A Complete Loss Ratio Indication Calculation

From Experience to an Indicated Change

A loss ratio indication asks whether the premium at the current rate level is adequate for the losses expected at a future rate level. The comparison is meaningful only after putting losses and premium on consistent bases: losses must represent the expected cost in the prospective period, and premium must represent the experience exposures priced at a common rate level.

The calculation combines several adjustments. Earned premium is on-leveled to the current rate level. Experience losses are brought to ultimate, if they are not already ultimate, and trended from the experience period to the prospective period. The adjusted losses are then compared with the amount of premium required under the permissible loss ratio.

Definition: A loss ratio indication is the percentage change in the current rate level implied by comparing projected losses with the premium needed to support those losses at the permissible loss ratio. The loss measure and premium basis must be consistent, including the treatment of loss adjustment expenses (LAE).

For a single experience period, let $P$ be earned premium, $F_{\text{on-level}}$ the factor that adjusts that premium to the current rate level, $L$ the experience-period losses, $F_{\text{dev}}$ the factor to develop those losses to ultimate, and $F_{\text{trend}}$ the factor to trend them to the prospective period. Let $r_{\text{perm}}$ be the permissible loss ratio.

P_{\text{on-level}}=P F_{\text{on-level}}, \qquad L_{\text{projected}}=L F_{\text{dev}}F_{\text{trend}}

The indicated rate-level factor is projected losses divided by the on-level premium allowed for losses. Subtracting 1 converts that factor into a percentage change.

I=\frac{L_{\text{projected}}}{P_{\text{on-level}}r_{\text{perm}}}-1

If the result is positive, the indicated rate level is above the current level; if it is negative, the indication is for a decrease. The formula assumes the indicated change scales premium proportionally while the exposure and coverage basis is held constant. It is not a substitute for reconsidering those assumptions when a proposed change alters coverage, mix, or other rating inputs.

Why the Formula Works

The permissible loss ratio is the share of premium available for the selected loss measure. Thus, if projected losses are $L_{\text{projected}}$, the premium required to support them is $L_{\text{projected}}/r_{\text{perm}}$. The existing on-level premium is $P_{\text{on-level}}$. Their ratio is the factor by which that existing premium would have to change to provide the required premium.

\text{Required premium}=\frac{L_{\text{projected}}}{r_{\text{perm}}}

\text{Indicated factor} =\frac{\text{Required premium}}{P_{\text{on-level}}} =\frac{L_{\text{projected}}}{P_{\text{on-level}}r_{\text{perm}}}

A factor of 1.0747, for example, means the indicated rate level is 1.0747 times the current level, or an increase of 7.47%. This derivation also shows why the permissible loss ratio belongs in the denominator: it converts a loss amount into the premium needed to fund that loss amount.

Worked Indications

Worked Example: One Experience Period From Start to Finish

A hypothetical insurer has earned premium of $1,200,000. The experience-period average rate level was 0.96 relative to the current rate level of 1.00. Incurred losses, including the LAE being analyzed, are $720,000. The selected development-to-ultimate factor is 1.10. Loss trend is 5% per year from the experience-period midpoint to the prospective-period midpoint, two years later. The permissible loss ratio is 0.65. Calculate the indication.

Method. On-level premium to the current rate level, develop losses to ultimate, trend those losses to the prospective midpoint, and compare them with premium at the permissible loss ratio.

1. On-level the premium. The current rate level divided by the historical rate level gives the on-level factor:

F_{\text{on-level}}=\frac{1.00}{0.96}=1.0416667

Therefore, on-level earned premium is $1,200,000 $\times 1.0416667$ = $1,250,000.

2. Develop losses to ultimate. Developed losses are $720,000 $\times 1.10$ = $792,000.

3. Trend losses. With annual compound trend $t=0.05$ over $n=2$ years, the factor is $(1+t)^n=(1.05)^2=1.1025$. Projected losses are $792,000 $\times 1.1025$ = $873,180.

4. Calculate the premium required for losses. At a permissible loss ratio of 0.65, the on-level premium provides $1,250,000 $\times 0.65$ = $812,500 for losses.

5. Calculate and check the indication. The indicated factor is $873,180 / $812,500 = 1.074683. Subtracting 1 gives an indicated change of +7.47%, rounded to two decimal places. As a check, premium increased by this factor would provide approximately $873,180 $\div 0.65$ in total premium, the amount required to support the projected losses at the permissible ratio.

Common mistake avoided: The historical rate level is 0.96 of current, so the premium must be multiplied by $1.00/0.96$, not by 0.96. Multiplying by 0.96 would produce only $1,152,000 of on-level premium and a false indication of about +16.61%.

Worked Example: Combining Two Experience Years

For a hypothetical coverage, Year 1 has $800,000 of earned premium at a rate level of 0.90 of current, and $480,000 of incurred losses. Its development factor is 1.15 and its trend factor to the prospective period is 1.10. Year 2 has $1,200,000 of earned premium at a rate level of 0.96 of current, and $780,000 of incurred losses. Its development factor is 1.05 and its trend factor is 1.05. The permissible loss ratio is 0.60. Calculate the aggregate indication.

Method. Adjust each year on its own basis, then sum the on-level premiums and projected losses before forming the ratio. This preserves the premium weighting of the experience.

1. On-level each year's premium. For Year 1, the factor is $1/0.90=1.111111$, giving $800,000 $\times 1.111111$ = $888,888.89. For Year 2, the factor is $1/0.96=1.0416667$, giving $1,200,000 $\times 1.0416667$ = $1,250,000. Total on-level premium is $888,888.89 + $1,250,000 = $2,138,888.89.

