Structural Parameter Estimates From Unequal Exposures

Why Unequal Exposures Change the Estimation

In a Bühlmann-Straub portfolio, each risk may contribute a different amount of experience. A risk observed for more exposure units generally has a more stable experience mean than one observed for fewer units. Structural parameter estimates must reflect that difference: within-risk variation is weighted by exposure, and between-risk variation is corrected for the sampling noise in each risk’s mean.

The quantities to estimate are the EPV, the expected process variance within a risk, and the VHM, the variance of risk-specific means across the portfolio. Once estimated, they determine the credibility constant and the credibility factor for each risk.

For risk \(i\), let \(w_i=\sum_j w_{ij}\) be its total exposure and define its exposure-weighted mean by \(\bar X_i=\sum_j w_{ij}X_{ij}/w_i\). Let \(n_i\) be its number of observed periods, \(I\) the number of risks, \(n=\sum_i n_i\), and \(W=\sum_i w_i\). The portfolio weighted mean is \(\bar X=\sum_i w_i\bar X_i/W\). These are weighted by exposure, not by the number of observations.

Estimating EPV From Within-Risk Variation

For each risk, compare each observation with that risk’s own weighted mean. Summing the squared deviations after weighting by exposure gives the within-risk sum of squares. Pool these sums over the portfolio and divide by the residual degrees of freedom.

A justification also clarifies which independence claims are needed. Within one risk, transform each centered observation error by multiplying it by \(\sqrt{w_{ij}}\). Its conditional variance is then \(\sigma^2(\Theta_i)\). Removing the fitted weighted mean amounts to projecting away one mean direction for that risk, leaving a residual projection of rank \(n_i-1\). For a projection matrix \(M\), the expected squared residual is \(\operatorname{tr}(M\operatorname{Cov}(e\mid\Theta_i))\). Here that trace is \((n_i-1)\sigma^2(\Theta_i)\); taking expectations over \(\Theta_i\) gives \((n_i-1)\phi\). Summing over risks yields \((n-I)\phi\), which justifies the estimator.

The residual contrasts need not be independent. In general, fitting a mean makes residuals dependent; with two equally weighted errors, for example, the residuals are opposites. The trace argument establishes the expected sum of squares without assuming independent residual contrasts. Joint normality would give independence for suitable orthogonal contrasts, but normality is not required for this moment calculation.

Estimating VHM With Unequal Risk Exposures

Risk means differ for two reasons: the underlying risks have different expected outcomes, and their observed means contain process noise. The weighted between-risk sum of squares contains both sources. Its expected process-noise contribution must be removed before estimating the VHM.

To see the correction, write \(\bar X_i=\mu(\Theta_i)+e_i\), where \(e_i\) is the estimation error in the risk mean. The weighted centered sum of squares of the true risk means has expectation \(\tau^2D\). For the errors, the expected weighted centered sum is \(\sum_i w_i E[e_i^2]-W E[(\sum_i w_i e_i/W)^2]\). Since \(E[w_i e_i^2]=\phi\) and the errors are uncorrelated across risks, this equals \(I\phi-\phi=(I-1)\phi\). The cross-product between the true means and errors has expectation zero because each error has conditional mean zero. Thus \(E[B]=\tau^2D+(I-1)\phi\), giving the stated correction.

The correction remains \((I-1)\phi\) even when the conditional process variance differs from risk to risk: averaging across identically distributed risks gives \(E[\sigma^2(\Theta_i)]=\phi\). Unequal total exposures enter through \(D\), not by replacing \(I-1\) with a quantity based on total exposure.

From Structural Estimates to Credibility Premiums

The estimated structural parameters give the credibility constant \(K=\phi/\tau^2\). Risk \(i\), with total exposure \(w_i\), receives credibility factor \(Z_i=w_i/(w_i+K)\). Its premium for one unit of future exposure is the credibility-weighted average of its own experience mean and the portfolio mean. This calculation uses the structural estimates, the risk’s exposure, and the two means; it does not treat the number of periods as a substitute for exposure.

Common Estimation Traps

A frequent error is to use the raw between-risk sum \(B\) as though it measured only true differences between risks. In the earlier calculation, that would give \(390/6.6=59.09\), rather than the corrected estimate \(362/6.6=54.85\). The raw estimate is too high because it includes the sampling variation of the observed risk means.

Another error is to use unweighted averages when exposures differ. For risk A in the first example, the ordinary mean of 10 and 14 is 12, but the exposure-weighted mean is 13 because the observation of 14 represents three exposure units. Using 12 would distort both its within-risk residuals and the between-risk comparison.

Finally, keep the roles of the two denominators distinct. The EPV denominator is \(n-I\), based on the count of observed periods and fitted risk means. The VHM denominator is \(D=W-\sum_iw_i^2/W\), based on the distribution of total exposures across risks. Neither denominator is interchangeable with the other.