ch2_brief

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



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



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)



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



(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)



(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)



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



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



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



(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)



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



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



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



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



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)



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



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)



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



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%



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 )



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



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



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



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



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



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



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



End of Slide SetEnd of Slide Set

ch2_brief

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