2002, Prentice Hall, Inc. Ch. 9: Capital Budgeting Decision Criteria.
Ch. 6 - The Time Value of Money , Prentice Hall, Inc.
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Transcript of Ch. 6 - The Time Value of Money , Prentice Hall, Inc.
The Time Value of Money
Compounding and Compounding and Discounting Single SumsDiscounting Single Sums
We know that receiving $1 today is worth We know that receiving $1 today is worth moremore than $1 in the future. This is due than $1 in the future. This is due toto opportunity costsopportunity costs..
The opportunity cost of receiving $1 in The opportunity cost of receiving $1 in the future is thethe future is the interestinterest we could have we could have earned if we had received the $1 sooner.earned if we had received the $1 sooner.
Today Future
If we can measure this opportunity cost, we can:
Translate $1 today into its equivalent in the futureTranslate $1 today into its equivalent in the future (compounding)(compounding)..
If we can measure this opportunity cost, we can:
Translate $1 today into its equivalent in the futureTranslate $1 today into its equivalent in the future (compounding)(compounding)..
Today
?
Future
If we can measure this opportunity cost, we can:
Translate $1 today into its equivalent in the futureTranslate $1 today into its equivalent in the future (compounding)(compounding)..
Translate $1 in the future into its equivalent todayTranslate $1 in the future into its equivalent today (discounting)(discounting)..
Today
?
Future
If we can measure this opportunity cost, we can:
Translate $1 today into its equivalent in the futureTranslate $1 today into its equivalent in the future (compounding)(compounding)..
Translate $1 in the future into its equivalent todayTranslate $1 in the future into its equivalent today (discounting)(discounting)..
?
Today Future
Today
?
Future
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 =PV = FV = 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:Calculator Solution:
P/Y = 1P/Y = 1 I = 6I = 6
N = 1 N = 1 PV = -100 PV = -100
FV = FV = $106$106
00 1 1
PV = -100PV = -100 FV = 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:Calculator Solution:
P/Y = 1P/Y = 1 I = 6I = 6
N = 1 N = 1 PV = -100 PV = -100
FV = FV = $106$106
00 1 1
PV = -100PV = -100 FV = FV = 106106
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:Mathematical Solution:
FV = PV (FVIF FV = PV (FVIF i, ni, n ))
FV = 100 (FVIF FV = 100 (FVIF .06, 1.06, 1 ) (use FVIF table, or)) (use FVIF table, or)
FV = PV (1 + i)FV = PV (1 + i)nn
FV = 100 (1.06)FV = 100 (1.06)1 1 = = $106$106
00 1 1
PV = -100PV = -100 FV = FV = 106106
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?
00 5 5
PV =PV = FV = 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:Calculator Solution:
P/Y = 1P/Y = 1 I = 6I = 6
N = 5 N = 5 PV = -100 PV = -100
FV = FV = $133.82$133.82
00 5 5
PV = -100PV = -100 FV = 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:Calculator Solution:
P/Y = 1P/Y = 1 I = 6I = 6
N = 5 N = 5 PV = -100 PV = -100
FV = FV = $133.82$133.82
00 5 5
PV = -100PV = -100 FV = FV = 133.133.8282
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:Mathematical Solution:
FV = PV (FVIF FV = PV (FVIF i, ni, n ))
FV = 100 (FVIF FV = 100 (FVIF .06, 5.06, 5 ) (use FVIF table, or)) (use FVIF table, or)
FV = PV (1 + i)FV = PV (1 + i)nn
FV = 100 (1.06)FV = 100 (1.06)5 5 = = $$133.82133.82
00 5 5
PV = -100PV = -100 FV = FV = 133.133.8282
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?
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?
0 ?
PV =PV = FV = FV =
Calculator Solution:Calculator Solution:
P/Y = 4P/Y = 4 I = 6I = 6
N = 20 N = 20 PV = PV = -100-100
FV = FV = $134.68$134.68
00 20 20
PV = -100PV = -100 FV = 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?
Calculator Solution:Calculator Solution:
P/Y = 4P/Y = 4 I = 6I = 6
N = 20 N = 20 PV = PV = -100-100
FV = FV = $134.68$134.68
00 20 20
PV = -100PV = -100 FV = FV = 134.134.6868
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?
Mathematical Solution:Mathematical Solution:
FV = PV (FVIF FV = PV (FVIF i, ni, n ))
FV = 100 (FVIF FV = 100 (FVIF .015, 20.015, 20 ) ) (can’t use FVIF table)(can’t use FVIF table)
FV = PV (1 + i/m) FV = PV (1 + i/m) m x nm x n
FV = 100 (1.015)FV = 100 (1.015)20 20 = = $134.68$134.68
00 20 20
PV = -100PV = -100 FV = FV = 134.134.6868
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?
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?
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?
0 ?
PV =PV = FV = FV =
Calculator Solution:Calculator Solution:
P/Y = 12P/Y = 12 I = 6I = 6
N = 60 N = 60 PV = PV = -100-100
FV = FV = $134.89$134.89
00 60 60
PV = -100PV = -100 FV = FV =
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?
