Actuary Certification Credibility Theory 2

Question 1

In the Bühlmann-Straub credibility model, the credibility factor for risk i with total exposure mᵢ is:

Accepted Answer

Zᵢ = mᵢ / (mᵢ + k)

Answer

In Bühlmann-Straub, Zᵢ = mᵢ/(mᵢ + k), where mᵢ is the total exposure weight — this mirrors the basic Bühlmann formula with n replaced by exposure.

Question 2

A risk has n = 5 years of observations with mean X̄ = 200. The portfolio mean is μ = 160 and Bühlmann's k = 5. What is the Bühlmann credibility premium?

Accepted Answer

180

Answer

Z = 5/(5+5) = 0.50; premium = 0.50(200) + 0.50(160) = 100 + 80 = 180.

Question 3

For the Poisson/Gamma conjugate model (Λ ~ Gamma(α, β) with E[Λ] = α/β, and X|Λ ~ Poisson(Λ)), the Bayesian posterior mean after observing S total claims in n periods equals:

Accepted Answer

(α + S) / (β + n)

Answer

For Poisson/Gamma, the posterior is Gamma(α+S, β+n) (rate parameterization), so the posterior mean is (α+S)/(β+n), which also equals the Bühlmann credibility estimate.

Question 4

In the Bühlmann-Straub model, the grand mean estimator μ̂ is computed as:

Accepted Answer

Σ(Zᵢ × X̄ᵢ) / ΣZᵢ

Answer

The grand mean μ̂ is the credibility-weighted average Σ(Zᵢ X̄ᵢ)/ΣZᵢ, using each risk's credibility factor as its weight.

Question 5

In empirical Bayes non-parametric credibility, the EVPV (v) is estimated by:

Accepted Answer

The average within-risk sample variance across all risks

Answer

The EVPV is estimated by averaging each risk's within-group sample variance, since EVPV represents the typical within-risk process variability.

Actuary Certification Practice Test

Actuary Certification Practice Test

Actuary Certification Credibility Theory 2