Why the Selection Requires Judgment
An expected loss ratio is an a priori assumption about the relationship between ultimate losses and premium for a specified portfolio, coverage, and premium basis. Historical ultimate loss experience can inform that assumption, but a historical ratio is evidence, not automatically the right prospective selection. The actuary must assess whether the experience represents the intended coverage and whether the premium denominator matches the premium basis to which the selected ratio will be applied.
For example, a ratio calculated for broader coverage may overstate the expected loss ratio for narrower coverage. A ratio using total billed amounts that include non-risk fees may understate the ratio for a premium basis that excludes those fees. Even when the data are comparable, individual years can vary, so selection should consider the full experience rather than mechanically choosing the latest observation.
Model and Selection Framework
The basic model assumes that, for the portfolio being considered, expected ultimate loss is proportional to premium on the stated basis. The selected expected loss ratio is the proportionality factor. This is a useful aggregate model when the experience and target business are sufficiently comparable; it does not by itself correct for changed coverage, exposure mix, or a premium definition that differs between the historical data and the intended application.
Here, \(L_{\mathrm{ult}}\) denotes ultimate loss and \(P\) is premium on the specified basis. Dividing both sides by \(P\), when \(P>0\), gives \(ELR=E[L_{\mathrm{ult}}]/P\). This explains why the ratio is appropriate for translating a premium amount into expected ultimate losses: multiplying the premium by the selected ratio restores the expected-loss amount. The formula does not make incomparable historical losses and premiums comparable; that assessment is part of selecting the input.
A standard historical indication for several periods is the aggregate ultimate loss divided by aggregate premium. It is equivalent to weighting each period’s loss ratio by that period’s share of total premium. A simple average gives each period equal weight instead, and can be useful when that is the intended judgment, but it does not generally equal the aggregate indication.
The aggregate calculation assumes each period’s losses and premium are on a comparable basis. If they are not, first identify the basis differences and determine whether the data can support a comparable indication. A precise arithmetic result from mismatched data is not a sound selection.
Calculate and Compare Historical Indications
Worked Example: Comparing Annual and Aggregate Indications
A hypothetical commercial equipment portfolio has the following three years of ultimate losses and earned premium, both already stated on a common coverage and premium basis. Calculate each annual loss ratio, the simple average, and the aggregate indication. Recommend a starting candidate based on these data.
Method: calculate annual ratios, then compare the equal-weighted average with the premium-weighted aggregate. The annual ratios are:
Year 1: \(4.8/8.0=0.600\), or 60.0%. Year 2: \(6.3/9.0=0.700\), or 70.0%. Year 3: \(5.4/9.0=0.600\), or 60.0%. Losses and premiums are in millions of dollars.
The simple average is \((60.0\%+70.0\%+60.0\%)/3=190.0\%/3=63.333\ldots\%\), or 63.3% to one decimal place.
Total ultimate losses are \(4.8+6.3+5.4=16.5\) million. Total earned premium is \(8.0+9.0+9.0=26.0\) million. The aggregate indication is \(16.5/26.0=0.634615\ldots\), or 63.5% to one decimal place.
The difference between 63.3% and 63.5% is small here. A defensible starting candidate is about 63.5% if the aim is to represent the aggregate premium volume; the simple average is a nearby reasonableness check. The calculation does not establish that 63.5% is automatically appropriate for future business if its coverage or premium basis differs.
Check: Applying the aggregate ratio to the 26.0 million total premium gives \(0.634615\ldots\times26.0=16.5\) million, the stated total ultimate loss. Common mistake: calling the simple average the aggregate loss ratio. Equal-weighting the three years gives 63.3%, whereas the aggregate calculation weights years according to premium and gives 63.5%.
Assess the Intended Coverage
Historical experience is relevant only to the extent that it represents the intended coverage. When a problem supplies a factor describing the relationship between expected losses under the target coverage and those under the historical coverage, apply that factor to the historical loss indication, holding the stated premium basis fixed. The factor’s direction matters: a factor below 1.00 means the target coverage is expected to produce lower losses, all else equal.
Worked Example: Translating a Historical Ratio to Target Coverage
A hypothetical insurer observes 14.4 million of historical ultimate losses on a specified premium basis of 22.0 million. The historical coverage is broader than the intended coverage. For this problem, assume expected losses under the target coverage are 92% of losses under the historical coverage, with premium held on the same stated basis. Calculate a target-coverage expected loss ratio.
Method: calculate the historical ratio, then apply the supplied target-to-historical loss factor. The historical ratio is \(14.4/22.0=0.654545\ldots\), or about 65.45%.
The target-coverage loss amount on this basis is \(14.4\times0.92=13.248\) million. Dividing by the same premium basis gives \(13.248/22.0=0.6021818\ldots\). Equivalently, \(0.654545\ldots\times0.92=0.6021818\ldots\). The target-coverage ratio is therefore 60.2182%, which rounds to 60.2% to one decimal place.