Calculator Solution:Calculator Solution:
P/Y = 12P/Y = 12 I = 6I = 6
N = 60 N = 60 PV = PV = -100-100
FV = FV = $134.89$134.89
00 60 60
PV = -100PV = -100 FV = FV = 134.134.8989
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?
Mathematical Solution:Mathematical Solution:
FV = PV (FVIF FV = PV (FVIF i, ni, n ))
FV = 100 (FVIF FV = 100 (FVIF .005, 60.005, 60 ) ) (can’t use FVIF table)(can’t use FVIF table)
FV = PV (1 + i/m) FV = PV (1 + i/m) m x nm x n
FV = 100 (1.005)FV = 100 (1.005)60 60 = = $134.89$134.89
00 60 60
PV = -100PV = -100 FV = FV = 134.134.8989
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?
Future Value - continuous compoundingWhat is the FV of $1,000 earning 8% with continuous compounding, after 100 years?
Future Value - continuous compoundingWhat is the FV of $1,000 earning 8% with continuous compounding, after 100 years?
0 ?
PV =PV = FV = FV =
Mathematical Solution:Mathematical Solution:
FV = PV (e FV = PV (e inin))
FV = 1000 (e FV = 1000 (e .08x100.08x100) = 1000 (e ) = 1000 (e 88) )
FV = FV = $2,980,957.$2,980,957.9999
00 100 100
PV = -1000PV = -1000 FV = FV =
Future Value - continuous compoundingWhat is the FV of $1,000 earning 8% with continuous compounding, after 100 years?
00 100 100
PV = -1000PV = -1000 FV = FV = $2.98m$2.98m
Future Value - continuous compoundingWhat is the FV of $1,000 earning 8% with continuous compounding, after 100 years?
Mathematical Solution:Mathematical Solution:
FV = PV (e FV = PV (e inin))
FV = 1000 (e FV = 1000 (e .08x100.08x100) = 1000 (e ) = 1000 (e 88) )
FV = FV = $2,980,957.$2,980,957.9999
Present Value - single sumsIf you receive $100 one year from now, what is the
PV of that $100 if your opportunity cost is 6%?
Present Value - single sumsIf you receive $100 one year from now, what is the
PV of that $100 if your opportunity cost is 6%?
0 ?
PV =PV = FV = FV =
Calculator Solution:Calculator Solution:
P/Y = 1P/Y = 1 I = 6I = 6
N = 1 N = 1 FV = FV = 100100
PV = PV = -94.34-94.34
00 1 1
PV = PV = FV = 100 FV = 100
Present Value - single sumsIf you receive $100 one year from now, what is the
PV of that $100 if your opportunity cost is 6%?
Calculator Solution:Calculator Solution:
P/Y = 1P/Y = 1 I = 6I = 6
N = 1 N = 1 FV = FV = 100100
PV = PV = -94.34-94.34
Present Value - single sumsIf you receive $100 one year from now, what is the
PV of that $100 if your opportunity cost is 6%?
PV = PV = -94.-94.3434 FV = 100 FV = 100
00 1 1
Mathematical Solution:Mathematical Solution:
PV = FV (PVIF PV = FV (PVIF i, ni, n ))
PV = 100 (PVIF PV = 100 (PVIF .06, 1.06, 1 ) (use PVIF table, or)) (use PVIF table, or)
PV = FV / (1 + i)PV = FV / (1 + i)nn
PV = 100 / (1.06)PV = 100 / (1.06)1 1 = = $94.34$94.34
Present Value - single sumsIf you receive $100 one year from now, what is the
PV of that $100 if your opportunity cost is 6%?
PV = PV = -94.-94.3434 FV = 100 FV = 100
00 1 1
Present Value - single sumsIf you receive $100 five years from now, what is the
PV of that $100 if your opportunity cost is 6%?
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 ?
PV =PV = FV = FV =
Calculator Solution:Calculator Solution:
P/Y = 1P/Y = 1 I = 6I = 6
N = 5 N = 5 FV = FV = 100100
PV = PV = -74.73-74.73
00 5 5
PV = PV = FV = 100 FV = 100
Present Value - single sumsIf you receive $100 five years from now, what is the
PV of that $100 if your opportunity cost is 6%?
Calculator Solution:Calculator Solution:
P/Y = 1P/Y = 1 I = 6I = 6
N = 5 N = 5 FV = FV = 100100
PV = PV = -74.73-74.73
Present Value - single sumsIf you receive $100 five years from now, what is the
PV of that $100 if your opportunity cost is 6%?
00 5 5
PV = PV = -74.-74.7373 FV = 100 FV = 100
Mathematical Solution:Mathematical Solution:
PV = FV (PVIF PV = FV (PVIF i, ni, n ))
PV = 100 (PVIF PV = 100 (PVIF .06, 5.06, 5 ) (use PVIF table, or)) (use PVIF table, or)
PV = FV / (1 + i)PV = FV / (1 + i)nn
PV = 100 / (1.06)PV = 100 / (1.06)5 5 = = $74.73$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%?
00 5 5
PV = PV = -74.-74.7373 FV = 100 FV = 100
Present Value - single sumsWhat is the PV of $1,000 to be received 15 years
from now if your opportunity cost is 7%?
