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Chapter 5 - The TimeValue of Money
2005, Pearson Prentice Hall
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The Time Value of Money
Compounding and
Discounting Single Sums
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We know that receiving $1 today is worth
more than $1 in the future. This is due
toopportunity costs.The opportunity cost of receiving $1 in
the future is theinterestwe could have
earned if we had received the $1 sooner.Today Future
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If we can measure this opportunitycost, we can:
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If we can measure this opportunitycost, we can:
Translate $1 today into its equivalent in the future(compounding).
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If we can measure this opportunitycost, we can:
Translate $1 today into its equivalent in the future(compounding).
Today
?
Future
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If we can measure this opportunitycost, we can:
Translate $1 today into its equivalent in the future(compounding).
Translate $1 in the future into its equivalent today
(discounting).
Today
?
Future
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If we can measure this opportunitycost, 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
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Compound Interest
and Future Value
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Future Value - single sums
If you deposit $100 in an account earning 6%, how
much would you have in the account after 1 year?
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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 =
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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 (FVIFi, 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
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Future Value - single sums
If you deposit $100 in an account earning 6%, how
much would you have in the account after 5 years?
i
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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 =
F V l i l
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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
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Future Value - single sumsIf you deposit $100 in an account earning 6% with
quarterly compounding, how much would you have
in the account after 5 years?
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0 ?
PV = FV =
Future Value - single sumsIf you deposit $100 in an account earning 6% with
quarterly compounding, how much would you have
in the account after 5 years?
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Mathematical Solution:
FV = PV (FVIF i, n)
FV = 100 (FVIF .015, 20) (cant 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 sumsIf you deposit $100 in an account earning 6% with
quarterly compounding, how much would you have
in the account after 5 years?
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Future Value - single sumsIf you deposit $100 in an account earning 6% with
monthly compounding, how much would you have
in the account after 5 years?
F t V l i l
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Future Value - single sumsIf you deposit $100 in an account earning 6% withmonthly compounding, how much would you have
in the account after 5 years?
0 ?
PV = FV =
F t V l i l
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Mathematical Solution:
FV = PV (FVIFi, n
)
FV = 100 (FVIF .005, 60) (cant 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 sumsIf you deposit $100 in an account earning 6% withmonthly compounding, how much would you have
in the account after 5 years?
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Future Value - continuous compoundingWhat is the FV of $1,000 earning 8% with
continuous compounding, after 100 years?
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Future Value - continuous compoundingWhat is the FV of $1,000 earning 8% with
continuous compounding, after 100 years?
0 ?
PV = FV =
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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 compoundingWhat is the FV of $1,000 earning 8% with
continuous compounding, after 100 years?
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0 100
PV = -1000 FV = $2.98m
Future Value - continuous compoundingWhat 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
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Present Value
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Present Value - single sumsIf you receive $100 one year from now, what is the
PV of that $100 if your opportunity cost is 6%?
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0 ?
PV = FV =
Present Value - single sumsIf you receive $100 one year from now, what is the
PV of that $100 if your opportunity cost is 6%?
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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 sumsIf you receive $100 one year from now, what is the
PV of that $100 if your opportunity cost is 6%?
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Present Value - single sumsIf you receive $100 five years from now, what is the
PV of that $100 if your opportunity cost is 6%?
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0 ?
PV = FV =
Present Value - single sumsIf you receive $100 five years from now, what is the
PV of that $100 if your opportunity cost is 6%?
i
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Mathematical Solution:
PV = FV (PVIFi, 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 sumsIf 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
P V l i l
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Present Value - single sumsWhat is the PV of $1,000 to be received 15 years
from now if your opportunity cost is 7%?
P V l i l
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0 15
PV = FV =
Present Value - single sumsWhat is the PV of $1,000 to be received 15 years
from now if your opportunity cost is 7%?
P t V l i l
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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 sumsWhat 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
P t V l i l
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Present Value - single sumsIf you sold land for $11,933 that you bought 5 years
ago for $5,000, what is your annual rate of return?
