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© 2003 The McGraw-Hill Companies, Inc. All rights
Discounted Cash
Flow Valuation
(Formulas)
Chapter
Six
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6.2 Key Concepts and Skills
• Be able to compute the future value ofmultiple cash flows
• Be able to compute the present value of
multiple cash flows• Be able to compute loan payments
• Be able to find the interest rate on a loan
• Understand how loans are amortized or paidoff
• Understand how interest rates are quoted
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6.3 Chapter Outline
• Future and resent !alues of "ultiple #ashFlows
• !aluin$ %evel #ash Flows& 'nnuities and
erpetuities• #omparin$ (ates& )he *ffect of #ompoundin$
eriods
• %oan )ypes and %oan 'mortization
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6.+ Multiple Cash Flows –Future Value Eample !"#
• Find the value at year 3 of each cash flowand add them to$ether. , )oday -year /& F! 0 1-./3 0 41.5
, ear & F! 0 +4-./2 0 +4667.6
, ear 2& F! 0 +4-./ 0 +432
, ear 3& value 0 +4
, )otal value in 3 years 0 1.5 8 +667.6 8 +32
8 + 0 243.7• !alue at year + 0 243.7-./ 0 2347+1.1
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6.7 Multiple Cash Flows – FV Eample $
• 9uppose you invest :7 in a mutual fundtoday and :6 in one year. ;f the fund pays
5< annually4 how much will you have in two
years= , F! 0 7-.5/2 8 6-.5/ 0 2+.7
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6.6 Multiple Cash Flows – Eample $ Continued
• >ow much will you have in 7 years if youma?e no further deposits=
• First way& , F! 0 7-.5/7 8 6-.5/+ 0 66.26
• 9econd way , use value at year 2&
, F! 0 2+.7-.5/3 0 66.26
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6.1 Multiple Cash Flows – FV Eample %
• 9uppose you plan to deposit : into anaccount in one year and :3 into the account
in three years. >ow much will be in the
account in five years if the interest rate is
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6. Multiple Cash Flows – &resent Value Eample !"%
• Find the ! of each cash flows and add them , ear #F& 2 @ -.2/ 0 1.71
, ear 2 #F& + @ -.2/2 0 3.
, ear 3 #F& 6 @ -.2/
3
0 +21.1 , ear + #F& @ -.2/+ 0 7.+
, )otal ! 0 1.71 8 3. 8 +21.1 8 7.+ 0
+32.53
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6.5 Eample !"% 'imeline
2 3 +
2 + 6 1.71
3.
+21.1
7.+
+32.53
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6. Multiple Cash Flows sin a Spreadsheet
• ou can use the ! or F! functions in *Acelto find the present value or future value of a
set of cash flows
• 9ettin$ the data up is half the battle , if it isset up properly4 then you can ust copy the
formulas
•#lic? on the *Acel icon for an eAample
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6. Multiple Cash Flows – &V *nother Eample
• ou are considerin$ an investment that will pay you : in one year4 :2 in two years
and :3 in three years. ;f you want to earn
< on your money4 how much would you be
willin$ to pay=
, ! 0 @ -./ 0 55.5
, ! 0 2 @ -./2 0 672.5
, ! 0 3 @ -./3 0 2273.5+
, ! 0 55.5 8 672.5 8 2273.5+ 0 +7.53
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6.2 Multiple ne+en Cash Flows – sinthe Calculator
•'nother way to use the financial calculator foruneven cash flows is to use the cash flow ?eys
, )eAas ;nstruments B'C;; lus
• ress #F and enter the cash flows be$innin$ with year .
• ou have to press the D*nterE ?ey for each cash flow• Use the down arrow ?ey to move to the neAt cash flow
• )he DFE is the number of times a $iven cash flow occurs in
consecutive years
• Use the ! ?ey to compute the present value by enterin$the interest rate for ;4 pressin$ the down arrow and then
compute
• #lear the cash flow ?eys by pressin$ #F and then #%( Gor?
