Chapter 4Future Value. Present Value and InterestRatesProblems and Solutions1.Compute the futurevalue of s100 at an 8 percent interest ratefive,ten and fifteenyears into the future.Answer:Futurevalue in5years=$100*(1.08)=$146.93Future value in 10 years=$100*(1.08)i0=$215.89Future value in 15 years = $100*(1.08)15= $317.222.Suppose that over the past 20 years the average annual return on investments has been10.7percent.For eachdollar invested at thebeginning of theperiod, howmuchmoney would investors have at the end?What if they had kept the investment foronly10years?For30years?Answer:For 20 years: $1*(1.107)20 = $7.64For 10 years: $1*(1.107)10 = $2.76For 30 years: $1*(1.107)30 = $21.113.Compute the present value of a s100 investment made six months, five years, and tenyearsfromnowat4percent interest.Answer:6 months:Present Value=100/(1.04)0.5=$98.065 years: Present Value =100/(1.04)= $82.1910 years: Present Value=100/(1.04)0= $67.564.Youhave$1000to invest,andareconsideringtwooptions:a.5 percent for two years, followed by 7 percent for two years.b.6percentfor four yearsWhich option would you choose?Provide the calculations to justify your answer.Answer:Future Value for option (a)= $1000*(1.05)2 *(1.07)=$1262.25Future Value for option (bi)= $1000*(1.06)*= $1262.481Instructor's Manual t/a Cecchetti:Money,Banking,andFinancial Markets
Instructor’s Manual t/a Cecchetti: Money, Banking, and Financial Markets 1 Chapter 4 Future Value, Present Value and Interest Rates Problems and Solutions 1. Compute the future value of $100 at an 8 percent interest rate five, ten and fifteen years into the future. Answer: Future value in 5 years = $100*(1.08)5 = $146.93 Future value in 10 years = $100*(1.08)10 = $215.89 Future value in 15 years = $100*(1.08)15 = $317.22 2. Suppose that over the past 20 years the average annual return on investments has been 10.7 percent. For each dollar invested at the beginning of the period, how much money would investors have at the end? What if they had kept the investment for only 10 years? For 30 years? Answer: For 20 years: $1*(1.107)20 = $7.64 For 10 years: $1*(1.107)10 = $2.76 For 30 years: $1*(1.107)30 = $21.11 3. Compute the present value of a $100 investment made six months, five years, and ten years from now at 4 percent interest. Answer: 6 months: Present Value = 100/(1.04)0.5 = $98.06 5 years: Present Value = 100/(1.04)5 = $82.19 10 years: Present Value = 100/(1.04)10 = $67.56 4. You have $1000 to invest, and are considering two options: a. 5 percent for two years, followed by 7 percent for two years. b. 6 percent for four years Which option would you choose? Provide the calculations to justify your answer. Answer: Future Value for option (a) = $1000*(1.05)2 *(1.07)2 = $1262.25 Future Value for option (bi) = $1000*(1.06)4 = $1262.48
Chapter 4Future Value, Present Value and Interest RatesYou will chooseoption (bi)5. You have purchased a $1000 certificate of deposit (CD) that matures in 10 years.Assuming that interest is paid annually and reinvested, what will be the value of theCD at maturity if the interest rate is 5 percent?What if the interest rate were 10percent?Answer:If i=5%, Future Value = $1000*(1.05)0= $1628.89Ifi=-10%,FutureValue=$1000*(1.10)10=$2593.746.1If the annual interest rate is 5 percent, which of the following has a higher presentvalue?a.Twopayments ofs50,onein sixmonthsandthesecond in twelvemonthsb.Onepaymentofs100inninemonthsWhat ifthe interest rate is 4 percent?Confirm your answer with calculationsAnswer:If i=5%, Present Value of option (a)= $50/(1.05)0.5 + $50/(1.05) = $96.41Present Value of option (bi)=$100/(1.05)0.75= $96.41Option (a)and option (b)have the samepresent value.If i=4%, Present Value of option (a) = $50/(1.04).5 + $50/(1.04) = $97.11Present Value of option (bi) = $100/(1.04)0.75 = $97.10Option (a) has a higherpresent value7.Assuming that the current interest rate is 3 percent, compute the value ofa five-year5 percent coupon bond with a face value of s1000. What happens when the interestrate goes to 4 percent?Answer:Present Value for 5-year 5 percent coupon bond