3-1

Chapter 3Chapter 3

Time Value of Time Value of MoneyMoney

Time Value of Time Value of MoneyMoney

3-2

The Time Value of MoneyThe Time Value of MoneyThe Time Value of MoneyThe Time Value of Money

The Interest Rate

Simple Interest

Compound Interest

Amortizing a Loan

The Interest Rate

Simple Interest

Compound Interest

Amortizing a Loan

3-3

Obviously, $10,000 today$10,000 today.

You already recognize that there is TIME VALUE TO MONEYTIME VALUE TO MONEY!!

The Interest RateThe Interest RateThe Interest RateThe Interest Rate

Which would you prefer -- $10,000 today $10,000 today or $10,000 in 5 years$10,000 in 5 years?

Interest is the money paid for the use of money.

3-4

TIMETIME allows you the opportunity to postpone consumption and earn

INTERESTINTEREST.

Why TIME?Why TIME?Why TIME?Why TIME?

Why is TIMETIME such an important element in your decision?

3-5

Types of InterestTypes of InterestTypes of InterestTypes of Interest

Compound InterestCompound Interest

Interest paid (earned) on any previous interest earned, as well as on the principal borrowed (lent).

Simple InterestSimple Interest

Interest paid (earned) on only the original amount, or principal borrowed (lend).

3-6

Simple Interest FormulaSimple Interest FormulaSimple Interest FormulaSimple Interest Formula

FormulaFormula SI = P0(i)(n)

SI: Simple Interest

P0: Deposit today (t=0)

i: Interest Rate per Period

n: Number of Time Periods

3-7

SI = P0(i)(n)= $1,00(.08)(10)= $80$80

Simple Interest ExampleSimple Interest ExampleSimple Interest ExampleSimple Interest Example

Assume that you deposit $1,00 in an account earning 8% simple interest for 10 years. What is the accumulated interest at the end of the 10nd year?

3-8

FVFV = P0 + SI = $1,00 + $80

= $180$180

Future ValueFuture Value is the value at some future time of a present amount of money, or a series of payments, evaluated at a given interest rate.

Simple Interest (FV)Simple Interest (FV)Simple Interest (FV)Simple Interest (FV)

What is the Future Value Future Value (FVFV) of the deposit?

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The Present Value is simply the $1,00 you originally deposited. That is the value today!

Present ValuePresent Value is the current value of a future amount of money, or a series of payments, evaluated at a given interest rate.

Simple Interest (PV)Simple Interest (PV)Simple Interest (PV)Simple Interest (PV)

What is the Present Value Present Value (PVPV) of the previous problem?

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Compound interestCompound interest

Interest that is earned on a given deposit and has become part of principal at the end of a specified period

Future value of a present amount at a future date, found by applying compound interest over a specified period of time.

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The equation for future valueThe equation for future value

FV=PV*(1+i)n

If Fred places $100 in a savings account paying 8% interest compounded annually, at the end of 1 year he will have $108 in the account.<100*(1.08)=$108>

If Fred were to leave this money in the account for another year, he would be paid interest at the rate of 8% on the new principal of $108.At the end of this second year there would be $116.64 in the account.<108*(1.08)=116.64> or <100*(1.08)2=116.64>

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FVFV11 = P0(1+i)1

FVFV22 = P0(1+i)2

General Future Value Future Value Formula:

FVFVnn = P0 (1+i)n

or FVFVnn = P0 (FVIFFVIFi,n) -- See Table ISee Table I

General Future General Future Value FormulaValue FormulaGeneral Future General Future Value FormulaValue Formula

etc.

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FVIFFVIFi,n is found on Table I at the end

of the book or on the card insert.

Valuation Using Table IValuation Using Table IValuation Using Table IValuation Using Table I

Period 6% 7% 8%1 1.060 1.070 1.0802 1.124 1.145 1.1663 1.191 1.225 1.2604 1.262 1.311 1.3605 1.338 1.403 1.469

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FVFV22 = $1,000 (FVIFFVIF7%,2)= $1,000 (1.145)

= $1,145$1,145 [Due to Rounding]

Using Future Value TablesUsing Future Value TablesUsing Future Value TablesUsing Future Value Tables

Period 6% 7% 8%1 1.060 1.070 1.0802 1.124 1.145 1.1663 1.191 1.225 1.2604 1.262 1.311 1.3605 1.338 1.403 1.469

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Julie Miller wants to know how large her deposit of $10,000$10,000 today will become at a compound annual interest rate of 10% for 5 years5 years.

