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```					             Topics Today
•   Conversion for Arithmetic Gradient Series
•   Conversion for Geometric Gradient Series
•   Quiz Review
•   Project Review

1
Series and Arithmetic Series
• A series is the sum of the terms of a
sequence.
• The sum of an arithmetic progression (an
arithmetic series, difference between one
and the previous term is a constant)
sn  a  (a  d )  (a  2d )  (a  3d )  ...  (a  (n  1)d )

• Can we find a formula so we don’t have to
add up every arithmetic series we come
across?

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Sum of terms of a finite AP
S n  a  (a  d )  (a  2d )  ... [a  (n  2)d ]  [a  (n  1)d ]
S n  a  (a  d )  (a  2d )  ... (a  nd  2d )  (a  nd  d )
S n  (a  nd  d )  (a  nd  2d )  ... (a  2d )  (a  d )  a
2S n  a  (2a  nd )  (2a  nd )  ... (2a  nd )  (2a  nd )  a
There are (n) 2a terms  2a  n  2an;
There are (n - 1) nd terms  nd  (n - 1)  nd (n - 1) ; Therefore,
2Sn  2an  nd (n  1)
2S n  n[2a  (n  1)d ]
n
S n  [2a  (n  1)d ]
2
3
• A series of N receipts or disbursements that increase
by a constant amount from period to period.
• Cash flows: 0G, 1G, 2G, ..., (N–1)G at the end of
periods 1, 2, ..., N
• Cash flows for arithmetic gradient with base annuity:
A', A’+G, A'+2G, ..., A'+(N–1)G at the end of
periods 1, 2, ..., N where A’ is the amount of the base
annuity

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• Finds A, given G, i and N
• The future amount can be “converted” to an
equivalent annuity. The factor is:
1       N
( A / G, i , N )  
i (1  i )N  1
• The annuity equivalent (not future value!)
to an arithmetic gradient series is A =
G(A/G, i, N)

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• The annuity equivalent to an arithmetic
gradient series is A = G(A/G, i, N)
• If there is a base cash flow A', the base
annuity A' must be included to give the
overall annuity:
Atotal = A' + G(A/G, i, N)
• Note that A' is the amount in the first year
and G is the uniform increment starting in
year 2.
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Base Annuity

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Example 3-8
• A lottery prize pays \$1000 at the end
of the first year, \$2000 the second,
\$3000 the third, etc., for 20 years. If
there is only one prize in the lottery,
10 000 tickets are sold, and you can
invest your money elsewhere at 15%
interest, how much is each ticket
worth, on average?

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• Method 1: First find annuity value of prize
and then find present value of annuity.
A' = 1000, G = 1000, i = 0.15, N = 20
A = A' + G(A/G, i, N) = 1000 + 1000(A/G,
15%, 20)
= 1000 + 1000(5.3651) = 6365.10

• Now find present value of annuity:
P = A (P/A, i, N) where A = 6365.10, i =
15%, N = 20
P = 6365.10(P/A, 15, 20)
= 6365.10(6.2593) = 39 841.07

• Since 10 000 tickets are to be sold, on
average each ticket is worth (39
841.07)/10,000 = \$3.98.                       9
(to Uniform Series)

• The arithmetic gradient conversion factor (to
uniform series) is used when it is necessary
to convert a gradient series into a uniform
series of equal payments.

