Chapter 4 The Time Value of Money
Fundamentals of Corporate Finance 3rd ed Jonathan Berk Ch4
Chapter 4Time Value of Money:Valuing Cash Flow StreamsNote: All problems in this chapter are available in MyFinanceLab. An asterisk (*) indicates problemswith a higher level of difficulty. Editor’s Note: As we move forward to more complex financial analysis, the student will notice that someproblems may contain a large amount of data from different time periods that require more complicated and intensive analysis. Modern information technology has evolved in the form of financial calculators with built-in analysis functions and Time Value ofMoney functions that are built into computer-based electronic spreadsheet software such as Excel. The solutions for many data- and computationally intensive problems will be presented in formula form with a solution, as well as with the appropriate financialcalculator commands and Excel functions to produce the same correct answer. In this way, it is our expectation that students will develop proficiency in solving financial analysis problems by mathematical calculation as well as by using financial calculators and electronic spreadsheets.1. a.Year 1 2 3 4 5 CF1020304050()()()()23451020304050106.531.10 1.10 1.10 1.10 1.10PV =++++= b.()()()()23455040302010120.921.10 1.10 1.10 1.10 1.10PV =++++= Year 1 2 3 4 5 CF5040302010Chapter 4 Time Value of Money: Valuing Cash Flow Streams 27c. The present value is different because the timing of the cash flows is different. In the secondset, you get the larger cash flows earlier, so it is more valuable to you and shows up as a higher PV.2.Year 1 2 3 4 CF100-100200-200()()()23410010020020028.491.15 1.15 1.15 1.15PV =-+-=3.Year 1 2 3 4 5 CF10000-2000-3000-3500-3975()()()2342,0003,0003,5003,97510,000.121.085 1.085 1.085 1.085PV =+++=, so yes, the PV of your payments would just cover the loan amount (actually exceeding it by 12 cents). Thus, the maximum you could borrow would be 10,000.124. Month1 2 3 4 5 6 7 8 CF500 550 600 650 700 750 800 850()()()()()()()23456785005506006507007508008505,023.751.015 1.015 1.015 1.015 1.015 1.015 1.015 1.015PV =+++++++=So, the PV of your payments will exceed the loan balance. This means that your planned payments will be enough to payoff your credit card.5.From the bank’s perspective, the timeline is the same except all the signs are reversed.28 Berk/DeMarzo/Harford •Fundamentals of Corporate Finance, Third Edition, Global Edition6.0 1 2 3 4 48–300 –300 –300 –300 –300From the bank’s perspective, the timeline would be identical except with opposite signs.7. Plan: Draw the timeline and then compute the FV of these two cash flows.Execute:Timeline (because we are computing the future value of the account, we will treat the cashflows as positive—going into the account):800 600FV = ?FV = 800(1.05) + 600 = 1,440Evaluate:The timeline helps us organize our work so that we get the number of periods of compoundingcorrect. The first cash flow will have 1 year of compounding, but the second cash flow will bedeposited at the end of period 1, so it receives no compounding.8. Editor’s Note: In several previous problems, we used a financial calculator to solve a timevalue of money problem. Problems could be solved quickly and easily by manipulating the N,I/Y, PV, PMT, and FV keys. In each of these problems, there was a series of payments of equal amount over time, i.e., an annuity. All you had to do to input this series was enter the payment(PMT) and the number of payment (N).Many financial analysis problems involve a series of equal payments, but others involve aseries of unequal payments. A financial calculator can be used to evaluate an unequal series ofcash flows (using the cash flow (CF) key), but the process is cumbersome because each cashflow must be entered individually. I urge each student to study the Chapter 4 Appendix: “Usinga Financial Calculator,” as well as instructional materials that are prod uced by themanufacturers of the financial calculator.Here we will solve a problem with uneven cash flows mathematically and with a financialcalculator.Plan: It is wonderful that you will receive this windfall from your investment in your friend’sbusiness. Because the cash flow payments to you are of different amounts and paid over threeyears, there are different ways in which you can think about how much money you arereceiving.Chapter 4 Time Value of Money: Valuing Cash Flow Streams 29Execute:a. 2310,00020,00030,000PV 1.035 1.035 1.0359,66218,67027,05855,390=++=++=The Texas Instrument BA II PLUS calculator has a cash flow worksheet accessed with the CF key. To clear all previous values that might be stored in the calculator, press the CF, second, and CE/C buttons. The screen should show CF 0 = asking for the cash flow at time 0, which in this problem is 0. Press 0, then press enter, and then the down key button. Thescreen should show CO1 asking for the cash flow at time 1, which in this problem is 10,000. Input 10,000 followed by the enter key, followed by the down button. The screen should show FO1 = 1.0 asking for the frequency of this cash flow. Because it occurs only once, it is correct, and we press the down key. The screen now has CO2 asking for the time 2 cash flow, which is 20,000, which we input, followed by the enter key and the down key. The screen now has FO2 = 1.0, which is correct. Enter the down key, which asks for the third cash flow, which is 30,000. Input 30,000, followed by the enter and down keys. Now press the NPV key, and the calculator will display I = asking for the interest rate, which is 3.5. Input 3.5, press the enter key, and press the down key, and the screen will display NPV =. Then press the CPT button, and the screen should display 55,390.33, the Net Present Value.b. 3FV 55,390 1.03561,412=⨯=Evaluate: You may ask: “How much better off am I because of this windfall?” There are several answers to this question. The value today (i.e., the present value) of the cash you will receive over three years is $55,390. If you decide to reinvest the cash flows as you receive them, then in three years you will have $61,412 (i.e., future value) from your windfall.9. Plan: Use Eq. 4.3 to compute the PV of this stream of cash flows and then use Eq. 4.1 to compute the FV of that present value. To answer part (c), you need to track the new deposit made each year along with the interest on the deposits already in the bank.Execute: a. and b.233100100100257.71(1.08)(1.08)(1.08)257.71(1.08)324.64PV FV =++===30 Berk/DeMarzo/Harford • Fundamentals of Corporate Finance, Third Edition, Global Editionc. ()11Year 1: 100Year 2: 1001.08100208Year 3: 208(1.08)100324.64+=+=Evaluate:By using the PV and FV tools, we are able to keep track of our balance as well as quicklycalculate the balance at the end. Whether we compute it step by step as in part (c) or directly as in part (b), the answer is the same.10. Plan: First, create a timeline to understand when the cash flows are occurring.Second, calculate the present value of the cash flows.Once you know the present value of the cash flows, compute the future value (of this present value) at date 3.Execute: 231,0001,0001,000PV 1.05 1.05 1.0595********,723=++=++=NI/Y PV PMT FV Excel FormulaGiven: 35.00% -1,000 0Solve for PV:2,723.25=PV(0.05,3,-1,000,0)33FV 2,723 1.053,152=⨯=NI/Y PV PMT FV Excel FormulaGiven: 35.00% -2,723.43Solve for FV:3,152.71=FV(0.05,3,0,-2,723.43)Evaluate: Because of the bank’s offer, you now hav e two choices as to how you will repay this loan. Either you will pay the bank $1,000 per year for the next three years as originallypromised, or you can decide to skip the three annual payments of $1,000 and pay $3,152 in year three.You now have the information to make your decision.11. a. The FV of 100,000 invested for 35 years at 9% is 100,000 × (1.09)35 = 2,041,397b. The FV of 100,000 invested for 25 years at 9% is 100,000 × (1.09)25 = 862,308Chapter 4 Time Value of Money: Valuing Cash Flow Streams 31c. The difference is so large because of the effect of compounding, which is exponential. Inthe scenario where you invest earlier, those 10 years are critical to the compound growthyou achieve on your investment.12. Plan: This scholarship is a perpetuity. The cash flow is $5,000 and the discount rate is 6%.Execute:Timeline:5,000 5,000 …PV 5,000/0.06 = 83,333.33Evaluate:With a donation of $83,333.33 today and 6% interest, the university can withdraw the interest every year ($5,000) and leave the endowment intact to generate the next year’s $5,000. It can keep doing this forever.13. P lan: This is a deferred perpetuity. Here is the timeline:Do this in two steps:1. Calculate the value of the perpetuity in year 9, when it will start in only one year(we already did this in problem 12).2. Discount that value back to the present.Execute:The value in year 9 is 5,000/0.06 = 83,333.33.The value today is 83,333.33/1.069 = 49,324.87Evaluate:Because your endowment will have 10 years to earn interest before making its first payment, you can endow the scholarship for much less. The value of your endowment must reach$83,333.33 the year before it starts (in 9 years). If you donate $49,324.87 today, it will grow at 6% interest for 9 years, just reaching $83,333.33, one year before the first payment.32 Berk/DeMarzo/Harford • Fundamentals of Corporate Finance, Third Edition, Global Edition14. The timeline for this investment is:a. The value of the bond is equal to the present value of the cash flows. By the perpetuity formula, which assumes the first payment is at period 1, the value of the bond is:PV = 1,000 /0.08 = £12,500 b. The value of the bond is equal to the present value of the cash flows. The first payment will be received at time zero. The cash flows are the perpetuity plus the payment that will be received immediately.PV = 1,000 /0.08 + 1,000 = £13,50015. Plan: Draw the timeline of the cash flows for the investment opportunity. Compute the NPV ofthe investment opportunity at 7% interest per year to determine its value.Execute: The cash flows are a 100-year annuity, so by the annuity formula:1001,0001PV 10.07 1.0714,269.25⎛⎫=- ⎪⎝⎭=NI/Y PV PMT FV Excel FormulaGiven: 1007.00% 1,000 0Solve for PV:(14,269.25)=PV(0.07,100,1000,0)Evaluate: The PV of $1,000 to be paid every year for 100 years discount to the present at 7% is $14,269.25.16. Plan: Prepare a timeline of your grandmother’s deposits.The deposits are an 18-year annuity. Use Eq. 4.6 to calculate the future value of the deposits.Execute: 1811(1)11,000(1.03)123,414.430.03N FV C r r =⨯+-=-=⎡⎤⎡⎤⎣⎦⎣⎦Chapter 4 Time Value of Money: Valuing Cash Flow Streams 33NI/Y PV PMT FV Excel FormulaGiven: 183.00% 0 1,000Solve for FV:(23,414.43)=FV(0.03,18,1000,0)At age 18, you will have $23,414.43 in your account.Evaluate:The interest on the deposits and interest on that interest adds more than $5,414 to the account.17. a.First, we need to calculate the PV of $160,000 in 18 years.18160,000PV (1.08)40,039.84==NI/Y PV PMT FV Excel FormulaGiven: 188.00% 0 160,000Solve for PV:(40,039.84)=PV(0.08,18,0,160000)In order for the parents to have $160,000 in your college account by your 18th birthday, the 18-year annuity must have a PV of $40,039.84. Solving for the annuity payments:1840,039.841110.08 1.08$4,272.33C =⎛⎫- ⎪⎝⎭=which must be saved each year to reach the goal.NI/Y PV PMT FV Excel FormulaGiven: 188.00% 40,039.84Solve for PMT:(4,272)=PMT(0.08,18,40039.84,0)b. First, we need to calculate the PV of $200,000 in 18 years.18200,000PV (1.08)50,049.81==34 Berk/DeMarzo/Harford • Fundamentals of Corporate Finance, Third Edition, Global EditionNI/Y PV PMT FV Excel FormulaGiven: 188.00% 0 200,000Solve for PV:(50,049.81)=PV(0.08,18,0,200000)In order for the parents to have $200,000 in your college account by your 18th birthday, the 18-year annuity must have a PV of $50,049.81. Solving for the annuity payments:18$50,049.811110.08 1.08$5340.42C =⎛⎫- ⎪⎝⎭=which must be saved each year to reach the goal.N I/Y PV PMT FV Excel FormulaGiven: 18.00 0.08 50,049.810.00Solve for PMT:-5340.42=PMT(0.08,18,50049.81,0)*18. Plan:a. Draw the timeline of the cash flows for the loan.1 2 3 4 5To pay off the loan, you must repay the remaining balance. The remaining balance is equal to the present value of the remaining payments. The remaining payments are a four-year annuity, so:b.4 5Execute:a. 45,0001PV 10.06 1.0617,325.53⎛⎫=- ⎪⎝⎭=N I/Y PV PMT FV Excel FormulaGiven: 4 6.00% 5000 0Solve for PV:(17,325.53)=PV(0.06,4,5000,0)Chapter 4 Time Value of Money: Valuing Cash Flow Streams 35b. 5,000PV 1.064,716.98==Evaluate: To pay off the loan after owning the vehicle for one year will require $17,325.53. To pay off the loan after owning the vehicle for four years will require $4,716.98.19. Plan: This is a deferred annuity. The cash flow timeline is:Calculate the value of the annuity in year 17, one period before it starts using Eq. 4.5 and then discount that value back to the present using Eq. 4.2.The value of the annuity in year 17, one period before it is to start is:41100,000111331,212.68(1)0.08(1.08)n CF PV r r ⎡⎤⎡⎤=-=-=⎢⎥⎢⎥+⎣⎦⎣⎦To get its value today, we need to discount that lump sum amount back 17 years to the present:17331,212.68$89,516.50(1.08)=Evaluate:Even though the cash flows are a little unusual (an annuity starting well into the future), we can still value them by combining the PV of annuity and PV of a FV tool. If we invest $89,516.50 today at an interest rate of 8%, it will grow to be enough to fund an annuity of $100,000 per year by the time it is needed for college expenses.20. Plan: This is a deferred annuity. The cash flow timeline is:Calculate the value of the annuity in year 44, one period before it starts using Eq. 4.5 and then discount that value back to the present using Eq. 4.2.Execute:The value of the annuity in year 44, one period before it is to start is:16140,000111377,865.94(1)0.07(1.07)n CF PV r r r ⎡⎤⎡⎤=-=-=⎢⎥⎢⎥+⎣⎦⎣⎦To get its value today, we need to discount that lump sum amount back 44 years to the present:44377,865.94$19,250.92(1.07)=Evaluate:Even though the cash flows are a little unusual (an annuity starting well into the future), we can still value them by combining the PV of annuity and PV of a FV tool. The total value to you today of Social Security’s promise is less than $20,000.