Story Problem ExampleStory Problem ExampleStory Problem ExampleStory Problem Example

0 1 2 3 4 55

$10,000$10,000

FVFV55

10%

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Calculation based on Table I:FVFV55 = $10,000 (FVIFFVIF10%, 5)

= $10,000 (1.611)= $16,110$16,110 [Due to Rounding]

Story Problem SolutionStory Problem SolutionStory Problem SolutionStory Problem Solution

Calculation based on general formula:FVFVnn = P0 (1+i)n

FVFV55 = $10,000 (1+ 0.10)5

= $16,105.10$16,105.10

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Present value of a single Present value of a single amountamount

The current dollar value of a future amount-the amount of money that would have to be invested today at a given interest rate over a specified period to equal the future amount.

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Concept of present valueConcept of present value

The process of finding present value is often referred to as discounting cash flows. It is concerned with answering the following question:" if I can earn i percent on my money, what is the most I would be willing to pay now for an opportunity to receive FV n dollars n periods from today?”

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Equation for calculating PVEquation for calculating PV

PV*(1+i)n=FV

PV=FV/(1+i)n

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Assume that you need $1,000$1,000 in 2 years.2 years. Let’s examine the process to determine how much you need to deposit today at a discount rate of 7% compounded annually.

0 1 22

$1,000$1,000

7%

PV1PVPV00

Present ValuePresent Value Single Deposit (Graphic)Single Deposit (Graphic)Present ValuePresent Value Single Deposit (Graphic)Single Deposit (Graphic)

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PVPV00 = FVFV22 / (1+i)2 = $1,000$1,000 / (1.07)2 =

FVFV22 / (1+i)2 = $873.44$873.44

Present Value Present Value Single Deposit (Formula)Single Deposit (Formula)Present Value Present Value Single Deposit (Formula)Single Deposit (Formula)

0 1 22

$1,000$1,000

7%

PVPV00

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PVPV00 = FVFV11 / (1+i)1

PVPV00 = FVFV22 / (1+i)2

General Present Value Present Value Formula:

PVPV00 = FVFVnn / (1+i)n

or PVPV00 = FVFVnn (PVIFPVIFi,n) -- See Table IISee Table II

General Present General Present Value FormulaValue FormulaGeneral Present General Present Value FormulaValue Formula

etc.

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PVIFPVIFi,n is found on Table II at the end

of the book or on the card insert.

Valuation Using Table IIValuation Using Table IIValuation Using Table IIValuation Using Table II

Period 6% 7% 8% 1 .943 .935 .926 2 .890 .873 .857 3 .840 .816 .794 4 .792 .763 .735 5 .747 .713 .681

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PVPV22 = $1,000$1,000 (PVIF7%,2)= $1,000$1,000 (.873)

= $873$873 [Due to Rounding]

Using Present Value TablesUsing Present Value TablesUsing Present Value TablesUsing Present Value Tables

Period 6% 7% 8%1 .943 .935 .9262 .890 .873 .8573 .840 .816 .7944 .792 .763 .7355 .747 .713 .681

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Julie Miller wants to know how large of a deposit to make so that the money will grow to $10,000$10,000 in 5 years5 years at a discount rate of 10%.

Story Problem ExampleStory Problem ExampleStory Problem ExampleStory Problem Example

0 1 2 3 4 55

$10,000$10,000PVPV00

10%

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Calculation based on general formula: PVPV00 = FVFVnn / (1+i)n

PVPV00 = $10,000$10,000 / (1+ 0.10)5

= $6,209.21$6,209.21

Calculation based on Table I:PVPV00 = $10,000$10,000 (PVIFPVIF10%, 5)

= $10,000$10,000 (.621)= $6,210.00$6,210.00 [Due to Rounding]

Story Problem SolutionStory Problem SolutionStory Problem SolutionStory Problem Solution

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Graphical view of FVGraphical view of FV

GRAPHICAL VIEW OF FV

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Future value relationshipFuture value relationship

Higher the interest rates, higher the future value

Longer the period of time, higher the future value

For an interest rate of 0% the FV is always equal to its PV(1.00). But for any interest rate greater than zero, future value is greater than the present value

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A graphical view of present A graphical view of present valuevalue

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Present value relationshipPresent value relationship

The higher the discount rate, the lower the present value

The longer the period of time, the lower the present value

At the discount rate 0%,the present value is always equal to its future value

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Types of AnnuitiesTypes of AnnuitiesTypes of AnnuitiesTypes of Annuities

Ordinary AnnuityOrdinary Annuity: Payments or receipts occur at the end of each period.

