International Finance: Putting Theory Into Practice

154 CHAPTER 4. UNDERSTANDING FORWARD EXCHANGE RATES FOR CURRENCY

Example 4.22

Let (T−t) =1/2, and the compound interest rate 10.25 percentp.a.; then

1 +rt,T= 1. 10251 /^2 =

√

1 .1025 = 1. 05. (4.34)

Compound interest is the standard method for zero-coupon loans and invest-

ments (without interim interest payments) exceeding one year.

3. Banks may also compound the interest every quarter, every month, or even
   every day. The result is an odd mixture of linear and exponential methods. If
   the interest rate for a six-month investment isip.a., compoundedmtimes per
   year, the bank awards youi/mper subperiod of 1/myear. For instance, the
   p.a.interest rate may bei= 6 percent, compounded four times per year. This
   means you get 6/4 = 1.5 percent per quarter. Your investment has a maturity
   of six months, which corresponds to two capitalization periods of one quarter
   each. After compounding over these two quarters, an initial investment of
   100 grows to 100×(1.015)^2 = 103.0225, implying an effective rate of return
   of 3.0225 percent. Thus, the effective return is computed from the quoted
   interest rate as:

1 +rt,T=

(

1 +

[quoted interest rate]

m

)(T−t)m

. (4.35)

Example 4.23

Let (T−t) =1/2, and the compound interest rate 9.878% with quarterly

compounding; then

1 +rt,T= (1 + 0. 098781 /4)^1 /^2 ×^4 = 1. 05. (4.36)

You may wonder why this byzantine mixture of linear and exponential is used

at all. In the real world it is used when the bank has a good reason to

understate the effective interest rate. This is generally the case for loans. For

deposits, the reason may be that the quoted rate is capped (by law, like the

U.S.’ former Regulations Q and M; or because of a cartel agreement amongst

banks). In finance theory, the mixture of linear and exponential is popular in

its limit form, the continuously compounded rate:

4. In the theoretical literature, the frequency of compounding is often carried to
   the limit (“continuous compounding”,i.e. m→ ∞). From your basic math
   course, you may remember that:

lim

m→∞

(1 +x/m)m=ex, (4.37)