Balancing Rating Relativities

Why Rating Relativities Need Balancing

Proposed rating relativities describe how rates should vary from class to class. But even if each proposed factor seems reasonable on its own, the set of factors can raise or lower the total premium collected from the portfolio. A balancing procedure scales the factors so their premium-weighted average preserves the intended overall rate level.

The weights matter: a class contributing a large share of current premium has more influence on the portfolio-level result than a class contributing a small share. The calculation therefore uses the current premium distribution for the portfolio being rated. It is not generally correct to give every class equal weight or to use exposure counts as weights.

Definition: For class $j$, let $E_j$ be its current number of exposures, $R_j$ its current rate per exposure, and $P_j=E_jR_j$ its current premium. Let $r_j$ be a proposed rating relativity. The current premium weight is $w_j=P_j/\sum_{k=1}^K P_k$, so $\sum_{j=1}^K w_j=1$. The premium-weighted average proposed relativity is $\bar r_P=\sum_{j=1}^K w_jr_j$.

A relativity set is balanced to a neutral overall level when its premium-weighted average is one. To achieve that, divide every proposed relativity by $\bar r_P$. If instead the intended overall premium change is a factor $G$, scale every proposed relativity by $G/\bar r_P$, making the premium-weighted average equal to $G$.

r_j^{\text{bal}}=\frac{G}{\bar r_P}r_j \qquad\text{where}\qquad \bar r_P=\sum_{j=1}^K w_jr_j.

For neutral balancing, set $G=1$. Another common convention is to normalize the relativities to a weighted average of one and apply the overall rate change separately. Both conventions give the same final class rates when used consistently. Do not apply the overall change twice.

Why the Premium-Weighted Average Controls the Rate Level

Suppose the new rate for class $j$ is its current rate multiplied by a factor $b_j$. With exposures held fixed, the new portfolio premium is the sum of exposure times new rate across classes. The ratio of new premium to current premium is therefore the premium-weighted average of the class factors.

\frac{\sum_{j=1}^K E_jR_jb_j}{\sum_{j=1}^K E_jR_j} = \sum_{j=1}^K \frac{P_j}{\sum_{k=1}^K P_k}b_j = \sum_{j=1}^K w_jb_j.

This identity justifies the balancing formula. If the proposed factors $r_j$ have weighted average $\bar r_P$, multiplying each by $G/\bar r_P$ gives new factors whose weighted average is

\sum_{j=1}^K w_j\left(\frac{G}{\bar r_P}r_j\right) = \frac{G}{\bar r_P}\sum_{j=1}^K w_jr_j = G.

Thus, provided the exposures and class mix used for the weights remain fixed, the aggregate premium changes by exactly the intended factor $G$. This is a balancing identity, not a claim that the portfolio's actual future premium will change by that amount if exposures or mix change.

Balancing to Preserve the Current Overall Rate Level

Worked Example: Normalize Three Proposed Class Relativities

A hypothetical insurer has 100 exposures in class A at a current rate of $500, 50 in class B at $800, and 25 in class C at $1,200. Proposed relativities are 0.90, 1.10, and 1.30, respectively. Balance them to preserve the current total premium.

Method. Calculate the current premium weights, find the weighted average of the proposed relativities, and divide each proposed relativity by that average. The common mistake to avoid is taking the simple, unweighted average of the three factors.

Step 1: Calculate current class premiums. Class A has $100\times \$500=\$50{,}000$. Class B has $50\times \$800=\$40{,}000$. Class C has $25\times \$1{,}200=\$30{,}000$. Total current premium is $\$120{,}000$, giving weights $50/120$, $40/120$, and $30/120$, with premiums measured in thousands of dollars.

Step 2: Find the premium-weighted average proposed relativity.

\bar r_P =\frac{50(0.90)+40(1.10)+30(1.30)}{120} =\frac{45+44+39}{120} =\frac{128}{120} =\frac{16}{15} =1.0667.

Step 3: Balance each factor to a neutral overall level. Here $G=1$, so divide each proposed factor by $16/15$:

r_A^{\text{bal}}=0.90\left(\frac{15}{16}\right)=0.84375,\quad r_B^{\text{bal}}=1.10\left(\frac{15}{16}\right)=1.03125,\quad r_C^{\text{bal}}=1.30\left(\frac{15}{16}\right)=1.21875.

Check. The resulting premiums are $\$50{,}000(0.84375)=\$42{,}187.50$, $\$40{,}000(1.03125)=\$41{,}250.00$, and $\$30{,}000(1.21875)=\$36{,}562.50$. Their sum is $\$120{,}000.00$, exactly the current total premium. The common scaling changes the class distribution while preserving the portfolio total.

Balancing to a Specified Overall Change

The target need not be neutral. If the intended overall premium change is, for example, a 5% increase, use $G=1.05$. The balancing step then makes the weighted average of the final class factors 1.05, rather than one. This is useful when the class-level pattern is proposed separately from the portfolio-level indication.

Worked Example: Apply a Target Overall Increase

A hypothetical portfolio has current premium weights of 50%, 30%, and 20% for classes A, B, and C. Proposed class factors are 0.80, 1.10, and 1.40. The intended overall premium increase is 5%. Find the balanced factors.

Method. First calculate the weighted average of the proposed factors. Then multiply every factor by the ratio of the target $1.05$ to that average. This keeps the relative pattern among classes while setting the portfolio change to the target.

