SOA Exam ALTAM Spring 2026, Question 6

Study guide: Exam ALTAM

An insurer issues a 10-year equity-linked contract to (70). The single premium is 100,000. You are given the following information and assumptions.

(i) There is a Guaranteed Minimum Maturity Benefit (GMMB) of 100% of the premium.

(ii) An initial expense charge of 500 is deducted at the start of the contract.

(iii) Management charges of 2% of the policyholder's account value are deducted at the start of each year of the contract including the first (after the initial expense deduction).

(iv) The underlying asset values of the policyholder's account follow a lognormal process with parameters $\mu = 0.05$ and $\sigma = 0.25$ .

(v) Mortality is assumed to follow the Standard Ultimate Survival Model (SUSM).

(vi) There are no lapses.

(vii) The risk free rate of interest is $r = 0.04$ per year, compounded continuously.

Part a (3 points)

Show that the cost of the GMMB is 13,540 to the nearest 10. You should calculate the cost to the nearest 1.

Part b (1.5 points)

(i) Determine the value of the stock part of the initial delta hedge portfolio (that is, the value of the investment in the underlying asset).

(ii) Explain briefly why the GMMB hedge portfolio takes a short position in the underlying assets.

Part c (2.5 points)

Calculate the probability that the GMMB matures in-the-money (that is, the probability, under the real world measure, that the contract is in force at maturity, and that the policyholder's fund value at that time is less than the GMMB).

Part d (2 points)

The contract has an additional Guaranteed Minimum Income Benefit (GMIB). At time 10 the policyholder may annuitize the greater of the policyholder's fund value at maturity, and the policy's Benefit Base at maturity. You are given the following additional information.

The Benefit Base is defined as 100% of the premium with interest at 4% per year.
The guaranteed annuitization rate at age 80 is 10.5%.
The annuity benefit is paid annually at the start of each year.

(i) Calculate the guaranteed minimum amount of annual annuity income payable from age 80, based on the information available at the policy inception.

(ii) The insurer calculated the guaranteed annuitization rate using the SUSM at a rate of interest of $j$ per year effective. State with reasons whether $j$ is greater than or less than 5%.

Part e (1 point)

Explain briefly why the cost to the insurer of the GMIB at maturity could be less than the Benefit Base at maturity, even if the GMIB is exercised.

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