Worked Class Loss-Cost Indications With Multiple Complements

Why the Complement Matters

A credibility-weighted class indication combines a class’s observed loss cost with a complement: an estimate used for the portion of the indication not supported by the class’s own experience. The complement might be based on the full portfolio or on a broader group of classes judged more comparable to the class being rated. Those choices can produce different indications even when the class experience and credibility are unchanged.

The calculation is simple once the complement is defined. The actuarial work is to make its scope clear, calculate any pooled loss cost using the appropriate exposure weights, and interpret the difference without attributing more influence to the complement than the credibility formula gives it. The methods below apply to hypothetical class data on a consistent basis, such as losses adjusted to the same period and exposure measure. They produce loss-cost indications, not complete rates that include expenses or other provisions.

Definition: For class $j$, let $E_j$ be its exposure, $L_j$ its adjusted losses, $m_j=L_j/E_j$ its observed loss cost, and $Z_j$ its credibility weight, with $0\leq Z_j\leq1$. Given a specified complement $B$, the class’s credibility-weighted loss-cost indication is $$ I_j(B)=Z_jm_j+(1-Z_j)B. $$ The complement $B$ is the loss-cost estimate used for the portion $1-Z_j$ of the indication.

A complement is not defined just by a label such as “portfolio” or “group.” The actuary must specify which classes contribute, which losses and exposures are included, and whether the subject class is part of the benchmark. Unless a problem states otherwise, pooled loss costs are exposure-weighted: total adjusted losses divided by total exposure. An arithmetic average of class loss costs gives each class equal weight and generally does not equal the pooled loss cost.

Calculating and Comparing Complements

A portfolio-wide complement can be calculated by pooling the classes in the portfolio. A broader-group complement is calculated by pooling the defined group relevant to the class. For either benchmark, the basic calculation is

B=\frac{\sum_{k\in\mathcal{S}}L_k}{\sum_{k\in\mathcal{S}}E_k} =\frac{\sum_{k\in\mathcal{S}}E_km_k}{\sum_{k\in\mathcal{S}}E_k},

where $\mathcal{S}$ is the set of classes included in that complement. The set may include the subject class or exclude it; the choice should be stated and applied consistently. Including the subject class is common for a portfolio average, but it means that the class’s own experience also affects its complement. Excluding it creates a peer benchmark that is based only on other classes, when enough suitable peer data are available.

With the same class experience and credibility weight, the effect of changing complements follows directly by subtraction:

$$ I_j(B_G)-I_j(B_P)=(1-Z_j)(B_G-B_P), $$

where $B_G$ is the broader-group complement and $B_P$ is the portfolio-wide complement. The class experience term $Z_jm_j$ cancels. Thus a complement difference is not passed through in full: only the portion weighted by $1-Z_j$ changes the indication.

Worked Indications

Worked Example: Comparing Portfolio and Group Complements

A hypothetical insurer has four classes with adjusted losses and exposures on the same basis. Use the portfolio-wide average as one complement and the average of each two-class group as the other. The groups are Classes 1–2 and Classes 3–4. Use the supplied credibility weights to calculate both indications for each class, then compare them.

Class	Exposure	Observed loss cost	Adjusted losses	Credibility
1	100	$120	$12,000	0.20
2	200	$100	$20,000	0.40
3	300	$80	$24,000	0.60
4	400	$70	$28,000	0.80

Step 1: Calculate the portfolio complement. Total adjusted losses are $12{,}000+20{,}000+24{,}000+28{,}000=\$84{,}000$, and total exposure is $100+200+300+400=1{,}000$. Therefore,

B_P=\frac{\$84{,}000}{1{,}000}=\$84.

Step 2: Calculate the group complements. For Classes 1–2, losses total $\$12{,}000+\$20{,}000=\$32{,}000$ and exposure totals $300$, so $B_{G,1}=\$32{,}000/300=\$106.6667$. For Classes 3–4, losses total $\$52{,}000$ and exposure totals $700$, so $B_{G,2}=\$52{,}000/700=\$74.2857$. These group averages include the subject class.

Step 3: Calculate the indications and differences. For Class 1, the portfolio indication is $0.20(120)+0.80(84)=24+67.20=\$91.20$. Its group indication is $0.20(120)+0.80(106.6667)=24+85.3334=\$109.33$, rounded to cents. The difference, group minus portfolio, is $\$18.13$.

