L3: Economic Equivalence
ECON 320 Engineering Economics
Mahmut Ali GOKCE
Industrial Systems Engineering
Computer Sciences
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Economic Equivalence
 What do we mean by “economic
equivalence?”
 Why do we need to establish an economic
equivalence?
 How do we establish an economic
equivalence?
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Economic Equivalence
 Economic equivalence exists between cash
flows that have the same economic effect
and could therefore be traded for one
another.
 Even though the amounts and timing of the
cash flows may differ, the appropriate interest
rate makes them equal.
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Equivalence from Personal Financing
Point of View
F
 If you deposit P dollars
today for N periods at
i, you will have F
dollars at the end of
period N.
PF
F  P(1  i) N
0
N
P
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Alternate Way of Defining Equivalence
P
 F dollars at the end of
period N is equal to a
single sum P dollars
now, if your earning
power is measured in
terms of interest rate i.
0
N
F
P  F (1  i)
0
N
N
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Practice Problem
At 8% interest, what is the equivalent worth
of $2,042 now 5 years from now?
If you deposit $2,042 today in a savings
account that pays 8% interest annually.
how much would you have at the end of
5 years?
$2,042
0
1
2
33
4
5
F
=
0
5
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Solution
F  $2,042(1  0.08)
5
 $3,000
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Example 2.2
At what interest rate
would these two amounts be equivalent?
$2,042
0
i=?
$3,000
5
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Equivalence Between Two Cash Flows
 Step 1: Determine the
$2,042
base period, say, year 5.
 Step 2: Identify the
interest rate to use.
 Step 3: Calculate
equivalence value.
0
$3,000
5
i  6%, F  $2,042(1  0.06)5  $2,733
i  8%, F  $2,042(1  0.08)5  $3,000
i  10%, F  $2,042(1  0.10)5  $3,289
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Example - Equivalence
Various dollar amounts that will be economically
equivalent to $3,000 in 5 years, given an interest
rate of 8%.
P
$3,000
 $2,042
5
(1  0.08)
P
F
$2,042 $2,205
0
1
$2,382
$2,572
2
3
$2,778
4
$3,000
5
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Example 2.3
V
$200
$150
$120
$100
$100
=
$80
0
1
2
3
4
5
0
1
2
3
4
5
Compute the equivalent lump-sum amount at n = 3 at 10% annual interest.
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Approach
V
$200
$150
$120
$100
$100
$80
0
1
2
3
4
5
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V3  $511.90  $264.46 V
 $776.36
$200
$200(1  0.10)1  $100(1  0.10)2
 $264.46
$150
$120
$100
$100
$80
0
1
2
3
4
5
100(1  0.10)3  $80(1  0.10)2  $120(1  0.10)  $150
 $511.90
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Practice Problem
2P
 How many years would
it take an investment to
double at 10% annual
interest?
0
N=?
P
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Solution
2P
F  2 P  P (1  0.10) N
2  1.1
N
0
N=?
P
log 2  N log1.1
log 2
N
log1.1
 7.27 years
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Rule of 72
Approximating
72
N
how long it will
interest rate (%)
take for a sum of
money to double
72

10
 7.2 years
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Practice Problem
$1,000
$500
Given: i = 10%,
A
0
Find: C that makes the
two cash flow streams
to be indifferent
1
2
C
C
3
B
0
1
2
3
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Approach
 Step 1: Select the base
period to use, say n =
2.
 Step 2: Find the
equivalent lump sum
value at n = 2 for both A
and B.
 Step 3: Equate both
equivalent values and
solve for unknown C.
$1,000
$500
A
0
1
2
C
C
3
B
0
1
2
3
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Solution
$1,000
 For A:
V2  $500(1  0.10)  $1,000(1  0.10)
2
 $1,514.09
$500
1
A
0
 For B:
V2  C (1  0.10)  C
 2.1C
 To Find C:
2.1C  $1, 514.09
C  $721
1
2
C
C
3
B
0
1
2
3
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Practice Problem
$1,000
At what interest rate
would you be
indifferent between the
two cash flows?
$500
A
0
1
2
3
$502 $502 $502
B
0
1
2
3
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Approach
 Step 1: Select the base
period to compute the
equivalent value (say, n
= 3)
A
 Step 2: Find the net
worth of each at n = 3.
$1,000
$500
0
1
2
$502
$502
3
$502
B
0
1
2
3
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Establish Equivalence at n = 3
Option A : F3  $500(1  i)  $1, 000
3
Option B : F3  $502(1  i) 2  $502(1  i)  $502
 Find the solution by trial and error, say i = 8%
Option A : F3  $500(1.08)3  $1, 000
 $1, 630
Option B : F3  $502(1.08) 2  $502(1.08)  $502
 $1, 630
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