《金融學》答案第四章貨幣的時間價值與現金流貼現分析

1、CHAPTER 4THE TIME VALUE OF MONEY AND DISCOUNTED CASH FLOW ANALYSISObjectives? To explain the concepts of compounding and discounting, future value and present value.? To show how these concepts are applied to making financial decisions.Outline4.1 Compounding4.2 The Frequency of Compounding4.3 Presen

2、t Value and Discounting4.4 Alternative Discounted Cash Flow Decision Rules4.5 Multiple Cash Flows4.6 Annuities4.7 Perpetual Annuities4.8 Loan Amortization4.9 Exchange Rates and Time Value of Money4.10 Inflation and Discounted Cash Flow Analysis4.11 Taxes and Investment DecisionsSummary? Compounding

3、is the process of going from present value (PV) to future value (FV). The future value of $1 earning interest at rate i per period for n periods is (1+i) n.? Discounting is finding the present value of some future amount. The present value of $1 discounted at rate i per period for n periods is 1/(1+

4、i) n.? One can make financial decisions by comparing the present values of streams of expected future cash flows resulting from alternative courses of action. The present value of cash inflows less the present value of cash outflows is called net present value ( NPV). If a course of action has a pos

5、itive NPV, it is worth undertaking.? In any time value of money calculation, the cash flows and the interest rate must be denominated in the same currency.? Never use a nominal interest rate when discounting real cash flows or a real interest rate when discounting nominal cash flows.I nstructors Man

6、ualChapter 4Page 69How to Do TVM Calculations in MS ExcelAssume you have the following cash flows set up in a spreadsheet:AB1tCF20-1003150426053706NPV7IRRMove the cursor to cell B6 in the spreadsheet. Click the function wizard fx in the tool bar and when a menu appears, select financial and then NPV

7、 . Then follow the instructions for inputting the discount rate and cash flows.You can input the column of cash flows by selecting and moving it with your mouse. Ultimately cell B6 should contain the following:=NPV(0.1,B3:B5)+B2The first variable in parenthesis is the discount rate. Make sure to inp

8、ut the discount rate as a decimal fraction (i.e., 10% is .1). Note that the NPV function in Excel treats the cash flows as occurring at the end of each period, and therefore the initial cash flow of 100 in cell B2 is added after the closing parenthesis. When you hit the ENTER key, the result should

9、be $47.63.Now move the cursor to cell B7 to compute IRR. This time select IRR from the list of financial functions appearing in the menu. Ultimately cell B7 should contain the following:=IRR(B2:B5)When you hit the ENTER key, the result should be 34%.Your spreadsheet should look like this when you ha

10、ve finished:AB1tCF20-1003150426053706NPV47.637IRR34%Solutions to Problems at End of Chapter1. If you invest $1000 today at an interest rate of 10% per year, how much will you have 20 years from now, assuming no withdrawals in the interim?SOLUTION:20101000?0FV =6,727.502. a. If you invest $100 every

11、year for the next 20 years, starting one year from today and you earn interest of 10% per year, how much will you have at the end of the 20 years?b. How much must you invest each year if you want to have $50,000 at the end of the 20 years?SOLUTION:a. 20100?100FV = 5,727.50b. 2010050,000?PMT = 872.98

12、3. What is the present value of the following cash flows at an interest rate of 10% per year?a. $100 received five years from now.b. $100 received 60 years from now.c. $100 received each year beginning one year from now and ending 10 years from now.d. $100 received each year for 10 years beginning n

13、ow.e. $100 each year beginning one year from now and continuing forever.SOLUTION:a. 510?1000PV = $62.09b. 6010?1000PV = $.3284c. 1010?0100 ordinaryPV = $614.46d. 1010?0100 immediatePV = $675.90e. Perpetuity10?0100 ordinarySee belowe. PV = $100 = $1,000.104. You want to establish a“wasting fund which

14、 will provide you with1000 per year for four years, atwhich time the fund will be exhausted. How much must you put in the fund now if you can earn 10% interest per year?SOLUTION:5. You take a one-year installment loan of $1000 at an interest rate of 12% per year (1% per month) to be repaid in 12 equ

15、al monthly payments.a. What is the monthly payment?b. What is the total amount of interest paid over the 12-month term of the loan?SOLUTION:1211,0000?PMT = $88.85a. PMT = $88.85b. 12 x $88.85 - $1,000 = $66.206. You are taking out a $100,000 mortgage loan to be repaid over 25 years in 300 monthly pa

16、yments.a. If the interest rate is 16% per year what is the amount of the monthly payment?b. If you can only afford to pay $1000 per month, how large a loan could you take?c. If you can afford to pay $1500 per month and need to borrow $100,000, how many months would it take to pay off the mortgage?d.

