Lecture4 TVM
Are you indifferent between receiving $1,000 today and receiving $1,000 one year from today? If your
intuition prefers receiving the funds today, rather than one year from today, then your intuition recognizes the
time value of money. Owners of cash can permit borrowers to rent the use of their cash.
Interest
is payment
for the use of cash.
Expenditures for an investment most often precede the receipts produced by that investment. Cash received
later has less value than cash received sooner. The difference in timing affects whether making an investment
will earn a profit. Amounts of cash received at different times have different values. We use interest
calculations to make valid comparisons among amounts of cash paid or received at different times.
CONCEPTS
Businesses typically state interest cost as a percentage of the amount borrowed per unit of time. Examples are
12 percent per year and 1 percent per month. When the statement of interest cost includes no time period, then
the rate applies to a year; thus "interest at the rate of 12 percent" means 12 percent per year.
The amount borrowed or loaned is the
principal
.
Compound interest
means that the amount of interest earned
during a period increases the principal, which is then larger for the next interest period.
If you deposit $1,000 in a savings account that pays compound interest at the rate of 6 percent per year, you
will earn $60 by the end of one year. If you do not withdraw the $60, then $1,060 will earn interest during the
second year. During the second year your principal of $1,060 will earn $63.60 interest; $60 on the initial
deposit of $1,000 and $3.60 on the $60 earned the first year. By the end of the second year, you will have
$1,123.60. Compounded annually at 8 percent, cash doubles itself in nine years. If a twentyfiveyear old
invests $2,000 each year which earns 8 percent a year, the retirement fund will grow to more than $425,000 by
the time that person reaches age sixtyfive.
When only the original principal earns interest during the entire life of the loan, the interest due at the time the
borrower repays the loan is
simple interest
. Simple interest calculations ignore interest on previously earned
interest. Nearly all economic calculations, however, involve compound interest.
Problems involving the time value of money generally fall into two groups:
1.
We want to know the
future value
of cash invested or loaned today.
2.
We want to know the
present value
, or today's value, of cash to be received or paid at later dates.
FUTURE VALUE
If you invest $1 today at 12 percent compounded annually, it will grow to $1.12000 at the end of one year,
$1.25440 at the end of two years, $1.40493 at the end of three years, and so on, according to the formula:
Fn;
=
P
(1 +
r
)
n
where
F
n
= accumulation or future value
P
= onetime investment today
r
= interest rate per period
n
= number of periods from today
The amount
F
n
is the
future value
of the present payment,
P
, compounded at
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 Spring '11
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