Why Limits Change the Portfolio Average
A selected class relativity can imply a large premium change for a particular class. A rating plan may therefore impose a cap, a maximum permitted factor, and a floor, a minimum permitted factor. These limits protect against extreme changes, but applying them can alter the portfolio’s weighted average relativity. The average matters because it determines the aggregate premium change relative to the premium basis used to weight the classes.
The key calculation is not generally an unweighted average. A class representing a larger share of base premium has more effect on the portfolio average than a small class. In the examples below, weights are shares of total base premium. They may be calculated from exposure and the applicable base rate; exposure shares alone are appropriate only when the class base rates are the same.
The limits may be common to all classes or class-specific. Unless otherwise stated, they apply to the final class factors being calculated. The weights must also match the intended comparison: changing the weighting basis, for example by using current premium shares when the target is defined on base premium, can produce a different aggregate result.
Apply the Limits, Then Measure the Effect
Clipping each indication to its permitted range is straightforward. What is not guaranteed is that the clipped factors retain the original weighted average or meet a required target. The following example isolates that effect.
Worked Example: Measuring the Effect of Direct Clipping
A hypothetical rating plan has four classes with base-premium weights \(0.20, 0.30, 0.30,\) and \(0.20\). Their indicated relativities are \(0.60, 0.90, 1.10,\) and \(1.50\). Apply a common floor of \(0.75\) and cap of \(1.30\), then calculate the weighted average before and after limiting.
Method. Clip each factor to the interval from \(0.75\) to \(1.30\), then take the weighted average using the given base-premium weights.
Before limiting, the weighted average is \(0.20(0.60)+0.30(0.90)+0.30(1.10)+0.20(1.50)\). The four contributions are \(0.12, 0.27, 0.33,\) and \(0.30\), so the total is \(1.02\).
The first indication is below the floor and becomes \(0.75\); the last is above the cap and becomes \(1.30\). The other two factors remain unchanged. The clipped factors are therefore \(0.75, 0.90, 1.10,\) and \(1.30\).
After limiting, the weighted average is \(0.20(0.75)+0.30(0.90)+0.30(1.10)+0.20(1.30)\). The contributions are \(0.15, 0.27, 0.33,\) and \(0.26\), which sum to \(1.01\).
The portfolio average has fallen from \(1.02\) to \(1.01\). Thus, direct clipping does not preserve the original \(1.02\) average and does not produce a target average of \(1.00\). Common mistake avoided: assuming that limits affect only the classes they bind for, without affecting the aggregate calculation.
Rebalancing With a Common Multiplier
One systematic approach is to multiply every indicated factor by a common positive number \(c\), then apply each class’s floor and cap. The multiplier is chosen so that the resulting bounded factors have the target weighted average. A class at a floor or cap stays at that bound; a class between its limits takes the scaled indicated factor.
This method is useful when the limits are intended to constrain the final factors, not merely the initial indications. The multiplier moves factors in a coordinated way; the clipping step ensures that none crosses its permitted range. A practical calculation identifies which classes are at bounds, writes the weighted-average equation for the remaining factors, and checks that the resulting \(c\) is consistent with those bound assignments.
Worked Example: Finding a Multiplier With Binding Limits
Four hypothetical classes have weights \(0.20, 0.30, 0.30,\) and \(0.20\), and indicated factors \(0.50, 0.80, 1.20,\) and \(2.00\). Every class has a floor of \(0.70\) and a cap of \(1.40\). Find factors with weighted average \(R=1.00\) using a common multiplier followed by clipping.
Method. Use \(x_j(c)=\min(1.40,\max(0.70,c r_j))\). We first test whether the first class is at the floor and the fourth is at the cap. Under those conditions, the weighted-average equation is \(0.20(0.70)+0.30(0.80c)+0.30(1.20c)+0.20(1.40)=1.00\).
The fixed contributions from the first and fourth classes are \(0.14\) and \(0.28\). The two interior contributions are \(0.24c\) and \(0.36c\), so the equation becomes \(0.42+0.60c=1.00\). Therefore \(c=0.58/0.60=29/30\), or approximately \(0.9667\).
The scaled indications are \(0.50(29/30)=0.4833\), \(0.80(29/30)=0.7733\), \(1.20(29/30)=1.1600\), and \(2.00(29/30)=1.9333\). After applying the floor and cap, the factors are \(0.70, 0.7733, 1.1600,\) and \(1.40\).
Check the assumed statuses: \(0.4833\) is below the floor, \(0.7733\) and \(1.1600\) lie between the bounds, and \(1.9333\) is above the cap. The weighted average is \(0.20(0.70)+0.30(0.7733)+0.30(1.1600)+0.20(1.40)\), which is approximately \(0.1400+0.2320+0.3480+0.2800=1.0000\). The factors meet the target, to four decimal places. Common mistake avoided: scaling the factors and stopping before checking which ones must be clipped.
Why the Multiplier Equation Works
The weighted-average equation has a useful structure. For fixed positive indications and bounds, each \(x_j(c)\) is nondecreasing in \(c\): increasing \(c\) cannot reduce \(c r_j\), and clipping at the floor or cap only makes the result constant after a bound is reached. Since \(w_j\geq 0\), the total \(F(c)=\sum_j w_j x_j(c)\) is also nondecreasing. It is continuous, including where a scaled indication first reaches a bound.
As \(c\) approaches zero, each positive indicated factor is eventually below its floor, so \(F(c)\) approaches \(\sum_j w_jL_j\). As \(c\) becomes sufficiently large, each factor reaches its cap, so \(F(c)\) approaches \(\sum_j w_jU_j\). Continuity therefore means every target strictly between those limiting averages is attained for some positive \(c\); the endpoint averages are attained once all classes are at the relevant bounds.
