1. Methods for Finding the Rate of Return
Lecture No. 24
Chapter 7
Contemporary Engineering Economics
Copyright © 2016
We may find i* by several procedures, each of which has its advantages and disadvantages. To facilitate the process of finding the rate of return for an investment project, we will first classify various types of investment cash flow.

2. Simple versus Nonsimple Investments
Definition: An investment with only one sign change in the net cash flow series
Definition: An investment in which more than one sign change occurs in the net cash flow series
We can classify an investment project by counting the number of sign changes in its net cash-flow sequence. A change from either “+” to “-”, or “-” to “+” is counted as one sign change. (We ignore a zero cash flow.) Then

3. Example 7.1: Investment Classification
Period n
Net Cash Flow
Project A
Project B
Project C 1 2 3 4
-1,000
-500
800
1,500
2,000
3,900
-5,030
2,145
1,000
-450
Given: Preceding cash flow sequences. Find: Classify the investments shown as either simple or nonsimple investments. • Project A represents many common simple investments. That is, many investment projects involve a series of cash outflows followed by a series of cash inflows. This situation reveals the NPW profile shown in Figure 7.1(a). The curve crosses the i-axis only once. • Project B represents a nonsimple investment. The NPW profile for this investment has the shape shown in Figure 7.1(b). The i-axis is crossed at 10%, 30%, and 50%. • Project C represents neither a simple nor a nonsimple investment, even though only one sign change occurs in the cash flow sequence. Since the first cash flow is positive, this is a simple borrowing cash flow, not an investment flow. Figure 7.1(c) depicts the NPW profile for this type of investment. COmments: Not all NPW profiles for nonsimple investments have multiple crossings of the i-axis. Clearly then, we should place a high priority on discovering this situation early in our analysis of a project’s cash flows. The quickest way to predict multiple i*s is to generate an NPW profile on a computer and check whether it crosses the horizontal axis more than once. In the next section, we illustrate when to expect such multiple crossings by examining types of cash flows.
Project A: a simple investment
Project B: a nonsimple investment
Project C: a simple borrowing

4. Predicting Multiple i*s
Net Cash Flow Rule of Signs
The number of real i*s that are greater than −100% for a project with N periods is never greater than the number of sign changes in the sequence of the cash flows. A zero cash flow is ignored.
Accumulated Cash Flow Sign Test
If the sequence of accumulated cash flow series starts negatively and changes sign only once, then a unique positive i* exists.
As hinted at in Example 7.1, for certain series of project cash flows, we may uncover the complication of multiple i* values that satisfy PW Equation (7.1). By analyzing and classifying cash flows, we may anticipate this difficulty and adjust our analysis approach later. Here we will focus on the initial problem of whether we can predict a unique i* for a project by examining its cash flow pattern. Two useful rules allow us to focus on sign changes: (1) in net cash flows; and (2) in accounting net profit (accumulated net cash flows).

5. Predicting the Number of i*s
Net Cash-Flow Rule of Signs
Accumulated Cash-Flow Sign Test
İkinci için:
If we let An represent the net cash flow in period n and Sn represent the accumulated cash flow (the accounting sum) up to period n, we have the data in Table 7.3. We then examine the sequence of accumulated cash flows (S0, S1, S2, S3, c, SN) to determine the number of sign changes. If the series Sn starts negatively and changes sign only once, then a unique positive i* exists. This cumulative cash flow sign rule is a more discriminating test for identifying the uniqueness of positive i* than the previously described method.
Number of real i*s ≤ 3.
This implies that the project could have (0, 1, 2, or 3) i*s, but NOT more than 3.
Number of sign changes = 1, indicating a unique i*.
i* = 10.46%

6. Computational Methods
Using Excel’s financial command
Direct solution method
Trial-and-error method
Works only for simple investment
Cash flow analyzer
Online financial calculator

7. Finding Rate of Return on Excel
Period (N)
Cash Flow
-$1,000 1
-500 2
800 3
1,500 4
2,000
=IRR(cell range, guess)
=IRR(B3:B7,10%)

8. Other Computational Methods
Direct Solution
Trial and Error Method
Computer Solution Method
Log
Quadratic n
Project A
Project B
Project C
Project D
−$1,000
−$2,000
−$75,000
−$10,000 1
1,300
24,400
20,000 2
1,500
27,340 3
55,760
25,000 4

9. Direct Solution Methods
Project A
Project B
Since an interest rate less than -100% has no economic significance, we find that the project’s i* is 25%. COmments: In both projects, one sign change occurred in the net cash-flow series, so we expected a unique i*. Also, these projects had very simple cash flows. When cash flows are more complicated, we generally use a trial-and-error method or a computer to find i*.

10. Trial and Error Method: Project C
Step 4: Once you bracket the solution, then you use a linear interpolation to approximate the solution.
Step 1: Guess an interest rate, say, i = 15%.
Step 2: Compute PW(i) at the guessed i value.
PW (15%) = $3,553
Step 3: If PW(i) > 0, then increase i. If PW(i) < 0, then decrease i.
PW(18%) = −$749
$3,553
−$749 i
15%
18%
Note: This method works only for finding i* for simple investments.

11. Multiple Rates of Return Problem
Project cash flows
$1,000
$2,300
$1,320 2 1

12. NPW Plot
At Issue: If your MARR = 15%, which ROR do you use to make an accept/reject decision—10% or 20%?