Present Value - single sumsWhat is the PV of $1,000 to be received 15 years
from now if your opportunity cost is 7%?
00 15 15
PV = PV = FV = FV =
Calculator Solution:Calculator Solution:
P/Y = 1P/Y = 1 I = 7I = 7
N = 15 N = 15 FV = FV = 1,0001,000
PV = PV = -362.45-362.45
Present Value - single sumsWhat is the PV of $1,000 to be received 15 years
from now if your opportunity cost is 7%?
00 15 15
PV = PV = FV = 1000 FV = 1000
Calculator Solution:Calculator Solution:
P/Y = 1P/Y = 1 I = 7I = 7
N = 15 N = 15 FV = FV = 1,0001,000
PV = PV = -362.45-362.45
Present Value - single sumsWhat is the PV of $1,000 to be received 15 years
from now if your opportunity cost is 7%?
00 15 15
PV = PV = -362.-362.4545 FV = 1000 FV = 1000
Mathematical Solution:Mathematical Solution:
PV = FV (PVIF PV = FV (PVIF i, ni, n ))
PV = 100 (PVIF PV = 100 (PVIF .07, 15.07, 15 ) (use PVIF table, or)) (use PVIF table, or)
PV = FV / (1 + i)PV = FV / (1 + i)nn
PV = 100 / (1.07)PV = 100 / (1.07)15 15 = = $362.45$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%?
00 15 15
PV = PV = -362.-362.4545 FV = 1000 FV = 1000
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?
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?
00 5 5
PV = PV = FV = FV =
Calculator Solution:Calculator Solution:
P/Y = 1P/Y = 1 N = 5N = 5
PV = -5,000 PV = -5,000 FV = 11,933FV = 11,933
I = I = 19%19%
00 5 5
PV = -5000PV = -5000 FV = 11,933 FV = 11,933
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?
Mathematical Solution:Mathematical Solution:
PV = FV (PVIF PV = FV (PVIF i, ni, n ) )
5,000 = 11,933 (PVIF 5,000 = 11,933 (PVIF ?, 5?, 5 ) )
PV = FV / (1 + i)PV = FV / (1 + i)nn
5,000 = 11,933 / (1+ i)5,000 = 11,933 / (1+ i)5 5
.419 = ((1/ (1+i).419 = ((1/ (1+i)55))
2.3866 = (1+i)2.3866 = (1+i)55
(2.3866)(2.3866)1/51/5 = (1+i) = (1+i) i = i = .19.19
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?
Hint for single sum problems:
In every single sum future value and In every single sum future value and present value problem, there are 4 present value problem, there are 4 variables: variables:
FVFV, , PVPV, , ii, and , and nn When doing problems, you will be When doing problems, you will be
given 3 of these variables and asked to given 3 of these variables and asked to solve for the 4th variable.solve for the 4th variable.
Keeping this in mind makes “time Keeping this in mind makes “time value” problems much easier!value” problems much easier!
The Time Value of Money
Compounding and DiscountingCompounding and Discounting
Cash Flow StreamsCash Flow Streams
0 1 2 3 4
Annuities
Annuity:Annuity: a sequence of a sequence of equalequal cash cash flowsflows, occurring at the , occurring at the endend of each of each period.period.
Annuity:Annuity: a sequence of a sequence of equalequal cash cash flows, occurring at the end of each flows, occurring at the end of each period.period.
0 1 2 3 4
Annuities
Examples of Annuities:
If you buy a bond, you will If you buy a bond, you will receive equal semi-annual coupon receive equal semi-annual coupon interest payments over the life of interest payments over the life of the bond.the bond.
If you borrow money to buy a If you borrow money to buy a house or a car, you will pay a house or a car, you will pay a stream of equal payments.stream of equal payments.
If you buy a bond, you will If you buy a bond, you will receive equal semi-annual coupon receive equal semi-annual coupon interest payments over the life of interest payments over the life of the bond.the bond.
If you borrow money to buy a If you borrow money to buy a house or a car, you will pay a house or a car, you will pay a stream of equal payments.stream of equal payments.
Examples of Annuities:
Future Value - annuityIf you invest $1,000 each year at 8%, how much
would you have after 3 years?
0 1 2 3
Calculator Solution:Calculator Solution:
P/Y = 1P/Y = 1 I = 8I = 8 N = 3N = 3
PMT = -1,000 PMT = -1,000
FV = FV = $3,246.40$3,246.40
Future Value - annuityIf you invest $1,000 each year at 8%, how much
would you have after 3 years?
0 1 2 3
10001000 10001000 1000 1000
Calculator Solution:Calculator Solution:
P/Y = 1P/Y = 1 I = 8I = 8 N = 3N = 3
PMT = -1,000 PMT = -1,000
FV = FV = $3,246.40$3,246.40
Future Value - annuityIf you invest $1,000 each year at 8%, how much
would you have after 3 years?
0 1 2 3
10001000 10001000 1000 1000
Mathematical Solution:Mathematical Solution:
Future Value - annuityIf you invest $1,000 each year at 8%, how much
would you have after 3 years?
Mathematical Solution:Mathematical Solution:
FV = PMT (FVIFA FV = PMT (FVIFA i, ni, n ))
Future Value - annuityIf you invest $1,000 each year at 8%, how much
would you have after 3 years?
