DISCOUNTEDCASHFLOWVALUATION
CHAPTER5 Answers to Concepts Review and Critical Thinking Questions 3. 7.
1
Assuming positive cash flows and a positive interest rate, the present value will fall, and the future value will rise. Yes, they should. APRs generally dont provide the relevant rate. The only advantage is that they are easier to compute, but, with modern computing equipment, that advantage is not very important.
Solutions to Questions and Problems NOTE: All end-of-chapter problems were solved using a spreadsheet. Many problems require multiple steps. Due to space and readability constraints, when these intermediate steps are included in this solutions manual, rounding may appear to have occurred. However, the final answer for each problem is found without rounding during any step in the problem. 2. To find the PVA, we use the equation: PVA = C({1 [1/(1 + r)]t } / r ) At a 5 percent interest rate: X@5%: PVA = $4,000{[1 (1/1.05)10 ] / .05 } = $30,886.94 Y@5%: PVA = $8,000{[1 (1/1.05)4 ] / .05 } = $28,367.60 And at a 15 percent interest rate: X@15%: PVA = $4,000{[1 (1/1.15)10 ] / .15 } = $20,075.07 Y@15%: PVA = $8,000{[1 (1/1.15)4 ] / .15 } = $22,839.83 7. Here we need to find the FVA. The equation to find the FVA is: FVA = C{[(1 + r)t 1] / r} FVA for 20 years = $2,000[(1.07520 1) / 0.075] FVA for 20 years = $86,609.36 FVA for 40 years = $2,000[(1.07540 1) / 0.075] FVA for 40 years = $451,513.04 Notice that doubling the number of periods does not double the FVA.
SolutionsManualt/aEssentialofCorporateFinance1ebyRoss
2 8. Here we have the FVA, the length of the annuity, and the interest rate. We want to calculate the annuity payment. Using the FVA equation: FVA = C{[(1 + r)t 1] / r} $50,000 = $C[(1.03757 1) / 0.0375] We can now solve this equation for the annuity payment. Doing so, we get: C = $50,000 / 7.8386 C = $6,378.68 13. Here we are given the EAR and need to find the APR. Using the equation for discrete compounding: EAR = [1 + (APR / m)]m 1 We can now solve for the APR. Doing so, we get: APR = m[(1 + EAR)1/m 1] EAR = 0.09 = [1 + (APR / 2)]2 1 EAR = 0.19 = [1 + (APR / 12)]12 1 EAR = 0.08 = [1 + (APR / 52)]52 1 EAR = 0.15 = [1 + (APR / 365)]365 1 APR = 2[(1.09)1/2 1] APR = 12[(1.19)1/12 1] APR = 52[(1.08)1/52 1] APR = 365[(1.15)1/365 1] = 8.81% = 17.52% = 7.70% = 13.98%
17. Forthisproblem,wesimplyneedtofindtheFVofalumpsumusingtheequation: FV=PV(1+r)t Itisimportanttonotethatcompoundingoccursdaily.Toaccountforthis,wewilldividetheinterest rateby365(thenumberofdaysinayear,ignoringleapyear),andmultiplythenumberofperiodsby 365.Doingso,weget: FVin5years =$5,000[1+(0.026/365)]5(365)=$5,694.12 FVin10years=$5,000[1+(0.026/365)]10(365)=$6,484.59 FVin20years=$5,000[1+(0.026/365)]20(365)=$8,409.88 20. Wefirstneedtofindtheannuitypayment.WehavethePVA,thelengthoftheannuity,andthe interestrate.UsingthePVAequation: PVA=C({1[1/(1+r)]t}/r) $52,350=$C[1{1/[1+0.086/12)]60}/(0.086/12)] SolutionsManualt/aEssentialofCorporateFinance1ebyRoss
3 Solvingforthepayment,weget: C=$52,350/48.6267 C=$1,076.57 TofindtheEAR,weusetheEARequation: EAR=[1+(APR/m)]m1 EAR=[1+0.086/12)]121 EAR=8.95% 21. Hereweneedtofindthelengthofanannuity.Weknowtheinterestrate,thePV,andthepayments. UsingthePVAequation: PVA=C({1[1/(1+r)]t}/r) $11,652=$400{[1(1/1.014)t]/0.014} Nowwesolvefort: 1/1.014t=1[($11,652)(.014)/($400)] 1.014t=1/(0.5922)=1.6887 t=ln1.6887/ln1.014 t=37.69months 22. FV=PV(1+r) HerewearetryingtofindtheinterestratewhenweknowthePVandFV.UsingtheFVequation: $4=$3(1+r) r=$4/$31 r=0.3333or33.33%perweek Theinterestrateis33.33%perweek.TofindtheAPR,wemultiplythisratebythenumberofweeks inayear,so: APR=(52)33.33%=1,733.33% AndusingtheequationtofindtheEAR,wefind: EAR=[1+(APR/m)]m1 EAR=[1+0.3333]521 SolutionsManualt/aEssentialofCorporateFinance1ebyRoss
