Balancing Revised Factors to the Overall Rate Level

Why Balance Revised Class Factors?

Proposed class factors describe how rates vary across classes relative to a common base rate. After reviewing indicated and current factors, an actuary may select a revised set that changes the relative rates among classes. But those relative changes can also change the portfolio’s average rate level. Balancing adjusts the factors so that their average, using the specified portfolio weights, matches a target overall level.

The adjustment is a common multiplicative rescaling. It changes the average level without changing the ratios between the proposed factors. This is useful when the intended classification changes are to be retained, but the overall level represented by the factors must be brought to a specified target.

Definition: Let $N_j$ be the exposure count for class $j$, and let $r_j$ be its proposed factor. When each exposure has the same common base rate before applying the factor, the proposed exposure-weighted average factor is $$ \bar r=\frac{\sum_j N_j r_j}{\sum_j N_j}. $$ If the target average factor is $T$, the balanced factors are $r_j^*=c r_j$, where $$ c=\frac{T}{\bar r}. $$

Set the Weighting Basis Before Calculating

The weights represent the portfolio over which the average is intended to hold. In the basic setup here, the weights are class exposures and the base rate per exposure is common across classes. A class with more exposures therefore contributes more to the portfolio average. The weights must refer to the same portfolio basis used to define the target.

If exposures have different base rates, exposure counts alone may not represent their contributions to premium. In that setting, the appropriate weights are the projected premiums before applying the class factors, or equivalent weights that correctly reflect the rate base. The exposure-weighted formula above does not apply automatically when a common base rate assumption is false.

The target $T$ is the desired weighted average of the factors on the stated basis. It may be 1.00, but it need not be: a target such as 1.10 means the balanced factors average 1.10 relative to the chosen base. Balancing factors does not, by itself, determine whether that target is appropriate; it only makes the selected factors consistent with it.

The Balancing Formula and Its Justification

First calculate the weighted average of the proposed factors. Then scale each proposed factor by the same multiplier. The multiplier is chosen so the scaled weighted average equals $T$.

\frac{\sum_j N_j r_j^*}{\sum_j N_j} = \frac{\sum_j N_j(c r_j)}{\sum_j N_j} = c\frac{\sum_j N_j r_j}{\sum_j N_j} = c\bar r.

Thus, setting $c\bar r=T$ gives $c=T/\bar r$, provided $\bar r\ne 0$. For the usual positive class factors and a positive target, the multiplier is positive. Substituting it back gives the balance identity:

\frac{\sum_j N_j r_j^*}{\sum_j N_j} = \frac{T}{\bar r}\bar r=T.

The same multiplier also preserves every pairwise factor ratio:

\frac{r_j^*}{r_k^*} = \frac{c r_j}{c r_k} = \frac{r_j}{r_k}.

Consequently, balancing changes the common scale, not the relative relationship between any two proposed class factors. It does not make a proposed factor set more or less differentiated across classes; it aligns the set’s weighted average with the target.

Confirm the portfolio basis.
List the exposure weights and verify that they correspond to the target average and to the common-base-rate assumption.

Calculate the proposed average.
Multiply each class factor by its weight, add the weighted amounts, and divide by total weight.

Calculate the common multiplier.
Divide the target average by the proposed weighted average.

Rescale and verify.
Multiply each proposed factor by the common multiplier. Check the weighted average using unrounded values, then assess any effect from displayed rounding.

Worked Example: Balancing Three Proposed Factors

A hypothetical program has 200 exposures in Class A, 500 in Class B, and 300 in Class C. The proposed factors are 0.70, 1.00, and 1.50, respectively. The target exposure-weighted average is 1.10. Find the balanced factors and verify the result. Use four decimal places for the reported factors.

Method. Calculate the proposed exposure-weighted average, then multiply every proposed factor by the same target-to-average ratio.

Step 1: Calculate total exposure and weighted factor total.

\sum_j N_j=200+500+300=1{,}000,

\sum_j N_jr_j =200(0.70)+500(1.00)+300(1.50) =140+500+450 =1{,}090.

Step 2: Calculate the proposed average and multiplier.

\bar r=\frac{1{,}090}{1{,}000}=1.09, \qquad c=\frac{1.10}{1.09}=\frac{110}{109}\approx1.0091743.

Step 3: Rescale each factor.

r_A^*=0.70\left(\frac{110}{109}\right)\approx0.7064220,

r_B^*=1.00\left(\frac{110}{109}\right)\approx1.0091743, \qquad r_C^*=1.50\left(\frac{110}{109}\right)\approx1.5137615.

Rounded to four decimal places, the balanced factors are 0.7064, 1.0092, and 1.5138. Verification using unrounded factors: their weighted average is $c\bar r=(110/109)(1.09)=1.10$, exactly the target. Using the displayed rounded factors gives $(200(0.7064)+500(1.0092)+300(1.5138))/1{,}000=1.10002$; the small difference from 1.10 is due to rounding. A common mistake this method avoids is basing $c$ on an incorrectly summed weighted total: the total here is 1,090, not 1,120.

Worked Example: A Target Below the Proposed Average

A hypothetical portfolio has 1,000 exposures in Class A, 600 in Class B, and 400 in Class C. Proposed factors are 0.85, 1.10, and 1.35. The target weighted average is 1.02. Calculate the factors balanced to the target and verify the result. Report factors to four decimal places.

Step 1: Calculate the weighted average. Total exposure is $1{,}000+600+400=2{,}000$. The weighted factor total is

$$ 1{,}000(0.85)+600(1.10)+400(1.35) =850+660+540 =2{,}050. $$

Therefore, $\bar r=2{,}050/2{,}000=1.025$. In particular, the average is 1.025, not 1.03; rounding it before computing the multiplier would alter the result.

