Exposure Mix Effects on Relativities

Why Exposure Mix Changes Average Premium

An insurer can have unchanged premiums for every rating class and still see its average premium per exposure change. This happens when the proportions of exposures in the classes change. For example, if more policies are written in a class with a higher relativity, the average premium can rise even though no class rate has changed.

To measure this effect, keep the class relativities separate from the distribution of exposures across classes. The relativities describe how class premiums compare with a base premium; the exposure weights describe how much each class contributes to the portfolio. This distinction is aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.

Definition: Suppose there are $J$ rating classes. Let $w_j$ be the proportion of exposures in class $j$, with $w_j\geq 0$ and $\sum_{j=1}^{J}w_j=1$. Let $r_j$ be that class's relativity, and let $B$ be a common base premium per exposure. The class premium is $B r_j$, and the portfolio's average premium per exposure is $$ \bar{P}=B\sum_{j=1}^{J}w_jr_j. $$ The quantity $\sum_j w_jr_j$ is the exposure-weighted average relativity.

This calculation assumes that exposures are measured on a consistent basis across classes and periods, and that the class premiums refer to the same coverage and policy term. If the number of policies, insured units, or exposure duration differs, first use the exposure basis appropriate to the premium being compared. A simple count of policies is not a suitable weight if the rates are stated per vehicle-year but the policies cover different fractions of a year.

The weights are exposure proportions, not premium proportions. Since premium itself depends on the relativity, using premium shares as weights would give higher-relativity classes extra weight a second time. To calculate an average premium per exposure, weight the premium per exposure by the share of exposures in each class.

Measuring a Pure Mix Effect

A pure exposure mix effect compares two distributions of exposures while holding the class relativities and base premium fixed. Write $w_j^{(0)}$ and $w_j^{(1)}$ for the old and new exposure weights, respectively. If the base premium is $B$ and the relativities do not change, then

\bar{P}^{(0)}=B\sum_j w_j^{(0)}r_j,\qquad \bar{P}^{(1)}=B\sum_j w_j^{(1)}r_j.

Thus the dollar change in average premium is $B\sum_j (w_j^{(1)}-w_j^{(0)})r_j$. The percentage change is the new average divided by the old average, minus one. The percentage change is not generally equal to the change in average relativity: it must be divided by the old average relativity.

Worked Example: A Shift Toward Higher-Relativity Classes

A hypothetical insurer charges a base annual premium of $500 for a specified coverage and policy term. Its three class relativities are 0.80, 1.20, and 2.00. The old exposure shares are 50%, 30%, and 20%; the new shares are 30%, 40%, and 30%. Find the average premium before and after the shift, assuming the base premium and relativities are unchanged.

Method. Calculate each exposure-weighted average relativity, then multiply by the common base premium.

Old average relativity:

$$ 0.50(0.80)+0.30(1.20)+0.20(2.00) =0.40+0.36+0.40 =1.16. $$

The old average premium is $500(1.16)=\$580.00$ per exposure.

New average relativity:

$$ 0.30(0.80)+0.40(1.20)+0.30(2.00) =0.24+0.48+0.60 =1.32. $$

The new average premium is $500(1.32)=\$660.00$ per exposure. The increase is $\$660-\$580=\$80.00$, or $80/580=0.137931\ldots$, which is 13.79% to two decimal places. No class relativity or base premium changed: the entire increase is due to the change in exposure mix.

Check. The shares in each period sum to one. The new portfolio assigns more exposure to the class with relativity 2.00, so a higher average premium is consistent with the direction of the shift. The common mistake this calculation avoids is attributing the increase to a rate change when the class rates were held fixed.

Why a Balanced Set of Relativities Can Stop Averaging to One

Relativities are often balanced so that their exposure-weighted average is one for a selected distribution. That balancing is a statement about both the relativities and the weights used in the calculation; it does not imply that the same relativities average to one under every possible exposure mix.

\sum_j w_j^{(0)}r_j=1 \quad\not\Rightarrow\quad \sum_j w_j^{(1)}r_j=1.

The reason is that the weights change the contribution of each class to the sum. A shift toward classes above the old weighted average tends to raise the new average relativity; a shift toward classes below it tends to lower it. Whether a revised set of relativities should be balanced again depends on the intended premium basis. The arithmetic change in average premium should be measured before making any such adjustment.

Worked Example: Balancing Under One Mix, Not Another

A hypothetical three-class program uses relativities 0.75, 1.50, and 2.50, with exposure shares 60%, 30%, and 10%. The base premium is $600 per exposure. Calculate the average premium under those shares and under a new mix of 40%, 40%, and 20%, with no changes to the base premium or relativities.

Old mix. The average relativity is

$$ 0.60(0.75)+0.30(1.50)+0.10(2.50) =0.45+0.45+0.25 =1.15. $$

The old average premium is $600(1.15)=\$690.00$.

