Credibility on Pure Premium Versus Its Components

Why the Level of Credibility Weighting Matters

A class pure premium can be estimated from aggregate losses per exposure, or by estimating claim frequency and average severity separately and multiplying the results. Both approaches combine class experience with a collective complement, but they do not generally produce the same indication. The difference matters because a credibility weight is tied to the quantity being estimated: a weight calibrated for frequency is not automatically appropriate for aggregate loss cost.

For a class with $E$ exposures, $N$ claims, and total loss $L$, the observed frequency is $N/E$, the observed severity is $L/N$ when $N>0$, and the observed pure premium is $L/E$. Thus, when $N>0$, observed pure premium equals observed frequency times observed severity. That identity does not mean the same identity holds after separately credibility-weighting the two components.

Definition: A direct pure-premium credibility estimate blends observed aggregate loss cost with a collective pure-premium complement. A component credibility estimate separately blends observed frequency and observed severity with their collective complements, then multiplies the two weighted components.

The Two Estimation Methods

For equal-sized exposure units, let $Y_j$ be aggregate loss per exposure unit for unit $j$, and let $\bar Y$ be its average over $n$ units. In the Bühlmann model, the conditional mean of $Y_j$ for risk parameter $\Theta$ is $m(\Theta)=E[Y_j\mid\Theta]$. The collective pure-premium complement is $\mu_p=E[m(\Theta)]$. With $EPV=E[\operatorname{Var}(Y_j\mid\Theta)]$ and $VHM=\operatorname{Var}(m(\Theta))$, the Bühlmann factor is $Z_p=n/(n+EPV/VHM)$, provided $VHM>0$. The direct estimate is $Z_p\bar Y+(1-Z_p)\mu_p$. If $VHM=0$, there is no between-risk variation in conditional means, so the Bühlmann estimate is the collective mean rather than an experience-driven blend.

For the separate approach, write the observed frequency and severity as $\hat f$ and $\hat s$, with collective complements $\mu_f$ and $\mu_s$. If their respective credibility factors are $Z_f$ and $Z_s$, the component estimate is

\hat p_{\mathrm{comp}} = \bigl[Z_f\hat f+(1-Z_f)\mu_f\bigr] \bigl[Z_s\hat s+(1-Z_s)\mu_s\bigr].

The pure-premium complement for the direct method is $\mu_p=E[FS]$, where $F$ and $S$ are the underlying frequency and severity for a risk. It equals $\mu_f\mu_s$ only when the relevant frequency and severity parameters have zero covariance (or when the model otherwise makes that equality hold). In particular, a product of collective averages is not always the collective average of the product.

Why the Indications Can Differ

The direct method uses one weight for observed aggregate loss cost. The component method uses two weights, and then multiplies two blended values. Even if both component weights happen to be the same, multiplication introduces an interaction between the frequency and severity deviations from their complements.

Result: If $\mu_p=\mu_f\mu_s$ and both component weights equal $z$, then the component estimate minus the direct estimate using weight $z$ is $-z(1-z)(\hat f-\mu_f)(\hat s-\mu_s)$. Thus, with a common weight, the estimates agree if either component equals its complement or if $z=0$ or $z=1$. Otherwise, the interaction can make them differ.

To justify the result, let $a=\hat f-\mu_f$ and $b=\hat s-\mu_s$. The component product expands to $(\mu_f+za)(\mu_s+zb)=\mu_f\mu_s+z\mu_s a+z\mu_f b+z^2ab$. The direct estimate with complement $\mu_f\mu_s$ is $\mu_f\mu_s+z[(\mu_f+a)(\mu_s+b)-\mu_f\mu_s]=\mu_f\mu_s+z\mu_s a+z\mu_f b+zab$. Subtracting gives $(z^2-z)ab=-z(1-z)ab$, as stated.

Worked Example: Different Weights Give Different Indications

A hypothetical class has 100 exposure units, 12 claims, and total losses of $60,000. Its observed frequency is 0.12 claims per exposure, its observed severity is $5,000, and its observed pure premium is $600. The collective frequency complement is 0.10 and the collective severity complement is $5,500. The frequency and severity factors, calibrated separately under Bühlmann assumptions, are $Z_f=0.60$ and $Z_s=0.50$. The direct pure-premium factor is $Z_p=0.40$, with collective pure-premium complement $550. Assume independent exposure units, conditional Poisson claim counts, and positive claim amounts with an exponential severity model; the stated credibility factors are supplied for this calculation. There is no discounting or mortality assumption because these are one-period loss costs.

