Why Combine More Than One Experience Year?
A single experience year can be affected by random variation, unusual claims, or incomplete information. A multi-year indication uses results from several years to develop one view of the loss experience. The calculation is meaningful only when the losses and premiums have been made comparable: losses must represent a consistent level of development and a common cost level, while premiums must be stated at a consistent rate level.
The key is to prepare each year on the same basis before combining the years. Selected ultimate losses address the expected ultimate cost of claims from each experience period. Trend adjustments bring those losses to a common projection date. On-leveling adjusts earned premiums to the selected current rate level. These adjustments serve different purposes: trend changes the loss cost level over time, while on-leveling changes the premium rate level.
The \(T_j\) factors may be supplied directly or calculated from a trend assumption and the time from each year's loss measurement date to the common projection date. If the annual trend rate is \(t\) and the period is \(n\) years, the factor is \((1+t)^n\), provided the rate and time unit are consistent and the trend assumption applies over that period. A monthly trend rate, for example, must not be applied as though it were an annual rate.
The premium denominator is the sum of the experience premiums after rate-level adjustment, not the sum of premiums trended as if they were losses. The numerator and denominator must also represent the same coverage, exposure population, and loss components. For example, if the selected losses include loss adjustment expense, the intended comparison must use a permissible loss ratio and premium basis consistent with that treatment.
Why the Adjustments Must Be Consistent
A multi-year ratio is intended to compare the total projected losses from the selected experience with the premium that would be associated with those experience exposures at the chosen rate level. Start with the total-loss-to-total-premium definition of a ratio. For each year, the loss amount used in that total is its selected ultimate loss after trend to the common projection date; each premium amount is the earned premium after on-leveling to the common rate level. This gives:
The common projection date is essential. Suppose losses from one year are projected to one date and losses from another year to a later date. The numerator then combines dollars at different expected cost levels. Trending every year's losses to the same target removes that inconsistency. Similarly, premiums from different historical rate levels cannot be compared directly: on-leveling puts them on the selected rate basis.
The indicated factor follows from the loss ratio method. At the selected premium level, the adjusted losses are \(LR_{\text{multi}}\) times the on-level premium. If a rate change multiplies that premium by \(F\), the resulting expected loss ratio is \(LR_{\text{multi}}/F\). Set that ratio equal to the permissible loss ratio and solve:
This formula assumes the permissible loss ratio is appropriate to the proposed rate level and that the selected losses and premium have a consistent scope. It does not itself determine the selected ultimate losses, trend assumptions, or on-level factors; those inputs must be selected separately and applied consistently.
A Practical Calculation Sequence
Specify the coverage and loss components, the common loss projection date, the premium rate level, and the permissible loss ratio.
Use the selected ultimate loss for each experience year and apply that year's trend factor to the common projection date.
Apply the relevant on-level factor or method to put earned premium at the selected rate level.
Add adjusted losses and on-level premiums separately, divide the totals, and compare the resulting loss ratio with the permissible loss ratio.
Worked Example: Combining Two Years With Supplied Trend Factors
A hypothetical specialty equipment insurer is preparing an indication. For Year 1, selected ultimate losses are $840,000, the trend factor to the common projection date is 1.08, and on-level earned premium is $1,200,000. For Year 2, the corresponding amounts are $990,000, 1.04, and $1,350,000. The permissible loss ratio is 65%. Calculate the multi-year loss ratio and indicated rate change.
Method. Apply each supplied trend factor to that year's selected ultimate losses, then combine the adjusted losses and on-level premiums. The premium values are already on-level; do not apply the loss trend factors to them.
Year 1 adjusted losses are:
Year 2 adjusted losses are:
The total adjusted losses are \(\$907{,}200+\$1{,}029{,}600=\$1{,}936{,}800\). The total on-level premium is \(\$1{,}200{,}000+\$1{,}350{,}000=\$2{,}550{,}000\). Therefore:
The indicated factor is \(0.7595294/0.65=1.1685068\). The indicated rate change is \(1.1685068-1=0.1685068\), or a 16.85% increase when rounded to two decimal places. As a check, the adjusted loss ratio exceeds the permissible ratio, so an increase is indicated.
Common mistake avoided: Applying trend to premium as well as losses would change the denominator without justification. Trend the losses to the projection date; on-level the earned premium to the rate level.
Worked Example: Calculating Different Trend Periods to One Target
A hypothetical marine cargo program has two experience years. Year A has selected ultimate losses of $500,000 measured at July 1, 2022; Year B has selected ultimate losses of $700,000 measured at July 1, 2023. Use an annual loss trend rate of 6% and project both years to July 1, 2025. Their on-level premiums are $800,000 and $1,000,000, respectively. The permissible loss ratio is 70%. Calculate the indication.
Method. Calculate the elapsed annual trend period separately for each year's loss date. Year A is three years from its measurement date to the target date; Year B is two years. Both trend periods end on July 1, 2025.
