Matching Expected Loss Rates to the Experience Period

Why the Basis of an Expected Loss Rate Matters

An expected loss is meaningful only when its rate and exposure refer to compatible conditions. If an experience plan uses exposures from an earlier period but applies rates from a later rate level or classification system, the resulting benchmark may not represent the risks that generated those exposures. A calculation can multiply correctly and still be wrong because its inputs are on different bases.

This tutorial focuses on matching expected loss rates to the experience period’s rate level and classification basis. It does not adjust losses for changes in claim costs over time, claim development, or other changes in loss conditions. Those are distinct adjustments. In experience rating, plan rules determine which rate basis to use and which adjustments are prescribed. The method below is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.

Definition: A rate-level adjustment converts a rate from one premium rate level to another. A classification-basis adjustment converts a rate from one classification system or class definition to another. An expected loss calculation is basis-matched when each rate is expressed on the rate-level and classification basis required for the exposure to which it is applied.

Suppose $r_0$ is a rate at rate-level index $L_0$, and $L_1$ is the target index. If the rate levels differ proportionally and the plan prescribes index-based adjustment, the rate at the target level is

r_1=r_0\frac{L_1}{L_0}.

The direction matters. A rate at index 120 converted to index 100 is multiplied by $100/120$; converting from index 100 to 120 uses $120/100$. The index is a scale for the premium rate level, not a measure of claim-cost trend. A classification factor must likewise be applied in its stated direction: a factor of 1.05 that converts a source-class rate to a target-class rate means multiply by 1.05.

Set the Target Basis Before Calculating

Before calculating expected losses, identify the target basis specified by the plan. It might be the rate level and classification basis applicable during the experience period, or another basis explicitly required by the plan. Do not assume that “current rates” are automatically the correct rates for historical exposures.

Once the target is stated, apply the relevant prescribed adjustments to the rates, then multiply each adjusted rate by its corresponding exposure. Keep exposures separated whenever the rate level, classification, coverage, or another applicable rating basis differs. The basic calculation is

\text{Expected losses}=\sum_j e_j r_j^{\text{target}},

where $e_j$ is exposure on segment $j$, and $r_j^{\text{target}}$ is the rate matched to that segment and expressed on the required basis. As in the earlier material on calculating expected losses from rates and exposures, the product of rate and exposure gives an expected loss amount; here the key additional task is ensuring the rate is appropriate for the period and class.

State the target.
Record the rate-level index and classification basis that the plan requires for the calculation.

Break out the experience.
Separate exposures by time segment and class whenever their applicable rates or conversion factors differ.

Convert the rates.
Apply the rate-level ratio and any plan-prescribed classification factor in the direction from the source basis to the target basis.

Multiply and reconcile.
Multiply each matched rate by its corresponding exposure, sum the segment amounts, and check that no exposure was omitted or adjusted twice.

Worked Calculations

Worked Example: Converting a Current Rate to a Historical Rate Level

A hypothetical equipment-services account has 300 exposure units in its experience period. The current expected loss rate is $48 per exposure unit at rate-level index 125. The experience period is to be evaluated at index 100. The classification basis is unchanged. Find the expected losses on the experience-period rate-level basis.

Method. Convert the current rate to the target index before multiplying by the experience-period exposure. The plan’s index-based adjustment applies, and there are no class changes.

Step 1: Calculate the rate-level ratio. The target index is 100 and the source index is 125:

\frac{100}{125}=0.8.

Step 2: Convert the rate.

\$48(0.8)=\$38.40\text{ per exposure unit}.

Step 3: Calculate expected losses.

300(\$38.40)=\$11{,}520.

Check. The historical rate is 80% of the current rate because index 100 is 80% of index 125. The expected loss amount is therefore 80% of $300(\$48)=\$14{,}400$, or $\$11{,}520$. A common mistake is to use $125/100$, which converts in the opposite direction and gives an unjustifiably higher amount.

Worked Example: An Experience Period Spanning Two Rate Levels

A hypothetical repair contractor has 100 exposure units during a period at rate-level index 100 and 200 units during a later segment at index 110. The current expected loss rate is $60 per unit at index 120. The plan requires expected losses at each segment’s applicable rate level. The classification basis is unchanged. Calculate total expected losses.

Method. Convert the current rate separately to each segment’s rate level, then multiply by that segment’s exposures. Segment-specific calculation avoids treating all exposure as if it were written at one rate level.

Step 1: Convert the rate for the index-100 segment.

\$60\frac{100}{120}=\$50\text{ per unit}.

The expected losses for that segment are $100(\$50)=\$5{,}000$.

Step 2: Convert the rate for the index-110 segment.

\$60\frac{110}{120}=\$55\text{ per unit}.

The expected losses for that segment are $200(\$55)=\$11{,}000$.

Step 3: Add the segment amounts.

\$5{,}000+\$11{,}000=\$16{,}000.

Check. The exposure-weighted index for the period is $(100(100)+200(110))/300=106.6667$. Converting the current rate to that index gives $\$60(106.6667/120)=\$53.3333$ per unit; multiplied by 300, that is $\$16{,}000$. The weighted-index shortcut agrees here because the rate-level relationship is proportional and the same rate and classification basis apply throughout. Rounding is not needed for the total.

