From Earned Premium to Expected Ultimate Loss
When an expected loss ratio has been selected for a period, the premium-based expected loss calculation is direct: multiply the period’s earned premium by its expected loss ratio. The arithmetic is simple, but the result is only meaningful if the premium, ratio, period, and loss measure are consistent. In particular, a percentage must be converted to a decimal for multiplication, and a premium stated in thousands must not be treated as dollars.
This tutorial applies the premium-based expected loss formula across several periods and shows how to check both period calculations and aggregate totals. The calculations are aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.
The calculation treats the selected expected loss ratio as an input assumption. It does not require a mortality assumption, an interest rate, or a specified probability distribution for \(U_j\). Unless a problem explicitly calls for discounting, the calculation is undiscounted; no interest adjustment is made. The word “expected” identifies the estimate produced by the selected actuarial assumption, not a claim that the actual ultimate loss will equal that amount.
Why the Formula Works
An expected loss ratio is a loss-to-premium relationship. For a period with positive earned premium, its definition is expected ultimate losses divided by earned premium. Multiplying both sides by earned premium gives the expected loss formula.
This relationship requires the selected ratio and the premium to refer to a compatible period and loss scope. For example, a ratio intended for one coverage should not be applied to premium for a different coverage without an appropriate basis. If \(EP_j=0\), the ratio definition by division is not available for that period; multiplying zero premium by a selected ratio would mechanically produce zero, but it would not establish that the period has no expected losses.
A Reliable Calculation Sequence
For multiple periods, calculate each expected ultimate loss separately before totaling. This makes it easier to spot a period-specific arithmetic error and ensures that each selected ratio is applied to its own premium base.
Identify the period, earned premium amount and units, selected expected loss ratio, and intended loss scope.
Write a percentage as a decimal. For example, \(62\%=0.62\).
Multiply that period’s earned premium by its selected decimal ratio. Keep the premium’s units throughout.
Add the period estimates. If calculating an aggregate loss ratio, divide total expected losses by total earned premium, rather than taking a simple average of the period ratios.
Worked Example: Calculating Estimates for Three Accident Periods
A hypothetical insurer has the following earned premiums and selected expected loss ratios. Calculate the expected ultimate losses for each accident period and the total across all three periods. All amounts are in dollars, and the stated ratios apply to the same loss scope as the requested estimates.
| Accident period | Earned premium | Selected expected loss ratio |
|---|---|---|
| 2023 | $2,400,000 | 62% |
| 2024 | $2,750,000 | 58% |
| 2025 | $3,100,000 | 61% |
Assumptions. Use the selected ratios as given. No interest adjustment or mortality assumption applies, and no probability distribution is specified or needed for these deterministic multiplications.
2023. Convert \(62\%\) to \(0.62\), then multiply: $$ \$2{,}400{,}000\times0.62=\$1{,}488{,}000. $$
2024. Convert \(58\%\) to \(0.58\): $$ \$2{,}750{,}000\times0.58 =\$2{,}750{,}000\times(0.60-0.02) =\$1{,}650{,}000-\$55{,}000 =\$1{,}595{,}000. $$
2025. Convert \(61\%\) to \(0.61\): $$ \$3{,}100{,}000\times0.61 =\$3{,}100{,}000\times(0.60+0.01) =\$1{,}860{,}000+\$31{,}000 =\$1{,}891{,}000. $$
Total estimate. Add the three period estimates: $$ \$1{,}488{,}000+\$1{,}595{,}000+\$1{,}891{,}000 =\$4{,}974{,}000. $$ As a check, total earned premium is \(\$2{,}400{,}000+\$2{,}750{,}000+\$3{,}100{,}000=\$8{,}250{,}000\). The aggregate implied ratio is \(\$4{,}974{,}000/\$8{,}250{,}000=0.602909\ldots\), or about \(60.29\%\). This lies between the smallest and largest period ratios, as it should for a premium-weighted average.
Method and mistake avoided. This is the premium-based expected loss method applied period by period. The check confirms both the arithmetic and the units: the estimates and premiums are dollars, and the aggregate ratio is dimensionless.
Keep Premium Units Consistent
Premium may be presented in dollars, thousands of dollars, or another stated unit. Multiplication preserves that unit. If premium is reported in thousands, the product is first an amount in thousands; multiply by 1,000 only if the answer is requested in dollars. Converting too early is acceptable if done consistently, but mixing units midway through the calculation is not.
Worked Example: Summing Quarterly Premiums Reported in Thousands
A hypothetical insurer reports earned premium by quarter, in thousands of dollars, for three periods. Apply each period’s selected expected loss ratio, then report total expected ultimate losses in both thousands and dollars.
| Period | Quarterly earned premiums ($000) | Selected expected loss ratio |
|---|---|---|
| A | 420, 510, 475, 395 | 72.5% |
| B | 530, 560, 610, 500 | 68% |
| C | 600, 625, 640, 635 | 64.4% |
Assumptions. The quarterly amounts are earned premiums in thousands of dollars, and each selected ratio applies to its period’s summed premium. No interest adjustment, mortality assumption, or probability distribution is used.
Period A premium. Sum the four quarters: $$ 420+510+475+395=1{,}800\text{ ($000)}. $$ Convert \(72.5\%\) to \(0.725\), then calculate: $$ 1{,}800\times0.725=1{,}305\text{ ($000)}. $$ Thus Period A’s expected ultimate losses are \(\$1{,}305{,}000\).