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6.3 Decisions, Decisions
• our bro?er calls you and tells you that he has this$reat investment opportunity. ;f you invest :today4 you will receive :+ in one year and :17 in twoyears. ;f you require a 7< return on investments of
this ris?4 should you ta?e the investment= , Use the #F ?eys to compute the value of the
investment
• #FH #F 0 H # 0 +H F 0 H #2 0 17H F2 0
• !H ; 0 7H #) ! 0 5.+5 , o , the bro?er is char$in$ more than you would be
willin$ to pay.
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6.+ Sa+in For -etirement
• ou are offered the opportunity to put somemoney away for retirement. ou will receive
five annual payments of :274 each
be$innin$ in + years. >ow much would you
be willin$ to invest today if you desire an
interest rate of 2
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6.7 Sa+in For -etirement 'imeline
2 I 35 + + +2 +3 ++
I 27J 27J 27J 27J 27J
otice that the year cash flow 0 -#F 0 /
)he cash flows years , 35 are -# 0 H F 0 35/
)he cash flows years + , ++ are 274 -#2 0 274H
F2 0 7/
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6.6 .uick .ui/ – &art 0
• 9uppose you are loo?in$ at the followin$ possible cash flows& ear #F 0 :H ears
2 and 3 #Fs 0 :2H ears + and 7 #Fs 0
:3. )he required discount rate is 1<
• Ghat is the value of the cash flows at year 7=
• Ghat is the value of the cash flows today=
• Ghat is the value of the cash flows at year 3=
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6.1 *nnuities and &erpetuities De1ined
• 'nnuity , finite series of equal payments thatoccur at re$ular intervals
, ;f the first payment occurs at the end of the period4
it is called an ordinary annuity
, ;f the first payment occurs at the be$innin$ of the
period4 it is called an annuity due
• erpetuity , infinite series of equal payments
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6. *nnuities and &erpetuities – 2asic Formulas
• erpetuity& ! 0 # @ r • 'nnuities&
−+=
+−=
r
r C FV
r
r C PV
t
t
2/2-
/2-22
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6.5 *nnuities and the Calculator
• ou can use the ") ?ey on the calculator forthe equal payment
• )he si$n convention still holds
• Krdinary annuity versus annuity due , ou can switch your calculator between the two
types by usin$ the 2nd BL 2nd 9et on the ); B'C;;
lus
, ;f you see DBLE or DBe$inE in the display of
your calculator4 you have it set for an annuity due
, "ost problems are ordinary annuities
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6.2 *nnuity – Eample !"3
• ou borrow money )KM' so you need tocompute the present value.
, + H ;@H C632 ")H #) ! 0 234555.7+
-:2+4/
• Formula&
7+.555423.
/.-
632+
=
−
= PV
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6.2 *nnuity – Sweepstakes Eample
• 9uppose you win the ublishers#learin$house : million sweepsta?es. )he
money is paid in equal annual installments of
:3334333.33 over 3 years. ;f the appropriate
discount rate is 7
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6.22 2uyin a 4ouse
• ou are ready to buy a house and you have :24
for a down payment and closin$ costs. #losin$ costs
are estimated to be +< of the loan value. ou have an
annual salary of :364 and the ban? is willin$ to
allow your monthly mort$a$e payment to be equal to2< of your monthly income. )he interest rate on the
loan is 6< per year with monthly compoundin$ -.7<
per month/ for a 3Cyear fiAed rate loan. >ow much
money will the ban? loan you= >ow much can youoffer for the house=
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6.23 2uyin a 4ouse 5 Continued
• Ban? loan , "onthly income 0 364 @ 2 0 34
, "aAimum payment 0 .2-34/ 0 +
, ! 0 +N , @.736O @ .7 0 +47
• )otal rice
, #losin$ costs 0 .+-+47/ 0 746+
, Mown payment 0 24 , 76+ 0 +4356
, )otal rice 0 +47 8 +4356 0 7+47
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6.2+ *nnuities on the Spreadsheet 5 Eample
• )he present value and future value formulas ina spreadsheet include a place for annuity
payments
•#lic? on the *Acel icon to see an eAample
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6.27 .uick .ui/ – &art 00
• ou ?now the payment amount for a loan andyou want to ?now how much was borrowed.Mo you compute a present value or a futurevalue=
