Finansal Yönetim - II The Capital Budgeting Process Part 4 The Capital Budgeting Process A second method for finding the present value of a deferred annuity is to: -t .-- Firrdthe-pres- ent value factotof an annuity for the total time period~ -In-this-- ---- ­ case, where n = 8, i = 8%, the PV IFA is 5.747. 2. Find the present value factor of an annuity for the total time period (8) minus the deferred annuity period (5). 8 - 5 = 3 n = 3, i = 8% The PV IFA value is 2.577. 3. Subtract the value in step 2 from the value in step 1, and multiply by A. 5.747 -2.577 3.170 3.170 X $1,000 = $3,170 (present value of the annuity) $3,170 is the same answer for the present value of the apIluity as that reached by the first method. The present value of the five-year annuity may now be added to the pres­ ent value of the inflows over the first three years to arrive at the total value. $5,022 Present value of first three period flows + 3, 170 Present value of five-year annuity . , $8;192 Total present value In working a time~value-of-money problem, the student should determine first whether the problem deals with future value or present value and second whether a single sum or an annuity is involved. The major calculations in Chapter 9 are sum- marized below. . 1. F1!-ture value of a single amount. Formula: FV = PV X FVIF Table: 9-1 'or Appendix A. When to use: In determining the future value of a single amount. Sampie problem: You invest $1,000 for four years at 10 percent interest. What is the value at the end of the fourth year? 2. Present value of a single amount. Formula: PV = FV X PV IF Table: 9-2 or Appendix B. < When to use: In determining the present value of an amount to be received in the future. " . Sample problem: You will receive $1,000 after four years at a discount rate of 1 0 percent. How much is this worth today? 3. Future value of an annuity. Formula: FV A - A X FVIFA Chapter 9 The Time Value of Money - Table: 9-3 or Appendix C . . When to use: In determining the future value of a series of consecutive, equal payments (an annuity). . . Sample proble~: You will receive $1,000 at the end of each period for four peri­ ods. What is ~e accumulated value (future worth) at the end of the fourth period if money grows at 10 percent? Present value of an annuity . . ,Formula: PYA ' = A X PV IFA Table: 9-4 or Appendix D. When to use: In determining the present worth of an annuity. Sample problem~ You will receive $1,000 at the end of each period for four years. At a discount rate of 10 percent, what is this cash flow currently worth? Annuity equaling a future value. ---- ' <, , ' FVA Formula: A = -~ ,', FViFA Table: 9~3 or AppendixC. , , . ' When to'use: 'In detennining the size of an annuity that will equal a future value. - .. , . S' ample' problem:' You need '$1 ,OOO-after:four,-periods;-:With'an-~te~es~rate qf . , - . , .. " must be set aside at the e~d of each period to accumulate .• ; 6. " AnnuitY i.,qualinga present value~ ': . -- PYA .... . .. ,' .... -.... ' , f,orm u1a : . A" ... PVIFA' , .,....,.,.... ........... -"" ...... .. ; ~ ". ,.- ...... .. :. . Table: '9,-4 or Appendix D. ·. ~. -'Wliento-use:-Iri~deteiTIlliiiiiglne-'size"ofan-annuiry-e1tliartO)r-given:-pre~-~nt value. . '. .' .: '.:' :-. . .... . . • • ...... ', - ", '., ... , •• ' ........ " .• ,. I ......... ~ __ ,~_.'"_ ••• • _, ••. .• ____ ._, •• Sample problems: a. What four-year annuity is the equivalent of $1,000 today with an interestrate of 10 percent? h. You deposit $1,000 today and wish to' withdraw funds equally'over'JoUT ., .... , .. . ' years. How much can you withdraw at the end of each year if funds earn 10 percent? c. You borrow $1,000 for four years at 10 percent interest. How much must be ......... ---.. ----.- .... ..:. 'fepaicrar toe eiio'of- eaclryear?---·- -_ .. _ .. · ...... -_···- - ,-.-.-.,- .... -... -~ .. _'.,i~ _ _ _ . ___ .~ .... - . Determining the yield on an investment. Formulas PV a. P"IF = - Tables 9-2, App'endix B 9-4, Appendix D .