The Time Value of Money Compounding and Discounting Single Sums.
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The Time Value of MoneyThe 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 more than $1 in the future. This is duemore 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 If we can MEASURE this opportunity cost, we can:opportunity cost, we can:
?
Translate $1 today into its equivalent in Translate $1 today into its equivalent in the futurethe future (COMPOUNDING)(COMPOUNDING)..
Today Future
If we can MEASURE this If we can MEASURE this opportunity cost, we can:opportunity cost, we can:
Translate $1 today into its equivalent in Translate $1 today into its equivalent in the futurethe future (COMPOUNDING)(COMPOUNDING)..
Translate $1 in the future into its Translate $1 in the future into its equivalent todayequivalent today (DISCOUNTING)(DISCOUNTING)..
?
?
Today Future
Today Future
Future ValueFuture Value
Future Value - single sumsFuture Value - single sums
If you deposit $100 in an account earning 6%, how If you deposit $100 in an account earning 6%, how much would you have in the account after 1 year?much would you have in the account after 1 year?
0 1
PV =PV = FV = FV =
Future Value - single sumsFuture Value - single sums
If you deposit $100 in an account earning 6%, how If you deposit $100 in an account earning 6%, how much would you have in the account after 1 year?much would you have in the account after 1 year?
Calculator Solution:Calculator Solution:
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 sumsFuture Value - single sums
If you deposit $100 in an account earning 6%, how If you deposit $100 in an account earning 6%, how much would you have in the account after 1 year?much would you have in the account after 1 year?
Calculator Solution:Calculator Solution:
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 sumsFuture Value - single sums
If you deposit $100 in an account earning 6%, how If you deposit $100 in an account earning 6%, how much would you have in the account after 1 year?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 sumsFuture Value - single sums
If you deposit $100 in an account earning 6%, how If you deposit $100 in an account earning 6%, how much would you have in the account after 5 years?much would you have in the account after 5 years?
00 5 5
PV =PV = FV = FV =
Future Value - single sumsFuture Value - single sums
If you deposit $100 in an account earning 6%, how If you deposit $100 in an account earning 6%, how much would you have in the account after 5 years?much would you have in the account after 5 years?
Calculator Solution:Calculator Solution:
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 sumsFuture Value - single sums
If you deposit $100 in an account earning 6%, how If you deposit $100 in an account earning 6%, how much would you have in the account after 5 years?much would you have in the account after 5 years?
Calculator Solution:Calculator Solution:
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 sumsFuture Value - single sums
If you deposit $100 in an account earning 6%, how If you deposit $100 in an account earning 6%, how much would you have in the account after 5 years?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 sumsFuture Value - single sumsIf you deposit $100 in an account earning 6% with If you deposit $100 in an account earning 6% with quarterly compoundingquarterly compounding, how much would you have , how much would you have
in the account after 5 years?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 = 100100
FV = FV = $134.68$134.68
00 20 20
PV = 100PV = 100 FV = FV =
Future Value - single sumsFuture Value - single sumsIf you deposit $100 in an account earning 6% with If you deposit $100 in an account earning 6% with quarterly compoundingquarterly compounding, how much would you have , how much would you have
in the account after 5 years?in the account after 5 years?
Calculator Solution:Calculator Solution:
P/Y = 4P/Y = 4 I = 6I = 6
N = 20 N = 20 PV = PV = 100100
FV = FV = $134.68$134.68
00 20 20
PV = 100PV = 100 FV = FV = 134.134.6868
Future Value - single sumsFuture Value - single sumsIf you deposit $100 in an account earning 6% with If you deposit $100 in an account earning 6% with quarterly compoundingquarterly compounding, how much would you have , how much would you have
in the account after 5 years?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, 20.06, 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 sumsFuture Value - single sumsIf you deposit $100 in an account earning 6% with If you deposit $100 in an account earning 6% with quarterly compoundingquarterly compounding, how much would you have , how much would you have
in the account after 5 years?in the account after 5 years?
Future Value - single sumsFuture Value - single sumsIf you deposit $100 in an account earning 6% with If you deposit $100 in an account earning 6% with monthly compoundingmonthly compounding, how much would you have , how much would you have
in the account after 5 years?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 = 100100
FV = FV = $134.89$134.89
00 60 60
PV = 100PV = 100 FV = FV =
Future Value - single sumsFuture Value - single sumsIf you deposit $100 in an account earning 6% with If you deposit $100 in an account earning 6% with monthly compoundingmonthly compounding, how much would you have , how much would you have
in the account after 5 years?in the account after 5 years?
