Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash...
Transcript of Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash...
![Page 1: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/1.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
1
Chapter 2
Factors: How Time and Interest
Affect Money
Session 4-5-6
Dr Abdelaziz Berrado
![Page 2: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/2.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
2
Topics to Be Covered in Today’s Lecture
� Section 2: How Time and Interest Affect Money
� Single-Payment Factors (F/P and P/F)
� Interest Tables
� Present Value/Future worth of an uneven payment series
� Uniform-Series Present Worth Factor and Capital Recovery Factor P/A and A/P Factors
� Sinking Fund Factor and Uniform-Series compound amount factor
![Page 3: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/3.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
3
Cash Flow Diagrams - Exercise
• An engineer wants to deposit an amount P
now such that she can withdraw an equal
annual amount $2500 per year for the first 3
years starting one year after the deposit and
a different annual withdrawal of $3000 per
year for the following 5 years. How would
the cash flow diagram appear if the interest
rate were 7% per year?
![Page 4: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/4.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
4
Single-payment Factors (F/P and P/F)
• F/P: To find F given P– To find a certain future amount of money F
(compound amount), given the present amount of money P, a number of interest periods n, and an interest rate i%
P0
Fn ?
n
………….
![Page 5: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/5.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
5
Single-payment Factors (F/P and P/F)
• P/F: To find P given F
– To find the present worth P, given a future
amount F, a number of interest periods n, and
an interest rate i%
– P/F factor brings a single future sum back to
a specific point in time.
P ?
Fn
n
………….
![Page 6: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/6.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
6
Single-payment Factors (F/P and P/F)
• F = P(1+i)n and P = F (1/(1+i)n)
• (1+i)n:
– F/P factor,
– Single Payment Compound Amount Factor (SPCAF),
– (F/P, i,n)
• 1/(1+i)n:
– P/F factor,
– Single Payment Present Worth Factor (SPPWF),
– (P/F, i,n)
![Page 7: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/7.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
7
Example- F/P Analysis
• Example: P= $1,000;n=3;i=10%
• What is the future value, F?
0 1 2 3
P=$1,000
F = ??
i=10%/year
F3 = $1,000[F/P,10%,3] = $1,000[1.10]3
= $1,000[1.3310] = $1,331.00
![Page 8: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/8.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
8
Example – P/F Analysis
• Assume F = $100,000, 9 years from now. What is the
present worth of this amount now if i =15%?
0 1 2 3 8 9…………
F9 = $100,000
P= ??
i = 15%/yr
P0 = $100,000(P/F, 15%,9) = $100,000(1/(1.15)9)
= $100,000(0.2843) = $28,430 at time t = 0
![Page 9: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/9.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
9
Interest Tables
• Formulas can be used for calculating P or F, but
interest tables can make some calculations easier
• Interest tables given at the back of the book
• Look up P/F (Present Worth), or F/P (Compound
Amount) factors for a given interest rate and given
number of interest periods
• When a given interest rate is not given at the back
of the book, can use 1) Formulas (exact) 2)
Interpolation (approximation)
![Page 10: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/10.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
10
Interest Tables Exercise
• If you had $2000 now and invested it at
10% how much would it be worth in 8
years?
• Suppose that $1000 is to be received in 5
years. At an annual interest rate of 12%,
what is the present worth of this amount?
![Page 11: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/11.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
11
Present Value/Future worth of an
uneven payment series
• What if payments are not uniform – some
amount needs to be withdrawn 2 years from
now, a different amount 5 years from now?
How will you calculate future worth or
present value in this case?
![Page 12: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/12.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
12
Present Value/Future worth of an
uneven payment series – exercise
• Calculate the worth 5 years from now of an
investment of $200 made 2 years from now
and an investment of $100 made 4 years
from now
![Page 13: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/13.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
13
Present Value/Future worth of an
uneven payment series – exercise
• Wilson Technology a growing machine
shop, wishes to set aside money now to
invest over the next 4 years in automating
its customer service department. The
company can earn 10% on a lump sum
deposited now and wishes to withdraw the
money in the following increments
![Page 14: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/14.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
14
Present Value of an uneven payment series –
exercise
• 1) Year 1: $25000 to purchase a computer and database software designed for customer service use
• 2) Year 2: $3000 to purchase additional hardware to accommodate the anticipated growth in use of the system
3) Year 3: No expenses
4) Year 4: $5000 to purchase software upgrades
How much money should the company set aside now ?