Period B premium. $$ 530+560+610+500=2{,}200\text{ ($000)}. $$ Then: $$ 2{,}200\times0.68=1{,}496\text{ ($000)} =\$1{,}496{,}000. $$
Period C premium. $$ 600+625+640+635=2{,}500\text{ ($000)}. $$ Then: $$ 2{,}500\times0.644=1{,}610\text{ ($000)} =\$1{,}610{,}000. $$
Total and check. Total expected ultimate losses are $$ 1{,}305+1{,}496+1{,}610=4{,}411\text{ ($000)}=\$4{,}411{,}000. $$ Total earned premium is \(1{,}800+2{,}200+2{,}500=6{,}500\) ($000). The implied aggregate ratio is \(4{,}411/6{,}500=0.678615\ldots\), or about \(67.86\%\). It is consistent with the period ratios of \(72.5\%\), \(68\%\), and \(64.4\%\).
Method and mistake avoided. This calculation first totals the quarterly earned premium within each period, then applies the period’s ratio. A common error is to multiply by \(72.5\) instead of \(0.725\), which would make the result 100 times too large.
Aggregating Ratios Correctly
When several periods have different earned premiums, their expected loss ratios do not generally have equal influence on the overall result. The aggregate ratio is total expected losses divided by total earned premium. Substituting the period formula shows that it is a premium-weighted average of the period ratios:
The weights are the periods’ shares of total earned premium. A simple arithmetic average gives each period equal weight regardless of its premium volume, so it is generally not the correct ratio for the combined premium base.
Worked Example: Finding the Correct Aggregate Ratio
A hypothetical program has three periods. Period A has earned premium of \(\$1{,}600{,}000\) and a selected expected loss ratio of \(55\%\); Period B has \(\$2{,}400{,}000\) and \(70\%\); Period C has \(\$1{,}000{,}000\) and \(64\%\). Find each expected ultimate loss, their total, and the aggregate implied ratio. Then compare with the simple average of the three ratios.
Assumptions. Each ratio applies to its stated period premium and the same loss scope. No discounting or mortality model applies, and no probability distribution is needed.
Period A: \(\$1{,}600{,}000\times0.55=\$880{,}000\).
Period B: \(\$2{,}400{,}000\times0.70=\$1{,}680{,}000\).
Period C: \(\$1{,}000{,}000\times0.64=\$640{,}000\).
Total expected ultimate losses: $$ \$880{,}000+\$1{,}680{,}000+\$640{,}000 =\$3{,}200{,}000. $$ Total earned premium is $$ \$1{,}600{,}000+\$2{,}400{,}000+\$1{,}000{,}000 =\$5{,}000{,}000. $$ Therefore, the aggregate implied ratio is $$ \frac{\$3{,}200{,}000}{\$5{,}000{,}000}=0.64=64\%. $$
Common wrong calculation. The simple average of the ratios is \((55\%+70\%+64\%)/3=63\%\). Applying that to the total premium gives \(0.63\times\$5{,}000{,}000=\$3{,}150{,}000\), which is \(\$50{,}000\) below the correct total.
Correction and check. The periods have different premium volumes, so calculate each period’s expected loss and add, or use the premium-weighted ratio. Period B has the greatest premium and the highest ratio; it therefore has more influence on the aggregate than a simple average allows. The corrected total is \(\$3{,}200{,}000\), and dividing by total premium returns \(64\%\), as required.
Method and mistake avoided. This is period-by-period premium-based estimation followed by a premium-weighted aggregation. It avoids treating a small-premium period as equally important as a large-premium period.
Exam Checks and Common Traps
Before accepting a result, verify that each multiplication uses earned premium, not a different premium measure; that the selected ratio is in decimal form; and that the ratio and premium have compatible period and loss scope. Check the result’s scale: for a positive ratio below \(100\%\), expected losses should be less than the corresponding premium. A ratio above \(100\%\) can produce expected losses above premium, so that inequality is a check only when the selected ratio is below \(100\%\).
- Using a whole-number percentage. Multiplying by \(62\) instead of \(0.62\) overstates the result by a factor of 100.
- Mixing units. A product calculated in thousands remains in thousands until explicitly converted.
- Using the wrong period’s ratio. Apply each selected ratio to the earned premium for the period to which that ratio pertains.
- Taking an unweighted average. The aggregate implied ratio weights each period by earned premium.
- Rounding too early. Keep the stated ratios and intermediate amounts at full precision, then round the final answer as requested.
Check Your Understanding
For each problem, show the period calculations and retain the stated units through the final step.
- A hypothetical accident period has earned premium of \(\$1{,}850{,}000\) and a selected expected loss ratio of \(63.2\%\). What are its expected ultimate losses? (Answer: \(\$1{,}169{,}200\))
- Quarterly earned premiums for a hypothetical period are \(315\), \(425\), and \(460\), all in thousands of dollars. The selected expected loss ratio is \(71\%\). Find expected ultimate losses in thousands and dollars. (Answer: \(852\) thousand dollars, or \(\$852{,}000\))
- Period A has earned premium of \(\$2{,}200{,}000\) and a \(59\%\) selected ratio; Period B has \(\$2{,}800{,}000\) and a \(66\%\) ratio. Find each expected loss, the combined expected loss, and the aggregate implied ratio. (Answer: \(\$1{,}298{,}000\) and \(\$1{,}848{,}000\); total \(\$3{,}146{,}000\); aggregate ratio \(62.92\%\))
- A hypothetical period has earned premium of \(\$3{,}750{,}000\) and expected ultimate losses of \(\$2{,}475{,}000\). What expected loss ratio is implied by these amounts? (Answer: \(66\%\))
- Period A has premium of \(\$900{,}000\) and a \(74\%\) ratio; Period B has premium of \(\$1{,}500{,}000\) and a \(62\%\) ratio. Find total expected ultimate losses and the aggregate implied ratio. (Answer: \(\$1{,}596{,}000\) and \(66.5\%\))