Ch,5 - The Time Value
of Money
? 2002,Prentice Hall,Inc,
The Time Value of Money
Compounding and
Discounting Single Sums
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,
If we can measure this opportunity cost,
we can,
? Translate $1 today into its equivalent in the future
(compounding),
If we can measure this opportunity cost,
we can,
? Translate $1 today into its equivalent in the future
(compounding),
Today
Future
If we can measure this opportunity cost,
we can,
? Translate $1 today into its equivalent in the future
(compounding),
? Translate $1 in the future into its equivalent today
(discounting),
Today
Future
If we can measure this opportunity cost,
we can,
? Translate $1 today into its equivalent in the future
(compounding),
? Translate $1 in the future into its equivalent today
(discounting),
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?
Future Value - single sums
If you deposit $100 in an account earning 6%,how
much would you have in the account after 1 year?
0 1
PV = FV =
Future Value - single sums
If you deposit $100 in an account earning 6%,how
much would you have in the account after 1 year?
Calculator Solution,
P/Y = 1 I = 6
N = 1 PV = -100
FV = $106
0 1
PV = -100 FV =
Future Value - single sums
If you deposit $100 in an account earning 6%,how
much would you have in the account after 1 year?
Calculator Solution,
P/Y = 1 I = 6
N = 1 PV = -100
FV = $106
0 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 1 year?
Mathematical Solution,
FV = PV (FVIF i,n )
FV = 100 (FVIF,06,1 ) (use FVIF table,or)
FV = PV (1 + i)n
FV = 100 (1.06)1 = $106
0 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?
Future Value - single sums
If you deposit $100 in an account earning 6%,how
much would you have in the account after 5 years?
0 5
PV = FV =
Future Value - single sums
If you deposit $100 in an account earning 6%,how
much would you have in the account after 5 years?
Calculator Solution,
P/Y = 1 I = 6
N = 5 PV = -100
FV = $133.82
0 5
PV = -100 FV =
Future Value - single sums
If you deposit $100 in an account earning 6%,how
much would you have in the account after 5 years?
Calculator Solution,
P/Y = 1 I = 6
N = 5 PV = -100
FV = $133.82
0 5
PV = -100 FV = 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?
Mathematical Solution,
FV = PV (FVIF i,n )
FV = 100 (FVIF,06,5 ) (use FVIF table,or)
FV = PV (1 + i)n
FV = 100 (1.06)5 = $133.82
0 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?
0?
PV = FV =
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?
Calculator Solution,
P/Y = 4 I = 6
N = 20 PV = -100
FV = $134.68
0 20
PV = -100 FV =
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?
Calculator Solution,
P/Y = 4 I = 6
N = 20 PV = -100
FV = $134.68
0 20
PV = -100 FV = 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?
Mathematical Solution,
FV = PV (FVIF i,n )
FV = 100 (FVIF,015,20 ) (can’t use FVIF table)
FV = PV (1 + i/m) m x n
FV = 100 (1.015)20 = $134.68
0 20
PV = -100 FV = 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?
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?
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?
0?
PV = FV =
Calculator Solution,
P/Y = 12 I = 6
N = 60 PV = -100
FV = $134.89
0 60
PV = -100 FV =
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?
Calculator Solution,
P/Y = 12 I = 6
N = 60 PV = -100
FV = $134.89
0 60
PV = -100 FV = 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?
Mathematical Solution,
FV = PV (FVIF i,n )
FV = 100 (FVIF,005,60 ) (can’t use FVIF table)
FV = PV (1 + i/m) m x n
FV = 100 (1.005)60 = $134.89
0 60
PV = -100 FV = 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?
Future Value - continuous compounding
What is the FV of $1,000 earning 8% with
continuous compounding,after 100 years?
Future Value - continuous compounding
What is the FV of $1,000 earning 8% with
continuous compounding,after 100 years?
0?
PV = FV =
Mathematical Solution,
FV = PV (e in)
FV = 1000 (e,08x100) = 1000 (e 8)
FV = $2,980,957.99
0 100
PV = -1000 FV =
Future Value - continuous compounding
What is the FV of $1,000 earning 8% with
continuous compounding,after 100 years?
0 100
PV = -1000 FV = $2.98m
Future Value - continuous compounding
What is the FV of $1,000 earning 8% with
continuous compounding,after 100 years?
Mathematical Solution,
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%?
0?
PV = FV =
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%?
Calculator Solution,
P/Y = 1 I = 6
N = 1 FV = 100
PV = -94.34
0 1
PV = FV = 100
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%?
Calculator Solution,
P/Y = 1 I = 6
N = 1 FV = 100
PV = -94.34
PV = -94.34 FV = 100
0 1
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%?
Mathematical Solution,
PV = FV (PVIF i,n )
PV = 100 (PVIF,06,1 ) (use PVIF table,or)
PV = FV / (1 + i)n
PV = 100 / (1.06)1 = $94.34
PV = -94.34 FV = 100
0 1
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%?
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%?
0?
PV = FV =
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%?
Calculator Solution,
P/Y = 1 I = 6
N = 5 FV = 100
PV = -74.73
0 5
PV = FV = 100
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%?
Calculator Solution,
P/Y = 1 I = 6
N = 5 FV = 100
PV = -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%?
0 5
PV = -74.73 FV = 100
Mathematical Solution,
PV = FV (PVIF i,n )
PV = 100 (PVIF,06,5 ) (use PVIF table,or)
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%?
0 5
PV = -74.73 FV = 100
Present Value - single sums
What is the PV of $1,000 to be received 15 years
from now if your opportunity cost is 7%?
0 15
PV = FV =
Present Value - single sums
What is the PV of $1,000 to be received 15 years
from now if your opportunity cost is 7%?
Calculator Solution,
P/Y = 1 I = 7
N = 15 FV = 1,000
PV = -362.45
Present Value - single sums
What is the PV of $1,000 to be received 15 years
from now if your opportunity cost is 7%?
0 15
PV = FV = 1000
Calculator Solution,
P/Y = 1 I = 7
N = 15 FV = 1,000
PV = -362.45
Present Value - single sums
What is the PV of $1,000 to be received 15 years
from now if your opportunity cost is 7%?
