Credibility Theory: Bayesian and Empirical Bayes Credibility Premiums

What this chapter covers

Credibility theory: Bayesian and empirical Bayes credibility premiums: topics in the order to study them

1. 1Credibility Theory Basics and Limited FluctuationStart here to learn the idea of weighting own experience against a collective mean, and the full and partial credibility rules, before the heavier models.
2. 2Bayesian Estimation and Posterior DistributionsYou need prior, likelihood and posterior working, including conjugate pairs, before you can derive any Bayesian premium.
3. 3Bayesian Credibility PremiumThis applies the posterior mean as a premium and shows that it often takes the form Z × X̄ + (1 − Z) × μ, which links directly to the next model.
4. 4Buhlmann Credibility ModelIt generalises the Bayesian result to a linear estimate that needs only E[s²(θ)] and Var[m(θ)], with Z = n ÷ (n + k).
5. 5Buhlmann-Straub ModelIt extends Bühlmann to unequal exposures, so learn the basic model first. Then change n to the total exposure and use the exposure-weighted mean claim per unit of exposure as X̄_i.
6. 6Empirical Bayes Credibility TheoryStudy this last because it estimates the Bühlmann parameters from data, so it uses everything that came before.

How to prepare Credibility theory: Bayesian and empirical Bayes credibility premiums

1. Write the premium formula Z × X̄ + (1 − Z) × μ on one page and define each symbol. Keep it in view for every question.
2. Learn limited fluctuation first. For Poisson claim numbers, full credibility needs expected claims of at least n_F = (y ÷ k)². Here k is the maximum proportional deviation you allow, p is the probability you choose, and y is the standard normal value with P(|N(0,1)| ≤ y) = (1 + p) ÷ 2. With n expected claims, partial credibility uses Z = min(1, √(n ÷ n_F)). Be careful with the letter k. In limited fluctuation it is the tolerance. In Bühlmann models it is the variance ratio E[s²(θ)] ÷ Var[m(θ)]. They are different quantities.
3. Practise conjugate posteriors until they are automatic. For a Poisson claim count with a Gamma prior of shape α and rate β, the posterior is Gamma(α + Σx, β + n), and its mean can be written as Z × X̄ + (1 − Z) × α ÷ β with Z = n ÷ (n + β).
4. For Bühlmann, always compute three things in order: μ = E[m(θ)], E[s²(θ)] and Var[m(θ)]. Then k = E[s²(θ)] ÷ Var[m(θ)] and Z = n ÷ (n + k).
5. For Bühlmann-Straub, let P_i be the total exposure for risk i and use Z_i = P_i ÷ (P_i + k), with k = E[s²(θ)] ÷ Var[m(θ)]. Here s²(θ) is the process variance per unit of exposure. X̄_i must be the exposure-weighted mean claim per unit of exposure for risk i, which is the total claims divided by the total exposure P_i. The overall mean μ is estimated in the same exposure-weighted way. Compare your working with the basic model to see exactly what changed.
6. For empirical Bayes, practise the estimates from a table of data. Let N be the number of risks and n the number of years of data for each risk. The overall mean X̄ estimates μ. The average of the sample variances, s², estimates E[s²(θ)]. Estimate Var[m(θ)] = (1 ÷ (N − 1)) Σ(X̄_i − X̄)² − s² ÷ n. Set a negative estimate of Var[m(θ)] to zero and note what that means.
7. Finish with timed mixed questions. Write the model assumptions, the formula, the working and a one-line interpretation of Z in each answer.

Common mistakes in Credibility theory: Bayesian and empirical Bayes credibility premiums

• Mixing up E[s²(θ)] and Var[m(θ)] when forming k

  Fix: Remember that k is the expected process variance divided by the variance of the hypothetical means. Compute each separately, with labels, before dividing.

• Using the wrong n, or the wrong exposure, in Z

  Fix: Identify the unit of observation first. In Bühlmann, n is the number of years of data for the risk. In Bühlmann-Straub, use the total exposure for the risk, and use the exposure-weighted mean claim per unit of exposure as X̄_i.

• Forgetting to take the prior mean as μ

  Fix: The collective mean is the prior expectation of the claim. Write it down at the start of every question.

