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IAI Actuarial Core Principles · Actuarial Statistics · Credibility theory: Bayesian and empirical Bayes credibility premiums

In the Bühlmann credibility model, the credibility factor is Z = n/(n + k). Which expression gives k in terms of the process variance E[s²(θ)] and the variance of the hypothetical means Var[m(θ)]?

k equals the expected process variance divided by the variance of the hypothetical means. Z = n/(n+k) then rises when risks differ strongly between policyholders and falls when within-risk variability is large. The inverted ratio would give the opposite behaviour.

  1. Ak = Var[m(θ)] / E[s²(θ)]
  2. Bk = E[s²(θ)] / Var[m(θ)]Correct
  3. Ck = E[s²(θ)] × Var[m(θ)]
  4. Dk = E[s²(θ)] + Var[m(θ)]
  5. k = sqrt(E[s²(θ)] / Var[m(θ)])

Explanation

In the Bühlmann model k = E[s²(θ)]/Var[m(θ)], the expected process variance divided by the variance of hypothetical means. A larger k means less weight on the individual experience. Option 0 inverts the ratio and would make Z fall as policyholder differences grow.

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