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3.1 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer. Chapter 3 Chapter 3 The Time The Time Value of Value of Money Money

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Page 1: Chapter  03  imp

3.1 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

Chapter 3Chapter 3

The Time Value The Time Value of Moneyof Money

The Time Value The Time Value of Moneyof Money

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3.2 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

After studying Chapter 3, After studying Chapter 3, you should be able to:you should be able to:

1. Understand what is meant by "the time value of money."

2. Understand the relationship between present and future value.

3. Describe how the interest rate can be used to adjust the value of cash flows – both forward and backward – to a single point in time.

4. Calculate both the future and present value of: (a) an amount invested today; (b) a stream of equal cash flows (an annuity); and (c) a stream of mixed cash flows.

5. Distinguish between an “ordinary annuity” and an “annuity due.”

6. Use interest factor tables and understand how they provide a shortcut to calculating present and future values.

7. Use interest factor tables to find an unknown interest rate or growth rate when the number of time periods and future and present values are known.

8. Build an “amortization schedule” for an installment-style loan.

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3.3 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

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

• Compounding More Than Once per Year

• The Interest Rate

• Simple Interest

• Compound Interest

• Amortizing a Loan

• Compounding More Than Once per Year

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3.4 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

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 $10,000 today today or $10,000 in 5 years$10,000 in 5 years?

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3.5 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

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?

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3.6 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

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 (lent).

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3.7 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

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

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3.8 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

• SI = P0(i)(n)= $1,000(0.07)(2)= $140$140

Simple Interest ExampleSimple Interest ExampleSimple Interest ExampleSimple Interest Example

• Assume that you deposit $1,000 in an account earning 7% simple interest for 2 years. What is the accumulated interest at the end of the 2nd year?

Page 9: Chapter  03  imp

3.9 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

FVFV = P0 + SI = $1,000 + $140= $1,140$1,140

• 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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3.10 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

The Present Value is simply the $1,000 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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3.11 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

0

5000

10000

15000

20000

1st Year 10thYear

20thYear

30thYear

Future Value of a Single $1,000 Deposit

10% SimpleInterest

7% CompoundInterest

10% CompoundInterest

Why Compound Interest?Why Compound Interest?Why Compound Interest?Why Compound Interest?

Fu

ture

Va

lue

(U

.S. D

olla

rs)

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3.12 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

Assume that you deposit $1,000$1,000 at a compound interest rate of 7% for

2 years2 years.

Future ValueFuture ValueSingle Deposit (Graphic)Single Deposit (Graphic)Future ValueFuture ValueSingle Deposit (Graphic)Single Deposit (Graphic)

0 1 22

$1,000$1,000

FVFV22

7%

Page 13: Chapter  03  imp

3.13 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

FVFV11 = PP00 (1 + i)1 = $1,000$1,000 (1.07)= $1,070$1,070

Compound Interest

You earned $70 interest on your $1,000 deposit over the first year.

This is the same amount of interest you would earn under simple interest.

Future ValueFuture ValueSingle Deposit (Formula)Single Deposit (Formula)Future ValueFuture ValueSingle Deposit (Formula)Single Deposit (Formula)

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3.14 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

FVFV11 = PP00 (1 + i)1 = $1,000$1,000 (1.07) = $1,070$1,070

FVFV22 = FV1 (1 + i)1 = PP0 0 (1 + i)(1 + i) = $1,000$1,000(1.07)

(1.07) = PP00 (1 + i)2 =

$1,000$1,000(1.07)2 = $1,144.90$1,144.90

You earned an EXTRA $4.90$4.90 in Year 2 with compound over simple interest.

Future ValueFuture ValueSingle Deposit (Formula)Single Deposit (Formula)Future ValueFuture ValueSingle Deposit (Formula)Single Deposit (Formula)

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3.15 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

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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3.16 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

FVIFFVIFi,n is found on Table I

at the end of the book.