• Example: What would be the equal annual
series, A, that would have the same net
present value at 20% interest per year to a
five year gradient series that started at \$1000
10
and increased \$150 every year thereafter?
(to Uniform Series)
1       2        3       4       5       1           2           3           4           5

\$1000

\$1150
A        A       A       A       A
\$1300
\$1450

(1  i ) n  (1  ni)                                                 \$1600
A  Ag  G
i[(1  i ) n  1]
(1  0.20 ) 5  (1  5 * 0.20 )
 \$1,000  \$150
0.20[(1  0.20 ) 5  1]
 \$1,246
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(to Present Value)

• This factor converts a series of cash
amounts increasing by a gradient value,
G, each period to an equivalent present
value at i interest per period.
• Example: A machine will require \$1000
in maintenance the first year of its 5
year operating life, and the cost will
increase by \$150 each year. What is
the present worth of this series of
maintenance costs if the firm’s minimum
attractive rate of return is 20%?       12
(to Present Value)
\$1600
\$1450
\$1300
\$1150
\$1000
1           2           3           4           5

P
(1  i ) n  1    1  (1  ni)(1  i )  n
PA                G
i (1  i ) n
i2
(1  0.20 ) 5  1         1  (1  5 * 0.20 )(1  0.20 ) 5
 \$1,000                      \$150
0.20 (1  0.20 )  5
(0.20 ) 2
 \$3,727
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• A series of cash flows that increase or decrease
by a constant proportion each period
• Cash flows: A, A(1+g), A(1+g)2, …, A(1+g)N–1
at the end of periods 1, 2, 3, ..., N
• g is the growth rate, positive or negative
percentage change
• Can model inflation and deflation using
geometric series
14
Geometric Series
• The sum of the consecutive terms of a
geometric sequence or progression is
called a geometric series.
• For example:
Sn  a  ak  ak 2  ak 3  ....  ak n  2  ak n 1
Is a finite geometric series with quotient
k.
• What is the sum of the n terms of a finite
geometric series
15
Sum of terms of a finite GP
Sn  a  ak  ak 2  .... ak n  2  ak n 1
kSn  ak  ak 2  .... ak n  2  ak n 1  ak n
Sn  kSn  a  0  0  ..... 0  0  ak n
Sn (1  k )  a (1  k n )
(1  k n )
Sn  a
(1  k )

• Where a is the first term of the geometric progression,
k is the geometric ratio, and n is the number of terms
in the progression.
16
Present Worth
• The present worth of a geometric series is:
A       A(1  g )     A(1  g )N 1
P                    
(1  i ) (1  i )2
(1  i )N
• Where A is the base amount and g is the
growth rate.
• Before we may get the factor, we need what
is called a growth adjusted interest rate:
1 i              1      1 g
i          1 so that       
1 g             1 i  1 i
17
Factor: (P/A, g, i, N)
(1  i  )N  1 1 
(P / A, g, i , N )                    
        N 1 g
i (1  i )       
(P/A,i ,N)

( 1  g)
Four cases:
(1) i > g > 0:         i° is positive  use tables or formula
(2) g < 0:     i° is positive  use tables or formula
(3) g > i > 0: i° is negative  Must use formula
(4) g = i > 0: i° = 0                            A 
P  N     
1 g 

18
Compound Interest Factors
Discrete Cash Flow, Discrete Compounding

To Find   Given       Name of Factor          Factor
Compound Amount
F        P      Factor (single payment)     (1  i) n
Present Worth Factor
P        F      (single payment)           (1  i )  n

Compound Amount           (1  i ) n  1
F        A      Factor (uniform series)         i
i
A        F      Sinking Fund Factor       (1  i ) n  1

19
Compound Interest Factors
Discrete Cash Flow, Discrete Compounding

To Find   Given      Name of Factor                 Factor
i (1  i ) n
A        P      Capital Recovery Factor        (1  i ) n  1
(1  i) n  1
Present Worth Factor
P        A      (uniform series)                 i (1  i) n
(1  i ) n  (1  ni )
Conversion Factor (to
A        G      uniform series)              i[(1  i ) n  1]
Conversion Factor (to     1  (1  ni)(1  i )  n
P        G      present value)                      i2
20
Compound Interest Factors
Discrete Cash Flow, Continuous Compounding

To Find   Given       Name of Factor          Factor
Compound Amount
F        P      Factor (single payment)     e rn
Present Worth Factor
P        F      (single payment)           e  rn

