Deriving the Basic Pure Premium Indication Formula

From Projected Experience to an Indicated Rate

A pure premium indication starts with a simple quantity: projected losses per exposure. The calculation is useful because it translates a total projected loss amount into an expected cost for one unit of insurance exposure. To turn that cost into an indicated rate, the calculation must also allow for any supplied expenses and profit or contingency provision.

The key is to keep the quantities on a consistent basis. Projected losses and projected exposures must refer to the same coverage, population, and period, and the exposure unit must be the one to which the rate applies. Earlier material on matching losses and exposures addresses that alignment; here, the aligned amounts are used to derive the indication.

Definition: Let $L_P$ be projected losses for the rate period and $E_P$ be the corresponding projected number of exposure units. The projected loss cost, also called the pure premium before any supplied expense and profit provisions, is $$ \text{Loss cost}=\frac{L_P}{E_P}. $$ It is the projected loss amount per exposure unit. It is not, by itself, a premium rate when additional provisions must be funded from premium.

For example, if projected losses are stated in dollars and projected exposures are vehicle-years, the quotient is dollars per vehicle-year. If the exposure base is a different unit, the quotient and the eventual rate are expressed per that unit instead. The arithmetic does not make unlike bases comparable: the loss and exposure inputs must already match.

Deriving the Basic Indication

Suppose the indicated rate is $R$ dollars per exposure unit. With $E_P$ projected exposures, the corresponding premium is $R E_P$. Now assume that expenses and the target profit or contingency provision are each specified as percentages of premium. Let $e$ be the expense proportion and $p$ the profit and contingency proportion. The remaining share of premium available for losses is $1-e-p$.

Set up the premium balance.
Premium must fund projected losses, the expense provision, and the profit or contingency provision. Under the stated percentage-of-premium model, $R E_P=L_P+eR E_P+pR E_P$.

Collect the premium terms.
Subtract the percentage-based provisions from both sides to obtain $R E_P(1-e-p)=L_P$.

Solve per exposure.
Divide by $E_P$, then by $1-e-p$, to obtain the indicated rate from the projected loss cost.

$$ R E_P=L_P+eR E_P+pR E_P $$

R=\frac{L_P/E_P}{1-e-p} =\frac{\text{projected loss cost}}{1-e-p}.

This derivation explains why the percentage provisions appear in the denominator rather than as simple additions to the loss cost. They are percentages of the premium being solved for, not percentages of losses. The formula requires $E_P>0$ and $e+p<1$; if $e+p$ is at least 1, no finite positive rate can leave a positive share of premium to fund losses under this model.

Model assumptions: The projected loss and exposure amounts are on a consistent basis; the rate is uniform for the exposures being indicated; the expense and profit or contingency proportions are percentages of premium; and the provisions are additive shares of premium. The formula does not apply unchanged if a supplied provision is a fixed dollar amount per policy or exposure, or if the expense percentages are defined on a different base.

Worked Applications

Worked Example: Percentage Provisions on Projected Loss Cost

A hypothetical insurer projects $1{,}248{,}000$ dollars of losses for a rating period and $24{,}000$ matching exposure units. The supplied variable expense provision is 18% of premium, and the profit and contingency provision is 5% of premium. Find the indicated rate per exposure unit. Use the basic pure premium indication method.

Step 1: Calculate projected loss cost. $$ \frac{1{,}248{,}000}{24{,}000}=52.00 $$ The projected loss cost is $52.00 per exposure unit.

Step 2: Combine the premium-based provisions. The total share of premium allocated to these provisions is $0.18+0.05=0.23$. The share available for losses is $1-0.23=0.77$.

Step 3: Solve for the indicated rate. $$ R=\frac{52.00}{0.77}=67.532467\ldots $$ Rounded to the nearest cent, the indicated rate is $67.53 per exposure unit.

Check. Using the unrounded rate, the loss-funded share is $0.77(67.532467\ldots)=52.00$. The remaining share, $0.23(67.532467\ldots)=15.532467\ldots$, funds the stated expense and profit provisions. The shares sum to the unrounded rate. This method avoids treating 18% and 5% as percentages of losses.

Worked Example: Why the Denominator Matters

For a hypothetical equipment policy, projected losses are $760{,}000$ dollars across $10{,}000$ matching exposure units. Expenses are 12% of premium and the target profit provision is 5% of premium. Calculate the indicated rate and compare it with the result from incorrectly adding 17% to loss cost.

Step 1: Find loss cost. $$ \frac{760{,}000}{10{,}000}=76.00 $$ The loss cost is $76.00 per exposure unit.

Step 2: Apply the percentage-of-premium formula. The total provision proportion is $0.12+0.05=0.17$, so the loss share of premium is $1-0.17=0.83$. $$ R=\frac{76.00}{0.83}=91.566265\ldots $$ The indicated rate, rounded to the nearest cent, is $91.57 per exposure unit.

