rigtltlY oF BII$IISSS rIrD Egol{omcsco8FogfiE flIlrrrcE (rnv*. roos)

.&s5rciflMf-Nr fuurt g/2olo

The Problans and p4ge numbers refer to problems in the texf Corporate FinanceFundamentals, byRWJ, McGraw-Hill, 96 ed. zoto.

la Problern 6-2. (Present Value and Multiple Cash Flows, p.179).

lb. Problem 6-3. (Future Value and Multiple Cash Flows, p.179)

lc. Problem G38 (GrowingAnruity, p.183)

ld. Problem 642. (Calculating Loan Payment with balloon paymen! p.l33)

2u hoblem 6-52. (Calculating hesent Values ofAnnuities, p.l8a)

2b. Problem 6-66, (Annuity Paynents, Part (a) only (Savings Plan) , p.1S6-7)

3a- Problem 6-12 (Calculating EAR given APR, p.180)

3b. Problem 6-13 (CalculatingAPR given EAR, p.180)

3c. Problem 6-51 (Calculating EA& p.18a)

4a. hoblem 6-53 (CalculatingAnnuities Due, p.18a)

4b. Problem 6-55 (Amortization with Equal Payments, p.l8a)

4c. Problem 6-56 (Amortization with Equal Principal Paynents, p.185)

5. Financial Planning, Calculating Annuity Values

David is 30 years old and wants to save to meet 3 objectives:

(1) He would like to retire 30 years from now with retirement income of $35,000 per

month for 20 years, with the 'first

paynent received 30 years and I month from now.

(2) He would like to purchase a house in China 10 years from now for about

$600,000.

(3) He would like to accumulate by age 80 an amount of $1,000,000 which he can pass

on to his relatives.

a. He can afford to save $6,000 per month for the next l0 years. Ifhe can earn an 8

percent EAR before he retires and 5 percent EAR after he retires, how much will he

have to save each month in years 11 through 30?

b. Apart from the 3 objectives, what other things has David missed that he should

consider in his retirement planning?

10.

C H A PTE R 6 Discounted Cash Flow Valuation

APR and EAR [LO4] Should lending laws be changed to require lenders to report

EARs instead of APRs? Why or whY not?

Time Value [LOf l On subsidized Stafford loans, a common source of financial aid

for college students, interest does not begin to accme until repayment begins. Who

receives a bigger subsidy, a freshman or a senior? Explain. In words, how'wouldyou go about valuing the subsidy on a subsidized Stafford loan?

Time Value [LO1] Eligibiliry for a subsidized Stafford loan is based on curent

financial need. However, both subsidized and unsubsidized Stafford loans are repaid

out of future income. Given this, do you see a possible objection to having two types?

Time Value [LO1] Aviatical settlement is a lump sum of money given to a

terminally ill individual in exchange for his life insurance policy. When the insured

person dies, the purchaser receives the payout from the life insurance policy. What

factors determine the value of the viatical settlement? Do you think such

settlements are ethical? Why or why not?

Present value and Multiple cash Flows [LOll seaborn co. has identified an

investment project with the following cash flows. If the discount rate is 10 percent,

what is the present value of these cash flows? What is the present value at

l8 percent? At24 percent?

BASIC(Questions 1-28)

Year Cash Flow

1 $9502 1,040

3 1 ,130

4 1,075

' Present Value and Multiple Cash Flows [LO'l] Investment X offers to pay you

::,$6,000 per year for nine years, whereas lnvestment Y offers to pay you $8'000 per

; year for siryears. Which of these cash flow streams has the higher present value ifi,iir Aft""*, *." it S percent? If the discount rate is 15 percent?

Future Value and Multiple Cash Flows [LOil Paradise, Inc', has identified an

investment project with the following cash flows. If the discount rate is 8 percent,

what is the future value of these cash flows in year 4? What is the future value at a

z.

discountrate of 11 percent? At24 percent?