Annuity DueAnnuity Due: Payments or receipts occur at the beginning of each period.

An AnnuityAn Annuity represents a series of equal payments (or receipts) occurring over a specified number of equidistant periods.

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Examples of AnnuitiesExamples of Annuities

Student Loan Payments

Car Loan Payments

Insurance Premiums

Mortgage Payments

Retirement Savings

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Parts of an AnnuityParts of an AnnuityParts of an AnnuityParts of an Annuity

0 1 2 3

$100 $100 $100

(Ordinary Annuity)EndEnd of

Period 1EndEnd of

Period 2

Today EqualEqual Cash Flows Each 1 Period Apart

EndEnd ofPeriod 3

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Parts of an AnnuityParts of an AnnuityParts of an AnnuityParts of an Annuity

0 1 2 3

$100 $100 $100

(Annuity Due)BeginningBeginning of

Period 1BeginningBeginning of

Period 2

Today EqualEqual Cash Flows Each 1 Period Apart

BeginningBeginning ofPeriod 3

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FVAFVAnn = R(1+i)n-1 + R(1+i)n-2 + ... + R(1+i)1 + R(1+i)0

Overview of an Overview of an Ordinary Annuity -- FVAOrdinary Annuity -- FVAOverview of an Overview of an Ordinary Annuity -- FVAOrdinary Annuity -- FVA

R R R

0 1 2 n n n+1

FVAFVAnn

R = Periodic Cash Flow

Cash flows occur at the end of the period

i% . . .

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FVAFVA33 = $1,000(1.07)2 + $1,000(1.07)1 + $1,000(1.07)0

= $1,145 + $1,070 + $1,000 = $3,215$3,215

Example of anExample of anOrdinary Annuity -- FVAOrdinary Annuity -- FVAExample of anExample of anOrdinary Annuity -- FVAOrdinary Annuity -- FVA

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

0 1 2 3 3 4

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

7%

$1,070

$1,145

Cash flows occur at the end of the period

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Future value interest factor Future value interest factor for an ordinary annuityfor an ordinary annuity

FVIFi,n=1/i*<(1+i)n-1>

FVA=PMT*(FVIFAi,n)

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Hint on Annuity ValuationHint on Annuity Valuation

The future value of an ordinary annuity can be viewed as

occurring at the endend of the last cash flow period, whereas the future value of an annuity due can be viewed as occurring at the beginningbeginning of the last cash

flow period.

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FVAFVAnn = R (FVIFAi%,n) FVAFVA33 = $1,000 (FVIFA7%,3)

= $1,000 (3.215) = $3,215$3,215

Valuation Using Table IIIValuation Using Table IIIValuation Using Table IIIValuation Using Table III

Period 6% 7% 8%1 1.000 1.000 1.0002 2.060 2.070 2.0803 3.184 3.215 3.2464 4.375 4.440 4.5065 5.637 5.751 5.867

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FVADFVADnn = R(1+i)n + R(1+i)n-1 + ... + R(1+i)2 +

R(1+i)1 = FVAFVAn n (1+i)

Overview View of anOverview View of anAnnuity Due -- FVADAnnuity Due -- FVADOverview View of anOverview View of anAnnuity Due -- FVADAnnuity Due -- FVAD

R R R R R

0 1 2 3 n-1n-1 n

FVADFVADnn

i% . . .

Cash flows occur at the beginning of the period

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numericalnumerical

Martin has $10000 that she can deposit in any of three saving counts for a 3 year period. Bank A compounds interest on an annual basis, bank B compounds interest twice each year, Bank C compounds interest each quarter. All three banks have a stated annual interest rate of 4%

What amount would Ms.Martin have at the end of third year?

On the basis of your findings in banks, which bank should she prefer.

