Why the Complement Matters
A partially credible experience indication contains useful information about the risk being rated, but it is not the only relevant information. The complement supplies the portion of the indication that the subject experience cannot support by itself. Choosing that complement is therefore a substantive actuarial judgment: a mathematically correct credibility blend can still be misleading if its two inputs do not describe comparable risks on a consistent basis.
This tutorial focuses on selecting and applying the complement after the credibility weight has been determined. The experience indication should already reflect the applicable development, trend, premium, exposure, and loss adjustments. The complement must be suitable for the same prospective rate decision. The methods and judgments here are aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.
Let \(Z\) be the credibility assigned to the subject experience, where \(0 \leq Z \leq 1\). If \(I_E\) is the adjusted experience indication and \(I_C\) is the selected complementary indication, both expressed on the same basis, the indicated result is
The inputs need not be expressed in the same units as long as they are converted to a common, meaningful measure first. For example, loss ratios can be blended with loss ratios, and loss costs per exposure can be blended with loss costs per exposure. A rate level can be blended with another rate level when both refer to the same coverage and rating basis.
How to Select a Relevant Complement
A strong complement is relevant to the risk being rated and already reflects the basis on which the new rate is intended to apply. Begin with the subject class’s coverage, territory or risk characteristics, exposure definition, and prospective period. Then assess the candidate data or expected result against those features.
- Coverage: Check that the complement represents the same coverage or that any coverage difference has been adjusted. A different deductible or limit can change expected losses materially.
- Risk characteristics: Prefer a broader group that is reasonably comparable to the subject class. If it differs in mix, adjust for the difference where supportable.
- Time basis: Use a complement appropriate to the prospective period. Historical experience may need development and trend adjustments before it is relevant.
- Exposure and premium basis: Confirm that a rate, loss cost, or loss ratio uses the same unit and premium basis as the subject indication. Convert units explicitly rather than treating them as interchangeable.
- Included components: Match the components included in the complement to those included in the experience indication. For example, do not blend a loss-only measure with an indication that includes loss adjustment expenses unless the difference is handled.
A complement from a larger group is not necessarily better merely because it has more observations. A large but poorly matched group may be less informative than a smaller, well-adjusted one. Conversely, a subject class’s own older experience is not automatically the best complement: it may be stale, based on a different coverage, or not projected to the period being rated.
State the coverage, exposure unit, measure being indicated, and prospective period.
Reject or adjust candidates that differ materially in risk mix, coverage, time period, or included components.
Express the complement and experience indication in the same units, such as dollars per exposure or loss ratios.
Apply \(Z\) to the experience indication, \(1-Z\) to the complement, and translate the result into a rate level or rate change as required.
Blending Rate Levels and Rate Changes
Suppose the current rate is \(R_0\), and the experience and complementary indications are rate levels \(R_E\) and \(R_C\), respectively. If both refer to the same current rate base, the indicated rate level is
The indicated percentage change is \(R/R_0-1\). The same result can be obtained by blending the percentage changes, but only when both changes use that same \(R_0\) as their base. To see why, write \(R_E=R_0(1+c_E)\) and \(R_C=R_0(1+c_C)\), where \(c_E\) and \(c_C\) are the respective changes. Then
The cancellation depends on the common rate base. If one indication is a percentage change from a different rate level, convert it to the target rate basis before blending. This is one reason to state exactly what each indication represents instead of combining percentages without checking their denominators.
Worked Example: Choosing Between Candidate Rate Complements
A hypothetical coverage has a current rate of $1,200 per exposure. After the appropriate adjustments, its experience indication is $1,380 per exposure, with credibility \(Z=0.35\). Two proposed complements are available: a prospective indication of $1,260 per exposure from a broader group with comparable coverage and exposure, and an older $1,050 indication for the subject class that has not been brought to the prospective period. Select a complement and calculate the indicated rate and rate change.
Method and selection. Use the $1,260 complement. It is stated on the same per-exposure basis, is prospective, and comes from a group with comparable coverage. The $1,050 figure has not been adjusted to the prospective period, so it is not ready to use as a complement on this basis.
The complement weight is \(1-Z=1-0.35=0.65\). The weighted experience component is
The weighted complementary component is
Thus, the indicated rate is \(483+819=\$1{,}302\) per exposure. The change from the current rate is
Check. The indicated rate lies between $1,260 and $1,380, as it must for a weighted average with weights between zero and one. The calculated 8.5% increase is also \(1{,}302-1{,}200=\$102\), divided by $1,200.
Common mistake avoided. Using the unadjusted $1,050 complement would give \(0.35(1{,}380)+0.65(1{,}050)=\$1{,}165.50\), implying a decrease. That calculation is arithmetically correct but uses a complement that has not been put on the prospective basis. It should not be treated as a valid indication for this decision.
Complementing a Loss Ratio Indication
A complement can be expressed as an expected loss ratio when the experience indication is also expressed as a loss ratio. The loss ratio is losses divided by the corresponding premium on the stated basis. When the final objective is a rate change, compare the credibility-weighted loss ratio with the target or permissible loss ratio specified for that indication. The premium and loss bases must be consistent.
This ratio is meaningful only when the weighted loss ratio and target loss ratio use compatible definitions of losses, premium, coverage, and period. A target loss ratio is part of the stated ratemaking assumptions in the example; the formula does not make an incompatible target suitable.