0 15
PV = -362.45 FV = 1000
Mathematical Solution,
PV = FV (PVIF i,n )
PV = 100 (PVIF,07,15 ) (use PVIF table,or)
PV = FV / (1 + i)n
PV = 100 / (1.07)15 = $362.45
Present Value - single sums
What is the PV of $1,000 to be received 15 years
from now if your opportunity cost is 7%?
0 15
PV = -362.45 FV = 1000
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?
0 5
PV = FV =
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?
Calculator Solution,
P/Y = 1 N = 5
PV = -5,000 FV = 11,933
I = 19%
0 5
PV = -5000 FV = 11,933
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 (PVIF i,n )
5,000 = 11,933 (PVIF?,5 )
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?
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?
0
PV = FV =
Calculator Solution,
? P/Y = 12 FV = 500
? I = 9.6 PV = -100
? 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?
0?
PV = -100 FV = 500
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
Hint for single sum problems,
? In every single sum future value and
present value problem,there are 4
variables,
? FV,PV,i,and n
? When doing problems,you will be
given 3 of these variables and asked to
solve for the 4th variable,
? Keeping this in mind makes,time
value” problems much easier!
The Time Value of Money
Compounding and Discounting
Cash Flow Streams
0 1 2 3 4
Annuities
? Annuity,a sequence of equal cash
flows,occurring at the end of each
period,
? Annuity,a sequence of equal cash
flows,occurring at the end of each
period,
0 1 2 3 4
Annuities
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,
? 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,
Examples of Annuities,
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?
Calculator Solution,
P/Y = 1 I = 8 N = 3
PMT = -1,000
FV = $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
1000 1000 1000
Calculator Solution,
P/Y = 1 I = 8 N = 3
PMT = -1,000
FV = $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
1000 1000 1000
Future Value - annuity
If you invest $1,000 each year at 8%,how much
would you have after 3 years?
Mathematical Solution,
Future Value - annuity
If you invest $1,000 each year at 8%,how much
would you have after 3 years?
Mathematical Solution,
FV = PMT (FVIFA i,n )
Future Value - annuity
If you invest $1,000 each year at 8%,how much
would you have after 3 years?
Mathematical Solution,
FV = PMT (FVIFA i,n )
FV = 1,000 (FVIFA,08,3 ) (use FVIFA table,or)
Future Value - annuity
If you invest $1,000 each year at 8%,how much
would you have after 3 years?
Mathematical Solution,
FV = PMT (FVIFA i,n )
FV = 1,000 (FVIFA,08,3 ) (use FVIFA table,or)
FV = PMT (1 + i)n - 1
i
Future Value - annuity
If you invest $1,000 each year at 8%,how much
would you have after 3 years?
Mathematical Solution,
FV = PMT (FVIFA i,n )
FV = 1,000 (FVIFA,08,3 ) (use FVIFA table,or)
FV = PMT (1 + i)n - 1
i
FV = 1,000 (1.08)3 - 1 = $3246.40
,08
Future Value - annuity
If you invest $1,000 each year at 8%,how much
would you have after 3 years?
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%?
Calculator Solution,
P/Y = 1 I = 8 N = 3
PMT = -1,000
PV = $2,577.10
0 1 2 3
1000 1000 1000
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%?
Calculator Solution,
P/Y = 1 I = 8 N = 3
PMT = -1,000
PV = $2,577.10
0 1 2 3
1000 1000 1000
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%?
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,
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,
PV = PMT (PVIFA i,n )
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,
PV = PMT (PVIFA i,n )
PV = 1,000 (PVIFA,08,3 ) (use PVIFA table,or)
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,
PV = PMT (PVIFA i,n )
PV = 1,000 (PVIFA,08,3 ) (use PVIFA table,or)
1
PV = PMT 1 - (1 + i)n
i
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,
PV = PMT (PVIFA i,n )
PV = 1,000 (PVIFA,08,3 ) (use PVIFA table,or)
1
PV = PMT 1 - (1 + i)n
i
1
PV = 1000 1 - (1.08 )3 = $2,577.10
,08
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%?
Other Cash Flow Patterns
0 1 2 3
The Time Value of Money
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 an
annuity that goes on forever,
Present Value of a
Perpetuity
? When we find the PV of an annuity,
we think of the following
relationship,
Present Value of a
Perpetuity
? When we find the PV of an annuity,
we think of the following
relationship,
PV = PMT (PVIFA i,n )
Mathematically,
Mathematically,
(PVIFA i,n ) =
Mathematically,
(PVIFA i,n ) =
1 -
1
(1 + i) n
i
Mathematically,
(PVIFA i,n ) =
We said that a perpetuity is an
annuity where n = infinity,What
happens to this formula when n
gets very,very large?
1 -
1
(1 + i) n
i
When n gets very large,
When n gets very large,
1 -
1
(1 + i) n
i
When n gets very large,
this becomes zero,
1 -
1
(1 + i) n
i
When n gets very large,
this becomes zero,
So we’re left with PVIFA =
1
i
1 -
1
(1 + i) n
i
? So,the PV of a perpetuity is very
simple to find,
Present Value of a Perpetuity
PMT
i PV =
? So,the PV of a perpetuity is very
simple to find,
Present Value of a Perpetuity
What should you be willing to pay in
order to receive $10,000 annually
forever,if you require 8% per year
on the investment?
What should you be willing to pay in
order to receive $10,000 annually
forever,if you require 8% per year
on the investment?
PMT $10,000
i,08 PV = =
What should you be willing to pay in
order to receive $10,000 annually
forever,if you require 8% per year
on the investment?
