Rate of Return Analysis
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Rate of Return Analysis
ECON 320 Engineering EconomicsMahmut Ali GOKCEIndustrial Systems EngineeringComputer Sciences
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Chapter 7Rate of Return Analysis Rate of Return Methods for Finding
ROR Internal Rate of Return
(IRR) Criterion Incremental Analysis Mutually Exclusive
Alternatives
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Chapter Opening Story – What is worth of getting a college degree? Education beyond High
School College Degree: 14.5%
return on investment Post graduate degree:
6.5% return on investment
Source: The Age, January 29, 2003
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What’s A Degree Worth?The Class of 2004
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Rate of Return
• Definition: A relative percentage method which measures the yield as a percentage of investment over the life of a project
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In 1970, when Wal-Mart Stores, Inc. went public, an investment of 100 shares cost $1,650. That investment would have been worth $12,283,904 on January 31, 2002.
What is the rate of return on that investment?
Example: Meaning of Rate of Return
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Solution:
0
32
$12,283,904
$1,650
Given: P = $1,650 F = $12,283,904 N = 32
Find i:
$12,283,904 = $1,650 (1 + i )32
i = 32.13%
NiPF )1(
Rate of Return
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Suppose that you invested that amount ($1,650) in a savings account at 6% per year. Then, you could have only $10,648 on January, 2002.
What is the meaning of this 6% interest here?
This is your opportunity cost if putting money in savings account was the best you can do at that time!
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So, in 1970, as long as you earn more than 6% interest in another investment, you will take that investment.
Therefore, that 6% is viewed as a minimum attractive rate of return (or required rate of return).
So, you can apply the following decision rule, to see if the proposed investment is a good one.
ROR (32.13%) > MARR(6%)
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Why ROR measure is so popular?
• This project will bring in a 15% rate of return on investment.
• This project will result in a net surplus of $10,000 in NPW.
Which statement is easier to understand?
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Return on Investment
• Definition 1: Rate of return (ROR) is defined as the interest rate earned on the unpaid balance of an installment loan.
• Example: A bank lends $10,000 and receives annual payment of $4,021 over 3 years. The bank is said to earn a return of 10% on its loan of $10,000.
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Loan Balance Calculation:
A = $10,000 (A/P, 10%, 3)= $4,021
Unpaid Return on Unpaid balance unpaid balanceat beg. balance Payment at the end
Year of year (10%) received of year
0123
-$10,000-$10,000-$6,979-$3,656
-$1,000 -$698 -$366
+$4,021+$4,021+$4,021
-$10,000-$6,979-$3,656
0
A return of 10% on the amount still outstanding at the beginning of each year
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Rate of Return:
• Definition 2: Rate of return (ROR) is the break-even interest rate, i*, which equates the present worth of a project’s cash outflows to the present worth of its cash inflows.
• Mathematical Relation:
PW i PW i PW i( ) ( ) ( )* * *
cash inflows cash outflows
0
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Methods for Finding Rate of Return
Using Excel’s Financial Command Direct Solution Method Trial-and-Error Method Cash Flow Analyzer – Online Financial
Calculator
Excel command to find the rate of return: =IRR(cell range, guess) e.g., =IRR(C0:C7, 10%)
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Period (N)
Project A
0 -$1,000
1 -500
2 800
3 1,500
4 2,000
1234567891011
A B CPeriod Cash Flow
0 -10001 -5002 8003 15004 2000
IRR = 44%
=IRR(B3:B7,10%)
Finding Rate of Return on Excel
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Other Computational Methods
Direct Solution
Direct Solution
Trial & Error Method
Computer Solution Method
Log 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
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Direct Solution Methods
• 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%
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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
%45.17
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Using Cash Flow Analyzer – Project D
Input data
Output
196%
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Basic Decision Rule:
If ROR > MARR, Accept
This rule does not work for a situation where an investment has multiple rates of return