Method: direct credibility. Blend observed pure premium with its stated complement:

0.40(600)+0.60(550)=240+330=\$570.

Method: separate component credibility. First weight frequency: $0.60(0.12)+0.40(0.10)=0.072+0.040=0.112$ claims per exposure. Then weight severity: $0.50(5{,}000)+0.50(5{,}500)=2{,}500+2{,}750=\$5{,}250$. Multiply: $0.112(5{,}250)=\$588$.

The component estimate is $588, compared with the direct estimate of $570. The difference is not an arithmetic error: the two methods use different credibility weights and apply them to different quantities. A common mistake is to use the frequency factor on aggregate loss cost, obtaining $550+0.60(600-550)=\$580$. That is not the specified direct estimate; the appropriate direct factor is 0.40, giving $570. Check: the observed aggregate loss cost $60{,}000/100$ is $600, consistent with $0.12(5{,}000)$.

Worked Example: The Interaction From Multiplying Two Blends

A hypothetical class has observed frequency 0.14 and observed severity $6,000. The collective complements are 0.10 and $5,000. Assume independent exposure units with conditional Poisson counts and exponential claim severity; the class and collective values are supplied, and both component factors are $z=0.50$. Assume the collective pure-premium complement is the product of the component complements. There is no discounting or mortality assumption.

Component method. The weighted frequency is $0.50(0.14)+0.50(0.10)=0.12$. The weighted severity is $0.50(6{,}000)+0.50(5{,}000)=\$5{,}500$. Their product is $0.12(5{,}500)=\$660$.

Direct comparison using the same weight. The collective pure premium is $0.10(5{,}000)=\$500$, and the observed pure premium is $0.14(6{,}000)=\$840$. Direct weighting gives $0.50(840)+0.50(500)=\$670$.

The component result is $10 lower. The interaction formula verifies it: $z(1-z)(\hat f-\mu_f)(\hat s-\mu_s)=0.50(0.50)(0.04)(1{,}000)=\$10$, and the component estimate is lower because both deviations are positive. Check: $660 is $10 below $670.

Aggregate Process Variation Also Matters

An aggregate loss-cost observation reflects both claim-count variation and variation in claim amounts. Consequently, the direct Bühlmann factor need not equal the frequency factor. The following derivation shows this in a simple model where severity has no risk-level variation but still has claim-to-claim randomness.

Specify the model.
For each exposure, conditional on $\lambda$, the claim count is Poisson with mean $\lambda$. Claim amounts are independent exponential random variables with mean $5,000 and variance $25{,}000{,}000$, independent of the count and of $\lambda$. Let $\lambda=0.08$ or $0.12$, each with probability 0.50. Thus $E[\lambda]=0.10$ and $\operatorname{Var}(\lambda)=0.0004$.

Find the aggregate pure-premium moments.
For one exposure, conditional mean aggregate loss is $5{,}000\lambda$. The exponential severity second moment is $25{,}000{,}000+25{,}000{,}000=50{,}000{,}000$. A compound Poisson sum has conditional variance $\lambda E[X^2]$, so $EPV=0.10(50{,}000{,}000)=5{,}000{,}000$. Also, $VHM=5{,}000^2(0.0004)=10{,}000$.

Calculate direct credibility for 250 exposures.
The Bühlmann parameter is $K=EPV/VHM=5{,}000{,}000/10{,}000=500$, so $Z_p=250/(250+500)=1/3$. The collective pure premium is $0.10(5{,}000)=\$500$.

Compare the frequency factor.
The observed frequency has conditional variance $\lambda/250$, whose expectation is $0.10/250$. Its $VHM$ is $0.0004$, so $K_f=(0.10/250)/0.0004=1$, and $Z_f=250/(250+1)$. This differs from the aggregate factor $1/3$, because aggregate loss cost also includes claim-severity process variation.

The frequency calculation in Step 4 needs a correction: $K_f=(0.10/250)/0.0004=1$, so the frequency factor is $250/(250+1)$, as written. This high factor follows from the specified low frequency process variance relative to between-risk variation; it is not the $n/(n+K)$ value one would get by confusing $K_f$ with the aggregate $K$. The example demonstrates why each factor must be calculated for its own response variable.