PMT $10,000
i,08
= $125,000
PV = =
Ordinary Annuity
vs,
Annuity Due
$1000 $1000 $1000
4 5 6 7 8
Begin Mode vs,End Mode
1000 1000 1000
4 5 6 7 8
Begin Mode vs,End Mode
1000 1000 1000
4 5 6 7 8 year year year 5 6 7
Begin Mode vs,End Mode
1000 1000 1000
4 5 6 7 8 year year year 5 6 7
PV
in
END
Mode
Begin Mode vs,End Mode
1000 1000 1000
4 5 6 7 8 year year year 5 6 7
PV
in
END
Mode
FV
in
END
Mode
Begin Mode vs,End Mode
1000 1000 1000
4 5 6 7 8 year year year 6 7 8
Begin Mode vs,End Mode
1000 1000 1000
4 5 6 7 8 year year year 6 7 8
PV
in
BEGIN
Mode
Begin Mode vs,End Mode
1000 1000 1000
4 5 6 7 8 year year year 6 7 8
PV
in
BEGIN
Mode
FV
in
BEGIN
Mode
Earlier,we examined this
“ordinary” annuity,
Earlier,we examined this
“ordinary” annuity,
0 1 2 3
1000 1000 1000
Earlier,we examined this
“ordinary” annuity,
Using an interest rate of 8%,we
find that,
0 1 2 3
1000 1000 1000
Earlier,we examined this
“ordinary” annuity,
Using an interest rate of 8%,we
find that,
? The Future Value (at 3) is
$3,246.40,
0 1 2 3
1000 1000 1000
Earlier,we examined this
“ordinary” annuity,
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,
0 1 2 3
1000 1000 1000
What about this annuity?
? 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.”
0 1 2 3
1000 1000 1000
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?
Calculator Solution,
Mode = BEGIN P/Y = 1 I = 8
N = 3 PMT = -1,000
FV = $3,506.11
0 1 2 3
-1000 -1000 -1000
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
-1000 -1000 -1000
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?
Calculator Solution,
Mode = BEGIN P/Y = 1 I = 8
N = 3 PMT = -1,000
FV = $3,506.11
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,
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)
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)
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)
FV = PMT (1 + i)n - 1
i
(1 + i)
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)
FV = PMT (1 + i)n - 1
i
FV = 1,000 (1.08)3 - 1 = $3,506.11
,08
(1 + i)
(1.08)
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
Calculator Solution,
Mode = BEGIN P/Y = 1 I = 8
N = 3 PMT = 1,000
PV = $2,783.26
0 1 2 3
1000 1000 1000
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%?
Calculator Solution,
Mode = BEGIN P/Y = 1 I = 8
N = 3 PMT = 1,000
PV = $2,783.26
0 1 2 3
1000 1000 1000
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%?
Present Value - annuity due
Mathematical Solution,
Present Value - annuity due
Mathematical Solution,Simply compound the FV of the
ordinary annuity one more period,
Present Value - annuity due
Mathematical Solution,Simply compound the FV of the
ordinary annuity one more period,
PV = PMT (PVIFA i,n ) (1 + i)
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)
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
PV = PMT 1 - (1 + i)n
i
(1 + i)
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
PV = PMT 1 - (1 + i)n
i
1
PV = 1000 1 - (1.08 )3 = $2,783.26
,08
(1 + i)
(1.08)
? Is this an annuity?
? How do we find the PV of a cash flow
stream when all of the cash flows are
different? (Use a 10% discount rate),
Uneven Cash Flows
0 1 2 3 4
-10,000 2,000 4,000 6,000 7,000
? Sorry! There’s no quickie for this one,
We have to discount each cash flow
back separately,
0 1 2 3 4
-10,000 2,000 4,000 6,000 7,000
Uneven Cash Flows
? Sorry! There’s no quickie for this one,
We have to discount each cash flow
back separately,
0 1 2 3 4
-10,000 2,000 4,000 6,000 7,000
Uneven Cash Flows
? Sorry! There’s no quickie for this one,
We have to discount each cash flow
back separately,
0 1 2 3 4
-10,000 2,000 4,000 6,000 7,000
Uneven Cash Flows
? Sorry! There’s no quickie for this one,
We have to discount each cash flow
back separately,
0 1 2 3 4
-10,000 2,000 4,000 6,000 7,000
Uneven Cash Flows
? Sorry! There’s no quickie for this one,
We have to discount each cash flow
back separately,
0 1 2 3 4
-10,000 2,000 4,000 6,000 7,000
Uneven Cash Flows
period CF PV (CF)
0 -10,000 -10,000.00
1 2,000 1,818.18
2 4,000 3,305.79
3 6,000 4,507.89
4 7,000 4,781.09
PV of Cash Flow Stream,$ 4,412.95
0 1 2 3 4
-10,000 2,000 4,000 6,000 7,000
Annual Percentage Yield (APY)
Which is the better loan,
? 8% compounded annually,or
? 7.85% compounded quarterly?
? We can’t compare these nominal (quoted)
interest rates,because they don’t include the
same number of compounding periods per
year!
We need to calculate the APY,
Annual Percentage Yield (APY)
Annual Percentage Yield (APY)
APY = ( 1 + ) m - 1 quoted rate m
Annual Percentage Yield (APY)
? Find the APY for the quarterly loan,
APY = ( 1 + ) m - 1 quoted rate m
Annual Percentage Yield (APY)
? Find the APY for the quarterly loan,
APY = ( 1 + ) m - 1 quoted rate m
APY = ( 1 + ) 4 - 1,0785 4
Annual Percentage Yield (APY)
? Find the APY for the quarterly loan,
APY = ( 1 + ) m - 1 quoted rate m
APY = ( 1 + ) 4 - 1
APY =,0808,or 8.08%
.0785
4
Annual Percentage Yield (APY)
? Find the APY for the quarterly loan,
? The quarterly loan is more expensive than
the 8% loan with annual compounding!
APY = ( 1 + ) m - 1 quoted rate m
APY = ( 1 + ) 4 - 1
APY =,0808,or 8.08%
.0785
4
Practice Problems
Example
? Cash flows from an investment are
expected to be $40,000 per year at the
end of years 4,5,6,7,and 8,If you
require a 20% rate of return,what is
the PV of these cash flows?
Example
0 1 2 3 4 5 6 7 8
$0 0 0 0 40 40 40 40 40
? Cash flows from an investment are
expected to be $40,000 per year at the
end of years 4,5,6,7,and 8,If you
require a 20% rate of return,what is
the PV of these cash flows?
? This type of cash flow sequence is
often called a,deferred annuity.”
