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Engineering Economy
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Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved2-1
Developed By:
Dr. Don Smith, P.E.
Department of Industrial Engineering
Texas A&M University
College Station, Texas
Executive Summary Version
Chapter 2Chapter 2
Factors: How Time Factors: How Time and Interest Affect and Interest Affect
MoneyMoney
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved2-2
LEARNING OBJECTIVESLEARNING OBJECTIVES1. F/P and P/F
factors
2. P/A and A/P factors
3. Interpolate for factor values
4. P/G and A/G factors
5. Geometric gradient
6. Calculate i
7. Calculate n
8. Spreadsheets
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved2-3
Sct 2.1 Single-Payment FactorsSct 2.1 Single-Payment Factors(F/P and P/F)(F/P and P/F)
Objective:Derive factors to determine the present or future
worth of a cash flow
Cash Flow Diagram – basic format
0 1 2 3 n-1 n
P0
Fn
i% / period
P0 = Fn1/(1+i)n →(P/F,i%,n) factor: Excel: =PV(i%,n,,F)
Fn = P0(1+i)n →(F/P,i%,n) factor: Excel: =FV(i%,n,,P)
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved2-4
Sct 2.2 Uniform-Series: Present Worth Sct 2.2 Uniform-Series: Present Worth Factor (P/A) and Factor (P/A) and
Capital Recovery Factor(A/P)Capital Recovery Factor(A/P) Cash flow profile for P/A factor
. . . . 0 1 2 3 n-2 n-1 n$A per interest period
i% per interest period
Required: To find P given A
Cash flows are equal, uninterrupted and flow at the end of each interest period
Find P
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved2-5
(P/A) Factor Derivation(P/A) Factor Derivation Setup the following:
Multiply by to obtain a second equation…
Subtract (1) from (2) to yield…
1 2 1
1 1 1 1..
(1 ) (1 ) (1 ) (1 )n nP A
i i i i
2 3 1
1 1 1 1..
1 (1 ) (1 ) (1 ) (1 )n n
PA
i i i i i
1
(1+i)
(1)
(2)
1
1 1
1 (1 ) (1 )n
iP Ai i i
(3)
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved2-6
(P/A) and (A/P) Factor Formulas(P/A) and (A/P) Factor Formulas Simplify (3) to yield…
Solve (4) for A to get (A/P) factor
(1 ) 1 0
(1 )
n
n
iP A for i
i i
(4)
(1 )
(1 ) 1
n
n
i iA P
i
(P/A,i%,n) factor
Excel: =PV(i%,n,A)
(A/P,i%,n) factor
Excel: =PMT(i%,n,P)(5)
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved2-7
ANSI Standard Notation for ANSI Standard Notation for Interest FactorsInterest Factors
Standard notation has been adopted to represent the various interest factors
Consists of two cash flow symbols, the interest rate, and the number of time periods
General form: (X/Y,i%,n) X represents what is unknown Y represents what is known i and n represent input parameters; can be known or
unknown depending upon the problem
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved2-8
Notation - continuedNotation - continued
Example: (F/P,6%,20) is read as:To find F, given P when the interest rate is 6% and
the number of time periods equals 20.
In problem formulation, the standard notation is often used in place of the closed-form equivalent relations (factor)
Tables at the back of the text provide tabulations of common values for i% and n
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved2-9
Sct 2.3 Sinking Fund Factor and Uniform Sct 2.3 Sinking Fund Factor and Uniform Series Compound Amount FactorSeries Compound Amount Factor
(A/F and F/A)(A/F and F/A)
Cash flow diagram for (A/F) factor
Start with what has already been developed1 (1 )
(1 ) (1 ) 1
n
n n
i iA F
i i
. . . . 0 1 2 3 n-2 n-1 nA=? per interest period
i% per interest period
F = given
Find A, given F
(1 ) 1n
iA F
i
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved2-10
(F/A) factor from (A/F)(F/A) factor from (A/F)
Given:
Solve for F in terms of A to yield
(1 ) 1n
iA F
i
(1 ) 1niF A
i
(A/F,i%,n) factor
Excel: =PMT(i%,n,,F)
(F/A,i%,n) factor
Excel: =FV(i%,n,A)
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved2-11
Sct 2.4 Interpolation in Interest TablesSct 2.4 Interpolation in Interest Tables When using tabulated interest tables one
might be forced to approximate a factor that is not tabulated
Can apply linear interpolation to approximateSee Table 2-4Factors are nonlinear functions, hence linear
interpolation will yield errors in the 2-4% rangeUse a spreadsheet model to calculate the factor
precisely
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved2-12
Sct 2.5 Arithmetic Gradient FactorsSct 2.5 Arithmetic Gradient Factors(P/G) and (A/G)(P/G) and (A/G)
Cash flow profile
0 1 2 3 n-1 n
A1+G
A1+2G
A1+(n-2)G
A1+(n-1)G
Find P, given gradient cash flow G
CFn = A1 ± (n-1)G
Base amount = A1
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved2-13
Gradient ExampleGradient Example
0 1 2 3 4 5 6 7
$100
$200
$300
$400
$500
$600
$700
Gradients have two components:
1. The base amount and the gradient
2. The base amount (above) = $100/time period
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved2-14
Gradient ComponentsGradient Components
…….. 0 1 2 3 n-2 n-1 n
Base amount = A / period
0G
1G 2G
(n-3)G (n-2)G (n-1)G
Present worth point is 1 period to the left of the 0G cash flow
For present worth of the base amount, use the P/A factor (already known)
For present worth of the gradient series, use the P/G factor (to be derived)
Find P of gradient series
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved2-15
Gradient DecompositionGradient Decomposition As we know, arithmetic gradients are
comprised of two components1. Gradient component2. Base amount
When working with a cash flow containing a gradient, the (P/G) factor is only for the gradient component
Apply the (P/A) factor to work on the base amount component
P = PW(gradient) + PW(base amount)
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved2-16
Derivation Summary for (P/G)Derivation Summary for (P/G) Start with:
Multiply (1) by (1+i)1 to create a second equationSubtract (1) from the second equation and simplifyYields…
( / , , 2) 2 ( / , ,3) 3 ( / , , 4) ...
