Transcript No Slide Title

Chapter
4
The Time Value Of Money
Copyright ©2003 South-Western/Thomson Learning
Introduction
• This chapter introduces the concepts
and skills necessary to understand
the time value of money and its
applications.
Notation
I
i
n
denotes simple interest
denotes the interest rate per period
denotes the number of periods
PMT denotes cash payment (annuities only)
PV denotes the present value dollar amount
T
denotes the tax rate
t
denotes time
PV0 = principal amount at time 0
FVn = future value n time periods from time 0
Simple Interest
• Simple Interest
– Interest paid on the principal sum only
– I = PV0  i  n
– FVn = PV0 + I = PV0 + PV0  i  n
Compound Interest
• Compound Interest
– Interest paid on the principal and on prior
interest that has not been paid or withdrawn
– FV1 = PV0(1+i)1
FV2 = FV1(1+i)1 = PV0(1+i)2
FV3 = FV2(1+i)1 = PV0(1+i)2(1+i)1 = PV0(1+i)3
Future Value of a Cash Flow
• At the end of year n for a sum compounded at
interest rate i is FVn = PV0(1 + i)n
Formula
– See Figure 4.1.
• In Table I in the text, (FVIFi,n) shows the future
value of $1 invested for n years at interest
rate i: FVIFi,n = (1 + i)n
Table I
• When using the table, FVn = PV0(FVIFi,n)
– See Figure 4.2.
Future Value of a Cash
• FVn = PV0(1 + i)n
FVn
n
 (1  i )  0
PV0
FVn
 PV0  n  (1  i ) n 1  0
i
FVn
n
 PV0  ln(1  i )  (1  i )  0
n
Tables Have Three Variables
• Interest factors (IF)
• Time periods (n)
• Interest rates per period (i)
• If you know any two, you can solve
algebraically for the third variable.
Present Value of a Cash Flow
• PV0 = FVn[1/(1+i)n]
Formula
• PVIFi, n = [1/(1+i)n]
Table II
• PV0 = FVn(PVIFi, n)
Table II
• See Figure 4.3.
Present Value of a Cash Flow
• PV0 = FVn[1/(1+i)n]
PV0
n  FVn
 n 1
 FVn  ( n)  (1  i )

0
n 1
i
(1  i )
PV0
n
 FVn  ln(1  i )  (1  i )  (1)
n
FVn  ln(1  i )

0
n
(1  i )
PV0
1
n
 (1  i ) 
0
n
FVn
(1  i )
Example Using Formula
• What is the PV of $100 one year from
now with 12 percent (annual) interest
compounded monthly?
PV0 = $100  1/(1 + .12/12)(12 1)
= $100  1/(1.126825)
= $100  (.88744923)
= $ 88.74
Example Using Table II
• PV0 = FVn(PVIFi, n)
= $100(.887)
= $ 88.70
From Table II
Annuity
• A series of equal dollar CFs for a
specified number of periods
• Ordinary annuity is where the CFs occur
at the end of each period.
• Annuity due is where the CFs occur at
the beginning of each period.
Future Value of an Ordinary
Annuity
• FVIFAi , n
(1  i ) n  1

i
• FVANn = PMT(FVIFAi, n)
Formula for IF
Table III
Future Value of an Ordinary
Annuity
• Suppose Ms. Jefferson receives a threeyear ordinary annuity of $1,000 per year
and deposits the money in a savings
account at the end of each year. The
account earns interest at a rate of 6%
compounded annually. How much will
her account be worth at the end of the
three-year period?
Future Value of an Ordinary
Annuity
• See Figure 4.6.
• FVAN3 = PMT(FVIFA0.06, 3)
= $1,000(3.184) = $3,184
Present Value of an Ordinary
Annuity
• PVIFA i , n
1
1
(1  i ) n

