Chapter 2
FixedIncome Securities and
Interest Rates
We now begin a systematic study of fixedincome securities and interest rates. By a
fixedincome security
we mean a financial instrument that promises fixed (or definite)
payments at prescribed future dates. In some cases only one payment is made at
maturity; in other cases, periodic payments are made. The payments are usually
characterized by an
interest rate
which, in some sense, is simply a rental rate for
money.
In practice, there is sometimes ambiguity regarding what
is
or
is not
a fixed
income security because the payments for some securities that are classified as being
fixedincome are actually tied to quantities such as (variable) interest rates and credit
ratings that can fluctuate, and consequently the payments are not known exactly at
the time the security is purchased. In this chapter we focus primarily on situations
where payments and interest rates are described precisely in advance and we assume
that there is no risk of default. We assume first that interest rates are constant; in
particular that they are independent of both the date on which the investment is
made and the length (or term) of the investment.
2.1
Basic Interest Rate Mechanics
Interest may be either
simple
or
compound
. With simple interest, only the original
principal accrues interest and no interest is paid on the interest. On the other hand,
with compound interest, interest is paid on the previously accrued interest as well as
on the principal. Unless stated otherwise, the time
t
is measured in years and interest
rates are given as annualized rates.
In order to describe the basic mechanics of interest calculations, we shall assume
that an initial amount
A >
0 is invested at time 0, that no additional investments (or
withdrawals) are made, and that the annual interest rate is
r
≥
0. We are interested
in the value
V
t
of the investment at future times
t
. Analogous formulas apply to the
case of a loan. (Indeed, a loan can be considered as an investment of an amount
A <
0.)
33
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2.1.1
Simple Interest
The interest earned up to time
t
is simply equal to
rtA
. (This is just the familiar
formula “Interest = (Principal)
×
(rate)
×
(time)”.) Therefore, we have
V
t
=
A
+
rtA
=
A
(1 +
rt
)
.
Notice that with simple interest the value of the account grows linearly with time. It
is sometimes convenient to express the simpleinterest rule as a differential equation
together with an initial condition, namely,
d
dt
V
t
=
rA
;
V
0
=
A.
In practice, the simple interest convention is used only for investments or loans of
relatively short maturity (one year or less).
Example 2.1.
Suppose that you invest $100 in an account that pays 8% simple
interest.
(a) At the end of three months, you will have
V
.
25
= $100(1 + (
.
08)
×
.
25) = $102
in the account.
(b) At the end of one year you will have
V
1
= $100(1 + (
.
08)
×
1) = $108
in the account.
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 Spring '10
 SCHAFFER
 Interest Rates, The Land

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