Credibility Premium after a Safety Programme Change
A workers’ compensation pool has 25 employers with between one and six years of experience. For employer A, four years of exposure produce 38 claims; the manual rate predicts 30 claims for the same exposure. Pool-level analysis estimates the expected process variance at 18.5 and the variance of hypothetical means at 6.2. The standard Bühlmann credibility factor is Z = n/(n + k), where k is the ratio of expected process variance to variance of hypothetical means. Employer A has just changed its safety programme, and the most recent year has 6 claims versus 10–12 in earlier years. The pool manager wants a credibility premium for the next year and asks whether the Bühlmann result is appropriate. A colleague suggests a Bayesian gamma–Poisson model with a prior mean of 30 and prior variance 6.2.
1. Calculate k, Z, and the Bühlmann credibility premium for employer A. Interpret each quantity.
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The credibility constant is k = expected process variance / variance of hypothetical means = 18.5/6.2 = 2.984. With n = 4 years, Z = 4/(4 + 2.984) = 0.573. The credibility premium is Z times the employer’s own experience plus (1 − Z) times the manual prior: 0.573 × 38 + 0.427 × 30 = 21.77 + 12.81 = 34.58 claims over four years, or about 8.65 claims per year. The value k = 2.98 means that about three years of exposure would be needed for 50% credibility; employer A has four years, so it receives 57% weight on its own experience. A large between-employer variance relative to process variance increases Z because the employer’s experience is more informative about its true mean. The calculation assumes that the four years are exchangeable and that the variance components are estimated on the same exposure scale as n; if they are not, the result is not interpretable.
2. The colleague proposes a gamma–Poisson model with prior mean 30 and prior variance 6.2. Compute the exact posterior mean if the prior is on the four-year total, and compare with the Bühlmann result. What does the comparison reveal?
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If the prior is on the four-year total claim count μ, a gamma prior with mean m = 30 and variance v = 6.2 has shape α = m²/v = 900/6.2 = 145.16 and rate β = m/v = 30/6.2 = 4.839. With a Poisson likelihood for the total count y = 38 and exposure 1 on the four-year scale, the posterior is gamma(α + 38, β + 1), with mean (145.16 + 38)/(4.839 + 1) = 183.16/5.839 = 31.37. This is lower than the Bühlmann premium of 34.58. If instead the prior variance is annual, convert: prior mean 7.5 per year, variance 6.2, α = 7.5²/6.2 = 9.07, β = 7.5/6.2 = 1.21, and the posterior mean per year is (9.07 + 38)/(1.21 + 4) = 9.03, or 36.1 over four years, which is closer to Bühlmann. The discrepancy shows that the units and variance-component definitions matter. Also, if the total mean is 30, a Poisson process variance should be about 30, not 18.5; the EPV being below the mean suggests either annual-scale quantities or a non-Poisson model.
3. The safety programme changed and the latest year has 6 claims versus 10–12 before. Should the credibility premium be used? Recommend an alternative and state the key assumption that is violated.
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The Bühlmann premium should not be used mechanically. Credibility theory assumes exchangeability: each year’s experience is drawn from the same employer-level mean, so the four years can be pooled with equal weight. A safety programme may change the mean, so the earlier years are not representative of the next year. Using all four years with equal weights overstates the old claim level and understates the improvement. Alternatives include a Bühlmann–Straub or credibility model with declining time weights, a state-space or change-point model that allows the employer mean to shift after the programme, or using only the post-change year with a much lower credibility factor. With one year, Z = 1/(1 + 2.984) = 0.251, giving an annual premium of 0.251 × 6 + 0.749 × 7.5 = 7.12 claims per year, but this is based on very little data. The key violated assumption is parameter homogeneity over time; the actuary should present a range, monitor the next year, and avoid claiming a definitive premium reduction until the new regime has enough experience.