?The McGraw-Hill Companies,Inc.,2001
3- 1
Irwin/McGraw-Hill
Chapter 3Fundamentals of Corporate FinanceThird Edition
The Time
Value of
Money
Brealey Myers Marcus
slides by Matthew Will
?The McGraw-Hill Companies,Inc.,2001
3- 2
Irwin/McGraw-Hill
Topics Covered
?Future Values
?Present Values
?Multiple Cash Flows
?Perpetuities and Annuities
?Inflation & Time Value
?Effective Annual Interest Rate
?The McGraw-Hill Companies,Inc.,2001
3- 3
Irwin/McGraw-Hill
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.
?The McGraw-Hill Companies,Inc.,2001
3- 4
Irwin/McGraw-Hill
Future Values
Future Value of $100 = FV
FV r t? ? ?$100 ( )1
?The McGraw-Hill Companies,Inc.,2001
3- 5
Irwin/McGraw-Hill
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?
?The McGraw-Hill Companies,Inc.,2001
3- 6
Irwin/McGraw-Hill
Future Values
FV r t? ? ?$100 ( )1
Example - FV
What is the future value of $100 if interest is
compounded annually at a rate of 6% for five years?
82.1 3 3$)06.1(1 0 0$ 5 ????FV
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3- 7
Irwin/McGraw-Hill
0
10
20
30
40
50
60
70
0 2 4 6 8
10 12 14 16 18 20 22 24 26 28 30
N u m b e r o f Ye a r s
F
V
o
f
$
1
0%
5%
10%
15%
Future Values with Compounding
Interest Rates
?The McGraw-Hill Companies,Inc.,2001
3- 8
Irwin/McGraw-Hill
Manhattan Island Sale
Peter Minuit bought Manhattan Island for $24 in 1626,
Was this a good deal?
t r i l l i o n
FV
9 7 9.75$
)08.1(24$ 3 7 4
?
???
To answer,determine $24 is worth in the year 2000,
compounded at 8%,
FYI - The value of Manhattan Island land is
well below this figure.
?The McGraw-Hill Companies,Inc.,2001
3- 9
Irwin/McGraw-Hill
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.
?The McGraw-Hill Companies,Inc.,2001
3- 10
Irwin/McGraw-Hill
Present Values
P r e s e n t V a l u e = PV
PV =
F u t u r e V a l u e a f t e r t p e r i o d s
( 1 + r)
t
?The McGraw-Hill Companies,Inc.,2001
3- 11
Irwin/McGraw-Hill
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?
5 7 2,2$2)08.1( 3 0 0 0 ??PV
?The McGraw-Hill Companies,Inc.,2001
3- 12
Irwin/McGraw-Hill
Present Values
Discount Factor = DF = PV of $1
?Discount Factors can be used to compute
the present value of any cash flow.
DF
r t
?
?
1
1( )
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3- 13
Irwin/McGraw-Hill
?The PV formula has many applications,
Given any variables in the equation,you
can solve for the remaining variable,
PV FV r t? ? ?11( )
Time Value of Money
(applications)
?The McGraw-Hill Companies,Inc.,2001
3- 14
Irwin/McGraw-Hill
?Value of Free Credit
?Implied Interest Rates
?Internal Rate of Return
?Time necessary to accumulate funds
Time Value of Money
(applications)
?The McGraw-Hill Companies,Inc.,2001
3- 15
Irwin/McGraw-Hill
PV of 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 is
8%,which do you prefer?
$ 1 5,1 3 3, 0 6 P VT o t al
36.4 2 9,3
70.7 0 3,3
8,0 0 0, 0 0
2
1
)08.1(
0 0 0,4
2
)08.1(
0 0 0,4
1
p ay m en t Im m e d i a t e
?
??
??
?
?
PV
PV
?The McGraw-Hill Companies,Inc.,2001
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Irwin/McGraw-Hill
PV of Multiple Cash Flows
?PVs can be added together to evaluate
multiple cash flows.
PV C
r
C
r
? ? ?
? ?
1
1
2
21 1( ) ( ),,,,
?The McGraw-Hill Companies,Inc.,2001
3- 17
Irwin/McGraw-Hill
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.
?The McGraw-Hill Companies,Inc.,2001
3- 18
Irwin/McGraw-Hill
Perpetuities & Annuities
PV of Perpetuity Formula
C = cash payment
r = interest rate
PV Cr?
