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CAS Exam 8 · Multiplicative Rating Structures · Tutorial 66 of 989

Comparing Two Risks by Cancelling Shared Factors

Compare two risks efficiently by cancelling the rating factors they share, then interpret the remaining ratio as a relative rate change.

Beginner 11 min read

What You'll Learn

  • Write each risk’s indicated rate as a product of its base rate and applicable relativities
  • Divide two rates and cancel factors that are identical for both risks
  • Identify which differing factors explain the final rate ratio
  • Convert a rate ratio into a percentage increase or decrease
  • Distinguish a rate comparison from a comparison of total premiums
  • Avoid reversing the comparison or subtracting relativities instead of dividing them

Why Compare Rates by Their Ratio?

A multiplicative rating plan expresses a risk’s indicated rate as a base rate multiplied by the relativities that apply to that risk. When comparing two risks, much of that calculation may be shared. Dividing their rates can remove the common terms and leave a compact expression showing exactly which rating factors account for the difference.

The ratio also gives a direct interpretation: if Risk B’s rate divided by Risk A’s rate is 1.08, then B’s rate is 8% higher than A’s. The direction matters, so state which risk is in the numerator and which is in the denominator before interpreting the result.

Definition: Suppose risks \(A\) and \(B\) are rated under the same multiplicative plan. Write their indicated rates per unit of exposure as \(R_A=B_0\prod_{k=1}^{n} r_{A,k}\) and \(R_B=B_0\prod_{k=1}^{n} r_{B,k}\), where \(B_0\) is their common base rate and \(r_{A,k}\), \(r_{B,k}\) are the applicable relativities for factor \(k\). Their rate ratio, comparing B with A, is \(R_B/R_A\). A factor cancels from this ratio when its relativity is identical for both risks.

This comparison assumes both rates are calculated on the same basis: the same base rate, coverage definition, and unit of exposure. Each applicable relativity must be positive for the rate ratio to be defined. If the risks have different base rates or are rated under different plans, those differences must remain in the comparison; they cannot simply be cancelled.

Deriving the Cancellation

A useful way to organize the calculation is to separate the factors into two groups: those with the same relativity for both risks and those with different relativities. Let \(C\) be the set of shared factors and \(D\) the set of differing factors. Dividing the two product formulas gives the result directly.

$$ \frac{R_B}{R_A} = \frac{B_0\left(\prod_{k\in C}r_k\right) \left(\prod_{k\in D}r_{B,k}\right)} {B_0\left(\prod_{k\in C}r_k\right) \left(\prod_{k\in D}r_{A,k}\right)} = \prod_{k\in D}\frac{r_{B,k}}{r_{A,k}}. $$

The justification is that division of products pairs each numerator factor with its denominator counterpart. The common base rate and each common relativity form a ratio of a quantity to itself, which equals 1. Only the differing relativities affect the rate ratio. This cancellation is an algebraic simplification; it does not mean that shared factors have no effect on either rate. They still multiply both rates, but do not explain the difference between them.

If \(R_B/R_A=q\), the percentage change in the rate from A to B is \(100(q-1)\%\). A positive result is an increase; a negative result is a decrease. For example, \(q=0.94\) means B’s rate is 6% lower than A’s, not 6% higher.

1
Write both rates on the same basis.
Include the common base rate and every applicable factor, so it is clear which entries correspond.
2
Set up the requested direction.
For “B compared with A,” form \(R_B/R_A\). Keep the same direction throughout the calculation.
3
Cancel identical terms.
Remove the common base and any factor whose relativity is the same for both risks.
4
Calculate and interpret.
Evaluate the remaining ratios, then subtract 1 to express the relative change as a percentage.

Worked Comparisons

Worked Example: Isolating Two Differing Relativities

A hypothetical rating plan has a base rate of $840 per unit of exposure. Two risks have the same usage relativity, 1.30. Risk X has territory and vehicle relativities of 1.15 and 0.80; Risk Y has territory and vehicle relativities of 0.95 and 1.05. Find the rate ratio \(R_Y/R_X\) and the percentage change from X to Y.