Year Cash Flow

1 $9402 1,090

3 1,340

q 1,405:;

ilculatingAnnuity present Value [LOl] An investment offers $5,300 per year

: 15 years" with the first payment occurring one year from now' If the required -

irrn is Z percent, what is the value of the investment? What would the value be ifpayments occurred for 40 years? For 75 years? Forever?

tc:ulatingAnnuity cash Flows [Lol] If you put up $34,000 today in exchange

/.65 percent, 15-year annuity, what will the annual cash flow be?

ir.

a

PART 3 Valuationof FutureCashFtows

6.carcuratingAnnuitvvarueslt"l."-?:ffi#iJll"f lf:l'ii;:g:;nt#"*"'ur my;:"*:rl;.x 3i"T;:H:ffiil'n" present varue or the

savings?

7 . carcurating Annuitv varues L?11 11 fi jjff L:J,ff fi,fi::fl *"T:j $t,|-" t"-izo y"ars into an account paymg r-r'r

vou have r *" J"i""r-t io v"Lri no*"r"J *i11 you have if you make deposits

8.Hfl 'l",il:Annu*y*"":l:?:1,.:"#T$1 jiltlil,3i''#"i."*'ii':.

account l0 years from now' and vou;;3"'"iL*in"tr 6.8 p"r""nt interest, what

il" """""t' " the end of each Yll-"o *

e ffi$Hli:::;lH:::'ia 3'ffi""1#;ff:lffJ[1.;?l?;:;il.;;-

; :*ff*:'"::'ffittii[?"'j n" "'*0"

Pav Lire Insurance co is -

tryinsto*ur"',T?i",**;"u*i:l;iii-nli:i*W#ili*':f Ji'iiJt?o'"u"t' it ttre required return on tnrs lnvtrlurvrrr ru ' '- r

,,1*nt$il:'ffi y::"::.ti?#,,'eH"fffi:niJ:"'f;T;"T'J"nT':*:r^' Jr*.iut" totd yoo the policy costs $J /

rz2. tfi:;*l*"AR

[Lo4] Ftnd the EAR tn each orthe r:1]o\:::;:;::

1,3. CalculatingAPR[LO41CASES:

Find the APR, or shted rate' in each of the following

*

n 14.

n ls.

ililfiii"ifi:::[:ffi$ffL';;;,aty on"its business-ntial borrower, which o"J'*5rra you go to for a new

semiannually' As a potenuar uvrrv"Y" "

loan? ,- ^-^,tir (.orn- wants to earn an effecdve :

ili*ruti,,e,q.pl rT1]":11ffi::;.i:::I;ffi; ;;. *e b ank use s daly

;;;;ffi; on its consum:1,i"."'T::i'"'ffi;il5il;"quired uv. 11* to report

l'ffi"*r "" g::ffi;Jlf;fiffi'-t',', *,;;;"rio an uninrormed

io fot ntiur borrowers? Exptatn wrry u'r r*'- --

borrower' .r n,rr \xrhqr is the future value of $2'100 in 17 years

;;;;t"t Future values tLo 1 I wha1 tl}t""u"u semiannuallv'r

assuming an interest rate of 8'4 percent compo'16.

8o/o

16

1216

QuadertY

Monthly

DailY

lnfinite

Hutft-t o r't99 cotggYndol8.60/o

19.8

9.4

16.5

SemiannuallY

MonthlY

WeeklY

lnfinite

37.

38.

CHAPTER 5 Discounted Cash FtowValuation

GrowingAnnuity [LOIJ You have just won the lottery and will receive$1,000,000 in one year. You will receive payments for 30 years, which willincrease 5 percent per year. If the appropriate discount rate is 8 percent, what isthe present value of your winnings?

Growing Annuity [LO1] Your job pays you only once a year for all the work you didover the previous 12 months. Today, December 31, you just received your salary of$50,000 and you plan to spend all of it. Howevet you want to start saving forrethement beginning next year. You have decided that one year from today you willbegin depositing 5 percent ofyour annual salary in an account that will earn 11 percentper year. Your salary will increase at 4 percent per year throughout your career. Howmuch money will you have on the date of your retirement 40 years from today?

