Let's look at the example a little differently.

The bank pays an interest of 10% a year, but computes interest every six months.

This means that the bank will pay you an interest of 5% every six months

on your deposit of $100.

So if you deposit a $100 today,

your balance will increase to $105 after six months.

The $105 will earn interest at 5% over the next six months.

Interest earned, it will be $105 x 0.05 which equals $5.25.

After one year, you will have $105 + $5.25 which equals $110.25.

This is $0.25 more than in the previous example where interest

was compounded only once a year.

The difference is because of compounding interest more frequently.

The more frequently interest is compounded,

the higher is the future value of your investment.

Conversely, the more frequently compounding is done,

the lower is the present value of your investment.

We can modify all the present and future value formulas we saw earlier

by replacing r with r/m,

where m is the number of times interest is compounded each year.

And now n is redefined as the total number of payments made in an annuity or

the total number of periods over which the cashflow is discounted.

One interesting thing to note in our example is that when the bank paid

interest annually, your $100 became $110, which is a return of

$110 minus $100 divided by $100, which is a 10% return.

But when it paid interest every six months, your $100 became

$110.25 after a year, which is a return of $100.25

minus $100 divided by $100, which equals 10.25%.

This 10.25% is the effective annual rate, EAR, of your investment.

The following formula relates EAR to the interest rate R.

(EAR) = (1 + r/m), the whole thing raised to the power of m- 1.

With annual compounding, EAR = r.

But for all other compounding frequencies, EAR will be greater than r.

r is usually referred to as the Annual Percentage Rate, APR.