AP_Krugman_Textbook

module 24 The Time Value of Money 241

they are realized; costs are indicated by a minus sign. The fourth column shows
the equations used to convert the flows of dollars into their present value, and the
fifth column shows the actual amounts of the total net present value for each of the
three projects.
For instance, to calculate the net present value of project B, we need to calculate
the present value of $115 received in one year. The present value of $1 received in
one year would be $1/(1 +r). So the present value of $115 is equal to 115 ×$1/(1+r);
that is, $115/(1 + r). The net present value of project B is the present value
of today’s and future benefits minus the present value of today’s and future costs:
−$10+$115/(1+r).
From the fifth column, we can immediately see which is the preferred project—it is
project C. That’s because it has the highest net present value, $100.82, which is higher
than the net present value of project A ($100) and much higher than the net present
value of project B ($94.55).
This example shows how important the concept of present value is. If we had failed
to use the present value calculations and instead simply added up the dollars generated
by each of the three projects, we could have easily been misled into believing that proj-
ect B was the best project and project C was the worst.

How Big Is That Jackpot, Anyway?

For a clear example of present value at work,

consider the case of lottery jackpots.

On March 6, 2007, Mega Millions set the

record for the largest jackpot ever in North

America, with a payout of $390 million. Well, sort

of. That $390 million was available only if you

chose to take your winnings in the form of an

“annuity,” consisting of an annual payment for

the next 26 years. If you wanted cash up front,

the jackpot was only $233 million and change.

Why was Mega Millions so stingy about quick

payoffs? It was all a matter of present value. If

the winner had been willing to take the annuity,

the lottery would have invested the jackpot

money, buying U.S. government bonds (in effect

lending the money to the federal government).

The money would have been invested in such a

way that the investments would pay just enough

to cover the annuity. This worked, of course, be-

cause at the interest rates prevailing at the

time, the present value of a $390 million annuity

spread over 26 years was just about $233 mil-

lion. To put it another way, the opportunity cost

to the lottery of that annuity in present value

terms was $233 million.

So why didn’t they just call it a $233 million

jackpot? Well, $390 million sounds more im-

pressive! But receiving $390 million over 26

years is essentially the same as receiving $233

million today.

fyi

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Module 24 AP Review

Check Your Understanding

1. Consider the three hypothetical projects shown in Table 24.1.
   This time, however, suppose that the interest rate is only 2%.
   a. Calculate the net present values of the three projects. Which
   one is now preferred?

b. Explain why the preferred choice is different with a 2%

interest rate from with a 10% interest rate.

Solutions appear at the back of the book.