Worked Example: Direct Credibility With Severity Process Risk

Use the model from the preceding derivation for a hypothetical class of 250 exposure units. Suppose the class has 30 claims and $180,000 in total losses. Conditional counts are Poisson with $\lambda=0.08$ or $0.12$, equally likely, and claim amounts are exponential with mean $5,000 and variance $25{,}000{,}000$. There is no discounting or mortality assumption.

The observed frequency is $30/250=0.12$. The observed severity is $180{,}000/30=\$6{,}000$, and the observed pure premium is $180{,}000/250=\$720$, also $0.12(6{,}000)$. From the preceding calculation, $Z_p=1/3$, and the collective pure premium is $500. Therefore, direct credibility gives

\frac{1}{3}(720)+\frac{2}{3}(500)=240+333.333\ldots=\$573.33.

Suppose severity is treated as having a known fixed mean of $5,000, with no between-risk severity variation. Then the separate frequency estimate uses $Z_f=250/251$: $(250/251)(0.12)+(1/251)(0.10)=0.1199203$, which gives a component pure premium of $0.1199203(5{,}000)=\$599.60$. The large frequency weight is reasonable under this model, but it does not reproduce direct credibility: the direct method also accounts for claim-severity process variance. Check: aggregate experience is $720 per exposure, and blending it one-third with the $500 complement yields $573.33, between those values.

Choosing and Interpreting the Indication

Neither method is universally preferable from the formulas alone. The method should match the rating question and the assumptions behind its credibility model. Direct credibility treats aggregate loss cost as the response and calibrates its factor to aggregate process variance and between-risk variation. Component credibility models frequency and severity separately, which can be useful when the components have distinct collective complements or distinct volumes of information. Their product, however, is a constructed indication, not generally the Bühlmann estimate for aggregate pure premium.

Use a direct pure-premium factor only for the aggregate loss-cost quantity for which it was calibrated.
Use separate factors only with their corresponding frequency and severity observations and complements.
Check whether the collective pure-premium complement is actually the product of the component complements.
When $N=0$, observed severity $L/N$ is undefined; do not silently assign it a value and multiply. A separate approach needs an explicit treatment of that case.
When comparing methods, report the assumptions and weights, not only the final indications.

A frequent exam trap is to assume that matching the collective pure-premium complement makes the methods equivalent. It does not: different weights, an interaction from multiplying blended components, and aggregate severity process variation can all change the result. This comparison is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.

Check Your Understanding

For each problem, identify the requested method and keep each credibility factor attached to the quantity it was calibrated for.

A hypothetical class has observed frequency 0.10 and severity $5,000. The collective complements are 0.08 and $4,000, with $Z_f=0.40$ and $Z_s=0.50$. Assume independent exposures, conditional Poisson counts, and exponential severities; the factors are supplied. There is no discounting or mortality assumption. Find the component pure-premium estimate. (Answer: $396.00)
Observed pure premium is $600, the collective pure-premium complement is $480, and the direct factor is 0.25. Counts are conditionally Poisson and severity is exponential; the factor is supplied. Find the direct estimate. There is no discounting or mortality assumption. (Answer: $510.00)
Observed frequency and severity are 0.15 and $3,600; complements are 0.12 and $3,000. Both component factors are 0.60, and the collective pure premium is the product of complements. Assume the supplied factors apply to independent-exposure Poisson-frequency and exponential-severity models. Compare the component estimate with a direct estimate using factor 0.60. There is no discounting or mortality assumption. (Answer: component $463.68; direct $468.00; component is $4.32 lower.)
A class has observed frequency 0.08 and severity $5,500. Its collective complements are 0.10 and $5,000. The supplied component factors are $Z_f=0.50$ and $Z_s=0.40$. The direct complement is $500 and the direct factor is 0.30. Assume conditional Poisson counts and exponential severities, with all factors supplied and applicable to their stated quantities. There is no discounting or mortality assumption. Calculate both indications. (Answer: component $468.00; direct $482.00.)
In a hypothetical model, $\lambda$ is 0.16 or 0.24 with equal probability; counts per exposure are Poisson conditional on $\lambda$. Severity is exponential with mean $2,500 and variance $6{,}250{,}000$, independent of counts. For 100 exposures, observed aggregate loss cost is $300. Find direct Bühlmann credibility using the model moments. There is no discounting or mortality assumption. (Answer: $EPV=2{,}500{,}000$, $VHM=10{,}000$, $Z_p=2/7$, and estimate $442.86.)

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