Calculator Solution:Calculator Solution:
P/Y = 12P/Y = 12 I = 6I = 6
N = 60 N = 60 PV = PV = 100100
FV = FV = $134.89$134.89
00 60 60
PV = 100PV = 100 FV = FV = 134.134.8989
Future Value - single sumsFuture Value - single sumsIf you deposit $100 in an account earning 6% with If you deposit $100 in an account earning 6% with monthly compoundingmonthly compounding, how much would you have , how much would you have
in the account after 5 years?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, 60.06, 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 sumsFuture Value - single sumsIf you deposit $100 in an account earning 6% with If you deposit $100 in an account earning 6% with monthly compoundingmonthly compounding, how much would you have , how much would you have
in the account after 5 years?in the account after 5 years?
Present ValuePresent Value
Present Value - single sumsPresent Value - single sumsIf you will receive $100 one year from now, what is If you will receive $100 one year from now, what is the PV of that $100 if your opportunity cost is 6%?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 = 94.3494.34
00 1 1
PV = PV = FV = 100 FV = 100
Present Value - single sumsPresent Value - single sumsIf you will receive $100 one year from now, what is If you will receive $100 one year from now, what is the PV of that $100 if your opportunity cost is 6%?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 = 94.3494.34
00 1 1
PV = PV = 94. 94.3434 FV = 100 FV = 100
Present Value - single sumsPresent Value - single sumsIf you will receive $100 one year from now, what is If you will receive $100 one year from now, what is the PV of that $100 if your opportunity cost is 6%?the PV of that $100 if your opportunity cost is 6%?
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
00 1 1
PV = PV = 94. 94.3434 FV = 100 FV = 100
Present Value - single sumsPresent Value - single sumsIf you will receive $100 one year from now, what is If you will receive $100 one year from now, what is the PV of that $100 if your opportunity cost is 6%?the PV of that $100 if your opportunity cost is 6%?
Present Value - single sumsPresent Value - single sumsIf you will receive $100 5 years from now, what is If you will receive $100 5 years from now, what is the PV of that $100 if your opportunity cost is 6%?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 = 74.7374.73
00 5 5
PV = PV = FV = 100 FV = 100
Present Value - single sumsPresent Value - single sumsIf you will receive $100 5 years from now, what is If you will receive $100 5 years from now, what is the PV of that $100 if your opportunity cost is 6%?the PV of that $100 if your opportunity cost is 6%?
00 5 5
PV = PV = FV = 100 FV = 100
Present Value - single sumsPresent Value - single sumsIf you will receive $100 5 years from now, what is If you will receive $100 5 years from now, what is the PV of that $100 if your opportunity cost is 6%?the PV of that $100 if your opportunity cost is 6%?
74.74.7373
Calculator Solution:Calculator Solution:
P/Y = 1P/Y = 1 I = 6I = 6
N = 5 N = 5 FV = FV = 100100
PV = PV = 74.7374.73
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
00 5 5
PV = PV = 74. 74.7373 FV = 100 FV = 100
Present Value - single sumsPresent Value - single sumsIf you will receive $100 5 years from now, what is If you will receive $100 5 years from now, what is the PV of that $100 if your opportunity cost is 6%?the PV of that $100 if your opportunity cost is 6%?
Present Value - single sumsPresent Value - single sumsWhat is the PV of $1,000 to be received 15 years What is the PV of $1,000 to be received 15 years
from now if your opportunity cost is 7%?from now if your opportunity cost is 7%?
0 ?
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 = 362.45362.45
00 15 15
PV = PV = FV = 1000 FV = 1000
Present Value - single sumsPresent Value - single sumsWhat is the PV of $1,000 to be received 15 years What is the PV of $1,000 to be received 15 years
from now if your opportunity cost is 7%?from now if your opportunity cost is 7%?
Calculator Solution:Calculator Solution:
P/Y = 1P/Y = 1 I = 7I = 7
N = 15 N = 15 FV = FV = 1,0001,000
PV = PV = 362.45362.45
00 15 15
PV = PV = 362.362.4545 FV = 1000 FV = 1000
Present Value - single sumsPresent Value - single sumsWhat is the PV of $1,000 to be received 15 years What is the PV of $1,000 to be received 15 years
from now if your opportunity cost is 7%?from now if your opportunity cost is 7%?
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
00 15 15
PV = PV = 362. 362.4545 FV = 1000 FV = 1000
Present Value - single sumsPresent Value - single sumsWhat is the PV of $1,000 to be received 15 years What is the PV of $1,000 to be received 15 years
from now if your opportunity cost is 7%?from now if your opportunity cost is 7%?