Present Value and Interest Rates [LOIJ What is the relationship between thevalue of an annuity and the level of interest rates? Suppose you just bought al5-year annuity of $9,000 per year at the current interest rate of 10 percent per year.What happens to the value of your investment if interest rates suddenly drop to5 percent? What if interest rates suddenly rise to l5 percent?

Calculating the Number of Payments [LO2] You're prepared to make monthlypayments of $340, beginning at the end of this month, into an account that pays6 percent interest compounded monthly. How many payments will you have madewhen your account balance reaches $20,000?

CalculatingAnnuity Present Values [LO2J You want to borrow $73,000 fromyour local bank to buy a new sailboat. You can afford to make monthly payments of$1,450, but no more. Assuming monthly compounding, what is the highest rate youcan afford on a 60-month APR loan?

Calculating Loan Payments [LOzl You need a 30-year, fixed-rate mortgage to nbuy a new home for $240,000. Your mortgage bank will lend you the money at a6.35 percent APR for this 360-month loan. However, you can afford monthlypayments of only $1,150, so you offer to pay off any remaining loan balance at theend of the loan in the form of a single balloon payment. How large will this balloonpayment have to be for you to keep your monthly payments at $1,150?

Present and Future Values [LO1] The present value of the following cash flowstream is $6,550 when discounted at l0 percent annually. What is the value of themissing cash flow?

Year Gash Flow

1 $1,7002?3 2,100

4 2,800

':Calculating Present Values [LOl] You just won the TVM Lottery. You willreceive $l million today plus another 10 annual payments that increase by

,000 per year. Thus, in one year, you receive $1.5 million. In two years youget $2 million, and so on. If the appropriate interest rate is 9 percent, what is the

value of your winnings?

versus APR [LO4] You have just purchased a new warehouse. To finance i21,1j

purchase, you've arranged for a 30-year mortgage loan for 80 percent ofthe),000 purchase price. The monthly payment on this loan will be $15,000.is the APR on this loan? The EAR?

39.

40.

41.

I

PART 3 Valuationof FutureCashFlows

a 46 ffi Tl!::lt,t#,*-l**::X$],$t{1,"$,'t*g;today' Given a relevant discount l1'fi;;;;" does the firm j

frrm make a Profit on this asset'l At

47. Present "ii"H;"n*'" *oI

$4'000 per vear' at a discount t::::;;t5;;"ived 25 vears

8 years rroi *-* and the last Luy^o , r-u"* annuiry pays $1,500 P"l3"1l:d

4s. vu'iunt" Interest Rates [Lo1]"^.:i-ilH;:'iitri"Li"'"" 'u*firt'1rffi::ilonthrv.,o'

::'*:mffihn!:"T:1T,'"-"T;*'""il17perce ,thereafter' what is the present "t::;;;;;" vour choice of 1'1TI"jS'"

4e.compu,inicashF,ows"::T:i+ji j"illti{5it}'f"Hll-,''lr"jtr;

accounts. Iivestment A is a t:-v1i.i p"lcent comej':T'vesrment,

atso good rorp"v*"I. and has an interest rate;;;'fu

Jo*l r:T,,:B

today for it to be wonhii i' u" 8 percent continuornlY::il;;;;"d to invest in

15 years' How much money lilt"ttt* ""-tu, *o"tt as investment A l5 years u*'r,'ri*

ftOtl Given an {tllll1"^?t-. ,a s0 :ru*#ffi tfffi'"'tt-*##5"T:i:*:-:;"

o"'*"*:*1Tff ll::T:,i"=.'-onon.":epp,Ty--q::-',;,i1fr |"J:*T'IT:"'*"s 1 i**:;:x ll* t"".fl'..t';fu"r',iig,1; r;q;:: l";41;"195;""

i+poo'"""uuse vou must t"nlin,ootii, or $2,a16'67Gurr,

have ro be quored?

