Data Requirements of Pure Premium and Loss Ratio Methods

Why Data Requirements Matter

A rate indication is only as meaningful as the data behind it. The pure premium method organizes experience around losses per exposure; the loss ratio method organizes it around losses relative to premium. Both can use the same projected losses, but they need different denominator data and answer the calculation in different units.

The method that is more directly calculable depends on what records are available and how reliable they are. A data set with well-defined exposure counts but incomplete premium detail may support a pure premium indication directly. A data set with credible on-level premium but no usable exposure count may support a loss ratio indication more directly. Neither method makes missing information irrelevant: the selected losses, coverage, experience period, and expense assumptions still need to be appropriate.

Definition: Let $L^P$ denote projected losses for the experience being used, $E^P$ the corresponding projected exposures, and $P^{OL}$ the earned premium from that experience adjusted to the current rate level. A pure premium is projected losses per exposure, $L^P/E^P$. A projected loss ratio is projected losses divided by on-level premium, $L^P/P^{OL}$. Losses must be defined consistently in both calculations, including whether loss adjustment expense is included.

What Each Method Needs

The pure premium method requires a suitable exposure base and projected losses for the same exposures and period. The exposure unit might be a vehicle-year, policy-year, payroll unit, or another defined measure. Its denominator is not premium. The method can also require a current rate or average premium per exposure when the desired result is a percentage change rather than a dollar rate.

The loss ratio method requires projected losses and earned premium adjusted to the current rate level. It also requires the permissible loss ratio under the chosen expense and profit assumptions. Its denominator is premium, not exposure. A premium total at an old rate level cannot simply be substituted for on-level premium: rate changes affect premium even when the underlying exposure and coverage have not changed.

Input or decision	Pure premium method	Loss ratio method
Projected losses	Required	Required
Exposure count and definition	Required to calculate losses per exposure	Not required for the basic loss ratio calculation
Earned premium at current rate level	Not the pure premium denominator; may be needed to express a percentage change	Required to calculate the projected loss ratio
Expense and profit assumptions	Needed to convert loss cost into an indicated rate	Needed to determine the permissible loss ratio
Typical direct output	Indicated dollars per exposure	Indicated factor or percentage change to the rate level

In either method, “projected losses” must refer to the same coverage and experience basis being priced. If the intended rate covers losses and loss adjustment expense, for example, a loss amount that excludes that expense is not a complete numerator unless the omitted amount is handled separately. Development and trend selections may be supplied as inputs; this tutorial treats the selected projected losses as given rather than deriving them.

Formulas Under a Simplified Expense Model

First consider a model with variable expenses equal to proportion $V$ of premium and a target profit and contingency provision equal to proportion $Q$ of premium. Assume there are no fixed expenses, the same provisions apply to the indicated business, and projected losses and exposures are on a consistent basis. Under these assumptions, the pure premium indicated rate per exposure is:

R_{\mathrm{PP}}=\frac{L^P/E^P}{1-V-Q}.

For the loss ratio method, assume the same proportional expense and profit model and define the permissible loss ratio as $1-V-Q$. If $P^{OL}$ is positive, the projected loss ratio is $L^P/P^{OL}$, and the indicated rate-level factor is:

F_{\mathrm{LR}}=\frac{L^P/P^{OL}}{1-V-Q}.

The indicated percentage change is $F_{\mathrm{LR}}-1$. These simplified formulas do not, by themselves, handle fixed dollar expenses. Such expenses require an explicit treatment and the additional data needed to estimate their effect; they should not be silently treated as a percentage of premium.

Why the pure premium formula holds: For one exposure, let $R$ be the indicated premium rate and let $c=L^P/E^P$ be projected loss cost. The premium must fund loss cost $c$, variable expenses $VR$, and profit provision $QR$. Thus $R=c+VR+QR$. Rearranging gives $R(1-V-Q)=c$, so $R=c/(1-V-Q)$, provided $1-V-Q>0$. If that denominator is zero or negative, this model cannot produce a finite positive indicated rate.

The two formulas can be checked against each other when the average current rate per exposure is $R_0=P^{OL}/E^P$, the same losses and exposures are used, and the assumptions are consistent. Then $L^P/P^{OL}=(L^P/E^P)/R_0$. Consequently, the loss ratio factor equals $R_{\mathrm{PP}}/R_0$. This check is not a reason to force agreement when the data bases or assumptions differ.

