Why Convert Class Rates Into Relativities?
A class-rate table gives the rate assigned to each level of a classification. A relativity expresses each class rate in comparison with one designated reference class. The conversion is a ratio, but the choice of denominator matters: using the wrong class rate changes every relativity and reverses the intended comparison.
This calculation is useful when a rating plan is presented as a reference rate and a set of class factors rather than as a list of dollar rates. It is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus. The central task here is narrow: convert rates on a consistent basis into relativities and check that the conversion preserves the original rates.
Conditions for a Valid Conversion
The division is simple, but the rates must be comparable. Before calculating, confirm that every rate in the table uses the same coverage, time period, exposure unit, and other rating assumptions. If the table is for annual rates per insured vehicle, for example, the reference rate and every class rate must use that same annual per-vehicle basis.
For a one-factor class table, other rating characteristics are treated consistently across the classes. If the entries differ in coverage or in another rating dimension as well as the class being compared, their ratio cannot be interpreted as the relativity for that class alone without first accounting for those differences. The reference rate must also be nonzero; a zero denominator does not define a relativity. In ordinary rate tables, rates are nonnegative, and a positive reference rate is required for this conversion.
The calculation does not require a probability distribution, interest assumption, or mortality assumption. It uses the rates in the table as given. It also does not determine whether those rates are appropriate; it only expresses their relative size on the stated basis.
Formula and Justification
Suppose the table is to be represented by one reference rate and a factor for each class. For class \(j\), write the rate as the reference rate multiplied by that class's relativity:
Solving this equality for the factor gives the conversion formula:
This also explains why the reference class must have relativity \(1\). Substituting \(j=\mathrm{ref}\) into the rate representation gives \(r_{\mathrm{ref}}=r_{\mathrm{ref}}\rho_{\mathrm{ref}}\). Since the reference rate is positive, division by it yields \(\rho_{\mathrm{ref}}=1\). Multiplying each calculated relativity by the reference rate must recover that class's original rate; this is the most direct check on the conversion.
The relativity describes a ratio, not a dollar difference. A relativity of \(1.20\) means the class rate is 1.20 times the reference rate, or 20% greater. A relativity of \(0.80\) means the class rate is 0.80 times the reference rate, or 20% lower. A class rate that is 20 dollars above the reference rate does not necessarily have a relativity of \(1.20\); the ratio depends on the size of the reference rate.
Check that the class rates and reference rate apply to the same coverage, period, and exposure unit.
Use the reference rate explicitly designated in the problem; do not assume it is the first or lowest rate.
Use the same denominator for every class, including the reference class.
Confirm that the reference relativity is 1 and that multiplying each relativity by the reference rate reproduces its class rate.
Worked Conversions
Worked Example: Converting a Drone-Equipment Rate Table
A hypothetical insurer has annual rates per insured drone for four equipment classes. All rates use the same coverage and annual exposure basis. The standard-equipment class is the reference class. Convert the table to relativities and check the result.
| Equipment class | Annual class rate |
|---|---|
| Compact | $252 |
| Standard (reference) | $315 |
| Rugged | $378 |
| Industrial | $441 |
Method and assumptions. Divide each annual class rate by the standard class rate, \(r_{\mathrm{ref}}=\$315\). No interest, mortality, or loss distribution is used; this is a deterministic comparison of rates on the same basis.
For Compact: $$ \rho_{\mathrm{Compact}}=\frac{252}{315}=0.80. $$
For the reference Standard class: $$ \rho_{\mathrm{Standard}}=\frac{315}{315}=1.00. $$
For Rugged: $$ \rho_{\mathrm{Rugged}}=\frac{378}{315}=1.20. $$
For Industrial: $$ \rho_{\mathrm{Industrial}}=\frac{441}{315}=1.40. $$
Check. Multiplying each relativity by the $315 reference rate gives \(0.80(315)=\$252\), \(1.00(315)=\$315\), \(1.20(315)=\$378\), and \(1.40(315)=\$441\). Each original rate is recovered. The reference relativity is 1, as required.
Common mistake avoided. The difference between Rugged and Standard is $63, but $63 is a dollar difference, not a relativity. The relativity is the ratio \(378/315=1.20\).
Worked Example: Using the Designated Reference Rate
A hypothetical insurer publishes annual class rates per policy for three service territories. The Central territory is explicitly designated as the reference, even though it is not the first row. Convert the rates to relativities, round the final relativities to four decimal places, and verify them.
| Territory | Annual class rate |
|---|---|
| Harbor | $768 |
| Central (reference) | $640 |
| Ridge | $544 |
| Valley | $896 |
Method and assumptions. All entries are annual rates per policy on the same coverage basis. Divide each by the Central rate of $640. No interest, mortality, or loss distribution is involved.
The calculations before rounding are: $$ \rho_{\mathrm{Harbor}}=\frac{768}{640}=1.2,\qquad \rho_{\mathrm{Central}}=\frac{640}{640}=1,\qquad \rho_{\mathrm{Ridge}}=\frac{544}{640}=0.85,\qquad \rho_{\mathrm{Valley}}=\frac{896}{640}=1.4. $$
Rounded to four decimal places, the relativities are 1.2000, 1.0000, 0.8500, and 1.4000, respectively.
