Why Directly Blending Class Relativities Can Mislead

Why the Order of Calculation Matters

A class relativity is a ratio, not a loss cost. A credibility-weighted loss cost combines two amounts on a common dollar scale; a relativity then compares the resulting indicated amounts. Blending relativities directly can instead combine ratios formed with different reference bases. Those operations need not give the same result.

This distinction matters when class experience is sparse and an actuary uses a collective complement. A direct blend can look reasonable while implying a different relationship among classes from the relationship in the credibility-weighted loss costs. The key questions are what each relativity is divided by, whether those denominators are the same, and whether the credibility weights are applied consistently. This comparison is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.

Definition: For class $j$, let $L_j$ be its observed loss cost, $C_j$ its collective complement loss cost, and $z_j$, with $0\leq z_j\leq1$, its credibility weight. The credibility-weighted loss-cost indication is $I_j=z_jL_j+(1-z_j)C_j$. A relativity is a loss cost divided by a stated reference amount. The reference amount must be identified to interpret or compare relativities.

The formula for $I_j$ assumes the observed and complement loss costs are expressed on the same basis: for example, the same coverage, loss adjustment, trend and development basis, and exposure unit. If they are not comparable, blending them does not repair the mismatch. Credibility weights also need to be the weights selected for the intended class loss-cost estimate; they are not automatically transferable to a ratio with a different denominator.

Two Different Calculations

Suppose class $j$ is compared with reference class $b$. Blending costs first gives the indicated relativity $R_j=I_j/I_b$, where both class indications are calculated on the stated common loss-cost basis. By contrast, a direct blend might combine the observed relativity $L_j/L_b$ and collective relativity $C_j/C_b$, using the target class weight $z_j$. The first ratio uses two credibility-weighted costs; the second averages two ratios.

R_j=\frac{z_jL_j+(1-z_j)C_j}{z_bL_b+(1-z_b)C_b} \qquad\text{whereas}\qquad D_j=z_j\frac{L_j}{L_b}+(1-z_j)\frac{C_j}{C_b}.

The expressions have different denominators. $R_j$'s denominator is the blended loss cost for the reference class. In $D_j$, the observed and collective ratios use their respective reference-class loss costs before they are blended. In addition, $D_j$ uses the target class weight $z_j$ for both ratios; it does not account for how the reference class itself was blended. Consequently, no general identity makes $R_j=D_j$.

Key condition: If all observed and complement costs are converted to relativities using the same fixed positive base $K$, then blending the costs and converting to relativities gives the same result as blending the corresponding relativities with that class's weight: $I_j/K=z_j(L_j/K)+(1-z_j)(C_j/K)$. This is a linear rescaling by a common denominator. It does not generally hold when the observed and collective relativities use different bases, or when the relativity is formed against a reference class whose own indicated cost is blended.

Why a Common Fixed Base Works

The fixed-base result follows by substituting the loss-cost blend into the relativity. Since division by the same positive constant distributes over addition, the two components retain their original weights and become their respective relativities to $K$.

\begin{aligned} \frac{I_j}{K} &=\frac{z_jL_j+(1-z_j)C_j}{K}\\ &=z_j\frac{L_j}{K}+(1-z_j)\frac{C_j}{K}. \end{aligned}

This is a limited but useful equivalence. It requires the same $K$ in both component relativities, and the weight applied to each component must be $z_j$. If the observed table is divided by its own average and the collective table by a different average, the denominators are not a shared fixed base. If a reference class is blended and used as the denominator after the cost blend, the denominator itself depends on credibility. Those changes break the simple linear argument.

Worked Example: Different Reference-Class Costs

A hypothetical insurer compares a coastal class A with reference class B. Observed loss costs are $L_A=\$1{,}500$ and $L_B=\$1{,}000$; collective complements are $C_A=\$1{,}200$ and $C_B=\$800$. The credibility weights are $z_A=0.60$ and $z_B=0.20$. Calculate the ratio of indicated costs and compare it with a direct blend of the observed and collective A-to-B relativities. State why the answers differ.

Method: blend each class's loss cost first. For class A, $I_A=0.60(\$1{,}500)+0.40(\$1{,}200)=\$900+\$480=\$1{,}380$. For class B, $I_B=0.20(\$1{,}000)+0.80(\$800)=\$200+\$640=\$840$. Thus the indicated A-to-B relativity is $R_A=\$1{,}380/\$840=1.642857\ldots$.

