Stability of Relativities Across Experience Periods

Why Relativity Stability Matters

A class factor is intended to describe a relative difference in expected cost, not simply to summarize the latest period’s observed losses. A factor that changes between periods may reflect a genuine change in underlying risk. It may instead reflect a different mix of risks within a class, a few unusually large claims, or limited experience. These explanations have different implications: reacting to random fluctuation can produce unstable prices, while overlooking a persistent change can leave factors misaligned.

A meaningful comparison starts with consistent definitions. The periods should use the same class boundaries, exposure basis, loss treatment, and base class. Losses should also be adjusted to a comparable coverage and cost level when needed. For example, if one period’s losses are trended to the proposed effective date but another period’s losses are not, differences between the resulting factors may reflect the inconsistent adjustment rather than risk.

Definition: A class relativity is a measure of a class’s expected cost relative to a stated reference, such as a base class or a weighted average across classes. Relativity stability means that comparisons made on consistent definitions and bases show no material change in those relative differences across experience periods. Stability is a conclusion supported by analysis, not a requirement that observed factors be numerically identical.

Make Period Comparisons on the Same Basis

Let $L_{ct}$ be the adjusted loss cost for class $c$ in period $t$: adjusted losses divided by the chosen exposure measure. When class $b$ is the base, the period-specific relativity is

R_{ct}=\frac{L_{ct}}{L_{bt}}.

The ratio compares class $c$ with the base class in the same period. Dividing by the base helps separate relative differences from a general change in the overall loss level. If every class’s loss cost increases by the same proportion, each class-to-base relativity is unchanged.

For a set of classes normalized to a weighted average of 1.00, use the same reference weights $w_c$ in each period, where $w_c\geq 0$ and $\sum_c w_c=1$:

R_{ct}=\frac{L_{ct}}{\sum_j w_j L_{jt}}.

Changing the weights between periods changes the benchmark as well as the observed experience. Thus, a comparison can move even when no class’s own loss cost changes. When the purpose is to examine class risk rather than the current portfolio’s composition, a fixed reference basis is usually more informative.

Worked Example: Separating a Common Level Change from Relativity Change

A hypothetical insurer compares three classes in two periods. Losses have already been adjusted to the same coverage and price level, and the exposure measure is unchanged. The base class is B. In Period 1, the loss costs are A: $120, B: $100, and C: $80 per exposure. In Period 2, they are A: $132, B: $110, and C: $88 per exposure. Determine whether the class relativities changed.

Method and assumptions. Calculate each class’s loss cost divided by the same-period base-class loss cost. Treat the provided adjusted loss costs as comparable; no interest or mortality assumption is relevant.

Period 1. The base loss cost is $100. Therefore, $R_{A1}=120/100=1.20$, $R_{B1}=100/100=1.00$, and $R_{C1}=80/100=0.80$.

Period 2. The base loss cost is $110. Therefore, $R_{A2}=132/110=1.20$, $R_{B2}=110/110=1.00$, and $R_{C2}=88/110=0.80$.

Each class’s loss cost rose by 10%, since $132/120=110/100=88/80=1.10$. The same-period ratios did not change, so the data show a common level increase, not a change in these relativities. Check: multiplying each Period 1 loss cost by 1.10 gives exactly the Period 2 loss cost.

Common mistake avoided: Comparing A’s $132 directly with its prior $120 and calling the 10% increase a relativity change confuses a change in cost level with a change relative to the base.

Separate Class Differences from Mix Shifts

A class may contain subgroups with different expected costs. If the share of exposures in those subgroups changes, the class’s observed average loss cost can change even when each subgroup’s loss cost is constant. This is a composition effect, not necessarily a change in the class’s underlying relative risk.

Suppose class $c$ is divided into subgroups $k$. Let $L_{ckt}$ be the adjusted loss cost for subgroup $k$, and $p_{ckt}$ its share of class $c$’s exposure in period $t$. The observed class loss cost is

L_{ct}=\sum_k p_{ckt}L_{ckt}.

To compare periods at a fixed mix, choose reference shares $q_k$, with $q_k\geq 0$ and $\sum_k q_k=1$, and calculate

L^{\mathrm{std}}_{ct}=\sum_k q_k L_{ckt}.

Apply the same reference shares to each class and period, then compare the standardized class costs or relativities. The fixed shares answer a specific question: what would each class’s cost be if its subgroup composition matched the selected reference mix?

Why standardization works: The observed difference between two periods can arise from changed subgroup costs, changed subgroup shares, or both. Holding the shares fixed removes the effect of changing composition from the comparison. It does not remove every possible source of bias: the subgroups must capture the mix differences relevant to the analysis, and their loss costs must be measured on a consistent basis.

