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CAS Exam 5 · Pure Premium Indication Method · Tutorial 273 of 989

Worked Example: Single-Period Pure Premium Indication

Learn to turn one period’s projected loss cost and supplied premium-based loadings into an indicated rate, then compare it with the current rate.

Intermediate 10 min read

What You'll Learn

  • Calculate a single-period pure premium from selected projected losses and matching exposures
  • Convert a pure premium to an indicated rate when loadings are expressed as shares of premium
  • Separate the indicated rate calculation from the percentage change calculation
  • Check whether the loading assumptions and exposure basis fit the formula
  • Recognize and correct common errors involving the loading denominator

Purpose of a Single-Period Pure Premium Indication

A pure premium indication begins with projected losses per exposure rather than with the current premium level. For a single experience period, the selected projected losses are divided by the corresponding exposures to estimate the loss cost per exposure. A premium-based loading is then allowed for expenses, profit, and contingencies to obtain an indicated rate.

The calculation has two distinct outputs. The indicated rate is the rate per exposure implied by the projected loss cost and supplied loading. The percentage change compares that indicated rate with the current rate. The current rate is needed for the comparison, but it is not used to calculate the loss cost itself.

This tutorial treats the loss, exposure, and loading inputs as already selected for the indication. It does not make new selections for development, trend, exposure basis, or experience-period weighting. The method is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Definition: For one experience period, the pure premium is the selected projected losses divided by the matching number of exposures. It is the projected loss cost per exposure, before the supplied premium-based loading. The indicated rate is the rate per exposure that covers this pure premium after allowing for the loading.

Formula and Assumptions

Let \(L\) be the selected projected losses for the period, \(E\) the corresponding exposure count, and \(r_c\) the current rate per exposure. Let \(v\) be the total loading fraction expressed as a share of premium. For example, if expenses, profit, and contingencies are each given as percentages of premium, add those percentages to obtain \(v\).

The formula below assumes that \(L\) and \(E\) refer to the same coverage, population, period, and exposure definition; that the projected losses are expressed on the basis intended for the indication; and that the loading components are percentages of premium. It also assumes there is no separate fixed-dollar-per-exposure provision. If any of these conditions is not met, the inputs or formula must be adjusted before calculating an indicated rate.

$$ \text{Pure premium}=\frac{L}{E} $$
$$ r_{\text{ind}}=\frac{L/E}{1-v} $$

The loading fraction must satisfy \(0\leq v<1\). If \(v\geq 1\), the denominator is zero or negative, so this formula cannot produce a meaningful positive indicated rate. For the percentage change, compare the indicated rate with the current rate:

$$ \%\text{ change}=\frac{r_{\text{ind}}-r_c}{r_c}\times 100\% $$

A positive result is an indicated increase; a negative result is an indicated decrease. Keep the unrounded indicated rate for this comparison, and round only the final reported results.

Why the Loading Is Divided Out

The indicated rate must pay for both losses and the loading. Under the stated assumption that the loading is a share of premium, the remaining share of premium available for losses is \(1-v\). The pure premium is the loss amount per exposure, so it must equal this remaining share of the indicated rate.

$$ \frac{L}{E}=r_{\text{ind}}(1-v) $$

Solving this equation for the indicated rate gives the formula. This balance explains why one divides by \(1-v\), rather than multiplying the pure premium by \(1+v\): the loading is defined as a fraction of the final premium, not as a fraction of the loss cost.

$$ r_{\text{ind}}=\frac{L/E}{1-v} $$

Worked Applications

Worked Example: Indicating an Increase From One Period

A hypothetical equipment-protection segment has 8,000 exposures and selected projected losses of $286,000 for the experience period. The current rate is $44.00 per exposure. The supplied loading is 18% for expenses, 4% for profit, and 3% for contingencies, each expressed as a share of premium. Calculate the indicated rate and percentage change. Report the rate to the nearest cent and the change to two decimal places.

Method. This is a single-period pure premium indication. First calculate the projected loss cost per exposure, then add the premium-based loading through the denominator.

Step 1: Calculate the pure premium. Divide projected losses by exposures:

$$ \frac{\$286{,}000}{8{,}000}=\$35.75 \text{ per exposure} $$

Step 2: Total the loading. The three components are all shares of premium, so add them:

$$ v=0.18+0.04+0.03=0.25 $$

Thus, 75% of the indicated premium remains for losses. The indicated rate is:

$$ r_{\text{ind}}=\frac{\$35.75}{1-0.25} =\frac{\$35.75}{0.75} =\$47.666666\ldots $$

The indicated rate rounded to the nearest cent is $47.67 per exposure.

Step 3: Compare with the current rate. Use the unrounded indicated rate:

$$ \frac{\$47.666666\ldots-\$44.00}{\$44.00}\times100\% =\frac{\$3.666666\ldots}{\$44.00}\times100\% =8.333333\ldots\% $$

The indicated change is an 8.33% increase. As a check, 75% of the unrounded indicated rate is $35.75, the calculated pure premium.

Worked Example: Indicating a Decrease

A hypothetical small-business policy segment has 16,000 exposures and selected projected losses of $512,000. Its current rate is $42.00 per exposure. The supplied loading is 12% for expenses, 6% for profit, and 2% for contingencies, all as shares of premium. Find the indicated rate and percentage change. Report the rate to the nearest cent and the change to two decimal places.

