INFO 631
Prof. Glenn Booker
Week 3 – Chapters 7-9
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Equivalence
Chapter 7
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Equivalence
Outline
•
•
•
•
Simple comparison of two proposals
Equivalence, defined
Simple equivalence
Equivalence with varying cash-flow
instances
• Equivalence with varying interest rates
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Simple Comparison of Two Proposals
• Your company sells a product for $20,000
– A customer offers to pay $2500 at the end of each of the next 10 years
instead. Is this a good deal?
End of Year
0
1
2
3
4
5
6
7
8
9
10
Total
Pay now
$20,000
$0
$0
$0
$0
$0
$0
$0
$0
$0
$0
$20,000
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Pay later
$0
$2500
$2500
$2500
$2500
$2500
$2500
$2500
$2500
$2500
$2500
$25,000
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Simple Comparison of Two
Proposals (cont)
• How do I evaluate?
• Impact of time?
– Interest
– What does 0% interest mean?
– Is this realistic?
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Simple Comparison of Two Proposals (cont)
• That analysis assumed 0% interest
– The interest rate is unlikely to be 0%
• What if we use a more reasonable interest
rate, say 9%?
P/A, 9%, 10
P = $2500 ( 6.4177 ) = $16,044
A/P, 9%, 10
A = $20,000 ( 0.1558 ) = $3116
P/A = equal-payment-series present-worth
A/P = equal-payment capital-recovery
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Recall - Naming Conventions in Interest
Formulas
•
P
– “Principal Amount”—how much is the money worth right now?
– Also known as “present value” or “present worth”
•
F
– “Final Amount”—how much will the money be worth at a later time?
– Also known as the “future value” or “future worth”
•
i
– Interest rate per period
– Assumed to be an annual rate unless stated otherwise
•
n
– Number of interest periods between the two points in time
•
A
– “Annuity”—a stream of recurring, equal payments that would be due at the end of
each interest period
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Equivalence
“Two or more different cash-flow instances
(or cash-flow streams) are equivalent at a
given interest rate only when they equal
the same amount of money at a common
point in time. More specifically, comparing
two different cash flows makes sense only
when they are expressed in the same time
frame”
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Equivalence (cont)
• Equivalence at one time means equivalence
at all other times
– Equivalence (or more appropriately the lack of it)
can be used as a basis of choice
– Basis of decision making
• If both proposal are equivalent, doesn't matter which
one we choose
• If different, one is better than the other
• Economic comparisons need to be made on
an equivalent basis
– Or you could make the wrong decision
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Simple Equivalence
• The compound interest formulas are statements of
simple equivalence (single payment compound
amount (F/P))
n
F  P1  i 
– If i% interest is fair, you would be indifferent to getting $P
now compared to getting $F after n interest periods
– Note: Fair is important word. Why?
• Simple equivalence in action
– Fast food joint pays its contest winner $2 million, as $200k
annually for 10 years
– Using an interest rate of 7%, that’s really
P/A, 7%, 10
P = $200k ( 7.0236 ) = $1.4 million
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Equivalence with Varying
Cash-Flow Instances
• Equivalence applied to entire cash flow
stream
– Each instance translated into common
reference time frame, then add them up
• Two approaches
– Elegant Approach
• Better when done by hand
• Hard to automate
– Brute Force Approach
• Easy to automate
• A lot of computations if done by hand
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Equivalence With Varying Cash-Flow
Instances (Elegant Approach)
Steps (pg 101)
1. Choose the reference time frame
2. Break cash flow stream into segments
