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TRANSCRIPT
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Ch. 5 - The Time Value
of Money
2002, Prentice Hall, Inc.
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What is r ?
Discount Rate
Compound rate
Interest rate
Rate of return
Required rate of return Expected rate of return
Opportunity cost of funds
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Cont.
Weighted average cost of capital (WACC)
Internal rate of return (IRR)
Yield to maturity (YTM)
Accounting rate of return
Economic rate of return Social rate of return
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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
morethan $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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I f we can measure this opportunity cost,
we can:
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I f we can measure this opportunity cost,
we can:
Translate $1 today into its equivalent in the future(compounding).
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I f we can measure this opportunity cost,
we can:
Translate $1 today into its equivalent in the future(compounding).
Today
?
Future
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I f 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
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I f 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
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Future Value
F t V l i l
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Future Value - single sums
I f you deposit $100 in an account earning 6%, how
much would you have in the account after 1 year?
F t V l i l
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Future Value - single sums
I f you deposit $100 in an account earning 6%, how
much would you have in the account after 1 year?
0 1
PV = FV =
F t V l i l
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Future Value - single sums
I f 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 =
F t V l i l
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Future Value - single sums
I f 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
F t V l i l
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Future Value - single sums
I f 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
F t V l i l
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Future Value - single sums
I f you deposit $100 in an account earning 6%, how
much would you have in the account after 5 years?
F t V l i l
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Future Value - single sums
I f you deposit $100 in an account earning 6%, how
much would you have in the account after 5 years?
0 5
PV = FV =
F t re Val e single s ms
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Future Value - single sums
I f 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
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Future Value - single sums
I f 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
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Future Value - single sums
I f 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 sumsI f you deposit $100 in an account earning 6% with
quarterly compounding, how much would you have
in the account after 5 years?
F t V l i l
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0 ?
PV = FV =
Future Value - single sumsI f you deposit $100 in an account earning 6% with
quarterly compounding, how much would you have
in the account after 5 years?
F t V l i l
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Calculator Solution:
P/Y = 4 I = 6
N = 20 PV = -100
FV = $134.68
0 20
PV = -100 FV =
Future Value - single sumsI f you deposit $100 in an account earning 6% with
quarterly compounding, how much would you have
in the account after 5 years?
F t V l i l
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Calculator Solution:
P/Y = 4 I = 6
N = 20 PV = -100
FV = $134.68
0 20
PV = -100 FV = 134.68
Future Value - single sumsI f 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 sums
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Future Value - single sumsI f 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
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Future Value - single sumsI f 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 =
Future Value single sums
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Calculator Solution:
P/Y = 12 I = 6
N = 60 PV = -100
FV = $134.89
0 60
PV = -100 FV =
Future Value - single sumsI f 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
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Calculator Solution:
P/Y = 12 I = 6
N = 60 PV = -100
FV = $134.89
0 60
PV = -100 FV = 134.89
Future Value - single sumsI f 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
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Mathematical Solution:
FV = PV (FVIF i, 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 sumsI f you deposit $100 in an account earning 6% with
monthly 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?
F V l i di
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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 sumsI f you receive $100 one year from now, what is the
PV of that $100 if your opportuni ty cost is 6%?
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0 ?
PV = FV =
Present Value - single sumsI f you receive $100 one year from now, what is the
PV of that $100 if your opportuni ty cost is 6%?
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Calculator Solution:
P/Y = 1 I = 6
N = 1 FV = 100
PV = -94.34
0 1
PV = FV = 100
Present Value - single sumsI f you receive $100 one year from now, what is the
PV of that $100 if your opportuni ty cost is 6%?
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Calculator Solution:
P/Y = 1 I = 6
N = 1 FV = 100
PV = -94.34
PV = -94.34 FV = 100
0 1
Present Value - single sumsI f you receive $100 one year from now, what is the
PV of that $100 if your opportuni ty cost is 6%?
