Why Credibility Is a Filing Selection
An insurer’s own experience may be directly relevant to a filed indication, but a limited or volatile data set can give an unstable estimate of the future. A complement provides another estimate of the same target, typically informed by broader data or an established actuarial selection. Credibility is the weight assigned to the insurer’s experience in combining those estimates.
The weight is not a measure of how much business an insurer writes, nor is it a judgment about whether its data are “good” in the abstract. It should reflect the data’s relevance and predictive reliability for the indication being developed. The calculation is useful only when the experience and complement measure the same quantity on a comparable basis.
For example, a loss-cost estimate should be complemented by a loss-cost estimate on the same basis, not blended directly with a percentage rate change. Alternatively, two indicated percentage changes can be blended if both use the same current-rate base and the same adjustments and projection basis. On-leveling, development, trend, and exposure choices should be made consistently before comparing the estimates.
What Makes a Complement Defensible?
A complement is not automatically suitable just because it is available. It should be selected for its relevance to the target and documented sufficiently for another reviewer to understand the choice. Depending on the situation, a complement might reflect broader relevant experience, an established expected value, or an actuarial selection supported by external information. A complement based on a materially different mix of risks may need adjustment before it is comparable.
A filing explanation should make the selection traceable. It should state the experience period and measure, the volume and stability of the insurer’s data, the source and basis of the complement, the credibility method, and any material limitations. If the selected weight involves judgment, explain what evidence informed that judgment and how plausible alternatives affect the result.
- Relevance: Does the experience reflect the risks and coverage to which the indication applies?
- Volume and stability: How much exposure or how many observations support the estimate, and how much does the estimate vary across reasonable subdivisions or periods?
- Comparability: Are the experience and complement on the same measurement and projection basis?
- Limitations: Are there known gaps, unusual concentrations, or changes that make the observed data less representative of the future target?
Data volume alone does not settle credibility. A large amount of experience may still be poorly matched to the target, while a smaller but highly relevant data set may add useful information. Conversely, a relevant estimate can remain too variable to rely on fully. The rationale should connect the selected weight to both the data and the purpose of the indication.
Credibility Approaches and Their Conditions
Several approaches can inform a credibility selection. They answer related questions but do not have identical assumptions, and their outputs should not be treated as interchangeable without justification.
| Approach | What informs the weight | Important limitation |
|---|---|---|
| Judgmental selection | Relevance, volume, stability, and documented actuarial reasoning | Requires a transparent rationale; the selected weight is not automatically a statistical estimate |
| Bühlmann credibility | Within-risk process variation, between-risk variation, and exposure volume | Requires assumptions about a population of risks and the variation in their underlying means |
| Uncertainty-based blending | Uncertainty of both estimates, under a specified error model | A simple formula requires unbiased estimates with independent errors targeting the same quantity |
| Limited-fluctuation credibility | A specified probability tolerance and a count or other assumed sampling model | The model and tolerance must fit the quantity being measured; count credibility alone does not capture every source of loss-cost uncertainty |
In the Bühlmann model, let \(n\) denote the number of equal exposure units, let EPV be the expected process variance within a risk, and let VHM be the variance of hypothetical means across risks. Under the model’s assumptions, \(K=\mathrm{EPV}/\mathrm{VHM}\), and the credibility of the experience mean is
This formula describes a specific model, not a universal rule for filed indications. It reflects that credibility increases with exposure volume and decreases when within-risk fluctuation is large relative to differences between risk means. When exposure units are unequal, the corresponding Bühlmann-Straub form uses total exposure \(W\), with \(Z=W/(W+K)\), subject to that model’s assumptions. A filing should identify the model and the meaning of its inputs rather than report a weight without its basis.