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NUMERICALSNUMERICALS

Ramish wishes to choose the better of two equally costly cash flow streams: annuity X and annuity Y.X is an annuity due with a cash inflow of $9000 for each of 6 years is an ordinary annuity with cash inflow of 410000 or each of 6 years. Assume that he can earn 15% on his investment.

On a subject basis, which annuity do you think is more attractive and why?

Find the future value at the end of year 6,for both annuity x and Y.

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NUMERICALSNUMERICALS

what is the present value of $ 6000 to be received at the end of 6 years if the discount rate is 12%?

$100 at the end of three years is worth how much today, assuming a discount rate of

100%

10%

0%

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FVADFVAD33 = $1,000(1.07)3 + $1,000(1.07)2 + $1,000(1.07)1

= $1,225 + $1,145 + $1,070 = $3,440$3,440

Example of anExample of anAnnuity Due -- FVADAnnuity Due -- FVADExample of anExample of anAnnuity Due -- FVADAnnuity Due -- FVAD

$1,000 $1,000 $1,000 $1,070

0 1 2 3 3 4

$3,440 = FVAD$3,440 = FVAD33

7%

$1,225

$1,145

Cash flows occur at the beginning of the period

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FVADFVADnn = R (FVIFAi%,n)(1+i)

FVADFVAD33 = $1,000 (FVIFA7%,3)(1.07)= $1,000 (3.215)(1.07) =

$3,440$3,440

Valuation Using Table IIIValuation Using Table IIIValuation Using Table IIIValuation Using Table III

Period 6% 7% 8%1 1.000 1.000 1.0002 2.060 2.070 2.0803 3.184 3.215 3.2464 4.375 4.440 4.5065 5.637 5.751 5.867

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Present value of ordinary Present value of ordinary annuityannuity

PVA=PMT(PVIFAi,n)

PVIF= 1 - 1

(1+i)n

i

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PRESENT VALUE OF PRESENT VALUE OF ANNUITY DUEANNUITY DUE

PVIF=PVIFA*(1+i)

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PV of an ordinary annuityPV of an ordinary annuity

Braden company a small producer of toys wants to determine the most it should pay to purchase a particular ordinary annuity. the annuity consist of cash flows of 700v at the end of each year for 5 years. The firm requires the annuity to provide a minimum return of 8%.

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Long method for finding the present value of an ordinary annuity

Year CF PVIF8%,n PV

(1) (2) (1*2) 1 700 0.926 648.20 2 700 0.857 599.90 3 700 0.794 555.80 4 700 0.735 514.50 5 700 0.681 476.70 present value of annuity=2795.10 PVIF=1/i*(1-1/(1+i)n)

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PVAPVAnn = R/(1+i)1 + R/(1+i)2

+ ... + R/(1+i)n

Finding present value of anFinding present value of anOrdinary Annuity -- PVAOrdinary Annuity -- PVAFinding present value of anFinding present value of anOrdinary Annuity -- PVAOrdinary Annuity -- PVA

R R R

0 1 2 n n n+1

PVAPVAnn

R = Periodic Cash Flow

i% . . .

Cash flows occur at the end of the period

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PVAPVA33 = $1,000/(1.07)1 + $1,000/(1.07)2 +

$1,000/(1.07)3

= $934.58 + $873.44 + $816.30 = $2,624.32$2,624.32

Example of anExample of anOrdinary Annuity -- PVAOrdinary Annuity -- PVAExample of anExample of anOrdinary Annuity -- PVAOrdinary Annuity -- PVA

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

0 1 2 3 3 4

$2,624.32 = PVA$2,624.32 = PVA33

7%

$ 934.58$ 873.44 $ 816.30

Cash flows occur at the end of the period

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Hint on Annuity ValuationHint on Annuity Valuation

The present value of an ordinary annuity can be viewed as

occurring at the beginningbeginning of the first cash flow period, whereas the present value of an annuity due can be viewed as occurring at the endend of the first cash flow

period.