• ou want to receive 7 per month inretirement. ;f you can earn .17< per monthand you eApect to need the income for 27
years4 how much do you need to have in youraccount at retirement=
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6.26 Findin the &ayment
• 9uppose you want to borrow :24 for anew car. ou can borrow at < per year4
compounded monthly -@2 0 .66661< per
month/. ;f you ta?e a + year loan4 what is your
monthly payment=
, 24 0 #N , @ .66661+O @ .66661
, # 0 +.26
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6.21 Findin the &ayment on a Spreadsheet
• 'nother )!" formula that can be found in aspreadsheet is the payment formula
, ")-rate4nper4pv4fv/
, )he same si$n convention holds as for the ! and
F! formulas
• #lic? on the *Acel icon for an eAample
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6.2 Findin the 6um7er o1 &ayments – Eample !"!
• 9tart with the equation and remember yourlo$s.
, 0 2- , @.7t/ @ .7
, .17 0 , @ .7t
, @ .7t 0 .27
, @ .27 0 .7t
, t 0 [email protected]/ @ ln-.7/ 0 53. months 0 1.17
years
• 'nd this is only if you donPt char$e anythin$
more on the cardQ
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6.25 Findin the 6um7er o1 &ayments – *notherEample
• 9uppose you borrow :2 at 7< and you are$oin$ to ma?e annual payments of :13+.+2.
>ow lon$ before you pay off the loan=
, 2 0 13+.+2- , @.7t/ @ .7
, .36665 0 , @.7t
, @.7t 0 .6333
, .7162+21 0 .7t
, t 0 ln-.7162+21/ @ ln-.7/ 0 3 years
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6.3 Findin the -ate
• 9uppose you borrow :4 from your parents to buy a car. ou a$ree to pay :21.7
per month for 6 months. Ghat is the
monthly interest rate=
, 9i$n convention mattersQQQ
, 6
, 4 !
, C21.7 ")
, #) ;@ 0 .17<
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6.3 *nnuity – Findin the -ate 8ithout aFinancial Calculator
• )rial and *rror rocess , #hoose an interest rate and compute the ! ofthe payments based on this rate
, #ompare the computed ! with the actual loan
amount , ;f the computed ! R loan amount4 then the
interest rate is too low
, ;f the computed ! S loan amount4 then theinterest rate is too hi$h
, 'dust the rate and repeat the process until thecomputed ! and the loan amount are equal
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6.32 .uick .ui/ – &art 000
•ou want to receive :7 per month for theneAt 7 years. >ow much would you need todeposit today if you can earn .17< per month=
• Ghat monthly rate would you need to earn if
you only have :24 to deposit=• 9uppose you have :24 to deposit and
can earn .17< per month.
, >ow many months could you receive the:7 payment=
, >ow much could you receive every month for7 years=
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6.33 Future Values 1or *nnuities
•9uppose you be$in savin$ for your retirement by depositin$ :2 per year in an ;('. ;f the
interest rate is 1.7
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6.3+ *nnuity Due
•ou are savin$ for a new house and you put:4 per year in an account payin$ ow much
will you have at the end of 3 years=
, F! 0 4N-.3 , / @ .O-./ 0 3746.2
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6.37 *nnuity Due 'imeline
2 3
324+6+
3746.2
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6.36 &erpetuity – Eample !"9
•erpetuity formula& ! 0 # @ r
• #urrent required return&
, + 0 @ r
, r 0 .27 or 2.7< per quarter • Mividend for new preferred&
, 0 # @ .27
, # 0 2.7 per quarter
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6.31 .uick .ui/ – &art 0V
• ou want to have : million to use for retirement in
37 years. ;f you can earn < per month4 how much
do you need to deposit on a monthly basis if the first
payment is made in one month=
• Ghat if the first payment is made today=• ou are considerin$ preferred stoc? that pays a
quarterly dividend of :.7. ;f your desired return is
3< per quarter4 how much would you be willin$ to
pay=
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6.3 8ork the 8e7 Eample
•'nother online financial calculator can befound at "oney#himp
• #lic? on the web surfer and wor? the
followin$ eAample
, #hoose calculator and then annuity
, ou ust inherited :7 million. ;f you can earn 6<
on your money4 how much can you withdraw each
year for the neAt + years=
, Matachimp assumes annuity dueQQQ
, ayment 0 :334+51.
