For you who study about finance should have known well about compounding. What is compounding? For an example, Jane make a loan $100 with the interest (r) 9%. At the end of the first year, Jane owes the lender:

$100 x (1 + r) = $1 x 1.09 = $109

Also, at the end of the first year, the lender have two choice, to take that $109 and spend it or invest it again for next 1 year. If the lender then choose to invest it again, it is called **compounding**. To illustrate about compounding, by the end of second year that $1 will be:

$100 x (1 + r) x (1 + r) = $100 x (1 + 0.09) x (1 + 0.09)

$100 x 1.09^{2} = $100 x 1.1881 = $118.81

If the cash is invested at compound interest, then each interest payment is reinvested. Try to change that $100 to $1000 or bigger, that compound interest can add up bigger dollar amounts.

The equation above applies to compounding annually (1x in a year). What if the compounding is semi annually? For example, Jane owes the lender $100 compounding semi-annually for two years with the same interest 9% per year:

= $100 x [1 + (9%/2)]^{^2×2}

= $100 x (1,045)^{^4}

= $100 x 1.1925

= $119.25

What if compounding quarterly? Quarterly means compounding 4 times/year.

= $100 x [1 + (9%/4)]^{^2×4}

= $100 x (1,0225)^{^8}

= $100 x 1.1948

= $119.48

What if compounding monthly? Monthly means compounding 12 times/year.

= $100 x [1 + (9%/12)]^{^2×12}

= $100 x (1,0075)^{^24}

= $100 x 1.1964

= $119.64

Jane will paid more for more compounding.

What about continous compounding? Suppose Jane invested $1,000 at a continuously compounding rate of 10% for one year. What is the value at the end of the first year?

$1,000 x e^{.010} = $1,000 x 1.1052 =$1,105.20

source: Corporate Finance 10th edition by Ross, Westerfield, Jaffe

What if Jane invested $1,000 at a continuously compounding rate of 10% for two year?

$1,000 x e^{.010 x 2} = $1,221.40

What if Jane will get $1,000 at the end of four years with continuously compounding rate of interest 8%?

$1,000 x [1 / (e^{.08 x 4}) = $1,000 x (1 / 1.3771) = $726.16

To better understand about the difference between annual, semi annual, or continuous compounding, see the graph below:

source: Corporate Finance 10th edition by Ross, Westerfield, Jaffe

Some really interesting info , well written and broadly speaking user genial.