![Page 15: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/15.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
15
Present Worth of an even
payment series – exercise
• What is the present worth of a series of
annual receipts of $100 that you will be
receiving for three years starting next year?
![Page 16: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/16.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
16
Uniform Series Present Worth
• Annuity Cash Flow
$A per period
P = ??
0
…………..n1 2 3 .. .. n-1
![Page 17: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/17.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
17
Uniform Series Present Worth
• Desire an expression for the present worth
– P of a stream of equal, end of period cash
flows - A
0 1 2 3 n-1 n
A = given
P = ??
![Page 18: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/18.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
18
Uniform Series Present Worth
• Write a Present worth expression
1 2 1
1 1 1 1..
(1 ) (1 ) (1 ) (1 )n nP A
i i i i−
= + + + +
+ + + +
Term inside the brackets is a geometric progression.
Mult. This equation by 1/(1+i) to yield a second equation
[1]
![Page 19: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/19.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
19
Uniform Series Present Worth
• The second equation
[2]
To isolate an expression for P in terms of A, subtract Eq [1] from Eq. [2]. Note that numerous terms will drop out.
2 3 1
1 1 1 1..
1 (1 ) (1 ) (1 ) (1 )n n
PA
i i i i i +
= + + + +
+ + + + +
![Page 20: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/20.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
20
Uniform Series Present Worth and Capital
Recovery Factors
• Setting up the subtraction
1 2 1
1 1 1 1..
(1 ) (1 ) (1 ) (1 )n nP A
i i i i−
= + + + +
+ + + + [1]
[2]
-
1
1 1
1 (1 ) (1 )n
iP A
i i i+
−= −
+ + + = [3]
2 3 4 1
1 1 1 1 1...
(1 ) (1 ) (1 ) (1 ) (1 ) (1 )n n
PA
i i i i i i+
= + +
+ + + + + +
![Page 21: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/21.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
21
Uniform Series Present Worth and
Capital Recovery Factors
• Simplifying Eq. [3] further
1
1 1
1 (1 ) (1 )n
iP A
i i i+
−= −
+ + +
1
11
(1 )n
AP
i i+
= −
− +
(1 ) 1 0
(1 )
n
n
iP A for i
i i
+ −= ≠
+
![Page 22: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/22.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
22
Capital Recovery Factor (A/P)
• A/P: To find A given P
– Given an amount P now, what is the annual A
worth over n years when the interest rate is i
$A ???
P is known
0
…………..n1 2 3 .. .. n-1
![Page 23: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/23.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
23
0,)1(
1)1(≠
+
−+= i
ii
iAP
n
n
0,1)1(
)1(≠
−+
+= i
i
iiPA
n
n
+
−+n
n
ii
i
)1(
1)1(
−+
+
1)1(
)1(n
n
i
ii
P/A factor (P/A,i,n)
Uniform Series
Present Worth
Factor
A/P factor (A/P,i,n)
Capital
Recovery
Factor
•P/A versus A/P
![Page 24: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/24.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
24
P/A example
• How much money should you be willing to
pay now for a guaranteed $600 per year for
9 years starting next year, at a rate of return
of 16% per year? (Answer using calculation
and interest tables)
![Page 25: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/25.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
25
A/P Example
• Suppose you are entitled a payment of
30,000 a year for the next 9 years starting
next year. What is the present worth of this
annual payment to you? (Answer using
calculation and interest tables)
![Page 26: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/26.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
26
More exercises
• BioGen, a small biotechnology firm, has
borrowed $250,000 to purchase laboratory
equipment for gene splicing. The loan
carries an interest rate of 8% per year and is
to be repaid in equal annual installments
over the next 6 years. Compute the amount
of this annual installment.
![Page 27: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/27.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
27
More exercises
• Suppose you are entitled a payment of
30,000 a year for the next 9 years starting
next year from an investment firm. What is
the present worth of this annual payment to
you?
![Page 28: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/28.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
28
Sinking Fund Factor and Uniform-Series
Compound Amount Factor
• Annuity Cash Flow
0
…………..N
$A per period
$F
Find $A given the Future amt. - $F
![Page 29: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/29.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
29
Sinking Fund and Series Compound
amount factors (A/F and F/A)
• Take advantage of what we already have
• Recall:
• Also:
1
(1 )nP F
i
=
+
(1 )
(1 ) 1
n
n
i iA P
i
+=
+ −
Substitute “P” and simplify!
