# Time Value of Money (Part – 2) : Discounting or Present Value Technique

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## Discounting or Present Value Technique

Discounting is the process of finding the present value of a cash flow or a series of cash slows; discounting is reverse of compounding. Present value of future cash flow is calculated by multiplying it with a discounting factor.

**Discounting technique can be explained for calculating**:

- Present value of a future sum.
- Present value of a series of unequal cash flows.
- Present value of a series of equal cash flows.

### Present Value of a Future Sum

Present value of the money received in future will be less than the value of same money in hand today. This is because the money in hand can be invested and its absolute value can be increased at a future date. Formula for calculating present value of a future sum is:

PV

here, PV = Present value

FV = Future value of Cash Flow

r = Rate of interest

n = No. of years or periods

= present value interest factor (PVIF)

**Example**: Mr. X is supposed to get ₹ 10,000 after 3 years. Let the existing rate of interest is 10 % , find the present value of this sum.

**Solution**: We know, PV

Where, FV = ₹ 10,000

r = 10 % p. a.

n = 3 years

Hence, PV

= 10,000 x 0.7513

= ₹ 7,513 (approx.)

### Present Value of a Series of Unequal Cash Flows

In most of the proposals, it is observed that unequal return from these are spread over a number of years. The Present value of a series of unequal cash flows can be calculated by using the following formula:

PV … or

Where, PV = Sum of the present value of each future cashflows

A_{1,} A_{2,} A_{3} = Cash flows occurring after period 1,2, 3 …

r = Rate of interest or discounting rate per period

n = No. of years or periods

**Example**: Mr. X has to receive ₹ 10,000, ₹ 5,000 and ₹ 2,000 at the end of 1^{st}, 2^{nd} and 3^{rd} year from the investment proposal. Calculate the present value of his future cash flows from this investment proposals if rate of interest is 10 % per annum.

**Solution**: Formula for calculating the present value of a series of unequal cash flows is:

PV

Here, = ₹ 10,000 , = ₹ 5,000 , = ₹ 2,000

r = 10 %

So, PV

= 9,090 + 4,130 + 1,502

= ₹ 14,722

### Present Value of a Series of Equal Cash Flows (Annuity)

When a business or individual gets constant cash inflows over a period of equal time intervals, such cash inflows are known as an annuity. We can find the present value of an annuity (PVA) by calculating the sum of present values of equal cash inflow occurring over a period of equal time intervals.

Formal for calculating present value of an annuity (PVA_{n}) is:

Where, A is annuity amount

r = rate of interest per time period

n = No. of time periods

**Example**: Mr. X wishes to calculate the present value of the annuity consisting of a cash inflow of ₹ 10,000 per year for 3 years. The rate of interest he can earn is 10 % .

**Solution**: We know,

PVA_{n}

= 10,000 X 2.487

= ₹ 24,870

## Present Value of a Perpetuity

A perpetuity is defined as an infinite series of equal cash flows occurring at regular time intervals. The perpetuity concept is quite useful in finding out the sum of money to be set aside for giving scholarships or awards of equal amount year after year in an educational institution. This type of annuity continues for an infinite period of time.

Formula for calculating present value of a perpetuity is:

PV_{p}

Where, PV_{p} = Present value of a perpetuity,

A = Amount or equal cash flow per period

r = rate of interest per period

= PVIF of perpetuity

## Future Value and Present Value of an Annuity Due

Till now we have calculated the FV and PV of an ordinary annuity on the assumption that equal cash flows occur at the end of each period. However, if equal cash flows occur at the starting of each period, then such an annuity is called as annuity due.

**Formula for calculating FV and PV of an annuity due are as follows**:

### Future Value of an Annuity Due (FVA Due)

When equal amount is invested at the beginning of each period, such an annuity is known as annuity due. Formula for calculating FVA due is:

FVA due x (1 + r) or

A x FVIFA x (1 + r)

Where, FVIFA Future value interest factor of an annuity for given r and n

r rate of interest per time period

n No. of years

A is annuity amount

### Present Value of an Annuity Due

When equal amount is received at the beginning of each period, such an annuity is also known as annuity due. Since cash inflows in annuity due occur one period in advance, the present value of an annuity due can be calculated by using the following formula.

PVA due x (1 + r) or

PVA due = A x PVIFA x (1 + r)

Where, PVIFA = Future value interest factor of an annuity for given r and

n or (discounting factor of an annuity at a given value of r

and n

r = Rate of interest

A is annuity amount

## Perpetuity Growing at a Constant Rate

An infinite series of periodic cash flows which grow at a constant rate per period is known as growing perpetuity. Present value of growing perpetuity can be calculated as follows:

PV

Where, CF_{1} = Cash flow (either inflow or outflow) at the end of 1^{st}

period

r = Rate of interest

g = Constant rate of growth in perpetuity amount per period

However, this formula cannot be used when g > r.

**Example**: A college wants to award a scholarship of ₹ 1,000 per year to its meritorious students. The first scholarship will be awarded after one year and the amount of scholarship will increase at a constant rate of 5 % every year to offset the inflation. Find the value of this scholarship if rate of interest is 10 % .

**Solution**: Present value of growing perpetuity =

**=**

**=** **=** ₹ 20,000

## Present Value of an Annuity Growing at a Constant Rate

A finite series of periodic cash flows which grow at a constant rate per period is known as growing annuity. Present value of a growing annuity can be calculated as follows:

PV ^{n}

Where, CF_{1} = Cash flow at the end of period one

r = Rate of interest per period

g = Rate of growth in annuity amount per period

n = No. of periods in annuity

However, this formula cannot be used if r = g, when r = g, then PV of a growing annuity can be calculated as follows:

PV = CF_{1} x

## Calculation of Doubling Period

Investor generally wants to know the period in which their investment becomes double. There is a rule of thumb for this purpose. This rule is known as rule of 72. According to this rule the doubling period is obtained by dividing 72 by the interest rate.

Hence, Doubling period = Where, r = Rate of interest

There is another more accurate rule of thumb known as rule of 69 for calculating doubling period. According to this rule, doubling period can be calculated as follows:

Doubling period = 0.35 +

-Manishika