Why Adjustments Must Be Combined
A historical loss ratio compares losses with the premium earned during an experience period. It is not automatically a suitable measure of the relationship between expected losses and premium at a future rate level. Loss costs may change over time, while the rates underlying earned premium may also change. A loss ratio indication therefore needs to place losses and premium on the intended, consistent bases before comparing them.
For the combined adjustment considered here, trend losses to the prospective period and on-level premium to the selected rate level. Then divide total adjusted losses by total adjusted premium. The factors do different jobs: a loss trend factor changes the time basis of losses, while an on-level factor changes the rate-level basis of premium. Their values and directions should be determined from the experience and prospective periods, not chosen merely because a factor is greater or less than one.
This setup assumes that the supplied losses are the amounts to which the stated loss trend applies, and that the earned premium has not already been adjusted by the stated on-level factors. Losses and premium must also be defined on compatible coverage and experience bases. If a component has already been adjusted to the required basis, do not apply the same adjustment again.
From an Adjusted Loss Ratio to a Rate Change
The adjusted loss ratio describes the loss cost supported by the adjusted premium. To turn it into a rate indication, compare it with the permissible loss ratio, the share of premium available for losses after the applicable provisions for expenses, profit, and contingencies. In the simple ratio method, these provisions are expressed as proportions of premium.
For example, an adjusted loss ratio of 0.78 against a permissible loss ratio of 0.70 means that the adjusted losses are 0.78 per dollar of adjusted premium, while the target is 0.70. The indicated rate-level factor is \(0.78/0.70=1.1143\), or an indicated increase of 11.43%. The percentage change is not the difference between the two ratios; it is their relative difference.
Why the Combined Formula Works
For each period, trending changes the loss amount from \(L_j\) to \(L_jT_j\), and on-leveling changes the earned premium from \(P_j\) to \(P_jO_j\). Across periods, the appropriate overall loss ratio compares the sum of the adjusted loss dollars with the sum of the adjusted premium dollars:
This ratio-of-sums follows from the meaning of a loss ratio: total losses divided by total premium. Averaging period loss ratios equally would give the same weight to a small and a large period, even when their premium volumes differ. The ratio of sums instead weights each period through its adjusted premium.
The same logic explains why, when factors differ by period, it is generally not correct to apply one average trend factor and one average on-level factor to the unadjusted totals. The adjusted dollars must be calculated period by period unless the relevant factors are identical or a valid weighting calculation has been established.
Worked Example: One Experience Period and a Target Ratio
A hypothetical insurer has $4,800,000 of losses and $6,000,000 of earned premium for an experience period. The losses must be trended by a factor of 1.05, and earned premium must be on-leveled by a factor of 1.08. The permissible loss ratio is 0.70. Calculate the adjusted loss ratio and indicated rate change.
Method. Adjust the losses and premium separately, then form their ratio. This avoids applying the premium on-level factor to losses or treating a historical loss ratio as if it were already on the prospective basis.
First, trend the losses: $$ \$4{,}800{,}000 \times 1.05 = \$5{,}040{,}000. $$
Next, on-level the premium: $$ \$6{,}000{,}000 \times 1.08 = \$6{,}480{,}000. $$
The adjusted loss ratio is $$ \frac{\$5{,}040{,}000}{\$6{,}480{,}000} =0.777777\ldots \approx 0.7778. $$
Compare this with the permissible loss ratio: $$ \text{Indicated rate-level factor} = \frac{0.777777\ldots}{0.70} =1.111111\ldots, $$ $$ \text{Indicated rate change} = 1.111111\ldots-1 =0.111111\ldots. $$
Rounded to two decimal places as a percentage, the indicated change is an 11.11% increase. Check: multiplying the permissible ratio by the indicated factor gives \(0.70\times1.111111\ldots=0.777777\ldots\), the adjusted loss ratio.
Worked Example: Combining Periods With Different Factors
A hypothetical insurer has two experience periods. In Period 1, losses are $2,400,000 and earned premium is $3,000,000; the loss trend factor is 1.12 and the premium on-level factor is 1.10. In Period 2, losses are $3,600,000 and earned premium is $4,500,000; the corresponding factors are 1.04 and 1.02. If the permissible loss ratio is 0.72, calculate the combined adjusted loss ratio and indicated rate change.
Method. Apply the period-specific factors before aggregating. This preserves the different adjustments for the two periods.
Period 1 adjusted losses and premium are $$ \$2{,}400{,}000\times1.12=\$2{,}688{,}000, \qquad \$3{,}000{,}000\times1.10=\$3{,}300{,}000. $$
Period 2 adjusted losses and premium are $$ \$3{,}600{,}000\times1.04=\$3{,}744{,}000, \qquad \$4{,}500{,}000\times1.02=\$4{,}590{,}000. $$
Total adjusted losses are $$ \$2{,}688{,}000+\$3{,}744{,}000=\$6{,}432{,}000, $$ and total adjusted premium is $$ \$3{,}300{,}000+\$4{,}590{,}000=\$7{,}890{,}000. $$
Therefore, $$ \text{Adjusted loss ratio} = \frac{\$6{,}432{,}000}{\$7{,}890{,}000} =0.815209\ldots \approx 0.8152. $$ The indicated rate-level factor is $$ \frac{0.815209\ldots}{0.72}=1.132235\ldots, $$ so the indicated rate change is \(1.132235\ldots-1=0.132235\ldots\), or a 13.22% increase rounded to two decimal places.
