《金融数学》第二版习题参考答案修订版

合集下载

《金融学(第二版)》讲义大纲及课后习题答案详解第15章

CHAPTER 15OPTIONS AND CONTINGENT CLAIMSObjectives•How to use options to modify one’s exposure to investment risk.•To understand the pricing relationships that exist among calls, puts, stocks and bonds.•To explain the binomial and Black-Scholes option-pricing models and apply them to the valuation of corporate bonds and other contingent claims.•To explore the range of financial decisions that can be fruitfully analyzed in terms of options.Outline15.1 How Options Work15.2 Investing with Options15.3 The Put-Call Parity Relation15.4 Volatility and Option Prices15.5 Two-State (Binomial) Option-Pricing15.6 Dynamic Replication and the Binomial Model15.7 The Black-Scholes Model15.8 Implied Volatility15.9 Contingent Claims Analysis of Corporate Debt and Equity15.10 Credit Guarantees15.11 Other Applications of Option-Pricing Methodologythe maturity of the option, and C the price of the call.•One can create a synthetic option from the underlying stock and the risk-free asset through a dynamic replication strategy that is self-financing after the initial investment. By the Law of One Price, the option’s price is given by the formula:where:C = price of the callS = price of the stockE = exercise pricer = risk-free interest rate (the annualized continuously compounded rate on a safe asset with the same maturity as the option)T = time to maturity of the option in yearsσ= standard deviation of the annualized continuously compounded rate of return on the stockd = continuous dividend yield on the stockln = natural logarithme = the base of the natural log function (approximately 2.71828)N(d1) = the probability that a random draw from a standard normal distribution will be less than d1.•The same methodology used to price options can be used to value many other contingent claims, including corporate stocks and bonds, loan guarantees, and the “real options” imbedded in investments in research and development and flexible manufacturing technology.Solutions to Problems at End of ChapterPayoff Diagrams1. Graph the payoff for a European put option with exercise price E, written on a stock with value S, when:a. You hold a long position (i.e., you buy the put)b. You hold a short position (i.e., you sell the put)SOLUTION: a.b.2.Graph the payoff to a portfolio holding one European call option and one European put option, each with the same expiration date and each with exercise price E, when both options are on a stock with value S.Investing with options3. The risk-free one –year rate of interest is 4%, and the Globalex stock index is at 100. The price of one-year European call options on the Globalex stock index with an exercise price of 104 is 8% of the current price of the index. Assume that the expected dividend yield on the stocks in the Globalex index is zero. You have $1million to invest for the next year. You plan to investenough of your money in one-year T-bills to insure that you will at least get back your original $1 million, and you will use the rest of your money to buy Globalex call options.a. Assuming that you can invest fractional amounts in Globalex options,show the payoff diagram for your investment. Measure the Globalex index on the horizontal axis and the portfolio rate of return on the vertical axis. What is the slope of the payoff line to the right of an index value of 104? b. If you think that there is a probability of .5 that the Globalex index a year from now will be up 12%, a probability of .25 that it will be up 40%, and a probability of .25 that it will be down 20%, what is the probability distribution of your portfolio rate of return?SOLUTION:a.To insure that you will at least get back your original $1 million, you need toinvestin T-bills.You can buy46.538,961$04.1000,000,1$1000,000,1$=+=+f r 69.4807854.461,38$10008.46.538,961$000,000,1$==⨯-options.The slope of the payoff line to the right of an index value of 104 is 4807.69, as seen from the graph:b.Put-Call Parity4.a.Show how one can replicate a one-year pure discount bond with a facevalue of $100 using a share of stock, a put and a call.b.Suppose that S=$100, P=$10, and C=$15. What must be the one-year interest rate?c.Show that if the one-year risk-free interest rate is lower than in youranswer to part b, there would be an arbitrage opportunity. (Hint: The price of the pure discount bond would be too high).SOLUTION:a.To replicate a one-year pure discount bond with a face value of $100, buy ashare of stock, and a European put with exercise price $100, and sell aEuropean call with an exercise price $100.b.S = $100, P = $10, and C = $15.E/(1+r) = S + P- C$100/(1+r) = $100 + $10 - $15 = $95r = 100/95 -1 = .053 or 5.3%c.If r = 4%, then one could make risk-free arbitrage profits by borrowing at 4%and investing in synthetic 1-year pure discount bonds consisting of a share of stock, a European put