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

Which statement about the Bühlmann-Straub credibility factor Z_i = m_i/(m_i + K) is correct, holding other quantities fixed?

Z rises when the variance of hypothetical means rises, because K falls. Risks then differ more between each other, so individual experience deserves more weight. Z also rises with exposure and stays below 1.

  1. AZ_i increases as the variance of the hypothetical means increasesCorrect
  2. BZ_i increases as the expected process variance increases
  3. CZ_i is independent of the exposure m_i
  4. DZ_i can exceed 1 for large exposure
  5. Z_i decreases as the exposure m_i increases

Explanation

K = E[s²]/Var[μ]. A larger Var[μ] lowers K, which raises Z. A larger process variance raises K and lowers Z. Z increases with m_i and lies below 1.

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