Slides

1
Topics Today

• Conversion for Arithmetic Gradient Series
• Conversion for Geometric Gradient Series
• Quiz Review
• Project Review

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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)
• Can we find a formula so we dont have to add up
  every arithmetic series we come across?

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Sum of terms of a finite AP
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Arithmetic Gradient Series

• A series of N receipts or disbursements that
  increase by a constant amount from period to
  period.
• Cash flows 0G, 1G, 2G, ..., (N1)G at the end of
  periods 1, 2, ..., N
• Cash flows for arithmetic gradient with base
  annuity
• A', AG, A'2G, ..., A'(N1)G at the end of
  periods 1, 2, ..., N where A is the amount of
  the base annuity

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Arithmetic Gradient to Uniform Series

• Finds A, given G, i and N
• The future amount can be converted to an
  equivalent annuity. The factor is
• The annuity equivalent (not future value!) to an
  arithmetic gradient series is A G(A/G, i, N)

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Arithmetic Gradient to Uniform Series

• 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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Arithmetic Gradient Series with 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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Example 3-8 Answer

• 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.

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Arithmetic Gradient Conversion Factor(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 and increased 150
  every year thereafter?

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Arithmetic Gradient Conversion Factor(to Uniform
Series)
1
2
3
4
5
1
2
3
4
5
1000
1150
A
A
A
A
A
1300
1450
1600
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Arithmetic Gradient Conversion Factor(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 firms
  minimum attractive rate of return is 20?

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Arithmetic Gradient Conversion Factor(to Present
Value)
1600
1450
1300
1150
1000
1
2
3
4
5
P
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Geometric Gradient Series

• A series of cash flows that increase or decrease
  by a constant proportion each period
• Cash flows A, A(1g), A(1g)2, , A(1g)N1 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

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Geometric Series

• The sum of the consecutive terms of a geometric
  sequence or progression is called a geometric
  series.
• For example
• Is a finite geometric series with quotient k.
• What is the sum of the n terms of a finite
  geometric series

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Sum of terms of a finite GP

• 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.

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Geometric Gradient to Present Worth

• The present worth of a geometric series is
• 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

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Geometric Gradient to Present Worth Factor (P/A,
g, i, N)

• Four cases
• (1) i gt g gt 0 i is positive ? use tables or
  formula
• (2) g lt 0 i is positive ? use tables or formula
  
• (3) g gt i gt 0 i is negative ? Must use formula
• (4) g i gt 0 i 0 ?

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Compound Interest FactorsDiscrete Cash Flow,
Discrete Compounding
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Compound Interest FactorsDiscrete Cash Flow,
Discrete Compounding
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Compound Interest FactorsDiscrete Cash Flow,
Continuous Compounding
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Compound Interest FactorsDiscrete Cash Flow,
Continuous Compounding
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Compound Interest FactorsContinuous Uniform Cash
Flow, Continuous Compounding
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Quiz---When and Where

• Quiz Tuesday, Sept. 27, 2005
• 1130 - 1220 (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_at_uvic.ca
• Group 1 (Last NameA-M) ELL 168
• LeYang             yangle_at_ece.uvic.ca      
• Group 2 (Last Name N-Z) ELL 061

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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

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Quiz---Based on Chapter 1.2.3.

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

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Quiz---Important Points

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

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Quiz---Books in Library!!!
Engineering Economics in Canada, 3/E
Niall M. Fraser, University of WaterlooElizabeth
M. Jewkes, University of WaterlooIrwin
Bernhardt, University of WaterlooMay Tajima,
University of Waterloo 
Economics Canada in the Global Environment by
Michael Parkin and Robin Bade.
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Calculator Talk

• No programmable
• No economic function
• Simple the best
• Trust your ability
• Trust your teaching group

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• Questions?
• (Sorry I forget the problems)

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Project----Time Table

• Find your group Mid-October
• Select Topic End of October
• Survey finished End of October
• Project November (3 Weeks)
• Project Report Due Final Quiz

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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

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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

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Project----Recourse

• Not your teaching group
• No spoon feed Independent work
• Example Government Web
• Example Library, Database, Google
• Example Economics Faculty
• Example Newspaper, TV
• Example Friends

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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