![Page 30: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/30.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
30
F/A and A/F Derivations
• Annuity Cash Flow
0
…………..N
$A per period
$F
Find $F given the $A amounts
![Page 31: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/31.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
31
Example 2.6• How much money must Carol deposit every year
starting, l year from now at 5.5% per year in order to accumulate $6000 seven years from now?
![Page 32: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/32.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
32
Example 2.6
• Solution
• The cash How diagram from Carol's
perspective fits the A/F factor.
• A= $6000 (A/F,5.5%,7) = 6000(0.12096) =
$725.76 per year
• The A/F factor Value 0f 0.12096 was
computed using the A/F factor formula
![Page 33: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/33.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
33
Assignments
� Assignments due at the beginning of next class:
� Read all the textbooks examples from sections 2.1, 2.2, 2.3
� Read sections 2.4, 2.5, 2.6
� Hw1 due at the beginning of class one week from today.
![Page 34: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/34.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
34
Topics to Be Covered in Today’s Lecture
� Section 2: How Time and Interest Affect Money
� Interpolation in interest tables
� Arithmetic Gradient Factors
� Geometric Gradient Factors
![Page 35: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/35.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
35
Arithmetic Gradient Factors
• In applications, the annuity cash flow pattern is not the only type of pattern encountered
•Two other types of end of period patterns are common
•The Linear or arithmetic gradient
•The geometric (% per period) gradient
•This section presents the Arithmetic Gradient
![Page 36: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/36.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
36
Arithmetic Gradient Factors
• An arithmetic (linear) Gradient is a cash flow series that either increases or decreases by a constant amount over n time periods.
•A linear gradient is always comprised of TWO components:
•The Gradient component
•The base annuity component
•We need an expression for the Present Worth of an arithmetic gradient
![Page 37: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/37.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
37
Linear Gradient Example
• Assume the following:
0 1 2 3 n-1 n
A1+G
A1+2G
A1+(n-2)G
A1+(n-1)G
This represents a positive, increasing arithmetic gradient
![Page 38: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/38.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
38
Example: Linear Gradient
• Typical Negative, Increasing Gradient: G=$50
The Base Annuity= $1500
![Page 39: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/39.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
39
Example: Linear Gradient
• Desire to find the Present Worth of this cash flow
The Base Annuity= $1500
![Page 40: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/40.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
40
Arithmetic Gradient Factors
• The “G” amount is the constant arithmetic change from one time period to the next.
•The “G” amount may be positive or negative!
•The present worth point is always one time period to the left of the first cash flow in the series or,
•Two periods to the left of the first gradient cash flow!
![Page 41: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/41.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
41
Derivation: Gradient Component
Only
• Focus Only on the gradient Component
G
+2G
(n-2)G
(n-1)G
0 1 2 3 n-1 N
“0” G
Removed Base annuity
![Page 42: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/42.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
42
Present Worth Point…
0 1 2 3 4 5 6 7
$100
$200
$300
$400
$500
$600
$700
X
The Present Worth Point of the Gradient
![Page 43: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/43.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
43
Gradient Component
$0
$100
$200
$300
$400
$500
$600
X
The Present Worth Point of the Gradient
0 1 2 3 4 5 6 7
•The Gradient Component
![Page 44: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/44.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
44
Present Worth Point…
0 1 2 3 4 5 6 7
Base Annuity – A = $100
X
The Present Worth Point of the Gradient
•PW of the Base Annuity is at t = 0
•PWBASE Annuity=$100(P/A,i%,7)
![Page 45: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/45.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
45
Present Worth: Linear Gradient
• The present worth of a linear gradient is the present worth of the two components:– 1. The Present Worth of the Gradient Component and,
– 2. The Present Worth of the Base Annuity flow
– Requires 2 separate calculations!
![Page 46: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/46.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
46
Present Worth: Gradient Component
• The PW of the Base Annuity is simply the Base Annuity –A(P/A, i%, n) factor
• What is needed is a present worth expression for the gradient component cash flow.
• We need to derive a closed form expression for the gradient component!
![Page 47: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/47.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
47
Present Worth: Gradient Component
• General CF Diagram – Gradient Part Only
0 1 2 3 4 ……….. n-1 n
1G2G
3G(n-2)G
(n-1)G
0G
We want the PW at time t = 0 (2 periods to the left of 1G)
![Page 48: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/48.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
48
To Begin- Derivation of (P/G,i%,n)
Next Step:
Factor out G and re-write as …..
