Page 281 - Krugmans Economics for AP Text Book_Neat

Rearranging Equation 24-2 to solve for $X, the amount you need today in order to re-
The present value of $1 realized one
ceive $1 one year from now is:
year from now is equal to $1/(1 + r ):
the amount of money you must lend out
(24-3) Amount lent today in order to receive $1 one year from now = today in order to have $1 in one year. It
$X = $1/(1 + r) is the value to you today of $1 realized
one year from now. Section 5 The Financial Sector
This means that you would be willing to accept today the amount $X defined by
Equation 24-3 for every $1 to be paid to you one year from now. The reason is that if
you were to lend out $X today, you would be assured of receiving $1 one year from now.
Returning to our original question, this also means that if someone promises to pay
you a sum of money one year in the future, you are willing to accept $X today in place
of every $1 to be paid one year from now.
Now let’s solve Equation 24-3 for the value of $X. To do this we simply need to use the
actual value of r (a value determined by the financial markets). Let’s assume that the ac-
tual value of r is 10%, which means that r = 0.10. In that case:

(24-4) Value of $X when r = 0.10: $X = $1/(1 + 0.10) = $1/1.10 = $0.91

So you would be willing to accept $0.91 today in exchange for every $1 to be paid to
you one year from now. Economists have a special name for $X—it’s called the present
value of $1. Note that the present value of any given amount will change as the interest
rate changes.
To see that this technique works for evaluating future costs as
well as evaluating future benefits, consider the following example.
Suppose you enter into an agreement that obliges you to pay $1 one
year from now—say, to pay off a car loan from your parents when
you graduate in a year. How much money would you need today to
ensure that you have $1 in a year? The answer is $X, the present
value of $1, which in our example is $0.91. The reason $0.91 is the
right answer is that if you lend it out for one year at an interest rate
of 10%, you will receive $1 in return at the end. So if, for example,
you must pay back $5,000 one year from now, then you need to de-
posit $5,000 × 0.91 = $4,550 into a bank account today earning an mptvimages.com
interest rate of 10% in order to have $5,000 one year from now.
(There is a slight discrepancy due to rounding.) In other words,
In the 1971 movie Willy Wonka and the
today you need to have the present value of $5,000, which equals $4,550, in order to Chocolate Factory, Veruca Salt appreci-
be assured of paying off your debt in a year. ated the added value of having things in
These examples show us that the present value concept provides a way to calculate the present. She wanted a “golden-egg-
the value today of $1 that is realized in a year—regardless of whether that $1 is realized laying-goose NOW!”
as a benefit (the bonus) or a cost (the car loan payback). To evaluate a project today
that has benefits, costs, or both to be realized in a year, we just use the relevant interest
rate to convert those future dollars into their present values. In that way we have “fac-
tored out” the complication that time creates for decision making.
Below we will use the present value concept to evaluate a project. But before we do
that, it is worthwhile to note that the present value method can be used for projects in
which the $1 is realized more than a year later—say, two, three, or even more years. Sup-
pose you are considering a project that will pay you $1 two years from today. What is
the value to you today of $1 received two years into the future? We can find the answer
to that question by expanding our formula for present value.
Let’s call $V the amount of money you need to lend today at an interest rate of r
in order to have $1 in two years. So if you lend $V today, you will receive $V × (1 + r) in
one year. And if you re-lend that sum for another year, you will receive $V × (1 + r) ×
2
(1 + r) = $V × (1 + r) at the end of the second year. At the end of two years, $V will be
2
worth $V × (1 + r) . In other words:
(24-5) Amount received in one year from lending $V = $V × (1 + r)

module 24 The Time Value of Money 239