Calculating Class Relativities From Loss Costs

Why Class Relativities Require More Than Loss Costs

A class relativity expresses the indicated rate for one class as a multiple of the rate for a selected base class. It is tempting to calculate that multiple from the classes’ loss costs alone. That works only when the other rate components affect both classes in the same proportion. If fixed expenses or premium-based provisions differ by class, the indicated rates must be calculated first.

The calculation is aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus. The focus here is translating prospective loss costs into class-specific indicated rates and then comparing those rates. It does not determine how a set of relativities should be balanced across a portfolio.

Definition: For class $c$, let $L_c$ be the prospective loss cost per exposure unit, $E_c$ the fixed expense per exposure unit, $v_c$ the variable expense proportion of premium, and $q_c$ the profit and contingency proportion of premium. Under the assumptions that these percentage provisions are additive and are both measured as proportions of premium, the indicated rate is $$ R_c=\frac{L_c+E_c}{1-v_c-q_c}. $$ For a chosen base class $b$, the relativity of class $c$ to the base is $$ r_c=\frac{R_c}{R_b}. $$

All quantities must use a consistent basis. For example, if the rate is stated per annual policy, the loss cost and fixed expense must also be per annual policy. The loss cost should be prospective and should represent the same coverage and exposure basis as the indicated rate. If loss adjustment expenses are included in the loss cost, they should not also be added a second time as a separate expense.

This setup assumes $v_c+q_c<1$, so a positive share of premium remains to fund losses and fixed expenses. It also assumes the percentage provisions are fractions of premium, not percentages of the loss cost. Different rating conventions may define provisions differently; in that case, use the convention stated in the problem rather than applying this formula automatically.

From Indicated Rates to Relativities

The calculation has two stages. First, gross up each class’s loss cost plus fixed expense for the share of premium consumed by percentage provisions. Second, divide each class’s indicated rate by the indicated rate of the chosen base. Keeping these stages separate makes it easier to spot a class-specific expense or provision that changes the relationship between loss costs and rates.

Check the basis.
Confirm that loss costs and fixed expenses are per the same exposure unit, and identify which provisions are measured as percentages of premium.

Calculate each class’s indicated rate.
Add the loss cost and fixed expense, then divide by one minus the sum of the applicable premium-based proportions.

Choose the base and form the ratios.
Divide each indicated class rate by the indicated base-class rate. The base class therefore has relativity 1.

The formula follows from the premium balance. If $R_c$ is the premium per exposure unit, the variable expense is $v_cR_c$ and the profit and contingency provision is $q_cR_c$. The portion left to fund the loss cost and fixed expense must equal $L_c+E_c$. Thus

R_c-v_cR_c-q_cR_c=L_c+E_c, \qquad R_c(1-v_c-q_c)=L_c+E_c.

Solving for $R_c$ gives the indicated-rate formula. The relativity then follows by definition: it is the class rate divided by the selected base rate. This also shows why using just $L_c/L_b$ is not generally valid: the fixed expenses and gross-up factors may not be proportional across the two classes.

Worked Example: Different Fixed Expenses and Percentage Provisions

A hypothetical insurer is comparing two annual policy classes. Class R has a prospective loss cost of $500 per policy, fixed expenses of $25 per policy, a variable expense provision of 10% of premium, and a profit provision of 5% of premium. Class S has a loss cost of $540, fixed expenses of $10, a variable expense provision of 15%, and a profit provision of 5%. Find S’s relativity to R.

Method and assumptions. The loss costs and fixed expenses are on the same per-policy basis. Both percentage provisions are additive proportions of premium, so the indicated rate is $(L+E)/(1-v-q)$.

Class R rate. The amount needed before percentage provisions is $500+25=\$525$. The combined premium-based provision is $0.10+0.05=0.15$, leaving $1-0.15=0.85$ of premium. Therefore

R_R=\frac{500+25}{1-0.10-0.05} =\frac{525}{0.85} =617.6470588\ldots =\$617.65.

Class S rate. The amount needed before percentage provisions is $540+10=\$550$. The combined provision is $0.15+0.05=0.20$, leaving $0.80$ of premium. Therefore

R_S=\frac{540+10}{1-0.15-0.05} =\frac{550}{0.80} =\$687.50.

Relativity. Divide the unrounded indicated rates, or equivalently retain the exact expressions until the final step:

r_S=\frac{R_S}{R_R} =\frac{550/0.80}{525/0.85} =1.113095238\ldots =1.1131.

Thus S’s indicated rate is about 1.1131 times R’s, or about 11.31% higher. As a check, $1.113095238\ldots\times 617.6470588\ldots=687.50$. The calculation avoids the common mistake of carrying forward an incorrect base-class rate; the correct R rate is $617.65, not $656.25.

When Loss-Cost Ratios Do and Do Not Work

If two classes have the same fixed expense per exposure and the same percentage provisions, the common terms may cancel in the rate ratio. In particular, with common $E$, $v$, and $q$,

\frac{R_c}{R_b} = \frac{(L_c+E)/(1-v-q)}{(L_b+E)/(1-v-q)} = \frac{L_c+E}{L_b+E}.

Even in this case, the relativity is generally not $L_c/L_b$ unless the fixed expense is zero or otherwise has no effect on the ratio. If the percentage provisions differ, their gross-up factors do not cancel either. A useful exam check is to ask whether the quantities being compared are loss costs or fully indicated rates.

Worked Example: Common Provisions but Different Loss Costs

A hypothetical portfolio has two annual classes with the same variable expense provision of 12% of premium and profit provision of 3%. Class A has a prospective loss cost of $720 and fixed expense of $30 per policy. Class B has a loss cost of $810 and fixed expense of $15 per policy. Calculate the rate for each class and B’s relativity to A.

