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CHAPTER 8Portfolio Theory and the Capital Asset Pricing ModelAnswers to Problem Sets1. a. 7%b. 27% with perfect positive correlation; 1% with perfect negative correlation;19.1% with no correlationc. See Figure 1 belowd. No, measure risk by beta, not by standard deviation.2. a. Portfolio A (higher expected return, same risk)b. Cannot say (depends on investor’s attitude -toward risk)c. Portfolio F (lower risk, same -expected return).3. Sharpe ratio = 7.1/20.2 = .3514. a. Figure 8.13b: Diversification reduces risk (e.g., a mixture of portfolios Aand B would have less risk than the average of A and B).b. Those along line AB in Figure 9.13a.c. See Figure 2 below5. a. See Figure 3 belowb. A, D, Gc. Fd. 15% in Ce. Put 25/32 of your money in F and lend 7/32 at 12%: Expected return = 7/32X 12 + 25/32 X 18 = 16.7%; standard deviation = 7/32 X 0 + (25/32) X 32 =25%. If you could borrow without limit, you would achieve as high anexpected return as you’d like, with correspondingly high risk, of course.6. a. 4 + (1.41 X 6) = 12.5%b. Amazon: 4 + (2.16 X 6) = 17.0%c. Campbell Soup: 4 + (.30 X 6) = 5.8%d. Lower. If interest rate is 4%, r = 4 + (1.75 X 6) = 14.5%; if rate = 6%, r = 6 +(1.75 X 4) = 13.0%e. Higher. If interest rate is 4%, r = 4 + (.55 X 6) = 7.3%; if rate = 6%, r = 6 +(.55 X 4) 5 8.2%7. a. Trueb. False (it offers twice the market risk premium)c. False8. a. 7%b. 7 + 1(5) + 1(-1) + 1(2) = 13%c. 7 + 0(5) + 2(-1) + 0(2) = 5%d. 7 + 1(5) + (-1.5)(-1) + 1(2) = 15.5%.9. a. False – investors demand higher expected rates of return on stocks withmore nondiversifiable risk.b. False – a security with a beta of zero will offer the risk-free rate of return.c. False – the beta will be: (1/3 ⨯ 0) + (2/3 ⨯ 1) = 0.67d. Truee. True10. In the following solution, security one is Campbell Soup and security two isBoeing. Then:r1 = 0.031 σ1 = 0.158r2 = 0.095 σ2 = 0.237Further, we know that for a two-security portfolio:r p = x1r1 + x2r2σp2 = x12σ12 + 2x1x2σ1σ2ρ12 + x22σ22 Therefore, we have the following results:x1x2r pσπ σπwhen ρ = 0 when ρ = 0.51 0 3.10% 0.02496 0.02496 0.9 0.1 3.74% 0.02078 0.02415 0.8 0.2 4.38% 0.01822 0.02422 0.7 0.3 5.02% 0.01729 0.02515 0.6 0.4 5.66% 0.01797 0.02696 0.5 0.5 6.30% 0.02028 0.02964 0.4 0.6 6.94% 0.02422 0.03320 0.3 0.7 7.58% 0.02977 0.03763 0.2 0.8 8.22% 0.03695 0.04294 0.1 0.9 8.86% 0.04575 0.04912 0 1 9.50% 0.05617 0.0561711. a.Portfolio r1 10.0% 5.1%2 9.0 4.63 11.0 6.4b. See the figure below. The set of portfolios is represented by the curvedline. The five points are the three portfolios from Part (a) plus thefollowing two portfolios: one consists of 100% invested in X and the otherconsists of 100% invested in Y.c. See the figure below. The best opportunities lie along the straight line.From the diagram, the optimal portfolio of risky assets is portfolio 1, andso Mr. Harrywitz should invest 50 percent in X and 50 percent in Y.12. a. Expected return = (0.6 ⨯ 15) + (0.4 ⨯ 20) = 17%Variance = (0.62⨯ 202) + (0.42⨯ 222) + 2(0.6)(0.4)(0.5)(20)(22) = 327.04Standard deviation = 327.04(1/2) = 18.08%b. Correlation coefficient = 0 ⇒ Standard deviation = 14.88%Correlation coefficient = –0.5 ⇒ Standard deviation = 10.76%c. His portfolio is better. The portfolio has a higher expected return and alower standard deviation.13. a.Average SD Sharpe RatioSauros 14.6 15.2 0.77S&P500 14.4 10.5 1.09Risk Free 2.9 1.7b. We can calculate the Beta of her investment as follows (see Table 7.7):TOTALDeviationfrom Averagemkt return 17.2 -1.9 -8.0 1.4 -8.8Deviationfrom AverageSauros return 24.5 -3.6 -12.0 3.4 -12.3SquaredDeviationfrom Averagemarketreturn 296.5 3.5 63.7 2.0 77.1 442.8Product ofDeviationsfrom Averagereturns 421.9 6.8 95.8 4.8 108.0 637.2Marketvariance 88.6Covariance 127.4Beta 1.4To construct a portfolio with a beta of 1.4, we will borrow .4 at the risk free rateand invest this in the market portfolio. This gives us annual returns as follows:Average1.4 times market 20.1less 0.4 time rf -1.2net return 19.0The Sauros portfolio does not generate sufficient returns to compensate for itsrisk.14. a. The Beta of the first portfolio is 0.714 and offers an average return of 5.9%b. We can devise a superior portfolio with a blend of 85.5% Campbell’s Soupand 14.5% Amazon.15.a.b. Market risk premium = r m – r f = 0.12 – 0.04 = 0.08 = 8.0%c.Use the security market line:r = r f + β(r m – r f )r = 0.04 + [1.5 ⨯ (0.12 – 0.04)] = 0.16 = 16.0%d.For any investment, we can find the opportunity cost of capital using the security market line. With β = 0.8, the opportunity cost of capital is:r = r f + β(r m – r f )r = 0.04 + [0.8 ⨯ (0.12 – 0.04)] = 0.104 = 10.4%The opportunity cost of capital is 10.4% and the investment is expected to earn 9.8%. Therefore, the investment has a negative NPV.e.Again, we use the security market line:r = r f + β(r m – r f )0.112 = 0.04 + β(0.12 – 0.04) ⇒ β = 0.916.a.Percival’s current portfolio provides an expected return of 9% with anannual standard deviation of 10%. First we find the portfolio weights for a combination of Treasury bills (security 1: standard deviation = 0%) and the index fund (security 2: standard deviation = 16%) such that portfolio standard deviation is 10%. In general, for a two security portfolio:σP 2 = x 12σ12 + 2x 1x 2σ1σ2ρ12 + x 22σ22 (0.10)2 = 0 + 0 + x 22(0.16)2 x 2 = 0.625 ⇒ x 1= 0.375Further:r p = x1r1 + x2r2r p = (0.375 ⨯ 0.06) + (0.625 ⨯ 0.14) = 0.11 = 11.0%Therefore, he can improve his expected rate of return without changingthe risk of his portfolio.b. With equal amounts in the corporate bond portfolio (security 1) and theindex fund (security 2), the expected return is:r p = x1r1 + x2r2r p = (0.5 ⨯ 0.09) + (0.5 ⨯ 0.14) = 0.115 = 11.5%σP2 = x12σ12 + 2x1x2σ1σ2ρ12 + x22σ22σP2 = (0.5)2(0.10)2 + 2(0.5)(0.5)(0.10)(0.16)(0.10) + (0.5)2(0.16)2σP2 = 0.0097σP = 0.985 = 9.85%Therefore, he can do even better by investing equal amounts in thecorporate bond portfolio and the index fund. His expected return increasesto 11.5% and the standard deviation of his portfolio decreases to 9.85%. 17. First calculate the required rate of return (assuming the expansion assets bearthe same level of risk as historical assets):r = r f + β(r m– r f)r = 0.04 + [1.4 ⨯ (0.12 – 0.04)] = 0.152 = 15.2%The use this to discount future cash flows; NPV = -25.2918. a. True. By definition, the factors represent macro-economic risks thatcannot be eliminated by diversification.b. False. The APT does not specify the factors.c. True. Different researchers have proposed and empirically investigateddifferent factors, but there is no widely accepted theory as to what thesefactors should be.d. True. To be useful, we must be able to estimate the relevant parameters.If this is impossible, for whatever reason, the model itself will be oftheoretical interest only.19. Stock P: r = 5% + (1.0 ⨯ 6.4%) + [(–2.0) ⨯ (–0.6%)] + [(–0.2) ⨯ 5.1%] = 11.58%Stock P2: r = 5% + (1.2 ⨯ 6.4%) + [0 ⨯ (–0.6%)] + (0.3 ⨯ 5.1%) = 14.21%Stock P3: r = 5% + (0.3 ⨯ 6.4%) + [0.5 ⨯ (–0.6%)] + (1.0 ⨯ 5.1%) = 11.72%20. a. Factor risk exposures:b1(Market) = [(1/3)⨯1.0] + [(1/3)⨯1.2] + [(1/3)⨯0.3] = 0.83b2(Interest rate) = [(1/3)⨯(–2.0)] +[(1/3)⨯0] + [(1/3)⨯0.5] = –0.50b3(Yield spread) = [(1/3)⨯(–0.2)] + [(1/3)⨯0.3] + [(1/3)⨯1.0] = 0.37b. r P = 5% + (0.83⨯6.4%) + [(–0.50)⨯(–0.6%)] + [0.37⨯5.1%] = 12.50%21. r Boeing = 0.2% + (0.66 ⨯ 7%) + (01.19 ⨯ 3.6%) + (-0.76 ⨯ 5.2%) = 5.152%R J&J = 0.2% + (0.54 ⨯ 7%) + (-0.58 ⨯ 3.6%) + (0.19 ⨯ 5.2%) = 2.88%R Dow = 0.2% + (1.05 ⨯ 7%) + (–0.15 ⨯ 3.6%) + (0.77 ⨯ 5.2%) = 11.014%r Msft= 0.2% + (0.91 ⨯ 7%) + ( 0.04 × 3.6%) + (–0.4 ⨯ 5.2%) = 4.346%22. In general, for a two-security portfolio:σp2 = x12σ12 + 2x1x2σ1σ2ρ12 + x22σ22and:x1 + x2 = 1Substituting for x2 in terms of x1 and rearranging:σp2 = σ12x12 + 2σ1σ2ρ12(x1– x12) + σ22(1 – x1)2Taking the derivative of σp2 with respect to x1, setting the derivative equal to zero and rearranging:x1(σ12– 2σ1σ2ρ12 + σ22) + (σ1σ2ρ12–σ22) = 0Let Campbell Soup be security one (σ1 = 0.158) and Boeing be security two(σ2 = 0.237). Substituting these numbers, along with ρ12 = 0.18, we have:x1 = 0.731Therefore:x2 = 0.26923. a. The ratio (expected risk premium/standard deviation) for each of the fourportfolios is as follows:Portfolio A: (22.8 – 10.0)/50.9 = 0.251Portfolio B: (10.5 – 10.0)/16.0 = 0.031Portfolio C: (4.2 – 10.0)/8.8 = -0.659Therefore, an investor should hold Portfolio A.b. The beta for Amazon relative to Portfolio A is identical.c. If the interest rate is 5%, then Portfolio C becomes the optimal portfolio, asindicated by the following calculations:Portfolio A: (22.8 – 5.0)/50.9 = 0.35Portfolio B: (10.5 – 5.0)/16.0 = 0.344Portfolio C: (4.2 – 5.0)/8.8 = -0.091The results do not change.24. Let r x be the risk premium on investment X, let x x be the portfolio weight of X (andsimilarly for Investments Y and Z, respectively).a. r x = (1.75⨯0.04) + (0.25⨯0.08) = 0.09 = 9.0%r y = [(–1.00)⨯0.04] + (2.00⨯0.08) = 0.12 = 12.0%r z = (2.00⨯0.04) + (1.00⨯0.08) = 0.16 = 16.0%b. This portfolio has the following portfolio weights:x x = 200/(200 + 50 – 150) = 2.0x y = 50/(200 + 50 – 150) = 0.5x z = –150/(200 + 50 – 150) = –1.5The portfolio’s sensitivities to the factors are:Factor 1: (2.0⨯1.75) + [0.5⨯(–1.00)] – (1.5⨯2.00) = 0Factor 2: (2.0⨯0.25) + (0.5⨯2.00) – (1.5⨯1.00) = 0Because the sensitivities are both zero, the expected risk premium is zero.c. This portfolio has the following portfolio weights:x x = 80/(80 + 60 – 40) = 0.8x y = 60/(80 + 60 – 40) = 0.6x z = –40/(80 + 60 – 40) = –0.4The sensitivities of this portfolio to the factors are:Factor 1: (0.8⨯1.75) + [0.6⨯(–1.00)] – (0.4⨯2.00) = 0Factor 2: (0.8⨯0.25) + (0.6⨯2.00) – (0.4⨯1.00) = 1.0The expected risk premium for this portfolio is equal to the expected riskpremium for the second factor, or 8 percent.d. This portfolio has the following portfolio weights:x x = 160/(160 + 20 – 80) = 1.6x y = 20/(160 + 20 – 80 ) = 0.2x z = –80/(160 + 20 – 80) = –0.8The sensitivities of this portfolio to the factors are:Factor 1: (1.6⨯1.75) + [0.2⨯(–1.00)] – (0.8⨯2.00) = 1.0Factor 2: (1.6⨯0.25) + (0.2⨯2.00) – (0.8⨯1.00) = 0The expected risk premium for this portfolio is equal to the expected riskpremium for the first factor, or 4 percent.e. The sensitivity requirement can be expressed as:Factor 1: (x x)(1.75) + (x y)(–1.00) + (x z)(2.00) = 0.5In addition, we know that:x x + x y + x z = 1With two linear equations in three variables, there is an infinite number of solutions. Two of these are:1. x x = 0 x y = 0.5 x z = 0.52. x x = 6/11 x y = 5/11 x z = 0The risk premiums for these two funds are:r1 = 0⨯[(1.75 ⨯ 0.04) + (0.25 ⨯ 0.08)]+ (0.5)⨯[(–1.00 ⨯ 0.04) + (2.00 ⨯ 0.08)]+ (0.5)⨯[(2.00 ⨯ 0.04) + (1.00 ⨯ 0.08)] = 0.14 = 14.0%r2 = (6/11)⨯[(1.75 ⨯ 0.04) + (0.25 ⨯ 0.08)]+(5/11)⨯[(–1.00 ⨯ 0.04) + (2.00 ⨯ 0.08)]+0 ⨯ [(2.00 ⨯ 0.04) + (1.00 ⨯ 0.08)] = 0.104 = 10.4%These risk premiums differ because, while each fund has a sensitivity of0.5 to factor 1, they differ in their sensitivities to factor 2.f. Because the sensitivities to the two factors are the same as in Part (b),one portfolio with zero sensitivity to each factor is given by:x x = 2.0 x y = 0.5 x z = –1.5The risk premium for this portfolio is:(2.0⨯0.08) + (0.5⨯0.14) – (1.5⨯0.16) = –0.01Because this is an example of a portfolio with zero sensitivity to eachfactor and a nonzero risk premium, it is clear that the Arbitrage PricingTheory does not hold in this case.A portfolio with a positive risk premium is:x x = –2.0 x y = –0.5 x z = 1.5。
Corporate Finance 第十版课程设计
一、课程简介
本课程是针对财务学专业的本科学生设置的，主要内容涵盖企业财务决策、投
资决策、资本结构等方面内容。
通过课程学习，学生将了解企业财务管理的基本概念和工具，掌握企业的融资、投资和分配决策的方法和技能，逐步形成批判性思维和问题解决能力。
二、教学目标
•理解企业财务管理的目标和作用
•掌握企业财务决策的基本框架和方法
•熟悉投资决策的基本理论和实践操作
•理解资本结构和成本的概念，掌握调整资本结构的技巧
•培养分析企业财务报表和业绩的能力
三、教学方式
本课程采用以理论为主导、实践为主的教学方式。
在课程中将采用案例、实例、讨论和小组演示等多种教学方法，对学生的分析能力、判断能力、团队协作和沟通能力进行培养和锻炼。
四、教学内容
1. 企业财务管理基础
•企业财务管理的概念和目标
•财务报表的理解和分析
•现金流量和时间价值的概念
1。
CHAPTER 2How to Calculate Present ValuesAnswers to Problem Sets1. If the discount factor is .507, then .507*1.126 = $12. 125/139 = .8993. PV = 374/(1.09)9 = 172.204. PV = 432/1.15 + 137/(1.152) + 797/(1.153) = 376 + 104 + 524 = $1,0035. FV = 100*1.158 = $305.906. NPV = -1,548 + 138/.09 = -14.67 (cost today plus the present value oftheperpetuity)7. PV = 4/(.14-.04) = $408. a. PV = 1/.10 = $10b. Since the perpetuity will be worth $10 in year 7, and since thatis roughlydouble the present value, the approximate PV equals $5.PV = (1 / .10)/(1.10)7 = 10/2= $5 (approximately)c. A perpetuity paying $1 starting now would be worth $10, whereas aperpetuity starting in year 8 would be worth roughly $5. Thedifference between these cash flows is therefore approximately $5.PV = 10 – 5= $5 (approximately)d. PV = C/(r-g) = 10,000/(.10-.05) = $200,000.9. a. PV = 10,000/(1.055) = $7,835.26 (assuming the cost of the car doesnotappreciate over those five years).b. You need to set aside (12,000 × 6-year annuity factor) = 12,000× 4.623 =$55,476.c. At the end of 6 years you would have 1.086 × (60,476 - 55,476) = $7,934.10. a. FV = 1,000e .12x5 = 1,000e .6 = $1,822.12.b. PV = 5e -.12 x 8 = 5e -.96 = $1.914 millionc. PV = C (1/r – 1/re rt ) = 2,000(1/.12 – 1/.12e .12 x15) = $13,912 11.a. FV = 10,000,000x(1.06)4 = 12,624,770b. FV = 10,000,000x(1 + .06/12)(4x12) = 12,704,892c. FV = 10,000,000xe (4x.06) = 12,712,492 12.a. PV = $100/1.0110= $90.53b. PV = $100/1.1310 = $29.46c. PV = $100/1.2515 = $ 3.52d.PV = $100/1.12 + $100/1.122 + $100/1.123 = $240.1813. a. ⇒=+=0.905r 11DF 11r 1 = 0.1050 = 10.50%b. 0.819(1.105)1)r (11DF 2222==+=c. AF 2 = DF 1 + DF 2 = 0.905 + 0.819 = 1.724d.PV of an annuity = C [Annuity factor at r% for t years]Here: $24.65 = $10[AF 3]AF= 2.4653e. AF3= DF 1 + DF 2 + DF 3 = AF 2 + DF 32.465 = 1.724 + DF 3 DF 3 = 0.74114. The present value of the 10-year stream of cash inflows is:6$886,739.6(1.14)0.1410.141 $170,000PV 10=⎥⎦⎤⎢⎣⎡⨯-⨯= Thus:NPV = –$800,000 + $886,739.66 = +$86,739.66At the end of five years, the factory’s value will be the present value of the five remaining $170,000 cash flows:6$583,623.7(1.14)0.1410.141 $170,000PV 5=⎥⎦⎤⎢⎣⎡⨯-⨯= 15.$23,696.151.12$50,0001.12$68,0001.12$80,0001.12$92,0001.12$92,000 1.12$85,0001.12$80,0001.12$75,0001.12$57,0001.12$50,000$380,000(1.12)C NPV 109876543210t t t =++++++++++-==∑=016. a.Let S t = salary in year t∑=-=301t t1t (1.08)(1.05)40,000PV 3$760,662.5(1.08).05)-(.08(1.05).05)-(.081 43030=⎥⎦⎤⎢⎣⎡⨯-⨯=000,0b.PV(salary) x 0.05 = $38,033.13Future value = $38,018.96 x (1.08)30 = $382,714.30c.$38,980.30 (1.08)0.0810.081$C (1.08)0.0810.081 C $r)(1r 1r 1 C PV 2020t =⎥⎥⎦⎤⎢⎢⎣⎡⨯-=⎥⎦⎤⎢⎣⎡⨯-⨯=⎥⎦⎤⎢⎣⎡+⨯-⨯=382,714.30382,714.3017.Period Present Value400,000.01 +100,000/1.12 = + 89,285.71 2 +200,000/1.122 = +159,438.783 +300,000/1.123 = +213,534.07Total = NPV = $62,258.5618. We can break this down into several different cash flows, such that thesum of these separate cash flows is the total cash flow. Then, the sum of the present values of the separate cash flows is the present value of the entire project. (All dollar figures are in millions.)▪ Cost of the ship is $8 million PV = $8 million▪Revenue is $5 million per year, operating expenses are $4 million. Thus, operating cash flow is $1 million per year for 15 years.million $8.559(1.08)0.0810.081 million 1$PV 15=⎥⎦⎤⎢⎣⎡⨯-⨯= ▪Major refits cost $2 million each, and will occur at times t = 5 and t = 10.PV = ($2 million)/1.085 + ($2 million)/1.0810= $2.288 million▪Sale for scrap brings in revenue of $1.5 million at t = 15.PV = $1.5 million/1.0815 = $0.473 millionAdding these present values gives the present value of the entire project:NPV = $8 million + $8.559 million $2.288 million + $0.473 millionNPV = $1.256 million19. a. PV = $100,000b. PV = $180,000/1.125 = $102,136.83c. PV = $11,400/0.12 = $95,000d. 4$107,354.2(1.12)0.1210.121 $19,000PV 10=⎥⎦⎤⎢⎣⎡⨯-⨯= e.PV = $6,500/(0.120.05) = $92,857.14Prize (d) is the most valuable because it has the highest present value.20. Mr. Basset is buying a security worth $20,000 now. That is its presentvalue. The unknown is the annual payment. Using the present value of an annuity formula, we have:$2,653.90 (1.08)0.0810.081000,20$C (1.08)0.0810.081 C 000,20$r)(1r 1r 1 C PV 1212t =⎥⎥⎦⎤⎢⎢⎣⎡⨯-=⎥⎦⎤⎢⎣⎡⨯-⨯=⎥⎦⎤⎢⎣⎡+⨯-⨯=21. Assume the Zhangs will put aside the same amount each year. Oneapproach to solving this problem is to find the present value of the cost of the boat and then equate that to the present value of the money saved. From this equation, we can solve for the amount to be put aside each year.PV(boat) = $20,000/(1.10)5 = $12,418PV(savings) = Annual savings ⎥⎦⎤⎢⎣⎡⨯-⨯5(1.10)0.1010.101 Because PV(savings) must equal PV(boat):Annual savings 418,12$ =⎥⎦⎤⎢⎣⎡⨯-⨯5(1.10)0.1010.101Annual savings$3,276(1.10)0.1010.101$12,4185=⎥⎦⎤⎢⎣⎡⨯-= Another approach is to use the future value of an annuity formula:$20,0005.10)(1 savings Annual =⎥⎥⎦⎤⎢⎢⎣⎡-+⨯10.1Annual savings = $ 3,27622. The fact that Ka ngaroo Autos is offering “free credit” tells us whatthe cash payments are; it does not change the fact that money has time value. A 10% annual rate of interest is equivalent to a monthly rate of 0.83%:r monthly = r annual /12 = 0.10/12 = 0.0083 = 0.83%The present value of the payments to Kangaroo Autos is:8$8,93(1.0083)0.008310.00831$300$1,00030=⎥⎦⎤⎢⎣⎡⨯-⨯+ A car from Turtle Motors costs $9,000 cash. Therefore, KangarooAutos offers the better deal, i.e., the lower present value of cost.23. The NPVs are:at 5% $25,011(1.05)$320,0001.05$100,000$170,000NPV 2=+--=⇒at 10% $3,554(1.10)320,0001.10$100,000$170,000NPV 2=+--=⇒ at 15% $14,991(1.15)320,0001.15$100,000$170,000NPV 2-=+--=⇒ The figure below shows that the project has zero NPV at about 11%.As a check, NPV at 11% is:$371(1.11)320,0001.11$100,000$170,000NPV 2-=+--=$1,428.570.07$100r C PV ===b.This is worth the PV of stream (a) plus the immediate payment of$100:PV = $100 + $1,428.57 = $1,528.57c.The continuously compounded equivalent to a 7% annually compounded rate is approximately 6.77%, because:e 0.0677 = 1.0700 Thus:$1,477.100.0677$100r C PV ===Note that the pattern of payments in part (b) is more valuable than the pattern of payments in part (c). It is preferable to receive cash flows at the start of every year than to spread thereceipt of cash evenly over the year; with the former pattern of payment, you receive the cash more quickly.25. a.PV = $1 billion/0.08 = $12.5 billionb. PV = $1 billion/(0.08 – 0.04) = $25.0 billionc.billion $9.818(1.08)0.0810.081billion $1PV 20=⎥⎦⎤⎢⎣⎡⨯-⨯= d.The continuously compounded equivalent to an 8% annually compounded rate is approximately 7.7% , because:e 0.0770 = 1.0800 Thus:billion $10.203e 0.07710.0771billion $1PV )(0.077)(20=⎥⎦⎤⎢⎣⎡⨯-⨯= This result is greater than the answer in Part (c) because theendowment is now earning interest during the entire year.26. With annual compounding: FV = $100(1.15)20 = $1,636.65With continuous compounding: FV = $100e (0.15×20) = $2,008.5527. One way to approach this problem is to solve for the present value of:(1) $100 per year for 10 years, and (2) $100 per year in perpetuity, with the first cash flow at year 11.If this is a fair deal, these present values must be equal, and thus we can solve for the interest rate (r).The present value of $100 per year for 10 years is:⎥⎦⎤⎢⎣⎡+⨯-⨯=10r)(1(r)1r 1$100PV The present value, as of year 10, of $100 per year forever, with thefirst payment in year 11, is: PV 10 = $100/r At t = 0, the present value of PV 10 is:⎥⎦⎤⎢⎣⎡⨯⎥⎦⎤⎢⎣⎡+=r $100r)(11PV 10 Equating these two expressions for present value, we have:⎥⎦⎤⎢⎣⎡⨯⎥⎦⎤⎢⎣⎡+=⎥⎦⎤⎢⎣⎡+⨯-⨯r $100r)(11r)(1(r)1r 1$1001010 Using trial and error or algebraic solution, we find that r = 7.18%.28. Assume the amount invested is one dollar.Let A represent the investment at 12%, compounded annually.Let B represent the investment at 11.7%, compounded semiannually. Let C represent the investment at 11.5%, compounded continuously.After one year:FV A = $1 (1 + 0.12)1 = $1.1200 FV B = $1 (1 + 0.0585)2 = $1.1204FV C = $1e (0.1151)= $1.1219After five years:FV A = $1 (1 + 0.12)5 = $1.7623 FV B = $1 (1 + 0.0585)10 = $1.7657FV C = $1e (0.1155)= $1.7771After twenty years:FV A = $1 (1 + 0.12)20 = $9.6463 FV B = $1 (1 + 0.0585)40 = $9.7193FV C = $1e (0.11520)= $9.9742The preferred investment is C.29. Because the cash flows occur every six months, we first need tocalculate the equivalent semi-annual rate. Thus, 1.08 = (1 + r/2)2 => r = 7.85 semi-annually compounded APR. Therefore the rate for six months is 7.85/2 or 3.925%:081,846039251039250103925010001000001009$).