4 EAR=314,916,515.69% 23. HereweneedtofindtheinterestratethatequatestheperpetuitycashflowswiththePVofthecash flows.UsingthePVofaperpetuityequation: PV=C/r $130,000=$2,000/r Wecannowsolvefortheinterestrateasfollows: r=$2,000/$130,000 r=0.0154or1.54%permonth Theinterestrateis1.54%permonth.TofindtheAPR,wemultiplythisratebythenumberof monthsinayear,so: APR=(12)1.54% APR=18.46% AndusingtheequationtofindtheEAR,wefind: EAR=[1+(APR/m)]m1 EAR=[1+0.0154]121 EAR=0.2011or20.11% 30. WeneedtousethePVAdueequation,whichis: PVAdue=(1+r)PVA Usingthisequation: PVAdue=$68,000=[1+(0.095/12)]C[{11/[1+(0.095/12)]60}/(0.095/12) $67,465.90=$C{1[1/(1+0.095/12)60]}/(0.0995/12) C=$1416.91 Notice,whenwefindthepaymentforthePVAdue,wesimplydiscountthePVoftheannuitydue backoneperiod.WethenusethisvalueasthePVofanordinaryannuity. 32. We will calculate the time we must wait if we deposit in the bank that pays simple interest. The interest amount we will receive each year in this bank will be: SolutionsManualt/aEssentialofCorporateFinance1ebyRoss
5 Interest = $72,000 (0.05) Interest = $3,600 per year The deposit will have to increase by the difference between the amount we need by the amount we originally deposit divided by the interest earned per year, so the number of years it will take in the bank that pays simple interest is: Years to wait = ($72,000 100,000) / $3,600 Years to wait = 7.78ears To find the number of years it will take in the bank that pays compound interest, we can use the future value equation for a lump sum and solve for the periods. Doing so, we find: FV=PV(1+r)t $100,000 = $72,000 [1 + (0.05/12)]t t = 80 months or 6.6 years 53. HerewearegiventhePVA,numberofperiods,andtheamountoftheannuity.Weneedtosolvefor theinterestrate.Wemustbecarefultousethecashflowsoftheloan.Usingthepresentvalueofan annuityequation,wefind: PVA=C({1[1/(1+r)]t}/r) $20,000=$1,916.67[{1[1/(1+r)]12}/r] Tofindtheinterestrate,weneedtosolvethisequationonafinancialcalculator,usingaspreadsheet, orbytrialanderror.Ifyouusetrialanderror,rememberthatincreasingtheinterestratelowersthe PVA,andincreasingtheinterestratedecreasesthePVA.Usingaspreadsheet,wefind: r=0.02219or2.219% Thisisthemonthlyinterestrate.TofindtheAPRwithamonthlyinterestrate,wesimplymultiply themonthlyrateby12,sotheAPRis: APR=0.0221912 APR=0.2662or26.62% AndtheEARis: EAR=[1+(APR/m)]m1 EAR=[1+0.02219]121 EAR=0.3012or30.12%
SolutionsManualt/aEssentialofCorporateFinance1ebyRoss
6 54. To solve this problem, we must find the FV of each cash flow and add them. To find the FV of a lump sum, we use: FV = PV(1 + r)t FV = $30,000(1.102)3 + $45,000(1.102)2 + $75,000 FV = $169,796.38 Notice, since we are finding the value at Year 5, the cash flow at Year 5 is simply added to the FV of the other cash flows. In other words, we do not need to compound this cash flow. To find the value in Year 10, we simply need to find the future value of this lump sum. Doing so, we find: FV = PV(1 + r)t FV = $169,796.38 (1.102)5 FV = $275,953.81
SolutionsManualt/aEssentialofCorporateFinance1ebyRoss