Method. Write each rate as the product of its base, usage, territory, and vehicle terms. The base and usage terms are shared, so they will cancel.

$$ R_X=840(1.30)(1.15)(0.80)=\$1{,}004.64 $$
$$ R_Y=840(1.30)(0.95)(1.05)=\$1{,}089.27 $$

The direct rate ratio is \(1{,}089.27/1{,}004.64\). Cancelling the shared terms gives the same result with fewer operations:

$$ \frac{R_Y}{R_X} = \frac{0.95}{1.15}\frac{1.05}{0.80} = \frac{0.9975}{0.92} = 1.08423913\ldots $$

Thus, Y’s rate is higher. The percentage change from X to Y is \(100(1.08423913\ldots-1)=8.423913\ldots\%\), or about 8.42% higher. As a check, the difference between the rates is $1,089.27 − $1,004.64 = $84.63, and $84.63/$1,004.64 is approximately 0.08424.

Common mistake avoided: The territory and vehicle comparisons must each be formed as Y’s relativity divided by X’s. Reversing just one ratio, or rounding the intermediate values too early, changes the result. The displayed ratio uses the requested Y-over-X direction throughout.

Worked Example: Shared Factors Do Not Explain the Difference

Consider two hypothetical property risks with a common base rate of $1,250. Both have an occupancy relativity of 1.20. Risk A has territory and protection relativities of 0.92 and 0.85; Risk B has corresponding relativities of 1.15 and 0.70. Calculate each indicated rate, compare B with A, and identify the factor that contributes most to the ratio.

First calculate the rates, retaining all factors:

$$ R_A=1{,}250(1.20)(0.92)(0.85) =1{,}500(0.782) =\$1{,}173.00 $$
$$ R_B=1{,}250(1.20)(1.15)(0.70) =1{,}500(0.805) =\$1{,}207.50 $$

The base and occupancy relativity are shared. Cancelling them leaves

$$ \frac{R_B}{R_A} = \frac{1.15}{0.92}\frac{0.70}{0.85} = 1.25\left(\frac{14}{17}\right) = \frac{17.5}{17} = 1.02941176\ldots $$

B’s indicated rate is \(100(1.02941176\ldots-1)=2.941176\ldots\%\), or about 2.94% higher than A’s. The territory ratio is 1.25, a 25% increase going from A’s territory relativity to B’s. The protection ratio is \(0.70/0.85=0.823529\ldots\), a decrease that offsets much of the territory effect. The overall ratio reflects both effects multiplied together, not either factor considered alone.

Check. The rate difference is $1,207.50 − $1,173.00 = $34.50; dividing by A’s $1,173.00 gives approximately 0.02941. This agrees with the ratio calculation.

Common mistake avoided: Comparing only territory would suggest a 25% increase, but that ignores the lower protection relativity for B. The total rate comparison must include every factor that differs.

Worked Example: Rate Ratio Versus Total Premium Ratio

A hypothetical commercial coverage has a common base rate of $2,400 per unit of exposure and a shared limit relativity of 1.25. Risk A has safety and deductible relativities of 0.90 and 1.12; Risk B has relativities of 1.00 and 0.95. A buys 40 exposure units and B buys 55. Find the rate ratio \(R_B/R_A\), then compare their total indicated premiums.

The rates per exposure unit are

$$ R_A=2{,}400(1.25)(0.90)(1.12) =3{,}000(1.008) =\$3{,}024.00 $$
$$ R_B=2{,}400(1.25)(1.00)(0.95) =3{,}000(0.95) =\$2{,}850.00 $$

The common base and limit relativity cancel from the rate ratio:

$$ \frac{R_B}{R_A} = \frac{1.00}{0.90}\frac{0.95}{1.12} = \frac{0.95}{1.008} = 0.94246032\ldots $$

B’s rate per exposure unit is \(100(0.94246032\ldots-1)=-5.753968\ldots\%\), or about 5.75% lower than A’s. Total indicated premiums also depend on exposure. A’s is \(40(\$3{,}024)=\$120{,}960\), while B’s is \(55(\$2{,}850)=\$156{,}750\). Therefore, the total premium ratio is

$$ \frac{55R_B}{40R_A} = \frac{55}{40}\frac{R_B}{R_A} = 1.375(0.94246032\ldots) = 1.29588294\ldots $$

So B has a total indicated premium about 29.59% higher, despite having the lower rate per exposure unit. The premium check gives \(\$156{,}750/\$120{,}960=1.29588294\ldots\).

