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Chapter 3Fundamentals of Corporate FinanceThird Edition

The Time Value of Money

Brealey Myers Marcus

slides by Matthew Will

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Topics Covered

Future ValuesPresent ValuesMultiple Cash FlowsPerpetuities and AnnuitiesInflation & Time ValueEffective Annual Interest Rate

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Future Values

Future Value - Amount to which an investment will grow after earning interest.

Compound Interest - Interest earned on interest.

Simple Interest - Interest earned only on the original investment.

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Future Values

Example - Simple Interest

Interest earned at a rate of 6% for five years on a principal balance of $100.

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Future Values

Example - Simple Interest

Interest earned at a rate of 6% for five years on a principal balance of $100.

Interest Earned Per Year = 100 x .06 = $ 6

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Future Values

Example - Simple Interest

Interest earned at a rate of 6% for five years on a principal balance of $100.

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Future Values

Example - Simple Interest

Interest earned at a rate of 6% for five years on a principal balance of $100.

Today Future Years

1 2 3 4 5

Interest Earned

Value 100

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Future Values

Example - Simple Interest

Interest earned at a rate of 6% for five years on a principal balance of $100.

Today Future Years

1 2 3 4 5

Interest Earned 6

Value 100 106

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Future Values

Example - Simple Interest

Interest earned at a rate of 6% for five years on a principal balance of $100.

Today Future Years

1 2 3 4 5

Interest Earned 6 6

Value 100 106 112

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Future Values

Example - Simple Interest

Interest earned at a rate of 6% for five years on a principal balance of $100.

Today Future Years

1 2 3 4 5

Interest Earned 6 6 6

Value 100 106 112 118

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Future Values

Example - Simple Interest

Interest earned at a rate of 6% for five years on a principal balance of $100.

Today Future Years

1 2 3 4 5

Interest Earned 6 6 6 6

Value 100 106 112 118 124

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Future ValuesExample - Simple Interest

Interest earned at a rate of 6% for five years on a principal balance of $100.

Today Future Years

1 2 3 4 5

Interest Earned 6 6 6 6 6

Value 100 106 112 118 124 130

Value at the end of Year 5 = $130

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Future Values

Example - Compound Interest

Interest earned at a rate of 6% for five years on the previous year’s balance.

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Future Values

Example - Compound Interest

Interest earned at a rate of 6% for five years on the previous year’s balance.

Interest Earned Per Year =Prior Year Balance x .06

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Future Values

Example - Compound Interest

Interest earned at a rate of 6% for five years on the previous year’s balance.

Today Future Years

1 2 3 4 5

Interest Earned

Value 100

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Future Values

Example - Compound Interest

Interest earned at a rate of 6% for five years on the previous year’s balance.

Today Future Years

1 2 3 4 5

Interest Earned 6.00

Value 100 106.00

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Future Values

Example - Compound Interest

Interest earned at a rate of 6% for five years on the previous year’s balance.

Today Future Years

1 2 3 4 5

Interest Earned 6.00 6.36

Value 100 106.00 112.36

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Future Values

Example - Compound Interest

Interest earned at a rate of 6% for five years on the previous year’s balance.

Today Future Years

1 2 3 4 5

Interest Earned 6.00 6.36 6.74

Value 100 106.00 112.36 119.10

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Future Values

Example - Compound Interest

Interest earned at a rate of 6% for five years on the previous year’s balance.

Today Future Years

1 2 3 4 5

Interest Earned 6.00 6.36 6.74 7.15

Value 100 106.00 112.36 119.10 126.25

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Future Values

Example - Compound Interest

Interest earned at a rate of 6% for five years on the previous year’s balance.

Today Future Years

1 2 3 4 5

Interest Earned 6.00 6.36 6.74 7.15 7.57

Value 100 106.00 112.36 119.10 126.25 133.82

Value at the end of Year 5 = $133.82

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Future Values

Future Value of $100 = FV

FV r t $100 ( )1

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Future Values

FV r t $100 ( )1

Example - FV

What is the future value of $100 if interest is compounded annually at a rate of 6% for five years?

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Future Values

FV r t $100 ( )1

Example - FV

What is the future value of $100 if interest is compounded annually at a rate of 6% for five years?

82.133$)06.1(100$ 5 FV

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0

10

20

30

40

50

60

70

Number of Years

FV

of

$1

0%

5%

10%

15%

Future Values with Compounding

Interest Rates

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Manhattan Island Sale

Peter Minuit bought Manhattan Island for $24 in 1626. Was this a good deal?

trillion

FV

979.75$

)08.1(24$ 374

To answer, determine $24 is worth in the year 2000, compounded at 8%.