http://www.datachimp.com/
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6.+ E11ecti+e *nnual -ate (E*-)
• )his is the actual rate paid -or received/ after
accountin$ for compoundin$ that occurs durin$ the
year
• ;f you want to compare two alternative investments
with different compoundin$ periods you need tocompute the *'( and use that for comparison.
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6.+ *nnual &ercentae -ate
•)his is the annual rate that is quoted by law• By definition '( 0 period rate times thenumber of periods per year
• #onsequently4 to $et the period rate werearran$e the '( equation& , eriod rate 0 '( @ number of periods per year
• ou should *!*( divide the effective rate
by the number of periods per year , it will K) $ive you the period rate
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6.+2 Computin *&-s
• Ghat is the '( if the monthly rate is .7
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6.+3 'hins to -emem7er
•ou '%G'9 need to ma?e sure that theinterest rate and the time period match.
, ;f you are loo?in$ at annual periods4 you needan annual rate.
, ;f you are loo?in$ at monthly periods4 youneed a monthly rate.
• ;f you have an '( based on monthly
compoundin$4 you have to use monthly periods for lump sums4 or adust the interestrate appropriately if you have payments otherthan monthly
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6.++ Computin E*-s 5 Eample
• 9uppose you can earn < per month on : investedtoday.
, Ghat is the '(= -2/ 0 2<
, >ow much are you effectively earnin$=
•F! 0 -./
2
0 .26• (ate 0 -.26 , / @ 0 .26 0 2.6<
• 9uppose if you put it in another account4 you earn 3< per quarter.
, Ghat is the '(= 3-+/ 0 2< , >ow much are you effectively earnin$=
• F! 0 -.3/+ 0 .277
• (ate 0 -.277 , / @ 0 .277 0 2.77<
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6.+7 E*- 5 Formula
m
'( *'(
m
−
+=
(emember that the '( is the quoted rate
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6.+6 Decisions, Decisions 00
• ou are loo?in$ at two savin$s accounts. Kne
pays 7.27
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6.+1 Decisions, Decisions 00 Continued
• %etPs verify the choice. 9uppose you invest
: in each account. >ow much will you
have in each account in one year=
, First 'ccount&
• Maily rate 0 .727 @ 367 0 .+33762
• F! 0 -.+33762/367 0 7.35
, 9econd 'ccount&
• 9emiannual rate 0 .735 @ 2 0 .267• F! 0 -.267/2 0 7.31
• ou have more money in the first account.
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6.+ Computin *&-s 1rom E*-s
• ;f you have an effective rate4 how can you
compute the '(= (earran$e the *'(
equation and you $et&
+= C*'(/-m'( m
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6.+5 *&- 5 Eample
• 9uppose you want to earn an effective rate of
2< and you are loo?in$ at an account that
compounds on a monthly basis. Ghat '(
must they pay=
[ ]
.35
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6.7 Computin &ayments with *&-s
• 9uppose you want to buy a new computer system and
the store is willin$ to sell it to allow you to ma?e
monthly payments. )he entire computer system costs
:37. )he loan period is for 2 years and the interest
rate is 6.5< with monthly compoundin$. Ghat isyour monthly payment=
, "onthly rate 0 .65 @ 2 0 .+333333
, umber of months 0 2-2/ 0 2+
, 37 0 #N , @ .+333333/2+O @ .+333333
, # 0 12.