Step 1: Calculate the proposed weighted average.

\bar r_P=0.50(0.80)+0.30(1.10)+0.20(1.40) =0.40+0.33+0.28=1.01.

Step 2: Calculate the common balancing multiplier.

c=\frac{G}{\bar r_P}=\frac{1.05}{1.01}=\frac{105}{101}=1.039604.

Step 3: Apply the multiplier to each proposed factor.

r_A^{\text{bal}}=0.80\left(\frac{105}{101}\right)=\frac{84}{101}=0.831683, $$ $$ r_B^{\text{bal}}=1.10\left(\frac{105}{101}\right)=\frac{115.5}{101}=1.143564, \qquad r_C^{\text{bal}}=1.40\left(\frac{105}{101}\right)=\frac{147}{101}=1.455446.

Check. Using the exact fractions, the weighted average is \[ 0.50\left(\frac{84}{101}\right)+0.30\left(\frac{115.5}{101}\right)+0.20\left(\frac{147}{101}\right) =\frac{42+34.65+29.4}{101} =\frac{106.05}{101} =1.05. \] The balanced factors therefore produce the intended 5% increase under the stated fixed weights. The common mistake is to balance to one and then forget to include the intended 5% increase; the correction is to target $G=1.05$, or to apply that increase separately after neutral balancing, but not both.

Choose Weights That Match the Premium Basis

For a premium-weighted balancing calculation, weights are based on current premium, not simply on the number of policies, exposures, or classes. An exposure-weighted average is appropriate only if the intended quantity being balanced is itself exposure-weighted. When class rates differ, exposure weights and premium weights can produce materially different answers.

Worked Example: Avoid Substituting Exposure Weights

A hypothetical insurer has 120 exposures in class A at a rate of $250 and 40 exposures in class B at a rate of $750. Proposed factors are 0.80 and 1.40. Balance them to preserve current total premium.

Method. Calculate each class's current premium before weighting the proposed factors. The trap is to weight by the exposure counts, which would give class A three times the influence of class B even though the two classes currently contribute equal premium.

Step 1: Calculate current premiums and weights. Class A contributes $120\times \$250=\$30{,}000$. Class B contributes $40\times \$750=\$30{,}000$. Total premium is $\$60{,}000$, so both premium weights are $0.50$.

Step 2: Calculate the premium-weighted average proposed factor.

\bar r_P=0.50(0.80)+0.50(1.40)=0.40+0.70=1.10.

Step 3: Normalize to preserve total premium. Divide each proposed factor by 1.10:

r_A^{\text{bal}}=\frac{0.80}{1.10}=\frac{8}{11}=0.727273,\qquad r_B^{\text{bal}}=\frac{1.40}{1.10}=\frac{14}{11}=1.272727.

Check. The new class premiums are $\$30{,}000(8/11)=\$21{,}818.18$ and $\$30{,}000(14/11)=\$38{,}181.82$, rounded to cents. Together they equal $\$60{,}000.00$.

If exposure weights were incorrectly used, the proposed average would be $(120(0.80)+40(1.40))/160=0.95$. Normalizing by 0.95 would not preserve premium: the resulting premium-weighted average would be $(0.80/0.95+1.40/0.95)/2=1.1579$, approximately a 15.79% increase. The correct weights follow the premium basis being balanced.

Common Traps and Practical Checks

Balancing is a short calculation, but the interpretation and weights must be explicit. Before using the result, confirm the following:

Use current premium weights. For each class, compute $P_j=E_jR_j$ on the same exposure and rate basis used for the portfolio total.
State the target. A neutral relativity set has weighted average one; a set carrying an overall increase $G-1$ has weighted average $G$.
Do not mix conventions. Either balance the class factors directly to $G$, or balance to one and apply $G$ separately. Applying both approaches at once double-counts the overall change.
Check the weighted average after rounding. Use unrounded factors for the calculation and round only for presentation. Rounded displayed factors may not reproduce the target exactly.
Keep the portfolio basis in view. The balancing identity assumes the weights represent the portfolio to which the factors will be applied. If the exposure distribution changes, the weighted average may change as well.

The central exam technique is to translate the desired overall result into a weighted-average condition before calculating. Once the weights, raw factors, and target are clear, one common multiplier balances every class without changing the relative pattern of the proposed factors. This treatment is aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each hypothetical portfolio, use the stated current premium weights and target to find the balanced factors. Round factors to four decimal places where needed.

Two classes have current premium weights of 60% and 40%, with proposed factors 0.90 and 1.20. Balance to a neutral overall level. (Answer: 0.8824 and 1.1765)
Two classes have current premium weights of 25% and 75%, with proposed factors 0.80 and 1.20. The target overall change is a 10% increase. Find the balanced factors. (Answer: 0.8000 and 1.2000)
A portfolio has current premiums of $20,000, $30,000, and $50,000 in three classes. Proposed factors are 1.20, 0.90, and 1.10. Balance to a neutral overall level. (Answer: 1.1321, 0.8491, and 1.0377)
Two classes each contribute $40,000 of current premium. Their proposed factors are 0.70 and 1.30, and the intended overall increase is 4%. Find the balanced factors. (Answer: 0.7280 and 1.3520)
Three classes have current premium weights of 10%, 20%, and 70%, with proposed factors 0.60, 1.00, and 1.20. Balance to preserve total premium. (Answer: 0.5455, 0.9091, and 1.0909)

What You'll Learn