# 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 2^{nd} 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 1^{st}, 2^{nd} and 3^{rd} year. Calculate the compound value of his investment at the end of 3^{rd} 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:

FVA_{n}

Where, FVA_{n} 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 1^{st}, 2^{nd} and 3^{rd} year. Calculate the compound value of his investment at the end of 3^{rd} year if interest is provided at the rate of 10 % compounded annually.

Solution: The compounded value of annuity will be:

FVA_{n}

= 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

-Manishika