Worked Example: Applying Class-Specific Crosswalk Factors

A hypothetical account has 200 exposure units in source class M and 100 in source class N. The expected loss rates on the source basis are $40 for M and $60 for N. The plan prescribes factors converting each source-class rate to the target classification basis: 0.95 for M and 1.05 for N. Rate levels are already matched. Calculate expected losses on the target basis.

Method. Apply each class’s own source-to-target factor to its rate, then multiply by that class’s exposure. Do not apply one class’s factor to another class.

Step 1: Convert the M rate.

\$40(0.95)=\$38\text{ per unit}.

Expected losses for M are $200(\$38)=\$7{,}600$.

Step 2: Convert the N rate.

\$60(1.05)=\$63\text{ per unit}.

Expected losses for N are $100(\$63)=\$6{,}300$.

Step 3: Add the target-basis amounts.

\$7{,}600+\$6{,}300=\boxed{\$13{,}900}.

Check. The two adjusted rates are $38 and $63, and their exposure-weighted total is $200(\$38)+100(\$63)=\$13{,}900$. The classification factors do not change the exposure counts; they change the rates used for those exposures.

Why the Exposure-Weighted Rate Works

When several experience segments have different rate-level indices, one can either calculate expected losses segment by segment or, under specific conditions, first form an exposure-weighted rate-level index. The equivalence follows directly from the expected loss formula.

Assume that a common source rate $r_0$ applies to all segments, the rate at index $L_j$ is $r_0L_j/L_0$, and the exposure in segment $j$ is $e_j$. Assume also that classifications and all other rating conditions are unchanged, and that the plan calls for the rate level applicable to each segment. Then total expected losses are

\sum_j e_j r_0\frac{L_j}{L_0} =\frac{r_0}{L_0}\sum_j e_jL_j.

Let $E=\sum_j e_j$, and define the exposure-weighted index $\bar L=\sum_j e_jL_j/E$. Substituting $\sum_j e_jL_j=E\bar L$ gives

\sum_j e_j r_0\frac{L_j}{L_0} =E r_0\frac{\bar L}{L_0}.

Thus, applying the rate corresponding to the exposure-weighted index to total exposure gives the same result as calculating each segment separately. This is a computational shortcut, not a reason to discard segment detail. It does not generally apply if different classes have different source rates, if adjustment factors vary by segment beyond the stated indices, or if the plan prescribes a different basis.

Combining Adjustments and Avoiding Traps

If both the rate level and classification basis change, apply the plan-prescribed conversions to each applicable rate. When the adjustments are independent multiplicative factors, a source rate $r$ can be converted using

r^{\text{target}} =r\left(\frac{L_{\text{target}}}{L_{\text{source}}}\right)C,

where $C$ is specifically defined as the factor converting the source classification rate to the target classification rate. If the plan defines a factor in the reverse direction, use its reciprocal instead. When rate-level and class factors are not independent or the plan gives a prescribed combined conversion, follow that method rather than assuming the factors can be combined.

A particularly tempting error is to apply a single class factor to every class. In the crosswalk example, applying 0.95 to both rates would give

200(\$40)(0.95)+100(\$60)(0.95) =\$7{,}600+\$5{,}700 =\$13{,}300.

That result is wrong because the prescribed factor for class N is 1.05, not 0.95. The corrected N amount is $100(\$60)(1.05)=\$6{,}300$, so the correct total is $\$7{,}600+\$6{,}300=\$13{,}900$.

Reversing an index ratio: Write source and target indices beside the rate before forming the ratio. The target index belongs in the numerator.
Using a blended rate without the right weights: Rate levels should be weighted by the relevant exposure, not by the number of segments or by an unweighted average of indices.
Mixing class bases: Match each class’s exposure with its own rate and conversion factor, unless the plan explicitly calls for a class aggregation.
Adjusting the wrong item: Rate-level and classification conversions apply to rates as specified; do not also alter exposure or observed losses unless the plan calls for a separate adjustment.
Assuming a conversion: A rate-level index is not a substitute for plan rules. Confirm the target basis and prescribed factors before calculating.

Basis matching matters because expected losses serve as a benchmark in an experience plan. An inconsistent benchmark can distort the comparison between expected and actual experience, even when the arithmetic is otherwise flawless. State the basis, show each conversion, retain the segment detail, and perform a total check.

Check Your Understanding

For each problem, state or apply the required basis conversion before calculating expected losses.

A hypothetical policy has 250 exposure units. Its current rate is $72 at index 120, while the plan requires index 108. The class is unchanged. Find the expected losses. (Answer: $16,200)
An account has 80 units at index 90 and 120 units at index 105. A current rate of $40 applies at index 120. Find expected losses at each segment’s index and then total them. (Answer: $6,600)
On a matched rate level, source class A has 100 units at $50 and source class B has 50 units at $80. The prescribed source-to-target factors are 1.10 for A and 0.90 for B. Find target-basis expected losses. (Answer: $9,100)
Current rates at index 125 are $40 for class A and $70 for class B. An experience period at index 100 has 200 units in A and 100 in B. The source-to-experience-class factors are 1.05 for A and 0.95 for B. Apply both adjustments and calculate the total. (Answer: $12,040)
A rate of $90 applies at index 150. An experience period has 100 units at index 120 and 200 units at index 135. Calculate the expected losses using the two separate period segments, not an unweighted average index. (Answer: $23,400)

What You'll Learn