Using Selected Ultimate Losses in an Indication

Introduction

A loss ratio indication depends on two central selections: the losses expected from the experience period and the premium those policies would have earned at the current rate level. The loss estimate is not necessarily the amount paid to date, or even the amount currently reported. It is the selected ultimate loss: the estimate of the total loss cost for the experience period after all losses have been accounted for.

This tutorial uses selected ultimate losses with on-level earned premium to calculate a loss ratio indication. It focuses on the loss basis and its relationship to the indicated factor. It does not include trend adjustments; those are separate adjustments to align experience with the period for which rates are being set.

Definition: Selected ultimate losses are the selected estimates of total losses for the experience being analyzed, including losses already paid and the estimated amount remaining to be paid. On-level earned premium is the earned premium adjusted to the chosen current rate level. The loss ratio indication compares selected ultimate losses with on-level earned premium, then compares that ratio with the permissible loss ratio.

From Selected Losses to an Indicated Factor

Let $L$ denote selected ultimate losses, $P_{\text{OL}}$ denote on-level earned premium, and $r_p$ denote the permissible loss ratio. The adjusted loss ratio is $L/P_{\text{OL}}$. The permissible loss ratio represents the share of premium available for losses and loss adjustment expenses under the assumed expense and profit provisions. For the calculation below, the loss and expense items included in $L$ must be consistent with those included in $r_p$.

\text{Adjusted loss ratio}=\frac{L}{P_{\text{OL}}}

The indicated factor is the multiplicative change from the current rate level needed to bring the adjusted loss ratio to the permissible loss ratio. A factor above 1.000 indicates an increase; a factor below 1.000 indicates a decrease. The corresponding percentage change is the factor minus 1.

F=\frac{L/P_{\text{OL}}}{r_p} =\frac{L}{P_{\text{OL}}r_p}, \qquad \text{indicated percentage change}=F-1

The formula follows from the target condition. If rates are multiplied by $F$, then, under the indication’s assumption that the premium base scales proportionally with rates, the premium at the indicated level is $F P_{\text{OL}}$. The losses remain $L$ for this calculation. To meet the permissible loss ratio, the ratio of losses to premium at the indicated level must equal $r_p$:

\frac{L}{F P_{\text{OL}}}=r_p \quad\Longrightarrow\quad F=\frac{L}{P_{\text{OL}}r_p}

This justification also shows why the selected ultimate estimate determines the indication’s loss basis: changing $L$ changes the loss ratio and, all else equal, changes $F$ in the same proportion. The calculation assumes the selected losses are appropriate for the experience being measured, the on-level premium corresponds to that experience, and the permissible loss ratio is appropriate for the same loss and expense definition. It does not by itself adjust losses for a future trend period, changes in exposure mix, or other separately selected adjustments.

Applying the Calculation

The calculation is concise, but its inputs require care. Use the selected ultimate estimate rather than a partial measure such as paid losses or reported losses when the indicated method calls for ultimate losses. Use on-level earned premium rather than unadjusted historical earned premium. Both quantities must refer to the same experience base and be expressed on a consistent coverage and expense basis.

Confirm the loss selection.
Identify the selected ultimate losses for the experience under consideration and verify what loss and loss adjustment expense components they include.

Confirm the premium basis.
Use earned premium adjusted to the specified current rate level. Do not substitute raw earned premium if it has not been on-leveled.

Calculate the adjusted loss ratio.
Divide selected ultimate losses by on-level earned premium.

Calculate and interpret the factor.
Divide the adjusted loss ratio by the permissible loss ratio, then subtract 1 from the factor to express the indicated percentage change.

Worked Example: Calculating an Indication from a Selected Loss Total

A hypothetical insurer reviews one experience period and selects ultimate losses of $3,360,000 for one coverage and $840,000 for a second coverage. The two coverages are being analyzed together, and the selected losses and permissible loss ratio use consistent definitions. On-level earned premium is $6,000,000, and the permissible loss ratio is 0.650. Calculate the adjusted loss ratio, indicated factor, and indicated percentage change.

Method. Aggregate the selected ultimate losses, divide by on-level earned premium, and compare the resulting loss ratio with the permissible loss ratio.

