Credibility Weighting in Rate Indications

Why Credibility Weighting Is Used

A rate indication based on a particular block of experience may be informative without being fully representative of the future. A small or volatile experience base, for example, can produce an indication that changes substantially from one period to another. A complementary indication provides another estimate to consider. Credibility weighting combines the two, assigning a specified share to each.

This tutorial concerns the calculation and interpretation of that blend. It takes the credibility weight and the two comparable indications as given. The indications must be on the same basis: they must refer to the same coverage, prospective period, rate level or rate-change measure, and relevant adjustments. A weighted average cannot repair indications that measure different things.

Definition: Let $I_E$ be an experience-based rate indication, $I_C$ a complementary indication, and $Z$ the credibility assigned to the experience indication, where $0\leq Z\leq 1$. The credibility-weighted indication is $$ I=Z I_E+(1-Z)I_C. $$ Thus, $Z$ is the weight on the experience indication, and $1-Z$ is the weight on the complementary indication. The formula applies when both indications are expressed in the same units and on a consistent basis.

The formula may be applied to indicated percentage changes or to indicated rate levels. The units of $I_E$, $I_C$, and $I$ must match. If the indications are percentage changes, the result is a percentage change—not a dollar rate. If they are dollar rates, the result is a dollar rate. Do not average a percentage change with a rate level.

Blending Indicated Percentage Changes

When each indication is stated as a change from the same current rate level, blend the changes first. A positive change represents an increase, and a negative change represents a decrease. After calculating the selected change, apply it to the current rate to obtain the selected rate level.

c=Zc_E+(1-Z)c_C, \qquad R_{\text{selected}}=R_{\text{current}}(1+c),

Here $c_E$ and $c_C$ are decimal changes. For instance, an increase of $6\%$ is $0.06$. If changes are entered as percentage points, keep that convention consistent throughout the weighted average and convert to a decimal before multiplying a rate.

Worked Example: Blending Two Percentage Changes

A hypothetical coverage has a current rate of $800 per exposure. The experience-based indication is a $12\%$ increase, the complementary indication is a $4\%$ increase, and the specified credibility weight on experience is $Z=0.65$. Find the selected change and selected rate.

Method. Blend the changes because both are stated relative to the same current rate. The weight on the complementary indication is $1-0.65=0.35$.

Experience contribution: $0.65(12\%)=7.8\%$.

Complementary contribution: $0.35(4\%)=1.4\%$.

Selected change: $7.8\%+1.4\%=9.2\%$, or $0.092$ as a decimal.

Selected rate: $$800(1.092)=$873.60$.

Check. The selected change lies between $4\%$ and $12\%$, as it must for a weighted average with weights between zero and one. The selected rate is therefore between $$800(1.04)=$832.00$ and $$800(1.12)=$896.00$. The result of $873.60 is within those bounds. This method avoids treating the blended $9.2\%$ change as a dollar amount.

Why the Weighted Average Has This Form

The credibility weight can be understood as allocating the selected indication between the two sources. The experience indication contributes the fraction $Z$ of the result, while the complementary indication contributes the remaining fraction. The coefficients sum to one:

$$ Z+(1-Z)=1. $$

This normalization ensures that if both sources give the same indication $a$, the blend remains $a$: $Za+(1-Z)a=a$. It also ensures that the result is between the two indications when $0\leq Z\leq1$. To see this, suppose $I_E\leq I_C$. Then

I-I_E=(1-Z)(I_C-I_E)\geq 0, \qquad I_C-I=Z(I_C-I_E)\geq 0.

Consequently $I_E\leq I\leq I_C$. The same argument applies with the endpoints reversed if $I_C\leq I_E$. This is a useful reasonableness check. If a computed blend falls outside the two indications, check the arithmetic, the weights, and whether the inputs were comparable.

Interpretation: A larger $Z$ moves the selected indication toward the experience indication; a smaller $Z$ moves it toward the complementary indication. At $Z=1$, the selected indication equals $I_E$. At $Z=0$, it equals $I_C$. Intermediate values produce a blend. The formula alone does not determine $Z$; use the weight specified or established for the problem.

Blending Indicated Rate Levels

Sometimes the two indications are given directly as rates rather than as percentage changes. If both dollar rates apply to the same exposure and prospective basis, combine the rate levels directly. Do not first convert them to percentage changes unless the problem asks for that representation.

Worked Example: Combining Indicated Rate Levels

A hypothetical service policy has a current rate of $1,080 per policy. After the relevant adjustments, the experience indication is $1,160 per policy and the complementary indication is $1,100 per policy. The specified weight on experience is $Z=0.40$. Calculate the selected rate and its change from the current rate.

Method. Both indications are already dollar rates per policy on the same basis, so blend the rate levels.

The complementary weight is $1-0.40=0.60$. The weighted experience amount is $0.40($1,160)=$464$. The weighted complementary amount is $0.60($1,100)=$660$.

Selected rate: $$464+$660=$1,124$ per policy.

Change from current rate: $$1,124-$1,080=$44$ per policy. As a percentage of the current rate, \[ \frac{$44}{$1,080}=0.0407407\ldots \] so the selected change is approximately $4.07\%$.

Check. The selected rate of $1,124 lies between the two indicated rates, $1,100 and $1,160. Relative to the current rate, the complementary indication is an increase of $$20/$1,080$, approximately $1.85\%$, while the experience indication is an increase of $$80/$1,080$, approximately $7.41\%$. The selected increase of approximately $4.07\%$ lies between them. This direct rate-level method avoids introducing rounding by converting each indication to a percentage first.