0 1 2 3 4 5 6 7 8
$0 0 0 0 40 40 40 40 40
How to solve,
1) Discount each cash flow back to
time 0 separately,
0 1 2 3 4 5 6 7 8
$0 0 0 0 40 40 40 40 40
How to solve,
1) Discount each cash flow back to
time 0 separately,
0 1 2 3 4 5 6 7 8
$0 0 0 0 40 40 40 40 40
How to solve,
1) Discount each cash flow back to
time 0 separately,
0 1 2 3 4 5 6 7 8
$0 0 0 0 40 40 40 40 40
How to solve,
1) Discount each cash flow back to
time 0 separately,
0 1 2 3 4 5 6 7 8
$0 0 0 0 40 40 40 40 40
How to solve,
1) Discount each cash flow back to
time 0 separately,
0 1 2 3 4 5 6 7 8
$0 0 0 0 40 40 40 40 40
How to solve,
1) Discount each cash flow back to
time 0 separately,
0 1 2 3 4 5 6 7 8
$0 0 0 0 40 40 40 40 40
How to solve,
1) Discount each cash flow back to
time 0 separately,
Or,
0 1 2 3 4 5 6 7 8
$0 0 0 0 40 40 40 40 40
2) Find the PV of the annuity,
PV,End mode; P/YR = 1; I = 20;
PMT = 40,000; N = 5
PV = $119,624
0 1 2 3 4 5 6 7 8
$0 0 0 0 40 40 40 40 40
2) Find the PV of the annuity,
PV3,End mode; P/YR = 1; I = 20;
PMT = 40,000; N = 5
PV3= $119,624
0 1 2 3 4 5 6 7 8
$0 0 0 0 40 40 40 40 40
119,624
0 1 2 3 4 5 6 7 8
$0 0 0 0 40 40 40 40 40
Then discount this single sum back to
time 0,
PV,End mode; P/YR = 1; I = 20;
N = 3; FV = 119,624;
Solve,PV = $69,226
119,624
0 1 2 3 4 5 6 7 8
$0 0 0 0 40 40 40 40 40
69,226
0 1 2 3 4 5 6 7 8
$0 0 0 0 40 40 40 40 40
119,624
? The PV of the cash flow
stream is $69,226,
69,226
0 1 2 3 4 5 6 7 8
$0 0 0 0 40 40 40 40 40
119,624
Retirement Example
? After graduation,you plan to invest
$400 per month in the stock market,
If you earn 12% per year on your
stocks,how much will you have
accumulated when you retire in 30
years?
Retirement Example
? After graduation,you plan to invest
$400 per month in the stock market,
If you earn 12% per year on your
stocks,how much will you have
accumulated when you retire in 30
years?
0 1 2 3,,, 360
400 400 400 400
0 1 2 3,,, 360
400 400 400 400
? Using your calculator,
P/YR = 12
N = 360
PMT = -400
I%YR = 12
FV = $1,397,985.65
0 1 2 3,,, 360
400 400 400 400
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?
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?
Mathematical Solution,
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?
Mathematical Solution,
FV = PMT (FVIFA i,n )
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?
Mathematical Solution,
FV = PMT (FVIFA i,n )
FV = 400 (FVIFA,01,360 ) (can’t use FVIFA table)
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?
Mathematical Solution,
FV = PMT (FVIFA i,n )
FV = 400 (FVIFA,01,360 ) (can’t use FVIFA table)
FV = PMT (1 + i)n - 1
i
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?
Mathematical Solution,
FV = PMT (FVIFA i,n )
FV = 400 (FVIFA,01,360 ) (can’t use FVIFA table)
FV = PMT (1 + i)n - 1
i
FV = 400 (1.01)360 - 1 = $1,397,985.65
,01
If you borrow $100,000 at 7% fixed
interest for 30 years in order to
buy a house,what will be your
monthly house payment?
House Payment Example
House Payment Example
If you borrow $100,000 at 7% fixed
interest for 30 years in order to
buy a house,what will be your
monthly house payment?
0 1 2 3,,, 360
? Using your calculator,
P/YR = 12
N = 360
I%YR = 7
PV = $100,000
PMT = -$665.30
0 1 2 3,,, 360
House Payment Example
Mathematical Solution,
House Payment Example
Mathematical Solution,
PV = PMT (PVIFA i,n )
House Payment Example
Mathematical Solution,
PV = PMT (PVIFA i,n )
100,000 = PMT (PVIFA,07,360 ) (can’t use PVIFA table)
House Payment Example
Mathematical Solution,
PV = PMT (PVIFA i,n )
100,000 = PMT (PVIFA,07,360 ) (can’t use PVIFA table)
1
PV = PMT 1 - (1 + i)n
i
House Payment Example
Mathematical Solution,
PV = PMT (PVIFA i,n )
100,000 = PMT (PVIFA,07,360 ) (can’t use PVIFA table)
1
PV = PMT 1 - (1 + i)n
i
1
100,000 = PMT 1 - (1.005833 )360 PMT=$665.30
,005833
Team Assignment
Upon retirement,your goal is to spend 5
years traveling around the world,To
travel in style will require $250,000 per
year at the beginning of each year,
If you plan to retire in 30 years,what are
the equal monthly payments necessary
to achieve this goal? The funds in your
retirement account will compound at
10% annually,
? How much do we need to have by
the end of year 30 to finance the
trip?
? PV30 = PMT (PVIFA,10,5) (1.10) =
= 250,000 (3.7908) (1.10) =
= $1,042,470
27 28 29 30 31 32 33 34 35
250 250 250 250 250
Using your calculator,
Mode = BEGIN
PMT = -$250,000
N = 5
I%YR = 10
P/YR = 1
PV = $1,042,466
27 28 29 30 31 32 33 34 35
250 250 250 250 250
? Now,assuming 10% annual
compounding,what monthly
payments will be required for
you to have $1,042,466 at the end
of year 30?
