Segment-Specific Complements For Heterogeneous Risks

Why the Complement Matters

A credibility premium combines observed experience with a complement: an estimate of the expected cost when the risk's own experience is incomplete or unstable. For a homogeneous group, the choice of complement may be straightforward. For a heterogeneous portfolio, however, segments can have materially different expected costs. A premium that blends a segment's experience with its own class expectation can differ from one that blends it with a common portfolio expectation, even when the experience and credibility weight stay unchanged.

The complement is not a minor computational detail. It determines the direction and size of the pull applied to a segment's observed experience. A portfolio-wide complement draws higher-cost segments downward and lower-cost segments upward relative to class-specific complements. Whether that is appropriate depends on the purpose and design of the rating method, as well as whether the segments are intended to retain distinct expected-cost levels.

The comparisons below are aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus. All examples describe hypothetical insurance portfolios.

Definition: For segment $j$, let $m_j$ be its class-specific complement, $m_P$ the portfolio-level complement, $\bar X_j$ its observed loss cost, and $Z_j$ its credibility weight. Using complement $m$, the indicated premium or loss cost is $P_j(m)=Z_j\bar X_j+(1-Z_j)m$. A class-specific complement uses $m=m_j$; a portfolio complement uses the same $m=m_P$ for every segment.

Here, loss cost and premium refer to the component being credibility-rated; expenses, profit provisions, and other rating adjustments are outside these calculations. The credibility weight is held fixed when comparing complements. That isolates the effect of the complement, rather than mixing it with a change in the amount of credibility.

Constructing a Portfolio Complement

A portfolio complement should represent the expected cost for the portfolio being rated. If the segments have expected prospective exposure shares $q_j$, with $\sum_j q_j=1$, and class complements $m_j$ on a consistent per-unit basis, then the exposure-weighted portfolio complement is

m_P=\sum_j q_jm_j.

The shares should reflect the portfolio mix appropriate to the prospective rating period, not automatically the number of policies or the distribution of historical observations. For example, if the segments have different average exposures per policy, policy counts alone do not give the correct exposure shares. The complements and shares must also use the same coverage basis and loss-cost unit.

The resulting portfolio complement is a weighted average of the segment expectations. It is not generally the simple average of those expectations. Nor is it necessarily appropriate to use a portfolio complement merely because it is mathematically easy to calculate: doing so deliberately replaces each segment's own expected-cost level with a common portfolio expectation.

Worked Example: Comparing Two Complements

A hypothetical insurer rates two segments. Segment 1 has a class complement of $140 and a prospective exposure share of 40%; Segment 2 has a class complement of $60 and a share of 60%. Their observed loss costs are $120 and $40, respectively. The credibility weights are 0.25 for Segment 1 and 0.75 for Segment 2. Calculate both premiums under class-specific complements and under the portfolio complement.

Step 1: Calculate the portfolio complement. Weight each class complement by the prospective exposure share:

$$ m_P=0.40(140)+0.60(60)=56+36=92. $$

The portfolio complement is $92.

Step 2: Use the class-specific complements. Segment 1 receives 25% weight on its observed cost and 75% on its class complement:

$$ P_1(m_1)=0.25(120)+0.75(140)=30+105=135. $$

Segment 2 receives 75% weight on its observed cost and 25% on its class complement:

$$ P_2(m_2)=0.75(40)+0.25(60)=30+15=45. $$

Step 3: Use the common portfolio complement. The credibility weights and observed costs do not change:

$$ P_1(m_P)=0.25(120)+0.75(92)=30+69=99, $$

$$ P_2(m_P)=0.75(40)+0.25(92)=30+23=53. $$

The results are Segment 1: $135 with its class complement and $99 with the portfolio complement; Segment 2: $45 and $53, respectively. The first segment's portfolio-based result is pulled below its class-based result because $92 is below its class complement of $140. The second is pulled upward because $92 is above its class complement of $60. The common mistake is to compute Segment 1's class premium as $30+105=125$; the correct sum is $135$.

Credibility Weights with Unequal Exposure

In the Bühlmann-Straub model, the segment's observed experience is commonly represented by a weighted average $\bar X_j$ based on total exposure $W_j$. Under the model, conditional observations have a common segment-specific mean, and their conditional variance decreases with exposure. The standard credibility form is $Z_j\bar X_j+(1-Z_j)m$, where $Z_j=W_j/(W_j+K)$ and $K$ is the ratio of expected process variance to variance of hypothetical means. This weight depends on the experience exposure and model parameters, not on which complement is selected.

