Rate
of Return
The rate of return of an investment is a measure
that expresses the merit of an investment as a percentage. Think
of an investment as a bank account. Expenditures are deposits
and receipts are withdrawals. For a simple investment, the Rate
or Return (ROR) is the compounded interest rate at which the
schedule of withdrawals will exactly use up the deposits plus
compounded interest. Because there are several measures that describe
the return for an investment we use the term rate of return for
the general concept and explain other measures as they arise.
In the current discussion we compute two rates, the internal
rate of return (IRR) and the return on invested capital (RIC).


Goals 



Be able to find the IRR
of a simple investment or a simple loan by trial
and error using equivalencyfactor tables or
cashflow calculators.

Use the IRR to make accept/reject
decisions.

Recognize and compute IRR for
simple cases.

Be able to identify nonsimple
investments and loans.

Use the Economics addin to find IRR and
RIC values.




Text 



3.4.5
Internal Rate of Return Method 




Computing the Internal Rate of
Return 

Click the icon to see an introductory movie.
We compute the IRR by expressing the NPW as a function
of the interest rate, i. The NPW is determined by multiplying the values of the cash flow components
by equivalency factors as discussed in Lesson 10 and in Chapter
3 of the textbook. These factors are nonlinear functions of the
interest rate i, so the NPW is also a nonlinear function
of i.
In some fashion, usually by
trial and error, we solve for the value of i that results
in NPW(i) = 0. Equivalently,
we can solve for NAW(i)
= 0. The solution is the IRR. The IRR is the nonnegative root of the NPW(i)
function.

The IRR is the interest rate for
which the NPW (or NAW) is exactly
zero.
If the IRR ≥ MARR, accept
the investment. Otherwise, reject the investment.


Finding the IRR is not too difficult for a simple
investment since the NPW is a decreasing function of i.
While searching for the solution, if one computes a positive
NPW for some i,
the IRR must be greater than i. If one computes a negative
NPW, the IRR must be smaller than i.
When solving for the IRR using equivalencyfactor
tables it is appropriate to use the tables to find the greatest
rate with NPW
> 0. In the figure below the rate is x with NPW = A.
We also find the smallest interest rate with NPW < 0. This
is point y with NPW = B. Then linear interpolation
is used to find the approximate root. The formulas at the left
show the solution for i, the approximate IRR.
For a simple investment the NPW(i) function
has at most one positive root. If the total revenues are less
than the total expenditures, the function has no positive roots
and the IRR will not exist. The project should certainly
be rejected. The problem of finding the IRR becomes more complex
for nonsimple investments because then there may be multiple
roots. We consider this on the next page of this lesson. For
simple investments, the IRR method results in the same accept
or reject decision as the NPW or NAW methods.
In the case of a simple loan where total revenues
are less than the total expenditures, the IRR exists and is unique.
In this case of a loan however, the IRR should be called the
cost of borrowing rather than the rate of return. 

Example 

To illustrate the IRR computations
we use the cash flow below. It is a simple investment because
expenditures precede receipts.
For a trialanderror analysis using equivalency factors, we
identify the following three components of the cash flow. The
cash flow consists of the single payment is at time 0, the uniform
series with the base value of 10 that runs through 5 years, and
a gradient begins at time 2 with value of 4. Recall that the
gradient equivalency factor assumes that the first gradient payment
(with value 0) is at time 2.
Component 
1 
2 
3 
Type(range) 
Single(0) 
Uniform(1,5) 
Gradient (2,5) 
Amount 
50 
10 
4 
To find the IRR, we write the NPW as a function
of the interest rate and set it to zero.
NPW(i) = 50 + 10(P/A, i, 5) + 4(P/G, i,
4)(P/F, i,
1) = 0
The table below shows the NPW function evaluated
in a trial and error fashion.
Interest, i 
CF1 
CF2 
CF3 
NPW(i) 
0% 
50 
50 
24 
24 
5% 
50 
43.29 
19.44 
12.73 
10% 
50 
37.91 
15.92 
3.83 
15% 
50 
33.52 
13.17 
3.31 
14% 
50 
34.33 
13.67 
2.00 
12% 
50 
36.05 
14.74 
0.79 
13% 
50 
35.17 
14.19 
0.64 
The evaluation at i = 0 shows that the
project returns a profit of 24. This confirms that there will
be a positive IRR. Evaluations at 5% and 10% yield positive values
for the NPW. The negative NPW at i = 15% signals that
the IRR has been exceeded. Subsequent evaluations at 12% and
13% bracket the value within a single percentage point. If tables
are used to evaluate the equivalency factors we must stop and
interpolate if more accuracy is required.
Using Excel or a calculator to compute factor values
makes it unnecessary to perform interpolation when fractional
values of the interest rate are allowed. We illustrate some alternatives
below.