Worked Example: Calculating a Pure Premium Indication From Exposure Data

A hypothetical equipment insurer selects projected losses of $1,080,000 for 12,000 exposure units. Assume variable expenses are 25% of premium and the target profit and contingency provision is 5% of premium. The current average rate is $140 per exposure. Calculate the indicated rate and percentage change. Assume no fixed expenses.

Method and assumptions. This is a pure premium calculation because projected exposure count is available. The losses, exposure units, and current average rate all refer to the same coverage and period. The expense and profit proportions total $V+Q=0.25+0.05=0.30$, leaving $1-V-Q=0.70$ of premium for losses.

Step 1: Calculate projected loss cost per exposure.

\frac{L^P}{E^P}=\frac{\$1{,}080{,}000}{12{,}000}=\$90.00.

Step 2: Gross up for expenses and profit.

R_{\mathrm{PP}}=\frac{\$90.00}{0.70}=\$128.571428\ldots.

Rounded to cents only at the final step, the indicated rate is $128.57 per exposure.

Step 3: Compare with the current average rate.

\frac{\$128.571428\ldots}{\$140}-1=-0.0816326\ldots.

The indicated change is -8.16%, rounded to two decimal places. As a check, the indicated premium per exposure allocates $90.00 to losses and 30% of $128.571428… (approximately $38.57) to expenses and profit, totaling approximately $128.57. The common mistake avoided is using premium as the pure premium denominator; here the denominator is 12,000 exposure units.

Worked Example: Calculating a Loss Ratio Indication From Premium Data

A hypothetical small-business insurer has projected losses of $2,700,000 and earned premium of $3,600,000 after adjustment to the current rate level. Assume variable expenses are 25% of premium, profit and contingency provisions are 5% of premium, and there are no fixed expenses. Calculate the indication.

Method and assumptions. This is a loss ratio calculation because current-rate-level premium is supplied. Assume the premium and losses cover the same business and period.

Step 1: Calculate the projected loss ratio.

\frac{L^P}{P^{OL}}= \frac{\$2{,}700{,}000}{\$3{,}600{,}000} =0.75.

Step 2: Calculate the permissible loss ratio.

$$ 1-V-Q=1-0.25-0.05=0.70. $$

Step 3: Calculate the factor and percentage change.

F_{\mathrm{LR}}=\frac{0.75}{0.70}=1.071428\ldots, \qquad F_{\mathrm{LR}}-1=0.071428\ldots.

The indicated factor is 1.071, and the indicated change is +7.14%, rounded to three and two decimal places, respectively. A check is that the projected loss ratio, 75%, is above the permissible 70%, so the indication must be an increase. The common mistake avoided is dividing by historical premium before adjusting it to the current rate level.

Choosing the More Direct Calculation

A method is more directly calculable when its required denominator is already available and suitable, not merely when its formula looks shorter. Exposure records may be especially useful when premium is distorted by rate changes, discounts, or incomplete historical rate-level detail. On the other hand, if reliable earned premium at the current rate level is available but exposure counts are missing or not consistently defined, the loss ratio method may be more direct.

“More direct” does not mean “automatically more accurate.” Exposure counts can be incomplete, measured on different bases, or inconsistent with the coverage attached to the losses. Premium can be misclassified, written rather than earned, or inadequately adjusted for changes in rate level. The analyst should establish that the numerator and denominator match before deciding which method to calculate.

Worked Example: Checking Both Methods Against the Same Data

A hypothetical property program has two experience segments. Segment A has 5,000 exposures and projected losses of $520,000; Segment B has 3,000 exposures and projected losses of $440,000. Earned premiums at the old rate level are $600,000 for A and $680,000 for B. Both premium amounts receive a rate-level adjustment factor of 1.20. Assume variable expenses are 25% of premium, profit and contingency provisions are 5%, and there are no fixed expenses. The current average rate is $192 per exposure. Calculate the pure premium and loss ratio indications and check their consistency.

Method and assumptions. Calculate each method using the same combined projected losses and experience. For the loss ratio calculation, first put both premium amounts on the current rate level.

Step 1: Combine projected losses and exposures.

