Compounding More Than Once A Year

Compounding More Than Once A Year                       

To begin, suppose that interest is paid semiannually. If you deposit 100 dollars in a saving account at a nominal 8 percent annual nominal interest rate, the future value at the end of six months would be:

FV0.5= $100[1 + (0.08/2)]

FV0.5= $104

Where FV0.5 = Future value after half year or after six months

And so interest is also divided in half i.e. 0.08/2 = 0.04

Nominal Interest Rate:

A rate of interest that has not been adjusted for frequency of compounding.

So it means if interest is compounding more than once year the effective interest rate will be higher than the nominal interest rate.

Now let’s see

We take data from our previous example

So if we calculate compounding once in a year our future value would be:

FV1 = $100(1 + 0.08) power 1

FV1 =$108

(Calculate Future Value of your amount here)

Now If we calculate compounding more than once in a year our future value would be:

FV1= $100[1 + (0.08/2)] power 2

FV1=108.16

The difference $0.16 is caused by interest being earned at semi annual basis.

We have learnt one thing from here:

"The more times interest is compounded in the year, the greater the future value will be"

Continuous Compounding

The number of times a year that interest is compounded, approaches infinity (∞)

So Future Value of continuous compounding amount would be:

FV = PV (e) power (in)

Where e is approximately 2.71828

And in= interest and number of years

For example, the future value of a $100 deposit at the end of three years with continuous compounding at 5 percent would be:

FV3= $100(2.71828) power (0.05)(3)

FV3= $116.18

Effective Annual Interest Rate

It is the actual rate of interest earned after adjusting the nominal rate for number of compounding periods per year.

So

Effective annual interest rate = (1 + [i/m]) power m -1

Where m denotes the compounding number of periods

For example

If a saving plan offered a nominal interest rate of 5 percent compounded quarterly

Then the effective annual interest would be:

[{1 + (0.05/4)} power 4]  -1

Effective annual interest rate =   0.051

Here we have learnt one thing that,

“The more the compounding periods, the greater the future values of the deposit athe greater the effective annual interest rate”.