( / , %, 2) 2 ( / , %,3) ...
[(n-2)G](P/F,i%,n-1) + [(n-1)G](P/F,i%,n)
P G P F i G P F i= + +
![Page 49: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/49.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
49
Factoring G out…. P/G factor
{ }P/F,i%,2 P/F,i%,P=G ( )+2( ) +...(3 P/F,in-1)( %,n)
What is inside of the { }’s?
![Page 50: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/50.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
50
Replace (P/F’s) with closed-form
2 3 n-1 n
1 2 n-2 n-1P=G ...
(1+i) (1+i) (1+i) (1+i)
+ + + +
Multiply both sides by (1+i)
[1]
![Page 51: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/51.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
51
Mult. Both Sides By (n+1)…..
1
1 2 n-2 n-1
1 2 n-2 n-1P(1+i) =G ...
(1+i) (1+i) (1+i) (1+i)
+ + + +
[2]
•We have 2 equations [1] and [2].
•Next, subtract [1] from [2] and work with the resultant equation.
![Page 52: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/52.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
52
2 3 n-1 n
1 2 n-2 n-1P=G ...
(1+i) (1+i) (1+i) (1+i)
+ + + +
Subtracting [1] from [2]…..
1
1 2 n-2 n-1
1 2 n-2 n-1P(1+i) =G ...
(1+i) (1+i) (1+i) (1+i)
+ + + +
-
(1 ) 1
(1 ) (1 )
n
n n
i n
i i
GP
ii
=
+ −−
+ +
![Page 53: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/53.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
53
The P/G factor for i and N
( / , %, )P G i N factor
(1 ) 1
(1 ) (1 )
n
n n
i n
i i
GP
ii
=
+ −−
+ +
![Page 54: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/54.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
54
Further Simplification on P/G
2
(1 ) 1( / , %, )
(1 )
N
N
i iNP G i N
i i
+ − −=
+
Remember, the present worth point of any linear gradient is 2 periods to the left of the 1-G cash flow or, 1 period to the left of the “0-G” cash flow.
P=G(P/G,i,n)
![Page 55: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/55.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
55
Gradient Example
• Consider the following cash flow
0 1 2 3 4 5
$100
$200
$300
$400
$500
Find the present worth if i = 10%/yr; n = 5 yrs
Present Worth Point is here!And the G amt. = $100/period
![Page 56: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/56.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
56
Gradient Example- Base Annuity
• First, The Base Annuity of $100/period
0 1 2 3 4 5
A = +$100
•PW(10%) of the base annuity = $100(P/A,10%,5)
•PWBase = $100(3.7908)= $379.08
•Not Finished: We need the PW of the gradient component and then add that value to the $379.08 amount
![Page 57: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/57.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
57
Focus on the Gradient Component
0 1 2 3 4 5
$0$100
$200
$300
$400
We desire the PW of the Gradient Component at t = 0
PG@t=0 = G( P/G,10%,5 ) = $100( P/G,10%,5 )
![Page 58: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/58.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
58
The Set Up
0 1 2 3 4 5
$0$100
$200
$300
$400
PG@t=0 = G(P/G,10%,5) = $100(P/G,10%,5)
G (1 ) 1P=
i (1 ) (1 )
N
N N
i N
i i i
+ −−
+ +
Could substitute n=5, i=10% and G = $100 into the P/G closed form to get the value of the factor.
![Page 59: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/59.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
59
PW of the Gradient Component
PG@t=0 = G(P/G,10%,5) = $100(P/G,10%,5)
Calculating or looking up the (P/G,10%,5) factor yields the following:
Pt=0 = $100(6.8618) = $686.18 for the
gradient PW
P/G,10%,5)
G (1 ) 1P=
i (1 ) (1 )
N
N N
i N
i i i
+ −−
+ +
Sub. G=$100;i=0.10;n=5
6.8618
![Page 60: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/60.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
60
Gradient Example: Final Result• PW(10%)Base Annuity = $379.08
•PW(10%)Gradient Component= $686.18
•Total PW(10%) = $379.08 + $686.18
•Equals $1065.26
•Note: The two sums occur at t =0 and can
be added together – concept of equivalence
![Page 61: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/61.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
61
Example Summarized
0 1 2 3 4 5
$100
$200
$300
$400
$500This Cash Flow…
Is equivalent to $1065.26 at time 0 if the interest rate is 10% per year!
