Why Exposure-Weighted Pure Premiums Differ From Simple Averages

Why the Choice of Average Matters

When experience from several periods is combined, there are two tempting summaries: average the period pure premiums, or pool the losses and exposures first. These calculations answer different questions when the periods have different exposure volumes. A period with many exposures represents more of the combined experience than a period with few exposures, so the pooled pure premium gives the periods different weights.

This distinction matters in a pure premium indication because the pure premium is loss cost per exposure. To combine periods on a consistent basis, each period must use the same kind of exposure unit and losses measured on the same basis. The arithmetic below assumes those conditions hold; it does not adjust for differences in coverage, trend, development, or exposure mix.

Definition: For period $j$, let $L_j$ be the losses assigned to that period, $E_j$ its exposures, and $PP_j=L_j/E_j$ its period pure premium. Assume $E_j>0$ for each included period. The pooled pure premium is total losses divided by total exposures: $$ PP_{\text{pooled}}=\frac{\sum_{j=1}^{n} L_j}{\sum_{j=1}^{n} E_j}. $$ The simple average of period pure premiums is $\frac{1}{n}\sum_{j=1}^{n}PP_j$, which gives each period equal weight.

Deriving the Exposure-Weighted Average

The relationship follows directly from the definition of each period's pure premium. Since $L_j=E_jPP_j$, substitute this expression for every period's losses into the pooled calculation.

Write total losses in period terms.
For each period, $L_j=E_jPP_j$. Therefore, total losses are $\sum_j E_jPP_j$.

Substitute into the pooled ratio.
Dividing the sum of period losses by the sum of period exposures gives $PP_{\text{pooled}}=\frac{\sum_j E_jPP_j}{\sum_j E_j}$.

Identify the weights.
Define $w_j=E_j/\sum_k E_k$. These weights are positive and sum to one, so $PP_{\text{pooled}}=\sum_j w_jPP_j$.

PP_{\text{pooled}} =\frac{\sum_{j=1}^{n}L_j}{\sum_{j=1}^{n}E_j} =\frac{\sum_{j=1}^{n}E_jPP_j}{\sum_{j=1}^{n}E_j} =\sum_{j=1}^{n}w_jPP_j, \qquad w_j=\frac{E_j}{\sum_{k=1}^{n}E_k}.

Because each weight is an exposure share, a period with twice as many exposures as another period receives twice as much weight in the pooled result. The pooled result is therefore not an arbitrary adjustment to the simple average; it is exactly the loss-per-exposure ratio for all included periods combined.

Result: With positive exposures, the pooled pure premium is a weighted average of the period pure premiums, using each period's share of total exposure as its weight. It must lie between the smallest and largest period pure premiums. It equals the simple average when the exposures are equal; it can also equal the simple average with unequal exposures if the differences happen to balance.

Worked Calculations

Worked Example: Unequal Exposure Volumes

A hypothetical small-equipment insurer has two periods of experience on the same coverage and exposure basis. Period 1 has 500 exposures and a pure premium of $80. Period 2 has 1,500 exposures and a pure premium of $100. Calculate the pooled pure premium and compare it with the simple average. The method is to reconstruct losses, pool losses and exposures, and then calculate the unweighted average for comparison.

Step 1: Calculate each period's losses. Period 1 losses are $500\times \$80=\$40{,}000$. Period 2 losses are $1{,}500\times \$100=\$150{,}000$.

Step 2: Pool the experience. Total losses are $\$40{,}000+\$150{,}000=\$190{,}000$. Total exposures are $500+1{,}500=2{,}000$. Thus, $$ PP_{\text{pooled}}=\frac{\$190{,}000}{2{,}000}=\$95.00\text{ per exposure}. $$

Step 3: Calculate the simple average. The simple average is $(\$80+\$100)/2=\$90.00$ per exposure. The weights in the pooled average are $500/2{,}000=0.25$ and $1{,}500/2{,}000=0.75$, giving $0.25(\$80)+0.75(\$100)=\$95.00$.

Check. The pooled value of $95.00 is between $80 and $100 and is closer to $100, the pure premium for the period with more exposure. The simple average gives each period equal weight and is therefore $5.00 lower. The common mistake this method avoids is averaging the two period values without accounting for their different exposure volumes.