Step 2: Calculate the multiplier.

c=\frac{1.02}{1.025} =\frac{204}{205} \approx0.9951220.

Step 3: Rescale the factors.

r_A^*=0.85\left(\frac{204}{205}\right)\approx0.8458537, \qquad r_B^*=1.10\left(\frac{204}{205}\right)\approx1.0946341,

r_C^*=1.35\left(\frac{204}{205}\right)\approx1.3434146.

The reported factors are 0.8459, 1.0946, and 1.3434. Verification: before rounding, their weighted average is $(204/205)(1.025)=1.02$. Using the four-decimal factors gives a weighted total of $845.9+656.76+537.36=2{,}040.02$, or an average of $2{,}040.02/2{,}000=1.02001$. The displayed factors therefore verify the target to the expected rounding precision. The multiplier is below 1 because the proposed average, 1.025, exceeds the target, 1.02.

Worked Example: Balancing Four Classes and Checking Relative Factors

A hypothetical plan has exposures of 400, 300, 200, and 100 in four classes. Their proposed factors are 0.60, 0.90, 1.20, and 1.80. The target weighted average is 1.02. Find the balanced factors to four decimal places, verify the average, and check the ratio of the highest to lowest factors.

Step 1: Find the proposed weighted average. Total exposure is $400+300+200+100=1{,}000$, and the weighted factor total is

$$ 400(0.60)+300(0.90)+200(1.20)+100(1.80) =240+270+240+180 =930. $$

Thus $\bar r=930/1{,}000=0.93$. Step 2: Find the multiplier:

c=\frac{1.02}{0.93}=\frac{34}{31}\approx1.0967742.

Step 3: Rescale:

0.60c=\frac{20.4}{31}\approx0.6580645,\quad 0.90c=\frac{30.6}{31}\approx0.9870968,

1.20c=\frac{40.8}{31}\approx1.3161290,\quad 1.80c=\frac{61.2}{31}\approx1.9741935.

To four decimals, the factors are 0.6581, 0.9871, 1.3161, and 1.9742. Verification: the unrounded weighted average is $c(0.93)=1.02$. The highest-to-lowest ratio is unchanged: before balancing it is $1.80/0.60=3$, and after balancing it is $(1.80c)/(0.60c)=3$. The common multiplier raises every factor because the proposed average is below the target.

Common Mistakes and Limits of the Method

The most important practical checks are that the weighting basis matches the target and that the weighted arithmetic is correct. Balancing cannot repair an incorrectly defined target or inappropriate weights; it only rescales the factors under the stated assumptions.

Using an unweighted mean. Classes with different exposure counts do not contribute equally to an exposure-weighted average. In the first example, the simple mean is $(0.70+1.00+1.50)/3=1.0667$, not the correct weighted average of 1.09. Scaling to 1.10 using that simple mean would give a weighted average of about $1.10(1.09/1.0667)=1.1241$, not 1.10.
Rounding too early. In the second example, replacing 1.025 by 1.03 before finding $c$ gives $1.02/1.03$, rather than $1.02/1.025$. Keep full precision through the calculations and round only the reported factors.
Applying different multipliers by class. Separate multipliers generally change the ratios between proposed factors. A single common $c$ is what ensures the proposed relative relationships remain intact.
Using exposure counts when base rates differ. If a unit of exposure does not represent the same base premium across classes, the exposure-weighted factor average need not be the relevant premium-weighted average. Select weights that represent the intended portfolio average.

If a separate requirement prevents some factors from moving proportionally, such as class-specific restrictions or fixed factors, a single common multiplier may not meet all requirements at once. The formula here applies when all factors can be rescaled together and the target is defined on the stated weighting basis.

Key takeaway: Calculate the proposed weighted average on the correct portfolio basis, multiply every proposed factor by $T/\bar r$, and verify the target with unrounded values. The common scaling preserves all pairwise factor ratios; rounding may create a small displayed discrepancy.

Check Your Understanding

For each problem, calculate the weighted average before balancing, apply one common multiplier, and check the result. Answers report factors to four decimal places where needed.

A hypothetical portfolio has 100 exposures with factor 0.80 and 300 exposures with factor 1.20. Balance to a target average of 1.05. What are the balanced factors? (Answer: $\bar r=1.10$, $c=21/22$; factors 0.7636 and 1.1455.)
Class exposures are 250, 250, and 500, with proposed factors 0.70, 1.00, and 1.40. Balance to a target average of 0.99. (Answer: $\bar r=1.125$, $c=0.88$; factors 0.6160, 0.8800, and 1.2320.)
A program has exposures of 120, 280, and 600 and proposed factors of 0.75, 1.05, and 1.25. Balance to 1.08 and verify the unrounded average. (Answer: $\bar r=1.134$, $c=20/21$; factors 0.7143, 1.0000, and 1.1905; unrounded weighted average 1.08.)
Two classes have 800 and 200 exposures and proposed factors 0.90 and 1.30. The target average is 1.00. Find the multiplier and factors, then state whether the factors rise or fall. (Answer: $\bar r=0.98$, $c=50/49\approx1.0204082$; factors 0.9184 and 1.3265; both rise.)
Three classes have exposures of 100, 200, and 700 and proposed factors of 0.80, 1.00, and 1.60. Balance to 1.20. Verify that the highest-to-lowest ratio is unchanged. (Answer: $\bar r=1.40$, $c=6/7$; factors 0.6857, 0.8571, and 1.3714; the ratio remains $1.60/0.80=2$.)

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