Experience-Rated Estimates For Individual Risks

From Multi-Period Experience to a Prospective Loss Cost

An individual risk may have several years of experience, with different amounts of exposure in each year. A prospective loss cost should reflect that experience, but it should not treat a small amount of experience as if it were fully reliable. Bühlmann-Straub credibility addresses this by combining the risk’s exposure-weighted historical loss cost with a collective complement.

The result is an estimate of the risk’s expected loss cost per exposure unit for a future period, under the assumption that the risk’s underlying expected loss cost is stable. It is not simply the historical average, nor is it a separate rate for each past period. The credibility factor determines how much weight the risk’s own experience receives.

Definition: For risk $r$ and period $j$, let $w_{rj}>0$ be the exposure and $X_{rj}$ the observed loss cost per exposure unit. The risk’s total exposure is $W_r=\sum_j w_{rj}$, and its exposure-weighted experience mean is $\overline{X}_r=\sum_j w_{rj}X_{rj}/W_r$. Let $m$ be the collective mean loss cost, $a$ the process-variance parameter per unit exposure, and $v$ the variance of the underlying risk means. The structural constant is $K=a/v$, and the Bühlmann-Straub credibility factor is $Z_r=W_r/(W_r+K)$. For positive exposure and $v>0$, the risk-specific prospective estimate is $Z_r\overline{X}_r+(1-Z_r)m$.

The observations $X_{rj}$ are loss costs, not total losses: if the period’s total loss is $L_{rj}$, then $X_{rj}=L_{rj}/w_{rj}$. The model allows unequal period exposures. It assumes that, conditional on the risk’s underlying mean $\Theta_r$, period observations are independent, have conditional mean $\Theta_r$, and have conditional variance $a/w_{rj}$. Across risks, $\Theta_r$ has mean $m$ and variance $v$. Thus, more exposure reduces the process variance of a risk’s experience mean, while greater variation in underlying risk means supports giving more weight to the risk’s own experience.

The model assumes that the periods being combined represent a comparable risk and coverage basis. If loss costs require adjustment to a common basis, that adjustment must be made before combining them. The formula is a linear credibility estimate under the stated moment assumptions; without additional distributional assumptions, it should not be described as an exact posterior mean.

Why the Estimate Has This Form

The risk’s experience mean is more informative when its total exposure is large. Under the model, the conditional variance of that weighted mean is $a/W_r$. This follows because its weighted-average error is a sum of independent period errors, each scaled by its exposure weight:

\operatorname{Var}(\overline{X}_r\mid\Theta_r) =\frac{1}{W_r^2}\sum_j w_{rj}^2\frac{a}{w_{rj}} =\frac{a}{W_r}.

To justify the credibility weight, consider a linear estimate $C=Z\overline{X}_r+(1-Z)m$ of $\Theta_r$. The experience error $\overline{X}_r-\Theta_r$ has conditional mean zero, so it is uncorrelated with $\Theta_r-m$. The mean squared error of $C$ is therefore the sum of the two weighted error variances:

E[(C-\Theta_r)^2] =Z^2\frac{a}{W_r}+(1-Z)^2v.

Differentiate with respect to $Z$ and set the derivative equal to zero:

2Z\frac{a}{W_r}-2(1-Z)v=0 \quad\Longrightarrow\quad Z\left(\frac{a}{W_r}+v\right)=v \quad\Longrightarrow\quad Z=\frac{W_r}{W_r+a/v} =\frac{W_r}{W_r+K}.

This derivation shows why exposure and the structural constant appear together. As $W_r$ increases, experience is more precise and receives more weight. As $K=a/v$ increases, process variation is larger relative to between-risk variation, so the collective mean receives more weight. The derivation assumes the model parameters are known; practical estimation of those parameters is a separate step.