27 28 29 30 31 32 33 34 35
250 250 250 250 250
1,042,466
? Using your calculator,
Mode = END
N = 360
I%YR = 10
P/YR = 12
FV = $1,042,466
PMT = -$461.17
27 28 29 30 31 32 33 34 35
250 250 250 250 250
1,042,466
? So,you would have to place $461.17 in
your retirement account,which earns
10% annually,at the end of each of the
next 360 months to finance the 5-year
world tour,
of Money
? 2002,Prentice Hall,Inc,
The Time Value of Money
Compounding and
Discounting Single Sums
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,
If we can measure this opportunity cost,
we can,
? Translate $1 today into its equivalent in the future
(compounding),
If we can measure this opportunity cost,
we can,
? Translate $1 today into its equivalent in the future
(compounding),
Today
Future
If we can measure this opportunity cost,
we can,
? Translate $1 today into its equivalent in the future
(compounding),
? Translate $1 in the future into its equivalent today
(discounting),
Today
Future
If we can measure this opportunity cost,
we can,
? Translate $1 today into its equivalent in the future
(compounding),
? Translate $1 in the future into its equivalent today
(discounting),
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?
Future Value - single sums
If you deposit $100 in an account earning 6%,how
much would you have in the account after 1 year?
0 1
PV = FV =
Future Value - single sums
If you deposit $100 in an account earning 6%,how
much would you have in the account after 1 year?
Calculator Solution,
P/Y = 1 I = 6
N = 1 PV = -100
FV = $106
0 1
PV = -100 FV =
Future Value - single sums
If you deposit $100 in an account earning 6%,how
much would you have in the account after 1 year?
Calculator Solution,
P/Y = 1 I = 6
N = 1 PV = -100
FV = $106
0 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 1 year?
Mathematical Solution,
FV = PV (FVIF i,n )
FV = 100 (FVIF,06,1 ) (use FVIF table,or)
FV = PV (1 + i)n
FV = 100 (1.06)1 = $106
0 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?
Future Value - single sums
If you deposit $100 in an account earning 6%,how
much would you have in the account after 5 years?
0 5
PV = FV =
Future Value - single sums
If you deposit $100 in an account earning 6%,how
much would you have in the account after 5 years?
Calculator Solution,
P/Y = 1 I = 6
N = 5 PV = -100
FV = $133.82
0 5
PV = -100 FV =
Future Value - single sums
If you deposit $100 in an account earning 6%,how
much would you have in the account after 5 years?
Calculator Solution,
P/Y = 1 I = 6
N = 5 PV = -100
FV = $133.82
0 5
PV = -100 FV = 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?
Mathematical Solution,
FV = PV (FVIF i,n )
FV = 100 (FVIF,06,5 ) (use FVIF table,or)
FV = PV (1 + i)n
FV = 100 (1.06)5 = $133.82
0 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?
0?
PV = FV =
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?
Calculator Solution,
P/Y = 4 I = 6
N = 20 PV = -100
FV = $134.68
0 20
PV = -100 FV =
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?
Calculator Solution,
P/Y = 4 I = 6
N = 20 PV = -100
FV = $134.68
0 20
PV = -100 FV = 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?
Mathematical Solution,
FV = PV (FVIF i,n )
FV = 100 (FVIF,015,20 ) (can’t use FVIF table)
FV = PV (1 + i/m) m x n
FV = 100 (1.015)20 = $134.68
0 20
PV = -100 FV = 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?
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?
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?
0?
PV = FV =
Calculator Solution,
P/Y = 12 I = 6
N = 60 PV = -100
FV = $134.89
0 60
PV = -100 FV =
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?
Calculator Solution,
P/Y = 12 I = 6
N = 60 PV = -100
FV = $134.89
0 60
PV = -100 FV = 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?
Mathematical Solution,
FV = PV (FVIF i,n )
FV = 100 (FVIF,005,60 ) (can’t use FVIF table)
FV = PV (1 + i/m) m x n
FV = 100 (1.005)60 = $134.89
0 60
PV = -100 FV = 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?
Future Value - continuous compounding
What is the FV of $1,000 earning 8% with
continuous compounding,after 100 years?
Future Value - continuous compounding
What is the FV of $1,000 earning 8% with
continuous compounding,after 100 years?
0?
PV = FV =
Mathematical Solution,
FV = PV (e in)
FV = 1000 (e,08x100) = 1000 (e 8)
FV = $2,980,957.99
0 100
PV = -1000 FV =
Future Value - continuous compounding
What is the FV of $1,000 earning 8% with
continuous compounding,after 100 years?
0 100
PV = -1000 FV = $2.98m
Future Value - continuous compounding
What is the FV of $1,000 earning 8% with
continuous compounding,after 100 years?
Mathematical Solution,
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%?
0?
PV = FV =
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%?
Calculator Solution,
P/Y = 1 I = 6
N = 1 FV = 100
PV = -94.34
0 1
PV = FV = 100
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%?
Calculator Solution,
P/Y = 1 I = 6
N = 1 FV = 100
PV = -94.34
PV = -94.34 FV = 100
0 1
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%?
Mathematical Solution,
PV = FV (PVIF i,n )
PV = 100 (PVIF,06,1 ) (use PVIF table,or)
PV = FV / (1 + i)n
PV = 100 / (1.06)1 = $94.34
PV = -94.34 FV = 100
0 1
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%?
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%?
0?
PV = FV =
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%?
Calculator Solution,
P/Y = 1 I = 6
N = 5 FV = 100
PV = -74.73
0 5
PV = FV = 100
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%?
Calculator Solution,
P/Y = 1 I = 6
N = 5 FV = 100
PV = -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%?
0 5
PV = -74.73 FV = 100
Mathematical Solution,
PV = FV (PVIF i,n )
PV = 100 (PVIF,06,5 ) (use PVIF table,or)
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%?
0 5
PV = -74.73 FV = 100
Present Value - single sums
What is the PV of $1,000 to be received 15 years
from now if your opportunity cost is 7%?
0 15
PV = FV =
Present Value - single sums
What is the PV of $1,000 to be received 15 years
from now if your opportunity cost is 7%?
Calculator Solution,
P/Y = 1 I = 7
N = 15 FV = 1,000
PV = -362.45
Present Value - single sums
What is the PV of $1,000 to be received 15 years
from now if your opportunity cost is 7%?
0 15
PV = FV = 1000
Calculator Solution,
P/Y = 1 I = 7
N = 15 FV = 1,000
PV = -362.45
Present Value - single sums
What is the PV of $1,000 to be received 15 years
from now if your opportunity cost is 7%?
