Severity Data And Credibility

Severity Experience and Credibility

Frequency and severity describe different parts of claim experience. Frequency concerns how often claims occur per exposure; severity concerns the amount associated with a claim. This tutorial focuses on the average amount per claim and on why that average can be more or less stable depending on claim-size variation. A credible severity estimate requires both enough claims and a clearly defined, comparable claim amount.

Definition: For a specified group and experience period, observed average severity is total claim amount divided by the number of claims, using the same defined amount for every claim. If the claim amounts are $X_1,\ldots,X_n$, where $n>0$, then $\bar{X}=(X_1+\cdots+X_n)/n$. Severity is a claim-level measure; dividing total loss by exposures instead gives loss per exposure, which combines frequency and severity.

The amount used must be defined before calculating or comparing averages. It might be paid loss or incurred loss, and it must consistently reflect the treatment of expenses, deductibles, limits, claim status, and any trend adjustment relevant to the analysis. Those choices can change the measured severity. A historical average is not comparable to a target or complement unless both refer to a consistent basis.

Why Claim-Size Variation Affects Stability

To describe sampling uncertainty, suppose a fixed number $n$ of claims are observed and their amounts $X_1,\ldots,X_n$ are independent and identically distributed, with finite mean $\mu=E[X]$ and variance $\sigma^2=\operatorname{Var}(X)$. The observed average $\bar{X}$ estimates the expected claim amount $\mu$. Under these assumptions, its standard error is $\sigma/\sqrt{n}$. This is a model-based measure of how much the average would vary across repeated samples of $n$ claims; it is not a guarantee that any one estimate lies within that distance of the mean.

\operatorname{Var}(\bar{X}) =\operatorname{Var}\left(\frac{1}{n}\sum_{j=1}^{n}X_j\right) =\frac{1}{n^2}\sum_{j=1}^{n}\operatorname{Var}(X_j) =\frac{\sigma^2}{n}, \qquad \operatorname{SE}(\bar{X})=\frac{\sigma}{\sqrt{n}}.

The justification uses independence: the variance of the sum is the sum of the $n$ individual variances. Each variance is $\sigma^2$; dividing the sum by $n$ divides its variance by $n^2$. Thus, more claims reduce the standard error, but a larger spread of individual claim amounts increases it. For relative stability, compare standard error to the expected amount. The coefficient of variation is $\sigma/\mu$, and the relative standard error of the average is $\sigma/(\mu\sqrt{n})$, when $\mu>0$.

Important condition: The standard-error formula above applies to a fixed number of independent, identically distributed claim amounts with finite variance. It can be misleading when claims are dependent, the claim population changes materially, or the amounts are not measured on a consistent basis. If the variance is unknown, it must be estimated from the observed claims; the resulting standard error is itself uncertain, especially for a small sample.

Worked Example: Calculating an Average and Its Model-Based Standard Error

A hypothetical small commercial portfolio has nine claims, with amounts in thousands of dollars of 2, 6, 2, 6, 6, 2, 6, 2, and 6. For illustration, assume each claim amount is an independent draw from a model that assigns probability 0.5 to $2,000 and probability 0.5 to $6,000. Calculate the observed average, the model mean, and the standard error of the average.

Observed average. The total is $2+6+2+6+6+2+6+2+6=38$ thousand dollars. There are nine claims, so the observed average is $38/9=4.2222$ thousand dollars, or $4,222.22 to the cent.

Model mean. Each possible amount is equally likely, so $E[X]=0.5(2)+0.5(6)=4$ thousand dollars.

Model variance and standard error. The deviations from the mean are $-2$ and $2$ thousand dollars. Therefore, $\operatorname{Var}(X)=0.5(2^2)+0.5(2^2)=4$ thousand-dollars-squared, and $\sigma=2$ thousand dollars. For nine independent claims, $\operatorname{SE}(\bar{X})=2/\sqrt{9}=0.6667$ thousand dollars, or $666.67.

The observed average is $222.22 above the model mean, less than one model-based standard error. This comparison describes sampling variation under the stated model; it does not prove that the model is appropriate. The arithmetic also checks: nine claims with a mean of $4,222.22 have a total of $38,000, subject to rounding.

Method and common mistake avoided. This is a claim-level average, so its denominator is nine claims. If the portfolio had 100 exposures, total loss divided by 100 would be $380 per exposure, a different measure that reflects both claim frequency and claim size.

Comparing Severity and Frequency Uncertainty

A useful comparison keeps the assumptions and amount of data visible. Consider a hypothetical set of $m$ independent exposure units, each of which has either zero or one claim, with claim probability $p$. The estimated frequency is the number of claims divided by $m$. Under this Bernoulli model, its standard error is $\sqrt{p(1-p)/m}$, and, for $p>0$, its relative standard error is $\sqrt{(1-p)/(mp)}$. For severity, with a fixed number $n$ of independent claims from a distribution with positive mean $\mu$ and finite standard deviation $\sigma$, the relative standard error is $\sigma/(\mu\sqrt{n})$. The formulas measure uncertainty in different quantities, so the comparison should be interpreted rather than treated as an automatic ranking.

Worked Example: Comparing Frequency and High-Variation Severity

A hypothetical program has 400 independent exposure units, each with a 0.05 probability of exactly one claim. For a separate severity calculation, suppose there are exactly 20 independent claims. Each claim amount is $1,000 with probability 0.9 and $19,000 with probability 0.1. Compare the relative standard errors of the estimated frequency and average severity under these assumptions.