?The McGraw-Hill Companies,Inc.,2001
3- 19
Irwin/McGraw-Hill
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,,
?The McGraw-Hill Companies,Inc.,2001
3- 20
Irwin/McGraw-Hill
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 0001 10 3 315,,(, ) $751,
?The McGraw-Hill Companies,Inc.,2001
3- 21
Irwin/McGraw-Hill
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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Irwin/McGraw-Hill
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( )
?The McGraw-Hill Companies,Inc.,2001
3- 23
Irwin/McGraw-Hill
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
1
10
1
10 1 10
3,
$9,.
., (, )
?The McGraw-Hill Companies,Inc.,2001
3- 24
Irwin/McGraw-Hill
Perpetuities & Annuities
Applications
?Value of payments
?Implied interest rate for an annuity
?Calculation of periodic payments
?Mortgage payment
?Annual income from an investment payout
?Future Value of annual payments
? ?FV C PVAF r t? ? ? ?( )1
?The McGraw-Hill Companies,Inc.,2001
3- 25
Irwin/McGraw-Hill
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
1
10
1
10 1 10
20
20,(, )
$229,
., (, )
?The McGraw-Hill Companies,Inc.,2001
3- 26
Irwin/McGraw-Hill
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.
?The McGraw-Hill Companies,Inc.,2001
3- 27
Irwin/McGraw-Hill
Inflation
1 ? r e a l i n t e r e s t r a t e = 1 + n o m i n a l i n t e r e s t r a t e1 + i n f l a t i o n r a t e
approximation formula
R e a l i n t, r a t e n o m i n a l i n t,r a t e - i n f l a t i o n r a t e?
?The McGraw-Hill Companies,Inc.,2001
3- 28
Irwin/McGraw-Hill
Inflation
Example
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?
2, 8 %o r, 0 2 8=,0 2 2-,0 5 0=io nA p p r o x i m a t
2, 7 %o r, 0 2 7 = r a t ein t e r e s t r e a l
1, 0 2 7 =r a t ein t e r e s t r e a l1
=r a t ein t e r e s t r e a l1
,0 2 2+1
,0 5 0+1
?
?
Savings
Bond
?The McGraw-Hill Companies,Inc.,2001
3- 29
Irwin/McGraw-Hill
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.
?The McGraw-Hill Companies,Inc.,2001
3- 30
Irwin/McGraw-Hill
Effective Interest Rates
example
Given a monthly rate of 1%,what is the Effective
Annual Rate(EAR)? What is the Annual
Percentage Rate (APR)?
E A R = (1 +, 0 1 ) - 1 = r
E A R = (1 +, 0 1 ) - 1 =, 1 2 6 8 o r 1 2, 6 8 %
A P R =, 0 1 x 1 2 =, 1 2 o r 1 2, 0 0 %
12
12
?The McGraw-Hill Companies,Inc.,2001
3- 31
Irwin/McGraw-Hill
基本財務數學摘要
?現值與未來值
?年金（ Annuity） 的未來值和現值
?貸款常數 （ mortgage constant） 與沉入
基金因子 （ sinking fund factor）
?The McGraw-Hill Companies,Inc.,2001
3- 32
Irwin/McGraw-Hill
現值與未來值
?現值與未來值之關係
FVn=PV(1+r)n FVn=PV(1+r/m)nm
?Effective Annual Yield (EAY)
EAY=(1+r/m)m -1
?未來值因子,FV(r/m,nm)=(1+r/m)nm
?現值因子,PV(r/m,nm)=1/(1+r/m)nm
?The McGraw-Hill Companies,Inc.,2001
3- 33
Irwin/McGraw-Hill
年金的未來值和現值（ I）
?永久年金 (Annuity)的現值：
?年金現值因子 (Annuity Present Value Factor)
為：
r)+(1
1-1
r
1=n)P V A ( r,
n ??
?
??
?
r
A
PV
.,,,,,
r)(1
A
r)(1
A
r1
A
PV
32
?
?
?
?
?
?
?
?
?The McGraw-Hill Companies,Inc.,2001
3- 34
Irwin/McGraw-Hill
年金的未來值和現值（ II）
?年金未來值因子為
? ?1)r1()
r
1
(
r)1(
r)+(1
1
-1)
r
1
(=
r)+(1n)P V A ( r,=n)F V A ( r,
n
n
n
n
???
?
?
?
?
?
?
?
?
?The McGraw-Hill Companies,Inc.,2001
3- 35
Irwin/McGraw-Hill
貸款常數 (mortgage constant)與
沉入基金因子 (sinking fund
factor)
?貸款常數
PV=A×PVA(r,n) A=PV×[1/PVA(r,n]
?MC=1×[1/PVA(r,n]
?沉入基金因子
FV=A× FVA(r,n) A=FV× [1/FVA(r,n]
?SFF=1×[1/FVA(r,n]
3- 1
Irwin/McGraw-Hill
Chapter 3Fundamentals of Corporate FinanceThird Edition
The Time
Value of
Money
Brealey Myers Marcus
slides by Matthew Will
?The McGraw-Hill Companies,Inc.,2001
3- 2
Irwin/McGraw-Hill
Topics Covered
?Future Values
?Present Values
?Multiple Cash Flows
?Perpetuities and Annuities
?Inflation & Time Value
?Effective Annual Interest Rate
?The McGraw-Hill Companies,Inc.,2001
3- 3
Irwin/McGraw-Hill
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.
