One-Period Bühlmann Premium Calculations

One Observation, Two Sources of Information

A Bühlmann premium balances a risk’s own observed experience against the collective mean. With only one period of experience, the balance depends on two features of the collective model: the expected variation within a risk and the variation in expected outcomes between risks. This tutorial calculates the premium directly from those quantities and interprets the resulting credibility weight.

The outcome must be defined on a consistent basis. For example, if the observation is a claim count per exposure, the collective mean is a claim count per exposure and the variance parameters describe that same claim-count measure. If the observation is an aggregate loss, the mean and variance parameters must instead correspond to aggregate loss on the same basis.

Definition: In the Bühlmann model, let $\Theta$ represent a risk’s underlying risk characteristics and let $X$ be its outcome for one period. The collective mean is $m=E[X]$. The expected process variance (EPV) is $v=E[\operatorname{Var}(X\mid\Theta)]$, the average variation within a risk. The variance of hypothetical means (VHM) is $a=\operatorname{Var}(E[X\mid\Theta])$, the variation between risks in their conditional expected outcomes.

The standard one-period Bühlmann premium uses the best linear predictor of a risk’s expected outcome, $E[X\mid\Theta]$, based on its one observed outcome $X=x$. It has the form $Zx+(1-Z)m$. Here $Z$ is the credibility assigned to the individual observation, while $1-Z$ is the credibility assigned to the collective mean.

Formula: When $v$ and $a$ are finite and $a+v>0$, the one-period Bühlmann credibility weight and premium are

Z=\frac{a}{a+v} \qquad\text{and}\qquad \text{Bühlmann premium}=Zx+(1-Z)m.

The parameters $a$ and $v$ must be variances on the same outcome basis. If $a=0$, all risks have the same conditional mean and $Z=0$; individual experience receives no weight under this model. If $a+v=0$, the formula is undefined because there is no variation of either kind to determine a credibility weight.

Why the Credibility Weight Has This Form

The weight follows from the best linear prediction of a risk’s conditional mean using the single observation. Write a linear predictor as $m+b(X-m)$, where $b$ is chosen to minimize mean squared prediction error. The least-squares coefficient is $\operatorname{Cov}(E[X\mid\Theta],X)/\operatorname{Var}(X)$. Both parts can be evaluated from the Bühlmann model.

First, the law of total variance gives the variance of the observed outcome. Second, conditioning on $\Theta$ shows that the covariance between the risk’s conditional mean and the observation is the VHM. Substituting these quantities gives the coefficient $Z$.

Calculate the observation variance.
By the law of total variance, $\operatorname{Var}(X)=E[\operatorname{Var}(X\mid\Theta)]+\operatorname{Var}(E[X\mid\Theta])=v+a$.

Calculate the relevant covariance.
Let $\mu(\Theta)=E[X\mid\Theta]$. Then $\operatorname{Cov}(\mu(\Theta),X)=\operatorname{Cov}(\mu(\Theta),E[X\mid\Theta])=a$, since the conditional covariance of $\mu(\Theta)$ with $X$, given $\Theta$, is zero.

Find the best linear coefficient.
The least-squares slope is $b=\operatorname{Cov}(\mu(\Theta),X)/\operatorname{Var}(X)=a/(a+v)$. Thus $b=Z$, and the predictor is $m+Z(x-m)=Zx+(1-Z)m$.

This derivation establishes a best linear predictor. It does not, by itself, establish that the premium equals the exact conditional expectation $E[E[X\mid\Theta]\mid X=x]$ for every possible distribution. That stronger equality requires additional assumptions about the model.

Calculating a One-Period Premium

The calculation is most reliable when the inputs are identified before substituting them. Use the observed outcome $x$, not a deviation from the collective mean, in the premium formula. The premium can also be written $m+Z(x-m)$, which makes clear that it adjusts the collective mean by a fraction $Z$ of the difference between the observation and that mean.

Worked Example: One Period of Aggregate Loss Experience

A hypothetical insurer is pricing one period of aggregate loss per policy for a particular risk. The collective mean is $m=\$1{,}200$, the EPV is $v=900{,}000$ dollars squared, and the VHM is $a=300{,}000$ dollars squared. The risk’s observed aggregate loss per policy is $x=\$1{,}500$. Calculate the Bühlmann premium and interpret the weight. Use the standard one-period Bühlmann method.

Step 1: Check the basis. The observation and collective mean are both aggregate loss per policy, and both variance parameters are in squared dollars on that same basis.

Step 2: Calculate credibility.

Z=\frac{a}{a+v} =\frac{300{,}000}{300{,}000+900{,}000} =\frac{300{,}000}{1{,}200{,}000} =0.25.

Step 3: Calculate the premium.

\begin{aligned} \text{Premium} &=Zx+(1-Z)m\\ &=0.25(1{,}500)+0.75(1{,}200)\\ &=375+900\\ &=\$1{,}275. \end{aligned}

Interpretation and check. The observed experience receives 25% weight and the collective mean receives 75% weight. The observation is $\$300$ above the collective mean, so the premium is $0.25(\$300)=\$75$ above it: $\$1{,}200+\$75=\$1{,}275$. The method avoids the common mistake of treating $Z$ itself as the premium.

