Actuary Certification Credibility Theory 2 — Questions and Answers
Question 1: In the Bühlmann-Straub credibility model, the credibility factor for risk i with total exposure mᵢ is:
- Zᵢ = mᵢ / (mᵢ + k) (Correct answer)
- Zᵢ = k / (mᵢ + k)
- Zᵢ = √(mᵢ / k)
- Zᵢ = mᵢ / (mᵢ + 1)
Correct answer: Zᵢ = mᵢ / (mᵢ + k)
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?
- 160
- 170
- 180 (Correct answer)
- 200
Correct answer: 180
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:
- (α + S) / (β + n) (Correct answer)
- S / n
- α / β
- (α + n) / (β + S)
Correct answer: (α + S) / (β + n)
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:
- Σ(Zᵢ × X̄ᵢ) / ΣZᵢ (Correct answer)
- Simple average of all individual risk means X̄ᵢ
- Σ(mᵢ × X̄ᵢ) / Σmᵢ
- Σ X̄ᵢ / r where r is the number of risks
Correct answer: Σ(Zᵢ × X̄ᵢ) / ΣZᵢ
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:
- The between-group variance of risk means
- The average within-risk sample variance across all risks (Correct answer)
- The total variance of all observations pooled together
- The variance of the estimated credibility premiums
Correct answer: The average within-risk sample variance across all risks
The EVPV is estimated by averaging each risk's within-group sample variance, since EVPV represents the typical within-risk process variability.
Question 6: Which is a core assumption of the Bühlmann credibility model?
- All risks in the portfolio share the same risk parameter Θ
- Observations from the same risk are conditionally independent given Θ (Correct answer)
- The prior distribution of Θ must be a normal distribution
- All risks must have the same number of observations
Correct answer: Observations from the same risk are conditionally independent given Θ
A fundamental assumption is that X₁, ..., Xₙ are conditionally i.i.d. given Θ; the unconditional positive correlation between same-risk observations arises entirely from the shared unknown Θ.
Question 7: As the number of observations n from a single risk increases toward infinity, the Bühlmann credibility factor Z approaches:
Correct answer: 1
Z = n/(n+k) → 1 as n → ∞ because the observed mean X̄ becomes a perfect estimator of the risk's true mean, so full weight is given to observed data.
