Time Value of Money, Part-6

Mr. A plans to invest 10% of his yearly salary for next 25 years, salary increases by 5% p.a. The rate of return is 12% p.a. His current salary is Rs.2,00,000 p.a. and investments are made at the end of the year.

This is the question of growing annuity where amount of investment is growing at 5% rate year on year and invested at 12% rate. We will have to use following equation as this question cannot be solved using pre-set formula of calculator.

FV = A×[(1+r)ⁿ – (1+g)ⁿ]/(r-g)

Where:

FV = Future value

A = Initial amount of investment

r = Rate of return

g = Growth rate

n = Number of years

So, in the above question amount of money Mr. A accumulates at the end of 25 years will be,

FV = 20000×[(1+0.12)²⁵ – (1+0.05)²⁵]/(0.12-0.05)

Hence, he will accumulate Rs.38,89,631 at the end of 25 years from now.

Now let’s assume that Mr. A starts investing from the beginning of the year, he will accumulate.. 

FV = A×(1+r)×[(1+r)ⁿ – (1+g)ⁿ]/ r-g

FV = 20000×1.12×[(1+0.12)²⁵ – (1+0.05)²⁵]/(0.12-0.05)

Rs. 43,56,387 is the answer. This formula of growing annuity is useful when investments are growing at a constant rate. However in case where r=g, this formula is not useful, we may have to switch over to excel.

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Time Value of Money, Part-5

If you want 1crore rupees after 35 year, what amount of money will you have to invest from today? (Assume rate of return as 12% p.a.)

This is the question of future value of annuity where we will have to find payments

FV = 10000000

N = 35

i/y = 12

PMT =? (Begin mode)

Answer: 20684

Let say you want to invest on monthly basis for the same amount, then

FV = 10000000

N = 35×12 = 420

i/y = 12/12 = 1

PMT =? (Begin mode)

Answer: 1540

 Now another topic, let say you have taken a loan @ 12% for 10 years of Rs.10 lakh, what will be the EMI?

This is the question of present value of annuity and we are finding payments here.

PV = 1000000 (Inflow hence positive)

N = 10×12 = 120

i/y = 12/12 =1

PMT = ?

Answer: 14347.

Remember: EMIs are always paid at the end of the period, hence no need to assume begin mode.

 Present value of annuity is useful to find current value of all the future payments.

Mr. A asks you to lend some money to him. He agrees to pay you Rs.1000 per month over a period of next 5 years. Prevailing rate of interest is 12% per annum. What amount of money should you lend him today?

PMT = 1000

N = 5×12 = 60

i/y = 12/12 = 1

PV = ?

Answer: 44955.