Contemporary Engineering Economics, 4th edition, © 2007

Methods for Finding the Rate of Return

Lecture No. 25Chapter 7Contemporary Engineering EconomicsCopyright © 2006

Contemporary Engineering Economics, 4th edition, © 2007

Simple versus Nonsimple Investments Simple Investment: The project with only one

sign change in the net cash flow

Nonsimple investment: an investment in which more than one sign change occurs in the net cash flow series

Contemporary Engineering Economics, 4th edition, © 2007

Example 7.1 Investment Classification

Period

n

Net Cash FlowProject

AProject

BProject

C

0

1

2

3

4

-1,000

-500

800

1,500

2,000

-1,000

3,900

-5,030

2,145

1,000

-450

-450

-450

Project A: a simple investmentProject B: a nonsimple investmentProject C: a simple borrowing

Contemporary Engineering Economics, 4th edition, © 2007

Predicting Multiple i*s

Net Cash Flow Rule of SignsThe 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 TestIf the sequence of accumulated cash flow series starts negatively and changes sign only once, then a unique

positive i* exists

Contemporary Engineering Economics, 4th edition, © 2007

Net Cash Flow Rule of Signs

- 100% < i *< infinity

• Net Cash Flow Rule of Signs

No. of real RORs (i*s)

<

No. of sign changes in the project cash flows

Contemporary Engineering Economics, 4th edition, © 2007

Example

n Net Cash flow Sign Change

0123456

-$100-$20$50

0$60

-$30$100

1

11

• No. of real i*s <= 3• This implies that the project could have (0, 1, 2, or 3) i*s but NOT more than 3.

Contemporary Engineering Economics, 4th edition, © 2007

Accumulated Cash Flow Sign TestStep 1: Find the accounting sum of net cash

flows at the end of each period over the life of the project

Period Cash Flow Sum(n) (An ) Sn

Step 2: If the series Sn starts negatively and changes sign ONLY ONCE, there exists a unique positive i*.

S A

S S A

S S A

S S AN N N

0 0

1 0 1

2 1 2

1

A

A

A

AN

0

1

2

0

1

2

N

Contemporary Engineering Economics, 4th edition, © 2007

Example

n An Sn Sign change

0123456

-$100-$20$50

0$60

-$30$100

-$100-$120-$70-$70-$10-$40$60 1

• No of sign change = 1, indicating a unique i*.• i* = 10.46%

Contemporary Engineering Economics, 4th edition, © 2007

Practice Problem

$2,145$3,900

$5,030

$1,000

0 12

3

• Is this a simple investment?• How many RORs (i*s) can you expect from examining the cash flows?• Can you tell whether or not this investment has a unique rate of return?

Contemporary Engineering Economics, 4th edition, © 2007

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

Excel command to find the rate of return:

=IRR(cell range, guess)

e.g., =IRR(C0:C7, 10%)

Contemporary Engineering Economics, 4th edition, © 2007

Period (N)

Cash Flow

0 -$1,000

1 -500

2 800

3 1,500

4 2,000

Finding Rate of Return on Excel

1234567891011

A B CPeriod Cash Flow

0 -10001 -5002 8003 15004 2000

IRR = 44%

=IRR(B3:B7,10%)

Contemporary Engineering Economics, 4th edition, © 2007

Other Computational MethodsDirect Solution

Direct Solution

Trial & Error Method

Computer Solution MethodLog Quadratic

n Project A Project B Project C Project D

0 -$1,000 -$2,000 -$75,000 -$10,000

1 0 1,300 24,400 20,000

2 0 1,500 27,340 20,000

3 0 55,760 25,000

4 1,500

Contemporary Engineering Economics, 4th edition, © 2007

• Project A • Project B

PW ii i

xi

PW i x x

x

x

i

ii

ii

i i

( ) $2,$1,

( )

$1,

( )

,

( ) , , ,

:

.

. .

, *

000300

1

500

10

1

12 000 1 300 1500

08

081

125%, 1667

1

1160%

100% 25%.

2

2

Let then

Solve for

or -1.667

Solving for yields

Since the project' s

$1, $1, ( / , , )

$1, $1, ( )

. ( )

ln .ln( )

. ln( ).

.

000 500 4

000 500 1

0 6667 1

0 6667

41

0101365 1

1

1

4

4

0 101365

0 101365

P F i

i

i

i

i

e i

i e

10.67%

Direct Solution Methods

Contemporary Engineering Economics, 4th edition, © 2007

Trial and Error Method – Project C

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

Step 4: If you bracket the solution, you use a linear interpolation to approximate the solution

3,5530

-749

15% i 18%

749553,3

553,3%3%15i

%45.17Note: This method works onlyfor finding i* for simple investments.

Contemporary Engineering Economics, 4th edition, © 2007

Graphical Method

Step 1: Create the NPWprofile.

Step 2: Find the point at which the curve crossesthe horizontal axis closelyapproximates i*

Contemporary Engineering Economics, 4th edition, © 2007

Using Cash Flow Analyzer – Project D

Input data

Output

196%