L^P=\$520{,}000+\$440{,}000=\$960{,}000, \qquad E^P=5{,}000+3{,}000=8{,}000.

Step 2: Calculate the pure premium indication.

\frac{L^P}{E^P}=\frac{\$960{,}000}{8{,}000}=\$120.00, \qquad R_{\mathrm{PP}}=\frac{\$120.00}{0.70}=\$171.428571\ldots.

Relative to the current average rate of $192, the indicated factor is $171.428571\ldots/192=0.892857\ldots$, a decrease of 10.71%.

Step 3: On-level the premium and calculate the loss ratio indication.

P^{OL}=(\$600{,}000)(1.20)+(\$680{,}000)(1.20) =\$720{,}000+\$816{,}000=\$1{,}536{,}000.

The projected loss ratio is $\$960{,}000/\$1{,}536{,}000=0.625$. Therefore:

F_{\mathrm{LR}}=\frac{0.625}{0.70}=0.892857\ldots, \qquad F_{\mathrm{LR}}-1=-0.107142\ldots.

The loss ratio indication is also a decrease of 10.71%, rounded to two decimal places. This agreement is a useful check: on-level premium per exposure is $\$1{,}536{,}000/8{,}000=\$192$, the stated current average rate. The common mistake avoided is comparing a pure premium result with a loss ratio result based on premium that was not adjusted to the same rate level.

Common Data Traps

Using premium as exposure. Premium reflects both the amount of insured exposure and the price charged. It is the loss ratio denominator, not the pure premium denominator.
Using written premium without checking the basis. The basic loss ratio comparison here uses earned premium adjusted to the current rate level. Written premium or unadjusted historical premium may not represent that basis.
Mixing periods or populations. Losses for one set of policies cannot be divided by exposures or premium for a different set and treated as a matched indication.
Assuming an exposure count is interchangeable across definitions. A policy-year and a vehicle-year are different measures unless the data and intended rate basis establish how they relate.
Ignoring expense-model differences. The simplified formulas assume proportional expense and profit provisions and no fixed expense. A fixed dollar cost needs an explicit allowance rather than being inserted as though it were a premium percentage.
Treating agreement as proof that inputs are sound. Methods can agree because they share a mistaken loss selection or an inappropriate assumption. Check the data and the calculation independently.

Key takeaway: Pure premium data are organized around projected losses and matching exposures; loss ratio data are organized around projected losses and matching on-level earned premium. Choose the method whose denominator is available and defensible, then verify that the losses, period, coverage, and expense assumptions align.

Check Your Understanding

For each problem, identify the data basis and apply the stated assumptions. Round percentage changes to two decimal places.

A hypothetical insurer has projected losses of $480,000 on 4,000 exposures. Variable expenses are 20% of premium and profit and contingency provisions are 5%. The current average rate is $140 per exposure. Find the pure premium indicated rate and percentage change. Assume no fixed expenses. (Answer: $160.00 per exposure; +14.29%.)
A hypothetical insurer has projected losses of $1,260,000 and on-level earned premium of $1,800,000. Variable expenses are 22% and profit and contingency provisions are 8% of premium. Find the loss ratio indication. (Answer: projected loss ratio 70.00%; permissible loss ratio 70.00%; no change.)
A data extract contains projected losses of $450,000 and earned premium of $600,000 at the current rate level, but no usable exposure count. With variable expenses of 25% and profit and contingency provisions of 5%, which method can be calculated directly from these fields, and what is its indicated change? (Answer: Loss ratio method; +7.14%. The pure premium cannot be calculated without exposures.)
A hypothetical insurer has projected losses of $720,000 on 6,000 matching exposures. Variable expenses are 30% and profit and contingency provisions are 5%. The current average rate is $175 per exposure. Find the indicated rate and percentage change. Assume no fixed expenses. (Answer: $184.62 per exposure; +5.49%.)
Two hypothetical segments have earned premiums of $500,000 and $400,000. Their respective current-rate-level adjustment factors are 1.10 and 1.00. Projected losses for the combined business are $665,000. Variable expenses are 25% and profit and contingency provisions are 5%. Calculate the loss ratio indication, using the adjusted premium. (Answer: On-level premium is $950,000; projected loss ratio is 70.00%; permissible loss ratio is 70.00%; no change.)

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