Present Value
Suppose that someone will give you a gift of $100, and
will give it to you either now or in four years. Which is
better, the money now or the money four years from now? The
rule that gifts with restrictions are
of less value than gifts without restrictions suggests
that money now is worth more than money in the future.
Anything that one can do with the gift of $100 four years
from now one can do with $100 now simply by saving it for
four years. But there are many things that one can do with
money now that one cannot do with money four years from now.
Therefore, $100 promised four years from now is not worth
$100 right now, but a smaller amount.
One of the things that can be done with money now is to
invest it so that it will earn interest. Because this cannot
be done with money four years from now, this option of
foregone interest is a cost of waiting for the money. When
this cost is measured, one sees the amount by which money in
the future must be discounted to obtain its
present value.
If the interest rate is 10%, $100 now can be turned into
$110 one year from now. Thus, $100 now and $110 a year from
now have the same value. (You may have to think about this
for a while.) This simple idea is vital in business and
governmental decisions because a great many decisions have
costs and benefits spread over time, and it is often
necessary to compare sums in different time periods.
Computing the present value of future sums is nothing
more than working compound interest problems backward. The
formula for finding the future value of a present sum after
one period is
(1) P + Pr = F
or
(2) P(1 + r) = F
where P is the present sum, r is the interest rate in
decimal form, and F is the future sum. (Try the formula for
P = $100 and r = .10. You should get F = $110.)
After two years the amount of money will be
(3) F1(1 + r) = F2
where F1 is the amount of money one year from now, and F2
is the amount of money two years from now. This may be
rewritten as
(4) (P(1 + r))(1 + r) = P(1 + r)^{2} = F2.
(Try this formula for P = $100 and r = .10. F2 should be
$121.) Using the same logic, the future value three years
from now will be
(5) P(1 + r)^{3} = F3
and for any arbitrary period n, it will be
(6) P(1 + r)^{n} = Fn.
Simple algebra allows us to solve this equation if we
have the time period and two of the three remaining
variables. (Logarithms help a lot if we are solving for r.)
In particular, if we have a future sum of money and want to
find its present value, the last equation can be rewritten
as
(7) P = F / (1 + r)^{n}
Using this formula in our case of $100 four years from
now and an interest rate of 10%, the present value is
P = 100/(1.10)^{4} = 100/1.464 = 68.30
This means that $68.30 invested at 10% will grow to $100
four years from now. Therefore, $68.30 now and $100 four
years in the future have equivalent value if the interest
rate is 10%.
Present value analysis explains bond
prices.
Copyright
Robert Schenk
