©The McGraw-Hill Companies, Inc.,2001 3- 1 Irwin/McGraw-Hill Irwin/McGraw-Hill Chapter 3 Fundamentals of Corporate Finance Third Edition The Time Value of Money Brealey Myers Marcus slides by Matthew Will Irwin/McGraw-Hill ©The McGraw-Hill Companies, Inc.,2001
© The McGraw-Hill Companies, Inc.,2001 3- 1 Irwin/McGraw-Hill Chapter 3 Fundamentals of Corporate Finance Third Edition The Time Value of Money Brealey.
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©The McGraw-Hill Companies, Inc.,2001
3- 1
Irwin/McGraw-HillIrwin/McGraw-Hill
Chapter 3Fundamentals of Corporate FinanceThird Edition
The Time Value of Money
Brealey Myers Marcus
slides by Matthew Will
Irwin/McGraw-Hill ©The McGraw-Hill Companies, Inc.,2001
©The McGraw-Hill Companies, Inc.,2001
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Irwin/McGraw-Hill
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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