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

Quantitative Analysis

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Quantitative Analysis is needed for good Decision Making

Benefits Need to Exceed Costs!

Need to Quantify the Benefits

Need to Quantity the CostsUnderstand the Risks Involved

Need to Know the Assumptions

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Business Discipline Specific

Strategy – Determine a competitor’s response to a price increase to the industry

Operations – Evaluate if modernization of equipment will decrease production costs

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Business Discipline Specific

Marketing – Determine revenue increases resulting from an advertising campaign

Organizational Behavior – Determine the effects of change on productivity

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Chapter Five ConceptsFuture Value & Compound Interest Calculating Present ValueFinding Interest RatesCalculating Multiple Cash FlowsPerpetuities and AnnuitiesEffective Annual Interest Rates Inflation

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Time Value of Money

Why? Economic Theory! (ECO201)

What?Calculation of Present Value, Future Value and Interest Rates.

How? Textbook Page 121

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Adam Smith

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1776 “The Wealth of Nations”

All people in a society consume goods and services (not just aristocracy / business)

The “invisible hand” guides the market – results in goods the society wants, in the quantity society desires for the price that society is prepared to pay.

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Key Terms

Competitive Market – A market in which the good can be bought and sold at the same price (due to market forces).

Valuation Principle – When the value of the benefits exceeds the value of the costs, the decision will increase the market value of the firm.

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Time Value of Money

Need for an Apple-to-Apples Comparison

Example 1 – $10,000 needed later

Example 2 – Delay of Sony PlayStation 3

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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.

Example: $10,000 Needed Later

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Future ValuesLet r = annual interest rate

Let t = # of years

Simple Interest vs. Compound Interest

FV = Initial investment (1 )tCompound 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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Time Value of Money

Video: “The Big Picture”

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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 & APREffective Annual Interest (EAR): Interest rate that is annualized using compound interest.

Annual Percentage Rate (APR): Interest rate that is annualized using simple interest.

5-5-3434*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