"e*nTl:"?llfi:i",:TJ:l"li#ru;at rate wourd'

o*ili[T;T'llll'1JH;;;i'","1]A5_yearan*::r.:jfiJ":i"H#il;TlH

sz. galcuratine Present valu"t lii"i";,;iil,n:1t.:-po, monthty, what rs m" y{ul

puy**,, witt uegin 8 years tt;j'i;;;""nt.compounc rhree years from now, wt1.

from now. lf the- discount:1?Hi*il is the valut

"t *t ""tJtt

fi'u" Y?*^ji^o'1""''' --:"^ rtlo ooo o"'i

56.

s3 ix$ffiT*ffi':-;:i}4fi*dffi,'**l'f.;-,;

*1ffi,'ffii:ilJil:""jJ53:"ffi:"'"*;"*'" *t ;*"'iib' Suppose

'ou n*n to inn"'t th"

9'1t^Th;.ny? What if the paymeno t" - l"

value if the payments are an ordtnary '-' 'r"o? Whl

;ltr,ii ili-.- ,- i1plesent value, rhe ordinarv *Tn' or annuitv *t*jc' which lT'*:.T.t'.i3.ii"J'*nt this alwavs be rue?

has the highesttuturt- rr A{r vn,, **, ;;;; ;w sports "1.t:::"YH" #;;;itn"st future value? *1'y:;;;;;"t a new sports car from ltr'\$'i

s 4 :^*riltrU{ni=ffi",1;'::nlH"$m'* ::.:: :#''{' *ru,'.X',1"1'* ;:l'fi

il;tt'l Ji'r:"#y; T";.,ff

,il: fji#-.ss lffiT*"HJ:ft#$Tqi:'.;1#:J [lhti r: illli,"*''ow 1

lhT*::ll$l?ffi't$[]t:":*' i

CHAPTER Discounted Cash Flow Va{uation

56.

57.

Amortization with Equal Principal Payments ILOAj Rework Problem 55

assuming that the loan agreement calls for a principal reduction of $8,400 every

year instead of equal annual payments.

CalculatingAnnuity Values [LO1] Bilbo Baggins wants to save money to meet

three objectives. First, he would like to be able to retire 30 years from now withretirement income of $20,000 per month for 25 years, with the first payment

received 30 years and I month from now. Second, he would like to purchase a

cabin in Rivendell in l0 years at an estimated cost of $380,000. Third, after he

passes on at the end of the 25 years of withdrawals, he would like to leave an

inheritance of $900,000 to his nephew Frodo. He can afford to save $2,500 per

month for the next l0 years. Ifhe can earn a l0 percent EAR before he retires and a

7 percent EAR after he retires, how much will he have to save each month in years

1l through 30?

Calculating Annuity Yalues [LO1] After deciding to buy a new car, you can

either lease the car or purchase it on a three-year loan. The car you wish to buy

costs $32,000. The dealer has a special leasing arrangement where you pay $99

today and $450 per month for the next three years. Ifyou purchase the car, you willpay it off in monthly payments over the next three years at a 7 percent APR. You

believe you will be able to sell the car for $23,000 in three years. Should you buy

or lease the car? What break-even resale price in three years would make you

indifferent between buying and leasing?

CalculatingAnnuity Values [LOl] An All-Pro defensive lineman is in contract

negotiations. The team has offered the following salary structure:

Time Salary

0 $7,000,000

1 $4,500,000

2 $5,000,000

3 $6,000,000

4 $6,800,000

5 $7,900,000

6 $8,800,000

All salaries are to be paid in lump sums. The player has asked you as his agent to

renegotiate the terms. He wants a $9 million signing bonus payable today and a

value increase of $1,400,fi)0. He also wants an equal salary paid every

trhree months, with the first paycheck three months from now. If the interest rate is

.5 percent compounded daily, what is the amount of bis quarterly check? Assume

days in a year.

lnterest Loans [LOal This question illustrates what is known as

interest.Imagine you are discussing a loan with a somewhat unscrupulous

. You want to borrow $25,000 for one year. The interest rate is 15 percent.

and the lender agree that the interest on the loan will be .15 X $25,000 =750. So the lender deducts this interest amount from the loan up front and gives

r'$21,250. ln this case, we say that the discount is $3,750. What's wrong here?