Direct-blend calculation. The observed A-to-B relativity is $\$1{,}500/\$1{,}000=1.50$. The collective A-to-B relativity is $\$1{,}200/\$800=1.50$. Using A's weight, the direct blend is $D_A=0.60(1.50)+0.40(1.50)=0.90+0.60=1.50$.

The results differ: $1.642857\ldots$ versus $1.50$. The direct calculation blends ratios whose reference costs are $\$1{,}000$ and $\$800$; the indicated-cost ratio instead divides by B's blended amount of $\$840$. The calculation also uses B's own weight, $0.20$, in that blended denominator. The direct blend misses this movement in the reference-class cost.

Check. Both class indications lie between their observed and collective costs: $\$1{,}380$ is between $\$1{,}200$ and $\$1{,}500$, and $\$840$ is between $\$800$ and $\$1{,}000$. Their ratio is above 1, consistent with A's indicated cost exceeding B's. A common mistake is to report 1.50 as the indicated-cost ratio merely because both source relativities equal 1.50; the computed indicated costs show why that shortcut fails.

Normalization Does Not Generally Restore Equivalence

A further source of discrepancy occurs when observed and collective relativities have each been normalized to their own table average before blending. A separate scale factor has then been applied to each table. Blending the resulting values class by class does not, in general, reproduce the relativities obtained from blended loss costs. Normalizing a direct blend afterward may change its overall level, but it does not necessarily restore the class relationships in the indicated costs.

Worked Example: Blending Separately Normalized Tables

Two hypothetical classes, A and B, have equal exposure. Their observed loss costs are $L_A=\$2{,}000$ and $L_B=\$1{,}000$; the collective complement is $C_A=C_B=\$1{,}000$. Their weights are $z_A=0.50$ and $z_B=0.25$. Compare the class relativities from blended loss costs with the result of directly blending the separately normalized observed and collective relativities, then normalizing that direct blend to an equal-exposure average of 1.

Blend costs first. The indicated costs are $I_A=0.50(\$2{,}000)+0.50(\$1{,}000)=\$1{,}500$, and $I_B=0.25(\$1{,}000)+0.75(\$1{,}000)=\$1{,}000$. Their equal-exposure mean is $(\$1{,}500+\$1{,}000)/2=\$1{,}250$. The indicated relativities are therefore A: $\$1{,}500/\$1{,}250=1.20$; B: $\$1{,}000/\$1{,}250=0.80$. They have an equal-exposure average of 1.

Normalize the component tables, then blend. The observed mean is $(\$2{,}000+\$1{,}000)/2=\$1{,}500$, so its relativities are A: $1.333333\ldots$ and B: $0.666667\ldots$. The collective mean is $\$1{,}000$, so its relativities are $1.00$ for each class.

Apply each class's own credibility weight to its normalized components. For A, the direct blend is $0.50(1.333333\ldots)+0.50(1.00)=1.166667\ldots$. For B, it is $0.25(0.666667\ldots)+0.75(1.00)=0.916667\ldots$. The equal-exposure average of these direct blends is $(1.166667\ldots+0.916667\ldots)/2=1.041667\ldots$. After rescaling to average 1, A is $1.166667\ldots/1.041667\ldots=1.12$ and B is $0.916667\ldots/1.041667\ldots=0.88$.

Check. Both pairs of final relativities average to 1, but they differ: blended-cost relativities are 1.20 and 0.80, while the normalized direct blend gives 1.12 and 0.88. The direct procedure applied different base amounts to the observed and collective tables and then used unequal class weights. A shared average of 1 at the end cannot undo those class-specific differences.

When Direct Blending Is Safe—and When It Is Not

The exact fixed-base condition is often the clearest way to test a proposed calculation. If the observed and collective costs are each divided by the same fixed $K$, and each class's two resulting relativities are blended with that class's cost credibility $z_j$, the result equals $I_j/K$. If the intended answer is a ratio to another class's indicated cost, calculate both indicated costs and form the ratio afterward.

A direct blend using observed and collective class-to-reference ratios can happen to equal the ratio of indicated costs for particular data. That numerical coincidence is not a general method. Equality may occur if relevant weights and costs happen to align, but unless the proposed procedure satisfies a valid common-base identity, the analyst should not assume it. Verify equality from the definitions rather than relying on a plausible-looking relativity table.