?The McGraw-Hill Companies,Inc.,2001
3- 4
Irwin/McGraw-Hill
Future Values
Future Value of $100 = FV
FV r t? ? ?$100 ( )1
?The McGraw-Hill Companies,Inc.,2001
3- 5
Irwin/McGraw-Hill
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?
?The McGraw-Hill Companies,Inc.,2001
3- 6
Irwin/McGraw-Hill
Future Values
FV r t? ? ?$100 ( )1
Example - FV
What is the future value of $100 if interest is
compounded annually at a rate of 6% for five years?
82.1 3 3$)06.1(1 0 0$ 5 ????FV
?The McGraw-Hill Companies,Inc.,2001
3- 7
Irwin/McGraw-Hill
0
10
20
30
40
50
60
70
0 2 4 6 8
10 12 14 16 18 20 22 24 26 28 30
N u m b e r o f Ye a r s
F
V
o
f
$
1
0%
5%
10%
15%
Future Values with Compounding
Interest Rates
?The McGraw-Hill Companies,Inc.,2001
3- 8
Irwin/McGraw-Hill
Manhattan Island Sale
Peter Minuit bought Manhattan Island for $24 in 1626,
Was this a good deal?
t r i l l i o n
FV
9 7 9.75$
)08.1(24$ 3 7 4
?
???
To answer,determine $24 is worth in the year 2000,
compounded at 8%,
FYI - The value of Manhattan Island land is
well below this figure.
?The McGraw-Hill Companies,Inc.,2001
3- 9
Irwin/McGraw-Hill
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.
?The McGraw-Hill Companies,Inc.,2001
3- 10
Irwin/McGraw-Hill
Present Values
P r e s e n t V a l u e = PV
PV =
F u t u r e V a l u e a f t e r t p e r i o d s
( 1 + r)
t
?The McGraw-Hill Companies,Inc.,2001
3- 11
Irwin/McGraw-Hill
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?
5 7 2,2$2)08.1( 3 0 0 0 ??PV
?The McGraw-Hill Companies,Inc.,2001
3- 12
Irwin/McGraw-Hill
Present Values
Discount Factor = DF = PV of $1
?Discount Factors can be used to compute
the present value of any cash flow.
DF
r t
?
?
1
1( )
?The McGraw-Hill Companies,Inc.,2001
3- 13
Irwin/McGraw-Hill
?The PV formula has many applications,
Given any variables in the equation,you
can solve for the remaining variable,
PV FV r t? ? ?11( )
Time Value of Money
(applications)
?The McGraw-Hill Companies,Inc.,2001
3- 14
Irwin/McGraw-Hill
?Value of Free Credit
?Implied Interest Rates
?Internal Rate of Return
?Time necessary to accumulate funds
Time Value of Money
(applications)
?The McGraw-Hill Companies,Inc.,2001
3- 15
Irwin/McGraw-Hill
PV of 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 is
8%,which do you prefer?
$ 1 5,1 3 3, 0 6 P VT o t al
36.4 2 9,3
70.7 0 3,3
8,0 0 0, 0 0
2
1
)08.1(
0 0 0,4
2
)08.1(
0 0 0,4
1
p ay m en t Im m e d i a t e
?
??
??
?
?
PV
PV
?The McGraw-Hill Companies,Inc.,2001
3- 16
Irwin/McGraw-Hill
PV of Multiple Cash Flows
?PVs can be added together to evaluate
multiple cash flows.
PV C
r
C
r
? ? ?
? ?
1
1
2
21 1( ) ( ),,,,
?The McGraw-Hill Companies,Inc.,2001
3- 17
Irwin/McGraw-Hill
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.
?The McGraw-Hill Companies,Inc.,2001
3- 18
Irwin/McGraw-Hill
Perpetuities & Annuities
PV of Perpetuity Formula
C = cash payment
r = interest rate
PV Cr?