0 15
PV = -362.45 FV = 1000
Mathematical Solution,
PV = FV (PVIF i,n )
PV = 100 (PVIF,07,15 ) (use PVIF table,or)
PV = FV / (1 + i)n
PV = 100 / (1.07)15 = $362.45
Present Value - single sums
What is the PV of $1,000 to be received 15 years
from now if your opportunity cost is 7%?
0 15
PV = -362.45 FV = 1000
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?
0 5
PV = FV =
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?
Calculator Solution,
P/Y = 1 N = 5
PV = -5,000 FV = 11,933
I = 19%
0 5
PV = -5000 FV = 11,933
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 (PVIF i,n )
5,000 = 11,933 (PVIF?,5 )
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?
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?
0
PV = FV =
Calculator Solution,
? P/Y = 12 FV = 500
? I = 9.6 PV = -100
? 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?
0?
PV = -100 FV = 500
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
Hint for single sum problems,
? In every single sum future value and
present value problem,there are 4
variables,
? FV,PV,i,and n
? When doing problems,you will be
given 3 of these variables and asked to
solve for the 4th variable,
? Keeping this in mind makes,time
value” problems much easier!
The Time Value of Money
Compounding and Discounting
Cash Flow Streams
0 1 2 3 4
Annuities
? Annuity,a sequence of equal cash
flows,occurring at the end of each
period,
? Annuity,a sequence of equal cash
flows,occurring at the end of each
period,
0 1 2 3 4
Annuities
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,
? 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,
Examples of Annuities,
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?
Calculator Solution,
P/Y = 1 I = 8 N = 3
PMT = -1,000
FV = $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
1000 1000 1000
Calculator Solution,
P/Y = 1 I = 8 N = 3
PMT = -1,000
FV = $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
1000 1000 1000
Future Value - annuity
If you invest $1,000 each year at 8%,how much
would you have after 3 years?
Mathematical Solution,
Future Value - annuity
If you invest $1,000 each year at 8%,how much
would you have after 3 years?
Mathematical Solution,
FV = PMT (FVIFA i,n )
Future Value - annuity
If you invest $1,000 each year at 8%,how much
would you have after 3 years?
Mathematical Solution,
FV = PMT (FVIFA i,n )
FV = 1,000 (FVIFA,08,3 ) (use FVIFA table,or)
Future Value - annuity
If you invest $1,000 each year at 8%,how much
would you have after 3 years?
Mathematical Solution,
FV = PMT (FVIFA i,n )
FV = 1,000 (FVIFA,08,3 ) (use FVIFA table,or)
FV = PMT (1 + i)n - 1
i
Future Value - annuity
If you invest $1,000 each year at 8%,how much
would you have after 3 years?
Mathematical Solution,
FV = PMT (FVIFA i,n )
FV = 1,000 (FVIFA,08,3 ) (use FVIFA table,or)
FV = PMT (1 + i)n - 1
i
FV = 1,000 (1.08)3 - 1 = $3246.40
,08
Future Value - annuity
If you invest $1,000 each year at 8%,how much
would you have after 3 years?
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%?
Calculator Solution,
P/Y = 1 I = 8 N = 3
PMT = -1,000
PV = $2,577.10
0 1 2 3
1000 1000 1000
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%?
Calculator Solution,
P/Y = 1 I = 8 N = 3
PMT = -1,000
PV = $2,577.10
0 1 2 3
1000 1000 1000
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%?
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,
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,
PV = PMT (PVIFA i,n )
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,
PV = PMT (PVIFA i,n )
PV = 1,000 (PVIFA,08,3 ) (use PVIFA table,or)
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,
PV = PMT (PVIFA i,n )
PV = 1,000 (PVIFA,08,3 ) (use PVIFA table,or)
1
PV = PMT 1 - (1 + i)n
i
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,
PV = PMT (PVIFA i,n )
PV = 1,000 (PVIFA,08,3 ) (use PVIFA table,or)
1
PV = PMT 1 - (1 + i)n
i
1
PV = 1000 1 - (1.08 )3 = $2,577.10
,08
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%?
Other Cash Flow Patterns
0 1 2 3
The Time Value of Money
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 an
annuity that goes on forever,
Present Value of a
Perpetuity
? When we find the PV of an annuity,
we think of the following
relationship,
Present Value of a
Perpetuity
? When we find the PV of an annuity,
we think of the following
relationship,
PV = PMT (PVIFA i,n )
Mathematically,
Mathematically,
(PVIFA i,n ) =
Mathematically,
(PVIFA i,n ) =
1 -
1
(1 + i) n
i
Mathematically,
(PVIFA i,n ) =
We said that a perpetuity is an
annuity where n = infinity,What
happens to this formula when n
gets very,very large?
1 -
1
(1 + i) n
i
When n gets very large,
When n gets very large,
1 -
1
(1 + i) n
i
When n gets very large,
this becomes zero,
1 -
1
(1 + i) n
i
When n gets very large,
this becomes zero,
So we’re left with PVIFA =
1
i
1 -
1
(1 + i) n
i
? So,the PV of a perpetuity is very
simple to find,
Present Value of a Perpetuity
PMT
i PV =
? So,the PV of a perpetuity is very
simple to find,
Present Value of a Perpetuity
What should you be willing to pay in
order to receive $10,000 annually
forever,if you require 8% per year
on the investment?
What should you be willing to pay in
order to receive $10,000 annually
forever,if you require 8% per year
on the investment?
PMT $10,000
i,08 PV = =
What should you be willing to pay in
order to receive $10,000 annually
forever,if you require 8% per year
on the investment?