• Using the wrong divisor in empirical Bayes variance estimates

  Fix: Use n − 1 for each sample variance and N − 1 for the variance of risk means, where N is the number of risks and n is the number of years. Then estimate Var[m(θ)] as (1 ÷ (N − 1)) Σ(X̄_i − X̄)² minus s² ÷ n, where s² is the average of the sample variances.

• Leaving a negative estimate of Var[m(θ)] unchanged

  Fix: A variance cannot be negative. Set the estimate to zero. Then k is infinite, Z = n ÷ (n + k) tends to 0, and the premium equals the overall mean X̄ for every risk. Say so in your answer.

• Treating limited fluctuation and Bayesian credibility as the same method

  Fix: Limited fluctuation chooses Z from a precision rule on the data alone. Bayesian and Bühlmann methods derive Z from the variance structure of the risk parameter. Do not confuse the tolerance k in limited fluctuation with the variance ratio k in Bühlmann.

Last-day revision: Credibility theory: Bayesian and empirical Bayes credibility premiums

• ✓Credibility premium = Z × own experience + (1 − Z) × collective mean, with 0 ≤ Z ≤ 1.
• ✓Z increases as the amount of data increases.
• ✓Bühlmann: Z = n ÷ (n + k), where k = E[s²(θ)] ÷ Var[m(θ)].
• ✓μ = E[m(θ)], where m(θ) = E[X | θ] and s²(θ) = Var[X | θ].
• ✓Total variance of X = E[s²(θ)] + Var[m(θ)].
• ✓Bühlmann-Straub: let P_i be the total exposure for risk i. Then Z_i = P_i ÷ (P_i + k), with k = E[s²(θ)] ÷ Var[m(θ)], where s²(θ) is the process variance per unit of exposure. X̄_i is the exposure-weighted mean claim per unit of exposure for risk i (total claims ÷ P_i).
• ✓Poisson-Gamma: posterior is Gamma(α + Σx, β + n), and Z = n ÷ (n + β) with a rate β.
• ✓The Bayesian premium equals the Bühlmann premium when the posterior mean is linear in the data, as for Poisson-Gamma, Normal-Normal with known variance, and Binomial-Beta. Binomial-Beta needs a fixed number of trials per observation. In general, the Bühlmann premium is the best linear approximation to the Bayesian premium in the least-squares sense.
• ✓Limited fluctuation, for Poisson claim numbers: the full credibility standard is n_F = (y ÷ k)² expected claims, where P(|N(0,1)| ≤ y) = (1 + p) ÷ 2. Partial credibility is Z = min(1, √(n ÷ n_F)), with n in expected claims.
• ✓The letter k means different things: in limited fluctuation it is the tolerance (maximum deviation), and in Bühlmann it is the variance ratio E[s²(θ)] ÷ Var[m(θ)].
• ✓Empirical Bayes: use sample variances with divisor n − 1. Set a negative Var[m(θ)] estimate to zero, which gives Z = 0 and a premium equal to the overall mean X̄ for every risk.
• ✓A large k in Bühlmann means the data is noisy relative to the spread between risks, so Z is small.
• ✓Always state what the premium is for, such as the next year's expected claim per unit of exposure.

Credibility theory: Bayesian and empirical Bayes credibility premiums practice questions

• 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 thei…
• In empirical Bayes credibility Model 1 with a fixed number of years n per risk, which statement about the credibility factor Z = n/(n+k), wi…
• Which statement about the Bühlmann-Straub credibility factor Z_i = m_i/(m_i + K) is correct, holding other quantities fixed?
• 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…
• Claim counts per year are Poisson(λ) with prior λ ~ Gamma(α = 3, rate β = 6). After n years of experience with sample mean x̄, the posterior…
• In a Poisson–Gamma model the prior mean claim frequency is 0.2 per year and the prior Gamma shape is 2 (rate 10). A policyholder has 0 claim…
• Given θ, claim amounts X are Normal(θ, variance 400). The prior for θ is Normal(mean 1000, variance 100). One observation x = 1100 is made. …
• A policyholder's claim amount X in a year has conditional mean μ(θ) and variance σ²(θ). Across risks, E[σ²(θ)] = 400 and Var[μ(θ)] = 100. Th…