*21. Plan: Clearly, Mr. Rodriguez’s contract is complex, calling for payments over many years.Assume that an appropriate discount rate for A-Rod to apply to the contract payments is 7% per year.a. Calculate the true promised payments under this contract, including the deferred payments with interest.b. Draw a timeline of all of the payments.c. Calculate the present value of the contract.d. Compare the present value of the contract to the quoted value of $252 million. What explains the difference?Execute: Determine the PV of each of the promised payments discounted to the present at 7%.2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 $18M 19M19M19M21M19M23M27M27M27M20112012 2013 2014 2015 2016 2017 2018 2019 2020 6.7196M 5.3757M 4.0317M 4.0317M 4.0317M 4.0317M 4.0317M 4.0317M 4.0317M 4.0317MThe PV of the promised cash flows is $165.77 million.Evaluate: The PV of the contract is much less than $252 million. The $252 million value does not discount the future cash flows or adjust deferred payments for accrued interest. *22. a.0 1 2343The amount in the retirement account in 43 years would be:43435,000FV (1.10)10.10$2,962,003.46=-⎡⎤⎣⎦=NI/Y PV PMT FV Excel FormulaGiven: 4310.00%0.00 -5,000Solve for FV:2,962,003.46=FV(0.1,43,-5000,0)b. To solve for the lump sum amount today, find the PV of the $2,962,003.46.432,962,003.46PV (1.10)$49,169.99==N I/Y PV PMT FV Excel FormulaGiven: 43 10.00%0 2,962,003Solve for PV:(49,169.99) =PV(0.1,43,0,2962003.46) c.Solve for the annuity cash flow that, after 20 years, exactly equals the starting value of the account.202,962,003.461110.10 1.10347,915.81C =⎛⎫-⎪⎝⎭=N I/Y PV PMT FV Excel FormulaGiven: 20.00 0.10 2,962,003.460.00Solve for PMT:-347,915.81=PMT(0.1,20,2962003.46,0)d.We want to solve for N , which is the length of time in which the PV of annual payments of $300,000 will equal $2,962,003.46. Setting up the PV of an annuity formula and solving for N :300,000112,962,003.460.10 1.1012,962,003.460.1010.98733451.10300,000110.98733450.01266551.101.1078.95456Log(78.95456)45.84Log(1.10)N N NN N ⎛⎫-= ⎪⎝⎭⨯⎛⎫-== ⎪⎝⎭=-==== e. If we can only invest $1,000 per year, then set up the PV formula using $1 million as the FV and $1,000 as the annuity payment. 431,000111,000,000(1)r r ⎛⎫-= ⎪+⎝⎭To solve for r , we can either guess or use the annuity calculator. You can check and see that r = 11.74291% solves this equation. So the required rate of return must be 11.74291%.N I/Y PV PMT FV Excel FormulaGiven: 43 0.00 -1,000 1,000,000Solve for Rate: 11.74291%=RATE(43,-1000,0,1000000)23. Plan: The bequest is a perpetuity growing at a constant rate. The bequest is identical to a firmthat pays a dividend that grows forever at a constant rate. We can use the constant dividend growth model to determine the value of the bequest.Execute: a.Using the formula for the PV of a growing perpetuity gives1,000PV 0.120.0825,000⎛⎫= ⎪-⎝⎭=which is the value today of the bequest.b.1 2 3 4Using the formula for the PV of a growing perpetuity gives:1,000(1.08)PV 0.120.0827,000=-=which is the value of the bequest after the first payment is made.Evaluate: The bequest is worth $25,000 today and will be worth $27,000 in 1 year’s time. *24. Plan: The machine will produce a series of savings that are growing at a constant rate. The rateof growth is negative, but the constant growth model can still be used.Execute:The timeline for the saving would look as follows:We must value a growing perpetuity with a negative growth rate of – 0.03:2,000PV 0.08(0.03)$18,181.82=--= Evaluate: The value of the savings produced by the machine is worth $18,181.82 today.25. Plan: Nobel’s bequest is a perpetu ity. The total amount is 5 ⨯ $45,000 = $225,000. With a cashflow of $225,000 and an interest rate of 7% per year, we can use Eq. 4.4 to solve for the total amount he would need to use to endow the prizes. In part (b), we will need to use the formula for a growing perpetuity (Eq. 4.7) to find the new value he would need to leave. Finally, in part (c), we can use the FV equation (Eq. 4.1) to solve for the future value his descendants would have had if they had kept the money and invested it at 7% per year.a. In order to endow a perpetuity of $225,000 per year with a 7% interest rate per year, he would need to leave $225,000/0.07 = $3,214,285.71.b. In order to endow a growing perpetuity with an interest rate of 7% and a growth rate of 4% and an initial cash flow of $225,000, he would have to leave:1225,0007,500,0000.070.04CF PV r g ===-- c. FV = PV (1 + r )n = 7,500,000(1.07)118 = $ 21,996,168,112Evaluate:The prizes that bear Nobel’s name were very expensive to endow—$3 million was an enormous sum in 1896. However, Nobel’s endowment has been able to generate enough interest each year to fund the prizes, which now have a cash award of approximately $1,500,000 each!26. Plan: The drug will produce 17 years of cash flows that will grow at 5% annually. The value ofthis stream of cash flows today must be determined. We can use the formula for a growing annuity (Eq. 4.8) or Excel to solve this. C = 2, r = 0.10, g = 0.05, N = 17Execute: 171 1.052121.860.100.05 1.10PV ⎛⎫⎛⎫⎛⎫=-= ⎪ ⎪ ⎪ ⎪-⎝⎭⎝⎭⎝⎭Because the cash flows from this investment will continue for 17 years, we decided to solve for the Net Present Value by using the NPV function in Excel. This is shown on the next page. The 17 cash flows are presented in columns C, D,…S. The initial cash flow of $2M is presented in cell C8, and each subsequent cash flow grows at 5% until $4.365749M is presented in year 17 in cell S8. (Note that columns G through Q are not presented.) The NPV of the project is calculated using the NPV formula = NPV(C3,C8:S8) in cell B10. The NPV of the future cash flows is $21.86M.ABCDEFR S 1 2 1+g 1.05 3 r 0.1 4 5 6 T 0 1 2 3 4 16 17 7 8 2 2.1 2.205 2.31525 4.1578564.3657499 10 NPV $21.86 11 12 EXCEL NPV FORMULA =NPV(C3,C8:S8) 13Evaluate: The value today of the cash flows produced by the drug over the next 17 years is $21.86 million. Because the cash flows are expected to grow at a constant rate, we can use the growing annuity formula as a shortcut.27. Plan: Your rich aunt is promising you a series of cash flows over the next 20 years. You mustdetermine the value of those cash flows today. This is a growing annuity and we can use Eq.4.8 to solve it, or we can also solve it in Excel. C= 5, r= 0.03, g= 0.05 and N= 20Execute:201 1.035179.820.050.03 1.05PV⎛⎫⎛⎫⎛⎫=-=⎪⎪ ⎪⎪-⎝⎭⎝⎭⎝⎭Because the cash flows from this investment will continue for 20 years, we decided to solve for the Net Present Value by using the NPV function in Excel. This is shown on the next page. The 20 cash flows are presented in columns C, D,…V. The initial cash flow of $5,000 is presented in cell C8, and each subsequent cash flow grows at 3% until $8,767.53 is presented in year 20 in cell V8. (Note that columns G through R are not presented.) The NPV of the project is calculated using the NPV formula =NPV(C3,C8:V8) in cell B10. The NPV of the future cash flows is $79,824.A B C D E …S T U V12 1 + g 1.033 r 0.05456 T 0 1 2 3 17 18 19 2078 5 5.15 5.3045 8.023532 8.264238 8.512165 8.76753910 NPV $79.821112 EXCEL NPV FORMULA =NPV(C3,C8:V8)13Evaluate: Because your aunt will be increasing what she gives each year at a constant rate, we can use the growing perpetuity formula as a shortcut to value the stream of cash flows. Her gift is quite generous: It is equivalent to giving you almost $80,000 today!28. Plan: This problem is asking us to solve for the rate of return (r). Because there are norecurring payments, we can use Eq. 4.1 to represent the problem and then just solvealgebraically for r. We have FV = 200, PV = 100, n = 10.Execute:FV = PV(1 +r)n200 = 100(1 +r)10, so r = (200/100) 1/10– 1 = 0.072 or 7.2%Evaluate:The implicit return we earned on the savings bond was 7.2%. Our money doubled in 10 years, which by the rule of 72 meant that we earned about 72/10 = 7.2% and our calculationconfirmed that.29. Plan: This problem is again asking us to solve for r. We will represent the investment with Eq.4.1 and solve for r. We have PV = 2,000, FV = 10,000, n = 10. The second part of the problemasks us to change the rate of return going forward and calculate the FV in another 10 years.Execute:a. FV= PV(1 + r)n10,000 = 2,000(1 +r)10, so r = (10,000/2,000) 1/10– 1 = 0.1746, or 17.46%b.FV = 10,000(1.12)10= $31,058.4830. Plan: Draw a timeline and determine the IRR of the investment.Execute:–IRR is the r that solves:15,000 = 20,000/(1 +r), so r = (20,000/15,000) – 1 = 33.33%Evaluate: You are making a 33.33% IRR on this investment.31. Plan: Draw a timeline to demonstrate when the cash flows will occur. Then solve the problemto determine the payments you will receive.Execute:–P = C/r,So, C = P × r= 500 × 0.08= $40Evaluate: You will receive $40 per year into perpetuity.32. Plan: Draw a timeline to determine when the cash flows occur. Solve the problem to determinethe annual payments. Timeline (from the perspective of the bank):Execute:–203,000,000$240,727.761110.05 1.05==⎛⎫- ⎪⎝⎭Cwhich is the annual payment.NI/YPVPMTFVExcel FormulaGiven: 205.00% -3000000Solve for PMT:$240727.76=PMT(0.05,20,-3000000,0)Evaluate: You will have to pay the bank $240,727.76 per year for 20 years in mortgage payments.*33. Plan: Draw a timeline to demonstrate when the cash flows will occur. Determine the annualpayments.Execute:0 2 4 6 20–This cash flow stream is an annuity. First, calculate the two-year interest rate: The one-year rate is 4%, and $1 today will be worth (1.04)2 = 1.0816 in two years, so the two-year interest rate is 8.16%. Using the equation for an annuity payment:1050,0001110.0816(1.0816)$7,505.34C =⎛⎫- ⎪⎝⎭=which is the payment you must make every two years.N I/Y PV PMT FV Excel Formula Given: 10 8.16% -50,000.00Solve for PMT: $7,505.34=PMT(0.0816,10,-50000,0)Evaluate: You must pay the art dealer $7505.34 every two years for 20 years.*34. Plan: Draw a timeline to determine when the cash flows occur. Timeline (where X is the balloon payment):0 1 2330–+ X Note that the PV of the loan payments must be equal to the amount borrowed.Execute:303023,5001300,00010.07 1.07(1.07)X⎛⎫=-+ ⎪⎝⎭ Solving for X :303023,5001300,0001(1.07)0.07 1.07$63,848X ⎡⎛⎫⎤=-- ⎪⎢⎥⎣⎝⎭⎦= NI/Y PV PMT FV Excel Formula Given: 307.00%-23,500Solve for PV:291,612.47=PV(0.07,30,-23500,0)The present value of the annuity is $291,612.47, which is $8,387.53 less than the $300,000.00. To make up for this shortfall with a balloon payment in year 30 would require a payment of $63,848.02.N I/YPVPMT FVExcel FormulaGiven: 30 7.00% 8,387.53 0Solve for FV:(63,848.02) =FV(0.07,30,0,8387.53)Evaluate: At the end of 30 years you would have to make a $63,848 single (balloon) payment to the bank.*35. Plan: Draw a timeline to demonstrate when the cash flows occur. We know that you intend tofund your retirement with a series of annuity payments and the future value of that annuity is $2 million.22 23 24 25 65 0 1 2343CC CCCExecute: FV = $2 million.The PV of the cash flows must equal the PV of $2 million in 43 years. The cash flows consist of a 43-year annuity, plus the contribution today, so the PV is:()431PV 10.05 1.05C C ⎛⎫=-+ ⎪⎝⎭The PV of $2 million in 43 years is:432,000,000$245,408.80(1.05)=N I/Y PV PMT FV Excel Formula Given: 43 5.00% 0 2,000,000Solve for PV: (245,408.80)=PV(0.05,43,0,2000000)Setting these equal gives434311245,408.800.05(1.05)245,408.80$13,232.5011110.05(1.05)C C C ⎛⎫-+= ⎪⎝⎭⇒==⎛⎫-+ ⎪⎝⎭We need $245,408.80 today to have $2,000,000 in 43 years. If we do not have $245,408.80 today, but wish to make 44 equal payments (the first payment is today, making the payments an annuity due) then the relevant Excel command is:=PMT(rate,nper,pv,(fv),type =PMT(.05,44,245,408.80,0,1) = 13,232.50Type is set equal to 1 for an annuity due as opposed to an ordinary annuity.Evaluate: You would have to put aside $13,232.50 annually to have the $2 million you wish to have in retirement.36. Plan: This problem is asking you to solve for n . You can do this mathematically using logs, orwith a financial calculator or Excel. Because the problem happens to be asking how long it will。
财金英语教程参考答案
财金英语教程参考答案Chapter 1: Introduction to Finance1. What is finance?- Finance is the management of money and includesactivities such as investing, borrowing, lending, budgeting, saving, and forecasting.2. What are the three main functions of finance?- The three main functions of finance are planning, acquiring, and managing financial resources.3. What is the time value of money?- The time value of money is the concept that a sum of money is worth more now than the same sum in the future dueto its potential earning capacity.4. How does inflation affect the value of money?- Inflation erodes the purchasing power of money over time, meaning that the same amount of money will buy fewer goodsand services in the future.5. What is the difference between a bond and a stock?- A bond is a debt instrument where an investor lends money to an entity in exchange for interest payments, while a stock represents ownership in a company and offers thepotential for capital gains and dividends.Chapter 2: Financial Statements1. What are the four main financial statements?- The four main financial statements are the balance sheet, income statement, cash flow statement, and statement of changes in equity.2. What is the purpose of a balance sheet?- The balance sheet provides a snapshot of a company's financial position at a specific point in time, showing its assets, liabilities, and equity.3. How is net income calculated?- Net income is calculated by subtracting all expensesfrom the total revenue of a company during a specific period.4. What does the cash flow statement show?- The cash flow statement shows the inflow and outflow of cash within a business over a period of time, categorizedinto operating, investing, and financing activities.5. What is the statement of changes in equity?- The statement of changes in equity shows the changes in the equity accounts of a company over a period of time, including retained earnings, capital contributions, and other comprehensive income.Chapter 3: Financial Analysis1. What are the main types of financial analysis?- The main types of financial analysis are ratio analysis,horizontal analysis, vertical analysis, and trend analysis.2. What is the purpose of ratio analysis?- Ratio analysis is used to evaluate a company's financial health by comparing various financial ratios such asliquidity, profitability, and leverage ratios.3. What is horizontal analysis?- Horizontal analysis involves comparing financial statement items over multiple periods to identify trends and changes in performance.4. What is vertical analysis?- Vertical analysis, also known as common-size analysis,is a method of financial statement analysis where each itemis expressed as a percentage of a base figure, typicallytotal assets or total revenue.5. What is trend analysis?- Trend analysis involves examining the historical data of financial metrics over time to predict future trends and performance.Chapter 4: Risk Management1. What is risk management?- Risk management is the process of identifying, assessing, and prioritizing potential risks to an investment or project, and taking steps to mitigate or avoid these risks.2. What are the types of risks in finance?- The types of risks in finance include market risk,credit risk, liquidity risk, operational risk, and legal risk.3. What is diversification?- Diversification is a risk management strategy that involves spreading investments across various financial instruments, industries, or geographic regions to reduce overall risk.4. What is hedging?- Hedging is a risk management technique used to reducethe risk of price fluctuations in an asset by taking an offsetting position in a related security.5. What is the role of insurance in risk management?- Insurance is a risk management tool that providesfinancial protection against potential losses or damages by transferring the risk to an insurance company in exchange for a premium.Chapter 5: Investment Strategies1. What are the different types of investment strategies?- Types of investment strategies include passive investing, active investing, value investing, growth investing, and income investing.2. What is the difference between passive and active investing?- Passive investing involves a "set it and forget it" approach, typically using index funds, while active investingrequires regular buying and selling of individual securities based on market research and analysis.3. What is value investing?- Value investing is an investment strategy that involves buying stocks that are considered undervalued by the market, with the expectation that their true value will eventually be recognized.4. What is growth investing?- Growth investing focuses on companies that are expected to grow at an above-average rate compared to the market, often investing in companies with strong competitive advantages and high growth potential.5. What is income investing?- Income investing is an investment strategy aimed at generating a steady stream of income from investments, typically through dividends or interest payments.Chapter 6: International Finance1. What is international。