(..,$,$PV =⎥⎦⎤⎢⎣⎡⨯-⨯+=30. a. Each installment is: $9,420,713/19 = $495,827$4,761,724(1.08)0.0810.081$495,827PV 19=⎥⎦⎤⎢⎣⎡⨯-⨯=b.If ERC is willing to pay $4.2 million, then:⎥⎦⎤⎢⎣⎡+⨯-⨯=19r)(1r 1r 1$495,827$4,200,000 Using Excel or a financial calculator, we find that r = 9.81%.31. a. 3$402,264.7(1.08)0.0810.081$70,000PV 8=⎥⎦⎤⎢⎣⎡⨯-⨯=b.Year Beginning-of-Year Balance Year-end Interest on Balance Total Year-end Payment Amortization of Loan End-of-Year Balance 1 402,264.73 32,181.18 70,000.00 37,818.82 364,445.91 2 364,445.91 29,155.67 70,000.00 40,844.33 323,601.58 3 323,601.58 25,888.13 70,000.00 44,111.87 279,489.71 4 279,489.71 22,359.18 70,000.00 47,640.82 231,848.88 5 231,848.88 18,547.91 70,000.00 51,452.09 180,396.79 6 180,396.79 14,431.74 70,000.00 55,568.26 124,828.54 7 124,828.54 9,986.28 70,000.00 60,013.72 64,814.82864,814.825,185.1970,000.0064,814.810.0132. This is an annuity problem with the present value of the annuityequal to $2 million (as of your retirement date), and theinterest rate equal to 8% with 15 time periods. Thus, your annual level of expenditure (C) is determined as follows:$233,659 (1.08)0.0810.081$2,000,000C (1.08)0.0810.081 C $2,000,000r)(1r 1r 1 C PV 1515t =⎥⎥⎦⎤⎢⎢⎣⎡⨯-=⎥⎦⎤⎢⎣⎡⨯-⨯=⎥⎦⎤⎢⎣⎡+⨯-⨯=With an inflation rate of 4% per year, we will still accumulate $2million as of our retirement date. However, because we want to spend a constant amount per year in real terms (R, constant for all t), the nominal amount (Ct ) must increase each year. For each year t: R = Ct/(1 + inflationrate)tTherefore:PV [all Ct] = PV [all R (1 + inflation rate)t] = $2,000,000$2,000,0000.08)(1.04)0(1....08)0(10.04)(10.08)(1.04)0(1R 15152211=⎥⎦⎤⎢⎣⎡+++++++++⨯ R[0.9630 + 0.9273 + . . . + 0.5677] = $2,000,000 R 11.2390 = $2,000,000R = $177,952 Alternatively, consider that the real rate is03846.1=-++0.04)(1.08)0(1. Then, redoing the steps above using the real rate gives a real cash flow equal to:$177,952=⎥⎥⎦⎤⎢⎢⎣⎡⨯-= (1.03846)0.0384610.038461$2,000,000C 15Thus C 1 = ($177,9521.04) = $185,070, C 2 = $192,473, etc.33. a.9$430,925.8(1.055)0.05510.0551$50,000PV 12=⎥⎦⎤⎢⎣⎡⨯-⨯=b. The annually compounded rate is 5.5%, so the semiannual rate is:(1.055)(1/2) – 1 = 0.0271 = 2.71%Since the payments now arrive six months earlier than previously:PV = $430,925.89 × 1.0271 = $442,603.9834. In three years, the balance in the mutual fund will be:FV = $1,000,000 × (1.035)3 = $1,108,718The monthly shortfall will be: $15,000 – ($7,500 + $1,500) = $6,000Annual withdrawals from the mutual fund will be: $6,000 × 12 = $72,000 Assume the first annual withdrawal occurs three years from today, when the balance in the mutual fund will be $1,108,718. Treating thewithdrawals as an annuity due, we solve for t as follows: r)(1r)(1r 1r 1C PV t +⨯⎥⎦⎤⎢⎣⎡+⨯-⨯=1.035(1.035)0.03510.0351$72,000$1,108,718t ⨯⎥⎦⎤⎢⎣⎡⨯-⨯= Using Excel or a financial calculator, we find that t = 22.5 years.35. a. PV = 2/.12 = $16.667 millionb. PV = 939.14$=⎥⎦⎤⎢⎣⎡⨯-⨯20(1.12)0.1210.121$2millionc. PV = 2/(.12-.03) = $22.222 milliond. PV = 061.18$=⎥⎦⎤⎢⎣⎡⨯-⨯2020(1.12).03)-(0.12 1.03.03)-(0.121$2million36. a. Using the Rule of 72, the time for money to double at 12% is72/12, or 6 years. More precisely, if x is the number ofyears for money to double, then:(1.12)x = 2Using logarithms, we find:x (ln 1.12) = ln 2x = 6.12 yearsb. With continuous compounding for interest rate r and time period x:e r x = 2Taking the natural logarithm of each side:r x = ln(2) = 0.693Thus, if r is expressed as a percent, then x (the time for moneyto double) is: x = 69.3/(interest rate, in percent).37. Spreadsheet exercise.38. a. This calls for the growing perpetuity formula with a negativegrowth rate (g = –0.04):million $14.290.14million $20.04)(0.10million $2PV ==--=b. The pipeline’s value at year 20 (i.e., at t = 20), assuming itscash flows last forever, is:gr g)(1C g r C PV 2012120-+=-= With C 1 = $2 million, g = –0.04, and r = 0.10:million $6.3140.14million $0.8840.140.04)(1million)($2PV 2020 ==-⨯= Next, we convert this amount to PV today, and subtract it from the answer to Part (a):million $13.35(1.10)million $6.314million $14.29PV 20=-=（注：可编辑下载，若有不当之处，请指正，谢谢!）。
PrinciplesofCorporateFinance英⽂第⼗版习题解答Chap005CHAPTER 5Net Present Value and Other Investment CriteriaAnswers to Problem Sets1. a. A = 3 years, B = 2 years, C = 3 yearsb. Bc. A, B, and Cd. B and C (NPV B = $3,378; NPV C = $2,405)e. Truef. It will accept no negative-NPV projects but will turn down some withpositive NPVs. A project can have positive NPV if all future cash flows areconsidered but still do not meet the stated cutoff period.2. Given the cash flows C0, C1, . . . , C T, IRR is defined by:It is calculated by trial and error, by financial calculators, or by spreadsheetprograms.3. a. $15,750; $4,250; $0b. 100%.4. No (you are effectively “borrowing” at a rate of interest highe r than theopportunity cost of capital).5. a. Twob. -50% and +50%c. Yes, NPV = +14.6.6. The incremental flows from investing in Alpha rather than Beta are -200,000;+110,000; and 121,000. The IRR on the incremental cash flow is 10% (i.e., -200 + 110/1.10 + 121/1.102 = 0). The IRR on Beta exceeds the cost of capital and so does the IRR on the incremental investment in Alpha. Choose Alpha.7. 1, 2, 4, and 68. a. $90.91.10)(1$10001000NP V A -=+-=$$4,044.7310)(1.$1000(1.10)$1000(1.10)$4000(1.10)$1000(1.10)$10002000NP V 5432B +=+++++-=$$39.4710)(1.$1000.10)(1$1000(1.10)$1000(1.10)$10003000NP V 542C +=++++-=$ b.Payback A = 1 year Payback B = 2 years Payback C = 4 years c. A and Bd.$909.09.10)(1$1000P V 1A ==The present value of the cash inflows for Project A never recovers the initial outlay for the project, which is always the case for a negative NPV project. The present values of the cash inflows for Project B are shown in the third row of the table below, and the cumulative net present values are shown in the fourth row:C 0 C 1C 2C 3C 4C 5-2,000.00 +1,000.00 +1,000.00 +4,000.00 +1,000.00 +1,000.00-2,000.00909.09 826.45 3,005.26 683.01 620.92-1,090.91 -264.46 2,740.803,423.81 4,044.73Since the cumulative NPV turns positive between year two and year three,the discounted payback period is:years 2.093,005.26264.462=+The present values of the cash inflows for Project C are shown in the third row of the table below, and the cumulative net present values are shown in the fourth row:C 0C 1C 2C 3 C 4C 5-3,000.00 +1,000.00 +1,000.00 0.00 +1,000.00 +1,000.00-3,000.00 909.09 826.450.00 683.01 620.92 -2,090.91 -1,264.46-1,264.46-581.45 39.47Since the cumulative NPV turns positive between year four and year five,the discounted payback period is:years 4.94620.92581.454=+e. Using the discounted payback period rule with a cutoff of three years, the firm would accept only Project B.9. a.When using the IRR rule, the firm must still compare the IRR with the opportunity cost of capital. Thus, even with the IRR method, one must specify the appropriate discount rate.b.Risky cash flows should be discounted at a higher rate than the rate used to discount less risky cash flows. Using the payback rule is equivalent to using the NPV rule with a zero discount rate for cash flows before the payback period and an infinite discount rate for cash flows thereafter. 10.The two IRRs for this project are (approximately): –17.44% and 45.27% Between these two discount rates, the NPV is positive. 11.a.The figure on the next page was drawn from the following points: Discount Rate0% 10% 20%NPV A +20.00 +4.13 -8.33 NPV B +40.00 +5.18 -18.98b.From the graph, we can estimate the IRR of each project from the point where its line crosses the horizontal axis:IRR A = 13.1% and IRR B = 11.9%c. The company should accept Project A if its NPV is positive and higherthan that of Project B; that is, the company should accept Project A if thediscount rate is greater than 10.7% (the intersection of NPV A and NPV B on the graph below) and less than 13.1%.d. The cash flows for (B – A) are:C0 = $ 0C1 = –$60C2 = –$60C3 = +$140Therefore:Discount Rate0% 10% 20%NPV B-A +20.00 +1.05 -10.65IRR B-A = 10.7%The company should accept Project A if the discount rate is greater than10.7% and less than 13.1%. As shown in the graph, for these discountrates, the IRR for the incremental investment is less than the opportunitycost of capital.12. a.Because Project A requires a larger capital outlay, it is possible that Project A has both a lower IRR and a higher NPV than Project B. (In fact, NPV A is greater than NPV B for all discount rates less than 10 percent.) Because the goal is to maximize shareholder wealth, NPV is the correct criterion.b.To use the IRR criterion for mutually exclusive projects, calculate the IRR for the incremental cash flows:C 0C 1C 2IRRA -B -200 +110 +121 10%Because the IRR for the incremental cash flows exceeds the cost of capital, the additional investment in A is worthwhile. c.81.86$(1.09)$3001.09$250400NPV 2A =++-=$ $79.10(1.09)$1791.09$140200NPV 2B =++-=$ 13.Use incremental analysis:C 1C 2 C 3Current arrangement -250,000 -250,000 +650,000 Extra shift -550,000 +650,000 0 Incremental flows -300,000+900,000 -650,000The IRRs for the incremental flows are (approximately): 21.13% and 78.87% If the cost of capital is between these rates, Titanic should work the extra shift.14. a..82010,0008,18210,000)( 1.1020,00010,000PI D ==--+-=.59020,00011,81820,000)( 1.1035,00020,000PI E ==--+-=b.Each project has a Profitability Index greater than zero, and so both areacceptable projects. In order to choose between these projects, we must use incremental analysis. For the incremental cash flows:0.3610,0003,63610,000)( 1.1015,00010,000PI D E ==--+-=-The increment is thus an acceptable project, and so the larger project should be accepted, i.e., accept Project E. (Note that, in this case, the better project has the lower profitability index.)15.Using the fact that Profitability Index = (Net Present Value/Investment), we find:Project Profitability Index 1 0.22 2 -0.02 3 0.17 4 0.14 5 0.07 6 0.18 70.12Thus, given the budget of $1 million, the best the company can do is to acceptProjects 1, 3, 4, and 6.If the company accepted all positive NPV projects, the market value (compared to the market value under the budget limitation) would increase by the NPV of Project 5 plus the NPV of Project 7: $7,000 + $48,000 = $55,000Thus, the budget limit costs the company $55,000 in terms of its market value. 16.The IRR is the discount rate which, when applied to a project’s cash flows, yields NPV = 0. Thus, it does not represent an opportunity cost. However, if eachproject’s cash flows could be invested at that project’s IRR, then the NPV of each project would be zero because the IRR would then be the opportunity cost of capital for each project. The discount rate used in an NPV calculation is the opportunity cost of capital. Therefore, it is true that the NPV rule does assume that cash flows are reinvested at the opportunity cost of capital.17.a.C 0 = –3,000 C 0 = –3,000 C 1 = +3,500 C 1 = +3,500 C 2 = +4,000 C 2 + PV(C 3) = +4,000 – 3,571.43 = 428.57 C 3 = –4,000 MIRR = 27.84%b.2321 1.12C 1.12xC xC -=+(1.122)(xC 1) + (1.12)(xC 2) = –C 3 (x)[(1.122)(C 1) + (1.12C 2)] = –C 3)()21231.12C )(C (1.12C -x +=0.4501.12)(4,00)(3,500(1.124,000x 2=+=)()0IRR)(1x)C -(1IRR)(1x)C -(1C 2210=++++0IRR)(10)0.45)(4,00-(1IRR)(10)0.45)(3,50-(13,0002=++++- Now, find MIRR using either trial and error or the IRR function (on afinancial calculator or Excel). We find that MIRR = 23.53%.It is not clear that either of these modified IRRs is at all meaningful.Rather, these calculations seem to highlight the fact that MIRR really has no economic meaning.18.Maximize: NPV = 6,700x W + 9,000x X + 0X Y – 1,500x Z subject to:10,000x W + 0x X + 10,000x Y + 15,000x Z ≤ 20,000 10,000x W + 20,000x X – 5,000x Y – 5,000x Z ≤ 20,000 0x W - 5,000x X– 5,000x Y – 4,000x Z ≤ 20,000 0 ≤ x W ≤ 1 0 ≤ x X ≤ 1 0 ≤ x Z ≤ 1Using Excel Spreadsheet Add-in Linear Programming Module:Optimized NPV = $13,450with x W = 1; x X = 0.75; x Y = 1 and x Z = 0If financing available at t = 0 is $21,000:Optimized NPV = $13,500with x W = 1; x X = (23/30); x Y = 1 and x Z = (2/30)Here, the shadow price for the constraint at t = 0 is $50, the increase in NPV for a $1,000 increase in financing available at t = 0.In this case, the program viewed x Z as a viable choice even though the NPV of Project Z is negative. The reason for this result is that Project Z provides a positive cash flow in periods 1 and 2.If the financing available at t = 1 is $21,000:Optimized NPV = $13,900with x W = 1; x X = 0.8; x Y = 1 and x Z = 0Hence, the shadow price of an additional $1,000 in t =1 financing is $450.。
CHAPTER 19Financing and ValuationAnswers to Problem Sets1. Market values of debt and equity are D = .9 X 75 = $67.5 million and E = 42 X2.5= $105 million.D/V = .39.WACC = .09(1 - .35).39 + .18(.61) = .1325, or 13.25%.2. Step 1: r = .09(.39) + .18(.61) = .145.Step 2: r D = .086,r E = .145 + (.145 - .086)(15/85) = .155.Step 3: WACC = .086(1 - .35).15 + .155(.85) = .14.3. a. Falseb. Truec. True4. The method values the equity of a company by discounting cash flows tostockholders at the cost of equity. See Section 19.2 for more details. The method assumes that the debt-to-equity ratio will remain constant.5. a. Trueb. False, if interest tax shields are valued separatelyc. True6. APV 5 base-case NPV ± PV financing side effectsa. APV = 0 - .15(500,000) = -75,000b. APV = 0 + 76,000 = +76,0007. a. 12%, of course.b. r E = .12 + (.12 - .075)(30/70) = .139, WACC = .075(1 - .35)(.30) + .139(.70)= .112, or 11.2%.8. a. Base-case NPV = -1,000 + 1200/1.20 = 0b. PV tax shield = (.35 X .1 X .3(1000))/1.1 = 9.55. APV = 0 + 9.55 = $9.559. No. The more debt you use, the higher rate of -return equity investors willrequire. (Lenders may demand more also.) Thus there is a hidden cost of the“cheap” debt: It makes equity more expensive.10. Patagonia does not have 90% debt capacity. KCS is borrowing $45 million partlyon the strength of its existing assets. Also the decision to raise bank finance for the purchase does not mean that KCS has changed its target debt ratio. An APV valuation of Patagonia would probably assume a 50% debt ratio.11. If the bank debt is treated as permanent financing, the capital structureproportions are:Bank debt (r D = 10 percent) $280 9.4%Long-term debt (r D = 9 percent) 1800 60.4Equity (r E = 18 percent, 90 x 10 million shares) 900 30.2$2980 100.0% WACC* = [0.10⨯(1 - 0.35)⨯0.094] + [0.09⨯(1 - 0.35)⨯0.604] + [0.18⨯0.302]= 0.096 = 9.6%12. Forecast after-tax incremental cash flows as explained in Section 6.1. Interest isnot included; the forecasts assume an all-equity financed firm.13. Calculate APV by subtracting $4 million from base-case NPV.14. We make three adjustments to the balance sheet:∙Ignore deferred taxes; this is an accounting entry and represents neither a liability nor a source of funds∙…Net out‟ accounts payable against current assets∙Use the market value of equity (7.46 million x $46)Now the right-hand side of the balance sheet (in thousands) is:Short-term debt $75,600Long-term debt 208,600Shareholders‟ equity343,160Total $627,360The after-tax weighted-average cost of capital formula, with oneelement for each source of funding, is:WACC = [r D-ST⨯(1 – T c)⨯(D-ST/V)]+[r D-LT⨯(1 – T c)⨯(D-LT/V)]+[r E⨯(E/V)]WACC = [0.06⨯(1 – 0.35)⨯(75,600/627,360)]+[0.08⨯(1 – 0.35)⨯(208,600/627,360)] + [0.15⨯(343,160/627,360)]= 0.004700 + 0.017290 + 0.082049 = 0.1040 = 10.40%15. Assume that short-term debt is temporary. From Problem 14:Long-term debt $208,600Share holder equity 343,160Total $551,760Therefore:D/V = $208,600/$551,760 = 0.378E/V = $343,160/$551,760 = 0.622Step 1:r = r D (D/V) + r E (E/V) = (0.08 ⨯ 0.378) + (0.15 ⨯ 0.622) = 0.1235 Step 2:r E = r + (r – r D) (D/E) = 0.1235 + (0.1235 – 0.08) ⨯ 0.403 = 0.1410 Step 3:WACC = [r D⨯ (1 – T C) ⨯ (D/V)] + [r E⨯ (E/V)]= (0.08 ⨯ 0.65 ⨯ 0.287) + (0.1410 ⨯ 0.713) = 0.1155 = 11.55%16. Base case NPV = –$1,000 + ($600/1.12) + ($700/1.122) = $93.75 or $93,750YearDebtOutstanding atStart Of Year InterestInterestTax ShieldPV(Tax Shield)1 300 24 7.20 6.672 150 12 3.60 3.09APV = $93.75 + $6.67 + $3.09 = 103.5 or $103,50017. a. Base-case NPV = –$1,000,000 + ($95,000/0.10) = –$50,000PV(tax shields) = 0.35 ⨯ $400,000 = $140,000APV = –$50,000 + $140,000 = $90,000b. PV(tax shields, approximate) = (0.35 ⨯ 0.07 ⨯ $400,000)/0.10 =$98,000APV = –$50,000 + $98,000 = $48,000The present value of the tax shield is higher when the debt is fixed andtherefore the tax shield is certain. When borrowing a constant proportion of the market value of the project, the interest tax shields are as uncertain as the value of the project, and therefore must be discounted at the project‟s opportunity cost of capital. 18.The immediate source of funds (i.e., both the proportion borrowed and the expected return on the stocks sold) is irrelevant. The project would not be any more valuable if the university sold stocks offering a lower return. If borrowing is a zero-NPV activity for a tax-exempt university, then base-case NPV equals APV, and the adjusted cost of capital r* equals the opportunity cost of capital with all-equity financing. Here, base-case NPV is negative; the university should not invest.19.a.Base-case $0.111.12$1.7510NP V 101t t-=+-=∑=$or – $110,000 APV = Base-case NPV + PV(tax shield)PV(tax shield) is computed from the following table: Year Debt Outstanding at Start of Year InterestInterest Tax Shield Present Valueof Tax Shield 1 $5,000 $400 $140 $129.632 4,500 360 126 108.023 4,000 320 112 88.914 3,500 280 98 72.035 3,000 240 84 57.176 2,500 200 70 44.117 2,000 160 56 32.68 8 1,500 120 42 22.69 9 1,000 80 28 14.01 10 5004014 6.48Total575.74APV = –$110,000 + $575,740 = $465,740b. APV = Base-case NPV + PV(tax shield) – equity issue costs= –$110,000 + $575,740 – $400,000 = $65,74020. Spreadsheet exercise; answers will vary.21.Note the following:∙ The costs of debt and equity are not 8.5% and 19%, respectively. These figures assume the issue costs are paid every year, not just at issue.∙ The fact that Bunsen can finance the entire cost of the project with debt is irrelevant. The cost of capital does not depend on the immediate source of funds; what matters is the project‟s contribution to the firm‟s overall borrowing power.∙ The project is expected to support debt in perpetuity. The fact that the first debt issue is for only 20 years is irrelevant. Assume the project has the same business risk as the firm‟s other assets.Because it is a perpetuity , we can use the firm‟s weighted -average cost of capital. If we ignore issue costs:WACC = [r D ⨯ (1 – T C ) ⨯ (D/V)] + [r E ⨯ (E/V)]WACC = [0.07 ⨯ (1 – 0.35) ⨯ 0.4] + [0.14 ⨯ 0.6] = 0.1022 = 10.22%Using this discount rate:$272,0160.1022$130,000$1,000,000NPV =+-=The issue costs are:Stock issue: 0.050 ⨯ $1,000,000 = $50,000 Bond issue:0.015 ⨯ $1,000,000 = $15,000Debt is clearly less expensive. Project NPV net of issue costs is reduced to: ($272,016 – $15,000) = $257,016. However, if debt is used, the firm‟s debt rat io will be above the target ratio, and more equity will have to be raised later. If debt financing can be obtained using retaining earnings, then there are no other issue costs to consider. If stock will be issued to regain the target debt ratio, an additional issue cost is incurred.A careful estimate of the issue costs attributable to this project would require a comparison of Bunsen‟s financial plan …with‟ as compared to …without‟ this project.22.Disagree. The Goldensacks calculations are based on the assumption that the cost of debt will remain constant, and that the cost of equity capital will not change even though the firm‟s financial structure has changed. The former assumption is appropriate while the latter is not.23.LatestyearForecast0 1 2 3 4 5 1. Sales 40,123.0 36,351.0 30,155.0 28,345.0 29,982.0 30,450.0 2. Cost of Goods Sold 22,879.0 21,678.0 17,560.0 16,459.0 15,631.0 14,987.0 3. Other Costs 8,025.0 6,797.0 5,078.0 4,678.0 4,987.0 5,134.0 4. EBITDA (1 – 2 – 3) 9,219.0 7,876.0 7,517.0 7,208.0 9,364.0 10,329.0 5. Depreciation and Amortization 5,678.0 5,890.0 5,670.0 5,908.0 6,107.0 5,908.0 6. EBIT (Pretax profit) (4 – 5) 3,541.0 1,986.0 1,847.0 1,300.0 3,257.0 4,421.0 7. Tax at 35% 1,239.4 695.1 646.5 455.0 1,140.0 1,547.4 8. Profit after tax (6 – 7) 2,301.7 1,290.9 1,200.6 845.0 2,117.1 2,873.79. Change in working capital 784.0 -54.0 -342.0 -245.0127.0235.010. Investment(change in Gross PP&E) 6,547.0 7,345.0 5,398.0 5,470.0 6,420.0 6,598.0 11. Free Cash Flow (8 + 5 – 9 – 10) 648.7 -110.11,814.61,528.0 1,677.1 1,948.7PV Free cash flow, years 1-4 3,501.6 Horizon value in year 4PV Horizon value 15,480.0 24,358.1PV of company 18,981.7The total value of the equity is: $18,981.7 – $5,000 = $13,981.7 Value per share = $13,981.7/865 = $16.1624. a.For a one-period project to have zero APV:Rearranging gives:For a one-period project, the left-hand side of this equation is the projectIRR. Also, (D/ -C 0) is the project‟s debt capacity. Therefore, the minimum acceptable return is:b. .094051.061.09840.20)0.06(0.350.0984r *=⎪⎭⎫⎝⎛⨯⨯-=0r 1D)r (T r 1C C AP V DD C A 10=+⨯⨯+++=⎪⎪⎭⎫⎝⎛++⎪⎪⎭⎫ ⎝⎛-⨯-=--D A 0D C 01r 1r 1 C D )r T (r 1C C ⎪⎪⎭⎫ ⎝⎛++⨯⨯-=D AD C A r1r 1)L r T (r *r25. Fixed debt levels, without rebalancing, are not necessarily better for stockholders.Note that, when the debt is rebalanced, next year‟s inte rest tax shields are fixedand, thus, discounted at a lower rate. The following year‟s interest is not knownwith certainty for one year and, hence, is discounted for one year at the higherrisky rate and for one year at the lower rate. This is much more realistic since itrecognizes the uncertainty of future events.26. The table below is a modification of Table 19.1 based on the assumption that,after year 7:∙Sales remain constant (that is, growth = 0%);∙Costs remain at 76.0% of sales;∙Depreciation remains at 14.0% of net fixed assets;∙Net fixed assets remain constant at 93.8;∙Working capital remains at 13.0% of sales.TABLE 20.1 Free cash flow projections and company value for Rio Corporation ($ millions)Latestyear Forecast0 1 2 3 4 5 6 7 81. Sales 83.6 89.5 95.8 102.5 106.6 110.8 115.2 118.7 118.72. Cost of goods sold 63.1 66.2 71.3 76.3 79.9 83.1 87.0 90.2 90.23. EBITDA (1 - 2) 20.5 23.3 24.4 26.1 26.6 27.7 28.2 28.5 28.54. Depreciation 3.3 9.9 10.6 11.3 11.8 12.3 12.7 13.1 13.15. Profit before tax (EBIT) (3 - 4) 17.2 13.4 13.8 14.8 14.9 15.4 15.5 15.4 15.46. Tax 6.0 4.7 4.8 5.2 5.2 5.4 5.4 5.4 5.47. Profit after tax (5 - 6) 11.2 8.7 9.0 9.6 9.7 10.0 10.1 10.0 10.08. Investment in fixed assets 11.0 14.6 15.5 16.6 15.0 15.6 16.2 15.9 13.19. Investment in working capital 1.0 0.5 0.8 0.9 0.5 0.6 0.6 0.4 0.010. Free cash flow (7 + 4 - 8 - 9) 2.5 3.5 3.2 3.4 5.9 6.1 6.0 6.8 10.0PV Free cash flow, years 1-7 24.0 (Horizon value in year 7) PV Horizon value 60.7 110.9PV of company 84.7Assumptions:Sales growth (percent) 6.7 7.0 7.0 7.0 4.0 4.0 4.0 3.0 0.0 Costs (percent of sales) 75.5 74.0 74.5 74.5 75.0 75.0 75.5 76.0 76.0 Working capital(% of sales) 13.3 13.0 13.0 13.0 13.0 13.0 13.0 13.0 13.0 Net fixed assets (% of sales) 79.2 79.0 79.0 79.0 79.0 79.0 79.0 79.0 79.0 Depreciation (% net fixed assets) 5.0 14.0 14.0 14.0 14.0 14.0 14.0 14.0 14.0 Tax rate, % 35.0Cost of debt, % (r D) 6.0Cost of equity, % (r E) 12.4Debt ratio (D/V) 0.4WACC, % 9.0Long-term growth forecast, % 0.0Fixed assets and working capitalGross fixed assets 95.0 109.6 125.1 141.8 156.8 172.4 188.6 204.5 217.6 Less accumulated depreciation29.0 38.9 49.5 60.8 72.6 84.9 97.6 110.7 123.9 Net fixed assets 66.0 70.7 75.6 80.9 84.2 87.5 91.0 93.8 93.8 Net working capital 11.1 11.6 12.4 13.3 13.9 14.4 15.0 15.4 15.4 Appendix Problems1. The award is risk-free because it is owed by the U.S. government. The after-taxamount of the award is: 0.65 × $16 million = $10.40 millionThe after-tax discount rate is: 0.65 × 0.055 = 0.03575 = 3.575%The present value of the award is: $10.4 million/1.03575 = $10.04 million2. The after-tax cash flows are: 0.65 × $100,000 = $65,000 per yearThe after-tax discount rate is: 0.65 × 0.09 = 0.0585 = 5.85%The present value of the lease is equal to the present value of a five-year annuityof $65,000 per year plus the immediate $65,000 payment:$65,000 × [annuity factor, 5.85%, 5 years] + $65,000 =($65,000 × 4.2296) + $65,000 = $339,924。
