McGraw-Hill/Irwin Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.

5-5-22

Money has a time value. It can be expressed in multiple ways:

A dollar today held in savings will grow.

A dollar received in a year is not worth as much as a dollar received today.

Time Value of Money

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Future ValuesFuture Value: Amount to which an investment will grow

after earning interest.

Let r = annual interest rate

Let t = # of years

Simple Interest Compound InterestFV = Initial investment (1 )t

Compound r FV = Initial investment (1 )Simple r t

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Simple Interest: ExampleInterest earned at a rate of 7% for five years on a

principal balance of $100.

Example - Simple InterestToday Future Years 1 2 3 4 5

Interest Earned Value 100

Value at the end of Year 5: $135

7107

7114

7121

7128

7135

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Interest earned at a rate of 7% for five years on the previous year’s balance.

Example - Compound InterestToday Future Years 1 2 3 4 5

Interest EarnedValue 100

Compound Interest: Example

7107

7.49114.49

8.01122.50

8.58131.08

9.18140.26

Value at the end of Year 5 = $140.26

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The Power of CompoundingInterest earned at a rate of 7% for the first forty years

on the $100 invested using simple and compound interest.

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Present ValueWhat is it?

Why is it useful?

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Present Value

Present Value:

Discount Rate:

Discount Factor:

1(1 )tr

PV FV

1(1 )tr

DF

Recall: t = number of years

r

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Present Value: Example

Always ahead of the game, Tommy, at 8 years old, believes he will need $100,000 to pay for college. If he can invest at a rate of 7% per year, how much money should he ask his rich Uncle GQ to give him?

101

(1.07)

1$100,000 $50,835

(1 )tPV FV

r

Note: Ignore inflation/taxes

$100,000 10 7%FV t yrs r

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The PV formula has many applications. Given any variables in the equation, you can solve for

the remaining variable.

1(1 )tr

PV FV

Time Value of Money(applications)

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0

20

40

60

80

100

120

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Number of Years

PV

of

$100

0%

5%

10%

15%

Present Values: Changing Discount Rates

Discount Rates

The present value of $100 to be received in 1 to 20 years at varying discount rates:

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PV of Multiple Cash Flows

1 21 2(1 ) (1 ) (1 )

.... tt

CC C

r r rPV

The present value of multiple cash flows can be calculated:

1

2

:

The cash flow in year 1

The cash flow in year 2

The cash flow in year t (with any number of cash flows in between)t

Denote

C

C

C

Recall: r = the discount rate

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Multiple Cash Flows: Example

Your auto dealer gives you the choice to pay $15,500 cash now or make three payments: $8,000 now and $4,000 at the end of the following two years. If your cost of money (discount rate) is 8%, which do you prefer?

1

2

4,0001 (1 .08)

4,0002 (1 .08)

Initial Payment* 8,000.00

3,703.70

3,429.36

Total PV $15,133.06

PV of C

PV of C

* The initial payment occurs immediately and therefore would not be discounted.

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Perpetuities

Let C = Yearly Cash PaymentPV of Perpetuity:

CrPV

What are they?

Recall: r = the discount rate

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Perpetuities: ExampleIn order to create an endowment, which pays $185,000 per year forever, how much money must be set aside today if the rate of interest is 8%?

What if the first payment won’t be received until 3 years from today?

185,000.08 $2,312,500PV

2

2,312,500

(1 .08)$1,982,596PV

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AnnuitiesWhat are they?Annuities are equally-spaced, level streams of cash flows lasting for a limited period of time.

Why are they useful?

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Present Value of an AnnuityLet:C = yearly cash payment r = interest rate t = number of years cash payment is received

1 1(1 )tr r r

PV C

The terms within the brackets are collectively called the “annuity factor.”

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Annuities: Example

You are purchasing a home and are scheduled to make 30 annual installments of $10,000 per year. Given an interest rate of 5%, what is the price you are paying for the house (i.e. what is the present value)?

301 1

.05 .05(1 .05)$10,000

$153,724.51

PV

PV

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Future Value of Annuities: Example

You plan to save $4,000 every year for 20 years and then retire. Given a 10% rate of interest, how much will you have saved by the time you retire?

20

201 1.10 .10(1 .10)

$4,000 (1 .10)

$229,100

FV

FV

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Annuity Due

How does it differ from an ordinary annuity?

What is it?

Recall: r = the discount rate

(1 )Annuity Due AnnuityFV FV r

How does the future value differ from an ordinary annuity?

(1 )Annuity Due AnnuityPV PV r

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Annuities Due: Example

)1( rFVFV AnnuityAD

Example: Suppose you invest $429.59 annually at the beginning of each year at 10% interest. After 50 years, how much would your investment be worth?

000,550$

)10.1()000,500($

)1(

AD

AD

AnnuityAD

FV

FV

rFVFV

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Interest Rates: EAR & APRWhat is EAR?

What is APR?

How do they differ?

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*where MR = monthly interest rate

EAR & APR Calculations

1)1( 12 MREAR

Effective Annual Interest Rate (EAR):

Annual Percentage Rate (APR):

12MRAPR

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EAR and APR: Example

Given a monthly rate of 1%, what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)?

%00.12)12()01.0(

%68.121)01.1( 12

APR

EAR

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InflationWhat is it?

What determines inflation rates?

What is deflation?

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Inflation and Real Interest

1+nominal interest rate1+inflation rate1 real interest rate=

Exact calculation:

Approximation: rateinflation -rateinterest nominalrateinterest Real

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Inflation: ExampleIf the nominal interest rate on your interest-bearing savings account is 2.0% and the inflation rate is 3.0%, what is the real interest rate?

1+.021+.031 real interest rate =

1 real interest rate = 0.9903

real interest rate = -.0097 or -.97%

Approximation = .02-.03 = .01 1%

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Appendix A: InflationAnnual U.S. Inflation Rates from 1900 - 2010