PMT $10,000
i,08
= $125,000
PV = =
Ordinary Annuity
vs,
Annuity Due
$1000 $1000 $1000
4 5 6 7 8
Begin Mode vs,End Mode
1000 1000 1000
4 5 6 7 8
Begin Mode vs,End Mode
1000 1000 1000
4 5 6 7 8 year year year 5 6 7
Begin Mode vs,End Mode
1000 1000 1000
4 5 6 7 8 year year year 5 6 7
PV
in
END
Mode
Begin Mode vs,End Mode
1000 1000 1000
4 5 6 7 8 year year year 5 6 7
PV
in
END
Mode
FV
in
END
Mode
Begin Mode vs,End Mode
1000 1000 1000
4 5 6 7 8 year year year 6 7 8
Begin Mode vs,End Mode
1000 1000 1000
4 5 6 7 8 year year year 6 7 8
PV
in
BEGIN
Mode
Begin Mode vs,End Mode
1000 1000 1000
4 5 6 7 8 year year year 6 7 8
PV
in
BEGIN
Mode
FV
in
BEGIN
Mode
Earlier,we examined this
“ordinary” annuity,
Earlier,we examined this
“ordinary” annuity,
0 1 2 3
1000 1000 1000
Earlier,we examined this
“ordinary” annuity,
Using an interest rate of 8%,we
find that,
0 1 2 3
1000 1000 1000
Earlier,we examined this
“ordinary” annuity,
Using an interest rate of 8%,we
find that,
? The Future Value (at 3) is
$3,246.40,
0 1 2 3
1000 1000 1000
Earlier,we examined this
“ordinary” annuity,
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,
0 1 2 3
1000 1000 1000
What about this annuity?
? 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.”
0 1 2 3
1000 1000 1000
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?
Calculator Solution,
Mode = BEGIN P/Y = 1 I = 8
N = 3 PMT = -1,000
FV = $3,506.11
0 1 2 3
-1000 -1000 -1000
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
-1000 -1000 -1000
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?
Calculator Solution,
Mode = BEGIN P/Y = 1 I = 8
N = 3 PMT = -1,000
FV = $3,506.11
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,
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)
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)
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)
FV = PMT (1 + i)n - 1
i
(1 + i)
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)
FV = PMT (1 + i)n - 1
i
FV = 1,000 (1.08)3 - 1 = $3,506.11
,08
(1 + i)
(1.08)
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
Calculator Solution,
Mode = BEGIN P/Y = 1 I = 8
N = 3 PMT = 1,000
PV = $2,783.26
0 1 2 3
1000 1000 1000
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%?
Calculator Solution,
Mode = BEGIN P/Y = 1 I = 8
N = 3 PMT = 1,000
PV = $2,783.26
0 1 2 3
1000 1000 1000
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%?
Present Value - annuity due
Mathematical Solution,
Present Value - annuity due
Mathematical Solution,Simply compound the FV of the
ordinary annuity one more period,
Present Value - annuity due
Mathematical Solution,Simply compound the FV of the
ordinary annuity one more period,
PV = PMT (PVIFA i,n ) (1 + i)
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)
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
PV = PMT 1 - (1 + i)n
i
(1 + i)
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
PV = PMT 1 - (1 + i)n
i
1
PV = 1000 1 - (1.08 )3 = $2,783.26
,08
(1 + i)
(1.08)
? Is this an annuity?
? How do we find the PV of a cash flow
stream when all of the cash flows are
different? (Use a 10% discount rate),
Uneven Cash Flows
0 1 2 3 4
-10,000 2,000 4,000 6,000 7,000
? Sorry! There’s no quickie for this one,
We have to discount each cash flow
back separately,
0 1 2 3 4
-10,000 2,000 4,000 6,000 7,000
Uneven Cash Flows
? Sorry! There’s no quickie for this one,
We have to discount each cash flow
back separately,
0 1 2 3 4
-10,000 2,000 4,000 6,000 7,000
Uneven Cash Flows
? Sorry! There’s no quickie for this one,
We have to discount each cash flow
back separately,
0 1 2 3 4
-10,000 2,000 4,000 6,000 7,000
Uneven Cash Flows
? Sorry! There’s no quickie for this one,
We have to discount each cash flow
back separately,
0 1 2 3 4
-10,000 2,000 4,000 6,000 7,000
Uneven Cash Flows
? Sorry! There’s no quickie for this one,
We have to discount each cash flow
back separately,
0 1 2 3 4
-10,000 2,000 4,000 6,000 7,000
Uneven Cash Flows
period CF PV (CF)
0 -10,000 -10,000.00
1 2,000 1,818.18
2 4,000 3,305.79
3 6,000 4,507.89
4 7,000 4,781.09
PV of Cash Flow Stream,$ 4,412.95
0 1 2 3 4
-10,000 2,000 4,000 6,000 7,000
Annual Percentage Yield (APY)
Which is the better loan,
? 8% compounded annually,or
? 7.85% compounded quarterly?
? We can’t compare these nominal (quoted)
interest rates,because they don’t include the
same number of compounding periods per
year!
We need to calculate the APY,
Annual Percentage Yield (APY)
Annual Percentage Yield (APY)
APY = ( 1 + ) m - 1 quoted rate m
Annual Percentage Yield (APY)
? Find the APY for the quarterly loan,
APY = ( 1 + ) m - 1 quoted rate m
Annual Percentage Yield (APY)
? Find the APY for the quarterly loan,
APY = ( 1 + ) m - 1 quoted rate m
APY = ( 1 + ) 4 - 1,0785 4
Annual Percentage Yield (APY)
? Find the APY for the quarterly loan,
APY = ( 1 + ) m - 1 quoted rate m
APY = ( 1 + ) 4 - 1
APY =,0808,or 8.08%
.0785
4
Annual Percentage Yield (APY)
? Find the APY for the quarterly loan,
? The quarterly loan is more expensive than
the 8% loan with annual compounding!
APY = ( 1 + ) m - 1 quoted rate m
APY = ( 1 + ) 4 - 1
APY =,0808,or 8.08%
.0785
4
Practice Problems
Example
? Cash flows from an investment are
expected to be $40,000 per year at the
end of years 4,5,6,7,and 8,If you
require a 20% rate of return,what is
the PV of these cash flows?
Example
0 1 2 3 4 5 6 7 8
$0 0 0 0 40 40 40 40 40
? Cash flows from an investment are
expected to be $40,000 per year at the
end of years 4,5,6,7,and 8,If you
require a 20% rate of return,what is
the PV of these cash flows?
? This type of cash flow sequence is
often called a,deferred annuity.”
