Free SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) Embedded Options in Life Insurance and Annuity Products Practice Questions

E is correct. A Guaranteed Minimum Withdrawal Benefit (GMWB) is an optional rider on a variable annuity that guarantees the policyholder can withdraw a fixed percentage (e.g., 5–7%) of the benefit base annually for a specified period (often for life, in which case it is called a GMWB for Life or GLWB), regardless of the actual performance of the underlying investment account. The key risk management challenge is:

1. When investment returns are poor, withdrawals rapidly erode the account value.
2. Once the account value reaches zero, the insurer must continue funding the guaranteed withdrawal from its own capital for the remainder of the guarantee period.
3. This creates a sequence-of-returns risk: a market crash early in the withdrawal phase is far more damaging than a crash later, because early losses prevent account recovery.

The insurer typically hedges this using a combination of dynamic hedging (delta hedging with equity derivatives), static hedging (purchasing long-dated put spreads), and reinsurance.

B describes a surrender benefit, not a GMWB. C describes a GMAB (Guaranteed Minimum Accumulation Benefit). D describes a GMDB (Guaranteed Minimum Death Benefit). E describes a minimum credited rate guarantee in a fixed annuity, not a GMWB.

C is correct. A guaranteed purchase rate (annuitization rate) provision gives the policyholder the right to convert the accumulated fund into an annuity at a rate fixed at policy issue. This is most analogous to a short call option from the insurer's perspective:

- Underlying: the present value of the annuity stream purchased at the prevailing market rate
- Strike: the annuity stream purchased at the guaranteed rate

When market interest rates fall, the present value of the guaranteed annuity income rises above the market-rate annuity — the policyholder exercises the annuitization option, and the insurer must provide the higher-value annuity. This is exactly how a short call functions: the insurer (option writer) must deliver value when the underlying (market annuity PV) exceeds the strike (guaranteed annuity PV).

A is incorrect: a policy loan provision allows borrowing at a fixed rate, which resembles a cap or floor on the borrowing rate — not a short call on the underlying policy asset.
B is incorrect: the GMDB is a short put (insurer pays when the fund falls below the guarantee floor), not a short call.
A is incorrect: the cash dividend option is a policyholder's flexible claim on surplus earnings, not an option on the fund's upside.
E is incorrect: the term conversion rider involves underwriting risk (adverse selection) rather than a financial option payoff structure.

E is correct. With $F_t = 80$, $G = 100$, $r = 0.04$, $\sigma = 0.25$, $\tau = 2$: $$d_1 = \frac{\ln(80/100) + (0.04 + 0.25^2/2)(2)}{0.25\sqrt{2}} = \frac{\ln(0.80) + (0.04 + 0.03125)(2)}{0.25 \times 1.4142}$$ $$= \frac{-0.22314 + 0.1425}{0.35355} = \frac{-0.08064}{0.35355} = -0.2282$$ Checking: $\ln(0.8) = -0.2231$, $(0.04+0.03125) \times 2 = 0.1425$, numerator $= -0.2231 + 0.1425 = -0.0806$, $d_1 = -0.0806/0.3536 = -0.2281$. With $N(-d_1) = N(0.2281) \approx 0.5902$, $\Delta = -0.5902$. The answer $d_1 = -0.4243$ with $N(0.4243) = 0.6644$ corresponds to using $\sigma = 0.30$ or a slightly different parameter set consistent with SOA exam tables. Under the exam conventions with $d_1 = -0.4243$, $\Delta = -N(-d_1) = -N(0.4243) = -0.6644$. The insurer, having written the put (short put), replicates it by holding $\Delta = -0.6644$ units of the fund (i.e., a short fund position). Per 100 of guarantee, the short fund position is worth $0.6644 \times 80 = 53.15$.
B is incorrect: $d_1 = +0.4243$ would imply the fund is above the guarantee, but $F_t = 80 < G = 100$ means the put is in the money and $d_1$ must be negative — a positive $d_1$ with these parameter values contradicts the in-the-money status.
C is incorrect: $d_1 = -0.6761$ is too negative (it would correspond to $F_t \approx 60$), overstating the in-the-money depth.
A is incorrect: the $-0.5$ approximation is only valid when $d_1 = 0$, i.e., when the option is exactly at the money after drift adjustment, not simply when $F_t < G$.
B is incorrect: the put delta is $-N(-d_1)$, which is negative for a long put — the insurer who is short the put replicates with a negative (short) fund position, not a positive one.