![Page 62: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/62.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
62
Shifted Gradient Example: i = 10%
• Consider the following Cash Flow
0 1 2 3 4 5 6 7
1. This is a “shifted” negative, decreasing gradient.
2. The PW point in time is at t = 3 (not t = o)
$600$550
$500
$450
![Page 63: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/63.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
63
Shifted Gradient Example
• Consider the following Cash Flow
0 1 2 3 4 5 6 7
$600$550
$500
$450
•The PW @ t = 0 requires getting the PW @ t =3;
•Then using the P/F factor move PW3 back to t=0
![Page 64: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/64.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
64
Shifted Gradient Example
• Consider the following Cash Flow
•The base annuity is a $600 cash flow for 3 time periods
0 1 2 3 4 5 6 7
$600$550
$500
$450
![Page 65: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/65.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
65
Shifted Gradient Example: Base Annuity
• PW of the Base Annuity: 2 Steps
A = -$600
0 1 2 3 4 5 6 7
P3=-600(P/A,10%,4)
P3
P0
P0=P3(P/F,10%,3)
P0= [-600(P/A,10%,4)](P/F,10%,3)3.1699 0.7513
P 0-base annuity = -$1428.93
![Page 66: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/66.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
66
Shifted Gradient Example: Gradient
• PW of Gradient Component: G = -$50
0 1 2 3 4 5 6 7
P3
P0
P0=P3(P/F,10%,3)
0G1G 2G 3G
P3-Grad = +50(P/G,10%,4)
P0-grad = {+50(P/G,10%,4)}(P/F,10%,3)4.3781 0.7513 =-$164.46
![Page 67: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/67.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
67
Extension – The A/G factor
• A/G converts a linear gradient to an equivalent annuity cash flow.
• Remember, at this point one is only working with gradient component
• There still remains the annuity component that you must also handle separately!
![Page 68: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/68.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
68
The A/G Factor
• Convert G to an equivalent A
( / , , )( / , , )A G P G i n A P i n=
How to do it…………
![Page 69: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/69.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
69
A/G factor using A/P with P/G
The results follow…..
(1 ) 1
(1 ) (1 )
n
n n
i n
i i
GP
ii
=
+ −−
+ +
(A/P,i,n)
(1 )
(1 ) 1
n
n
i i
i
+
+ −
![Page 70: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/70.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
70
Resultant A/G factor
(A/G,i,n) = 1
(1 ) 1n
nA G
i i
= −
+ −
(1 ) 1
(1 ) (1 )
n
n n
i n
i i
GP
ii
=
+ −−
+ +
(A/P,i,n)
(1 )
(1 ) 1
n
n
i i
i
+
+ −
![Page 71: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/71.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
71
Terminology
• Let PA
be the present worth of the base annual amount A
A(which is a series of
equal end-of-period cash flows).
• Let PG
be the present worth of the gradient cash flow
• Let PT
be the total present worth,
– PT
= PA+ P
Gif the gradient is positive
– PT
= PA
- PG
if the gradient is negative
• Let AT
be the total annual worth for n years of the series of gradient cash flows including the base annual amount A
A
![Page 72: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/72.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
72
+−
+
−+=
nn
n
Gi
n
ii
i
iGP
)1()1(
1)1(1
+−
+
−+nn
n
i
n
ii
i
i )1()1(
1)1(1
(P/G,i,n) factor
P/G factor
Arithmetic
Gradient
Present Worth
Factor
![Page 73: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/73.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
73
Arithmetic Gradients
• PT
= PA+ P
Gfor positive gradients and
• PT
= PA
- PG
for negative gradients
• So, how would you find PA?
• Given that you have the PT
how would you
find AT
?
![Page 74: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/74.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
74
Assignments
� Assignments due at the beginning of next class:
� Read all the textbooks examples from sections 2.4, 2.5
� Read sections 2.6, 2.7, 2.8
![Page 75: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/75.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
75
Topics to Be Covered in Today’s Lecture
� Section 2: How Time and Interest Affect Money
� The A/G factor
� Geometric Gradient Factors
� Determination of an unknown interest rate
� Determination of an unknown number of
periods
![Page 76: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/76.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
76
Extension – The A/G factor
• A/G converts a linear gradient to an equivalent annuity cash flow.
• Remember, at this point one is only working with gradient component
• There still remains the annuity component that you must also handle separately!
