568 Chapter 14 Evaluating Projected Cash Flows
bills, and that each machine will last 10 years. Assuming she requires an 8% rate of
return on her investments, what would the present value of the savings be?In this case using a perpetuity would be inappropriate, because we have a specifi c period of
time that we expect the investment to continue paying off. Thus:PV PMT a n (^) |i
PV $245a 10 | (^) .08
PV $245(6.7100814)
PV $1,643.97
Knowing this present value can help the laundromat owner decide whether the new washers
are a good investment. If the cost of upgrading the washers is less than $1,643.97, then doing
so would result in an investment return that is better than the 8% that she normally requires.
If the upgrades would cost more than that amount, there still may be reasons to do it, and the
investment may still provide a positive return, but it will be below the desired 8%.
More Complicated Projections
In the examples considered above, it was assumed that the payoff from an investment took
the form of either a simple lump sum or an annuity. This assumption is not always realistic.
As an investor in a business venture you not only hope that the business provides a return
on your investment in the form of profits, but you no doubt hope that the business itself will
grow and produce increasing profits.
Suppose that you are considering investing in two different businesses, and are trying to
figure out which is the better option. One choice is the opportunity to buy in to a profitable local
bowling alley for $100,000. Right now, your share of the profits would be $8,000 per year.
Another choice is the chance to buy in on a residential solar electric installation business, also
for $100,000. Right now, your share of the profits would be $5,000 per year. However, the bowl-
ing alley business is a mature business—solid, but not growing especially quickly. You believe
that profits can probably grow at a 2% rate in the future. The solar business, though, is growing
rapidly, and you believe its profits will grow at a 10% rate. Which is the better choice?
On the one hand, the bowling alley is more profitable right now, but its slower growth rate
means that the more rapidly growing solar business will eventually overtake it. The question,
then, is which business has the higher present value. Unfortunately, since the payments are
increasing, we no longer have a true annuity. (Readers who have covered Chapter 7 may recall
that we were faced with a similar challenge when dealing with the impact of inflation.)
Suppose that you require a 12% rate of return from your investments. Then the bowling
alley business gives you 2% of this from the growth of the payments themselves, so we cal-
culate the present value of its profits as though they were not increasing, using a 10% rate.^1
We do the same for the solar business, except in this case we use a 12% 8% 4% rate.
Assuming that both businesses will continue to operate according to these assumptions
for the foreseeable future, we calculate their present values to be:
Bowling alley: PV PMT_____i
$8,000
___0.10 $80,000
Solar business: PV PMT_____i
$5,000
___0.04 $125,000
Since we said that either business would require a $100,000 investment, we can see from these
calculations that the bowling alley would not be worth its $100,000 up-front cost. On the other
hand, the solar business would actually be worth more than its $100,000 investment.
Note that these present values depend on the required rate of return. If the required rate
of return were different, we might put a different value on these businesses.Example 14.1.4 Determine the present values for both the bowling alley’s and solar
installation business’s projected profi t, assuming that the required rate of return is 18%.(^1) As discussed in Chapter 7, this may not be precisely mathematically correct, but in most cases it provides an
acceptable approximation.