Credibility for Class Frequency and Severity

Why Separate Frequency and Severity?

A class’s expected loss cost per exposure has two useful components: the expected number of claims per exposure and the expected cost per claim. Credibility can be applied to each component, allowing the available information to differ. A class may have substantial exposure volume but relatively few claims, or many claims whose average cost is still uncertain. The relevant volume—and potentially the credibility weight—will therefore differ for frequency and severity.

The method in this tutorial uses Bühlmann credibility for each component. It produces a frequency indication and a severity indication, then multiplies them to obtain an indicated loss cost. The exposure basis must be consistent: a frequency stated per policy-year, for example, yields a loss cost per policy-year when multiplied by severity in dollars per claim.

Definition: Let $e$ be a class’s exposure, $N$ its claim count, and $X_1,\ldots,X_n$ the observed costs of its $n$ claims. The observed class frequency is $N/e$, and the observed average severity is $\bar X=\sum_{j=1}^{n}X_j/n$. A component credibility indication blends each observed component with its corresponding collective mean: $f_I=Z_f(N/e)+(1-Z_f)\mu_f$ and $s_I=Z_s\bar X+(1-Z_s)\mu_s$. The indicated class loss cost is $L_I=f_I s_I$.

Credibility for Frequency and Severity

The collective frequency $\mu_f$ is the portfolio-level expected claim count per unit of exposure, and $\mu_s$ is the portfolio-level expected cost per claim. In Bühlmann notation, each is the expected value of its component’s class-specific risk parameter. The frequency and severity components can have different process variation and between-class variation, so they generally have different credibility parameters.

For frequency, suppose class frequency observations have conditional mean equal to the class’s underlying frequency parameter. Assume the conditional variance of the observed frequency based on exposure $e$ is the frequency process variance $v_f$ divided by $e$. Let $a_f$ be the between-class variance of the underlying frequency parameter. For severity, assume the claim costs are conditionally independent with a common class-specific mean, conditional variance $v_s$, and between-class variance $a_s$ for the class-specific severity mean. These are Bühlmann assumptions; other frequency or severity models may require a different variance structure.

K_f=\frac{v_f}{a_f}, \qquad Z_f=\frac{e}{e+K_f}

K_s=\frac{v_s}{a_s}, \qquad Z_s=\frac{n}{n+K_s}

Here $K_f$ is measured in exposure units and $K_s$ in claim counts. Each $K$ is the volume at which the corresponding credibility weight equals one-half. The parameters must be estimated or supplied for the relevant portfolio and component; they are not calculated from a single class’s observed mean alone.

Why the Bühlmann Weights Have This Form

The credibility weights follow from balancing process variance against variation among class means. For frequency, the conditional variance of the observed rate is $v_f/e$, while the variance of the class-specific mean is $a_f$. The Bühlmann weight on the observed rate is the between-class variance divided by the sum of the between-class variance and the expected conditional variance:

Z_f=\frac{a_f}{a_f+v_f/e} =\frac{a_f e}{a_f e+v_f} =\frac{e}{e+v_f/a_f} =\frac{e}{e+K_f}.

The same argument applies to the average of $n$ claim severities. Conditional independence gives $\operatorname{Var}(\bar X\mid \text{class parameter})=v_s/n$. Replacing the frequency quantities with their severity counterparts yields $Z_s=a_s/(a_s+v_s/n)=n/(n+K_s)$. Thus, more exposure makes the observed frequency more informative, while more claims make the observed average severity more informative.

Calculate observed frequency.
Use total claims divided by the total matching exposure. If experience spans periods with different exposure volumes, aggregate claims and exposure before dividing.

Calculate frequency credibility.
Use the frequency exposure volume and $K_f$ to find $Z_f$, then blend observed frequency with $\mu_f$.

Calculate severity credibility.
Use the number of observed claims and $K_s$ to find $Z_s$, then blend observed average severity with $\mu_s$.

Form the loss cost indication.
Multiply indicated frequency per exposure by indicated severity per claim. The result is loss cost per the same exposure unit.

Worked Frequency and Severity Applications

Worked Example: Credibility-Weighting Frequency Across Two Periods

A hypothetical class has 18 claims in a period with 120 policy-years and 33 claims in a second period with 180 policy-years. Its collective frequency is 0.14 claims per policy-year, and $K_f=200$ policy-years. Find its indicated frequency. Name the method and a common error to avoid.

Method. Apply Bühlmann credibility to the aggregate frequency, using the combined exposure as the volume. The common error is averaging the two period frequencies without weighting for their different exposures.

The total claim count is $18+33=51$, and total exposure is $120+180=300$ policy-years. Therefore the observed frequency is $51/300=0.17$ claims per policy-year. The credibility weight is $Z_f=300/(300+200)=300/500=0.60$.

The indicated frequency is $0.60(0.17)+0.40(0.14)=0.102+0.056=0.158$ claims per policy-year. The weight on the collective mean is $1-0.60=0.40$. As a check, the indication lies between 0.14 and 0.17, as a weighted average must.

An unweighted average of the period rates would be $(18/120+33/180)/2=(0.15+0.18333\ldots)/2=0.16667\ldots$, which is not the aggregate frequency. It gives the smaller-exposure period the same influence as the larger one.

