Weighted Risk Means And Portfolio Means

Why Two Different Means Matter

In the Bühlmann-Straub setting, experience is observed for several risks over periods that may have different exposures. Two averages are useful, but they answer different questions. A risk mean summarizes the experience for one risk, giving more weight to its periods with greater exposure. A portfolio mean summarizes the experience across all risks, giving more weight to risks with greater total exposure.

The portfolio mean is often used as the observed collective complement in credibility calculations. It should not be confused with the mean of the risk means, which gives each risk equal weight regardless of its exposure. This tutorial focuses on calculating and interpreting these means; it does not derive a credibility estimate.

Definition: Let $X_{ij}$ be the observed cost per exposure unit for risk $i$ in period $j$, and let $w_{ij}$ be that period’s exposure. The exposure-weighted mean for risk $i$ is

\bar X_i=\frac{\sum_j w_{ij}X_{ij}}{\sum_j w_{ij}}.

The total exposure for risk $i$ is $W_i=\sum_j w_{ij}$. For a group of risks, the exposure-weighted portfolio mean is

\bar X_{\mathrm{port}} =\frac{\sum_i W_i\bar X_i}{\sum_i W_i} =\frac{\sum_i\sum_j w_{ij}X_{ij}}{\sum_i\sum_j w_{ij}}.

Model and Data Conditions

In the Bühlmann-Straub model, each risk $i$ has an underlying expected cost per exposure unit, denoted by $\Theta_i$. Period observations for that risk are treated as experience around the same risk-specific mean: $E[X_{ij}\mid\Theta_i]=\Theta_i$. The collective mean is $m=E[\Theta_i]$. The observed portfolio mean is a data-based summary that can serve as a complement for the collective mean; in general, it is not exactly equal to the unknown $m$.

The weighting formulas require that the exposures measure comparable units and that the cost rates have been put on a consistent basis. For example, if periods have different coverage terms or have not been adjusted to a common rating level, their rates may not be directly comparable. Exposure should reflect the amount of risk represented by each observation, not merely the number of rows in a data table.

If the data give a period’s total loss $L_{ij}$, rather than its cost per exposure unit, first note that $X_{ij}=L_{ij}/w_{ij}$. Substituting this into the risk-mean formula gives a useful equivalent form:

\bar X_i =\frac{\sum_j w_{ij}(L_{ij}/w_{ij})}{\sum_j w_{ij}} =\frac{\sum_j L_{ij}}{\sum_j w_{ij}}.

Thus a risk’s mean is its total loss divided by its total exposure. The same logic applies across risks: the portfolio mean is total portfolio loss divided by total portfolio exposure, provided the loss totals and exposures cover the same data.

Why the Portfolio Formula Is Exposure-Weighted

The portfolio rate should represent the total cost divided by the total amount of exposure. Each risk mean is already a rate, not a total cost. Multiplying that rate by the risk’s total exposure recovers the corresponding total experience. Adding those amounts and dividing by all exposure gives the aggregate rate:

\frac{\sum_i W_i\bar X_i}{\sum_i W_i} = \frac{\sum_i W_i\left(\frac{\sum_j w_{ij}X_{ij}}{W_i}\right)} {\sum_i W_i} = \frac{\sum_i\sum_j w_{ij}X_{ij}}{\sum_i\sum_j w_{ij}}.

The cancellation in the numerator shows why the formula is consistent: it aggregates the exposure-weighted experience across all periods and risks. An unweighted average of risk means would instead answer a different question: what is the average of the observed risk-level rates when each risk counts once?

Worked Example: Period Rates for Two Risks

A hypothetical equipment insurer records cost rates and exposures for two risks. Risk A has a rate of $240 per exposure unit on 1.5 units, and $200 per unit on 2.5 units. Risk B has a rate of $300 per unit on 1 unit, and $180 per unit on 3 units. Calculate both risk means and the portfolio mean. Use the exposure-weighted method and identify the error avoided.

Risk A. Its exposure-weighted experience is $1.5(240)=360$ and $2.5(200)=500$. The total is $360+500=860$, over $1.5+2.5=4$ exposure units. Therefore,

\bar X_A=\frac{860}{4}=\$215.00\text{ per exposure unit}.

Risk B. Its weighted experience is $1(300)=300$ and $3(180)=540$. The total is $300+540=840$, over $1+3=4$ units. Therefore,

\bar X_B=\frac{840}{4}=\$210.00\text{ per exposure unit}.

Portfolio mean. The total weighted experience is $860+840=1{,}700$, and total exposure is $4+4=8$. Thus,

\bar X_{\mathrm{port}}=\frac{1{,}700}{8} =\$212.50\text{ per exposure unit}.

As a check, weighting the risk means by their exposures gives $(4(215)+4(210))/8=(860+840)/8=\$212.50$. The method avoids averaging period rates without regard to their exposures.

Worked Example: Unequal Total Exposures

A hypothetical specialty liability portfolio has two risks. Risk Cedar has rates of $100 on 1 exposure unit and $140 on 1 unit. Risk Maple has rates of $160 on 6 units and $220 on 2 units. Find each risk mean and the portfolio mean. Explain why the simple average of the two risk means is not the portfolio mean.

Risk Cedar. Its weighted experience is $1(100)+1(140)=240$, and its total exposure is $1+1=2$. Its mean is $240/2=\$120.00$ per unit.

Risk Maple. Its weighted experience is $6(160)+2(220)=960+440=1{,}400$, and its exposure is $6+2=8$. Its mean is $1{,}400/8=\$175.00$ per unit.