The same argument gives a feasibility check. Under independent class-specific bounds and weights summing to one, a target is possible only if \(\sum_j w_jL_j\leq R\leq\sum_j w_jU_j\). It is also sufficient for factors that can be chosen freely within those intervals, because the weighted sum can vary continuously between its minimum and maximum. If some factors are required to remain fixed, calculate the feasible range using those fixed values and the bounds on the remaining classes. A solution need not be unique if the weighted average is flat over a range of multipliers, such as when all positive-weight classes are already at bounds.
For a practical calculation, one can solve the equation by examining the classes that are interior at a trial multiplier. If \(B\) is the set of classes held at a bound and \(I\) is the set of interior classes, then \(\sum_{j\in B} w_j x_j+\sum_{j\in I} w_j c r_j=R\). This is a linear equation for \(c\). The proposed solution must then be checked: each class assigned to \(I\) must remain within its bounds, and each class assigned to \(B\) must indeed be at the stated bound after scaling. If the check fails, revise the assumed sets and solve again. A numerical search can also be used because \(F(c)\) is continuous and nondecreasing.
Balancing With a Selected Class Factor
A common multiplier is not the only option. Sometimes a rating decision holds factors fixed for classes already at limits and adjusts one or more other classes to meet the target. The adjustment must still be made on the stated weights, and every adjusted factor must remain within its own permitted range.
Worked Example: Adjusting One Interior Class
A hypothetical portfolio has base premiums of \(\$40{,}000\), \(\$35{,}000\), and \(\$25{,}000\) for three classes. The indicated factors are \(0.72, 1.05,\) and \(1.40\). The first class has a floor of \(0.80\), the third has a cap of \(1.30\), and the middle class may be adjusted between \(0.80\) and \(1.30\). Keep the first and third classes at their limits and find the middle factor that gives the portfolio a weighted average of \(1.00\).
Method. Total base premium is \(\$40{,}000+\$35{,}000+\$25{,}000=\$100{,}000\), giving weights \(0.40, 0.35,\) and \(0.25\). Let \(x\) be the adjusted middle factor. The target equation is \(0.40(0.80)+0.35x+0.25(1.30)=1.00\).
The fixed contributions are \(0.40(0.80)=0.32\) and \(0.25(1.30)=0.325\). Thus \(0.35x=1.00-0.32-0.325=0.355\), and \(x=0.355/0.35=1.0142857\). Rounded to four decimal places, the middle factor is \(1.0143\), which lies within its permitted interval.
The resulting premium amounts are \(\$40{,}000(0.80)=\$32{,}000\), \(\$35{,}000(1.0142857)\approx \$35{,}500\), and \(\$25{,}000(1.30)=\$32{,}500\). Their sum is \(\$100{,}000\), equal to total base premium; the target weighted average is \(1.00\). Common mistake avoided: adjusting the middle factor by the full shortfall in the portfolio average rather than dividing that shortfall by the middle class’s weight.
Normalization Traps and Practical Checks
A tempting shortcut is to clip all factors and then divide each clipped factor by their weighted average. That does produce a weighted average of one, but it can violate the original limits. For example, in the multiplier example, direct clipping of the original indications gives \(0.70, 0.80, 1.20,\) and \(1.40\), whose weighted average is \(1.02\). Dividing every factor by \(1.02\) changes the first factor to \(0.70/1.02\approx0.6863\), below the \(0.70\) floor. The corrected approach is to solve the bounded weighted-average equation, which gave \(0.70, 0.7733, 1.1600,\) and \(1.40\).
Before finalizing a selection, check the following:
- Confirm that weights are nonnegative, sum to one, and represent the intended base-premium mix.
- Apply each class’s own floor and cap, rather than assuming all limits are identical.
- Recalculate the weighted average using the final factors, not the unbounded indications.
- Verify that every factor respects its limit and that any proposed target lies within the feasible range.
- State the target basis. An average of one is not automatic if the target is a different aggregate change.
These checks distinguish two separate decisions: which class factors are acceptable individually, and what aggregate premium change the selected factors produce. A cap or floor can be appropriate for the first decision while still requiring an explicit balancing step for the second. This distinction is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.
Check Your Understanding
Use the stated weights and limits for each problem. Give factors to four decimal places where needed.
- A three-class portfolio has weights \(0.25, 0.50,\) and \(0.25\), with indicated factors \(0.60, 1.00,\) and \(1.60\). Apply a floor of \(0.80\) and cap of \(1.40\). What is the weighted average after clipping? (Answer: \(1.0500\))
- Two classes have weights \(0.40\) and \(0.60\). The first factor is fixed at its floor of \(0.80\); the second may vary from \(0.90\) to \(1.30\). What second factor gives a weighted average of \(1.00\)? (Answer: \(1.1333\))
- Four equally weighted classes have indications \(0.50, 0.90, 1.30,\) and \(1.80\), with a common floor of \(0.70\) and cap of \(1.50\). Using a common multiplier followed by clipping, find the multiplier for target average \(1.00\) and the final factors. (Answer: \(c=0.8250\); factors \(0.7000, 0.7425, 1.0725, 1.4850\))
- Two classes have weights \(0.60\) and \(0.40\), with factor ranges \(0.90\) to \(1.10\) and \(0.80\) to \(1.30\), respectively. Can they achieve a weighted average of \(0.85\)? (Answer: No; the minimum feasible average is \(0.60(0.90)+0.40(0.80)=0.8600\))
- Two classes have base premiums of \(\$50{,}000\) and \(\$30{,}000\), and final factors \(0.90\) and \(1.10\). Calculate the total premium after applying the factors and the weighted-average relativity. (Answer: \(\$78{,}000\); weighted-average relativity \(78{,}000/80{,}000=0.9750\))