ch2_brief (2)

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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 IndustrialEngineering

Texas A&M University

College Station, Texas

Executive Summary Version

Chapter 2Factors: How Timeand Interest Affect

Money

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LEARNING OBJECTIVES

1. F/P and P/Ffactors

2. P/A and A/Pfactors

3. Interpolate forfactor values

4. P/G and A/G

factors

5. Geometricgradient

6. Calculate i

7. Calculate n

8. Spreadsheets

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Sct 2.1 Single-Payment Factors(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)

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Sct 2.2 Uniform-Series: Present WorthFactor (P/A) and

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 ofeach interest period

Find P

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(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 n P A

i i i i

2 3 1

1 1 1 1..

1 (1 ) (1 ) (1 ) (1 )n n

P A

i i i i i

1

(1+i)

(1)

(2)

1

1 1

1 (1 ) (1 )n

i P A

i i i

(3)

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(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

i P A for i

i i

(4)

(1 )

(1 ) 1

n

n

i i A P i

(P/A,i%,n) factor

Excel: =PV(i%,n,A)

(A/P,i%,n) factorExcel: =PMT(i%,n,P)

(5)

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ANSI Standard Notation forInterest Factors

Standard notation has been adopted torepresent the various interest factors

Consists of two cash flow symbols, theinterest 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 orunknown depending upon the problem

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Notation - 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-formequivalent relations (factor)

Tables at the back of the text providetabulations of common values for i% and n

S t 2 3 Si ki F d F t d U if

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Sct 2.3 Sinking Fund Factor and UniformSeries Compound Amount Factor

(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 i A 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

i A F

i

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(F/A) factor from (A/F)

Given:

Solve for F in terms of A to yield

(1 ) 1ni A F i

(1 ) 1ni F A

i

(A/F,i%,n) factorExcel: =PMT(i%,n,,F)

(F/A,i%,n) factor

Excel: =FV(i%,n,A)

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Sct 2.4 Interpolation in Interest Tables

When using tabulated interest tables onemight be forced to approximate a factor that isnot tabulated

Can apply linear interpolation to approximate

See Table 2-4

Factors are nonlinear functions, hence linearinterpolation will yield errors in the 2-4% range

Use a spreadsheet model to calculate the factorprecisely

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Sct 2.5 Arithmetic Gradient Factors(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= A

1

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Gradient 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

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Gradient Components

……..

0 1 2 3 n-2 n-1 n

Base amount = A / period

0G

1G2G

(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

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Gradient Decomposition As we know, arithmetic gradients are

comprised of two components1. Gradient component

2. Base amount

When working with a cash flow containing agradient, the (P/G) factor is only for thegradient component

Apply the (P/A) factor to work on the baseamount component

P = PW(gradient) + PW(base amount)

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Derivation Summary for (P/G) Start with:

Multiply (1) by (1+i)1 to create a second equation

Subtract (1) from the second equation and simplify

Yields…

( / , , 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) factor

No Excel relation exists

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Use 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 ofgradient series

A A A . . . A A

A = G(A/G,i,n)

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Sct 2.6 Geometric Gradient Series Factor

Geometric Gradient

Cash flow series that starts with a base amount A1

Increases or decreases from period to period by aconstant 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 timeperiod to the next

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Typical 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%

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Basic Derivation: Geometric Gradient2 1

1 1 1 1

1 2 3

(1 ) (1 ) (1 )...

(1 ) (1 ) (1 ) (1 )

n

g n

A A g A g A g P

i i i i

1 2 1

1 2 3

1 (1 ) (1 ) (1 )...

(1 ) (1 ) (1 ) (1 )

n

g n

g g g P A

i i i i

(1)

Subtract Eq. (2 ) from Eq. (3 ) to yield

1 11+g (1 ) 111+i (1 ) 1

n

g n

g P Ai 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 g P A

i i i i

(3)

1

11

1 g i

n

g

g

i P A

i g

Multiply by (1+g)/(1+i) to obtain Eq. (3 )

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Two Forms to Consider…

1

(1 ) g

nA P i

1

1

1 1 g i

n

g

g

i P 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

S t 2 7 D t i ti f

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Sct 2.7 Determination ofUnknown 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 otherparameters is not difficult

For annuity and gradient type problems,

solving for i% can be tediousTrial and error method

Apply spreadsheet models

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The IRR Spreadsheet Function

Define the total cash flow as a column ofvalues 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

S t 2 8 D t i ti f

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Sct 2.8 Determination ofUnknown Number of Years

Class of problems where the number of timeperiods (years) is the unknown

In single payment type problems, solving forn is straight forward

In other types of cash flow profiles, solvingfor 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

S t 2 9 S d h t A li ti B i

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Sct 2.9 Spreadsheet Application – BasicSensitivity Analysis

Sensitivity Analysis is a process ofdetermining 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 toformulate sensitivity analysis for a givenproblem

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Summary

Interest factors exist to aid in determiningeconomic equivalence of various cash flowpatterns

Notation is introduced that is appliedthroughout the remainder of the text

Introduction of important Excel spreadsheetfinancial functions to aid in evaluation ofengineering economy problems

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