Topics Today • Conversion for Arithmetic Gradient Series • Conversion for Geometric Gradient Series • Quiz Review • Project Review
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 don’t have to add up every arithmetic series we come across?
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, ..., (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
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)
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.
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?
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.
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?
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
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 firm’s minimum attractive rate of return is 20%?
Arithmetic Gradient Conversion Factor(to Present Value) $1600 $1450 $1300 $1150 $1000 1 2 3 4 5 P
Geometric Gradient Series • 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
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
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.
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:
Geometric Gradient to Present Worth Factor: (P/A, g, i, N) 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
Compound Interest FactorsContinuous Uniform Cash Flow, Continuous Compounding
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)
Quiz---Who will be there • U, U, U, U, and U!!!! • CraigTipping email@example.com • Group 1 (Last NameA-M) ELL 168 • LeYang firstname.lastname@example.org • Group 2 (Last Name N-Z) ELL 061
Quiz---Problems, Solutions • Do not argue with your TA! • Question? Problems? TAWei • Solutions will be given on Tutorial • Bring: Blank Letter Paper, Pen, Formula Sheet, Calculator, Student Card • Write: Name, Student No. and Email
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)
Quiz---Important Points • Simple Interests • Compound Interests • Future Value • Present Value • Key: Compound Interest • Key: Understand the Question
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.
Calculator Talk • No programmable • No economic function • Simple the best • Trust your ability • Trust your teaching group
Questions? • (Sorry I forget the problems)
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
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
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
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
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