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Mathematical Solution:
PV = FV (PVIFi, 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 sumsI f you receive $100 one year from now, what is the
PV of that $100 if your opportuni ty cost is 6%?
P t V l i l
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Present Value - single sumsI f you receive $100 five years from now, what is the
PV of that $100 if your opportuni ty cost is 6%?
P t V l i l
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0 ?
PV = FV =
Present Value - single sumsI f you receive $100 five years from now, what is the
PV of that $100 if your opportuni ty cost is 6%?
Present Value single sums
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Calculator Solution:
P/Y = 1 I = 6
N = 5 FV = 100
PV = -74.73
0 5
PV = FV = 100
Present Value - single sumsI f you receive $100 five years from now, what is the
PV of that $100 if your opportuni ty cost is 6%?
Present Value single sums
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Calculator Solution:
P/Y = 1 I = 6
N = 5 FV = 100
PV = -74.73
Present Value - single sumsI f you receive $100 five years from now, what is the
PV of that $100 if your opportuni ty cost is 6%?
0 5
PV = -74.73 FV = 100
Present Value single sums
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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 sumsI f you receive $100 five years from now, what is the
PV of that $100 if your opportuni ty cost is 6%?
0 5
PV = -74.73 FV = 100
Present Value single sums
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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%?
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Present Value single sums
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Calculator Solution:
P/Y = 1 I = 7
N = 15 FV = 1,000
PV = -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 = FV = 1000
Present Value single sums
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Calculator Solution:
P/Y = 1 I = 7
N = 15 FV = 1,000
PV = -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
Present Value single sums
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Mathematical Solution:
PV = FV (PVIFi, 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
Present Value single sums
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Present Value - single sumsI f 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
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0 5
PV = FV =
Present Value - single sumsI f 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
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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 sumsI f 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
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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 sumsI f 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
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gSuppose you placed $100 in an account that pays
9.6% interest, compounded monthly. How long wil l
i t take for your account to grow to $500?
0
PV = FV =
Present Value - single sums
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Calculator Solution:
P/Y = 12 FV = 500
I = 9.6 PV = -100
N = 202 months
gSuppose you placed $100 in an account that pays
9.6% interest, compounded monthly. How long wil l
i t take for your account to grow to $500?
0 ?
PV = -100 FV = 500
Present Value - single sums
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gSuppose you placed $100 in an account that pays
9.6% interest, compounded monthly. How long wil l
i t 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
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H int for single sum problems:
In every single sum future value andpresent 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!
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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 of equalcash
flows, occurring at the endof each
period.
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Annuity: a sequence of equalcash
flows, occurring at the end of each
period.
0 1 2 3 4
Annuities
E l f A iti
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Examples of Annuities:
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.
E l f A iti
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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 - annui ty
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yI f you invest $1,000 each year at 8%, how much
would you have after 3 years?
Future Value - annui ty
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0 1 2 3
yI f you invest $1,000 each year at 8%, how much
would you have after 3 years?
Future Value - annui ty
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Calculator Solution:
P/Y = 1 I = 8 N = 3
PMT = -1,000
FV = $3,246.40
yI f 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 - annui ty
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Calculator Solution:
P/Y = 1 I = 8 N = 3
PMT = -1,000
FV = $3,246.40
yI f 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 - annui ty
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yI f you invest $1,000 each year at 8%, how much
would you have after 3 years?
Future Value - annui ty
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Mathematical Solution:
yI f you invest $1,000 each year at 8%, how much
would you have after 3 years?
Future Value - annui ty
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Mathematical Solution:
FV = PMT (FVIFA i, n)
yI f you invest $1,000 each year at 8%, how much
would you have after 3 years?
Future Value - annui ty
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Mathematical Solution:
FV = PMT (FVIFA i, n)
FV = 1,000 (FVIFA .08, 3) (use FVIFA table, or)
yI f you invest $1,000 each year at 8%, how much
would you have after 3 years?