-l' .... Yield-present value of a single amount Yield-present value of an annuity . ',' ...•. _._, .... _.--... -.. - .-~---..•.. .. ~ ... ,- ... -. -.. - " ," ~--~----~~~--~.~---,v When to use: In detennining the interest rate (i)' that will equate an investment ------ --: w -=-:j '7 th LZ ::"; t :::- ur :::: e :-t:": e :;;:: n ::; efi ts . Sample problem: You invest $1,000 now, and the funds are expected to increase 1st of Tenns Iscusslon uestlons to $1 ,360 after fq 1.,!rperiods. . ' Whatis the yield on the investment? Use PV PV IF =­ FV .. -.- ..... ------.--~----- '-'- .---~ .... --- .. ---- .~---.---.. -~ --_ .. __ . _------_._----- - " 8. Less than annual compounding periods. Semiannual Quarterly Monthly Mu~ , Multiply.!0-1 Multiply~ Divide iby~ , Divide,.i.QY..! Divide i by 12 ------ [ thenuse1 normal formula When to use: If the compounOing period is more (or perhaps less) frequent than .~ once a year ", _' -~,. -'---------00--­ Sample problem: 'Yoti invest $1,000 compounded semiannually at 8 percent per annum over four yei1.rs; Detimnine the future value. 9. Pattem S'oTJJliYTii" eni2(Jej e'frelf liluiiiity. .. ,,'~.' .. ' Formulas P~A. x 'PV 1 FA ' , PV. =, Bl2.< PV 1F Tables 9-1, Appendix D } ;, Method 1 9-2, ,i\ppenc;iix B '. , . When to use: If an annuity begins in the future. Sample problem: You will receive $1,000 per period, starting at the end of the , fourth period and IUIiniilg through the end of the eighth period. With a discount rate of8 percent, determine the present value. 'Th~ 'shident is encouraged to work the many problems found at the end ~f the chapter. , future v~iue 256 ,interest factor . 257 present value 257 discount rate 257 annuity 258 future value of an annuity 258 pres~nt value of an annuity 260 yield 270 compounded semiannually 272 1. How is the future value (Appendix A) related t (Appendix B)? (L01) '. " 0 the present value of a single sqrn 2. How is the present value of a single sum (A' . . value, of an annuity (Appendix D)? (L03) ppendix B) related to the present 3. Why does money have a rill1e vlI luc' (LOI) ' h ore than a . . k' I IIII' 1 f\(I"y 1\1(1\ I 111 . 4. Does inflation have anything 10 do wl lh II HI 1\1 )1 /I , I' , dollar tomorrow? (LOJ) 2 t for . . I ' , I' nO llnl fl i J percen 5. Adjust the annual fOIlllUla for u fUlure vll lll e' II /I ~III .\ ~ II I . . ' I \' I I I ' III ' inl eresl factors 10 years to a semiannual compoulldtn 1 IMIlIlI II , ,,1ft \I c; (FV IF ) before and after? Why are they dinct't' lIt ' (l,0 5) . 6. If, as an investor, you had a choice r dnil y, mOlllhl )" or qu nrt erl y co mpoundm g, which would you choose? Why ? (LO. ) 7. What is a deferred annuity? (L04) 8. List five different financiaJ applicution~ of Ihe tim" vn lll or money. (L01) 1. a. You invest $12,000 today 1119 p'rcent per OJ', How mu h will you have after 15 years? b. What is the current vaJue of $100,000 afler ]0 y 1\1', if the di scount rate is 12 percent? c. You invest $2,000 a year for 20 y /u·s at 11 p ent. How much will you have after 20 years? 2. a. How much must Katie Wilson s t a id . a h y ar to accumulate $80,000 after 15 years? The interest rate is lOp rcent. b. How much must Josh Thomp on repay each year for'fiveyeadrto'pay off a' $20,000 loan that he just took out. The interest rate is 8 percen~, ' , \. , ' S~IUti~ns---'~'-~ 1. a, Thi!' is the future val1;le of a single arno~1, FV = PV X FVIF (n = 15, i = 9%) .---- --fV.-=- $12;.000 X 3.642 _ = $43,7.04 b, This is the present value of a single amoun1, PV = FV X PV IF (n = 10, i = 12%) PV = $100,000 X ,322 = $32,200 c. This is the future value of an annuity. FVA = A X lNIFA (n = 20, i = 11 %) FYA _ =:. $2,000, X _6:J- ,20~ = $128,406 Appendix A Appendix B AppendixC Practice problems and Solutions Future value Present value Future value (L02&3) Solving for an annuity 'Solving for an annuity (L04) " ... ~. -- .. 