CHAPTER 3Valuing BondsAnswers to Problem Sets1. a. Does not changeb. Price fallsc. Yield rises.2. a. If the coupon rate is higher than the yield, then investors must beexpecting a decline in the capital value of the bond over its remaining life.Thus, the bond’s price must be greater than its face value.b. Conversely, if the yield is greater than the coupon, the price will be belowface value and it will rise over the remaining life of the bond.3. The yield over 6 months is 3.965/2 = 1.79825%.Therefore, PV = 3/1.0179825 + 3/1.01798252+…. + 103/1.017982534 = 130.37 4. Yields to maturity are about 4.3% for the 2% coupon, 4.2% for the 4% coupon,and 3.9% for the 8% coupon. The 8% bond had the shortest duration (7.65years), the 2% bond the longest (9.07 years).5. a. Fall (e.g., 1-year 10% bond is worth 110/1.1 5 100 if r 5 10% and is worth110/1.15 = 95.65 if r = 15%).b. Less (e.g., See 5a).c. Less (e.g., with r = 5%, 1-year 10% bond is worth 110/1.05 = 104.76).d. Higher (e.g., if r = 10%, 1-year 10% bond is worth 110/1.1 = 100, while 1-year 8% bond is worth 108/1.1 = 98.18).e. No, low-coupon bonds have longer durations (unless there is only oneperiod to maturity) and are therefore more volatile (e.g., if r falls from 10%to 5%, the value of a 2-year 10% bond rises from 100 to 109.3 (a rise of9.3%). The value of a 2-year 5% bond rises from 91.3 to 100 (a rise of9.5%).6. a. Spot interest rates. Yield to maturity is a complicated average of the separate spot rates of interest.b. Bond prices. The bond price is determined by the bond’s cash flows andthe spot rates of interest. Once you know the bond price and the bond’s cash flows, it is possible to calculate the yield to maturity.7. a. 4%b.PV = $1,075.448. a. PV ()2211050,1150r r +++=b.PV ()21050,1150y y +++= c. Less (it is between the 1-year and 2-year spot rates).9. a. r 1 = 100/99.423 – 1 = .58%; r 2 = (100/97.546).5 – 1 = 1.25%; r 3 = (100/94.510).33 -.1 = 1.90%; r 4 = (100/90.524).25 – 1 = 2.52%.b. Upward-sloping.c.Lower (the yield on the bond is a complicated average of the separate spot rates).10. a. Price today is 108.425; price after 1 year is 106.930.b. Return = (106.930 1 8)/108.425 - 1 = .06, or 6%.c. If a bond’s yield to maturity is unchanged, the return to the bondholder is equal to the yield.11. a. False. Duration depends on the coupon as well as the maturity.b. False. Given the yield to maturity, volatility is proportional to duration.c. True. A lower coupon rate means longer duration and therefore higher volatility.d.False. A higher interest rate reduces the relative present value of (distant) principal repayments.12.13. 7.01% (the extra return that you earn for investing for two years rather than oneis 1.062/1.05 – 1 = .0701).14. a. Real rate = 1.10/1.05 – 1 = .0476, or 4.76%b. The real rate does not change. The nominal rate increases to 1.0476 x1.07 – 1 = .1209, or 12.9%.15. With annual coupon payments:=+⎥⎦⎤⎢⎣⎡⨯-⨯=1010(1.06)100(1.06)0.0610.0615PV €92.6416. a.$10,231.64(1.026)10,000(1.026)0.02610.0261275PV 2020=+⎥⎦⎤⎢⎣⎡⨯-⨯=b.Interest rate PV of Interest PV of Face value PV of Bond 1.0% $5,221.54 $9,050.63 $14,272.17 2.0% 4,962.53 8,195.44 13,157.97 3.0% 4,721.38 7,424.70 12,146.08 4.0% 4,496.64 6,729.71 11,226.36 5.0% 4,287.02 6,102.71 10,389.73 6.0% 4,091.31 5,536.76 9,628.06 7.0% 3,908.41 5,025.66 8,934.07 8.0% 3,737.34 4,563.87 8,301.21 9.0% 3,577.18 4,146.43 7,723.61 10.0% 3,427.11 3,768.89 7,196.00 11.0% 3,286.36 3,427.29 6,713.64 12.0% 3,154.23 3,118.05 6,272.28 13.0% 3,030.09 2,837.97 5,868.06 14.0% 2,913.35 2,584.19 5,497.54 15.0%2,803.492,354.135,157.6217.Purchase price for a 6-year government bond with 5 percent annual coupon:1,108.34(1.03)1,000(1.03)0.0310.03150PV 66$=+⎥⎦⎤⎢⎣⎡⨯-⨯= Price one year later (yield = 3%):1,091.59(1.03)1,000(1.03)0.0310.03150PV 55$=+⎥⎦⎤⎢⎣⎡⨯-⨯= Rate of return = [$50 + ($1,091.59 – $1,108.34)]/$1,108.34 = 3.00% Price one year later (yield = 2%):1,141.40(1.02)1,000(1.02)0.0210.02150PV 55$=+⎥⎦⎤⎢⎣⎡⨯-⨯= Rate of return = [$50 + ($1,141.40 – $1,108.34)]/$1,108.34 = 7.49%18.The key here is to find a combination of these two bonds (i.e., a portfolio ofbonds) that has a cash flow only at t = 6. Then, knowing the price of the portfolio and the cash flow at t = 6, we can calculate the 6-year spot rate.We begin by specifying the cash flows of each bond and using these and their yields to calculate their current prices:Investment Yield C 1 . . . C 5 C 6 Price 6% bond 12% 60 . . . 60 1,060 $753.32 10% bond 8%100. . .1001,100$1,092.46From the cash flows in years one through five, we can see that buying two 6% bonds produces the same annual payments as buying 1.2 of the 10% bonds. To see the value of a cash flow only in year six, consider the portfolio of two 6% bonds minus 1.2 10% bonds. This portfolio costs:($753.32 × 2) – (1.2 ⨯ $1,092.46) = $195.68The cash flow for this portfolio is equal to zero for years one through five and, for year 6, is equal to:(1,060 × 2) – (1.2 ⨯ 1,100) = $800Thus:$195.68 ⨯ (1 + r 6)6 = 800 r 6 = 0.2645 = 26.45%19. Downward sloping. This is because high coupon bonds provide a greaterproportion of their cash flows in the early years. In essence, a high coupon bond is a ‘shorter’ bond than a low coupon bond of the same maturity. 20.a.Year Discount Factor Forward Rate 1 1/1.05 = 0.9522 1/(1.054)2 = 0.900 (1.0542/1.05) – 1 = 0.0580 = 5.80% 3 1/(1.057)3 = 0.847 (1.0573 /1.0542 ) – 1 = 0.0630 = 6.30% 4 1/(1.059)4 = 0.795 (1.0594 /1.0573 ) – 1 = 0.0650 = 6.50% 51/(1.060)5 = 0.747(1.0605 /1.0594 ) – 1 = 0.0640 = 6.40%b.i.5%, two-year note:$992.79(1.054)$10501.05$50PV 2=+=ii. 5%, five-year bond:$959.34(1.060)$1050(1.059)$50(1.057)$50(1.054)$501.05$50PV 5432=++++=iii. 10%, five-year bond:$1,171.43(1.060)$1100(1.059)$100(1.057)$100(1.054)$1001.05$100PV 5432=++++=c. First, we calculate the yield for each of the two bonds. For the 5% bond, this means solving for r in the following equation:5432r)(1$1050r)(1$50r)(1$50r)(1$50r 1$50$959.34+++++++++=r = 0.05964 = 5.964% For the 10% bond:5432r)(1$1100r)(1$100r)(1$100r)(1$100r 1$100$1171.43+++++++++=r = 0.05937 = 5.937%The yield depends upon both the coupon payment and the spot rate at thetime of the coupon payment. The 10% bond has a slightly greater proportion of its total payments coming earlier, when interest rates are low, than does the 5% bond. Thus, the yield of the 10% bond is slightly lower.d. The yield to maturity on a five-year zero coupon bond is the five-year spotrate, here 6.00%.e. First, we find the price of the five-year annuity, assuming that the annual payment is $1:Now we find the yield to maturity for this annuity:r = 0.05745 = 5.745%f.The yield on the five-year note lies between the yield on a five-year zero-coupon bond and the yield on a 5-year annuity because the cash flows of the Treasury bond lie between the cash flows of these other two financial instruments during a period of rising interest rates. That is, the annuity has fixed, equal payments, the zero-coupon bond has one payment at the end, and the bond’s payments are a combination of these.21. Assuming we are calculating the durations as of February 2009, the strip’sduration equals$4.2417(1.060)1.059)(11.057)(11.054)(111.051PV 5432=++++=5432r)(11r)(11r)(11r)(11r 114.2417+++++++++=6uration ==88.7456x88.745D , and the modified duration equals 6/1.02 = 5.88. At asemi-annual yield of 1.5%, the price of the 6-year strips equals 91.424, and at a semi-annual yield of 2.5%, the price of the strips equals 86.151. The difference in the prices, 5.273 is 5.94% of the price of the strips. This is close to the 5.88 duration, and the difference is due to the first-order approximation of the price change provided by duration.To calculate the duration for the 4% bonds, consider the following table similar toTotal PV = 111.26, and the duration = 5.432. The modified duration equals 5.432/1.02 = 5.325.The price of the 4% coupon bond at 1.5% and 2.5% equals 114.294 and 108.310, respectively. This price difference, 5.984, is 5.38% of the original price, which is very close to the 5.33 duration.22. Table 3.3 can be flipped and recalculated as follows:This makes sense as we are now receiving smaller payments early in the life of the bond.Payments at the end of the bonds life are discounted more heavily, resulting in a greater fraction of total value being paid early.23. The duration of a perpetual bond is: [(1 + yield)/yield]The duration of a perpetual bond with a yield of 5% is:D5 = 1.05/0.05 = 21 yearsThe duration of a perpetual bond yielding 10% is:D10 = 1.10/0.10 = 11 yearsBecause the duration of a zero-coupon bond is equal to its maturity, the 15-year zero-coupon bond has a duration of 15 years.Thus, comparing the 5% perpetual bond and the zero-coupon bond, the 5%perpetual bond has the longer duration. Comparing the 10% perpetual bond and the 15 year zero, the zero has a longer duration.24. Answers will differ. Generally, we would expect yield changes to have thegreatest impact on long-maturity and low coupon bonds.25. The calculations are shown in the table below:26. We will borrow $1000 at a five year loan rate of 2.5% and buy a 4-year strippaying 4%. We may not know what interest rates we will earn on the last year(4 5) but our $1000 will come due and we put it in under our mattress earning0% if necessary to pay off the loan.Let’s turn to present value calculations: As shown above, the cost of the strip is $854.80. We will receive proceeds from the 2.5% loan = 1000 / (1.025)^5 =$883.90. Pocket the difference of $29.10, smile, and repeat.The minimum sensible value would set the discount factors used in year 5 equal to that of year 4, which would assume a 0% interest rate from year 4 to 5. Wecan solve for the interest rate where 1/ (1+r)5 = 0.8548, which is roughly 3.19%27.a. If the expectations theory of term structure is right, then we can determinethe expected future one year spot rate (at t=3) as follows: investing $100in a 3-year instrument at 4.2% gives us 100 * (1+.042)^3 = 113.136.Investing $100 in a 4-year instrument at 4.0% gives us 100 * (1+.04)^4 =116.986. This reveals a one year spot rate from year 3 to 4 of (116.98 –113.136) / 113.136 = 3.4%b. If investing in long-term bonds carries additional risks, then the riskequivalent one year spot rate in year three would be even less (reflectingthe fact that some risk premium must be built into this 3.4% spot rate).28.a. Your nominal return will be1.082 -1 = 16.64% over the two years. Yourreal return is (1.08/1.03) x (1.08/1.05) - 1 = 7.85%b. With the TIPS, the real return will remain at 8% per year, or 16.64% overtwo years. The nominal return on the TIPS will equal (1.08x1.03) x(1.08x1.05) – 1 = 26.15%.29. The bond price at a 5.41% yield is:1,196.49(1.0541)1,000(1.0541)0.054110.05411100PV 55$=+⎥⎦⎤⎢⎣⎡⨯-⨯= If the yield increases to 8.47%, the price would decline to:1,060.34(1.0847)1,000(1.0847)0.084710.08471100PV 55$=+⎥⎦⎤⎢⎣⎡⨯-⨯=30. Spreadsheet problem; answers will vary.31. Arbitrage opportunities can be identified by finding situations where the impliedforward rates or spot rates are different.We begin with the shortest-term bond, Bond G, which has a two-year maturity. Since G is a zero-coupon bond, we determine the two-year spot rate directly by finding the yield for Bond G. The yield is 9.5 percent, so the implied two-year spot rate (r 2) is 9.5 percent. Using the same approach for Bond A, we find that the three-year spot rate (r 3) is 10.0 percent.Next we use Bonds B and D to find the four-year spot rate. The followingposition in these bonds provides a cash payoff only in year four:a long position in two of Bond B and a short position in Bond D.Cash flows for this position are:[(–2 ⨯ $842.30) + $980.57] = –$704.03 today[(2 ⨯ $50) – $100] = $0 in years 1, 2 and 3[(2 ⨯ $1050) – $1100] = $1000 in year 4 We determine the four-year spot rate from this position as follows:4)4r (1$1000$704.03+= r 4 = 0.0917 = 9.17%Next, we use r 2, r 3 and r 4 with one of the four-year coupon bonds to determine r 1. For Bond C:978.74r 1$120(1.0917)$1120(1.100)$120(1.095)$120r 1$120$1,065.2814321$++=++++= r 1 = 0.3867 = 38.67%Now, in order to determine whether arbitrage opportunities exist, we use these spot rates to value the remaining two four-year bonds. This produces thefollowing results: for Bond B, the present value is $854.55, and for Bond D, thepresent value is $1,005.07. Since neither of these values equals the currentmarket price of the respective bonds, arbitrage opportunities exist. Similarly, the spot rates derived above produce the following values for the three-year bonds: $1,074.22 for Bond E and $912.77 for Bond F.32.We begin with the definition of duration as applied to a bond with yield r and an annual payment of C in perpetuityWe first simplify by dividing both the numerator and the denominator by C:The denominator is the present value of a perpetuity of $1 per year, which isequal to (1/r). To simplify the numerator, we first denote the numerator S and then divide S by (1 + r):Note that this new quantity [S/(1 + r)] is equal to the square of denominator in the duration formula above, that is:Therefore:Thus, for a perpetual bond paying C dollars per year:33.We begin with the definition of duration as applied to a common stock with yield r and dividends that grow at a constant rate g in perpetuity: ++++++++++++++++++=t 32t 32r)(1C r)(1C r)(1C r 1C r)(1tC r)(13C r)(12C r 11C DUR ++++++++++++++++++=t 32t 32r)(11r)(11r)(11r 11r)(1t r)(13r)(12r)(11DUR+++++++++=++1t 432r)(1t r)(13r)(12r)(11r)(1S 2t 32r)(11r)(11r)(11r 11r)(1S ⎪⎪⎭⎫ ⎝⎛+++++++++=+ 22r r 1S r 1r)(1S +=⇒⎪⎭⎫ ⎝⎛=+r r 1r)/(11r r 1DUR 2+=⨯+= ++++++++++++++++++++++++++=t t3322tt3322r)(1g)C(1r)(1g)C(1r)(1g)C(1r 1g)C(1r)(1g)tC(1r)(1g)3C(1r)(1g)2C(1r 1g)1C(1DURWe first simplify by dividing each term by [C(1 + g)]:The denominator is the present value of a growing perpetuity of $1 per year,which is equal to [1/(r - g)]. To simplify the numerator, we first denote thenumerator S and then divide S by (1 + r):Note that this new quantity [S/(1 + r)] is equal to the square of denominator in the duration formula above, that is:Therefore:Thus, for a perpetual bond paying C dollars per year:34. a. We make use of the one-year Treasury bill information in order to determine the one-year spot rate as follows:1r 1$100$93.46+= r 1 = 0.0700 = 7.00% The following position provides a cash payoff only in year two:++++++++++++++++++++++++=--t 1t 322t 1t 322r)(1g)(1r)(1g)(1r)(1g 1r 11r)(1g)t(1r)(1g)3(1r)(1g)2(1r 11DUR ++++++++++++=++-1t 2t 4232r)(1g)t(1r)(1g)3(1r)(1g)2(1r)(11 r)(1S 2t 1t 322r)(1g)(1r)(1g)(1r)(1g 1r 11r)(1S ⎪⎪⎭⎫ ⎝⎛++++++++++++=+- 22g)(r r 1S g r 1r)(1S -+=⇒⎪⎪⎭⎫ ⎝⎛-=+g r r 1g)](r /[11g)(r r 1DUR 2-+=-⨯-+=a long position in twenty-five two-year bonds and a short position in oneone-year Treasury bill. Cash flows for this position are:[(–25 ⨯ $94.92) + (1 ⨯ $93.46)] = –$2,279.54 today[(25 ⨯ $4) – (1 ⨯ $100)] = $0 in year 1(25 ⨯ $104) = $2,600 in year 2 We determine the two-year spot rate from this position as follows:2)2r (1$2,600$2,279.54+= r 2 = 0.0680 = 6.80%The forward rate f 2 is computed as follows:f 2 = [(1.0680)2/1.0700] – 1 = 0.0660 = 6.60%The following position provides a cash payoff only in year three:a long position in the three-year bond and a short position equal to (8/104)times a package consisting of a one-year Treasury bill and a two-yearbond. Cash flows for this position are:[(–1 ⨯ $103.64) + (8/104) ⨯ ($93.46 + $94.92)] = –$89.15 today[(1 ⨯ $8) – (8/104) ⨯ ($100 + $4)] = $0 in year 1[(1 ⨯ $8) – (8/104) ⨯ $104] = $0 in year 21 ⨯ $108 = $108 in year 3 We determine the three-year spot rate from this position as follows:3)3r (1$108$89.15+= r 3 = 0.0660 = 6.60%The forward rate f 3 is computed as follows:f 3 = [(1.0660)3/(1.0680)2] – 1 = 0.0620 = 6.20%b. We make use of the spot and forward rates to calculate the price of the4 percent coupon bond:The actual price of the bond ($950) is significantly greater than the pricededuced using the spot and forward rates embedded in the prices of theother bonds ($931.01). Hence, a profit opportunity exists. In order to takeadvantage of this opportunity, one should sell the 4 percent coupon bondshort and purchase the 8 percent coupon bond.$931.01(1.062)(1.066)(1.07)1040(1.066)(1.07)40(1.07)40P =++=35.a. Bond D allows us to calculate the four year spot rate by solving for the YTM using the 841.78 price of Bond D: 432r)(1$1000r)(1$0r)(1$0r 1$0$841.78+++++++=r 4 = 4.4%We can then set up the following three equations using the prices of bonds A, B, and C: Using bond A: 2r2)(1$1080r11$80$1075.82+++=Using bond B: 32r3)(1$1110r2)(1$110r11$110$1189.10+++++=Using bond C: 432r4)(1$1060r3)(1$60r2)(1$60r11$60$1058.76+++++++=We know r4 = 4.4% so we can substitute that into the last equation. Nowwe have three equations and three unknowns and can solve this withvariable substitution or linear programming to get r1 = 3.5%, r2 = 4%; r3 =4.2%; r4 = 4.4%b.We will want to invest in the underpriced C and borrow money at thecurrent spot market rates to construct an offsetting position. For example,we might borrow $60 at the 1 year rate of 3.5%, 60 at the 2 year rate of4%, 60 at the 3 year rate of 4.2% and 1060 at the 4 year rate of 4.4%. Ofcourse the PV amount we will receive on these loans is $1058.76. Now wepurchase the discounted bond C at $1040, use the proceeds of this bondto repay our loans as they come due. We can pocket the difference of$18.76, smile, and repeat.。
