Measuring Premium Impact of Relativity Changes

Why Measure Premium Impact?

A revised relativity changes the premium charged to a class, but class-level changes do not by themselves reveal the effect on the whole portfolio. A substantial increase for a small class may have little total impact, while a modest change for a class with many exposures may be more consequential. The aggregate result depends on both the revised factors and the distribution of exposures across classes.

To isolate the effect of the factors, compare old and revised relativities on the same exposure distribution. This is a fixed-mix comparison: it answers what premium would change if the factors changed but the exposures and all other rating elements did not. It does not forecast a future portfolio whose class mix may itself change.

Definition: For class $i$, let $E_i$ be the number of exposures, $b_i$ the premium per exposure before applying the classification relativity, $r_i^{old}$ the old relativity, and $r_i^{new}$ the revised relativity. With all other rating elements held fixed, the class premiums are $P_i^{old}=E_i b_i r_i^{old}$ and $P_i^{new}=E_i b_i r_i^{new}$. The portfolio premiums are the sums of the class premiums across all classes.

The premium per exposure $b_i$ may be the same across classes, as when one common base rate applies. It can also differ, for example, if the exposures have different unchanged coverage amounts or other fixed rating elements. In that case, the relevant class weights are the premiums before applying the relativity, $E_i b_i$, rather than exposure counts alone.

Portfolio Impact Formulas

For a portfolio with $n$ classes, define its premium impact in dollars as revised premium minus old premium. Define its percentage impact using old premium as the denominator:

\Delta P=\sum_{i=1}^{n} E_i b_i(r_i^{new}-r_i^{old}), \qquad \%\Delta P=\frac{\Delta P}{\sum_{i=1}^{n}E_i b_i r_i^{old}}.

The percentage change for a single class is $r_i^{new}/r_i^{old}-1$, provided $r_i^{old}>0$. This class percentage is not generally the portfolio percentage. The portfolio percentage weights each class's percentage change by its share of old premium.

Result: If $P_i^{old}>0$ for each included class, then $\%\Delta P=\sum_i w_i g_i$, where $w_i=P_i^{old}/\sum_j P_j^{old}$ is the class's share of old premium and $g_i=r_i^{new}/r_i^{old}-1$ is its percentage change. The weights sum to 1. This is a premium-weighted average of class percentage changes, not an unweighted average.

To see why, write each revised class premium as $P_i^{new}=P_i^{old}(1+g_i)$. Then the portfolio difference is $\sum_i(P_i^{new}-P_i^{old})=\sum_iP_i^{old}g_i$. Dividing by total old premium gives $\sum_i [P_i^{old}/\sum_jP_j^{old}]g_i$, which is the weighted-average expression above. The derivation assumes a fixed exposure distribution and unchanged base premiums and other rating elements.

When $b_i=b$ is common to all classes, it cancels from the portfolio percentage. Equivalently, calculate the old and new exposure-weighted average relativities, then compare them:

\bar r^{old}=\frac{\sum_i E_i r_i^{old}}{\sum_i E_i}, \qquad \bar r^{new}=\frac{\sum_i E_i r_i^{new}}{\sum_i E_i}, \qquad \%\Delta P=\frac{\bar r^{new}}{\bar r^{old}}-1.

These formulas compare premiums at the indicated fixed mix. If exposures change, a separate comparison is needed to distinguish the effect of a changed factor schedule from the effect of a changed class distribution.

Worked Premium Comparisons

Worked Example: Class Changes and a Common Base Rate

A hypothetical insurer has 1,000 exposure-years in Class A, 600 in Class B, and 400 in Class C. The annual base premium is $500 per exposure in every class. Old relativities are 0.80, 1.00, and 1.50, respectively; revised relativities are 0.75, 1.10, and 1.60. Calculate each class's premium change and the portfolio premium impact, holding exposures fixed.

Method and assumptions. Apply each relativity to the same class exposure and $500 base premium. The comparison is deterministic; no interest, mortality, or probability model is involved.

Class A. Old premium: $1{,}000 \times \$500 \times 0.80=\$400{,}000$. Revised premium: $1{,}000 \times \$500 \times 0.75=\$375{,}000$. Change: $\$375{,}000-\$400{,}000=-\$25{,}000$, or $-\$25{,}000/\$400{,}000=-6.2500\%$.