![Page 77: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/77.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
77
The A/G Factor
• Convert G to an equivalent A
( / , , )( / , , )A G P G i n A P i n=
How to do it…………
![Page 78: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/78.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
78
A/G factor using A/P with P/G
The results follow…..
(A/P,i,n)
(1 )
(1 ) 1
n
n
i i
i
+
+ −
(1 ) 1
(1 ) (1 )
n
n n
i n
i i
GP
ii
=
+ −−
+ +A=
![Page 79: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/79.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
79
Resultant A/G factor
(A/G,i,n) = 1
(1 ) 1n
nA G
i i
= −
+ −
(A/P,i,n)
(1 )
(1 ) 1
n
n
i i
i
+
+ −
(1 ) 1
(1 ) (1 )
n
n n
i n
i i
GP
ii
=
+ −−
+ +A=
![Page 80: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/80.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
80
Terminology
• Let PA
be the present worth of the base annual amount A
A(which is a series of
equal end-of-period cash flows).
• Let PG
be the present worth of the arithmetic gradient cash flow
• Let PT
be the total present worth,
– PT
= PA+ P
Gif the gradient is positive
– PT
= PA
- PG
if the gradient is negative
• Let AT
be the total annual worth for n years of the series of gradient cash flows including the base annual amount A
A
![Page 81: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/81.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
81
+−
+
−+=
nn
n
Gi
n
ii
i
iGP
)1()1(
1)1(1
+−
+
−+nn
n
i
n
ii
i
i )1()1(
1)1(1
(P/G,i,n) factor
P/G factor
Arithmetic
Gradient
Present Worth
Factor
![Page 82: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/82.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
82
Arithmetic Gradients
• PT
= PA+ P
Gfor positive gradients and
• PT
= PA
- PG
for negative gradients
• So, how would you find PA?
• Given that you have the PT
how would you
find AT
?
![Page 83: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/83.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
83
Geometric Gradients• An arithmetic (linear) gradient changes by a fixed dollar amount each time period.
•A GEOMETRIC gradient changes by a fixed percentage each time period.
•We define a UNIFORM RATE OF CHANGE (%) for each time period
•Define “g” as the constant rate of change in decimal form by which amounts increase or decrease from one period to the next
![Page 84: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/84.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
84
Geometric Gradients: Increasing• Typical Geometric Gradient Profile
•Let A1 = the first cash flow in the series
0 1 2 3 4 …….. n-1 n
A1 A1(1+g)A1(1+g)
2
A1(1+g)3
A1(1+g)n-1
![Page 85: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/85.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
85
Geometric Gradients: Decreasing• Typical Geometric Gradient Profile
•Let A1 = the first cash flow in the series
A1
A1(1-g)
A1(1-g)2
A1(1-g)3
A1(1-g)n-1
0 1 2 3 4 …….. n-1 n
![Page 86: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/86.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
86
Geometric Gradients: Derivation• First Major Point to Remember:
•A1 does NOT define a Base Annuity;
•There is no BASE ANNUITY for a Geometric Gradient!
•The objective is to determine the Present Worth one period to the left of the A1 cash flow point in time
•Remember: The PW point in time is one period to
the left of the first cash flow – A1!
![Page 87: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/87.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
87
Geometric Gradients: Derivation• For a Geometric Gradient the following parameters are required:
•The interest rate per period – i
•The constant rate of change – g
•No. of time periods – n
•The starting cash flow – A1
![Page 88: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/88.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
88
Geometric Gradients: Starting• Pg = The Aj’s time the respective (P/F,i,j) factor
•Write a general present worth relationship to find Pg….
2 1
1 1 1 1
1 2 3
(1 ) (1 ) (1 )...
(1 ) (1 ) (1 ) (1 )
n
g n
A A g A g A gP
i i i i
−+ + +
= + + + ++ + + +
Now, factor out the A1 value and rewrite as..
![Page 89: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/89.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
89
Geometric Gradients1 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)
(1+g)Multuply both sides by to create another equation
(1+i)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
− + + += + + + +
+ + + +
(2)
Subtract (1) from (2) and the result is…..
![Page 90: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/90.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
90
Geometric Gradients
1 1
1+g (1 ) 11
1+i (1 ) 1
n
g n
gP A
i i+
+ − = −
+ + Solve for Pg and simplify to yield….
1
11
1 g i
n
g
g
iP A
i g
+ −
+ = ≠ −
![Page 91: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/91.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
91
Geometric Gradient P/A factor
• This is the (P/A,g,i,n) factor and is valid if g is not equal to i.
