Actuary Certification ACTUARY Actuarial Models 2 — Questions and Answers
Question 1: Under the Cox-Ingersoll-Ross (CIR) model, which property ensures interest rates remain non-negative?
- Mean reversion with a floor at zero
- The condition 2αθ ≥ σ² (Correct answer)
- Constant volatility assumption
- Bounded drift term
Correct answer: The condition 2αθ ≥ σ²
The Feller condition 2αθ ≥ σ² ensures the process never reaches zero, keeping rates non-negative.
Question 2: In a select-and-ultimate mortality table, what does the 'select' period represent?
- The period after age 65
- The initial period when recent underwriting affects mortality (Correct answer)
- The period of highest mortality rates
- The ultimate steady-state mortality
Correct answer: The initial period when recent underwriting affects mortality
The select period captures the effect of recent underwriting or selection, during which mortality differs from the population average.
Question 3: A compound Poisson process has Poisson frequency λ and severity distribution F(x). What is the variance of aggregate losses?
- λ·E[X]
- λ·E[X²] (Correct answer)
- λ·Var(X)
- λ·(E[X])²
Correct answer: λ·E[X²]
The variance of a compound Poisson process equals λ·E[X²], which combines frequency variance and second moment of severity.
Question 4: Which interpolation method for fractional ages assumes a uniform distribution of deaths (UDD)?
- Hyperbolic (Balducci) assumption
- Linear interpolation of l_x (Correct answer)
- Constant force of mortality
- De Moivre's law
Correct answer: Linear interpolation of l_x
Under UDD, the number of survivors l_{x+t} is linear in t between integer ages, implying a uniform distribution of deaths.
Question 5: In the Black-Scholes framework, what is the risk-neutral drift of a non-dividend-paying stock?
- The expected return μ
- The risk-free rate r (Correct answer)
- Zero
- μ minus the risk-free rate
Correct answer: The risk-free rate r
Under the risk-neutral measure, all assets earn the risk-free rate r, replacing the real-world drift μ.
Question 6: For a fully continuous whole life insurance, which equation defines the net premium reserve at time t using the prospective method?
- PV(future benefits) + PV(future premiums)
- PV(future benefits) − PV(future premiums) (Correct answer)
- PV(past premiums) − PV(past benefits)
- PV(future premiums) only
Correct answer: PV(future benefits) − PV(future premiums)
The prospective reserve equals the present value of future benefits minus the present value of future net premiums.
Question 7: What does the 'credibility complement' represent in the Bühlmann credibility framework?
- The pure Bayesian estimate
- The grand mean used when credibility weight is zero (Correct answer)
- The variance of the hypothetical means
- The expected value of the process variance
Correct answer: The grand mean used when credibility weight is zero
When the credibility weight Z equals zero, the estimate defaults entirely to the complement, which is the overall mean of the portfolio.
Actuary Certification ACTUARY Actuarial Models 2
Under the Cox-Ingersoll-Ross (CIR) model, which property ensures interest rates remain non-negative?