Why Derive Relativities from Class Loss Costs?
A class relativity compares the expected loss cost of one classification with that of a selected reference classification. When the comparison is based on class loss costs, the calculation starts with losses per exposure—not with the premium charged to each class. This provides a direct comparison of the classes’ estimated loss costs on a common basis.
The method is useful only when the losses and exposures being compared are defined consistently. For example, losses must be on a comparable valuation and trend basis, and all classes must use the same type of exposure unit. The resulting factors describe relative expected loss costs; by themselves, they are not complete premium rates. Expense provisions, profit and other rate components are outside this calculation.
Calculation and Interpretation
First calculate each class’s loss cost using the same exposure definition. Then divide each class loss cost by the loss cost of the selected reference class. A relativity above 1 means the class’s indicated loss cost is higher than the reference class’s; a relativity below 1 means it is lower. For example, a relativity of 1.20 indicates a loss cost 20% higher than the reference loss cost, while 0.85 indicates one 15% lower.
The reference class is a denominator for comparison. Choosing a different reference changes the numerical factors but not the underlying pairwise comparisons, provided the same loss costs are used. Relativities from this calculation are not automatically an overall-average-normalized set of factors, nor do they determine the final base rate. Those are separate rating decisions.
Worked Example: Comparing Three Vehicle Classes
A hypothetical insurer has the following losses and earned vehicle-year exposures for three classes. Losses are already on the same valuation and trend basis. Calculate the loss costs using Class B as the reference, then interpret the Class A and Class C factors.
| Class | Losses | Vehicle-year exposures |
|---|---|---|
| A | $1,260,000 | 30,000 |
| B | $1,680,000 | 24,000 |
| C | $1,575,000 | 21,000 |
Assumptions. Each exposure is one vehicle-year, and the losses are measured on a consistent basis across classes. No interest rate or mortality assumption applies. The supplied aggregate losses are used directly; no parametric probability distribution is assumed.
Step 1: Calculate each class loss cost.
For Class A, \(LC_A=\$1{,}260{,}000/30{,}000=\$42\) per vehicle-year. For Class B, \(LC_B=\$1{,}680{,}000/24{,}000=\$70\) per vehicle-year. For Class C, \(LC_C=\$1{,}575{,}000/21{,}000=\$75\) per vehicle-year.
Step 2: Divide each loss cost by the reference loss cost. Class B is the reference, so its relativity is \(R_B=70/70=1.000\). Class A has \(R_A=42/70=0.600\), and Class C has \(R_C=75/70=1.071\) when rounded to three decimals.
Step 3: Interpret. On this loss-cost basis, Class A’s indicated loss cost is 40% below Class B’s. Class C’s is approximately 7.1% above Class B’s.
Check. Multiplying the reference loss cost by each factor reproduces the corresponding loss cost: \(\$70(0.600)=\$42\), \(\$70(1.000)=\$70\), and \(\$70(75/70)=\$75\). The calculation avoids confusing a class’s total losses with its loss cost per exposure.
Why the Ratio Represents Relative Expected Loss Cost
The justification follows from the meaning of a loss cost. Suppose class \(c\) has an expected loss cost of \(\lambda_c\) dollars per exposure unit, and the exposures are comparable units. Under the assumption that the class’s expected loss cost applies across its exposures, the expected aggregate losses are \(E_c\lambda_c\). Dividing by exposures gives the expected loss cost:
The same reasoning applies to reference class \(r\). Thus the ratio of the two expected loss costs is \(\lambda_c/\lambda_r\). In practice, actual observed losses are used to estimate those expectations, so \(LC_c/LC_r\) is an empirical estimate of the relative expected loss cost.
This justification depends on comparable exposure units and losses measured on comparable bases. It does not claim that observed class loss costs equal their true expected values exactly. Random variation, limited exposure, changes in the mix of risks within a class, or inconsistent loss adjustments can affect the estimates. A factor calculated from class loss costs should therefore be interpreted as an indication from the available experience, not as a guarantee about every individual risk.
Worked Comparisons on Different Scales
A loss cost may be expressed per exposure, per 100 exposures, or per 1,000 exposures. The choice changes the displayed dollar amounts but not the relativity, as long as every class uses the same scale. The following example shows this explicitly.
Worked Example: Using a Per-1,000-Exposure Scale
A hypothetical insurer studies two classes of a small commercial property product. Class K has $2,160,000 in losses and 36,000 earned location-years. Class M has $1,620,000 in losses and 30,000 earned location-years. The insurer wants Class K’s relativity to Class M.
Assumptions. A location-year is the common exposure unit, and both loss totals use the same valuation and trend basis. No interest or mortality assumption applies, and the aggregate losses are not modelled with a specified probability distribution.
Step 1: Calculate loss costs per location-year. Class K has \(LC_K=\$2{,}160{,}000/36{,}000=\$60\). Class M has \(LC_M=\$1{,}620{,}000/30{,}000=\$54\).
Step 2: Express both on a per-1,000 scale. Class K’s loss cost is \(\$60(1{,}000)=\$60{,}000\) per 1,000 location-years. Class M’s is \(\$54(1{,}000)=\$54{,}000\) per 1,000 location-years.
Step 3: Calculate the relativity. Using Class M as the reference, \(R_K=\$60{,}000/\$54{,}000=60/54=10/9=1.111\) to three decimals. Equivalently, using the per-location-year figures gives \(\$60/\$54=1.111\).
