Question

Suppose you are investing some money into a savings account bearing 10% return in the following manner:

1. You started your first deposit at the end of year 1 with an amount of $1,000, increasing by $1,000 every year through year 30.

2. For years 31 through 40, you did nothing.

3. Then you started to withdraw the entire amount equally from years 41 through 60 with nothing left in the account after the last withdrawal.

Draw the cash flow diagram showing your investment scheme, then calculate:

a. The Present value of your deposits at year 0;

b. The Future value of your deposits at year 30;

c. The Annual worth of your deposits for 30 years;

d. The annual amount of withdrawal for years 41 through 60;

e. If you would want that $x (calculated from part d) can be withdrawn forever beginning years 41 , what additional amount need to be deposited at year 40?

Answer 1

a. The Present value of your deposits at year 0

First convert the gradient cash flow into Uniform Annual Deposits

A = A1 + G (A/G, 10%, 30)

A = $1,000 + $1,000 (8.1762) = $9,176.2

Present Value = A (P/A, 10%, 30)

Present Value = $9,176.2 (9.4269) = 86,513

b. The Future value of your deposits at year 30

Uniform Annual Deposits

A = A1 + G (A/G, 10%, 30)

A = $1,000 + $1,000 (8.1762) = $9,176.2

Future Value = A (F/A, 10%, 30)

Future Value = $9,176.2 (164.4940) = $1,509,430

c. The Annual worth of your deposits for 30 years;

Convert the gradient cash flow into Uniform Annual Deposits (Annual Worth)

A = A1 + G (A/G, 10%, 30)

A = $1,000 + $1,000 (8.1762) = $9,176.2

d. The annual amount of withdrawal for years 41 through 60

Future Value of $1,509,430 at year 40^th

Future Value = $1,509,430 (F/P, 10%, 10)

Future Value = $1,509,430 (2.5937) = $3,915,009

Let $3,915,009 at year 40 is the present value of all equal amounts that will spread from years 41 through 60 with nothing left in the account after the last withdrawal.

Annual withdrawal = P (A/P, 10%, 20)

Annual withdrawal = $3,915,009 (0.1175) = $460,014

e. If you would want that $x (calculated from part d) can be withdrawn forever beginning years 41 , what additional amount need to be deposited at year 40?

Annual Deposit (calculated from part d) = $460,014

If it is withdrawn forever, the present worth of those deposits will be called as capitalized cost.

Capitalized cost = Annual withdrawals / rate of interest

Capitalized cost = $460,014 / 0.10 = $4,600,140

Previous deposit at 40^th year = $3,915,009

If forever is taken, the deposit in 40^th year = $4,600,140

Additional deposit required = $4,600,140 – $3,915,009 = 685,131