When segment-level class complements are available, the usual comparison substitutes $m_j$ or $m_P$ into the same credibility formula. This is a comparison of complement choices, not a claim that the model has estimated the two complements in the same way. In practice, $K$ and the credibility weights need to be supported by an appropriate model and data.

The reason the complement changes the premium can be shown directly. For a fixed segment and fixed $Z_j$, subtract the class-complement premium from the portfolio-complement premium:

\begin{aligned} P_j(m_P)-P_j(m_j) &=Z_j\bar X_j+(1-Z_j)m_P-\left[Z_j\bar X_j+(1-Z_j)m_j\right]\\ &=(1-Z_j)(m_P-m_j). \end{aligned}

The observed-experience terms cancel. Thus the change is the complement gap multiplied by the portion of the premium not supported by the segment's own experience. If $Z_j=1$, the complement has no effect. If $Z_j=0$, the premium is entirely the selected complement. This also explains why segments with smaller credibility weights can experience larger changes when the complement changes.

Worked Example: Unequal Exposures and Bühlmann-Straub Weights

A hypothetical portfolio has segments A and B. Segment A has total experience exposure $W_A=6$, observed loss cost $150, and class complement $130. Segment B has $W_B=2$, observed loss cost $50, and class complement $70. Their Bühlmann-Straub parameters give $K_A=4$ and $K_B=8$. The prospective portfolio exposure shares are 20% for A and 80% for B. Find the weights, the portfolio complement, and the premiums under both complement choices.

Step 1: Calculate each credibility weight. Use $Z_j=W_j/(W_j+K_j)$:

Z_A=\frac{6}{6+4}=0.60,\qquad Z_B=\frac{2}{2+8}=0.20.

Step 2: Calculate the portfolio complement. Apply the stated prospective exposure shares:

$$ m_P=0.20(130)+0.80(70)=26+56=82. $$

Step 3: Calculate premiums using class complements.

$$ P_A(m_A)=0.60(150)+0.40(130)=90+52=142, $$

$$ P_B(m_B)=0.20(50)+0.80(70)=10+56=66. $$

Step 4: Substitute the portfolio complement while retaining the weights.

$$ P_A(m_P)=0.60(150)+0.40(82)=90+32.80=122.80, $$

$$ P_B(m_P)=0.20(50)+0.80(82)=10+65.60=75.60. $$

The portfolio-complement premiums are $122.80 for A and $75.60 for B. For a check, the premium differences are $(1-0.60)(82-130)=-19.20$ for A, taking its class premium from $142 to $122.80, and $(1-0.20)(82-70)=9.60$ for B, taking its class premium from $66 to $75.60. A frequent error is to use $W_A$ and $W_B$ as the portfolio mix; those values determine the credibility weights here, while the problem separately supplies the prospective mix for $m_P$.

Bayesian Complements and Interpretation

A Bayesian model gives another precise interpretation of a complement. In a Poisson-Gamma model, suppose a segment's claim count $N$, conditional on its claim frequency $\lambda$, is Poisson with mean $E\lambda$, where $E$ is exposure. Let the prior distribution of $\lambda$ be Gamma with shape $\alpha$ and rate $\beta$, so its prior mean is $\alpha/\beta$. With this parameterization, the posterior distribution has shape $\alpha+N$ and rate $\beta+E$; its posterior mean is the credibility estimate of the segment's frequency.

The posterior mean can be written as a weighted average of the observed frequency $N/E$ and the prior mean. The weight on observed frequency is $E/(\beta+E)$; the weight on the prior mean is $\beta/(\beta+E)$. This representation follows by putting the two terms over the common denominator $\beta+E$:

\frac{\alpha+N}{\beta+E} =\frac{E}{\beta+E}\frac{N}{E} +\frac{\beta}{\beta+E}\frac{\alpha}{\beta}.

The equality assumes $E>0$; if $E=0$, the posterior mean remains the prior mean. In a segment-specific Bayesian model, the prior mean is a natural class complement. Replacing that mean with a portfolio mean while keeping the original weight is a useful comparison, but it is not the posterior mean under the original segment prior. To call the result a Bayesian posterior under a new prior, the prior distribution itself must be specified consistently.

Worked Example: A Portfolio Complement in a Poisson-Gamma Comparison

For hypothetical Segment C, claim count $N=90$ is observed over exposure $E=2$. Its frequency prior is Gamma with shape $\alpha=40$ and rate $\beta=1$. A separate segment D has prior mean frequency 20. In the prospective portfolio, C has 25% of exposure and D has 75%. Calculate C's posterior mean under its own prior and the credibility estimate obtained by substituting the portfolio complement while keeping C's weight fixed.

