Worked Example of a Multi-Period Indication Setup

Why a Multi-Period Setup Needs Care

An indication using more than one experience period requires a consistent comparison across periods. Premium and loss selections must refer to matching experience, be adjusted to the intended basis, and receive the weights specified for the analysis. If any of these steps is inconsistent, the resulting factor may be arithmetically correct but answer the wrong question.

This tutorial focuses on the setup and calculation: combine aligned premiums and selected losses, then compare the resulting loss amount with the loss amount allowed by the target loss ratio. It is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Definition: For experience period $j$, let $P_j$ be the premium on the selected common rate and exposure basis, and let $L_j$ be the selected loss amount projected to the intended prospective basis. Let $w_j$ be the weight assigned to the period, with $w_j\geq 0$ and $\sum_j w_j=1$. The weighted premium and weighted selected losses are $\sum_j w_jP_j$ and $\sum_j w_jL_j$, respectively.

For a common target loss ratio $T$, the overall indication factor is the weighted selected loss amount divided by the target loss ratio applied to the weighted premium. The factor is relative to the premium basis represented by the aligned premiums: a factor above 1 indicates an increase, and a factor below 1 indicates a decrease. The corresponding percentage change is the factor minus 1.

I=\frac{\sum_j w_jL_j}{T\sum_j w_jP_j}

The weights apply to both sides of the comparison. They describe how much each period contributes to the combined experience; they are not, in general, the weights to use when averaging the periods’ individual indication factors. The examples below show how to set up the inputs, carry out the calculation, and check the result.

Deriving the Combined Indication Relationship

The relationship follows from the meaning of the target loss ratio. On a common basis, the target loss amount associated with premium $P$ is $TP$. For multiple weighted periods, the combined premium is $\sum_j w_jP_j$, so the target loss amount on that premium is $T\sum_j w_jP_j$. The experience selection to be supported is $\sum_j w_jL_j$. An indicated factor $I$ scales the target loss amount to equal that selected loss amount:

I\left(T\sum_j w_jP_j\right)=\sum_j w_jL_j

Dividing both sides by the positive target loss amount gives the formula for $I$. This derivation assumes that the same target loss ratio applies to every period and that all premium and loss inputs are aligned to the same intended basis. If target loss ratios differ by period, the weighted target loss amount must instead be calculated as $\sum_j w_jT_jP_j$; using one common $T$ would not represent those period-specific targets.

The calculation also presumes that the selected losses and premiums are the appropriate inputs for the indication being made. In particular, a premium must not be paired with losses from a different experience period or a different coverage basis. Adjustments such as loss projection and premium on-leveling should be applied to the relevant inputs before they are combined, not silently omitted or applied twice.

Worked Multi-Period Setups

Worked Example: Combining Two Aligned Periods

A hypothetical insurer selects two experience periods for a product. The premiums are already on the same rate and exposure basis, and the losses are already projected to the same prospective period. Period 1 has premium of $800,000 and selected losses of $520,000; period 2 has premium of $1,200,000 and selected losses of $720,000. The respective weights are 40% and 60%, and the common target loss ratio is 65%. Calculate the overall indicated factor and percentage change.

Method. Weight the aligned premium and loss amounts separately, then divide weighted losses by the target loss ratio times weighted premium. This avoids treating period weights as dollar amounts or applying the target ratio to only one period.

Weighted premium. The period 1 contribution is $0.40\times \$800{,}000=\$320{,}000$. The period 2 contribution is $0.60\times \$1{,}200{,}000=\$720{,}000$. Thus:

\sum_j w_jP_j=\$320{,}000+\$720{,}000=\$1{,}040{,}000

Weighted selected losses. The contributions are $0.40\times \$520{,}000=\$208{,}000$ and $0.60\times \$720{,}000=\$432{,}000$. Therefore:

\sum_j w_jL_j=\$208{,}000+\$432{,}000=\$640{,}000

Target loss amount and factor. The target loss amount is $0.65\times \$1{,}040{,}000=\$676{,}000$. The indicated factor is:

I=\frac{\$640{,}000}{\$676{,}000}=0.946745\ldots\approx 0.9467

The percentage change is $0.946745\ldots-1=-0.053255\ldots$, or approximately a 5.33% decrease. As a check, the indicated target loss amount is $0.946745\ldots\times \$676{,}000=\$640{,}000$, matching the weighted selected losses. The factor is rounded to four decimal places and the percentage change to two decimal places.

