Materials & labor
Depreciation & amortization (1)
Net amount after tax
Depreciation & amortization
Net cash flow
Net Present Value
Internal Rate of Return
(1) Depreciation & Amortization:
Depreciation of equipment
Amortization of start-up costs
Dec 31, 2018
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Analysis of investments in long-term projects is referred to as capital budgeting. Long-term projects are worthy of
special attention because of the fact that they frequently require large initial investments, and because the cash outlay
to start such projects often precedes the receipt of cash inflows by a significant period of time. In such cases, we are
interested in being able to predict the profitability of the project. We want to be sure that the profits from the project
are greater than what we could have received from alternative investments or uses of our money.
This chapter focuses on how managers can evaluate long-term projects and determine whether the expected return
from the projects is great enough to justify taking the risks that are inherent in long-term investments. Several different
approaches to capital budgeting are discussed: the payback method, the net present value method, and the internal rate
of return method. The latter two of these methods require us to acknowledge the implications of the time value of
The time value of money refers to the fact that money received at different points in time is not equally valuable.
To give a rather elementary example, suppose that someone offered to buy your product for $250, and that they are
willing to pay you either today or one year from today. You will certainly prefer to receive the $250 today. At the very
least, you could put the $250 in a bank and earn interest in the intervening year.
Suppose, however, that the buyer offered you $250 today or $330 in twenty-two months. Now your decision is
much more difficult. How sure are you that the individual will pay you twenty-two months from now? Perhaps he or
she will be bankrupt by then. What could we do with the money if we received it today? Would we put the $250 in
some investment that would yield us more than $330 twenty-two months from today? These are questions that we
have to be able to answer in order to evaluate long-term investment opportunities. But first let’s discuss some basic
issues of investment analysis.
The first step that must be taken in investment analysis is to identify the investment opportunity.
Such opportunities fall into two major classes: new project investments, and replacement or
reinvestment in existing projects. New project ideas can come from a variety of sources. They may
be the result of research and development activity or exploration. Your firm may have a department
solely devoted to new product development. Ideas may come from outside of the firm.
Reinvestment is often the result of production managers pointing out that certain equipment needs
to be replaced. Such replacement should not be automatic. If a substantial outlay is required, it
may be an appropriate time to reevaluate the product or project to determine if the profits being
generated are adequate to justify continued additional investment.
The data needed to evaluate an investment opportunity are the expected cash flows related to the
investment. Many projects have a large initial cash outflow as we acquire plant and equipment and
incur start-up costs prior to actual production and sale of our new product. In the years that follow,
there will be receipt of cash from the sale of our product (or service) and there will be cash
expenditures related to the expenses of production and sales. We refer to the difference between
each year’s cash receipts and cash expenditures as the net cash flow for that year.
You’re probably wondering why we have started this discussion with cash flow instead of net
income for each year. There are several important reasons. First, net income, even if it were a
perfect measure of profitability, doesn’t consider the time value of money. For instance, suppose
that we have two alternative projects. The first project requires that we purchase a machine for
$10,000 in cash. The machine has a ten-year useful life. Depreciation expense is $1,000 per year.
A totally different project requires that we lease a machine for $1,000 a year for ten years, with
lease payments at the start of each year. Are the two alternative projects equal? No, they aren’t.
Even though they both have an expense of $1,000 per year for ten years, one project requires us to
spend $10,000 at the beginning. The other project requires an outlay of only $1,000 in the first
year. In this second project, we could hold on to $9,000 that had been spent right away in the first
project. That $9,000 can be invested and can earn additional profits for the firm before the next
lease payment is due.
The data needed for investment or project analysis includes cash flow information for each of
the years of the investment’s life. Naturally we cannot be 100 percent certain about how much the
project will cost and how much we will eventually receive. There is no perfect solution for the fact
that we have to make estimates. However, we must be aware at all times that, because our estimates
may not be fully correct, there is an element of risk. Project analysis must be able to assess whether
the expected return can compensate for the risks we are taking. It should also include consideration
of any taxes that will have to be paid.
THE PAYBACK METHOD
The payback method of analysis evaluates projects based on how long it takes to recover the
amount of money put into the project. The shorter the payback period, the better. There is intuitive
appeal to this method. The sooner we get our money out of the project, the lower the risk. If we
have to wait a number of years for a project to “pay off,” all kinds of things can go wrong.
Exhibit 11-1 presents an example of the payback method. In the exhibit, four alternative projects
are being compared. In each project, the initial outlay is $400. By the end of 2020, projects one
and two have recovered the initial $400 investment. Therefore, they have a payback period of three
years. Projects three and four do not recover the initial investment until the end of 2021. Their
payback period is four years, and they are therefore considered to be inferior to the other two
Payback Method—Alternative Projects
PROJECT CASH FLOWS
It is not difficult at this point to see one of the principal weaknesses of the payback method. It
ignores what happens after the payback period. The total cash flow for project four is much greater
than the cash received from any of the other projects, yet it is considered to be one of the worst of
the projects. In a situation in which cash flows extend for twenty or thirty years, this problem might
not be as obvious, but it could cause us to choose incorrectly.