CHAPTER 4The Value of Common StocksAnswers to Problem Sets1. a. Trueb. True2. Investors who buy stocks may get their return from capital gains as well asdividends. But the future stock price always depends on subsequent dividends.There is no inconsistency.3. P0 = (5 + 110)/1.08 = $106.484. r = 5/40 = .125.5. P0 = 10/(.08 - .05) = $333.33.6. By year 5, earnings will grow to $18.23 per share. Forecasted price per share atyear 4 is 18.23/.08 = $227.91.7. 15/.08 + PVGO = 333.33; therefore PVGO = $145.83.8. Z’s forecasted dividends and prices grow as follows:Calculate the expected rates of return:From year 0 to 1: 08.33.333)33.333350(10=-+From year 1 to 2: 08.350)35050.367(50.10=-+From year 2 to 3:08.50.367)50.36788.385(03.11=-+Double expects 8% in each of the first 2 years. Triple expects 8% in each of the first 3 years.9. a. Falseb. True.10. PVGO = 0, and EPS 1 equals the average future earnings the firm could generate under no-growth policy.11. Free cash flow is the amount of cash thrown off by a business after allinvestments necessary for growth. In our simple examples, free cash flow equals operating cash flow minus capital expenditure. Free cash flow can be negative if investments are large.12. The value at the end of a forecast period. Horizon value can be estimated using the constant-growth DCF formula or by using price –earnings or market – book ratios for similar companies.13. If PVGO = 0 at the horizon date H , horizon value = earnings forecasted for H + 1 divided by r .14. Newspaper exercise, answers will vary15.Expected Future Values Present Values Horizon Period (H) Dividend (DIV t ) Price (P t ) Cumulative Dividends FuturePrice Total0 100.00 100.00100.001 10.00 105.00 8.70 91.30 100.002 10.50 110.25 16.64 83.36 100.003 11.03 115.76 23.88 76.12 100.004 11.58 121.55 30.50 69.50 100.00 10 15.51 162.89 59.74 40.26 100.00 20 25.27 265.33 83.79 16.21 100.00 50 109.21 1,146.74 98.94 1.06 100.00 100 1,252.39 13,150.13 99.99 0.01 100.00Assumptions 1. Dividends increase at 5% per year compounded. 2. Capitalization rate is 15%.16.$100.000.10$10r DIV P 1A ===$83.33.0400.10$5g r DIV P 1B =-=-=⎪⎭⎫⎝⎛⨯++++++=67665544332211C 1.1010.10DIV 1.10DIV 1.10DIV 1.10DIV 1.10DIV 1.10DIV 1.10DIV P $104.501.1010.1012.441.1012.441.1010.371.108.641.107.201.106.001.105.00P 6654321C =⎪⎭⎫⎝⎛⨯++++++=At a capitalization rate of 10%, Stock C is the most valuable. For a capitalization rate of 7%, the calculations are similar. The results are:P A = $142.86 P B = $166.67 P C = $156.48Therefore, Stock B is the most valuable.17. a. $21.90.027500.095 1.0275$1.35$1.35g r DIV DIV P 100=-⨯+=-+=b.First, compute the real discount rate as follows:(1 + r nominal ) = (1 + r real ) ⨯ (1 + inflation rate)1.095 = (1 + r real ) ⨯ 1.0275(1 + r real ) = (1.095/1.0275) – 1 = .0657 = 6.57%In real terms, g = 0. Therefore:$21.900.0657$1.35$1.35g r DIV DIV P 100=+=-+= 18.a. Plowback ratio = 1 – payout ratio = 1.0 – 0.5 = 0.5Dividend growth rate = g= Plowback ratio × ROE = 0.5 × 0.14 = 0.07 Next, compute EPS 0 as follows:ROE = EPS 0 /Book equity per share 0.14 = EPS 0 /$50 ⇒ EPS 0 = $7.00Therefore: DIV 0 = payout ratio × EPS 0 = 0.5 × $7.00 = $3.50 EPS and dividends for subsequent years are:YearEPS DIV0 $7.00$7.00 × 0.5 = $3.501 $7.00 × 1.07 = $7.4900 $7.4900 × 0.5 = $3.50 × 1.07 = $3.74502 $7.00 × 1.072 = $8.0143 $8.0143 × 0.5 = $3.50 × 1.072 = $4.0072 3 $7.00 × 1.073 = $8.5753 $8.5753 × 0.5 = $3.50 × 1.073 = $4.28774 $7.00 × 1.074 = $9.1756$9.1756 × 0.5 = $3.50 × 1.074 = $4.58785 $7.00 × 1.074 × 1.023 = $9.3866 $9.3866 × 0.5 = $3.50 × 1.074 × 1.023 = $4.6933EPS and dividends for year 5 and subsequent years grow at 2.3% per year, as indicated by the following calculation:Dividend growth rate = g = Plowback ratio × ROE = (1 – 0.08) × 0.115 = 0.023b.⎪⎭⎫ ⎝⎛⨯++++=45443322110 1.11510.115DIV 1.115DIV 1.115DIV 1.115DIV 1.115DIV P $45.651.1010.023-0.1154.6931.1154.5881.1154.2881.1154.0071.1153.74544321=⎪⎭⎫⎝⎛⨯++++=The last term in the above calculation is dependent on the payout ratioand the growth rate after year 4.19. a.11.75%0.11750.0752008.5g P DIV r 01==+=+=b.g = Plowback ratio × ROE = (1 − 0.5) × 0.12 = 0.06 = 6.0%The stated payout ratio and ROE are inconsistent with the security analysts’ forecasts. With g = 6.0% (and assuming r remains at 11.75%) then:pesos 147.830.06- 0.11758.5g r DIV P 10==-=20.The security analyst’s forecast is wrong because it a ssumes a perpetual constant growth rate of 15% when, in fact, growth will continue for two years at this rate and then there will be no further growth in EPS or dividends. The value of the company’s stock is the present value of the expected divi dend of $2.30 to be paid in 2020 plus the present value of the perpetuity of $2.65 beginning in 2021. Therefore, the actual expected rate of return is the solution for r in the following equation:r)r(1$2.65r 1$2.30$21.75+++=Solving algebraically (using the quadratic formula) or by trial and error, we find that: r = 0.1201= 12.01% 21.a.An Incorrect Application . Hotshot Semiconductor’s earnings anddividends have grown by 30 percent per year since the firm’s founding ten years ago. Current stock price is $100, and next year’s dividend is projected at $1.25. Thus:31.25%.31250.3001001.25g P DIV r 01==+=+=This is wrong because the formula assumes perpetual growth; it is notpossible for Hotshot to grow at 30 percent per year forever.A Correct Application. The formula might be correctly applied to the Old Faithful Railroad, which has been growing at a steady 5 percent rate for decades. Its EPS 1 = $10, DIV 1 = $5, and P 0 = $100. Thus:10.0%.100.0501005g P DIV r 01==+=+=Even here, you should be careful not to blindly project past growth into thefuture. If Old Faithful hauls coal, an energy crisis could turn it into a growth stock.b.An Incorrect Application. Hotshot has current earnings of $5.00 per share. Thus:5.0%.0501005P EPS r 01====This is too low to be realistic. The reason P 0 is so high relative to earningsis not that r is low, but rather that Hotshot is endowed with valuable growth opportunities. Suppose PVGO = $60:PVGO rEPS P 10+=60r5100+=Therefore, r = 12.5%A Correct Application. Unfortunately, Old Faithful has run out of valuable growth opportunities. Since PVGO = 0:PVGO r EPS P 10+=0r10100+=Therefore, r = 10.0%22.gr NPVr EPS price Share 1-+= Therefore:0.15)(r NP V r E P S Ραα1αα-+=α0.08)(r NP V r E P S Ρββββ1β-+=The statement in the question implies the following:⎪⎪⎭⎫⎝⎛-+->⎪⎪⎭⎫ ⎝⎛-+-0.15)(r NPV r EPS 0.15)(r NPV 0.08)(r NPV r EPS 0.08)(r NPV αααα1ααββββ1ββ Rearranging, we have:β1βββα1αααE P S r 0.08)(r NP V E P S r0.15)(r NP V ⨯-<⨯- a.NPV α < NPV β, everything else equal.b. (r α - 0.15) > (r β - 0.08), everything else equal.c.0.08)(r NP V 0.15)(r NP V ββαα-<-, everything else equal. d. β1βα1αE P S r E P S r <, everything else equal. 23.a.Growth-Tech’s stock price should be:23.81.08)0(0.12$1.24(1.12)1(1.12)$1.15(1.12)$0.60(1.12)$0.50P 332$=⎪⎪⎭⎫ ⎝⎛-⨯+++=b.The horizon value contributes:$22.07.08)0(0.12$1.24(1.12)1)P V(P 3H =-⨯=c.Without PVGO, P 3 would equal earnings for year 4 capitalized at 12 percent:$20.750.12$2.49= Therefore: PVGO = $31.00 – $20.75 = $10.25d.The PVGO of $10.25 is lost at year 3. Therefore, the current stock price of $23.81 will decrease by:$7.30(1.12)$10.253= The new stock price will be: $23.81 – $7.30 = $16.5124.a.Here we can apply the standard growing perpetuity formula with DIV 1 = $4, g = 0.04 and P 0 = $100:8.0%.080.040$100$4g P DIV r 01==+=+=The $4 dividend is 60 percent of earnings. Thus:EPS 1 = 4/0.6 = $6.67Also:PVGO rEPS P 10+=PVGO 0.08$6.67$100+=PVGO = $16.63b.DIV 1 will decrease to: 0.20 ⨯ 6.67 = $1.33However, by plowing back 80 percent of earnings, CSI will grow by 8 percent per year for five years. Thus: Year 1 2 3 4 5 6 7, 8 . . . DIV t 1.33 1.44 1.55 1.68 1.81 5.88 Continued growth at EPS t6.677.207.788.409.079.80 4 percentNote that DIV 6 increases sharply as the firm switches back to a 60 percent payout policy. Forecasted stock price in year 5 is:$147.0400.085.88g r DIV P 65=-=-=Therefore, CSI’s stock price will increase to:$106.211.081471.811.081.681.081.551.081.441.081.33P 54320=+++++=25.a.First, we use the following Excel spreadsheet to compute net income (or dividends) for 2009 through 2013:2009 2010 2011 2012 2013 Production (million barrels) 1.8000 1.6740 1.5568 1.4478 1.3465 Price of oil/barrel 65 60 55 50 52.5 Costs/barrel 25 25 25 25 25 Revenue 117,000,000 100,440,000 85,625,100 72,392,130 70,690,915 Expenses 45,000,000 41,850,000 38,920,500 36,196,065 33,662,340 Net Income (= Dividends) 72,000,000 58,590,000 46,704,600 36,196,065 37,028,574Next, we compute the present value of the dividends to be paid in 2010,2011 and 2012:=++=320 1.0936,196,0651.0946,704,6001.0958,590,000P $121,012,624 The present value of dividends to be paid in 2013 and subsequent yearscan be computed by recognizing that both revenues and expenses can be treated as growing perpetuities. Since production will decrease 7%per year while costs per barrel remain constant, the growth rate of expenses is: –7.0%To compute the growth rate of revenues, we use the fact that production decreases 7% per year while the price of oil increases 5% per year, so that the growth rate of revenues is:[1.05 × (1 – 0.07)] – 1 = –0.0235 = –2.35%Therefore, the present value (in 2012) of revenues beginning in 2013 is:45$622,827,4(-0.0235)-0.0970,690,915PV 2012==Similarly, the present value (in 2012) of expenses beginning in 2013 is:25$210,389,6(-0.07)-0.0933,662,340P V 2012==Subtracting these present values gives the present value (in 2012) of net income, and then discounting back three years to 2009, we find that the present value of dividends paid in 2013 and subsequent years is: $318,477,671The total value of the company is:$121,012,624 + $318,477,671 = $439,490,295Since there are 7,000,000 shares outstanding, the present value per share is:$439,490,295 / 7,000,000 = $62.78b. EPS 2009 = $72,000,000/7,000,000 = $10.29 EPS/P = $10.29/$62.78 = 0.16426.[Note: In this problem, the long-term growth rate, in year 9 and all later years, should be 8%.]The free cash flow for years 1 through 10 is computed in the following table:Year1 2 3 4 5 6 7 8 9 10 Asset value 10.00 12.00 14.40 17.28 20.74 23.12 25.66 28.36 30.63 33.08 Earnings 1.20 1.44 1.73 2.07 2.49 2.77 3.08 3.40 3.68 3.97 Investment 2.00 2.40 2.88 3.46 2.38 2.54 2.69 2.27 2.45 2.65 Free cash flow -0.80-0.96-1.15-1.380.100.230.381.131.23 1.32 Earnings growth from previous period20.0% 20.0% 20.0% 20.0% 20.0% 11.5% 11.0% 10.5%8.0%8.0%Computing the present value of the free cash flows, following the approach from Section 4.5, we find that the present value of the free cash flows occurring in years 1 through 7 is:654321 1.100.231.100.101.101.38-1.101.15-1.100.96-1.100.80-PV +++++==+71.100.38-$2.94 The present value of the growing perpetuity that begins in year 8 is:29.10.08)0(0.101.1343(1.10)1PV 7$=⎪⎪⎭⎫ ⎝⎛-⨯= Therefore, the present value of the business is:-$2.94 + $29.10 = $26.16 million27.From the equation given in the problem, it follows that:bROE )/(r b1ROE )(b r b)(1ROE BVP S P 0--=⨯--⨯= Consider three cases:ROE < r ⇒ (P 0/BVPS) < 1 ROE = r ⇒ (P 0/BVPS) = 1 ROE > r ⇒ (P 0/BVPS) > 1Thus, as ROE increases, the price-to-book ratio also increases, and, when ROE = r, price-to-book equals one. 28.Assume the portfolio value given, $100 million, is the value as of the end of the first year. Then, assuming constant growth, the value of the contract is given by the first payment (0.5 percent of portfolio value) divided by (r – g). Also:r = dividend yield + growth rateHence:r – growth rate = dividend yield = 0.05 = 5.0%Thus, the value of the contract, V, is:million $100.05million$1000.005V =⨯=For stocks with a 4 percent yield:r – growth rate = dividend yield = 0.04 = 4.0%Thus, the value of the contract, V, is:Chapter 04 - The Value of Common Stocks 4-11 million $12.50.04million $1000.005V =⨯=29. If existing stockholders buy newly issued shares to cover the $3.6 millionfinancing requirement, then the value of Concatco equals the discounted value of the cash flows (as computed in Section 4.5): $18.8 million.Since the existing stockholders own 1 million shares, the value pershare is $18.80.Now suppose instead that the $3.6 million comes from new investors, who buy shares each year at a fair price. Since the new investors buy shares at a fair price, the value of the existing stockholders’ shares must remain at $18.8 million. Since existing stockholders expect to earn 10% on their investment, the expected value of their shares in year 6 is:$18.8 million × (1.10) 6 = $33.39 millionThe total value of the firm in year 6 is:$1.59 million / (0.10 – 0.06) = $39.75 millionCompensation to new stockholders in year 6 is:$39.75 million – $33.39 million = $6.36 millionSince existing stockholders own 1 million shares, then in year 6, newstockholders will own:($6.36 million / $33.39 million) × 1,000,000 = 190,300 sharesShare price in year 6 equals:$39.75 million / 1.1903 million = $33.39。
CHAPTER 1Goals and Governance of the FirmAnswers to Problem Sets1. a. realb. executive airplanesc. brand namesd. financiale. bondsf. investmentg. capital budgetingh. financing2. c, d, e, and g are real assets. Others are financial.3. a. Financial assets, such as stocks or bank loans, are claims held byinvestors. Corporations sell financial assets to raise the cash to invest inreal assets such as plant and equipment. Some real assets are intangible.b. Capital budgeting means investment in real assets. Financing meansraising the cash for this investment.c. The shares of public corporations are traded on stock exchanges and canbe purchased by a wide range of investors. The shares of closely heldcorporations are not traded and are not generally available to investors.d. Unlimited liability: investors are responsible for all the firm’s debts. A soleproprietor has unlimited liability. Investors in corporations have limitedliability. They can lose their investment, but no more.e. A corporation is a separate legal “person” with unlimited life. Its ownershold shares in the business. A partnership is a limited-life agreement toestablish and run a business.4. c, d.5. b, c.6. Separation of ownership and management typically leads to agency problems,where managers prefer to consume private perks or make other decisions fortheir private benefit -- rather than maximize shareholder wealth.7. a. Assuming that the encabulator market is risky, an 8% expected return onthe F&H encabulator investments may be inferior to a 4% return on U.S.government securities.b. Unless their financial assets are as safe as U.S. government securities,their cost of capital would be higher. The CFO could consider what theexpected return is on assets with similar risk.8. Shareholders will only vote for (a) maximize shareholder wealth. Shareholderscan modify their pattern of consumption through borrowing and lending, matchrisk preferences, and hopefully balance their own checkbooks (or hire a qualified professional to help them with these tasks).9. If the investment increases the firm’s wealth, it will increase the value of the firm’sshares. Ms. Espinoza could then sell some or all of these more valuable shares in order to provide for her retirement income.10. As the Putnam example illustrates, the firm’s value typically falls by significantlymore than the amount of any fines and settlements. The firm’s reput ation suffers in a financial scandal, and this can have a much larger effect than the fines levied.Investors may also wonder whether all of the misdeeds have been contained. 11. Managers would act in shareholders’ interests because they have a legal dut y toact in their interests. Managers may also receive compensation, either bonusesor stock and option payouts whose value is tied (roughly) to firm performance.Managers may fear personal reputational damage that would result from notacting in shareho lders’ interests. And m anagers can be fired by the board ofdirectors, which in turn is elected by shareholders. If managers still fail to act inshareholders’ interests, shareholders may sell their shares, lowering the stockprice, and potentially creating the possibility of a takeover, which can again lead tochanges in the board of directors and senior management.12. Managers that are insulated from takeovers may be more prone to agencyproblems and therefore more likely to act in their own interests rather than in shareholders’. If a firm instituted a new takeover defense, we might expect to see the value of its shares decline as agency problems increase and lessshareholder value maximization occurs. The counterargument is that defensive measures allow managers to negotiate for a higher purchase price in the face of a takeover bid – to the benefit of shareholder value.Appendix Questions :1. Both would still invest in their friend’s business. A invests and receives $121,000for his investment at the end of the year (which is greater than the $120,000 it would receive from lending at 20%). G also invests, but borrows against the $121,000 payment, and thus receives $100,833 today.2. a. He could consume up to $200,000 now (foregoing all future consumption) orup to $216,000 next year (200,000*1.08, foregoing all consumption this year). To choose the same consumption (C) in both years, C = (200,000 – C) x 1.08 or C = $103,846.203,704200,000220,000216,000Dollars Next YearDollars Nowb. He should invest all of his wealth to earn $220,000 next year. If he consumesall this year, he can now have a total of $203,703.7 (200,000 x 1.10/1.08) this year or $220,000 next year. If he consumes C this year, the amount available for next year’s consumption is (203,703.7 – C) x 1.08. To get equal consumption in both years, set the amount consumed today equal to the amount next year:C = (203,703.7 – C) x 1.08C = $105,769.2。
CHAPTER 9Risk and the Cost of CapitalAnswers to Problem Sets1. Overestimate2. Company cost of capital = 10 x .4 + (10 + .5 x 8) x .6 = 12.4%After-tax WACC = (1 - .35) x 10 x .4 + (10 + .5 x 8) x .6 = 11.0%3. .297, or 29.7% of variation was due to market movements; .703 or 70.3% of thevariation was diversifiable. Diversifiable risk shows up in the scatter about thefitted line. The standard error of the estimated beta was unusually high at .436.If you said that the true beta was 2 x .436 = .872 either side of your estimate, you would have a 95% chance of being right.4. a. The expected return on debt. If the debt has very low default risk, this isclose to its yield to maturity.b. The expected return on equity.c. A weighted average of the cost of equity and the after-tax cost of debt,where the weights are the relative market values of the firm’s debt andequity.d. The change in the return of the stock for each additional 1% change in themarket return.e. The change in the return on a portfolio of all the firm’s securities (debt andequity) for each additional 1% change in the market return.f. A company specializing in one activity that is similar to that of a division ofa more diversified company.g. A certain cash flow occurring at time t with the same present value as anuncertain cash flow at time t.5. Beta of assets = .5 × .15 + .5 × 1.25 = .76. A diversifiable risk has no affect on the risk of a well-diversified portfolio andtherefore n o affect on the project’s beta. If a risk is diversifiable it does notchange the cost of capital for the project. However, any possibility of bad outcomesdoes need to be factored in when calculating expected cash flows.7. Suppose that the expected cash flow in Year 1 is 100, but the project proposerprovides an estimate of 100 × 115/108 = 106.5. Discounting this figure at 15% gives the same result as discounting the true expected cash flow at 8%.Adjusting the discount rate, therefore, works for the first cash flow but it does not do so for later cash flows (e.g., discounting a 2-year cash flow of 106.5 by 15% is not equivalent to discounting a 2-year flow of 100 by 8%).8. a. A (higher fixed cost)b. C (more cyclical revenues).9. a. Falseb. Falsec. True10. a. PV =[]2)(1121)(1110f m f f m f r r r r r r ++++-++ββ = 200$10.112110.11102=+b.CEQ 1/1.05 = 110/1.10, CEQ 1 = $105; CEQ 2 /1.052 = 121/1.102, CEQ 2 = $110.25.c.Ratio 1 = 105/110 = .95; Ratio 2 = 110.25/121 = .91.11.a.r equity = r f + β ⨯ (r m – r f ) = 0.04 + (1.5 ⨯ 0.06) = 0.13 = 13% b. ⎪⎭⎫ ⎝⎛⨯+⎪⎭⎫ ⎝⎛⨯=+=0.13$10million $6million 0.04$10million $4million r V E r V D r equity debt assets r assets = 0.094 = 9.4%c. The cost of capital depends on the risk of the project being evaluated.If the risk of the project is similar to the risk of the other assets of thecompany, then the appropriate rate of return is the company cost of capital.Here, the appropriate discount rate is 9.4%.d. r equity = r f + β ⨯ (r m – r f ) = 0.04 + (1.2 ⨯ 0.06) = 0.112 = 11.2%⎪⎭⎫ ⎝⎛⨯+⎪⎭⎫ ⎝⎛⨯=+=0.112$10million $6million 0.04$10million $4million r V E r V D r equity debt assets r assets = 0.0832 = 8.32%12. a.0.836n $439millio n $299millio 1.20n $439millio $40million 0.20n $439millio n $100millio 0V C βV P βV D ββcommon preferred debt assets =⎪⎭⎫ ⎝⎛⨯⨯⎪⎭⎫ ⎝⎛⨯+⎪⎭⎫ ⎝⎛⨯=⎪⎭⎫ ⎝⎛⨯+⎪⎭⎫ ⎝⎛⨯+⎪⎭⎫ ⎝⎛⨯=b.r = r f + β ⨯ (r m – r f ) = 0.05 + (0.836⨯ 0.06) = 0.10016 = 10.016%13. a. The R 2 value for Toronto Dominion was 0.25, which means that 25% oftotal risk comes from movements in the market (i.e., market risk).Therefore, 75% of total risk is unique risk.The R 2 value for Canadian Pacific was 0.30, which means that 30% oftotal risk comes from movements in the market (i.e., market risk).Therefore, 70% of total risk is unique risk.b. The variance of Toronto Dominion is: (25)2 = 625Market risk for Toronto Dominion: 0.25 × 625 = 156.25Unique risk for Alcan: 0.75 × 625 = 468.75c.The t-statistic for βCP is: 1.04/0.20 = 5.20This is significant at the 1% level, so that the confidence level is 99%.d.r TD = r f + βTD ⨯ (r m – r f ) = 0.05 + [0.82 ⨯ (0.12 – 0.05)] = 0.1074 = 10.74%e.r TD = r f + βTD ⨯ (r m – r f ) = 0.05 + [0.82 ⨯ (0 – 0.05)] = 