Why Compare Credibility Weights?
In the Bühlmann-Straub model, a risk’s experience does not receive the same weight in every circumstance. Its credibility weight depends on its total exposure relative to the credibility constant. This makes it possible to compare how strongly two risks’ experience affects their premiums, even when the risks have different exposure histories.
A useful comparison holds the deviation from the collective mean fixed. If two risks’ observed experience rates are each 20 dollars above the collective mean, the risk with the larger credibility factor receives the larger premium increase. This comparison concerns the response of the premium to experience; it does not, by itself, say which risk has the higher premium when their observed rates differ.
The exposure units used for \(w_i\) must be consistent with the units used to define \(a\). For example, if exposure is measured in policy-years and \(a\) is the expected process variance for one policy-year, then \(K\) is also expressed in policy-years. The formula uses a risk’s total exposure across the observed periods, not its number of periods and not an unweighted average of its exposure amounts.
How Exposure Changes the Weight
The credibility factor can be written as \(Z_i=w_i/(w_i+K)\). With \(K\) fixed, increasing \(w_i\) increases \(Z_i\). Since \(1-Z_i=K/(w_i+K)\), increasing exposure reduces the weight on the collective mean by exactly the same amount that it increases the weight on the risk’s experience.
To make a direct comparison, define a risk’s experience deviation as \(d_i=\bar X_i-m\). Subtracting \(m\) from the premium gives
Thus the premium’s movement away from the collective mean is the credibility factor multiplied by the observed deviation. When two risks have the same deviation, their premium movements are in the same direction and are proportional to their credibility factors. When their deviations differ, compare the products \(Z_i d_i\), not the weights alone.
Why the Credibility Formula Has This Form
The formula’s dependence on total exposure follows from the balance between variation among risks and process variation within a risk. The weighted experience mean has conditional expectation \(\mu(\Theta_i)\). Under the model assumptions, its expected conditional variance is \(a/w_i\): independent period variances add after weighting, and larger total exposure reduces the noise in the weighted mean.
Consider a linear predictor \(m+z(\bar X_i-m)\) for the underlying mean \(\mu(\Theta_i)\). Its mean squared error, averaged over risks and experience, is
The first term is the squared-error contribution from shrinking away from the risk’s underlying mean; the second is the contribution from noisy experience. Differentiating with respect to \(z\) and setting the result equal to zero gives
This derivation explains the comparison: greater exposure reduces process noise \(a/w_i\), so the experience mean can receive more weight. The result is the Bühlmann-Straub credibility factor under the stated model; it is not a general rule for arbitrary data with dependent observations or a different exposure-variance relationship.
Worked Example: Same Deviation, Different Total Exposure
Consider two hypothetical risks with a common collective mean of 100 dollars per exposure unit. Assume a Bühlmann-Straub model with conditionally independent normal observations, \(X_{it}\mid\Theta_i\sim N(\mu(\Theta_i),s^2(\Theta_i)/w_{it})\), and structural parameters \(v=300\) and \(a=900\). There is no interest or mortality adjustment. Risk A has total exposure 2, and Risk B has total exposure 8. Each risk’s experience mean is 112. Calculate both premiums and compare their movements from the collective mean.
Method. Calculate \(K\), then each \(Z_i\), and finally apply \(P_i=m+Z_i(\bar X_i-m)\).
First, the credibility constant is
The deviation for each risk is \(112-100=12\). For Risk A,
For Risk B,
Rounded to the nearest cent, the premiums are 104.80 dollars and 108.73 dollars. Their movements above the collective mean are 4.80 dollars and 8.73 dollars, respectively. Risk B’s greater total exposure gives its experience mean more influence. Check: both premiums lie between the collective mean of 100 and the observed mean of 112, and Risk B’s premium is closer to 112.
Common mistake avoided: using the number of periods in place of total exposure. The calculation requires \(w_A=2\) and \(w_B=8\), not a period count that might be identical for the two risks.
Unequal Exposures Across Periods
When exposures vary by period, first compute the exposure-weighted experience mean and the total exposure. A period with twice the exposure receives twice the weight in the experience mean. The credibility factor then uses the sum of exposures; it does not use the arithmetic mean of the period observations or the largest single-period exposure.
Worked Example: Calculating Weighted Experience Before Comparing Premiums
A hypothetical portfolio uses a collective mean of 500 dollars per exposure unit. Assume conditionally independent normal observations with conditional variance \(s^2(\Theta_i)/w_{it}\), structural parameters \(a=1200\) and \(v=200\), and no interest or mortality adjustment. Risk A has observations 550 and 570 at exposures 1 and 3. Risk B has observations 520 and 530 at exposures 2 and 8. Find each weighted experience mean, credibility factor, and premium.
Method. Sum exposure by risk, calculate each weighted mean, and then apply the credibility formula using \(K=a/v\).
The credibility constant is
For Risk A, total exposure is \(w_A=1+3=4\), and its weighted mean is
Its credibility factor and premium are
For Risk B, total exposure is \(w_B=2+8=10\), and its weighted mean is
Therefore,
The premiums are 526.00 dollars for Risk A and 517.50 dollars for Risk B. Risk A has the larger premium even though its credibility factor is smaller, because its experience mean is much farther above the collective mean: its deviation is 65, compared with 28 for Risk B. Check: the premium deviations are \(0.4(65)=26\) and \(0.625(28)=17.5\), matching \(526-500\) and \(517.50-500\).