Class B. Old premium: $600 \times \$500 \times 1.00=\$300{,}000$. Revised premium: $600 \times \$500 \times 1.10=\$330{,}000$. Change: $\$30{,}000$, or $\$30{,}000/\$300{,}000=10.0000\%$.

Class C. Old premium: $400 \times \$500 \times 1.50=\$300{,}000$. Revised premium: $400 \times \$500 \times 1.60=\$320{,}000$. Change: $\$20{,}000$, or $\$20{,}000/\$300{,}000=6.6667\%$.

Portfolio. Total old premium is $\$400{,}000+\$300{,}000+\$300{,}000=\$1{,}000{,}000$. Total revised premium is $\$375{,}000+\$330{,}000+\$320{,}000=\$1{,}025{,}000$. The dollar impact is $+\$25{,}000$; the percentage impact is $\$25{,}000/\$1{,}000{,}000=2.5000\%$. The portfolio premium therefore increases by 2.5000%, even though Class A decreases.

Check and common mistake avoided. The old-premium weights are 40%, 30%, and 30%. Weighting the class changes gives $0.40(-6.2500\%)+0.30(10.0000\%)+0.30(6.6667\%)=2.5000\%$, allowing for rounding. A simple average of the three class percentage changes would be 3.4722%, which is wrong because it gives each class equal weight regardless of its premium volume.

Worked Example: When Classes Have Different Base Premiums

A hypothetical equipment policy portfolio includes 120 exposures with a $500 base premium each in Class D and 80 exposures with a $1,000 base premium each in Class E. The old factors are 0.90 for D and 1.20 for E; the revised factors are 1.00 and 1.10. Calculate the portfolio impact on this fixed mix.

Method and assumptions. The exposure counts and base premiums stay unchanged. Because the base premium differs by class, calculate class premiums directly rather than weighting factors by exposure count alone. No interest, mortality, or probability model is involved.

Class D. Its premium before applying the factor is $120 \times \$500=\$60{,}000$. Old premium is $\$60{,}000 \times 0.90=\$54{,}000$; revised premium is $\$60{,}000 \times 1.00=\$60{,}000$. Its increase is $\$6{,}000$, or $\$6{,}000/\$54{,}000=11.1111\%$.

Class E. Its premium before applying the factor is $80 \times \$1{,}000=\$80{,}000$. Old premium is $\$80{,}000 \times 1.20=\$96{,}000$; revised premium is $\$80{,}000 \times 1.10=\$88{,}000$. Its decrease is $-\$8{,}000$, or $-\$8{,}000/\$96{,}000=-8.3333\%$.

Portfolio. Old premium is $\$54{,}000+\$96{,}000=\$150{,}000$, and revised premium is $\$60{,}000+\$88{,}000=\$148{,}000$. Thus $\Delta P=-\$2{,}000$, and $\%\Delta P=-\$2{,}000/\$150{,}000=-1.3333\%$. The revised schedule lowers premium by 1.3333% on this fixed mix.

Check and common mistake avoided. The old-premium weights are $54/150=36\%$ for D and $96/150=64\%$ for E. Their weighted class impact is $0.36(11.1111\%)+0.64(-8.3333\%)=-1.3333\%$, subject to rounding. Averaging the two factors using exposure counts alone would ignore the different base premiums and would not give the portfolio premium impact.

Worked Example: A Small Class With a Large Factor Increase

A hypothetical commercial portfolio has 900 exposure-years in Class F and 100 in Class G. The base premium is $400 per exposure for both. Old factors are 1.00 for F and 1.50 for G; revised factors are 1.04 and 1.80. Find the class-level and total impact. Report dollar results to the cent and percentage results to four decimal places.

Method and assumptions. Hold the exposure-years and base premium fixed. Compute each class's old and revised premium, then use total old premium as the denominator for the portfolio percentage. This is a deterministic comparison, with no interest, mortality, or probability assumption.

