Answer to Question #182136 in Economics of Enterprise for Angelo

Question

Engr. Hermo, believing that life begins at 40, decided to retire and start enjoying life at age 40. he wishes to have upon his retirement the sum of P5,000,000. on his 21stbirthday, he deposited a certain amount and increased his deposit by 15% each year until he will be 40. if the money is deposited in a super savings account which earns 15% interest compounded annually. How much was his initial deposit?

Answer 1

An initial deposit A is made at the age of 21. The deposits are increased by 15% every year up till the age of 40. It is assumed that the 15% increase in deposit every year is with respect to the previous year's deposit. A future value of 5000000 is expected at the end of age 40.

The cash flow for the problem is shown below.

Gradient G = 0.15 (15%)

interest rate i = 0.15 (15%)

Future value FV = 5000000

The present value at the end of year 21 of the year and FV 5000000 at the end of year 40 should be equal.

The present value at the end of the year 21 of the annual deposit

The gradient and the interest rate are the same:

rate of increase in deposits = discounting rate

The PV of all deposits discounted back by 15% will be equal to the initial deposit.

PV of annual deposit = (initial deposit) $\times$ (number of deposits)

Number of deposists = 20

PV_i=$A \times 20$

Present value at the end of the year 21 of single payment 5000000:

FV=5000000

i = 0.15

period n = 40-21 = 19

$PV_{ii} = \frac{FV}{(1+i)^n}= \frac{5000000}{(1+0.15)^19}=351326.604$

The present value $PV_i$ and $PV_{ii}$ should be equal.

$PV_i = PV_{ii} \\ A \times 20 = 351326.604 \\ A = 17566.33$

The initial deposit amount is 17566.33

The updated cash flow: