566 Chapter 14 Evaluating Projected Cash Flows
The mathematics agrees with our commonsense assessment about which investment would
have the higher value; using a 7% rate the $10,000 per year for 3 years investment would
be worth $26,243, while the higher payoff far in the future is only worth $10,337. Note that
this calculation is equivalent to asking how much money we would need to invest at a 7%
rate of return to generate either of these future payouts.
Using present values not only allows us to compare these investments, though, it also
allows us to decide how much we would be willing to pay for either one. If we are pre-
sented with the opportunity to invest in the first one for $20,000, we can see that this would
represent a much better return than the 7% that we require, since $20,000 is less than what
we would have valued the opportunity at the 7% rate. On the other hand, if we were given
the opportunity to invest in the second option for $20,000, we could see that this would
represent a much poorer investment than 7% rate that we require, since $20,000 is far more
than the present value of this opportunity at 7%.Example 14.1.1 Using present values and an 8.5% rate, would you prefer a business
opportunity that could be expected to earn $5,000 per year for 5 years or an
investment that would pay $32,000 all at once at the end of 5 years?The present value of the $5,000 per year works out to be:PV PMTa n (^) | (^) i
PV $5,000a 5 | (^) .085
PV $5,000(3.94064208)
PV $19,703
The present value of the $32,000 lump sum works out to:
FV PV(1 i)n
$32,000 PV(1.085)^5
$32,000 PV(1.50365669)
PV $21,281
For these present values, the lump sum is large enough to be worth waiting for. Since it has
a higher present value, we would prefer it over the $5,000 per year option.
Perpetuities
Suppose that you are considering an investment in part of a small business that you expect
would earn you profits of $7,500 per year. The rate of return that you would require is 8%.
Using the present value method, what value would you place on this business?
Unfortunately, there is one detail we are missing: what is the term? In order to find the future
value, we need to make some assumption about how long the profits will keep on flowing in.
Realistically, though, in many cases that is just about impossible to do. If the business will stop
earning a profit in the near term, we probably wouldn’t be all that interested in investing in it;
if the business is likely to keep running profitably well into the future, any sort of prediction
about how long it will keep on going would be nothing more than a wild guess.
Suppose that we assume a 20-year term. Using this, we can calculate that the present
value would be $73,636. This might be reasonable if you plan to hold onto the investment
for only 20 years—except that at the end of the 20 years you would presumably be able to
sell your share to someone else for the present value of the expected profits at that point.
Just finding the present value of the next 20 years’ payouts ignores the fact that the business
can be sold at the end of the 20 years. For this reason, we can’t ignore the profits farther
into the future, even if we don’t expect to be the owners of the business at that point.
But how far into the future should we go? Suppose that we use a 50-year term. Then
the present value works out to be $91,751. But why stop at 50? Why not 75 years, or 100,
or 1,000? In fact, what if we just assume that the business keeps running indefinitely into
the future? A stream of payments which continues indefinitely into the future is called
a perpetuity. You might think that since the payments of a perpetuity go on forever this
would lead to an infinite present value, but in fact it doesn’t.