Wk2 tvom

The Time Value of MoneyThe Time Value of Money
Compounding andCompounding and
Discounting Single SumsDiscounting Single Sums

We know that receiving $1 today is worthWe know that receiving $1 today is worth
moremore than $1 in the future. This is duethan $1 in the future. This is due
toto opportunity costsopportunity costs..
The opportunity cost of receiving $1 inThe opportunity cost of receiving $1 in
the future is thethe future is the interestinterest we could havewe could have
earned if we had received the $1earned if we had received the $1
sooner.sooner.
Today Future

If we can measure this opportunityIf we can measure this opportunity
cost, we can: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

Compound InterestCompound Interest
and Future Valueand Future Value

Future Value - single sumsFuture Value - single sums
If you deposit $100 in an account earning 6%, howIf 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 (FVIFFV = PV (FVIF i,ni,n ))
FV = 100 (FVIFFV = 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)11
== $106$106
00 11
PV = -100PV = -100 FV =FV = 106106

If you deposit $100 in an account earning 6%, howIf 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?
FV = 100 (FVIFFV = 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)55
== $$133.82133.82
00 55
PV = -100PV = -100 FV =FV = 133.133.8282

FV = 100 (FVIFFV = 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) mxnmxn
FV = 100 (1.015)FV = 100 (1.015)2020
== $134.68$134.68
00 2020
PV = -100PV = -100 FV =FV = 134.134.6868
If you deposit $100 in an account earning 6% withIf 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?

FV = 100 (FVIFFV = 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) mxnmxn
FV = 100 (1.005)FV = 100 (1.005)6060
== $134.89$134.89
00 6060
PV = -100PV = -100 FV =FV = 134.134.8989
If you deposit $100 in an account earning 6% withIf 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?

00 100100
PV = -1000PV = -1000 FV =FV = $2.98m$2.98m
Future Value - continuous compoundingFuture Value - continuous compounding
What is the FV of $1,000 earning 8% withWhat is the FV of $1,000 earning 8% with
continuous compoundingcontinuous compounding, after 100 years?, after 100 years?
FV = PV (eFV = PV (e inin
))
FV = 1000 (eFV = 1000 (e .08x100.08x100
) = 1000 (e) = 1000 (e 88
))
FV =FV = $2,980,957.$2,980,957.9999

PV = FV (PVIFPV = FV (PVIF i,ni,n ))
PV = 100 (PVIFPV = 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)11
== $94.34$94.34
PV =PV = -94.-94.3434
FV = 100FV = 100
00 11
Present Value - single sumsPresent Value - single sums
If you receive $100 one year from now, what is theIf you receive $100 one year from now, what is the
PV of that $100 if your opportunity cost is 6%?PV of that $100 if your opportunity cost is 6%?

PV = FV / (1 + i)PV = FV / (1 + i)nn
PV = 100 / (1.06)PV = 100 / (1.06)55
== $74.73$74.73
If you receive $100 five years from now, what is theIf you receive $100 five years from now, what is the
PV of that $100 if your opportunity cost is 6%?PV of that $100 if your opportunity cost is 6%?
00 55
PV =PV = -74.-74.7373
FV = 100FV = 100

PV = FV / (1 + i)PV = FV / (1 + i)nn
PV = 100 / (1.07)PV = 100 / (1.07)1515
== $362.45$362.45
What is the PV of $1,000 to be received 15 yearsWhat 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%?
00 1515
PV =PV = -362.-362.4545
FV = 1000FV = 1000

00 55
PV =PV = FV =FV =
If you sold land for $11,933 that you bought 5If you sold land for $11,933 that you bought 5
years ago for $5,000, what is your annual rate ofyears ago for $5,000, what is your annual rate of
return?return?

5,000 = 11,933 (PVIF5,000 = 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)55
.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
If you sold land for $11,933 that you bought 5If you sold land for $11,933 that you bought 5
years ago for $5,000, what is your annual rate ofyears ago for $5,000, what is your annual rate of
return?return?

Suppose you placed $100 in an account that paysSuppose you placed $100 in an account that pays
9.6% interest, compounded monthly. How long9.6% 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 =

Suppose you placed $100 in an account that paysSuppose you placed $100 in an account that pays
9.6% interest, compounded monthly. How long9.6% 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?
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 present value andIn every single sum present value and
future value problem, there are fourfuture value problem, there are four
variables:variables:
FVFV,, PVPV,, ii andand nn..
 When doing problems, you will be givenWhen doing problems, you will be given
three variables and you will solve for thethree variables and you will solve for the
fourth variable.fourth variable.
 Keeping this in mind makes solving timeKeeping this in mind makes solving 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

 Annuity:Annuity: a sequence ofa sequence of equalequal cashcash
flows, occurring at the end of eachflows, occurring at the end of each
period.period.
0 1 2 3 4
AnnuitiesAnnuities

 If you buy a bond, you willIf you buy a bond, you will
receive equal semi-annual couponreceive equal semi-annual coupon
interest payments over the life ofinterest payments over the life of
the bond.the bond.
 If you borrow money to buy aIf you borrow money to buy a
house or a car, you will pay ahouse or a car, you will pay a
stream of equal payments.stream of equal payments.
Examples of Annuities:Examples of Annuities:

0 1 2 3
Future Value - annuityFuture Value - annuity
If you invest $1,000 each year at 8%, how muchIf you invest $1,000 each year at 8%, how much
would you have after 3 years?would you have after 3 years?

Calculator Solution:Calculator Solution:
P/Y = 1P/Y = 1 I = 8I = 8 N = 3N = 3
PMT = -1,000PMT = -1,000
FV =FV = $3,246.40$3,246.40
0 1 2 3
10001000 10001000 10001000

FV = PMT (FVIFAFV = PMT (FVIFA i,ni,n ))
FV = 1,000 (FVIFAFV = 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

0 1 2 3
Present Value - annuityPresent Value - annuity
What is the PV of $1,000 at the end of each of theWhat 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%?

P/Y = 1P/Y = 1 I = 8I = 8 N = 3N = 3
PMT = -1,000PMT = -1,000
PV =PV = $2,577.10$2,577.10
0 1 2 3
10001000 10001000 10001000

PV = PMT (PVIFAPV = PMT (PVIFA i, ni, n ))
PV = 1,000 (PVIFAPV = 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

Other Cash Flow PatternsOther Cash Flow Patterns
0 1 2 3
The Time Value of Money

PerpetuitiesPerpetuities
 Suppose you will receive a fixedSuppose you will receive a fixed
payment every period (month, year,payment every period (month, year,
etc.) forever. This is an example ofetc.) forever. This is an example of
a perpetuity.a perpetuity.
 You can think of a perpetuity as anYou can think of a perpetuity as an
annuityannuity that goes onthat goes on foreverforever..

Present Value of aPresent Value of a
PerpetuityPerpetuity
 When we find the PV of anWhen we find the PV of an annuityannuity,,
we think of the followingwe think of the following
relationship:relationship:
PV = PMT (PVIFAPV = PMT (PVIFA i, ni, n ))

Mathematically,Mathematically,
(PVIFA i, n ) =(PVIFA i, n ) =
We said that a perpetuity is anWe said that a perpetuity is an
annuity where n = infinity. Whatannuity where n = infinity. What
happens to this formula whenhappens 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

PMT
i
PV =
 So, the PV of a perpetuity is verySo, the PV of a perpetuity is very
simple to find:simple to find:
Present Value of a Perpetuity

What should you be willing to pay inWhat should you be willing to pay in
order to receiveorder to receive $10,000$10,000 annuallyannually
forever, if you requireforever, if you require 8%8% per yearper year
on the investment?on the investment?
PMT $10,000PMT $10,000
i .08i .08
= $125,000= $125,000
PV = =PV = =

Ordinary AnnuityOrdinary Annuity
vs.vs.
Annuity DueAnnuity Due
$1000 $1000 $1000$1000 $1000 $1000
4 5 6 7 8

Begin Mode vs. End ModeBegin Mode vs. End Mode
1000 1000 10001000 1000 1000
4 5 6 7 84 5 6 7 8
year year year
5 6 7

1000 1000 10001000 1000 1000
4 5 6 7 84 5 6 7 8
year year year
5 6 7
PVPV
inin
ENDEND
ModeMode
FVFV
inin
ENDEND
ModeMode

1000 1000 10001000 1000 1000
4 5 6 7 84 5 6 7 8
year year year
6 7 8

1000 1000 10001000 1000 1000
4 5 6 7 84 5 6 7 8
year year year
6 7 8
PVPV
inin
BEGINBEGIN
ModeMode
FVFV
inin
BEGINBEGIN
ModeMode