Interpretation and check. Class K’s estimated loss cost is \(1.111-1=0.111\), or approximately 11.1%, above Class M’s. The common scale factor of 1,000 cancels in the ratio, confirming that the chosen reporting scale does not affect the relativity.
Adjustments Must Be Consistent Across Classes
Loss experience may need to be adjusted before class loss costs are compared. For example, a calculation may use losses brought to a common development or trend basis. The key requirement for this tutorial’s ratio is that each class loss cost represents the same target period and valuation basis. Applying different adjustment factors may be appropriate when the class data warrant them, but the analyst must be clear about what each adjustment represents and apply it to the correct class data.
The exposure denominator must also match the losses being used. A total for one coverage or period should not be divided by exposures for a different coverage or period. Likewise, if a class’s experience is expressed per policy-year while another is expressed per vehicle-year, their dollar amounts per exposure do not provide a direct comparison unless the exposure definitions are made compatible.
Worked Example: Adjusting Class Losses Before Comparing
A hypothetical insurer has two classes of equipment coverage. Class P has $540,000 of reported losses, an indicated development factor of 1.20, a trend factor of 1.05, and 10,000 exposure units. Class Q has $420,000 of reported losses, an indicated development factor of 1.10, a trend factor of 1.08, and 8,000 exposure units. All exposures use the same definition. Calculate Class P’s relativity to Class Q using the adjusted losses.
Assumptions. The supplied development and trend factors are applicable to their respective class losses and bring both classes to a common basis. No interest rate or mortality assumption applies. The provided aggregate losses are adjusted deterministically; no parametric distribution is assumed.
Step 1: Adjust Class P’s losses. First, development gives \(\$540{,}000(1.20)=\$648{,}000\). Applying the trend factor gives \(\$648{,}000(1.05)=\$680{,}400\).
Step 2: Adjust Class Q’s losses. Development gives \(\$420{,}000(1.10)=\$462{,}000\). Applying the trend factor gives \(\$462{,}000(1.08)=\$498{,}960\).
Step 3: Calculate adjusted loss costs. For P, \(\$680{,}400/10{,}000=\$68.04\) per exposure. For Q, \(\$498{,}960/8{,}000=\$62.37\) per exposure.
Step 4: Calculate the relativity. Taking Q as the reference, \(R_P=\$68.04/\$62.37=68.04/62.37=1.090909\ldots\), or 1.091 to three decimals.
Interpretation and check. Class P’s adjusted loss cost is about 9.1% higher than Class Q’s. Multiplying Q’s loss cost by the unrounded factor gives \(\$62.37(1.090909\ldots)=\$68.04\), as required. Adjusting losses but omitting exposures would compare aggregate dollars rather than loss costs.
Common Mistakes and Exam Traps
The most important calculation check is to write the requested comparison in words before dividing: “class of interest divided by reference class.” If Class A has a loss cost of $42 and the reference has $56, then:
A common incorrect calculation reverses the ratio: \(56/42=1.333\). That number describes the reference loss cost relative to Class A, not Class A relative to the reference. The correction is to put the class named in the requested relativity in the numerator and the chosen reference in the denominator.
- Using total losses instead of loss costs. A class with more exposures may have more total losses even if its expected loss per exposure is lower. Divide losses by exposures before comparing classes.
- Mixing exposure units. The numerator and denominator of each loss cost must use the same type and scale of exposure. A consistent scale change cancels in the relativity; an inconsistent one does not.
- Confusing the factor with a percentage change. A factor of 0.80 means 20% lower than the reference, not 80% lower. A factor of 1.20 means 20% higher.
- Treating the relativity as a complete rate. It compares loss costs. It does not, by itself, add expense provisions, profit, or other rate components.
- Ignoring the loss basis. Ratios based on losses adjusted to inconsistent periods or valuation bases may not represent a meaningful comparison of expected loss costs.
Check Your Understanding
For each problem, calculate the requested class loss cost or relativity. Unless stated otherwise, assume class losses and exposures are on consistent bases. Round final relativities to four decimals where needed.
- A hypothetical delivery-vehicle class has $594,000 of losses and 9,000 vehicle-year exposures. A reference class has $750,000 of losses and 15,000 vehicle-year exposures. What is the first class’s relativity to the reference? (Answer: 1.3200)
- Class U has $1,260,000 of losses over 18,000 exposure units; Class V, the reference, has $1,500,000 over 25,000 exposure units. Calculate both loss costs and U’s relativity to V. (Answer: \(LC_U=\$70\), \(LC_V=\$60\), and \(R_U=1.1667\))
- Class C has $880,000 of losses, a development factor of 1.10, a trend factor of 1.05, and 16,000 exposures. Reference Class D has $720,000 of losses, a development factor of 1.05, a trend factor of 1.10, and 12,000 exposures. Calculate C’s adjusted loss-cost relativity to D. (Answer: \(LC_C=\$63.525\), \(LC_D=\$69.30\), and \(R_C=0.9167\))
- A class loss cost is $96 per exposure and its selected reference loss cost is $120 per exposure. State the class relativity and interpret it as a percentage difference. (Answer: 0.8000; the class loss cost is 20% below the reference)
- Class X has losses of $1,080,000 and 12,000 exposures; reference Class Y has losses of $1,260,000 and 18,000 exposures. A candidate reports \(R_X=1.1667\). Check the calculation and state the correct factor and interpretation. (Answer: \(LC_X=\$90\), \(LC_Y=\$70\), so \(R_X=1.2857\); X’s loss cost is about 28.6% above Y’s)