For Year A, the factor is \(1.06^3=1.191016\), so adjusted losses are:
For Year B, the factor is \(1.06^2=1.1236\), so adjusted losses are:
Total adjusted losses are \(\$595{,}508+\$786{,}520=\$1{,}382{,}028\), and total on-level premium is \(\$800{,}000+\$1{,}000{,}000=\$1{,}800{,}000\). Thus:
The indicated factor is \(0.7677933/0.70=1.0968476\), giving an indicated increase of \(1.0968476-1=0.0968476\), or 9.68% to two decimal places. The check is consistent: the adjusted loss ratio is above 70%, so the indicated factor is above 1.
Common mistake avoided: Giving both years the same number of trend years merely because there are two experience years would be wrong. Each factor must reflect the time from that year's loss measurement date to the specified common target.
Check the Direction and the Basis
The arithmetic is usually straightforward once the inputs are aligned. Most errors arise before division: a trend factor may end at the wrong date, a premium may remain at an old rate level, or a premium may be trended instead of on-leveled. A useful audit is to write down the date represented by every adjusted loss and the rate level represented by every premium before adding any amounts.
| Item | Adjustment in this calculation | Question to check |
|---|---|---|
| Selected ultimate loss | Trend to common projection date | Do all years' adjusted losses represent the same target date? |
| Earned premium | On-level to selected rate level | Do all premium amounts represent the same rate level? |
| Loss ratio | Total adjusted losses divided by total on-level premium | Do numerator and denominator refer to the same coverage and experience? |
| Indicated factor | Multi-year loss ratio divided by permissible loss ratio | Is an above-permissible loss ratio producing an increase? |
For example, if a loss ratio is below the permissible ratio, the indicated factor is below 1 and the indicated change is a decrease. If it is above the permissible ratio, the factor exceeds 1 and the indicated change is an increase. This direction check will not detect every input error, but it often catches a reversed ratio or a sign error.
A further practical check is to calculate totals only after the year-by-year adjustments are visible. Retaining a row for each year makes it easier to trace an unusual factor or a large contribution to the total. Keep adequate precision through the calculations and round only the reported indication; early rounding can slightly alter the final result.
Worked Example: Auditing Trend Dates Before Combining
A hypothetical small commercial insurer has selected ultimate losses of $300,000 and $420,000 for two experience years. The actuarial analysis supplies trend factors of 1.10 and 1.06, respectively, and confirms that both factors bring losses to the same projection date. On-level premiums are $450,000 and $600,000. The permissible loss ratio is 68%. Calculate the indicated change and identify the error in an analyst's proposed method of adding the unadjusted losses.
Method. First use the supplied factors because they share a common target. Do not combine the raw selected losses and apply one average factor: the factors differ by year, and the appropriate operation is to adjust each year's loss amount separately.
The adjusted losses are \( \$300{,}000(1.10)=\$330{,}000\) and \(\$420{,}000(1.06)=\$445{,}200\). Their total is \(\$775{,}200\). Total on-level premium is \(\$450{,}000+\$600{,}000=\$1{,}050{,}000\). Then:
The indicated factor is \(0.7382857/0.68=1.0857143\), so the indicated change is 8.57% upward to two decimal places. As a check, 73.83% is greater than the 68% permissible ratio, and the factor is greater than 1.
The analyst's unadjusted-loss calculation would use \((\$300{,}000+\$420{,}000)/\$1{,}050{,}000=0.6857143\), producing only a 0.84% increase. It fails to project the losses to the common target date. The corrected approach applies each year-specific factor before summing.
Common mistake avoided: Summing losses before applying year-specific trend factors loses the distinction between the experience years' cost levels. Adjust each year's loss first, then total.
Check Your Understanding
For each problem, calculate the indicated change using the stated common projection basis. Report rate changes to two decimal places.
- Two years have selected ultimate losses of $240,000 and $360,000, trend factors of 1.10 and 1.05 to the same date, and on-level premiums of $400,000 and $600,000. If the permissible loss ratio is 65%, what is the multi-year loss ratio and indicated change? (Answer: 64.20% loss ratio; 1.23% decrease.)
- A hypothetical insurer has losses of $500,000 and $650,000 with trend factors of 1.08 and 1.04 to a common date. The corresponding on-level premiums are $700,000 and $900,000. Find the indicated change if the permissible loss ratio is 68%. (Answer: 11.76% increase.)
- For two years, selected ultimate losses are $400,000 and $500,000, common-date trend factors are 1.10 and 1.05, and on-level premiums are $600,000 and $700,000. Calculate the multi-year loss ratio and indicated change at a 70% permissible loss ratio. (Answer: 74.23% loss ratio; 6.04% increase.)
- Losses of $300,000 and $450,000 are measured three years and one year before the common projection date. The annual trend rate is 4%. On-level premiums are $500,000 and $650,000. With a 65% permissible loss ratio, calculate the indicated change. (Answer: 70.04% loss ratio; 7.75% increase.)
- Two years have selected ultimate losses of $200,000 and $300,000, common-date trend factors of 1.21 and 1.10, and on-level premiums of $400,000 and $500,000. Calculate the indicated change at a 70% permissible loss ratio. (Answer: 9.21% decrease.)