CFA考试投资分析的数量方法投资工具
第一章：货币的时间价值Chapter ⒈ The Time Value of Money§⒈解释利息率是对投资者的不同风险予以回报的实际无风险利率和风险溢价的总和利息率和折现率（Interest Rates and Discount Rates）货币时间价值概念的基础：收益率（rates of return）、利息率（interest rate）、要求的收益率（required rates of return）、折现率（discount rates）、机会成本（opportunity costs）、通货膨胀（inflation）和风险（risk）。
货币的时间价值，反映了时间、现金流量和利息率三者之间的关系。
投资者偏好现在消费。
利息率是投资者推迟现在消费的回报。
在确定世界，利息率被认为是无风险（risk-free）利率。
一般是国家的短期债券，如美国的国库券（Treasury-bills, T-bills）。
在不确定的世界，有两个因素影响利息率：①通货膨胀。
贷款者承担通货溢价（inflation premium）和推迟消费的机会成本。
因此，货币的名义成本（nominal cost of money），由实际利率（real rate）和通货溢价组成。
②风险。
贷款者还承担了不履行风险（default risk）。
因此，利息率包括：名义的无风险利率和不履行风险溢价。
利息率的意义：①收益要求率。
即促使投资者放弃现在消费所要求的收益。
②折现率（利息率和折现率可以交互使用）。
③机会成本。
即投资者按某一选择行为而放弃其他选择所失去的价值。
影响利息率最重要的因素是：资金的供求关系。
§⒉计算整笔现金的终值（FV）和现值（PV）单一现金流量的终值（The Future Value of a Single Cash Flow）整笔现金流（或lump-sum investment）的终值计算公式（N的初值为0）：基本概念：①简单利息（simple interest），即利息率乘原始本金。
财务管理基础课件：The Time Value of Money
Where FV = Future value PV = Present value i = Interest rate n = Number of periods;
In the previous case, PV = $1,000, i = 10%, n = 4, hence;
1st year……$1,000 X 1.10 = $1,100 2nd year…...$1,100 X 1.10 = $1,210 3rd year……$1,210 X 1.10 = $1,331 4th year……$1,331 X 1.10 = $1,464
9-4
Future Value – Single Amount (Cont’d)
• The time value of money is used to determine whether future benefits are sufficiently large to justify current outlays
• Mathematical tools of the time value of money are used in making capital allocation decisions
annuity
9-25
Yield – Present Value of a Single Amount
• To calculate the yield on an investment producing $1,464 after 4 years having a present value of $1,000:
Table 9–5
财务管理专业英语-The Time Value of Money
5 years: PV = 500 / (1.1)5 = 310.46 10 years: PV = 500 / (1.1)10 = 192.77
4
Effects of Compounding
Simple interest Compound interest Consider the previous example
FV with simple interest = 1000 + 50 + 50 = 1100 FV with compound interest = 1102.50 The extra 2.50 comes from the interest of .05(50)
12
PV – One Period Example
Suppose you need $10,000 in one year for the down payment on a new car. If you can earn 7% annually, how much do you need to invest today?
8
Future Values – Example 3
Suppose you had a relative deposit $10 at 5.5% interest 200 years ago. How much would the investment be worth today?
FV = 10(1.055)200 = 447,189.84
16
《金融学》答案第四章 货币的时间价值与现金流贴现分析
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.Outline4.1Compounding4.2The Frequency of Compounding4.3Present Value and Discounting4.4Alternative Discounted Cash Flow Decision Rules4.5Multiple Cash Flows4.6Annuities4.7Perpetual Annuities4.8Loan Amortization4.9Exchange Rates and Time Value of Money4.10Inflation and Discounted Cash Flow Analysis4.11Taxes and Investment DecisionsSummary∙Compounding 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+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 positive 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.How to Do TVM Calculations in MS ExcelAssume you have the following cash flows set up in a spreadsheet:A B1t CF20-1003150426053706NPV7IRRMove the cursor to cell B6 in the spreadsheet. Click the function wizard f x in the tool bar and when a menu appears, select financial and then NPV. 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 B6should contain the following:=NPV(0.1,B3:B5)+B2The first variable in parenthesis is the discount rate. Make sure to input 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 be $47.63.Now move the cursor to cell B7to 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 have finished:A B1t CF20-1003150426053706NPV47.637IRR34%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?2. a. If you invest $100 every year for the next 20 years, starting one year from today and you earninterest 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?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 now.e.$100 each year beginning one year from now and continuing forever.e.PV = $100 = $1,000.104.You want to establish a “wasting” fund which will provide you with $1000 per year for four years, at which 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 equal 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:b. 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 payments.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 taketo pay off the mortgage?d.If you can pay $1500 per month, need to borrow $100,000, and want a 25 year mortgage, what is thehighest interest rate you can pay?SOLUTION:a.Note: Do not round off the interest rate when computing the monthly rate or you will not get the same answerreported 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 Island 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:8.You win a $1 million lottery which 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:9.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 per year out of this inheritance, how long will the money last?SOLUTION:10.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 APR on the mortgage loan?SOLUTION:If 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:11.Suppose that the mortgage loan described 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 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 between 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 will most individuals make? What likely reaction will Bank B have? SOLUTION:$500 x (1.07) = $535Formula:$500 x (1.06) = $530a.You will decide to save your money in Bank A because you will have more money at the end of the year. Youmade an extra $5 because of your savings decision. That is an increase in value of 1%. Because interestcompounded 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 individuals will make the same decision and eventually Bank B will have to raise its rates. However, it isalso 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 order 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 increase her wealth to $5,000?b.Exactly how long does it actually take?SOLUTION:a.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.b.At 10% interestFormula:$2,500 x (1.10)n = $5,000Hence, (1.10)n = 2.0n log 1.10 = log 2.0n = .693147 = 7.27 Years.095310rry’s bank account has a “floating” interest rate on certain 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 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% 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 decision?SOLUTION:a.Effective Annual Rate: BankAnnual = 8%.Effective Annual Rate BankDaily = [1 + .075]365 - 1 = .07788 = 7.788%365Based on effective annual rates, you would prefer BankAnnual (you will earn more money.)b.If BankAnnual’s 8% annual return is conditioned upon leaving the money in for one 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 might need the money, 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) = $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 - 1ma.(1 + .12)12 - 1 = .1268 = 12.68%12b.(1 + .10)- 1 = .10 = 10%1c.(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?EAR=(1.029302)4-1=12.25%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 years 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 inBankOne? BankTwo?SOLUTION:PV = $2,500= $2,311.39(1.04)2PV = $2,500= $2,289.32(1.045)2You would prefer BankTwo because you earn more; therefore, you can deposit fewer dollars today in order to reach your goal of $2,500 two years from today.b.PV = $2,500= $2,222.49(1.04)3PV = $2,500= $2,190.74(1.045)3Again, you would prefer BankTwo because you earn more; therefore, you can deposit fewer dollars today 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 gift 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:a. Future Value of gift from great-uncle is simply equal to what she will receive one year from today ($1000). Sheearns no interest as she doesn’t receive the money until next year.b. Future Value of gift from great-aunt: $900 x (1.12) = $1,008.c. She should choose the gift from her great-aunt because it has future value of $1008 one year from today. Thegift from 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.d. 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 would 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 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 receive it) what would be alogical 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 netpresent value of that investment? Should you invest in the project?c.What would 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:a.Because alternative investments are earning 4%, a logical choice would be to discount the project’s cash flowsat 4%. This is because 4% can be considered as your opportunity cost for taking the project; hence, it is your cost of funds.b.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 the project.c.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.d.You would be indifferent between the two investments when the bank is paying the following one-year interestrate:$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 from 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: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 flows? 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 the bank pays 5% annually?SOLUTION:a.$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.b.Upon learning that the bank is paying 3.5%, you would tell her to choose the bond because it is earning a higherrate of return of 3.82% .c.If the bank were paying 5% per year, you would tell her to deposit her money in the bank. She would earn ahigher rate of return.5.92% is higher than the rate the bank is paying (5%); hence, she should choose to buy the bond.23.You and your sister have just inherited $300 and a US savings bond from your great-grandfather who had left them in a safe deposit box. Because you are the oldest, you get to choose whether you want the cash or the bond. The bond has only four years left to maturity at which time it will pay the holder $500.a.If you took the $300 today and invested it at an interest rate 6% per year, how long (in years) would ittake for your $300 to grow to $500? (Hint: you want to solve for n or number of periods. Given these circumstances, which are you going to choose?b.Would your answer change if you could invest the $300 at 10% per year? At 15% per year? What otherDecision Rules could you use to analyze this decision?SOLUTION:a.$300 x (1.06)n = $500(1.06)n = 1.6667n log 1.06 = log 1.6667n = .510845 = 8.77 Years.0582689You would choose the bond because it will increase in value to $500 in 4 years. If you tookthe $300 today, it would take more than 8 years to grow to $500.b.You could also analyze this decision by computing the NPV of the bond investment at the different interest rates:In the calculations of the NPV, $300 can be considered your “cost” for acquiring the bond since you will give up $300 in cash by choosing the bond. Note that the first two interest rates give positive NPVs for the bond, i.e. you should go for the bond, while the last NPV is negative, hence choose the cash instead. These results confirm the previous method’s results.24.Suppose you have three personal loans outstanding to your friend Elizabeth. A payment of $1,000 is due today, a $500 payment is due one year from now and a $250 payment is due two years from now. You would like to consolidate the three loans into one, with 36 equal monthly payments, beginning one month from today. Assume the agreed interest rate is 8% (effective annual rate) per year.a.What is the annual percentage rate you will be paying?b.How large will the new monthly payment be?SOLUTION:a.To find the APR, you must first compute the monthly interest rate that corresponds to an effective annual rate of8% and then multiply it by 12:1.08 = (1+ i)12Take 12th root of both sides:1.006434 = 1+ ii = .006434 or .6434% per monthOr using the financial calculator:b.The method is to first compute the PV of the 3 loans and then compute a 36 month annuity payment with thesame PV. Most financial calculators have keys which allow you to enter several cash flows at once. This approach will give the user the PV of the 3 loans.Note: The APR used to discount the cash flows is the effective rate in this case, because this method is assuming annual compounding.25.As CEO of ToysRFun, you are offered the chance to participate, without initial charge, in a project that produces cash flows of $5,000 at the end of the first period, $4,000 at the end of the next period and a loss of $11,000 at the end of the third and final year.a.What is the net present value if the relevant discount rate (the company’s cost of capital) is 10%?b.Would you accept the offer?c.What is the internal rate of return? Can you explain why you would reject a project which has aninternal rate of return greater than its cost of capital?SOLUTION:At 10% discount rate:Net Present Value = - 0 + $5,000 + $4,000 - $11,000 = - 413.22(1.10)(1.10)2 (1.10)3c.This example is a project with cash flows that begin positive and then turn negative--it is like a loan. The 13.6% IRR is therefore like an interest rate on that loan. The opportunity to take a loan at 13.6% when the cost of capital is only 10% is not worthwhile.26.You must pay a creditor $6,000 one year from now, $5,000 two years from now, $4,000 three years from now, $2,000 four years from now, and a final $1,000 five years from now. You would like to restructure the loan into five equal annual payments due at the end of each year. If the agreed interest rate is 6% compounded annually, what is the payment?SOLUTION:Since there are a number of different cash flows, it is easiest to do the first step of this problem using cash flow keys on the calculator. To find the present value of the current loan payments:27.Find the future value of the following ordinary annuities (payments begin one year from today and all interest rates compound annually):a.