Predicting Aggregate Loss At Future Exposure

From Credibility-Adjusted Rate to Future Total

An experience-rated estimate is often a loss cost per exposure unit, not a dollar amount of total losses. To predict expected aggregate loss, first determine the credibility-adjusted loss cost on the appropriate basis, then multiply it by the exposure expected in the future period. Keeping those two quantities distinct is essential: the rate describes expected loss per unit, while the aggregate prediction describes expected dollars over all future units.

For example, a loss cost per vehicle-year can be multiplied by projected vehicle-years. It should not be multiplied by the number of policies unless the exposure unit is a policy-year and the rate is expressed per policy-year. The exposure basis and the level of the loss-cost data must match.

Definition: Let $w_{ij}$ be risk $i$'s exposure in historical period $j$, and let $L_{ij}$ be its aggregate loss in that period. The observed loss cost is $X_{ij}=L_{ij}/w_{ij}$. The total historical exposure is $W_i=\sum_j w_{ij}$, and the exposure-weighted observed loss cost is $\bar X_i=\sum_j w_{ij}X_{ij}/W_i=\sum_jL_{ij}/W_i$. If the Bühlmann-Straub credibility estimate is $\hat\theta_i=Z_i\bar X_i+(1-Z_i)\hat m$, then, under the future-period assumptions stated below, the expected aggregate loss for future exposure $e_f$ is $e_f\hat\theta_i$.

The Rate Estimate and Its Conversion

In the Bühlmann-Straub model, each risk has an underlying expected loss cost $\theta_i$. Historical loss costs are assumed to be conditionally independent given $\theta_i$, with $E[X_{ij}\mid\theta_i]=\theta_i$ and $\operatorname{Var}(X_{ij}\mid\theta_i)=\sigma^2/w_{ij}$. Across risks, assume $E[\theta_i]=m$ and $\operatorname{Var}(\theta_i)=\tau^2$. Here, $\sigma^2$ is the expected process variance per unit of exposure, and $\tau^2$ is the variance of the underlying risk means. Under these assumptions, the credibility constant is $K=\sigma^2/\tau^2$, and the credibility assigned to the risk's observed mean is $Z_i=W_i/(W_i+K)$.

When $m$ and $K$ are treated as given, the credibility estimate is a weighted average of the risk's observed mean and the collective mean. In applications, these structural quantities may themselves be estimated from a portfolio. For this tutorial, calculations either specify them or supply them directly.

\hat\theta_i = Z_i\bar X_i+(1-Z_i)\hat m, \qquad Z_i=\frac{W_i}{W_i+K}, \qquad K=\frac{\sigma^2}{\tau^2}.

The estimate $\hat\theta_i$ remains a rate: its units are dollars per exposure unit. To convert it into an expected dollar total, specify the future exposure $e_f$ in the same units. If future losses have the same expected loss cost per unit as the rate being estimated, then

\widehat{E[S_f\mid\text{historical data}]} = e_f\hat\theta_i,

where $S_f$ is aggregate loss in the future period. This is an estimate of the expected aggregate loss, not a prediction that the realized total will equal that amount.

Why Multiplying by Future Exposure Works

Suppose the future period can be divided into exposure units on the same basis as the historical data. Conditional on the risk's underlying mean $\theta_i$, assume the expected loss per exposure unit is $\theta_i$. For a fixed future exposure $e_f$, linearity of expectation gives $E[S_f\mid\theta_i]=e_f\theta_i$. If future outcomes are conditionally independent of historical outcomes given $\theta_i$, then

E[S_f\mid\text{historical data}] = E\!\left[E[S_f\mid\theta_i,\text{historical data}] \mid\text{historical data}\right] = e_f E[\theta_i\mid\text{historical data}].

Bühlmann-Straub credibility estimates the risk mean using the linear credibility estimate $\hat\theta_i$. Substituting that estimate gives $e_f\hat\theta_i$. This argument explains both the multiplication and its limits: the future exposure must be measured on a compatible basis, and the expected loss cost per unit must remain applicable. The credibility estimate is a linear predictor; it need not equal an exact Bayesian posterior mean for every possible distribution of risk parameters.

Put historical experience on the intended basis.
Use loss costs and exposures measured consistently. If historical losses have been adjusted to the prospective price or coverage level, confirm that the stated estimate is already on that level.

