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.

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).

E is correct.

The put delta in the Black-Scholes framework is $\Delta =-N(-d_1)$, where

$$d_1=\frac{\ln (F_t/G)+\left(r+\frac{1}{2}{\sigma }^2\right)\tau }{\sigma \sqrt{\tau }}.$$

Substitute the given parameters $F_t=80$, $G=100$, $r=0.04$, $\sigma =0.25$, $\tau =2$. First the variance and drift terms:

$$\frac{1}{2}{\sigma }^2=\frac{1}{2}(0.25{)}^2=0.03125,r+\frac{1}{2}{\sigma }^2=0.04+0.03125=\mathrm{0.07125.}$$

The numerator is

$$\ln (0.80)+(0.07125)(2)=-0.22314+0.14250=-0.08064,$$

and the denominator is

$$\sigma \sqrt{\tau }=0.25\sqrt{2}=0.25\times 1.41421=\mathrm{0.35355.}$$

Therefore

$$d_1=\frac{-0.08064}{0.35355}=-\mathrm{0.2281.}$$

The put delta uses $-N(-d_1)$. Since $d_1=-0.2281$, we have $-d_1=0.2281$, so

$$\Delta =-N(-d_1)=-N(0.2281)=-\mathrm{0.5902.}$$

This is the put delta: $\Delta =-0.5902$.

For the replicating portfolio, the put delta is negative, so the insurer holds a short fund position. The number of fund units held short per 100 of guarantee is $0.5902$, with market value

$$0.5902\times 80=47.22$$

in the fund per 100 of guarantee.

Final result: $d_1=-0.2281$ and $\Delta =-0.5902$.