The Time Value of Money
We know that receiving $1 today is
worth more than $1 in the future.
This is due to opportunity costs.
The opportunity cost of receiving
$1 in the future is the interest we
could have earned if we had
received the $1 sooner.
Today
Future
If we can measure this
opportunity cost, we can:

Translate $1 today into its equivalent in
the future (compounding).
Today
Future
?

Translate $1 in the future into its
equivalent today (discounting).
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
0
Mathematical Solution:
(Arithmetic Method)
FV = PV (1 + i)n
FV = 100 (1.06)1 = $106
FV = 106
1
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
Mathematical Solution:
(Arithmetic Method)
FV = PV (1 + i)n
FV = 100 (1.06)5 = $133.82
5
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
Mathematical Solution:
(Arithmetic Method)
FV = PV (1 + i/m) m x n
FV = 100 (1.015)20 = $134.68
20
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
Mathematical Solution:
(Arithmetic Method)
FV = PV (1 + i/m) m x n
FV = 100 (1.005)60 = $134.89
60
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
Mathematical Solution:
(Arithmetic Method)
PV = FV / (1 + i)n
PV = 100 / (1.06)1 = $94.34
1
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
Mathematical Solution:
(Arithmetic Method)
PV = FV / (1 + i)n
PV = 100 / (1.06)5 = $74.73
5
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
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
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
i%
1
2
3
PMT
PMT
PMT
1
2
3
PMT
PMT
Annuity Due
0
i%
PMT
Ordinary Annuity

a sequence of equal cash flows,
occurring at the end of each
period.
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
house or a car, you will pay a
stream of equal payments.

Future Value - annuity
If you invest $1,000 each year at 8%, how
much would you have after 3 years?
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
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












Ordinary Annuity
vs.
Annuity Due
4
$1000
$1000
$1000
5
6
7
8
Begin Mode vs. End Mode
4
1000
1000
1000
5
6
7
8
Begin Mode vs. End Mode
1000
4
year
5
5
1000
year
6
6
1000
year
7
7
8
Begin Mode vs. End Mode
1000
4
PV
in
END
Mode
year
5
5
1000
year
6
6
1000
year
7
7
8
Begin Mode vs. End Mode
1000
4
year
5
5
1000
year
6
6
1000
year
7
7
PV
FV
in
END
Mode
in
END
Mode
8
Begin Mode vs. End Mode
1000
4
5
1000
year
6
6
1000
year
7
7
year
8
8
Begin Mode vs. End Mode
1000
4
5
1000
year
6
PV
in
BEGIN
Mode
6
1000
year
7
7
year
8
8
Begin Mode vs. End Mode
1000
4
5
1000
year
6
6
1000
year
7
7
year
8
8
PV
FV
in
BEGIN
Mode
in
BEGIN
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?
1000
1000
1000
0
1
2
3
Same 3-year time line,
 Same 3 $1000 cash flows, but
 The cash flows occur at the
beginning of each year, rather
than at the end of each year.
 This is an “annuity due.”

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?
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)
 (1  i ) n  1
FV  PMT 
 (1  i )
i


 (1.08) 3  1
FV  1,000 
 (1.08)
 0.08 
= $3,506.11
use FVIFA table or
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)
1

1


(1  i ) n
PV  PMT 
i



1

1  (1.08) 3
PV  1,000 
0.08



= $2,783.26


 (1  i )





 (1.08)



use PVIFA table or
Practice Problems
 2002, Prentice Hall, Inc.
Retirement Example

If you invest $400 at the end of each
month for the next 30 years at 12%,
how much would you have at the end
of year 30?