Base Classes and Reference Risks

Introduction

A classification rating plan needs a point of reference. The rate for that reference risk anchors the plan; other class rates are expressed relative to it. This role can be easy to misread: a base rate is not automatically the average rate paid by the insurer’s current portfolio.

The distinction matters whenever a portfolio contains multiple classes. Its average rate depends both on the rate assigned to each class and on how much exposure is in each class. A base class, by contrast, identifies the risk to which the rating plan assigns a relativity of 1.00. It need not be the most common class, the least risky class, or even a class with current exposure.

Definition: A base class is the classification used as the reference point for a rating plan. Its risk, described here as the reference risk, is assigned a relativity of 1.00. The base rate is the rate assigned to that reference risk. A class relativity expresses another class’s rate as a multiple of the base rate.

The Base Rate and the Portfolio Average

For a simple rating plan, let $R_0$ be the base rate per exposure unit, and let $r_j$ be the relativity for class $j$. The rate for that class is $R_j=R_0r_j$. The base class has $r_j=1.00$, so its rate is $R_0$. This relationship assumes the same rate basis and exposure unit for all classes, and that no other rating factors or adjustments are being applied.

To calculate an average rate for a portfolio, the class rates must be weighted by the corresponding exposures. Let $E_j$ denote the number of exposure units in class $j$. The portfolio average rate is the total premium at the stated class rates divided by total exposure. It is not generally the base rate, nor is it generally the unweighted average of the class rates.

\bar{R} = \frac{\sum_j E_jR_j}{\sum_j E_j} = R_0\frac{\sum_j E_jr_j}{\sum_j E_j}

The quantity $\frac{\sum_j E_jr_j}{\sum_j E_j}$ is the exposure-weighted average relativity for the particular portfolio. It equals 1.00 only when the portfolio’s exposure mix produces a weighted average relativity of 1.00. The base-class relativity being 1.00 does not make that true by itself.

Key distinction: The base rate is the rate for the reference risk. The portfolio average rate is the exposure-weighted average of the class rates represented in the portfolio. They coincide only when the portfolio’s weighted average relativity is 1.00.

Why the Exposure-Weighted Formula Holds

The formula follows directly from the definition of an average rate. In class $j$, applying rate $R_j$ to $E_j$ exposure units produces premium $E_jR_j$. Adding those amounts gives the total premium across the classes. Dividing that total by the total number of exposure units gives premium per exposure unit for the portfolio.

Substituting $R_j=R_0r_j$ into the average shows exactly where the base rate and portfolio mix enter:

\bar{R} = \frac{\sum_j E_j(R_0r_j)}{\sum_j E_j} = R_0\frac{\sum_j E_jr_j}{\sum_j E_j}

The weights $E_j/\sum_k E_k$ are nonnegative and sum to 1, so the portfolio’s average rate is a weighted average of its class rates. This interpretation assumes a common exposure unit and the stated class rates apply directly to all counted exposures. If actual premiums include policy fees, minimum premiums, caps, or other adjustments, dividing actual premium by exposure may not equal this simple weighted average.

Worked Example: Calculating a Portfolio Average

A hypothetical equipment policy uses an annual rate of $720 per equipment-year for its standard reference risk. The plan assigns a relativity of 0.80 to a lower-rated class and 1.35 to a higher-rated class. The portfolio has 50 equipment-years in the standard class, 30 in the lower-rated class, and 20 in the higher-rated class. Find each class rate, the total premium at these rates, and the portfolio average rate.

Model and assumptions. The rates are annual and all exposures are equipment-years. Premium is calculated deterministically as rate multiplied by exposure, with no additional adjustments. No mortality assumption, interest rate, or probability distribution is used.

Method: rate each class, then divide total premium by total exposure. The standard class has relativity 1.00, so its rate is $720 per equipment-year. The lower-rated class rate is $720(0.80)=\$576$, and the higher-rated class rate is $720(1.35)=\$972$.

The premiums by class are:

Standard: $50(\$720)=\$36{,}000$.
Lower-rated: $30(\$576)=\$17{,}280$.
Higher-rated: $20(\$972)=\$19{,}440$.

