SOA Exam ALTAM Fall 2024, Question 6
Study guide: Exam ALTAM
A 10-year equity-linked policy is issued to a life age 60. The policyholder pays a single premium of 1000. The policy offers a guaranteed minimum maturity benefit (GMMB) of 100% of the single premium. The insurer deducts a management charge at a rate of 2% per year, payable continuously, from the policyholder's funds. There are no other expense deductions.
The insurer constructs a hedge portfolio at the issue date based on the Black-Scholes option valuation. You are given the following assumptions.
(i) The risk free rate is r = 0.03 continuously compounded.
(ii) The volatility is \sigma = 0.20
(iii) Mortality follows the Standard Ultimate Mortality Model.
(iv) There are no exits other than death.
(v) \bar{a}_{60:\overline{10}|\,\delta=0.02} = 8.8569
Part a (4 points)
(i) Show that the GMMB hedge portfolio value at the issue date is 150 to the nearest 10. You should calculate the value to the nearest 0.01.
(ii) Let c denote the cost of the hedge portfolio expressed as a continuous risk premium. Show that c = 0.017 to the nearest 0.001. You should calculate the value to the nearest 0.0001.
Part b (3 points)
Over the first 3 years, the price of the underlying stocks in the policyholder's fund increases by 50%.
(i) Calculate the policyholder's fund value at t = 3.
(ii) Calculate the value of the total GMMB hedge portfolio required at t = 3.
(iii) You are given that \bar{a}_{63:\overline{7}|\,\delta=0.02} = 6.4021. State with reasons whether the charge determined in (a)(ii) is sufficient to pay the future costs of the GMMB as of t = 3.
Part c (2 points)
The insurer is considering introducing a reset option. The minimum contract term is 10 years. Resets would be allowed at any policy anniversary.
(i) Assume that the policyholder exercised the reset option at t = 3. Explain in words the effect of the reset on the policy.
(ii) Describe one advantage and one disadvantage of the reset option from the insurer's perspective.
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