17、 If you can pay $1500 per month, need to borrow $100,000, and want a 25 year mortgage, what is the highest interest rate you can pay?SOLUTION:a. 30016/12100,0000?PMT =$1358.89b. 30016/12?01,000PV = $73,590c. ?16/12100,00001,500n = 166d. 300?100,00001,500i = 1.482% per montha. Note: Do not round off

18、the interest rate when computing the monthly rate or you will not get the same answer reported here. Divide 16 by 12 and then press the i key.b. Note: You must input PMT and PV with opposite signs.c. Note: You must input PMT and PV with opposite signs.7. In 1626 Peter Minuit purchased Manhattan Isla

19、nd from the Native Americans for about $24 worth of trinkets. If the tribe had taken cash instead and invested it to earn 6% per year compounded annually, how much would the Indians have had in 1986, 360 years later?SOLUTION:360624?0FV = 3.091010FV = 30,925,930,0008. You win a $1 million lottery whi

20、ch pays you $50,000 per year for 20 years, beginning one year from now. How much is your prize really worth assuming an interest rate of 8% per year?SOLUTION:208?050,000PV = $490,9079. Your great-aunt left you $20,000 when she died. You can invest the money to earn 12% per year. If you spend $3,540

21、per year out of this inheritance, how long will the money last?SOLUTION:1220,0003,540n = 10 years10. You borrow $100,000 from a bank for 30 years at an APR of 10.5%. What is the monthly payment? If you must pay two points up front, meaning that you only get $98,000 from the bank, what is the true AP

22、R on the mortgage loan?SOLUTION:360.875100,0000?PMT = $914.74If you must pay 2 points up front, the bank is in effect lending you only $98,000. Keying in 98000 as PV and computing i, we get:360?98,000914.74i = .89575.8957510.75%i =.89575% per month; APR = 1211. Suppose that the mortgage loan describ

23、ed in question 10 is a one-year adjustable rate mortgage (ARM), which means that the 10.5% interest applies for only the first year. If the interest rate goes up to 12% in the second year of the loan, what will your new monthly payment be?SOLUTION:Step 1 is to compute the remaining balance after the

24、 first 12 payments:348.875?0914.74PV = $ 99499.57Step 2 is to compute the new monthly payment at an interest rate of 1% per month:12. You just received a gift of $500 from your grandmother and you are thinking about saving this money for graduation which is four years away. You have your choice betw

25、een Bank A which is paying 7% for one-year deposits and Bank B which is paying 6% on one-year deposits. Each bank compounds interest annually. What is the future value of your savings one year from today if you save your money in Bank A? Bank B? Which is the better decision? What savings decision wi

26、ll most individuals make? What likely reaction will Bank B have?SOLUTION:Future Value in Bank A:17-$500Solve0$535Formula:$500 x (1.07) = $535Future Value in Bank B:16-$500Solve$530Formula:$500 x (1.06) = $530a. You will decide to save your money in Bank A because you will have more money at the end

27、of the year. You made an extra $5 because of your savings decision. That is an increase in value of 1%. Because interest compounded only once per year and your money was left in the account for only one year, the increase in value is strictly due to the 1% difference in interest rates.b. Most indivi

28、duals will make the same decision and eventually Bank B will have to raise its rates. However, it is also possible that Bank A is paying a high rate just to attract depositors even though this rate is not profitable for the bank. Eventually Bank A will have to lower its rate to Bank B s rate in orde

29、r to make money.13. Sue Consultant has just been given a bonus of $2,500 by her employer. She is thinking about using the money to start saving for the future. She can invest to earn an annual rate of interest of 10%.a. According to the Rule of 72, approximately how long will it take for Sue to incr