$100 per year for 10 years at 9%.b.$500 per year for 8 years at 15%.c.$800 per year for 20 years at 7%.d.$1,000 per year for 5 years at 0%.e.Now find the present values of the annuities in a-d.f.What is the relationship between present values and future values?SOLUTION:Future Value of Annuity:e.f.The relationship between present value and future value is the following:FV = PV x (1+i)n28.Suppose you will need $50,000 ten years from now. You plan to make seven equal annual deposits beginning three years from today in an account that yields 11% compounded annually. How large should the annual deposit be?SOLUTION:You will be making 7 payments beginning 3 years from today. So, we need to find the value of an immediate annuity with 7 payments whose FV is $50,000:29.Suppose an investment offers $100 per year for five years at 5% beginning one year from today.a.What is the present value? How does the present value calculation change if one additional payment isadded today?b.What is the future value of this ordinary annuity? How does the future value change if one additionalpayment is added today?SOLUTION:$100 x [(1.05)5] - 1 = $552.56.05If you were to add one additional payment of $100 today, the future value would increase by:$100 x (1.05)5 = $127.63. Total future value = $552.56 + $127.63 = $680.19.Another way to do it would be to use the BGN mode for 5 payments of $100 at 5%, find the future value of that, and then add $100. The same $680.19 is obtained.30.You are buying a $20,000 car. The dealer offers you two alternatives: (1) pay the full $20,000 purchase price and finance it with a loan at 4.0% APR over 3 years or (2) receive $1,500 cash back and finance the rest at a bank rate of 9.5% APR. Both loans have monthly payments over three years. Which should you choose? SOLUTION:31.You are looking to buy a sports car costing $23,000. One dealer is offering a special reduced financing rate of 2.9% APR on new car purchases for three year loans, with monthly payments. A second dealer is offering a cash rebate. Any customer taking the cash rebate would of course be ineligible for the special loan rate and would have to borrow the balance of the purchase price from the local bank at the 9%annual rate. How large must the cash rebate be on this $23,000 car to entice a customer away from the dealer who is offering the special 2.9% financing?SOLUTION:of the 2.9% financing.32.Show proof that investing $475.48 today at 10% allows you to withdraw $150 at the end of each of the next 4 years and have nothing remaining.SOLUTION:You deposit $475.48 and earn 10% interest after one year. Then you withdraw $150. The table shows what happensAnother way to do it is simply to compute the PV of the $150 annual withdrawals at 10% : it turns out to be exactly $475.48, hence both amounts are equal.33.As a pension manager, you are considering investing in a preferred stock which pays $5,000,000 per year forever beginning one year from now. If your alternative investment choice is yielding 10% per year, what is the present value of this investment? What is the highest price you would be willing to pay for this investment? If you paid this price, what would be the dividend yield on this investment?SOLUTION:Present Value of Investment:PV = $5,000,000 = $50,000,000.10Highest price you would be willing to pay is $50,000,000.Dividend yield = $5,000,000 = 10%.$50,000,00034. A new lottery game offers a choice for the grand prize winner. You can receive either a lump sum of $1,000,000 immediately or a perpetuity of $100,000 per year forever, with the first payment today. (If you die, your estate will still continue to receive payments). If the relevant interest rate is 9.5% compounded annually, what is the difference in value between the two prizes?SOLUTION:The present value of the perpetuity assuming that payments begin at the end of the year is:$100,000/.095 = $1,052,631.58If the payments begin immediately, you need to add the first payment. $100,000 + 1,052,632 = $1,152,632.So the annuity has a PV which is greater than the lump sum by $152,632.35.Find the future value of a $1,000 lump sum investment under the following compounding assumptions:a.7% compounded annually for 10 yearsb.7% compounded semiannually for 10 yearsc.7% compounded monthly for 10 yearsd.7% compounded daily for 10 yearse.7% compounded continuously for 10 yearsa.$1,000 x (1.07)10 = $1,967.15b.$1,000 x (1.035)20 = $1,989.79c.$1,000 x (1.0058)120 = $2,009.66d.$1,000 x (1.0019178)3650 = $2,013.62e.$1,000 x e.07x10 = $2,013.7536.Sammy Jo charged $1,000 worth of merchandise one year ago on her MasterCard which has a stated interest rate of 18% APR compounded monthly. She made 12 regular monthly payments of $50, at the end of each month, and refrained from using the card for the past year. How much does she still owe? SOLUTION:Sammy Jo has taken a $1,000 loan at 1.5% per month and is paying it off in monthly installments of $50. We could work out the amortization schedule to find out how much she still owes after 12 payments, but a shortcut on the financial calculator is to solve for FV as follows:37.Suppose you are considering borrowing $120,000 to finance your dream house. The annual percentage rate is 9% and payments are made monthly,a.If the mortgage has a 30 year amortization schedule, what are the monthly payments?b.What effective annual rate would you be paying?c.How do your answers to parts a and b change if the loan amortizes over 15 years rather than 30?EFF = [1 + .09]1238.Suppose last year you took out the loan described in problem #37a. Now interest rates have declined to 8% per year. Assume there will be no refinancing fees.a.What is the remaining balance of your current mortgage after 12 payments?b.What would be your payment if you refinanced your mortgage at the lower rate for 29 years? SOLUTION:Exchange Rates and the Time Value of Money39.The exchange rate between the pound sterling and the dollar is currently $1.50 per pound, the dollar interest rate is 7% per year, and the pound interest rate is 9% per year. You have $100,000 in a one-year account that allows you to choose between either currency, and it pays the corresponding interest rate.a.If you expect the dollar/pound exchange rate to be $1.40 per pound a year from now and are indifferentto risk, which currency should you choose?b.What is the “break-even” value of the dollar/pound exchange rate one year from now?SOLUTION:a.You could invest $1 today in dollar-denominated bonds and have $1.07 one year from now. Or you couldconvert the dollar today into 2/3 (i.e., 1/1.5) of a pound and invest in pound-denominated bonds to have .726667(i.e., 2/3 x 1.09) pounds one year from now. At an exchange rate of $1.4 per pound, this would yield 0.726667(1.4) = $1.017 (this is lower than $1.07), so you would choose the dollar currency.b.For you to break-even the .726667 pounds would have to be worth $1.07 one year from now, so the break-evenexchange rate is $1.07/.726667 or $1.4725 per pound. So for exchange rates lower than $1.4725 per pound one year from now, the dollar currency will give a better return.。
Time value of money课后习题答案
2. a Present value of $5,000 today = $5,000 b. Present value of $15,000 received in 5 years at 9%: PV0 = $15,000(PVIF.09,5) = $15,000 (0.650) = $9,750 (calculator solution = $9,748.97) c. Present value of a 15 year, $1,000 annuity at 9%: PVAN0 = $1,000 (PVIFA.09,15) = $1,000(8.061) = $8,061 (calculator solution = $8,060.69) Therefore, you prefer $15,000 in five years because it has the highest present value.
CHAPTER 4 THE TIME VALUE The investment paying five percent compound interest is more attractive because you will receive interest not only on the principal amount each year, but interest will be earned on the previous year's interest as well. 2. The future value interest factor for 10 percent and two years is 1.210, whereas the present value interest factor for 10 percent and two years is 0.826. 3. As the interest rate increases, any annuity amount is being discounted by a higher value, thereby reducing the present value of the annuity. This can be seen in Table IV by looking across any row of successively higher interest rates. In contrast, the future value of an annuity increases as the interest (compounding) rate increases. (See Table III.) 4. Daily compounding is preferred because you will earn interest on the interest earned in the account each day. Table 4-6 illustrates this. 5. Annuity due computations are common for lease contracts and insurance policies, where payments are generally made at the beginning of each period. 6. As can be seen in Table 4-7, the more frequent the compounding period, the lower the present values. 7. a A marketing manager might use present value concepts to evaluate the success of an advertising or other promotional campaign, the benefits of which are likely to extend beyond one year in time. Also, a firm selling capital goods must be familiar with the type of present value economic analysis that customers will use to evaluate purchases.
罗斯公司理财Chap004全英文题库及答案
Chapter 04 Discounted Cash Flow Valuation Answer KeyMultiple Choice Questions1. An annuity stream of cash flow payments is a set of:A. level cash flows occurring each time period for a fixed length of time.B. level cash flows occurring each time period forever.C. increasing cash flows occurring each time period for a fixed length of time.D. increasing cash flows occurring each time period forever.E. arbitrary cash flows occurring each time period for no more than 10 years.Difficulty level: EasyTopic: ANNUITYType: DEFINITIONS2. Annuities where the payments occur at the end of each time period are called _____, whereas _____ refer to annuity streams with payments occurring at the beginning of each time period.A. ordinary annuities; early annuitiesB. late annuities; straight annuitiesC. straight annuities; late annuitiesD. annuities due; ordinary annuitiesE. ordinary annuities; annuities dueDifficulty level: EasyTopic: ANNUITIES DUEType: DEFINITIONS3. An annuity stream where the payments occur forever is called a(n):A. annuity due.B. indemnity.C. perpetuity.D. amortized cash flow stream.E. amortization table.Difficulty level: EasyTopic: PERPETUITYType: DEFINITIONS4. The interest rate expressed in terms of the interest payment made each period is called the _____ rate.A. stated annual interestB. compound annual interestC. effective annual interestD. periodic interestE. daily interestDifficulty level: EasyTopic: STATED INTEREST RATESType: DEFINITIONS5. The interest rate expressed as if it were compounded once per year is called the _____ rate.A. stated interestB. compound interestC. effective annualD. periodic interestE. daily interestDifficulty level: EasyTopic: EFFECTIVE ANNUAL RATEType: DEFINITIONS6. The interest rate charged per period multiplied by the number of periods per year is called the _____ rate.A. effective annualB. annual percentageC. periodic interestD. compound interestE. daily interestDifficulty level: EasyTopic: ANNUAL PERCENTAGE RATEType: DEFINITIONS7. Paying off long-term debt by making installment payments is called:A. foreclosing on the debt.B. amortizing the debt.C. funding the debt.D. calling the debt.E. None of the above.Difficulty level: EasyTopic: AMORTIZATIONType: DEFINITIONS8. You are comparing two annuities which offer monthly payments for ten years. Both annuities are identical with the exception of the payment dates. Annuity A pays on the first of each month while annuity B pays on the last day of each month. Which one of the following statements is correct concerning these two annuities?A. Both annuities are of equal value today.B. Annuity B is an annuity due.C. Annuity A has a higher future value than annuity B.D. Annuity B has a higher present value than annuity A.E. Both annuities have the same future value as of ten years from today.Difficulty level: MediumTopic: ORDINARY ANNUITY VERSUS ANNUITY DUEType: CONCEPTS9. You are comparing two investment options. The cost to invest in either option is the same today. Both options will provide you with $20,000 of income. Option A pays five annual payments starting with $8,000 the first year followed by four annual payments of $3,000 each. Option B pays five annual payments of $4,000 each. Which one of the following statements is correct given these two investment options?A. Both options are of equal value given that they both provide $20,000 of income.B. Option A is the better choice of the two given any positive rate of return.C. Option B has a higher present value than option A given a positive rate of return.D. Option B has a lower future value at year 5 than option A given a zero rate of return.E. Option A is preferable because it is an annuity due.Difficulty level: MediumTopic: UNEVEN CASH FLOWS AND PRESENT VALUEType: CONCEPTS10. You are considering two projects with the following cash flows:Which of the following statements are true concerning these two projects?I. Both projects have the same future value at the end of year 4, given a positive rate of return. II. Both projects have the same future value given a zero rate of return.III. Both projects have the same future value at any point in time, given a positive rate of return. IV. Project A has a higher future value than project B, given a positive rate of return.A. II onlyB. IV onlyC. I and III onlyD. II and IV onlyE. I, II, and III onlyDifficulty level: MediumTopic: UNEVEN CASH FLOWS AND FUTURE VALUEType: CONCEPTS11. A perpetuity differs from an annuity because:A. perpetuity payments vary with the rate of inflation.B. perpetuity payments vary with the market rate of interest.C. perpetuity payments are variable while annuity payments are constant.D. perpetuity payments never cease.E. annuity payments never cease.Difficulty level: EasyTopic: PERPETUITY VERSUS ANNUITYType: CONCEPTS12. Which one of the following statements concerning the annual percentage rate is correct?A. The annual percentage rate considers interest on interest.B. The rate of interest you actually pay on a loan is called the annual percentage rate.C. The effective annual rate is lower than the annual percentage rate when an interest rate is compounded quarterly.D. When firms advertise the annual percentage rate they are violating U.S. truth-in-lending laws.E. The annual percentage rate equals the effective annual rate when the rate on an account is designated as simple interest.Difficulty level: MediumTopic: ANNUAL PERCENTAGE RATEType: CONCEPTS13. Which one of the following statements concerning interest rates is correct?A. The stated rate is the same as the effective annual rate.B. An effective annual rate is the rate that applies if interest were charged annually.C. The annual percentage rate increases as the number of compounding periods per year increases.D. Banks prefer more frequent compounding on their savings accounts.E. For any positive rate of interest, the effective annual rate will always exceed the annual percentage rate.Difficulty level: MediumTopic: INTEREST RATESType: CONCEPTS14. Which of the following statements