Common mistake avoided: Comparing $640,000 of weighted losses with $676,000 of target losses gives the factor directly. Reversing that ratio would produce a factor above 1 and incorrectly imply an increase.

Worked Example: Adjusting Inputs Before Weighting

For a hypothetical coverage, an analyst has three experience periods. The premiums require on-level factors of 1.10, 1.00, and 0.95. The selected ultimate losses require projection factors of 1.08, 1.04, and 1.02. The raw premiums and selected ultimate losses are, respectively: period A, $600,000 and $390,000; period B, $900,000 and $585,000; period C, $1,400,000 and $910,000. The period weights are 20%, 30%, and 50%, and the common target loss ratio is 68%. Find the indicated factor.

Method. First align each premium and loss to the stated basis. Then apply each period's weight to both aligned amounts. This avoids weighting raw inputs that are not yet comparable.

Align the premiums. Period A: $\$600{,}000\times1.10=\$660{,}000$. Period B: $\$900{,}000\times1.00=\$900{,}000$. Period C: $\$1{,}400{,}000\times0.95=\$1{,}330{,}000$.

Project the losses. Period A: $\$390{,}000\times1.08=\$421{,}200$. Period B: $\$585{,}000\times1.04=\$608{,}400$. Period C: $\$910{,}000\times1.02=\$928{,}200$.

Weight and sum premiums. The weighted contributions are $0.20\times \$660{,}000=\$132{,}000$, $0.30\times \$900{,}000=\$270{,}000$, and $0.50\times \$1{,}330{,}000=\$665{,}000$. Thus the weighted premium is $\$132{,}000+\$270{,}000+\$665{,}000=\$1{,}067{,}000$.

Weight and sum losses. The contributions are $0.20\times \$421{,}200=\$84{,}240$, $0.30\times \$608{,}400=\$182{,}520$, and $0.50\times \$928{,}200=\$464{,}100$. Thus weighted selected losses equal $\$84{,}240+\$182{,}520+\$464{,}100=\$730{,}860$.

Calculate the indication. The target loss amount is $0.68\times \$1{,}067{,}000=\$725{,}560$. Therefore:

I=\frac{\$730{,}860}{\$725{,}560}=1.0073047\ldots\approx 1.0073

The implied change is $1.0073047\ldots-1=0.0073047\ldots$, or approximately a 0.73% increase. Checking, $1.0073047\ldots\times \$725{,}560=\$730{,}860$, subject only to displayed rounding. The factor is rounded to four decimal places and the percentage change to two decimal places.

Common mistake avoided: Applying the loss projection factors to premiums, or the premium on-level factors to losses, would adjust the wrong quantities. Each factor is applied only to the input and purpose stated in the problem.

Worked Example: Why a Ratio of Weighted Totals Is Not an Average of Factors

A hypothetical insurer has two aligned periods. Period 1 has premium of $400,000 and a loss ratio of 55%; period 2 has premium of $1,600,000 and a loss ratio of 70%. The respective period weights are 25% and 75%, and the common target loss ratio is 65%. Calculate the overall factor from weighted amounts, then compare it with the weighted average of the two individual factors.

Method. Convert each loss ratio to a loss amount using that period's premium. Apply the given weights to premium and loss dollars before forming the overall ratio. The common trap is to assume that averaging period factors necessarily gives the same result.

Find period losses. Period 1 losses are $0.55\times \$400{,}000=\$220{,}000$. Period 2 losses are $0.70\times \$1{,}600{,}000=\$1{,}120{,}000$.