Calculating a Risk-Specific Estimate

For an individual risk, first calculate its exposure-weighted experience mean, then calculate credibility using its total exposure, and finally blend that mean with the collective complement. Keep the exposure unit consistent across the history and the structural constant. The estimate remains a loss cost per exposure unit.

Worked Example: Three Periods With Unequal Exposure

A hypothetical equipment insurer evaluates one account over three periods. The account’s loss costs are $80, $100, and $120 per exposure unit, with exposures 1, 2, and 1 units, respectively. The collective complement is $90 per unit, and the structural constant is $K=4$ exposure units. Find the prospective estimate. No interest or mortality assumption applies. No parametric probability distribution is specified; the credibility calculation uses the stated means, exposures, and structural constant.

Step 1: Calculate total exposure. The account has $W_r=1+2+1=4$ exposure units.

Step 2: Calculate its weighted experience mean. The exposure-weighted loss-cost total is $1(80)+2(100)+1(120)=80+200+120=400$ exposure-cost dollars. Thus, $\overline{X}_r=400/4=\$100$ per exposure unit.

Step 3: Calculate credibility. $Z_r=4/(4+4)=4/8=0.5$.

Step 4: Blend experience and complement. The estimate is $0.5(100)+0.5(90)=50+45=\$95.00$ per exposure unit.

The result is below the account’s historical mean because the complement is lower, and the two receive equal weight. As a check, the weights sum to one and the estimate, $95.00, lies between $90 and $100. The method avoids treating the four exposure units as fully credible.

Worked Example: A Different Exposure Mix

A hypothetical commercial account has loss costs of $120, $100, and $80 per exposure unit in three periods, with exposures 0.5, 1, and 1.5 units. Its collective complement is $110 per unit and $K=3$ exposure units. Calculate the prospective loss cost. No interest or mortality assumption applies. No parametric distribution is assumed; use the specified credibility model and parameters.

Step 1: Total exposure. $W_r=0.5+1+1.5=3$ units.

Step 2: Exposure-weighted mean. The weighted total is $0.5(120)+1(100)+1.5(80)=60+100+120=280$. Therefore, $\overline{X}_r=280/3=\$93.333\ldots$ per unit.

Step 3: Credibility factor. $Z_r=3/(3+3)=0.5$.

Step 4: Prospective estimate. $0.5(280/3)+0.5(110)=140/3+55=46.666\ldots+55=101.666\ldots$. Rounded only at the final step to the nearest cent, the estimate is $101.67 per exposure unit.

The estimate exceeds the historical mean because the collective complement is higher. As a check, the result lies between $93.33 and $110. Using the simple average of the three period loss costs would ignore their unequal exposures.

Worked Example: Forming the Collective Complement From Other Risks

A hypothetical insurer has a subject risk with exposure-weighted historical loss costs of $70, $100, and $130 per unit, over exposures 1, 2, and 2 units. Three other risks provide the collective complement: risk B has exposure 2 and mean loss cost $90; risk C has exposure 3 and mean loss cost $120; risk D has exposure 5 and mean loss cost $100. Use the other risks’ pooled exposure-weighted mean as $m$, and use $K=5$ exposure units for the subject risk. Calculate the subject risk’s prospective estimate. No interest or mortality assumption applies. No parametric distribution is specified; the calculation uses the stated exposures, loss costs, and structural constant.

Step 1: Pool the other risks for the complement. Their weighted total is $2(90)+3(120)+5(100)=180+360+500=1{,}040$. Their total exposure is $2+3+5=10$. Hence $m=1{,}040/10=\$104$ per unit. The subject risk is excluded from this complement calculation.

Step 2: Calculate the subject risk’s experience mean. Its total exposure is $W_r=1+2+2=5$. Its weighted total is $1(70)+2(100)+2(130)=70+200+260=530$. Thus, $\overline{X}_r=530/5=\$106$ per unit.

Step 3: Calculate the subject risk’s credibility. $Z_r=5/(5+5)=0.5$.