Calculate the weighted historical loss cost.
Divide total historical loss by total historical exposure, or equivalently weight each period's loss cost by its exposure.

Apply credibility.
Blend the risk's weighted historical loss cost with the collective mean using the supplied or calculated credibility factor.

Calculate the future expected total.
Multiply the credibility-adjusted loss cost by the specified future exposure, keeping the units and scale consistent.

Worked Applications

Worked Example: Vehicle-Year Rate to Expected Loss

A hypothetical fleet has historical loss costs of $280 per vehicle-year over 2 vehicle-years and $220 per vehicle-year over 3 vehicle-years. The collective mean is $240 per vehicle-year, and the Bühlmann-Straub constant is $K=4$ vehicle-years. The fleet is expected to have 7.5 vehicle-years of exposure next year. Assume losses are on the same coverage and price level in both periods, the model's conditional mean and variance assumptions hold, and the stated future exposure is fixed. No interest or mortality assumptions are relevant. Find the expected aggregate loss for next year.

Method. Calculate the exposure-weighted historical rate, credibility-adjust it, and multiply by future vehicle-years.

The historical exposure is $W=2+3=5$ vehicle-years. The historical aggregate losses represented by the two rates are $2(280)=560$ dollars and $3(220)=660$ dollars. Thus total loss is $560+660=1{,}220$ dollars, and

\bar X=\frac{1{,}220}{5}=\$244\text{ per vehicle-year}.

The credibility factor is $Z=5/(5+4)=5/9$. The credibility-adjusted rate is

\hat\theta = \frac{5}{9}(244)+\frac{4}{9}(240) = \frac{1{,}220+960}{9} = \$242.2222\ldots\text{ per vehicle-year}.

For 7.5 future vehicle-years, the expected aggregate loss is $7.5(242.2222\ldots)=\$1{,}816.6666\ldots$. Rounded to the nearest cent, the answer is $1,816.67.

Check and common mistake. The rate lies between the observed rate of $244 and the collective rate of $240, as a credibility-weighted average should. Multiplying the unadjusted observed rate by future exposure gives $7.5(244)=\$1{,}830$, which incorrectly omits the credibility adjustment. The requested total uses the adjusted rate.

Worked Example: Unequal Period Exposures and Future Machinery Units

A hypothetical machinery account has aggregate losses of $2,100 over 1.5 machine-years and $3,000 over 2.5 machine-years. Its collective loss cost is $1,250 per machine-year, and $K=6$ machine-years. Next year, the account is expected to have 3.2 machine-years. Assume the historical losses are already adjusted to the same prospective coverage and price level, future exposure is fixed, and the Bühlmann-Straub assumptions apply. No interest or mortality assumptions are relevant. Calculate the expected aggregate loss.

Method. Form the aggregate historical loss cost using total losses and total exposure, then apply the credibility factor and future exposure.

The total historical loss is $2{,}100+3{,}000=\$5{,}100$, and total exposure is $1.5+2.5=4$ machine-years. The period loss costs are $2{,}100/1.5=\$1{,}400$ and $3{,}000/2.5=\$1{,}200$ per machine-year. Weighting them gives

\bar X = \frac{1.5(1{,}400)+2.5(1{,}200)}{4} = \frac{2{,}100+3{,}000}{4} = \$1{,}275\text{ per machine-year}.

The credibility factor is $Z=4/(4+6)=0.4$. Therefore,

\hat\theta = 0.4(1{,}275)+0.6(1{,}250) = 510+750 = \$1{,}260\text{ per machine-year}.

For 3.2 future machine-years, expected aggregate loss is $3.2(1{,}260)=$ $4,032.

Check and common mistake. The adjusted rate of $1,260 is between $1,250 and $1,275. A simple unweighted average of the period rates, $(1{,}400+1{,}200)/2=\$1{,}300$, would be wrong because the periods have unequal exposure. The correct historical mean weights each rate by its machine-years.

Worked Example: Payroll-Based Rate and a Larger Future Book

A hypothetical contractor's historical losses are $360,000 over 4 units of exposure and $540,000 over 6 units. One exposure unit is $1,000,000 of payroll. The collective loss cost is $75,000 per exposure unit, and $K=5$ exposure units. The next year's projected payroll is $12,000,000. Assume losses have been placed on the prospective coverage and price level, the exposure definition is unchanged, and the Bühlmann-Straub assumptions apply. No interest or mortality assumptions are relevant. Find expected aggregate loss.