concerning the effective annual rate are correct?I. When making financial decisions, you should compare effective annual rates rather than annual percentage rates.II. The more frequently interest is compounded, the higher the effective annual rate.III. A quoted rate of 6% compounded continuously has a higher effective annual rate than if the rate were compounded daily.IV. When borrowing and choosing which loan to accept, you should select the offer with the highest effective annual rate.A. I and II onlyB. I and IV onlyC. I, II, and III onlyD. II, III, and IV onlyE. I, II, III, and IVDifficulty level: MediumTopic: EFFECTIVE ANNUAL RATEType: CONCEPTS15. The highest effective annual rate that can be derived from an annual percentage rate of 9% is computed as:A. .09e - 1.B. e.09 ⨯ q.C. e ⨯ (1 + .09).D. e.09 - 1.E. (1 + .09)q.Difficulty level: MediumTopic: CONTINUOUS COMPOUNDINGType: CONCEPTS16. The time value of money concept can be defined as:A. the relationship between the supply and demand of money.B. the relationship between money spent versus money received.C. the relationship between a dollar to be received in the future and a dollar today.D. the relationship between interest rate stated and amount paid.E. None of the above.Difficulty level: EasyTopic: TIME VALUEType: CONCEPTS17. Discounting cash flows involves:A. discounting only those cash flows that occur at least 10 years in the future.B. estimating only the cash flows that occur in the first 4 years of a project.C. multiplying expected future cash flows by the cost of capital.D. discounting all expected future cash flows to reflect the time value of money.E. taking the cash discount offered on trade merchandise.Difficulty level: EasyTopic: CASH FLOWSType: CONCEPTS18. Compound interest:A. allows for the reinvestment of interest payments.B. does not allow for the reinvestment of interest payments.C. is the same as simple interest.D. provides a value that is less than simple interest.E. Both A and D.Difficulty level: EasyTopic: INTERESTType: CONCEPTS19. An annuity:A. is a debt instrument that pays no interest.B. is a stream of payments that varies with current market interest rates.C. is a level stream of equal payments through time.D. has no value.E. None of the above.Difficulty level: EasyTopic: ANNUITYType: CONCEPTS20. The stated rate of interest is 10%. Which form of compounding will give the highest effective rate of interest?A. annual compoundingB. monthly compoundingC. daily compoundingD. continuous compoundingE. It is impossible to tell without knowing the term of the loan.Difficulty level: EasyTopic: COMPOUNDINGType: CONCEPTS21. The present value of future cash flows minus initial cost is called:A. the future value of the project.B. the net present value of the project.C. the equivalent sum of the investment.D. the initial investment risk equivalent value.E. None of the above.Difficulty level: EasyTopic: PRESENT VALUEType: CONCEPTS22. Find the present value of $5,325 to be received in one period if the rate is 6.5%.A. $5,000.00B. $5,023.58C. $5,644.50D. $5,671.13E. None of the above.Difficulty level: EasyTopic: PRESENT VALUE - SINGLE SUMType: PROBLEMS23. If you have a choice to earn simple interest on $10,000 for three years at 8% or annually compounded interest at 7.5% for three years which one will pay more and by how much?A. Simple interest by $50.00B. Compound interest by $22.97C. Compound interest by $150.75D. Compound interest by $150.00E. None of the above.Simple Interest = $10,000 (.08)(3) = $2,400;Compound Interest = $10,000((1.075)3 - 1) = $2,422.97;Difference = $2,422.97 - $2,400 = $22.97Difficulty level: EasyTopic: SIMPLE & COMPOUND INTERESTType: PROBLEMS24. Bradley Snapp has deposited $7,000 in a guaranteed investment account with a promised rate of 6% compounded annually. He plans to leave it there for 4 full years when he will make a down payment on a car after graduation. How much of a down payment will he be able to make?A. $1,960.00B. $2,175.57C. $8,960.00D. $8,837.34E. $9,175.57$7,000 (1.06)4 = $8,837.34Difficulty level: EasyTopic: FUTURE VALUE - SINGLE SUMType: PROBLEMS25. Your parents are giving you $100 a month for four years while you are in college. At a 6% discount rate, what are these payments worth to you when you first start college?A. $3,797.40B. $4,167.09C. $4,198.79D. $4,258.03E. $4,279.32Difficulty level: EasyTopic: ORDINARY ANNUITY AND PRESENT VALUEType: PROBLEMS26. You just won the lottery! As your prize you will receive $1,200 a month for 100 months. If you can earn 8% on your money, what is this prize worth to you today?A. $87,003.69B. $87,380.23C. $87,962.77D. $88,104.26E. $90,723.76Difficulty level: EasyTopic: ORDINARY ANNUITY AND PRESENT VALUEType: PROBLEMS27. Todd is able to pay $160 a month for five years for a car. If the interest rate is 4.9%, how much can Todd afford to borrow to buy a car?A. $6,961.36B. $8,499.13C. $8,533.84D. $8,686.82E. $9,588.05Difficulty level: EasyTopic: ORDINARY ANNUITY AND PRESENT VALUEType: PROBLEMS28. You are the beneficiary of a life insurance policy. The insurance company informs you that you have two options for receiving the insurance proceeds. You can receive a lump sum of $50,000 today or receive payments of $641 a month for ten years. You can earn 6.5% on your money. Which option should you take and why?A. You should accept the payments because they are worth $56,451.91 today.B. You should accept the payments because they are worth $56,523.74 today.C. You should accept the payments because they are worth $56,737.08 today.D. You should accept the $50,000 because the payments are only worth $47,757.69 today.E. You should accept the $50,000 because the payments are only worth $47,808.17 today.Difficulty level: MediumTopic: ORDINARY ANNUITY AND PRESENT VALUEType: PROBLEMS29. Your employer contributes $25 a week to your retirement plan. Assume that you work for your employer for another twenty years and that the applicable discount rate is 5%. Given these assumptions, what is this employee benefit worth to you today?A. $13,144.43B. $15,920.55C. $16,430.54D. $16,446.34E. $16,519.02Difficulty level: MediumTopic: ORDINARY ANNUITY AND PRESENT VALUEType: PROBLEMS30. You have a sub-contracting job with a local manufacturing firm. Your agreement calls for annual payments of $50,000 for the next five years. At a discount rate of 12%, what is this job worth to you today?A. $180,238.81B. $201,867.47C. $210,618.19D. $223,162.58E. $224,267.10Difficulty level: MediumTopic: ORDINARY ANNUITY AND PRESENT VALUEType: PROBLEMS31. The Ajax Co. just decided to save $1,500 a month for the next five years as a safety net for recessionary periods. The money will be set aside in a separate savings account which pays 3.25% interest compounded monthly. It deposits the first $1,500 today. If the company had wanted to deposit an equivalent lump sum today, how much would it have had to deposit?A. $82,964.59B. $83,189.29C. $83,428.87D. $83,687.23E. $84,998.01Difficulty level: MediumTopic: ANNUITY DUE AND PRESENT VALUEType: PROBLEMS32. You need some money today and the only friend you have that has any is your ‘miserly' friend. He agrees to loan you the money you need, if you make payments of $20 a month for the next six months. In keeping with his reputation, he requires that the first payment be paid today. He also charges you 1.5% interest per month. How much money are you borrowing?A. $113.94B. $115.65C. $119.34D. $119.63E. $119.96Difficulty level: MediumTopic: ANNUITY DUE AND PRESENT VALUEType: PROBLEMS33. You buy an annuity which will pay you $12,000 a year for ten years. The payments are paid on the first day of each year. What is the value of this annuity today at a 7% discount rate?A. $84,282.98B. $87,138.04C. $90,182.79D. $96,191.91E. $116,916.21Difficulty level: MediumTopic: ANNUITY DUE AND PRESENT VALUEType: PROBLEMS34. You are scheduled to receive annual payments of $10,000 for each of the next 25 years. Your discount rate is 8.5%. What is the difference in the present value if you receive these payments at the beginning of each year rather than at the end of each year?A. $8,699B. $9,217C. $9,706D. $10,000E. $10,850Difference = $111,040.97 - $102,341.91 = $8,699.06 = $8,699 (rounded)Note: The difference = .085 $102,341.91 = $8,699.06Difficulty level: MediumTopic: ORDINARY ANNUITY VERSUS ANNUITY DUEType: PROBLEMS35. You are comparing two annuities with equal present values. The applicable discount rate is 7.5%. One annuity pays $5,000 on the first day of each year for twenty years. How much does the second annuity pay each year for twenty years if it pays at the end of each year?A. $4,651B. $5,075C. $5,000D. $5,375E. $5,405Because each payment is received one year later, then the cash flow has to equal: $5,000 (1 + .075) = $5,375Difficulty level: MediumTopic: ORDINARY ANNUITY VERSUS ANNUITY DUEType: PROBLEMS36. Martha receives $100 on the first of each month. Stewart receives $100 on the last day of each month. Both Martha and Stewart will receive payments for five years. At an 8% discount rate, what is the difference in the present value of these two sets of payments?A. $32.88B. $40.00C. $99.01D. $108.00E. $112.50Difference = $4,964.72 - $4,931.84 = $32.88Difficulty level: MediumTopic: ORDINARY ANNUITY VERSUS ANNUITY DUEType: PROBLEMS37. What is the future value of $1,000 a year for five years at a 6% rate of interest?A. $4,212.36B. $5,075.69C. $5,637.09D. $6,001.38E. $6,801.91Difficulty level: EasyTopic: ORDINARY ANNUITY AND FUTURE VALUEType: PROBLEMS38. What is the future value of $2,400 a year for three years at an 8% rate of interest?A. $6,185.03B. $6,847.26C. $7,134.16D. $7,791.36E. $8,414.67Difficulty level: EasyTopic: ORDINARY ANNUITY AND FUTURE VALUEType: PROBLEMS39. Janet plans on saving $3,000 a year and expects to earn 8.5%. How much will Janet have at the end of twenty-five years if she earns what she expects?A. $219,317.82B. $230,702.57C. $236,003.38D. $244,868.92E. $256,063.66Difficulty level: EasyTopic: ORDINARY ANNUITY AND FUTURE VALUEType: PROBLEMS40. Toni adds $3,000 to her savings on the first day of each year. Tim adds $3,000 to his savings on the last day of each year. They both earn a 9% rate of return. What is the difference in their savings account balances at the end of thirty years?A. $35,822.73B. $36,803.03C. $38,911.21D. $39,803.04E. $40,115.31Difference = $445,725.65 - $408,922.62 = $36,803.03Note: Difference = $408,922.62 .09 = $36,803.03Difficulty level: MediumTopic: ANNUITY DUE VERSUS ORDINARY ANNUITYType: PROBLEMS41. You borrow $5,600 to buy a car. The terms of the loan call for monthly payments for four years at a 5.9% rate of interest. What is the amount of each payment?A. $103.22B. $103.73C. $130.62D. $131.26E. $133.04Difficulty level: EasyTopic: ORDINARY ANNUITY PAYMENTSType: PROBLEMS42. You borrow $149,000 to buy a house. The mortgage rate is 7.5% and the loan period is 30 years. Payments are made monthly. If you pay for the house according to the loan agreement, how much total interest will you pay?A. $138,086B. $218,161C. $226,059D. $287,086E. $375,059Total interest = ($1,041.83 ⨯ 30 ⨯ 12) - $149,000 = $226,058.80 = $226,059 (rounded) Difficulty level: MediumTopic: ORDINARY ANNUITY PAYMENTS AND COST OF INTERESTType: PROBLEMS43. The Great Giant Corp. has a management contract with its newly hired president. The contract requires a lump sum payment of $25 million be paid to the president upon the completion of her first ten years of service. The company wants to set aside an equal amount of funds each year to cover this anticipated cash outflow. The company can earn 6.5% on these funds. How much must the company set aside each year for this purpose?A. $1,775,042.93B. $1,798,346.17C. $1,801,033.67D. $1,852,617.25E. $1,938,018.22Difficulty level: EasyTopic: ORDINARY ANNUITY PAYMENTS AND FUTURE VALUEType: PROBLEMS44. You retire at age 60 and expect to live another 27 years. On the day you retire, you have $464,900 in your retirement savings account. You are conservative and expect to earn 4.5% on your money during your retirement. How much can you withdraw from your retirement savings each month if you plan to die on the day you spend your last penny?A. $2,001.96B. $2,092.05C. $2,398.17D. $2,472.00E. $2,481.27Difficulty level: MediumTopic: ORDINARY ANNUITY PAYMENTS AND PRESENT VALUEType: PROBLEMS45. The McDonald Group purchased a piece of property for $1.2 million. It paid a down payment of 20% in cash and financed the balance. The loan terms require monthly payments for 15 years at an annual percentage rate of 7.75% compounded monthly. What is the amount of each mortgage payment?A. $7,440.01B. $8,978.26C. $9,036.25D. $9,399.18E. $9,413.67Amount financed = $1,200,000 (1 - .2) = $960,000Difficulty level: MediumTopic: ORDINARY ANNUITY PAYMENTS AND PRESENT VALUEType: PROBLEMS46. You estimate that you will have $24,500 in student loans by the time you graduate. The interest rate is 6.5%. If you want to have this debt paid in full within five years, how much must you pay each month?A. $471.30B. $473.65C. $476.79D. $479.37E. $480.40Difficulty level: MediumTopic: ORDINARY ANNUITY PAYMENTS AND PRESENT VALUEType: PROBLEMS47. You are buying a previously owned car today at a price of $6,890. You are paying $500 down in cash and financing the balance for 36 months at 7.9%. What is the amount of each loan payment?A. $198.64B. $199.94C. $202.02D. $214.78E. $215.09Amount financed = $6,890 - $500 = $6,390Difficulty level: MediumTopic: ORDINARY ANNUITY PAYMENTS AND PRESENT VALUEType: PROBLEMS48. The Good Life Insurance Co. wants to sell you an annuity which will pay you $500 per quarter for 25 years. You want to earn a minimum rate of return of 5.5%. What is the most you are willing to pay as a lump sum today to buy this annuity?A. $26,988.16B. $27,082.94C. $27,455.33D. $28,450.67E. $28,806.30Difficulty level: MediumTopic: ORDINARY ANNUITY PAYMENTS AND PRESENT VALUEType: PROBLEMS49. Your car dealer is willing to lease you a new car for $299 a month for 60 months. Payments are due on the first day of each month starting with the day you sign the lease contract. If your cost of money is 4.9%, what is the current value of the lease?A. $15,882.75B. $15,906.14C. $15,947.61D. $16,235.42E. $16,289.54Difficulty level: MediumTopic: ANNUITY DUE PAYMENTS AND PRESENT VALUEType: PROBLEMS50. Your great-aunt left you an inheritance in the form of a trust. The trust agreement states that you are to receive $2,500 on the first day of each year, starting immediately and continuing for fifty years. What is the value of this inheritance today if the applicable discount rate is 6.35%?A. $36,811.30B. $37,557.52C. $39,204.04D. $39,942.42E. $40,006.09Difficulty level: MediumTopic: ANNUITY DUE PAYMENTS AND PRESENT VALUEType: PROBLEMS51. Beatrice invests $1,000 in an account that pays 4% simple interest. How much more could she have earned over a five-year period if the interest had compounded annually?A. $15.45B. $15.97C. $16.65D. $17.09E. $21.67Ending value at 4% simple interest = $1,000 + ($1,000 ⨯ .04 ⨯ 5) = $1,200.00; Ending value at 4% compounded annually = $1,000 ⨯ (1 +.04)5 = $1,216.65;Difference = $1,216.65 - $1,200.00 = $16.65Difficulty level: EasyTopic: SIMPLE VERSUS COMPOUND INTERESTType: PROBLEMS52. Your firm wants to save $250,000 to buy some new equipment three years from now. The plan is to set aside an equal amount of money on the first day of each year starting today. The firm can earn a 4.7% rate of return. How much does the firm have to save each year to achieve its goal?A. $75,966.14B. $76,896.16C. $78,004.67D. $81.414.14E. $83,333.33Difficulty level: MediumTopic: ANNUITY DUE PAYMENTS AND FUTURE VALUEType: PROBLEMS53. Today is January 1. Starting today, Sam is going to contribute $140 on the first of each month to his retirement account. His employer contributes an additional 50% of the amount contributed by Sam. If both Sam and his employer continue to do this and Sam can earn a monthly rate of ½ of 1 percent, how much will he have in his retirement account 35 years from now?A. $199,45.944B. $200,456.74C. $249,981.21D. $299,189.16E. $300,685.11Difficulty level: MediumTopic: ANNUITY DUE PAYMENTS AND FUTURE VALUEType: PROBLEMS54. You are considering an annuity which costs $100,000 today. The annuity pays $6,000 a year. The rate of return is 4.5%. What is the length of the annuity time period?A. 24.96 yearsB. 