SOA Exam ALTAM Spring 2023, Question 6

Part a (2 points)

In the context of a Variable Annuity policy, describe the difference between a Guaranteed Minimum Maturity Benefit (GMMB) and a Guaranteed Minimum Income Benefit (GMIB).

Part b (4 points)

An insurer issues a 10-year Variable Annuity with a single premium of 10,000 to (50).

You are given:

(i) The GMMB is 90% of the single premium. There are no other guarantees.

(ii) The initial expense charge is 6% of the premium.

(iii) The management charge is 0.2% of the policyholder's fund, deducted at the end of each month, including at time 10.

(iv) The underlying fund asset value follows a geometric Brownian motion, with volatility $\sigma = 0.25$ .

(v) The risk-free rate is $r = 4\%$ per year, compounded continuously.

(vi) There are no exits other than death.

(vii) Mortality is assumed to follow the Standard Ultimate Life Table.

The insurer uses the Black-Scholes formula to construct a hedge portfolio of 10-year zero-coupon bonds and the underlying fund assets.

(i) (3 points) Show that the value of the hedge portfolio at time 0 is 1400 to the nearest 100. You should calculate the value to the nearest 1.

(ii) (1 point) Write down the value of the bond portion of the hedge portfolio at time 0.

Part c (4 points)

The insurer does not rebalance the hedge portfolio until time 1. At that time the policy is still in force and the price of the underlying fund assets has increased by 5%.

Calculate the rebalancing cost for the hedge portfolio.

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SOA Exam ALTAM Spring 2023, Question 6

Part a (2 points)

Part b (4 points)

Part c (4 points)

Concepts tested

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