Worked Example: More Exposure in the Lower-Pure-Premium Period

A hypothetical specialty policy has two periods. The first has 1,800 exposures at a pure premium of $60, and the second has 200 exposures at a pure premium of $120. Find the pooled and simple averages, and explain why their ordering differs from the prior example. Use exposure weights for the pooled calculation.

Step 1: Find the period losses. Period 1 losses are $1{,}800\times \$60=\$108{,}000$. Period 2 losses are $200\times \$120=\$24{,}000$.

Step 2: Calculate the pooled value. Total losses are $\$108{,}000+\$24{,}000=\$132{,}000$, and total exposures are $1{,}800+200=2{,}000$. Therefore, $$ PP_{\text{pooled}}=\frac{\$132{,}000}{2{,}000}=\$66.00\text{ per exposure}. $$ The same result follows from the weights $1{,}800/2{,}000=0.90$ and $200/2{,}000=0.10$: $0.90(\$60)+0.10(\$120)=\$66.00$.

Step 3: Calculate the simple average. The simple average is $(\$60+\$120)/2=\$90.00$ per exposure. It is greater than the pooled value because it gives the small, high-pure-premium period half the weight, while that period represents only 10% of total exposure.

Check. The pooled value of $66.00 is between $60 and $120 and lies much closer to $60, the value with 90% of the exposure. The common mistake this method avoids is assuming that the pooled result must be close to the simple average simply because there are two periods.

Worked Example: Three Periods and a Balanced Average

A hypothetical property program has three periods with 400, 1,000, and 600 exposures, respectively. Their period pure premiums are $75, $90, and $120. Calculate the pooled pure premium, the simple average, and each exposure weight. The method is to calculate period losses and then compare the weighted and unweighted summaries.

Step 1: Calculate period losses. The losses are $400\times \$75=\$30{,}000$, $1{,}000\times \$90=\$90{,}000$, and $600\times \$120=\$72{,}000$.

Step 2: Pool losses and exposures. Total losses are $\$30{,}000+\$90{,}000+\$72{,}000=\$192{,}000$. Total exposures are $400+1{,}000+600=2{,}000$. Thus, the pooled pure premium is $\$192{,}000/2{,}000=\$96.00$ per exposure.

Step 3: Verify using weights. The weights are $400/2{,}000=0.20$, $1{,}000/2{,}000=0.50$, and $600/2{,}000=0.30$. The weighted value is $0.20(\$75)+0.50(\$90)+0.30(\$120)=\$15+\$45+\$36=\$96.00$.

Step 4: Compare with the simple average. The simple average is $(\$75+\$90+\$120)/3=\$285/3=\$95.00$ per exposure. In this case it is only $1.00 lower than the pooled result, but the two methods still do not give the same answer.

Check. The weights sum to $0.20+0.50+0.30=1.00$, and the pooled value of $96.00 lies between $75 and $120. The common mistake this method avoids is assuming that a small difference between the two averages means the averaging methods are interchangeable.

When Do the Two Averages Agree?

If all periods have equal exposures, each exposure weight is $1/n$. The weighted average then reduces directly to the simple average. With unequal exposures, equality is possible but not guaranteed: a period's high or low pure premium may be offset by the exposure weights assigned to other periods.

One way to see the condition is to compare the exposure-weighted average with the unweighted average. Let $\bar E$ be the arithmetic mean of the exposures and $\overline{PP}$ the arithmetic mean of the period pure premiums. Define the population covariance across the $n$ periods as $\operatorname{Cov}(E,PP)=\frac{1}{n}\sum_j(E_j-\bar E)(PP_j-\overline{PP})$. Then:

PP_{\text{pooled}} =\frac{\frac{1}{n}\sum_j E_jPP_j}{\bar E} =\frac{\bar E\,\overline{PP}+\operatorname{Cov}(E,PP)}{\bar E} =\overline{PP}+\frac{\operatorname{Cov}(E,PP)}{\bar E}.

The middle equality follows by expanding the covariance: $\frac{1}{n}\sum_jE_jPP_j=\bar E\,\overline{PP}+\operatorname{Cov}(E,PP)$. Since $\bar E>0$, the pooled and simple averages are equal exactly when this covariance is zero. If larger exposures tend to coincide with larger period pure premiums, the covariance is positive and the pooled value exceeds the simple average. If larger exposures tend to coincide with smaller pure premiums, it is negative and the pooled value is below the simple average. Zero covariance is the precise balancing condition; equal exposures are one clear way to satisfy it.