2. a, This calls for solving for an annuity to equal a future value. A = ~: (n = 15, i 10%) Appendix c, $38~~~~0 =_$_~,~l?~ V\\;~ \- ~~LAf.t .: " '---'--~~~~~--~ A b\ \'0 () " 3. Why does money have a time value? (LO}) . than a . . ki d lIa ' !Odoy worth \1101 e 4. Does inflation have anything to do wIth rna ng a 0 I . dollar tomorrow? (LO} ) t for . . al f . I am unt at J percell 5 Ad iust the annu al formula for a future v ue 0 a SlIlg e - . , . ill Wh t th intere t factors 10 years to a semi annual compoundmg foml a. . a are (FVlF) before and after? Why are they different . (L05) 6. If, as an investor, you had a choice of daily, monthly, or quarterl compounding, whi ch would you choose? Why? (L05) 7. What is a deferred annuity? (L04) 8. List five different fi nancial applications of the time value of money. (LO}) L a. b. c. 2. a. b. You invest $12,000 today at 9 percent per year. How much will you have after 15 years? c. What is the current value of $100;000 after 10 years if the discount rate is 12 percent? You invest $2,000 a year for 20 years at 11 percent. How much will you have after 20 years? How much must Katie Wilson set aside each year to accumulate $80,000 after 15 years? The interest rate is 10 percent. How much must Jos b Thompsoffrepay-eaclr year-foriive-years to]lay off a $20,000 loan that he just took out. The interest rate is 8 -percent. '----- SoluUons 1. a. This is the future val~e of a single am()unt ~ FV = PV ,X FVIF (n = 15, i = 9%) - Appendix A . .' ~ FV =- $12,.000 X .3.64L= _$43,.1.04 ________ _ b. This is the present value of a single amount. PV = FV X PV IF (n = 10, i = 12%) PV = $100,000 X .322 = $32,200 c. This is the future value of an annuity. FVA = A X lNIFA (n = 20, i = 11 %) FYA . =::.-$2LQQQ_.0_9.4,~QL'=_$128, 406 Appendix B AppendixC ~----.-----.........--.- 2. a. This calls for solving for an annuity to equal a future value. 'I A = FV A { (n = 15, i = 10%) A Practice Problems and Solutions Future value Present value Future value (L02&3) Solving for an annuity . Solving for an annuity (L04) tf~54,r 10.1' , ~~Ift~~ FV IFA ppendix C A =--~3 81~~~~0 = $2,517.94 V\ \ '; () . ~ ~ ~<.A f.t )o(U.~ ·~\~O d 278 Problems Present value (L03) . Present value . (L03) Present value (L03) luture value £02) The Capital Budgeting Process h. . PV '" A = _ A _ (n = 5, i = 8~) PV IFA A = $20,000 = ' $5,008.7'7 _. _" . ___ ... 3.993- .--.--. -.-'---.. - .. --.-.-------.-- rTT1 no til reface for more Information. ~ All Problems are available In Homework Manager. Please sea e p ~ You invest $3,000 a year for three years at 12 percent. .' ~ a. Wh~~ is the value of your investment after ones ear? Multiply $3,000 X 1.12. b. What is the value of your investment after two years? Multiply'your answer to part a by 1.12. '.':.' c. your after three years? Multiply your answer ' to part b by 1.12. Tbi,s .gives your final answer. . ,_ d. Co~ that your finalariswer is correct by going to AppeiidiX'A (future value of $1), and looking up the future value for n = 3, and i = 12 per­ cent. Multiply this tabular value by $3,000 and compare your answer to the answer in part c .. There may be a slight difference due to rounding . . m Whatis~epresentvalueof: .' _ ----,.---:.-~ ... --:. 'l\1it.\1r~ "-Y a. $9,000 ill 7 y ears- it 8 perc- e nt? r V ~ x= J b. $20,000 in 5 years at 10 percent? W- -l-- ' '" t\ c\G s,t\..\ r v:.J $10,000 in 25 years at 6 percent? . • t -I- \ I d. $1,000 in 50 years at 16 percent? 3. You will receive $5,000 three years from now. The discount ~ate is 8 percent .. a. What is the value of your investment two years from now? Multiply . r $5,000 X .'926 (one year's discount rate at 8 percent). . b. What is the value of your investment one year from now? Multiply 'your answer to part a by .926 (one year's discount rate at 8 percent). c. What is the value of your investment today? Multiply your ~swer to p~ b by .926 (one year's discount rate at 8 percent). ' d. Confirm that your answer to part c is correct by going to.Appendix,B (pres­ ent value of $1) for n = 3 and i = 8 percen .t. Multiply this tabular value by $5,000 and compare your answer to part c. There may be a slight difference due to rounding. , . o If you invest $9,000 today, how much will you have: a. In 2 years at 9 percent? b. In 7 years at 12 percent? c In 25 ears at 14 pe "nt'. )? . mpQunded semiannually . d. In 25 ears at 14 . Present value . ears or $95 today. If money IS 5. Your uncle offers y U:l h . 