with face value of 1000 (i-3%) =$50/(1.03) + $50/(1.03)2 + $50/(1.03) +$50/(1.03)+ $50/(1.03)= $1091.59Present Value for 5-year 5 percent coupon bond with face value of $1000 (i=4%)=$50/(1.04) + $50/(1.04)2 + $50/(1.04)+ $50/(1.04)4 + $50/(1.04)°= $1044.52The present value falls when the interest rate rises to 4 percent.8.A financial institution offers you a one-year certificate of deposit with an interest rateof5percent.You expect the inflation ratetobe3percent.What is thereal returnonyourdeposit?Answer:real interestrate=5%-3%=2%You are a manager in charge of a factory that makes automobile tires. A new9.production process has been invented, and you want to purchase new machines totake advantage of it.a. Describe howyou would convince your company's presidentto purchase themachines.b.At the end of the discussion, you conclude that the real rate of return on theinvestment in 10 percent, so it is worth undertaking.The president responds that2Instructor's Manual t/a Cecchetti:Money,Banking,and Financial Markets
Chapter 4 Future Value, Present Value and Interest Rates Instructor’s Manual t/a Cecchetti: Money, Banking, and Financial Markets 2 You will choose option (bi) 5. You have purchased a $1000 certificate of deposit (CD) that matures in 10 years. Assuming that interest is paid annually and reinvested, what will be the value of the CD at maturity if the interest rate is 5 percent? What if the interest rate were 10 percent? Answer: If i=5%, Future Value = $1000*(1.05)10 = $1628.89 If i=10%, Future Value = $1000*(1.10)10 = $2593.74 6. If the annual interest rate is 5 percent, which of the following has a higher present value? a. Two payments of $50, one in six months and the second in twelve months b. One payment of $100 in nine months What if the interest rate is 4 percent? Confirm your answer with calculations. Answer: If i=5%, Present Value of option (a) = $50/(1.05)0.5 + $50/(1.05) = $96.41 Present Value of option (bi) = $100/(1.05)0.75 = $96.41 Option (a) and option (b) have the same present value. If i=4%, Present Value of option (a) = $50/(1.04)0.5 + $50/(1.04) = $97.11 Present Value of option (bi) = $100/(1.04)0.75 = $97.10 Option (a) has a higher present value. 7. Assuming that the current interest rate is 3 percent, compute the value of a five-year, 5 percent coupon bond with a face value of $1000. What happens when the interest rate goes to 4 percent? Answer: Present Value for 5-year 5 percent coupon bond with face value of $1000 (i=3%) = $50/(1.03) + $50/(1.03)2 + $50/(1.03)3 + $50/(1.03)4 + $50/(1.03)5 = $1091.59 Present Value for 5-year 5 percent coupon bond with face value of $1000 (i=4%) = $50/(1.04) + $50/(1.04)2 + $50/(1.04)3 + $50/(1.04)4 + $50/(1.04)5 = $1044.52 The present value falls when the interest rate rises to 4 percent. 8. A financial institution offers you a one-year certificate of deposit with an interest rate of 5 percent. You expect the inflation rate to be 3 percent. What is the real return on your deposit? Answer: real interest rate = 5% - 3% = 2% 9. You are a manager in charge of a factory that makes automobile tires. A new production process has been invented, and you want to purchase new machines to take advantage of it. a. Describe how you would convince your company’s president to purchase the machines. b. At the end of the discussion, you conclude that the real rate of return on the investment in 10 percent, so it is worth undertaking. The president responds that
Chapter4Future Value, Present Value and Interest Ratesin the current financial environment, he cannot borrow for less than 12 percent, sohe can't justify the investment.How would you counter this argument?Answer:a. The new machines will allow the company to produce tires more cheaply, thefuture revenuefromthe machineswill exceed thecost of purchasingthemachinesb.In general, an investment is profitable if the internal rate of return exceeds thecost of borrowing, so the president may be right.However, using the new machinescould allow the company to save on labor costs, maintenance costs, etc.if thosecostsarelargeenough,theinvestmentcould stillbeworthwhile.10.You decide you would like to retire at age 65, and expect to live until you are 85(assume there is no chance you will die younger or live longer).You figure that youcan live nicely on $50,000per year.Describethe calculation you need to make to determine how much you must savea.to purchase an annuity paying $50,000 per year for the rest your life.Assumethe interest rate is 7 percent.b.How would your calculation change if you expected inflation to average 2 percentfor the rest of your life?Answer:$50,000/(1.07)+$50,000/(1.07)2+$50,000/(1.07)3+...