0.0090 = 0.90%14. The total market value of outstanding debt is $300,000. The cost of debtcapital is 8 percent. For the common stock, the outstanding market value is: $50 ⨯ 10,000 = $500,000. The cost of equity capital is 15 percent. Thus,Lorelei’s weighted -average cost of capital is:0.15500,000300,000500,0000.08500,000300,000300,000r assets ⨯⎪⎪⎭⎫ ⎝⎛++⨯⎪⎪⎭⎫ ⎝⎛+= r assets = 0.124 = 12.4%15.a. r BN = r f + βBN ⨯ (r m – r f ) = 0.05 + (01.01 ⨯ 0.07) = 0.1207 = 12.07% r IND = r f + βIND ⨯ (r m – r f ) = 0.05 + (01.24 ⨯ 0.07) = 0.1368 = 13.68%b. No, we can not be confident that Burlington’s true beta is not the industryaverage. The difference between βBN and βIND (0.23) is less than twotimes the standard error (2 ⨯ 0.19 = 0.38), so we cannot reject thehypothesis that βBN = βIND with 95% confidence.c.Burlington’s beta might be different from the industry beta for a variety ofreasons. For example, Burlington’s business might be more cyclical thanis the case for the typical firm in the industry. Or Burlington might havemore fixed operating costs, so that operating leverage is higher. Anotherpossibility is that Burlington has more debt than is typical for the industryso that it has higher financial leverage.16. Financial analysts or investors working with portfolios of firms may use industrybetas. To calculate an industry beta we would construct a series of industryportfolio investments and evaluate how the returns generated by this portfolio relate to historical market movements.17. We should use the market value of the stock, not the book value shown on the annual report. This gives us an equity value of 500,000 shares times $18 = $9 million. So Binomial Tree Farm has a debt/value ratio of 5/14 = 0.36 and anequity /value ratio of 9/14 = 0.64.18. a. If you agree to the fixed price contract, operating leverage increases.Changes in revenue result in greater than proportionate changes in profit.If all costs are variable, then changes in revenue result in proportionatechanges in profit. Business risk, measured by βassets , also increases as aresult of the fixed price contract. If fixed costs equal zero, then:βassets = βrevenue . However, as PV(fixed cost) increases, βassets increases.b.With the fixed price contract:PV(assets) = PV(revenue) – PV(fixed cost) – PV(variable cost)9(1.09)(0.09)$10million )6%,10years factor annuity $10million 0.09$20million PV(assets)⨯-⨯-=( PV(assets) = $97,462,710 Without the fixed price contract: PV(assets) = PV(revenue) – PV(variable cost) 0.09$10million 0.09$20million PV(as s ets )-= PV(assets) = $111,111,11119. a.The threat of a coup d’état means that the expected cash flow is less than $250,000. The threat could also increase the discount rate, but only if it increases market risk.b.The expected cash flow is: (0.25 ⨯ 0) + (0.75 ⨯ 250,000) = $187,500 Assuming that the cash flow is about as risky as the rest of the company’s business: PV = $187,500/1.12 = $167,41120. a.Expected daily production = (0.2 ⨯ 0) + 0.8 ⨯ [(0.4 x 1,000) + (0.6 x 5,000)] = 2,720 barrels Expected annual cash revenues = 2,720 x 365 x $15 = $49,640,000b.The possibility of a dry hole is a diversifiable risk and should not affect the discount rate. This possibility should affect forecasted cash flows, however. See Part (a).21. a.Using the Security Market Line, we find the cost of capital: r = 0.07 + [1.5 ⨯ (0.16 – 0.07)] = 0.205 = 20.5% Therefore:b.CEQ 1 = 40⨯(1.07/1.205) = 35.52CEQ 2 = 60⨯(1.07/1.205)2 = 47.31 CEQ 3 = 50⨯(1.07/1.205)3 = 35.01103.091.205501.205601.20540PV 32=++=c.a 1 = 35.52/40 = 0.8880a 2 = 47.31/60 = 0.7885a 3 = 35.01/50 = 0.7002d.Using a constant risk-adjusted discount rate is equivalent to assuming that a t decreases at a constant compounded rate.22.At t = 2, there are two possible values for the project’s NPV:Therefore, at t = 0:23. It is correct that, for a high beta project, you should discount all cash flows at ahigh rate. Thus, the higher the risk of the cash outflows, the less you should worry about them because, the higher the discount rate, the closer the present value of these cash flows is to zero. This result does make sense. It is better to have a series of payments that are high when the market is booming and low when it is slumping (i.e., a high beta) than the reverse.The beta of an investment is independent of the sign of the cash flows. If an investment has a high beta for anyone paying out the cash flows, it must have a high beta for anyone receiving them. If the sign of the cash flows affected the discount rate, each asset would have one value for the buyer and one for the seller, which is clearly an impossible situation.24. a. Since the risk of a dry hole is unlikely to be market-related, we can usethe same discount rate as for producing wells. Thus, using the SecurityMarket Line:r nominal = 0.06 + (0.9 ⨯ 0.08) = 0.132 = 13.2%We know that:(1 + r nominal ) = (1 + r real ) ⨯ (1 + r inflation ) Therefore:8.85% 0.0885 1 1.041.132r real ==-=0)successful not is test if (NP V 2=$833,3330.12700,0005,000,000)s ucces s ful is tes t if (NPV 2=+-=$244,8981.40833,333).60(00)(0.40500,000NPV 20-=⨯+⨯+-=b.c. Expected income from Well 1: [(0.2 ⨯ 0) + (0.8 ⨯ 3 million)] = $2.4 millionExpected income from Well 2: [(0.2 ⨯ 0) + (0.8 ⨯ 2 million)] = $1.6 million Discounting at 8.85 percent gives:d. For Well 1, one can certainly find a discount rate (and hence a “fudgefactor”) that, when applied to cash flows of $3 million per year for 10 years, will yield the correct NPV of $5,504,600. Similarly, for Well 2, one can find the appropriate discount rate. However, these two “fudge factors” will be different. Specifically, Well 2 will have a smaller “fudge factor” because its cash flows are more distant. With more distant cash flows, a smalleraddition to the discount rate has a larger impact on present value.(3.1914)](3million)million 101.28853million million 10NP V 101t t 1⨯+-=+-=∑=[$425,800NP V 1-=(3.3888)](2million)[10million 1.28852million million 10NP V 151t t 2⨯+-=+-=∑=$3,222,300NP V 2-=(6.4602)]n)(2.4millio [million 101.08852.4million million 10NP V 101t t 1⨯+-=+-=∑=$5,504,600NP V 1=(8.1326)]n)(1.6millio [10million 1.08851.6million million 10NP V 151t t 2⨯+-=+-=∑=$3,012,100NP V 2=。
CHAPTER 7Introduction to Risk and ReturnAnswers to Problem Sets1. Expected payoff is $100 and expected return is zero. Variance is 20,000 (%squared) and standard deviation is 141%.2. a. Standard deviation = 19.3%b. Average real return = -2.2%3. Ms. Sauros had a slightly higher average return (14.6% vs. 14.4% for the market).However, the fund also had a higher standard deviation (13.6% vs. 9.4% for themarket).4. a. Falseb. Truec. Falsed. Falsee. Falsef. Trueg. Trueh. False5. d6.7. a. 26% b. zero c. .75d. L ess than 1.0 (the portfolio’s risk is the same as the market, but some ofthis risk is unique risk).8. 1.3 (Diversification does not affect market risk.) 9. A, 1.0; B, 2.0; C, 1.5; D, 0; E, 21.0 10.Recall from Chapter 4 that:(1 + r nominal ) = (1 + r real ) ⨯ (1 + inflation rate)Therefore:r real = [(1 + r nominal )/(1 + inflation rate)] – 1a.The real return on the stock market in each year was: 1929: -14.7% 1930: -23.7% 1931: -38.0% 1932: 0.5% 1933:56.5%b. From the results for Part (a), the average real return was: -3.89%c.The risk premium for each year was: 1929: -19.3% 1930: -30.7% 1931: -45.0% 1932: -10.9% 1933:57.0%d. From the results for Part (c), the average risk premium was: –9.78%e.The standard deviation (σ) of the risk premium is calculated as follows:[22220.0978))0.450(0978))0 0.307(0.0978))0.193(151σ---+---+---⨯⎪⎪⎭⎫ ⎝⎛-=(.((0.1557390.0978))(0.5700.0978))0.109(22=--+---+]((39.46%0.3946370.155739σ===11.a.A long-term United States government bond is always absolutely safe in terms of the dollars received. However, the price of the bond fluctuates as interest rates change and the rate at which coupon payments received can be invested also changes as interest rates change. And, of course, the payments are all in nominal dollars, so inflation risk must also be considered.b.It is true that stocks offer higher long-run rates of return than do bonds, but it is also true that stocks have a higher standard deviation of return. So, which investment is preferable depends on the amount of risk one iswilling to tolerate. This is a complicated issue and depends on numerous factors, one of which is the investment time horizon. If the investor has a short time horizon, then stocks are generally not preferred.c. Unfortunately, 10 years is not generally considered a sufficient amount of time for estimating average rates of return. Thus, using a 10-year average is likely to be misleading.12.The risk to Hippique shareholders depends on the market risk, or beta, of the investment in the black stallion. The information given in the problem suggests that the horse has very high unique risk, but we have no information regarding the horse’s market risk. So, the best estimate is that t his horse has a market risk about equal to that of other racehorses, and thus this investment is not a particularly risky one for Hippique shareholders.13.In the context of a well-diversified portfolio, the only risk characteristic of a single security that matters is the security’s contribution to the overall portfolio risk. This contribution is measured by beta. Lonesome Gulch is the safer investment for a diversified investor because its beta (+0.10) is lower than the beta ofAmalgamated Copper (+0.66). For a diversified investor, the standard deviations are irrelevant. 14.x I = 0.60 σI = 0.10 x J = 0.40 σJ = 0.20a.1ρIJ =)]σσρx 2(x σx σx [σJ I IJ J I 2J 2J 2I 2I 2p ++=b.c. 15.a.Refer to Figure 7.13 in the text. With 100 securities, the box is 100 by 100. The variance terms are the diagonal terms, and thus there are 100variance terms. The rest are the covariance terms. Because the box has (100 times 100) terms altogether, the number of covariance terms is:1002 – 100 = 9,900Half of these terms (i.e., 4,950) are different.b.Once again, it is easiest to think of this in terms of Figure 7.13. With 50 stocks, all with the same standard deviation (0.30), the same weight in the portfolio (0.02), and all pairs having the same correlation coefficient (0.40), the portfolio variance is:σ2 = 50(0.02)2(0.30)2 + [(50)2 – 50](0.02)2(0.40)(0.30)2 =0.03708 σ = 0.193 = 19.3%c.For a fully diversified portfolio, portfolio variance equals the average covariance:σ2 = (0.30)(0.30)(0.40) = 0.036 σ = 0.190 = 19.0%16.a.Refer to Figure 7.13 in the text. For each different portfolio, the relative weight of each share is [one divided by the number of shares (n) in the portfolio], the standard deviation of each share is 0.40, and the correlation between pairs is 0.30. Thus, for each portfolio, the diagonal terms are the same, and the off-diagonal terms are the same. There are n diagonal terms and (n 2 – n) off-diagonal terms. In general, we have:Variance = n(1/n)2(0.4)2 + (n 2 – n)(1/n)2(0.3)(0.4)(0.4)0.0196]0)(0.20)40)(1)(0.12(0.60)(0.(0.20)0.40)((0.10)(0.60)[2222=++=0ρij=0.0148])0.10)(0.2040)(0.50)(2(0.60)(0.(0.20)0.40)((0.10)(0.60)[2222=++=0.50ρIJ =0.0100]0)(0.20)40)(0)(0.12(0.60)(0.(0.20)0.40)((0.10)(0.60)[2222=++=)]σσρx 2(x σx σx [σJ I IJ J I 2J 2J 2I 2I 2p ++=)]σσρx 2(x σx σx [σJ I IJ J I 2J 2J 2I 2I 2p ++=For one share: Variance = 1(1)2(0.4)2 + 0 = 0.160000For two shares:Variance = 2(0.5)2(0.4)2 + 2(0.5)2(0.3)(0.4)(0.4) = 0.104000 The results are summarized in the second and third columns of thetable below.b. (Graphs are on the next page.) The underlying market risk that can not bediversified away is the second term in the formula for variance above:Underlying market risk = (n2 - n)(1/n)2(0.3)(0.4)(0.4)As n increases, [(n2 - n)(1/n)2] = [(n-1)/n] becomes close to 1, so that theunderlying market risk is: [(0.3)(0.4)(0.4)] = 0.048c. This is the same as Part (a), except that all of the off-diagonal terms arenow equal to zero. The results are summarized in the fourth and fifthcolumns of the table below.(Part a) (Part a) (Part c) (Part c)No. of Standard StandardShares Variance Deviation Variance Deviation1 .160000 .400 .160000 .4002 .104000 .322 .080000 .2833 .085333 .292 .053333 .2314 .076000 .276 .040000 .2005 .070400 .265 .032000 .1796 .066667 .258 .026667 .1637 .064000 .253 .022857 .1518 .062000 .249 .020000 .1419 .060444 .246 .017778 .13310 .059200 .243 .016000 .126Graphs for Part (a):Graphs for Part (c):17. The table below uses the format of Figure 7.13 in the text in order to calculate theportfolio variance. The portfolio variance is the sum of all the entries in the matrix. Portfolio variance equals: 0.03664355518.“Safest” means lowest risk; in a portfolio context, thi s means lowest variance of return. Half of the portfolio is invested in Canadian Pacific stock, and half of the portfolio must be invested in one of the other securities listed. Thus, we calculate the portfolio variance for seven different portfolios to see which is the lowest. The safest attainable portfolio is comprised of Canadian Pacific and Nestle.19.⨯ change inchange is +1.25%.b. “Safest” implies lowest risk. Assuming the well-diversified portfolio isinvested in typical securities, the portfolio beta is approximately one. Thelargest reduction in beta is achieved by investing the $20,000 in a stockwith a negative beta. Answer (iii) is correct.20. Expected portfolio return = x A E[R A ] + x B E[R B ] = 12% = 0.12Let x B = (1 – x A )x A (0.10) + (1 – x A) (0.15) = 0.12 ⇒ x A = 0.60 and x B = 1 – x A = 0.40Portfolio variance = x A2σA2 + x B2σB2 +2(x A x B ρAB σA σB )= (0.60 2 ) (20 2 ) + (0.40 2 ) (40 2 ) + 2(0.60)(0.40)(0.50)(20)(40) = 592 Standard deviation = 24.33%592σ==21. a. In general:Portfolio variance = σP2 = x12σ12 + x22σ22 + 2x1x2ρ12σ1σ2Thus:σP2 = (0.52)(30.92)+(0.52)(17.22)+2(0.5)(0.5)(0.31)(30.9)(17.2)σP2 = 395.942Standard deviation = σP = 19.88%b. We can think of this in terms of Figure 7.13 in the text, with threesecurities. One of these securities, T-bills, has zero risk and, hence, zerostandard deviation. Thus:σP2 = (1/3)2(30.92)+(1/3)2(17.22)+2(1/3)(1/3)(0.31)(30.9)(17.2)σP2 = 175.574Standard deviation = σP = 13.25%Another way to think of this portfolio is that it is comprised of one-thirdT-Bills and two-thirds a portfolio which is half Dell and half McDonalds.Because the risk of T-bills is zero, the portfolio standard deviation is two-thirds of the standard deviation computed in Part (a) above:Standard deviation = (2/3)(19.88) = 13.25%c. With 50% margin, the investor invests twice as much money in theportfolio as he had to begin with. Thus, the risk is twice that found in Part(a) when the investor is investing only his own money:Standard deviation = 2 ⨯ 19.88% = 39.76%d. With 100 stocks, the portfolio is well diversified, and hence the portfoliostandard deviation depends almost entirely on the average covariance ofthe securities in the portfolio (measured by beta) and on the standarddeviation of the market portfolio. Thus, for a portfolio made up of 100stocks, each with beta = 1.41, the portfolio standard deviation isapproximately: 1.41 ⨯ 15% = 21.15%For stocks like McDonalds, it is: 0.77 ⨯ 15% = 11.55%22. For a two-security portfolio, the formula for portfolio risk is:Portfolio variance = x12σ12 + x22σ22 + 2x1x2ρρ12σ1σ2If security one is Treasury bills and security two is the market portfolio, then σ1 is zero, σ2 is 20%. Therefore:Portfolio variance = x22σ22 = x22(0.20)2Standard deviation = 0.20x2Portfolio expected return = x1(0.06) + x2(0.06 + 0.85)Portfolio expected return = 0.06x1 + 0.145x2Portfolio X1X2ExpectedReturnStandardDeviation1 1.0 0.0 0.060 0.0002 0.8 0.2 0.077 0.0403 0.6 0.4 0.094 0.0804 0.4 0.6 0.111 0.1205 0.2 0.8 0.128 0.1606 0.0 1.0 0.145 0.20023. The matrix below displays the variance for each of the eight stocks along thediagonal and each of the covariances in the off-diagonal cells:The covariance of BP with the market portfolio (σBP, Market) is the mean of the eight respective covariances between BP and each of the eight stocks in the portfolio. (The covariance of BP with itself is the variance of BP.) Therefore, σBP, Market is equal to the average of the eight covariances in the first row or, equivalently, the average of the eight covariances in the first column. Beta for BP is equal to the covariance divided by the market variance, which we calculated at 0.03664 in problem 17. The covariances and betas are displayed in the table below:。
CHAPTER 17Does Debt Policy Matter?Answers to Problem Sets1.Note the market value of Copperhead is far in excess of its book value:Ms. Kraft owns .625% of the firm, which proposes to increase common stock to $17 million and cut short-term debt. Ms. Kraft can offset this by (a)borrowing .00625 X 1,000,000 = $6,250, and (b) buying that much more Copperhead stock. 2. a. 55.5.⨯+E r % = 12.5%; E r = 20%b. 12.5%c. E/P = 20%; P/E = 5d. $50e..5 X E β + .5 X 0 = 1.0; E β= 2.0.3.E xpected return on assets is r A = .08 X 30/80 + .16 X 50/80 = .13. The new return on equity will be r E = .13 + (20/60)(.13 - .08) = .147.4. a.b. ⎪⎭⎫⎝⎛⨯++⎪⎭⎫⎝⎛⨯+=E D A E D E E D D βββ.8 = (.25 x 0) + (.75 x βE)βE = 1.075. a. Trueb. True (as long as the return earned by the company is greater than theinterest payment, earnings per share increase, but the PyE falls to reflectthe higher risk).c. False (the cost of equity increases with the ratio D/E).d. False (the formula r E = r A + (D/E)(r A - r D) does not require r D to beconstant).e. False (debt amplifies variations in equity income).f. False (value increases only if clientele is not satisfied).6. a. r A = .15, r E = .175b. βA= .6 (unchanged), βD= .3, βE= .9.7. See Figure 17.3.8. Currently r A = r E = .14, or 14%. From proposition 2 the leverage causes r E toincrease to r E = r A + (r A– r D)(D/E) = .14 + (.14 - .095) X (45/55) = .1768, or17.68%After-tax WACC = .095 X (1 - .40) X .45 + .1768 X .55 = .1229, or 12.29%.9. a. The two firms have equal value; let V represent the total value of the firm.Rosencrantz could buy one percent of Company B’s equity and borrow anamount equal to:0.01 ⨯ (D A - D B) = 0.002VThis investment requires a net cash outlay of (0.007V) and provides a netcash return of:(0.01 ⨯ Profits) – (0.003 ⨯ r f⨯ V)where r f is the risk-free rate of interest on debt. Thus, the two investmentsare identical.b. Guildenstern could buy two percent of Company A’s equity and lend anamount equal to:0.02 ⨯ (D A - D B) = 0.004VThis investment requires a net cash outlay of (0.018V) and provides a netcash return of:(0.02 ⨯ Profits) – (0.002 ⨯ r f⨯ V)Thus the two investments are identical.c. The expected dollar return to Rosencrantz’ original investment in A is:(0.01 ⨯ C) – (0.003 ⨯ r f⨯ V A)where C is the expected profit (cash flow) generated by the firm’s assets.Since the firms are the same except for capital structure, C must also bethe expected cash flow for Firm B. The dollar return to Rosencrantz’alternative strategy is:(0.01 ⨯ C) – (0.003 ⨯ r f⨯ V B)Also, the cost of the original strategy is (0.007V A) while the cost of thealternative strategy is (0.007V B).If V A is less than V B, then the original strategy of investing in Company Awould provide a larger dollar return at the same time that it would cost lessthan the alternative. Thus, no rational investor would invest in Company Bif the value of Company A were less than that of Company B.10. When a firm issues debt, it shifts its cash flow into two streams. MM’sProposition I states that this does not affect firm value if the investor canreconstitute a firm’s cash flow stream by creati ng personal leverage or byundoing the effect of the firm’s leverage by investing in both debt and equity.It is similar with Carruther’s cows. If the cream and skim milk go into the samepail, the cows have no special value. (If an investor holds both the debt andequity, the firm does not add value by splitting the cash flows into the twostreams.) In the same vein, the cows have no special value if a dairy cancostlessly split up whole milk into cream and skim milk. (Firm borrowing does not add value if investors can borrow on their own account.) Carruther’s cows willhave extra value if consumers want cream and skim milk and if the dairy cannot split up whole milk, or if it is costly to do so.11. a. The market price of the stock is not affected by the announcement.b. Since the market price of the shares is $10, the company can buy back:$160 million/$10 = 16 million sharesc. After the change in capital structure, the market value of the firm isunchanged:Equity + Debt = (9 million ⨯ $10) + $160 million = $250 milliond. After the change in structure, the debt ratio is:Debt/(Debt + Equity) = $160 million/$250 million = 0.64e. No one gains or loses. (See the answer to the next question.)12. a. The market value of the firm’s equity incr eases by $30 million, the amountof the decrease in the market value of the firm’s existing debt. Therefore,the price of the stock increases to:($150 million + $30 million)/15 million shares = $12b. Since the market price of the shares is $12, the company can buy back:$60 million/$12 = 5 million sharesc. After the change in capital structure, the market value of the firm isunchanged:Equity + Debt = (10 million ⨯ $12) + $130 million = $250 milliond. After the change in structure, the debt ratio is:Debt/(Debt + Equity) = $130 million/$250 million = 0.52e. The investors in the existing debt lose $30 million while the shareholdersgain this $30 million. The value of each share increases by:$30 million/15 million shares = $213. The company cost of capital is:r A = (0.8 ⨯ 0.12) + (0.2⨯ 0.06) = 0.108 = 10.8%Under Proposition I, this is unaffected by capital structure changes. With thebonds remaining at the 6% default-risk free rate, we have:Debt-Equityr E r ARatio0.00 0.108 0.1080.10 0.113 0.1080.50 0.132 0.1081.00 0.156 0.1082.00 0.204 0.1083.00 0.252 0.108See figure on next page.14. This is not a valid objection. MM’s Proposition II explicitly allows for the rates ofreturn for both debt and equity to increase as the proportion of debt in the capital structure increases. The rate for debt increases because the debt-holders aretaking on more of the risk of the firm; the rate for common stock increasesbecause of increasing financial leverage. See Figure 17.2 and theaccompanying discussion.15. a. Under Proposition I, the firm’s cost of capital (r A) is not affected by thechoice of capital structure. The reason the quoted statement seems to betrue is that it does not account for the changing proportions of the firmfinanced by debt and equity. As the debt-equity ratio increases, it is truethat both the cost of equity and the cost of debt increase, but a smallerproportion of the firm is financed by equity. The overall effect is to leavethe firm’s cost of capital unchanged.b. Moderate borrowing does not significantly affect the probability of financialdistress, but it does increase the variability (and market risk) borne bystockholders. This additional risk must be offset by a higher averagereturn to stockholders.16. a. If the opportunity were the firm’s only asset, this would be a good deal.Stockholders would put up no money and, therefore, would havenothing to lose. However, rational lenders will not advance 100% ofthe asset’s value for an 8% promised return unless other assets areput up as collateral.Sometimes firms find it convenient to borrow all the cash required for aparticular investment. Such investments do not support all of theadditional debt; lenders are p rotected by the firm’s other assets too.In any case, if firm value is independent of leverage, then any asset’scontribution to firm value must be independent of how it is financed. Notealso that the statement ignores the effect on the stockholders of anincrease in financial leverage.b. This is not an important reason for conservative debt levels. So long asMM’s Proposition I holds, the company’s overall cost of capital isunchanged despite increasing interest rates paid as the firm borrows more.