# Compounding and Discounting

Money

In this tutorial we will learn about Compounding and Discounting.

## Important terms

Present Value = It is the value of a sum of money today.

Future Value = It is the value of a sum of money in the future.

Compounding = Finding the future value from present value.

Discounting = Finding the present value from future value.

## Compounding

Compounding helps us to find the future value of a present value (or amount) that is compounded for a given interest rate for a given number of years.

Let's say we have \$10,000 and we want to find its future value when the amount is invested for 10 years at 10% interest rate compounded annually. To calculate this we use Compounding.

Compounding uses Compound Interest concept.

## Types of Compounding

Mostly compounding is done annually. But here are some of the common compounding type.

• Interest compounded annually - This means we are calculating the interest once per year.
• Interest compounded half-yearly - This means we are calculating the interest twice per year.
• Interest compounded quarterly - This means we are calculating the interest 4 times per year.
• Interest compounded monthly - This means we are calculating the interest every month per year.

## Formula for Compounding annually

When we calculate interest once per year then it is called compounding annually.

The formula we use to calculate this is given below.

``````FV = PV(1 + r)n

Where,
FV = Future Value
PV = Present Value

r = Rate of interest
So, if rate is 10%
Then r = 10/100 = 0.1

n = Number of years
``````

### Example

Let's say we have \$10,000 today and we want to find the future value if the amount is invested for 10 years at 10% interest rate compounded annually.

``````We have

PV = 10000

r = 10%
= 10/100
= 0.1

n = 10

So, future value

FV = PV(1 + r)n
= 10000(1 + 0.1)10
= 25937.42
``````

So, after 10 years the future value will be \$25,937.42 when the amount is compounded annually.

## Generalised formula for Compounding

Following is the generalised formula for compounding.

``````FV = PV(1 + r/f)fn

Where,
FV = Future Value
PV = Present Value

f = Number of times interest is calculate per year
f = 1 for yearly
f = 2 for half yearly
f = 4 for quarterly
f = 12 for monthly

r = Rate of interest
So, if rate is 10%
Then r = 10/100 = 0.1

n = Number of years
``````

## Compounding half yearly

When we calculate interest twice per year then it is called compounding half yearly.

### Example

Let's say we have the same amount \$10,000 today and we want to find the future value if the amount is invested for 10 years at 10% interest rate compounded half yearly.

``````We have

PV = 10000

r = 10%
= 10/100
= 0.1

f = 2 (as we are compounding half yearly)

r/f = 0.1/2
= 0.05

n = 10

fn = 2x10
= 20

So, future value

FV = PV(1 + r/f)fn
= 10000(1 + 0.05)20
= 26532.98
``````

So, after 10 years the future value will be \$26,532.98 when the amount is compounded half yearly.

## Discounting

Discounting helps us to find the present value or present worth of money for a given future value (or amount).

Using discounting we can figure out the present value.

\$100 is worth more now then 10 years from now. This is because of inflation. Purchasing power of money falls as time passes by. We can buy more stuff today with \$100 then 10 years from now.

## Formula for Discounting annually

Following is the formula for annual discounting.

``````PV = FV/(1+r)n

Where,

PV = Present Value

FV = Future Value

r = Discount rate
So, if rate is 10%
Then r = 10/100 = 0.1

n = Number of years
``````

### Example

Let's calculate the present value if we are told that the discount rate is 10% and future value 10 years from now is \$50,000.

``````FV = 50000

r = 10%
= 10/100
= 0.1

n = 10

So, present value will be

PV = FV/(1+r)n
= 50000/(1+0.1)10
= 19277.16
``````

So, present value is \$19,277.16 for future value \$50,000 at 10% discount rate 10 years from now.

## Generalised formula for Discounting

Following is the generalised formula for discounting.

``````PV = FV/(1 + r/f)fn

Where,
FV = Future Value
PV = Present Value

f = Number of times interest is calculated per year
f = 1 for yearly
f = 2 for half yearly
f = 4 for quarterly
f = 12 for monthly

r = Discount rate
So, if rate is 10%
Then r = 10/100 = 0.1

n = Number of years
``````

Alright, this brings us to the end of this tutorial. Don't forget to share this tutorial. Thanks for reading. See you in the next tutorial. Have fun.