Method. Calculate the historical rate per exposure unit, credibility-adjust it, and express projected payroll in those units before multiplying.

The two period rates are $360{,}000/4=\$90{,}000$ and $540{,}000/6=\$90{,}000$ per exposure unit. Total historical exposure is $4+6=10$, so the weighted historical rate is also

\bar X = \frac{360{,}000+540{,}000}{10} = \$90{,}000\text{ per exposure unit}.

The credibility factor is $Z=10/(10+5)=2/3$. The adjusted rate is

\hat\theta = \frac{2}{3}(90{,}000)+\frac{1}{3}(75{,}000) = 60{,}000+25{,}000 = \$85{,}000\text{ per exposure unit}.

Since one unit represents $1,000,000 of payroll, projected exposure is $12{,}000{,}000/1{,}000{,}000=12$ units. Expected aggregate loss is $12(85{,}000)=$ $1,020,000.

Check and common mistake. The estimated rate is in dollars per $1,000,000 of payroll, not per dollar of payroll. Multiplying it directly by $12,000,000 would apply the exposure scale incorrectly; first convert payroll to 12 exposure units.

Conditions, Interpretation, and Exam Traps

The multiplication $e_f\hat\theta_i$ is appropriate when the future exposure uses the same definition as the rate, and when the estimated mean remains relevant to the future risk. Examples of a basis mismatch include applying a rate per vehicle-year to a count of policies with no vehicle-year conversion, or using a rate per payroll unit while multiplying by raw payroll dollars. If the future coverage, loss definition, or price level differs from the basis of the rate, the rate must first be made appropriate to that basis; the multiplication alone does not account for such changes.

A further distinction matters in interpretation. The credibility estimate is a rate-level estimate of the risk's expected loss per exposure unit. Multiplying it by future exposure gives a point estimate of expected total loss. It does not provide the distribution of the future total, a probability that losses will exceed a threshold, or a measure of predictive uncertainty. Those require additional assumptions and calculations.

Do not multiply by historical exposure. Historical exposure determines the weight $Z_i$; specified future exposure scales the predicted future total.
Do not skip credibility. Unless full credibility is specified, the risk's observed mean is blended with the collective mean before calculating the future total.
Do not average unequal-period rates equally. The historical loss cost is weighted by exposure, equivalently total loss divided by total exposure.
Do not confuse a rate with a total. State the rate's units, convert future exposure to those units, and then calculate the expected aggregate.

Key takeaway: First find the Bühlmann-Straub credibility estimate on a clearly stated loss-cost basis. Then multiply that rate by the specified future exposure on the same basis to estimate expected aggregate loss.

Check Your Understanding

For each problem, assume the Bühlmann-Straub assumptions apply, future exposure is fixed, and historical loss costs are on the stated prospective basis. No interest or mortality assumptions are relevant.

A risk has loss costs of $180 per unit over 2 units and $240 per unit over 1 unit. The collective mean is $210 per unit, $K=3$ units, and future exposure is 4 units. Find the expected aggregate loss. (Answer: $820)
A service account has losses of $1,000 over 2 equipment-years and $2,800 over 4 equipment-years. Its collective rate is $600 per equipment-year, $K=2$ equipment-years, and future exposure is 1.5 equipment-years. Find the expected aggregate loss. (Answer: $937.50)
A hypothetical risk has $550,000 of loss over 5 exposure units, where each unit is $500,000 of payroll. The collective rate is $100,000 per unit, $K=5$ units, and future payroll is $4,000,000. Find the expected aggregate loss. (Answer: $840,000)
A risk's loss costs are $400 per unit over 1 unit and $200 per unit over 3 units. The collective mean is $250 per unit, $K=2$ units, and future exposure is 2.4 units. Find the expected aggregate loss. (Answer: $600)
A risk has rates of $90 per unit over 3 units and $110 per unit over 5 units. The collective rate is $100 per unit, $K=4$ units, and future exposure is 15 units. Find the expected aggregate loss, rounded to the nearest cent. (Answer: $1,525.00)

What You'll Learn