(However, the increasing interest rates may signal an increasingprobability of financial distress—and that can be important.)17. Examples of such securities are given in the text and include unbundled stockunits, preferred equity redemption cumulative stock and floating-rate notes. Note that, in order to succeed, such securities must both meet regulatory requirements and appeal to an unsatisfied clientele.18. a.As leverage is increased, the cost of equity capital rises. This is the same as saying that, as leverage is increased, the ratio of the income afterinterest (which is the cash flow stockholders are entitled to) to the value of equity increases. Thus, as leverage increases, the ratio of the market value of the equity to income after interest decreases.b.(i) Assume MM are correct. The market value of the firm is determinedby the income of the firm, not how it is divided among the firm’s security holders. Also, the firm’s income before interest isindependent of the firm’s financing. Th us, both the value of the firm and the value of the firm’s income before interest remain constant as leverage is increased. Hence, the ratio is a constant. (ii) Assume the traditionalists are correct. The firm’s income beforeinterest is independent of levera ge. As leverage increases, the firm’s cost of capital first decreases and then increases; as a result, themarket value of the firm first increases and then decreases. Thus, the ratio of the market value of the firm to firm income before interest first increases and then decreases, as leverage increases.19.We begin with r E and the capital asset pricing model:r E = r f + βE (r m – r f ) = 0.10 + 1.5 (0.18 – 0.10) = 0.22 = 22.0%Similarly for debt:r D = r f + βD (r m – r f )0.12 = 0.10 + βD (0.18 – 0.10) βD = 0.25Also, we know that:17.0%0.170.22)(0.50.12)(0.5E D r E D E r E D DrA==⨯+⨯⎪⎪⎭⎫⎝⎛⎪⎪⎭⎫⎝⎛=⨯++⨯+= To solve for βA , use the following:0.8751.5)(0.50.25)(0.5β E D E β E D D βE D A =⨯+⨯=⎪⎪⎭⎫ ⎝⎛⨯++⎪⎪⎭⎫ ⎝⎛⨯+=20.We know from Proposition I that the value of the firm will not change. Also, because the expected operating income is unaffected by changes in leverage, the firm’s overall cost of capital will not change. In other words, r A remains equal to 17% and βA remains equal to 0.875. However, risk and, hence, the expected return for equity and for debt, will change. We know that r D is 11%, so that, for debt:r D = r f + βD (r m – r f )0.11 = 0.10 + βD (0.18 – 0.10) βD = 0.125For equity:⎪⎪⎭⎫⎝⎛⎪⎪⎭⎫⎝⎛⨯++⨯+=ED rE D E r E D DrA0.17 = (0.3 ⨯ 0.11) + (0.7 ⨯ r E ) r E = 0.196 = 19.6%Also:r E = r f + βE (r m – r f )0.196 = 0.10 + βE (0.18 – 0.10) βE = 1.2021.[Note: In the following solution, we have assumed that $200 million of long-term bonds have been issued.] a.E = $55 ⨯ 10 million = $550 millionV = D + E = $200 million + $550 million = $750 million0.267n$750millio n $200millio V D == 0.733n$750millio n $550millio V E == After-tax WACC = VErV D )T (1r E CD+- 10.01%0.10010.733)(0.120.2670.35)(10.07==⨯+⨯-⨯=b.The after-tax WACC would increase to the extent of the loss of the tax deductibility of the interest on debt. Therefore, the after-tax WACC would equal the opportunity cost of capital, computed from the WACC formula without the tax-deductibility of interest:10.67%0.10670.733)(0.120.267)(0.07VE r V D rWACC E D==⨯+⨯=+= 22.We make use of the basic relationship:⎪⎭⎫ ⎝⎛⎪⎭⎫ ⎝⎛⨯+-= V E r V D )T (1r r E CDASince overall beta (βA ) is not affected by capital structure or taxes, then:r A = r f + βA (r m – r f ) = 0.06 + (1.5 × 0.08) = 0.18The following table shows the value of r E for various values of D/E (and thecorresponding values of D/V), derived from the above formula. The graph is on the next page.D/E D/V r A r D r E 0.00 0.00000 0.18 0.0600 0.1800 0.05 0.04762 0.18 0.0600 0.1871 0.10 0.09091 0.18 0.0600 0.1941 0.15 0.13043 0.18 0.0600 0.2012 0.20 0.16667 0.18 0.0600 0.2082 0.25 0.20000 0.18 0.0600 0.2153 0.30 0.23077 0.18 0.0610 0.2221 0.35 0.25926 0.18 0.0620 0.2289 0.40 0.28571 0.18 0.0630 0.2356 0.45 0.31034 0.18 0.0640 0.2423 0.50 0.33333 0.18 0.0650 0.2489 0.55 0.35484 0.18 0.0660 0.2554 0.60 0.37500 0.18 0.0670 0.2619 0.65 0.39394 0.18 0.0680 0.2683 0.70 0.41176 0.18 0.0690 0.2746 0.75 0.42857 0.18 0.0690 0.2814 0.80 0.44444 0.18 0.0700 0.2876 0.85 0.45946 0.18 0.0725 0.2929 0.90 0.47368 0.18 0.0750 0.2981 0.95 0.48718 0.18 0.0775 0.3031 1.000.500000.180.08000.308023. Assume the election is near so that we can safely ignore the time value ofmoney.Because one, and only one, of three events will occur, the guaranteed payofffrom holding all three tickets is $10. Thus, the three tickets, taken together, could never sell for less than $10. This is true whether they are bundled into onecomposite security or unbundled into three separate securities.However, unbundled they may sell for more than $10. This will occur if theseparate tickets fill a need for some currently unsatisfied clientele. If this isindeed the case, then Proposition I fails. The sum of the parts is worth more than the whole.24. Some shoppers may want only the chicken drumstick. They could buy a wholechicken, cut it up, and sell off the other parts in the supermarket parking lot. This is costly. It is far more efficient for the store to cut up the chicken and sell thepieces separately. But this also has some cost, hence the observation thatsupermarkets charge more for chickens after they have been cut.The same considerations affect financial products, but:a. The proportionate costs to companies of repackaging the cash flow streamare generally small.b. Investors can also repackage cash flows cheaply for themselves. In fact,specialist financial institutions can often do so more cheaply than thecompanies can do it themselves.25. Firms that are ab le to identify an ‘unsatisfied’ clientele and then design a financialservice or instrument that satisfies the demands of this clientele can, in violationof MM’s capital-structure irrelevance theory, enhance firm value. However, if this is done successfully by one financial innovator, others will follow, eventuallyrestoring the validity of the MM irrelevance theory.If the financial innovation can be patented, the creator of the innovation can restrict the use of the innovation by other financial managers and therebycontinue to use the innovation to create value. Consequently, MM’s capital-structure irrelevance theory would potentially be violated during the life of the patent.。
曼昆的第十版《经济学原理》英文原名为“Principles of Economics，10th Edition”，简称Economics 10th。
这本书是经济学领域的经典著作之一，系统地介绍了经济学的基本概念和分析方法。
它涵盖了微观经济学和宏观经济学两个方面的内容，包括供给与需求、市场结构、生产者行为、消费者行为、市场失灵、宏观经济政策、经济增长、通货膨胀等主题。
曼昆的写作风格通俗易懂，深入浅出地解释了经济学原理和应用，使得读者可以更好地理解和应用这些原理。
此外，这本书还提供了大量的案例和数据，帮助读者更好地理解经济学原理的实际应用。
曼昆的第十版《经济学原理》是一本非常优秀的经济学教材，适合经济学初学者和有一定经济学基础的读者阅读。
CHAPTER 2How to Calculate Present ValuesAnswers to Problem Sets1.If the discount factor is .507, then .507* = $12.125/139 = .8993. PV = 374/9 =4. PV = 432/ + 137/ + 797/ = 376 + 104 + 524 = $1,0035.FV = 100* = $6. NPV = -1,548 + 138/.09 = (cost today plus the present value of theperpetuity)7.PV = 4/(. = $408. a. PV = 1/.10 = $10b. Since the perpetuity will be worth $10 in year 7, and since thatis roughlydouble the present value, the approximate PV equals $5.PV = (1 / .10)/7 = 10/2= $5 (approximately)c. A perpetuity paying $1 starting now would be worth $10, whereas aperpetuity starting in year 8 would be worth roughly $5. Thedifference between these cash flows is therefore approximately $5.PV = 10 – 5= $5 (approximately)d. PV = C/(r-g) = 10,000/(. = $200,000.9. a. PV = 10,000/ = $7, (assuming the cost of the car does notappreciate over those five years).b. You need to set aside (12,000 × 6-year annuity factor) = 12,000× =$55,476.c. At the end of 6 years you would have × (60,476 - 55,476) = $7,934.10. a. FV = 1, = 1, = $1,.b.PV = x 8 = = $ millionc.PV = C (1/r – 1/re rt ) = 2,000(1/.12 – 1/.12e .12 x15) = $13,91211.a. FV = 10,000,000x4 = 12,624,770b. FV = 10,000,000x(1 + .06/12)(4x12) = 12,704,892c. FV = 10,000,000xe = 12,712,49212.a.PV = $100/ = $ b.PV = $100/ = $ c.PV = $100/ = $ d.PV = $100/ + $100/ + $100/ = $13. a.⇒=+=0.905r 11DF 11r 1 = = %b.0.819(1.105)1)r (11DF 2222==+=c.AF 2 = DF 1 + DF 2 = + =d. PV of an annuity = C [Annuity factor at r% for t years]Here:$ = $10 [AF 3]AF=3e. AF 3 = DF 1 + DF 2 + DF 3 = AF 2 + DF 3= + DF 3DF 3 =14. The present value of the 10-year stream of cash inflows is:6$886,739.6(1.14)0.1410.141 $170,000PV 10=⎥⎦⎤⎢⎣⎡⨯-⨯= Thus:NPV = –$800,000 + $886, = +$86,At the end of five years, the factory’s value will be the present value of the five remaining $170,000 cash flows:6$583,623.7(1.14)0.1410.141 $170,000PV 5=⎥⎦⎤⎢⎣⎡⨯-⨯= 15.$23,696.151.12$50,0001.12$68,0001.12$80,0001.12$92,0001.12$92,000 1.12$85,000 1.12$80,0001.12$75,0001.12$57,0001.12$50,000$380,000(1.12)C NPV 109876543210t t t =++++++++++-==∑=016. a. Let S t = salary in year t∑=-=301t t1t (1.08)(1.05)40,000PV 3$760,662.5(1.08).05)-(.08(1.05).05)-(.081 43030=⎥⎦⎤⎢⎣⎡⨯-⨯=000,0b.PV(salary) x = $38,Future value = $38, x 30 = $382,c.$38,980.30 (1.08)0.0810.081$C (1.08)0.0810.081 C $r)(1r 1r 1 C PV 2020t =⎥⎥⎦⎤⎢⎢⎣⎡⨯-=⎥⎦⎤⎢⎣⎡⨯-⨯=⎥⎦⎤⎢⎣⎡+⨯-⨯=382,714.30382,714.3017.PeriodPresent Value400, 1+100,000/ = + 89, 2+200,000/ = +159, 3+300,000/ = +213, Total = NPV = $62,18. We can break this down into several different cash flows, such that thesum of these separate cash flows is the total cash flow. Then, the sum of the present values of the separate cash flows is the present value of the entire project. (All dollar figures are in millions.)Cost of the ship is $8 million PV = $8 million Revenue is $5 million per year, operating expenses are $4 million.Thus, operating cash flow is $1 million per year for 15 years. million $8.559(1.08)0.0810.081 million 1$PV 15=⎥⎦⎤⎢⎣⎡⨯-⨯= Major refits cost $2 million each, and will occur at times t = 5and t = 10.PV = ($2 million)/ + ($2 million)/ = $ millionSale for scrap brings in revenue of $ million at t = 15.PV = $ million/ = $ millionAdding these present values gives the present value of the entire project:NPV = $8 million + $ million $ million + $ millionNPV = $ million19. a. PV = $100,000b. PV = $180,000/ = $102,c. PV = $11,400/ = $95,000d.4$107,354.2(1.12)0.1210.121 $19,000PV 10=⎥⎦⎤⎢⎣⎡⨯-⨯=e. PV = $6,500/ = $92,Prize (d) is the most valuable because it has the highest present value.20. Mr. Basset is buying a security worth $20,000 now. That is its presentvalue. The unknown is the annual payment. Using the present value of an annuity formula, we have:$2,653.90 (1.08)0.0810.081000,20$C (1.08)0.0810.081 C 000,20$r)(1r 1r 1 C PV 1212t =⎥⎥⎦⎤⎢⎢⎣⎡⨯-=⎥⎦⎤⎢⎣⎡⨯-⨯=⎥⎦⎤⎢⎣⎡+⨯-⨯=21. Assume the Zhangs will put aside the same amount each year. Oneapproach to solving this problem is to find the present value of the cost of the boat and then equate that to the present value of the money saved. From this equation, we can solve for the amount to be put aside each year.PV(boat) = $20,000/5 = $12,418PV(savings) = Annual savings ⎥⎦⎤⎢⎣⎡⨯-⨯5(1.10)0.1010.101 Because PV(savings) must equal PV(boat):Annual savings 418,12$ =⎥⎦⎤⎢⎣⎡⨯-⨯5(1.10)0.1010.101Annual savings $3,276(1.10)0.1010.101$12,4185=⎥⎦⎤⎢⎣⎡⨯-= Another approach is to use the future value of an annuity formula:$20,0005.10)(1 savings Annual =⎥⎥⎦⎤⎢⎢⎣⎡-+⨯10.1Annual savings = $ 3,27622. The fact that Kangaroo Autos is offering “free credit” tells us whatthe cash payments are; it does not change the fact that money has time value. A 10% annual rate of interest is equivalent to a monthly rate of %:r monthly = r annual /12 = 12 = = %The present value of the payments to Kangaroo Autos is:8$8,93(1.0083)0.008310.00831$300$1,00030=⎥⎦⎤⎢⎣⎡⨯-⨯+ A car from Turtle Motors costs $9,000 cash. Therefore, Kangaroo Autos offers the better deal, ., the lower present value of cost.23. The NPVs are:at 5% $25,011(1.05)$320,0001.05$100,000$170,000NPV 2=+--=⇒ at 10% $3,554(1.10)320,0001.10$100,000$170,000NPV 2=+--=⇒at 15% $14,991(1.15)320,0001.15$100,000$170,000NPV 2-=+--=⇒ The figure below shows that the project has zero NPV at about 11%.As a check, NPV at 11% is:$371(1.11)320,0001.11$100,000$170,000NPV 2-=+--=$1,428.570.07$100r C PV ===b.This is worth the PV of stream (a) plus the immediate payment of$100:PV = $100 + $1, = $1,c.The continuously compounded equivalent to a 7% annually compounded rate is approximately %, because:=Thus:$1,477.100.0677$100r C PV ===Note that the pattern of payments in part (b) is more valuable than the pattern of payments in part (c). It is preferable to receive cash flows at the start of every year than to spread thereceipt of cash evenly over the year; with the former pattern of payment, you receive the cash more quickly.25. a.PV = $1 billion/ = $ billion b. PV = $1 billion/ – = $ billionc.billion $9.818(1.08)0.0810.081billion $1PV 20=⎥⎦⎤⎢⎣⎡⨯-⨯= d.The continuously compounded equivalent to an 8% annually compounded rate is approximately % , because:= Thus:billion $10.203e 0.07710.0771billion $1PV )(0.077)(20=⎥⎦⎤⎢⎣⎡⨯-⨯= This result is greater than the answer in Part (c) because theendowment is now earning interest during the entire year.26. With annual compounding: FV = $10020 = $1,With continuous compounding: FV = $100e ×20) = $2,27. One way to approach this problem is to solve for the present value of:(1) $100 per year for 10 years, and (2) $100 per year in perpetuity, with the first cash flow at year 11.If this is a fair deal, these present values must be equal, and thus we can solve for the interest rate (r).The present value of $100 per year for 10 years is:⎥⎦⎤⎢⎣⎡+⨯-⨯=10r)(1(r)1r 1$100PV The present value, as of year 10, of $100 per year forever, with thefirst payment in year 11, is: PV 10 = $100/r At t = 0, the present value of PV 10 is:⎥⎦⎤⎢⎣⎡⨯⎥⎦⎤⎢⎣⎡+=r $100r)(11PV 10 Equating these two expressions for present value, we have:⎥⎦⎤⎢⎣⎡⨯⎥⎦⎤⎢⎣⎡+=⎥⎦⎤⎢⎣⎡+⨯-⨯r $100r)(11r)(1(r)1r 1$1001010 Using trial and error or algebraic solution, we find that r = %.28. Assume the amount invested is one dollar.Let A represent the investment at 12%, compounded annually. Let B represent the investment at %, compounded semiannually. Let C represent the investment at %, compounded continuously.After one year:FV A = $1 (1 + 1 = $ FV B = $1 (1 + 2 = $ FV C = $1e1)= $After five years:FV A = $1 (1 + 5 = $ FV B = $1 (1 + 10 = $ FV C = $1e5)= $After twenty years:FV A = $1 (1 + 20 = $ FV B = $1 (1 + 40 = $ FV C = $1e20)= $The preferred investment is C.29. Because the cash flows occur every six months, we first need tocalculate the equivalent semi-annual rate. Thus, = (1 + r/2)2 => r = semi-annually compounded APR. Therefore the rate for six months is 2 or %:081,846039251039250103925010001000001009$).(..,$,$PV =⎥⎦⎤⎢⎣⎡⨯-⨯+=30. a. Each installment is: $9,420,713/19 = $495,827$4,761,724(1.08)0.0810.081$495,827PV 19=⎥⎦⎤⎢⎣⎡⨯-⨯=b.If ERC is willing to pay $ million, then:⎥⎦⎤⎢⎣⎡+⨯-⨯=19r)(1r 1r 1$495,827$4,200,000 Using Excel or a financial calculator, we find that r = %.31. a. 3$402,264.7(1.08)0.0810.081$70,000PV 8=⎥⎦⎤⎢⎣⎡⨯-⨯=b.Year Beginning-of-Year Balance Year-end Interest on Balance Total Year-end Payment Amortization of Loan End-of-Year Balance1 402, 32, 70, 37, 364,2 364, 29, 70, 40, 323,3 323, 25, 70, 44, 279,4 279, 22, 70, 47, 231,5 231, 18, 70, 51, 180,6 180, 14, 70, 55, 124,7 124, 9, 70, 60, 64,864,5,70,64,32. This is an annuity problem with the present value of the annuityequal to $2 million (as of your retirement date), and theinterest rate equal to 8% with 15 time periods. Thus, your annual level of expenditure (C) is determined as follows:$233,659 (1.08)0.0810.081$2,000,000C (1.08)0.0810.081 C $2,000,000r)(1r 1r 1 C PV 1515t =⎥⎥⎦⎤⎢⎢⎣⎡⨯-=⎥⎦⎤⎢⎣⎡⨯-⨯=⎥⎦⎤⎢⎣⎡+⨯-⨯=With an inflation rate of 4% per year, we will still accumulate $2million as of our retirement date. However, because we want to spend a constant amount per year in real terms (R, constant for all t), thenominal amount (Ct ) must increase each year. For each year t: R = Ct/(1 + inflation rate)tTherefore:PV [all Ct] = PV [all R (1 + inflation rate)t] = $2,000,000$2,000,0000.08)(1.04)0(1....08)0(10.04)(10.08)(1.04)0(1R 15152211=⎥⎦⎤⎢⎣⎡+++++++++⨯ R [ + + . . . + ] = $2,000,000 R= $2,000,000R = $177,952Alternatively, consider that the real rate is03846.1=-++0.04)(1.08)0(1. Then,redoing the steps above using the real rate gives a real cash flow equal to:$177,952=⎥⎥⎦⎤⎢⎢⎣⎡⨯-= (1.03846)0.0384610.038461$2,000,000C 15Thus C 1 = ($177,952 = $185,070, C 2 = $192,473, etc.33. a. 9$430,925.8(1.055)0.05510.0551$50,000PV 12=⎥⎦⎤⎢⎣⎡⨯-⨯=b.The annually compounded rate is %, so the semiannual rate is:(1/2)– 1 = = %Since the payments now arrive six months earlier than previously:PV = $430, × = $442,34. In three years, the balance in the mutual fund will be:FV = $1,000,000 × 3 = $1,108,718The monthly shortfall will be: $15,000 – ($7,500 + $1,500) = $6,000 Annual withdrawals from the mutual fund will be: $6,000 × 12 = $72,000 Assume the first annual withdrawal occurs three years from today, when the balance in the mutual fund will be $1,108,718. Treating the withdrawals as an annuity due, we solve for t as follows:r)(1r)(1r 1r 1C PV t +⨯⎥⎦⎤⎢⎣⎡+⨯-⨯=1.035(1.035)0.03510.0351$72,000$1,108,718t ⨯⎥⎦⎤⎢⎣⎡⨯-⨯=Using Excel or a financial calculator, we find that t = years.35. a. PV = 2/.12 = $ millionb. PV = 939.14$=⎥⎦⎤⎢⎣⎡⨯-⨯20(1.12)0.1210.121$2millionc. PV = 2/(. = $ milliond. PV = 061.18$=⎥⎦⎤⎢⎣⎡⨯-⨯2020(1.12).03)-(0.12 1.03.03)-(0.121$2million36. a. Using the Rule of 72, the time for money to double at 12% is72/12, or 6 years. More precisely, if x is the number ofyears for money to double, then:x = 2Using logarithms, we find:x (ln = ln 2x = yearsb. With continuous compounding for interest rate r and time period x:e r x = 2Taking the natural logarithm of each side:r x = ln(2) =Thus, if r is expressed as a percent, then x (the time for money to double) is: x = (interest rate, in percent).37. Spreadsheet exercise.38. a. This calls for the growing perpetuity formula with a negativegrowth rate (g = –:million $14.290.14million $20.04)(0.10million $2PV ==--=b.The pipeline’s value at year 20 ., at t = 20), assuming its cashflows last forever, is:gr g)(1C g r C PV 2012120-+=-= With C 1 = $2 million, g = –, and r = :million $6.3140.14million $0.8840.140.04)(1million)($2PV 2020 ==-⨯= Next, we convert this amount to PV today, and subtract it from the answer to Part (a):million $13.35(1.10)million $6.314million $14.29PV 20=-=。