Is that the only problem with this method? No. Another obvious problem stems from the fact
that according to this method, projects one and two are equally attractive because they both have
a three-year payback period. Although their total cash flows are the same, the timing is different.
Project one provides $1 in 2019, and then $399 during 2020. Project two generates $399 in 2019
and only $1 in 2020. Are these two projects equally as good because their total cash flows are the
same? No. The extra $398 received in 2019 from project two is available for investment in other
profitable opportunities for one extra year, as compared to project one. Therefore, it is clearly
superior to project one. The problem is that the payback method doesn’t formally take into account
the time value of money.
This deficiency is obvious when looking at project three as well. Project three appears to be less
valuable than projects one or two on two counts. First, its payback is four years rather than three,
and second, its total cash flow is less than either project one or two. But if we consider the time
value of money, then project three is better than either project one or two. With project three, we
get the $399 right away. The earnings on that $399 during 2019 and 2020 will more than offset
the shorter payback and larger cash flow of projects one and two.
Although payback is commonly used for a quick and dirty project evaluation, problems
associated with the payback method are quite serious. There are several methods commonly
referred to as discounted cash flow models that overcome these problems. Later in this chapter,
we will discuss the most commonly used of these methods, net present value and internal rate of
return. However, before we discuss them, we need to specifically consider the issues and
mechanics surrounding time value of money calculations.
THE TIME VALUE OF MONEY
It is very easy to think of projects in terms of total dollars of cash received. Unfortunately, this
tends to be misleading. Consider a project in which we invest $400 and in return we receive $520
after three years. We have made a cash profit of $120. Because the profit was earned over a threeyear period, it is a profit of $40 per year. Because $40 is 10 percent of the initial $400 investment,
we have apparently earned a 10 percent annual rate of return on our money. While this is true, that
10 percent is calculated based on simple interest.
Consider putting money into a bank that pays a 10 percent return “compounded annually.” The
term compounded annually means that the bank calculates interest at the end of each year and adds
the interest onto the initial amount deposited. In future years, interest is earned not only on the
initial deposit, but also on interest earned in prior years. If we put $400 in the bank at 10 percent
compounded annually, we would earn $40 of interest in the first year. At the beginning of the
second year we would have $440. The interest on the $440 would be $44. At the beginning of the
third year, we would have $484 (the $400 initial deposit plus the $40 interest from the first year,
plus the $44 interest from the second year). The interest for the third year would be $48.40. We
would have a total of $532.40 at the end of three years.
The 10 percent compounded annually gives a different result from the 10 percent simple interest. We have $532.40
instead of $520 from the project. The reason for this difference is that in the case of the project, we did not get any
cash flow until the end of the project. In the case of the bank, we were given a cash flow at the end of each year. We
reinvested that cash to earn additional interest.
Consider a cash amount of $ 100 today. We refer to it as a present value (PV). How much could this cash amount
accumulate to if we invested it at an interest rate (i) of 10 percent for a period of time (N) equal to two years? Assuming
that we compound annually, the $100 would earn $10 in the first year (10 percent of $100). This $10 would be added
to the $100. In the second year our $110 would earn $11 (that is, 10 percent of $110). The future value (FV) is $121.
That is, two years into the future we would have $121.
Mechanically this is a simple process—multiply the interest rate times the initial investment to find the interest for
the first period. Add the interest to the initial investment. Then multiply the interest rate times the initial investment
plus all interest already accumulated to find the interest for the second year.
While this is not complicated, it can be rather tedious. To simplify this process, mathematical formulas have been
developed to solve a variety of time value of money problems. The most basic of these formulas states that:
FV = PV (1 + i)N
This formula has been built into both business calculators and computer spreadsheet programs. If we supply the
appropriate raw data, the calculator or spreadsheet software performs all of the necessary interest computations.
For instance, if we wanted to know what $ 100 would grow to in two years at 10 percent, we would simply tell our
calculator that the present value, PV, equals $100; the interest rate, i, equals 10 percent; and the number of periods,
N, equals 2. Then we would ask the calculator to compute FV, the future value.
Can we use this method if compounding occurs more frequently than once a year? Bonds often pay interest twice
a year. Banks often compound monthly to calculate mortgage payments. Using our example of $100 invested for two
years at 10 percent, we could easily adjust the calculation for semiannual, quarterly, or monthly compounding. For
example, for quarterly compounding, N equals 8 (four quarters per year for two years) and i equals 21/2 percent (10
percent per year divided by four quarters per year). For monthly compounding, N equals 24 and i equals 10/12. Thus,
for monthly compounding, we would tell the calculator that PV = $100, i = 10/12, and N = 24. Then we would tell the
calculator to compute FV. We need a calculator designed to perform present value functions in order to do this. Note
that most calculators will interpret 10 input as i to be 10%. Computer spreadsheet programs require that you include
the % sign.