Worked Example: Blending an Expected Loss Ratio
For a hypothetical class, adjusted prospective losses are $540,000 and the matching on-level premium is $900,000. The experience indication therefore has a loss ratio of 0.60. Its credibility is \(Z=0.40\). A comparable prospective group supports a complementary loss ratio of 0.64, and the target loss ratio for this indication is 0.60. Calculate the indicated rate change.
Method. Blend the subject and complementary loss ratios first, then compare that result with the target loss ratio.
The experience loss ratio is
The complement weight is \(1-0.40=0.60\). The weighted experience component is \(0.40(0.60)=0.24\), and the weighted complementary component is \(0.60(0.64)=0.384\). Therefore, the credibility-weighted loss ratio is
Relative to the target loss ratio, the indicated rate factor is \(0.624/0.60=1.04\). The indicated rate change is \(1.04-1=0.04\), or a 4.00% increase.
Check. The blended loss ratio of 0.624 is between the experience ratio of 0.60 and the complement of 0.64. Since it exceeds the target of 0.60, the indicated change is positive. Using the target as the denominator is essential: subtracting the target from the blended ratio would produce a difference in loss-ratio points, not the percentage rate change.
Converting a Complement to the Subject Exposure Basis
A complement can be relevant in substance but stated per a different exposure unit. In that case, convert it before blending. The conversion should follow from the exposure definitions in the problem, not from an informal assumption that the units are equivalent.
Worked Example: Adjusting a Pure Premium Complement to the Target Unit
A hypothetical class has a current rate of $840 per policy exposure. Its adjusted experience loss cost is $540 per policy exposure, with credibility \(Z=0.40\). A comparable portfolio provides a prospective loss cost of $480 per vehicle-year. Each policy exposure in the target class represents 1.1 vehicle-years. The target loss ratio is 0.60. Calculate the indicated rate and rate change.
Method. Convert the complement from dollars per vehicle-year to dollars per policy exposure, blend it with the experience loss cost, and divide by the target loss ratio to obtain the indicated rate.
The complement on the target exposure basis is
The complement weight is \(1-0.40=0.60\). The weighted experience loss cost is \(0.40(\$540)=\$216\), and the weighted complementary loss cost is \(0.60(\$528)=\$316.80\). The blended loss cost is
Under the stated target loss ratio, the indicated rate is
Relative to the current rate of $840, the indicated change is
Check. The blended loss cost of $532.80 is between $528 and $540, and the indicated rate of $888 exceeds the current $840. The final percentage is rounded to two decimal places.
Common mistake avoided. Blending the unconverted $480 directly with $540 would give \(0.40(\$540)+0.60(\$480)=\$504\) per policy exposure and an indicated rate of \(\$504/0.60=\$840\), or no change. This is invalid because dollars per vehicle-year and dollars per policy exposure are different units. The stated 1.1 conversion is what makes the complement comparable.
How the Complement Influences the Result
For fixed \(Z\), the indicated result is pulled toward the complement in proportion to \(1-Z\). If the complement is above the experience indication, the blended indication is above the experience indication; if it is below, the blend is below it. The amount of the pull is
This identity follows by subtracting \(I_E\) from \(I=ZI_E+(1-Z)I_C\):
It makes the selection consequential even when the calculation is straightforward. A complement close to the experience indication will have little effect on the result; a distant complement can move it substantially, especially when experience credibility is low. This is a reason to document why a proposed complement is relevant, rather than selecting one merely because it is readily available.
Common errors include using an unadjusted historical result as though it were prospective, confusing the complement weight with the experience weight, blending figures stated per different exposure units, and reporting a loss-ratio-point difference as a percentage rate change. For every calculation, label the basis and units of both indications, show that the weights sum to one, and check that the blend lies between its two inputs.
Check Your Understanding
For each problem, select or use the stated complement, show the basis conversion if needed, and check whether the resulting indication is between its two blended inputs.
- A hypothetical class has a current rate of $900 per exposure, an adjusted experience indication of $930, and credibility \(Z=0.20\). A comparable prospective group supports a complement of $880 per exposure. Calculate the indicated rate and percentage change. (Answer: $890 per exposure; a 1.11% decrease.)
- A subject class has an adjusted experience loss ratio of 0.72 and credibility \(Z=0.25\). A comparable prospective group supports a complementary loss ratio of 0.66, and the target loss ratio is 0.60. Calculate the indicated rate change. (Answer: 12.50% increase.)
- A class has a current rate of $650 per policy exposure and an experience loss cost of $490 per policy exposure, with \(Z=0.40\). A complement is $520 per vehicle-year, and each policy exposure represents 0.9 vehicle-years. The target loss ratio is 0.70. Calculate the indicated rate and rate change. (Answer: $681.14 per policy exposure; a 4.79% increase.)
- A class’s adjusted experience indication is 1.08 times its current rate, and \(Z=0.30\). Candidate A is a prospective indication of 1.02 times the same current rate for comparable coverage. Candidate B is a raw historical indication of 0.96 times the current rate that has not been projected to the prospective period. Which complement is suitable on the stated information, and what rate factor results? If the current rate is $750, give the indicated rate and change. (Answer: Candidate A; rate factor 1.038; $778.50, a 3.80% increase.)
- A subject experience indication is $1,100 per exposure and has credibility \(Z=0.15\). A well-matched prospective complement is $1,020 per exposure. Calculate the blended indication and the amount by which it differs from the experience indication. (Answer: $1,032 per exposure; $68 below the experience indication.)