Actuary Certification Credibility Theory 3

Question 1

The full credibility standard for pure premiums requires more expected claims than for claim frequency alone because:

Accepted Answer

Pure premiums depend on both frequency and severity, adding severity variance to total variance

Answer

Pure premiums combine frequency and severity, so total variance includes both sources; the standard becomes n₀ = (z/r)²(1 + CV²_S) where CV_S is the severity coefficient of variation.

Question 2

If claim severity follows an exponential distribution (CV_S = 1.0), the full credibility standard for pure premiums at 90% confidence with ±5% accuracy requires approximately:

Accepted Answer

2,164 expected claims

Answer

n₀ = 1,082 × (1 + CV²_S) = 1,082 × (1 + 1²) = 2,164 expected claims for full credibility of pure premiums.

Question 3

Within a given risk (conditional on its parameter Θ), successive annual observations X₁, X₂, ..., Xₙ in the Bühlmann model are:

Accepted Answer

Conditionally independent and identically distributed given Θ

Answer

Given Θ, observations are conditionally i.i.d.; unconditional positive correlation between same-risk observations arises only from the unknown shared Θ, not from serial dependence.

Question 4

The exact Bayesian credibility estimate equals the Bühlmann linear credibility estimate when:

Accepted Answer

The likelihood is from the exponential family with a conjugate prior (e.g., Poisson/Gamma, Normal/Normal)

Answer

For exponential family likelihoods paired with conjugate priors, the Bayesian posterior mean is linear in the observed data, making it identical to the Bühlmann estimate.

Question 5

A credibility premium is computed as P = Z × 250 + (1−Z) × 200 = 230. What is the credibility factor Z?

Accepted Answer

0.60

Answer

Setting Z(250) + (1−Z)(200) = 230 gives 200 + 50Z = 230, so 50Z = 30, thus Z = 0.60.

Actuary Certification Practice Test

Actuary Certification Practice Test

Actuary Certification Credibility Theory 3