30、ease her wealth to $5,000?b. Exactly how long does it actually take?SOLUTION:c. According to the Rule of 72: n = 72/10 = 7.2 yearsIt will take approximately 7.2 years for Sue s $2,500 to double to $5,000 at 10% interest.d. At 10% interestIMIMISolve10-$2,500$5,0007.27 YearsFormula:$2,500 x (1.10) n =

31、 $5,000Hence, (1.10)n = 2.0n log 1.10 = log 2.0n = .693147 = 7.27 Years .09531014. Larry s bank account has a a floating interestcertaion deposits. Every year the interest rate is adjusted. Larry deposited $20,000 three years ago, when interest rates were 7% (annual compounding). Last year the rate

32、was only 6%, and this year the rate fell again to 5%. How much will be in his account at the end of this year?SOLUTION:$20,000 x 1.07 x 1.06 x 1.05 = $23,818.2015. You have your choice between investing in a bank savings account which pays 8% compounded annually (BankAnnual) and one which pays 7.5%

33、compounded daily (BankDaily).a. Based on effective annual rates, which bank would you prefer?b. Suppose BankAnnual is only offering one-year Certificates of Deposit and if you withdraw your moneyearly you lose all interest. How would you evaluate this additional piece of information when making your

34、 decision?SOLUTION:c. Effective Annual Rate: BankAnnual = 8%.Effective Annual Rate BankDaily = 1 + .075365 - 1 = .07788 = 7.788%365Based on effective annual rates, you would prefer BankAnnual (you will earn more money.)d. If BankAnnual s 8% annual returndsnditioned upon leaving the money in for one

35、full year, I would need tobe sure that I did not need my money within the one year period. If I were unsure of when I mightneed themoney, it might be safer to go for BankDaily. The option to withdraw my money whenever I might need it will cost me the potential difference in interest:FV (BankAnnual)

36、= $1,000 x 1.08 = $1,080FV (BankDaily) = $1,000 x 1.07788 = $1,077.88Difference = $2.12.16. What are the effective annual rates of the following:a. 12% APR compounded monthly?b. 10% APR compounded annually?c. 6% APR compounded daily?SOLUTION:Effective Annual Rate (EFF) = 1 + APR m - 1md. (1 + .12) 1

37、2 1 = .1268 = 12.68%12e. (1 + .10) - 1 = .10 = 10%1f. (1 + .06) 365 - 1 = .0618 = 6.18%36517. Harry promises that an investment in his firm will double in six years. Interest is assumed to be paid quarterly and reinvested. What effective annual yield does this represent?SOLUTION:EAR=(1.029302) 4-1=1

38、2.25%18. Suppose you know that you will need $2,500 two years from now in order to make a down payment on a car.a. BankOne is offering 4% interest (compounded annually) for two-year accounts, and BankTwo is offering4.5% (compounded annually) for two-year accounts. If you know you need $2,500 two yea

39、rs from today, how much will you need to invest in BankOne to reach your goal? Alternatively, how much will you need to invest in BankTwo? Which Bank account do you prefer?b. Now suppose you do not need the money for three years, how much will you need to deposit today in BankOne? BankTwo?SOLUTION:a

40、. Present Value of Deposit in BankOne:Formula:PV = $2,500= $2,311.39(1.04)2Present Value of Deposit in BankTwo:24.5?$2,5000PV = $2,289.32Formula:PV = $2,500= $2,289.32一2(1.045)2You would prefer BankTwo because you earn more; therefore, you can deposit fewer dollars today in order to reach your goal

41、of $2,500 two years from today.b.Present Value of Deposit in BankOne:34?$2,5000PV= $2,222.49Formula:PV = $2,500= $2,222.49(1.04)3Present Value of Deposit in BankTwo:Formula:PV = $2,500= $2,190.74(1.045)3Again, you would prefer BankTwo because you earn more; therefore, you can deposit fewer dollars t

42、oday in order to reach your goal of $2,500 three years from today.19. Lucky Lynn has a choice between receiving $1,000 from her great-uncle one year from today or $900 from her great-aunt today. She believes she could invest the $900 at a one-year return of 12%.a. What is the future value of the gif

