Question

A man wishes to provide a fund for his retirement such that from his 60^th to 70^th birthdays he will be able to withdraw equal sums of ₱18,000 for his yearly expenses. He invests equal amounts for his 41^st to 59^th birthdays in a fund earning 10% compounded annually. How much should each of these amounts be?

Answer 1

Let say amount saved each year from 41 to 59 is X

I am calculating future value of money saved when the man is 60 and present value of withdrawal when the man is 60 and equate them.

Money saved when the person turns 41 will be saved for 19 years whose future worth would be X * 1.1^¹⁹

Money saved when the person turns 42 will be saved for 18 years whose future worth would be X * 1.1^¹⁸

Money saved when the person turns 43 will be saved for 17 years whose future worth would be X * 1.1^¹⁷

and so on till year 59

Sum of money saved = [X * 1.1^¹⁹] + [X * 1.1^¹⁸] + [X * 1.1^¹⁷] + ......... + [X * 1.1^¹] = 56.275X

Present value of Money withdrawed when the person turns 60 is [18,000 / 1.1^⁰]

Present value of Money withdrawed  when the person turns 61 is [18,000 / 1.1^¹]

....

Present value of Money withdrawed when the person turns 70 is [18,000 / 1.1^¹⁰]

Sum of present value of money withdrawed = [18,000 / 1.1^⁰] + [18,000 / 1.1^¹] + ........... + [18,000 / 1.1^¹⁰] = 128,602.20

56.275X = 128,602.20

X = 2,285.24

Each deposit must be 2,285.24