One-Way Relativities and Portfolio Mix

Why Portfolio Mix Matters

A one-way relativity compares classes defined by one classification variable, such as vehicle use or building type. But policies in those classes may also differ in other characteristics. If those other characteristics are associated with loss costs, a class’s observed average reflects both its own experience and the mix of risks within it. The factor can therefore describe the portfolio being observed without representing a pure, stand-alone effect of the classification variable.

This distinction matters when interpreting indicated factors. A class with a higher marginal loss cost may contain more high-cost risks on another dimension, even if it has lower loss costs within every comparable subgroup. The one-way calculation is not necessarily wrong; the danger is treating its result as though the underlying mix were irrelevant.

Definition: A one-way loss cost for class $j$ is the total adjusted losses for exposures assigned to that class, divided by those exposures, without separately controlling for other classification characteristics. A one-way relativity compares that class loss cost with the loss cost of a designated base class. It is a marginal comparison: the risks in each class retain their actual portfolio mix.

For a precise calculation, let $k$ index categories of another risk characteristic. Let $e_{jk}$ be the number of exposures in target class $j$ and category $k$, and let $L_{jk}$ be the corresponding loss cost per exposure. The class’s total adjusted losses are $\sum_k e_{jk}L_{jk}$, and its total exposures are $\sum_k e_{jk}$. Its one-way loss cost is therefore:

L_j=\frac{\sum_k e_{jk}L_{jk}}{\sum_k e_{jk}} =\sum_k w_{jk}L_{jk}, \qquad w_{jk}=\frac{e_{jk}}{\sum_k e_{jk}}.

The weights $w_{jk}$ are the proportions of the class’s exposures in each subgroup. They sum to 1. If $b$ denotes the base class, the one-way relativity is $r_j^{(1)}=L_j/L_b$, provided the numerator and denominator use a consistent exposure basis and comparable loss treatment. If losses require adjustments such as trend or development, those adjustments must be applied consistently before comparing the resulting loss costs.

Why the Weighted Average Is the Right Calculation

The formula follows directly from the definition of loss cost. In subgroup $k$, a loss cost of $L_{jk}$ across $e_{jk}$ exposures corresponds to total losses of $e_{jk}L_{jk}$. Adding subgroup losses gives the class total; adding subgroup exposures gives the class exposure count. Dividing the totals yields the exposure-weighted average.

\frac{\text{total losses in class }j}{\text{total exposures in class }j} = \frac{\sum_k e_{jk}L_{jk}}{\sum_k e_{jk}} = \sum_k\left(\frac{e_{jk}}{\sum_h e_{jh}}\right)L_{jk}.

Thus the marginal result depends on two things: the subgroup loss costs and the exposure weights. If the subgroup costs stay fixed but the weights change, the one-way loss cost can change. Conversely, two classes with different subgroup costs can have similar marginal loss costs if their exposure mixes differ in offsetting ways.

This identity assumes the subgroup loss costs and exposure counts describe the same population and are measured consistently. It does not establish that the observed differences are caused by the target classification variable. It is an accounting identity for the aggregate experience, not a causal conclusion.

Worked Examples

Worked Example: A Reversed One-Way Comparison

Method: calculate each class’s marginal loss cost from subgroup totals, then compare it with the base class. A hypothetical delivery-vehicle portfolio separates vehicles into two use classes, X and Y. Each class has low- and high-mileage risks. The adjusted loss costs are $100 for low-mileage X risks, $300 for high-mileage X risks, $120 for low-mileage Y risks, and $320 for high-mileage Y risks. Class X has 900 low-mileage and 100 high-mileage exposures; class Y has 100 low-mileage and 900 high-mileage exposures. Find the one-way relativity of X to Y and compare the within-mileage comparisons.

Class X. Low-mileage losses are $900\times \$100=\$90{,}000$. High-mileage losses are $100\times \$300=\$30{,}000$. Total losses are $\$90{,}000+\$30{,}000=\$120{,}000$, and total exposures are $900+100=1{,}000$. Therefore $L_X=\$120{,}000/1{,}000=\$120$.