'f 0,000 III 50 y ? ~ (L~~) 7 discounted at 12 percellt, whi h &h uld you choose. /II ~} ~1~. Jane is . Present value h. Your aunt offers you a . b i f 0,000 in 40 year; or $85~ today. r r- y . ~ (L03) 0!.J discounted at 11 percent whi ,h should you choose. rv ~ fV:}l( .... LJ" i.l}tlt. I . t Y Present value o You are going to recei ]00.000 in 50 years. What is the difference m presen /' (L03) value between using a di ~t rat of 14 percent versus 4' percent? L-- ~~ l 'fJ;, ~esent value C9 How much would you hav t im t today to receive: ~r L03) a. $15,000 in years at 1 percent? "!; ..:> b. $20,000 in 12 ye..ars at 1 percent? .> c. $6,000 each year fur 10 lears at 9 percent? d. $50,000 each year fur 0 'ears at 7 percent? @rryOU invest $2,000 a year in a retirement account, how much will you have: a. In 5 years at 6 percent? , Future value (L02) • b. In 20 years at 10 percent? fr__ c. In 40 years at 12 percent? . I 2-~ You invest a single amountof$lO~OOO fo~ 5 ~~~ ~-io ;;;~ent. At the ~nd of ~' 5 years you take the proceeds and invest them for 12 yeats at 15 percent. How [. , much will you have after 17 years? ~ ~~ 11. Jean Splicing will receive $8,500 a year for the next 15 years from her trust. !:_I . I' if a 7 - percent interest rate is applied, what is the current 'value of-die future payments? ;:; ' 12. PhilGoode will receive $175 000 iii. 50 years-:tIis fiienos - are Veryjea:lous of ' ~, him. If the funds are discounted back at a rate of 14 percent, w1!.~! is ,~~_presellt value of his future "pot of gold"? 13. Polly Graham will receive $12,000 a year for the next 15 years as a result of her patent. If a 9 percent rate is applied, should she be willing to sell out her-future rights'now for $100,000? . ~ Carrie Tune will receive $19,500 for the next 20 years as 'a payment for a new song- she-bas written.-If a 10.pen:ent ratei.S apIilied,_ sl:lOUld she be willing to sell out her future rights now for $ 16O,000? n \ l\ ~ . R ~- .--- .. -- ---- - 15. The Clearinghouse Sweepstakes has just inform~you that you have won $1 million. The amount is to be paid out at the rate of $20,000 a year.for the next 50 years. With a discount rate of 10 percent, what is the present value of your prize? Future value (L03) Present value , (L03) Present value (L03) Present value ·~L03) " Present value (L03) Present-value-­ (L03) t1 . Part 4 The Capital Budgetillg Process • ' A ' \ _ Q ,,1. Vb P II A < ;;- " \ ~ "b (\dV 'j l._~.'''\.t AO> ~.Ii6)Toan Lucky won the $80 millio~ lottery. She is to receive $1 million a year for j LO .JJ __ ~he next 50 years pllls_an.alddiltiorla1JlUIIlp_S:uIlll_pa.¥mlent~of..$~IO_ITLilli0n_aft()r----:--~ -- 50 years. The discount rate is 12 percent. What is the ~urrent value of,her ~d Cj> Present value Future value (L02) Future value (L02) Future value (L02) Present value · (L03) winnings? ) J'O'A '8 Al Rosen invests $25,000 in a mint condition 1952 Mickey Mantle Topps base- .... £{ Q .. .... ball card. He expects the card to increase in value 12 percent per year for the {\ t!J . next 10 years. How much wjJ] his card be worth after 10 years? 18. Dr. Ruth has been secretly deposiiiiii$2 : :So1fiii her savings - account -every-----·­ December starting in 1999. Her account earns 5 percent compounded annually. How much wjJ] she have in December 2008? (Assume that a deposit is made in the year 2008.) Make sure to carefully count the years. ' . . ~ At a growth (interest) rate ~f 9 percent annually, how long will it take for a s~ to double? To triple? Select the year that is closest to the correct answer. . ( ' 20. If you owe $40,000 payable at the end of seven years, what amount should your creditor accept in payment immediately if she could earn 12 percent on her money? ----.