Cash Flow Calculator 

This website has two Flashbased calculators
that make the trial and error process much easier.
The Cash Flow Calculator computes the present
worth, uniform worth and future worth for problems that have
no more than 10 periods. Individual cash flow amounts must
be integers between 99 and +99. Period entries must be positive
integers. Enter the cash flow, interest rate and the periods
for the uniform and future worth. Then press the enter button.
The results are computed near the bottom of the display.
The IRR is found by trying different interest
rates until the Present Worth is near 0. The interest rate
changes in 0.1% intervals. This calculator is limited by the
number of periods, the small range of values for the cash flows,
and the limitation to integer cash flow values.
Click on the tool icon to open the cash
flow calculator. Try to duplicate the results shown above.


Project Calculator 

The Project Calculator is illustrated
below with data entered for the example. Note that the start value
for the gradient is the period of the first nonzero amount,
differing from the factor assumption in this respect.
The inputs to the Project Calculator are the separate
components of a complicated cash flow. Components may be single
payments, uniform series or gradient series. Up to 10 components
may be entered. Cash flow amounts are any positive or negative
numbers. The project life (duration) must be an integer.
The MARR is an integer percentage. For a single payment, the start field
tells when the payment occurs. For a uniform series payment,
the start field indicates when the first payment occurs,
and the end field indicates when the last payment occurs.
For the gradient series, the start field indicates when
the first nonzero payment occurs and the end field
indicates when the last payment occurs.
To find the IRR change the value for the MARR until
the Present Net Worth value is near 0. Only integer values of
the MARR are allowed, so interpolation is necessary for more
accurate answers.
The calculator is programmed in Flash and it takes
a little while to download and open. Click on the tool icon
to open the project calculator. Try to duplicate the example.


Economics Addin 

A project defined by the Economics addin
is dynamic in that the formulas in the yellow cells are immediately
evaluated when a number in the white cells of the form is changed.
Because the NPW values in column I depend on the MARR in cell F2,
we can search for the value in F2 that results in a zero value
for the NPW. The example is illustrated below with the MARR equal
to the IRR. The NPW in I2 is computed as 0.00. More decimal accuracy
would show that the value is not really zero, but for our purposes
that is close enough. The IRR value shown in cell I5 is not dynamic,
as indicated by its green color. The IRR cell holds the original
default value of 0.00 when the form is created, but it must be
updated by the user as described in the next paragraph.
The addin changes the IRR cell when the Compute
Rates command is selected from the addin menu.
A dialog is presented with a list of the projects defined in
the workbook (there is only one for the example). The cells
at the bottom of the dialog set the range of search for the
IRR and the initial guess. These are important if there are
multiple roots to the NPW equation. Clicking OK initiates the
search.
Unseen by the user, the addin manipulates the
MARR cell (F2) until the NPW cell (I2) is within some tolerance
of zero. The result is reported in the IRR cell (I5), and the
MARR cell is returned to its original value. We color the cell
green to indicate that it holds the numerical result of an algorithm.
The result is very accurate. To show greater accuracy
on the form, change the number format of the IRR cell. If the
data changes, the IRR cell will not change unless the user recalculates
it with the Compute
Rates command.
The addin is very general as to the number of
items, the range of times, and the cash flow amounts. The method
also works for problems with taxes and inflation. The link is
to documentation on this feature of the addin.


Economics
Addin: Compute Rates 




Simple Cases 

There are some cases where it is easy to find
the IRR with either no or only minimal calculations. This
is useful for the student looking for quick answers, and the
cases help to explain the IRR measure. In the examples, N periods
separate the first and last payment. The amount invested is P and
when there is an annual return it is A.
Case 
IRR 
Payback 
Comment 
Total Investment is equal to total return.

IRR = 0 
N 
The project makes no profit. 
There is an annual payment and the total
investment is also returned with the last payment.

IRR = A/P 
Min{P/A,
N} 
This is like a savings account for which
interest is withdrawn annually and the principal is returned
with the final interest payment. 
The there is an annual payment forever.

IRR = A/P 
P/A 
When interest earned is withdrawn annually, the amount
invested remains the same. 
There is an annual payment
for a fixed number of years with NA ≥ P.

A=P(A/P,
IRR, N)
(A/P, IRR, N) = A/P

P/A 
Compute A/P and find the interest
rate that has this value for
(A/P, IRR, N). 


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