Actuary Certification Actuary Certification Loss Models 2 — Questions and Answers
Question 1: The Bühlmann credibility formula for the credibility estimate is:
- Z · X̄ + (1−Z) · μ (Correct answer)
- Z · μ + (1−Z) · X̄
- Z · X̄ / μ
- Z · σ² + (1−Z) · μ
Correct answer: Z · X̄ + (1−Z) · μ
The Bühlmann credibility estimate is Z·X̄ + (1−Z)·μ, a weighted blend of the observed mean X̄ and the prior mean μ, where Z = n/(n+k) is the credibility factor.
Question 2: In the Bühlmann credibility model, the credibility factor Z = n/(n+k) where k equals:
- Expected process variance / variance of hypothetical means (Correct answer)
- Variance of hypothetical means / expected process variance
- Number of years of data
- The prior mean μ
Correct answer: Expected process variance / variance of hypothetical means
k = v/a where v is the expected value of the process variance (EVPV) and a is the variance of the hypothetical means (VHM), so k = EVPV/VHM.
Question 3: Which reinsurance arrangement has the reinsurer paying losses above an attachment point up to a maximum layer?
- Quota share reinsurance
- Excess of loss (per-occurrence) reinsurance (Correct answer)
- Aggregate stop-loss reinsurance
- Proportional surplus reinsurance
Correct answer: Excess of loss (per-occurrence) reinsurance
Per-occurrence excess of loss reinsurance covers losses above the cedant's retention (attachment point) up to a specified maximum per single occurrence.
Question 4: The method of moments estimator equates sample moments to theoretical moments. For a one-parameter distribution, this means setting:
- Sample variance = theoretical variance
- Sample mean = theoretical mean (Correct answer)
- Sample median = theoretical median
- Maximum sample value = theoretical maximum
Correct answer: Sample mean = theoretical mean
With a single parameter, method of moments sets the sample mean equal to the theoretical mean E[X] and solves for the parameter.
Question 5: Which goodness-of-fit test statistic compares the squared differences between observed and expected frequencies divided by expected frequencies?
- Kolmogorov-Smirnov statistic
- Anderson-Darling statistic
- Chi-square statistic (Correct answer)
- Log-likelihood ratio
Correct answer: Chi-square statistic
The chi-square goodness-of-fit statistic is Σ(Oi − Ei)²/Ei, measuring the discrepancy between observed and expected cell frequencies.
Question 6: The Kolmogorov-Smirnov test compares:
- Observed vs. expected cell frequencies
- Maximum absolute difference between empirical and fitted CDFs (Correct answer)
- Likelihood of the data under the null model
- Mean and variance of residuals
Correct answer: Maximum absolute difference between empirical and fitted CDFs
The K-S statistic is Dn = sup|Fn(x) − F(x)|, the maximum vertical distance between the empirical CDF and the hypothesized CDF.
Actuary Certification Actuary Certification Loss Models 2
The Bühlmann credibility formula for the credibility estimate is: