chap5 lecture

8/6/2019 Chap5 Lecture

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Chapter 5Chapter 5 -- The TimeThe TimeValue of MoneyValue of Money

2005, Pearson Prentice Hall


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The Time Value of MoneyThe Time Value of Money

Compounding andCompounding and

Discounting Single SumsDiscounting Single Sums


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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 $1 sooner.earned if we had received the $1 sooner.

Today Future


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If we can measure this opportunityIf we can measure this opportunity

cost, we can:cost, we can:


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Translate $1 today into its equivalent in the futureTranslate $1 today into its equivalent in the future(compounding)(compounding)..

Today

?

Future


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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


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Compound InterestCompound Interest

and Future Valueand Future Value


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Future ValueFuture Value -- single sumssingle 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?


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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 (FVIFi, 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 =PV = --100100 FV =FV = 106106


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much would you have in the account after 5 years?much would you have in the account after 5 years?


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much would you have in the account after 5 years?much would you have in the account after 5 years?


FV = PV (FVIFFV = PV (FVIF i, ni, n ))

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 =PV = --100100 FV =FV = 133.133.8282


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Future ValueFuture Value -- single sumssingle sumsIf 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 havein the account after 5 years?in the account after 5 years?


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FV = PV (FVIFFV = PV (FVIFi, ni, n

))

FV = 100 (FVIFFV = 100 (FVIF .015, 20.015, 20 )) (cantuse FVIFtable)(cantuse FVIFtable)

FV = PV (1 + i/m)FV = PV (1 + i/m) m x nm x n

FV = 100 (1.015)FV = 100 (1.015)2020 == $134.68$134.68

00 2020

PV =PV = --100100 FV =FV = 134.134.6868


quarterly compoundingquarterly compounding, how much would you have, how much would you havein the account after 5 years?in the account after 5 years?


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monthly compoundingmonthly compounding, how much would you have, how much would you havein the account after 5 years?in the account after 5 years?


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FV = PV (FVIFFV = PV (FVIF i, ni, n ))

FV = 100 (FVIFFV = 100 (FVIF .005, 60.005, 60 )) (cantuse FVIFtable)(cantuse FVIFtable)

FV = PV (1 + i/m)FV = PV (1 + i/m) m x nm x n

FV = 100 (1.005)FV = 100 (1.005)6060 == $134.89$134.89

00 6060

PV =PV = --100100 FV =FV = 134.134.8989


monthlycompoundingmonthly

compounding, how much would you have, how much would you havein the account after 5 years?in the account after 5 years?


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Future ValueFuture Value -- continuous compoundingcontinuous compoundingWhat is the FV of $1,000 earning 8% withWhat is the FV of $1,000 earning 8% with

continuouscompoundingcontinuouscompounding, after 100 years?, after 100 years?


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00 100100

PV =PV = --10001000 FV =FV = $2.98m$2.98m

Future ValueFuture Value -- continuous compoundingcontinuous compoundingWhat is the FV of $1,000 earning 8% withWhat is the FV of $1,000 earning 8% with

continuouscompoundingcontinuouscompounding, 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


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Present ValuePresent Value


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Present ValuePresent Value -- single sumssingle sumsIf 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%?


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PV = FV (PVIFPV = FV (PVIFi, 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 ValuePresent Value -- single sumssingle sumsIf you receive $100 one year from now, what is theIf you receive $100 one year from now, what is the



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Present ValuePresent Value -- single sumssingle sumsIf you receive $100 five years from now, what is theIf you receive $100 five years from now, what is the



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PV = FV (PVI

FPV = FV (PVI

F i, ni, n ))PV = 100 (PVIFPV = 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)55 == $74.73$74.73

Present ValuePresent Value -- single sumssingle sumsIf you receive $100 five years from now, what is theIf you receive $100 five years from now, what is the


00 55

PV =PV = --74.74.7373 FV = 100FV = 100


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Present ValuePresent Value -- single sumssingle sumsWhat 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%?


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PV = FV (PVIFPV = FV (PVIF i, ni, n ))

PV = 100 (PVIFPV = 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)1515 == $362.45$362.45

Present ValuePresent Value -- single sumssingle sumsWhat 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


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Present ValuePresent Value -- single sumssingle sumsIf you sold land for $11,933 that you bought 5 yearsIf 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?


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PV = FV (PVIFPV = FV (PVIF i, ni, n ))

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

Present ValuePresent Value -- single sumssingle sumsIf you sold land for $11,933 that you bought 5 yearsIf 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?


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Present ValuePresent Value -- single sumssingle sumsSuppose 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 longwill 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


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Hint for single sum problems:Hint for single sum problems:

In every single sum present value andIn every single sum present value andfuture value problem, there are fourfuture value problem, there are fourvariables:variables:

FVFV,, PVPV,, ii andand nn..

When doing problems, you will be givenWhen doing problems, you will be giventhree 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!


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The Time Value of MoneyThe Time Value of Money

Compounding and DiscountingCompounding and Discounting

Cash Flow StreamsCash Flow Streams

0 1 2 3 4


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Annuity:Annuity: a sequence ofa sequence ofequalequal cashcash

flows, occurring at the end of eachflows, occurring at the end of each

period.period.

0 1 2 3 4

AnnuitiesAnnuities


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If you buy a bond, you willIf you buy a bond, you will

receive equal semireceive equal semi--annual couponannual 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 ofAnnuities:Examples ofAnnuities:


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Future ValueFuture Value -- annuityannuityIf 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?


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FV = PMT (FVIFAFV = PMT (FVIFAi, 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 -- 11

ii

FV = 1,000 (1.08)FV = 1,000 (1.08)33 -- 1 =1 = $3246.40$3246.40

.08.08

Future ValueFuture Value -- annuityannuityIf 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?


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Present ValuePresent Value -- annuityannuityWhat 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%?


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PV = PMT (PVIFAPV = PMT (PVIFAi, ni, n ))

PV = 1,000 (PVIFAPV = 1,000 (PVIFA.08, 3.08, 3

) (use PVIFA table, or)) (use PVIFA table, or)

11

PV = PMT 1PV = PMT 1 -- (1 + i)(1 + i)nn

ii

11

PV = 1000 1PV = 1000 1 -- (1.08 )(1.08 )33 == $2,577.10$2,577.10

.08.08

Present ValuePresent Value -- annuityannuityWhat 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%?


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Other Cash Flow PatternsOther Cash Flow Patterns

0 1 2 3

The Time Value of Money


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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 anannuityannuity that goes onthat goes on foreverforever..


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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 followingrelationship:relationship:



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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?

11 --11

(1 + i)(1 + i)nn

ii


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When n gets very large,When n gets very large,

this becomes zero.this becomes zero.

So were left with PVIFA =So were left with PVIFA =

1

i

1 -1

(1 + i)n

i


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PMT

iPV =

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


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What should you be willing to pay inWhat should you be willing to pay in

order to receiveorder to receive $10,000$10,000 annuallyannuallyforever, if you requireforever, if you require 8%8% per yearper year

on the investment?on the investment?


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What should you be willing to pay inWhat should you be willing to pay in

order to receiveorder to receive $10,000$10,000 annuallyannuallyforever, 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 = =


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Ordinary AnnuityOrdinary Annuityvs.vs.

A

nnuity DueA

nnuity Due

$1000 $1000 $1000$1000 $1000 $1000

4 5 6 7 8


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Begin Mode vs. End ModeBegin Mode vs. End Mode

1000 1000 10001000 1000 1000

4 5 6 7 84 5 6 7 8year year year

5 6 7

PVPV

inin

ENDEND

ModeMode

FVFV

inin

ENDEND

ModeMode


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Begin Mode vs. End ModeBegin Mode vs. End Mode

1000 1000 10001000 1000 1000

4 5 6 7 84 5 6 7 8year year year

6 7 8

PVPV

inin

BEGINBEGIN

ModeMode

FVFV

inin

BEGINBEGIN

ModeMode


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Earlier, we examined thisEarlier, we examined this

ordinary annuity:ordinary annuity:

0 1 2 3

10001000 10001000 10001000


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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


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What about this annuity?What about this annuity?

Same 3Same 3--year time line,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, ratherthan 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

F t V lF t V l it dit d


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Future ValueFuture Value -- annuity dueannuity dueIf 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?

Mathematical Solution:Mathematical Solution: Simply compound the FV of theSimply compound the FV of theordinary annuity one more period:ordinary annuity one more period:

FV = PMT (FVIFAFV = PMT (FVIFAi, 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)

FV = PMT (1 + i)FV = PMT (1 + i)nn

-- 11ii

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)


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Present ValuePresent Value -- annuity dueannuity dueWhat 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%?

0 1 2 3


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Present ValuePresent Value -- annuity dueannuity due

Mathematical Solution:Mathematical Solution: Simply compound the FV of theSimply compound the FV of theordinary annuity one more period:ordinary annuity one more period:

PV = PMT (PVIFAPV = PMT (PVIFAi, 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 1PV = PMT 1 -- (1 + i)(1 + i)nn

ii

11

PV = 1000 1PV = 1000 1 -- (1.08 )(1.08 )33 == $2,783.26$2,783.26

.08.08

(1 + i)(1 + i)

(1.08)(1.08)


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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 aredifferent? (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 7,00010,000 2,000 4,000 6,000 7,000


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Sorry! Theres no quickie for this one.Sorry! Theres 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 7,00010,000 2,000 4,000 6,000 7,000

Uneven Cash FlowsUneven Cash Flows

10 000 2 000 4 000 6 000 7 00010 000 2 000 4 000 6 000 7 000


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periodperiod CFCF PV (CF)PV (CF)

00 --10,00010,000 --10,000.0010,000.0011 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 7,00010,000 2,000 4,000 6,000 7,000


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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 cant compare these nominal (quoted)We cant compare these nominal (quoted)

interest rates, because they dont include theinterest rates, because they dont include the

same number ofcompounding periods persame number ofcompounding periods per

year!year!

We need to calculate the APY.We need to calculate the APY.


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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 -- 11quoted ratequoted ratemm

APY =APY = (( 1 +1 + )) 44 -- 11

APY = .0808, or 8.08%APY = .0808, or 8.08%

.0785.0785

44


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Practice ProblemsPractice Problems


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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 areexpected 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 isthe PV of these cash flows?the PV of these cash flows?


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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


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Howtosolve:Howtosolve: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


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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


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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

R ti t E lR ti t E l


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Retirement ExampleRetirement Example

After graduation, you plan to investAfter graduation, you plan to invest

$400$400 per monthper month in the stock market.in the stock market.

If you earnIf you earn 12%12% per yearper year on youron your

stocks, how much will you havestocks, how much will you have

accumulated when you retire inaccumulated when you retire in 3030

yearsyears??

R ti t E lR ti t E l


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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



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

next 30 years at 12%, how much would you have atnext 30 years at 12%, how much would you have at

the end of year 30?the end of year 30?


FV = PMT (FVIFAFV = PMT (FVIFAi, ni, n ))

FV = 400 (FVIFAFV = 400 (FVIFA.01, 360.01, 360 )) (cant use FVIFA table)(cant use FVIFA table)

FV = PMT (1 + i)FV = PMT (1 + i)nn

-- 11ii

FV = 400 (1.01)FV = 400 (1.01)360360 -- 1 =1 = $1,397,985.65$1,397,985.65

.01.01

H P t E lH P t E l


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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?

H P t E lH P t E l


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House Payment ExampleHouse Payment Example



100,000 = PMT (PVIFA100,000 = PMT (PVIFA.07, 360.07, 360 )) (cant use PVIFA table)(cant use PVIFA table)

11

PV = PMT 1PV = PMT 1 -- (1 + i)(1 + i)nn

ii

11

100,000 = PMT 1100,000 = PMT 1 -- (1.005833 )(1.005833 )360360 PMT=$665.30PMT=$665.30

.005833.005833


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Team AssignmentTeam Assignment

Upon retirement, your goal is to spendUpon retirement, your goal is to spend 55years 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 yourretirement account will compound atretirement account will compound at

10%10% annually.annually.

250 250 250 250 250250 250 250 250 250


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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

250 250 250 250 250250 250 250 250 250


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Now, assuming 10% annualNow, assuming 10% annualcompounding, 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 ofyear 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


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So, you would have to placeSo, you would have to place $461.17$461.17 inin

your retirement account, which earnsyour retirement account, which earns10% annually, at the end of each of the10% annually, at the end of each of the

next 360 months to finance the 5next 360 months to finance the 5--yearyear

chap5 lecture

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