Question

Problem 4-33
Required Annuity Payments

Assume that your father is now 50 years old, that he plans to retire in 10 years, and that he expects to live for 25 years after he retires - that is, until he is 85. He wants his first retirement payment to have the same purchasing power at the time he retires as $50,000 has today. He wants all his subsequent retirement payments to be equal to his first retirement payment. (Do not let the retirement payments grow with inflation: Your father realizes that the real value of his retirement income will decline year by year after he retires). His retirement income will begin the day he retires, 10 years from today, and he will then get 24 additional annual payments. Inflation is expected to be 6% per year from today forward. He currently has $100,000 saved up; and he expects to earn a return on his savings of 7% per year with annual compounding. To the nearest dollar, how much must he save during each of the next 10 years (with equal deposits being made at the end of each year, beginning a year from today) to meet his retirement goal? (Note: Neither the amount he saves nor the amount he withdraws upon retirement is a growing annuity.) Do not round intermediate steps.

Answer 1

Amount of annual payment required = 50,000(1.06)¹⁰

= $89,542.3848

Total amount needed on the day of retirement = 89,542.3848 + 89,542.3848*PVAF(7%, 24 years)

= $1,116,533.9032

Value of 100,000 already saved up on the day of retirement

= 100,000 (1.07)¹⁰

= $196,715.1357

Additional Amount required = $1,116,533.9032 - $196,715.1357

= $919,818.7675

Let the annual deposit be x

X*[{(1.07)¹⁰ -1}/0.07] = 919,818.7675

13.816447X = 919,818.7675

X = $66,574.19

Hence, amount required to be deposited each year = $66,574.19