New mix. The average relativity is

$$ 0.40(0.75)+0.40(1.50)+0.20(2.50) =0.30+0.60+0.50 =1.40. $$

The new average premium is $600(1.40)=\$840.00$. The increase is $\$150.00$ per exposure, or $150/690=0.217391\ldots$, which is 21.74% to two decimal places.

If the relativities had been balanced to average one under the old mix, the calculation would have produced an old average of $600. Here the specified relativities average 1.15, so the old average is $690; the calculation does not presume a balance that the data do not show. In either case, changing the weights can change the average. The common mistake is to assume that a set of relativities must continue to average to one after the exposure mix changes.

Check. Both exposure mixes sum to one, and the new mix places 10 percentage points more exposure in the highest-relativity class while reducing the lowest class by 20 points. The direction of the result is plausible.

Separating Mix Changes From Relativity Changes

In practice, both the exposure weights and the relativities may change between periods. These effects can be separated by calculating averages under combinations of old and new inputs. Let $A(w,r)=\sum_j w_jr_j$ denote an average relativity. The exact total change is

A(w^{(1)},r^{(1)})-A(w^{(0)},r^{(0)}) = \sum_j\bigl(w_j^{(1)}-w_j^{(0)}\bigr)r_j^{(0)} +\sum_j w_j^{(1)}\bigl(r_j^{(1)}-r_j^{(0)}\bigr).

This version first changes the mix while retaining the old relativities, then changes the relativities under the new mix. The same total can instead be calculated by first changing relativities under the old mix and then changing the mix under the new relativities:

A(w^{(1)},r^{(1)})-A(w^{(0)},r^{(0)}) = \sum_j w_j^{(0)}\bigl(r_j^{(1)}-r_j^{(0)}\bigr) +\sum_j\bigl(w_j^{(1)}-w_j^{(0)}\bigr)r_j^{(1)}.

These are both exact decompositions, but the individual allocations to “mix” and “relativity” can differ. When both change, their combined effect depends on the interaction between the changes. State which order is used rather than presenting a path-dependent allocation as the only possible one.

Worked Example: Decomposing a Combined Change

A hypothetical program has three classes. Old exposure weights are 0.60, 0.30, and 0.10, and old relativities are 0.75, 1.50, and 2.50. New weights are 0.40, 0.40, and 0.20, and new relativities are 0.80, 1.40, and 2.60. The base premium is $600 in both periods. Find the old and new average premiums and decompose the change by first applying the mix change at old relativities.

Step 1: Old average.

A(w^{(0)},r^{(0)}) =0.60(0.75)+0.30(1.50)+0.10(2.50) =0.45+0.45+0.25 =1.15.

The old average premium is $600(1.15)=\$690.00$.

Step 2: Apply the new mix to old relativities.

A(w^{(1)},r^{(0)}) =0.40(0.75)+0.40(1.50)+0.20(2.50) =0.30+0.60+0.50 =1.40.

The mix effect on average relativity is $1.40-1.15=0.25$, equivalent to $600(0.25)=\$150.00$ per exposure.

Step 3: Apply new relativities to the new mix.

A(w^{(1)},r^{(1)}) =0.40(0.80)+0.40(1.40)+0.20(2.60) =0.32+0.56+0.52 =1.40.

The relativity effect at the new mix is $1.40-1.40=0$. Thus the new average premium is $600(1.40)=\$840.00$, and the total increase is $\$840-\$690=\$150.00$, exactly matching the mix effect plus the relativity effect on this path.

Check the alternative order. At the old mix, the new relativities give $0.60(0.80)+0.30(1.40)+0.10(2.60)=0.48+0.42+0.26=1.16$. The relativity change on that path is $1.16-1.15=0.01$. Applying the new mix to those new relativities gives 1.40, so the subsequent mix change is $1.40-1.16=0.24$. The total is still $0.01+0.24=0.25$. The common mistake is to expect each component to be identical under both orders; only the total change must agree.

Derivation and Practical Interpretation

The average-premium formula follows by conditioning on the rating class of a randomly selected exposure. That exposure is in class $j$ with probability $w_j$, and its premium per exposure in that class is $Br_j$. The law of total expectation therefore gives

E[\text{premium per exposure}] =\sum_j P(\text{class }j)\,E[\text{premium per exposure}\mid \text{class }j] =\sum_j w_j(Br_j) =B\sum_j w_jr_j.

This derivation explains why the relevant weights are exposure proportions: the class is selected according to the distribution of exposures. It also clarifies the limits of the formula. If the base premium differs by class for reasons outside the relativity, or if different coverage terms are being compared, use each class's actual premium per exposure rather than forcing it into one common base $B$.

For a portfolio with $N$ total exposures, the total premium is $N\bar{P}$, provided the same exposure unit is used throughout. If $N$ is unchanged, the percentage change in total premium equals the percentage change in average premium. If $N$ also changes, separate the change in exposure volume from the change in average premium; a larger total premium alone does not establish a mix effect.