3. For each segment, apply appropriate
formula to translate it into the reference
time frame
4. Sum up all the results
 Represents the net equivalent value of chase
flow stream in terms of reference time frame
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Equivalence With Varying Cash-Flow
Instances (Elegant Approach)
End of
Year
Partial present equivalent amounts
0
$2657.70
$878.36
+ $1386.76
- $156.11 + $548.69
1
P/F,12,3
2
3 $1234
$1234 ( 0.7118 )
P/A,12,5
P/F,12,5
4
5
$678 ( 3.6048 ) = $2444.05
$2444.05 ( 0.5674 )
6
$678
7
$678
8
$678
9
$678
P/F,12,11
10
$678
11 -$543
-$543 ( 0.2875 )
12
13
$890
F/A,12,3
P/F,12,15
14
$890
15
$890
$890 ( 3.3744 ) = $3003.21
$3003.21 ( 0.1827 )
Assume Interest = 12%, 15 Year,
P/F, i, n
Using Single Payment Present Worth Value = P = F (
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Recall - Compound Interest
Formulas
• Six different compound interest formulas
– Single-payment compound-amount (F/P)
– Single-payment present-worth (P/F)
– Equal-payment-series compound-amount
(F/A)
– Equal-payment-series sinking-fund (A/F)
– Equal-payment-series capital-recovery (A/P)
– Equal-payment-series present-worth (P/A)
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Equivalence With Varying Cash-Flow
Instances (Brute Force)
Translate each
cash flow into
reference time
frame (now)
using Single
Payment
Compound
Interest
Year
n
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Net cash-flow
at end of year
$0
$0
$1234
$0
$0
$678
$678
$678
$678
$678
-$543
$0
$890
$890
$890
Present-worth factor
(P/F,12%,n)
0.8929
0.7972
0.7118
0.6355
0.5674
0.5066
0.4523
0.4039
0.3606
0.3220
0.2875
0.2567
0.2292
0.2046
0.1827
Total
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Equivalent value
at end of year 0
$0
$0
$878.36
$0
$0
$343.47
$306.66
$273.84
$244.49
$218.32
-$156.11
$0
$203.99
$182.09
$162.60
$2657.71
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Equivalence With Varying
Cash-Flow Instances
• Last scenario assumed a single interest
rate
• Is that always the correct assumption?
– In general yes.
• Interest rates do usually change over time
• Most business decision are based on “nominal”
interest rate
• What happens if interest rate varies?
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Equivalence With Varying Interest Rates
(Elegant Approach)
End of
Year
Partial present equivalent amounts
0
$2706.90
$878.36 + $1168.44 + $222.26 - $161.82 + $599.66
1
2
P /F,12,3
3 $1234
$1234 ( 0.7118 )
4
P /A,12,4
P /F,12,5
12%
5
$678 ( 3.0373 ) = $2059.29
$2059.29 ( 0.5674 )
6
$678
7
$678
8
$678
P /F,12,9
P /F,12,9
P /F,12,9
9
$678
P /F,10,1
$616.37 ( 0.3606 ) -$448.74 ( 0.3606 ) $1662.96 ( 0.3606 )
10 $678
$678 ( 0.9091 )
P /F,10,2
11 -$543
-$543 ( 0.8264 )
12
10%
13 $890
14 $890
F/A,10,3
P /F,10,6
15 $890
$890 ( 3.3100 ) = $2945.90
$2945.90 ( 0.5645 )
Notice the interest rate is now 12% above the red line, 10% below it
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Equivalence With Varying Interest Rates
(cont) (Brute Force)
• In the 12% region
Year
n
1
2
3
4
5
6
7
8
9
Net cash-flow
at end of year
$0
$0
$1234
$0
$0
$678
$678
$678
$678
Present-worth factor
(P/F,12%,n)
0.8929
0.7972
0.7118
0.6355
0.5674
0.5066
0.4523
0.4039
0.3606
Total
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Equivalent value
at end of year 0
$0
$0
$878.36
$0
$0
$343.47
$306.66
$273.84
$244.49
$2046.82
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Equivalence With Varying Interest Rates
(more) (Brute Force)
• In the 10% region
Year
n
1
2
3
4
5
6
Net cash-flow
at end of year
$678
-$543
$0
$890
$890
$890
Present-worth factor
(P/F,10%,n)
0.9091
0.8264
0.7513
0.6830
0.6209
0.5645
Total
Equivalent value
at end of year 0
$616.37
-$448.74
$0
$607.87
$552.60
$502.41
$1830.51
• Translating that to the beginning of the 12% region
P/F, 12%, 9
P = $1830.51 ( 0.3603 ) = $660.08
• Adding that to the previous sum for the 12% region
$2046.82 + $660.08 = $2706.90