If we expect to receive $121 in two years, can we calculate how much that is worth today? This question calls for
a reversal of the compounding process. Suppose we would normally expect to earn a return on our money of 10
percent. What we are really asking here is, “How much would we have to invest today at 10 percent, to get $121 in
two years?” The answer requires unraveling compound interest. If we calculate how much of the $121 to be received
in two years is interest earned on our original investment, then we know the present value of that $121. This process
of removing or unraveling the interest is called discounting. The 10 percent rate is referred to as a discount rate. Using
the calculator, this is a simple process. We again supply the i and the N, but instead of telling the calculator the PV
and asking for the FV, we tell it the FV and ask it to calculate the PV.
Earlier in this chapter, we posed a problem of whether to accept $250 today or $330 in twenty-two months. Assume
that we can invest money in a project with a 10 percent return and monthly compounding. Which choice is better? We
can tell our calculator (by the way, if you have access to a business-oriented calculator, you can work out these
calculations as we go) that FV = $330, N = 22, and i = 10/12. If we then ask it to compute PV, we find that the present
value is $275. This means that if we invest $275 today at 10 percent compounded monthly for twenty-two months, it
accumulates to $330. That is, receiving $330 in twenty-two months is equivalent to having $275 today. Think of the
problem as “would you prefer to have $250 today or $275 today?” Clearly you would prefer $275. Well, in that case
you should choose the alternative of waiting twenty-two months for the $330, because $330 in twenty-two month is
worth the same to us as $275 today. This assumes there is no risk of default. Looking at this problem another way,
how much would our $250 grow to if we invested it for twenty-two months at 10 percent? Here we have PV = $250,
N = 22, and i = 10/12. Our calculation indicates that the FV = $300. If we wait, we will get $330 twenty-two months
from now. If we take $250 today and invest it at 10 percent, we only have $300 twenty-two months from now. Twentytwo months from now would you rather have $300 or $330? Clearly, $330. So, we find that we are better off waiting
for the $330, assuming we are sure that we will receive it.
Are we limited to solving for only the present or future value? No, this methodology is quite flexible. Assume, for
example, that we wish to put $100,000 aside today to pay off a $1,000,000 loan in fifteen years. What rate of return
must be earned, compounded annually, for our $100,000 to grow to $1,000,000? Here we have the present value, or
$100,000; the number of periods, fifteen years; and the future value, or $1,000,000. It is a simple process to determine
the required rate of return. If we simply supply our calculator with the PV, FV, and N, the calculator readily supplies
the i, which is 16.6 percent in this case.
Or, for that matter, if we had $100,000 today and knew that we could earn a 13 percent return, we would calculate
how long it would take to accumulate $1,000,000. Here we know PV, FV, and i, and we wish to find N. In this case,
N = 18.8 years. Given any three of our four basic components, PV, FV, N, and i, we can solve for the fourth. This is
because the calculator is simply using our basic formula stated earlier and solving for the missing variable.
So, far, however, we have considered only one single payment. Suppose that we don’t have $100,000 today, but
we are willing to put $10,000 aside every year for fifteen years. If we earn 12 percent, will we have enough to repay
$1,000,000 at the end of the fifteen years? There are two ways to solve this problem. We can determine the future
value, fifteen years from now, of each of the individual payments. We would have to do fifteen separate calculations
because each succeeding payment earns interest for one year less. We would then have to sum the future value of each
of the payments. This is rather tedious. A second way to solve this problem is by using a formula that accumulates the
payments for us. The formula is:
In this formula, PMT represents the payment made each period, or annuity payment. Although you may think of
annuities as payments made once a year, an annuity simply means payments that are exactly the same in amount, and
are made at equally spaced intervals of time, such as monthly, quarterly, or annually.
To solve problems with a series of identical payments, we have five variables instead of the previous four. We now
have FV, PV, N, i, and PMT.
Annuity formulas are built into business calculators and computer spread-sheet programs such as Excel.
Annuity formulas provide you with a basic framework for solving many problems concerning receipt or payment
of cash in different time periods. Keep in mind that the annuity method can be used only if the amount of the payment
is the same each period. If that isn’t the case, each payment must be evaluated separately.
Using Computer Spreadsheets for TVM Computations
As noted above, electronic spreadsheet software programs can be used to solve TVM problems.
Excel has TVM formulas built into the program.1 Excel uses the following abbreviations for the
Nper—Number of periods
PMT—Periodic or annuity payment
Consider the problem of finding the future value of $100 invested for two years at 6 percent.
Using Excel, begin clicking on a cell where you want the FV to appear. Then type:
As soon as you do that, Excel will show the variables that should be inserted, and the order in
which they have to be entered. See Exhibit 11-2.
Using Excel to Solve for FV. Initial View.
Note that we are trying to solve for the FV in Cell A1. When we start the form ...
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