+$50,000/(1.07)20If you want to have s50,000 in purchasing power for each year of your retirement, youwould need to calculate:$50,000/(1.07) + $50,000*(1.02)/(1.07) + $50,000*(1.02)/(1.07)* +.. +$50,000*(1.02)1%(1.07)2011. A company offers you a job and tells you that you may choose either a $100,000signing bonus plus a $90,000 salary or a $110,000 salary.If the interest rate is 6percent, how many years will you need to work for the company in order to justifytaking the higher salary?Answer: You need to find the number of years that equates the present value of bothYou can do this usingspreadsheet software.The answer is 6+ years.options.YearPVof$100,000+PVof$110.000$90,000 salarysalary1184905.6604103773.58492265005.34201673.19333340571.0755294031.31444411859.5051381161.61745479112.7407463360.01646542559.1893540905.67597602414.3296614061.95848658881.443683077.31929712152.3047748186.150210762407.8346809609.575711809818.7119867556.20343Instructor's Manual t/a Cecchetti:Money, Banking, and Financial Markets
Chapter 4 Future Value, Present Value and Interest Rates Instructor’s Manual t/a Cecchetti: Money, Banking, and Financial Markets 3 in the current financial environment, he cannot borrow for less than 12 percent, so he can’t justify the investment. How would you counter this argument? Answer: a. The new machines will allow the company to produce tires more cheaply; the future revenue from the machines will exceed the cost of purchasing the machines. b. In general, an investment is profitable if the internal rate of return exceeds the cost of borrowing, so the president may be right. However, using the new machines could allow the company to save on labor costs, maintenance costs, etc.—if those costs are large enough, the investment could still be worthwhile. 10. You decide you would like to retire at age 65, and expect to live until you are 85 (assume there is no chance you will die younger or live longer). You figure that you can live nicely on $50,000 per year. a. Describe the calculation you need to make to determine how much you must save to purchase an annuity paying $50,000 per year for the rest your life. Assume the interest rate is 7 percent. b. How would your calculation change if you expected inflation to average 2 percent for the rest of your life? Answer: $50,000/(1.07) + $50,000/(1.07)2 + $50,000/(1.07)3 +.+ $50,000/(1.07)20 If you want to have $50,000 in purchasing power for each year of your retirement, you would need to calculate: $50,000/(1.07) + $50,000*(1.02)/(1.07)2 + $50,000*(1.02)2 /(1.07)3 +.+ $50,000*(1.02)19/(1.07)20 11. A company offers you a job and tells you that you may choose either a $100,000 signing bonus plus a $90,000 salary or a $110,000 salary. If the interest rate is 6 percent, how many years will you need to work for the company in order to justify taking the higher salary? Answer: You need to find the number of years that equates the present value of both options. You can do this using spreadsheet software. The answer is 6+ years. Year PV of $100,000 + $90,000 salary PV of $110,000 salary 1 184905.6604 103773.5849 2 265005.34 201673.1933 3 340571.0755 294031.3144 4 411859.5051 381161.6174 5 479112.7407 463360.0164 6 542559.1893 540905.6759 7 602414.3296 614061.9584 8 658881.443 683077.3192 9 712152.3047 748186.1502 10 762407.8346 809609.5757 11 809818.7119 867556.2034