Class Y. Low-mileage losses are $100\times \$120=\$12{,}000$. High-mileage losses are $900\times \$320=\$288{,}000$. Total losses are $\$12{,}000+\$288{,}000=\$300{,}000$, and total exposures are $100+900=1{,}000$. Therefore $L_Y=\$300{,}000/1{,}000=\$300$.

Taking Y as the base, the one-way relativity for X is $\$120/\$300=0.400$. Yet within the low-mileage subgroup, X’s cost relative to Y is $\$100/\$120=0.8333$; within the high-mileage subgroup it is $\$300/\$320=0.9375$. X has lower cost in both subgroups, but its marginal relativity is much lower because X has many more low-mileage exposures and Y has many more high-mileage exposures.

Check: The subgroup totals reconcile to the class totals: $900+100=1{,}000$ exposures in each class, with losses of $\$120{,}000$ for X and $\$300{,}000$ for Y. The calculation avoids the mistake of interpreting the 0.400 marginal factor as a direct comparison between otherwise identical risks.

Worked Example: Comparing Observed and Common-Mix Costs

Method: calculate observed one-way costs, then recompute both classes using the same subgroup weights as a diagnostic. A hypothetical equipment insurer compares two equipment classes, A and B, across low- and high-duty usage. The subgroup costs are $70 and $210 for A, and $90 and $230 for B, respectively. In A, 80% of exposures are low-duty; in B, 20% are low-duty. Calculate B’s observed relativity to A. Then calculate the comparison if both classes had a 50% low-duty, 50% high-duty mix.

Observed A cost. The weighted cost is $0.80(\$70)+0.20(\$210)=\$56+\$42=\$98$.

Observed B cost. The weighted cost is $0.20(\$90)+0.80(\$230)=\$18+\$184=\$202$. Thus B’s observed one-way relativity to A is $\$202/\$98=2.0612$, or about 2.061 to four significant figures.

Common-mix diagnostic. With equal weights, A’s cost is $0.50(\$70)+0.50(\$210)=\$35+\$105=\$140$. B’s cost is $0.50(\$90)+0.50(\$230)=\$45+\$115=\$160$. The common-mix comparison is $\$160/\$140=1.1429$, or about 1.143 to four significant figures.

The observed factor is larger because B has a greater share of high-duty exposures. The common-mix calculation shows how the comparison changes when both classes are evaluated at the same illustrative mix. It is a diagnostic, not automatically a selected rating factor: using it for pricing requires an appropriate rating approach and sufficient data.

Check: Each pair of weights sums to 1, and the common-mix costs are the arithmetic averages of the two subgroup costs. The calculation avoids the mistake of assuming the observed 2.061 relativity is independent of the classes’ different usage distributions.

Worked Example: A Changing Portfolio Average With Stable Class Costs

Method: calculate the exposure-weighted portfolio average from fixed class costs under each portfolio mix. A hypothetical property portfolio has a low-risk class with an indicated loss cost of $100 per exposure and a high-risk class with an indicated loss cost of $160 per exposure. The class costs do not change. For a portfolio of 10,000 exposures, compare total indicated losses when the mix is 70% low-risk and 30% high-risk with total indicated losses when it is 40% low-risk and 60% high-risk.

First mix. Low-risk exposures are $10{,}000(0.70)=7{,}000$; high-risk exposures are $10{,}000(0.30)=3{,}000$. Indicated losses are $7{,}000(\$100)=\$700{,}000$ and $3{,}000(\$160)=\$480{,}000$. Total indicated losses are $\$700{,}000+\$480{,}000=\$1{,}180{,}000$. The average loss cost is $\$1{,}180{,}000/10{,}000=\$118$ per exposure.