A common limited-fluctuation illustration uses a Poisson count with expected count \(\lambda\). Under a normal approximation, a full-credibility count threshold for relative error tolerance \(r\) and central confidence level \(1-\alpha\) is \(N_0=z_{1-\alpha/2}^2/r^2\). A square-root partial-credibility rule may then use \(Z=\min(1,\sqrt{N/N_0})\), where \(N\) is the observed count. This is tied to the count model and stated tolerance; it is not, by itself, a general credibility formula for a loss ratio or a severity estimate.
Why an Uncertainty-Based Blend Has This Form
A useful justification for blending is to ask which weight minimizes mean squared error. Suppose the target is a fixed quantity \(\theta\). Assume the experience estimate \(E\) and complement \(C\) are both unbiased for \(\theta\), their errors are independent, and their variances are \(v_E\) and \(v_C\), respectively. These assumptions are strong; in particular, both estimates must target the same quantity.
Write \(E=\theta+\varepsilon_E\) and \(C=\theta+\varepsilon_C\), where both errors have mean zero. For \(I=ZE+(1-Z)C\), the error is \(I-\theta=Z\varepsilon_E+(1-Z)\varepsilon_C\). Independence makes the covariance term zero, so the mean squared error is
Setting the derivative to zero gives \(Z(v_E+v_C)=v_C\). Since the second derivative is \(2(v_E+v_C)>0\) when the estimates have positive variance, this stationary point is the minimum. Thus the minimizing weight on the experience is
The direction of the weighting matters: a more uncertain experience estimate (larger \(v_E\)) receives less weight, while a more uncertain complement (larger \(v_C\)) permits more weight on experience. If the estimates are correlated, biased, or aimed at different targets, this result does not apply as written. In particular, treating a complement as exact is an additional assumption, not a consequence of using one.
Worked Example: Blending Comparable Indicated Changes
A hypothetical insurer has completed the same stated adjustments to its own experience and to a selected complement. Both results are indicated changes to the same current-rate base for the same coverage and projection period. The experience indicates a 14% increase, the complement indicates a 6% increase, and the documented credibility selection for experience is 35%. Calculate the blended indication and identify a weighting error to avoid.
Method. Since both inputs are percentage changes on a common base, apply \(I=ZE+(1-Z)C\). Here \(Z=0.35\), so the complement weight is \(1-0.35=0.65\).
Experience contribution. \(0.35\times14\%=4.90\) percentage points.
Complement contribution. \(0.65\times6\%=3.90\) percentage points.
Blend and check. \(I=4.90\%+3.90\%=8.80\%\). The result lies between 6% and 14%, as a convex blend must.
A common mistake is to give the complement the 35% weight and the experience the 65% weight. That incorrect calculation is \(0.35(6\%)+0.65(14\%)=2.10\%+9.10\%=11.20\%\). The correction is to apply \(Z\) to experience, as defined, giving 8.80%. The example’s weight is stipulated; it would still need a documented rationale in an actual filing.
Worked Example: Applying Bühlmann Credibility to a Loss-Cost Estimate
A hypothetical classification has 30 equal exposure units. For this model, the expected process variance is 144 squared dollars per exposure unit, and the variance of hypothetical means is 16 squared dollars. The experience loss-cost estimate is $540 per exposure unit, and the complement is $480 on the same basis. Use the Bühlmann model to calculate the credibility-weighted loss-cost selection. Assume the model’s population and variance assumptions are appropriate for this illustration.
Step 1: Calculate the structural parameter. \(K=\mathrm{EPV}/\mathrm{VHM}=144/16=9\) exposure units.
Step 2: Calculate credibility. \(Z=n/(n+K)=30/(30+9)=30/39=10/13\), or 0.7692 to four decimals. The complement weight is \(1-Z=3/13=0.2308\).
Step 3: Blend the estimates. The experience contribution is \((10/13)\times\$540=\$5{,}400/13=\$415.3846\). The complement contribution is \((3/13)\times\$480=\$1{,}440/13=\$110.7692\). Their sum is \(\$6{,}840/13=\$526.1538\).