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Key Points
• Comparisons must be made on an
equivalent basis
• Interest formulas are statements of simple
equivalence
• Different cash-flow instances can be
translated into an equivalent basis
– This can be done across different interest
rates
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Sample Exercise
$100
$80
$50
0
1
2
3
4
5
6
7
8
9
-$20
-$100
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Exercise Answer
P/F, 6,1
P/A,6,4
P/F,6,3
P/F,6,8
P/F,6,9
-$100 + -$20 (0.9434) + $50 (3.4651)(0.8396) + $80 (0.6247) + $100 (0.5919)
-$100 - $19 + $145 + $50 + $59 = $135
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Bases for Comparison
Chapter 8
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Bases for Comparison
Outline
•
•
•
•
•
•
•
•
•
•
Basis for comparison defined
An example
Present worth
Future worth
Annual equivalent
Internal rate of return
Payback period
Discounted payback period
Project balance
Capitalized equivalent amount
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Basis for Comparison
• Last chapter discusses how cash-flow
instances can be added, subtracted,
compare at the same time frame
• Will expand to different cash-flow streams:
– A common frame of reference for comparing
two or more cash-flow streams in a consistent
way
• Basically, all streams are converted into the same
basis, such as Present Worth
• Then compared
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Basis for Comparison con’t
• Six Bases
– Present worth
– Future worth
– Annual equivalent
– Internal rate of return
– Payback period
– Capitalized equivalent amount
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Comparing Cash-Flow Streams
• Need to be converted into same basis
• After all proposal expressed in same basis
for comparison
– Best one obvious
– Mechanics of actual choice in Chapter 9
• Caution
– Always use the same
• Interest (i)
• Study Period (n)
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An Example - Project: Automated Test
Equipment (ATE)
Taken from Lecture Ch 3
•
One person-year = $125k
•
Initial investment
– $300k for test hardware and development equipment (Year 0)
– 20 person-years of software development staff (Year 1)
– 10 person-years of software development staff (Year 2)
•
Operating and maintenance costs
– $30k per year for test hardware and dev equipment (Years 1-10)
– 5 person-years of software maintenance staff (Years 3-10)
•
Sales income
– None
•
Cost avoidance
– $1.3 million in reduced factory staffing (Years 2-10)
•
Salvage value
– Negligible
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Automated Test Equipment (ATE) Simple
Example
Year
0
1
2
3
4
5
6
7
8
9
10
Dev Staff
0
-$2.5M
-$1.25M
-$625K
-$625K
-$625K
-$625K
-$625K
-$625K
-$625K
-$625K
Equipment
-$300K
0
0
0
0
0
0
0
0
0
0
O&M
0
-$30K
-$30K
-$30K
-$30K
-$30K
-$30K
-$30K
-$30K
-$30K
-$30K
Savings
0
0
$1.3M
$1.3M
$1.3M
$1.3M
$1.3M
$1.3M
$1.3M
$1.3M
$1.3M
Total
-$300K
-$2.53M
$20K
$645K
$645K
$645K
$645K
$645K
$645K
$645K
$645K
Example from Ch 3 Lecture Slides
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The ATE Example – Cash Flow Stream
$645K
0
1
$20K
2
3
4
5
6
7
8
9
10
-$300K
-$2.53M
Example from Ch 3 Lecture Slides
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Present Worth, PW(i)
• How much is the future cash-flow
stream worth (equivalent to) right now at
interest rate, i?
– Reference time for PW(i) =
• Beginning of first period (end of period
0)
• Also called Net Present Value (NPV)
– How much is the cash-flow stream worth
today?

NOTE: “Present” - can be any arbitrary point in time as
appropriate for decision
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Present Worth, PW(i)
• Formula
– Uses single-payment present-worth (P/F,i,n) to translate
each individual net-cash flow
– Then sum all amounts
n
PW(i)   Ft 1  i 
t
t 0
Ft = net-cash flow instance in period t
Notes: Except for Year 0, PW values are always < original cash flow.