43、t from her great-uncle upon receipt? From her great-aunt?b. Which gift should she choose?c. How does your answer change if you believed she could invest the $900 from her great-aunt at only 10%?At what rate is she indifferent?SOLUTION:d. Future Value of gift from great-uncle is simply equal to what

44、she will receive one year from today ($1000). She earns no interest as she doesn t receive the money until next year.e. Future Value of gift from great-aunt: $900 x (1.12) = $1,008.f. She should choose the gift from her great-aunt because it has future value of $1008 one year from today. The gift fr

45、om her great-uncle has a future value of $1,000. This assumes that she will able to earn 12% interest on the $900 deposited at the bank today.g. If she could invest the money at only 10%, the future value of her investment from her great-aunt would only be $990: $900 x (1.10) = $990. Therefore she w

46、ould choose the $1,000 one year from today. Lucky Lynn would be indifferent at an annual interest rate of 11.11%:$1000 = $900 or (1+i) = 1,000 = 1.1111(1+i)900i = .1111 = 11.11%20. As manager of short-term projects, you are trying to decide whether or not to invest in a short-term project that pays

47、one cash flow of $1,000 one year from today. The total cost of the project is $950. Your alternative investment is to deposit the money in a one-year bank Certificate of Deposit which will pay 4% compounded annually.a. Assuming the cash flow of $1,000 is guaranteed (there is no risk you will not rec

48、eive it) what would be a logical discount rate to use to determine the present value of the cash flows of the project?b. What is the present value of the project if you discount the cash flow at 4% per year? What is the net present value of that investment? Should you invest in the project?c. What w

49、ould you do if the bank increases its quoted rate on one-year CDs to 5.5%?d. At what bank one-year CD rate would you be indifferent between the two investments?SOLUTION:e. Because alternative investments are earning 4%, a logical choice would be to discount the projectat 4%. This is because 4% can b

50、e considered as your opportunity cost for taking the project; hence, it is your cost of funds.f. Present Value of Project Cash Flows:PV = $1,000= $961.54(1.04)The net present value of the project = $961.54 - $950 (cost) = $11.54The net present value is positive so you should go ahead and invest in t

51、he project.g. If the bank increased its one-year CD rate to 5.5%, then the present value changes to:PV = $1,000= $947.87(1.055)Now the net present value is negative: $947.87 - $950 = - $2.13. Therefore you would not want to invest in the project.h. You would be indifferent between the two investment

52、s when the bank is paying the following one-year interest rate:$1,000 = $950 hence i = 5.26% (1+i)21. Calculate the net present value of the following cash flows: you invest $2,000 today and receive $200 one year from now, $800 two years from now, and $1,000 a year for 10 years starting four years f

53、rom now. Assume that the interest rate is 8%.SOLUTION:Since there are a number of different cash flows, it is easiest to do this problem using cash flow keys on the calculator:0-$2,000Cfi1$200Cfi2$800Cfi3$0CfiRefer to note102nd F Ni41,000Cfi8iNPV=$4,197.74Note: This enables you to instruct the calcu

54、lator that the next cash flow occurs for N times (here N =10).22. Your cousin has asked for your advice on whether or not to buy a bond for $995 which will make one payment of $1,200 five years from today or invest in a local bank account.a. What is the internal rate of return on the bond s cash flo

55、ws? What additional information do you need tomake a choice?b. What advice would you give her if you learned the bank is paying 3.5% per year for five years (compounded annually?)c. How would your advice change if the bank were paying 5% annually for five years? If the price of thebond were $900 and

56、 the bank pays 5% annually?SOLUTION:a. Internal Rate of Return:5?-$995$12000i =3.82%Formula:$995 x (1+i) 5 = $1,200.(1+i) 5 = $1,200$995Take 5th root of both sides:(1+i) =1.0382i = .0382 = 3.82%In order to make a choice, you need to know what interest rate is being offered by the local bank.d. Upon learning that the bank is paying 3.5%, you would tell her to choose the bond because it is earning a higher rate of return of 3.82% .e. If the bank were paying 5% per