Class A. Its amount before percentage provisions is $720+30=\$750$. The combined provision is $0.12+0.03=0.15$, so

R_A=\frac{750}{1-0.15}=\frac{750}{0.85} =882.3529412\ldots=\$882.35.

Class B. Its amount before percentage provisions is $810+15=\$825$. The percentage provisions are unchanged, so

R_B=\frac{825}{0.85} =970.5882353\ldots=\$970.59.

Relativity and check. The common gross-up factor cancels:

r_B=\frac{R_B}{R_A} =\frac{825/0.85}{750/0.85} =\frac{825}{750} =1.10.

B’s rate is 1.10 times A’s. The ratio of loss costs alone would be $810/720=1.125$, which is not the answer because the classes have different fixed expenses. Multiplying $1.10$ by the unrounded A rate gives $970.5882353\ldots$, confirming the result.

Class-Specific Loads Can Change the Ordering

A class with the larger loss cost does not necessarily have the same rate ranking or relative difference after expenses and premium provisions are included. The relevant comparison is between the full indicated rates. In particular, higher percentage provisions increase the gross-up needed to fund a given amount of loss cost and fixed expense.

Worked Example: Comparing Classes With Different Loads

A hypothetical insurer sets rates per annual contract. Class C has a prospective loss cost of $400, fixed expenses of $20, variable expenses equal to 10% of premium, and profit and contingency equal to 5% of premium. Class D has a loss cost of $460, fixed expenses of $5, variable expenses equal to 15% of premium, and profit and contingency equal to 5%. Find D’s indicated rate and relativity to C.

Class C rate. The amount before percentage provisions is $400+20=\$420$. Its total premium-based proportion is $0.10+0.05=0.15$, hence

R_C=\frac{420}{1-0.10-0.05} =\frac{420}{0.85} =494.1176471\ldots=\$494.12.

Class D rate. Its amount before provisions is $460+5=\$465$. Its total proportion is $0.15+0.05=0.20$, hence

R_D=\frac{465}{1-0.15-0.05} =\frac{465}{0.80} =\$581.25.

Relativity. Using the unrounded C rate,

r_D=\frac{581.25}{420/0.85} =\frac{581.25(0.85)}{420} =\frac{494.0625}{420} =1.1763392857\ldots =1.1763.

D’s indicated rate is 1.1763 times C’s. As a check, $1.1763392857\ldots\times 494.1176471\ldots=581.25$, up to displayed precision. The method avoids comparing only $460/400$: that loss-cost ratio omits both classes’ fixed expenses and their different gross-up factors.

Common Exam Trap: Applying a Premium Load to the Wrong Base

A variable expense or profit provision stated as a percentage of premium must be funded from premium. It is not generally correct to add that percentage directly to the loss cost plus fixed expense. For example, in the first worked example, the incorrect calculation for class R would be

(500+25)(1+0.10+0.05)=525(1.15)=\$603.75.

That calculation treats 15% of the pre-load amount as the provision. But the stated provision is 15% of the final premium. The correct balance is $R_R-0.15R_R=525$, giving $R_R=525/0.85=\$617.65$. The distinction is between adding a percentage of a known cost base and grossing up for a percentage of the unknown premium.

Another frequent error is to use the wrong base class in the denominator. Always write the selected base rate explicitly, such as $r_S=R_S/R_R$, before substituting numbers. The base class’s own relativity is exactly 1 because $R_b/R_b=1$.

Key takeaway: Calculate each class’s indicated rate using its own loss cost, fixed expense, and premium-based provisions. Then divide by the indicated rate of the selected base class. Loss-cost ratios are a valid shortcut only when the rate components permit the relevant terms to cancel.

Check Your Understanding

For each problem, treat the stated variable expense and profit provisions as additive percentages of premium. Round indicated rates to cents and relativities to four decimal places.

Class A is the base, with loss cost $300, fixed expense $12, variable expense 10%, and profit provision 5%. Class B has loss cost $330 and no fixed expense; its percentage provisions are the same as A’s. Find B’s relativity to A. (Answer: $1.0577$)
Class K is the base, with loss cost $640, fixed expense $40, variable expense 12%, and profit provision 3%. Class M has loss cost $700 and fixed expense $20, with the same percentage provisions. Find both indicated rates and M’s relativity. (Answer: $R_K=\$800.00$, $R_M=\$847.06$, $r_M=1.0588$)
Class P is the base, with loss cost $500, fixed expense $25, variable expense 10%, and profit provision 4%. Class Q has loss cost $560, fixed expense $10, variable expense 14%, and profit provision 4%. Find Q’s indicated rate and relativity. (Answer: $R_Q=\$695.12$, $r_Q=1.1387$)
Class U is the base, with loss cost $900, fixed expense $30, variable expense 8%, and profit provision 2%. Class V has loss cost $840, fixed expense $60, variable expense 10%, and profit provision 5%. Find both rates and V’s relativity. (Answer: $R_U=\$1,033.33$, $R_V=\$1,058.82$, $r_V=1.0247$)
Class B is the base, with loss cost $250, fixed expense $20, variable expense 8%, and profit provision 2%. Class C has loss cost $285, fixed expense $15, variable expense 12%, and profit provision 3%; class D has loss cost $310, no fixed expense, variable expense 10%, and profit provision 5%. Find C’s and D’s relativities to B. (Answer: $r_C=1.1765$, $r_D=1.2157$)

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