Worked Example: A Dramatic Raw Change Caused by Mix

A hypothetical insurer compares class X with base class B. Each has low- and high-cost subgroups. In Period 1, X has subgroup loss costs of $80 and $200, with exposure shares of 75% low-cost and 25% high-cost. In Period 2, the subgroup costs are unchanged, but the shares are 25% low-cost and 75% high-cost. For B, subgroup costs are $60 and $150 in both periods; its shares are 50% low-cost and 50% high-cost in Period 1, then 75% low-cost and 25% high-cost in Period 2. Standardize both classes to a 50%/50% mix.

Observed Period 1 costs. For X, $0.75(80)+0.25(200)=60+50=\$110$. For B, $0.50(60)+0.50(150)=30+75=\$105$. The raw relativity is $110/105=1.0476$, or about 1.05.

Observed Period 2 costs. For X, $0.25(80)+0.75(200)=20+150=\$170$. For B, $0.75(60)+0.25(150)=45+37.50=\$82.50$. The raw relativity is $170/82.50=2.0606$, or about 2.06.

Standardized costs. With 50% in each subgroup, X costs $0.50(80)+0.50(200)=\$140$ in either period. B costs $0.50(60)+0.50(150)=\$105$ in either period. The standardized relativity is $140/105=1.3333$, or about 1.33, in both periods.

The raw factor moved from about 1.05 to 2.06 because X shifted toward its high-cost subgroup while B shifted toward its low-cost subgroup. The subgroup costs themselves did not change. Check: the fixed-mix factor is identical across periods, as the unchanged subgroup costs imply.

Common mistake avoided: Treating the raw change from 1.05 to 2.06 as evidence of a new class effect ignores the changing shares. The correction is to compare at a common mix, while recognizing that the chosen subgroups and reference shares must be appropriate to the question.

Check Sensitivity to Individual Claims

A class-period total can be strongly affected by one claim, especially when exposure is limited. Compare the indicated relativity with and without a material claim as a sensitivity test. This does not mean that a claim should automatically be excluded from the selected experience: the treatment must follow the applicable loss definition and analytical purpose. The point is to identify whether the conclusion depends heavily on one observation.

Worked Example: Testing Whether One Claim Drives a Change

In a hypothetical product, adjusted losses for class A are $96,000 on 120 exposures in Period 1 and $150,000 on 125 exposures in Period 2. The Period 2 total includes one $50,000 claim. Base-class losses are $80,000 on 100 exposures in Period 1 and $100,000 on 125 exposures in Period 2. Calculate the class-to-base relativities, then test the effect of the $50,000 claim.

Method and assumptions. Divide adjusted losses by exposure to calculate each loss cost, then divide A’s cost by B’s same-period cost. Treat the claim removal as a sensitivity test only. All amounts are already on a comparable basis; interest and mortality assumptions are not applicable.

Period 1. A’s loss cost is $96{,}000/120=\$800$. B’s is $80{,}000/100=\$800$. Therefore $R_{A1}=800/800=1.00$.

Period 2, including the claim. A’s loss cost is $150{,}000/125=\$1{,}200$. B’s is $100{,}000/125=\$800$. Therefore $R_{A2}=1{,}200/800=1.50$.

Sensitivity calculation. Subtract the claim from A’s Period 2 losses: $150{,}000-50{,}000=\$100{,}000$. A’s adjusted sensitivity loss cost is $100{,}000/125=\$800$. Relative to B’s $800 loss cost, the sensitivity relativity is $800/800=1.00$.

The observed increase to 1.50 is entirely driven by this claim under the stated sensitivity calculation. That is evidence of claim sensitivity, not proof that the claim should be excluded from a rate indication. Check: the remaining $100,000 of A losses equals $125\times\$800$, so its sensitivity loss cost is $800.

Common mistake avoided: Dividing the full $150,000 by exposure after claiming to remove the large claim leaves the original factor unchanged. The correction is to subtract the claim first, then recalculate the loss cost and relativity.

Distinguish Instability from Limited Experience

Even when definitions and mix are consistent, observed factors can vary because losses are random. A small number of claims or exposures usually provides less information about an underlying difference than a large, stable body of experience. A point estimate alone does not show how uncertain that estimate is.

A simple diagnostic illustrates this issue. Suppose independent claim counts in two classes follow Poisson distributions, exposures are fixed, and every claim has the same known severity. Under those assumptions, the estimated frequency ratio is also the loss-cost ratio. For counts $N_A$ and $N_B$, a large-sample approximation for the standard error of the log count ratio is

\operatorname{SE}\!\left(\log\frac{N_A/E_A}{N_B/E_B}\right) \approx \sqrt{\frac{1}{N_A}+\frac{1}{N_B}}.