P t V l i l
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0 5
PV = FV =
Present Value - single sumsIf you sold land for $11,933 that you bought 5 years
ago for $5,000, what is your annual rate of return?
P t V l i l
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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 sumsIf you sold land for $11,933 that you bought 5 years
ago for $5,000, what is your annual rate of return?
P t V l i l
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Present Value - single sumsSuppose you placed $100 in an account that pays9.6% interest, compounded monthly. How long
will it take for your account to grow to $500?
0
PV = FV =
P t V l i l
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Present Value - single sumsSuppose you placed $100 in an account that pays9.6% 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)Nln 5 = N ln (1.008)
1.60944 = .007968 N N = 202 months
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Hint for single sum problems:
In every single sum present value andfuture value problem, there are fourvariables:
FV, PV, i and n. When doing problems, you will be given
three variables and you will solve for the
fourth variable. Keeping this in mind makes solving time
value problems much easier!
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The Time Value of Money
Compounding and Discounting
Cash Flow Streams
0 1 2 3 4
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Annuities
Annuity: a sequence ofequal cash
flows, occurring at the end of each
period.
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Annuity: a sequence ofequal cash
flows, occurring at the end of each
period.
0 1 2 3 4
Annuities
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If you buy a bond, you willreceive 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
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Future Value - annuityIf you invest $1,000 each year at 8%, how much
would you have after 3 years?
Future Value annuity
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0 1 2 3
Future Value - annuityIf you invest $1,000 each year at 8%, how much
would you have after 3 years?
Future Value - annuity
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Mathematical Solution:
Future Value - annuityIf you invest $1,000 each year at 8%, how much
would you have after 3 years?
Future Value - annuity
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Mathematical Solution:
FV = PMT (FVIFA i, n)
Future Value - annuityIf you invest $1,000 each year at 8%, how much
would you have after 3 years?
Future Value - annuity
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Mathematical Solution:
FV = PMT (FVIFA i, n)
FV = 1,000 (FVIFA .08, 3) (use FVIFA table, or)
Future Value - annuityIf you invest $1,000 each year at 8%, how much
would you have after 3 years?
Future Value - annuity
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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 - annuityIf you invest $1,000 each year at 8%, how much
would you have after 3 years?
Future Value - annuity
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Mathematical Solution:
FV = PMT (FVIFA i, n)
FV = 1,000 (FVIFA .08, 3) (use FVIFA table, or)
FV = PMT (1 + i)n - 1
iFV = 1,000 (1.08)3 - 1 = $3246.40
.08
Future Value - annuityIf you invest $1,000 each year at 8%, how much
would you have after 3 years?
Present Value - annuity
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Present Value - annuityWhat 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
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0 1 2 3
Present Value annuityWhat 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
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Mathematical Solution:
Present Value annuityWhat 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
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Mathematical Solution:
PV = PMT (PVIFA i, n)
Present Value annuityWhat 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
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Mathematical Solution:
PV = PMT (PVIFA i, n)
PV = 1,000 (PVIFA .08, 3) (use PVIFA table, or)
Present Value annuityWhat 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
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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 annuityWhat 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
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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 annuityWhat is the PV of $1,000 at the end of each of the
next 3 years, if the opportunity cost is 8%?
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Other Cash Flow Patterns
0 1 2 3
The Time Value of Money
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Perpetuities
Suppose you will receive a fixed
payment every period (month, year,
etc.) forever. This is an example ofa perpetuity.
You can think of a perpetuity as anannuity that goes on forever.
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Present Value of a
Perpetuity
When we find the PV of an annuity,
we think of the following
relationship:
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Present Value of a
Perpetuity
When we find the PV of an annuity,
we think of the following
relationship:
PV = PMT (PVIFA i, n)
M th ti ll
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Mathematically,
Mathematicall
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Mathematically,
(PVIFA i, n ) =
Mathematically
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Mathematically,
(PVIFA i, n ) = 1 -
1
(1 + i)n
i
Mathematically
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Mathematically,
(PVIFA i, n ) =
We said that a perpetuity is an
annuity where n = infinity. What
happens to this formula when ngets very, very large?