Future Value - annui ty
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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
I f you invest $1,000 each year at 8%, how much
would you have after 3 years?
Future Value - annui ty
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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
I f you invest $1,000 each year at 8%, how much
would you have after 3 years?
Present Value - annuity
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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
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0 1 2 3
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
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Calculator Solution:
P/Y = 1 I = 8 N = 3
PMT = -1,000
PV = $2,577.10
0 1 2 3
1000 1000 1000
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
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Calculator Solution:
P/Y = 1 I = 8 N = 3
PMT = -1,000
PV = $2,577.10
0 1 2 3
1000 1000 1000
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
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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
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Mathematical Solution:
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
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Mathematical Solution:
PV = PMT (PVIFA i, n)
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
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Mathematical Solution:
PV = PMT (PVIFA i, n)
PV = 1,000 (PVIFA .08, 3) (use PVIFA table, or)
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
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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
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
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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
What 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 F low 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 an
annuitythat goes on forever.
Present Value of a
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Present Value of a
Perpetuity
When we find the PV of an annuity,
we think of the following
relationship:
Present Value of a
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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)
Mathematically,
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Mathematically,
Mathematically,
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at e at ca y,
(PVIFA i, n ) =
Mathematically,
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y,
(PVIFA i, n ) = 1 -
1
(1 + i)n
i
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When n gets very large,
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g y g
When n gets very large,
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g y g
1 -1
(1 + i)n
i
When n gets very large,
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this becomes zero.1 -
1(1 + i)
n
i
When n gets very large,
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this becomes zero.
So were left with PVIFA =
1
i
1 -1
(1 + i)n
i
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Cont.
FV = PV + PV (r)
x+100 = x + x (r)
x (r) = 100
x =
PV =
100
r
C
r
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What should you be willing to pay in
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What should you be willing to pay in
order to receive $10,000annually
forever, if you require 8%per year
on the investment?
PMT $10,000i .08
PV = =
What should you be willing to pay in
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What should you be willing to pay in
order to receive $10,000annually
forever, if you require 8%per year
on the investment?
PMT $10,000i .08
= $125,000
PV = =
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Ordinary Annui ty
vs.
Annui ty Due
$1000 $1000 $1000
4 5 6 7 8
Begin Mode vs. End Mode
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1000 1000 1000
4 5 6 7 8
Begin Mode vs. End Mode
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1000 1000 1000
4 5 6 7 8year year year
5 6 7
Begin Mode vs. End Mode
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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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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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1000 1000 1000
4 5 6 7 8year year year
6 7 8
Begin Mode vs. End Mode
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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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1000 1000 1000
4 5 6 7 8year year year
6 7 8
PV
inBEGIN
Mode
FV
inBEGIN
Mode
Earlier, we examined this
di i
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ordinary annuity:
Earlier, we examined this
di i
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ordinary annuity:
0 1 2 3
1000 1000 1000
Earlier, we examined this
di i
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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
di i
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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
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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
beginningof each year, ratherthan at the endof each year.
This is an annuity due.
0 1 2 3
1000 1000 1000
Future Value - annui ty dueI f you invest $1,000 at the beginning of each of the
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0 1 2 3
y g g
next 3 years at 8%, how much would you have at the
end of year 3?
Future Value - annui ty dueI f you invest $1,000 at the beginning of each of the
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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
y g g
next 3 years at 8%, how much would you have at the
end of year 3?
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Future Value - annui ty dueI f you invest $1,000 at the beginning of each of the
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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 - annui ty dueI f you invest $1,000 at the beginning of each of the
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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 - annui ty dueI f you invest $1,000 at the beginning of each of the
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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 - annui ty dueI f you invest $1,000 at the beginning of each of the
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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 - annui ty dueI f you invest $1,000 at the beginning of each of the
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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 dueWhat is the PV of $1,000 at the beginning of each of
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$ , g g
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 of
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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
, g g
the next 3 years, if your opportunity cost is 8%?
