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Warm-Up ReviewWarm-Up Review

• Homepage• Rule of 72• Single Sum Compounding • Annuities

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Homepage is ImportantHomepage is Important• Homepage has everything

– All Slides ……… (important)– Lecture outline– Quiz and project ..time. location– Class/Quiz/Tutorial information– Teaching Group (1+4) contact

• www.ece.uvic.ca/~wli

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The Rule of 72The Rule of 72• Estimates how many years an investment

will take to double in value

• Number of years to double =

72 / annual compound interest rate• Example -- 72 / 8 = 9 therefore, it will

take 9 years for an investment to double in value if it earns 8% annually

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Approx. Years to Double = 7272 / i%

7272 / 12% = 6 Years 6 Years

[Actual Time is 6.12 Years]

Quick! How long does it take to double $5,000 at a compound rate of 12% per year?

Example: Double Your Money!!!Example: Double Your Money!!!

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Given:• Amount of deposit today (PV):

$50,000• Interest rate: 11%• Frequency of compounding: Annual • Number of periods (5 years): 5

periodsWhat is the future value of this single sum?FVn = PV(1 + i)n

$50,000 x (1.68506) = $84,253

Single Sum Problems: Future ValueSingle Sum Problems: Future Value

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Given:• Amount of deposit end of 5 years:

$84,253• Interest rate (discount) rate: 11%• Frequency of compounding: Annual • Number of periods (5 years): 5 periodsWhat is the present value of this single sum?• FVn = PV(1 + i)n

$84,253 x (0.59345) = $50,000

Single Sum Problems: Present ValueSingle Sum Problems: Present Value

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An annuity requires that:• the periodic payments or receipts

(rents) always be of the same amount,

• the interval between such payments or receipts be the same, and

• the interest be compounded once each interval.

Annuity ComputationsAnnuity Computations

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If one saves $1,000 a year at the end of every year for three years in an account earning 7% interest, compounded annually, how much will one have at the end of the third year?

Example of AnnuityExample of AnnuityExample of AnnuityExample of Annuity

$1,000 $1,000 $1,000

0 1 2 3 3 4

$3,215 = FVA$3,215 = FVA33

End of Year

7%

$1,070

$1,145

FVA3 = $1,000(1.07)2 + $1,000(1.07)1 + $1,000(1.07)0 = $3,215

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Given:• Deposit made at the end of each

period: $5,000• Compounding: Annual• Number of periods: Five• Interest rate: 12%What is future value of these deposits?F = A[(1+i)n - 1] / i

$5,000 x (6.35285) = $ 31,764.25

Annuities: Future ValueAnnuities: Future Value

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Given:• Rental receipts at the end of each

period: $6,000• Compounding: Annual• Number of periods (years): 5• Interest rate: 12%

What is the present value of these receipts?

F = A[(1+i)n - 1] / i$6,000 x (3.60478) = $ 21,628.68

Annuities: Present Annuities: Present ValueValue

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Key of Annuity CalculationKey of Annuity Calculation

Fv = Pv[(1+i)n - 1] / i

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• Single Payment, Present/Future Value Factor• Sinking Factor, Capital Recovery Factor• Conversion for Arithmetic Gradient Series• Conversion for Geometric Gradient Series

Topics Today

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Compound Amount Factor Compound Amount Factor (Single Payment)(Single Payment)

• This factor finds the equivalent future worth, F, of a present investment, P, held for n periods at i rate of interest.

• Example: What is the value in 9 years of $1,200 invested now at 10% interest

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Compound Amount Factor Compound Amount Factor (Single Payment)(Single Payment)

P = $1,200

1 2 3 4 9

F

829,2$

)10.01(200,1$

)1(9

niPF

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Present Worth Factor Present Worth Factor (Single Payment)(Single Payment)

• This factor finds the equivalent present value, P, of a single future cash flow, F, occurring at n periods in the future when the interest rate is i per period.

• Example: What amount would you have to invest now to yield $2,829 in 9 years if the interest rate per year is 10%?

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Present Worth Factor Present Worth Factor (Single Payment)(Single Payment)

P

1 2 3 4 9

F = $2,829

200,1$

)10.01(829,2$

)1(9

niFP

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Compound Amount FactorCompound Amount Factor(Uniform Series)(Uniform Series)

• This factor finds the equivalent future value, F, of the accumulation of a uniform series of equal annual payments, A, occurring over n periods at i rate of interest per period.

• Example: What would be the future worth of an annual year-end cash flow of $800 for 6 years at 12% interest per year?

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Compound Amount Factor Compound Amount Factor (Uniform Series)(Uniform Series)

1 2 3 4 6

F

492,6$12.0

1)12.01(800$

1)1(

6

i

iAF

n

5

$800 $800$800 $800 $800 $800

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Sinking Fund FactorSinking Fund Factor• This factor determines how much must be

deposited each period in a uniform series, A, for n periods at i interest per period to yield a specified future sum.

• Example: If a $1.2 million bond issue is to be retired at the end of 20 years, how much must be deposited annually into a sinking fund at 7% interest per year?

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Sinking Fund FactorSinking Fund Factor

1 2 3 4 20

F = $1,200,000

272,29$

1)07.01(

07.0000,200,1$

1)1(

20

ni

iFA

A AA A A A

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Capital Recovery FactorCapital Recovery Factor• This factor finds an annuity, or uniform

series of payments, over n periods at i interest per period that is equivalent to a present value, P.

• Example: What savings in annual manufacturing costs over an 8 year period would justify the purchase of a $120,000 machine if the firm’s minimum attractive rate of return (MARR) were 25%?

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Capital Recovery FactorCapital Recovery Factor

$120,000

1 2 3 8

A A A A A

048,36$

1)25.01(

)25.01(25.0000,120$

1)1(

)1(

8

8

n

n

i

iiPA

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Present Worth FactorPresent Worth Factor(Uniform Series)(Uniform Series)

• This factor finds the equivalent present value, P, of a series of end-of-period payments, A, for n periods at i interest per period.

• Example: What lump sum payment would be required to provide $50,000 per year for 30 years at an annual interest rate of 9%?

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Present Worth FactorPresent Worth Factor(Uniform Series)(Uniform Series)

P

1 2 3 30

$50,000 $50,000 $50,000 $50,000 $50,000

452,620$

)07.01(07.0

1)07.01(000,50$

)1(

1)1(

30

30

n

n

ii

iAP

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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

45

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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Calculator TalkCalculator Talk

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

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SummarySummary

• Single Sum Compounding • Annuities• Conversion for Arithmetic Gradient Series• Conversion for Geometric Gradient Series

• Key: Compound Interests Calculation