Why Credibility Depends on the Measure
Within one rating class, an actuary may estimate claim frequency, average claim severity, or aggregate loss cost. These measures describe related aspects of loss, but they are not the same quantity and do not draw their information from the same number of observations. Frequency is informed by exposure and claim counts; severity is informed by the number and sizes of claims; aggregate loss cost combines both sources of variation.
Consequently, credibility should be assessed for the particular measure being estimated. A large number of policy-years may support a stable frequency estimate while producing comparatively few claims for estimating severity. Conversely, a class with many claims can provide a substantial severity sample even if its exposure base is modest. Aggregate loss cost uses the observed total losses and exposure, but its variation reflects both claim counts and claim sizes.
The earlier material on defining claim measures and selecting a relevant credibility complement matters here: the observed data and the complement must refer to the same measure. A frequency estimate should be compared with a frequency complement, for example, not directly with an aggregate loss cost. The calculations below use the Bühlmann credibility form as a disciplined way to compare measures. They are model illustrations, not universal credibility standards.
A Common Model, Different Observation Units
In the Bühlmann model, observations are conditionally independent given a risk parameter. Risks are drawn from a common population, and the target is the risk's conditional mean. Let \(v\) denote the expected process variance of one observation and \(a\) the variance between risks in their conditional means. For \(n\) comparable observations, the credibility-weighted estimate is
where \(\bar{X}\) is the observed mean and \(m\) is the population mean used as the complement. This form presumes equal-weight observations under the stated model; unequal exposures, changing conditions, dependence, or a different target require a suitable model rather than mechanical use of the same \(n\) and \(K\).
The expression for \(Z\) follows by balancing the two sources of uncertainty. If the complement mean is treated as known, the variance contribution from sampling the experience mean is \(v/n\), while variation among risk means is \(a\). For an estimate \(Z\bar{X}+(1-Z)m\), the mean squared error attributable to these terms is
Differentiating with respect to \(Z\) and setting the result to zero gives \(2Zv/n-2(1-Z)a=0\). Thus \(Z(v/n+a)=a\), so \(Z=a/(a+v/n)=n/(n+v/a)\). This explains why more observations increase credibility, while greater process variation reduces it and greater between-risk variation increases it. It also explains why the relevant \(v\), \(a\), and \(n\) depend on the measure.
Worked Examples by Rate Component
Worked Example: Credibility for Claim Frequency
A hypothetical vehicle class has 500 vehicle-years of experience and 48 claims. Assume the observations are equal one-year exposure units, claim counts are conditionally Poisson with a risk-specific frequency, and the risk frequencies follow a common population distribution. For this model, the population mean frequency is 0.080 claims per vehicle-year, the expected process variance per unit is \(v=0.080\), and the between-risk variance is \(a=0.0004\). Calculate the credibility estimate. State the method and the mistake it avoids.
Method: Bühlmann credibility applied to annual claim counts. There are \(n=500\) exposure-unit observations. The experience frequency is
The hypothetical model's complement constant is \(K=v/a=0.080/0.0004=200\) exposure units. Therefore,
Weight the experience frequency and the complement using this credibility:
The estimate is 0.0914 claims per vehicle-year to four significant figures. As a check, it lies between the observed frequency, 0.096, and the complement, 0.080, and is closer to the observed value because \(Z>0.5\). The mistake avoided is treating 48 claims as 48 exposure observations: frequency credibility here uses 500 vehicle-years, because the observations are defined as annual exposure units.
Worked Example: Credibility for Average Severity
For a hypothetical class, the 15 observed claims have an average severity of $1,500. Assume individual claim severities are conditionally independent with a risk-specific mean; the expected process variance of a claim severity is \(v=\$360{,}000\) squared, and the between-risk variance in mean severity is \(a=\$40{,}000\) squared. The population mean severity is $1,400. Find the credibility estimate. Name the method and the mistake it avoids.
Method: Bühlmann credibility applied to claim severities. The observation count is the number of claims, \(n=15\), not the number of policies or exposure-years. The complement constant and credibility are
The estimated severity is
The answer is $1,462.50 per claim. It is between the experience and complement, and is closer to experience because its weight is 0.625. This calculation avoids using exposure-years as the severity sample size: each claim contributes one severity observation under the stated model. A real analysis would also check that claim amounts have a comparable coverage, loss basis, and treatment of large claims.
Worked Example: Credibility for Aggregate Loss Cost
A hypothetical class has aggregate losses averaging $160 per exposure-year over 10 equal exposure units. A Bühlmann model for annual aggregate loss cost specifies population mean \(m=\$120\), expected process variance \(v=\$208{,}800\) squared per unit, and between-risk variance in conditional mean \(a=\$14{,}400\) squared. Treat these as model inputs. Calculate the estimate and explain why its credibility need not equal either component's credibility.
