CHAPTER 3 How to Calculate Present Values. Answers to Practice Questions

Transcription

1 CHAPTER 3 How o Calculae Presen Values Answers o Pracice Quesions. a. PV $00/.0 0 $90.53 b. PV $00/.3 0 $9.46 c. PV $00/.5 5 $ 3.5 d. PV $00/. + $00/. + $00/. 3 $40.8. a. DF + r r % b. DF 0.89 ( + r ) (.05) c. AF DF + DF d. PV of an annuiy C [Annuiy facor a r% for years] Here: $4.65 $0 [AF3] AF3.465 e. AF3 DF + DF + DF3 AF + DF DF3 DF The presen value of he 0-year sream of cash inflows is: PV $70,000 $886, (.4) Thus: 3-

2 NPV $800,000 + $886, $86, A he end of five years, he facory s value will be he presen value of he five remaining $70,000 cash flows: PV $70,000 $583, (.4) 4. NPV 0 0 C (.) $380,000 + $50,000. $57,000 $75,000 $80, $85, $9,000 $9,000 $80,000 $68,000 $50, $3, a. Le S salary in year PV 30 S (.08) 30 40,000 (.05) (.08) 30 (40,000/.05) (.08 /.05) 30 38,095.4 (.086) 38,095.4 $760, (.086) b. PV(salary) x 0.05 $38,08.96 Fuure value $38,08.96 x (.08) 30 $38,57.75 c. PV C r (+ $38,57.75 C (.08) 0 C $38,57.75 (.08) 0 $38,

3 6. Period Presen Value 0 400, ,000/. + 89, ,000/. +59, ,000/. 3 +3, Toal NPV $6, We can break his down ino several differen cash flows, such ha he sum of hese separae cash flows is he oal cash flow. Then, he sum of he presen values of he separae cash flows is he presen value of he enire projec. (All dollar figures are in millions.) Cos of he ship is $8 million PV $8 million Revenue is $5 million per year, operaing expenses are $4 million. Thus, operaing cash flow is $ million per year for 5 years. PV $million $8.559 million 5 (.08) Major refis cos $ million each, and will occur a imes 5 and 0. PV ($ million)/ ($ million)/.08 0 $.88 million Sale for scrap brings in revenue of $.5 million a 5. PV $.5 million/.08 5 $0.473 million Adding hese presen values gives he presen value of he enire projec: NPV $8 million + $8.559 million $.88 million + $0.473 million NPV $.56 million 8. a. PV $00,000 b. PV $80,000/. 5 $0,36.83 c. PV $,400/0. $95,000 d. PV $9,000 $07, (.) e. PV $6,500/( ) $9,

4 Prize (d) is he mos valuable because i has he highes presen value. 9. Mr. Basse is buying a securiy worh $0,000 now. Tha is is presen value. The unknown is he annual paymen. Using he presen value of an annuiy formula, we have: PV C r (+ $0,000 C (.08) C $0,000 (.08) $, Assume he Zhangs will pu aside he same amoun each year. One approach o solving his problem is o find he presen value of he cos of he boa and hen equae ha o he presen value of he money saved. From his equaion, we can solve for he amoun o be pu aside each year. PV(boa) $0,000/(.0) 5 $,48 PV(savings) Annual savings (.0) 5 Because PV(savings) mus equal PV(boa): Annual savings $, (.0) Annual savings $,48 $3, (.0) Anoher approach is o find he value of he savings a he ime he boa is purchased. Because he amoun in he savings accoun a he end of five years mus be he price of he boa ($0,000) we can solve for he amoun o be pu aside each year. If x is he amoun o be pu aside each year, hen: 3-4

5 x(.0) 4 + x(.0) 3 + x(.0) + x(.0) + x $0,000 x( ) $0,000 x(6.05) $0,000 x $ 3,76. The fac ha Kangaroo Auos is offering free credi ells us wha he cash paymens are; i does no change he fac ha money has ime value. A 0% annual rae of ineres is equivalen o a monhly rae of 0.83%: rmonhly rannual / 0.0/ % The presen value of he paymens o Kangaroo Auos is: $,000 + $300 $8, (.0083) A car from Turle Moors coss $9,000 cash. Therefore, Kangaroo Auos offers he beer deal, i.e., he lower presen value of cos.. a. This is he usual perpeuiy, and hence: C $00 PV $,48.57 r 0.07 b. This is worh he PV of sream (a) plus he immediae paymen of $00: PV $00 + $,48.57 $,58.57 c. The coninuously compounded equivalen o a 7% annually compounded rae is approximaely 6.77%, because: Thus: e C $00 PV $,477.0 r Noe ha he paern of paymens in par (b) is more valuable han he paern of paymens in par (c). I is preferable o receive cash flows a he sar of every year han o spread he receip of cash evenly over he year; wih he former paern of paymen, you receive he cash more quickly. 3-5

