Selecting a Complement for Territory Credibility

Why the Complement Matters

A territory’s own experience may be too volatile to use by itself, even after it has been standardized and expressed on a consistent basis. Credibility weighting addresses that uncertainty by combining the territory’s indication with a complement: an estimate that supplies information when the territory’s own data are less reliable. The complement is not just a calculation input. Its choice determines what broader information the territorial estimate is being pulled toward.

A statewide indication can provide a stable benchmark, but it may not reflect meaningful differences among parts of the state. A regional indication can reflect some of those differences, while being less stable if it uses fewer observations. A model-based indication can account for relevant characteristics, but its reliability depends on the model, its data, and how well it predicts the territory being estimated. The appropriate choice depends on the target being estimated and the quality of each candidate complement.

Definition: Let $I$ be a territory’s own indication of the quantity being estimated, and let $C$ be a complement indication for that same quantity, on the same basis. A credibility-weighted estimate is $\hat{\theta}=ZI+(1-Z)C$, where $0\leq Z\leq1$. The complement receives weight $1-Z$. The indication and complement must be defined consistently, for example as loss costs per exposure for the same coverage, period, and exposure basis.

Here, the credibility weight $Z$ is treated as already selected or as calculated from an explicit error model. The question in this tutorial is how to select the complement and understand how that selection affects the weighted result. Any resulting territorial indications or factors may still require appropriate balancing to the rating plan’s required average.

Three Common Complement Choices

Complement	Potential advantage	Question to examine
Statewide indication	Uses a broad base of experience and may be relatively stable	Does the statewide estimate represent the territory’s expected level?
Regional indication	Can reflect broad geographic differences within the state	Is the regional grouping sufficiently supported and appropriate for this territory?
Model-based indication	Can incorporate relevant variables beyond a simple geographic grouping	Is the model suitable, validated, and supported for predictions in this territory?

These are alternative sources of information, not automatically ordered levels of quality. A regional complement is not necessarily better than a statewide one, and a sophisticated model is not necessarily more reliable than a simple benchmark. The candidate complements should estimate the same target. For example, one should not combine a territory’s loss cost per exposure with a statewide loss ratio unless both are first put on a compatible basis.

A model-based complement should represent a prediction for the territory being estimated, not a fitted value that simply reuses the territory’s own experience as if it were independent information. If the territory’s data helped fit the model, the apparent agreement between its indication and the model complement may overstate the amount of independent information. The actuary should account for that dependence when estimating uncertainty.

Choosing a Complement With an Error Model

A useful comparison starts by treating $I$ and $C$ as estimates of a common target $\theta$. Define their errors as $e_I=I-\theta$ and $e_C=C-\theta$. If both estimates are unbiased, then $E[e_I]=E[e_C]=0$. Write $\sigma_I^2=\operatorname{Var}(e_I)$, $\sigma_C^2=\operatorname{Var}(e_C)$, and $\operatorname{Cov}(e_I,e_C)$ for the error covariance.

The variance of the weighted estimate depends on both error variances and their covariance. When the errors are independent, the covariance is zero. When they are not independent, using the independent-error formula can give the wrong weight. For an unconstrained weight, minimizing the variance gives the following expression; a credibility weight used in practice is usually constrained to the interval from zero to one.

\operatorname{Var}(\hat{\theta}) =Z^2\sigma_I^2+(1-Z)^2\sigma_C^2 +2Z(1-Z)\operatorname{Cov}(e_I,e_C)

To justify the optimal weight, differentiate this variance with respect to $Z$. Setting the derivative equal to zero yields the unconstrained solution below. If this solution lies outside $[0,1]$, the variance-minimizing constrained choice is the nearer endpoint: $Z=0$ or $Z=1$. The denominator is the variance of the difference between the errors, so it cannot be negative.

Z^* = \frac{\sigma_C^2-\operatorname{Cov}(e_I,e_C)} {\sigma_I^2+\sigma_C^2-2\operatorname{Cov}(e_I,e_C)} \qquad \text{if } \sigma_I^2+\sigma_C^2-2\operatorname{Cov}(e_I,e_C)>0

For independent errors, this reduces to $Z^*=\sigma_C^2/(\sigma_I^2+\sigma_C^2)$, provided $\sigma_I^2+\sigma_C^2>0$. A less uncertain complement has a smaller error variance and therefore receives greater weight under this model. If both error variances are zero, every $Z\in[0,1]$ minimizes the variance: the displayed independent-error ratio is undefined in that case and must not be used.

