ACTUARY Actuarial Models Flashcards
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A reinsurance treaty uses an excess-of-loss arrangement with retention M per occurrence. Which formula gives the reinsurer's expected payment per loss?
Answer: E[max(X − M, 0)]
The reinsurer pays max(X−M, 0) per loss, so its expected payment is E[max(X−M, 0)], also called the excess loss mean.
In actuarial notation, what does ä_x represent?
Answer: A whole life annuity-due paying 1 per year while alive
ä_x (double-dot a_x) denotes the expected present value of a whole life annuity-due, paying 1 at the beginning of each year while the insured survives.
Panjer's recursion formula is used to compute aggregate loss distributions when the claim count follows an (a, b, 0) class. Which distributions belong to this class?
Answer: Poisson, Binomial, Negative Binomial
The (a,b,0) class includes Poisson, Binomial, and Negative Binomial distributions, for which p_k/p_{k-1} = a + b/k.
What is the primary purpose of the 'floor' in a guaranteed minimum accumulation benefit (GMAB) rider in a variable annuity?
Answer: To ensure the account value never falls below a guaranteed amount at a specified date
A GMAB guarantees that at the end of the accumulation period, the account value will be at least a specified floor amount regardless of investment performance.
In the Heston stochastic volatility model, the variance process v(t) follows which type of dynamics?
Answer: Cox-Ingersoll-Ross (square-root) process
The Heston model specifies that variance v(t) follows a CIR square-root process, ensuring non-negative variance while allowing mean reversion.
Under a fully discrete whole life insurance with unit benefit, the recursion relation for reserves is: ₜV + π = v·q_{x+t}·(1 − ₜ₊₁V) + v·ₜ₊₁V. What does this equation represent?
Answer: The year-t accumulation of the reserve plus premium funds the death benefit and next-period reserve
This recursion says the reserve plus premium, accumulated for one year, exactly funds the expected death claim and the expected reserve carried forward.
In extreme value theory, the Generalized Extreme Value (GEV) distribution unifies which three limiting distributions?
Answer: Gumbel (Type I), Fréchet (Type II), and Weibull (Type III)
The GEV distribution encompasses Gumbel (ξ=0), Fréchet (ξ>0), and reverse Weibull (ξ<0) as special cases parameterized by the shape parameter ξ.