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© 2003 The McGraw-Hill Companies, Inc. All rights

Discounted Cash

Flow Valuation

(Formulas)

Chapter

Six


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McGraw-Hill!Irwin © 2003 The McGraw-Hill Companies, Inc. All rights

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


7/60McGraw-Hill!Irwin © 2003 The McGraw-Hill Companies, Inc. All rights

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



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



6.5 Eample !"% 'imeline

2 3 +

2 + 6 1.71

3.

+21.1

7.+

+32.53



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



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



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?



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.



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



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/



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=



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



6. *nnuities and &erpetuities – 2asic Formulas

• erpetuity& ! 0 # @ r • 'nnuities&

−+=

+−=

r

r C FV

r

r C PV

t

t

2/2-

/2-22



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



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



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

http://www.cmbmortgage.com/


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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=

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