How an Added Experience Year Changes the Indication

Why an Added Experience Year Changes the Result

A combined indication summarizes experience from more than one period. When a new experience year is included, the combined result depends on two things: the indication from that year and the weight it receives relative to the experience already included. A year with a high indication tends to pull the combined result upward; a year with a low indication tends to pull it downward. The size of that movement depends on the year’s weight.

This tutorial focuses on that update. It assumes the annual indications have already been placed on a comparable basis: they use consistent loss and premium measures and reflect a common prospective period and coverage basis. The earlier material on combining multiple experience periods and choosing weights develops those setup decisions. Here, the key question is what changes mathematically when one more year is added.

Definition: Let $I_j$ be the indication for experience year $j$, and let $w_j$ be its positive weight. The combined indication for a set of years is the weighted average $$ I=\frac{\sum_j w_j I_j}{\sum_j w_j}. $$ If the indication is a loss-to-premium ratio, the weight is the corresponding premium denominator, so the combined indication is also the ratio of total losses to total premium. More generally, the weights must be appropriate to the quantity being combined and the annual indications must share a consistent basis.

Updating the Indication When One Year Is Added

Suppose the existing experience has total weight $W$ and combined indication $I_{\text{old}}$. A new year has weight $w$ and indication $I_{\text{new}}$. The weighted amount represented by the existing years is $W I_{\text{old}}$, and the new year contributes $w I_{\text{new}}$. Adding those amounts and dividing by the new total weight gives the updated indication.

I_{\text{updated}} =\frac{W I_{\text{old}}+wI_{\text{new}}}{W+w} =I_{\text{old}}+\frac{w}{W+w}(I_{\text{new}}-I_{\text{old}}).

The second form shows the direction and size of the change directly. The fraction $w/(W+w)$ is the new year’s share of the total weight after it is added. Because that fraction is between zero and one, the updated indication moves from the old indication toward the new year’s indication. It does not move past the new year’s indication.

Why the Update Formula Holds

The formula follows by rewriting the weighted average. Starting with the definition,

\begin{aligned} I_{\text{updated}} &=\frac{W I_{\text{old}}+wI_{\text{new}}}{W+w}\\ &=\frac{(W+w)I_{\text{old}}+w(I_{\text{new}}-I_{\text{old}})}{W+w}\\ &=I_{\text{old}}+\frac{w}{W+w}(I_{\text{new}}-I_{\text{old}}). \end{aligned}

This derivation also gives a useful check: the change is the difference between the new and old indications, multiplied by the new year’s share of total weight. If the new indication equals the old one, the difference is zero and the combined result does not change. If the added year has very little weight, its effect is small; if it has a large weight, its effect is larger.

Key result: For positive weights, $I_{\text{updated}}$ lies between $I_{\text{old}}$ and $I_{\text{new}}$. If $I_{\text{new}}>I_{\text{old}}$, the combined indication increases; if $I_{\text{new}}<I_{\text{old}}$, it decreases. This conclusion concerns the weighted average itself. It does not say that adding a year must increase or decrease a final rate change, which may depend on other components of the indication.

Worked Example: Updating an Indication With a Heavier New Year

A hypothetical insurer has an existing combined indication of 0.8500, based on total weight 5. A newly available experience year has an indication of 1.1000 and weight 2. Calculate the updated combined indication and explain the direction of change.

Method. Use the weighted-update formula. The old weighted amount is $5(0.8500)=4.2500$. The new year contributes $2(1.1000)=2.2000$.

Combine the weighted amounts. The total weighted amount is $4.2500+2.2000=6.4500$, and the total weight is $5+2=7$. Therefore,

I_{\text{updated}}=\frac{6.4500}{7}=0.921428\ldots

Rounded to four decimal places, the updated indication is 0.9214. Its increase from the old indication is $0.921428\ldots-0.8500=0.071428\ldots$.

Check. The new year’s share of total weight is $2/7$. The difference between its indication and the old indication is $1.1000-0.8500=0.2500$. The update formula gives $0.8500+(2/7)(0.2500)=0.921428\ldots$, matching the ratio of weighted totals. The answer is between 0.8500 and 1.1000, as it must be.

Common mistake avoided. Do not give the new year the same influence as the entire old experience just because it is one year. Its weight is 2, while the existing experience has weight 5; its share of the updated total weight is only $2/7$.

Connecting Weights to Loss and Premium Totals

For a loss-ratio-style calculation, let $L_j$ be the projected loss amount for year $j$ on the selected basis, and let $P_j$ be the corresponding premium at the relevant rate level. The year’s indication is $I_j=L_j/P_j$, and its weight is $P_j$. The combined indication is then

\frac{\sum_j P_j I_j}{\sum_j P_j} =\frac{\sum_j P_j(L_j/P_j)}{\sum_j P_j} =\frac{\sum_j L_j}{\sum_j P_j}.

This equality explains why the ratio of totals and the premium-weighted average of annual ratios agree. It also makes an important condition visible: the loss and premium amounts being combined must correspond to the same periods and a consistent basis. If the weights are exposure counts instead, the quantities being averaged must be costs per exposure on a comparable basis.

