Actuary Certification Credibility Theory 1 — Questions and Answers
Question 1: Under limited fluctuation credibility, approximately how many expected claims are needed for full credibility of claim frequency at 90% confidence with ±5% accuracy?
- 271
- 1,082 (Correct answer)
- 384
- 543
Correct answer: 1,082
Applying n₀ = (z/r)² = (1.645/0.05)² ≈ 1,082, where z = 1.645 for 90% confidence and r = 0.05 for ±5% accuracy.
Question 2: In the Bühlmann credibility premium P = Z × X̄ + (1−Z) × μ, the credibility factor Z equals:
- n / (n + k) (Correct answer)
- k / (n + k)
- √(n / k)
- n / k
Correct answer: n / (n + k)
Z = n/(n+k) is the Bühlmann credibility factor, where n is the number of observations and k = EVPV/VHM.
Question 3: Under classical credibility, if full credibility requires n₀ = 1,082 claims and a risk has n = 271 claims, the partial credibility factor Z is:
- 0.25
- 0.50 (Correct answer)
- 0.71
- 0.27
Correct answer: 0.50
Using the square-root formula: Z = √(n/n₀) = √(271/1082) = √(0.25) = 0.50.
Question 4: In the Bühlmann model, the parameter k is defined as:
- VHM / EVPV
- EVPV / VHM (Correct answer)
- Total Variance / VHM
- VHM / Total Variance
Correct answer: EVPV / VHM
k = EVPV/VHM (Expected Value of Process Variance divided by Variance of Hypothetical Means); a larger k means more observations are needed to earn credibility.
Question 5: The Expected Value of Process Variance (EVPV) in credibility theory is formally defined as:
- Var_Θ[E(X|Θ)]
- E_Θ[Var(X|Θ)] (Correct answer)
- Var(X) + Var(Θ)
- E[X²] − (E[X])²
Correct answer: E_Θ[Var(X|Θ)]
EVPV = E_Θ[Var(X|Θ)], the expected within-risk process variance averaged over the prior distribution of risk parameters.
Question 6: The Variance of Hypothetical Means (VHM) in credibility theory is formally defined as:
- E_Θ[Var(X|Θ)]
- Var_Θ[E(X|Θ)] (Correct answer)
- Var(X) − EVPV
- E[Var(X)]
Correct answer: Var_Θ[E(X|Θ)]
VHM = Var_Θ[E(X|Θ)], measuring how much the true risk means vary across different risks in the portfolio.
Question 7: By the law of total variance, the total variance of a single observation X in the Bühlmann model equals:
- EVPV − VHM
- VHM − EVPV
- EVPV + VHM (Correct answer)
- EVPV × VHM
Correct answer: EVPV + VHM
Var(X) = E[Var(X|Θ)] + Var(E[X|Θ]) = EVPV + VHM, by direct application of the law of total variance.