with exercise price $100, and a short position in aEuropean call with an exercise price $100. The synthetic bond would cost $95 and pay off $100 at maturity in 1 year. The principal and interest on the $95 it costs to buy this synthetic bond would be $95 x 1.04 = $98.8. Thus there would be a pure arbitrage profit of $1.20 per bond a year from now with zero initial outlay of funds.5. A 90-day European call option on a share of the stock of Toshiro Corporation is currently trading at 2,000 yen whereas the current price of the share itself is 2, 400 yen. 90-day zero-coupon securities issued by the government of Japan are selling for 9, 855 yen per 10, 000 yen face value. Infer the price of a 90-day European put option on this stock if both the call and put have a common exercise price of 500 yen.SOLUTION:Using the expression for put-call parity, P =-S + E/(1+r)T + CS is the share price, P is the price of the put, C is the price of the call and E is the common exercise price.Since government bonds are selling at . 9855 per 1 yen of face value, this is thediscount factor for computing the PV of the exercise price. There is no need to compute the riskless rate, r.Substituting in the parity equation we get:P = -2,400 + 500 x .9855 +2,000 = 92.75 yen6. Gordon Gekko has assembled a portfolio consisting of ten 90-day US Treasury bills, each having a face value of $1, 000 and a current price of $990.10, and 200 90-day European call options, each written on a share of Paramount stock and having an exercise price of $50.00. Gekko is offering to trade you this portfolio for 300 shares of Paramount stock, which is currently valued at $215.00 a share.If 90-day European put options on Paramount stock with a $50.00 exercise price are currently valued at $25.00,a.Infer the value of the calls in Gekko’s portfolio.b.Determine whether you should accept Gekko’s offer.SOLUTION:ing put-call parity, the current price of a call is found to be approximately$190.50 as follows:C =S - E/(1+r)T + P= $215 - $50 x .9901 + $25 = $190.495b.The v alue of Gekko’s portfolio is 10 x $990.10 + 200 x $190.495 = $48,000But the value of 300 shares is $64,500. We should reject Gekko’s offer.7. The stock of Kakkonen, Ltd., a hot tuna distributor, currently lists for $500.00 a share, whereas one-year European call options on this stock, with an exercise price of $200.00, sell for $400.00 and European put options with a similar expiration date and exercise price sell for $84.57.a.Infer the yield on a one-year, zero-coupon U.S. government bond soldtoday.b.If this yield is actually at 9%, construct a profitable trade to exploit thepotential for arbitrage.SOLUTION:ing put-call parity, we can infer the riskless yield to be approximately 8.36%as follows:A portfolio consisting of a share of the stock, a put, and a short position in acall is equivalent to a 1-year T-bill with a face value of E. Therefore the price of such a T-bill would have to be $184.57:E/(1+r)T = S + P- C = $500.00 +$84.57 -$400.00 = $184.571+r= 200/184.57 = 1.0836r= .0836 or 8.36%b.There are many ways to exploit the violation of the Law of One Price to makearbitrage profits. Since the risk-free interest rate is 9%, and the implied interest rate on the replicating portfolio is 8.36%, we could go short the replicating portfolio and invest the proceeds in T-bills. For example, at current prices,short-sell a “unit” portfolio, which consists of long positions in one put and one share and writing one call, to earn immediate revenue of $184.57. The portfolio you sold short requires payment of $200 one year from now. If you invest the $184.57 in one-year T-bills you will have 1.09x $184.57 = $201.18 a year from now. Thus you will earn a risk-free arbitrage profit of $1.18 with no outlay of your own money.Two-State Option Pricing8. Derive the formula for the price of a put option using the two-state model. SOLUTION:To price the put option, we create a synthetic option by selling short a fraction (denote the fraction “a”) and len d $b in risk-free asset. Denote the price of the stock S, the price of the put option P, the stock price when the next period is an “up” to be uS, the stock price when the next period is a “down” to be dS, the payoffs of the put option in each state Pu and Pd, and the risk-free interest r.We solve for (a, b) in 1) and 2) We find:andSo that the price of the put optionuP b r a S u =⨯++⨯⨯-)1(d P b r a S d =⨯++⨯⨯-)1(Sd u P P a d u ⨯---=)()1()(r d u P d P u b u d +⨯-⨯-⨯=⎥⎦⎤⎢⎣⎡⨯⎪⎭⎫ ⎝⎛---+⨯⎪⎭⎫ ⎝⎛--+⨯+=+⨯-=d u P d u r u P d u d r r b S a P 1)1)1(19. The share value of Drummond, Griffin and McNabb, a New Orleanspublishing house, is currently trading at $100.00 but is expected, 90 days from today, to rise to $150.00 or to decline to $50.00, depending on critical reviews of its new biography of Ezra Pound. Assuming the risk-free interest rate over the next 90 days is 1%, can