Worked Example: Adding a Year With a Different Premium Weight

A hypothetical book has existing projected losses of $1,080,000 and corresponding premium of $1,200,000. A newly included year has projected losses of $247,500 and corresponding premium of $225,000. Find the combined indication, and show how the added year changes the old result.

Find the existing indication. The existing loss-to-premium ratio is $1{,}080{,}000/1{,}200{,}000=0.9000$. The existing weight is $1,200,000.

Find the new year’s indication. Its ratio is $247{,}500/225{,}000=1.1000$. Its weight is $225,000.

Combine the periods. Total projected losses are $1{,}080{,}000+247{,}500=\$1{,}327{,}500$. Total premium is $1{,}200{,}000+225{,}000=\$1{,}425{,}000$. Thus,

I_{\text{updated}}=\frac{1{,}327{,}500}{1{,}425{,}000} =0.931578\ldots

Rounded to four decimal places, the combined indication is 0.9316. The update formula gives the same result: the new year’s share of total premium is $225{,}000/1{,}425{,}000=0.1578947\ldots$, so $0.9000+0.1578947\ldots(1.1000-0.9000)=0.931578\ldots$.

Check. The new year has a higher ratio than the existing experience, so the combined result should rise above 0.9000. It does, but remains below 1.1000 because the new year is only part of the total premium weight.

Common mistake avoided. A simple average of the two annual ratios would be $(0.9000+1.1000)/2=1.0000$. That incorrectly gives each year half the influence, even though their premium weights differ. The ratio of totals, 0.9316, is the appropriate premium-weighted result.

Different Costs and Exposure Weights

The same update principle applies when the annual quantity is a cost per exposure and the weights are exposure counts. In that case, the combined cost is the total projected cost divided by the total number of exposures. If the cost is compared with a common selected target cost, the combined indication is the combined cost divided by that target.

Worked Example: Adding a Recent Year With Higher Cost

A hypothetical program has 120 exposures in its existing experience, with a projected cost of $480 per exposure. A recent year adds 40 exposures with a projected cost of $600 per exposure. The selected prospective cost is $500 per exposure. Calculate the combined cost and indication.

Convert each cost to an indication. The existing indication is $480/500=0.9600$. The recent-year indication is $600/500=1.2000$. The corresponding weights are 120 and 40 exposures.

Calculate projected costs. The existing weighted cost is $120(\$480)=\$57{,}600$. The recent-year weighted cost is $40(\$600)=\$24{,}000$. Together, the projected cost is $\$57{,}600+\$24{,}000=\$81{,}600$ over $120+40=160$ exposures.

The combined cost is $\$81{,}600/160=\$510$ per exposure. Relative to the selected cost of $500, the combined indication is $\$510/\$500=\mathbf{1.0200}$.

Check. The weighted-average calculation gives $[120(0.9600)+40(1.2000)]/160=(115.2+48.0)/160=1.0200$. The answer is between 0.9600 and 1.2000. Its being closer to 0.9600 is consistent with the existing experience carrying three times the exposure weight of the added year.

Common mistake avoided. Do not average the two costs or indications equally. The new year has 40 exposures, not the same weight as the existing 120.

Conditions, Interpretation, and Practical Checks

The update is meaningful only when the annual values are comparable. If an added year’s losses, premium, coverage, or exposure basis has not been made consistent with the other years, a numerical weighted average can still be computed, but it may not represent the intended combined indication. The averaging formula does not correct mismatched inputs.

Before interpreting the change, check the following:

Same definition. Each annual indication must represent the same kind of measure, such as loss divided by corresponding premium or projected cost per exposure.
Compatible basis. The periods must be appropriately aligned to the common comparison basis intended for the indication.
Appropriate weights. Use premium weights for a ratio of losses to premium, or exposure weights for costs per exposure. Do not substitute a convenient weight without justification.
Positive total weight. The update requires $W+w>0$, and the usual interpretation assumes both the old total weight and the added weight are positive.

A quick diagnostic is to compare the new result with both inputs. With positive weights, it must lie between the old combined indication and the new year’s indication. A result outside that interval signals an arithmetic error, a sign error, or an inconsistent formula. Also verify that the weighted-average result agrees with the corresponding ratio of totals when both are available.

Understanding this movement matters because a newly available year can affect the combined indication without having the same importance as the older experience. The year’s result determines the direction of change; its weight determines how far the combined result moves. This distinction is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Check Your Understanding

For each hypothetical situation, use the stated weights and basis. Answers are rounded to four decimal places where an indication is requested.

An existing combined indication is 0.9200 with total weight 6. A new year has indication 1.1000 and weight 3. Find the updated indication. (Answer: 0.9800)
Existing projected losses are $720,000 and corresponding premium is $800,000. An added year has projected losses of $210,000 and corresponding premium of $200,000. Find the combined loss-to-premium indication. (Answer: 0.9300)
Existing experience has 50 exposures at a projected cost of $540 per exposure. An added year has 30 exposures at $450 per exposure. Find the combined cost per exposure. (Answer: $506.25)
An existing indication is 1.1500 with weight 4. A new year has indication 0.8500 and weight 1. Find the updated indication and state whether it rises or falls. (Answer: 1.0900; it falls)
An existing indication is 0.9500 with total weight 8. A new year has weight 2. What indication must the new year have for the combined indication to equal 1.0000? (Answer: 1.2000)

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