29.48 yearsC. 31.49 yearsD. 33.08 yearsE. 38.00 yearsDifficulty level: MediumTopic: ORDINARY ANNUITY TIME PERIODS AND PRESENT VALUEType: PROBLEMS55. Today, you signed loan papers agreeing to borrow $4,954.85 at 9% compounded monthly. The loan payment is $143.84 a month. How many loan payments must you make before the loan is paid in full?A. 29.89B. 36.00C. 38.88D. 40.00E. 41.03Difficulty level: MediumTopic: ORDINARY ANNUITY TIME PERIODS AND PRESENT VALUEType: PROBLEMS56. Winston Enterprises would like to buy some additional land and build a new factory. The anticipated total cost is $136 million. The owner of the firm is quite conservative and will only do this when the company has sufficient funds to pay cash for the entire expansion project. Management has decided to save $450,000 a month for this purpose. The firm earns 6% compounded monthly on the funds it saves. How long does the company have to wait before expanding its operations?A. 184.61 monthsB. 199.97 monthsC. 234.34 monthsD. 284.61 monthsE. 299.97 monthsDifficulty level: MediumTopic: ORDINARY ANNUITY TIME PERIODS AND FUTURE VALUEType: PROBLEMS57. Today, you are retiring. You have a total of $413,926 in your retirement savings and have the funds invested such that you expect to earn an average of 3%, compounded monthly, on this money throughout your retirement years. You want to withdraw $2,500 at the beginning of every month, starting today. How long will it be until you run out of money?A. 185.00 monthsB. 213.29 monthsC. 227.08 monthsD. 236.84 monthsE. 249.69 monthsDifficulty level: MediumTopic: ANNUITY DUE TIME PERIODS AND PRESENT VALUEType: PROBLEMS58. The Bad Guys Co. is notoriously known as a slow-payer. It currently needs to borrow $25,000 and only one company will even deal with Bad Guys. The terms of the loan call for daily payments of $30.76. The first payment is due today. The interest rate is 21% compounded daily. What is the time period of this loan?A. 2.88 yearsB. 2.94 yearsC. 3.00 yearsD. 3.13 yearsE. 3.25 yearsDifficulty level: MediumTopic: ANNUITY DUE TIME PERIODSType: PROBLEMS59. The Robertson Firm is considering a project which costs $123,900 to undertake. The project will yield cash flows of $4,894.35 monthly for 30 months. What is the rate of return on this project?A. 12.53%B. 13.44%C. 13.59%D. 14.02%E. 14.59%This can not be solved directly, so it's easiest to just use the calculator method to get an answer. You can then use the calculator answer as the rate in the formula just to verify that your answer is correct.Difficulty level: MediumTopic: ORDINARY ANNUITY INTEREST RATEType: PROBLEMS60. Your insurance agent is trying to sell you an annuity that costs $100,000 today. By buying this annuity, your agent promises that you will receive payments of $384.40 a month for the next 40 years. What is the rate of return on this investment?A. 3.45%B. 3.47%C. 3.50%D. 3.52%E. 3.55%This can not be solved directly, so it's easiest to just use the calculator method to get an answer. You can then use the calculator answer as the rate in the formula just to verify that you answer is correct.Difficulty level: MediumTopic: ORDINARY ANNUITY INTEREST RATEType: PROBLEMS61. You have been investing $120 a month for the last 15 years. Today, your investment account is worth $47,341.19. What is your average rate of return on your investments?A. 9.34%B. 9.37%C. 9.40%D. 9.42%E. 9.46%This can not be solved directly, so it's easiest to just use the calculator method to get an answer. You can then use the calculator answer as the rate in the formula just to verify that you answer is correct.Difficulty level: MediumTopic: ORDINARY ANNUITY INTEREST RATEType: PROBLEMS。
finance Time Value of Money(2)
Chapter 3 Time Value of Money: An IntroductionProblem 4Suppose Bank One offers a risk-free interest rate of 5.5% on both savings and loans, and Bank Enn offers a risk-free interest rate of 6% on both savings and loans.a.What arbitrage opportunity is available?b.Which bank would experience a surge in the demand for loans? Which bankwould receive a surge in deposits?c.What would you expect to happen to the interest rates the two banks areoffering?a.Take a loan from Bank One at 5.5% and save the money in Bank Enn at 6%.b.Bank One would experience a surge in the demand for loans, while Bank Ennwould receive a surge in deposits.c.Bank One would increase the interest rate, and/or Bank Enn would decrease itsrate.Problem 7Bubba is a shrimp farmer. In an ironic twist, Bubba is allergic to shellfish, so he cannot eat any shrimp. Each day he has one-ton supply of shrimp. The market price of shrimp is $10,000 per ton.a.What is the value of a ton of shrimp to him?b.Would this value change if he were not allergic to shrimp? Why or why not?a.The value of one ton of shrimp to Bubba is $10,000 because that is the marketprice.b.No. As long as he can buy or sell shrimp at $10,000 per ton, his personalpreference or use for shrimp is irrelevant to the value of the shrimp.Problem 11A friend asks to borrow $55 from you and in return will pay you $58 in one year. If your bank is offering a 6% interest rate on deposits and loans:a. How much would you have in one year if you deposited the $55 instead?b. How much money could you borrow today is you pay the bank $58 in one year?c. Should you loan the money to your friend or deposit it in the bank?a. I f you deposit the money in the bank today you will have:$1.06 in one year FV $55 today $58.30 in one year $ today ⎛⎫=⨯= ⎪⎝⎭b.If you lend the money to your friend for one year and borrow against the promised $58 repayment, then you could borrow:$1.06 in one year PV $58 in one year $54.72 today $ today ⎛⎫=÷= ⎪⎝⎭c. F rom a financial perspective, you should deposit the money in the bank, as it will result in more money for you at the end of the year.Problem 16Calculate the future value of $2000 ina. Five years at an interest rate of 5% per year.b. Ten years at an interest rate of 5% per year.c. Five years at an interest rate of 10% per year.d. Why is the amount of interest earned in part (a) less than half the amount of interest earned in part (b)?a. Timeline:0 1 2 555FV 2,000 1.052,552.56=⨯=b. Timeline:0 1 2 101010FV 2,000 1.053,257.79=⨯=c. Timeline:0 1 2 555FV 2,000 1.13,221.02=⨯=d. Because in the last 5 years you get interest on the interest earned in the first 5 years as well as interest on the original $2,000.Problem 22Your grandfather put some money in an account for you on the day you were born. You are now 18 years old and are allowed to withdraw the money for the first time. The account currently has $3996 in it and pays an 8% interest rate.a. How much money would be in the account if you left the money there until your 25th birthday?b. What if you left the money until your 65th birthday?c. How much money did your grandfather originally put in the account?a. Timeline:18 19 20 21 25 0 1 2 3 77FV 3,996(1.08)6,848.44==b. Timeline:18 19 20 21 65 0 1 2 3 4747FV 3,996(1.08)148,779==c. Timeline:0 1 2 3 4 18183,996PV 1,0001.08==Chapter 4 Time Value of Money: Valuing Cash Flow StreamsProblem 1You have just taken out a five-year loan from a bank to buy an engagement ring. The ring costs $5000. You plan to put down $1000 and borrow $4000. You will need to make annual payments of $1000 at the end of each year. Show the timeline of the loan from your perspective. How would the timeline differ if you created it from the bank’s perspective?0 1 2 3 4 5From the bank’s perspective, the timeline is the same except all the signs are reversed.Problem 9The British government has a consol bond outstanding paying £100 per year forever. Assume the current interest rate is 4% per year.a. What is the value of the bond immediately after a payment is made?b. What is the value of the bond immediately before a payment is made? Timeline:0 12 3a. The value of the bond is equal to the present value of the cash flows. By the perpetuity formula:100PV 2,500.0.04£==b. The value of the bond is equal to the present value of the cash flows. The cash flows are the perpetuity plus the payment that will be received immediately.PV =100+100=£2,600 Problem 30You are saving for retirement. To live comfortably, you decide you will need to save $2 million by the time you are 65. Today is your 30th birthday, and you decide, starting today and continuing on every birthday up to and including your 65th birthday, that you will put the same amount into a savings account. If the interest rate is 5%, how much must you set aside each year to make sure that you will have $2 million in the account on your 65th birthday? Timeline:30 31 32 33 65 0 12 3 35FV = $2 millionThe PV of the cash flows must equal the PV of $2 million in 35 years. The cash flows consist of a 35-year annuity, plus the contribution today, so the PV is:()35C 1PV 1 C.0.05 1.05=-+⎛⎫ ⎪⎝⎭The PV of $2 million in 35 years is()352,000,000$362,580.57.1.05=Setting these equal gives:()()3535C 11C 362,580.570.05 1.05362,580.57C $20,868.91.11110.05 1.05-+=⇒==-+⎛⎫⎪⎝⎭⎛⎫⎪⎝⎭。
THE TIME VALUE OF MONEY
O VERVIEWA dollar in the hand today is worth more than a dollar to be received in the future because, if you had it now, you could invest that dollar and earn interest. Of all the techniques used in finance, none is more important than the concept of the time value of money,or discounted cash flow (DCF) analysis. The principles of time value analysis that are developed in this chapter have many applications, ranging from setting up schedules for paying off loans to decisions about whether to acquire new equipment.Future value and present value techniques can be applied to a single cash flow (lump sum), ordinary annuities, annuities due, and uneven cash flow streams. Future and present values can be calculated using a regular calculator or a calculator with financial functions. When compounding occurs more frequently than once a year, the effective rate of interest is greater than the quoted rate.The cash flow time line is one of the most important tools in time value of money analysis. Cash flow time lines help to visualize what is happening in a particular problem. Cash flows are placed directly below the tick marks, and interest rates are shown directly above the time line; unknown cash flows are indicated by question marks. Thus, to find the future value of $100 after 5 years at 5 percent interest, the following cash flow time line can be set up: Time: 0 1 2 3 4 5| | | | | | Cash flows: -100 FV5 = ?◆ A cash outflow is a payment, or disbursement, of cash for expenses, investments, and so on.◆ A cash inflow is a receipt of cash from an investment, an employer, or other sources.5%CHAPTER 3: THE TIME VALUE OF MONEY40Compounding is the process of determining the value of a cash flow or series of cash flows some time in the future when compound interest is applied. The future value is the amount to which a cash flow or series of cash flows will grow over a given period of time when compounded at a given interest rate. The future value can be calculated asFV n = PV(1 + k)n,where PV = present value, or beginning amount; k = interest rate per period; and n = number of periods involved in the analysis. This equation can be solved in one of two ways: numerically or with a financial calculator. For calculations, assume the following data that were presented in the time line above: present value (PV) = $100, interest rate (k) = 5%, and number of years (n) = 5.◆Compounded interest is interest earned on interest.◆To solve numerically, use a regular calculator to find 1 + k = 1.05 raised to the fifth power,which equals 1.2763. Multiply this figure by PV = $100 to get the final answer of FV5 = $127.63.◆With a financial calculator, the future value can be found by using the time value of moneyinput keys, where N = number of periods, I = interest rate per period, PV = present value, PMT = annuity payment, and FV = future value. By entering N = 5, I = 5, PV = -100, and PMT = 0, and then pressing the FV key, the answer 127.63 is displayed.♦Some financial calculators require that all cash flows be designated as either inflows or outflows, thus an outflow must be entered as a negative number (for example, PV= -100 instead of PV = 100).♦Some calculators require you to press a “Compute” key before pressing the FV key.◆ A graph of the compounding process shows how any sum grows over time at variousinterest rates. The greater the rate of interest, the faster is the rate of growth.♦The interest rate is, in fact, a growth rate.♦The time value concepts can be applied to anything that is growing.Finding the present value of a cash flow or series of cash flows is called discounting, and it is simply the reverse of compounding. In general, the present value is the value today of a future cash flow or series of cash flows. By solving for PV in the future value equation, the present value, or discounting, equation can be developed and written in several forms:CHAPTER 3: THE TIME VALUE OF MONEY41rate, one can utilize either of the two solution methods:♦Numerical solution: Divide $127.63 by 1.05 five times to get PV = $100.♦Financial calculator solution: Enter N = 5, I = 5, PMT = 0, and FV = 127.63, and then press the PV key to get PV = -100.◆The opportunity cost rate is the rate of return on the best available alternative investment ofequal risk.◆ A graph of the discounting process shows how the present value of any sum to be receivedin the future diminishes and approaches zero as the payment date is extended farther into the future. At relatively high interest rates, funds due in the future are worth very little today, and even at a relatively low discount rate, the present value of a sum due in the very distant future is quite small.The compounding and discounting processes are reciprocals, or inverses, of one another. In addition, there are four variables in the time value of money equations: PV, FV, k, and n. If three of the four variables are known, you can find the value of the fourth.