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Chapter 04 Chapter 04

Time Value of Time Value of MoneyMoney

Time Value of Time Value of MoneyMoney

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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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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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Main Idea about Time Value of Main Idea about Time Value of MoneyMoney

Money that the firm or an individual has in its possession today is more valuable than future payments because the money it now can be invested and earn positive returns.

“A bird in hand worth more than two in the bush”

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Why Money has Time Value?Why Money has Time Value?

The existence of interest rates in the economy results in money with its time value.

Sacrificing present ownership requires possibility of having more future ownership.

Inflation in economy is one of the major cause.

Risk or uncertainty of favorable outcome.

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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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Major Importance of Time Valueof MoneyMajor Importance of Time Valueof Money

It is required for accounting accuracy for certain transactions such as loan amortization, lease payments, and bond interest.

In order to design systems that optimize the firm’s cash flows.

For better planning about cash collections and disbursements in a way that will enable the firm to get the greatest value from its money.

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Major Importance of Time Value of MoneyMajor Importance of Time Value of Money

Funding for new programs, products, and projects can be justified financially using time –value-of-money techniques.

Investments in new equipment, in inventory, and in production quantities are affected by time-value-of-money techniques.

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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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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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SI = P0(i)(n)= $1,000(.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?

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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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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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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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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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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.

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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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 time (N) ProblemSolving the time (N) ProblemSolving the time (N) ProblemSolving the time (N) Problem

N I/Y PV PMT FV

Inputs

Compute

12 -1,000 0 +2,000

6.12 years

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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.

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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Self-Test ProblemSelf-Test Problem

Practice Q 1:

Suppose you will receive $2000 after 10 years and now the interest rate is 8%, calculate the present value of this amount.

Practice Q 2:

You have $1500 to invest today at 9% interest compounded semi-annually. Find how much you will have accumulated in the account at the end of 6 years.

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Self Test Problems:Self Test Problems:Solving for interest rate (i) & period (n)Solving for interest rate (i) & period (n)

Practice Q.3 Suppose you can buy a security at a price of Tk78.35 that will pay you Tk100 after five years. What will be the rate of return, if you purchase the security?

Practice Q.4 Suppose you know that a security will provide a 10% return per year, its price is Tk68.30 and you will receive Tk.100 at maturity. How many years does the security take to mature?

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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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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 (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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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 future value of an annuity

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

period.

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Example for Annuity (cont.)Example for Annuity (cont.)

Example-5: (P. 166)

Cute Baby Company, a small producer of plastic toys, wants to determine the most it should pay to purchase a particular ordinary annuity. Find the present value if the annuity consists of cash flows of $700 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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PerpetuityPerpetuity

# A stream of equal payments expected to continue forever.

Formula:

Payment PMT

PV (Perpetuity) = =

Interest Rate i

The present value of this special type of annuity will be required when we value perpetual bonds and preferred stock.

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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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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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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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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%!

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

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Example-7Example-7

Mr. Mahin has Tk.10000 that he can deposit any of three savings accounts for 3-year period. Bank A compounds interest on an annual basis, bank B twice each year, and bank C each quarter. All 3 banks have a stated annual interest rate of 4%.

a. What amount would Mr. Mahin have at the end of the third year in each bank?

b. What effective annual rate (EAR) would he earn in each of the banks?

c. On the basis of your findings in a and b, which bank should Mr. Mahin deal with? Why?

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Example-8Example-8

A municipal savings bond can be converted to $100 at maturity 6 years from purchase. If this state bonds are to be competitive with B.D Government savings bonds, which pay 8% annual interest (compounded annually), at what price must the state sell its bonds? Assume no cash payments on savings bond prior to redemption.

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Solve for interest rateSolve for interest rate

You can buy a security at a price Tk. 7835, that will pay you Tk. 10,000 after three years. Calculate the interest rate, you will earn on your investment.

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Solve for time (n)Solve for time (n)

You know that a particular deposit will provide a return of 12% per year. It will cost Tk. 6830 and that you will receive Tk. 10,000 at maturity. How long it will take?

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Class WorkClass Work

If a firm’s earnings increase from Tk. 260 per share to Tk. 340 over a 6-year period, what is the rate of growth?

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Class WorkClass Work

A first talker representative from Prime Bank just comes and offers to you an attractive rate of return on a particular deposit at 9.5% per year. You need to pay Tk. 50,000 now and will receive Tk. 75,000 at the time of maturity. Calculate the maturity time to receive the stated amount.