Class F. Old premium is $900 \times \$400 \times 1.00=\$360{,}000$. Revised premium is $900 \times \$400 \times 1.04=\$374{,}400$. The increase is $\$14{,}400$, or $14{,}400/360{,}000=4.0000\%$.

Class G. Old premium is $100 \times \$400 \times 1.50=\$60{,}000$. Revised premium is $100 \times \$400 \times 1.80=\$72{,}000$. The increase is $\$12{,}000$, or $12{,}000/60{,}000=20.0000\%$.

Portfolio. Old premium totals $\$360{,}000+\$60{,}000=\$420{,}000$. Revised premium totals $\$374{,}400+\$72{,}000=\$446{,}400$. Therefore, the increase is $\$26{,}400.00$, and the percentage impact is $\$26{,}400/\$420{,}000=0.062857\ldots=6.2857\%$.

Check and common mistake avoided. The old-premium weights are $360/420=85.7143\%$ for F and $60/420=14.2857\%$ for G. The weighted class impact is approximately $0.857143(4\%)+0.142857(20\%)=6.2857\%$. Taking the unweighted average, 12.0000%, would substantially overstate the portfolio impact because the class with the larger percentage increase has much less old premium.

Interpretation and Practical Checks

A fixed-mix premium comparison is useful for quantifying the direct effect of a revised factor schedule on a defined book of exposures. It can also show where the change falls: one class may receive a premium increase while another receives a decrease, even if the portfolio total changes only slightly. For a larger set of rating cells, the same calculation applies to each cell; summing cell premiums gives the aggregate result.

Keep the comparison basis consistent. Use the same exposures, base premiums, coverage assumptions, and other unchanged rating elements for both factor schedules. If a class has zero old premium, its individual percentage change is not defined, although its dollar change can still be calculated. If total old premium is zero, the portfolio percentage change is likewise undefined.

A separate issue is whether revised factors were balanced to an overall level. That balancing concerns the factor schedule and a chosen weighting basis. The realized premium impact depends on the exposure and base-premium weights in the portfolio being measured. If those weights differ, a schedule balanced on one basis need not produce zero premium change on another. The calculation here measures the impact on the specified portfolio; it does not assume that the old and revised schedules have equal weighted averages on every possible mix.

Common exam trap: Do not average class percentage changes equally, and do not assume exposure counts are the correct weights when base premiums differ. Calculate old and revised class premiums on the same exposure basis, sum them, and divide the total change by total old premium.

Check Your Understanding

For each problem, hold the stated exposures and other rating elements fixed. Use total old premium as the denominator for the portfolio percentage impact.

A hypothetical portfolio has 500 exposures in Class A and 300 in Class B. The common base premium is $400. Old factors are 0.90 and 1.40; new factors are 1.00 and 1.30. Find the dollar and percentage portfolio impact. (Answer: old premium $348,000; new premium $356,000; increase $8,000, or 2.2989%.)
A portfolio has 1,000, 500, and 250 exposures in three classes, with a common $300 base premium. Old factors are 0.80, 1.00, and 1.60; new factors are 0.84, 0.95, and 1.50. Find the aggregate impact. (Answer: old premium $510,000; new premium $507,000; decrease $3,000, or 0.5882%.)
Class A has 120 exposures with a $500 base premium each; Class B has 80 with a $1,000 base premium each. Old factors are 0.90 and 1.20; new factors are 1.00 and 1.10. Calculate the portfolio percentage impact. (Answer: old premium $150,000; new premium $148,000; decrease 1.3333%.)
For a hypothetical portfolio with common $500 base premiums, the old factors are 0.80 for Class A and 1.20 for Class B; new factors are 0.90 and 1.10. Calculate the impact first for a current mix of 80 A and 120 B exposures, then for a forecast mix of 150 A and 50 B. (Answer: current mix, old $104,000 and new $102,000, a 1.9231% decrease; forecast mix, old $90,000 and new $95,000, a 5.5556% increase.)
Three classes have old premiums of $60,000, $90,000, and $50,000. Their revised factors imply class premium changes of +5%, -10%, and +20%, respectively. Find the total dollar and percentage portfolio impact. (Answer: increase of $4,000; revised premium $204,000; portfolio increase 2.0000%.)

What You'll Learn