+[(n-2)G](P/F,i,n-1)+[(n-1)G](P/F,i,n)
P G P F i G P F i G P F i (1)
2
G (1 ) 1 (1 ) 1P=
i (1 ) (1 ) (1 )
n n
n n n
i n i in
i i i i i
(P/G,i,n) factorNo Excel relation exists
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved2-17
Use of the (A/G) FactorUse of the (A/G) Factor
0 1 2 3 n-1 n
G
2G
(n-2)G
(n-1)G
Find A, given gradient cash flow G
CFn = (n-1)G
Equivalent A of gradient series
A A A . . . A A
A = G(A/G,i,n)
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved2-18
Sct 2.6 Geometric Gradient Series FactorSct 2.6 Geometric Gradient Series Factor
Geometric GradientCash flow series that starts with a base amount A1
Increases or decreases from period to period by a constant percentage amount
This uniform rate of change defines… A GEOMETRIC GRADIENT Notation:
g = the constant rate of change, in decimal form, by which future amounts increase or decrease from one time period to the next
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved2-19
Typical Geometric GradientTypical Geometric Gradient
A1
A1(1+g)A1(1+g)2
. . . .0 1 2 3 n-2 n-1 n
A1(1+g)n-1
Required: Find a factor (P/A,g%,i%,n) that will convert future cash flows to a single present worth value at time t = 0
Given A1, i%, and g%
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved2-20
Basic Derivation: Geometric GradientBasic Derivation: Geometric Gradient2 1
1 1 1 11 2 3
(1 ) (1 ) (1 )...
(1 ) (1 ) (1 ) (1 )
n
g n
A A g A g A gP
i i i i
1 2 1
1 2 3
1 (1 ) (1 ) (1 )...
(1 ) (1 ) (1 ) (1 )
n
g n
g g gP A
i i i i
(1)
Subtract Eq. (2 ) from Eq. (3 ) to yield
1 1
1+g (1 ) 11
1+i (1 ) 1
n
g n
gP A
i i
Solve for Pg and simplify to
yield….
Start with:
Factor out A1 out and re-write
(2)
1 2 1
1 2 3
(1+g) (1+g) 1 (1 ) (1 ) (1 )...
(1+i) (1+i) (1 ) (1 ) (1 ) (1 )
n
g n
g g gP A
i i i i
(3)
1
11
1 g i
n
g
gi
P Ai g
Multiply by (1+g)/(1+i) to obtain Eq. (3 )
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved2-21
Two Forms to Consider…Two Forms to Consider…
1
(1 )g
nAP
i
1
11
1 g i
n
g
g
iP A
i g
Case: g = i Case: g = i
A1 is the starting cash flow
There is NO base amount associated with a geometric gradient
The remaining cash flows are generated from the A1 starting value
No tables available to tabulate this factor…too many combinations of i% and g% to support tables
To use the (P/A,g%,i%,n) factor
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved2-22
Sct 2.7 Determination of Sct 2.7 Determination of Unknown Interest Rate Unknown Interest Rate
Class of problems where the interest rate, i%, is the unknown value
For simple, single payment problems (i.e., P and F only), solving for i% given the other parameters is not difficult
For annuity and gradient type problems, solving for i% can be tediousTrial and error method Apply spreadsheet models
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved2-23
The IRR Spreadsheet FunctionThe IRR Spreadsheet Function
Define the total cash flow as a column of values within Excel
Apply the IRR function:=IRR(first_cell:last_cell, guess value)
If the cash flow series is an A value then apply the RATE function:=RATE(number_years, A,P,F)
See examples 2.12 and 2.13
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved2-24
Sct 2.8 Determination of Sct 2.8 Determination of Unknown Number of YearsUnknown Number of Years
Class of problems where the number of time periods (years) is the unknown
In single payment type problems, solving for n is straight forward
In other types of cash flow profiles, solving for n requires trial and error or spreadsheet
In Excel, given A, P, and/or F, and i% values apply:=NPER(i%,A,P,F) to return the value of n
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved2-25
Sct 2.9 Spreadsheet Application – Basic Sct 2.9 Spreadsheet Application – Basic Sensitivity AnalysisSensitivity Analysis
Sensitivity Analysis is a process of determining what input variables really matter in a given problem formulation
Sensitivity analysis aids in evaluating certain what-If scenarios
Spreadsheet modeling is the best approach to formulate sensitivity analysis for a given problem
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved2-26
Sensitivity Analysis…Sensitivity Analysis…
See Example 2.15 Illustratesa what-if situation for receiving money in
three different time periodsTabulates the associated rate of returns for the
three situations
See Example 2.16The evaluation of non-sequential cash flows
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved2-27
SummarySummary
Interest factors exist to aid in determining economic equivalence of various cash flow patterns
Notation is introduced that is applied throughout the remainder of the text
Introduction of important Excel spreadsheet financial functions to aid in evaluation of engineering economy problems