Applying Credibility to Classification Relativities

From Credibility-Weighted Costs to Classification Relativities

A classification estimate is not automatically a relativity. Credibility blending produces an estimated cost for each class by combining its observed experience with a complement. A relativity expresses that class estimate relative to a selected overall level. To make the factors usable in a rating plan, the selected factors are commonly normalized to have a weighted average of one under a specified portfolio mix.

The target mix matters. A set of class estimates can produce different normalized relativities under different class weights, even though none of the underlying estimates changed. This tutorial develops the calculation and shows how to check that the final factors satisfy the stated average.

Definition: For class $j$, let $\bar X_j$ be its observed loss cost on a consistent per-exposure basis, $Z_j$ its credibility, and $m_{0j}$ its complement. The credibility-weighted estimate is $\hat m_j=Z_j\bar X_j+(1-Z_j)m_{0j}$. For target portfolio shares $q_j$, where $q_j\ge 0$ and $\sum_j q_j=1$, the target-mix average estimate is $M_q=\sum_j q_j\hat m_j$. The normalized relativity is $r_j=\hat m_j/M_q$.

The credibility factor $Z_j$ is between zero and one. When $Z_j=1$, the estimate is the class’s observed cost; when $Z_j=0$, it is the complement. For intermediate credibility, the estimate lies between those two values. In Bühlmann-Straub applications, the credibility factor reflects the class’s experience weight and the model’s variance components; here, use the factor supplied or calculated for each class.

The costs being blended must be comparable: for example, they should represent the same coverage, period basis, loss definition, and exposure unit. The target shares $q_j$ describe the portfolio mix for which the factors are to average to one. They are not credibility weights. Credibility determines each class estimate; target shares determine the normalization average.

Why Normalization Works

The desired condition is that the target-share-weighted average relativity equals one. Dividing every class estimate by the same target-mix average achieves that condition:

\sum_j q_jr_j =\sum_j q_j\frac{\hat m_j}{M_q} =\frac{\sum_jq_j\hat m_j}{M_q} =\frac{M_q}{M_q} =1.

This is also why the denominator must be calculated from the credibility-weighted estimates using the specified target shares. Using the raw observed costs, an unweighted average, or a different portfolio mix generally makes the weighted average of the resulting factors unequal to one.

Blend each class estimate.
Calculate $\hat m_j=Z_j\bar X_j+(1-Z_j)m_{0j}$, keeping the observed cost and complement on the same basis.

Apply the target mix.
Check that the stated shares sum to one, then calculate $M_q=\sum_jq_j\hat m_j$.

Calculate and check relativities.
Set $r_j=\hat m_j/M_q$, then verify $\sum_jq_jr_j=1$, using unrounded factors for the check.

Worked Calculations

Worked Example: Two Classes and a Specified Portfolio Mix

A hypothetical insurer estimates loss costs for two vehicle-use classes. Class A has observed cost $\$120$, credibility $0.60$, and complement $\$90$. Class B has a completed estimate of $\$80$. For the target portfolio, the shares are $q_A=0.40$ and $q_B=0.60$. Calculate the credibility-weighted class estimates and normalized relativities. Give relativities to four decimal places.

Method. Blend Class A’s observed cost with its complement, calculate the average estimate under the target shares, and divide each class estimate by that average.

Class A estimate. Its complement weight is $1-0.60=0.40$, so

\hat m_A=0.60(120)+0.40(90)=72+36=\$108.

Class B’s estimate is $\hat m_B=\$80$. The target shares sum to $0.40+0.60=1.00$. Thus the target-mix average estimate is

M_q=0.40(108)+0.60(80)=43.20+48.00=\$91.20.

The relativities are

r_A=\frac{108}{91.20}=1.1842105\ldots\approx1.1842, \qquad r_B=\frac{80}{91.20}=0.8771929\ldots\approx0.8772.

Check. Using unrounded factors, $0.40(108/91.20)+0.60(80/91.20)=91.20/91.20=1.0000$. The factors therefore average to one under the specified mix.

Common mistake avoided. The denominator is not $\$88$; that value does not equal $0.40(\$108)+0.60(\$80)$. Using the incorrect denominator would produce factors whose target-weighted average is not one. The correct average is $\$91.20$.

Worked Example: Three Classes with Different Credibilities

A hypothetical property program has three construction classes. The observed costs are $\$140$, $\$90$, and $\$60$; their respective credibilities are $0.50$, $0.40$, and $0.25$. For this calculation, each class uses the same portfolio complement of $\$100$. The target shares are $0.20$, $0.30$, and $0.50$, respectively. Find the blended estimates and normalized relativities, reporting factors to four decimal places.

Class 1. The complement weight is $1-0.50=0.50$:

\hat m_1=0.50(140)+0.50(100)=70+50=\$120.

Class 2. Its complement weight is $1-0.40=0.60$:

\hat m_2=0.40(90)+0.60(100)=36+60=\$96.

Class 3. Its complement weight is $1-0.25=0.75$:

\hat m_3=0.25(60)+0.75(100)=15+75=\$90.

