SOA Exam ALTAM Sample Question 55

Study guide: Exam ALTAM

An insurer issues a 3-year equity-linked insurance policy to ( $x$ ). The policyholder pays a single premium of 5000, and an additional policy fee of 350. The insurer invests 100% of the premium in a separate account, with management charges of 2.5% of the fund value deducted at the start of each year including the first.

The policy offers a Guaranteed Minimum Death Benefit and a Guaranteed Minimum Maturity Benefit of 100% of the initial premium (excluding the policy fee).

The insurer stress tests the profitability of the contract using the following assumptions:

Pre-contract expenses are 200
Renewal expenses are 30 at the start of the 2nd and 3rd years.
Fund management expenses are 2% of the policyholder's fund value at the start of the first year, and 1% of the policyholder's fund value at the start of the 2nd and 3rd years, before management charge deductions in each case.
$q_{x+t} = 0.05 + 0.01t$ for $t = 0, 1, 2$ .
There are no lapses
Interest of 5% per year is earned on the insurer's funds.
No reserve is held for the contract guarantees in the insurer's assets.
The insurer uses a risk discount rate of 10% per year.
The return on the assets in the policyholder's fund for each of the three years (before any management charge deductions) are:


Year 1	–14%
Year 2	–5%
Year 3	+ 4%

Part a

Show that the projected fund value at time 3 is 3940 to the nearest 10. You should calculate the value to the nearest 1.

Part b

Calculate the profit vector for the contract.

Part c

Show that the net present value of this contract, based on the stress test assumptions given, is –600 to the nearest 10. You should calculate the value to the nearest 1.

Part d

The insurer can purchase options on the underlying fund assets to hedge the guarantees, using the following parameters:

(i) Volatility, $\sigma = 0.25$
(ii) The risk free rate is 4% per year compounded continuously
(iii) Management charge deductions are 2.5% per year

Show that the price of a 3-year put option guaranteeing a return of premium is 0.136 per unit of premium, to the nearest 0.001. You should calculate the value to the nearest 0.0001.

Part e

You are given the following market prices at time 0 for 1 and 2 year put options on the underlying fund assets, based on the pricing assumptions above, and a premium of 5000.

Term	Put Option Price
1-year	446.7
2-year	590.8

Assume that mortality follows the profit test assumptions above. Show that the total cost at issue of hedging the guaranteed minimum death benefit and the guaranteed minimum maturity benefit is 665 to the nearest 5. You should calculate the value to the nearest 0.1.

Part f

Calculate the net present value of the contract under the same stress test as in (a), but assuming that the insurer hedges the guarantees by purchasing options at a cost of 665 at the start of the first year.

Part g

Describe one advantage and one disadvantage for the insurer of purchasing put options to hedge the guarantee costs.

Answer this question and see the worked solution