Introduction to Corporate Finance

3: The Time Value of Money

of an investment. Principal is the amount of money on which the interest is paid. To demonstrate, if the

investment in our previous example pays 5% simple interest, then the future value in any year equals

$100 plus the product of the annual interest payment and the number of years. In this case, its future

value would be $110 after two years [$100 + (2 × $5)], $115 in three years [$100 + (3 × $5)], $120 at

the end of the fourth year [$100 + (4 × $5)], and so on.

Compound interest is interest earned both on the initial principal and on the interest earned in previous

periods. To demonstrate compound interest, assume that you have the opportunity to deposit $100 into

an account paying 5% annual interest. After one year, your account will have a balance of $105. This sum

represents the initial principal of $100 plus 5% ($5) in interest. This future value is calculated as follows:

Future value after one year = $100 × (1 + 0.05) = $105

If you leave this money in the account for another year, the investment will pay interest at the rate of 5%

on the new principal of $105. In other words, you will receive 5% interest both on the initial principal of $100

and on the first year’s interest of $5. At the end of this second year, there will be $110.25 in your account,

representing the principal at the beginning of year 2 ($105) plus 5% of the $105, or $5.25, in interest.^1

The future value at the end of the second year is computed as follows:

Future value after two years = $105 × (1 + 0.05) = $110.25

Substituting the first equation into the second one yields the following:

Future value after two years = $100 × (1 + 0.05) × (1 + 0.05)

= $100 × (1 + 0.05)^2

= $110.25

Therefore, $100 deposited at 5% compound annual interest will be worth $110.25 at the end of two

years.

It is important to recognise the difference in future values resulting from compound versus simple

interest. Although the difference between the account balances for simple versus compound interest in

this example ($110 versus $110.25) seems rather trivial, the difference grows exponentially over time.

With simple interest, this account would have a balance of $250 after 30 years [$100 + (30 × $5)]; with

compound interest, the account balance after 30 years would be $432.19.

3-2b THE EQUATION FOR FUTURE VALUE

Because financial analysts routinely use compound interest, we generally use compound rather than

simple interest throughout this book. Equation 3.1 gives the general algebraic formula for calculating

the future value, at the end of n periods, of a lump sum invested today at an interest rate of r% per period:

Eq. 3.1 FV = PV × (1 + r)n

where

FV = future value of an investment,

PV = present value of an investment (the lump sum),

r = interest rate per period (typically 1 year),

n = number of periods (typically years) that the lump sum is invested.

1 Said differently, compound interest includes the beneficial effect of ‘earning interest on your interest’. In this example, during the second year

you earn $5 interest on your initial $100 principal, plus you earn another $0.25 interest on the interest you earned (and saved) the first period.

In total, in the second period you earn $5.25 in interest ($5 + $0.25).

compound interest

Interest earned both on the

initial principal and on the

interest earned in previous

periods

LO3.1

principal

The amount of money

borrowed on which interest

is paid