The Pure Premium Rate Indication Method

What the Pure Premium Method Indicates

A rate indication translates projected losses into a prospective amount per exposure. In the pure premium method, the starting point is projected losses divided by the corresponding exposure count. That result is the loss cost: the expected loss dollars per exposure. If a specified dollar expense allowance per exposure is also included, the result is an indicated rate under the assumptions of the calculation.

The distinction matters. A loss cost describes projected losses; a rate must also account for any included expenses. Comparing the loss cost alone with the current rate can therefore give the wrong indicated rate change. The calculations below assume the projected losses and exposure base have already been brought to the prospective basis intended for the indication.

Definition: Let $L$ be total projected losses for a book of business, $E$ its matching number of exposures, $e$ a specified dollar expense allowance per exposure, and $R_0$ the current average rate per exposure. The pure premium loss cost is $L/E$. If the stated expense allowance is included, the indicated rate is $R_I=L/E+e$, and the indicated rate change is $R_I/R_0-1$.

An exposure is the unit to which the rate applies in the calculation, such as one hypothetical one-year policy or one vehicle-year. The numerator and denominator must refer to the same portfolio and prospective period. The current rate must also be expressed per the same exposure unit, or converted to that basis before comparing it with $R_I$.

This tutorial uses dollar expenses per exposure. It does not add a percentage-of-premium expense or profit provision. If such a provision is part of a separate calculation, simply adding a dollar amount per exposure does not account for it.

Why the Formula Works

The calculation follows from balancing the prospective amount to be funded against the amount collected per exposure. Suppose $L$ is the total expected loss amount for $E$ exposures, and the expense allowance is $e$ dollars per exposure. Total expected losses and included expenses are $L+eE$. If the indicated rate is $R_I$ dollars per exposure, the corresponding total amount is $ER_I$. Equating these totals gives the formula.

ER_I=L+eE \quad\Longrightarrow\quad R_I=\frac{L+eE}{E}=\frac{L}{E}+e

The loss cost $L/E$ is therefore only the loss component of the indicated rate. The addition of $e$ is justified here because the expense allowance is given as a dollar amount for each exposure. The rate change then compares the indicated rate with the current rate: a positive result is an increase, and a negative result is a decrease.

Key distinction: $L/E$ is the projected loss cost per exposure. $L/E+e$ is the indicated rate per exposure when the only additional provision specified is the dollar expense allowance $e$. Do not call the loss cost the final indicated rate when expenses are intended to be included.

Worked Indications

Worked Example: Adding a Dollar Expense Allowance

A hypothetical insurer has 10,000 prospective one-year policy exposures. Projected losses for those exposures are $1,260,000. The indicated expense allowance is $18 per exposure, and the current average rate is $160 per exposure. Calculate the loss cost, indicated rate, and indicated rate change. Name the method and identify a common mistake.

Method. Use the pure premium method: divide projected losses by exposures, add the stated dollar expense per exposure, and compare the indicated rate with the current rate.

Step 1: Calculate the loss cost.

\frac{L}{E} = \frac{\$1{,}260{,}000}{10{,}000} = \$126.00\text{ per exposure}

Step 2: Include the expense allowance. The allowance is $18 per exposure, so the indicated rate is $126+18=\$144.00$ per exposure.

Step 3: Calculate the rate change.

\frac{R_I}{R_0}-1 = \frac{\$144.00}{\$160.00}-1 = 0.90-1 = -0.10

The indicated rate change is a 10.00% decrease. As a check, 90% of the current $160 rate is $144, matching the indicated rate.

Common mistake avoided. Comparing the loss cost alone with the current rate would give $126/160-1=-21.25\%$. That calculation omits the stated $18 expense allowance. It answers a loss-cost comparison, not the requested rate indication including expenses.

Worked Example: Combining Two Exposure Groups

A hypothetical book has two groups of one-year policy exposures. Group A has 4,800 exposures, projected losses of $691,200, and a current rate of $190 per exposure. Group B has 3,200 exposures, projected losses of $537,600, and a current rate of $200 per exposure. The indicated expense allowance is $22 per exposure for the combined book. Calculate the combined indicated rate and rate change. Explain why the current rate must be weighted by exposures.

Method. Combine projected losses and exposures before calculating the book loss cost. Calculate the current average rate as total current premium divided by total exposures, not as the simple average of the two group rates.

Step 1: Combine exposures and projected losses.

$$ E=4{,}800+3{,}200=8{,}000 $$

L=\$691{,}200+\$537{,}600=\$1{,}228{,}800

Step 2: Calculate the combined loss cost.

\frac{L}{E} = \frac{\$1{,}228{,}800}{8{,}000} = \$153.60\text{ per exposure}

Step 3: Add the expense allowance.

R_I=\$153.60+\$22.00=\$175.60\text{ per exposure}

Step 4: Find the current average rate on the same exposure basis. Current premium for Group A is $4{,}800\times\$190=\$912{,}000$. Current premium for Group B is $3{,}200\times\$200=\$640{,}000$. Thus total current premium is $\$912{,}000+\$640{,}000=\$1{,}552{,}000$, and the exposure-weighted current rate is $\$1{,}552{,}000/8{,}000=\$194.00$.

Step 5: Calculate the rate change.