1 -
1
(1 + i)n
i
When n gets very large
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When n gets very large,
When n gets very large
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When n gets very large,
1 -1
(1 + i)n
i
When n gets very large
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When n gets very large,
this becomes zero.1 -
1(1 + i)n
i
When n gets very large
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When n gets very large,
this becomes zero.
So were left with PVIFA =
1
i
1 -1
(1 + i)n
i
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So, the PV of a perpetuity is verysimple to find:
Present Value of a Perpetuity
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PMT
iPV =
So, the PV of a perpetuity is verysimple to find:
Present Value of a Perpetuity
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What should you be willing to pay in
order to receive $10,000 annually
forever, if you require 8% per year
on the investment?
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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 $10,000i .08
PV = =
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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 $10,000i .08
= $125,000
PV = =
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Ordinary Annuity
vs.
Annuity Due
$1000 $1000 $1000
4 5 6 7 8
Begin Mode vs. End Mode
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Begin Mode vs. End Mode
1000 1000 1000
4 5 6 7 8
Begin Mode vs. End Mode
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Begin Mode vs. End Mode
1000 1000 1000
4 5 6 7 8year year year
5 6 7
Begin Mode vs. End Mode
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Begin Mode vs. End Mode
1000 1000 1000
4 5 6 7 8year year year
5 6 7
PV
inEND
Mode
Begin Mode vs. End Mode
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g
1000 1000 1000
4 5 6 7 8year year year
5 6 7
PV
inEND
Mode
FV
inEND
Mode
Begin Mode vs. End Mode
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g
1000 1000 1000
4 5 6 7 8year year year
6 7 8
Begin Mode vs. End Mode
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g
1000 1000 1000
4 5 6 7 8year year year
6 7 8
PV
inBEGIN
Mode
Begin Mode vs. End Mode
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g
1000 1000 1000
4 5 6 7 8year year year
6 7 8
PV
inBEGIN
Mode
FV
inBEGIN
Mode
Earlier, we examined this
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,
ordinary annuity:
Earlier, we examined this
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,
ordinary annuity:
0 1 2 3
1000 1000 1000
Earlier, we examined this
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,
ordinary annuity:
Using an interest rate of 8%, wefind that:
0 1 2 3
1000 1000 1000
Earlier, we examined this
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,
ordinary annuity:
Using an interest rate of 8%, wefind that:
The Future Value (at 3) is
$3,246.40.
0 1 2 3
1000 1000 1000
Earlier, we examined this
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,
ordinary annuity:
Using an interest rate of 8%, wefind 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?
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Same 3-year time line,
Same 3 $1000 cash flows, but
The cash flows occur at the
beginning of each year, ratherthan at the end of each year.
This is an annuity due.
0 1 2 31000 1000 1000
Future Value - annuity dueIf you invest $1 000 at the beginning of each of the
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0 1 2 3
If you invest $1,000 at the beginning of each of thenext 3 years at 8%, how much would you have at
the end of year 3?
Future Value - annuity dueIf you invest $1 000 at the beginning of each of the
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Calculator Solution:
Mode = BEGIN P/Y = 1 I = 8N = 3 PMT = -1,000
FV = $3,506.11
0 1 2 3
-1000 -1000 -1000
If you invest $1,000 at the beginning of each of thenext 3 years at 8%, how much would you have at
the end of year 3?
Future Value - annuity dueIf you invest $1 000 at the beginning of each of the
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0 1 2 3
-1000 -1000 -1000
If you invest $1,000 at the beginning of each of thenext 3 years at 8%, how much would you have at
the end of year 3?