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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3.17 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

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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3.18 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

TVM on the CalculatorTVM on the Calculator

• Use the highlighted row of keys for solving any of the FV, PV, FVA, PVA, FVAD, and PVAD problems

N: Number of periodsI/Y: Interest rate per periodPV: Present valuePMT: Payment per periodFV: Future value

CLR TVM: Clears all of the inputs into the above TVM keys

Source: Courtesy of Texas Instruments

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3.19 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

Using The TI BAII+ CalculatorUsing The TI BAII+ CalculatorUsing The TI BAII+ CalculatorUsing The TI BAII+ Calculator

N I/Y PV PMT FV

Inputs

Compute

Focus on 3Focus on 3rdrd Row of keys (will be Row of keys (will be displayed in slides as shown above)displayed in slides as shown above)

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3.20 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

Entering the FV ProblemEntering the FV Problem

Press:

2nd CLR TVM

2 N

7 I/Y

–1000 PV

0 PMT

CPT FV

Source: Courtesy of Texas Instruments

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3.21 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

N: 2 Periods (enter as 2)

I/Y: 7% interest rate per period (enter as 7 NOT 0.07)

PV: $1,000 (enter as negative as you have “less”)

PMT: Not relevant in this situation (enter as 0)

FV: Compute (Resulting answer is positive)

Solving the FV ProblemSolving the FV ProblemSolving the FV ProblemSolving the FV Problem

N I/Y PV PMT FV

Inputs

Compute

2 7 –1,000 0

1,144.90

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3.22 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

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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3.23 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

• 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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3.24 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

Entering the FV ProblemEntering the FV Problem

Press:

2nd CLR TVM

5 N

10 I/Y

–10000 PV

0 PMT

CPT FV

Source: Courtesy of Texas Instruments

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3.25 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

The result indicates that a $10,000 investment that earns 10% annually

for 5 years will result in a future value of $16,105.10.

Solving the FV ProblemSolving the FV ProblemSolving the FV ProblemSolving the FV Problem

N I/Y PV PMT FV

Inputs

Compute

5 10 –10,000 0

16,105.10

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3.26 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

We will use the ““Rule-of-72Rule-of-72””..

Double Your Money!!!Double Your Money!!!Double Your Money!!!Double Your Money!!!

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

year (approx.)?

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3.27 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

Approx. Years to Double = 7272 / i%

7272 / 12% = 6 Years6 Years[Actual Time is 6.12 Years]

The “Rule-of-72”The “Rule-of-72”The “Rule-of-72”The “Rule-of-72”

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

year (approx.)?

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3.28 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

The result indicates that a $1,000 investment that earns 12% annually will double to $2,000 in 6.12 years.

Note: 72/12% = approx. 6 years

Solving the Period ProblemSolving the Period ProblemSolving the Period ProblemSolving the Period Problem

N I/Y PV PMT FV

Inputs

Compute

12 –1,000 0 +2,000

6.12 years

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3.29 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

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)

Page 30: Chapter  03  imp

3.30 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

PVPV00 = FVFV22 / (1 + i)2 = $1,000$1,000 / (1.07)2 = FVFV22 / (1 + i)2 = $873.44$873.44

0 1 22

$1,000$1,000

7%

PVPV00

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

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3.31 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

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

etc.

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

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3.32 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

PVIFPVIFi,n is found on Table II

at the end of the book.

Period 6% 7% 8% 1 0.943 0.935 0.926 2 0.890 0.873 0.857 3 0.840 0.816 0.794 4 0.792 0.763 0.735 5 0.747 0.713 0.681

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

Page 33: Chapter  03  imp

3.33 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

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

= $873$873 [Due to Rounding]

Period 6% 7% 8% 1 0.943 0.935 0.926 2 0.890 0.873 0.857 3 0.840 0.816 0.794 4 0.792 0.763 0.735 5 0.747 0.713 0.681

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

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3.34 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

N: 2 Periods (enter as 2)

I/Y: 7% interest rate per period (enter as 7 NOT 0.07)

PV: Compute (Resulting answer is negative “deposit”)

PMT: Not relevant in this situation (enter as 0)

FV: $1,000 (enter as positive as you “receive $”)

N I/Y PV PMT FV

Inputs

Compute

2 7 0 +1,000

–873.44

Solving the PV ProblemSolving the PV ProblemSolving the PV ProblemSolving the PV Problem

Page 35: Chapter  03  imp

3.35 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

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%.

0 1 2 3 4 55

$10,000$10,000PVPV00

10%

Story Problem ExampleStory Problem ExampleStory Problem ExampleStory Problem Example

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3.36 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

• 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 (0.621)= $6,210.00$6,210.00 [Due to Rounding]

Story Problem SolutionStory Problem SolutionStory Problem SolutionStory Problem Solution

Page 37: Chapter  03  imp

3.37 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

N I/Y PV PMT FV

Inputs

Compute

5 10 0 +10,000

–6,209.21

The result indicates that a $10,000 future value that will earn 10% annually for 5 years requires a $6,209.21 deposit

today (present value).