Annuity Values [LO1] You :ue serving on a jury. A plaintiff is suingbity for injuries sustained after a freak street sweeper accident. In the trial,

testified that it will be five years before the plaintiff is able to return to

CHALLENGE(Questions 57-78)

58.

PA RT ] Valuation of Future Cash Flows

workrhe juryn*"o:".1,^:;"ii::J}t#il"T,illi#i5;I"u",n""ff t?i'***"

*,n1rue1runn:::Jlli:f iii:iilffi ;n'rne-nlif i*'sannual

salarv for *" '*t"*i'"i' *""ta.n"""#; fip60;d $5b'000' respectivelv'

(b) The present ;;;;;i;"" v"tt' ru** 'urulv'

Yoo *'ot"^the salarv will be

$55,000n*,"*.iiiiidb'odor".p^**o*,utr"'ing.(d)$20,000forcourtcosts.Assume that the ,ui*v puv*.r,' *::Tf;,*i$irj f n: ll::iF:t

month

**mt?::fl':Tffi#inT'Ti'i'ii'-*;ut'igt'"'orrowerinterest rate? t^^lrinc at a olle-Y€?f loan of

62f ii:#tTf;J*,:#T:"",HiHti:'i:'iiiitsi:'*i";;";:r"-1"*is simply t p"r"Jnt 1o"" p"r"-ntug" t"t"'t;t thaloan *lit^3t"tes similar to

this one are common with home *otgu;""' The interest rate quotadon in this

example ."quir"i,ii i",,"*y, ,o -n1r.$;

** ;: P: ti1ll up rront and repav

theloanlaterwithSpercent**":..y;;i.ut"*outoyouactualiybepayinghere?63. Calculating ;; ffi;oto t'ool^The interest rate on a one-vear loan is

quoted""'ni"""t;;;tI"q;*d4n"'^t-"t"*ni"ot"m;'wtratistheEAR?

644.'#il*'il"tffi";ltl*:f*m:ry"*19:1,"i'$240'000monsages ",

oi pJ*"'ianf cfars! | i;;t"* ryp1t:f:llfi"tt""Tlilt"*

the applicatio;;#*"; tt iyt^1"* Bank and Trust rs re

appricarion,,u""iJ,*i:'."":*::#*tt"lX;*;t$fl :,T"il"J:ffi .".;-i*

ru;**'":ffi:{i3ii:i"i:n'"'J;i;"* ggR bu'l

'lhis is no'[

required*'*".*.i*oablefeestp,"**"urvbecauserefundablefeesarepart.l.of the loan 'J"itt'un

a fee)' what ;:ff;fi; iltr'"t" two loans? *out *

" '

6s. Hf#;* "AR

with Add'on rnterest [Loa] This problem illustrates

deceptive wav or q""'i"e interest tlt"' 1{'Ja ' u-* ii1'*tlj1:ttn" that vou see

anadvertisem"^';;i;;Judy's'stereo-iltytrt*readssomethingtikethis:..$1,000Instantcl"uiiiiu,ii*pr"rn.","",.idJ.i.l'*P-aylLow,Low'..:..:,Monthly puv-"o,'.?'v*;r" "",-"**rrv

*." *i* all th]s P"*:,*d somebody has

spilled ink over the APR on the loan """;;';oo utt ttt" manager t-"t-,^*".