Calculator Solution:
Mode = BEGIN P/Y = 1 I = 8N = 3 PMT = -1,000
FV = $3,506.11
Future Value - annuity dueIf you invest $1 000 at the beginning of each of the
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If you invest $1,000 at the beginning of each of thenext 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 dueIf you invest $1 000 at the beginning of each of the
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If you invest $1,000 at the beginning of each of thenext 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 dueIf you invest $1 000 at the beginning of each of the
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If you invest $1,000 at the beginning of each of thenext 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 dueIf you invest $1 000 at the beginning of each of the
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If you invest $1,000 at the beginning of each of thenext 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 dueIf you invest $1 000 at the beginning of each of the
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If you invest $1,000 at the beginning of each of thenext 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 dueWhat is the PV of $1 000 at the beginning of each
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What is the PV of $1,000 at the beginning of eachof the next 3 years, if your opportunity cost is 8%?
0 1 2 3
Present Value - annuity dueWhat is the PV of $1 000 at the beginning of each
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Calculator Solution:
Mode = BEGIN P/Y = 1 I = 8N = 3 PMT = 1,000
PV = $2,783.26
0 1 2 3
1000 1000 1000
What is the PV of $1,000 at the beginning of eachof the next 3 years, if your opportunity cost is 8%?
Present Value - annuity dueWhat is the PV of $1 000 at the beginning of each
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Calculator Solution:
Mode = BEGIN P/Y = 1 I = 8N = 3 PMT = 1,000
PV = $2,783.26
0 1 2 3
1000 1000 1000
What is the PV of $1,000 at the beginning of eachof the next 3 years, if your opportunity cost is 8%?
Present Value - annuity due
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Mathematical Solution:
Present Value - annuity due
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Mathematical Solution: Simply compound the FV of theordinary annuity one more period:
Present Value - annuity due
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Mathematical Solution: Simply compound the FV of theordinary annuity one more period:
PV = PMT (PVIFA i, n) (1 + i)
Present Value - annuity due
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Mathematical Solution: Simply compound the FV of theordinary 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
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Mathematical Solution: Simply compound the FV of theordinary annuity one more period:
PV = PMT (PVIFA i, n) (1 + i)
PV = 1,000 (PVIFA .08, 3) (1.08) (use PVIFA table, or)
1PV = PMT 1 - (1 + i)n
i(1 + i)
Present Value - annuity due
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Mathematical Solution: Simply compound the FV of theordinary annuity one more period:
PV = PMT (PVIFA i, n) (1 + i)
PV = 1,000 (PVIFA .08, 3) (1.08) (use PVIFA table, or)
1PV = PMT 1 - (1 + i)n
i
1
PV = 1000 1 - (1.08 )3 = $2,783.26
.08
(1 + i)
(1.08)
Uneven Cash Flows
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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.)
0 1 2 3 4
-10,000 2,000 4,000 6,000 7,000
Uneven Cash Flows
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Sorry! Theres 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
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Sorry! Theres 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
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Sorry! Theres 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
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Sorry! Theres 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
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Sorry! Theres 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
-10,000 2,000 4,000 6,000 7,000
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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 4,507.894 7,000 4,781.09
PV of Cash Flow Stream: $ 4,412.95
0 1 2 3 4
Annual Percentage Yield (APY)
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Annual Percentage Yield (APY)
Which is the better loan: 8% compounded annually, or
7.85% compounded quarterly?
We cant compare these nominal (quoted)
interest rates, because they dont include the
same number of compounding periods per
year!
We need to calculate the APY.
Annual Percentage Yield (APY)
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Annual Percentage Yield (APY)
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APY = ( 1 + ) m - 1quoted ratem
Annual Percentage Yield (APY)
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Find the APY for the quarterly loan:
APY = ( 1 + ) m - 1quoted ratem
Annual Percentage Yield (APY)
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Find the APY for the quarterly loan:
APY = ( 1 + ) m - 1quoted ratem
APY = ( 1 + ) 4 - 1.07854
Annual Percentage Yield (APY)
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Find the APY for the quarterly loan:
APY = ( 1 + ) m - 1quoted ratem
APY = ( 1 + ) 4 - 1
APY = .0808, or 8.08%
.0785
4
Annual Percentage Yield (APY)
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Find the APY for the quarterly loan:
The quarterly loan is more expensive than
the 8% loan with annual compounding!
