1. Topics Today Conversion for Arithmetic Gradient Series
Conversion for Geometric Gradient Series
Quiz Review
Project Review

2. 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?

3. Sum of terms of a finite AP

4. 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

5. 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)

6. 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.

7. Arithmetic Gradient Series with Base Annuity

8. 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, tickets are sold, and you can invest your money elsewhere at 15% interest, how much is each ticket worth, on average?

9. 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) = (A/G, 15%, 20)
= (5.3651) =
Now find present value of annuity:
P = A (P/A, i, N) where A = , i = 15%, N = 20
P = (P/A, 15, 20)
= (6.2593) =
Since tickets are to be sold, on average each ticket is worth ( )/10,000 = $3.98.

10. 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?

11. 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

12. 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%?

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

14. 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

15. 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

16. 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.

17. 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:

18. 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 

19. Compound Interest Factors Discrete Cash Flow, Discrete Compounding

20. Compound Interest Factors Discrete Cash Flow, Discrete Compounding

21. Compound Interest Factors Discrete Cash Flow, Continuous Compounding

22. Compound Interest Factors Discrete Cash Flow, Continuous Compounding

23. Compound Interest Factors Continuous Uniform Cash Flow, Continuous Compounding

24. Quiz---When and Where Quiz: Tuesday, Sept. 27, 2005
11: :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)

25. Quiz---Who will be there
U, U, U, U, and U!!!!
CraigTipping   
Group 1 (Last NameA-M) ELL 168
LeYang            
Group 2 (Last Name N-Z) ELL 061

26. 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

27. 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)

28. Quiz---Important Points
Simple Interests
Compound Interests
Future Value
Present Value
Key: Compound Interest
Key: Understand the Question

29. 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.

30. Calculator Talk No programmable No economic function Simple the best
Trust your ability
Trust your teaching group

31. Questions?
(Sorry I forget the problems)

32. 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

33. 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

34. 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

35. 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

36. 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