Chapter 4Future Value, Present Value and Interest Rates12854545.9546922222.833413896741.4666973795.125914936548.55341022448.23215974102.40891068347.389161009530.5741111648.48171042953.3721152498.566181074484.3131191036.383191104230.4841227392.814201132292.911261691.33412.Mostbusinessesreplace theircomputerseverytwoto threeyears.Assume thatacomputer costs $2000 and that it fully depreciates in three years, at which point it hasno resalevalue whatsoever and is thrown away.a.If the interestrate for financing the equipment equals i, show how to compute theminimum cash flow that a computer must generate to be worth the purchase. Youranswer will depend on i.b.Howmuchdifference would itmake ifthecomputerdid notfully depreciate,butstill had somevalue at the time it was replaced?Assuming its resale value iss$250, recompute your answer to part a.What if financing can only be had at 10 percent interest rate?Recomputeyourc.answerto parta.Answer:a.Ifx=minimumannual cashflow:$2000 = x/(1+i) + x/(1+i)2 + x/(1+i)3x=$2000/[1/(1+i)+1/(1+i)2+1/(1+i)b. $2000 = x/(1+i) + x/(1+i)2 + x/(1+i)3 + $250/(1+i)3x =[$2000 - $250/(1+i)}]/[1/(1+i) + 1/(1+i) + 1/(1+i)]]c.x=$2000/[1/(1+0.1) + 1/(1+0.1) + 1/(1+0.1)))=$804.2313.Some friends of yours have just had a child.Realizing the power ofcompound interest,theyare considering investingfortheir child's collegeeducation, which will begin in 18 years. Assume that the cost of acollege education today is $125,000; there is no inflation, and there arenotaxes on interest income that is used to pay college tuition andexpensesIf the interest rate is 5 percent, how much money will your friends need to put intotheir savings account today to have $125,000 in 18 years?What if the interest rate is 10 percent?The chance that a college education will cost the same 18 years from now as it doestodayseems remote.Assumingthatthepricewill rise3percentperyear,andthat today's interest rate is 8 percent, what will your friend's investment need tobe?4Instructor's Manual t/a Cecchetti:Money,Banking,andFinancial Markets
Chapter 4 Future Value, Present Value and Interest Rates Instructor’s Manual t/a Cecchetti: Money, Banking, and Financial Markets 4 12 854545.9546 922222.8334 13 896741.4666 973795.1259 14 936548.5534 1022448.232 15 974102.4089 1068347.389 16 1009530.574 1111648.48 17 1042953.372 1152498.566 18 1074484.313 1191036.383 19 1104230.484 1227392.814 20 1132292.91 1261691.334 12. Most businesses replace their computers every two to three years. Assume that a computer costs $2000 and that it fully depreciates in three years, at which point it has no resale value whatsoever and is thrown away. a. If the interest rate for financing the equipment equals i, show how to compute the minimum cash flow that a computer must generate to be worth the purchase. Your answer will depend on i. b. How much difference would it make if the computer did not fully depreciate, but still had some value at the time it was replaced? Assuming its resale value is $250, recompute your answer to part a. c. What if financing can only be had at 10 percent interest rate? Recompute your answer to part a. Answer: a. If x = minimum annual cash flow: $2000 = x/(1+i) + x/(1+i) 2 + x/(1+i) 3 x = $2000/[1/(1+i) + 1/(1+i) 2 + 1/(1+i) 3 ] b. $2000 = x/(1+i) + x/(1+i) 2 + x/(1+i) 3 + $250/(1+i) 3 x = [$2000 - $250/(1+i) 3 ]/[1/(1+i) + 1/(1+i) 2 + 1/(1+i) 3 ] c. x = $2000/[1/(1+0.1) + 1/(1+0.1)2 + 1/(1+0.1)3 ] = $804.23 13. Some friends of yours have just had a child. Realizing the power of compound interest, they are considering investing for their child’s college education, which will begin in 18 years. Assume that the cost of a college education today is $125,000; there is no inflation; and there are no taxes on interest income that is used to pay college tuition and expenses. If the interest rate is 5 percent, how much money will your friends need to put into their savings account today to have $125,000 in 18 years? What if the interest rate is 10 percent? The chance that a college education will cost the same 18 years from now as it does today seems remote. Assuming that the price will rise 3 percent per year, and that today’s interest rate is 8 percent, what will your friend’s investment need to be?
