Why Two Relativities Can Describe Different Things
A rating plan may express differences between classes as relativities, but the word relativity alone does not say what is being compared. One factor may compare expected loss costs; another may compare premiums. The distinction matters whenever a premium includes components that do not vary in the same way as expected losses.
For a class, expected loss cost per exposure is the expected claim cost allocated to one unit of exposure. Premium per exposure includes the amount charged to fund that loss cost and may also include expenses, contingencies, or other loads. When those additional components are purely proportional, the loss-cost and premium relativities can be equal. A fixed dollar charge generally makes them different.
Write \(L_j\) for expected loss cost per exposure in class \(j\), and \(P_j\) for premium per exposure. Then the two factors are \(L_j/L_0\) and \(P_j/P_0\), respectively. A factor above 1.00 means the class value is higher than the reference value for the quantity being compared; it does not, by itself, say whether that quantity is loss cost or premium.
Separate the Loss-Cost and Premium Bases
A loss-cost relativity compares losses before premium loads are added. A premium relativity compares amounts charged after the relevant premium components are included. If a rating table is described only as a set of “class relativities,” check what rate or amount forms its base. The same numeric factor can have different meanings depending on whether it multiplies a base loss cost or a base premium.
For a simple illustration of the components, suppose each exposure has expected loss cost \(L\), a fixed expense amount \(e_f\), and variable expense and profit loads represented by fractions \(e_v\) and \(q\) of premium. Under this stated pricing model, the premium satisfies
This is a simplified model, not a universal premium formula. It assumes the fixed expense is a dollar amount per exposure, the variable loads are percentages of premium, and the specified components are the only ones being included. If the expense basis or other rate components differ, the premium equation must reflect those differences.
Why Fixed Expenses Change the Ratio
The reason can be seen directly from the premium equation. With a common proportional-load denominator, the premium ratio is the ratio of loss costs after adding the fixed amount to each class. The fixed amount adds the same dollars to both classes; it does not multiply both loss costs by the same factor. As a result, it has a relatively larger effect on the lower-loss-cost class.
For example, if one class has twice the loss cost of another, adding the same positive fixed expense to each class makes the ratio of their totals less than two. The shared proportional loading does not undo that change, because it multiplies both totals by the same amount.
To verify the inequality, all denominators are positive, so cross-multiplication gives
The last inequality holds because \(e_f>0\) and \(L_j>L_0\). Thus, for a class above the reference class in loss cost, a common positive fixed expense makes its premium relativity smaller than its loss-cost relativity under this model.
Worked Example: Fixed Expense and Two Different Relativities
A hypothetical rating plan has a reference class with expected loss cost of $400 per exposure and another class with expected loss cost of $500 per exposure. The plan includes a fixed expense of $40 per exposure and combined variable expense and profit loads equal to 20% of premium. Calculate both the loss-cost and premium relativity of the second class to the reference class.
Method. Calculate each relativity using its own base: expected loss cost for the loss-cost factor and total premium for the premium factor.
Loss-cost relativity. The reference loss cost is $400, so the second class has \(500/400=1.25\). Its expected loss cost is 25% higher than the reference class.
Reference premium. The variable load fraction is \(0.20\), so \(1-0.20=0.80\). The reference premium is \((400+40)/0.80=440/0.80=\$550.00\).
Second-class premium. Its premium is \((500+40)/0.80=540/0.80=\$675.00\).
Premium relativity. The second class's premium relativity is \(675/550=1.227272\ldots\), or 1.2273 to four decimal places.
Check and trap avoided. The premium relativity is below the loss-cost relativity of 1.25, as expected when the same positive fixed expense is added to both classes. Treating the loss-cost factor 1.25 as the premium factor would incorrectly imply a second-class premium of \(550(1.25)=\$687.50\), rather than the correctly calculated $675.00.
When the Relativities Coincide
If premium is just a common proportional multiple of expected loss cost, the two relativities are the same. For instance, with no fixed expense and a common 20% variable load, \(P_j=L_j/0.80\) and \(P_0=L_0/0.80\). Dividing cancels the shared factor \(1/0.80\), leaving \(P_j/P_0=L_j/L_0\).
More generally, the relevant test is whether the premium relationship to loss cost is a common multiplicative factor across the classes being compared. Merely knowing that each premium contains “loads” is not enough to conclude the relativities coincide. A common fixed-dollar charge is not a common multiple of loss cost, and class-specific expenses can also change the relationship.
Worked Example: Proportional Loads Preserve the Relativity
A hypothetical insurer compares two classes. Their expected loss costs are $600 and $750 per exposure. Premium is set by adding variable expense and profit loads equal to 25% of premium, with no fixed expense per exposure. Calculate the loss-cost and premium relativities.
Method. First calculate the loss-cost ratio. Then calculate both premiums using the specified proportional-load equation and compare them.
Loss-cost relativity. The second class's factor is \(750/600=1.25\).
Premiums. With no fixed expense, \(P=L/(1-0.25)=L/0.75\). The reference premium is \(600/0.75=\$800.00\). The second-class premium is \(750/0.75=\$1{,}000.00\).
