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

Zᵢ = k / (mᵢ + k)

Answer

Zᵢ = √(mᵢ / k)

Answer

Zᵢ = mᵢ / (mᵢ + 1)

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

160

Answer

170

Answer

200

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

S / n

Answer

α / β

Answer

(α + n) / (β + S)

Question 4

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

Accepted Answer

Σ(Zᵢ × X̄ᵢ) / ΣZᵢ

Answer

Simple average of all individual risk means X̄ᵢ

Answer

Σ(mᵢ × X̄ᵢ) / Σmᵢ

Answer

Σ X̄ᵢ / r where r is the number of risks

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 between-group variance of risk means

Answer

The total variance of all observations pooled together

Answer

The variance of the estimated credibility premiums

Question 6

Which is a core assumption of the Bühlmann credibility model?

Accepted Answer

Observations from the same risk are conditionally independent given Θ

Answer

All risks in the portfolio share the same risk parameter Θ

Answer

The prior distribution of Θ must be a normal distribution

Answer

All risks must have the same number of observations

Question 7

As the number of observations n from a single risk increases toward infinity, the Bühlmann credibility factor Z approaches:

Accepted Answer

1

Answer

0

Answer

k

Answer

0.5