Why Two Averages Can Give Different Answers
When experience spans several years, each year has a loss ratio: its losses divided by its premium. A simple arithmetic average gives every year equal influence. The ratio of total losses to total premium instead gives more influence to years with more premium. Those measures answer different questions, and they need not agree when annual premium volumes differ.
For a pooled measure of the experience, the ratio of summed losses to summed premium is the relevant calculation. It describes total losses relative to total premium across the whole period. The simple average describes the average of the yearly ratios when each year is counted once, regardless of how much premium it represents.
In an indication analysis, the losses and premiums being compared must be on compatible bases. For example, if the analysis uses adjusted losses and on-level premiums, use those consistently in each year’s ratio and in the pooled ratio. The weighting identity below holds for any such consistently defined values. It does not by itself determine which adjustments are appropriate.
The Aggregate Ratio Is Premium-Weighted
Because \(L_t=P_t r_t\), the aggregate ratio can be rewritten as a weighted average of the annual ratios. Define the premium weight for year \(t\) as \(w_t=P_t/\sum_{j=1}^{n}P_j\). The weights sum to one, and a year with more premium receives a larger weight.
This identity follows directly by substituting \(L_t=P_t r_t\) into the aggregate ratio. It explains why pooling the dollars of loss and premium does not generally give the same result as averaging the yearly percentages: pooling implicitly weights each annual ratio by that year’s share of total premium.
The two measures are equal if the premiums are equal, since then every \(w_t=1/n\). They are also equal if all annual loss ratios are the same, whatever the premium distribution. With unequal premiums and different annual ratios, they may still happen to agree, but equality is not guaranteed. The exact condition is that the premium-weighted average of the annual ratios equals their unweighted average.
The expression on the right shows the source and direction of the difference. Years whose premium shares exceed \(1/n\) receive more weight in the aggregate ratio than in the simple average. If those years have higher loss ratios, the aggregate ratio tends to be higher; if they have lower ratios, it tends to be lower.
Worked Example: Two Years With Unequal Premium
A hypothetical specialty insurer reviews two years of experience. Year 1 has $100,000 of premium and $70,000 of losses. Year 2 has $300,000 of premium and $240,000 of losses. Calculate both the aggregate ratio and the simple average of the annual ratios. State which measure describes pooled loss experience.
Method. First calculate the annual ratios, then calculate each summary directly from its definition. This keeps the years’ percentages distinct from the ratio of the combined dollars.
Year 1 ratio: \(r_1=70{,}000/100{,}000=0.70=70\%\).
Year 2 ratio: \(r_2=240{,}000/300{,}000=0.80=80\%\).
Simple average: \((70\%+80\%)/2=75\%\). Each year contributes one-half to this calculation.
Aggregate ratio: total losses are \(70{,}000+240{,}000=310{,}000\), and total premium is \(100{,}000+300{,}000=400{,}000\). Therefore, \(310{,}000/400{,}000=0.775=77.50\%\), to two decimal places as a percentage.
As a check, the premium weights are \(100{,}000/400{,}000=0.25\) and \(300{,}000/400{,}000=0.75\). Their weighted ratio is \(0.25(70\%)+0.75(80\%)=17.50\%+60.00\%=77.50\%\), matching the pooled calculation. The aggregate ratio is the appropriate measure of the combined loss-to-premium relationship.
Common mistake avoided: reporting 75% as the pooled ratio just because it is the average of the two annual ratios. That calculation gives the smaller-premium year the same influence as the year with three times as much premium.
Worked Comparisons
The annual loss ratio alone does not show how much premium that year contributes to a multi-year result. Always retain the premium amounts, or the premium shares, when combining years. The following examples use hypothetical experience and illustrate different weighting patterns.
Worked Example: Three Years and Unequal Premium Shares
A hypothetical regional insurer has the following experience, with all amounts stated in thousands of dollars. Find each annual loss ratio, the simple average, and the aggregate ratio. Explain why they differ.
| Year | Premium | Losses |
|---|---|---|
| 1 | $80 | $48 |
| 2 | $120 | $108 |
| 3 | $300 | $225 |
Step 1: Annual ratios. Year 1 is \(48/80=0.60=60\%\). Year 2 is \(108/120=0.90=90\%\). Year 3 is \(225/300=0.75=75\%\).
Step 2: Simple average. \((60\%+90\%+75\%)/3=225\%/3=75\%\).
Step 3: Aggregate ratio. Total losses are \(48+108+225=381\) thousand dollars. Total premium is \(80+120+300=500\) thousand dollars. Thus the aggregate ratio is \(381/500=0.762=76.20\%\).
Step 4: Verify by weights. The premium shares are \(80/500=0.16\), \(120/500=0.24\), and \(300/500=0.60\). The weighted ratio is \(0.16(60\%)+0.24(90\%)+0.60(75\%)=9.60\%+21.60\%+45.00\%=76.20\%\).