For Class 2, the portfolio indication is $0.40(100)+0.60(84)=40+50.40=\$90.40$. Its group indication is $0.40(100)+0.60(106.6667)=40+64=\$104.00$. The difference is $\$13.60$.

For Class 3, the portfolio indication is $0.60(80)+0.40(84)=48+33.60=\$81.60$. Its group indication is $0.60(80)+0.40(74.2857)=48+29.7143=\$77.71$, rounded to cents. The difference is $-\$3.89$.

For Class 4, the portfolio indication is $0.80(70)+0.20(84)=56+16.80=\$72.80$. Its group indication is $0.80(70)+0.20(74.2857)=56+14.8571=\$70.86$, rounded to cents. The difference is $-\$1.94$.

Check. The signs agree with the complement differences: the group complement is higher than the portfolio average for Classes 1–2 and lower for Classes 3–4. Each difference is the complement gap multiplied by $1-Z_j$, subject to rounding.

Method and mistake avoided. Pool losses and exposures before calculating each complement, then blend the selected complement with the class cost. Taking the unweighted mean of the two class costs in a group would give the wrong group complement when their exposures differ.

Worked Example: Measuring the Effect of Including the Class

In a hypothetical portfolio, Classes A, B, and C form a broader group; Class D is outside it. Their data are: A has exposure 100 and loss cost $100; B has exposure 100 and loss cost $50; C has exposure 300 and loss cost $70; D has exposure 500 and loss cost $60. Class A has credibility $Z_A=0.40$. Calculate Class A’s indication using (1) the full portfolio and broader-group averages that include A and (2) corresponding leave-A-out averages. Compare the two results.

Step 1: Calculate the inclusive complements. Losses are A: $100(100)=\$10{,}000$; B: $100(50)=\$5{,}000$; C: $300(70)=\$21{,}000$; and D: $500(60)=\$30{,}000$. Total portfolio losses are $\$66{,}000$ on exposure $1{,}000$, giving $B_P=\$66$. The A–B–C group has losses $\$36{,}000$ on exposure $500$, giving $B_G=\$72$.

Step 2: Apply the credibility weight. Using the inclusive portfolio complement gives $0.40(100)+0.60(66)=40+39.60=\$79.60$. Using the inclusive group complement gives $0.40(100)+0.60(72)=40+43.20=\$83.20$. The group-based indication is $\$3.60$ higher.

Step 3: Recalculate the complements without A. The leave-A-out portfolio has losses $5{,}000+21{,}000+30{,}000=\$56{,}000$ and exposure $900$, so its complement is $\$56{,}000/900=\$62.2222$. The leave-A-out broader group has losses $5{,}000+21{,}000=\$26{,}000$ and exposure $400$, so its complement is $\$65$.

The leave-A-out portfolio indication is $0.40(100)+0.60(62.2222)=40+37.3333=\$77.33$, rounded to cents. The leave-A-out group indication is $0.40(100)+0.60(65)=40+39=\$79.00$. Their difference is $\$1.67$, rounded to cents.

Check. Removing A lowers both complements because A’s loss cost exceeds both inclusive averages. Since the complement receives weight $0.60$, the indication also falls. The inclusion convention changes the benchmarks and must be disclosed when indications are compared.

Method and mistake avoided. Recalculate both losses and exposures after removing the subject class; do not subtract its loss cost directly from a pooled average. A pooled mean changes according to the class’s loss dollars and exposure.

Worked Example: Translating a Complement Change Into Dollars

A hypothetical class has an observed loss cost of $132, credibility $Z=0.60$, and exposure 250. The portfolio complement is $92 and the broader-group complement is $107. Calculate both loss-cost indications, the indicated aggregate loss dollars under each, and the change between them.

Step 1: Calculate the portfolio-complement indication. The complement weight is $1-0.60=0.40$. Thus $I(B_P)=0.60(132)+0.40(92)=79.20+36.80=\$116$.

Step 2: Calculate the group-complement indication. $I(B_G)=0.60(132)+0.40(107)=79.20+42.80=\$122$.

Step 3: Convert to aggregate indicated loss dollars. At exposure 250, the portfolio-complement indication gives $250(116)=\$29{,}000$. The group-complement indication gives $250(122)=\$30{,}500$. The increase is $\$30{,}500-\$29{,}000=\$1{,}500$.