Premium relativity. The ratio is \(1{,}000/800=1.25\), the same as the loss-cost relativity.
Check and trap avoided. Both premiums are the corresponding loss costs multiplied by \(1/0.75\), so that common multiplier cancels in the ratio. Do not add a fixed expense that the problem does not specify; doing so would incorrectly make the two relativities differ.
Use the Denominator That Matches the Question
A reliable calculation begins by naming both the quantity and the reference class in the denominator. If a question asks for a loss-cost relativity, divide expected loss costs. If it asks for a premium relativity, divide premiums after including the stated premium components. Do not move between the two just because the factors concern the same classes.
Determine whether the requested factor compares expected loss costs or premiums per exposure.
Use the stated reference value in the denominator, keeping the exposure basis consistent.
For a premium comparison, calculate each premium using only the fixed and proportional components specified in the problem.
State whether the factor describes loss cost or premium and what a value above or below 1 means relative to the reference.
Premium relativities are especially sensitive to the definition of premium. A quoted premium might include a fixed policy charge, while another rate basis might exclude it or apply it separately. Before calculating, establish whether the amount is per exposure, per policy, or on another stated basis. A fixed charge per policy cannot automatically be treated as the same fixed amount per exposure if policies contain different numbers of exposures.
The following comparison summarizes the essential distinction.
| Relativity | Numerator and denominator | What it compares | Possible effect of a fixed expense |
|---|---|---|---|
| Loss-cost | \(L_j/L_0\) | Expected loss cost per exposure across classes | Not included in the loss-cost ratio |
| Premium | \(P_j/P_0\) | Premium per exposure across classes | Can change the ratio when the expense is added to premiums |
Worked Example: Class-Specific Fixed Expenses
A hypothetical insurer has a reference class with expected loss cost of $300 per exposure and a second class with expected loss cost of $450 per exposure. The reference class has a fixed expense of $60 per exposure; the second class has a fixed expense of $90 per exposure. Both classes have variable expense and profit loads equal to 25% of premium. Calculate the loss-cost and premium relativities of the second class.
Method. The class-specific fixed expenses must be used in the separate premium calculations. They are not a common dollar amount.
Loss-cost relativity. The second-class loss-cost factor is \(450/300=1.50\).
Reference premium. Its premium is \((300+60)/0.75=360/0.75=\$480.00\).
Second-class premium. Its premium is \((450+90)/0.75=540/0.75=\$720.00\).
Premium relativity. The second-class premium factor is \(720/480=1.50\).
Check and trap avoided. Here the premium relativity happens to equal the loss-cost relativity because each fixed expense is 20% of that class's loss cost: \(60/300=90/450=0.20\). This is a class-specific proportional relationship, not a general rule that fixed expenses preserve relativities. For example, assuming the reference fixed expense of $60 also applied to the second class would produce a different premium and a different ratio.
Common Exam Trap
A frequent error is to calculate a loss-cost relativity correctly and then apply it to a premium that contains a fixed charge. That silently assumes the premium factor equals the loss-cost factor. The assumption is valid under a common proportional multiplier, but it is not generally valid when a fixed amount is present.
For instance, in the first worked example, multiplying the $550 reference premium by the 1.25 loss-cost relativity gives $687.50. That is the wrong premium under the stated model. The correct second-class premium is found by applying the model to that class's loss cost: \((500+40)/0.80=\$675.00\). The resulting premium relativity is \(675/550=1.2273\), not 1.25.
Check Your Understanding
For each problem, identify the requested basis before calculating.
- A reference class has an expected loss cost of $500 per exposure and another class has an expected loss cost of $650. Premium is a common 10% multiple of loss cost, with no fixed charge. Find the second class's loss-cost and premium relativities. (Answer: Both are 1.3000.)
- Two classes have expected loss costs of $400 and $520. Each has a fixed expense of $30 per exposure, and variable expense and profit loads equal 20% of premium. Calculate the second class's premium relativity. (Answer: \(550/430=1.2791\).)
- The reference class has expected loss cost of $300, and a second class has expected loss cost of $450. Both have a $60 fixed expense per exposure and variable expense and profit loads of 25% of premium. Find both relativities. (Answer: Loss-cost relativity \(=1.5000\); premium relativity \(=680/480=1.4167\).)
- A reference premium is $500, consisting of expected loss cost plus a $50 fixed expense, with variable expense and profit loads equal to 20% of premium. A second class's premium is $600 under the same formula and fixed expense. Find both expected loss costs and the second class's loss-cost relativity. (Answer: Reference loss cost \(=$350\), second-class loss cost \(=$430\), loss-cost relativity \(=430/350=1.2286\).)
- A reference class has loss cost $250 and fixed expense $40; a second class has loss cost $400 and fixed expense $70. Variable expense and profit loads equal 20% of premium. Find the loss-cost and premium relativities, and state whether they are equal. (Answer: Loss-cost relativity \(=1.6000\); premiums are \(290/0.80=$362.50\) and \(470/0.80=$587.50\), so premium relativity \(=587.50/362.50=1.6207\); they are not equal.)