Compound Amount           e rn  1
F        A      Factor (uniform series)   er 1
er 1
A        F      Sinking Fund Factor       e rn  1

21
Compound Interest Factors
Discrete Cash Flow, Continuous Compounding

To Find   Given      Name of Factor                Factor
e rn (e r  1)
A        P      Capital Recovery Factor          e rn  1
e rn  1
Present Worth Factor
P        A      (uniform series)               e rn (e r  1)
Conversion Factor (to           rn
A        G      uniform series)
e 1 e 1
r

Conversion Factor (to     e rn  1  n(e r  1)
P        G      present value)                e rn (e r  1) 2
22
Compound Interest Factors
Continuous Uniform Cash Flow, Continuous Compounding

To Find   Given       Name of Factor        Factor
Sinking Fund Factor
r
(continuous, uniform
C        F      payments)                 e rn  1
Capital Recovery Factor
re rn
(continuous, uniform
C        P      payments)                 e rn  1
Compound Amount
e rn  1
Factor (continuous,
F        C      uniform payments)             r
Present Worth Factor
e rn  1
(continuous, uniform
P        C      payments)                  re rn
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Quiz---When and Where
•   Quiz: Tuesday, Sept. 27, 2005
•   11:30 - 12:20 (Quiz: 30 minutes)
•   Tutorial: Wednesday, Sept. 28, 2005
•   ELL 168 Group 1
•   (Students with Last Name A-M)
•   ELL 061 Group 2
•   (Students with Last Name N-Z)

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Quiz---Who will be there

•   U, U, U, U, and U!!!!
•   CraigTipping ctipping@uvic.ca
•   Group 1 (Last NameA-M) ELL 168
•   LeYang          yangle@ece.uvic.ca
•   Group 2 (Last Name N-Z) ELL 061

25
Quiz---Problems, Solutions

• Do not argue with your TA!
• Question? Problems? TAWei
• Solutions will be given on Tutorial
• Bring: Blank Letter Paper, Pen, Formula
Sheet, Calculator, Student Card
• Write: Name, Student No. and Email

26
Quiz---Based on Chapter 1.2.3.

•   Important: Wei’s Slides
•   Even More Important: Examples in Slides
•   1 Formula Sheet is a good idea
•   5 Questions for 1800 seconds.
•   Wei used 180 seconds (relax)

27
Quiz---Important Points

•   Simple Interests
•   Compound Interests
•   Future Value
•   Present Value
•   Key: Compound Interest
•   Key: Understand the Question

28
Quiz---Books in Library!!!

Niall M. Fraser, University of Waterloo
Elizabeth M. Jewkes, University of Waterloo
Irwin Bernhardt, University of Waterloo
May Tajima, University of Waterloo

Economics: Canada in the Global Environment
by Michael Parkin and Robin Bade.

29
Calculator Talk

•   No programmable
•   No economic function
•   Simple the best

30
• Questions?

• (Sorry I forget the problems)

31
Project----Time Table
•   Select Topic: End of October
•   Survey finished: End of October
•   Project: November (3 Weeks)
•   Project Report Due: Final Quiz

32
Project----Requirements

•   Group: 3-6 Students
•   Topic: Practical, Small
•   Report: On Time, Original
•   Marks: 1 make to 1 report
•   Report: 25 marks out of 100

33
Project Topic----What to do

•   You Find it
•   Practical
•   Example: Run a Pizza Shop
•   Example: Run a Store for computer renting
•   Example: Survey on the Tuition Increase
•   Example: Why ??? Company failed…..
•   Team Work

34
Project----Recourse
•   No spoon feed: Independent work
•   Example: Government Web
•   Example: Economics Faculty
•   Example: Newspaper, TV
•   Example: Friends
35
Summary
•   Conversion for Arithmetic Gradient Series
•   Conversion for Geometric Gradient Series
•   Quiz: My slides and the examples in slides
•   Project: Good Idea, be open, independent

36

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