The variance-minimizing result is not by itself a complete selection rule. It assumes that both estimates are unbiased for the same target. If a candidate complement is systematically high or low, its bias matters. With biases $b_I=E[I]-\theta$ and $b_C=E[C]-\theta$, the mean squared error of the weighted estimate is its variance plus its squared bias:

\operatorname{MSE}(\hat{\theta}) = Z^2\sigma_I^2+(1-Z)^2\sigma_C^2 +2Z(1-Z)\operatorname{Cov}(e_I,e_C) +\left(Zb_I+(1-Z)b_C\right)^2

Thus a complement with low variance may still be unsuitable if it is biased for the territory’s target. In practice, error variances, bias, and covariance are estimated or assessed using relevant historical data and professional judgment. A numerical weight is only as credible as those assumptions.

Worked Example: Comparing Statewide and Regional Complements

Method and assumptions. A hypothetical territory has an own loss-cost indication of $I=\$160$ per exposure. The candidate statewide complement is $C_S=\$140$, and the candidate regional complement is $C_R=\$150$. Assume both complements and the own indication are unbiased for the same territory loss cost. The error variances are $\sigma_I^2=900$, $\sigma_S^2=100$, and $\sigma_R^2=225$, in squared dollars per exposure. For each candidate, assume its error is independent of the territory’s own error. Compare the variance-minimizing estimate under each complement.

Statewide complement. For independent errors, the optimal weight on the own indication is the complement variance divided by the sum of the two variances:

Z_S=\frac{100}{900+100}=0.10, \qquad 1-Z_S=0.90

The credibility-weighted estimate is $0.10(\$160)+0.90(\$140)=\$16+\$126=\$142$ per exposure.

Regional complement. Its optimal weight is:

Z_R=\frac{225}{900+225}=0.20, \qquad 1-Z_R=0.80

The estimate is $0.20(\$160)+0.80(\$150)=\$32+\$120=\$152$ per exposure.

The regional complement has greater assumed error variance than the statewide complement, so it receives less weight than the statewide complement does in its own calculation. Yet the regional estimate is closer to the territory’s own indication, and the resulting weighted estimate is higher. The comparison illustrates that complement selection affects the estimate through both the complement value and the weight. Common mistake avoided: applying one complement’s optimal weight to a different complement without checking its error variance.

Worked Example: A Correlated Model-Based Complement

Method and assumptions. A hypothetical territory has own indication $I=\$130$ per exposure. A model-based complement for that territory is $C=\$118$. Assume both are unbiased for the same target. Their error variances are $\sigma_I^2=250$ and $\sigma_C^2=100$, and the error covariance is $\operatorname{Cov}(e_I,e_C)=30$, in squared dollars per exposure. Find the variance-minimizing weighted estimate, constraining $Z$ to $[0,1]$.

The denominator in the covariance formula is $250+100-2(30)=290$, which is positive. The numerator is $100-30=70$. Therefore:

Z^*=\frac{70}{290}=\frac{7}{29}\approx0.2414, \qquad 1-Z^*=\frac{22}{29}

This weight is in the permitted interval. The weighted estimate is:

\hat{\theta} =\frac{7}{29}(\$130)+\frac{22}{29}(\$118) =\frac{\$910+\$2{,}596}{29} =\frac{\$3{,}506}{29} \approx \$120.90

The arithmetic is rounded only at the final step, to the nearest cent. The positive covariance means the two errors share some variation; the covariance-adjusted weight is not the independent-error weight $100/(250+100)$. Common mistake avoided: treating a model complement as independent merely because it is presented separately. If the model used related experience, dependence should be considered.

Worked Example: Accounting for Complement Bias

Method and assumptions. For a hypothetical territory, suppose the true target loss cost for evaluating the method is $\theta=\$150$ per exposure. The observed own indication is $I=\$170$, and a statewide complement is $C=\$155$. Assume the own indication is unbiased with error variance $400$, the complement has bias $b_C=\$5$ and error variance $100$, and the errors are independent. Find the weight that minimizes mean squared error under these assumptions, then calculate the weighted estimate.