Worked Example: Keeping Dollar and Variance Units Straight

For a hypothetical coverage, one risk’s observed loss per exposure is $x=\$650$. The collective mean loss per exposure is $m=\$800$. On this same basis, the EPV is $v=160{,}000$ dollars squared and the VHM is $a=40{,}000$ dollars squared. Calculate the one-period Bühlmann premium. Use the standard one-period formula, checking that the variance inputs are not mistaken for dollar amounts.

Step 1: Calculate credibility. The two variance parameters have matching squared-dollar units, so their ratio is dimensionless.

Z=\frac{40{,}000}{40{,}000+160{,}000} =\frac{40{,}000}{200{,}000} =0.20.

Step 2: Apply both weights.

\begin{aligned} \text{Premium} &=0.20(650)+0.80(800)\\ &=130+640\\ &=\$770. \end{aligned}

Interpretation and check. The observation is $\$150$ below the collective mean. A 20% credibility weight moves the premium down by $0.20(\$150)=\$30$, giving $\$800-\$30=\$770$. A common error is to add or compare the dollar observation directly to a variance in dollars squared. The variances are used to calculate the dimensionless weight; only then are the dollar-valued experience and mean combined.

Worked Example: A High-Variation Between Risks

A hypothetical pool has one-period outcome mean $m=200$ units, EPV $v=2{,}500$ units squared, and VHM $a=7{,}500$ units squared. One risk has an observed outcome $x=260$ units. Find the one-period Bühlmann premium, and state how much the observation moves the premium from the collective mean.

Step 1: Calculate credibility.

Z=\frac{7{,}500}{7{,}500+2{,}500} =\frac{7{,}500}{10{,}000} =0.75.

Step 2: Calculate the premium.

\begin{aligned} \text{Premium} &=0.75(260)+0.25(200)\\ &=195+50\\ &=245\text{ units}. \end{aligned}

Step 3: Check the movement from the mean. The observation is $60$ units above the mean, and the premium moves $0.75(60)=45$ units above it. Therefore $200+45=245$ units, as calculated. The method uses the larger VHM relative to EPV to give the risk’s own experience greater weight; it does not assign 75% of the premium in units.

Interpreting the Weight and Avoiding Traps

Since $a$ and $v$ are nonnegative variance quantities, when $a+v>0$ the credibility weight lies between 0 and 1. A larger VHM relative to EPV increases $Z$: the risks’ expected outcomes differ more from one another relative to the random variation within a risk. A larger EPV relative to VHM decreases $Z$: one observation is less informative about which risk-specific mean applies.

The formula can also be expressed using the Bühlmann credibility constant $K=v/a$, when $a>0$. For one observation, $Z=1/(1+K)$. This is simply an alternative form of the same one-period weight, not a different premium method.

Exam trap: Reversing EPV and VHM gives the wrong weight. For example, with $a=300$ and $v=900$, the incorrect calculation $900/(300+900)=0.75$ uses EPV in the numerator. The correct calculation is $Z=300/(300+900)=0.25$, because the VHM is the between-risk variance that determines the weight on the risk’s observed experience. Then use $0.25x+0.75m$.

Other checks can catch errors before they reach the final premium:

Confirm the observation basis. A claim count, aggregate loss, and loss per exposure are different outcomes. Do not combine one with a mean or variance defined for another.
Use variance units correctly. EPV and VHM are variances, so they have squared outcome units. Their ratio in the formula for $Z$ is dimensionless.
Check whether the premium lies between $m$ and $x$. For $0\leq Z\leq1$, the premium is a weighted average, so it cannot lie outside the interval between the observation and collective mean.
Keep the weight separate from the premium. $Z$ is a dimensionless proportion; the premium is expressed in the units of the outcome.

One-period Bühlmann calculations are useful when a problem supplies structural parameters and asks how one risk’s experience should adjust the collective mean. The central reasoning is to calculate the share of individual experience from the VHM and EPV, then apply that share to the observed outcome—not to its variance or to its deviation alone.

Check Your Understanding

For each problem, use the one-period Bühlmann premium and keep the outcome basis consistent.

A hypothetical portfolio has collective mean $m=1{,}000$ dollars per exposure, EPV $v=900$ dollars squared, and VHM $a=300$ dollars squared. One risk’s observed outcome is $1{,}400$ dollars per exposure. Find its premium. (Answer: $1{,}100$ dollars per exposure.)
A one-period outcome has collective mean $m=240$ units, EPV $v=1{,}600$ units squared, and VHM $a=6{,}400$ units squared. The observed outcome is $300$ units. Calculate the credibility weight and premium. (Answer: $Z=0.8000$; premium $=288$ units.)
For loss per exposure, $m=\$500$, $v=3{,}600$ dollars squared, and $a=8{,}400$ dollars squared. The observed loss is $\$620$. Calculate the premium and its increase above the collective mean. (Answer: $Z=0.7000$; premium $=\$584$, an increase of $\$84$.)
A hypothetical risk has $m=75$ units, $v=25$ units squared, and $a=75$ units squared. Its observed outcome is $95$ units. Find the one-period premium. (Answer: $90$ units.)
A risk has collective mean $m=1{,}000$ dollars, EPV $v=400$ dollars squared, and VHM $a=600$ dollars squared. Its one-period Bühlmann premium is $1{,}120$ dollars. Infer the observed outcome. (Answer: $\$1{,}200$.)

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