Check: \(0.72\times1.132235\ldots=0.815209\ldots\). The calculation avoids assigning one period’s adjustment factors to the other period.
Worked Example: Already On-Level Premium and a Two-Year Trend
A hypothetical portfolio has $1,200,000 of losses and $1,500,000 of earned premium. The losses are to be projected forward two years at an annual compound trend rate of 4%. The premium is explicitly stated to be already on the selected rate level. The permissible loss ratio is 0.75. Calculate the adjusted loss ratio and indicated rate change.
Method. Compound the annual loss trend for two years. Since the premium is already on-level, its on-level factor is 1.00; applying an additional premium adjustment would change the stated basis incorrectly.
The two-year loss trend factor is $$ 1.04^2=1.0816. $$
Trended losses are $$ \$1{,}200{,}000\times1.0816=\$1{,}297{,}920. $$ Premium on the selected rate level remains $$ \$1{,}500{,}000\times1.00=\$1{,}500{,}000. $$
The adjusted loss ratio is $$ \frac{\$1{,}297{,}920}{\$1{,}500{,}000}=0.86528. $$ The indicated rate-level factor is $$ \frac{0.86528}{0.75}=1.153706\ldots, $$ so the indicated rate change is \(1.153706\ldots-1=0.153706\ldots\), or a 15.37% increase rounded to two decimal places.
Check: \(0.75\times1.153706\ldots=0.86528\). The common mistake this avoids is on-leveling premium a second time even though the problem states that it is already at the selected rate level.
Common Mistakes and Interpretation
A useful way to check a combined calculation is to label every factor with both its subject and its direction: for example, “losses, experience to prospective” and “premium, earned level to selected level.” The factor should be applied to the amount it adjusts, not just to whichever number appears next in the calculation.
- Reversing the ratio. The loss ratio is losses divided by premium, not premium divided by losses.
- Applying the wrong factor. Loss trend adjusts losses; the on-level factor adjusts premium.
- Using a simple average of period ratios. The combined ratio is total adjusted losses divided by total adjusted premium.
- Adjusting an already adjusted amount again. An amount already on the required basis should not receive the same adjustment a second time.
- Subtracting ratios to obtain a rate change. The indicated change is adjusted loss ratio divided by permissible loss ratio, minus one.
- Mixing bases. A numerator and denominator that refer to inconsistent periods or rate levels do not produce the intended prospective comparison.
The adjusted loss ratio is a diagnostic and an input to the indication, not a statement that every policyholder or class has that exact loss experience. The resulting rate change also depends on whether the permissible loss ratio and the underlying experience definitions are appropriate to the hypothetical indication.
Check Your Understanding
For each hypothetical indication, calculate the requested result. Round loss ratios to four decimal places and rate changes to two decimal places.
- A portfolio has losses of $750,000 and earned premium of $1,000,000. Apply a loss trend factor of 1.06 and a premium on-level factor of 1.04. If the permissible loss ratio is 0.70, what are the adjusted loss ratio and indicated rate change? (Answer: adjusted loss ratio 0.7644; indicated increase 9.20%.)
- Period 1 has losses of $1,100,000, premium of $1,400,000, trend factor 1.10, and on-level factor 1.05. Period 2 has losses of $1,500,000, premium of $2,000,000, trend factor 1.04, and on-level factor 1.02. Find the combined adjusted loss ratio and the indicated rate change if the permissible loss ratio is 0.72. (Answer: adjusted loss ratio 0.7892; indicated increase 9.61%.)
- A portfolio has $1,800,000 of losses and $2,400,000 of premium. Losses require a trend factor of 1.08, but premium is already on the selected rate level. If the permissible loss ratio is 0.75, calculate the adjusted loss ratio and indicated rate change. (Answer: adjusted loss ratio 0.8100; indicated increase 8.00%.)
- A hypothetical insurer has $900,000 of losses and $1,200,000 of earned premium. The loss trend factor is 1.05 and the premium on-level factor is 1.10. Calculate the adjusted loss ratio and indicated rate change against a permissible loss ratio of 0.70. (Answer: adjusted loss ratio 0.7159; indicated increase 2.27%.)
- Period 1 has $400,000 of losses and $500,000 of premium, with trend and on-level factors of 1.20 and 1.00. Period 2 has $900,000 of losses and $1,500,000 of premium, with factors of 1.00 and 1.20. Calculate the combined adjusted loss ratio and indicated rate change against a permissible loss ratio of 0.65. (Answer: adjusted loss ratio 0.6000; indicated decrease 7.69%.)