Earlier, we examined thisEarlier, we examined this
“ordinary” annuity:“ordinary” annuity:
Using an interest rate of 8%, weUsing an interest rate of 8%, we
find that:find that:
 TheThe Future ValueFuture Value (at 3) is(at 3) is
$3,246.40$3,246.40..
 TheThe Present ValuePresent Value (at 0) is(at 0) is
$2,577.10$2,577.10..
0 1 2 3
10001000 10001000 10001000

What about this annuity?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 theThe cash flows occur at the
beginningbeginning of each year, ratherof each year, rather
than at thethan at the endend of each year.of each year.
 This is anThis is an “annuity due.”“annuity due.”
0 1 2 3
10001000 10001000 10001000

0 1 2 3
-1000-1000 -1000-1000 -1000-1000
Future Value - annuity dueFuture Value - annuity due
If you invest $1,000 at the beginning of each of theIf you invest $1,000 at the beginning of each of the
next 3 years at 8%, how much would you have atnext 3 years at 8%, how much would you have at
the end of year 3?the end of year 3?
Mode = BEGIN P/Y = 1Mode = BEGIN P/Y = 1 I = 8I = 8
N = 3N = 3 PMT = -1,000PMT = -1,000
FV =FV = $3,506.11$3,506.11

Future Value - annuity dueFuture Value - annuity due
If you invest $1,000 at the beginning of each of theIf you invest $1,000 at the beginning of each of the
Mathematical Solution:Mathematical Solution: Simply compound the FV of theSimply compound the FV of the
ordinary annuity one more period:ordinary annuity one more period:
FV = PMT (FVIFAFV = PMT (FVIFA i,ni,n ) (1 + i)) (1 + i)
FV = 1,000 (FVIFAFV = 1,000 (FVIFA .08,3.08,3 ) (1.08)) (1.08) (use FVIFA table, or)(use FVIFA table, or)
- 1- 1
ii
FV = 1,000 (1.08)FV = 1,000 (1.08)33
- 1 =- 1 = $3,506.11$3,506.11
(1 + i)(1 + i)
(1.08)(1.08)

Mode = BEGIN P/Y = 1Mode = BEGIN P/Y = 1 I = 8I = 8
N = 3N = 3 PMT = 1,000PMT = 1,000
PV =PV = $2,783.26$2,783.26
0 1 2 3
10001000 10001000 10001000
Present Value - annuity duePresent Value - annuity due
What is the PV of $1,000 at the beginning of eachWhat is the PV of $1,000 at the beginning of each
of the next 3 years, if your opportunity cost is 8%?of the next 3 years, if your opportunity cost is 8%?

Present Value - annuity duePresent Value - annuity due
Mathematical Solution:Mathematical Solution: Simply compound the FV of theSimply compound the FV of the
ordinary annuity one more period:ordinary annuity one more period:
PV = PMT (PVIFAPV = PMT (PVIFA i,ni,n ) (1 + i)) (1 + i)
PV = 1,000 (PVIFAPV = 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 anIs this an annuityannuity??
 How do we find the PV of a cash flowHow do we find the PV of a cash flow
stream when all of the cash flows arestream when all of the cash flows are
different? (Use a 10% discount rate.)different? (Use a 10% discount rate.)
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

 Sorry! There’s no quickie for this one.Sorry! There’s no quickie for this one.
We have to discount each cash flowWe have to discount each cash flow
back separately.back separately.
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

periodperiod CFCF PV (CF)PV (CF)
00 -10,000-10,000 -10,000.00-10,000.00
11 2,0002,000 1,818.181,818.18
22 4,0004,000 3,305.793,305.79
33 6,0006,000 4,507.894,507.89
44 7,0007,000 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

Annual Percentage Yield (APY)Annual Percentage Yield (APY)
Which is the better loan:Which is the better loan:
 8%8% compoundedcompounded annuallyannually, or, or
 7.85%7.85% compoundedcompounded quarterlyquarterly??
 We can’t compare these nominal (quoted)We can’t compare these nominal (quoted)
interest rates, because they don’t include theinterest rates, because they don’t include the
same number of compounding periods persame number of compounding periods per
year!year!
We need to calculate the APY.We need to calculate the APY.