------. -------.. ---. ---"-.-- .. -- ... --- ... --_---~I,.·-·.---:-- Present value , rii1:\ Jack Hammer invests in a stock that will pay dividends of $2.00 at the e~d. ~f the _ · (L03) \...XJ) first year; $2.20 at the end of the second year; and $2.40 at the end of the third Present value · (L03) year. Also, he believes that at the end of the third. year 'he 'Will be able to sell the . stock for $33. What is the present value of all future benefits if a discount rate of 11 percent is apPlied? (Round all values to two places to the right of the decimal point.) 22. Les Moore retired 'as - presIdent i)f'Goodman Simck -Foods Company' but is cur" rently on a consulting contract for $35,000 per year for the next 10 years. a. If Mr. Moore's opportunity cost (potential return) is 10 percent, what is the present value of his consulting contract? - b. Assuming Mr. Moore will not retire for two more years and will not start to . recei.ve his 10 payments until the end of the third year, whaf would be the ~ J...... value' of his deferred annuity? . Compounding ~uan Garza invested $20,000 10 years ago a f\J 12 ercent, compounded quartegy iuarterly (L05) How much has he accumulated? "f=='V'= 'L:. V ' , ' \ . 'pecial \- r:£\ . \ ~ '\ \\~ , --') () \ tinR $2,000 per year ' five years ago and will continue to do so for five more years. H~cli~ore will your parents have to invest each year for the next five years to have the.nec- essary funds for Linda's education? Use 10 percent as the appropriate interest rate throughout this pr~blem (for dis~ou~ting or compounding). 45. Linda (from problem 44) is now 18 years old (five years have passed),'and she ." wants to get married instead of going to school. Your parents have accumulated the necessary funds for her education. ,~stea? .?_ t:.~eE schooling, your parents are paying $8,000 for her upcoming wedding and plan to take " year-end vaCiilioUs'cc)sfihg-$5;000'pei: yearforthellext' three years. 0 " 'Y' How much money will your parents have at the end of three years to help you with graduate school, which-yoll will start then? You plan to work on a master's. and perha~s a PhD. If graduate school costs $14,045 per year, approxi- mately how long WIll you be able ~o stay in school based on these funds? Use Annuity with changing interest rates (L04&5) Annuity consideration (L04) Special'- '.' considerations of annuities and time periods (£03) -----'--.-,. - - --' l-O-percent-as-the-apprepriate-mterest_ ratec:uu:o.ughout_ this.problem. ===-______ ~ ____ __ 284 \1edical Research :::orporation :Comprehensive :ime value of money) ~L03) Part 4 rile Capital Budgeting Process Dr. Harold Wolf of Medical Research Corporation (MRC) was thrilled with the res pan e h had received from drug companies for his latest discovery, a unique el~c­ tronic timulator that reduces the pain from arthritis. The process had yet to pass ng­ orou Federal Drug Administration (FDA) testing and was still in the early stages of developm nt, but the interest ~as intense. He received the three _~lf~s descJj.l]~4 below _ _ this paragraph. (A 10 percent interest rate should be - used throughout this analysis unless otherwi e specified.) Offer I $1,000,000 now' plus ~200,000 from year 6 through 15. Also if the product did over $100 million in cumulative sales by the end of year 15, he would receive an additional $3,000,000. Dc Wolf thought there was a 70 percent probability this would happen. Offer II Thirty percent of the buyer's gross profit on the product for the next four years. The buyer in this case was Zbay Pharmaceutical. Zbay's gross profit margin was 60 percent-cSales-in year one were-projected t