Check: \(0.6021818\ldots\times22.0=13.248\) million, matching the adjusted target-coverage loss amount. Common mistake: reversing the factor or multiplying incorrectly. Since target losses are stated to be lower, the ratio must decrease from 65.45% to about 60.22%, not increase. The loss adjustment is \(14.4\times0.92=13.248\), not a different amount.
A coverage factor should not be invented merely because the target coverage differs. If no credible quantitative relationship is supplied or supportable, the appropriate conclusion may be that the historical ratio is not directly transferable, rather than applying an arbitrary adjustment.
Check the Premium Denominator
A ratio’s denominator must match the premium amount to which the selected ratio will later be applied. Premium may be described in different ways for different analyses. Before accepting a candidate, identify what is included in the historical denominator and what is included in the intended premium basis. Do not treat billed totals, earned premium, and a specified rating premium as interchangeable without a stated reason.
Worked Example: Excluding a Non-Risk Charge From the Basis
A hypothetical policy group has 7.2 million of ultimate losses and 12.0 million of earned premium, excluding a separate 0.6 million administrative charge. A candidate is needed for a premium basis that also excludes administrative charges. Calculate the comparable ratio and explain why using the larger billed total would be inconsistent.
Method: match the denominator to the stated premium basis. Because the target basis excludes the administrative charge, use the 12.0 million earned premium. The ratio is \(7.2/12.0=0.60\), or 60.0%.
The billed total including the charge would be \(12.0+0.6=12.6\) million. Dividing by that amount gives \(7.2/12.6=0.571428\ldots\), or 57.1% to one decimal place. That is not the comparable candidate here: it includes a charge in the denominator that the intended premium basis excludes.
Check: Applying 60.0% to the specified 12.0 million basis gives \(0.60\times12.0=7.2\) million. Common mistake: using all amounts collected as premium without checking the target definition. The lower 57.1% calculation is arithmetically correct for 12.6 million, but it answers a different basis question.
Make a Defensible Prospective Selection
A candidate expected loss ratio is a prospective assumption, so the selection should be explained in terms of both numerical evidence and applicability. Historical ultimate ratios are a natural starting point because they include an estimate of losses for the period rather than only the amount paid or reported to date. However, a historical ultimate estimate is still an estimate, and the resulting ratio may reflect period-specific variation.
A disciplined selection process asks:
Name the intended coverage, portfolio, and premium basis. A ratio without these qualifications is incomplete.
Calculate period ratios and an aggregate indication where the underlying losses and premiums are comparable. Note whether an unusual period or data limitation materially affects the indication.
Determine whether the historical experience matches the target coverage and premium basis. Apply a supplied, justified adjustment when appropriate; do not conceal a mismatch by selecting more decimal places.
State a candidate at sensible precision and give its evidence and limitations. A rounded candidate near the aggregate indication may be reasonable, but the data and the intended application must support it.
The selection need not equal the latest year, the simple average, or the aggregate indication in every situation. Each is a different summary. The aggregate ratio is generally the direct answer to “what was total ultimate loss divided by total premium?” The simple average answers “what was the average of the period ratios with equal weight per period?” A prospective selection may consider both, but should state which evidence it emphasizes and why.
This selection is not a substitute for examining whether historical ultimate losses are on the correct loss scope or whether material changes make the experience nonrepresentative. If a candidate is later used to estimate expected losses, multiplying it by a premium amount is appropriate only when that premium amount is on the candidate’s stated basis.
Check Your Understanding
For each question, calculate the requested indication and keep the coverage and premium basis in view.
- A hypothetical portfolio has ultimate losses and earned premiums of 3.6 and 6.0 million, 5.6 and 8.0 million, and 4.8 and 8.0 million in three successive periods. Calculate the three annual ratios and the aggregate indication. (Answer: 60.0%, 70.0%, and 60.0%; aggregate \(14.0/22.0=63.6\%\).)
- Historical ultimate losses are 9.6 million on a premium basis of 16.0 million. Target coverage is expected to produce 105% of historical-coverage losses, with premium unchanged. What is the target-coverage ratio? (Answer: \(9.6/16.0=60.0\%\); \(60.0\%\times1.05=63.0\%\).)
- A hypothetical group has 8.4 million ultimate losses, 14.0 million earned premium excluding fees, and 1.0 million in fees. The target premium basis excludes fees. What ratio is comparable, and what ratio would result from incorrectly including fees? (Answer: comparable ratio \(8.4/14.0=60.0\%\); including fees gives \(8.4/15.0=56.0\%\), which uses the wrong basis for the stated target.)
- Three periods have ratios of 55%, 65%, and 75%, with premiums of 4, 8, and 8 million, respectively. Calculate the simple average and aggregate indication. (Answer: simple average \(65.0\%\); aggregate \((2.2+5.2+6.0)/20=67.0\%\).)
- A selected expected loss ratio is 61.5% on the stated premium basis. What expected ultimate loss does it imply for 18.0 million of premium on that same basis? (Answer: \(0.615\times18.0=11.07\) million.)