Cash Flow Analysis
To illustrate the models for economic analysis, consider
the following situation.
businessman is considering the purchase of an asset
that has an initial cost of $2000. The asset promises
an annual return of $600. It's operating cost is $100
the first year, $150 the second, and increases by $50
in each subsequent year. The salvage value for the asset
in 10 years is $400, or 20% of its initial cost. The
cash flow for this situation showing the individual
components explicitly is in the figure below. If the
businessman's minimal acceptable rate of return is 10%,
should he invest in this project?
The cash flow shows the amounts expended or received in each
period of the analysis. An arrow drawn downward is an expenditure
and an arrow drawn upward is a receipt. To provide a general
procedure for evaluation, we define the general cash outflow
or cost at time t as ,
and the general cash inflow or income at time t as
The time index t takes integer values between 1 and
n, where n is the time horizon of the analysis
and the units of time measurement are years (some other time
measure could be adopted). Cash flows within a year are accumulated
and, for analysis purposes, are assumed to occur at the end
of the year. The net inflow at time t is .
Time zero represents some arbitrary reference time and is
usually identified as the present time. Most analyses must
determine the appropriateness of some investment, and we will
define P to be the value of the investment made at
Although the figure shows a single investment at time 0,
a project may require investments at other times. Often time
0 represents the startup time a project and several outlays
during a construction phase would be represented by negative
cash flows at times with negative indices. Net cash flows
may be positive or negative in any year.
Cash flow analysis assumes that all flows over the life of
the project are known with certainty. The analysis will reduce
the cash flow to a single quantity that measures the profitability
of the project. That measure will be used to accept or reject
Minimum Rate of Return
||In order to combine cash flows that
occur at different points in time into a single measure, it is
necessary that some rate of return be specified. In most cases
this quantity is determined by management and represents the minimum
rate of return that an investment must earn if it is to be acceptable.
The rate of return is similar to the rate that a bank pays to
persons who invest their money with the bank. In the case of a
production system, the investment is the design and installation
costs, and the return is the stream of cash flows returned during
operation. We call this the minimum acceptable rate of return
(MARR) and use the notation i to represent it. For the
example, i = 10%.
Present Worth Analysis
The present worth (or present worth
cost) is a single measure that expresses the investment at time
zero and all cash flows that occur over n periods as
an equivalent amount at time zero. It is particularly appropriate
when the cash flows associated with a single project or several
competing alternatives vary over time.
The present worth of a single cash flow at time t is
When the numerator is positive, this could be
interpreted as the amount of money that must be invested at
time zero at interest rate i to allow a withdrawal
at time t of the amount .
If the numerator is negative, the present worth can be interpreted
as the negative of the amount that can be borrowed at time zero
at interest rate i such that the amount that must be
repaid is .
When operated upon in the manner of the equation above, the
cash flow is said to have been discounted to time zero. The
is called the discount factor for time t.
When i is positive, the discount factor is always less
than 1 for t greater than zero. The use of discounted
cash flows assigns more worth to cash flows that occur closer
to the present time than those that occur further in the future.
It recognizes the "time value of money." Money available
now is worth more than the same amount obtained at some time
in the future.
The present worth of a series of cash flows occurring
from times zero to n is the sum of the discounted cash
NPW stands for the net present worth
of the project.
When the NPW is positive, the investment results
in a rate of return greater than the minimum rate i.
In the absence of alternatives this would be an acceptable investment.
When the present worth is zero, the investment returns exactly
the minimum acceptable rate, and this too would indicate an
acceptable investment. When the present worth is negative, the
investment would not satisfy the requirement of the minimum
rate, and should be rejected.
Analysis with the Economics Add-in
||This site provides an add-in that
performs present worth analyses for arbitrary cash flows. The
figure below shows the form constructed by the add-in for the
The investment in row 9 as a negative
number. The Salvage is the percentage of the investment
that is returned at the end of the life of the asset (20%
= 400/2000). The Start value indicates when the
asset is put into service and the End value indicates
the end of its life.
