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?

Sum of terms of a finite AP

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.

Arithmetic Gradient Series with Base Annuity

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 Factors Discrete Cash Flow, Discrete Compounding

Compound Interest Factors Discrete Cash Flow, Discrete Compounding

Compound Interest Factors Discrete Cash Flow, Continuous Compounding

Compound Interest Factors Discrete Cash Flow, Continuous Compounding

Compound Interest Factors Continuous 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    ctipping@uvic.ca Group 1 (Last NameA-M) ELL 168 LeYang             yangle@ece.uvic.ca       Group 2 (Last Name N-Z) ELL 061

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

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 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.

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