6 3. a. PV $ billion/ $.5 billion b. PV $ billion/( 0.04) $5.0 billion c. PV $ billion $9.88 billion 0 (.08) d. The coninuously compounded equivalen o an 8% annually compounded rae is approximaely 7.7%, because: Thus: e PV $ billion ) e (0.077)(0 $0.03 billion This resul is greaer han he answer in Par (c) because he endowmen is now earning ineres during he enire year. 4. Wih annual compounding: FV $00 (.5) 0 $, Wih coninuous compounding: FV $00 e (0.5 0) $, One way o approach his problem is o solve for he presen value of: () $00 per year for 0 years, and () $00 per year in perpeuiy, wih he firs cash flow a year. If his is a fair deal, hese presen values mus be equal, and hus we can solve for he ineres rae (. The presen value of $00 per year for 0 years is: PV $00 ( (+ 0 The presen value, as of year 0, of $00 per year forever, wih he firs paymen in year, is: PV0 $00/r A 0, he presen value of PV0 is: (+ PV 0 $00 r 3-6

7 Equaing hese wo expressions for presen value, we have: $00 $ ( (+ (+ r Using rial and error or algebraic soluion, we find ha r 7.8%. 6. Assume he amoun invesed is one dollar. Le A represen he invesmen a %, compounded annually. Le B represen he invesmen a.7%, compounded semiannually. Le C represen he invesmen a.5%, compounded coninuously. Afer one year: FVA $ ( + 0.) $.00 FVB $ ( ) $.04 FVC $ e (0.5 ) $.9 Afer five years: FVA $ ( + 0.) 5 $.763 FVB $ ( ) 0 $.7657 FVC $ e (0.5 5) $.777 Afer weny years: FVA $ ( + 0.) 0 $ FVB $ ( ) 40 $9.793 FVC $ e (0.5 0) $9.974 The preferred invesmen is C. 7. The oal elapsed ime is 3 years. A 5%: FV $00 ( ) 3 $4,797 A 0%: FV $00 ( + 0.0) 3 $4,757,44 3-7

8 8. Because he cash flows occur every six monhs, we use a six-monh discoun rae, here 8%/, or 4%. Thus: PV $00,000 + $00,000 $843, (.04) 9. a. Each insallmen is: $9,40,73/9 $495,87 PV $495,87 $4,76,74 9 (.08) b. If ERC is willing o pay $4. million, hen: $4,00,000 $495,87 r (+ 9 Using Excel or a financial calculaor, we find ha r 9.8%. 0. a. PV $70,000 $40, (.08) b. Year Beginning-of- Year Balance Year-end Ineres on Balance Toal Year-end Paymen Amorizaion of Loan End-of-Year Balance 40, ,8.8 70, , , , , , , , , , , ,.87 79, ,489.7, , , , , , , , , , , , , , , , , , , ,84.8 5, , ,

9 . This is an annuiy problem wih he presen value of he annuiy equal o $ million (as of your reiremen dae), and he ineres rae equal o 8% wih 5 ime periods. Thus, your annual level of expendiure (C) is deermined as follows: PV C r (+ $,000,000 C (.08) 5 C $,000,000 (.08) 5 $33,659 Wih an inflaion rae of 4% per year, we will sill accumulae $ million as of our reiremen dae. However, because we wan o spend a consan amoun per year in real erms (R, consan for all ), he nominal amoun (C ) mus increase each year. For each year : Therefore: R C /( + inflaion rae) PV [all C ] PV [all R ( + inflaion rae) ] $,000,000 ( ) (+ ) ( ) + ( + ) ( ) (+ ) 5 R 5 R [ ] $,000,000 R.390 $,000,000 R $77,95 $,000,000 Thus C ($77,95.04) $85,070, C $9,473, ec.. a. PV $50,000 $430, (.055) b. The annually compounded rae is 5.5%, so he semiannual rae is: (.055) (/) % Since he paymens now arrive six monhs earlier han previously: PV $430, $44,

10 3. a. Using he Rule of 7, he ime for money o double a % is 7/, or 6 years. More precisely, if x is he number of years for money o double, hen: (.) x Using logarihms, we find: x (ln.) ln x 6. years b. Wih coninuous compounding for ineres rae r and ime period x: e r x Taking he naural logarihm of each side: r x ln() Thus, if r is expressed as a percen, hen x (he ime for money o double) is: x 69.3/(ineres rae, in percen). 4. a. This calls for he growing perpeuiy formula wih a negaive growh rae (g 0.04): $ million $ million PV 0.0 ( 0.04) 0.4 $4.9 million b. The pipeline s value a year 0 (i.e., a 0), assuming is cash flows las forever, is: PV 0 C C( + g) r g r g Wih C $ million, g 0.04, and r 0.0: PV 0 0 ($ million) ( 0.04) $0.884 million $ million Nex, we conver his amoun o PV oday, and subrac i from he answer o Par (a): $6.34 million $4.9 million $3.35 million (.0) PV 0 3-0