Mathematical Solution:Mathematical Solution:
FV = PMT (FVIFA FV = PMT (FVIFA i, ni, n ))
FV = 1,000 (FVIFA FV = 1,000 (FVIFA .08, 3.08, 3 ) ) (use FVIFA table, or)(use FVIFA table, or)
Future Value - annuityIf you invest $1,000 each year at 8%, how much
would you have after 3 years?
Mathematical Solution:Mathematical Solution:
FV = PMT (FVIFA FV = PMT (FVIFA i, ni, n ))
FV = 1,000 (FVIFA FV = 1,000 (FVIFA .08, 3.08, 3 ) ) (use FVIFA table, or)(use FVIFA table, or)
FV = PMT (1 + i)FV = PMT (1 + i)nn - 1 - 1
ii
Future Value - annuityIf you invest $1,000 each year at 8%, how much
would you have after 3 years?
Mathematical Solution:Mathematical Solution:
FV = PMT (FVIFA FV = PMT (FVIFA i, ni, n ))
FV = 1,000 (FVIFA FV = 1,000 (FVIFA .08, 3.08, 3 ) ) (use FVIFA table, or)(use FVIFA table, or)
FV = PMT (1 + i)FV = PMT (1 + i)nn - 1 - 1
ii
FV = 1,000 (1.08)FV = 1,000 (1.08)33 - 1 = - 1 = $3246.40$3246.40
.08 .08
Future Value - annuityIf you invest $1,000 each year at 8%, how much
would you have after 3 years?
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 - annuityWhat 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
Calculator Solution:Calculator Solution:
P/Y = 1P/Y = 1 I = 8I = 8 N = 3N = 3
PMT = -1,000 PMT = -1,000
PV = PV = $2,577.10$2,577.10
0 1 2 3
10001000 10001000 1000 1000
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%?
Calculator Solution:Calculator Solution:
P/Y = 1P/Y = 1 I = 8I = 8 N = 3N = 3
PMT = -1,000 PMT = -1,000
PV = PV = $2,577.10$2,577.10
0 1 2 3
10001000 10001000 1000 1000
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 - annuityWhat is the PV of $1,000 at the end of each of the
next 3 years, if the opportunity cost is 8%?
Mathematical Solution: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%?
Mathematical Solution:Mathematical Solution:
PV = PMT (PVIFA PV = PMT (PVIFA i, ni, 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%?
Mathematical Solution:Mathematical Solution:
PV = PMT (PVIFA PV = PMT (PVIFA i, ni, n ))
PV = 1,000 (PVIFA PV = 1,000 (PVIFA .08, 3.08, 3 ) (use PVIFA table, or)) (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%?
Mathematical Solution:Mathematical Solution:
PV = PMT (PVIFA PV = PMT (PVIFA i, ni, n ))
PV = 1,000 (PVIFA PV = 1,000 (PVIFA .08, 3.08, 3 ) (use PVIFA table, or)) (use PVIFA table, or)
11
PV = PMT 1 - (1 + i)PV = PMT 1 - (1 + i)nn
ii
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%?
Mathematical Solution:Mathematical Solution:
PV = PMT (PVIFA PV = PMT (PVIFA i, ni, n ))
PV = 1,000 (PVIFA PV = 1,000 (PVIFA .08, 3.08, 3 ) (use PVIFA table, or)) (use PVIFA table, or)
11
PV = PMT 1 - (1 + i)PV = PMT 1 - (1 + i)nn
ii
11
PV = 1000 1 - (1.08 )PV = 1000 1 - (1.08 )33 = = $2,577.10$2,577.10
.08.08
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%?
Perpetuities
Suppose you will receive a fixed Suppose you will receive a fixed payment every period (month, year, payment every period (month, year, etc.) forever. This is an example of etc.) forever. This is an example of a perpetuity.a perpetuity.
You can think of a perpetuity as an You can think of a perpetuity as an annuityannuity that goes on that goes on foreverforever..
Present Value of a Perpetuity
When we find the PV of an When we find the PV of an annuityannuity, , we think of the following we think of the following relationship:relationship:
Present Value of a Perpetuity
When we find the PV of an When we find the PV of an annuityannuity, , we think of the following we think of the following relationship:relationship:
PV = PMT (PVIFA PV = PMT (PVIFA i, ni, n ) )
Mathematically, Mathematically,
(PVIFA i, n ) = (PVIFA i, n ) =
We said that a perpetuity is an We said that a perpetuity is an annuity where n = infinity. What annuity where n = infinity. What happens to this formula when happens to this formula when nn gets very, very large? gets very, very large?
1 - 1 - 11
(1 + i)(1 + i)nn
ii
When n gets very large,When n gets very large,
this becomes zero.this becomes zero.
So we’re left with PVIFA =So we’re left with PVIFA =
1 i
1 - 1
(1 + i)n
i
So, the PV of a perpetuity is very So, the PV of a perpetuity is very simple to find:simple to find:
Present Value of a PerpetuityPresent Value of a Perpetuity
PMT i
PV =
So, the PV of a perpetuity is very So, the PV of a perpetuity is very simple to find:simple to find:
Present Value of a PerpetuityPresent Value of a Perpetuity
What should you be willing to pay in What should you be willing to pay in order to receive order to receive $10,000$10,000 annually annually forever, if you require forever, if you require 8%8% per year per year on the investment?on the investment?