Identify the intended output.
Decide whether the result is an indicated loss cost, a relativity to a fixed base, or a ratio to another class's indicated cost.

Check the bases.
Write the denominator for each observed and collective relativity. Different denominators mean the fixed-common-base equivalence does not apply.

Apply credibility on its intended scale.
For a cost indication, blend comparable costs using the class's selected weight. If a reference class is also being blended, calculate its indication with its own weight.

Form the requested relativity.
Divide the indicated costs, or divide by the stated common fixed base. Check that the resulting values answer the question asked.

Common exam trap. A solver sees observed and collective relativities and blends them using the target class's $z_j$, even though each table was normalized to a different base. That calculation may be a valid arithmetic average of two ratios, but it is not automatically the relativity of the credibility-weighted costs. The correction is to return to the dollar loss costs, blend each relevant class, and take the required ratio. If the problem explicitly specifies a common fixed base, direct blending is valid by the derivation above.

In practice, this distinction helps keep indicated class relationships interpretable. It also makes assumptions visible: whether the complement is comparable to the experience, which exposure basis supports the costs, and what base defines the relativity. The purpose is not to reject relativities, but to ensure that the scale on which credibility is applied matches the scale on which the final indication is defined.

Worked Example: A Common Fixed Base Gives the Same Answer

For two hypothetical classes, an actuary uses the fixed reference loss cost $K=\$1{,}000$. Class A has observed cost $L_A=\$1{,}800$, complement $C_A=\$1{,}200$, and $z_A=0.40$. Class B has $L_B=\$900$, $C_B=\$1{,}100$, and $z_B=0.70$. Calculate each relativity by both methods.

Blend costs, then divide by $K$. For A, $I_A=0.40(\$1{,}800)+0.60(\$1{,}200)=\$720+\$720=\$1{,}440$, so $I_A/K=\$1{,}440/\$1{,}000=1.44$. For B, $I_B=0.70(\$900)+0.30(\$1{,}100)=\$630+\$330=\$960$, so $I_B/K=\$960/\$1{,}000=0.96$.

Blend relativities to the same $K$. For A, the observed and collective relativities are 1.80 and 1.20. Their credibility blend is $0.40(1.80)+0.60(1.20)=0.72+0.72=1.44$. For B, they are 0.90 and 1.10; the blend is $0.70(0.90)+0.30(1.10)=0.63+0.33=0.96$.

Check. Both methods give A = 1.44 and B = 0.96. The agreement follows from dividing every component by the same $\$1{,}000$ before or after the weighted addition. No class-to-class denominator is being blended in this example.

Check Your Understanding

For each problem, identify the scale and reference base before deciding whether a direct blend represents the indicated-cost relativity.

A class has $L=\$2{,}100$, $C=\$1{,}500$, and $z=0.40$. Both are expressed relative to the same fixed base $K=\$1{,}200$. What is the relativity from blending costs first, and what is the direct blend? (Answer: Both are 1.45.)
Class A has $L_A=\$2{,}400$, $C_A=\$1{,}600$, $z_A=0.50$; reference class B has $L_B=\$1{,}200$, $C_B=\$1{,}000$, $z_B=0.25$. Find the ratio of indicated costs and the direct blend of A-to-B source relativities using $z_A$. (Answer: Indicated-cost ratio $=2{,}000/1{,}050=1.9048$; direct blend $=0.50(2.00)+0.50(1.60)=1.80$.)
Two equal-exposure classes have observed costs $\$1{,}800$ and $\$900$, collective costs $\$1{,}200$ each, and weights $z_A=0.25$, $z_B=0.75$. Find the relativities from blended costs, normalized to equal-exposure mean 1. (Answer: Indicated costs are $\$1{,}350$ and $\$975$; mean is $\$1{,}162.50$; relativities are 1.1613 and 0.8387.)
Using the data in question 3, normalize observed costs to their equal-exposure mean and collective costs to their equal-exposure mean; blend the resulting relativities by class weight, then normalize the blend to average 1. (Answer: Final direct-blend relativities are approximately 1.1818 for A and 0.8182 for B, not the blended-cost relativities in question 3.)
State the condition that guarantees direct blending of relativities equals blending costs first and then forming the relativity. (Answer: Both component costs must be divided by the same fixed positive base, and their corresponding relativities must use the same class-specific credibility weight.)

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