Selection. The blended loss-cost estimate is $526.15 per exposure unit, rounded to the nearest cent. It is between $480 and $540, which checks the weighted-average calculation. The method avoids assigning full credibility solely because the experience is an insurer’s own: the model’s credibility depends on both volume and the stated variance components. In practice, those inputs and the model’s fit require support.
Worked Example: Choosing a Weight From Estimate Uncertainty
For a hypothetical filing, the experience and complement are both estimates of the same indicated percentage change. Assume each is unbiased and their estimation errors are independent. The experience estimate is 11.0% with a standard error of 3.0 percentage points; the complement is 7.0% with a standard error of 1.5 percentage points. Use the minimum-mean-squared-error blend under these assumptions.
Step 1: Square the standard errors. The experience variance is \(v_E=3.0^2=9.00\) square percentage points. The complement variance is \(v_C=1.5^2=2.25\) square percentage points.
Step 2: Calculate the experience weight. \(Z=v_C/(v_E+v_C)=2.25/(9.00+2.25)=2.25/11.25=0.20\). The complement weight is 0.80.
Step 3: Blend. \(I=0.20(11.0\%)+0.80(7.0\%)=2.20\%+5.60\%=7.80\%\).
Check. The selection is closer to 7.0%, the estimate with the smaller standard error, and it lies between the two estimates. The minimum mean squared error under the stated assumptions is \(0.20^2(9.00)+0.80^2(2.25)=0.36+1.44=1.80\) square percentage points, compared with 9.00 for experience alone and 2.25 for the complement alone. The approach avoids treating a lower-uncertainty estimate as if it deserved less influence. If errors are correlated or either estimate is biased, this simple weight is not justified without further analysis.
Filing Practice and Common Traps
A credibility selection should be reproducible from the stated inputs. When a method produces a weight, retain the calculation and state the assumptions. When the weight is judgmental, explain what information supports it. It is often useful to show how the indication changes under reasonable alternative weights or complements; this makes the impact of the selection visible without implying that one formula resolves every judgment.
- Reversing the weights: In \(I=ZE+(1-Z)C\), \(Z\) is the experience weight. Label the inputs before substituting.
- Blending unlike quantities: Do not combine a pure premium and a percentage change directly. Convert them to a common target or select a consistent measure.
- Claiming a model supports more than it does: A count-based threshold addresses count fluctuation under its assumptions, not every source of uncertainty in a loss indication.
- Ignoring the complement’s basis: A complement that reflects a different population or period may not estimate the same target unless the difference is addressed.
- Reporting a weight without its rationale: A number alone does not explain data relevance, volume, stability, or model limitations.
Check Your Understanding
For each problem, use the stated method and assumptions; amounts are hypothetical.
- A filing uses an experience indication of 16% and a complement indication of 4%, both on the same base. Experience credibility is 25%. Calculate the blended change. (Answer: 7.00%)
- Under an applicable Bühlmann model, \(n=36\) equal exposure units, EPV is 240, and VHM is 20. Experience loss cost is $620 and the comparable complement is $500. Calculate \(K\), \(Z\), and the blended loss cost. (Answer: \(K=12\), \(Z=0.7500\), blended loss cost = $590.00)
- Two independent, unbiased estimates of one target are 10.0% from experience with standard error 5.0 percentage points, and 4.0% from a complement with standard error 2.0 percentage points. Find the minimum-mean-squared-error weight on experience and the blended estimate. (Answer: \(Z=4/(25+4)=0.1379\), blended estimate = 4.83%)
- For a Bühlmann-Straub application, total exposure is \(W=24\) and \(K=8\). The experience loss-cost estimate is $710 and the comparable complement is $650. Calculate the credibility and selection. (Answer: \(Z=24/(24+8)=0.75\), selected loss cost = $695.00)
- A proposed blend uses experience on a developed and trended pure-premium basis, but the complement is an unadjusted percentage rate change. Is the blend ready to calculate? State the required correction. (Answer: No; first express both estimates on the same target and measurement basis, or convert them consistently before applying credibility.)