Actuary Certification ACTUARY Actuarial Models 3 — Questions and Answers
Question 1: A disability income policy pays $1,000/month while disabled. Using a multiple-state model, which transition intensity governs recovery from disability?
- μ_ad (active to disabled)
- μ_da (disabled to active) (Correct answer)
- μ_dd (disabled to dead)
- μ_aa (active to active)
Correct answer: μ_da (disabled to active)
The recovery intensity μ_da governs the transition from the disabled state back to the active (healthy) state.
Question 2: Under the Vasicek interest rate model, the short rate r(t) follows dr = α(θ−r)dt + σdW. What is the long-run mean of r(t)?
- α
- σ²/(2α)
- θ (Correct answer)
- σ/α
Correct answer: θ
The mean-reverting drift pulls r toward θ, which is the long-run equilibrium (unconditional mean) of the process.
Question 3: In collective risk theory, the stop-loss premium for retention d equals which expression?
- E[S] − E[min(S, d)]
- E[max(S−d, 0)]
- E[S·1(S>d)]
- Both A and B are equivalent (Correct answer)
Correct answer: Both A and B are equivalent
The stop-loss premium E[max(S−d,0)] equals E[S] − E[min(S,d)], so both expressions A and B are equivalent definitions.
Question 4: Which property of the Pareto distribution makes it particularly useful for modeling heavy-tailed insurance losses?
- Finite moments of all orders
- Lack of memory property
- Slowly decaying power-law tail (Correct answer)
- Symmetric distribution around the mean
Correct answer: Slowly decaying power-law tail
The Pareto distribution's power-law tail F̄(x) ~ x^{-α} decays slowly, capturing rare but extremely large losses.
Question 5: In a multi-decrement table with causes of decrement j=1,2,...,m, what is the relationship between the single-decrement probabilities q'_x^(j) and the associated single-decrement tables?
- q'_x^(j) = q_x^(j) / q_x^(τ)
- q'_x^(j) is the probability of decrement j in a world where only cause j operates (Correct answer)
- q'_x^(j) = 1 − p_x^(τ)
- q'_x^(j) equals the cause-j force of decrement
Correct answer: q'_x^(j) is the probability of decrement j in a world where only cause j operates
The associated single-decrement probability q'_x^(j) is defined in a hypothetical world where only decrement j is active, eliminating competing risks.
Question 6: Under the lognormal model, if ln(S_T/S_0) ~ N(μT, σ²T), what is E[S_T]?
- S_0·e^(μT)
- S_0·e^(μT + σ²T/2) (Correct answer)
- S_0·e^(σ²T/2)
- S_0·e^(μT − σ²T/2)
Correct answer: S_0·e^(μT + σ²T/2)
For a lognormal variable, E[S_T] = S_0·exp(μT + σ²T/2), applying the moment generating function of the normal distribution.
Question 7: The Kolmogorov forward equations in a Markov chain model describe which relationship?
- How occupation probabilities evolve forward in time as a function of transition intensities (Correct answer)
- How to compute backward expectations from terminal conditions
- The stationary distribution of an ergodic chain
- The spectral decomposition of the generator matrix
Correct answer: How occupation probabilities evolve forward in time as a function of transition intensities
The Kolmogorov forward equations express the time derivative of transition probabilities p_{ij}(t) in terms of the current state j and its outgoing intensities.
Actuary Certification ACTUARY Actuarial Models 3
A disability income policy pays $1,000/month while disabled.
Using a multiple-state model, which transition intensity governs recovery from disability?