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PVAPVAnn = R (PVIFAi%,n) PVAPVA33 = $1,000 (PVIFA7%,3)

= $1,000 (2.624) = $2,624$2,624

Valuation Using Table IVValuation Using Table IVValuation Using Table IVValuation Using Table IV

Period 6% 7% 8%1 0.943 0.935 0.9262 1.833 1.808 1.7833 2.673 2.624 2.5774 3.465 3.387 3.3125 4.212 4.100 3.993

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PVADPVADnn = R/(1+i)0 + R/(1+i)1 + ... + R/(1+i)n-1

= PVAPVAn n (1+i)

Overview of anOverview of anAnnuity Due -- PVADAnnuity Due -- PVADOverview of anOverview of anAnnuity Due -- PVADAnnuity Due -- PVAD

R R R R

0 1 2 n-1n-1 n

PVADPVADnn

R: Periodic Cash Flow

i% . . .

Cash flows occur at the beginning of the period

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PVADPVADnn = $1,000/(1.07)0 + $1,000/(1.07)1 + $1,000/(1.07)2 = $2,808.02$2,808.02

Example of anExample of anAnnuity Due -- PVADAnnuity Due -- PVADExample of anExample of anAnnuity Due -- PVADAnnuity Due -- PVAD

$1,000.00 $1,000 $1,000

0 1 2 33 4

$2,808.02 $2,808.02 = PVADPVADnn

7%

$ 934.58$ 873.44

Cash flows occur at the beginning of the period

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PVADPVADnn = R (PVIFAi%,n)(1+i)

PVADPVAD33 = $1,000 (PVIFA7%,3)(1.07) = $1,000 (2.624)(1.07) =

$2,808$2,808

Valuation Using Table IVValuation Using Table IVValuation Using Table IVValuation Using Table IV

Period 6% 7% 8%1 0.943 0.935 0.9262 1.833 1.808 1.7833 2.673 2.624 2.5774 3.465 3.387 3.3125 4.212 4.100 3.993

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1. Read problem thoroughly

2. Determine if it is a PV or FV problem

3. Create a time line

4. Put cash flows and arrows on time line

5. Determine if solution involves a single CF, annuity stream(s), or mixed flow

6. Solve the problem

7. Check with financial calculator (optional)

Steps to Solve Time Value Steps to Solve Time Value of Money Problemsof Money ProblemsSteps to Solve Time Value Steps to Solve Time Value of Money Problemsof Money Problems

3-58

Julie Miller will receive the set of cash flows below. What is the Present Value Present Value at a discount rate of 10%10%?

Mixed Flows ExampleMixed Flows ExampleMixed Flows ExampleMixed Flows Example

0 1 2 3 4 55

$600 $600 $400 $400 $100$600 $600 $400 $400 $100

PVPV00

10%10%

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1. Solve a “piece-at-a-timepiece-at-a-time” by discounting each piecepiece back to

t=0.

2. Solve a “group-at-a-timegroup-at-a-time” by firstbreaking problem into groups of

annuity streams and any single cash flow group. Then discount each groupgroup back to t=0.

How to Solve?How to Solve?How to Solve?How to Solve?

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““Piece-At-A-Time”Piece-At-A-Time”““Piece-At-A-Time”Piece-At-A-Time”

0 1 2 3 4 55

$600 $600 $400 $400 $100$600 $600 $400 $400 $10010%

$545.45$545.45$495.87$495.87$300.53$300.53$273.21$273.21$ 62.09$ 62.09

$1677.15 $1677.15 = = PVPV00 of the Mixed Flowof the Mixed Flow

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““Group-At-A-Time” (#1)Group-At-A-Time” (#1)““Group-At-A-Time” (#1)Group-At-A-Time” (#1)

0 1 2 3 4 55

$600 $600 $400 $400 $100$600 $600 $400 $400 $100

10%

$1,041.60$1,041.60$ 573.57$ 573.57$ 62.10$ 62.10

$1,677.27$1,677.27 = = PVPV00 of Mixed Flow of Mixed Flow [Using Tables][Using Tables]

$600(PVIFA10%,2) = $600(1.736) = $1,041.60$400(PVIFA10%,2)(PVIF10%,2) = $400(1.736)(0.826) = $573.57

$100 (PVIF10%,5) = $100 (0.621) = $62.10

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““Group-At-A-Time” (#2)Group-At-A-Time” (#2)““Group-At-A-Time” (#2)Group-At-A-Time” (#2)

0 1 2 3 4

$400 $400 $400 $400$400 $400 $400 $400

PVPV00 equals

$1677.30.$1677.30.