Process of translating cash-flow backwards is referred to as “discounting”
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Present Worth, PW(i) (cont)
• Manual calculation of PW(10%) for ATE
Year
n
0
1
2
3
4
5
6
7
8
9
10
Net cash-flow
at end of year
-$300K
-$2,530K
$20K
$645K
$645K
$645K
$645K
$645K
$645K
$645K
$645K
Present-worth factor
Equivalent value
(P/F,10%,n)
at end of year 0
1.0000
-$300K
0.9091
-$2,300K
0.8264
$17K
0.7513
$485K
0.6830
$441K
0.6209
$400K
0.5645
$364K
0.5132
$331K
0.4665
$301K
0.4241
$274K
0.3855
$249K
PW(10%)
$260K
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Comments on PW(i)
• There is a single value of PW(i) for any i
– Generally, as i increases PW(i) decreases
Critical i,
where PW(i) = 0,
is IRR (slide 43)
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Comments on PW(i) (cont)
• 2nd most widely used basis for comparison
– Future Value is 1st
• PW(i) over -1 < i < oo is meaningful
– Only 0 < i < oo is important
– Negative interest rates almost impossible
• Graph shows several important things
– Equivalent profit or loss at any i
– What ranges of i would be profitable
– The “critical i” where PW(i)=0
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Future Worth, FW(i)
• Just like PW(i) except it’s referenced to a
future point in time
– Reference time for FW(i) =
• Usually the end of the cash-flow stream
– Answer the question:
» How much is this proposal worth in the
end-of-the-proposal time frame?
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Future Worth, FW(i) Con’t
• Formula
– Uses single-payment compound-amount
(F/P,i,n) to translate each individual net-cash
flow instance
– Then sum all amounts
n
FW(i)   Ft 1  i 
n t
t 0
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Present Worth, FW(i) (cont)
• Manual calculation of FW(10%) for ATE
Year
n
0
1
2
3
4
5
6
7
8
9
10
Net cash-flow
at end of year
-$300K
-$2,530K
$20K
$645K
$645K
$645K
$645K
$645K
$645K
$645K
$645K
Future-worth factor
Equivalent value
(F/P,10%,n)
at end of year
2.5937
-$778K
2.3579
-$5,965K
2.1436
$42K
1.9487
$1256K
1.7716
$1142K
1.6105
$1038K
1.4641
$944K
1.3310
$858K
1.2100
$780K
1.1000
$709K
1.0000
$645K
FW(10%)
$675K
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Comments on FW(i)
• The only difference between PW(i) and FW(i) is the time
frame
– PW(i) and FW(i) are mathematically related
– Number from class example above
F/P, i, n
FW(i) = PW(i) (
F/P, 10%, 10
)
FW(10%) = $260K ( 2.5937 ) = $675K
• For fixed i and n, FW(i) = PW(i) times a constant
– FW(i) = 0 when PW(i) = 0
• for the same value of “critical i”
– Comparing cash-flow streams in FW(i) terms will always lead to
the same conclusion as comparing with PW(i)
• Assuming used consistently for all cash-flow streams
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Annual Equivalent, AE(i)
• PW(i) and FW(i) represent the cash-flow stream as an
equivalent one-time cash-flow instance
– Either:
• at the beginning (PW) or
• at the end (FW) of the cash-flow stream
• AE(i) represents it as a series of equal cash-flow
instances over the life of the study
– AE(i) relates to PW(i) the same as A relates to P
A/P, i, n
AE(i) = PW(i) (
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Annual Equivalent, AE(i) (cont)
• Formula
 i (1  i ) n 
 n
t 
AE(i)   Ft 1  i    

n
(
1

i
)

1
 t 0
 

• Manual calculation of AE(10%)
– Start with PW(i) and multiple by equal-payment-series
capital recovery (A/P,i,n) factor.
A/P, 10%, 10
AE(10%) = PW(10%) (
) = $260K ( 0.1627 ) = $42.3K
Cash flow stream equivalent = $42.3 K at the end of each of the next 10 yrs
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Comments on AE(i)
• For fixed i and n, AE(i) = PW(i) times a constant
– AE(i) = 0 when PW(i) = 0
• for the same value of “critical i”
– Comparing cash-flow streams in AE(i) terms will always lead to
the same conclusion as comparing with PW(i)
• Assuming used consistently for all proposals
• Advantage
– AE(i) form is useful for repeating cash-flow streams
– Easy to represent as annual equivalents
• If the ATE project can be repeated, AE(i) = $42.3K over 20
years, or over 30 years, …
– Example: renewable bond
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Internal Rate of Return, IRR
• PW(i), FW(i), and AE(i) express the cashflow stream as equivalent dollar amounts
– IRR expresses the cash-flow stream as an
interest rate
• What interest rate would a bank have to pay to match your
payments and withdraws and end up with $0 at the end of the cashflow stream?
