Selecting a Severity Trend from Experience

Why Selecting a Severity Trend Requires Care

A severity trend estimates how the average cost per claim changes over time. It is an input to projecting historical claim costs to a future rating period, but a change in average severity is not automatically a change in the underlying cost of a typical claim. Claim counts, the losses included, coverage terms, and the definition of each experience period all affect the result.

The central calculation is simple: divide a period’s aggregate losses by the number of corresponding claims. The judgment lies in making sure the numerator and denominator describe the same claims on the same basis, and then deciding which observed changes are representative enough to use. This process is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Definition: For experience period $j$, let $L_j$ be the aggregate amount of loss on the selected basis and $N_j$ the number of claims corresponding to that loss amount. The observed average severity is $S_j=L_j/N_j$. A severity trend is the rate used to project this average cost per claim from one time point to another.

The loss measure must match the purpose of the analysis. For example, paid losses divided by a corresponding count of claims produce a paid severity; incurred losses divided by that count produce an incurred severity. Do not combine a loss measure from one basis with a count defined on another basis. Also keep the coverage and limit basis consistent across periods: a change in the losses included can look like a change in severity even when the underlying claim costs have not changed.

Annualizing a Change in Average Severity

Suppose two experience periods have average severities $S_1$ and $S_2$, and their midpoints are $t$ years apart. If the candidate model assumes a constant multiplicative annual change, the annual trend rate $r$ satisfies $S_2=S_1(1+r)^t$. The elapsed time is the distance between the period midpoints, not simply the number of periods in the data table.

r=\left(\frac{S_2}{S_1}\right)^{1/t}-1

This is a compound annual rate. It is not the total percentage change divided by the number of years. For example, a 20% increase over four years has a compound annual rate of $(1.20)^{1/4}-1$, not $20\%/4$.

Justification: Under the constant multiplicative trend assumption, each year multiplies severity by $1+r$. After $t$ years, the multiplier is $(1+r)^t$, so $S_2/S_1=(1+r)^t$. Taking the $t$-th root and subtracting 1 gives the annual rate above. The model is not appropriate as a description of every intervening year if the data show material nonconstant changes; in that case, it is a summary rate over the selected interval.

Worked Example: Annualizing Severity from Two Periods

A hypothetical insurer reviews losses from two four-year-separated annual experience periods. In the earlier period, 120 claims produced $600,000 of losses. In the later period, 150 claims produced $900,000. The periods’ midpoints are four years apart. Calculate the observed severities and the annualized change.

Method. Calculate each period’s average severity using its own matching claim count, then compound the ratio over the midpoint interval.

Earlier severity: $S_1=\$600{,}000/120=\$5{,}000$ per claim.

Later severity: $S_2=\$900{,}000/150=\$6{,}000$ per claim.

The severity ratio is $S_2/S_1=\$6{,}000/\$5{,}000=1.20$. Thus:

r=(1.20)^{1/4}-1=0.046635\ldots

The estimated annualized change is 4.66%, rounded to two decimal places. As a check, compounding for four years gives $\$5{,}000(1.046635)^4\approx \$6{,}000$. Aggregate losses rose by 50%, but that is not the severity change: the claim count also rose. The common mistake is to trend aggregate losses without dividing by the corresponding counts.

Comparing Candidate Trends Across Periods

A trend estimated from only two periods can be sensitive to the endpoints. When several periods are available, calculate severity separately for each period and inspect the sequence. Compare the pattern with plausible candidate rates over the same time intervals. A steady progression may support a constant trend; sharp reversals or isolated jumps call for investigation rather than automatic extrapolation.

For annual periods, consecutive period midpoints are usually one year apart. If their severities are $S_j$ and $S_{j+1}$, the one-year change is $S_{j+1}/S_j-1$. For nonconsecutive periods, use the actual midpoint separation and annualize. When summarizing multiple periods, be explicit about whether the estimate is an endpoint compound rate or a fitted trend through all observations; these are different summaries and need not agree.

Worked Example: Checking a Candidate Against Several Years

A hypothetical specialty insurer records the following losses and claim counts on a consistent basis. The annual periods have consecutive midpoints.

Period	Claims	Aggregate losses	Average severity
Year 1	80	$720,000	$9,000
Year 2	100	$954,000	$9,540
Year 3	120	$1,213,488	$10,112.40

Step 1: Verify the period severities. Year 1 is $\$720{,}000/80=\$9{,}000$. Year 2 is $\$954{,}000/100=\$9{,}540$. Year 3 is $\$1{,}213{,}488/120=\$10{,}112.40$.

Step 2: Calculate the successive changes. From Year 1 to Year 2, $\$9{,}540/\$9{,}000=1.06$, a 6% increase. From Year 2 to Year 3, $\$10{,}112.40/\$9{,}540=1.06$, also a 6% increase.

Step 3: Evaluate the candidate. A 6% annual trend reproduces both observed changes. Over the two-year interval, it gives $\$9{,}000(1.06)^2=\$10{,}112.40$, exactly the third-period severity. On these data, 6% is a coherent candidate.

Check and limitation. The arithmetic is consistent across all three periods, but a short sequence alone does not establish that 6% will continue into the future. The common mistake is to select a candidate solely because it connects the first and last observations, ignoring the intervening period. Here, checking the middle value confirms the same progression; in less regular data, the intermediate observations matter.

