Fixed Income Securities by Sunil Kumar Parameswaran

1 A Primer on the Time Value of Money
All of us have either paid and/or received interest at some point in our lives. Those of us who have
taken educational, housing, or automobile loans have paid interest to the lending institution. On
the other hand, those of us who have deposited funds with a bank in the form of a savings
account or a time deposit have received interest. The same holds true for people who have
bought bonds or debentures.
Interest may be defined as the compensation paid by the borrower of capital to the lender, for
permitting him to use his funds. An economist would define interest as the rent paid by the
borrower of capital to the lender, to compensate him for the loss of the opportunity to use the
funds when it is on loan. After all when we decide not to live in an apartment or house owned by
us, we typically let it out to a tenant. The tenant will pay us a monthly rental because as long as he
is occupying our property, we are deprived of an opportunity to use it ourselves. The same
principle is involved when it comes to a loan of funds. The difference is that the compensation in
the case of property is termed as rent, whereas when it comes to capital, we term it as interest.
Nominal and Effective Rates of Interest
The quoted rate of interest per period is called the nominal rate of interest. The nominal rate is
usually quoted on a per annum basis. The effective rate of interest may be defined as the interest
that a unit of currency invested at the beginning of a year would have earned by the end of the
year. Quite obviously the effective rate will be equal to the quoted or nominal rate if interest is
compounded once per annum. However, if the interest is compounded more frequently, then the
effective rate will exceed the nominal rate of interest. The term “effective” connotes that
compounding at the stated frequency using the quoted nominal rate, is equivalent to
compounding once a year at the effective rate of interest.
Variables and Terms to Be Used and the Corresponding Symbols
P ≡ amount of principal that is invested at the outset
N ≡ number of periods for which the investment is being made. It may be in terms of years, or
in terms of smaller intervals of time, such as a half-year or a quarter.
r ≡ nominal rate of interest per annum
i ≡ effective rate of interest per annum
m ≡ number of times interest is compounded per annum
P.V. ≡ present value of a stream of cash flows
F.V. ≡ future value of a stream of cash flows
The Concept of Simple Interest
Consider the case of an investment of $P that has been made for N years. According to the
principle of simple interest, the interest that will be earned every period is a constant. In other
words, interest is computed every period and credited only on the original principal, and no
interest is payable on any interest that has been accumulated at an intermediate stage.
If r is the quoted rate of interest per year, then an investment of $P will become dollars 
P(1 + r) after one year. In the second year, interest will be paid only on P and not on P(1 + r).
Consequently the accumulated value after two years will be P(1 + 2r). Extending the logic, the
terminal balance will be P(1 + rN). N need not be an integer: that is, investments may be made
for a fraction of a year.

Example 1.1.
Maureen Chen deposited $10,000 with First National Bank for a period of three years. If the
deposit earns simple interest at the rate of 10% per annum, how much will she have at the
end?
An investment of $10,000 will become

10, 000 × 1.10 = $11, 000

after one year. During the second year, only the original principal of $10,000 will earn
interest and not the accumulated value of $11,000. Consequently the accumulated value
after two years will be

10, 000 × 1.10 + 1, 000 = $12, 000

By the same logic the terminal balance after three years will be $13,000.

13, 000 = 10, 000 × (1 + 0.10 × 3) ≡ P(1 + rN)

Example 1.2.
Andrew Gordon deposited $10,000 with ABC Bank five years and six months ago, and wants
to withdraw the balance now. If the bank pays 8% interest per annum on a simple interest
basis, how much is he eligible to withdraw?

P(1 + rN) = 10, 000 × (1 + 0.08 × 5.5) = $14, 400

Thus Andrew can withdraw $14,400.

The Concept of Compound Interest
Consider the case of an investment of $P that has been made for N years. Assume that the
interest is compounded once per annum: that is, the quoted rate is equal to the effective rate.
According to the principle of compound interest, every time interest is earned it is
automatically reinvested at the same rate for the next period. Thus, the interest earned every year
will not be a constant like in the case of simple interest, but will steadily increase. In this case an
original investment of $P will become P(1 + r) dollars after one year. The difference is that
during the second year the entire amount will earn interest and consequently the balance at the
end of two years will be P(1 + r)2
. Extending the logic, the balance after N years will be 
P(1 + r)N
. Once again N need not be an integer.
Example 1.3.
Assume that Maureen Chen has deposited $10,000 with First National Bank for three years,
and that the bank pays 10% interest per annum compounded annually. How much will she

Fixed Income Securities by Sunil Kumar Parameswaran