• Also called Return on Investment (ROI)
• Occurs when “critical i” brings PW(i) to zero (next slide)
• Formula
n
0  PW(i * )   Ft 1  i 
t
t 0
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PW(i) = 0, Critical i at IRR
Critical i,
where PW(i) = 0,
discussed later IRR
Yup, the same figure from slide 34
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Internal Rate of Return, IRR (cont)
• To compute IRR, the cash-flow stream must have these
properties:
– First nonzero net cash-flow is negative (expense)
– That is followed by 0..n further expenses followed by incomes
from there on
• Only one sign change in the cash-flow stream
– The net cash-flow stream is profitable
• Sum of all income > sum of all expenses
• PW(0%)>$0
• If not met, do not use
– Criteria might not have IRR or
– Might have more then 1 IRR
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Computing IRR Algorithm
Given the cash flow stream with
the first non-zero cash flow being negative,
and only 1 sign change,
and PW(0%) > 0
Start with the estimated IRR = 0%
Assume we will move IRR in an increasing (+) direction
Assume an initial step amount (say, 10%)
Calculate PW(i=0%) and save the result
Move the IRR in the current direction by the step amount
repeat
recalculate the PW(i=IRR)
if the PW(i=IRR) is closer to $0.00 than before
then move the estimated IRR in the same direction
by the step amount
else switch direction and cut the step amount in
half
until the PW(i=IRR) is within a pre-determined range
of $0.00 (say, 50 cents)
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Computing IRR (cont)
1.
2.
3.
4.
5.
6.
7.
8.
9.
Start with IRR = PW(0%) = $2,350K, step = 10%, direction = increasing
Calculate PW(10%), it’s $260K. That’s closer to zero than $2,350K so move the estimated IRR
in the same direction (up) by another 10%. It’s now estimated to be 20%.
Calculate PW(20%), it’s -$676K. That’s farther from zero than -$260K so switch direction and
cut the step amount in half, to 5%. The estimated IRR is now 15%.
Calculate PW(15%), it’s -$296K. That’s closer to zero than -$676K so move the estimated IRR
in the same direction by another 5%. It’s now estimated to be 10%.
Calculate PW(10%), it’s $260K. That’s closer to zero than -$296K so move the estimated IRR in
the same direction (down) by another 5%. It’s now estimated to be 5%.
Calculate PW(5%), it’s $1090K. That’s farther from zero than -$260K so switch direction and cut
the step amount in half, to 2.5%. The estimated IRR is now 7.5%.
Calculate PW(7.5%), it’s $633K. That’s closer to zero than $1090K so move the estimated IRR
in the same direction by another 2.55%. It’s now estimated to be 10%.
Calculate PW(10%), it’s $260K. That’s closer to zero than $633K so move the estimated IRR in
the same direction (up) by another 2.5%. It’s now estimated to be 12.5%...
… and so on while the PW(i) at the estimated IRR converges on $0.00. When the PW(i) is
within +/- $0.50 of $0, the loop stops and the estimated IRR of 12.1% is returned.
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Payback Period, PP
• PW(i), FW(i), and AE(i) express the cash-flow stream as
equivalent dollar amounts and IRR expresses it as an
interest rate
– Payback period expresses the cash-flow stream as a time
• how long to recover the investment
• Like saying “This investment will pay for itself in 5 years”
• Formula
– Smallest n where
n
F
t 0
t
0
Ft = net-cash flow instance in period t
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Payback Period, PP (cont)
• Manual calculation of PP for ATE
Year
n
0
1
2
3
4
5
6
7
Net cash-flow
at end of year
-$300K
-$2,530K
$20K
$645K
$645K
$645K
$645K
$645K
Running sum
thru year n
-$300K
-$2,830K
-$2,810K
-$2,165K
-$1,520K
-$875K
-$230K
$415K
n=7
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Comments on Payback Period
• PW(i), FW(i), AE(i), and IRR
– Indicators of profitability
• Payback Period
– Indicator of liquidity
– Organization’s exposure to risk of financial loss
• Example
– If the project starts but gets canceled before the end of the
payback period, the organization loses money
• Payback = 5 is better then Payback = 10 yrs
– Less financial risk
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Discounted Payback Period, DPP(i)
• Payback period doesn’t address interest
– Discounted payback period does
– So DPP is a much more realistic measure!