Step 1: Find C's observed frequency and prior mean.

\frac{N}{E}=\frac{90}{2}=45,\qquad m_C=\frac{\alpha}{\beta}=\frac{40}{1}=40.

Step 2: Calculate the portfolio complement.

$$ m_P=0.25(40)+0.75(20)=10+15=25. $$

Step 3: Calculate the credibility weight and the posterior mean under C's own prior.

Z_C=\frac{E}{\beta+E}=\frac{2}{1+2}=\frac{2}{3}.

The conjugate posterior mean is:

E[\lambda\mid N]=\frac{\alpha+N}{\beta+E} =\frac{40+90}{1+2} =\frac{130}{3} =43.333\ldots

As a check, the credibility form gives $(2/3)(45)+(1/3)(40)=30+13.333\ldots=43.333\ldots$. The posterior mean is $43.33 per exposure unit, rounded to two decimals.

Step 4: Substitute the portfolio complement, retaining $Z_C=2/3$.

\frac{2}{3}(45)+\frac{1}{3}(25)=30+8.333\ldots=38.333\ldots

The portfolio-complement estimate is $38.33 per exposure unit, rounded to two decimals. The own-prior posterior remains $43.33; substituting 25 changes the complement-based estimate, not the posterior implied by the stated Gamma prior. The trap is to use $90/3=30$ as the own-prior posterior mean: this omits the prior shape parameter, which must be added to the count.

Choosing and Communicating the Complement

A class-specific complement preserves differences in expected cost between segments. A portfolio complement pools those expected-cost levels toward a common value. Neither choice is universally correct. The analyst should state which population the complement represents, how its weights were chosen, and whether the resulting pooling is intended. The comparison is especially useful when a portfolio has a mix of segments that differ in expected frequency or severity.

Before calculating, check the following:

Keep units consistent. A claim frequency, pure premium, and total expected loss are different quantities and cannot be mixed as complements.
Use the appropriate mix. Portfolio weights should represent the relevant prospective exposure distribution, not an unexamined count or historical loss share.
Separate the two roles of weights. The segment credibility weight $Z_j$ controls the blend of experience and complement; the portfolio share $q_j$ constructs the portfolio complement.
Hold weights fixed for a clean complement comparison. If the model or experience basis also changes, the premium difference cannot be attributed to the complement alone.
Interpret Bayesian substitutions carefully. A changed complement in a credibility formula is not automatically a posterior under the original prior.

A concise exam calculation can show the complement first, then each segment's credibility weight, and finally each premium. Reporting the components makes arithmetic errors easier to detect. As a final check, the difference between the two premiums for segment $j$ should equal $(1-Z_j)(m_P-m_j)$.

Check Your Understanding

For each hypothetical problem, calculate the requested portfolio complement or credibility premium. Answers use the units stated in each question.

Segments A and B have class complements of $160 and $90 and prospective exposure shares of 30% and 70%. Calculate the portfolio complement. (Answer: $111)
Two segments have class complements of $100 and $120, with prospective exposure shares of 25% and 75%. For Segment A, the observed cost is $125 and $Z_A=0.40$. Calculate its premium under its own class complement and under the portfolio complement. (Answer: portfolio complement $115; class premium $110; portfolio-complement premium $119)
A segment has exposure $W=3$, $K=5$, observed cost $80, class complement $110, and portfolio complement $90. Calculate its Bühlmann-Straub weight and both premiums. (Answer: $Z=0.375$; class-complement premium $98.75; portfolio-complement premium $86.25)
In a Poisson-Gamma model, a segment has $N=42$, $E=1$, and a Gamma prior with shape 18 and rate 2. Its portfolio complement is 15. Calculate the observed frequency, credibility weight, posterior mean under its own prior, and estimate using the portfolio complement with the same weight. (Answer: observed frequency 42; $Z=1/3$; own-prior posterior mean 20; portfolio-complement estimate 24)
Two segments have class complements of $240 and $120. Their prospective exposure shares are 20% and 80%. Segment A has observed cost $200 and $Z_A=0.50$; Segment B has observed cost $150 and $Z_B=0.25$. Calculate the portfolio complement and both portfolio-complement premiums. (Answer: portfolio complement $144; Segment A $172; Segment B $145.50)

What You'll Learn