Total premium is $\$36{,}000+\$17{,}280+\$19{,}440=\$72{,}720$. Total exposure is $50+30+20=100$ equipment-years. Therefore the portfolio average rate is $\$72{,}720/100=\$727.20$ per equipment-year.

Check. The exposure-weighted average relativity is $(50(1.00)+30(0.80)+20(1.35))/100=(50+24+27)/100=1.01$. Multiplying by the base rate gives $1.01(\$720)=\$727.20$, as calculated from total premium.

Common mistake avoided: Treating $720 as the portfolio average because the reference-class relativity is 1.00. The calculation shows that the portfolio mix has a weighted average relativity of 1.01, so its average rate is $727.20, not $720.

Portfolio Mix Can Change the Average

When the rates and relativities in a plan stay fixed, changing the exposure distribution across classes can change the portfolio average rate. This is a mix effect: the average changes because different proportions of the portfolio are charged different class rates. It does not mean the reference risk or its base rate has changed.

Worked Example: Comparing Two Portfolio Mixes

A hypothetical annual service plan has a base rate of $640 per service-year. Its reference class has relativity 1.00, a preferred class has relativity 0.70, and a higher-rated class has relativity 1.50. Portfolio P has 20, 60, and 20 service-years in those classes, respectively. Portfolio Q has 50, 20, and 30 service-years, respectively. Calculate and compare the average rates.

Model and assumptions. Both portfolios use the same annual rates and exposure unit. Premium equals rate times service-years, with no other charges or adjustments. These deterministic calculations require no mortality assumption, interest rate, or probability distribution.

Method: calculate each portfolio’s exposure-weighted average. The class rates are $640(1.00)=\$640$, $640(0.70)=\$448$, and $640(1.50)=\$960$ per service-year.

For Portfolio P, the class premiums are $20(\$640)=\$12{,}800$, $60(\$448)=\$26{,}880$, and $20(\$960)=\$19{,}200$. Total premium is $\$58{,}880$ over 100 service-years, giving an average rate of $\$58{,}880/100=\$588.80$.

The weighted relativity check for Portfolio P is $(20(1.00)+60(0.70)+20(1.50))/100=(20+42+30)/100=0.92$. Thus $0.92(\$640)=\$588.80$.

For Portfolio Q, the class premiums are $50(\$640)=\$32{,}000$, $20(\$448)=\$8{,}960$, and $30(\$960)=\$28{,}800$. Total premium is $\$69{,}760$ over 100 service-years, giving an average rate of $\$69{,}760/100=\$697.60$.

The weighted relativity check for Portfolio Q is $(50(1.00)+20(0.70)+30(1.50))/100=(50+14+45)/100=1.09$. Thus $1.09(\$640)=\$697.60$. Portfolio Q’s average is $\$697.60-\$588.80=\$108.80$ higher per service-year because its exposure mix gives more weight to the higher-rated classes.

Check. Each portfolio has 100 service-years, and the calculated premiums divided by 100 reproduce the stated average rates. The base rate remains $640 in both portfolios.

Common mistake avoided: Concluding that the portfolio average must remain $640 because the rating plan has not changed. The plan is unchanged, but its exposure mix is different.

The Reference Class Need Not Have Current Exposure

A reference class is a rating benchmark, not a requirement that the current portfolio contain exposures in that class. A plan can use a defined reference risk to anchor its class rates even when no policy currently occupies that class. The average for the actual portfolio is still calculated from the rates and exposures that are present.

Worked Example: A Base Class With No Exposure

A hypothetical annual equipment plan uses $1,050 per machine-year for its standard reference risk. A lower-rated class has relativity 0.65, and an elevated-risk class has relativity 1.40. The current book has no standard-class exposures, 80 machine-years in the lower-rated class, and 20 in the elevated-risk class. Find the portfolio average and assess the statement, “There is no standard-class exposure, so the base rate cannot be used.”

Model and assumptions. All rates are annual per machine-year; premiums equal rate times exposure, with no additional adjustments. No mortality assumption, interest rate, or probability distribution is used.

