Base Rates and the Reference Risk

Base Rate and Reference Risk

A multiplicative rating plan assigns a rate to each combination of rating characteristics. Its base rate is the rate assigned to one designated combination: the plan’s reference risk. It is not automatically the rate paid by a typical policyholder, nor is it generally the average rate across the portfolio. Keeping these quantities separate matters whenever a rate table is interpreted, summarized, or applied to a particular book of business.

The calculations below assume that all rates refer to the same coverage, rating period, and unit of exposure. A rate might be a pure premium or a rate including other components; whichever basis is used, it must be consistent across the table. The examples use hypothetical rating plans and data.

Definition: In a multiplicative rating plan, the base rate $B$ is the rate for the plan’s designated reference risk. If risk $j$ has applicable relativities $r_{j1}, r_{j2}, \ldots, r_{jk}$, its rate is $$ R_j = B\prod_{m=1}^{k} r_{jm}. $$ The reference risk has a factor of 1 for each rating dimension in this representation, so its rate is $B$. The portfolio average rate is instead the aggregate rate for the risks in a specified portfolio, calculated using their exposures as weights.

The reference risk is a rating-plan definition: it is the profile to which the base rate applies. The portfolio average depends on which risks are in the portfolio and how much exposure each contributes. A plan’s reference profile can be uncommon in the book, and a book’s mix can change without changing the plan’s base rate.

A rating table may display the base rate directly, or it may show rates for individual cells. If the reference-risk cell is identified, its rate is the base rate. If a non-reference cell’s rate is known along with all its factors, the base rate can be recovered by dividing that cell rate by the product of its factors.

Why the Portfolio Average Is Exposure-Weighted

Suppose cell $j$ has rate $R_j$ per unit of exposure and $E_j$ units of exposure. At that rate, the indicated total amount for the cell is $E_jR_j$. Adding those amounts across cells gives the indicated total for the portfolio. Dividing by total exposure gives the portfolio’s average rate per exposure unit.

\bar{R}_{\text{portfolio}} = \frac{\sum_j E_j R_j}{\sum_j E_j}

This formula follows from the meaning of a rate. Each cell contributes rate times exposure to the total; the portfolio rate must reproduce that same total when multiplied by total exposure. Thus, if $\bar{R}_{\text{portfolio}}\sum_j E_j=\sum_j E_jR_j$, solving for the portfolio rate yields the formula above. It is an arithmetic mean only when the cells have equal exposure. If exposures differ, averaging the cell rates without weights gives each cell equal influence regardless of how much business it represents.

When exposures are policy counts, the formula treats each policy as one equal exposure unit. If policies contribute different amounts of exposure, use those exposure amounts instead of policy counts. The resulting average describes the stated portfolio and exposure basis; it does not redefine the plan’s reference risk or base rate.

Calculating and Interpreting the Two Rates

Worked Example: A Reference-Cell Rate and a Portfolio Average

A hypothetical plan has three rating cells. Cell A is the designated reference-risk cell. Its rate is $600 per policy-year. Cell B has a rate of $720 per policy-year, and Cell C has a rate of $900 per policy-year. The portfolio has 100 policy-years in A, 300 in B, and 100 in C. Find the base rate and the portfolio average rate. The rates and exposure amounts are on the same basis.

Method. Identify the rate for the designated reference cell, then calculate total indicated amount and divide by total exposure. This avoids confusing the reference-risk rate with an average across cells.

Base rate. Cell A is the reference-risk cell, so its rate is the base rate: $B=\$600$ per policy-year.

Cell contributions. Cell A contributes $100\times \$600=\$60{,}000$. Cell B contributes $300\times \$720=\$216{,}000$. Cell C contributes $100\times \$900=\$90{,}000$.

The total indicated amount is $\$60{,}000+\$216{,}000+\$90{,}000=\$366{,}000$. Total exposure is $100+300+100=500$ policy-years. Therefore,

\bar{R}_{\text{portfolio}} = \frac{\$366{,}000}{500} = \$732 \text{ per policy-year}.

Check. Applying the portfolio average to all 500 policy-years gives $500\times \$732=\$366{,}000$, matching the sum of the cell contributions. The base rate is $\$600$, while the portfolio average is $\$732$.

Worked Example: Building Cell Rates From the Base Rate

A hypothetical rating plan has a base rate of $500 per unit of exposure for its reference risk. A policy in Cell B has a territory relativity of 1.20 and a vehicle-use relativity of 0.90. A policy in Cell C has factors of 0.80 and 1.10. The portfolio contains 40 exposure units in the reference cell, 100 in B, and 60 in C. Calculate the two non-reference cell rates and the portfolio average.

