Credibility Theory Flashcards
7 cards from real Actuary Certification practice questions. Tap to flip, then mark Knew It or Still Learning — missed cards come back until you master them.
Read the first 7 Credibility Theory flashcards as text
In the Bühlmann-Straub credibility model, the credibility factor for risk i with total exposure mᵢ is:
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.
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?
Answer: 180
Z = 5/(5+5) = 0.50; premium = 0.50(200) + 0.50(160) = 100 + 80 = 180.
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:
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.
In the Bühlmann-Straub model, the grand mean estimator μ̂ is computed as:
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.
In empirical Bayes non-parametric credibility, the EVPV (v) is estimated by:
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.
Which is a core assumption of the Bühlmann credibility model?
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.
As the number of observations n from a single risk increases toward infinity, the Bühlmann credibility factor Z approaches:
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.