Annual Percentage Yield (APY)Annual Percentage Yield (APY)
 Find the APY for the quarterly loan:Find the APY for the quarterly loan:
 The quarterly loan is more expensive thanThe quarterly loan is more expensive than
the 8% loan with annual compounding!the 8% loan with annual compounding!
APY =APY = (( 1 +1 + )) mm
- 1- 1quoted ratequoted rate
mm
APY =APY = (( 1 +1 + )) 44
- 1- 1
APY = .0808, or 8.08%APY = .0808, or 8.08%
.0785.0785
44

Practice ProblemsPractice Problems

ExampleExample
00 11 22 33 44 55 66 77 88
$0$0 00 00 00 4040 4040 4040 4040 4040
 Cash flows from an investment areCash flows from an investment are
expected to beexpected to be $40,000$40,000 per year at theper year at the
end of years 4, 5, 6, 7, and 8. If youend of years 4, 5, 6, 7, and 8. If you
require arequire a 20%20% rate of return, what israte of return, what is
the PV of these cash flows?the PV of these cash flows?

 This type of cash flow sequence isThis type of cash flow sequence is
often called aoften called a ““deferred annuitydeferred annuity.”.”
00 11 22 33 44 55 66 77 88
$0$0 00 00 00 4040 4040 4040 4040 4040

How to solve:How to solve:
1)1) Discount each cash flow back toDiscount each cash flow back to
time 0 separately.time 0 separately.
Or,Or,
00 11 22 33 44 55 66 77 88
$0$0 00 00 00 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 = 5PMT = 40,000; N = 5
PVPV33== $119,624$119,624
00 11 22 33 44 55 66 77 88
$0$0 00 00 00 4040 4040 4040 4040 4040

Then discount this single sum back toThen 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 00 00 00 4040 4040 4040 4040 4040

 The PV of the cash flowThe PV of the cash flow
stream isstream is $69,226$69,226..
69,22669,226
00 11 22 33 44 55 66 77 88
$0$0 00 00 00 4040 4040 4040 4040 4040
119,624119,624

Retirement ExampleRetirement Example
 After graduation, you plan to investAfter graduation, you plan to invest
$400$400 per month in the stock market.per month in the stock market.
If you earnIf you earn 12%12% per year on yourper year on your
stocks, how much will you havestocks, how much will you have
accumulated when you retire in 30accumulated 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 = 360N = 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

If you invest $400 at the end of each month for theIf you invest $400 at the end of each month for the

If you invest $400 at the end of each month for theIf you invest $400 at the end of each month for the
FV = PMT (FVIFAFV = PMT (FVIFA i,ni,n ))
FV = 400 (FVIFAFV = 400 (FVIFA .01,360.01,360 )) (can’t use FVIFA table)(can’t use FVIFA table)
- 1- 1
ii
FV = 400 (1.01)FV = 400 (1.01)360360
- 1 =- 1 = $1,397,985.65$1,397,985.65

House Payment ExampleHouse Payment Example
If you borrowIf you borrow $100,000$100,000 atat 7%7% fixedfixed
interest forinterest for 3030 years in order toyears in order to
buy a house, what will be yourbuy 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 ExampleHouse Payment Example
PV = PMT (PVIFAPV = PMT (PVIFA i,ni,n ))
100,000 = PMT (PVIFA100,000 = 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 AssignmentTeam Assignment
Upon retirement, your goal is to spendUpon retirement, your goal is to spend 55
years traveling around the world. Toyears traveling around the world. To
travel in style will requiretravel in style will require $250,000$250,000 perper
year at theyear at the beginningbeginning of each year.of each year.
If you plan to retire inIf you plan to retire in 3030 yearsyears, what are, what are
the equalthe equal monthlymonthly payments necessarypayments necessary
to achieve this goal? The funds in yourto achieve this goal? The funds in your
retirement account will compound atretirement account will compound at
10%10% annually.annually.

 How much do we need to have byHow much do we need to have by
the end of year 30 to finance thethe 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 250250 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 250250 250 250 250 250

 Now, assuming 10% annualNow, assuming 10% annual
compounding, what monthlycompounding, what monthly
payments will be required for youpayments will be required for you
to haveto have $1,042,466$1,042,466 at the end ofat the end of
year 30?year 30?
2727 2828 2929 3030 3131 3232 3333 3434 3535
250 250 250 250 250250 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 250250 250 250 250 250
1,042,4661,042,466

 So, you would have to placeSo, you would have to place $461.17$461.17 inin
your retirement account, which earnsyour retirement account, which earns
10% annually, at the end of each of the10% annually, at the end of each of the
next 360 months to finance the 5-yearnext 360 months to finance the 5-year
world tour.world tour.

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