2. Develop and trend each year's losses. Year 1 projected losses are $480,000 $\times 1.15 \times 1.10$ = $552,000 $\times 1.10$ = $607,200. Year 2 projected losses are $780,000 $\times 1.05 \times 1.05$ = $819,000 $\times 1.05$ = $859,950. Total projected losses are $607,200 + $859,950 = $1,467,150.

3. Calculate the indication. Premium available for losses is $2,138,888.89 $\times 0.60$ = $1,283,333.33. The indicated factor is $1,467,150 / $1,283,333.33 = 1.143234, so the indicated change is +14.32%, rounded to two decimal places.

Check and common mistake avoided: The indicated factor exceeds 1 because projected losses exceed the premium currently available for losses. Averaging the two years’ individual loss ratios or indication percentages would not, in general, equal this aggregate result. The calculation uses the total adjusted losses divided by the total on-level premium, not a simple average of yearly results.

Worked Example: Reading an Indication and a Selected Change

A hypothetical program has $1,000,000 of earned premium at the current rate level. Incurred losses are $800,000, the selected development factor is 1.05, and the trend factor to the prospective period is 1.21. The permissible loss ratio is 0.65. After reviewing the indication, the insurer selects a rate change of +20%. Calculate the indication, then determine the projected loss ratio at the selected change.

Method. First calculate the change implied by the permissible loss ratio. Then apply the selected change to the current premium and recompute the projected loss ratio. An indication is a benchmark; it is not automatically the selected change.

1. Project losses. Developed losses are $800,000 $\times 1.05$ = $840,000. Trended losses are $840,000 $\times 1.21$ = $1,016,400.

2. Calculate the indicated change. Premium available for losses at current rates is $1,000,000 $\times 0.65$ = $650,000. The indicated factor is $1,016,400 / $650,000 = 1.563692, so the indication is +56.37%, rounded to two decimal places.

3. Recalculate at the selected change. A selected +20% change produces premium of $1,000,000 $\times 1.20$ = $1,200,000. The resulting projected loss ratio is $1,016,400 / $1,200,000 = 0.8470, or 84.70%, to two decimal places as a percentage.

Check and common mistake avoided: The selected change is below the indicated change, so the projected loss ratio remains above the permissible 65%. Treating +20% as if it were the calculated indication would conceal this difference. This comparison is a mechanical check, not a conclusion about why the selection was made.

Checks, Assumptions, and Common Traps

A complete calculation depends on matching the underlying quantities. The earned premium should correspond to the experience exposures and the same coverage basis as the losses. Development should take the selected loss measure to ultimate; trend should project it from the experience period to the prospective period. Applying both factors is appropriate when the selected development factor and trend factor address those distinct adjustments. Do not apply either adjustment again if the input losses or factors already incorporate it.

When multiple experience periods are used, each period may require its own on-level, development, or trend factor. Adjust the period amounts first and then aggregate. A single ratio of aggregate amounts naturally gives more weight to periods with more premium than a simple average of period-level results.

Reversing the on-level factor. If historical rates are below current rates, the factor to current is greater than 1. The direction of the adjustment is a useful check.
Using the wrong trend interval. The exponent is the time between the selected experience and prospective trend anchors, not automatically the number of calendar years named in the data.
Applying the permissible ratio to the wrong amount. The premium allowed for losses is on-level premium multiplied by the permissible loss ratio.
Mixing loss bases. If the selected loss ratio basis includes LAE, the losses and permissible ratio must use a consistent treatment of LAE.
Calling the indication the selected change. The indication is the calculation’s result under its assumptions; the selected change may differ and should be evaluated on its own basis.

Key takeaway: On-level the earned premium, develop and trend losses to the prospective period, and compare aggregate projected losses with on-level premium multiplied by the permissible loss ratio. Subtract 1 from the resulting factor to express the indication as a percentage change.

Check Your Understanding

For each problem, calculate the requested indication or projected loss ratio. Assume the stated losses and factors use consistent bases.

A hypothetical book has $900,000 of earned premium at a historical rate level of 0.90 of current, and $540,000 of incurred losses. Use a development factor of 1.10, a trend factor of 1.05, and a permissible loss ratio of 0.65. What is the indication? (Answer: −4.05%)
A hypothetical account has $800,000 of current-level earned premium and $400,000 of incurred losses. Apply a 1.20 development factor and a 1.10 trend factor. If the permissible loss ratio is 0.60, what is the indication? (Answer: +10.00%)
For Year 1, earned premium is $600,000 at a rate level of 0.80 of current, losses are $300,000, and the development and trend factors are 1.20 and 1.10. For Year 2, earned premium is $900,000 at a rate level of 0.90 of current, losses are $500,000, and the factors are 1.10 and 1.04. With a permissible loss ratio of 0.64, calculate the aggregate indication. (Answer: −13.57%)
A hypothetical segment has $1,200,000 of on-level earned premium and $816,000 of projected losses. If variable expenses are 22% of premium and the profit-and-contingency provision is 4% of premium, calculate the permissible loss ratio and the indication. (Answer: Permissible loss ratio 0.74; indication −8.11%)
A hypothetical book has $1,000,000 of current-level earned premium, $700,000 of incurred losses, a development factor of 1.10, and a trend factor of 1.10. The permissible loss ratio is 0.70. If a +10% change is selected, what is the projected loss ratio at the selected level, and does that selected change reach the permissible loss ratio? (Answer: Projected loss ratio 77.00%; no.)

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