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

Four risks each have three years of data. The risk means are 20, 24, 16 and 28. The sums of squared deviations of the observations from their own risk mean are 6, 10, 4 and 12 respectively. Using empirical Bayes credibility (Model 1) with unbiased variance estimators, what is the credibility premium for the fourth risk?

The premium is 27.70. The expected process variance is 4, the sample variance of the means is 26.667, so the variance of hypothetical means is 25.333. This gives Z = 0.95, and the premium is 0.95×28 + 0.05×22 = 27.70.

  1. A26.60
  2. B27.30
  3. C27.70Correct
  4. D27.80
  5. 28.00

Explanation

Risk variances are 6/2, 10/2, 4/2, 12/2 = 3, 5, 2, 6, so E[s²] = 4. The overall mean is 22 and the sum of squared mean deviations is 80, so the sample variance of the means is 80/3 = 26.667. Then a = 26.667 − 4/3 = 25.333, k = 0.15789 and Z = 0.95. The premium is 0.95×28 + 0.05×22 = 27.70. Dividing the sums of squares by 3 instead of 2 gives 27.80.

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