WWWFinance -- Compounding

*Copyright 1996 by Campbell R. Harvey and Stephen Gray.
All rights reserved. No part of this lecture may be reproduced without
the permission of the author.*

Latest Revision: August 1996.

As we saw, for a given nominal rate, the more compounding periods per year, the higher the effective annual rate. When comparing the rates of alternative investments, it is important to look at the effective annual rates rather than the nominal rates.

Different investments use different conventions for reporting the rate.
In the U.S., most bonds are reported as a semiannual yield. This means
that they assume two compounding periods per year. A bond with a semiannual
yield of 10% will have an annual effective rate of ** (1+.10/2)^{2}-1**
or 10.25%

Not all bonds pay twice per year. Eurobonds (bonds issued outside the
U.S. which pay in U.S. dollars) usually pay interest once per year. The
yield of these bonds are sometimes expressed as an annual yield, and sometimes
as a semiannual yield. The reason for expressing it as a semiannual yield
is so that it can be compared to yields on other bonds without having to
calculate an effective annual yield on those bonds. Semiannual rates are
sometimes referred to as *Bond Equivalent Yields*.

Similarly, bonds backed by home mortgages pay 12 times per year. The nominal rate is sometimes expressed as a mortgage yield -- which assumes 12 compounding periods per year, and sometimes as a semiannual yield.

Perhaps the strangest convention for converting nominal rates is the
one used for bank deposits. They compound interest on a daily basis. For
a non leap year, interest is compounded 365 times. However, when determining
the periodic rate, the nominal rate is divided by 360, rather than 365.
So, if money is deposited in an account which earns 4% per year, the nominal
rate will be ** 1.04^{365/360} - 1 = 4.0567%**. This method
gives a nominal rate which is higher than continuous compounding!