i
• PVAN0 = PMT(PVIFAi, n)
Formula
Table IV
Present Value of an Ordinary
Annuity
• What is the present value of an ordinary
$1,000 annuity received at the end of
each year for five years discounted at a
6% rate?
• See Figure 4.8.
• PVAN0 = PMT(PVIFA0.06, 5)
= $1,000(4.212)
= $4,212
Annuity Due
• Future Value of an Annuity Due
– FVANDn = PMT(FVIFAi, n)(1 + i)
Table III
• Present Value of an Annuity Due
– PVAND0 = PMT(PVIFAi, n)(1 + i)
Table IV
Annuity Due
• Consider the case of Jefferson cited
earlier. If she deposits $1,000 in a
savings account at the beginning of each
year for the next three years and the
account earns 6% interest, compounded
annually, how much will be in the
account at the end of three years?
• See Figure 4.7.
• FVAND3 = PMT(FVIFA0.06, 3)(1 + 0.06)
= $1,000(3.375) = $3,375
Annuity Due
• Consider the case of a five-year annuity
of $1,000 each year, discounted at 6%
interest rate. What is the present value
of this annuity if each payment is
received at the beginning of each year?
• See Figure 4.9.
• PVAND0 = PMT(PVIFA0.06, 5)(1 + 0.06)
= $1,000(4.465)
= $4,465
Other Important Formulas
• Sinking Fund
– PMT = FVANn  (FVIFAi, n)
Table III
• Payments on a Loan
– PMT = PVAN0  (PVIFAi, n)
Table IV
• Present Value of a Perpetuity
– PVPER0 = PMT  i
Example: Sinking Fund
Problem
• Suppose the Omega Graphics
Company wishes to set aside an equal,
annual, end-of-year amount in a
“sinking fund account” earning 10% per
annum over the next five years. The
firm wants to have $5 million in the
account at the end of five years to retire
(pay off) $5 million in outstanding
bonds. How much must be deposited in
the account at the end of each year?
Solution Based on Table III
• $5,000,000 = PMT(FVIFA0.10, 5)
= PMT(6.105)
 PMT = $819,001
Solution Based on the Financial
Calculator
•
•
•
•
•
1. 5,000,000 → FV
2. 10 → %i
3. 5 → N
4. Compute
5. PMT (= -818,987.40)
Example: Payments on a Loan
• Suppose you borrowed $10,000 from the
ICBC. The loan is for a period of four
years at an interest rate of 10%. It
requires that you make four equal,
annual, end-of-year payments that
include both principal and interest on the
outstanding balance.
Solution Based on Table IV
• PMT = PVAN0  (PVIFAi, n)
= $10,000  (PVIFA0.10, 4)
= $10,000  3.170
= $3,155
Solution Based on the Financial
Calculator
•
•
•
•
•
1. 10,000 → PV
2. 10 → %i
3. 4 → N
4. Compute
5. PMT (= -3,154.71)
Present Value of a Perpetuity
• Assume that Kansas City Power & Light
series E preferred stock promises
payments of $4.50 per year forever and
that an investor requires a 10% rate of
return on this type of investment. How
much would the investor be willing to
pay for this security?
• PVPER0 = PMT  i
PVPER0 = $4.50  10% = $45
Present Value of an Uneven
Payment Stream
• Algebraically, the present value of an
uneven payment stream can be
represented as
PMT3
PMTn
PMT1 PMT2
PV0 