What should you be willing to pay in What should you be willing to pay in order to receive order to receive $10,000$10,000 annually annually forever, if you require forever, if you require 8%8% per year per year on the investment?on the investment?
PMT $10,000PMT $10,000 i .08 i .08
PV = =PV = =
What should you be willing to pay in What should you be willing to pay in order to receive order to receive $10,000$10,000 annually annually forever, if you require forever, if you require 8%8% per year per year on the investment?on the investment?
PMT $10,000PMT $10,000 i .08 i .08
= $125,000= $125,000
PV = =PV = =
Begin Mode vs. End Mode
1000 1000 10001000 1000 1000
4 5 6 7 8 4 5 6 7 8 year year year 5 6 7
PVPVinin
ENDENDModeMode
Begin Mode vs. End Mode
1000 1000 10001000 1000 1000
4 5 6 7 8 4 5 6 7 8 year year year 5 6 7
PVPVinin
ENDENDModeMode
FVFVinin
ENDENDModeMode
Begin Mode vs. End Mode
1000 1000 10001000 1000 1000
4 5 6 7 8 4 5 6 7 8 year year year 6 7 8
PVPVinin
BEGINBEGINModeMode
Begin Mode vs. End Mode
1000 1000 10001000 1000 1000
4 5 6 7 8 4 5 6 7 8 year year year 6 7 8
PVPVinin
BEGINBEGINModeMode
FVFVinin
BEGINBEGINModeMode
Earlier, we examined this “ordinary” annuity:
Using an interest rate of 8%, we Using an interest rate of 8%, we find that:find that:
0 1 2 3
10001000 10001000 1000 1000
Earlier, we examined this “ordinary” annuity:
Using an interest rate of 8%, we Using an interest rate of 8%, we find that:find that:
The The Future ValueFuture Value (at 3) is (at 3) is $3,246.40$3,246.40..
0 1 2 3
10001000 10001000 1000 1000
Earlier, we examined this “ordinary” annuity:
Using an interest rate of 8%, we Using an interest rate of 8%, we find that:find that:
The The Future ValueFuture Value (at 3) is (at 3) is $3,246.40$3,246.40..
The The Present ValuePresent Value (at 0) is (at 0) is $2,577.10$2,577.10..
0 1 2 3
10001000 10001000 1000 1000
What about this annuity?
Same 3-year time line,Same 3-year time line, Same 3 $1000 cash flows, butSame 3 $1000 cash flows, but The cash flows occur at the The cash flows occur at the
beginningbeginning of each year, rather of each year, rather than at the than at the endend of each year. of each year.
This is an This is an “annuity due.”“annuity due.”
0 1 2 3
10001000 1000 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:Calculator Solution:
Mode = BEGIN P/Y = 1Mode = BEGIN P/Y = 1 I = 8I = 8
N = 3N = 3 PMT = -1,000 PMT = -1,000
FV = FV = $3,506.11$3,506.11
0 1 2 3
-1000-1000 -1000 -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 -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:Calculator Solution:
Mode = BEGIN P/Y = 1Mode = BEGIN P/Y = 1 I = 8I = 8
N = 3N = 3 PMT = -1,000 PMT = -1,000
FV = FV = $3,506.11$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:Mathematical Solution: Simply compound the FV of the Simply compound the FV of the ordinary annuity one more period: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:Mathematical Solution: Simply compound the FV of the Simply compound the FV of the ordinary annuity one more period:ordinary annuity one more period:
FV = PMT (FVIFA FV = PMT (FVIFA i, ni, 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:Mathematical Solution: Simply compound the FV of the Simply compound the FV of the ordinary annuity one more period:ordinary annuity one more period:
FV = PMT (FVIFA FV = PMT (FVIFA i, ni, n ) (1 + i)) (1 + i)
FV = 1,000 (FVIFA FV = 1,000 (FVIFA .08, 3.08, 3 ) (1.08) ) (1.08) (use FVIFA table, or)(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:Mathematical Solution: Simply compound the FV of the Simply compound the FV of the ordinary annuity one more period:ordinary annuity one more period:
FV = PMT (FVIFA FV = PMT (FVIFA i, ni, n ) (1 + i)) (1 + i)
FV = 1,000 (FVIFA FV = 1,000 (FVIFA .08, 3.08, 3 ) (1.08) ) (1.08) (use FVIFA table, or)(use FVIFA table, or)
FV = PMT (1 + i)FV = PMT (1 + i)nn - 1 - 1
ii(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:Mathematical Solution: Simply compound the FV of the Simply compound the FV of the ordinary annuity one more period:ordinary annuity one more period:
FV = PMT (FVIFA FV = PMT (FVIFA i, ni, n ) (1 + i)) (1 + i)
FV = 1,000 (FVIFA FV = 1,000 (FVIFA .08, 3.08, 3 ) (1.08) ) (1.08) (use FVIFA table, or)(use FVIFA table, or)
FV = PMT (1 + i)FV = PMT (1 + i)nn - 1 - 1
ii
FV = 1,000 (1.08)FV = 1,000 (1.08)33 - 1 = - 1 = $3,506.11$3,506.11
.08 .08
(1 + i)(1 + i)
(1.08)(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:Calculator Solution:
Mode = BEGIN P/Y = 1Mode = BEGIN P/Y = 1 I = 8I = 8
