Compound Interest

Compound interest includes interest earned on the interest that was previously accumulated.

Compare, for example, a bond paying 6 percent semiannually (that is, coupons of 3 percent twice a year) with a certificate of deposit (GIC) that pays 6 percent interest once a year. The total interest payment is $6 per $100 par value in both cases, but the holder of the semiannual bond receives half the $6 per year after only 6 months (time preference), and so has the opportunity to reinvest the first $3 coupon payment after the first 6 months, and earn additional interest.

For example, suppose an investor buys $10,000 par value of a US dollar bond, which pays coupons twice a year, and that the bond's simple annual coupon rate is 6 percent per year. This means that every 6 months, the issuer pays the holder of the bond a coupon of 3 dollars per 100 dollars par value. At the end of 6 months, the issuer pays the holder:

${\frac {r\cdot B\cdot m}{n}}={\frac {6\%\times \$10\,000\times 1}{2}}=\$300$

Assuming the market price of the bond is 100, so it is trading at par value, suppose further that the holder immediately reinvests the coupon by spending it on another $300 par value of the bond. In total, the investor therefore now holds:

$\$10\,000+\$300=\left(1+{\frac {r}{n}}\right)\cdot B=\left(1+{\frac {6\%}{2}}\right)\times \$10\,000$

and so earns a coupon at the end of the next 6 months of:

${\begin{aligned}{\frac {r\cdot B\cdot m}{n}}&={\frac {6\%\times \left(\$10\,000+\$300\right)}{2}}\\&={\frac {6\%\times \left(1+{\frac {6\%}{2}}\right)\times \$10\,000}{2}}\\&=\$309\end{aligned}}$

Assuming the bond remains priced at par, the investor accumulates at the end of a full 12 months a total value of:

${\begin{aligned}\$10,000+\$300+\$309&=\$10\,000+{\frac {6\%\times \$10,000}{2}}+{\frac {6\%\times \left(1+{\frac {6\%}{2}}\right)\times \$10\,000}{2}}\\&=\$10\,000\times \left(1+{\frac {6\%}{2}}\right)^{2}\end{aligned}}$

and the investor earned in total:

${\begin{aligned}\$10\,000\times \left(1+{\frac {6\%}{2}}\right)^{2}-\$10\,000\\=\$10\,000\times \left(\left(1+{\frac {6\%}{2}}\right)^{2}-1\right)\end{aligned}}$

The formula for the annual equivalent compound interest rate is:

$\left(1+{\frac {r}{n}}\right)^{n}-1$

where

r is the simple annual rate of interest

n is the frequency of applying interest

For example, in the case of a 6% simple annual rate, the annual equivalent compound rate is:

$\left(1+{\frac {6\%}{2}}\right)^{2}-1=1.03^{2}-1=6.09\%$

Resources

• Simple and Compound Interest, Book Chapter
• Guided Notes