Solving the PV ProblemSolving the PV ProblemSolving the PV ProblemSolving the PV Problem

Page 38: Chapter  03  imp

3.38 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

• 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.

Types of AnnuitiesTypes of AnnuitiesTypes of AnnuitiesTypes of Annuities

Page 39: Chapter  03  imp

3.39 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

• Student Loan Payments

• Car Loan Payments

• Insurance Premiums

• Mortgage Payments

• Retirement Savings

Examples of AnnuitiesExamples of Annuities

Page 40: Chapter  03  imp

3.40 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

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

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

Page 41: Chapter  03  imp

3.41 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

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

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

Page 42: Chapter  03  imp

3.42 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

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

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% . . .

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

Page 43: Chapter  03  imp

3.43 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

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

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

0 1 2 33 4

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

7%

$1,070

$1,145

Cash flows occur at the end of the period

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

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3.44 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

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.

Hint on Annuity ValuationHint on Annuity Valuation

Page 45: Chapter  03  imp

3.45 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

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

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

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

Page 46: Chapter  03  imp

3.46 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

N: 3 Periods (enter as 3 year-end deposits)

I/Y: 7% interest rate per period (enter as 7 NOT 0.07)

PV: Not relevant in this situation (no beg value)

PMT: $1,000 (negative as you deposit annually)

FV: Compute (Resulting answer is positive)

N I/Y PV PMT FV

Inputs

Compute

3 7 0 –1,000

3,214.90

Solving the FVA Problem Solving the FVA Problem Solving the FVA Problem Solving the FVA Problem

Page 47: Chapter  03  imp

3.47 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

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

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

R R R R R

0 1 2 3 n–1n–1 n

FVADFVADnn

i% . . .

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

Cash flows occur at the beginning of the period

Page 48: Chapter  03  imp

3.48 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

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

$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

Example of anExample of anAnnuity Due – FVADAnnuity Due – FVADExample of anExample of anAnnuity Due – FVADAnnuity Due – FVADCash flows occur at the beginning of the period

Page 49: Chapter  03  imp

3.49 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

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

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

$3,440$3,440Period 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

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

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3.50 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

N I/Y PV PMT FV

Inputs

Compute

3 7 0 –1,000

3,439.94

Complete the problem the same as an “ordinary annuity” problem, except you must change the calculator setting

to “BGN” first. Don’t forget to change back!

Step 1: Press 2nd BGN keys

Step 2: Press 2nd SET keys

Step 3: Press 2nd QUIT keys

Solving the FVAD ProblemSolving the FVAD ProblemSolving the FVAD ProblemSolving the FVAD Problem

Page 51: Chapter  03  imp

3.51 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

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

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

R R R

0 1 2 n n n+1

PVAPVAnn

R = Periodic Cash Flow

i% . . .

Overview of anOverview of anOrdinary Annuity – PVAOrdinary Annuity – PVAOverview of anOverview of anOrdinary Annuity – PVAOrdinary Annuity – PVA

Cash flows occur at the end of the period

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3.52 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

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

$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

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

Cash flows occur at the end of the period

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3.53 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

The present value of an ordinary annuity can be viewed as

occurring at the beginningbeginning of the first cash flow period, whereas the future value of an annuity

due can be viewed as occurring at the endend of the first cash flow

period.

Hint on Annuity ValuationHint on Annuity Valuation

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3.54 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

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

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

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

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3.55 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

N: 3 Periods (enter as 3 year-end deposits)

I/Y: 7% interest rate per period (enter as 7 NOT .07)

PV: Compute (Resulting answer is positive)

PMT: $1,000 (negative as you deposit annually)

FV: Not relevant in this situation (no ending value)

N I/Y PV PMT FV

Inputs

Compute

3 7 –1,000 0

2,624.32

Solving the PVA ProblemSolving the PVA ProblemSolving the PVA ProblemSolving the PVA Problem

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3.56 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

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

R R R R

0 1 2 n–1n–1 n

PVADPVADnn

R: Periodic Cash Flow

i% . . .