Present Value - single sumsPresent Value - single sumsIf you sold land for $11,933 that you bought 5 years If you sold land for $11,933 that you bought 5 years ago for $5,000, what is your annual rate of return?ago for $5,000, what is your annual rate of return?
0 ?
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 = 5,000PV = 5,000 FV = 11,933 FV = 11,933
Present Value - single sumsPresent Value - single sumsIf you sold land for $11,933 that you bought 5 years If you sold land for $11,933 that you bought 5 years ago for $5,000, what is your annual rate of return?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 = .19i = .19
Present Value - single sumsPresent Value - single sumsIf you sold land for $11,933 that you bought 5 years If you sold land for $11,933 that you bought 5 years ago for $5,000, what is your annual rate of return?ago for $5,000, what is your annual rate of return?
Present Value - single sumsPresent Value - single sumsSuppose you placed $100 in an account that pays Suppose you placed $100 in an account that pays 9.6% interest, compounded monthly. How long 9.6% interest, compounded monthly. How long
will it take for your account to grow to $500?will it take for your account to grow to $500?
00
PV = PV = FV = FV =
Calculator Solution:Calculator Solution: P/Y = 12P/Y = 12 FV = 500FV = 500 I = 9.6I = 9.6 PV = -100PV = -100 N = N = 202 months202 months
Present Value - single sumsPresent Value - single sumsSuppose you placed $100 in an account that pays Suppose you placed $100 in an account that pays 9.6% interest, compounded monthly. How long 9.6% interest, compounded monthly. How long
will it take for your account to grow to $500?will it take for your account to grow to $500?
00 ? ?
PV = 100PV = 100 FV = 500 FV = 500
Present Value - single sumsPresent Value - single sumsSuppose you placed $100 in an account that pays Suppose you placed $100 in an account that pays 9.6% interest, compounded monthly. How long 9.6% interest, compounded monthly. How long
will it take for your account to grow to $500?will it take for your account to grow to $500?
Mathematical Solution:Mathematical Solution:
PV = FV / (1 + i)PV = FV / (1 + i)nn
100 = 500 / (1+ .008)100 = 500 / (1+ .008)NN
5 = (1.008)5 = (1.008)NN
ln 5 = ln (1.008)ln 5 = ln (1.008)NN
ln 5 = N ln (1.008)ln 5 = N ln (1.008)
1.60944 = .007968 N1.60944 = .007968 N N = 202 monthsN = 202 months
Hint for single sum problems:Hint for single sum problems:
In every single sum future value In every single sum future value and present value problem, there and present value problem, there are 4 variables: are 4 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 given 3 of these variables and asked to solve for the 4th variable.asked to 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 MoneyThe Time Value of Money
Compounding and DiscountingCompounding and Discounting
Cash Flow StreamsCash Flow Streams
0 1 2 3 4
AnnuitiesAnnuities
Annuity: a sequence of Annuity: a sequence of equal cash equal cash flowsflows, occurring at the , occurring at the endend of each of each period.period.
AnnuitiesAnnuities
Annuity: a sequence of equal cash Annuity: a sequence of equal cash flows, occurring at the end of each flows, occurring at the end of each period.period.
0 1 2 3 4
Examples of Annuities:Examples of Annuities:
If you buy a bond, you will If you buy a bond, you will receive equal coupon interest receive equal coupon interest payments over the life of the payments over the life of the bond.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:Examples of Annuities:
If you buy a bond, you will If you buy a bond, you will receive equal coupon interest receive equal coupon interest payments over the life of the payments over the life of the bond.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.
Future Value - annuityFuture Value - annuityIf you invest $1,000 at the end of the next 3 years, If you invest $1,000 at the end of the next 3 years, at 8%, how much would you have after 3 years?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 - annuityFuture Value - annuityIf you invest $1,000 at the end of the next 3 years, If you invest $1,000 at the end of the next 3 years, at 8%, how much would you have after 3 years?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 - annuityFuture Value - annuityIf you invest $1,000 at the end of the next 3 years, If you invest $1,000 at the end of the next 3 years, at 8%, how much would you have after 3 years?at 8%, how much would you have after 3 years?
0 1 2 3
10001000 10001000 1000 1000
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 - annuityFuture Value - annuityIf you invest $1,000 at the end of the next 3 years, If you invest $1,000 at the end of the next 3 years, at 8%, how much would you have after 3 years?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 - annuityFuture Value - annuityIf you invest $1,000 at the end of the next 3 years, If you invest $1,000 at the end of the next 3 years, at 8%, how much would you have after 3 years?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 - annuityFuture Value - annuityIf you invest $1,000 at the end of the next 3 years, If you invest $1,000 at the end of the next 3 years, at 8%, how much would you have after 3 years?at 8%, how much would you have after 3 years?
