Rebasing Relativities to a Chosen Base Class

Why Rebase Classification Relativities?

Classification relativities express how the rates for different classes compare. A set of factors may be normalized so its weighted portfolio average is 1.000, or it may be expressed relative to a chosen base class, whose factor is 1.000. These are different reference points for the same underlying class comparisons.

Rebasing changes the reference point, not the relative differences among classes. If the rate for a selected base class is known, factors relative to that class make it direct to calculate the other class rates. The central operation is simple—divide every factor by the factor for the selected base class—but its meaning, weighting implications, and rounding require care.

Definition: Let $r_j$ be the factor for class $j$, and let $b$ denote the chosen base class. The factor for class $j$ rebased to class $b$ is $$ r_j^{(b)}=\frac{r_j}{r_b}. $$ Thus $r_b^{(b)}=1$. The factors must be on a common multiplicative scale, and $r_b$ must be positive.

A set of factors is normalized to a portfolio average when a specified set of portfolio weights $w_j$ satisfies $\sum_j w_j=1$ and $\sum_j w_j r_j=1$. The weights represent the chosen averaging basis, such as exposure shares. The normalization says that the weighted average factor is 1 on that basis; it does not say that every class has factor 1.

Rebasing does not generally preserve that weighted-average normalization. Instead, $$ \sum_j w_j r_j^{(b)} =\sum_j w_j\frac{r_j}{r_b} =\frac{1}{r_b}\sum_j w_jr_j =\frac{1}{r_b}, $$ when the original factors have weighted average 1. That result is expected: after rebasing, the selected class—not the portfolio average—is set to 1.

Why Pairwise Comparisons Do Not Change

A class factor is meaningful through comparison with another class. Before rebasing, the ratio of the factor for class $j$ to the factor for class $k$ is $r_j/r_k$. After rebasing, both factors have been divided by the same positive number $r_b$, so that common scale cancels:

\frac{r_j^{(b)}}{r_k^{(b)}} = \frac{r_j/r_b}{r_k/r_b} = \frac{r_j}{r_k}.

This is the key justification for rebasing: it changes the units in which the factors are expressed, but not the underlying class-to-class relationship. The result assumes the rating plan is multiplicative, with each class rate equal to a common base rate multiplied by its class factor. It does not apply as a conversion between factors in a plan where classes are combined through additive adjustments or other nonmultiplicative rules.

Identify the existing factor scale.
Confirm that all class factors use the same reference and that the factors are positive. If they are portfolio-average normalized, identify the weights used for that normalization.

Choose the base class.
Record its existing factor $r_b$. The class may have any positive factor on the original scale; it becomes 1 after rebasing.

Divide every factor by the base factor.
Calculate $r_j^{(b)}=r_j/r_b$ for each class, including the base class.

Check the result.
Verify the base factor is 1 and compare selected pairwise ratios before and after rebasing. Use unrounded values for the check when available.

Apply the factors to the base-class rate.
For a multiplicative plan, the rate for class $j$ is the selected base-class rate multiplied by $r_j^{(b)}$.

Worked Applications

Worked Example: Rebasing Portfolio-Average Factors to the Lower-Factor Class

A hypothetical two-class program has portfolio-average-normalized factors $r_A=7/11$ and $r_B=13/11$. The portfolio weights are $w_A=1/3$ and $w_B=2/3$. Rebase the factors to class A, then verify the class comparison. Show what happens if the original factors are first displayed to four decimal places.

Method. Divide both portfolio-average factors by the factor for the chosen base class, $r_A=7/11$.

Step 1: Check the original normalization.

\sum_j w_jr_j =\frac{1}{3}\frac{7}{11}+\frac{2}{3}\frac{13}{11} =\frac{7}{33}+\frac{26}{33} =1.

Step 2: Rebase to class A.

r_A^{(A)}=\frac{7/11}{7/11}=1, \qquad r_B^{(A)}=\frac{13/11}{7/11}=\frac{13}{7}\approx 1.8571.

The exact ratio of class B to class A is $13/7$, so the comparison is unchanged. The weighted average of the rebased factors is $ (1/3)(1)+(2/3)(13/7)=11/7$, which is also $1/r_A$, not 1.

Step 3: Check the displayed rounded values. To four decimal places, the original factors are $0.6364$ and $1.1818$. Their displayed quotient is $$ \frac{1.1818}{0.6364}\approx 1.8570, $$ whereas the exact ratio $13/7\approx1.8571$. The small difference is due to rounding the two original factors before dividing.

Check. The base factor is 1, and the exact class B-to-A comparison remains $13/7$. The common mistake is to treat the quotient of rounded displayed factors as exact; retain full precision for calculations and round only the reported result.

Worked Example: Converting to a Base Class and Calculating Rates

A hypothetical insurer has three classes with portfolio-average factors $r_L=0.6$, $r_M=1.0$, and $r_H=1.4$, using weights $0.25$, $0.50$, and $0.25$, respectively. The insurer wants class H to be the base class, with a selected class H rate of $840 per exposure. Find the rebased factors and all three class rates, then verify the weighted average rate.

Method. First check the original weighted normalization. Then divide each factor by $r_H$, and multiply each rebased factor by the stated class H base rate.

Step 1: Verify the original factor average.

$$ 0.25(0.6)+0.50(1.0)+0.25(1.4) =0.15+0.50+0.35 =1.00. $$

Step 2: Rebase to class H. Since $r_H=1.4$,

r_L^{(H)}=\frac{0.6}{1.4}=\frac{3}{7}, \qquad r_M^{(H)}=\frac{1.0}{1.4}=\frac{5}{7}, \qquad r_H^{(H)}=\frac{1.4}{1.4}=1.