Worked Example: From Exposure Counts to Portfolio Premium

A hypothetical insurer has a base premium of $420 per policy-term and class relativities 0.70, 1.10, and 1.80. Initially it has 600, 300, and 100 exposures in the three classes. Later it has 300, 500, and 200 exposures. Assume the coverage and term are identical and that the base premium and relativities remain unchanged. Calculate each period's average and total premium and the percentage change in total premium.

Old period. There are $600+300+100=1{,}000$ exposures, so the weights are 0.60, 0.30, and 0.10. The average relativity is

$$ 0.60(0.70)+0.30(1.10)+0.10(1.80) =0.42+0.33+0.18 =0.93. $$

The average premium is $420(0.93)=\$390.60$. Total premium is $1{,}000(\$390.60)=\$390{,}600.00$.

New period. There are also $300+500+200=1{,}000$ exposures, so the weights are 0.30, 0.50, and 0.20. The average relativity is

$$ 0.30(0.70)+0.50(1.10)+0.20(1.80) =0.21+0.55+0.36 =1.12. $$

The average premium is $420(1.12)=\$470.40$. Total premium is $1{,}000(\$470.40)=\$470{,}400.00$.

The total premium increase is $\$470{,}400-\$390{,}600=\$79{,}800.00$. Its percentage is $79{,}800/390{,}600=0.204301\ldots$, or 20.43% to two decimal places. Because the exposure counts total 1,000 in both periods, this is also the percentage increase in average premium. The common mistake this avoids is comparing totals without checking whether exposure volume changed.

Check. The average premium difference is $\$470.40-\$390.60=\$79.80$; multiplying by 1,000 exposures gives $\$79{,}800$, the same total increase calculated directly.

Common Exam Traps

Using the wrong mix. A mix effect uses exposure weights from the period being evaluated. Mixing old weights with a new average, without stating that calculation as a deliberate comparison, can attribute changes incorrectly.
Assuming relativities average to one automatically. A mean of one applies only to the weights under which the relativities were balanced. Recalculate with the stated mix.
Confusing exposure weights with premium weights. For average premium per exposure, use exposure shares. Premium shares already reflect the class premium level.
Calling a composition change a class rate change. If $B$ and all $r_j$ are fixed, then any change in $B\sum_jw_jr_j$ comes from $w_j$, not from a change in the class prices.
Reporting an order-dependent decomposition without its order. When both weights and relativities change, report the sequence used. The total change is invariant, but the separate attributed components need not be.

Key takeaway: Average premium per exposure is the base premium multiplied by the exposure-weighted average relativity. Hold relativities fixed to isolate a pure mix effect; compare old and new relativities at a stated mix to measure a relativity effect.

Check Your Understanding

For each problem, use exposure shares as weights and show enough intermediate work to distinguish a mix change from a relativity change.

A hypothetical program has base premium $400 and fixed relativities 0.80, 1.20, and 2.00. Old exposure shares are 0.50, 0.30, and 0.20; new shares are 0.25, 0.50, and 0.25. Find both average premiums and the percentage change. (Answer: old $464.00; new $520.00; increase $56.00, or 12.07%.)
Class relativities are 0.60, 1.10, and 1.90. Exposure counts change from 400, 400, and 200 to 200, 500, and 300; the base premium is $500 and remains fixed. Find the old and new average premiums and the percentage change. (Answer: old average relativity 1.06 and premium $530.00; new average relativity 1.24 and premium $620.00; increase 16.98%.)
Exposure shares remain 0.50, 0.30, and 0.20, while relativities change from 0.80, 1.40, and 2.00 to 0.90, 1.20, and 2.10. With base premium $750, calculate the change in average premium attributable to the relativity revision. (Answer: average relativity changes from 1.22 to 1.23; average premium rises from $915.00 to $922.50, an increase of $7.50 or 0.82%.)
Old weights are 0.70, 0.20, and 0.10; new weights are 0.40, 0.40, and 0.20. Old relativities are 0.80, 1.30, and 2.40; new relativities are 0.90, 1.20, and 2.20. Calculate the old and new average relativities, then decompose the change by applying the mix change first at old relativities. (Answer: old average 1.06; mix-first intermediate average 1.32, a mix effect of 0.26; final average 1.28, a subsequent relativity effect of −0.04; total increase 0.22.)
A hypothetical portfolio has base premium $360, relativities 0.75, 1.25, and 2.00, and 1,000 exposures in each period. Counts shift from 500, 300, and 200 to 250, 450, and 300. Find the total premium in each period and its percentage change. (Answer: old average relativity 1.15 and total premium $414,000.00; new average relativity 1.35 and total premium $486,000.00; increase $72,000.00, or 17.39%.)

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