Limited Experience and the Effect of the Weight

The credibility weight controls how much the experience indication affects the selection; it does not change the experience indication itself. When experience is limited and the assigned $Z$ is smaller, the experience result receives less influence and the complementary result receives more. This can reduce the sensitivity of the selected rate to unusual results in the experience data.

The size of the effect is transparent from the formula. Holding the two indications fixed, the difference between the blended result and the complementary indication is

$$ I-I_C=Z(I_E-I_C). $$

Thus, the blend is $Z$ times the distance from the complementary indication to the experience indication. A smaller $Z$ reduces that distance. This is an interpretation of the specified weighting, not a rule for choosing a credibility weight. The weight may be supplied in a problem; determining an appropriate complementary indication is a separate task.

Worked Example: Interpreting a Lower Weight

A hypothetical equipment policy has a current rate of $500 per unit. Its experience-based indication is a $15\%$ increase, while its complementary indication is a $5\%$ increase. The specified credibility weight is $Z=0.20$. Calculate the selected rate and compare it with the rate that would result from assigning all weight to experience.

Method. Blend the changes, using $0.20$ on experience and $1-0.20=0.80$ on the complementary indication.

Experience contribution: $0.20(15\%)=3\%$.

Complementary contribution: $0.80(5\%)=4\%$.

Selected change: $3\%+4\%=7\%$.

Selected rate: $$500(1.07)=$535.00$.

If all weight were assigned to experience, the change would be $15\%$, producing $$500(1.15)=$575.00$. The specified blend is $40 lower. It is also $10 above the rate produced by the complementary indication alone, $$500(1.05)=$525.00$. This is consistent with a $20\%$ weight moving the result one fifth of the way from the complementary change toward the experience change: $5\%+0.20(15\%-5\%)=7\%$.

Check. The selected $7\%$ increase lies between $5\%$ and $15\%$. The smaller credibility weight keeps the selected rate closer to the complementary indication than to the experience indication.

Common Errors and Practical Checks

A credibility blend is simple to calculate, but errors in interpreting the inputs can produce a plausible-looking yet incorrect result. Before accepting a calculation, identify what each input represents and verify the selected indication against the endpoints.

Reversing the weights. The defined $Z$ is the weight on experience. The complementary weight is $1-Z$, not $Z$ again.
Applying the wrong operation to a rate change. A selected increase of $7\%$ means multiply the current rate by $1.07$, not by $0.07$.
Mixing percentage changes and rate levels. First express both indications as changes or both as rates on the same basis. Never take a weighted average of, for example, $8\%$ and $1,200.
Forgetting the sign. A decrease of $3\%$ is $-3\%$. It must be added as a negative change, not treated as a positive contribution.
Assuming limited experience determines the weight automatically. A small volume of experience may motivate less reliance on it, but this formula does not prescribe a weight from exposure count. Use the specified $Z$ unless the problem explicitly asks for another method.
Ignoring comparability. If the indications differ in coverage, period, exposure basis, or adjustments, a numerical blend may not be meaningful until they are placed on a consistent basis.

Worked Example: Correcting a Reversed-Weight Calculation

A hypothetical policy has a current rate of $1,250. The experience indication is a $9\%$ increase, the complementary indication is a $3\%$ decrease, and the specified credibility weight on experience is $Z=0.60$. Find the selected percentage change and rate.

Incorrect calculation. Reversing the weights gives $0.40(9\%)+0.60(-3\%)=3.6\%-1.8\%=1.8\%$. This is wrong because the problem assigns $0.60$, not $0.40$, to experience.

Correct calculation. The experience contribution is $0.60(9\%)=5.4\%$. The complementary weight is $1-0.60=0.40$, so its contribution is $0.40(-3\%)=-1.2\%$. Therefore the selected change is $5.4\%-1.2\%=4.2\%$.

The selected rate is $$1,250(1.042)=$1,302.50$.

Check. The selected change of $4.2\%$ lies between the complementary decrease of $-3\%$ and the experience increase of $9\%$. The selected rate is $52.50 above the current rate, which equals $0.042($1,250)$. The correction avoids both reversing the assigned weights and losing the cents when applying the change.

Check Your Understanding

For each problem, identify whether the inputs are percentage changes or rate levels before calculating. Use the stated weight on the experience indication.

A hypothetical rate is currently $600. The experience indication is an $8\%$ increase, the complementary indication is a $2\%$ increase, and $Z=0.50$. Find the selected change and rate. (Answer: $5\%$ increase; $630.00.)
A hypothetical class has a current rate of $900. Its experience indication is a $6\%$ decrease, its complementary indication is a $4\%$ increase, and $Z=0.30$. Find the selected change and rate. (Answer: $1\%$ increase; $909.00.)
Two comparable indications for a hypothetical coverage are rate levels: $740 and $680 per exposure, experience and complementary, respectively. If $Z=0.25$, find the selected rate. (Answer: $695.00 per exposure.)
A hypothetical policy rate is $1,250. The experience indication is a $9\%$ increase, the complementary indication is a $3\%$ decrease, and $Z=0.60$. Find the selected change and rate. (Answer: $4.2\%$ increase; $1,302.50.)
A hypothetical rate is $400. The experience indication is a $10\%$ decrease, the complementary indication is a $5\%$ increase, and $Z=0.80$. Find the selected rate and state which indication it is closer to. (Answer: $7\%$ decrease; $372.00, closer to the experience indication.)

Credibility weighting is a controlled blend, not a replacement for checking the basis of the indications. Confirm that the inputs are comparable, apply $Z$ to experience and $1-Z$ to the complementary indication, and verify that the result lies between the two inputs. These calculations and interpretations are aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.

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