Frequency. The expected claim probability is $p=0.05$. The standard error of the observed claim proportion is $\sqrt{0.05(0.95)/400}=\sqrt{0.00011875}=0.010897$, or about 1.09 percentage points. Its relative standard error is $0.010897/0.05=0.2179$, or 21.79%.

Severity mean. In thousands of dollars, the expected severity is $0.9(1)+0.1(19)=2.8$, or $2,800. The variance is $0.9(1-2.8)^2+0.1(19-2.8)^2=0.9(3.24)+0.1(262.44)=29.16$ thousand-dollars-squared. Thus the standard deviation is $\sqrt{29.16}=5.4$ thousand dollars.

With 20 claims, the standard error of average severity is $5.4/\sqrt{20}=1.2075$ thousand dollars, or $1,207.48 to the cent. Its relative standard error is $1.2075/2.8=0.4312$, or 43.12%, to four decimal places before converting to a percentage. Under this model, severity has about $0.4312/0.2179=1.98$ times the relative standard error of frequency.

Method and common mistake avoided. The severity calculation conditions on the stated 20 claims and uses claim amounts, not the 400 exposure units, as its sample size. A common error is to use 400 in the severity standard-error formula just because 400 exposures produced the experience. That would incorrectly treat each exposure as one observed claim amount.

This comparison shows how a small chance of a very large claim can make average severity less stable than frequency, even when the expected number of claims is 20. It does not establish that severity is always less stable. A narrow claim-size distribution, or substantially more claims, can make a severity average relatively stable. Conversely, a low claim probability can make frequency unstable. Both the amount of data and the relevant distribution of observations matter.

Using Severity Experience in a Credibility Estimate

Credibility combines observed experience with complementary information. The input should match the component being estimated: for severity, a claim-level average on the defined basis. If a problem supplies a credibility weight, apply it to that severity estimate and the stated complement. The weight is an input to the calculation, not something determined solely by the observed average or by its standard error.

Worked Example: Applying a Stated Credibility Weight to Severity

A hypothetical equipment insurer observes five claim amounts, in thousands of dollars: 2, 4, 6, 8, and 10. A supplied credibility weight is $Z=0.40$, and the complementary severity estimate is $5,000. Calculate the credibility-weighted severity. Also estimate the standard error of the observed average using the sample variance.

Observed average. The total is $2+4+6+8+10=30$ thousand dollars. Dividing by five gives an observed average of $6,000.

Weighted estimate. Use $Z\bar{X}+(1-Z)$ times the complement. The observed-experience component is $0.40(6,000)=$2,400$; the complementary component is $0.60(5,000)=$3,000$. The weighted severity is therefore $5,400.

Estimated standard error. The deviations from the observed average, in thousands, are $-4,-2,0,2,4$. The sum of squared deviations is $16+4+0+4+16=40$. The unbiased sample variance is $s^2=40/(5-1)=10$ thousand-dollars-squared. The estimated standard error of the average is $s/\sqrt{5}=\sqrt{10/5}=\sqrt{2}=1.4142$ thousand dollars, or $1,414.21 to the cent.

The weighted estimate is $5,400, between the observed average of $6,000 and the complement of $5,000, as it must be for a weight between zero and one. The standard error calculation describes the observed average under an independent, identically distributed claim model; it does not derive or validate the supplied $Z=0.40$.

Method and common mistake avoided. The credibility calculation weights two severity estimates. Do not weight total losses against a per-claim complement, or average claim amounts against a per-exposure complement: the two inputs must measure the same rate component and use consistent units.

Common Traps and Practical Checks

Severity credibility can be distorted by a simple denominator error or by treating unlike claim amounts as comparable. Before using an average, check the claim count, the claim definition, and the amount basis. A low claim count can also leave the observed average sensitive to a single large amount; the mean includes every claim in full and can move substantially when one observation changes.

Define the measure.
State exactly which claim amount is being averaged and the experience period it represents.

Count claims, not exposures.
For average severity, divide the total of the defined claim amounts by the number of claims with those amounts.

Assess variation.
When the independent, identical-claim model is reasonable, relate standard error to claim-count volume and claim-size spread.

Keep the comparison aligned.
Compare estimates of the same component, and distinguish a stated credibility weight from a sampling-error calculation.

The practical implication is not that a particular number of claims automatically makes severity credible. Rather, an estimate can be more stable with more claims and less stable when individual amounts are highly variable, provided the claim population and measurement basis remain comparable. This reasoning is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Check Your Understanding

For each problem, use the stated assumptions and keep claim-level severity distinct from loss per exposure.

A hypothetical book has four claims of $3,000, $5,000, $5,000, and $11,000 across 80 exposures. Find the observed average severity and total loss per exposure. (Answer: severity = $6,000; loss per exposure = $300.)
Claim amounts are independent with mean $4,000 and standard deviation $2,000. For 16 claims, calculate the standard error and relative standard error of the average. (Answer: standard error = $500; relative standard error = 0.1250, or 12.50%.)
For 200 independent exposure units with at most one claim per unit and claim probability 0.02, calculate the standard error and relative standard error of the estimated frequency. (Answer: standard error = 0.0099, approximately; relative standard error = 0.4950, or 49.50%.)
A hypothetical severity model pays $2,000 with probability 0.75 and $10,000 with probability 0.25. For 16 independent claims, find the expected severity, standard deviation, standard error of the sample average, and relative standard error. (Answer: expected severity = $4,000; standard deviation = $3,464.10; standard error = $866.03; relative standard error = 0.2165, or 21.65%.)
An observed average severity is $7,200, the complement is $6,000, and the supplied credibility weight is 0.25. Calculate the weighted severity and state whether it lies between the two inputs. (Answer: $6,300; yes, between $6,000 and $7,200.)

What You'll Learn