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 equivalency-factor tables or cash-flow calculators. Use the IRR to make accept/reject decisions. Recognize and compute IRR for simple cases. Be able to identify non-simple investments and loans. Use the Economics add-in to find IRR and RIC values.
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 3.4.5 Internal Rate of Return Method
 Computing the Internal Rate of Return

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 Rate of Return

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 equivalency-factor 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 non-simple 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 trial-and-error 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 Flash-based 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.

 Cash Flow Calculator
 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 non-zero 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.

 Project Calculator

A project defined by the Economics add-in 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 up-dated by the user as described in the next paragraph.

The add-in changes the IRR cell when the Compute Rates command is selected from the add-in 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 add-in 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 add-in 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 add-in.

 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).

Engineering Finance
by Paul A. Jensen