Method. Multiply the base rate by the factors for each cell, then aggregate using the given exposure weights. This avoids adding percentage changes or treating the base rate as a portfolio summary.

Reference cell. Its factors are both 1, so the rate is $\$500\times 1\times 1=\$500$ per exposure unit.

Cell B. The combined factor is $1.20\times 0.90=1.08$. Its rate is $\$500\times 1.08=\$540$ per exposure unit.

Cell C. The combined factor is $0.80\times 1.10=0.88$. Its rate is $\$500\times 0.88=\$440$ per exposure unit.

The three indicated amounts are $40\times \$500=\$20{,}000$, $100\times \$540=\$54{,}000$, and $60\times \$440=\$26{,}400$. Total indicated amount is $\$100{,}400$, and total exposure is $40+100+60=200$. Thus,

\bar{R}_{\text{portfolio}} = \frac{\$100{,}400}{200} = \$502 \text{ per exposure unit}.

Check. The average is close to, but not equal to, the $500 base rate because the portfolio’s exposure-weighted mix includes both a higher-rated cell and a lower-rated cell. Multiplying $200$ by $\$502$ recovers $\$100{,}400$.

Worked Example: Recovering the Base Rate From a Non-Reference Cell

A hypothetical cell has a rate of $720 per exposure unit. Its two applicable relativities are 1.25 and 0.96. The designated reference-risk cell has 50 exposure units and a rate that is not separately printed. The $720 cell has 30 exposure units. A third cell has rate $528 and 20 exposure units. Find the base rate and the portfolio average rate. Assume all three cell rates use the same basis.

Method. First divide the known non-reference rate by its combined factor to recover the base rate. Then use the cell rates and exposures to calculate the portfolio average. This avoids incorrectly labeling the known non-reference rate as the base.

The known cell’s combined factor is $1.25\times 0.96=1.20$. Since its rate equals base rate times combined factor,

B = \frac{\$720}{1.20} = \$600 \text{ per exposure unit}.

The reference cell therefore contributes $50\times \$600=\$30{,}000$. The known cell contributes $30\times \$720=\$21{,}600$, and the third cell contributes $20\times \$528=\$10{,}560$. Total indicated amount is $\$30{,}000+\$21{,}600+\$10{,}560=\$62{,}160$; total exposure is $50+30+20=100$.

\bar{R}_{\text{portfolio}} = \frac{\$62{,}160}{100} = \$621.60 \text{ per exposure unit}.

Check. The recovered base rate reproduces the known cell rate because $\$600\times 1.20=\$720$. The portfolio average reproduces the total indicated amount because $100\times \$621.60=\$62{,}160$.

Common Mistake: Averaging Cells Without Weights

In the first example, the unweighted average of the three displayed cell rates is $(\$600+\$720+\$900)/3=\$740$. That calculation assigns one-third of the result to each cell, despite Cell B having 300 exposure units while the other cells each have 100. The correct exposure-weighted average is $\$732$. An unweighted average would be appropriate only if the cells had equal exposure, or if the question specifically asked for the unweighted average of the listed cell rates.

Another common error is to call the portfolio average the base rate because it summarizes the book. The base rate is tied to the plan’s named reference risk; the portfolio average is tied to the portfolio’s exposure mix. They may happen to be equal, but equality is not part of the definition.

Key takeaway: Find the base rate by identifying the designated reference risk or undoing the factors for a known cell. Find the portfolio average by dividing total indicated amount by total exposure. These answer different questions.

Check Your Understanding

For each hypothetical rating plan, calculate the requested rate on the stated exposure basis.

A reference-risk rate is $725 per exposure unit. A cell has factors 1.10 and 0.80. What is its rate? (Answer: $638 per exposure unit)
A reference cell has rate $500 and 20 exposure units. Two other cells have rates $650 and $800, with 60 and 20 exposure units, respectively. Find the base rate and portfolio average. (Answer: Base rate $500; portfolio average $650 per exposure unit)
A plan’s base rate is $900. A cell has factors 0.95 and 1.20. The reference cell has 60 exposure units and this cell has 40. Find the cell rate and the portfolio average. (Answer: Cell rate $1,026; portfolio average $950.40 per exposure unit)
A cell rate is $756, and its two factors are 1.05 and 1.20. In a portfolio, the reference cell has 25 exposure units and another cell has 75 exposure units at the known $756 rate. Find the base rate and portfolio average. (Answer: Base rate $600; portfolio average $717 per exposure unit)
A hypothetical plan charges $420 per vehicle-year for its reference risk and $630 per vehicle-year for another cell. The portfolio has 80 vehicle-years in the reference cell and 20 in the other cell. Find the base rate and portfolio average. (Answer: Base rate $420; portfolio average $462 per vehicle-year)

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