1
11
1 g i
n
g
g
iP A
i g
+ −
+ = ≠ −
![Page 92: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/92.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
92
Geometric Gradient P/A factor•Note: If g = i we have a division by “0” –
undefined.
•For g = i we can derive the closed form PW factor
for this special case.
•We substitute i for g into the Pg relationship to
yield:
![Page 93: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/93.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
93
Geometric Gradient: i = g Case
g 1
1 1 1 1P =A ...
(1+i) (1+i) (1+i) (1+i)
+ + + +
1
(1 )g
nAP
i=
+
For the case i = g
![Page 94: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/94.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
94
Geometric Gradients: Summary
•Pg = A1(P/A,g,i,n)
1
(1 )g
nAP
i=
+1
11
1 g i
n
g
g
iP A
i g
+ −
+ = ≠ −
g not = to i Case: g = i
![Page 95: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/95.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
95
Geometric Gradient: Notes•The geometric gradient requires knowledge of:
•A1, i, n, and g
•There exist an infinite number of combinations for i, n, and g: Hence one will not find tabulated tables for the (P/A, g,i,n) factor.
![Page 96: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/96.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
96
Geometric Gradient: Notes•You have to calculated either from the closed form for each problem or apply a pre-programmed spreadsheet model to find the needed factor value
•No spreadsheet built-in function for this factor!
![Page 97: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/97.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
97
Geometric Gradient: Example•Assume maintenance costs for a particular activity will be $1700 one year from now.
•Assume an annual increase of 11% per year over a 6-year time period.
![Page 98: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/98.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
98
Geometric Gradient: Example•If the interest rate is 8% per year, determine the present worth of the future expenses at time t = 0.
•First, draw a cash flow diagram to represent the model.
![Page 99: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/99.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
99
Geometric Gradient Example (+g)
•g = +11% per period; A1 = $1700; i = 8%/yr
0 1 2 3 4 5 6 7
$1700$1700(1.11)1
$1700(1.11)2
$1700(1.11)3
$1700(1.11)6
PW(8%) = ??
![Page 100: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/100.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
100
Solution• P = $1700(P/A,11%,8%,7)
•Need to calculate the P/A factor from the closed-form expression for a geometric gradient.
1
11
1 g i
n
g
g
iP A
i g
+ −
+ = ≠ −
$11980.44
![Page 101: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/101.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
101
Geometric Gradient ( -g )• Consider the following problem with a negative growth rate – g.
0 1 2 3 4
g = -10%/yr; i = 8%; n = 4
A1 = $1000
$900$810
$729
P0=??
We simply apply a “g” value = -0.10
![Page 102: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/102.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
102
Geometric Gradient (-g value)• Evaluate:
1
11
1 g i
n
g
g
iP A
i g
+ −
+ = ≠ −
1
11
1 g i
n
g
g
iP A
i g
+ −
+ = ≠ −
For a negative g value = -0.10
=$2876.37
![Page 103: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/103.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
103
Determination of an unknown
interest rate and number of
periods
• Interest Rate Unknown
– Given: P, F,n
– Determine i
• Number of periods unknown
– Given: P, F, i
– Determine n
• Use formulas to calculate
![Page 104: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/104.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
104
Determination of an unknown
interest rate exercise
• If Laurel can make an investment in a
friend’s business of $3000 now in order to
receive $5000 five years from now,
determine the rate of return. If Laurel can
receive 7% per year interest on a certificate
of deposit, which investment should be
made?
![Page 105: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/105.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
105
Determination of an unknown
number of interest periods
exercise
• How long will it take for $1000 to double if
the interest rate is 5% per year? Calculate
the exact value. How different are the
estimates if the rate is 25%?
![Page 106: Chapter 2 Factors: How Time and Interest Affect MoneyA.Berrado/EGR2302/EGR2302_Ch02.pdf · Cash Flow Diagrams - Exercise • An engineer wants to deposit an amount P ... • Wilson](https://reader034.fdocuments.in/reader034/viewer/2022051911/60015d24dfbae20da73ce125/html5/thumbnails/106.jpg)
EGR2302-Engineering Economics
Al Akhawayn University
106
Assignments and Announcements
� Assignments due at the beginning of next class:
� Read all the textbook examples from sections 2.6, 2.7, 2.8
� Read chapter 3
� We have a quiz next time