Solutions Manual Fundamentals of Corporate Finance10th edition Ross, Westerfield, and Jordan06-25-2013Prepared byBrad JordanUniversity of KentuckyJoe SmoliraBelmont UniversityCHAPTER 1INTRODUCTION TO CORPORATE FINANCEAnswers to Concepts Review and Critical Thinking Questions1.Capital budgeting (deciding whether to expand a manufacturing plant), capital structure (decidingwhether to issue new equity and use the proceeds to retire outstanding debt), and working capital management (modifying the firm’s credit collection policy with its customers).2.Disadvantages: unlimited liability, limited life, difficulty in transferring ownership, difficulty inraising capital funds. Some advantages: simpler, less regulation, the owners are also the managers, sometimes personal tax rates are better than corporate tax rates.3.The primary disadvantage of the corporate form is the double taxation to shareholders of distributedearnings and dividends. Some advantages include: limited liability, ease of transferability, ability to raise capital, and unlimited life.4.In response to Sarbanes-Oxley, small firms have elected to go dark because of the costs ofcompliance. The costs to comply with Sarbox can be several million dollars, which can be a large percentage of a small firm’s profits. A major cost of going dark is less access to capital. Since the firm is no longer publicly traded, it can no longer raise money in the public market. Although the company will still have access to bank loans and the private equity market, the costs associated with raising funds in these markets are usually higher than the costs of raising funds in the public market.5.The treasurer’s office and the controller’s office are the two primary organizational groups thatreport directly to the chief financial officer. The controller’s office handles cost and financial accounting, tax management, and management information systems, while the treasurer’s office is responsible for cash and credit management, capital budgeting, and financial planning. Therefore, the study of corporate finance is concentrated within the treasury group’s fu nctions.6.To maximize the current market value (share price) of the equity of the firm (whether it’spubliclytraded or not).7. In the corporate form of ownership, the shareholders are the owners of the firm. The shareholderselect the directors of the corporation, who in turn appoint the firm’s management. This separation of ownership from control in the corporate form of organization is what causes agency problems to exist. Management may act in its own or someone else’s best interests, rather than th ose of the shareholders. If such events occur, they may contradict the goal of maximizing the share price of the equity of the firm.8. A primary market transaction.公司理财英文版第10版课后习题答案9.In auction markets like the NYSE, brokers and agents meet at a physical location (the exchange) tomatch buyers and sellers of assets. Dealer markets like NASDAQ consist of dealers operating at dispersed locales who buy and sell assets themselves, communicating with other dealers either electronically or literally over-the-counter.10.Such organizations frequently pursue social or political missions, so many different goals areconceivable. One goal that is often cited is revenue minimization; that is, provide whatever goods and services are offered at the lowest possible cost to society. A better approach might be to observe that even a not-for-profit business has equity. Thus, one answer is that the appropriate goal is to maximize the value of the equity.11.Presumably, the current stock value reflects the risk, timing, and magnitude of all future cash flows,both short-term and long-term. If this is correct, then the statement is false.12.An argument can be made either way. At the one extreme, we could argue that in a market economy,all of these things are priced. There is thus an optimal level of, for example, ethical and/or illegal behavior, and the framework of stock valuation explicitly includes these. At the other extreme, we could argue that these are noneconomic phenomena and are best handled through the political process. A classic (and highly relevant) thought question that illustrates this debate goes something like this: “A firm has estimated that the cost of improving the safety of one of its products is $30 million. However, the firm believes that improving the safety of the product will only save $20 million in product liability claims. What should the firm do?”13.The goal will be the same, but the best course of action toward that goal may be different because ofdiffering social, political, and economic institutions.14.The goal of management should be to maximize the share price for the current shareholders. Ifmanagement believes that it can improve the profitability of the firm so that the share price will exceed $35, then they should fight the offer from the outside company. If management believes that this bidder or other unidentified bidders will actually pay more than $35 per share to acquire the company, then they should still fight the offer. However, if the current management cannot increase the value of the firm beyond the bid price, and no other higher bids come in, then management is not acting in the interests of the shareholders by fighting the offer. Since current managers often lose their jobs when the corporation is acquired, poorly monitored managers have an incentive to fight corporate takeovers in situations such as this.15.We would expect agency problems to be less severe in countries with a relatively small percentageof individual ownership. Fewer individual owners should reduce the number of diverse opinions concerning corporate goals. The high percentage of institutional ownership might lead to a higher degree of agreement between owners and managers on decisions concerning risky projects. In addition, institutions may be better able to implement effective monitoring mechanisms on managers than can individual owners, based on the institutions’ deeper resources and experiences with their own management. The increase in institutional ownership of stock in the United States and the growing activism of these large shareholder groups may lead to a reduction in agency problems for U.S. corporations and a more efficient market for corporate control.16. How much is too much? Who is worth more, Lawrence Ellison or Tiger Woods? The simplestanswer is that there is a market for executives just as there is for all types of labor. Executive compensation is the price that clears the market. The same is true for athletes and performers.Having said that, one aspect of executive compensation deserves comment. A primary reason executive compensation has grown so dramatically is that companies have increasingly moved to stock-based compensation. Such movement is obviously consistent with the attempt to better align stockholder and management interests. In recent years, stock prices have soared, so management has cleaned up. It is sometimes argued that much of this reward is simply due to rising stock prices in general, not managerial performance. Perhaps in the future, executive compensation will be designed to reward only differential performance, that is, stock price increases in excess of general market increases.公司理财英文版第10版课后习题答案CHAPTER 2FINANCIAL STATEMENTS, TAXES, AND CASH FLOWAnswers to Concepts Review and Critical Thinking Questions1.Liquidity measures how quickly and easily an asset can be converted to cash without significant lossin value. It’s desirable for firms to have high liquidity so that they have a large factor of safety in meeting short-term creditor demands. However, since liquidity also has an opportunity cost associated with it—namely that higher returns can generally be found by investing the cash into productive assets—low liquidity levels are also desirable to the firm. It’s up to the firm’s financial management staff to find a reasonable compromise between these opposing needs.2.The recognition and matching principles in financial accounting call for revenues, and the costsassociated with producing those revenues, to be “booked” when the revenue process is essentially complete, not necessarily when the cash is collected or bills are paid. Note that this way is not necessarily correct; it’s the way accountants have chosen to do it.3.Historical costs can be objectively and precisely measured whereas market values can be difficult toestimate, and different analysts would come up with different numbers. Thus, there is a trade-off between relevance (market values) and objectivity (book values).4. Depreciation is a noncash deduction that reflects adjustments made in asset book values inaccordance with the matching principle in financial accounting. Interest expense is a cash outlay, but it’s a financing cost, not an operating cost.5.Market values can never be negative. Imagine a share of stock selling for –$20. This would meanthat if you placed an order for 100 shares, you would get the stock along with a check for $2,000.How many shares do you want to buy? More generally, because of corporate and individual bankruptcy laws, net worth for a person or a corporation cannot be negative, implying that liabilities cannot exceed assets in market value.6.For a successful company that is rapidly expanding, for example, capital outlays will be large,possibly leading to negative cash flow from assets. In general, what matters is whether the money is spent wisely, not whether cash flow from assets is positive or negative.7.It’s probably not a good sign for an established company, but it would be fairly ordinary for a start-up, so it depends.8.For example, if a company were to become more efficient in inventory management, the amount ofinventory needed would decline. The same might be true if it becomes better at collecting its receivables. In general, anything that leads to a decline in ending NWC relative to beginning would have this effect. Negative net capital spending would mean more long-lived assets were liquidated than purchased.9.If a company raises more money from selling stock than it pays in dividends in a particular period,its cash flow to stockholders will be negative. If a company borrows more than it pays in interest, its cash flow to creditors will be negative.10.The adjustments discussed were purely accounting changes; they had no cash flow or market valueconsequences unless the new accounting information caused stockholders to revalue the derivatives.11.Enterprise value is the theoretical takeover price. In the event of a takeover, an acquirer would haveto take on the company's debt but would pocket its cash. Enterprise value differs significantly from simple market capitalization in several ways, and it may be a more accurate representation of a firm's value. In a takeover, the value of a firm's debt would need to be paid by the buyer when taking overa company. This enterprise value provides a much more accurate takeover valuation because itincludes debt in its value calculation.12.In general, it appears that investors prefer companies that have a steady earnings stream. If true, thisencourages companies to manage earnings. Under GAAP, there are numerous choices for the way a company reports its financial statements. Although not the reason for the choices under GAAP, one outcome is the ability of a company to manage earnings, which is not an ethical decision. Even though earnings and cash flow are often related, earnings management should have little effect on cash flow (except for tax implications). If the market is “fooled” and prefers steady earnings, shareholder wealth can be increased, at least temporarily. However, given the questionable ethics of this practice, the company (and shareholders) will lose value if the practice is discovered. Solutions to Questions and ProblemsNOTE: All end of chapter problems were solved using a spreadsheet. Many problems require multiple steps. Due to space and readability constraints, when these intermediate steps are included in this solutions manual, rounding may appear to have occurred. However, the final answer for each problem is found without rounding during any step in the problem.Basic1.To find owners’ equity, we must construct a balance sheet as follows:Balance SheetCA $ 4,800 CL $ 4,200NFA 27,500 LTD 10,500OE ??TA $32,300 TL & OE $32,300We know that total liabilities and owners’ equity (TL & OE) must equal total assets of $32,300.We also know that TL & OE is equal to current liabilities plus long-term debt plus owners’ equity, so owners’ equity is:OE = $32,300 – 10,500 – 4,200 = $17,600NWC = CA – CL = $4,800 – 4,200 = $600公司理财英文版第10版课后习题答案2. The income statement for the company is:Income StatementSales $734,000Costs 315,000Depreciation 48,000EBIT $371,000Interest 35,000EBT $336,000Taxes (35%) 117,600Net income $218,4003.One equation for net income is:Net income = Dividends + Addition to retained earningsRearranging, we get:Addition to retained earnings = Net income – Dividends = $218,400 – 85,000 = $133,4004.EPS = Net income / Shares = $218,400 / 110,000 = $1.99 per shareDPS = Dividends / Shares = $85,000 / 110,000 = $0.77 per share5.To find the book value of current assets, we use: NWC = CA –CL. Rearranging to solve forcurrent assets, we get:CA = NWC + CL = $215,000 + 900,000 = $1,115,000The market value of current assets and fixed assets is given, so:Book value CA = $1,115,000 Market value CA = $1,250,000Book value NFA = $3,200,000 Market value NFA = $5,300,000Book value assets = $4,315,000 Market value assets = $6,550,0006.Taxes = 0.15($50,000) + 0.25($25,000) + 0.34($25,000) + 0.39($255,000 – 100,000) = $82,7007.The average tax rate is the total tax paid divided by taxable income, so:Average tax rate = $82,700 / $255,000 = .3243, or 32.43%The marginal tax rate is the tax rate on the next $1 of earnings, so the marginal tax rate = 39%.8.To calculate OCF, we first need the income statement:Income StatementSales $39,500Costs 18,400Depreciation 1,900EBIT $19,200Interest 1,400Taxable income $17,800Taxes (35%) 6,230Net income $11,570OCF = EBIT + Depreciation – Taxes = $19,200 + 1,900 – 6,230 = $14,870 capital spending = NFA end– NFA beg + DepreciationNet capital spending = $3,600,000 – 2,800,000 + 345,000Net capital spending = $1,145,00010. Change in NWC = NWC end– NWC begChange in NWC = (CA end– CL end) – (CA beg– CL beg)Change in NWC = ($3,460 – 1,980) – ($3,120 – 1,570)Change in NWC = $1,480 – 1,550 = –$7011.Cash flow to creditors = Interest paid – Net new borrowingCash flow to creditors = Interest paid – (LTD end– LTD beg)Cash flow to creditors = $190,000 – ($2,550,000 – 2,300,000)Cash flow to creditors = –$60,00012. Cash flow to stockholders = Dividends paid – Net new equityCash flow to stockholders = Dividends paid – [(Common end + APIS end) – (Common beg + APIS beg)] Cash flow to stockholders = $540,000 – [($715,000 + 4,700,000) – ($680,000 + 4,300,000)]Cash flow to stockholders = $105,000Note, APIS is the additional paid-in surplus.13. Cash flow from assets = Cash flow to creditors + Cash flow to stockholders= –$60,000 + 105,000 = $45,000Cash flow from assets = $45,000 = OCF – Change in NWC – Net capital spending= $45,000 = OCF – (–$55,000) – 1,300,000Operating cash flow = $45,000 – 55,000 + 1,300,000Operating cash flow = $1,290,000公司理财英文版第10版课后习题答案Intermediate14.To find the OCF, we first calculate net income.Income StatementSales $235,000Costs 141,000Other expenses 7,900Depreciation 17,300EBIT $ 68,800Interest 12,900Taxable income $ 55,900Taxes 19,565Net income $ 36,335Dividends $12,300Additions to RE $24,035a.OCF = EBIT + Depreciation – Taxes = $68,800 + 17,300 – 19,565 = $66,535b.CFC = Interest – Net new LTD = $12,900 – (–4,500) = $17,400Note that the net new long-term debt is negative because the company repaid part of its long-term debt.c.CFS = Dividends – Net new equity = $12,300 – 6,100 = $6,200d.We know that CFA = CFC + CFS, so:CFA = $17,400 + 6,200 = $23,600CFA is also equal to OCF – Net capital spending – Change in NWC. We already know OCF.Net capital spending is equal to:Net capital spending = Increase in NFA + Depreciation = $25,000 + 17,300 = $42,300Now we can use:CFA = OCF – Net capital spending – Change in NWC$23,600 = $66,535 – 42,300 – Change in NWCChange in NWC = $635This means that the company increased its NWC by $635.15.The solution to this question works the income statement backwards. Starting at the bottom:Net income = Dividends + Addition to retained earnings = $1,800 + 5,300 = $7,100Now, looking at the income statement:EBT – EBT × Tax rate = Net incomeRecognize that EBT × Tax rate is simply the calculation for taxes. Solving this for EBT yields:EBT = NI / (1– Tax rate) = $7,100 / (1 – 0.35) = $10,923Now you can calculate:EBIT = EBT + Interest = $10,923 + 4,900 = $15,823The last step is to use:EBIT = Sales – Costs – Depreciation$15,823 = $52,000 – 27,300 – DepreciationSolving for depreciation, we find that depreciation = $8,87716.The balance sheet for the company looks like this:Balance SheetCash $ 127,000 Accounts payable $ 210,000Accounts receivable 105,000 Notes payable 160,000Inventory 293,000 Current liabilities $ 370,000Current assets $ 525,000 Long-term debt 845,000Total liabilities $1,215,300 Tangible net fixed assets 1,620,000Intangible net fixed assets 630,000 Common stock ??Accumulated ret. earnings 1,278,000 Total assets $2,775,000 Total liab. & owners’ equity$2,755,000Total liabilities and owners’ equity is:TL & OE = CL + LTD + Common stock + Retained earningsSolving for this equation for equity gives us:Common stock = $2,755,000 – 1,215,300 – 1,278,000 = $282,00017.The market value of shareholders’ equity cannot be negative. A negative market value in this casewould imply that the company would pay you to own the stock. The market value of shareholders’ equity can be stated as: Shareholders’ equity = Max [(TA – TL), 0]. So, if TA is $7,100, equity is equal to $1,300, and if TA is $5,200, equity is equal to $0. We should note here that the book value of shareholders’ equity can be negative.公司理财英文版第10版课后习题答案18. a. Taxes Growth = 0.15($50,000) + 0.25($25,000) + 0.34($1,000) = $14,090Taxes Income = 0.15($50,000) + 0.25($25,000) + 0.34($25,000) + 0.39($235,000)+ 0.34($7,600,000 – 335,000)= $2,584,000b. Each firm has a marginal tax rate of 34% on the next $10,000 of taxable income, despite theirdifferent average tax rates, so both firms will pay an additional $3,400 in taxes.19.Income StatementSales $850,000COGS 610,000A&S expenses 110,000Depreciation 140,000EBIT –$10,000Interest 85,000Taxable income –$95,000Taxes (35%) 0 income –$95,000b.OCF = EBIT + Depreciation – Taxes = –$10,000 + 140,000 – 0 = $130,000 income was negative because of the tax deductibility of depreciation and interest expense.However, the actual cash flow from operations was positive because depreciation is a non-cash expense and interest is a financing expense, not an operating expense.20. A firm can still pay out dividends if net income is negative; it just has to be sure there is sufficientcash flow to make the dividend payments.Change in NWC = Net capital spending = Net new equity = 0. (Given)Cash flow from assets = OCF – Change in NWC – Net capital spendingCash flow from assets = $130,000 – 0 – 0 = $130,000Cash flow to stockholders = Dividends – Net new equity = $63,000 – 0 = $63,000Cash flow to creditors = Cash flow from assets – Cash flow to stockholdersCash flow to creditors = $130,000 – 63,000 = $67,000Cash flow to creditors = Interest – Net new LTDNet new LTD = Interest – Cash flow to creditors = $85,000 – 67,000 = $18,00021. a.Income StatementSales $27,360Cost of goods sold 19,260Depreciation 4,860EBIT $ 3,240Interest 2,190Taxable income $ 1,050Taxes (34%) 357Net income $ 693b.OCF = EBIT + Depreciation – Taxes= $3,240 + 4,860 – 357 = $7,743c.Change in NWC = NWC end– NWC beg= (CA end– CL end) – (CA beg– CL beg)= ($7,116 – 3,780) – ($5,760 – 3,240)= $3,336 – 2,520 = $816Net capital spending = NFA end– NFA beg + Depreciation= $20,160 – 16,380 + 4,860 = $8,640CFA = OCF – Change in NWC – Net capital spending= $7,743 – 816 – 8,640 = –$1,713The cash flow from assets can be positive or negative, since it represents whether the firmraised funds or distributed funds on a net basis. In this problem, even though net income andOCF are positive, the firm invested heavily in both fixed assets and net working capital; ithad to raise a net $1,713 in funds from its stockholders and creditors to make theseinvestments.d.Cash flow to creditors = Interest – Net new LTD = $2,190 – 0 = $2,190Cash flow to stockholders = Cash flow from assets – Cash flow to creditors= –$1,713 – 2,190 = –$3,903We can also calculate the cash flow to stockholders as:Cash flow to stockholders = Dividends – Net new equitySolving for net new equity, we get:Net new equity = $1,560 – (–3,903) = $5,463The firm had positive earnings in an accounting sense (NI > 0) and had positive cash flowfrom operations. The firm invested $816 in new net working capital and $8,640 in new fixedassets. The firm had to raise $1,713 from its stakeholders to support this new investment. Itaccomplished this by raising $5,463 in the form of new equity. After paying out $1,560 ofthis in the form of dividends to shareholders and $2,190 in the form of interest to creditors,$1,713 was left to meet the firm’s cash flow needs for investment.22. a.Total assets 2010 = $914 + 3,767 = $4,681Total liabilities 2010 = $365 + 1,991= $2,356Owners’ equity 2010 = $4,681 – 2,356 = $2,325Total assets 2011 = $990 + 4,536 = $5,526Total liabilities 2011 = $410 + 2,117 = $2,527Owners’ equity 2011 = $5,526 – 2,527 = $2,999b.NWC 2010 = CA10 – CL10 = $914 – 365 = $549NWC 2011 = CA11 – CL11 = $990 – 410 = $580Change in NWC = NWC11 – NWC10 = $580 – 549 = $31公司理财英文版第10版课后习题答案c.We can calculate net capital spending as:Net capital spending = Net fixed assets 2011 – Net fixed assets 2010 + DepreciationNet capital spending = $4,536 – 3,767 + 1,033 = $1,802So, the company had a net capital spending cash flow of $1,802. We also know that netcapital spending is:Net capital spending = Fixed assets bought – Fixed assets sold$1,802 = $1,890 – Fixed assets soldFixed assets sold = $1,890 – 1,802 = $88To calculate the cash flow from assets, we must first calculate the operating cash flow. Theincome statement is:Income StatementSales $11,592Costs 5,405Depreciation expense 1,033EBIT $ 5,154Interest expense 294EBT $ 4,860Taxes (35%) 1,701Net income $ 3,159So, the operating cash flow is:OCF = EBIT + Depreciation – Taxes = $5,154 + 1,033 – 1,701 = $4,486And the cash flow from assets is:Cash flow from assets = OCF – Change in NWC – Net capital spending.