Page 474 - Introduction to Business

448 PART 5 Finance

Technology and Business

The Time Value of Money: How Money Grows

Money declines in value over time. earned interest on interest. This compound interest is
Simply put, a dollar next year is not otherwise known as the force of interest.
worth as much as a dollar today. The To understand more about compound interest,
reason is that the prices of goods and services gener- imagine that we extended the loan agreement to
ally rise over time due to inflation. Higher prices 20 years. Using the same example,
mean that your purchasing power declines over time.
Future value $100 (1 r) 20
Returning to our example, a business borrows $100
from a bank at a 10 percent rate of interest. The $100 (1 0.10) 20
future value of the $100 is $100 (1 0.10) $110. $672.75
What if you wanted to know the future value of
which is the future value of $100 in 20 years using a
$100 two years from now. Let’s assume that the risk-
10 percent rate of return. Given that the initial princi-
less rate of interest and default premium stay the
pal was $100, the total interest paid is $572.75. If only
same in both the first and second years. So the nomi-
10 percent per year were earned on the $100 princi-
nal rate of interest is 10 percent. In the first year the
pal value of the loan each year, the total interest
bank makes a loan at the beginning of the year for
would have been only $200 ($10 per year 20 years).
$100 and is repaid $110 at the end of the year.
The difference between $572.75 and $200 is the com-
Assume further that the bank takes this $110 and
pound interest earned on the loan. In other words,
lends it out to another firm in the second year. At the
$372.75 was earned simply due to the force of inter-
end of the second year the bank would have $110
est. This example is used to construct Exhibit 13.5,
(0.10 $110) $110 $11 $121. After two years
which shows how interest grows over time.
the initial investment of $100 has grown to a value of
Compound interest causes borrowers to pay more
$121. We can write this example in mathematical
in interest payments than otherwise. Alternatively,
terms.
lenders that invest small amounts today can realize
$100 (1 0.10) (1 0.10) $121 large total returns in the future. This fact is an impor-
tant reason why investing money today can drasti-
or, more generally,
cally change future consumption. People invest
Present value (1 r) (1 r) future value money today by forgoing consumption now. In doing
so, they gain higher future levels of consumption
Dividing both sides by (1 r) , we can see that
2
from the force of interest.
Present value future value/(1 r) 2
That is, Questions
$100 $121/(1.10) 2 1. Assume that you borrowed $100 from a bank at a
rate of interest of 6 percent for two years. What
$121/1.21
will be the total amount that you will pay back to
An interesting result with regard to the above the bank?
loan is that we have more interest on the loan than 2. In the previous example, how much interest will
10 percent per year. If we made 10 percent in year you pay and how much of this interest payment is
one and 10 percent in year two, the total interest compound interest?
would be $10 $10 $20. But the total profit is $21, 3. What if you borrowed the $100 for five years
not $20. What happened? How did the bank earn an instead of only two years. Now what answers do
extra $1? Looking at the equations, we see that this you get for total amount paid, total interest paid,
extra $1 is the 10 percent interest earned in year two and amount of compound interest?
on the $10 interest earned in year one. The bank has

So, we would get a nominal interest rate of 10 percent (2 3 5 percent). The
total payment of principal and interest would rise from $105 to $110 [$100 (0.10
$100) $100 $10] due to default risk. If there were no default risk, the nominal

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