Range Checks and a Common Exam Trap

A weighted average with positive weights summing to one cannot exceed every value being averaged or fall below every value being averaged. This range check is useful before doing detailed arithmetic: if a proposed pooled pure premium is outside the range of the period pure premiums, it cannot be produced by any positive exposure volumes using those period values.

Worked Example: Testing an Impossible Pooled Target

A hypothetical repair-insurance portfolio has period pure premiums of $72 and $96 per exposure. The first period has 1,000 exposures. A proposed target pooled pure premium is $100. Determine whether any positive exposure amount in the second period can achieve that target, and calculate the simple average. The method is to test the target using the pooled equation and then check it against the range of the two period values.

Step 1: Set up the pooled equation. Let $x$ be the second period's exposure count. Its losses are $96x$, while the first period's losses are $72\times1{,}000=72{,}000$. To reach $100, the equation would have to be $$ \frac{72{,}000+96x}{1{,}000+x}=100. $$

Step 2: Solve and check the implied exposure. Multiplying by $1{,}000+x$ gives $72{,}000+96x=100{,}000+100x$. Rearranging gives $-28{,}000=4x$, so $x=-7{,}000$. This is not a positive exposure amount, so no valid second-period exposure can achieve the target.

Step 3: Calculate the simple average. The simple average is $(\$72+\$96)/2=\$84.00$ per exposure.

Check. With positive exposures, the pooled pure premium must lie between $72 and $96, so $100 is impossible regardless of the exposure amount. The algebra's negative solution agrees with that range check. The common mistake this method avoids is reporting a positive exposure as though it could raise a weighted average above both period values.

Practical Interpretation

The pooled calculation answers: “What were total losses per total exposure across these periods?” The simple average answers: “What is the arithmetic mean of the period-specific loss costs when each period counts once?” Neither calculation is inherently wrong, but they do not have the same interpretation. For a combined loss-per-exposure measure, pooling losses and exposures gives the exposure-weighted result by definition.

Before combining experience, confirm that the exposure units match and that the losses and exposures refer to the same periods and basis. If they do not, the ratio can be numerically computable yet not answer the intended ratemaking question. For the comparison in this tutorial, hold those bases constant; adjustments to the underlying experience require separate assumptions and methods.

Use the pooled ratio when the intended summary is total losses divided by total exposures.
Use a simple average only when the intended measure gives each period equal importance, or when equal exposures make it coincide with the pooled result.
Check the range of the period pure premiums before accepting a proposed pooled result.

This distinction is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus because it tests whether a candidate understands what the exposure denominator does, rather than treating averaging as a purely mechanical step.

Check Your Understanding

For each question, identify whether the requested result is pooled or simple, and check that the pooled result lies within the range of the period pure premiums.

A hypothetical program has 300 exposures at a pure premium of $40 and 700 exposures at $70. Calculate the pooled pure premium and the simple average. (Answer: Pooled $61.00 per exposure; simple average $55.00 per exposure.)
Two periods have 1,000 exposures at $55 and 500 exposures at $85. Calculate both averages and state which period has the larger pooled weight. (Answer: Pooled $65.00 per exposure; simple average $70.00 per exposure; the first period has the larger weight, 2/3 versus 1/3.)
Three periods have 200, 300, and 500 exposures, with pure premiums of $30, $50, and $90. Find the pooled pure premium, the simple average, and the exposure weights. (Answer: Pooled $66.00 per exposure; simple average $56.67 per exposure; weights 0.20, 0.30, and 0.50.)
Three periods have 100, 200, and 300 exposures, with pure premiums of $80, $100, and $80. Show whether the pooled and simple averages agree. (Answer: Yes; both are $86.67 per exposure.)
Two periods have pure premiums of $72 and $96; the first has 1,000 exposures. Can any positive exposure amount in the second period make the pooled pure premium $100? What is the simple average? (Answer: No; the pooled value must be between $72 and $96. The simple average is $84.00 per exposure.)

What You'll Learn