# Compounding and Interest Rate

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 1.092 = \$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

## One Reply to “Compounding and Interest Rate”

1. zvodretiluret says: Reply

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