Keep the Experience Basis and Periods Comparable

Before interpreting a calculated trend, check what is included in each observation. The count must correspond to the losses being measured: for instance, do not divide a total that includes a category of claims by a count that excludes it. Use the same loss basis in every period, such as paid with paid or incurred with incurred, and keep the treatment of limits and other coverage features consistent. If the basis changes, the ratio may measure a change in data definition rather than a change in severity.

Period definitions also matter. State whether experience is organized by accident date, report date, or another consistent claim-period definition, and use comparable period lengths. The time interval for annualizing is measured between the periods’ midpoints. If periods are of unequal length, simply treating each as one year can misstate the annual trend. Similarly, ensure that the latest observation has been measured on a basis comparable to earlier periods; otherwise, its apparent severity change may reflect differences in data maturity rather than underlying cost change.

Define the measure.
Specify which losses and claims are included and the basis on which they are recorded.

Align each numerator and denominator.
For every period, divide the aggregate loss amount by the corresponding claim count.

Set the time points.
Identify the period midpoints and calculate the actual elapsed time between them.

Compare candidate patterns.
Review the individual period severities and their successive changes; do not rely only on an endpoint comparison.

Assess unusual observations.
Check whether a large claim or a change in the data basis explains a jump before deciding how representative the observed rate is.

How a Large Claim Can Affect the Selected Trend

Average severity is an arithmetic mean, so each claim’s loss contributes to the period total. A single large claim can therefore move the mean substantially, especially when the claim count is small. That effect is real in the observed data, but it does not by itself demonstrate a persistent annual trend. Compare the all-claim result across periods, inspect the size and context of unusual claims, and consider how stable the candidate looks under a clearly described sensitivity check.

A sensitivity calculation is a diagnostic, not permission to remove a claim from the selected basis without a stated reason. The intended treatment of large claims must be consistent with the analysis being performed. This distinction prevents two opposite errors: treating every isolated large loss as recurring trend, or silently excluding a costly claim simply because it makes the result inconvenient.

Worked Example: Separating a Large-Claim Effect from the Observed Change

A hypothetical portfolio has 50 claims in each of two consecutive annual periods. Earlier-period losses total $250,000. In the later period, 49 claims average $5,100 each, and the remaining claim is $80,100. Calculate the all-claim severity change, then assess the effect of the large claim as a sensitivity check.

Earlier severity. $\$250{,}000/50=\$5{,}000$.

Later aggregate losses. The 49 other claims total $49\times \$5{,}100=\$249{,}900$. Adding the large claim gives $\$249{,}900+\$80{,}100=\$330{,}000$. Later all-claim severity is $\$330{,}000/50=\$6{,}600$.

All-claim change. The ratio is $\$6{,}600/\$5{,}000=1.32$, so observed severity increased by 32% from one year to the next.

Sensitivity check. The 49 other claims average $5,100, which is 2% above the earlier all-claim average of $5,000. This comparison shows that the $80,100 claim has a substantial effect on the later all-claim mean. It does not establish that excluding that claim is appropriate: the 2% comparison uses a different claim set and is only diagnostic.

Conclusion. The all-claim observed change is 32%; the sensitivity calculation indicates that one claim strongly influences it. The common mistake is to report 2% as the selected trend merely because it appears more typical, without defining and applying a consistent treatment of large claims across periods.

Common Selection Traps

Using aggregate loss growth as severity growth. Aggregate losses reflect both claim counts and average severity. Calculate $L_j/N_j$ before comparing periods.
Using the wrong time interval. Annualize over the elapsed time between period midpoints, not the number of rows or the number of calendar labels.
Mixing definitions. Different loss bases, claim counts, or coverage limits can create an artificial change in the average.
Letting one observation decide the trend. A large claim may materially affect severity. Investigate its influence and the consistency of the pattern across periods.
Confusing a historical summary with a forecast guarantee. A compound rate describes the selected interval under a constant-rate summary model; it does not prove that the same rate will persist.

Key takeaway: A defensible severity trend starts with comparable period averages, calculated from matching losses and claim counts. Annualize using the actual midpoint interval, compare the path across periods, and explain how unusual claims affect the evidence before selecting a candidate rate.

Check Your Understanding

For each problem, calculate the requested severity or trend using the stated period definitions.

A hypothetical insurer has $480,000 of losses on 80 claims in an earlier annual period and $660,000 on 100 claims in a later annual period. Their midpoints are two years apart. What is the compound annual severity trend, rounded to two decimal places? (Answer: 4.88% per year.)
Two experience periods have midpoints four years apart. The earlier period has 200 claims and $1,000,000 of losses; the later has 250 claims and $1,500,000 of losses. Calculate both average severities and the annualized trend, rounded to two decimal places. (Answer: $5,000 and $6,000; 4.66% per year.)
Three consecutive annual periods have average severities of $8,000, $8,800, and $9,680. Calculate each successive one-year change and state whether a constant 10% candidate matches the observations. (Answer: 10% in each interval; yes, the candidate matches.)
In two consecutive annual periods, a portfolio has 40 claims each. Earlier losses total $200,000. Later, 39 claims average $5,100 and one claim is $49,900. Calculate the later all-claim average and its percentage change from the earlier average. (Answer: $6,220; a 24.4% increase.)

What You'll Learn