Step 3: Check the premium balance. At the unrounded rate, the loss share is $0.83(91.566265\ldots)=76.00$, while expenses and profit together use $0.17(91.566265\ldots)=15.566265\ldots$. These sum to the indicated rate.

Common mistake and correction. Adding 17% to the loss cost gives $76.00(1.17)=88.92$, which is too low. That calculation makes the provisions 17% of the loss cost, but the problem specifies 17% of premium. The correct premium balance solves $R=76+0.17R$, giving $R=76/0.83$, not $76(1.17)$.

Worked Example: Adding a Fixed Expense per Exposure

A hypothetical commercial coverage has projected losses of $630{,}000$ dollars and $9{,}000$ matching exposure units. In addition to those losses, the rating plan calls for a fixed expense of $8 per exposure unit. Variable expenses are 15% of premium, and the profit and contingency provision is 5% of premium. Find the indicated rate per exposure unit.

Step 1: Calculate projected loss cost. $$ \frac{630{,}000}{9{,}000}=70.00 $$ The projected loss cost is $70.00 per exposure unit.

Step 2: Add the fixed per-exposure amount. Unlike a percentage of premium, the fixed expense is already stated per exposure. The amount that must be funded before the percentage-based provisions is $70.00+8.00=78.00$ dollars per exposure.

Step 3: Allow for premium-based provisions. Their combined proportion is $0.15+0.05=0.20$, leaving $0.80$ of premium to fund the loss cost plus fixed expense. $$ R=\frac{70.00+8.00}{1-0.15-0.05} =\frac{78.00}{0.80} =97.50 $$ The indicated rate is $97.50 per exposure unit.

Check. At $97.50, variable expenses are $0.15(97.50)=\$14.625$ per exposure and the profit and contingency provision is $0.05(97.50)=\$4.875$. Total premium funds $70.00 of losses, $8.00 of fixed expense, $14.625 of variable expenses, and $4.875 of provision: $70+8+14.625+4.875=97.50$. This method avoids incorrectly treating the fixed $8 as a percentage of premium or applying it after solving for the rate.

Conditions and Exam Traps

The basic calculation is short, but its inputs and bases determine whether the answer is meaningful. A quotient can be arithmetically correct and still be an invalid indication if the losses do not correspond to the exposures, or if a provision is applied using the wrong base.

Check	Why it matters	Correction if inconsistent
Losses and exposures match	The quotient is intended to represent loss dollars per relevant exposure unit.	Use amounts aligned to the same coverage, population, period, and exposure basis.
Expense provision base	A percentage of premium must be solved as a share of the unknown premium.	Use the denominator $1-e-p$ when both specified provisions are percentages of premium.
Fixed amount versus percentage	A fixed amount per exposure does not vary as a share of premium.	Add it to loss cost before dividing by the remaining premium share.
Exposure unit	The rate inherits the unit used in the exposure count.	State whether the result is, for example, dollars per vehicle-year or dollars per policy.

A useful algebra check is to multiply the proposed rate by the fraction of premium available for losses. Under the percentage-only model, that product must equal projected loss cost: $$ R(1-e-p)=\frac{L_P}{E_P}. $$ If it does not, either the arithmetic or the treatment of the provisions is wrong. When a fixed expense per exposure $f$ is also present, the corresponding check is $R(1-e-p)=L_P/E_P+f$.

Key takeaway: First calculate projected losses per matching exposure unit. Then solve the premium balance using the stated provision bases. For provisions that are percentages of premium, divide the per-exposure amount to be funded by the premium share remaining after those provisions.

Check Your Understanding

For each problem, state the indicated rate per exposure unit and keep the provision base in view.

A hypothetical policy has projected losses of $420,000 over 6,000 matching exposure units. Expenses are 20% of premium and profit is 5% of premium. Find the indicated rate. (Answer: $93.33 per exposure unit)
A hypothetical coverage has projected losses of $1,080,000 over 18,000 exposure units. Variable expenses are 14% of premium and the profit provision is 6% of premium. Find the indicated rate. (Answer: $75.00 per exposure unit)
Projected losses are $945,000 for 15,000 matching exposure units. The plan also requires a fixed expense of $4 per exposure, variable expenses of 16% of premium, and profit of 4% of premium. Find the indicated rate. (Answer: $83.75 per exposure unit)
A hypothetical book has projected losses of $540,000 and 9,000 matching exposure units. Expenses and profit together are 25% of premium. Find the indicated rate per exposure, then calculate the total indicated premium for 12,000 future exposures at that same rate. (Answer: $80.00 per exposure unit; $960,000 total premium)
For a hypothetical product, projected losses are $2,400,000 over 30,000 matching exposure units. There is a fixed expense of $6 per exposure, variable expenses of 12% of premium, and a profit provision of 8% of premium. Find the indicated rate. (Answer: $107.50 per exposure unit)

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