\frac{\$175.60}{\$194.00}-1 = -0.0948453608\ldots \approx -9.4845\%

The combined indicated rate is $175.60 per exposure, and the indicated change is a 9.4845% decrease. The calculation checks because the weighted current rate is based on the same 8,000 exposures as the indicated rate.

Common mistake avoided. Taking the unweighted mean of the current rates gives $(190+200)/2=\$195$, which gives equal importance to groups of different sizes. The correct current average is $194 because Group A has 4,800 exposures and Group B has 3,200.

Worked Example: Weighting Loss Costs and Matching Expenses

A hypothetical portfolio has 2,600 exposures in Group C with projected losses of $286,000, and 1,400 exposures in Group D with projected losses of $168,000. Projected dollar expenses for the combined 4,000 exposures are $45,000. The current average rate is $130 per exposure. Calculate the combined loss cost, indicated rate, and rate change. Round the rate change to four decimal places.

Method. Calculate the portfolio loss cost from the total projected losses divided by total exposures. Convert total expenses to a per-exposure amount before adding them.

Step 1: Total the exposures and losses.

$$ E=2{,}600+1{,}400=4{,}000 $$

L=\$286{,}000+\$168{,}000=\$454{,}000

Step 2: Calculate the combined loss cost.

\frac{L}{E} = \frac{\$454{,}000}{4{,}000} = \$113.50\text{ per exposure}

Step 3: Convert the expenses to dollars per exposure.

e=\frac{\$45{,}000}{4{,}000}=\$11.25\text{ per exposure}

Step 4: Calculate the indicated rate and change.

R_I=\$113.50+\$11.25=\$124.75

\frac{\$124.75}{\$130.00}-1 = -0.0403846153\ldots \approx -4.0385\%

The indicated rate is $124.75 per exposure, a 4.0385% decrease from the current rate. As a check, the included projected loss and expense amount is $\$454{,}000+\$45{,}000=\$499{,}000$, and $\$499{,}000/4{,}000=\$124.75$ per exposure.

Common mistake avoided. The $45,000 is a total expense, not a per-exposure amount. Adding it directly to the $113.50 loss cost would mix total dollars with dollars per exposure. First dividing expenses by exposures puts both components on the same basis.

Interpretation and Exam Traps

An indicated rate is not automatically a rate that will be charged. It is a calculation based on the selected projected losses, exposure count, expense allowance, and current-rate comparison. In this simplified method, the indicated rate includes only the loss cost and any stated dollar expense per exposure. The calculation does not by itself establish whether the underlying assumptions are appropriate or whether other provisions should be included.

Before reporting an indication, check that all quantities share a consistent basis:

Loss basis: Use the projected losses for the intended prospective period, not an unadjusted historical total if the problem supplies prospective losses.
Exposure basis: Divide by exposures corresponding to those losses. If the exposure unit changes, convert the quantities before calculating.
Expense basis: Confirm whether an expense is a total amount or an amount per exposure. Convert it before adding it to a loss cost.
Current-rate basis: Compare the indicated rate with the current average rate per matching exposure unit. For multiple groups, use total current premium divided by total exposures.
Change convention: Compute $R_I/R_0-1$. A value of $-0.04$ is a 4% decrease, not a 4% increase.

A frequent conceptual trap is to use “pure premium” and “indicated rate” as if they always mean the same quantity. In the notation here, the pure premium loss cost is $L/E$; the indicated rate after a specified dollar expense is $L/E+e$. Naming both components makes the calculation easier to audit and prevents expenses from being omitted or counted twice.

Key takeaway: Calculate projected loss cost by dividing projected losses by matching exposures. Add a dollar expense allowance only after expressing it per exposure. Then compare the resulting indicated rate with a current rate on the same basis.

Check Your Understanding

For each hypothetical indication, show the loss cost, indicated rate, and rate change as appropriate. Answers are given at the end of each question for checking.

A portfolio has 5,000 exposures, projected losses of $575,000, an expense allowance of $12 per exposure, and a current rate of $130. Find the indicated rate change. (Answer: loss cost $115; indicated rate $127; decrease of 2.3077%)
A book has 1,200 exposures at a current rate of $240 and 800 exposures at a current rate of $270. Total projected losses are $420,000, and the dollar expense allowance is $15 per exposure. Find the exposure-weighted current rate, indicated rate, and rate change. (Answer: current rate $252; indicated rate $225; decrease of 10.7143%)
For 3,600 exposures, projected losses are $468,000 and total projected expenses are $54,000. The current rate is $150 per exposure. Convert expenses to a per-exposure amount and calculate the indicated rate and rate change. (Answer: expense $15 per exposure; loss cost $130; indicated rate $145; decrease of 3.3333%)
Group A has 900 exposures and projected losses of $99,000. Group B has 2,100 exposures and projected losses of $273,000. The combined expense allowance is $18 per exposure and the current average rate is $145. Find the combined loss cost, indicated rate, and rate change. (Answer: loss cost $124; indicated rate $142; decrease of 2.0690%)
A hypothetical portfolio has 2,400 exposures with projected losses of $312,000 and an expense allowance of $20 per exposure. Its current rate is $155. A calculation reports a loss cost of $130 and an indicated rate change of $-16.1290\%$. Identify the omitted component and give the corrected change. (Answer: the $20 expense allowance was omitted; indicated rate $150; decrease of 3.2258%)

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