Common mistake avoided: averaging the two observations for each risk without exposure weights. For instance, Risk B’s unweighted average is 525, not the correct weighted mean of 528; the period with exposure 8 should contribute more than the period with exposure 2.
Comparing Responsiveness and Solving for Exposure
When two risks have the same deviation \(d\), the ratio of their premium movements is the ratio of their credibility factors, provided the deviation is nonzero. This gives a direct measure of relative responsiveness. The same formula can be rearranged to find the exposure needed for a target credibility factor \(z\), where \(0<z<1\):
This rearrangement is useful when an analysis specifies how much of a deviation should flow into the premium. It also makes clear that reaching a target credibility depends on both the target and the credibility constant.
Worked Example: Comparing Premium Response and Finding Target Exposure
Two hypothetical risks have the same experience deviation, 50 dollars per exposure unit, above a collective mean of 200 dollars per exposure unit. Assume conditionally independent normal observations with conditional variance \(s^2(\Theta_i)/w_{it}\), \(a=1000\), \(v=250\), and no interest or mortality adjustment. Risk C has total exposure 2; Risk D has total exposure 8. Compare their premium movements and find the total exposure required for a credibility factor of 0.75.
Method. Compute the credibility constant and weights, multiply each weight by the common deviation, and rearrange the credibility formula for the target exposure.
The credibility constant is
Risk C’s weight is \(Z_C=2/(2+4)=1/3\). Its premium movement is
Risk D’s weight is \(Z_D=8/(8+4)=2/3\). Its movement and premium are
Rounded to the nearest cent, the premiums are 216.67 dollars and 233.33 dollars. Risk D’s premium movement is twice Risk C’s because \(Z_D/Z_C=(2/3)/(1/3)=2\). For a target credibility of 0.75, the required exposure is
Check: \(12/(12+4)=12/16=0.75\), as required. The target exposure is 12 units, not 3; 3 is the numerator \(zK\) before division by \(1-z\).
Common mistake avoided: comparing premiums without first subtracting the collective mean. The premiums differ because both risks have the same collective mean but different weights; the response comparison is \(33.33/16.67=2\), not a ratio of the premium levels.
Common Exam Traps
The following checks help keep comparisons interpretable and calculations consistent:
- Reverse the ratio in the credibility constant. The convention here is \(K=a/v\), expected process variance divided by variance of hypothetical means. Using \(v/a\) changes both the units and the credibility factors.
- Use the wrong exposure total. Sum the risk’s exposure units across periods. Do not substitute the number of observations, the largest exposure, or an unweighted average of period experience.
- Compare weights without considering deviations. A higher \(Z_i\) means a stronger response to experience, but the actual premium movement is \(Z_i(\bar X_i-m)\). A smaller deviation can produce a smaller premium movement even with a larger weight.
- Lose the sign of the deviation. If \(\bar X_i<m\), then \(P_i-m\) is negative. The premium moves below the collective mean, not above it.
- Ignore model conditions. The formula relies on the specified Bühlmann-Straub variance structure and conditional independence. If observations are dependent or their conditional variances do not scale as \(1/w_{it}\), the stated calculation need not apply.
For the first trap, suppose \(K=3\) and \(w=2\). The correct factor is \(2/(2+3)=0.4\). Reversing the numerator and denominator of the structural ratio would instead use \(v/a=1/3\) as though it were \(K\), producing the incorrect factor \(2/(2+1/3)=6/7\), or approximately 0.8571. This incorrect result gives the low-exposure risk far too much weight.
Check Your Understanding
For each problem, calculate exposure totals before applying the credibility formula. Round money to the nearest cent where needed.
- A hypothetical risk has total exposure 5 and experience mean 30 dollars above the collective mean. If \(a=1000\) and \(v=200\), find its credibility factor and premium movement from the collective mean. (Answer: \(K=5\), \(Z=0.5000\), movement \(=15.00\) dollars.)
- Two risks have the same deviation of \(-20\) dollars from the collective mean. Let \(a=800\), \(v=200\), and their total exposures be 2 and 8. Find each premium movement. (Answer: \(K=4\); movements are \(-6.67\) dollars and \(-13.33\) dollars, respectively.)
- A hypothetical risk has observations 430 and 460 at exposures 2 and 4. The collective mean is 400, \(a=900\), and \(v=150\). Find its weighted experience mean, total exposure, credibility factor, and premium. (Answer: \(\bar X=450\), \(w=6\), \(K=6\), \(Z=0.5000\), premium \(=425.00\) dollars.)
- If the credibility constant is 9, what total exposure is required for a credibility factor of 0.70? (Answer: \(w=21\) exposure units.)
- Two risks each have an experience deviation of 18 dollars. If \(K=3\) and their total exposures are 3 and 9, find their premium movements and the ratio of the larger movement to the smaller. (Answer: movements are \(9.00\) dollars and \(13.50\) dollars; ratio \(=1.5\).)