you value a European call option written on a share of DGM stock if the option carries an exercise price of $85.00?SOLUTION:Should the value of DBM stock rise to $150 in 90 days, the call will be worth $65.00 at that date, but if DBM stock falls to $50.00 the call will be worthless at expiration. Using the 2-state option pricing model, we find that the call is worth $32.82:To replicate the call using the stock and risk-free borrowing we buy x shares ofstock and borrow y. We findthe values for x and y by setting up two equations, one for each of the possible payoffs of the call at expiration:The solution to this set of two equations is x = .65 and y = $32.50/1.01 = $32.18 Thus, we can replicate the call option by buying .65 of a share of stock (at a cost of $65) and borrowing $32.18.The price of the call option is $32.82: C = .65 x $100 - $32.18 = $32.82The Black-Scholes Formula 10.a. Use the Black-Scholes formula to find the price of a 3-month European call option on a non-dividend- paying stock with a current price of $50.Assume the exercise price is $51, the continuously compounded risk-free interest rate is 8% per year, and σ is .4.b. What is the composition of the initial replicating portfolio for this call option?c. Use the put-call parity relation to find the Black-Scholes formula for the price of the corresponding put option.SOLUTION:a. The price of the call is $3.987. Since the present value of the exercise price is01.1506501.1150=⨯-⨯=⨯-⨯y x y x 4$5.504.4.4.=⨯⨯⨯=⨯⨯⨯=T S C σapproximately equal to the current stock price, we could use the linearapproximation to the Black-Scholes formula:b.The hedge ratio, which is the number of shares of stock you must buy, equalsN(d1), and the amount to borrow is N(d2) times the PV of the exercise price.To find N(d1) and N(d2) you must compute d1 and d2 and then apply the N( ) function (i.e., the standard normal cumulative density function). You can either use NORMDIST in Excel or use a statistical table to do this. The hedge ratio is .54, which means you would buy .54 of a share of stock for $27. The amount to borrow is $23.c. From put-call parity: P = -S + E e0.02 + C = -$50 + $51.01 + $4 = $5.0111. As a financial analyst at Yew and Associates, a Singaporean investment house, you are asked by a client if she should purchase European call options on Rattan, Ltd. stock, which are currently selling in U.S. dollars for $30.00. These options have an exercise price of $50.00. Rattan stock currently exhibits a share price of $55.00, and the estimated rate of return variance of the stock is .04.If these options expire in 25 days and the risk-free interest rate over that period is 5% per year, what do you advise your client to do?SOLUTION:We can apply the Black-Scholes formula, where S = $55, E = $50, σ = .2, T =25/365, r = .05. We find thatC = $5.20. This is a lot less than $30, so clearly the options are not worth buying.Valuation of Corporate Securities with the Two-State Model12. Lorre and Greenstreet, Inc., a purveyor of antique statues, currently has corporate assets valued at$100,000 and must repay $50, 000, the aggregate face value of zero-coupon bonds sold to private investors, in 90 days. An independent appraisal of a newly acquired antique falcon from Malta will be publicly released at that time, and the value of the firm’s assets is expected to increase to $170,000 if the falcon is certified as genuine, but to decline to a mere $45, 000 if the antique is found to be a fake. The firm will declare bankruptcy in this latter circumstance and shareholders will surrender the assets of the firm to its creditors.a.Can you express the current aggregate value of equity in Lorre andGreenstreet as a contingent expression of the value of the firm’s assets and the face value of its outstanding debt?b.Is there a relation between the expression you have derived for equity anda 90-day European call option written upon the aggregate value of thefirm’s assets?c.Can you express the current aggregate value of the bonds issued by Lorreand Greenstreet in terms of the value of the firm’s assets and the facevalue of its outstanding debt?d.Is there a relation among the current value of the bonds the firm hasissued, the current value of riskless bonds with the same term to maturity and face value, and a European put option written on the aggregate value of the firm’s assets? What would the implication of such a relationship be for expressing the value of risky debt in terms of risk-free debt andcollateral?SOLUTION:a.The aggregate value of the firm’s equity in 90 d ays is E1 = max (V1 -B, 0)where E1 and V1 are, respectively, the aggregate values in 90 days of thefirm’s equity and assets, and where B is the aggregate face value of the firm’s debt.The current value of the equity can be expressed as:E = xV - ywhere x is the fraction of the value of the firm that one must purchase