Present Value - annuityPresent Value - annuityWhat is the PV of $1,000 at the end of each of the What is the PV of $1,000 at the end of each of the
next 3 years, if the opportunity cost is 8%?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 - annuityPresent Value - annuityWhat is the PV of $1,000 at the end of each of the What is the PV of $1,000 at the end of each of the
next 3 years, if the opportunity cost is 8%?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 - annuityPresent Value - annuityWhat is the PV of $1,000 at the end of each of the What is the PV of $1,000 at the end of each of the
next 3 years, if the opportunity cost is 8%?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 - annuityPresent Value - annuityWhat is the PV of $1,000 at the end of each of the What is the PV of $1,000 at the end of each of the
next 3 years, if the opportunity cost is 8%?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 - annuityPresent Value - annuityWhat is the PV of $1,000 at the end of each of the What is the PV of $1,000 at the end of each of the
next 3 years, if the opportunity cost is 8%?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 - annuityPresent Value - annuityWhat is the PV of $1,000 at the end of each of the What is the PV of $1,000 at the end of each of the
next 3 years, if the opportunity cost is 8%?next 3 years, if the opportunity cost is 8%?
Other Cash Flow PatternsOther Cash Flow Patterns
0 1 2 3
PerpetuitiesPerpetuities
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 PerpetuityPresent Value of a Perpetuity
When we find the PV of an annuity, When we find the PV of an annuity, we think of the following we think of the following relationship:relationship:
Present Value of a PerpetuityPresent 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 ) =
Mathematically, Mathematically,
(PVIFA i, n ) = (PVIFA i, n ) = 1 - 1 - 11
(1 + i)(1 + i)nn
ii
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,
When n gets very large,When n gets very large,
this becomes zero.this becomes zero.1 -
1
(1 + i)n
i
1 - 1
(1 + i)n
i
1 1 i i
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 =
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 PMT
iiPV =PV = ==
$10,000 $10,000
.08.08
= = $125,000$125,000
Is this an annuity?Is this an annuity? 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).
Cash FlowsCash Flows
00 11 22 33 44
-10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000
Uneven Cash FlowsUneven Cash Flows
00 11 22 33 44
-10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000
Uneven Cash FlowsUneven Cash Flows
00 11 22 33 44
-10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000
Uneven Cash FlowsUneven Cash Flows
00 11 22 33 44
-10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000
Uneven Cash FlowsUneven Cash Flows
00 11 22 33 44
-10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000
Uneven Cash FlowsUneven Cash Flows
00 11 22 33 44
-10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000
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 11 22 33 44
-10,000 2,000 4,000 6,000 7,000-10,000 2,000 4,000 6,000 7,000
ExampleExample
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 30 30 yearsyears??
Retirement ExampleRetirement 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
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 = $1,397,985.65FV = $1,397,985.65
00 11 22 33 . . . 360. . . 360
400 400 400 400400 400 400 400
Retirement ExampleRetirement Example If you invest $400 at the end of each month for the If you invest $400 at the end of each month for the next 30 years at 12%, how much would you have at next 30 years at 12%, how much would you have at
the end of year 30? 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 ExampleRetirement Example If you invest $400 at the end of each month for the If you invest $400 at the end of each month for the next 30 years at 12%, how much would you have at next 30 years at 12%, how much would you have at
the end of year 30? 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 ExampleRetirement Example If you invest $400 at the end of each month for the If you invest $400 at the end of each month for the next 30 years at 12%, how much would you have at next 30 years at 12%, how much would you have at
the end of year 30? 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 ) )
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 at 7%$100,000 at 7% fixed fixed interest for interest for 30 years30 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 ExampleHouse Payment Example
House Payment ExampleHouse Payment Example
If you borrow $100,000 at 7% fixed If you borrow $100,000 at 7% fixed interest for 30 years in order to interest for 30 years in order to buy a house, what will be your buy a house, what will be your
monthly house payment?monthly house payment?
0 1 2 3 . . . 360
? ? ? ?
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 = -$665.30PMT = -$665.30
00 11 22 33 . . . 360. . . 360
? ? ? ?? ? ? ?
House Payment ExampleHouse 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 ) )
House Payment ExampleHouse 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 ) )
11
PV = PMT 1 - (1 + i)PV = PMT 1 - (1 + i)nn
ii
House Payment ExampleHouse 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 ) )
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
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