Overview of anOverview of anAnnuity Due – PVADAnnuity Due – PVADOverview of anOverview of anAnnuity Due – PVADAnnuity Due – PVADCash flows occur at the beginning of the period

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3.57 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

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

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

0 1 2 33 4

$2,808.02 $2,808.02 = PVADPVADnn

7%

$ 934.58$ 873.44

Example of anExample of anAnnuity Due – PVADAnnuity Due – PVADExample of anExample of anAnnuity Due – PVADAnnuity Due – PVADCash flows occur at the beginning of the period

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3.58 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

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

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

$2,808$2,808Period 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

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

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3.59 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

N I/Y PV PMT FV

Inputs

Compute

3 7 –1,000 0

2,808.02

Complete the problem the same as an “ordinary annuity” problem, except you must change the calculator setting

to “BGN” first. Don’t forget to change back!

Step 1: Press 2nd BGN keys

Step 2: Press 2nd SET keys

Step 3: Press 2nd QUIT keys

Solving the PVAD ProblemSolving the PVAD ProblemSolving the PVAD ProblemSolving the PVAD Problem

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3.60 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

1. Read problem thoroughly

2. Create a time line

3. Put cash flows and arrows on time line

4. Determine if it is a PV or FV problem

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

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3.61 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

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

0 1 2 3 4 55

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

PVPV00

10%10%

Mixed Flows ExampleMixed Flows ExampleMixed Flows ExampleMixed Flows Example

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3.62 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

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 groups. Then discount each groupgroup back to t=0.

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

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3.63 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

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

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

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3.64 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

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

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

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3.65 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

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

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

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3.66 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

• Use the highlighted key for starting the process of solving a mixed cash flow problem

• Press the CF key and down arrow key through a few of the keys as you look at the definitions on the next slide

Solving the Mixed Flows Solving the Mixed Flows Problem using CF RegistryProblem using CF RegistrySolving the Mixed Flows Solving the Mixed Flows Problem using CF RegistryProblem using CF Registry

Source: Courtesy of Texas Instruments

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3.67 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

Defining the calculator variables:For CF0:This is ALWAYS the cash flow occurring at

time t=0 (usually 0 for these problems)

For Cnn:* This is the cash flow SIZE of the nth group of cash flows. Note that a “group” may only contain a single cash flow (e.g., $351.76).

For Fnn:* This is the cash flow FREQUENCY of the nth group of cash flows. Note that this is always a positive whole number (e.g., 1, 2, 20, etc.).

* nn represents the nth cash flow or frequency. Thus, the first cash flow is C01, while the tenth cash flow is C10.

Solving the Mixed Flows Solving the Mixed Flows Problem using CF RegistryProblem using CF RegistrySolving the Mixed Flows Solving the Mixed Flows Problem using CF RegistryProblem using CF Registry

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3.68 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

Steps in the ProcessStep 1: Press CF key

Step 2: Press 2nd CLR Work keys

Step 3: For CF0 Press 0 Enter ↓ keys

Step 4: For C01 Press 600 Enter ↓ keys

Step 5: For F01 Press 2 Enter ↓ keys

Step 6: For C02 Press 400 Enter ↓ keys

Step 7: For F02 Press 2 Enter ↓ keys

Solving the Mixed Flows Solving the Mixed Flows Problem using CF RegistryProblem using CF RegistrySolving the Mixed Flows Solving the Mixed Flows Problem using CF RegistryProblem using CF Registry

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3.69 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

Steps in the Process

Step 8: For C03 Press 100 Enter ↓ keys

Step 9: For F03 Press 1 Enter ↓ keys

Step 10: Press ↓ ↓ keys

Step 11: Press NPV key

Step 12: For I =, Enter 10 Enter ↓ keys

Step 13: Press CPT key

Result: Present Value = $1,677.15

Solving the Mixed Flows Solving the Mixed Flows Problem using CF RegistryProblem using CF RegistrySolving the Mixed Flows Solving the Mixed Flows Problem using CF RegistryProblem using CF Registry

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3.70 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

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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3.71 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

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 + [0.12/1])(1)(2)

= 1,254.401,254.40

Semi FV2 = 1,0001,000(1 + [0.12/2])(2)(2)

= 1,262.481,262.48

Impact of FrequencyImpact of FrequencyImpact of FrequencyImpact of Frequency

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3.72 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

Qrtly FV2 = 1,0001,000(1 + [0.12/4])(4)(2)

= 1,266.771,266.77

Monthly FV2 = 1,0001,000(1 + [0.12/12])(12)(2)

= 1,269.731,269.73

Daily FV2 = 1,0001,000(1 + [0.12/365])(365)

(2) = 1,271.201,271.20

Impact of FrequencyImpact of FrequencyImpact of FrequencyImpact of Frequency

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3.73 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

The result indicates that a $1,000 investment that earns a 12% annual

rate compounded quarterly for 2 years will earn a future value of $1,266.77.