Second mix. Low-risk exposures are $10{,}000(0.40)=4{,}000$; high-risk exposures are $10{,}000(0.60)=6{,}000$. Indicated losses are $4{,}000(\$100)=\$400{,}000$ and $6{,}000(\$160)=\$960{,}000$. Total indicated losses are $\$400{,}000+\$960{,}000=\$1{,}360{,}000$. The average loss cost is $\$1{,}360{,}000/10{,}000=\$136$ per exposure.

The average increases by $\$136-\$118=\$18$ per exposure, and total indicated losses increase by $\$1{,}360{,}000-\$1{,}180{,}000=\$180{,}000$. The class costs stayed fixed; the change comes entirely from the increased share of high-risk exposures.

Check: Each mix totals 10,000 exposures, and each total loss is the sum of the two class totals. The calculation avoids attributing a changing portfolio average to a change in class-level costs when only the portfolio composition changed.

Interpreting One-Way Factors Carefully

A one-way factor is useful for summarizing observed experience by one classification variable. It answers a specific question: what was the average adjusted loss cost in this class, given the actual composition of the class during the experience period? It does not, by itself, answer what the class cost would have been if every class had the same distribution of other characteristics.

That distinction is especially important when:

the target classes have substantially different exposure distributions across other risk characteristics;
the mix of risks changes between experience periods, making marginal factors hard to compare over time; or
a factor is interpreted as the effect of changing one characteristic while holding all others constant.

A common-mix calculation can help reveal the influence of mix: apply the same chosen weights to each class’s subgroup costs and compare the resulting averages. However, this calculation is not a substitute for assessing data quality, credibility, or the suitability of a rating plan. It also does not by itself establish which characteristics should be included in a final classification structure.

The key exam distinction is between a class’s observed marginal experience and a within-subgroup comparison. The first uses each class’s own weights; the second compares costs at a specified subgroup or common mix. Both calculations can be correct while answering different questions. This treatment is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Common Mistake: Averaging Subgroup Costs Without the Right Weights

Suppose a class has 900 low-risk exposures at a cost of $100 and 100 high-risk exposures at a cost of $300. An incorrect calculation takes the unweighted average, $(\$100+\$300)/2=\$200$, as the class loss cost. That would be correct only if the two subgroups had equal exposure counts. The actual class loss cost is:

\frac{900(\$100)+100(\$300)}{900+100} =\frac{\$90{,}000+\$30{,}000}{1{,}000} =\$120.

The error is not merely arithmetic: the unweighted calculation silently replaces the portfolio’s actual 90%/10% mix with a hypothetical 50%/50% mix. Always identify the weighting basis before combining subgroup results. A factor based on earned exposures, for example, should not be combined using equal weights unless the exposures are in fact equal.

Check Your Understanding

For each problem, calculate the requested marginal result using the stated exposure mix.

A hypothetical appliance portfolio has a class with 60 low-use exposures at a loss cost of $90 and 40 high-use exposures at a loss cost of $210. What is the class’s one-way loss cost? (Answer: $138 per exposure.)
In a hypothetical fleet, class U has low-mileage and high-mileage costs of $100 and $250, with 80% of its exposures low-mileage. Class V has corresponding costs of $120 and $270, with 30% of its exposures low-mileage. Calculate both marginal costs and V’s one-way relativity to U. (Answer: U = $130, V = $225, and V/U = 1.7308.)
A hypothetical insurer’s class A has subgroup costs of $80 and $200 and a 75%/25% mix. Class B has subgroup costs of $100 and $220 and a 25%/75% mix. Calculate the observed marginal costs and B’s relativity to A. (Answer: A = $110, B = $190, and B/A = 1.7273.)
Using the subgroup costs in question 3, calculate each class’s cost under a common 50%/50% mix, then calculate B’s common-mix relativity to A. (Answer: A = $140, B = $160, and B/A = 1.1429.)
A hypothetical book has 8,000 exposures. Its low-risk class costs $95 per exposure and its high-risk class costs $155. The low-risk share changes from 65% to 35%, with class costs fixed. Calculate the average loss cost and total indicated losses under each mix. (Answer: First mix: $116 per exposure and $928,000; second mix: $134 per exposure and $1,072,000.)

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