N = 3N = 3 PMT = 1,000 PMT = 1,000
PV = PV = $2,783.26$2,783.26
0 1 2 3
10001000 1000 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:Calculator Solution:
Mode = BEGIN P/Y = 1Mode = BEGIN P/Y = 1 I = 8I = 8
N = 3N = 3 PMT = 1,000 PMT = 1,000
PV = PV = $2,783.26$2,783.26
0 1 2 3
10001000 1000 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:Mathematical Solution: Simply compound the FV of the Simply compound the FV of the ordinary annuity one more period:ordinary annuity one more period:
Present Value - annuity due
Mathematical Solution:Mathematical Solution: Simply compound the FV of the Simply compound the FV of the ordinary annuity one more period:ordinary annuity one more period:
PV = PMT (PVIFA PV = PMT (PVIFA i, ni, n ) (1 + i)) (1 + i)
Present Value - annuity due
Mathematical Solution:Mathematical Solution: Simply compound the FV of the Simply compound the FV of the ordinary annuity one more period:ordinary annuity one more period:
PV = PMT (PVIFA PV = PMT (PVIFA i, ni, n ) (1 + i)) (1 + i)
PV = 1,000 (PVIFA PV = 1,000 (PVIFA .08, 3.08, 3 ) (1.08) ) (1.08) (use PVIFA table, or)(use PVIFA table, or)
Present Value - annuity due
Mathematical Solution:Mathematical Solution: Simply compound the FV of the Simply compound the FV of the ordinary annuity one more period:ordinary annuity one more period:
PV = PMT (PVIFA PV = PMT (PVIFA i, ni, n ) (1 + i)) (1 + i)
PV = 1,000 (PVIFA PV = 1,000 (PVIFA .08, 3.08, 3 ) (1.08) ) (1.08) (use PVIFA table, or)(use PVIFA table, or)
11
PV = PMT 1 - (1 + i)PV = PMT 1 - (1 + i)nn
ii(1 + i)(1 + i)
Present Value - annuity due
Mathematical Solution:Mathematical Solution: Simply compound the FV of the Simply compound the FV of the ordinary annuity one more period:ordinary annuity one more period:
PV = PMT (PVIFA PV = PMT (PVIFA i, ni, n ) (1 + i)) (1 + i)
PV = 1,000 (PVIFA PV = 1,000 (PVIFA .08, 3.08, 3 ) (1.08) ) (1.08) (use PVIFA table, or)(use PVIFA table, or)
11
PV = PMT 1 - (1 + i)PV = PMT 1 - (1 + i)nn
ii
11
PV = 1000 1 - (1.08 )PV = 1000 1 - (1.08 )33 = = $2,783.26$2,783.26
.08.08
(1 + i)(1 + i)
(1.08)(1.08)
Is this an Is this an annuityannuity?? How do we find the PV of a cash flow How do we find the PV of a cash flow
stream when all of the cash flows are stream when all of the cash flows are different? (Use a 10% discount rate).different? (Use a 10% discount rate).
Uneven Cash Flows
00 1 1 2 2 3 3 4 4
-10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000
Sorry! There’s no quickie for this one. Sorry! There’s no quickie for this one. We have to discount each cash flow We have to discount each cash flow back separately.back separately.
00 1 1 2 2 3 3 4 4
-10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000
Uneven Cash Flows
Sorry! There’s no quickie for this one. Sorry! There’s no quickie for this one. We have to discount each cash flow We have to discount each cash flow back separately.back separately.
00 1 1 2 2 3 3 4 4
-10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000
Uneven Cash Flows
Sorry! There’s no quickie for this one. Sorry! There’s no quickie for this one. We have to discount each cash flow We have to discount each cash flow back separately.back separately.
00 1 1 2 2 3 3 4 4
-10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000
Uneven Cash Flows
Sorry! There’s no quickie for this one. Sorry! There’s no quickie for this one. We have to discount each cash flow We have to discount each cash flow back separately.back separately.
00 1 1 2 2 3 3 4 4
-10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000
Uneven Cash Flows
Sorry! There’s no quickie for this one. Sorry! There’s no quickie for this one. We have to discount each cash flow We have to discount each cash flow back separately.back separately.
00 1 1 2 2 3 3 4 4
-10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000
Uneven Cash Flows
periodperiod CF CF PV (CF)PV (CF)
00 -10,000 -10,000 -10,000.00-10,000.00
11 2,000 2,000 1,818.181,818.18
22 4,000 4,000 3,305.793,305.79
33 6,000 6,000 4,507.894,507.89
44 7,000 7,000 4,781.094,781.09
PV of Cash Flow Stream: $ 4,412.95PV of Cash Flow Stream: $ 4,412.95
00 1 1 2 2 3 3 4 4
-10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000
Example
Cash flows from an investment are Cash flows from an investment are expected to be expected to be $40,000$40,000 per year at the per year at the end of years 4, 5, 6, 7, and 8. If you end of years 4, 5, 6, 7, and 8. If you require a require a 20%20% rate of return, what is rate of return, what is the PV of these cash flows?the PV of these cash flows?