N I/Y PV PMT FV

Inputs

Compute

2(4) 12/4 –1,000 0

1266.77

Solving the Frequency Solving the Frequency Problem (Quarterly)Problem (Quarterly)Solving the Frequency Solving the Frequency Problem (Quarterly)Problem (Quarterly)

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3.74 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

Press:

2nd P/Y 4 ENTER

2nd QUIT

12 I/Y

–1000 PV

0 PMT

2 2nd xP/Y N

CPT FV

Solving the Frequency Solving the Frequency Problem (Quarterly Altern.)Problem (Quarterly Altern.)Solving the Frequency Solving the Frequency Problem (Quarterly Altern.)Problem (Quarterly Altern.)

Source: Courtesy of Texas Instruments

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3.75 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

The result indicates that a $1,000 investment that earns a 12% annual

rate compounded daily for 2 years will earn a future value of $1,271.20.

N I/Y PV PMT FV

Inputs

Compute

2(365) 12/365 –1,000 0

1271.20

Solving the Frequency Solving the Frequency Problem (Daily) Problem (Daily)Solving the Frequency Solving the Frequency Problem (Daily) Problem (Daily)

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3.76 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

Press:

2nd P/Y 365 ENTER

2nd QUIT

12 I/Y

–1000 PV

0 PMT

2 2nd xP/Y N

CPT FV

Solving the Frequency Solving the Frequency Problem (Daily Alternative)Problem (Daily Alternative)Solving the Frequency Solving the Frequency Problem (Daily Alternative)Problem (Daily Alternative)

Source: Courtesy of Texas Instruments

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3.77 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

Effective Annual Interest Rate

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

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3.78 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

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 + 0.06 / 4 )4 – 1 = 1.0614 - 1 = 0.0614 or 6.14%!6.14%!

BWs Effective BWs Effective Annual Interest RateAnnual Interest RateBWs Effective BWs Effective Annual Interest RateAnnual Interest Rate

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3.79 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

Press:

2nd I Conv

6 ENTER

↓ ↓

4 ENTER

↑ CPT

2nd QUIT

Converting to an EARConverting to an EAR

Source: Courtesy of Texas Instruments

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3.80 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

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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3.81 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

Julie Miller is borrowing $10,000 $10,000 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)

$10,000 $10,000 = R (PVIFA 12%,5)

$10,000$10,000 = R (3.605)

RR = $10,000$10,000 / 3.605 = $2,774$2,774

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

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3.82 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

End ofYear

Payment Interest Principal EndingBalance

0 — — — $10,000

1 $2,774 $1,200 $1,574 8,426

2 2,774 1,011 1,763 6,663

3 2,774 800 1,974 4,689

4 2,774 563 2,211 2,478

5 2,775 297 2,478 0

$13,871 $3,871 $10,000

[Last Payment Slightly Higher Due to Rounding]

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

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3.83 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

The result indicates that a $10,000 loan that costs 12% annually for 5 years and will be completely paid off at that time

will require $2,774.10 annual payments.

N I/Y PV PMT FV

Inputs

Compute

5 12 10,000 0

–2774.10

Solving for the PaymentSolving for the PaymentSolving for the PaymentSolving for the Payment

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3.84 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

Press:

2nd Amort

1 ENTER

1 ENTERResults:

BAL = 8,425.90* ↓

PRN = –1,574.10* ↓

INT = –1,200.00* ↓

Year 1 information only

*Note: Compare to 3-82

Using the Amortization Using the Amortization Functions of the CalculatorFunctions of the Calculator

Source: Courtesy of Texas Instruments

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3.85 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

Press:

2nd Amort

2 ENTER

2 ENTERResults:

BAL = 6,662.91* ↓

PRN = –1,763.99* ↓

INT = –1,011.11* ↓

Year 2 information only

*Note: Compare to 3-82

Using the Amortization Using the Amortization Functions of the CalculatorFunctions of the Calculator

Source: Courtesy of Texas Instruments

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3.86 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

Press:

2nd Amort

1 ENTER

5 ENTERResults:

BAL = 0.00 ↓

PRN = – 10,000.00 ↓

INT = –3,870.49 ↓

Entire 5 Years of loan information(see the total line of 3-82)

Using the Amortization Using the Amortization Functions of the CalculatorFunctions of the Calculator

Source: Courtesy of Texas Instruments

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3.87 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

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.

Usefulness of Usefulness of AmortizationAmortization