Step 4: Combine the two means. The estimate is $0.5(106)+0.5(104)=53+52=\$105.00$ per exposure unit.

The result is between the subject’s $106 experience mean and the $104 complement, as expected. Excluding the subject from the complement here prevents its experience from appearing both as the subject’s experience and inside the collective mean.

Interpreting the Result and Avoiding Traps

The estimate is a credibility-weighted prospective loss cost for the individual risk, not a full premium indication. It can serve as a risk-specific expected cost per exposure unit, subject to the model’s stability and comparability assumptions. The complement provides a reference point when an individual history is limited or noisy; it does not replace the need to define the coverage, loss basis, and exposure unit consistently.

A frequent error is to reverse the weights while keeping the same value of $Z_r$. This tutorial uses $Z_r$ as the weight on the risk’s experience. For example, if $Z_r=0.25$, $\overline{X}_r=\$120$, and $m=\$80$, the correct estimate is $0.25(120)+0.75(80)=30+60=\$90$. The incorrect calculation $0.25(80)+0.75(120)=20+90=\$110$ puts the greater weight on experience and reverses the intended estimate. The alternate convention is valid only if its coefficient is redefined as $1-Z_r$, not if the same numerical coefficient is retained.

Put the experience on a consistent basis.
Confirm that period loss costs and exposures refer to comparable coverage and exposure units.

Weight by exposure.
Calculate $W_r=\sum_j w_{rj}$ and $\overline{X}_r=\sum_j w_{rj}X_{rj}/W_r$, not the unweighted average of period loss costs.

Calculate credibility.
Use $Z_r=W_r/(W_r+K)$, with total exposure and $K$ measured in consistent exposure units.

Apply each weight to its intended component.
Calculate $Z_r\overline{X}_r+(1-Z_r)m$: credibility multiplies the risk experience, and the remaining weight multiplies the complement.

Check the estimate.
For $0\leq Z_r\leq1$, the result must lie between $\overline{X}_r$ and $m$, and the two weights must sum to one.

Key takeaway: An individual risk’s Bühlmann-Straub prospective loss cost is $Z_r\overline{X}_r+(1-Z_r)m$. Unequal period exposures determine the risk’s weighted experience mean and total credibility exposure; the collective complement receives the remaining weight.

Check Your Understanding

For each problem, use $Z_r$ as the weight on the risk’s own experience. Report loss costs to the nearest cent where needed.

A risk has loss costs of $50 and $70 over exposures 2 and 3 units. The complement is $60 and $K=5$. Find its weighted experience mean, credibility, and prospective estimate. (Answer: experience mean $62.00; $Z_r=0.5000$; estimate $61.00 per unit.)
A risk has loss costs of $80, $100, and $110 over exposures 1, 1, and 2 units. The complement is $90 and $K=6$. Find the prospective estimate. (Answer: experience mean $100.00; $Z_r=0.4000$; estimate $94.00 per unit.)
A risk’s exposure-weighted experience mean is $120 over total exposure 3 units. Its complement is $100 and $K=2$. Find the prospective estimate and identify which component receives more weight. (Answer: $Z_r=0.6000$; estimate $112.00 per unit; experience receives more weight.)
Two risks other than the subject have exposures 2 and 4 units and mean loss costs of $80 and $110. The subject risk has loss costs $140 and $100 over exposures 1 and 3 units. With $K=4$, calculate the other risks’ pooled complement and the subject’s prospective estimate. (Answer: complement $100.00; subject experience mean $110.00; $Z_r=0.5000$; estimate $105.00 per unit.)
A risk has loss costs of $75, $90, and $105 over exposures 1, 2, and 3 units. Its complement is $110 and $K=3$. Calculate its estimate and explain whether it should be above or below its experience mean. (Answer: experience mean $95.00; $Z_r=0.6667$; estimate $100.00 per unit, above the experience mean because the complement is higher.)

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