toreplicate the payoffs from the equity, and y is the amount that must beborrowed. We find the values for x and y by setting up two equations, one for each of the possible payoffs of the equity 90 days from now:170000x - y(1+r) = 120,00045000x - y(1+r) = 0The solution to this set of two equations is x = 120/125 = .96 and y =$43,200/(1+r), where r is the risk-free 90-day interest rate. Thus, we canreplicate the equity by buying 96% of the firm’s assets (at a cost of $96,000) and borrowing the present value of $43,200. The current value of the equity is therefore:E = $96,000 -$43,200/(1+r)b.They are exactly analogous. The call value is analogous to the aggregate valueof equity, the share price is analogous to the aggregate value of the firm’sassets, and the exercise price is analogou s to the face value of the firm’s debt.In effect, the firm’s shareholders hold a call option on the firm’s assets, which they can exercise by repaying the face value of the debt.c.In the presence of limited corporate liability, the realized aggregate payoff tothe firm’s creditors in 90 days, D1, can be written as:D1 = min (V1,B).d.The difference in value between the firm’s bonds and the correspondingdefault-free bonds equals the value of a European put on the firm’s assets. This relation implies the limit ed liability that stockholders have to sell the firm’s assets at the debt’s face value, which also implies that the value of risk-free debt equals the sum of the value of the risky debt and the value of collateral.13. Gephardt, Army and Gore, a vaudeville booking agency, has issued zero-coupon corporate debt this week, consisting of 80 bonds, each with a face value of $1,000 and a term to maturity of one year. Industry analysts predict that the value of GAG assets will be $160,000 in one year if Rupert Murdoch succeeds in purchasing and converting the Washington Press Club into a comedy venue, $130,000 if Murdoch buys the club but retains its current scheduling, and $20,000 if Murdoch builds an alternative comedy venue in Washington. Industry analysts also predict that aggregate value of the assets of a second firm in the field of comedy entertainment, Yelstin Yuks, Ltd., will have the values of $100,000, $100,000, and $40,000 in these respective circumstances. Assuming that investors can purchase portfolios comprised of shares of the assets of GAG and YY Ltd., as well as buying or short –selling one-year, zero-coupon, government bonds at the risk-free annual rate of .10, thena.Infer the three alternative values for aggregate equity in GAG, one yearfrom today.b.Devise a portfolio that is a perfect substitute for the payoffs given by aportfolio composed only of equity in GAG.c.Determine the current market value of a share of equity in GAG, assuming10,000 shares of GAG stock are outstanding, the current market value of GAG assets is $120,000, and the current market value of YY Ltd., assets is $85,725.d.Determine the current market value of a bond issued by GAG, assuming80 bonds are issued, under these circumstances. What of the yield tomaturity on each such bond?SOLUTION:a.In the first circumstance, the value of aggregate equity in GAG is $160,000-$80,000=$80,000.In the second circumstance, the value of aggregate equity in GAG is $130,000-$80,000=$50,000.In the third circumstance, the value of aggregate equity in GAG is 0.b.Suppose that the replicating portfolio consists of buying x units of the GAGasset, y units of the YY asset, and z units of the government bond (with a face value of $100,000). The payoff of this portfolio will exactly match that of the GAG equity, that gives:160,000x + 100,000y + 100,000z = 80,000130,000x + 100,000y + 100,000z = 50,00020,000x + 40,000y +100,000z = 0solving the equations, we have x = 1, y = -1, z = 0.2c.We define three types of pure contingent claims: at the end of one year, AD1will pay off $1 if and only if Rupert Murdoch succeeds in purchasing andconverting the Washington Press Club into a comedy venue; AD2 will pay off $1 if and only if Murdoch buys the club but retains its current scheduling; AD3 will pay off $1 if and only if Murdoch builds an alternative comedy venue in Washington. Denote the prices as of today of the three pure contingent claims as p1, p2 and p3, respectively.From the value equation of the GAG asset, YY asset and the government bond, we have:GAG: 160,000p1 + 130,000p2 + 20,000p3 = 120,000YY: 100,000p1 + 100,000p2 + 40,000p3 = 85,725Government bond: 100,000p1 + 100,000p2 + 100,000p3 = 100,000/(1+10%) = 90,909Solve the equations, we have p1 = 0.3774, p2=0.4453, p3=0.0864The total value of GAG equity is80,000p1 + 50,000p2 = $52457The per share price of equity is $5.25.d.The value of the bond and the value of the equity add up to the value of theasset, so,V bond = V asset– V equity = 120,000 – 52547 = $67543The yield-to-maturity of bond is (80,000-67543)/67543 = 18.4%。

