SOA Exam ALTAM Spring 2025, Question 6

Study guide: Exam ALTAM

An insurer issues 10-year Variable Annuity policies, each with a single premium of $P = 100{,}000$ , to lives age 50 at issue.

After an initial expense charge of 3% of the premium, the remainder is invested in a separate account called the policyholder's fund. A management charge of $m = 2\%$ per year is deducted from the policyholder's fund continuously while the policy is in force.

The policy offers a Guaranteed Minimum Maturity Benefit (GMMB) of 100% of the premium. The insurer uses the Black-Scholes model to value the GMMB.

You are given the following valuation assumptions:

There are no lapses or surrenders.
Mortality follows the Standard Ultimate Mortality Model.
The asset volatility is $\sigma = 0.20$ .
The risk-free rate is $r = 0.04$ , compounded continuously.

Part a (3 points)

Show that the value of the GMMB option at issue is $\pi(0) = 12{,}360$ to the nearest 10. You should calculate the value to the nearest 1.

Part b (3 points)

At time 6, the policyholder's fund value is $F_6 = 140{,}000$ .

Calculate the value at time 6 of the GMMB, assuming no changes to the valuation assumptions.

Part c (1 point)

The policy also offers a Guaranteed Minimum Income Benefit (GMIB) at maturity. The GMIB is calculated using a Benefit Base, and a guaranteed annuitization rate of $\gamma = 8\%$ . The GMIB Benefit Base is equal to the maximum of the initial premium $P$ accumulated at 3% per year, with a step-up guarantee every 3 years.

You are given that the policyholder's fund value at time 3 was $F_3 = 103{,}000$ .

Calculate the guaranteed minimum annual income under the GMIB, based on the information available at time 6.

Part d (1 point)

At time 6, the insurer offers the opportunity to exchange the policy's existing GMIB for a Guaranteed Minimum Withdrawal Benefit (GMWB). The GMWB allows lifetime withdrawals of 6% of the GMWB Benefit Base, starting at time 10.

The GMWB Benefit Base is the greater of:

105% of the policyholder's fund value at time 6; and
the highest year-end value of the fund during the first 4 years after the exchange.

Calculate the minimum guaranteed withdrawal amount payable from age 60, based on the information at the time of the exchange offer.

Part e (1 point)

Describe one advantage and one disadvantage of the GMWB compared with the GMIB, from the perspective of the policyholder.

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