= $4,486 – 31 – 1,802 = $2,653 new borrowing = LTD11 – LTD10 = $2,117 – 1,991 = $126Cash flow to creditors = Interest – Net new LTD = $294 – 126 = $168Net new borrowing = $126 = Debt issued – Debt retiredDebt retired = $378 – 126 = $252Challenge capital spending = NFA end– NFA beg + Depreciation= (NFA end– NFA beg) + (Depreciation + AD beg) – AD beg= (NFA end– NFA beg)+ AD end– AD beg= (NFA end + AD end) – (NFA beg + AD beg)= FA end– FA beg24. a.The tax bubble causes average tax rates to catch up to marginal tax rates, thus eliminating thetax advantage of low marginal rates for high income corporations.b.Taxes = 0.15($50,000) + 0.25($25,000) + 0.34($25,000) + 0.39($235,000) = $113,900Average tax rate = $113,900 / $335,000 = 34%The marginal tax rate on the next dollar of income is 34 percent.For corporate taxable income levels of $335,000 to $10 million, average tax rates are equal tomarginal tax rates.Taxes = 0.34($10,000,000) + 0.35($5,000,000) + 0.38($3,333,333)= $6,416,667Average tax rate = $6,416,667 / $18,333,333 = 35%The marginal tax rate on the next dollar of income is 35 percent. For corporate taxableincome levels over $18,333,334, average tax rates are again equal to marginal tax rates.c.Taxes = 0.34($200,000) = $68,000$68,000 = 0.15($50,000) + 0.25($25,000) + 0.34($25,000) + X($100,000);X($100,000) = $68,000 – 22,250X = $45,750 / $100,000X = 45.75%25.Balance sheet as of Dec. 31, 2010Cash $ 6,067 Accounts payable $ 4,384Accounts receivable 8,034 Notes payable 1,171Inventory 14,283 Current liabilities $ 5,555Current assets $28,384Long-term debt $20,320 Net fixed assets $50,888 Owners' equity $53,397Total assets $79,272 Total liab. & equity $79,272Balance sheet as of Dec. 31, 2011Cash $ 6,466 Accounts payable $ 4,644Accounts receivable 9,427 Notes payable 1,147Inventory 15,288 Current liabilities $ 5,791Current assets $31,181Long-term debt $24,636 Net fixed assets $54,273 Owners' equity $55,027Total assets $85,454 Total liab. & equity $85,454公司理财英文版第10版课后习题答案2010 Income Statement 2011 Income Statement Sales $11,573.00Sales $12,936.00 COGS 3,979.00COGS 4,707.00 Other expenses 946.00Other expenses 824.00 Depreciation 1,661.00Depreciation 1,736.00 EBIT $ 4,987.00EBIT $ 5,669.00 Interest 776.00Interest 926.00 EBT $ 4,211.00EBT $ 4,743.00 Taxes (34%) 1,431.74Taxes (34%) 1,612.62 Net income $ 2,779.26Net income $ 3,130.38 Dividends $1,411.00Dividends $1,618.00 Additions to RE 1,368.26Additions to RE 1,512.3826.OCF = EBIT + Depreciation – Taxes = $5,669 + 1,736 – 1,612.62 = $5,792.38Change in NWC = NWC end– NWC beg = (CA – CL) end– (CA – CL) beg= ($31,181 – 5,791) – ($28,384 – 5,555)= $2,561Net capital spending = NFA end– NFA beg+ Depreciation= $54,273 – 50,888 + 1,736 = $5,121Cash flow from assets = OCF – Change in NWC – Net capital spending= $5,792.38 – 2,561 – 5,121 = –$1,889.62Cash flow to creditors = Interest – Net new LTDNet new LTD = LTD end– LTD begCash flow to creditors = $926 – ($24,636 – 20,320) = –$3,390Net new equity = Common stock end– Common stock begCommon stock + Retained earnings = Tota l owners’ equityNet new equity = (OE – RE) end– (OE – RE) beg= OE end– OE beg + RE beg– RE endRE end= RE beg+ Additions to RE08Net new equity = OE end– OE beg+ RE beg– (RE beg + Additions to RE11)= OE end– OE beg– Additions to RENet new equity = $55,027 – 53,397 – 1,512.38 = $117.62CFS = Dividends – Net new equityCFS = $1,618 – 117.62 = $1,500.38As a check, cash flow from assets is –$1,889.62.CFA = Cash flow from creditors + Cash flow to stockholdersCFA = –$3,390 + 1,500.38 = –$1,889.62公司理财英文版第10版课后习题答案CHAPTER 3WORKING WITH FINANCIAL STATEMENTSAnswers to Concepts Review and Critical Thinking Questions1. a.If inventory is purchased with cash, then there is no change in the current ratio. If inventory ispurchased on credit, then there is a decrease in the current ratio if it was initially greater than 1.0.b.Reducing accounts payable with cash increases the current ratio if it was initially greater than 1.0.c.Reducing short-term debt with cash increases the current ratio if it was initially greater than 1.0.d.As long-term debt approaches maturity, the principal repayment and the remaining interestexpense become current liabilities. Thus, if debt is paid off with cash, the current ratio increases if it was initially greater than 1.0. If the debt has not yet become a current liability, then paying it off will reduce the current ratio since current liabilities are not affected.e.Reduction of accounts receivables and an increase in cash leaves the current ratio unchanged.f.Inventory sold at cost reduces inventory and raises cash, so the current ratio is unchanged.g. Inventory sold for a profit raises cash in excess of the inventory recorded at cost, so the currentratio increases.2.The firm has increased inventory relative to other current assets; therefore, assuming current liabilitylevels remain unchanged, liquidity has potentially decreased.3. A current ratio of 0.50 means that the firm has twice as much in current liabilities as it does incurrent assets; the firm potentially has poor liquidity. If pressed by its short-term creditors and suppliers for immediate payment, the firm might have a difficult time meeting its obligations. A current ratio of 1.50 means the firm has 50% more current assets than it does current liabilities. This probably represents an improvement in liquidity; short-term obligations can generally be met completely with a safety factor built in. A current ratio of 15.0, however, might be excessive. Any excess funds sitting in current assets generally earn little or no return. These excess funds might be put to better use by investing in productive long-term assets or distributing the funds to shareholders.4. a.Quick ratio provides a measure of the short-term liquidity of the firm, after removing the effectsof inventory, generally the least liquid of the firm’s current assets.b.Cash ratio represents the ability of the firm to completely pay off its current liabilities with itsmost liquid asset (cash).c.Total asset turnover measures how much in sales is generated by each dollar of firm assets.d.Equity multiplier represents the degree of leverage for an equity investor of the firm; it measuresthe dollar worth of firm assets each equity dollar has a claim to.e.Long-term debt ratio measures the percentage of total firm capitalization funded by long-termdebt.。
CHAPTER 2How to Calculate Present ValuesAnswers to Problem Sets1. If the discount factor is .507, then .507*1.126 = $12. 125/139 = .8993. PV = 374/(1.09)9 = 172.204. PV = 432/1.15 + 137/(1.152) + 797/(1.153) = 376 + 104 + 524 = $1,0035. FV = 100*1.158 = $305.906. NPV = -1,548 + 138/.09 = -14.67 (cost today plus the present value of theperpetuity)7. PV = 4/(.14-.04) = $408. a. PV = 1/.10 = $10b. Since the perpetuity will be worth $10 in year 7, and since that is roughlydouble the present value, the approximate PV equals $5.PV = (1 / .10)/(1.10)7 = 10/2= $5 (approximately)c. A perpetuity paying $1 starting now would be worth $10, whereas aperpetuity starting in year 8 would be worth roughly $5. The differencebetween these cash flows is therefore approximately $5. PV = 10 – 5= $5(approximately)d. PV = C/(r-g) = 10,000/(.10-.05) = $200,000.9. a. PV = 10,000/(1.055) = $7,835.26 (assuming the cost of the car does notappreciate over those five years).b. You need to set aside (12,000 × 6-year annuity factor) = 12,000 × 4.623 =$55,476.c. At the end of 6 years you would have 1.086 × (60,476 - 55,476) = $7,934. 10. a. FV = 1,000e .12x5 = 1,000e .6 = $1,822.12. b. PV = 5e -.12 x 8 = 5e -.96 = $1.914 millionc. PV = C (1/r – 1/re rt ) = 2,000(1/.12 – 1/.12e .12 x15) = $13,91211.a. FV = 10,000,000x(1.06)4 = 12,624,770b. FV = 10,000,000x(1 + .06/12)(4x12) = 12,704,892c. FV = 10,000,000xe (4x.06) = 12,712,492 12.a. PV = $100/1.0110 = $90.53b. PV = $100/1.1310 = $29.46c. PV = $100/1.2515 = $ 3.52d.PV = $100/1.12 + $100/1.122 + $100/1.123 = $240.1813. a.⇒=+=0.905r 11DF 11r 1 = 0.1050 = 10.50%b. 0.819(1.105)1)r (11DF 2222==+=c. AF 2 = DF 1 + DF 2 = 0.905 + 0.819 = 1.724d.PV of an annuity = C ⨯ [Annuity factor at r% for t years] Here:$24.65 = $10 ⨯ [AF 3] AF 3 = 2.465e. AF 3 = DF 1 + DF 2 + DF 3 = AF 2 + DF 3 2.465 = 1.724 + DF 3 DF 3 = 0.74114.The present value of the 10-year stream of cash inflows is:6$886,739.6(1.14)0.1410.141 $170,000PV 10=⎥⎦⎤⎢⎣⎡⨯-⨯= Thus:NPV = –$800,000 + $886,739.66 = +$86,739.66At the end of five years, the factory’s value will be the present value of the fiveremaining $170,000 cash flows:6$583,623.7(1.14)0.1410.141 $170,000PV 5=⎥⎦⎤⎢⎣⎡⨯-⨯= 15.$23,696.151.12$50,0001.12$68,0001.12$80,0001.12$92,0001.12$92,000 1.12$85,0001.12$80,0001.12$75,0001.12$57,0001.12$50,000$380,000(1.12)C NPV 109876543210t t t =++++++++++-==∑=016.a.Let S t = salary in year t∑=-=301t t1t (1.08)(1.05)40,000PV 3$760,662.5(1.08).05)-(.08(1.05).05)-(.081 43030=⎥⎦⎤⎢⎣⎡⨯-⨯=000,0b.PV(salary) x 0.05 = $38,033.13Future value = $38,018.96 x (1.08)30 = $382,714.30c.$38,980.30 (1.08)0.0810.081$C (1.08)0.0810.081 C $r)(1r 1r 1 C PV 2020t =⎥⎥⎦⎤⎢⎢⎣⎡⨯-=⎥⎦⎤⎢⎣⎡⨯-⨯=⎥⎦⎤⎢⎣⎡+⨯-⨯=382,714.30382,714.3017.PeriodPresent Value0 -400,000.00 1 +100,000/1.12 = + 89,285.712 +200,000/1.122 = +159,438.783+300,000/1.123 = +213,534.07Total = NPV = $62,258.5618.We can break this down into several different cash flows, such that the sum ofthese separate cash flows is the total cash flow. Then, the sum of the present values of the separate cash flows is the present value of the entire project. (All dollar figures are in millions.) ▪ Cost of the ship is $8 million PV = -$8 million▪Revenue is $5 million per year, operating expenses are $4 million. Thus, operating cash flow is $1 million per year for 15 years.million $8.559(1.08)0.0810.081 million 1$PV 15=⎥⎦⎤⎢⎣⎡⨯-⨯= ▪ Major refits cost $2 million each, and will occur at times t = 5 and t = 10. PV = (-$2 million)/1.085 + (-$2 million)/1.0810 = -$2.288 million ▪Sale for scrap brings in revenue of $1.5 million at t = 15. PV = $1.5 million/1.0815 = $0.473 millionAdding these present values gives the present value of the entire project:NPV = -$8 million + $8.559 million - $2.288 million + $0.473 million NPV = -$1.256 million19. a. PV = $100,000b. PV = $180,000/1.125 = $102,136.83c. PV = $11,400/0.12 = $95,000d.4$107,354.2(1.12)0.1210.121 $19,000PV 10=⎥⎦⎤⎢⎣⎡⨯-⨯= e.PV = $6,500/(0.12 - 0.05) = $92,857.14Prize (d) is the most valuable because it has the highest present value.20.Mr. Basset is buying a security worth $20,000 now. That is its present value.The unknown is the annual payment. Using the present value of an annuity formula, we have:$2,653.90 (1.08)0.0810.081000,20$C (1.08)0.0810.081 C 000,20$r)(1r 1r 1 C PV 1212t =⎥⎥⎦⎤⎢⎢⎣⎡⨯-=⎥⎦⎤⎢⎣⎡⨯-⨯=⎥⎦⎤⎢⎣⎡+⨯-⨯=21.Assume the Zhangs will put aside the same amount each year. One approach to solving this problem is to find the present value of the cost of the boat and then equate that to the present value of the money saved. From this equation, we can solve for the amount to be put aside each year.PV(boat) = $20,000/(1.10)5 = $12,418PV(savings) = Annual savings ⎥⎦⎤⎢⎣⎡⨯-⨯5(1.10)0.1010.101 Because PV(savings) must equal PV(boat):Annual savings 418,12$ =⎥⎦⎤⎢⎣⎡⨯-⨯5(1.10)0.1010.101Annual savings $3,276(1.10)0.1010.101$12,4185=⎥⎦⎤⎢⎣⎡⨯-= Another approach is to use the future value of an annuity formula:$20,0005.10)(1 savings Annual =⎥⎥⎦⎤⎢⎢⎣⎡-+⨯10.1Annual savings = $ 3,27622.The fact that Kangaroo Autos is offering “free credit” tells us what the cash payments are; it does not change the fact that money has time value. A 10% annual rate of interest is equivalent to a monthly rate of 0.83%:r monthly = r annual /12 = 0.10/12 = 0.0083 = 0.83%The present value of the payments to Kangaroo Autos is:8$8,93(1.0083)0.008310.00831$300$1,00030=⎥⎦⎤⎢⎣⎡⨯-⨯+ A car from Turtle Motors costs $9,000 cash. Therefore, Kangaroo Autosoffers the better deal, i.e., the lower present value of cost. 23.The NPVs are: at 5%$25,011(1.05)$320,0001.05$100,000$170,000NPV 2=+--=⇒at 10% $3,554(1.10)320,0001.10$100,000$170,000NPV 2=+--=⇒ at 15% $14,991(1.15)320,0001.15$100,000$170,000NPV 2-=+--=⇒ The figure below shows that the project has zero NPV at about 11%.As a check, NPV at 11% is:$371(1.11)320,0001.11$100,000$170,000NPV 2-=+--=$1,428.570.07$100r C PV ===b. This is worth the PV of stream (a) plus the immediate payment of $100:PV = $100 + $1,428.57 = $1,528.57c.The continuously compounded equivalent to a 7% annually compoundedrate is approximately 6.77%, because:e 0.0677 = 1.0700 Thus:$1,477.100.0677$100r C PV ===Note that the pattern of payments in part (b) is more valuable than the pattern of payments in part (c). It is preferable to receive cash flows at the start of every year than to spread the receipt of cash evenly over the year; with the former pattern of payment, you receive the cash more quickly.25.a. PV = $1 billion/0.08 = $12.5 billionb. PV = $1 billion/(0.08 – 0.04) = $25.0 billionc. billion $9.818(1.08)0.0810.081billion $1PV 20=⎥⎦⎤⎢⎣⎡⨯-⨯= d.The continuously compounded equivalent to an 8% annually compounded rate is approximately 7.7% , because:e 0.0770 = 1.0800 Thus:billion $10.203e 0.07710.0771billion $1PV )(0.077)(20=⎥⎦⎤⎢⎣⎡⨯-⨯= This result is greater than the answer in Part (c) because the endowmentis now earning interest during the entire year.26. With annual compounding: FV = $100 ⨯ (1.15)20 = $1,636.65 With continuous compounding: FV = $100 ⨯ e (0.15×20) = $2,008.5527.One way to approach this problem is to solve for the present value of: (1) $100 per year for 10 years, and(2) $100 per year in perpetuity, with the first cash flow at year 11.If this is a fair deal, these present values must be equal, and thus we can solve for the interest rate (r).The present value of $100 per year for 10 years is:⎥⎦⎤⎢⎣⎡+⨯-⨯=10r)(1(r)1r 1$100PV The present value, as of year 10, of $100 per year forever, with the first payment in year 11, is: PV 10 = $100/r At t = 0, the present value of PV 10 is:⎥⎦⎤⎢⎣⎡⨯⎥⎦⎤⎢⎣⎡+=r $100r)(11PV 10 Equating these two expressions for present value, we have:⎥⎦⎤⎢⎣⎡⨯⎥⎦⎤⎢⎣⎡+=⎥⎦⎤⎢⎣⎡+⨯-⨯r $100r)(11r)(1(r)1r 1$1001010 Using trial and error or algebraic solution, we find that r = 7.18%.28.Assume the amount invested is one dollar.Let A represent the investment at 12%, compounded annually.Let B represent the investment at 11.7%, compounded semiannually. Let C represent the investment at 11.5%, compounded continuously. After one year:FV A = $1 ⨯ (1 + 0.12)1= $1.1200FV B = $1 ⨯ (1 + 0.0585)2 = $1.1204 FV C = $1 ⨯ e (0.115 ⨯ 1)= $1.1219After five years:FV A = $1 ⨯ (1 + 0.12)5= $1.7623FV B = $1 ⨯ (1 + 0.0585)10 = $1.7657 FV C = $1 ⨯ e (0.115 ⨯ 5)= $1.7771 After twenty years:FV A = $1 ⨯ (1 + 0.12)20= $9.6463FV B = $1 ⨯ (1 + 0.0585)40 = $9.7193 FV C = $1 ⨯ e (0.115 ⨯ 20)= $9.9742 The preferred investment is C. 29.Because the cash flows occur every six months, we first need to calculate the equivalent semi-annual rate. Thus, 1.08 = (1 + r/2)2 => r = 7.85 semi-annually compounded APR. Therefore the rate for six months is 7.85/2 or 3.925%:081,846039251039250103925010001000001009$).(..,$,$PV =⎥⎦⎤⎢⎣⎡⨯-⨯+=30. a. Each installment is: $9,420,713/19 = $495,827$4,761,724(1.08)0.0810.081$495,827PV 19=⎥⎦⎤⎢⎣⎡⨯-⨯=b.If ERC is willing to pay $4.2 million, then:⎥⎦⎤⎢⎣⎡+⨯-⨯=19r)(1r 1r 1$495,827$4,200,000 Using Excel or a financial calculator, we find that r = 9.81%.31. a.3$402,264.7(1.08)0.0810.081$70,000PV 8=⎥⎦⎤⎢⎣⎡⨯-⨯=b.Year Beginning-of-Year Balance Year-end Interest on Balance Total Year-end Payment Amortization of Loan End-of-Year Balance 1 402,264.73 32,181.18 70,000.00 37,818.82 364,445.91 2 364,445.91 29,155.67 70,000.00 40,844.33 323,601.58 3 323,601.58 25,888.13 70,000.00 44,111.87 279,489.71 4 279,489.71 22,359.18 70,000.00 47,640.82 231,848.88 5 231,848.88 18,547.91 70,000.00 51,452.09 180,396.79 6 180,396.79 14,431.74 70,000.00 55,568.26 124,828.54 7 124,828.54 9,986.28 70,000.00 60,013.72 64,814.82864,814.825,185.1970,000.0064,814.810.0132.This is an annuity problem with the present value of the annuity equal to $2 million (as of your retirement date), and the interest rate equal to 8% with 15 time periods. Thus, your annual level of expenditure (C) is determined as follows:$233,659 (1.08)0.0810.081$2,000,000C (1.08)0.0810.081 C $2,000,000r)(1r 1r 1 C PV 1515t =⎥⎥⎦⎤⎢⎢⎣⎡⨯-=⎥⎦⎤⎢⎣⎡⨯-⨯=⎥⎦⎤⎢⎣⎡+⨯-⨯=With an inflation rate of 4% per year, we will still accumulate $2 million as of ourretirement date. However, because we want to spend a constant amount per year in real terms (R, constant for all t), the nominal amount (C t ) must increase each year. For each year t: R = C t /(1 + inflation rate)t Therefore:PV [all C t ] = PV [all R ⨯ (1 + inflation rate)t ] = $2,000,000$2,000,0000.08)(1.04)0(1....08)0(10.04)(10.08)(1.04)0(1R 15152211=⎥⎦⎤⎢⎣⎡+++++++++⨯ R ⨯ [0.9630 + 0.9273 + . . . + 0.5677] = $2,000,000R ⨯ 11.2390 = $2,000,000R = $177,952 Alternatively, consider that the real rate is03846.1=-++0.04)(1.08)0(1. Then, redoing the steps above using the real rate gives a real cash flow equal to: $177,952=⎥⎥⎦⎤⎢⎢⎣⎡⨯-= (1.03846)0.0384610.038461$2,000,000C 15Thus C 1 = ($177,952 ⨯ 1.04) = $185,070, C 2 = $192,473, etc.33. a. 9$430,925.8(1.055)0.05510.0551$50,000PV 12=⎥⎦⎤⎢⎣⎡⨯-⨯=b. The annually compounded rate is 5.5%, so the semiannual rate is:(1.055)(1/2) – 1 = 0.0271 = 2.71%Since the payments now arrive six months earlier than previously:PV = $430,925.89 × 1.0271 = $442,603.9834. 34. In three years, the balance in the mutual fund will be:FV = $1,000,000 × (1.035)3 = $1,108,718The monthly shortfall will be: $15,000 – ($7,500 + $1,500) = $6,000Annual withdrawals from the mutual fund will be: $6,000 × 12 = $72,000Assume the first annual withdrawal occurs three years from today, when the balance in the mutual fund will be $1,108,718. Treating the withdrawals as an annuity due, we solve for t as follows: r)(1r)(1r 1r 1C PV t +⨯⎥⎦⎤⎢⎣⎡+⨯-⨯= 1.035(1.035)0.03510.0351$72,000$1,108,718t ⨯⎥⎦⎤⎢⎣⎡⨯-⨯=Using Excel or a financial calculator, we find that t = 21.38 years.35.a. PV = 2/.12 = $16.667 millionb. PV = 939.14$=⎥⎦⎤⎢⎣⎡⨯-⨯20(1.12)0.1210.121$2millionc. PV = 2/(.12-.03) = $22.222 milliond. PV = 061.18$=⎥⎦⎤⎢⎣⎡⨯-⨯2020(1.12).03)-(0.12 1.03.03)-(0.121$2million 36. a. Using the Rule of 72, the time for money to double at 12% is 72/12,or 6 years. More precisely, if x is the number of years for money todouble, then:(1.12)x = 2Using logarithms, we find:x (ln 1.12) = ln 2x = 6.12 yearsb. With continuous compounding for interest rate r and time period x:e r x = 2Taking the natural logarithm of each side:r x = ln(2) = 0.693Thus, if r is expressed as a percent, then x (the time for money to double)is: x = 69.3/(interest rate, in percent).37.Spreadsheet exercise.38. a. This calls for the growing perpetuity formula with a negative growth rate(g = –0.04):million $14.290.14million $20.04)(0.10million $2PV ==--=b. The pipeline’s value at year 20 (i.e., at t = 20), assuming its cash fl ows last forever, is:gr g)(1C g r C PV 2012120-+=-= With C 1 = $2 million, g = –0.04, and r = 0.10:million $6.3140.14million $0.8840.140.04)(1million)($2PV 2020 ==-⨯= Next, we convert this amount to PV today, and subtract it from the answer to Part (a):million $13.35(1.10)million $6.314million $14.29PV 20=-=。