Step 3: Calculate the rates. Multiply each rebased factor by $840:

\begin{aligned} R_L&=840\left(\frac{3}{7}\right)=\$360,\\ R_M&=840\left(\frac{5}{7}\right)=\$600,\\ R_H&=840(1)=\$840. \end{aligned}

Step 4: Check the weighted average rate.

0.25(\$360)+0.50(\$600)+0.25(\$840) =\$90+\$300+\$210 =\$600.

The weighted average rate is $600, equal to the selected class H rate divided by its original factor: $840/1.4=\$600$. Also, the H-to-L rate ratio is $840/360=7/3$, equal to the original factor ratio $1.4/0.6=7/3$. The common mistake is to set the rebased base factor to 1 and then assume the weighted average factor must also be 1.

Worked Example: Changing the Reference Without Changing Class Rates

A hypothetical program has portfolio-average-normalized factors $r_A=0.8$, $r_B=1.0$, and $r_C=1.4$, with portfolio weights $0.50$, $0.25$, and $0.25$. The portfolio-average rate is $500 per exposure. Convert the factors to class C as the base, calculate the class rates under both presentations, and verify the results.

Method. Confirm the normalization and calculate the original class rates. Then rebase all factors to C and use C’s original rate as the new base-class rate.

Step 1: Verify the weighted factor average.

$$ 0.50(0.8)+0.25(1.0)+0.25(1.4) =0.40+0.25+0.35 =1.00. $$

Step 2: Calculate rates from the portfolio-average presentation.

R_A=\$500(0.8)=\$400,\qquad R_B=\$500(1.0)=\$500,\qquad R_C=\$500(1.4)=\$700.

The weighted average is $0.50(\$400)+0.25(\$500)+0.25(\$700)=\$200+\$125+\$175=\$500$, as required.

Step 3: Rebase to C and calculate rates again. Divide each factor by $1.4$, and use the class C rate of $700 as the base-class rate:

r_A^{(C)}=\frac{0.8}{1.4}=\frac{4}{7},\qquad r_B^{(C)}=\frac{1.0}{1.4}=\frac{5}{7},\qquad r_C^{(C)}=1.

R_A=\$700\left(\frac{4}{7}\right)=\$400,\qquad R_B=\$700\left(\frac{5}{7}\right)=\$500,\qquad R_C=\$700(1)=\$700.

The rates are identical under both presentations. For an additional pairwise check, $r_C/r_A=1.4/0.8=7/4$, while $r_C^{(C)}/r_A^{(C)}=1/(4/7)=7/4$. The common mistake is to retain the old $500 portfolio-average rate as the base rate after rebasing. The new base rate must be the rate for class C, $700.

Common Errors and Practical Checks

Rebasing is a change in reference class, not a second estimation of the relativities. It preserves the relative rating structure only when every factor is divided by the same base factor and the rate schedule applies those factors multiplicatively.

Dividing in the wrong direction. To make the chosen base factor equal 1, divide each factor by $r_b$, not multiply by $r_b$. Check that $r_b/r_b=1$.
Renormalizing back to an average of 1. That would undo the chosen base-class reference. With original weighted-average factors of 1, the new weighted average is $1/r_b$, not generally 1.
Changing the base factor but not the base rate. If the old factors were paired with a portfolio-average rate, the new factors must be paired with the selected class’s rate. Otherwise the resulting class rates are scaled incorrectly.
Rounding too early. Dividing rounded factors can slightly change displayed ratios. Keep unrounded values through the conversion and rate calculations, then round money to the nearest cent if needed.
Comparing factors from different scales. A factor from one normalization cannot be directly compared with a factor from another without aligning their reference points.

A reliable final check is to compare selected class pairs using both sets of factors. If the original ratio is $r_j/r_k$, the rebased ratio must equal $r_j^{(b)}/r_k^{(b)}$. If it does not, check for a wrong divisor, inconsistent rounding, or an incorrectly copied factor.

Key takeaway: To rebase multiplicative classification factors to class $b$, calculate $r_j^{(b)}=r_j/r_b$ for every class. The base class becomes 1, pairwise ratios stay unchanged, and the weighted average generally no longer equals 1.

Check Your Understanding

Work each problem before checking its final answer. Treat all rates as hypothetical and use unrounded factors during calculations.

A three-class portfolio has factors $0.72$, $1.08$, and $1.20$. Rebase to the middle-factor class. Give all rebased factors and the ratio of the highest-factor class to the lowest-factor class. (Answer: $2/3,\ 1,\ 10/9$; highest-to-lowest ratio $5/3$.)
Factors $0.5$, $1.0$, and $1.5$ have weights $0.25$, $0.50$, and $0.25$. Rebase to the class with factor $0.5$. If its rate is $450, find all three class rates and the weighted-average rate. (Answer: factors $1,\ 2,\ 3$; rates $450, $900, and $1,350; weighted-average rate $900.)
Three classes have factors $0.9$, $1.2$, and $1.5$. Rebase to the class with factor $1.5$, and verify the ratio of the first class to the second class. (Answer: rebased factors $0.6,\ 0.8,\ 1$; first-to-second ratio $0.75$, unchanged.)
A rating table gives raw multiplicative factors $6$, $9$, and $15$, not normalized to an average. Rebase to the class with factor $9$. (Answer: $2/3,\ 1,\ 5/3$; the original pairwise ratios are preserved.)
Factors $0.7$ and $1.3$ have equal portfolio weights, so their weighted average is 1. Rebase to the class with factor $1.3$. What is the new weighted-average factor, and should it equal 1? (Answer: $10/13\approx0.7692$; no, the rebased weighted average is $1/1.3=10/13$, not 1.)

What You'll Learn