Actuarial Notes

SOA Exam ALTAM Sample Question 56

Study guide: Exam ALTAM

A 10-year equity-linked policy is issued to (x). The policy offers a guaranteed minimum maturity benefit of 100% of the initial single premium. After an initial expense charge of 5% of the premium, the remainder is invested in separate fund. The insurer deducts a 0.2% management charge from the policyholder's fund at the end of each month, if the policyholder survives.

You are given:

  • An investment at t = 0 of 1 in the underlying asset will have value S_t at t, where S_t follows a geometric Brownian motion, with volatility σ, and S_0 = 1.
  • \mathrm{BSP}(K) = \mathrm{E}_0^Q\left[e^{-10r}\left(K - S_{10}\right)^+\right] is the Black-Scholes option price formula for a put option on an investment of 1 in the underlying stock at time 0, with strike price K.
  • There are no exits other than death.

Part a

Show that the risk-neutral value of the GMMB option at issue is

\pi(0) = P\left({}_{10}p_x\,\xi\,\mathrm{BSP}(K^*)\right)

You should state any assumptions you need, and you should define the parameters ξ and K^* in terms of the information given about this specific contract.

Part b

Given σ = 0.25, x = 60, P = 100{,}000, Standard Ultimate Life Table mortality, and r = 0.04, show that \pi(0) = 17{,}100 to the nearest 100.

Part c

Assume that the insurer constructs a hedge portfolio at t = 0 based on the Black Scholes option valuation. The insurer does not rebalance the hedge until one month later. You are given that in the first month, the underlying asset values fell by 3%. Calculate the value of the hedge portfolio at the end of the first month, before rebalancing.

Part d

You are given that, at \delta = 2.40\%, a_{60:\overline{10|}}^{(12)} = 8.679. Calculate the cost of the option as a monthly deduction from the fund, assuming no lapses.

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Concepts tested

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