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Present Value - annuity due
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Mathematical Solution:
Present Value - annuity due
M h i l S l i
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Mathematical Solution: Simply compound the FV of theordinary annuity one more period:
Present Value - annuity due
M th ti l S l ti
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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
M th ti l S l ti
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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
M th ti l S l ti
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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)
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Uneven Cash F lows
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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 F lows
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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 F lows
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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 F lows
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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 F lows
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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
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-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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ua e ce age e d ( )
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 peryear!
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 rate
m
Annual Percentage Yield (APY)
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Find the APY for the quarterly loan:
APY= (1 + ) m - 1quoted rate
m
Annual Percentage Yield (APY)
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Find the APY for the quarterly loan:
APY= (1 + ) m - 1quoted rate
m
APY = (1 + ) 4 - 1.07854
Annual Percentage Yield (APY)
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Find the APY for the quarterly loan:
APY= (1 + ) m - 1quoted rate
m
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 rate
m
APY = (1 + ) 4 - 1
APY = .0808, or 8.08%
.0785
4
Practice Problems
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Practice Problems
Example
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Cash flows from an investment are
expected to be $40,000per 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 $40,000per 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
After graduation, you plan to invest
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After graduation, you plan to invest
$400per monthin the stock market.If you earn 12%per yearon your
stocks, how much will you have
accumulated when you retire in 30years?
Retirement Example
After graduation, you plan to invest
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After graduation, you plan to invest
$400per 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
0 1 360
400 400 400 400
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Using your calculator,
P/YR = 12
N = 360
PMT = -400I%YR = 12
FV = $1,397,985.65
0 1 2 3 . . . 360
Retirement ExampleI f 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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y y
the end of year 30?
Retirement ExampleI f 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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y y
the end of year 30?Mathematical Solution:
Retirement ExampleI f 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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y y
the end of year 30?Mathematical Solution:
FV = PMT (FVIFA i, n)
Retirement ExampleI f 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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the end of year 30?
Mathematical Solution:
FV = PMT (FVIFA i, n)FV = 400 (FVIFA .01, 360) (cant use FVIFA table)
Retirement ExampleI f 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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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 ExampleI f 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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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
If you borrow $100,000at 7%fixed
House Payment Example
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y ,
interest for 30yearsin order tobuy a house, what will be your
monthly house payment?
House Payment Example
If you borrow $100,000at 7%fixed
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y
interest for 30years in order tobuy a house, what will be your
monthly house payment?
0 1 2 3 360
? ? ? ?
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0 1 2 3 . . . 360
0 1 2 3 360
? ? ? ?
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Using your calculator,
P/YR = 12
N = 360
I%YR = 7PV = $100,000
PMT = -$665.30
0 1 2 3 . . . 360
House Payment Example
Mathematical Solution:
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House Payment Example
Mathematical Solution:
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PV = PMT (PVIFA i, n)
House Payment Example
Mathematical Solution:
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PV = PMT (PVIFA i, n)
100,000 = PMT (PVIFA .07, 360) (cant use PVIFA table)
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House Payment Example
Mathematical Solution:
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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
Team Assignment
Upon retirement your goal is to spend 5
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Upon retirement, your goal is to spend 5
years traveling around the world. To
travel in style will require $250,000per
year at the beginningof each year.
If you plan to retire in 30 years, what are
the equal monthlypayments necessary
to achieve this goal? The funds in yourretirement account will compound at
10%annually.
27 28 29 30 31 32 33 34 35
250 250 250 250 250
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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
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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
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Now, assuming 10% annualcompounding, what monthly
payments will be required for
you to have $1,042,466at the endof year 30?
1,042,466
27 28 29 30 31 32 33 34 35
250 250 250 250 250
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Using your calculator,
Mode = ENDN = 360
I%YR = 10
P/YR = 12
FV = $1,042,466
PMT = -$461.17
1,042,466
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So, you would have to place $461.17in
your retirement account, which earns