IAI Actuarial Core Principles · Actuarial Mathematics for Modelling · Means and variances of assurance and annuity payments
A 2-year increasing annuity-due on a life aged x pays 1 at time 0 and 2 at time 1 if the life is then alive. Given p_x = 0.9 and v = 0.8, what is the variance of the present value of the payments?
The variance is 0.23. Only the second payment is random, with discounted value 2 × 0.8 = 1.6 paid with probability 0.9, so the variance is 1.6² × 0.9 × 0.1 = 0.2304. The first payment at time 0 is certain and adds nothing.
- A0.14
- B0.23Correct
- C0.26
- D2.30
- 0.58
Explanation
The present value is 1 with probability 0.1 and 1 + 2(0.8) = 2.6 with probability 0.9. The mean is 2.44 and E[Y²] = 0.1 + 0.9(6.76) = 6.184, so Var = 6.184 − 5.9536 = 0.2304, which is 0.23. Using 0.1 instead of 0.09 for pq gives 0.26, which is wrong.
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