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

Topics Today

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Series and Arithmetic SeriesSeries 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?

))1((...)3()2()( dnadadadaasn

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Sum of terms of a finite APSum of terms of a finite AP

])1(2[2

])1(2[2

)1(22S

Therefore, ; 1)-(n nd1)-(n nd termsnd 1)-(n are There

2an;n2a terms2a (n) are There

)2()2(...)2()2(2

)()2(...)2()(

)()2(...)2()(

])1([])2([...)2()(

n

dnan

S

dnanS

nndan

andandandandaaS

adadadndadndaS

dndadndadadaaS

dnadnadadaaS

n

n

n

n

n

n

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Arithmetic Gradient SeriesArithmetic 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

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Arithmetic Gradient to Uniform SeriesArithmetic 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)

1)1(

1),,/(

Ni

Ni

NiGA

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Arithmetic Gradient to Uniform SeriesArithmetic 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 Arithmetic Gradient Series with Base AnnuityBase Annuity

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Example 3-8Example 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: AnswerExample 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 = 20A = 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 = 20P = 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 FactorArithmetic Gradient Conversion Factor(to Uniform Series)(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 FactorArithmetic Gradient Conversion Factor(to Uniform Series)(to Uniform Series)

1 2 3 4 51 2 3 4 5

A A A A A

$1000

$1150

$1300

$1450

$1600

246,1$

]1)20.01[(20.0

)20.0*51()20.01(150$000,1$

]1)1[(

)1()1(

5

5

n

n

g ii

niiGAA

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Arithmetic Gradient Conversion FactorArithmetic Gradient Conversion Factor(to Present Value)(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%?

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Arithmetic Gradient Conversion FactorArithmetic Gradient Conversion Factor(to Present Value)(to Present Value)

$1000

$1150$1300

$1450$1600

1 2 3 4 5

P

727,3$

)20.0(

)20.01)(20.0*51(1150$

)20.01(20.0

1)20.01(000,1$

)1)(1(1

)1(

1)1(

2

5

5

5

2

i

iniG

ii

iAP

n

n

n

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Geometric Gradient SeriesGeometric 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

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Geometric SeriesGeometric 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

1n2n32n akak....akakakaS

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Sum of terms of a finite GPSum 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.

)k1(

)k1(aS

)k1(a)k1(S

ak00.....00akSS

akakak....akakkS

akak....akakaS

n

n

nn

nnn

n1n2n2n

1n2n2n

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Geometric Gradient to Geometric Gradient to Present WorthPresent 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:

N

N

i

gA

i

gAi

AP

)1(

)1(

)1(

)1()1(

1

2

ig

igi

i

11

1

1 that so 1

11

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

g)(,N)(P/A,i

gii

iNigAP

N

N

1

11

)1(

1)1(),,,/(

gA

NP1

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Compound Interest FactorsCompound Interest FactorsDiscrete Cash Flow, Discrete CompoundingDiscrete Cash Flow, Discrete Compounding

To Find Given Name of Factor Factor

F PCompound Amount Factor (single payment)

P FPresent Worth Factor (single payment)

F ACompound Amount Factor (uniform series)

A F Sinking Fund Factor

ni)1(

ni )1(

i

i n 1)1(

1)1( ni

i

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Compound Interest FactorsCompound Interest FactorsDiscrete Cash Flow, Discrete CompoundingDiscrete Cash Flow, Discrete Compounding

To Find Given Name of Factor Factor

A P Capital Recovery Factor

P APresent Worth Factor (uniform series)

A G

Arithmetic Gradient Conversion Factor (to uniform series)

P G

Arithmetic Gradient Conversion Factor (to present value)

1)1(

)1(

n

n

i

ii

n

n

ii

i

)1(

1)1(

]1)1[(

)1()1(

n

n

ii

nii

2

)1)(1(1

i

ini n

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Compound Interest FactorsCompound Interest FactorsDiscrete Cash Flow, Continuous CompoundingDiscrete Cash Flow, Continuous Compounding

To Find Given Name of Factor Factor

F PCompound Amount Factor (single payment)

P FPresent Worth Factor (single payment)

F ACompound Amount Factor (uniform series)

A F Sinking Fund Factor

rne

rne

1

1

r

rn

e

e

1

1

rn

r

e

e

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Compound Interest FactorsCompound Interest FactorsDiscrete Cash Flow, Continuous CompoundingDiscrete Cash Flow, Continuous Compounding

To Find Given Name of Factor Factor

A P Capital Recovery Factor

P APresent Worth Factor (uniform series)

A G

Arithmetic Gradient Conversion Factor (to uniform series)

P G

Arithmetic Gradient Conversion Factor (to present value)

1

)1(

rn

rrn

e

ee

)1(

1

rrn

rn

ee

e

11

1

rnr e

n

e

2)1(

)1(1

rrn

rrn

ee

ene

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Compound Interest FactorsCompound Interest FactorsContinuous Uniform Cash Flow, Continuous CompoundingContinuous Uniform Cash Flow, Continuous Compounding

To Find Given Name of Factor Factor

C F

Sinking Fund Factor (continuous, uniform payments)

C P

Capital Recovery Factor (continuous, uniform payments)

F C

Compound Amount Factor (continuous, uniform payments)

P C

Present Worth Factor (continuous, uniform payments)

1rne

r

1rn

rn

e

re

r

ern 1

rn

rn

re

e 1

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Quiz---When and WhereQuiz---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 thereQuiz---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

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Quiz---Problems, SolutionsQuiz---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

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Quiz---Based on Chapter 1.2.3.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)

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Quiz---Important PointsQuiz---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!!!Quiz---Books in Library!!!

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

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 

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Calculator TalkCalculator 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 TableProject----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----RequirementsProject----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 doProject 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----RecourseProject----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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SummarySummary

• Conversion for Arithmetic Gradient Series

• Conversion for Geometric Gradient Series

• Quiz: My slides and the examples in slides

• Project: Good Idea, be open, independent