Step 1: Aggregate selected ultimate losses.

L=\$3{,}360{,}000+\$840{,}000=\$4{,}200{,}000

Step 2: Calculate the adjusted loss ratio.

\frac{L}{P_{\text{OL}}} =\frac{\$4{,}200{,}000}{\$6{,}000{,}000} =0.700

Step 3: Calculate the indicated factor.

F=\frac{0.700}{0.650}=1.076923\ldots

Step 4: Express the result as a percentage change. The unrounded change is $1.076923\ldots-1=0.076923\ldots$. Rounded at the final step to three decimal places for the factor and one decimal place for the percentage change, the indicated factor is 1.077 and the indicated change is a 7.7% increase.

Check. At the indicated level, the premium base would be approximately $\$6{,}000{,}000(1.076923)=\$6{,}461{,}538$. Dividing the selected losses by that premium gives approximately $\$4{,}200{,}000/\$6{,}461{,}538=0.650$, the permissible loss ratio. The calculation uses the combined selected ultimate loss total, not just one coverage’s loss estimate.

Why the Selection Matters

The selected ultimate loss estimate is the numerator of the adjusted loss ratio. If the selected estimate increases while premium and the permissible loss ratio remain unchanged, the indicated factor increases proportionally. For example, a 5% increase in selected losses, with all other inputs fixed, produces a 5% increase in the indicated factor. This is why the loss selection is not a clerical detail: it directly determines the indication’s loss basis.

A selected ultimate estimate can differ from losses paid or reported to date because some claims may remain unpaid, unreported, or otherwise not fully reflected in the observed amount. The selected estimate should therefore be the ultimate amount chosen for the indication, not an intermediate figure simply because it is readily available. The method for developing and selecting that estimate is distinct from the calculation in this tutorial.

Worked Example: Using On-Level Premium with Selected Ultimate Losses

A hypothetical portfolio has selected ultimate losses of $4,000,000. Its recorded earned premium is $7,200,000, and the applicable on-level adjustment factor is 1.040. The permissible loss ratio is 0.600. Calculate the indication. Assume the stated adjustment factor applies to all of the earned premium in this example.

Method. First put premium on-level, then use that premium with selected ultimate losses. The adjustment factor applies to premium, not to losses.

Step 1: Calculate on-level earned premium.

P_{\text{OL}}=\$7{,}200{,}000(1.040)=\$7{,}488{,}000

Step 2: Calculate the adjusted loss ratio.

\frac{L}{P_{\text{OL}}} =\frac{\$4{,}000{,}000}{\$7{,}488{,}000} =0.5341880342\ldots

Step 3: Calculate the indicated factor and percentage change.

F=\frac{0.5341880342\ldots}{0.600} =0.8903133903\ldots

The percentage change is $0.8903133903\ldots-1=-0.1096866097\ldots$. Rounded at the final step to four decimal places for the factor and one decimal place for the percentage change, the indicated factor is 0.8903, or an indicated 11.0% decrease.

Check. The indicated premium base is approximately $\$7{,}488{,}000(0.8903133903)=\$6{,}666{,}667$. The selected loss ratio on that base is approximately $\$4{,}000{,}000/\$6{,}666{,}667=0.600$. The result is below 1 because the adjusted loss ratio, about 0.5342, is below the permissible ratio of 0.600.

Worked Example: The Effect of Selecting Ultimate Rather Than Reported Losses

A hypothetical specialty program has reported losses to date of $2,400,000, but the selected ultimate losses for the same experience period are $2,850,000. On-level earned premium is $4,750,000, and the permissible loss ratio is 0.625. Calculate the indication using the selected ultimate losses, and quantify the result that would arise from mistakenly using reported losses instead.

Method. The correct calculation uses the selected ultimate estimate. The reported-loss calculation is shown only to measure the effect of the mistake.

Step 1: Calculate the adjusted loss ratio using selected ultimate losses.

\frac{\$2{,}850{,}000}{\$4{,}750{,}000}=0.600

Step 2: Calculate the correct indicated factor.