This approximation is a diagnostic, not a universal uncertainty formula for insurance loss costs. It does not account for variable claim severity, dependence, changing exposure quality, or adjustments to losses. With substantial severity variation, claim-level information or an appropriate model is needed to assess uncertainty.

Worked Example: A Noisy Relativity from Few Claims

A hypothetical portfolio has 100 exposures in each of two classes. Over a period, class A has 16 claims and class B has 25. Assume independent Poisson claim counts, fixed exposures, and a known constant severity of $1,000 for every claim. Estimate the A-to-B relativity and use the log-ratio approximation to assess its uncertainty.

Estimated loss costs. A’s frequency is $16/100=0.16$ claims per exposure, giving a loss cost of $0.16(\$1{,}000)=\$160$. B’s frequency is $25/100=0.25$, giving a loss cost of $0.25(\$1{,}000)=\$250$. The estimated relativity is $160/250=0.64$.

Approximate standard error. The constant severity cancels in the ratio, so $\operatorname{SE}(\log R)\approx\sqrt{1/16+1/25} =\sqrt{0.0625+0.0400} =\sqrt{0.1025}=0.3202$.

Approximate 95% interval. On the log scale, the estimate is $\log(0.64)=-0.4463$. The approximate endpoints are $-0.4463\pm1.96(0.3202)=-0.4463\pm0.6276$, or $-1.0739$ and $0.1813$. Exponentiating gives an interval of approximately $0.3417$ to $1.1987$.

Because the interval includes 1.00, these data do not sharply distinguish the classes under the stated approximation. This is not evidence that their true costs must be equal; it shows that the observed difference is uncertain with these counts. Check: the interval endpoints are positive after exponentiation, as a relativity must be.

Common mistake avoided: Treating 0.64 as a precise finding overlooks the small counts. The calculation also depends on the stipulated constant-severity Poisson model; it should not be applied unchanged when claim severities vary materially.

A Practical Stability Review

A useful review asks whether the apparent change remains after the main comparability issues have been addressed. The checks should be interpreted together; no single numerical threshold establishes stability in every situation.

Align the basis.
Use the same class definitions, exposure measure, loss treatment, coverage, and adjustment basis across periods. Confirm that the base class or reference weights are consistent.

Compare relative factors.
Calculate each period’s factor against its same-period base or against a common fixed weighted average. Separate common changes in loss level from changes in relative position.

Investigate mix and claim sensitivity.
Standardize relevant subgroup shares when composition changes, and test whether individual large claims dominate the observed difference.

Interpret the evidence in light of volume.
Consider the number and distribution of observations. A large factor movement based on limited or concentrated experience may be less persuasive than a consistent movement across periods and subgroups.

A common exam trap is to compare raw class loss costs across years and call any difference a relativity change. Another is to standardize one class to a fixed mix but leave its comparison class at its changing observed mix. Both comparisons use inconsistent bases. State the denominator, weights, and adjustments explicitly, then verify that the same choices are used for every period and class being compared.

Stability analysis matters because a classification plan should distinguish persistent risk differences from temporary experience variation. A stable factor across comparable periods is supportive evidence for a persistent difference; a changed factor warrants investigation into mix, claims, volume, and basis before it is treated as a change in expected relative cost. This treatment is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Check Your Understanding

For each problem, state the comparison basis and show the key calculation.

A hypothetical two-class portfolio has adjusted loss costs of A: $90 and B: $75 in Period 1, and A: $108 and B: $90 in Period 2. B is the base. Are the relativities stable? (Answer: Yes; A’s relativity is 1.20 in both periods.)
Class P has low- and high-subgroup costs of $50 and $150, with shares 80%/20% in Period 1 and 40%/60% in Period 2. Class Q has subgroup costs $40 and $120 in both periods. Standardize both classes to a 50%/50% mix. What is the standardized P-to-Q relativity? (Answer: P costs $100, Q costs $80, so the relativity is 1.25 in both periods.)
Class M has Period 2 adjusted losses of $84,000 on 100 exposures, including a $24,000 claim. Its base class has losses of $60,000 on 100 exposures. Calculate the relativity with the claim and in a sensitivity test without it. (Answer: With the claim, $840/600=1.40$; without it, $600/600=1.00$.)
In two periods, class C has loss costs of $180 and $198, while its base has loss costs of $150 and $165. Calculate C’s relativity in each period and identify the common level change. (Answer: The relativity is 1.20 in each period; both loss costs increased by 10%.)
Under independent Poisson claim counts with equal exposures and constant severity, two classes have counts 9 and 36. Estimate the first-to-second relativity and the approximate standard error of its log. (Answer: The relativity is $9/36=0.25$; the standard error is $\sqrt{1/9+1/36}=\sqrt{0.1389}\approx0.3727$.)

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