Worked Example: Credibility-Weighting Average Severity

For a hypothetical class, five observed claim costs are $4,800, $5,200, $6,000, $7,000, and $8,000. The collective average severity is $5,600 per claim, and $K_s=15$ claims. Find the indicated severity. State the method and the trap avoided.

Method. Apply Bühlmann credibility to the average of the five claim costs. The common trap is using policy exposure in the severity weight; the specified severity volume is the number of claims.

The sum of the observed claim costs is $4{,}800+5{,}200+6{,}000+7{,}000+8{,}000=31{,}000$ dollars. The observed average is $31{,}000/5=\$6{,}200$ per claim. The severity credibility is $Z_s=5/(5+15)=5/20=0.25$.

The indicated severity is $0.25(\$6{,}200)+0.75(\$5{,}600)=\$1{,}550+\$4{,}200=\$5{,}750$ per claim. The result is between the observed and collective means, and is closer to the collective mean because the credibility weight is only 0.25.

Worked Example: Combining Separate Component Indications

A hypothetical class has 35 claims in 250 policy-years. The collective frequency is 0.12 claims per policy-year and $K_f=250$ policy-years. Its eight claim costs are $6,000, $6,500, $6,800, $7,000, $7,200, $7,400, $8,000, and $8,700. The collective severity is $6,800 per claim and $K_s=12$ claims. Calculate the indicated loss cost per policy-year.

Method. Calculate two Bühlmann indications with their separate volume measures, then multiply. The common error is to use the frequency credibility weight for severity as well, or to multiply the raw observed components instead of the indicated components.

Observed frequency is $35/250=0.14$ claims per policy-year. Its credibility is $Z_f=250/(250+250)=0.50$, so indicated frequency is $0.50(0.14)+0.50(0.12)=0.13$ claims per policy-year.

The eight claim costs sum to $6{,}000+6{,}500+6{,}800+7{,}000+7{,}200+7{,}400+8{,}000+8{,}700=\$57{,}600$. Thus, observed severity is $\$57{,}600/8=\$7{,}200$ per claim. Its credibility is $Z_s=8/(8+12)=8/20=0.40$, giving indicated severity $0.40(\$7{,}200)+0.60(\$6{,}800)=\$2{,}880+\$4{,}080=\$6{,}960$ per claim.

The indicated loss cost is $0.13\times \$6{,}960=\$904.80$ per policy-year. The units check: claims per policy-year multiplied by dollars per claim gives dollars per policy-year. For comparison, the raw observed components would give $0.14\times \$7{,}200=\$1{,}008$, not the credibility-weighted indication.

Interpretation, Limits, and Common Traps

Separate component credibility is useful when frequency and severity have different data volumes or different levels of process and between-class variation. For example, frequency credibility is driven by exposure under the stated model, while severity credibility is driven by the number of claims with observed costs. A class with many policy-years but few claims can therefore have a more stable frequency estimate than severity estimate.

The product $f_I s_I$ is a component-based indicated loss cost. It is not generally equal to a credibility-weighted blend of the observed pure premium and a collective pure premium: multiplication of two blended components creates cross-products. Also, if a class’s underlying frequency and severity are dependent, the expected loss cost need not equal the product of their marginal expected values. The simple product is appropriate when the rating method uses these separate component indications as its selected frequency and severity assumptions; dependence or a direct loss-cost objective may call for a different model.

Key takeaway: Keep each component’s numerator, volume, collective mean, and credibility parameter aligned. Use exposure for frequency and claim count for average severity, then multiply the two indicated values only after their units and assumptions are clear.

Check Your Understanding

For each problem, calculate the requested component indication or loss cost. Answers are shown for checking.

A class has 24 claims in 160 exposure units. Its collective frequency is 0.13 claims per exposure unit and $K_f=240$ exposure units. Find the indicated frequency. (Answer: $Z_f=0.40$; indicated frequency $=0.138$ claims per exposure unit.)
Six claim costs are $4,800, $5,200, $5,600, $6,000, $6,400, and $8,000. The collective severity is $5,400 per claim and $K_s=18$ claims. Find the indicated severity. (Answer: observed mean $=\$6{,}000$; $Z_s=0.25$; indicated severity $=\$5{,}550$ per claim.)
A class has 44 claims in 400 exposure units; its collective frequency is 0.10 and $K_f=400$. Its 10 claims have average severity $7,500; collective severity is $7,000 and $K_s=10$. Find the indicated loss cost. (Answer: indicated frequency $=0.105$, indicated severity $=\$7{,}250$, loss cost $=\$761.25$ per exposure unit.)
A class has 8 claims in 80 exposure units in one period and 48 claims in 320 exposure units in another. The collective frequency is 0.12 claims per exposure unit and $K_f=400$ exposure units. Find the indicated frequency using the combined experience. (Answer: observed frequency $=56/400=0.14$; $Z_f=0.50$; indicated frequency $=0.13$ claims per exposure unit.)
A class has 72 claims in 600 exposure units. Its collective frequency is 0.11 claims per exposure unit and $K_f=300$ exposure units. Its 12 observed claims average $8,400; collective severity is $7,800 and $K_s=18$ claims. Find the indicated loss cost. (Answer: $Z_f=2/3$, indicated frequency $=0.11667$; $Z_s=0.40$, indicated severity $=\$8{,}040$; loss cost $=\$938.00$ per exposure unit.)

What You'll Learn