Method: Bühlmann credibility applied directly to annual aggregate loss cost. The observed mean is $160 per exposure-year. The complement constant is
Thus the credibility weight for ten exposure units is
Credibility-weight the aggregate loss cost:
The result is $136.33 per exposure-year, rounded to the nearest cent. It lies between the observed mean and complement, and is closer to the complement because \(Z<0.5\). The model's process variance reflects the fluctuation in total losses, including both claim counts and claim sizes; its risk variation concerns differences in the conditional aggregate mean.
For comparison only, multiplying the previous examples' frequency estimate by their severity estimate gives \(0.0914286\times \$1{,}462.50=\$133.7143\), or $133.71 per exposure-year. That differs from $136.33 because the separate component models use their own data counts, process variation, and credibility weights. The difference is not an arithmetic inconsistency. A common mistake is to assume direct aggregate credibility must equal the product of separately credibility-weighted frequency and severity; in general, it need not.
What Drives the Differences?
The table summarizes which observations and variation matter for each target. “Volume” means the number of relevant observations under the model, not simply the size of the policy file.
| Target | Typical observation unit | Main data volume | Process variation to consider |
|---|---|---|---|
| Frequency | Claim count per exposure unit | Exposure units, with counts observed within them | Random fluctuation in claim counts |
| Severity | Amount for one claim | Number of claims with usable severity data | Variation in claim amounts |
| Aggregate loss cost | Total loss per exposure unit | Exposure units and their total losses | Variation in both counts and amounts, including their joint behavior if dependent |
For a compound Poisson model, suppose a unit has claim count \(N\) with mean \(\lambda\) and variance \(\lambda\), and claim amounts are independent and identically distributed with mean \(E[X]\) and variance \(\operatorname{Var}(X)\), independent of \(N\). The aggregate loss \(S=\sum_{j=1}^{N}X_j\) has
The first variance term is the contribution from claim size variation at a given count; the second reflects count variation. This identity helps explain why aggregate loss cost can be volatile even where frequency and severity each appear reasonably stable. It relies on the stated independence and Poisson assumptions; other count models, dependence, or catastrophe effects can change the variance.
In practice, component estimates may also be based on different usable records. Claims with incomplete amounts may be excluded from a severity sample but still appear in count data; losses may be evaluated on paid or incurred bases; and large claims can dominate aggregate totals. The earlier material on data maturity, data quality, and large claims is relevant when deciding whether the model inputs describe the experience actually being weighted.
Common Traps and Interpretation
- Using the wrong denominator. Frequency is claims per exposure; severity is loss per claim; aggregate loss cost is loss per exposure. A number without its denominator is not a well-defined rate component.
- Using the wrong observation count. A severity sample of 20 claims has 20 claim observations, not automatically 20 policy-years. Exposure duration and claim count are not interchangeable.
- Assuming one credibility applies to every component. The process variance, between-risk variance, and effective observation volume can differ. A single weight is justified only by a model that supports it for the target being estimated.
- Confusing credibility with adequacy. A highly credible estimate can still be high or low relative to the indicated rate. Credibility describes how experience and complement are blended, not whether the resulting rate is sufficient.
- Multiplying weighted components without checking the target. The product of a credibility-weighted frequency and severity is not generally the same as a direct credibility estimate of aggregate loss cost.
Check Your Understanding
For each problem, use the stated Bühlmann assumptions and round dollar estimates to the nearest cent where needed.
- A hypothetical class has 300 exposure units and 27 claims. Under a frequency model, \(m=0.060\), \(v=0.060\), and \(a=0.0004\). Find \(K\), \(Z\), and the credibility-weighted frequency. (Answer: \(K=150\) exposure units, \(Z=0.6667\), estimated frequency \(=0.0800\) claims per exposure unit.)
- A class has eight usable claims with average severity $1,800. The severity complement is $1,500, \(v=\$48{,}000\) squared, and \(a=\$4{,}000\) squared. Calculate the credibility-weighted severity. (Answer: \(K=12\) claims, \(Z=0.4000\), estimated severity = $1,620.00.)
- A direct aggregate loss-cost model has \(m=\$100\), \(v=\$90{,}000\) squared per exposure unit, and \(a=\$10{,}000\) squared. Over 20 equal exposure units, the observed average is $130. Find \(K\), \(Z\), and the weighted estimate. (Answer: \(K=9\) exposure units, \(Z=0.6897\), estimated aggregate loss cost = $120.69 per exposure unit.)
- A frequency estimate is 0.050 claims per exposure unit, its complement is 0.040, and its credibility is 0.60. A severity estimate is $2,000, its complement is $1,800, and its credibility is 0.25. Calculate both weighted component estimates and their product. (Answer: weighted frequency \(=0.046\), weighted severity \(=\$1{,}850\), product \(=\$85.10\) per exposure unit.)
- In a compound Poisson model, \(\lambda=0.10\), \(E[X]=\$2{,}000\), and \(\operatorname{Var}(X)=\$1{,}000{,}000\). Under the stated independent count and severity assumptions, calculate the expected aggregate loss and its variance per exposure unit. (Answer: expected loss \(=\$200\); variance \(=\$500{,}000\) squared.)