0 1 2

$200 $200$200 $200

0 1 2 3 4 5

$100$100

$1,268.00$1,268.00

$347.20$347.20

$62.10$62.10

PlusPlus

PlusPlus

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General Formula:

FVn = PVPV00(1 + [i/m])mn

n: Number of Yearsm: Compounding Periods per

Yeari: Annual Interest RateFVn,m: FV at the end of Year n

PVPV00: PV of the Cash Flow today

Frequency of Frequency of CompoundingCompoundingFrequency of Frequency of CompoundingCompounding

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Julie Miller has $1,000$1,000 to invest for 2 years at an annual interest rate of

12%.

Annual FV2 = 1,0001,000(1+ [.12/1])(1)(2)

= 1,254.401,254.40

Semi FV2 = 1,0001,000(1+ [.12/2])(2)(2)

= 1,262.481,262.48

Impact of FrequencyImpact of FrequencyImpact of FrequencyImpact of Frequency

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Qrtly FV2 = 1,0001,000(1+ [.12/4])(4)(2)

= 1,266.771,266.77

Monthly FV2 = 1,0001,000(1+ [.12/12])(12)(2)

= 1,269.731,269.73

Daily FV2 = 1,0001,000(1+[.12/365])(365)

(2) = 1,271.201,271.20

Impact of FrequencyImpact of FrequencyImpact of FrequencyImpact of Frequency

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Present value of perpetuityPresent value of perpetuity

An annuity with an infinite life, providing continual annual cash flow

PVIF=1/i

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The annual rate of interest actually paid or earned

The actual rate of interest earned (paid) after adjusting the nominal

rate for factors such as the number of compounding periods per year.

(1 + [ i / m ] )m - 1

Effective Annual Effective Annual Interest RateInterest RateEffective Annual Effective Annual Interest RateInterest Rate

3-68

Nominal annual rateNominal annual rate

Contractual annual rate of interest charged by a lender or promised by a borrower.

3-69

Basket Wonders (BW) has a $1,000 CD at the bank. The interest rate is

6% compounded quarterly for 1 year. What is the Effective Annual

Interest Rate (EAREAR)?

EAREAR = ( 1 + 6% / 4 )4 - 1 = 1.0614 - 1 = .0614 or 6.14%!6.14%!

BW’s Effective BW’s Effective Annual Interest RateAnnual Interest RateBW’s Effective BW’s Effective Annual Interest RateAnnual Interest Rate

3-70

Loan amortizationLoan amortization

The determination of the equal periodic loan payments necessary to provide lender with a specified interest return and to repay the loan principal over a specified period.

3-71

1. Calculate the payment per period.

2. Determine the interest in Period t. (Loan balance at t-1) x (i% / m)

3. Compute principal payment principal payment in Period t.(Payment - interest from Step 2)

4. Determine ending balance in Period t.(Balance - principal payment principal payment from Step

3)

5. Start again at Step 2 and repeat.

Steps to Amortizing a LoanSteps to Amortizing a LoanSteps to Amortizing a LoanSteps to Amortizing a Loan

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Julie Miller is borrowing $22,000 $22,000 at a compound annual interest rate of 12%.

Amortize the loan if annual payments are made for 5 years.

Step 1: Payment

PVPV00 = R (PVIFA i%,n)

$22,000 $22,000 = R (PVIFA 12%,5)

$22,000$22,000 = R (3.605)

RR = $22,000$22,000 / 3.605 = $5351$5351

Amortizing a Loan ExampleAmortizing a Loan ExampleAmortizing a Loan ExampleAmortizing a Loan Example

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Amortizing a Loan ExampleAmortizing a Loan ExampleAmortizing a Loan ExampleAmortizing a Loan Example

End of Year

Payment Interest Principal (Pmt-int)

Ending Balance

0 --- --- --- $22,000

1 $5351 2640 2711 19289

2 5351 2315 3036 16253

3 5351 1951 3400 12853

4 5351 1542 3809 9044

5 5351 1085 4266 4778

6 5351 573 4778 0

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Usefulness of AmortizationUsefulness of Amortization

2.2. Calculate Debt Outstanding Calculate Debt Outstanding -- The quantity of outstanding debt

may be used in financing the day-to-day activities of the firm.

1.1. Determine Interest Expense Determine Interest Expense -- Interest expenses may reduce taxable income of the firm.