Because the own indication is unbiased, $b_I=0$. The mean squared error as a function of $Z$ is:

\operatorname{MSE}(Z) =400Z^2+100(1-Z)^2+\left(5(1-Z)\right)^2 =400Z^2+125(1-Z)^2

Differentiate and set the derivative to zero: $800Z-250(1-Z)=0$, so $1{,}050Z=250$, and $Z=5/21$. This lies in $[0,1]$. The complement weight is $16/21$, and the weighted estimate from the observed indications is:

\hat{\theta} =\frac{5}{21}(\$170)+\frac{16}{21}(\$155) =\frac{\$850+\$2{,}480}{21} =\frac{\$3{,}330}{21} \approx \$158.57

The estimate is rounded only at the final step, to the nearest cent. Ignoring the complement’s bias would instead give $Z=100/(400+100)=0.20$, which is not the mean-squared-error-minimizing weight under the stated assumptions. Common mistake avoided: selecting a complement by variance alone when the assumed bias is material.

A Practical Complement-Selection Process

A disciplined comparison makes the assumptions behind the final estimate visible. The following sequence helps distinguish a suitable complement from one that is merely convenient to calculate.

Define the target.
Specify the territory quantity being estimated and the basis of measurement. Confirm that the own indication and every candidate complement refer to that same target.

Assess representativeness.
Ask whether statewide, regional, or model-based information is relevant to the territory, and whether the complement may be systematically high or low for it.

Assess uncertainty and dependence.
Estimate or compare the error variance of each complement and consider whether its errors are correlated with the territory’s own indication.

Calculate and compare weighted estimates.
Use the stated weight assumptions consistently. Show how each candidate changes both the weight and the resulting indication.

Check limitations.
Reconsider the result if a candidate is biased, based on data that overlap the territory experience, or estimated with uncertainty not included in the calculation.

Common Traps and Interpretation

Assuming the broadest complement is always best. A statewide estimate may be stable but poorly representative of a territory. Stability and relevance are separate considerations.
Assuming a more granular complement is always better. A regional estimate may better reflect local conditions, but its smaller data base can make it less stable.
Ignoring covariance. A model-based complement and the territory indication may share data or underlying experience. If so, an independent-error calculation may misstate the appropriate weight.
Confusing the weight with the complement share. In $\hat{\theta}=ZI+(1-Z)C$, $Z$ multiplies the territory indication; $1-Z$ multiplies the complement.
Using a variance formula when bias matters. Variance minimization is appropriate under unbiasedness. A biased complement requires considering mean squared error or another justified selection criterion.
Dividing by zero in the independent-error formula. If $\sigma_I^2+\sigma_C^2=0$, every weight minimizes variance. The ratio for $Z$ is undefined, not a value to approximate.

Selecting a complement is consequential because the same territory indication can produce materially different estimates when paired with different complements. A defensible choice states what the complement represents, what uncertainty and dependence assumptions support its weight, and whether bias could affect the result. This treatment is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Check Your Understanding

For each problem, use the assumptions stated in the question and retain full precision until the final answer.

A hypothetical territory has $I=\$180$ per exposure and an unbiased statewide complement $C=\$150$. Their independent error variances are $500$ and $125$, respectively. Find the variance-minimizing weight on $I$ and the weighted estimate. (Answer: $Z=0.20$; estimate $=\$156$ per exposure.)
A territory indication is $I=\$130$, and a model complement is $C=\$118$, both unbiased. Their error variances are $250$ and $100$, and their error covariance is $30$. Calculate the constrained variance-minimizing $Z$ and the weighted estimate, rounded to the nearest cent. (Answer: $Z=7/29\approx0.2414$; estimate $=\$120.90$ per exposure.)
Suppose an own indication and a complement are both unbiased, each has error variance zero, and both equal $\$142$. What weight minimizes variance, and what is the weighted estimate? (Answer: Every $Z\in[0,1]$ minimizes variance; estimate $=\$142$ per exposure for every such $Z$.)
An own indication has observed value $I=\$160$, zero bias, and error variance $300$. A complement has observed value $C=\$150$, bias $b_C=\$4$, and error variance $100$. Assume independent errors. Find the mean-squared-error-minimizing weight on $I$, then calculate the weighted estimate to the nearest cent. (Answer: $Z=29/104\approx0.2788$; estimate $\approx\$152.79$ per exposure.)
For a hypothetical territory, $I=\$200$ and the candidate statewide and regional complements are $\$160$ and $\$180$. If the same specified weight $Z=0.25$ is used with either complement, calculate both weighted estimates and their factors relative to a $\$175$ reference loss cost, rounded to four decimal places. (Answer: Statewide estimate $=\$170$, factor $=0.9714$; regional estimate $=\$185$, factor $=1.0571$.)

What You'll Learn