Check. The complement gap is $\$107-\$92=\$15$. Its effect on the indicated loss cost is $0.40(15)=\$6$ per exposure; $250(6)=\$1{,}500$, matching the aggregate calculation.

Method and mistake avoided. Apply the complement change only to the noncredible portion, then multiply the per-exposure indication by exposure. Treating the entire $15 complement gap as the class’s indication change would overstate the effect.

Why the Difference Formula Holds

The comparison formula is a direct consequence of the credibility-weighted indication. Write the indication once using the group complement and once using the portfolio complement, keeping the observed class cost and credibility fixed:

\begin{aligned} I_j(B_G)-I_j(B_P) &=\left[Z_jm_j+(1-Z_j)B_G\right] -\left[Z_jm_j+(1-Z_j)B_P\right]\\ &=Z_jm_j-Z_jm_j+(1-Z_j)B_G-(1-Z_j)B_P\\ &=(1-Z_j)(B_G-B_P). \end{aligned}

The class experience cancels because it is identical in both calculations. If $Z_j=1$, the indication is entirely based on class experience and changing the complement has no effect. If $Z_j=0$, the indication is entirely the complement and its change passes through in full. For an intermediate credibility weight, the complement difference is scaled by $1-Z_j$.

Interpreting the Results and Avoiding Traps

The complement choice is a modeling and judgment decision, not a calculation that can be made from the class’s credibility weight alone. A broader group may offer a more relevant benchmark when its classes have similar risk characteristics, while a portfolio average may represent the collective level more broadly. The benchmark should be consistent with the intended rating basis and with the data’s adjustment and exposure definitions.

Do not confuse pooling with averaging class averages. Use total losses divided by total exposures unless the method explicitly calls for a different weighting.
State whether the subject class is included. Inclusion lets its experience affect both the observed class term and the complement; exclusion uses only the other included classes.
Keep the credibility weight attached to the class experience fixed when isolating complement effects. If $Z_j$ also changes, the difference is not explained by the complement formula alone.
Do not report the full complement gap as the indication change. The correct change is $(1-Z_j)(B_G-B_P)$.
Keep the scope of the result clear. These calculations indicate loss costs; they do not by themselves specify expenses, profit provisions, rate constraints, or a final filed rate.

A useful audit is to calculate the indication both directly and as a deviation from the complement:

$$ I_j(B)=B+Z_j(m_j-B). $$

Expanding the right-hand side gives $B+Z_jm_j-Z_jB=Z_jm_j+(1-Z_j)B$, the original formula. This form shows that credibility moves the complement toward the class experience by fraction $Z_j$. It also helps detect a reversed credibility weight: the class experience receives $Z_j$, not $1-Z_j$.

Key takeaway: A class indication is conditional on its complement. Calculate each proposed complement with its stated scope and exposure weights, then compare indications using $(1-Z_j)$ times the complement difference. Report the inclusion convention and the resulting effect by class.

Check Your Understanding

For each problem, state the complement calculation before applying credibility. Report dollar loss costs to the cent where needed.

A class has observed loss cost $80 and credibility 0.25. Its portfolio complement is $70 and its broader-group complement is $90. Calculate both indications and the group-minus-portfolio difference. (Answer: $72.50 and $87.50; difference $15.00.)
Two classes have losses of $18,000 on exposure 300 and $12,000 on exposure 100. Calculate their pooled loss cost and compare it with the unweighted average of their class loss costs. (Answer: pooled loss cost $75; unweighted average $90.)
A class has observed loss cost $110, credibility 0.60, and portfolio complement $90. A broader-group complement is $100. Calculate the two indications and the change in indicated loss cost. (Answer: $102 and $106; increase $4.)
A class with loss cost $75 and exposure 200 has credibility 0.40. Its inclusive portfolio complement is $65; the leave-class-out portfolio complement is $60. Calculate the indication under each convention and the indicated aggregate loss-dollar difference. (Answer: $69 and $66; decrease $600.)
A class has observed loss cost $140 and credibility 0.30. Its portfolio complement is $100, and its group complement is $120. Find both indications and verify the difference using the complement-difference formula. (Answer: $112 and $126; increase $14.)

What You'll Learn