The aggregate result is 1.20 percentage points higher than the simple average because the third year, with a 75% ratio, has a 60% premium share, while the second year’s higher 90% ratio has only a 24% share. The comparison is between 76.20% and 75.00%; do not describe the difference as a 1.20% relative change.
Common mistake avoided: averaging 60%, 90%, and 75% and calling that pooled experience. The arithmetic mean is correct for an equally weighted summary of years, but not for the ratio of the total losses to total premium.
Worked Example: A High-Volume Year Pulls the Aggregate Up
A hypothetical insurer records premium of $100,000, $100,000, and $800,000 over three years. The respective annual loss ratios are 50%, 70%, and 90%. Calculate the implied losses for each year, then compare the aggregate ratio with the simple average.
Step 1: Recover each year’s losses. Since \(L_t=P_t r_t\), the losses are:
- Year 1: \($100{,}000(0.50)=$50{,}000\).
- Year 2: \($100{,}000(0.70)=$70{,}000\).
- Year 3: \($800{,}000(0.90)=$720{,}000\).
Step 2: Calculate the simple average. \((50\%+70\%+90\%)/3=210\%/3=70\%\).
Step 3: Calculate the aggregate ratio. Total losses are \($50{,}000+$70{,}000+$720{,}000=$840{,}000\). Total premium is \($100{,}000+$100{,}000+$800{,}000=$1{,}000{,}000\). The aggregate ratio is \(840{,}000/1{,}000{,}000=84\%\).
Step 4: Check the premium weighting. The premium shares are 10%, 10%, and 80%. The weighted ratio is \(0.10(50\%)+0.10(70\%)+0.80(90\%)=5\%+7\%+72\%=84\%\), matching the direct calculation.
The simple average is 70%, but it gives the 90% ratio year the same one-third weight as each $100,000 year. In the pooled result, that year represents 80% of premium, so the aggregate ratio is much closer to 90%. The error to avoid is confusing an average across years with a ratio across the combined dollars.
Choosing and Reporting the Right Measure
For a question about combined experience, report the aggregate ratio and show its total-loss and total-premium basis. If an analysis specifically asks for the average annual ratio with years counted equally, report the simple average and label it as such. A clear label matters: both calculations are mathematically valid, but they summarize experience differently.
A useful calculation sequence is:
Confirm that losses and premiums cover the intended experience periods and use compatible definitions and adjustments.
For each year, divide that year’s losses by its corresponding premium.
For pooled experience, divide total losses by total premium. For an equal-year summary, average the annual ratios.
Confirm that the pooled ratio equals the premium-weighted average of annual ratios.
For a pooled loss ratio, a compact table with annual premium, annual losses, and annual ratio makes the calculation auditable. It also exposes a common source of errors: applying weights to ratios that do not correspond to the stated premium, or averaging ratios after omitting a year’s premium volume.
Exam trap: an instruction to “average the annual loss ratios” is not automatically the same as asking for the ratio of aggregate losses to aggregate premium. For example, using the three-year data in the preceding example, the incorrect pooled calculation would be \((50\%+70\%+90\%)/3=70\%\). The correction is to use the premium-weighted calculation, \(0.10(50\%)+0.10(70\%)+0.80(90\%)=84\%\), or equivalently \(840{,}000/1{,}000{,}000=84\%\). State which measure the question requests before calculating.
Check Your Understanding
For each problem, distinguish an equal-year average from a pooled ratio when the data call for it.
- A hypothetical insurer has premium of $150,000 and losses of $105,000 in Year 1, and premium of $450,000 and losses of $360,000 in Year 2. Find the simple average and aggregate loss ratio. (Answer: Simple average = 75%; aggregate ratio = 77.50%.)
- Three hypothetical years have premiums of $250,000, $250,000, and $500,000, and annual loss ratios of 72%, 84%, and 78%. Find both averages. (Answer: Simple average = 78%; aggregate ratio = 78%.)
- Premiums for three hypothetical years are $100,000, $200,000, and $700,000, with losses of $50,000, $150,000, and $630,000. Find the annual ratios, their simple average, and the aggregate ratio. (Answer: Annual ratios = 50%, 75%, and 90%; simple average = 71.67%; aggregate ratio = 83%.)
- Annual ratios are 60%, 75%, and 90%. The corresponding premium shares are \(1/7\), \(2/7\), and \(4/7\). Find the aggregate ratio and compare it with the simple average. (Answer: Aggregate ratio = 81.43%; simple average = 75%.)
- A hypothetical insurer has $100,000 of premium and $60,000 of losses in Year 1, $300,000 of premium and unknown losses in Year 2, and $100,000 of premium and $90,000 of losses in Year 3. The aggregate ratio is 78%. Find Year 2 losses and the simple average of the annual ratios. (Answer: Year 2 losses = $240,000; simple average = 76.67%.)