)
A risk has rates $\$40$ at exposure 2 and $\$80$ at exposure 1. If $\mu=\$50$ and $k=3$, find its premium rate. (Answer: $\$51.67$)
A risk has rates $\$100$ at exposure 1 and $\$140$ at exposure 4. If $\mu=\$120$ and $k=5$, find its premium rate and expected aggregate at future exposure 2. (Answer: Premium rate $\$126.00$; expected aggregate $\$252.00$)
A risk has rates $\$10$ at exposure 1 and $\$20$ at exposure 3. If $\mu=\$16$ and $k=2$, find its premium rate and expected aggregate at future exposure 6. (Answer: Premium rate $\$17.00$; expected aggregate $\$102.00$)
Use $\mu=\$500$ and $k=4$. Risk A has rates $\$400$ and $\$600$, each at exposure 1; its future exposure is 3. Risk B has a rate of $\$550$ at exposure 8; its future exposure is 2. Find each premium rate and the combined expected future aggregate. (Answer: A rate $\$500.00$, B rate $\$533.33$; combined expected aggregate $\$2{,}566.67$)

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