◆If we are given PV, FV, and n, we can determine k by substituting the known values intoeither the present value or future value equations, and then solving for k. Thus, if you can buy a security at a price of $78.35 which will pay you $100 after 5 years, what is the interest rate earned on the investment?♦Numerical solution: Use a trial and error process to reach the 5% value for k. This is a tedious and inefficient process. Alternatively, you could use algebra to solve the timevalue equation.♦Financial calculator solution: Enter N = 5, PV = -78.35, PMT = 0, and FV = 100, then press the I key, and I = 5 is displayed.◆Likewise, if we are given PV, FV, and k, we can determine n by substituting the knownvalues into either the present value or future value equations, and then solving for n. Thus, if you can buy a security with a 5 percent interest rate at a price of $78.35 today, how long will it take for your investment to return $100?♦Numerical solution: Use a trial and error process to reach the value of 5 for n. This is a tedious and inefficient process. The equation can also be solved algebraically.♦Financial calculator solution: Enter I = 5, PV = -78.35, PMT = 0, and FV = 100, then press the N key, and N = 5 is displayed.An annuity is a series of equal payments made at fixed intervals for a specified number ofCHAPTER 3: THE TIME VALUE OF MONEY 42periods. If the payments occur at the end of each period, as they typically do, the annuity is an ordinary, or deferred, annuity. If the payments occur at the beginning of each period, it is called an annuity due. ◆ The future value of an ordinary annuity, FVA n , is the total amount one would have at theend of the annuity period if each payment were invested at a given interest rate and held to the end of the annuity period.♦ Defining FVA n as the future value of an ordinary annuity of n years, and PMT as the periodic payment, we can writethe FV key, and 315.25 is displayed.♦ For an annuity due, each payment is compounded for one additional period, so the future value of the entire annuity is equal to the future value of an ordinary annuity compounded for one additional period. Thus:FVA (DUE)n = PMT ⎥⎦⎤⎢⎣⎡+⨯⎭⎬⎫⎩⎨⎧-+)k 1(k 1)k 1(n .♦ Most financial calculators have a switch, or key, marked “DUE” or “BEG” that permitsyou to switch from end-of-period payments (an ordinary annuity) to beginning-of-period payments (an annuity due). Switch your calculator to “BEG” mode, and calculate as you would for an ordinary annuity. Do not forget to switch your calculator back to “END” mode when you are finished. ◆The present value of an ordinary annuity, PVA n , is the single (lump sum) payment today that would be equivalent to the annuity payments spread over the annuity period. It is the amount today that would permit withdrawals of an equal amount (PMT) at the end (or beginning for an annuity due) of each period for n periods.♦ Defining PVA n as the present value of an ordinary annuity of n years and PMT as the periodic payment, we can writePVA n = PMT ⎥⎦⎤⎢⎣⎡⎪⎪⎭⎫ ⎝⎛+∑=n 1t t )k 1(1 = PMT ⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡+-k )k 1(11n = PMT ⎥⎦⎤⎢⎣⎡+--k )k 1(1n .CHAPTER 3: THE TIME VALUE OF MONEY43♦ Using a financial calculator, enter N = 3, I = 5, PMT = -100, and FV = 0, and then press the PV key, for an answer of $272.32.♦ One especially important application of the annuity concept relates to loans with constant payments, such as mortgages and auto loans. With these amortized loans the amount borrowed is the present value of an ordinary annuity, and the payments constitute the annuity stream. ◆The present value for an annuity due isPVA (DUE)n = PMT ⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡+⨯⎪⎪⎭⎪⎪⎬⎫⎪⎪⎩⎪⎪⎨⎧+-)k 1(k )k 1(11n .♦ Using a financial calculator, switch to the “BEG” mode, and then enter N = 3, I = 5,PMT = -100, and FV = 0, and then press PV to get the answer, $285.94. Again, do not forget to switch your calculator back to “END” mode when you are finished.◆You can solve for the interest rate (rate of return) earned on an annuity.♦ To solve numerically, you must use the trial-and-error process and plug in different values for k in the annuity equation to solve for the interest rate.♦ You can use the financial calculator by entering the appropriate values for N, PMT, and either FV or PV, and then pressing I to solve for the interest rate.◆You can solve for the number of periods (N) in an annuity.♦ To solve numerically, you must use the trial-and-error process and plug in different values for N in the annuity equation to solve for the number of periods.♦ You can use the financial calculator by entering the appropriate values for I, PMT, and either FV or PV, and then pressing N to solve for the number of periods.A perpetuity is a stream of equal payments expected to continue forever. ◆ The present value of a perpetuity is:PVP =kPMTrate Interest Payment =.♦ For example, if the interest rate were 12 percent, a perpetuity of $1,000 a year wouldhave a present value of $1,000/0.12 = $8,333.33. ◆ A consol is a perpetual bond issued by the British government to consolidate past debts; in general, any perpetual bond.CHAPTER 3: THE TIME VALUE OF MONEY44◆The value of a perpetuity changes dramatically when interest rates change.Many financial decisions require the analysis of uneven, or nonconstant, cash flows rather than a stream of fixed payments such as an annuity. An uneven cash flow stream is a series of cash flows in which the amount varies from one period to the next.◆The term payment, PMT, designates constant cash flows, while the term CF designates cashflows in general, including uneven cash flows.◆The present value of an uneven cash flow stream is the sum of the PVs of the individualcash flows of the stream.♦The PV is found by applying the following general present value equation:PV = ∑=⎪⎪⎭⎫⎝⎛+n1 ttt)k1(1 CF.♦With a financial calculator, enter each cash flow (beginning with the t = 0 cash flow) into the cash flow register, CF j, enter the appropriate interest rate, and then press theNPV key to obtain the PV of the cash flow stream.♦Be sure to clear the cash flow register before starting a new problem.◆Similarly, the future value of an uneven cash flow stream, or terminal value, is the sum ofthe FVs of the individual cash flows of the stream.♦The FV can be found by applying the following general future value equation:FV n = ∑=-+n 1 ttnt)k1(CF.♦Some calculators have a net future value (NFV) key which allows you to obtain the FV of an uneven cash flow stream.◆We generally are more interested in the present value of an asset’s cash flow stream than inthe future value because the prese nt value represents today’s value, which we can compare with the price of the asset.♦Once we know its present value, we can find the future value of an uneven cash flow stream by treating the present value as a lump sum amount and compounding it to thefuture period.◆If one knows the relevant cash flows, the effective interest rate can be calculated efficientlywith a financial calculator. Enter each cash flow (beginning with the t = 0 cash flow) into the cash flow register, CF j, and then press the IRR key to obtain the interest rate of anCHAPTER 3: THE TIME VALUE OF MONEY45uneven cash flow stream.♦ IRR stands for internal rate of return , which is the return on an investment.Annual compounding is the arithmetic process of determining the final value of a cash flow or series of cash flows when interest is added once a year. Semiannual, quarterly, and other compounding periods more frequent than on an annual basis are often used in financial transactions. Compounding on a nonannual basis requires an adjustment to both the compounding and discounting procedures discussed previously. Moreover, when comparing securities with different compounding periods, they need to be put on a common basis. This requires distinguishing between the simple, or quoted, interest rate and the effective annual rate. ◆ The simple , or quoted, interest rate is the contracted, or quoted, interest rate that is used tocalculate the interest paid per period. ◆ The periodic rate is the interest rate charged per period.Periodic rate = Stated annual interest rate/Number of periods per year.◆ The annual percentage rate, APR, is the periodic rate times the number of periods per year. ◆The effective annual rate, EAR, is the rate that would have produced the final compounded value under annual compounding. The effective annual rate is given by the following formula:Effective annual rate (EAR) = ,0.1m k 1mSIMPLE -⎪⎭⎫⎝⎛+where k SIMPLE is the simple, or quoted, interest rate (that is, the APR), and m is the numberof compounding periods (interest payments) per year. The EAR is useful in comparing securities with different compounding periods. ◆For example, to find the effective annual rate if the simple rate is 6 percent and semiannual compounding is used, we have:EAR = (1 + 0.06/2)2 – 1.0 = 6.09%.◆For annual compounding use the formula to find the future value of a single payment (lump sum):FV n = PV(1 + k)n .CHAPTER 3: THE TIME VALUE OF MONEY 46♦ When compounding occurs more frequently than once a year, use this formula:FV n = PV nm SIMPLE m k 1⨯⎪⎭⎫⎝⎛+.Here m is the number of times per year compounding occurs, and n is the number of years.◆The amount to which $1,000 will grow after 5 years if quarterly compounding is applied to a nominal 8 percent interest rate is found as follows:FV n = $1,000(1 + 0.08/4)(4)(5) = $1,000(1.02)20 = $1,485.95.♦ Financial calculator solution: Enter N = 20, I = 2, PV = -1000, and PMT = 0, and then press the FV key to find FV = $1,485.95.◆The present value of a 5-year future investment equal to $1,485.95, with an 8 percent nominal interest rate, compounded quarterly, is found as follows:.000,1$)02.1(95.485,1$PV /4)08.01(PV 95.485,1$20(4)(5)==+=♦ Financial calculator solution: Enter N = 20, I = 2, PMT = 0, and FV = 1485.95, and then press the PV key to find PV = -$1,000.00.◆In general, nonannual compounding can be handled one of two ways.♦ State everything on a periodic rather than on an annual basis. Thus, n = 6 periods rather than n = 3 years and k = 3% instead of k = 6% with semiannual compounding.♦ Find the effective annual rate (EAR) with the equation below and then use the EAR as the rate over the given number of years.EAR = .0.1m k 1mSIMPLE -⎪⎭⎫⎝⎛+An important application of compound interest involves amortized loans, which are paid offin equal installments over the life of the loan.◆The amount of each payment, PMT, is found using a financial calculator by entering N (number of years), I (interest rate), PV (amount borrowed), and FV = 0, and then pressing the PMT key to find the periodic payment.◆ Each payment consists partly of interest and partly of repayment of the amount borrowed (principal). This breakdown is often developed in a loan amortization schedule .CHAPTER 3: THE TIME VALUE OF MONEY47♦ The interest component is largest in the first period, and it declines over the life of the loan as the outstanding balance of the loan decreases.♦ The repayment of principal is smallest in the first period, and it increases thereafter.The text discussion has involved three different interest rates. It is important to understand their differences.◆The simple , or quoted, rate, k SIMPLE , is the interest rate quoted by borrowers and lenders. This quotation must include the number of compounding periods per year.♦ This rate is never shown on a time line, and it is never used as an input in a financial calculator unless compounding occurs only once a year. ♦ k SIMPLE = Periodic rate ⨯ m = Annual percentage rate = APR.◆The periodic rate, k PER , is the rate charged by a lender or paid by a borrower each interest period. Periodic rate = k PER = k SIMPLE /m.♦ The periodic rate is used for calculations in problems where two conditions hold: (1) payments occur on a regular basis more frequently than once a year, and (2) a payment is made on each compounding (or discounting) date.♦ The APR, or annual percentage rate, represents the periodic rate stated on an annual basis without considering interest compounding. The APR never is used in actual calculations; it is simply reported to borrowers. ◆The effective annual rate, EAR, is the rate with which, under annual compounding, we would obtain the same result as if we had used a given periodic rate with m compounding periods per year.♦ EAR is found as follows:EAR = .0.1m k 1mSIMPLE -⎪⎭⎫⎝⎛+In Appendix 3A we discuss using spreadsheets to solve time value of money problems.In Appendix 3B we discuss using interest tables to solve time value of money problems.In Appendix 3C we discuss how to generate a loan amortization schedule using a financial calculator.S ELF-TEST Q UESTIONSDefinitionalCHAPTER 3: THE TIME VALUE OF MONEY481.A(n) ______ _________ is a payment, or disbursement, of cash for expenses, investments,and so on.2.A(n) ______ ________ is a receipt of cash from an investment, an employer, or othersources.3._____________ is the process of determining the value of a cash flow or series of cashflows some time in the future.4.The ________ _______ is the amount to which a cash flow or series of cash flows willgrow over a given period of time when compounded at a given interest rate.5.The beginning value of an account or investment in a project is known as its ________________.ing a savings account as an example, the difference between the account’s present valueand its future value at the end of the period is due to __________ earned during the period.7.The expression PV(1 + k)n determines the ________ _______ of a sum at the end of ___periods.8.Finding the present value of a cash flow or series of cash flows is often referred to as_____________, and it is simply the reverse of the _____________ process.9.The _____________ ______ ______ is the rate of return on the best alternative investmentof equal risk.10. A series of equal payments at fixed intervals for a specified number of periods is a(n)_________. If the payments occur at the end of each period it is a(n) __________ annuity, while if the payments occur at the beginning of each period it is an annuity _____.11.A(n) ____________ is a stream of equal payments expected to continue forever.12.The term PMT designates __________ cash flows, while the term CF designates cash flowsin general, including ________ cash flows.13.The present value of an uneven cash flow stream is the _____ of the PVs of the individualcash flows of the stream.49 14.Since different types of investments use different compounding periods, it is important todistinguish between the quoted, or ________, interest rate and the ___________ annual interest rate, the rate that would have produced the final compound value under annual compounding.15.____________ time periods are used when payments occur within periods, instead of ateither the beginning or the end of periods.16.