Time Value of Money
Time Value of Money
Time Value of Money
References:
Keown, 2005, Financial Management: Principles and
Applications, 10th ed., Prentice Hall
Ross, Westerfield, and Jordan, 2006, Fundamentals of
Corporate Finance, 7th ed., McGraw-Hill
Brigham & Houston, Fundamentals of Financial
Management, 8th ed.
Prepared By:
Liem Pei Fun, S.E., MCom.
About me
1998-2002 Bachelor of Economics (Majoring in
Finance), Petra Christian University
2004-2005 Postgraduate Diploma in Finance,
The University of Melbourne
2005-2006 Master of Commerce (Finance), The
University of Melbourne
Pass CFA Exam Level I
Bloomberg Global Product Certification – Equity
Level One
Head of Finance Program, Faculty of Economics,
Petra Christian University
OUTLINE
Future Value Single Sum
Present Value Single Sum
Future Value Annuities
Present Value Annuities
Ordinary Annuity vs Annuity Due
Perpetuities
Practice Problems
The Time Value of Money
Today Future
Today Future
?
Future Value
Future Value - single sums
If you deposit $100 in an account earning
6%, how much would you have in the
account after 1 year?
PV = -100 FV = 106
0 1
Mathematical Solution:
(Arithmetic Method)
FV = PV (1 + i)n
FV = 100 (1.06)1 = $106
Calculator Keys
Texas Instruments BA-II Plus
FV = future value
PV = present value
I/Y = period interest rate
P/Y must equal 1 for the I/Y to be the period rate
Interest is entered as a percent, not a decimal
N = number of periods
Remember to clear the registers (CLR TVM)
after each problem
Other calculators are similar in format
Future Value - single sums
If you deposit $100 in an account earning
6%, how much would you have in the
account after 1 year?
PV = -100 FV = 106
0 1
Calculator Solution:
P/Y = 1 I/Y = 6
N=1 PV = -100
FV = $106
Future Value - single sums
If you deposit $100 in an account earning
6%, how much would you have in the
account after 5 years?
PV = -100 FV = 133.82
0 5
Mathematical Solution:
(Arithmetic Method)
FV = PV (1 + i)n
FV = 100 (1.06)5 = $133.82
Future Value - single sums
If you deposit $100 in an account earning
6%, how much would you have in the
account after 5 years?
PV = -100 FV = 133.82
0 5
Calculator Solution:
P/Y = 1 I/Y = 6
N=5 PV = -100
FV = $133.82
Future Value - single sums
If you deposit $100 in an account earning
6% with quarterly compounding, how
much would you have in the account after
5 years?
PV = -100 FV = 134.68
0 20
Mathematical Solution:
(Arithmetic Method)
FV = PV (1 + i/m) m x n
FV = 100 (1.015)20 = $134.68
Future Value - single sums
If you deposit $100 in an account earning
6% with quarterly compounding, how
much would you have in the account after
5 years?
PV = -100 FV = 134.68
0 20
Calculator Solution:
P/Y = 4 I/Y = 6
N = 20 PV = -100
FV = $134.68
Future Value - single sums
If you deposit $100 in an account earning
6% with monthly compounding, how
much would you have in the account after
5 years?
PV = -100 FV = 134.89
0 60
Mathematical Solution:
(Arithmetic Method)
FV = PV (1 + i/m) m x n
FV = 100 (1.005)60 = $134.89
Future Value - single sums
If you deposit $100 in an account earning
6% with monthly compounding, how
much would you have in the account after
5 years?
PV = -100 FV = 134.89
0 60
Calculator Solution:
P/Y = 12 I/Y = 6
N = 60 PV = -100
FV = $134.89
Future Value - continuous
compounding
What is the FV of $1,000 earning 8% with
continuous compounding, after 100 years?
PV = -1000 FV = $2.98m
0 100
Mathematical Solution:
(Arithmetic Method)
FV = PV (e in)
FV = 1000 (e .08x100) = 1000 (e 8)
FV = $2,980,957.99
Present Value
Present Value - single sums
If you receive $100 one year from now,
what is the PV of that $100 if your
opportunity cost is 6%?