• Formula
– Smallest n where
n
 Ft 1  i 
t
0
t 0
NOTE: DPP for next slide is before end of 9th year. Use linear
interpolation technique in Appendix C to find precise DPP.
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Discounted Payback Period, DPP(i) (cont)
• Manual calculation of DPP(10%) for ATE
Year
n
0
1
2
3
4
5
6
7
8
9
Net cash-flow
at end of year
-$300K
-$2,530K
$20K
$645K
$645K
$645K
$645K
$645K
$645K
$645K
Present-worth factor
(P/F,10%,n)
1.0000
0.9091
0.8264
0.7513
0.6830
0.6209
0.5645
0.5132
0.4665
0.4241
Equivalent value
at end of year 0
-$300K
-$2,300K
$17K
$485K
$441K
$400K
$364K
$331K
$301K
$274K
INFO631 Week 3
Running sum
through year n
-$300K
-$2,600K
-$2,583K
-$2,099K
-$1,658K
-$1,258K
-$894K
-$563K
-$262K
$12K
n=9
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Project Balance, PB(i)
• Not really a basis of comparison but
closely related to DPP(i)
– Simply continues DPP(i) calculations for the
life of the cash-flow stream
– PB(i) = profile that shows the equivalent
amount of dollars invested, or earned from,
the proposal at the end of time period over life
of cash-flow stream.
• Formula
PB(i )T 
T
 Ft 1  i 
T t
t 0
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Project Balance, PB(i) (cont)
• Manual calculation of PB(10%) for ATE
Year
n
0
1
2
3
4
5
6
7
8
9
10
Net cash-flow
at end of year
-$300K
-$2,530K
$20K
$645K
$645K
$645K
$645K
$645K
$645K
$645K
$645K
Present-worth factor
(P/F,10%,n)
1.0000
0.9091
0.8264
0.7513
0.6830
0.6209
0.5645
0.5132
0.4665
0.4241
0.3855
INFO631 Week 3
Equivalent value
at end of year 0
-$300K
-$2,300K
$17K
$485K
$441K
$400K
$364K
$331K
$301K
$274K
$249K
Running sum
through year n
-$300K
-$2,600K
-$2,583K
-$2,099K
-$1,658K
-$1,258K
-$894K
-$563K
-$262K
$12K
$260K
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Graph of PB(10%) for ATE
0
8
1
2
4
3
5
6
7
-$300K
$260K
$12K
9
-$262K
Net
Equiv $
Earned
10
-$563K
-$894K
-$1.26M
-$1.67M
Net
Equiv $
Exposed
Risk
-$2.10M
-$2.60M
-$2.58M
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Capitalized Equivalent Amount CE(i)
• Formal:
– CE(i) = dollar amount now, that at a given interest rate, will be
equivalent to the net difference of the income and payments if
the cash-flow pattern is repeated indefinitely
• Informal
– Amount to invest at interest rate i to produce an equivalent cashflow stream on interest alone
• Example
– Self-supporting endowments
• Formula
CE (i ) 
AE (i )
i
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Capitalized Equivalent Amount CE(i) for
ATE
Get AE(i) for project ATE from slide 41
$42.3K
CE (10%) 
 $423K
0.10
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Key Points
• A basis for comparison is a common frame of reference
– Use of equivalence
• Eight different bases were discussed:
– Present worth—how much is it worth today?
– Future worth—how much will it be worth later?
– Annual equivalent—how much as a set of equal cash-flow
instances?
– Internal rate of return—what’s the equivalent interest rate
– Payback period -- how long to recover the investment?
– Discounted payback period—how long to recover the
investment with interest?
– Project balance—what is the balance over time?
– Capitalized equivalent amount—how much capital is frozen?
