Portfolio Credibility Premiums and Aggregate Experience

From Risk-Level Rates to a Portfolio Premium

A credibility premium is often calculated separately for each risk, but an insurer may need one total premium for a portfolio containing several risks. The key is to distinguish the exposure used to assess the credibility of each risk’s history from the exposure that will be priced in the future. Those quantities need not be equal, and the portfolio premium must reflect the future mix of risks.

This tutorial uses the Bühlmann-Straub model. All scenarios and amounts are hypothetical. Assume the loss measures are expressed in dollars per exposure unit, and no discounting or other coverage adjustment is needed.

In this model, observations for a given risk are conditionally independent given its risk parameter. The expected loss rate for that risk is its hypothetical mean, and the variance of a historical rate based on exposure \(W_i\), conditional on the risk parameter, is \(\mathrm{EPV}/W_i\). Risks share the same collective mean and structural variance components. The standard credibility formula presumes finite variances and positive historical exposure for the risk being priced. It is not a claim that two risks with the same \(W_i\) and \(\bar X_i\) must have identical future losses; it is a formula for their model-based predictive rates.

The amount \(W_i\) determines how much weight the risk’s own experience receives. By contrast, \(e_i\) scales its predicted rate into a future premium. A large future exposure does not itself increase the credibility of the historical data.

Why the Credibility Weight Has This Form

The Bühlmann-Straub weight can be justified as the coefficient in the best linear predictor of a risk’s hypothetical mean from its historical rate. Write that hypothetical mean as \(\mu_i\). The historical average \(\bar X_i\) has conditional expectation \(\mu_i\), and conditional variance \(\mathrm{EPV}/W_i\). Across risks, \(\operatorname{Var}(\mu_i)=\mathrm{VHM}\) and \(E[\mu_i]=m\).

The best linear predictor of \(\mu_i\) from \(\bar X_i\) is \(m+b(\bar X_i-m)\), where \(b=\operatorname{Cov}(\mu_i,\bar X_i)/\operatorname{Var}(\bar X_i)\). Conditional unbiasedness gives \(\operatorname{Cov}(\mu_i,\bar X_i)=\mathrm{VHM}\). The law of total variance gives \(\operatorname{Var}(\bar X_i)=\mathrm{VHM}+\mathrm{EPV}/W_i\). Therefore:

Thus the predicted rate is \(m+b(\bar X_i-m)=Z_i\bar X_i+(1-Z_i)m\). If the future exposure is \(e_i\), the predicted aggregate loss for that risk is \(e_iP_i\), because the expectation of a sum is the sum of its expectations. Summing over the portfolio gives the portfolio premium.

Calculate Each Rate, Then Aggregate

The reliable procedure is to calculate a credibility weight and premium rate for each risk, multiply each rate by its own future exposure, and add the resulting premiums. If a single portfolio rate is useful, divide the aggregate premium by the total future exposure \(\sum_i e_i\). This portfolio rate is a future-exposure-weighted average of the risk-level rates, not necessarily the unweighted average.

Useful Portfolio Benchmarks

A collective-only benchmark ignores the individual histories and prices all future exposure at \(m\), giving \(m\sum_i e_i\). A raw-experience benchmark applies the historical portfolio average rate to the future portfolio exposure. The historical raw rate is \(\sum_i W_i\bar X_i/\sum_i W_i\), so this benchmark is \((\sum_i e_i)(\sum_i W_i\bar X_i/\sum_i W_i)\). It uses historical exposure weights to combine experience, but does not apply separate credibility weights.

A third calculation sometimes encountered is to pool the historical experience first and apply one credibility factor based on total historical exposure. This treats the entire portfolio as one risk. Its pooled rate is \(Z_{\mathrm{pool}}\bar X_{\mathrm{pool}}+(1-Z_{\mathrm{pool}})m\), where \(\bar X_{\mathrm{pool}}=\sum_iW_i\bar X_i/\sum_iW_i\) and \(Z_{\mathrm{pool}}=(\sum_iW_i)/(\sum_iW_i+K)\). Multiplying that rate by total future exposure gives the pooled premium. In general, this is not the same operation as separately credibility-rating the risks and applying their respective future exposures.

Common Exam Trap: Use the Correct Exposure in Each Step

Suppose \(m=\$100\), \(K=4\), Risk A has \(W_A=4\), \(\bar X_A=\$120\), and \(e_A=2\), while Risk B has \(W_B=12\), \(\bar X_B=\$90\), and \(e_B=1\). For A, \(Z_A=4/(4+4)=0.5\), giving \(P_A=0.5(\$120)+0.5(\$100)=\$110\). For B, \(Z_B=12/(12+4)=0.75\), giving \(P_B=0.75(\$90)+0.25(\$100)=\$92.50\). The aggregate premium is \(2(\$110)+1(\$92.50)=\$312.50\).

A common wrong calculation is to report \(\$318\), perhaps after using the wrong weight or arithmetic for one risk. Another error is to set A’s credibility weight to \(2/(2+4)\) because its future exposure is 2. That incorrectly substitutes \(e_A\) for historical exposure \(W_A\). A’s weight is \(4/(4+4)=0.5\), not \(2/6\). The collective-only comparison is \(3(\$100)=\$300\), which is separate from the risk-level premium.

A useful final check is to recompute the aggregate directly from the displayed rates and future exposures. Here \(2(110)+1(92.50)=220+92.50=312.50\). The implied portfolio rate is \(\$312.50/3=\$104.1667\ldots\), which lies between the risk-level rates of \(\$92.50\) and \(\$110\).

This portfolio-aggregation approach is aligned with the SOA Exam ASTAM (Advanced Short-Term Actuarial Mathematics) syllabus.