Example
00 11 22 33 44 55 66 77 88
$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040
Cash flows from an investment are Cash flows from an investment are expected to be expected to be $40,000$40,000 per year at the per year at the end of years 4, 5, 6, 7, and 8. If you end of years 4, 5, 6, 7, and 8. If you require a require a 20%20% rate of return, what is rate of return, what is the PV of these cash flows?the PV of these cash flows?
This type of cash flow sequence is This type of cash flow sequence is often called a often called a ““deferred annuitydeferred annuity.”.”
00 11 22 33 44 55 66 77 88
$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040
How to solve:How to solve:
1) 1) Discount each cash flow back to Discount each cash flow back to time 0 separately.time 0 separately.
00 11 22 33 44 55 66 77 88
$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040
How to solve:How to solve:
1) 1) Discount each cash flow back to Discount each cash flow back to time 0 separately.time 0 separately.
00 11 22 33 44 55 66 77 88
$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040
How to solve:How to solve:
1) 1) Discount each cash flow back to Discount each cash flow back to time 0 separately.time 0 separately.
00 11 22 33 44 55 66 77 88
$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040
How to solve:How to solve:
1) 1) Discount each cash flow back to Discount each cash flow back to time 0 separately.time 0 separately.
00 11 22 33 44 55 66 77 88
$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040
How to solve:How to solve:
1) 1) Discount each cash flow back to Discount each cash flow back to time 0 separately.time 0 separately.
00 11 22 33 44 55 66 77 88
$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040
How to solve:How to solve:
1) 1) Discount each cash flow back to Discount each cash flow back to time 0 separately.time 0 separately.
00 11 22 33 44 55 66 77 88
$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040
How to solve:How to solve:
1) 1) Discount each cash flow back to Discount each cash flow back to time 0 separately.time 0 separately.
Or,Or,
00 11 22 33 44 55 66 77 88
$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040
2) 2) Find the PV of the annuity:Find the PV of the annuity:
PVPV:: End mode; P/YR = 1; I = 20; End mode; P/YR = 1; I = 20; PMT = 40,000; N = 5 PMT = 40,000; N = 5
PV = PV = $119,624$119,624
00 11 22 33 44 55 66 77 88
$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040
2) 2) Find the PV of the annuity:Find the PV of the annuity:
PVPV3:3: End mode; P/YR = 1; I = 20; End mode; P/YR = 1; I = 20; PMT = 40,000; N = 5 PMT = 40,000; N = 5
PVPV33= = $119,624$119,624
00 11 22 33 44 55 66 77 88
$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040
Then discount this single sum back to Then discount this single sum back to time 0.time 0.
PV: End mode; P/YR = 1; I = 20; PV: End mode; P/YR = 1; I = 20;
N = 3; FV = 119,624; N = 3; FV = 119,624;
Solve: PV = Solve: PV = $69,226$69,226
119,624119,624
00 11 22 33 44 55 66 77 88
$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040
The PV of the cash flow The PV of the cash flow stream is stream is $69,226$69,226..
69,22669,226
00 11 22 33 44 55 66 77 88
$0$0 0 0 0 0 0 0 4040 4040 4040 4040 4040
119,624119,624
Retirement Example
After graduation, you plan to invest After graduation, you plan to invest $400$400 per month per month in the stock market. in the stock market. If you earn If you earn 12%12% per year per year on your on your stocks, how much will you have stocks, how much will you have accumulated when you retire in accumulated when you retire in 3030 yearsyears??
Retirement Example
After graduation, you plan to invest After graduation, you plan to invest $400$400 per month in the stock market. per month in the stock market. If you earn If you earn 12%12% per year on your per year on your stocks, how much will you have stocks, how much will you have accumulated when you retire in 30 accumulated when you retire in 30 years?years?
00 11 22 33 . . . 360. . . 360
400 400 400 400400 400 400 400
Using your calculator,Using your calculator,
P/YR = 12P/YR = 12
N = 360 N = 360
PMT = -400PMT = -400
I%YR = 12I%YR = 12
FV = FV = $1,397,985.65$1,397,985.65
00 11 22 33 . . . 360. . . 360
400 400 400 400400 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: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:Mathematical Solution:
FV = PMT (FVIFA FV = PMT (FVIFA i, ni, 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:Mathematical Solution:
FV = PMT (FVIFA FV = PMT (FVIFA i, ni, n ) )
FV = 400 (FVIFA FV = 400 (FVIFA .01, 360.01, 360 ) ) (can’t use FVIFA table)(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:Mathematical Solution:
FV = PMT (FVIFA FV = PMT (FVIFA i, ni, n ) )
FV = 400 (FVIFA FV = 400 (FVIFA .01, 360.01, 360 ) ) (can’t use FVIFA table)(can’t use FVIFA table)
FV = PMT (1 + i)FV = PMT (1 + i)nn - 1 - 1
ii
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:Mathematical Solution:
FV = PMT (FVIFA FV = PMT (FVIFA i, ni, n ) )
FV = 400 (FVIFA FV = 400 (FVIFA .01, 360.01, 360 ) ) (can’t use FVIFA table)(can’t use FVIFA table)
FV = PMT (1 + i)FV = PMT (1 + i)nn - 1 - 1
ii
FV = 400 (1.01)FV = 400 (1.01)360360 - 1 = - 1 = $1,397,985.65$1,397,985.65
.01 .01
If you borrow If you borrow $100,000$100,000 at at 7%7% fixed fixed interest for interest for 3030 years years in order to in order to buy a house, what will be your buy a house, what will be your
monthly house paymentmonthly house payment??