 ... 
2
3
(1  i ) (1  i )
(1  i )
(1  i ) n
n
PMTt
= 
t
(1

i
)
t 1
n
=
 PMT (PVIF
t
t =1
• See Figure 4.10.
i, t
)
Present Value of Deferred Annuities
• Suppose that you wish to provide for the
college education of your daughter. She
will begin college five years from now,
and you wish to have $15,000 available
for her at the beginning of each year in
college. How much must be invested
today at a 12% annual rate of return in
order to provide the four-year, $15,000
annuity for your daughter?
• See Figure 4.11.
Solution Based on Tables II and III
• First step: calculate the present value of
the four-year (ordinary) annuity
PVAN4 = $15,000(PVIFA0.12, 4)
= $15,000(3.037)
= $45,555
• Second step: calculate the present value
of the (ordinary) annuity
PVAN0 = PVAN4(PVIF0.12, 4)
= $45,555(0.636) = $28,973
Solution Based on the Financial
Calculator
•
•
•
•
•
•
•
•
•
•
1. 15,000 → PMT
2. 12 → %i
3. 4 → N
4. Compute
5. PV (= -45,560)
6. 45,560 → FV
7. 12 → %i
8. 4 → N
9. Compute
10. PV (= -28,954)
Interest Compounded More
Frequently Than Once Per Year
• Suppose:
– m = # of times interest is compounded per year
– n = # of years
Future Value
inom nm
FVn  PV0 (1 
)
m
Present Value
FVn
PV0 
inom nm
(1 
)
m
Compounding and Effective Rates
• Rate of interest per compounding period
1
inom
im 
 (1  ieff ) m  1
m
• Effective annual rate of interest
ieff
inom m
 (1 
) 1
m
Compounding and Effective Rates
• Suppose a bank offers you a loan an
annual nominal interest rate of 12%
compounded quarterly. What effective
annual interest rate is the bank charging
you?
inom m
ieff  (1 
) 1
m
12% 4
 (1 
) 1
4
 0.1255  12.55%
Example
• Mr. Moore is 45 years old today (Dec.
31) and is beginning to plan for his
retirement. He wants to set aside an
equal amount at the end of each of the
next 20 years so that he can retire at age
65. He expects to live to the maximum
age of 85 and wants to be able to
withdraw $25,000 per year from the
account on his 66th through 85th
birthdays.
Example
• The account is expected to earn 10
percent per annum for the entire period
of time. Determine the size of the annual
deposits that must be made by Mr.
Moore.
Solution Based on Tables III and IV
• PVAN = PMT(PVIFA0.10,20)
= $25,000(8.514)
= $212,850 (needed on 65th birthday)
• $212,850 = PMT(FVIFA0.10,20)
= PMT(57.275)
 PMT = $3,716.28
Solution Based on the Financial
Calculator
•
•
•
•
•
•
•
•
•
•
1. 25,000 → PMT
2. 10 → %i
3. 20 → N
4. Compute
5. PV (= -212,839)
6. 212,839 → FV
7. 10 → %i
8. 20 → N
9. Compute
10. PMT (= -3,716.08)
Example
• You sold 1000 shares of stock today for
$80.515 per share that you paid $50 for
5 years ago. Determine the average
annual rate of return on your investment,
assuming the stock paid no dividends.
Solution Based on Table II
• PV0 = FVn(PVIFi, n);
n = 5; PV0 = $50; FV5 = $80.515
$50 = $80.515(PVIFi, 5)
(PVIFi, 5) = 0.621; Therefore i = 10% from
Table II.
Example
• The Texas lottery agrees to pay the
winner $500,000 at the end of each year
for the next 20 years. What is the future
value of this lottery if you plan to put
each payment in an account earning 12
percent?
Solution Based on Table III
• FVAN20
= PMT(FVIFAi, n)
= $500,000(FVIFA0.12, 20)
= $500,000(72.052)
= $36,026,000
Solution Based on the Financial
Calculator
•
•
•
•
•
1. -500,000 → PMT
2. 12 → %i
3. 20 → N
4. Compute
5. FV (= 36,026,221.22)
Example
• Your firm has just leased a $32,000
BMW for you. The lease requires five
beginning of the year payments that will
fully amortize the cost of the car. What is
the amount of the payments if the
interest rate is 10 percent?
Solution Based on Table IV
• PVAND0 = PMT(PVIFAi, n)(1+i)
$32,000 = PMT(PVIFA0.10, 5)(1+10%)
$32,000 = PMT(3.791)(1.10)
 PMT = $7,673.68
Solution Based on the Financial
Calculator
•
•
•
•
•
•
1. 32,000 → PV
2. 10 → %i
3. 5 → N
4. Compute
5. PMT (= -8,441.52)
6. -8,441.52  1.10 = -7,674.11