PV = -94.34 FV = 100
0 1
Mathematical Solution:
(Arithmetic Method)
PV = FV / (1 + i)n
PV = 100 / (1.06)1 = $94.34
Present Value - single sums
If you receive $100 one year from now,
what is the PV of that $100 if your
opportunity cost is 6%?
PV = -94.34 FV = 100
0 1
Calculator Solution:
P/Y = 1 I/Y = 6
N=1 FV = 100
PV = -94.34
Present Value - single sums
If you receive $100 five years from now,
what is the PV of that $100 if your
opportunity cost is 6%?
PV = -74.73 FV = 100
0 5
Mathematical Solution:
(Arithmetic Method)
PV = FV / (1 + i)n
PV = 100 / (1.06)5 = $74.73
Present Value - single sums
If you receive $100 five years from now,
what is the PV of that $100 if your
opportunity cost is 6%?
PV = -74.73 FV = 100
0 5
Calculator Solution:
P/Y = 1 I/Y = 6
N=5 FV = 100
PV = -74.73
Present Value - single sums
If you sold land for $11,933 that you
bought 5 years ago for $5,000, what is your
annual rate of return?
Mathematical Solution:
PV = FV / (1 + i)n
5,000 = 11,933 / (1+ i)5
.419 = ((1/ (1+i)5)
2.3866 = (1+i)5
(2.3866)1/5 = (1+i)
i = .19
Present Value - single sums
If you sold land for $11,933 that you
bought 5 years ago for $5,000, what is your
annual rate of return?
PV = -5000 FV = 11,933
0 5
Calculator Solution:
P/Y = 1 N=5
PV = -5,000 FV = 11,933
I/Y = 19%
Present Value - single sums
Suppose you placed $100 in an account
that pays 9.6% interest, compounded
monthly. How long will it take for your
account to grow to $500?
Mathematical Solution:
PV = FV / (1 + i)N
100 = 500 / (1+ .008)N
5 = (1.008)N
ln 5 = ln (1.008)N
ln 5 = N ln (1.008)
1.60944 = .007968 N
N = 202 months
Present Value - single sums
Suppose you placed $100 in an account
that pays 9.6% interest, compounded
monthly. How long will it take for your
account to grow to $500?
PV = -100 FV = 500
0 ?
Calculator Solution:
P/Y = 12 FV = 500
I/Y = 9.6 PV = -100
N = 202 months
Compounding and
Discounting
Cash Flow Streams
0 1 2 3 4
What is the difference between an
ordinary annuity and an annuity due?
Ordinary Annuity
0 1 2 3
i%
0 1 2 3 4
Examples of Annuities:
If you buy a bond, you will
receive equal semi-annual
coupon interest payments over
the life of the bond.
If you borrow money to buy a
0 1 2 3
Future Value - annuity
If you invest $1,000 each year at 8%,
how much would you have after 3
years?
Mathematical Solution:
(1 i ) n 1
FV PMT
i
(1.08) 3 1
FV 1,000
0.08
= $3,246.40
Future Value - annuity
If you invest $1,000 each year at 8%, how
much would you have after 3 years?
0 1 2 3
Calculator Solution:
P/Y = 1 I/Y = 8
PMT = -1,000 N=3
FV = $3,246.40
Present Value - annuity
What is the PV of $1,000 at the end of each
of the next 3 years, if the opportunity cost
is 8%?
0 1 2 3
Present Value - annuity
What is the PV of $1,000 at the end of
each of the next 3 years, if the
opportunity cost is 8%?
Mathematical Solution:
1
1
(1 i ) n
PV PMT
i
1
1 (1.08) 3
PV 1,000
0.08
PV $2,577.10
Present Value - annuity
What is the PV of $1,000 at the end of each
of the next 3 years, if the opportunity cost
is 8%?