0 1 2 3 4 5 6 7 8
$0 0 0 0 40 40 40 40 40
How to solve,
1) Discount each cash flow back to
time 0 separately,
0 1 2 3 4 5 6 7 8
$0 0 0 0 40 40 40 40 40
How to solve,
1) Discount each cash flow back to
time 0 separately,
0 1 2 3 4 5 6 7 8
$0 0 0 0 40 40 40 40 40
How to solve,
1) Discount each cash flow back to
time 0 separately,
0 1 2 3 4 5 6 7 8
$0 0 0 0 40 40 40 40 40
How to solve,
1) Discount each cash flow back to
time 0 separately,
0 1 2 3 4 5 6 7 8
$0 0 0 0 40 40 40 40 40
How to solve,
1) Discount each cash flow back to
time 0 separately,
0 1 2 3 4 5 6 7 8
$0 0 0 0 40 40 40 40 40
How to solve,
1) Discount each cash flow back to
time 0 separately,
0 1 2 3 4 5 6 7 8
$0 0 0 0 40 40 40 40 40
How to solve,
1) Discount each cash flow back to
time 0 separately,
Or,
0 1 2 3 4 5 6 7 8
$0 0 0 0 40 40 40 40 40
2) Find the PV of the annuity,
PV,End mode; P/YR = 1; I = 20;
PMT = 40,000; N = 5
PV = $119,624
0 1 2 3 4 5 6 7 8
$0 0 0 0 40 40 40 40 40
2) Find the PV of the annuity,
PV3,End mode; P/YR = 1; I = 20;
PMT = 40,000; N = 5
PV3= $119,624
0 1 2 3 4 5 6 7 8
$0 0 0 0 40 40 40 40 40
119,624
0 1 2 3 4 5 6 7 8
$0 0 0 0 40 40 40 40 40
Then discount this single sum back to
time 0,
PV,End mode; P/YR = 1; I = 20;
N = 3; FV = 119,624;
Solve,PV = $69,226
119,624
0 1 2 3 4 5 6 7 8
$0 0 0 0 40 40 40 40 40
69,226
0 1 2 3 4 5 6 7 8
$0 0 0 0 40 40 40 40 40
119,624
? The PV of the cash flow
stream is $69,226,
69,226
0 1 2 3 4 5 6 7 8
$0 0 0 0 40 40 40 40 40
119,624
Retirement Example
? After graduation,you plan to invest
$400 per month in the stock market,
If you earn 12% per year on your
stocks,how much will you have
accumulated when you retire in 30
years?
Retirement Example
? After graduation,you plan to invest
$400 per month in the stock market,
If you earn 12% per year on your
stocks,how much will you have
accumulated when you retire in 30
years?
0 1 2 3,,, 360
400 400 400 400
0 1 2 3,,, 360
400 400 400 400
? Using your calculator,
P/YR = 12
N = 360
PMT = -400
I%YR = 12
FV = $1,397,985.65
0 1 2 3,,, 360
400 400 400 400
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?
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?
Mathematical Solution,
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?
Mathematical Solution,
FV = PMT (FVIFA i,n )
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?
Mathematical Solution,
FV = PMT (FVIFA i,n )
FV = 400 (FVIFA,01,360 ) (can’t use FVIFA table)
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?
Mathematical Solution,
FV = PMT (FVIFA i,n )
FV = 400 (FVIFA,01,360 ) (can’t use FVIFA table)
FV = PMT (1 + i)n - 1
i
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?
Mathematical Solution,
FV = PMT (FVIFA i,n )
FV = 400 (FVIFA,01,360 ) (can’t use FVIFA table)
FV = PMT (1 + i)n - 1
i
FV = 400 (1.01)360 - 1 = $1,397,985.65
,01
If you borrow $100,000 at 7% fixed
interest for 30 years in order to
buy a house,what will be your
monthly house payment?
House Payment Example
House Payment Example
If you borrow $100,000 at 7% fixed
interest for 30 years in order to
buy a house,what will be your
monthly house payment?
0 1 2 3,,, 360
? Using your calculator,
P/YR = 12
N = 360
I%YR = 7
PV = $100,000
PMT = -$665.30
0 1 2 3,,, 360
House Payment Example
Mathematical Solution,
House Payment Example
Mathematical Solution,
PV = PMT (PVIFA i,n )
House Payment Example
Mathematical Solution,
PV = PMT (PVIFA i,n )
100,000 = PMT (PVIFA,07,360 ) (can’t use PVIFA table)
House Payment Example
Mathematical Solution,
PV = PMT (PVIFA i,n )
100,000 = PMT (PVIFA,07,360 ) (can’t use PVIFA table)
1
PV = PMT 1 - (1 + i)n
i
House Payment Example
Mathematical Solution,
PV = PMT (PVIFA i,n )
100,000 = PMT (PVIFA,07,360 ) (can’t use PVIFA table)
1
PV = PMT 1 - (1 + i)n
i
1
100,000 = PMT 1 - (1.005833 )360 PMT=$665.30
,005833
Team Assignment
Upon retirement,your goal is to spend 5
years traveling around the world,To
travel in style will require $250,000 per
year at the beginning of each year,
If you plan to retire in 30 years,what are
the equal monthly payments necessary
to achieve this goal? The funds in your
retirement account will compound at
10% annually,
? How much do we need to have by
the end of year 30 to finance the
trip?
? PV30 = PMT (PVIFA,10,5) (1.10) =
= 250,000 (3.7908) (1.10) =
= $1,042,470
27 28 29 30 31 32 33 34 35
250 250 250 250 250
Using your calculator,
Mode = BEGIN
PMT = -$250,000
N = 5
I%YR = 10
P/YR = 1
PV = $1,042,466
27 28 29 30 31 32 33 34 35
250 250 250 250 250
? Now,assuming 10% annual
compounding,what monthly
payments will be required for
you to have $1,042,466 at the end
of year 30?
27 28 29 30 31 32 33 34 35
250 250 250 250 250
1,042,466
? Using your calculator,
Mode = END
N = 360
I%YR = 10
P/YR = 12
FV = $1,042,466
PMT = -$461.17
27 28 29 30 31 32 33 34 35
250 250 250 250 250
1,042,466
? So,you would have to place $461.17 in
your retirement account,which earns
10% annually,at the end of each of the
next 360 months to finance the 5-year
world tour,