Chapter 4Future Value, Present Value and Interest RatesReturn to part a, the case with a 5 percent interest rate and no inflation. Assume thatyour friendsdon't have enough financial resources to make the entire investmentat the beginning.Instead, they think they will be able to split their investmentinto two equal parts, one invested immediately and the second invested in fiveyears.How would you compute the required size of the two equal investmentsmade five years apart?Answer:a.PV=$125,000/(1.05)18=$51,940.08b.PV=$125,000/(1.10)18=$22,482.35c.If the price rises 3% per year, the cost of a college education in 18 years will be:$125,000*(1.03)18=$212,804.13PV=$212,804.13/(1.08)18 =$53,254.03d.If x is the size of each investment$125,000 =x(1.05)18 + x(1.05)13x = $125,000/[(1.05)18 + (1.05)3] = $29,122.1314.Using the data from the last page of The Economist magazine (alsoavailable at www.economist.com) compute the ex post real interest rateoverthepastyearforthe countries listedand comment onwhatyoufind.Answer: The data are on the last page of the print version, as well as “Emerging MarketIndicators"athttp://www.economist.com/markets/.15.You are considering buying a new house, and have found a 30-yearfixed-rate mortgage for $100,000 with an interest rate of 7 percent.Thismortgagerequires360monthlypayments ofapproximately $651eachIf the interest rate rises to 8 percent, what will happen to your monthlypayment?Comparethepercentagechangeinthemonthlypayment withthe percentage change in the interest rate. (You will need to use a formulafrom Appendix 4.)Answer:Iftheannual interest rate is 8%,thenthemonthlyrate is(1.08)/21=0.006434Using the equation from Appendix 4A:C=($100,000*0.006434)/[1-(1/(1.006434)360))=$714. Monthly payments haverisen by ($714-$651)/$651= 9.7% and the interest rate has risen by (8%-7%)/7%= 14.3%.5Instructor's Manual t/a Cecchetti:Money,Banking,and Financial Markets
Chapter 4 Future Value, Present Value and Interest Rates Instructor’s Manual t/a Cecchetti: Money, Banking, and Financial Markets 5 Return to part a, the case with a 5 percent interest rate and no inflation. Assume that your friends don’t have enough financial resources to make the entire investment at the beginning. Instead, they think they will be able to split their investment into two equal parts, one invested immediately and the second invested in five years. How would you compute the required size of the two equal investments made five years apart? Answer: a. PV = $125,000/(1.05)18 = $51,940.08 b. PV = $125,000/(1.10)18 = $22,482.35 c. If the price rises 3% per year, the cost of a college education in 18 years will be: $125,000*(1.03)18 = $212,804.13 PV = $212,804.13/(1.08)18 = $53,254.03 d. If x is the size of each investment: $125,000 = x(1.05)18 + x(1.05)13 x = $125,000/[(1.05)18 + (1.05)13] = $29,122.13 14. Using the data from the last page of The Economist magazine (also available at www.economist.com) compute the ex post real interest rate over the past year for the countries listed and comment on what you find. Answer: The data are on the last page of the print version, as well as “Emerging Market Indicators” at http://www.economist.com/markets/. 15. You are considering buying a new house, and have found a 30-year fixed-rate mortgage for $100,000 with an interest rate of 7 percent. This mortgage requires 360 monthly payments of approximately $651 each. If the interest rate rises to 8 percent, what will happen to your monthly payment? Compare the percentage change in the monthly payment with the percentage change in the interest rate. (You will need to use a formula from Appendix 4.) Answer: If the annual interest rate is 8%, then the monthly rate is (1.08)1/12 – 1 = 0.006434 Using the equation from Appendix 4A: C = ($100,000*0.006434)/[1 – (1/(1.006434)360)] = $714. Monthly payments have risen by ($714-$651)/$651 = 9.7% and the interest rate has risen by (8%-7%)/7% = 14.3%