Common mistake avoided: A rate ratio compares prices per unit of exposure; it is not automatically the ratio of total premiums. Include the exposure ratio when comparing totals.

Interpreting Results and Avoiding Traps

Cancellation makes a comparison easier to audit because it isolates the factors that differ. It also helps explain the result: each remaining term says how one risk’s relativity compares with the other’s for a particular factor. However, cancellation does not change the underlying rating plan, and it does not justify omitting a factor that differs.

A frequent exam error is to subtract relativities when the task is to compare rates. For instance, if one risk has a territory relativity of 1.20 and another has 0.80, the difference is 0.40, but that does not mean the first risk’s territory-based rate contribution is 40% higher. The ratio is \(1.20/0.80=1.50\), so the first relativity is 50% higher than the second, all else equal. Ratios are the appropriate tool for relative changes under a multiplicative plan.

  • Check the direction. \(R_B/R_A\) compares B with A. Its reciprocal answers the reverse question.
  • Cancel only equal factors. Similar names do not make factors common; the actual relativities must be equal.
  • Keep a consistent basis. A rate per exposure unit and a total premium are different quantities.
  • Interpret the ratio, not just its decimal. Subtract 1 and express the result as a percentage to state the increase or decrease.
  • Retain precision until the end. Round the final rate or percentage at the requested precision, not each intermediate ratio.
Key takeaway: For two risks rated on the same multiplicative basis, divide the rates in the requested order and cancel the common base and identical relativities. The remaining product of relativity ratios identifies how the differing factors combine to produce the rate difference.

Check Your Understanding

For each problem, form the ratio in the stated direction and keep rate-per-exposure comparisons separate from total-premium comparisons.

  1. A hypothetical plan has a base rate of $960 and a shared garage relativity of 1.10. Risk A has area and alarm relativities of 0.80 and 1.25; Risk B has area and alarm relativities of 1.04 and 0.75. Find both rates and the percentage change from A to B. (Answer: \(R_A=\$1{,}056.00\), \(R_B=\$823.68\), and B is 22.00% lower.)
  2. Two risks share a base rate of $1,600 and vehicle relativity of 1.05. A has terrain and use relativities of 0.90 and 1.10; B has 1.08 and 0.95. Find \(R_B/R_A\) and the percentage change. (Answer: \(R_B/R_A=1.03636\ldots\), so B’s rate is about 3.64% higher.)
  3. Two risks share a base rate of $700 and limit relativity of 1.30. A has safety and region relativities of 1.20 and 0.95; B has 1.05 and 1.10. A has 120 exposure units and B has 90. Find the rate ratio and total-premium ratio, and state the direction of each comparison. (Answer: \(R_B/R_A=1.01316\ldots\), so B’s rate is about 1.32% higher; the total-premium ratio is \(0.75987\ldots\), so B’s total premium is about 24.01% lower.)
  4. A plan has a $2,000 base rate and shared protection relativity 1.15. Risk A’s class and deductible relativities are 0.82 and 1.08. Risk B’s are 1.05 and 0.94. Calculate both rates and the percentage change from A to B. (Answer: \(R_A=\$2{,}036.88\), \(R_B=\$2{,}270.10\), and B’s rate is about 11.45% higher.)
  5. Two risks share a $900 base rate, occupancy relativity 1.25, and vehicle relativity 1.20. A’s region relativity is 0.90 and vehicle-type relativity is 1.15. B’s region relativity is 1.08; find the vehicle-type relativity B needs for its rate to be exactly 8% higher than A’s. (Answer: B’s vehicle-type relativity must be 1.035.)

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