Check. Using the unrounded factors, \(640(1.2)=\$768\), \(640(1)=\$640\), \(640(0.85)=\$544\), and \(640(1.4)=\$896\). The reconstruction matches every rate. The rounding has not changed any of these values.
Common mistake avoided. Dividing the Central rate by Harbor's rate gives \(640/768=0.8333\), but that is the inverse comparison. The requested Harbor relativity uses the Harbor rate in the numerator and the designated Central reference rate in the denominator: \(768/640=1.2\).
Worked Example: Completing and Checking a Relativity Table
A hypothetical insurer lists annual rates per covered unit for four machine classes. The reference class is Class B. Convert each listed rate into a relativity, use the supplied Class D relativity to find its missing rate, and check the completed table.
| Machine class | Annual class rate | Information supplied |
|---|---|---|
| Class A | $480 | Rate listed |
| Class B (reference) | $600 | Rate listed |
| Class C | $720 | Rate listed |
| Class D | Not listed | Relativity is 1.35 |
Method and assumptions. All rates are annual rates per covered unit on the same basis. Use \(r_{\mathrm{ref}}=\$600\). No interest, mortality, or loss distribution is used.
First convert the three listed rates: $$ \rho_A=\frac{480}{600}=0.80,\qquad \rho_B=\frac{600}{600}=1.00,\qquad \rho_C=\frac{720}{600}=1.20. $$
For Class D, the rate representation is \(r_D=r_{\mathrm{ref}}\rho_D\). Substitute the reference rate and supplied relativity: $$ r_D=600(1.35)=\$810. $$
Check. The completed rates reconstructed from their factors are \(600(0.80)=\$480\), \(600(1.00)=\$600\), \(600(1.20)=\$720\), and \(600(1.35)=\$810\). The listed rates are recovered, and the missing rate is consistent with the stated Class D relativity.
Common mistake avoided. The missing rate is found by multiplying the reference rate by 1.35, not by adding 1.35 dollars or 35 dollars. A relativity is a multiplier, so the calculation is \(600\times1.35\).
Common Errors and Practical Checks
A clean conversion should pass several checks before it is used in a rating table. These checks are especially valuable under time pressure because a result can look plausible even when the denominator is wrong.
- Using the wrong reference. If the problem designates a particular class, use its rate as the denominator for every class. The first row, lowest rate, or most common class is not automatically the reference.
- Reversing the ratio. The class being converted goes in the numerator, and the reference class goes in the denominator. Reversing them gives an inverse factor.
- Subtracting instead of dividing. A difference has monetary units; a relativity is a dimensionless ratio. A $50 difference is not itself a factor.
- Mixing rate bases. Comparing a monthly rate with an annual rate, or rates for different coverage limits, does not isolate the class difference. Put the rates on a consistent basis before interpreting their ratio.
- Rounding too early. Keep sufficient precision during calculations and round only the final factors. Then perform the reconstruction check with the unrounded values when possible.
For example, suppose one class rate is $850 and the designated reference rate is $680. The correct factor is \(850/680=1.25\). The incorrect reverse calculation is \(680/850=0.80\). Multiplying the incorrect factor by $680 gives \(0.80(680)=\$544\), not the original $850 class rate, so the reconstruction check exposes the error.
Check Your Understanding
For each problem, use the stated reference class, keep the rate basis consistent, and check the result by reconstruction where appropriate.
- A hypothetical bicycle insurer lists annual rates per insured bicycle of $330 for Basic (reference), $396 for Touring, and $264 for Cargo. Calculate all three relativities and verify the two nonreference rates. (Answer: Basic \(1.00\), Touring \(1.20\), Cargo \(0.80\); reconstructed rates are $396 and $264.)
- A hypothetical equipment plan has rates of $600 for Level 1, $480 for Level 2, and $720 for Level 3. Level 2 is the reference. Find each relativity and identify the class below reference. (Answer: Level 1 \(1.25\), Level 2 \(1.00\), Level 3 \(1.50\); no class is below reference.)
- A hypothetical annual rate table for three storage zones gives $540 for North, $675 for Central (reference), and $810 for South. Convert each rate, then state the percentage difference of North and South from Central. (Answer: North \(0.80\), Central \(1.00\), South \(1.20\); North is 20% lower and South is 20% higher.)
- A hypothetical class table has a reference rate of $525 per covered unit. A class has relativity 1.16. Find its class rate, then check by division. (Answer: Class rate \(=\$609\); check: \(609/525=1.16\).)
- A hypothetical annual policy-rate table lists $680 for Standard (reference), $850 for Extended, and $1,020 for Heavy-Duty. A candidate calculates the Extended factor as \(680/850\). State the correct three relativities and the specific error in that candidate's calculation. (Answer: Standard \(1.00\), Extended \(1.25\), Heavy-Duty \(1.50\); the candidate reversed the numerator and denominator, obtaining the inverse \(0.80\) instead of \(1.25\).)