APY = ( 1 + ) m - 1quoted ratem
APY = ( 1 + ) 4 - 1
APY = .0808, or 8.08%
.0785
4
The Effective Annual Rate
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122
EAR can be calculated for any compounding
period using the formula
Effect of more frequent compounding is greater
at higher interest rates
mnom
m
kEAR
1
Practice Problems
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Practice Problems
Example
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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
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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?
$0 0 0 0 40 40 40 40 40
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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119,624
0 1 2 3 4 5 6 7 8
$0 0 0 0 40 40 40 40 40
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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
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69,226
0 1 2 3 4 5 6 7 8
119,624
$0 0 0 0 40 40 40 40 40
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The PV of the cash flow
stream is $69,226.
69,226
0 1 2 3 4 5 6 7 8
119,624
Retirement Example
Aft d ti l t i t
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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 30years?
Retirement Example
Aft d ti l t i t
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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 30years?
0 1 2 3 . . . 360
400 400 400 400
400 400 400 400
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0 1 2 3 . . . 360
Retirement ExampleIf you invest $400 at the end of each month for the
next 30 years at 12% how much would you have at
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next 30 years at 12%, how much would you have at
the end of year 30?Mathematical Solution:
Retirement ExampleIf you invest $400 at the end of each month for the
next 30 years at 12% how much would you have at
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next 30 years at 12%, how much would you have at
the end of year 30?Mathematical Solution:
FV = PMT (FVIFA i, n)
Retirement ExampleIf you invest $400 at the end of each month for the
next 30 years at 12% how much would you have at
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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) (cant use FVIFA table)
Retirement ExampleIf you invest $400 at the end of each month for the
next 30 years at 12% how much would you have at
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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) (cant use FVIFA table)
FV = PMT (1 + i)n - 1
i
Retirement ExampleIf you invest $400 at the end of each month for the
next 30 years at 12% how much would you have at
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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) (cant use FVIFA table)
FV = PMT (1 + i)n - 1
i
FV = 400 (1.01)360 - 1 = $1,397,985.65
01
House Payment Example
If you borrow $100 000 at 7% fixed
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If you borrow $100,000 at 7% fixed
interest for 30 years in order tobuy a house, what will be your
monthly house payment?
? ? ? ?
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0 1 2 3 . . . 360
House Payment Example
Mathematical Solution:
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Mathematical Solution:
House Payment Example
Mathematical Solution:
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Mathematical Solution:
PV = PMT (PVIFA i, n)
House Payment Example
Mathematical Solution:
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Mathematical Solution:
PV = PMT (PVIFA i, n)
100,000 = PMT (PVIFA .07, 360) (cant use PVIFA table)
House Payment Example
Mathematical Solution:
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Mathematical Solution:
PV = PMT (PVIFA i, n)
100,000 = PMT (PVIFA .07, 360) (cant use PVIFA table)
1PV = PMT 1 - (1 + i)n
i
House Payment Example
Mathematical Solution:
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Mathematical Solution:
PV = PMT (PVIFA i, n)
100,000 = PMT (PVIFA .07, 360) (cant use PVIFA table)
1PV = PMT 1 - (1 + i)n
i
1
100,000 = PMT 1 - (1.005833 )360 PMT=$665.30
.005833
Exercise
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1) If you deposit $10,000 in a bank accountthat pay 10% interest annually, how much will
it be in your account after 5 years?
Compounded annually Compounded monthly
Compounded quarterly
Compounded semi-annually
Exercise
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What is the value of security that will pay$5,000 in 20 years if securities of equal risk
pay 7% annually?
Compounded annually Compounded monthly
Compounded quarterly
Compounded semi-annually