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6.7 Future Values with Monthly Compoundin
• 9uppose you deposit :7 a month into an
account that has an '( of 5ow much will you
have in the account in 37 years=
, "onthly rate 0 .5 @ 2 0 .17
, umber of months 0 37-2/ 0 +2
, F! 0 7N.17+2 , O @ .17 0 +145.22
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6.72 &resent Value with Daily Compoundin
• ou need :74 in 3 years for a new car. ;f
you can deposit money into an account that
pays an '( of 7.7< based on daily
compoundin$4 how much would you need to
deposit= , Maily rate 0 .77 @ 367 0 .76+53
, umber of days 0 3-367/ 0 57
, F! 0 74 @ -.76+53/57 0 241.76
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6.73 Continuous Compoundin
• 9ometimes investments or loans are fi$ured
based on continuous compoundin$
• *'( 0 eq ,
, )he e is a special function on the calculator
normally denoted by eA
• *Aample& Ghat is the effective annual rate of
1< compounded continuously=
, *'( 0 e.1 , 0 .127 or 1.27<
6 + . i k . i & V
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6.7+ .uick .ui/ – &art V
• Ghat is the definition of an '(=
• Ghat is the effective annual rate=
• Ghich rate should you use to compare
alternative investments or loans=• Ghich rate do you need to use in the time
value of money calculations=
6 77 & Di t : E l ! #$
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6.77 &ure Discount :oans – Eample !"#$
• )reasury bills are eAcellent eAamples of purediscount loans. )he principal amount is repaidat some future date4 without any periodicinterest payments.
• ;f a )Cbill promises to repay :4 in 2months and the mar?et interest rate is 1 percent4 how much will the bill sell for in themar?et= , ! 0 4 @ .1 0 53+7.15
6 76 0 t t O l : E l
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6.76 0nterest Only :oan 5 Eample
• #onsider a 7Cyear4 interest only loan with a
1< interest rate. )he principal amount is
:4. ;nterest is paid annually.
, Ghat would the stream of cash flows be=
• ears , +& ;nterest payments of .1-4/ 0 1
• ear 7& ;nterest 8 principal 0 41
• )his cash flow stream is similar to the cash
flows on corporate bonds and we will tal?about them in $reater detail later.
6 71 *morti/ed :oan with Fied &rincipal &ayment 5
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6.71 *morti/ed :oan with Fied &rincipal &ayment 5Eample
• #onsider a :744 year loan at <
interest. )he loan a$reement requires the firm
to pay :74 in principal each year plus
interest for that year.
• #lic? on the *Acel icon to see the amortization
table
6 7
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6.7 *morti/ed :oan with Fied &ayment 5 Eample
• *ach payment covers the interest eApense plus
reduces principal
• #onsider a + year loan with annual payments. )he
interest rate is < and the principal amount is :7.
, Ghat is the annual payment=• +
• ;@
• 7 !
• #) ") 0 C75.6
• #lic? on the *Acel icon to see the amortization table
6 75 8 k th 8 7 E l
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6.75 8ork the 8e7 Eample
• )here are web sites available that can easily
prepare amortization tables
• #lic? on the web surfer to chec? out the #"B
"ort$a$e site and wor? the followin$ eAample
• ou have a loan of :274 and will repay the
loan over 7 years at < interest.
, Ghat is your loan payment=
, Ghat does the amortization schedule loo? li?e=
6 6 . i k . i & t V0
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6.6 .uick .ui/ – &art V0
• Ghat is a pure discount loan= Ghat is a $ood
eAample of a pure discount loan=
• Ghat is an interest only loan= Ghat is a $ood
eAample of an interest only loan=
• Ghat is an amortized loan= Ghat is a $ood
eAample of an amortized loan=