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Developing Mutually Exclusive
Alternatives
Chapter 9
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Developing Mutually Exclusive Alternative
Outline
• Independent proposals
• Dependent proposals
– Co-dependent proposals
– Mutual exclusive proposals
– Contingent proposals
• Developing mutually-exclusive alternatives
• “Do-nothing” alternative
• Cash-flow streams for alternatives
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Independent Proposals
• A set of proposals are independent when
selecting any one from that set has no
effect on accepting any other
– Ignoring, for now, resource constraints
• Example
– A proposal to develop a system that predicts
the stock market vs. a proposal to develop a
system that plays chess
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Dependent Proposals
• A set of proposals are dependent when
selecting any one from that set can have
an effect on accepting any other
• Forms of dependency
– Co-dependent
– Mutually exclusive
– Contingent
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Co-Dependent Proposals
• A set of proposals are co-dependent when
selecting any one from that set requires
accepting another
– These proposals should be combined into one
• Example
– Upgrade to new operating system and buy
more memory
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Mutually Exclusive Proposals
• A pair of proposals are mutually exclusive
when selecting one from that pair negates
accepting the other
• Example
– Get Java compiler from Vendor A vs. Java
compiler from Vendor B
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Contingent Proposals
• A pair of proposals are contingent when
selecting one from that pair requires
accepting the other, but not the other way
• Example
– Using the Swing UI toolkit vs. switching to
Java
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Mutually Exclusive Alternatives
• Mutual exclusion among choices is easiest to
work with
• In many cases you’ll have resources to do more
than one proposal at the same time
• A systematic way of turning proposals, along
with their dependencies, into a set of mutually
exclusive possible courses of action would be
handy
– An alternative is a unique, mutually exclusive course
of action consisting of a set of zero or more proposals
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Developing Mutually Exclusive
Alternatives, Step 1
• Generate the set of all theoretically possible
combinations of proposals
– Build a matrix with a column for each proposal and a row for
each alternative
– Fill in the cells to form all potential alternatives
• “1” in cell (I,J) means Proposal(I) is in Alternative(J)
• “0” in cell (I,J) means it’s not
– Notice the binary counting
• Under Proposal(1) alternate 0,1,0,1,…
• Under Proposal(2) alternate 0,0,1,1,0,0,1,1,…
• Under Proposal(k) alternate 2 k-1 0’s followed by an equal
number of 1’s
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Example of Step 1
Alternative
A0
A1
A2
A3
A4
A5
A6
A7
P1
0
1
0
1
0
1
0
1
P2
0
0
1
1
0
0
1
1
P3
0
0
0
0
1
1
1
1
INFO631 Week 3
Meaning
“Do nothing”
P1 only
P2 only
P1 and P2
P3 only
P1 and P3
P2 and P3
All
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Developing Mutually Exclusive
Alternatives, Step 2
• Remove all invalid alternatives
– Any alternative containing mutually exclusive proposals
– Any alternative containing unsatisfied contingencies
– Any alternatives exceeding resource constraints
• Example, assume:
– P1 and P2 are mutually exclusive
– P3 is contingent on P2
– Can’t afford to do all three at same time
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Example of Step 2
Alternative
A0
A1
A2
A3
A4
A5
A6
A7
P1
0
1
0
1
0
1
0
1
P2
0
0
1
1
0
0
1
1
P3
0
0
0
0
1
1
1
1
INFO631 Week 3
Meaning
“do nothing”
P1 only
P2 only
P1 and P2
P3 only
P1 and P3
P2 and P3
All
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The “Do Nothing” Alternative
• Notice alternative A0 is called “do nothing”
– Doesn’t really mean doing nothing at all
– Only means that none of the proposals in the set being
considered are carried out
– Instead, money is put into other investments that give a predetermined rate of return
• Bonds, interest bearing accounts, a more profitable part of
the corporation, etc.
• “Do nothing” should always be considered except when
– You’re required to do something
• e.g., repair or replace broken equipment
– You’re working with “service alternatives” (see Chapter 11)
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The “Do Nothing” Alternative (cont)
• Sometimes even the best of the proposals is
worse than what could be achieved by investing
somewhere else
– When the “do nothing” alternative comes out the best,
it means the organization would be better off not
carrying out any of the proposals being considered
and should put the money into a more profitable
investment elsewhere
• A0 is assumed to have
– PW(i) = $0
– FW(i) = $0
– AE(i) = $0
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Cash-Flow Stream for Alternatives
• The cash-flow stream for any alternative
(other than A0) will be the sum of the
cash-flow streams of all proposals it
contains
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Key Points
• There are several forms of dependency between
proposals
• Decisions are easiest when choices are mutually
exclusive
• An alternative is a set of zero to many proposals
• There is a process for turning proposals with
dependencies into valid, mutually exclusive alternatives
• The “do nothing” alternative doesn’t really mean do
nothing at all, just none of the projects proposed
• The cash-flow stream for an alternative is the sum of the
cash-flow streams for all its proposals
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