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ENGINEERING ECONOMICS ISE460SESSION 3
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OUTLINE
• Questions?
• News?
• Recommendations – no obligation
• Chapter 3
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Overview
• EXCEL DEMO OF CHAPTER 3 MATERIAL
• CHAPTER 3 ESSENTIALS
– FORMULAS
– EXAMPLES 3.25, 3.26, 3.27
– TABLES
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The time value of money
• If you have money, you can either
– Lend it and get interest
– Invest it and hope you get interest and that you get it back
– Invest it and hope you can resell it for more
– Use it to buy something or a service
• In the first two instances, when you ask for it back in the future, you’ll get your money plus what it earned. In other words, it will have grown
• Unfortunately, during the same time period, money in general will have lost some of its value due to inflation
• But if you invest wisely, the net should be higher
• In this chapter we assume that the two effects are combined in the quoted rates
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Graphical Representation
Time
Income
Expense
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Graphical Representation - Example
Time
Borrowed Money
Repayment
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Simple Interest
• At 5 % per year (per annum)
Deposit $100
One Year
$5 $5 $5 $5
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Compound Interest
• At 5 % per year (per annum), compounded annually
Deposit $100
One Year
$5.00 105*.05=$5.25
110.25*.05=$5.5125
115.76*0.05 = $5.788
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Equivalence
• We can compare different cash flows by comparing their
value at the same time
• The choice of time is arbitrary, but is usually the present or
some date in the future
• Equivalence is valid for only one interest rate
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Equivalence - example
• @ 5% compound interest
• $100 held as cash for 3 years will be worth $86.38
• $100 loaned today can be paid back in 3 equal annual
installments of $36.72
• $100 promised to be paid 2 years from now is worth $90.70
today
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Uniform Series
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Linear Gradient
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Geometric Gradient
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Irregular Series
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Notation
• An = payment or receipt occurring at the end of period n
• A = a constant payment or receipt at end of each period
• N = number of periods
• i = interest rate per interest period
• P = a sum of money at time zero = Present value or worth
• F = a sum of money at some future date (usually the end of
the time span that you are analyzing
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CHAPTER 3 FORMULAS
Name Formula Excel Function
Single PaymentCompound Amount Factor (compounding) NiPF 1 FV(i, N, 0, P)
Present Worth Factor (discounting) NiFP 1 PV(i, N, 0, F)
Equal Payments
Compound Amount Factor
i
iAF
N 11FV(i, N, A)
Sinking Fund Factor 1
11
i
iFA
N
PMT(i, N, 0, F)
Capital Recovery Factor (Annuity Factor)
1
1
11
N
N
ii
iPA PMT(i, N, P)
Present Worth Factor
N
N
ii
iAP
1
11PV(i, N, A)
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CHAPTER 3 FORMULAS (CONTINUED)
Linear Gradient Series
Present Worth Factor
N
N
ii
iNiiGP
1
112
none
Gradient to equal payment conversion Factor
]11[
11N
N
ii
iNiGA none
Geometric Gradient Series
Present Worth Factor
giiNAP
gigi
igAP
NN
,)1/(
,)1()1(1
1
1 none
Future Worth Factor
giiNAF
gigi
giAF
N
NN
,)1/(
,)1()1(
11
1none
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Derivations
• DERIVATION OF SOME FORMULAS
– SINGLE CASH FLOW
– UNIFORM SERIES OF PAYMENTS
– LINEAR GRADIENT SERIES
– GEOMETRIC GRADIENT SERIES
– IRREGULAR SERIES
• GO OVER REMAINING CHAPTER 3 EXAMPLES
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Formulas
F – future value
P – present value
N – number of periods
i – interest rate per period
A – payment at the end of each period
G – increase in payment per period starting in the second
A1 – Initial payment
n – Index for periods
g –percentage change from payment to payment
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Formulas – Future value, Present value
iF
P
N
iN
iFP
iF
P
ip
F
ipF
N
N
N
N
1log
log )6
1logF
Plog )5
1 )4
1 )3
)1( )2
)1( )1
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Formulas – Payments –the last term in 7) should be raised to N-1
11 )11
11 )10
11 )9
11 )7)8
1....11i1 )8
1....11 )72
2
N
N
N
N
N
i
iFA
i
iAF
iAiF
iAAFiF
iAiAiAF
iAiAiAAF
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Formulas – Payments (continued)
11
1 )14
1
11 )13
111
4) into )12
N
N
N
N
NN
i
iiPA
ii
iAP
ii
iAP
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Formulas – Gradient Series
Gradient Series
Ni
i
i
GF
ii
iNiGA
ii
iNNiGP
iGjP
iGNiGiG
iAiAiAP
N
N
N
NG
N
j
NG
N
NTOT
11 19)
10) into )18
11
11 18)
)14 into )17
1
11 )17
11 )16
11......1210
1....11 )15
2
1
32
11
21
11
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Formulas -- Geometric Gradient Series
• Geometric Gradient Series
N
n
n
N
n
nn
nnn
nnn
nn
i
g
g
AP
igAP
igAP
iAP
gAA
1
1
1
11
11
11
1
1
1 )24
11 23)
:payments allover Summing
11 22)
21) into )20
1 )21
1 )20
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Formulas -- Geometric Gradient (continued)
giiNA
gigi
igA
P
x
xxaP
xxaxPP
xxxaxP
xxxaP
i
gx
g
Aa
NN
N
N
N
N
1/
,111
27)
1 cannot x Note 1
)(
)(
)25)26
...( 26)
:by x gmultiplyin
...( )25
1
1
1
1
1
1
1
132
2
1
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Formulas -- Geometric Gradient (continued)
giiNA
gigi
igA
F
NN
1
,11
28)
27) into )1
1-N1
1
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Tables
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Equivalence at any point in time
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Example 3.6 - solution
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3.25
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3.26
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3.26
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3.26
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Example 3.27
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3.27 – cash flows
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3.27 – part a)
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3.27 part b)
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3.27 part b)
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3.27 part b)
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3.27 part b)