1000 1000 1000
0 1 2 3
Calculator Solution:
P/Y = 1 I/Y = 8
PMT = -1,000 N=3
PV = $2,577.10
Ordinary Annuity
vs.
Annuity Due
4 5 6 7 8
Begin Mode vs. End Mode
4 5 6 7 8
Begin Mode vs. End Mode
PV
in
END
Mode
Begin Mode vs. End Mode
PV FV
in in
END END
Mode Mode
Begin Mode vs. End Mode
PV
in
BEGIN
Mode
Begin Mode vs. End Mode
PV FV
in in
BEGIN BEGIN
Mode Mode
Earlier, we examined this
“ordinary” annuity:
1000 1000 1000
0 1 2 3
Using an interest rate of 8%,
we find that:
The Future Value (at 3) is
$3,246.40.
The Present Value (at 0) is
$2,577.10.
What about this annuity?
0 1 2 3
Future Value - annuity due
If you invest $1,000 at the beginning
of each of the next 3 years at 8%, how
much would you have at the end of
year 3?
Mathematical Solution: Simply compound the FV of
the ordinary annuity one more period:
FV = PMT (FVIFA i, n ) (1 + i)
FV = 1,000 (FVIFA .08, 3 ) (1.08) use FVIFA table or
(1 i ) n 1
FV PMT (1 i )
i
(1.08) 3 1
FV 1,000 (1.08)
0.08
= $3,506.11
Present Value - annuity due
What is the PV of $1,000 at the beginning
of each of the next 3 years, if your
opportunity cost is 8%?
0 1 2 3
Present Value - annuity due
Mathematical Solution: Simply compound the FV of
the ordinary annuity one more period:
PV = PMT (PVIFA i, n ) (1 + i)
PV = 1,000 (PVIFA .08, 3 ) (1.08) use PVIFA table or
1
1
(1 i ) n
PV PMT (1 i )
i
1
1
(1.08) 3
PV 1,000 (1.08)
0.08
= $2,783.26
Other Cash Flow Patterns
0 1 2 3
Perpetuities
Suppose you will receive a fixed
payment every period (month,
year, etc.) forever. This is an
example of a perpetuity.
You can think of a perpetuity as
PV = PMT (PVIFA i, n )
Mathematically,
1
n
(PVIFA i, n ) = 1- (1 + i)
i
PMT
PV =
i
What should you be willing to pay
in order to receive $10,000
annually forever, if you require
8% per year on the investment?
PV = PMT = $10,000
i .08
= $125,000
Uneven Cash Flows
0 1 2 3 4
Is this an annuity?
How do we find the PV of a cash flow
0 1 2 3 4
0 1 2 3 4
0 1 2 3 4
0 1 2 3 4
0 1 2 3 4
0 1 2 3 4
period CF PV (CF)
0 -10,000 -10,000.00
1 2,000 1,818.18
2 4,000 3,305.79
3 6,000 4,507.89
4 7,000 4,781.09
PV of Cash Flow Stream: $ 4,412.95
Annual Percentage Yield
(APY)
Which is the better loan:
8% compounded annually, or
APY = ( 1+
.0785
4
) 4
- 1
Annual Percentage Yield
(APY)
APY = (1+ quoted rate
m
) m
- 1
Find the APY for the quarterly loan:
APY = ( 1+
.0785
4
) 4
- 1
APY = ( 1+
.0785
4
4
)
- 1
APY = .0808, or 8.08%
The quarterly loan is more expensive
than the 8% loan with annual
compounding!
Spreadsheet Example
Use the following formulas for TVM
calculations
FV(rate,nper,pmt,pv)
PV(rate,nper,pmt,fv)
RATE(nper,pmt,pv,fv)
NPER(rate,pmt,pv,fv)
The formula icon is very useful when you can’t
remember the exact formula
Click on the Excel icon to open a spreadsheet
containing four different examples.
Practice Problems
$0 0 0 0 40 40 40 40 40
0 1 2 3 4 5 6 7 8
Retirement Example