FYI - The value of Manhattan Island land is FYI - The value of Manhattan Island land is well below this figure.well below this figure.

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Present Values

Present Value

Value today of a future cash

flow.

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Present Values

Present Value

Value today of a future cash

flow.

Discount Factor

Present value of a $1 future payment.

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Present Values

Present Value

Value today of a future cash

flow.

Discount Rate

Interest rate used to compute

present values of future cash flows.

Discount Factor

Present value of a $1 future payment.

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Present Values

Present Value = PV

PV = Future Value after t periods

(1+r) t

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Present Values

Example

You just bought a new computer for $3,000. The payment terms are 2 years same as cash. If you can earn 8% on your money, how much money should you set aside today in order to make the payment when due in two years?

572,2$2)08.1(3000 PV

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Present Values

Discount Factor = DF = PV of $1

Discount Factors can be used to compute the present value of any cash flow.

DFr t

1

1( )

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

PV FVr t

1

1( )

Time Value of Money(applications)

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Value of Free CreditImplied Interest RatesInternal Rate of ReturnTime necessary to accumulate funds

Time Value of Money(applications)

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PV of Multiple Cash Flows

ExampleYour 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 is 8%, which do you prefer?

$15,133.06 PVTotal

36.429,3

70.703,3

8,000.00

2

1

)08.1(

000,42

)08.1(

000,41

payment Immediate

PV

PV

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PV of Multiple Cash Flows

PVs can be added together to evaluate multiple cash flows.

PV C

r

C

r

1

12

21 1( ) ( )....

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Perpetuities & Annuities

Perpetuity

A stream of level cash payments that never ends.

Annuity

Equally spaced level stream of cash flows for a limited period of time.

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Perpetuities & Annuities

PV of Perpetuity Formula

C = cash payment

r = interest rate

PV Cr

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Perpetuities & Annuities

Example - Perpetuity

In order to create an endowment, which pays $100,000 per year, forever, how much money must be set aside today in the rate of interest is 10%?

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Perpetuities & Annuities

Example - Perpetuity

In order to create an endowment, which pays $100,000 per year, forever, how much money must be set aside today in the rate of interest is 10%?

PV 100 00010 000 000,. $1, ,

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Perpetuities & Annuities

Example - continued

If the first perpetuity payment will not be received until three years from today, how much money needs to be set aside today?

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Perpetuities & Annuities

Example - continued

If the first perpetuity payment will not be received until three years from today, how much money needs to be set aside today?

PV

1 000 000

1 10 3 315, ,

( . )$751,

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Perpetuities & Annuities

PV of Annuity Formula

C = cash payment

r = interest rate

t = Number of years cash payment is received

PV C r r r t

1 11( )

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Perpetuities & Annuities

PV Annuity Factor (PVAF) - The present value of $1 a year for each of t years.

PVAF r r r t

1 11( )

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Perpetuities & Annuities

Example - Annuity

You are purchasing a car. You are scheduled to make 3 annual installments of $4,000 per year. Given a rate of interest of 10%, what is the price you are paying for the car (i.e. what is the PV)?

PV

PV

4 000

947 41

110

110 1 10 3,

$9, .

. . ( . )

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Perpetuities & Annuities

ApplicationsValue of paymentsImplied interest rate for an annuityCalculation of periodic payments

Mortgage paymentAnnual income from an investment payoutFuture Value of annual payments

FV C PVAF r t ( )1

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Perpetuities & Annuities

Example - Future Value of annual payments

You plan to save $4,000 every year for 20 years and then retire. Given a 10% rate of interest, what will be the FV of your retirement account?

FV

FV

4 000 1 10

100

110

110 1 10

2020, ( . )

$229,

. . ( . )

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Inflation

Inflation - Rate at which prices as a whole are increasing.

Nominal Interest Rate - Rate at which money invested grows.

Real Interest Rate - Rate at which the purchasing power of an investment increases.

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Inflation

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

approximation formula

Real int. rate nominal int. rate - inflation rate

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InflationExample

If the interest rate on one year govt. bonds is 5.0% and the inflation rate is 2.2%, what is the real interest rate?

Savings

Bond

1 + real interest rate = 1+.0501+.022

1 + real interest rate = 1.027

Real interest rate = .027 or 2.7%

Approximation = .050 - .022 or .028 or 2.8%

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Effective Interest Rates

Annual Percentage Rate - Interest rate that is annualized using simple interest.

Effective Annual Interest Rate - Interest rate that is annualized using compound interest.

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Effective Interest Rates

example

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

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Effective Interest Rates

example

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

EAR = (1+.01) - 1 = r

EAR = (1+.01) - 1 =.1268 or 12.68%

APR =.01 x 12 =.12 or 12.00%

12

12

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