# Time Value of Money (Part – 1) : Reasons of Preference to Receive Money Today

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## Topics to be Covered Are

• Meaning of time vale money
• Reasons of preference to receive money today
• Valuation technique – Compounding and Discounting technique
• Present value of a perpetuity
• Future value and Present value of an annuity due
• Perpetuity growing at a constant rate
• Present value of an annuity growing at a constant rate
• Calculation of doubling period

## Meaning

Time value of money (TVM) concept says that value of a certain sum of money received today is more than the same amount of money to be received at a future date.

## Reasons of Preference to Receive Money Today

• Inflation
• Future uncertainties
• Personal preference for present consumption
• Investment opportunities

## Valuation Techniques

• By Compounding the various cash flows to a future date or
• By Discounting the various cash flows to the present date.

### Compounding Technique

The compounding is the arithmetic process of determining the future value of a cash flow or a series of cash flows when compound interest is applied.

This technique can be explained for calculating:

• The future value of a single present cash flow.
• The future value of a series of unequal cash flows over a period of time.
• The future value of a series of equal cash flows over a period of time.
• The future value of a cash flow when compounding is done more than once in a year.

#### The Future Value of a Single Present Cash Flows

The mathematical formula of calculating compounding interest can be used in calculating the future value of a single present cash flow.

Hence, FV = P (1 + r) n or PV (1 + r) n

Where, FV = Future Value

P/PV = Principal (initial cash outflow) or present value

r = rate of interest per period

n = number of periods

(1 + r) = represents the compounding value (future value) of ₹ 1 for given values of r and n.

Example: Mr. Verma invested ₹ 10,000 at the beginning of the year one at the rate of 10 % compounded annually. What amount of money he will receive after the end of 2nd year?

Solution: FV or amount, Mr. Verma will receive after one year will be:

FV = 10,000 (1 + 0.10) 2

= 10,000 (1.10) 2

= ₹ 11,000

#### The Future Value of a Series of Unequal Cash Flows over a Period of Time

When an investor has invested his money in unequal instalments over a period of time, then the compounded value or future value can be calculated as:

Mr. X invested ₹ 10,000, ₹ 5,000 and ₹ 2,000 at the starting of 1st, 2nd and 3rd year. Calculate the compound value of his investment at the end of 3rd year when interest is provided at the rate of 10 % compounded annually.

Solution: (a) Compounded value or FV of ₹ 10,000 invested for 3 years

FV = 10,000 (1 + 0.10) 3 = ₹ 13,310

Compounded value or FV of ₹ 5,000 invested for 2 years

FV = 5,000 (1 + 0.10) 2 = ₹ 6,050

Compounded value or FV of ₹ 2,000 invested for 1 years

FV = 2,000 (1 + 0.10) 1 = ₹ 2,200

Hence, total compounded value of Mr. X՚s investments will be

= 13,310 + 6,050 + 2,200 = ₹ 21,560

#### The Future Value of a Series of Equal Cash Flows over a Period of Time

Series of equal cash flows (either inflows or outflows) occurring over a period of equal time intervals is known as annuity. We can find the future value of an annuity by calculating the sum of future values of equal amount occurring over a period of equal time interval as follows:

FVAn

Where, FVAn is the future value of an annuity.

A is annuity amount

r = rate of interest per time period

n = No. of time periods

Example: Mr. X has invested an amount of ₹ 10,000 each at the end of 1st, 2nd and 3rd year. Calculate the compound value of his investment at the end of 3rd year if interest is provided at the rate of 10 % compounded annually.

Solution: The compounded value of annuity will be:

FVAn

= 10,000 x 3.310

= ₹ 33,100

#### The Future Value of a Cash Flow when Compounding is Done More Than Once in a Year

In many cases, interest is compounded more than once in a year, say semi-annually, Quarterly, even monthly. Formula for calculating future value of a cash flow when compounding is done more than once in a year will be same i.e..

FV = P (1 + r) n or PV (1 + r) n

here, r = effective rate of interest per period. If r is given 10 % per annum, then for semi-annual compounding, effective rate of interest will be = 10/2 = 5 %

and n = no. of periods. If n is given 3 years, then for semi-annual compounding, no. of periods will be 3 x 2 = 6 years.

Example: Mr. X has invested ₹ 10,000 now for 3 years at the interest rate of 12 % per annum compounded quarterly. Find the amount he will get after 3 years.

Solution: We know, FV = PV (1 + r) n

where, P = Principal = ₹ 10,000

r = effective rate of interest per quarter = 12/4 = 3 %

n = No. of quarters = 3 x 4 = 12

Hence, FV = 10,000 (1 + 0.03) 12

= 10,000 x 1.426

= ₹ 14,260

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