Introduction to Corporate Finance

3: The Time Value of Money

As you should expect by now, the more frequently that interest compounds, the greater the amount

of money that accumulates.

A General Equation

We can generalise the preceding examples in a simple equation. Suppose that a lump sum, denoted by

PV, is invested at r% for n years. If m equals the number of times per year that interest compounds, the

future value grows as shown in the following equation:

Eq. 3.12 FV PV

r

m

=1+

mn

×









×

Notice that if m = 1, this reduces to Equation 3.1. The next several examples verify that this

equation yields the same ending account values after two years, as shown in Tables 3.1 and 3.2.

Continuous Compounding

As we switch from annual, to semiannual, to quarterly compounding, the interval during which interest

compounds gets shorter, while the number of compounding periods per year gets larger. Theoretically,

there is almost no limit to this process – interest could be compounded daily, hourly or second by second.

Continuous compounding, the most extreme case, occurs when interest compounds literally at every moment

as time passes. In this case, m in Equation 3.12 would approach infinity, and Equation 3.12 converges

to this expression:

Eq. 3.13 FV(continuous compounding) = PV × (er × n)

The number e is an irrational number, like the number π from geometry, which is useful in

mathematical applications involving quantities that grow continuously over time. The value of e is

approximately 2.7183. As before, increasing the frequency of compounding, in this case by compounding

as frequently as possible, increases the future value of an investment.

continuous compounding

Interest compounds literally at

every moment as time passes

We have calculated the amount that you would have
at the end of two years if you deposited $100 at 8%
interest compounded semiannually and quarterly.
For semiannual compounding, m = 2 in Equation
3.12; for quarterly compounding, m = 4. Substituting
the appropriate values for semiannual and quarterly
compounding into Equation 3.12 yields the following
results.

For semiannual compounding:

FV=×+











 =×+=

×

$

.

100 1 $(.)$

008

2

100 1004 11

22

(^4669). 9
For quarterly compounding:

FV=×+











 =×+=

×

$

.

100 1 $(.)$

008

2

100 1002 11

42

(^4771). 7
example