Step 1: Calculate the pure premium.

$$ \frac{\$512{,}000}{16{,}000}=\$32.00 \text{ per exposure} $$

Step 2: Calculate the total loading.

$$ v=0.12+0.06+0.02=0.20 $$

The share of premium available for losses is \(1-0.20=0.80\). Therefore:

$$ r_{\text{ind}}=\frac{\$32.00}{0.80}=\$40.00 \text{ per exposure} $$

Step 3: Calculate the percentage change.

$$ \frac{\$40.00-\$42.00}{\$42.00}\times100\% =\frac{-\$2.00}{\$42.00}\times100\% =-4.761904\ldots\% $$

The indicated rate is $40.00 per exposure, implying a 4.76% decrease. The sign is important: the indicated rate is below the current rate. As a check, \(0.80\times \$40.00=\$32.00\), which recovers the pure premium.

Worked Example: Avoiding the Wrong Loading Base

A hypothetical event-liability segment has 6,400 exposures, selected projected losses of $204,800, and a current rate of $40.00 per exposure. The loading is 15% for expenses, 5% for profit, and 5% for contingencies, each stated as a share of premium. Calculate the indicated rate and percentage change. Report the rate to the nearest cent and the change to two decimal places.

Step 1: Calculate the pure premium.

$$ \frac{\$204{,}800}{6{,}400}=\$32.00 \text{ per exposure} $$

Step 2: Add the premium shares.

$$ v=0.15+0.05+0.05=0.25 $$

The loss portion of the rate is 75%, so:

$$ r_{\text{ind}}=\frac{\$32.00}{0.75} =\$42.666666\ldots =\$42.67 \text{ per exposure, rounded} $$

Step 3: Compare to the current rate. Keep the unrounded indicated rate in the calculation:

$$ \frac{\$42.666666\ldots-\$40.00}{\$40.00}\times100\% =6.666666\ldots\% $$

The result is an indicated rate of $42.67 per exposure, or a 6.67% increase.

Common mistake and correction. A tempting but incorrect calculation is to multiply the pure premium by \(1+v\): \(\$32.00\times1.25=\$40.00\). That treats the 25% loading as a share of losses, which is not what the problem states. Because it is a share of premium, the loss portion is 75% of the indicated rate; the correct calculation is \(\$32.00/0.75=\$42.666666\ldots\). The check is \(0.75\times \$42.666666\ldots=\$32.00\).

Interpreting the Result and Avoiding Errors

A pure premium is a loss cost per exposure, not a complete premium rate when there are premium-based loadings. The indicated rate includes the amount needed for the supplied loading. Therefore, report the indicated rate and the pure premium as distinct quantities, even when both are stated in dollars per exposure.

The percentage change is also distinct from the indicated rate. It answers how far the indicated rate is above or below the current rate, relative to the current rate. Dividing the difference by the indicated rate instead would answer a different question and would not give the rate change from the current level.

Before finalizing a single-period calculation, check the following:

  • Matching basis: The projected losses and exposure count must describe the same coverage and period and use compatible exposure units.
  • Loading interpretation: Add the supplied components only when they are expressed on the same basis. Use \(1-v\) as the denominator when their total is a share of premium.
  • Current-rate role: Use the current rate in the percentage-change calculation, not in the pure premium calculation.
  • Direction: A positive difference \(r_{\text{ind}}-r_c\) indicates an increase; a negative difference indicates a decrease.
  • Rounding: Carry full precision through the indication and percentage-change calculations, and round only the final reported values.

A single-period indication is especially useful as a transparent calculation: every step can be traced from the selected losses and exposures to the final rate. Its result depends on those selected inputs and the stated loading assumptions; it should not be interpreted as a forecast independent of those choices.

Key takeaway: For one experience period, divide selected projected losses by matching exposures to obtain the pure premium. When the loading is a share of premium, divide that pure premium by one minus the total loading fraction. Then compare the indicated rate with the current rate using the current rate as the percentage-change denominator.

Check Your Understanding

For each problem, calculate the pure premium, indicated rate, and percentage change where requested. Round the indicated rate to the nearest cent and the percentage change to two decimal places.

  1. A hypothetical repair plan has 4,200 exposures and selected projected losses of $176,400. The loading is 14% for expenses, 4% for profit, and 2% for contingencies, each as a share of premium. The current rate is $50.00. What are the indicated rate and percentage change? (Answer: $52.50 per exposure; 5.00% increase)
  2. A hypothetical farm-equipment segment has 9,000 exposures, projected losses of $315,000, and a current rate of $48.00. Its total premium-based loading is 25%. Find the indicated rate and percentage change. (Answer: $46.67 per exposure; 2.78% decrease)
  3. A hypothetical contractor policy segment has 12,000 exposures and selected projected losses of $468,000. Expenses are 17% of premium, profit is 5%, and contingencies are 3%. The current rate is $50.00. Find the indicated rate and percentage change. (Answer: $52.00 per exposure; 4.00% increase)
  4. A hypothetical event policy segment has 5,500 exposures and selected projected losses of $231,000. The supplied premium-based loading is 10% for expenses, 7% for profit, and 3% for contingencies. The current rate is $55.00. Find the indicated rate and percentage change. (Answer: $52.50 per exposure; 4.55% decrease)
  5. A hypothetical equipment warranty segment has 20,000 exposures and selected projected losses of $850,000. Its loading is 15% for expenses, 4% for profit, and 1% for contingencies, all as shares of premium. The current rate is $49.00. Find the indicated rate and percentage change. (Answer: $53.13 per exposure; 8.42% increase)

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