金融数学课后习题答案 2

第一章习题答案1. 设总量函数为A(t) = t2 + 2t + 3 。

试计算累积函数a(t) 和第n 个时段的利息In 。

解: 把t = 0 代入得A(0) = 3 于是:a(t) =A(t)A(0)=t2 + 2t + 33In = A(n) − A(n −1)= (n2 + 2n + 3) −((n −1)2 + 2(n −1) + 3))= 2n + 12. 对以下两种情况计算从t 时刻到n(t < n) 时刻的利息: (1)Ir(0 < r <n); (2)Ir = 2r(0 < r < n).解:(1)I = A(n) − A(t)= In + In¡1 + ・・・+ It+1=n(n + 1)2− t(t + 1)2(2)I = A(n) − A(t)=Σnk=t+1Ik =Σnk=t+1Ik= 2n+1 −2t+13. 已知累积函数的形式为: a(t) = at2 + b 。

若0 时刻投入的100 元累积到3 时刻为172 元，试计算：5 时刻投入的100 元在10 时刻的终值。

第1 页解: 由题意得a(0) = 1, a(3) =A(3)A(0)= 1.72⇒ a = 0.08, b = 1∴A(5) = 100A(10) = A(0) ・ a(10) = A(5) ・ a(10)a(5)= 100 ×3 = 300.4. 分别对以下两种总量函数计算i5 和i10 :(1) A(t) = 100 + 5t; (2) A(t) = 100(1 + 0.1)t.解：(1)i5 =A(5) − A(4)A(4)=5120≈4.17%i10 =A(10) − A(9)A(9)=5145≈3.45%(2)i5 =A(5) − A(4)A(4)=100(1 + 0.1)5 −100(1 + 0.1)4100(1 + 0.1)4= 10%i10 =A(10) − A(9)A(9)=100(1 + 0.1)10 −100(1 + 0.1)9100(1 + 0.1)9= 10%第2 页5.设A(4) = 1000, in = 0.01n. 试计算A(7) 。

《金融数学》(第二版)练习题(修订版)

