10. An actuary will retire in 15 years' time. At the time of retirement, he wishes...

Transcribed Image Text:

10. An actuary will retire in 15 years' time. At the time of retirement, he wishes to buy Annuity Y with a single premium. Annuity Y provides payments at the end of each quarter for 25 years at an initial rate of $120,000 per annum. At each anniversary, the payments of Annuity Y will increase by $6,000 per annum. (a) Assuming an effective interest rate of 5% per annum, calculate the single premium required in 15 years' time to purchase Annuity Y. [5 marks] In order to buy Annuity Y immediately after retirement, the actuary plans to invest a lump sum immediately in an investment product so that it will have an accumulated value equal to the single premium calculated in (a) 15 years later. The annual effective returns from the investment product are independent and (1 + It) is lognormally distributed with It being the return in the tth year. The expected annual effective rate of return is 6% and the standard deviation of annual returns is 15%. 2542963.44 (b) Calculate the lump sum which the actuary should invest immediately in order to have a probability of 0.975 that the proceeds will be sufficient to purchase Annuity Y in 15 years' time.763-44 (c) Comment on your answer in (b). 52543 3585450.44 [7 marks] [3 marks] [Total: 15 marks] 10. An actuary will retire in 15 years' time. At the time of retirement, he wishes to buy Annuity Y with a single premium. Annuity Y provides payments at the end of each quarter for 25 years at an initial rate of $120,000 per annum. At each anniversary, the payments of Annuity Y will increase by $6,000 per annum. (a) Assuming an effective interest rate of 5% per annum, calculate the single premium required in 15 years' time to purchase Annuity Y. [5 marks] In order to buy Annuity Y immediately after retirement, the actuary plans to invest a lump sum immediately in an investment product so that it will have an accumulated value equal to the single premium calculated in (a) 15 years later. The annual effective returns from the investment product are independent and (1 + It) is lognormally distributed with It being the return in the tth year. The expected annual effective rate of return is 6% and the standard deviation of annual returns is 15%. 2542963.44 (b) Calculate the lump sum which the actuary should invest immediately in order to have a probability of 0.975 that the proceeds will be sufficient to purchase Annuity Y in 15 years' time.763-44 (c) Comment on your answer in (b). 52543 3585450.44 [7 marks] [3 marks] [Total: 15 marks]