Calculate weighted amounts. Weighted premium is $0.25\times \$400{,}000+0.75\times \$1{,}600{,}000=\$100{,}000+\$1{,}200{,}000=\$1{,}300{,}000$. Weighted losses are $0.25\times \$220{,}000+0.75\times \$1{,}120{,}000=\$55{,}000+\$840{,}000=\$895{,}000$.

The target loss amount is $0.65\times \$1{,}300{,}000=\$845{,}000$. The combined factor is:

I=\frac{\$895{,}000}{\$845{,}000}=1.0591716\ldots\approx 1.0592

This corresponds to an increase of approximately 5.92%. For comparison, the individual factors are $0.55/0.65=0.8461538\ldots$ and $0.70/0.65=1.0769231\ldots$. Their weighted average is $0.25(0.8461538\ldots)+0.75(1.0769231\ldots)=1.0192308\ldots$, which is not the combined factor. The specified method weights dollar amounts first; the periods have different premium volumes, so averaging their individual factors is a different calculation. As a check, $1.0591716\ldots\times \$845{,}000=\$895{,}000$. Factors are displayed to four decimal places.

Common mistake avoided: Do not substitute a weighted average of individual factors for the ratio of weighted losses to weighted target losses when the required setup is based on weighted dollar amounts.

Checks and Common Traps

Before accepting a multi-period indication, verify the setup as well as the arithmetic. A useful review is to confirm that each premium and loss pair represents the same period and coverage basis; each adjustment has been applied to the intended input; and the same stated weights have been used for the premiums and losses. Confirm also that the weights sum to 1. If weights are given as relative amounts, such as 2:3, normalize them to 0.40 and 0.60 before using the weighted-total formula. Multiplying all weights by the same positive constant would cancel in the ratio, but normalized weights make the combined amounts interpretable.

A second check is to compare weighted selected losses with weighted target losses. If selected losses exceed the target amount, the factor must exceed 1; if they are lower, the factor must be below 1. Finally, distinguish the indicated factor from the percentage change: a factor of 1.0592 means an increase of about 5.92%, not 105.92%.

Key takeaway: For a common target loss ratio and aligned inputs, combine the weighted loss dollars and weighted premium dollars first, then form the indication. Keep the period weights, premium basis, loss basis, and target provision consistent throughout the calculation.

Check Your Understanding

For each problem, combine the specified period inputs on the stated basis. Round indication factors to four decimal places and percentage changes to two decimal places when requested.

Two aligned periods have premiums of $700,000 and $1,300,000 and loss ratios of 60% and 72%. Their weights are 40% and 60%, and the common target loss ratio is 68%. Find the overall indication factor. (Answer: 1.0122)
Three periods have aligned premiums of $500,000, $800,000, and $1,200,000 and selected losses of $310,000, $520,000, and $780,000. Their weights are 20%, 30%, and 50%, and the target loss ratio is 65%. Find the factor and percentage change. (Answer: factor 0.9951; decrease of 0.49%)
Two aligned periods have premiums of $300,000 and $600,000, with loss ratios of 65% and 65%. Their weights are 50% and 50%, and the target loss ratio is 66%. Find the combined factor. (Answer: 0.9848)
Two periods have premiums of $400,000 and $1,000,000 and loss ratios of 50% and 75%. Their weights are 20% and 80%, and the target loss ratio is 65%. Find the factor from weighted amounts. Also calculate the weighted average of the individual period factors and state whether it agrees. (Answer: combined factor 1.1189; weighted average of individual factors 1.0769; they do not agree)
Weights are given in the ratio 2:3. The periods have aligned premiums of $200,000 and $500,000, selected losses of $130,000 and $350,000, and a common target loss ratio of 65%. Normalize the weights and find the overall factor. (Answer: normalized weights 0.40 and 0.60; factor 1.0607)

What You'll Learn