House Payment Example
House Payment Example
If you borrow If you borrow $100,000$100,000 at at 7%7% fixed fixed interest for interest for 3030 years in order to years in order to buy a house, what will be your buy a house, what will be your
monthly house payment?monthly house payment?
Using your calculator,Using your calculator,
P/YR = 12P/YR = 12
N = 360N = 360
I%YR = 7I%YR = 7
PV = $100,000PV = $100,000
PMT = PMT = -$665.30-$665.30
00 11 22 33 . . . 360. . . 360
? ? ? ?? ? ? ?
House Payment Example
Mathematical Solution:Mathematical Solution:
PV = PMT (PVIFA PV = PMT (PVIFA i, ni, n ))
House Payment Example
Mathematical Solution:Mathematical Solution:
PV = PMT (PVIFA PV = PMT (PVIFA i, ni, n ) )
100,000 = PMT (PVIFA 100,000 = PMT (PVIFA .07, 360.07, 360 ) ) (can’t use PVIFA table)(can’t use PVIFA table)
House Payment Example
Mathematical Solution:Mathematical Solution:
PV = PMT (PVIFA PV = PMT (PVIFA i, ni, n ) )
100,000 = PMT (PVIFA 100,000 = PMT (PVIFA .07, 360.07, 360 ) ) (can’t use PVIFA table)(can’t use PVIFA table)
11
PV = PMT 1 - (1 + i)PV = PMT 1 - (1 + i)nn
ii
House Payment Example
Mathematical Solution:Mathematical Solution:
PV = PMT (PVIFA PV = PMT (PVIFA i, ni, n ) )
100,000 = PMT (PVIFA 100,000 = PMT (PVIFA .07, 360.07, 360 ) ) (can’t use PVIFA table)(can’t use PVIFA table)
11
PV = PMT 1 - (1 + i)PV = PMT 1 - (1 + i)nn
ii
11
100,000 = PMT 1 - (1.005833 )100,000 = PMT 1 - (1.005833 )360360 PMT=$665.30PMT=$665.30
.005833.005833
Team Assignment
Upon retirement, your goal is to spend Upon retirement, your goal is to spend 55 years traveling around the world. To years traveling around the world. To travel in style will require travel in style will require $250,000$250,000 per per year at the year at the beginningbeginning of each year. of each year.
If you plan to retire in If you plan to retire in 30 30 yearsyears, what are , what are the equal the equal monthlymonthly payments necessary payments necessary to achieve this goal? The funds in your to achieve this goal? The funds in your retirement account will compound at retirement account will compound at 10%10% annually. annually.
How much do we need to have by How much do we need to have by the end of year 30 to finance the the end of year 30 to finance the trip?trip?
PVPV3030 = PMT (PVIFA = PMT (PVIFA .10, 5.10, 5) (1.10) =) (1.10) =
= 250,000 (3.7908) (1.10) == 250,000 (3.7908) (1.10) =
= = $1,042,470$1,042,470
2727 2828 2929 3030 3131 3232 3333 3434 3535
250 250 250 250 250 250 250 250 250 250
Using your calculator,Using your calculator,
Mode = BEGINMode = BEGIN
PMT = -$250,000PMT = -$250,000
N = 5N = 5
I%YR = 10I%YR = 10
P/YR = 1P/YR = 1
PV = PV = $1,042,466$1,042,466
2727 2828 2929 3030 3131 3232 3333 3434 3535
250 250 250 250 250 250 250 250 250 250
Now, assuming 10% annual Now, assuming 10% annual compounding, what monthly compounding, what monthly payments will be required for payments will be required for you to have you to have $1,042,466$1,042,466 at the end at the end of year 30?of year 30?
2727 2828 2929 3030 3131 3232 3333 3434 3535
250 250 250 250 250 250 250 250 250 250
1,042,4661,042,466
Using your calculator,Using your calculator,
Mode = ENDMode = END
N = 360N = 360
I%YR = 10I%YR = 10
P/YR = 12P/YR = 12
FV = $1,042,466FV = $1,042,466
PMT = PMT = -$461.17-$461.17
2727 2828 2929 3030 3131 3232 3333 3434 3535
250 250 250 250 250 250 250 250 250 250
1,042,4661,042,466
So, you would have to place So, you would have to place $461.17$461.17 in in your retirement account, which earns your retirement account, which earns 10% annually, at the end of each of the 10% annually, at the end of each of the next 360 months to finance the 5-year next 360 months to finance the 5-year world tour.world tour.