F=\frac{0.600}{0.625}=0.960

The indicated percentage change is $0.960-1=-0.040$, so the correct result is an indicated 4.0% decrease.

Step 3: Show the mistaken calculation. If reported losses are substituted for selected ultimate losses, the apparent adjusted loss ratio is

\frac{\$2{,}400{,}000}{\$4{,}750{,}000} =0.505263\ldots

The mistaken indicated factor would be $0.505263\ldots/0.625=0.808421\ldots$, implying a decrease of approximately 19.2%. That is not the indication based on the selected ultimate estimate.

Check. The correct ultimate-loss calculation gives a factor of 0.960, and $\$2{,}850{,}000/(\$4{,}750{,}000(0.960))=0.625$. The much larger decrease from the mistaken calculation comes from omitting $450,000 from the loss basis. It does not reflect a different rate formula; it reflects using the wrong loss input.

Common Mistakes and Exam Traps

Treating an adjusted loss ratio as the indicated factor. The adjusted loss ratio is $L/P_{\text{OL}}$; the factor is that ratio divided by $r_p$. A loss ratio of 0.700 with a permissible ratio of 0.650 does not mean a 70% rate increase. It gives a factor of $0.700/0.650=1.077$, or about a 7.7% increase.
Using reported or paid losses when the problem supplies selected ultimate losses. This changes the numerator and can materially change the indication, as the third example shows.
Using raw earned premium instead of on-level earned premium. If the premium has not been adjusted to the specified rate level, the resulting loss ratio is on the wrong premium basis.
Applying the on-level factor to losses. On-leveling adjusts premium for rate level; it does not convert the selected ultimate loss estimate into a premium amount.
Subtracting the factor from 1 in the wrong order. The percentage change is $F-1$. A factor of 0.960 is a 4.0% decrease, not a 96.0% decrease.
Mixing inconsistent definitions. If selected losses include loss adjustment expenses but the permissible loss ratio is defined on a different basis, the inputs do not support a consistent indication until reconciled.
Adding an adjustment not included in the stated task. The formula here uses selected ultimate losses and on-level earned premium. A trend adjustment or another adjustment should not be added unless the analysis calls for it and specifies the relevant basis.

Key takeaway: A loss ratio indication uses the selected ultimate estimate as its loss basis and on-level earned premium as its premium basis. Divide losses by premium to obtain the adjusted loss ratio, divide by the permissible loss ratio to obtain the indicated factor, and subtract 1 to express the percentage change. The result is meaningful only when the selected losses, premium, and permissible ratio are consistent.

Check Your Understanding

For each problem, calculate the adjusted loss ratio and indicated factor before interpreting the percentage change. Round only at the final step.

A hypothetical book has selected ultimate losses of $1,620,000, on-level earned premium of $2,700,000, and a permissible loss ratio of 0.600. What is the indicated factor and percentage change? (Answer: adjusted loss ratio 0.600; factor 1.000; no indicated change.)
A hypothetical insurer selects $2,250,000 of ultimate losses for a segment. On-level earned premium is $3,000,000 and the permissible loss ratio is 0.625. What is the indicated change? (Answer: adjusted loss ratio 0.750; factor 1.200; 20.0% increase.)
For a hypothetical coverage, the selected ultimate loss components are $1,400,000 and $600,000. On-level earned premium is $3,200,000, and the permissible loss ratio is 0.500. Calculate the indication. (Answer: selected ultimate losses $2,000,000; adjusted loss ratio 0.625; factor 1.250; 25.0% increase.)
A hypothetical portfolio has earned premium of $5,000,000 before on-leveling. The on-level factor is 1.080, selected ultimate losses are $2,970,000, and the permissible loss ratio is 0.625. Calculate the indication using on-level premium. (Answer: on-level earned premium $5,400,000; adjusted loss ratio 0.550; factor 0.880; 12.0% decrease.)
A hypothetical program has selected ultimate losses of $4,160,000, on-level earned premium of $6,500,000, and a permissible loss ratio of 0.625. Calculate the indicated factor and percentage change, and state whether the result is an increase or decrease. (Answer: adjusted loss ratio 0.640; factor 1.024; 2.4% increase.)

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