,,:ffi;il. . ,c hree years at 14 percent rnterest',in

Judy explains that rr you borrow $1'000 for t

121,^

three Years You will owe: I '

$1,000 X 1'14r = $t'000 X 1'41854: $1'481"54 "l

Now, Judy recognizes.that coming oq yith $l'481'54 all at once might be a strzun'

so she rets vou make "tow' rowmontnr::'t"T?*;*:ii: l's+rti = *t

::so she lets vou mautv l";; ;;;;;"okkeeping^work for her'

;;;, ""; though'hl't':i:,:.:::5:t"'::t ffi-i,,n"epn on this roan?

Is this a 14 percent roan? Why or why'not'/lrunatt:#:fr

is the EAR? whv do t"'Li""iliio"::, :1,:T-::*::":*".ent probremi, ,t" gnnf Why do you think thls ls sarrr nt problern. A ,

66. calculating A"d;;"*""".*: il Jhis is a classic'"tt'1T1, t'o;;;, *u*

time rine *lr n"rii,i roiuing ir. youl frie:d

is celebrating n"t j]tiil;I"il* u"_

and wants * ';;;;i;i;' T":':iTl,Ty,:il"1#:::;;'#;;;i9 ryfr i:;'ru'*T,Ji3:fl'*:"#'if 'il#'ilffiilont'eroombirthdavYour

CHAPTER 6 DiscountedCash FlowValuation

intends to invest her money in the local credit union, which offers 7 percent

per year. She wants to make equal annual payments on each birthday into

account established at the credit union for her retirement fund.

If she starts making these deposits on her 36th birthday and continues to make

deposits until she is 65 (the last deposit will be on her 65th birthday), what

amount must she deposit annually to be able to make the desired withdrawals at

iretirement?;i Suppose your friend has just inherited a large sum of money. Rather than making

il equat annual payments, she has decided to make one lump sum payment on her

35th birthday to cover her retirement needs. What amount does she have to

deposit?

i Suppose your friend's employer will contribute $1,500 to the accorrnt every year

i,i as part of the company's profit-sharing plan. In addition, your friend expects a

$150,000 distribution from a family trust fund on her 55th birthday, which she

will also put into the retirement account. What amount must she deposit

ir annually now to be able to make the desired withdrawals at retirement?

the Number of Periods [LO2] Your Christmas ski vacation was

but it unfortunately ran a bit over budget. AII is not lost: Youjust received an

in the mail to transfer your $10,000 balance from your current credit card,

ich charges an annual rate of 19.8 percent, to a new credit card charging a rate of2 percent. How much fastercould you pay the loan off by making your planned

payments of $200 with the new card? What if there was a2 percent fee

on any balances transferred?

Value and Multiple Cash Flows [LO1] An insurance company is offering

new policy to its customers. Typically, the policy is bought by a parent or

:andparent for a child at the child's birth. The details of the policy are as follows:

purchaser (say, the parent) makes the following six payments to the insurance

v:First birthday: $ 900

Second birthday: $ 900

Third birthday: $1 ,000

Fourth birthday: $1,000

Fifth birthday: $1 ,100

Sixth birthday: $1 ,100

After the child's sixth birthday, no more payments are made. When the child

ieaches'age 65, he or she receives $500,000. If the relevant interest rate is

12 percent for the first six years and 8 percent for all subsequent years, is the

policy worth buying?

ve

rrs:

r-Calculating a Balloon Payment [LO2] You have just atranged for a $750,000

.'mortgagetofinancethepurchaseofalargetractofland.Themortgagehasan8.1 pirient APR, and it calls for monthly payments over the next 30 years'

However, the loan has an eight-year balloon payment, meaning that the loan must

be paid off then. How big *iU m" balloon payment be?

,Calculating Intertst Rates [LO4l A financial planning service offers a college A'savings program. The plan calls for you to make six annual payments of $9,000

each, with the first payment occurring today, your child's 12th birthday. Beginning

on your child's l8th birthday, the plan will provide $20,000 per year for four years.

What return is this investment offering?