Method: apply the stated relativities, then average over actual exposure. The standard reference risk has rate $1,050, even though its exposure count is zero. The lower-rated class rate is $1{,}050(0.65)=\$682.50$, and the elevated-risk class rate is $1{,}050(1.40)=\$1{,}470$.

The standard class contributes $0(\$1{,}050)=\$0$ premium. The lower-rated class contributes $80(\$682.50)=\$54{,}600$, and the elevated-risk class contributes $20(\$1{,}470)=\$29{,}400$. Total premium is $\$84{,}000$; total exposure is $80+20=100$ machine-years. The portfolio average is $\$84{,}000/100=\$840$ per machine-year.

As a check, the weighted average relativity is $(80(0.65)+20(1.40))/100=(52+28)/100=0.80$, and $0.80(\$1{,}050)=\$840$. The statement is incorrect: the base rate remains the rate for the reference risk and can anchor the other class rates without current exposure in the reference class. It is not the average rate for this portfolio.

Common mistake avoided: Requiring a positive exposure count in the base class before using the base rate. The base class defines the comparison point; the portfolio average uses only the exposures actually present.

Common Interpretation Traps

A few distinctions help keep base-class calculations and portfolio summaries separate:

A relativity of 1.00 is a reference convention. It states that the base class rate equals the base rate. It does not state that the class has average risk relative to the portfolio.
The base rate is not necessarily a portfolio average. It equals the portfolio average only if the portfolio’s exposure-weighted average relativity is 1.00.
Do not average class rates equally unless the exposures are equal. The relevant weights are exposure units, not the number of class labels.
Use a consistent denominator. If rates are per policy-year, the exposure weights must be policy-years. Mixing exposure bases makes the resulting average uninterpretable.
Distinguish a rate average from a policy premium average. The formula here calculates premium per exposure unit. Average premium per policy requires policy counts and can differ when policies have different numbers of exposure units.

These checks are useful when reviewing a rating plan or interpreting an indicated average. A rate attached to a reference risk answers a different question from the average premium per exposure unit for a particular portfolio. A sound calculation identifies which quantity is requested, applies the correct exposure weights, and states the basis of the result. This treatment is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Check Your Understanding

For each problem, calculate the requested quantity using the stated exposure basis. Assume premiums equal class rate times exposure, with no additional adjustments.

A hypothetical annual plan has a base rate of $900 per unit-year. The reference class has relativity 1.00 and 25 unit-years; another class has relativity 0.80 and 75 unit-years. Find total premium and the portfolio average rate. (Answer: total premium $76,500; average rate $765 per unit-year.)
A base rate is $500 per contract-year. Class A has relativity 1.20 and 10 contract-years; Class B has relativity 0.60 and 30 contract-years. Calculate the portfolio average rate and its weighted average relativity. (Answer: average rate $375 per contract-year; weighted average relativity 0.75.)
A hypothetical equipment plan has a base rate of $1,200 per machine-year. The base class has relativity 1.00, Class L has relativity 0.75, and Class H has relativity 1.25. There are 0, 40, and 60 machine-years in those classes, respectively. Find the portfolio average rate and explain whether the base rate is still defined. (Answer: average rate $1,260 per machine-year; the base rate remains defined for the reference risk despite zero base-class exposure.)
A base rate is $800 per exposure-year, with relativities 1.00, 0.90, and 1.40 for the reference, lower-rated, and higher-rated classes. A portfolio has 30, 50, and 20 exposure-years, respectively. Calculate its total premium and average rate, then compare the average with the base rate. (Answer: total premium $82,400; average rate $824 per exposure-year, which is $24 above the base rate.)
A candidate says that a plan with class rates $700, $350, and $1,050 has an average rate of $700 because that is the base-class rate. The corresponding exposures are 80, 10, and 10 units. Calculate the correct portfolio average and identify the reasoning error. (Answer: average rate $700 per exposure unit; the conclusion happens to be correct because the exposure-weighted relativity is 1.00, not merely because one class has rate $700.)

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