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Chapter 5Chapter 5

Time Value of Time Value of MoneyMoney

Time Value of Time Value of MoneyMoney

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After studying Chapter 5 After studying Chapter 5 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 flow.

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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 Compounding More Than

Once per Year

The Interest Rate Simple Interest Compound Interest 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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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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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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examplesexamples

If u know that pv=1000

Interest rate = 6%

Future value = 1180

How u can caculate number of years for this investment?

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examplesexamples

If u know that during 3 years

Ur future value is 1180

Interest rate is 6%

Can u compute ur initial value for this investment

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examplesexamples

If u invest 1000 dollars by 6% simply interest rate during:

26 weeks

9 months

240 days

How u can calculate ur future value

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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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Distinguish invest 1000 Distinguish invest 1000 dollars by 6% simply or dollars by 6% simply or compounded rate for 3 compounded rate for 3 yearsyears

then mention which is bettre for investor ?

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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%

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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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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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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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ExampleExample

If you know that price of BMW always 20000 dollars. U like to buy one-currently u have only 7752 dollars-how many years will it take for your initial investment of 7752 dollars to grow 20000- if it is invested at 9% compounded annually using table of ( FVIF)

ALSO IF u have currently 11167 dollars invested for 10 years- what is interest rate u need to get price of BMW –using same table

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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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exampleexample

What is present value of an investment yields 500 dollars to be received in 5 years and 1000 dollars to be received in 10 years if u know that discount rate is 4% for both

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exampleexample

If present value=676 and future value= 1000 dollars and discount rate was =4% so number of years was 8 years using PVIF table

Mention: true or falce and why.

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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