Portfolio mean. Total weighted experience is $240+1{,}400=1{,}640$, and total exposure is $2+8=10$. Hence the portfolio mean is $1{,}640/10=\$164.00$ per unit. The risk-mean check is $(2(120)+8(175))/10=(240+1{,}400)/10=\$164.00$.

The simple average of the risk means is $(120+175)/2=\$147.50$. That is not the portfolio mean because it gives Cedar and Maple equal influence, even though Maple has four times Cedar’s exposure. The calculation avoids treating “average across risks” and “average across exposure” as interchangeable.

When the Data Are Total Losses

A frequent calculation error is to multiply a total loss by exposure a second time. Exposure weighting applies to a rate $X_{ij}$. If the observation is already a total loss $L_{ij}$, it is summed directly; divide the sum by total exposure to obtain a rate. The next example makes that distinction explicit.

Worked Example: Starting From Loss Totals

For a hypothetical property program, Risk Harbor has losses of $18,000 on 120 exposure units and $30,000 on 200 units. Risk Ridge has losses of $12,600 on 70 units and $27,000 on 150 units. Calculate the two risk means and the portfolio mean to the nearest cent per exposure unit.

Risk Harbor. Total loss is $18{,}000+30{,}000=\$48{,}000$, and exposure is $120+200=320$. Its mean is $48{,}000/320=\$150.00$ per unit.

Risk Ridge. Total loss is $12{,}600+27{,}000=\$39{,}600$, and exposure is $70+150=220$. Its mean is $39{,}600/220=\$180.00$ per unit.

Portfolio mean. Total loss is $48{,}000+39{,}600=\$87{,}600$, and exposure is $320+220=540$. Therefore $87{,}600/540=\$162.222\ldots$, which rounds to $162.22 per exposure unit to two decimal places.

The exposure-weighted check is $(320(150)+220(180))/540=(48{,}000+39{,}600)/540=\$162.22$ after rounding. The method avoids multiplying the already-aggregated loss totals by exposure.

Using the Portfolio Mean as a Collective Complement

In credibility applications, an individual risk’s experience may be compared with a collective benchmark. The portfolio mean is a natural observed benchmark when the risks are comparable, the portfolio experience is available, and the exposure weights are appropriate for the intended aggregate. It reflects the portfolio’s observed experience per exposure unit, rather than the average experience of a typical risk selected with equal probability.

This distinction matters when risks differ greatly in size. A small risk can have an unusually high or low rate, but its impact on the portfolio mean remains proportional to its exposure. Conversely, equal weighting of risk means may be appropriate for a different question—for example, describing the average risk-level result when each risk is deliberately given equal importance—but it does not calculate the aggregate portfolio rate.

Before using a portfolio mean as a complement, confirm that all included experience is on a consistent basis and that the portfolio definition matches the intended application. If risks or periods are missing, do not silently treat absent observations as zero-cost experience. The denominator should include only exposure represented by the corresponding included experience.

Identify the observation type.
Determine whether each datum is a rate per exposure unit or a total loss.

Find each risk’s total experience and exposure.
For rates, multiply each rate by its exposure and add; for loss totals, add the losses directly. Add the corresponding exposures.

Calculate each risk mean.
Divide that risk’s total experience by its total exposure.

Aggregate for the portfolio.
Divide total portfolio experience by total portfolio exposure, or weight each risk mean by its total exposure.

Check the answer.
Reconstruct portfolio experience from risk means times total exposures, then divide by total exposure.

Common Exam Trap

Suppose a candidate has correctly calculated two risk means, $120 and $175, for risks with total exposures 2 and 8. The incorrect calculation is $(120+175)/2=\$147.50$, followed by labeling the result the portfolio mean. This is the unweighted mean of risk means, not the exposure-weighted portfolio result.

The correction is $(2(120)+8(175))/(2+8)=(240+1{,}400)/10=\$164.00$ per exposure unit. A second trap is applying exposure weights to total losses rather than rates. In either case, first ask what each number represents, then make the numerator and denominator correspond to the same exposure base.

Key takeaway: A risk mean weights that risk’s periods by period exposure. A portfolio mean weights risk means by each risk’s total exposure. The portfolio mean is an observed aggregate complement, not an unweighted average of risk-level results and not necessarily the exact underlying collective mean.

Check Your Understanding

For each problem, calculate or identify the requested mean using the stated exposures.

Risk A has rates of $90 on 2 units and $150 on 1 unit. Risk B has a rate of $130 on 3 units. Find both risk means and the portfolio mean. (Answer: Risk A $110.00; Risk B $130.00; portfolio mean $120.00 per unit.)
Risk Elm has losses of $4,800 on 40 units and $9,000 on 60 units. Risk Ash has a loss of $21,000 on 150 units. Find each risk mean and the portfolio mean. (Answer: Elm $138.00; Ash $140.00; portfolio mean $139.20 per unit.)
Risk North has a rate of $400 on 0.5 units and $200 on 1.5 units. Risk South has a rate of $300 on 2 units. Calculate the portfolio mean and explain which risk receives more weight. (Answer: $275.00 per unit; both risks have total exposure 2 and receive equal weight.)
Three risks have means and total exposures as follows: A, $50 and 10 units; B, $70 and 30 units; C, $90 and 60 units. Calculate the portfolio mean and compare it with the unweighted average of the risk means. (Answer: portfolio mean $80.00 per unit; unweighted average $70.00 per unit.)
A portfolio has two risks. One has a mean of $75 over 12 vehicle-years, and the other has a mean of $120 over 8 vehicle-years. Calculate the aggregate portfolio mean, then state whether it is closer to the larger or smaller risk mean. (Answer: $93.00 per vehicle-year; it is closer to the smaller risk mean.)

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