The return is entered in row 13 as a uniform series
with value 600. The first payment time (at the end of year
1) is entered in the Start cell and the last payment
time (at the end of year 10) is entered in the End cell.
The uniform part of the operating cost is entered in row
14 as a negative number (-100). The operating cost is increasing
by an amount $50 per year. This is entered as a negative gradient
(-50) in row 15. The Start cell indicates when the
first non-zero payment occurs (2) and the End cell
indicates when the last payment occurs. The Parameter
for the gradient is the periodicity of the payments (1 means
that it occurs annually).
The add-in automatically computes the present worth for each
component of the cash flow in column S starting in row 9.
The total NPW for the project is in cell Q2 (81.93). The fact
that this is positive indicates that the project returns more
than 10%. In fact, the computed entry in cell Q5 indicates
that the project returns 11.13% on the initial investment
The form makes it easy to change features of the project and
to experiment with alternative assumptions concerning the cash
flow. For example if one changes the MARR to 15%, the NPW becomes
a negative number indicating that the project is not acceptable
if the investor's minimum rate were 15%.
add-in has many features available to evaluate single projects.
The add-in presents cash flows in either tabular or graphical
forms. It can analyze situations with inflation and/or taxes.
See the add-in instructions for more details.
Net Annual Worth
The number computed in Q3 is the
net annual worth (NAW) of the project. If
the NPW amount (81.93) were used to buy an annuity with 10 end
of the year payments, the annuity payments would be the NAW
(13.33). The NAW will always have the same sign as the NPW,
so it adds no new information regarding the acceptability of
the project. The NAW measure is useful when alternatives with
different lives are compared.
The annual worth is a single measure that expresses the investment
at time zero and all cash flows that occur over n periods
as a uniform annual amount. The annual equivalent of an investment
at time zero, P, that obtains a salvage value S
at time n is
The factor in the brackets is called the capital
recovery factor. It determines a uniform amount that includes
an allowance for the capital expended (P – S)
and the investment cost of the capital. The salvage value, since
it is the part of the investment recovered at the end of the
project, only contributes an investment cost (Si).
When the income and costs associated with a project
are uniform over the time horizon with values I and
C respectively, the uniform net annual worth of the
NAW = I – C
When incomes and costs are not uniform, one first
determines the NPW of the cash flows and then uses the capital
recovery factor to convert that to a uniform annual equivalent.
This formula is used to calculate the NAW on the
project form (Cell Q3 for the example).
The NAW can be used to evaluate a single project
or compare alternatives. When NAW is positive, the project has
sufficient profit to justify the investment given the MARR,
while if NAW is negative the project is not justified.
Internal Rate or Return
A third way to evaluate a cash
flow is by computing its internal rate of return (IRR). The
IRR of a project is the value of the interest rate that causes
the NPW and the NAW to be zero. To find the IRR we find the
that solves the following equation.
This is a useful measure, because it is common to consider
investments in terms of their rates of return. The fact that
for some unusual cash flows there may be more than one value
that yields zero NPW, makes IRR difficult to interpret for those
Except for very simple cash flows, there is no closed form
equation for computing the IRR. Rather the procedure is to use
trial and error to evaluate the NPW or NAW, whichever is more
convenient, for different values of i, until the result
becomes sufficiently close to zero.The add-in uses a binary
search procedure. For the example the IRR has the value 11.13
as presented in cell Q5.
With the IRR computed it is easy to make decisions regarding
a cash flow. The project is acceptable if its IRR is greater
than the MARR and not acceptable if its IRR is less than the
||We have described three measures
for representing a cash flow with a single quantity, the NPW,
the NAW and the IRR. If a project is judged acceptable with
one of the measures, it will also be acceptable under both of
the others. There is one additional measure the return on
(RIC). For simple investments the IRR has only one value that
causes the NPW to be zero, the IRR is equal to the RIC. For mixed
investments, where some incomes may preceed some expenditures,
the IRR and RIC may be different. The RIC is useful because it
can have at most one value for a given cash flow while for some
mixed cash flows the IRR may have more than one solution. It
is discussed more fully on the IRR/RIC
page in the Computations section.