The target shares sum to $0.20+0.30+0.50=1.00$. The target-mix average is

M_q=0.20(120)+0.30(96)+0.50(90) =24+28.80+45 =\$97.80.

Divide each estimate by $\$97.80$:

r_1=\frac{120}{97.80}=1.2269939\ldots\approx1.2270, \quad r_2=\frac{96}{97.80}=0.9815951\ldots\approx0.9816, \quad r_3=\frac{90}{97.80}=0.9202454\ldots\approx0.9202.

Check. With unrounded factors, the weighted sum is $(0.20(120)+0.30(96)+0.50(90))/97.80=97.80/97.80=1.0000$. The displayed factors are rounded; a check performed only with those rounded values may differ slightly in the last decimal.

Common mistake avoided. Do not give every class the same credibility simply because the complement is common. Each class’s own $Z_j$ determines its blend. Also, the target shares are applied after blending, when calculating the portfolio average.

Worked Example: The Target Mix Changes the Relativities

Two hypothetical classes have already been credibility-weighted, producing estimates of $\$120$ for Class A and $\$80$ for Class B. Calculate relativities first for a target mix with shares $0.25$ and $0.75$, then for a target mix with shares $0.50$ and $0.50$. Report factors to four decimal places and explain what changes.

First mix. The target average is $M_{q,1}=0.25(120)+0.75(80)=30+60=\$90$. Thus $r_{A,1}=120/90=1.3333$ and $r_{B,1}=80/90=0.8889$. Using unrounded factors, $0.25(120/90)+0.75(80/90)=90/90=1.0000$.

Second mix. The target average is $M_{q,2}=0.50(120)+0.50(80)=60+40=\$100$. Thus $r_{A,2}=120/100=1.2000$ and $r_{B,2}=80/100=0.8000$. The check is $0.50(1.20)+0.50(0.80)=0.60+0.40=1.0000$.

The estimates and the ratio of the two class estimates have not changed: $120/80=1.5$. The normalized factors do change because each target mix selects a different overall level. A common mistake is to reuse factors normalized to an old portfolio mix without checking the intended average under the new mix.

Interpretation and Exam Traps

A factor greater than one indicates an estimated class cost above the target-mix average; a factor below one indicates a cost below that average. These comparisons are relative, not absolute. A class’s factor can change when the target mix changes even if its credibility-weighted estimate stays fixed.

The normalization condition is about the portfolio average of the factors, weighted by the specified shares. It does not say that the unweighted arithmetic mean of the factors must equal one. Nor does it require the factors to have a particular total. For example, with two classes and unequal shares, a factor for the more heavily weighted class has more influence on the average.

Do not blend totals with per-exposure costs. The observed class experience and complement must use compatible measures and exposure bases.
Do not confuse credibility weights with portfolio shares. $Z_j$ blends a class’s experience and complement; $q_j$ weights class estimates in the target portfolio average.
Do not normalize before blending. First calculate the credibility-weighted class estimates, then obtain their target-mix average.
Do not round early. Keep unrounded estimates and factors for the weighted-average check, and round only the final reported factors.
Do not omit the target mix. A statement that factors “average to one” is incomplete unless the weights defining that average are known.

Key takeaway: Credibility determines each class estimate, while the selected portfolio mix determines the normalization level. Divide every credibility-weighted estimate by the target-share-weighted average of those estimates, then verify that the same target shares give a weighted average relativity of one.

Check Your Understanding

For each problem, calculate the credibility-weighted estimates before normalizing. Use the target shares stated in the problem and retain unrounded values for the normalization check.

A hypothetical equipment program has two classes. Class A has observed cost $\$150$, credibility $0.40$, and complement $\$100$; Class B’s completed estimate is $\$80$. Target shares are $0.30$ and $0.70$. Find both relativities to four decimals. (Answer: estimates $\$120$ and $\$80$; average $\$92$; factors $1.3043$ and $0.8696$.)
Three hypothetical classes have observed costs $\$160,\$100,\$70$, credibilities $0.50,0.20,0.60$, and respective complements $\$100,\$100,\$80$. Target shares are $0.20,0.50,0.30$. Find the blended estimates, the target average, and factors to four decimals. (Answer: estimates $\$130,\$100,\$74$; average $\$98.20$; factors $1.3238,1.0183,0.7536$.)
Two completed class estimates are $\$105$ and $\$75$, with target shares $0.40$ and $0.60$. Find the target average and normalized relativities to four decimals, and verify the weighted average. (Answer: average $\$87$; factors $1.2069$ and $0.8621$; weighted average $1.0000$ using unrounded factors.)
Two class estimates are $\$140$ and $\$70$. A candidate normalizes them using equal weights, but the specified target shares are $0.25$ and $0.75$. Calculate the correct target average and factors to four decimals. (Answer: average $\$87.50$; factors $1.6000$ and $0.8000$; their target-weighted average is $1.0000$.)
Three hypothetical classes have credibility-weighted estimates of $\$110,\$90,\$60$, with target shares $0.10,0.40,0.50$. Find the target average and all three factors to four decimals. (Answer: average $\$77$; factors $1.4286,1.1688,0.7792$; weighted average $1.0000$ using unrounded factors.)

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