Actuarial Notes

SOA Exam ALTAM Sample Question 61

Study guide: Exam ALTAM

Your company issues 2-year equity-linked policies to lives age 85. Policyholders pay a single premium of 10,000. The insurer deducts a management charge of 3% of the fund value at the start of each year, including the first. There are no other expense deductions. The fund price is 1 per share at inception. There is no guaranteed minimum maturity benefit.

If the policyholder dies during the contract term, the death benefit, payable at the end of the year of death, is the fund value at the end of the year, or the guaranteed minimum death benefit (GMDB) if larger. The GMDB is the premium accumulated at 6% compounded continuously.

Part a

Let \{S_t\}_{t>=0} denotes the stock price process. Calculate the highest ratio of S_2/S_0 such that the benefit payable for a death at time 1.3 would be the GMDB. Your answer should be 1.20 to the nearest 0.01. You should calculate the value to the nearest 0.0001.

Part b

The insurer constructs a hedge portfolio at the issue date, after deduction of the management charge, based on the Black-Scholes option valuation.

You are given the following assumptions:

  • (i) The risk-free rate is r = 0.04 continuously compounded.

  • (ii) The volatility is \sigma = 0.30.

  • (iii) Mortality follows the Standard Ultimate Mortality Model (as printed).

  • (iv) There are no exits other than death.

  • (i) Show that, in the calculation of the contribution of year 1 to the cost of the hedge portfolio at issue, d_1 is -0.018 to the nearest 0.001. You should calculate the value to the nearest 0.0001.

  • (ii) Show that the cost of the contribution of year 1 to the cost of the hedge portfolio at issue is 77 (as printed) to the nearest 1. You should calculate the value to the nearest 0.01.

  • (iii) Calculate the cost of the hedge portfolio at issue.

  • (iv) Let c denote the component of the annual 3% management charge that is required to fund the GMDB guarantee. You are given that c is a portion of the 3% annual charge at the start of each year, not an additional charge. Calculate c.

  • (v) Without further calculation, state with reason whether the cost of the hedge portfolio at issue would decrease if the fund price were 2 per share at inception.

Part c

At the end of year 1, the fund price per share has doubled from its initial value. The insurer offers, and the policyholder accepts, a reset. Future GMDBs are based on the fund value at the end of year 1, accumulated as before. The insurer calculates a new hedge portfolio

  • (i) Let D be the value of d1 in the calculation of the first year after reset to the cost of the hedge portfolio. Show that the ratio of D to your answer to (b)(i) is 1.00 to the nearest 0.01.
  • (ii) Calculate the value of the hedge portfolio at time 1.

Part d

The insurer decides it will also offer these policies with a GMMB as an option, either with no GMDB or with the existing GMDB.

Let ValueGMDB, ValueGMMB, and Valueboth denote the corresponding values of the hedge fund at time 0. Your colleague suggests that Valueboth < ValueGMDB + ValueGMMB, since the GMDB and GMMB payouts are negatively correlated.

State whether you agree with your colleague. Justify your response.

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Concepts tested

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