___________ loans are paid off in equal installments over their lifetime and are animportant application of compound interest.17.The __________ rate is equal to the simple interest rate divided by the number ofcompounding periods per year.Conceptual18.If a bank uses quarterly compounding for savings accounts, the simple interest rate will begreater than the effective annual rate (EAR).a. Trueb. False19.If money has time value (that is, k > 0), the future value of some amount of money willalways be more than the amount invested. The present value of some amount to be received in the future is always less than the amount to be received.a. Trueb. False20.You have determined the profitability of a planned project by finding the present value ofall the cash flows from that project. Which of the following would cause the project to look less appealing, that is, have a lower present value?a.The discount rate decreases.b.The cash flows are received in later years (further into the future).c.The discount rate increases.d.Statements b and c are correct.e.Statements a and b are correct.21.As the discount rate increases without limit, the present value of a future cash inflowa.Gets larger without limit.b.Stays unchanged.50c.Approaches zero.d.Gets smaller without limit; that is, approaches minus infinity.e.Goes to e k n.5122.Which of the following statements is correct?a.Except in situations where compounding occurs annually, the periodic interest rateexceeds the simple interest rate.b.The effective annual rate always exceeds the simple interest rate, no matter how few ormany compounding periods occur each year.c.If compounding occurs more frequently than once a year, and if payments are made attimes other than at the end of compounding periods, it is impossible to determinepresent or future values, even with a financial calculator. The reason is that under theseconditions, the basic assumptions of discounted cash flow analysis are not met.d.Assume that compounding occurs quarterly, that the simple interest rate is 8 percent,and that you need to find the present value of $1,000 due 10 months from today. Youcould get the correct answer by discounting the $1,000 at 8.2432 percent for 10/12ths ofa year.e.Statements a, b, c, and d are all false.S ELF-TEST P ROBLEMS(Note: In working these problems, you may get an answer which differs from ours by a few cents due to differences in rounding. This should not concern you; just choose the closest answer.)1.Assume that you purchase a 6-year, 8 percent savings certificate for $1,000. If interest iscompounded annually, what will be the value of the certificate when it matures?a.$630.17b. $1,469.33c. $1,677.10d. $1,586.87e. $1,766.332. A savings certificate similar to the one in the previous problem is available with theexception that interest is compounded semiannually. What is the difference between the ending value of the savings certificate compounded semiannually and the one compounded annually?a.The semiannual certificate is worth $14.16 more than the annual certificate.b.The semiannual certificate is worth $14.16 less than the annual certificate.c.The semiannual certificate is worth $21.54 more than the annual certificate.d.The semiannual certificate is worth $21.54 less than the annual certificate.e.The semiannual certificate is worth the same as the annual certificate.523. A friend promises to pay you $600 two years from now if you loan him $500 today. Whatannual interest rate is your friend offering?a. 7.55%b. 8.50%c. 9.54%d. 10.75%e. 11.25%4.At an inflation rate of 9 percent, the purchasing power of $1 would be cut in half in just over 8years (some calculators round to 9 years). How long, to the nearest year, would it take for the purchasing power of $1 to be cut in half if the inflation rate were only 4 percent?a. 12 yearsb. 15 yearsc. 18 yearsd. 20 yearse. 23 years5.You are offered an investment opportunity with the “guarantee” that your investment willdouble in 5 years. Assuming annual compounding, what annual rate of return would this investment provide?a. 40.00%b. 100.00%c. 14.87%d. 20.00%e. 18.74%6.You decide to begin saving toward the purchase of a new car in 5 years. If you put $1,000at the end of each of the next 5 years in a savings account paying 6 percent compounded annually, how much will you accumulate after 5 years?a. $6,691.13b. $5,637.09c. $1,338.23d. $5,975.32e. $5,731.947.Refer to Self-Test Problem 6. What would be the ending amount if the payments weremade at the beginning of each year?a. $6,691.13b. $5,637.09c.$1,338.23d. $5,975.32e. $5,731.948.Refer to Self-Test Problem 6. What would be the ending amount if $500 payments weremade at the end of each 6-month period for 5 years and the account paid 6 percent compounded semiannually?a. $6,691.13b.$5,637.09c. $1,338.23d. $5,975.32e. $5,731.949.Calculate the present value of $1,000 to be received at the end of 8 years. Assume aninterest rate of 7 percent.a. $582.01b. $1,718.19c. $531.82d. $5,971.30e. $649.3753 10.Jane Smith has $20,000 in a brokerage account, and she plans to contribute an additional$7,500 to the account at the end of every year. The brokerage account has an expected annual return of 8 percent. If Jane’s goal is to a ccumulate $375,000 in the account, how many years will it take for Jane to reach her goal?a. 5.20b. 10.00c. 12.50d. 16.33e. 18.4011. How much would you be willing to pay today for an investment that would return $800each year at the end of each of the next 6 years? Assume a discount rate of 5 percent.a. $5,441.53b. $4,800.00c. $3,369.89d. $4,060.55e. $4,632.3712.You have applied for a mortgage of $60,000 to finance the purchase of a new home. Thebank will require you to make annual payments of $7,047.55 at the end of each of the next20 years. Determine the interest rate in effect on this mortgage.a. 8.0%b. 9.8%c. 10.0%d. 5.1%e. 11.2%13.If you would like to accumulate $7,500 over the next 5 years, how much must you depositeach six months, starting six months from now, given a 6 percent interest rate and semiannual compounding?a. $1,330.47b. $879.23c. $654.23d. $569.00e. $732.6714. A company is offering bonds that pay $100 per year indefinitely. If you require a 12 percentreturn on these bonds (that is, the discount rate is 12 percent), what is the value of each bond?a. $1,000.00b. $962.00c. $904.67d. $866.67e.$833.3315.What is the present value (t = 0) of the following cash flows if the discount rate is 12percent?0 1 2 3 4 512%| | | | | |0 2,000 2,000 2,000 3,000 -4,000a.$4,782.43b. $4,440.51c. $4,221.79d. $4,041.23e. $3,997.9816.What is the effective annual percentage rate (EAR) of 12 percent compounded monthly?。
公司金融英文版教材
公司金融英文版教材Corporate Finance: English Edition TextbookIntroduction:Welcome to the English edition of the Corporate Finance textbook. This comprehensive guide is designed to help you understand the fundamental principles and concepts of corporate finance in a global business environment. Whether you are a student or a professional seeking to enhance your knowledge, this textbook will provide you with the necessary tools to excel in the field of corporate finance.Chapter 1: Introduction to Corporate Finance- Role and importance of corporate finance in business- Financial objectives of a firm and shareholder value maximization- Understanding the key financial decisions and their impact on the firmChapter 2: Financial Statements and Analysis- Understanding the income statement, balance sheet, and cash flow statement- Financial ratio analysis and its interpretation- Evaluating the financial health and performance of a firm Chapter 3: Time Value of Money- Understanding the concept of time value of money- Calculating present value, future value, and annuity payments- Applying time value of money principles in investment decision makingChapter 4: Capital Budgeting- Evaluating investment projects and capital budgeting techniques - Net present value (NPV), internal rate of return (IRR), and profitability index- Assessing risk and uncertainty in investment decisionsChapter 5: Cost of Capital- Determining the cost of debt, equity, and weighted average cost of capital (WACC)- Importance of cost of capital in investment decisions and firm valuation- Estimating the cost of capital using various approaches Chapter 6: Capital Structure- Understanding the capital structure and its impact on firm value - Modigliani-Miller theorem and its implications- Determining the optimal capital structure and the tradeoff between debt and equityChapter 7: Dividend Policy- Role and significance of dividend policy in corporate finance- Dividend theories and factors influencing dividend decisions- Dividend payout ratios, stock repurchases, and dividend reinvestment plansChapter 8: Working Capital Management- Managing short-term assets and liabilities- Cash conversion cycle and its optimization- Credit policies, inventory management, and cash flow forecastingChapter 9: Financial Planning and Forecasting- Importance of financial planning in corporate finance- Developing financial forecasts and budgeting processes- Variance analysis and monitoring financial performance Chapter 10: Corporate Valuation- Different approaches to valuing a firm: discounted cash flow (DCF), relative valuation, and market multiples- Understanding the concept of free cash flow and economic value added (EVA)- Valuation models and their application in mergers and acquisitionsConclusion:This Corporate Finance textbook provides a comprehensive understanding of the key principles and concepts in corporate finance. Whether you are a student aspiring to pursue a career in finance or a professional seeking to enhance your knowledge, this textbook will equip you with the necessary tools to make informed financial decisions and create value for the firm.。
货币的时间价值 chapter Time value of money
Buy or sell a bond you are holding?
……
3-1
Ch3 Time Value of Money
CHAPTER 3 Time Value of Money
Compounding and Discounting of Single Sums
Annuities Types of Interest Rates
Do you have these puzzles?
2020/3/3
Buy a mobile by lump-sum payment or by installment ?
Arrange savings for a future expenditure?
What kind of loans to apply for?
Future
?
2020/3/3
Ch3 Time Value of Money
3-10
Example
What is the future value (FV) of an initial $100 after 3 years, if I/YR = 10%?
FV can be solved by using the arithmetic, or Table A-3 (pp.A-6)
Today
?
2020/3/3
Future
Ch3 Time Value of Money
3-14
Example
What is the present value (PV) of $100 due in 3 years, if I/YR = 10%?
0
10%
PV = ?
time value of money
• How much money do you have at the end of the year?
• The number of compounding period (n) may not be the same as the number of years.
14
Periodic vs. Annual
• The annual rate must be converted to periodic rate if compounding is taking place more often than once a year.
• The number of Compounding periods (n).
6
SINGLE AMOUNT
• Consider only two cash flows, one at the beginning and one at the end of the period.
• Application - place some money in your saving account today and withdraw the proceeds in the future.
26
Present Value of Perpetuity
• Present Value of Perpetuity
PV = PMT/r
27
Annuity
• An annuity is a series of payments of equal amount.
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22
24
26
28
Number of Years
10
30
0
2
4
6
8
Manhattan Island Sale
Fundamentals of Corporate Finance
Peter Minuit bought Manhattan Island for $24 in 1626. Was this a good deal? To answer, determine $24 is worth in the year 2006, compounded at 8%.
—PV of Perpetuity Formula
PV
C r
22
Perpetuities
Fundamentals of Corporate Finance
PV
C1
( 1 r )
1
C2
( 1 r )
2
....
19
PV of Multiple Cash Flows
Fundamentals of Corporate Finance
Example
Your auto dealer gives you the choice to pay $15,500 cash now, or make three payments: $8,000 now and $4,000 at the end of the following two years. If your cost of money is 8%, which do you prefer?
Perpetuity A stream of level cash payments that never ends.
21
Perpetuities
Fundamentals of Corporate Finance
—A
stream of level cash payments that never ends.
Fundamentals of Corporate Finance
120 100
Interest Rates
5% 10% 15%
PV of $100
80 60 40 20 0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Number of Years
12
Present Values
Fundamentals of Corporate Finance
P resent V alue = P V
F uture V alue after t periods (1+ r)
t
PV =
13
Present Values
Fundamentals of Corporate Finance
5
9
Future Values with Compounding
Fundamentals of Corporate Finance
7000 6000 5000
Interest Rates
0% 5% 10% 15%
FV of $100
4000 3000 2000 1000 0
10
12
14
16
18
20
Simple Interest - Interest earned only on the original investment.
3
Future Values
Fundamentals of Corporate Finance
Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100.
Today Interest Earned Value 100
Future Years 1 2 3 4 5 6 6.36 6.74 7.15 7.57 106 112.36 119.10 126.25 133.82
Value at the end of Year 5 = $133.82
7
Future ues
The PV formula has many applications. Given any variables in the equation, you can solve for the remaining variable.
PV FV
1 t ( 1 r )
16
Present Values with Compounding
FV $ 24 (1 . 08 )
380
$ 120 . 57 trillion
FYI - The value of Manhattan Island land is well below this figure.
11
Present Values
Fundamentals of Corporate Finance
Immediate pay ment
8,000.00
PV 1 PV 2 Total PV
4 , 000 ( 1 . 08 ) 4 , 000 ( 1 . 08 )
2 1
3,703 . 70 3, 429 . 36
$15,133.06
20
Perpetuities
Fundamentals of Corporate Finance
18
Multiple Cash Flows
Fundamentals of Corporate Finance
Future Value of Multiple Cash Flows
Accumulating a future sum via unequal, periodic payments entails a combination of a series of single future value cash flows.
Example You just bought a new computer for $3,000. The payment terms are 2 years same as cash. If you can earn 8% on your money, how much money should you set aside today in order to make the payment when due in two years?
2
Future Values
Fundamentals of Corporate Finance
Future Value - Amount to which an investment will grow after earning interest. Compound Interest - Interest earned on interest.
FV C1 (1 r ) C 2 (1 r ) ....
1 2
Present Value of Multiple Cash Flows
Calculating the present value of a unequal series of future cash flows is determined by summing the present values of each discounted single future cash flow.
17
Finding the Interest Rate
Fundamentals of Corporate Finance
In the expression, PV = FV(1+r)t, when the PV, FV, and t are known, (1+r) may be solved mathematically. The discount rate calculated is also called the annual interest rate, growth rate, and internal rate of return, depending on the situation.
Interest Earned Per Year =Prior Year Balance x .06
6
Future Values
Fundamentals of Corporate Finance
Example - Compound Interest Interest earned at a rate of 6% for five years on the previous year’s balance.
PV
3000 2 ( 1 . 08 )
$ 2 ,572
14
Present Values
Fundamentals of Corporate Finance
Present Value Interest Factor=PVIF =Discount Factor = DF = PV of $1
DF
Fundamentals of Corporate Finance
Future Value of $100 = FV Future Value Interest Factor=FVIF
F V $100 (1 r )
t
8
Future Values
Fundamentals of Corporate Finance
1 t ( 1 r )
Discount Factors can be used to compute the present value of any cash flow.