?The McGraw-Hill Companies,Inc.,2001
3- 19
Irwin/McGraw-Hill
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,,
?The McGraw-Hill Companies,Inc.,2001
3- 20
Irwin/McGraw-Hill
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 0001 10 3 315,,(, ) $751,
?The McGraw-Hill Companies,Inc.,2001
3- 21
Irwin/McGraw-Hill
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( )
?The McGraw-Hill Companies,Inc.,2001
3- 22
Irwin/McGraw-Hill
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( )
?The McGraw-Hill Companies,Inc.,2001
3- 23
Irwin/McGraw-Hill
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
1
10
1
10 1 10
3,
$9,.
., (, )
?The McGraw-Hill Companies,Inc.,2001
3- 24
Irwin/McGraw-Hill
Perpetuities & Annuities
Applications
?Value of payments
?Implied interest rate for an annuity
?Calculation of periodic payments
?Mortgage payment
?Annual income from an investment payout
?Future Value of annual payments
? ?FV C PVAF r t? ? ? ?( )1
?The McGraw-Hill Companies,Inc.,2001
3- 25
Irwin/McGraw-Hill
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
1
10
1
10 1 10
20
20,(, )
$229,
., (, )
?The McGraw-Hill Companies,Inc.,2001
3- 26
Irwin/McGraw-Hill
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.
?The McGraw-Hill Companies,Inc.,2001
3- 27
Irwin/McGraw-Hill
Inflation
1 ? r e a l i n t e r e s t r a t e = 1 + n o m i n a l i n t e r e s t r a t e1 + i n f l a t i o n r a t e
approximation formula
R e a l i n t, r a t e n o m i n a l i n t,r a t e - i n f l a t i o n r a t e?
?The McGraw-Hill Companies,Inc.,2001
3- 28
Irwin/McGraw-Hill
Inflation
Example
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?
2, 8 %o r, 0 2 8=,0 2 2-,0 5 0=io nA p p r o x i m a t
2, 7 %o r, 0 2 7 = r a t ein t e r e s t r e a l
1, 0 2 7 =r a t ein t e r e s t r e a l1
=r a t ein t e r e s t r e a l1
,0 2 2+1
,0 5 0+1
?
?
Savings
Bond
?The McGraw-Hill Companies,Inc.,2001
3- 29
Irwin/McGraw-Hill
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.
?The McGraw-Hill Companies,Inc.,2001
3- 30
Irwin/McGraw-Hill
Effective Interest Rates
example
Given a monthly rate of 1%,what is the Effective
Annual Rate(EAR)? What is the Annual
Percentage Rate (APR)?
E A R = (1 +, 0 1 ) - 1 = r
E A R = (1 +, 0 1 ) - 1 =, 1 2 6 8 o r 1 2, 6 8 %
A P R =, 0 1 x 1 2 =, 1 2 o r 1 2, 0 0 %
12
12
?The McGraw-Hill Companies,Inc.,2001
3- 31
Irwin/McGraw-Hill
基本財務數學摘要
?現值與未來值
?年金（ Annuity） 的未來值和現值
?貸款常數 （ mortgage constant） 與沉入
基金因子 （ sinking fund factor）
?The McGraw-Hill Companies,Inc.,2001
3- 32
Irwin/McGraw-Hill
現值與未來值
?現值與未來值之關係
FVn=PV(1+r)n FVn=PV(1+r/m)nm
?Effective Annual Yield (EAY)
EAY=(1+r/m)m -1
?未來值因子,FV(r/m,nm)=(1+r/m)nm
?現值因子,PV(r/m,nm)=1/(1+r/m)nm
?The McGraw-Hill Companies,Inc.,2001
3- 33
Irwin/McGraw-Hill
年金的未來值和現值（ I）
?永久年金 (Annuity)的現值：
?年金現值因子 (Annuity Present Value Factor)
為：
r)+(1
1-1
r
1=n)P V A ( r,
n ??
?
??
?
r
A
PV
.,,,,,
r)(1
A
r)(1
A
r1
A
PV
32
?
?
?
?
?
?
?
?
?The McGraw-Hill Companies,Inc.,2001
3- 34
Irwin/McGraw-Hill
年金的未來值和現值（ II）
?年金未來值因子為
? ?1)r1()
r
1
(
r)1(
r)+(1
1
-1)
r
1
(=
r)+(1n)P V A ( r,=n)F V A ( r,
n
n
n
n
???
?
?
?
?
?
?
?
?
?The McGraw-Hill Companies,Inc.,2001
3- 35
Irwin/McGraw-Hill
貸款常數 (mortgage constant)與
沉入基金因子 (sinking fund
factor)
?貸款常數
PV=A×PVA(r,n) A=PV×[1/PVA(r,n]
?MC=1×[1/PVA(r,n]
?沉入基金因子
FV=A× FVA(r,n) A=FV× [1/FVA(r,n]
?SFF=1×[1/FVA(r,n]