1.21 投资者 A 今天在一项基金中存入 10，5 年后存入 30，已知此项基金按单利 11%计息；投资者 B 将进行同样数额的两笔存款，但是在 n 年后存入 10，在 2n 年后存入 30，已知此项基金按复利 9.15%计息。在第 10 年末，两基金的累积值相等。求 n。
1.22
已知利息力为 δ t
批注: 第二版书稿中有错
2.16 一项年金从 2000 年 1 月 1 日开始，每月末支付 100 元，支付 60 次；这项年金的价值等价于在第 K 月末支付一笔 6000 元的款项。每月复利一次的名义利率为 12%。求 k。
2.17 如果 a = x ， a = y ，试将 d 表示为 x 和 y 的函数。
2.22 投资者在 t=0 和 t=10 时分别向一项基金投资 12，这项基金以年实际利率 i 计息。利息在年末支付，并以 0.75i 的年实际利率进行再投资。在 t=20 时，再投资利息的累积值为 64，求 i。
2.23 如果利息力为常数 δ = 1 ，求 a 的表达式。
1+ t
n
5
2.24 给定利息力 δ = 1 ，0≤ t ≤5，请计算 s 。
1+ 0.5t
5
∫ 2.25
已知
8
a
0t
dt
= 100 。请计算 a 10
。
2.26
如果 3a (2) n|
=
2a (2) 2n|
=
45s ( 2 ) 1|
，试计算
i
是多少？
2.27 当 t 为多少时，在时刻 t 支付 1 元相当于将这 1 元在时刻 0 与 1 之间连续支付？
2.28
已知 a = 4 ， s =12 ，求利息力。

金融数学引论答案第二版

金融数学引论答案第二版【篇一：北大版金融数学引论第二章答案】>第二章习题答案1．某家庭从子女出生时开始累积大学教育费用5万元。

如果它们前十年每年底存款1000元，后十年每年底存款1000+x 元，年利率7%。

计算x 。

解：s = 1000s?7%+xs?7%20p10p20px = 50000 ? 1000s?7% = 651.72s?p7%102．价值10,000元的新车。

购买者计划分期付款方式：每月底还250元，期限4年。

月结算名利率18%。

计算首次付款金额。

解：设首次付款为x ，则有10000 = x + 250a?p1.5%48解得x = 1489.3613．设有n年期期末年金，其中年金金额为n，实利率i =n解：p v = na?npi= 1nn+2 =(n + 1)nn2n4．已知：a?pn= x，a?p2n= y 。

试用x和y 表示d 。

解： a?p2n= a?pn+ a?p (1 ? d)则nny ? xd = 1 ? ( x ) n5．已知：a?p7= 5.58238, a?= 7.88687, a?= 10.82760。

计算i。

11p18p解：a?p = a?p + a?p v718711解得=i = 6.0%10?p +a∞?p6.证明： 11?v10s。

解：p v = 100a?+ 100a20?8p3% p3% = 2189.7168．某人现年40岁，现在开始每年初在退休金帐号上存入1000元，共计25年。

然后，从65岁开始每年初领取一定的退休金，共计15年。

设前25年的年利率为8%，后15年的年利率7%。

计算每年的退休金。

解：设每年退休金为x，选择65岁年初为比较日=解得x = 8101.658。

金融学第二版课后习题答案

金融学第二版课后习题答案【篇一：王重润公司金融学第二版课后答案】业有几种组织方式？各有什么特点？（ 1）有两种，有限责任公司和股份有限责任公司（ 2）有限责任公司特点：有限责任公司是指股东以其出资额为限对公司承担责任，公司以其全部资产对公司的债务承担责任的企业法人；有限责任公司注册资本的最低限额为人民币3万元；其资本并不必分为等额股份，也不公开发行股票，股东持有的公司股票可以再公司内部股东之间自由转让，若向公司以外的人转让，须经过公司股东的同意；公司设立手续简便，而且公司无须向社会公开公司财务状况。

（ 3）股份有限责任公司特点：1、有限责任2、永续存在3、股份有限责任公司的股东人数不得少于法律规定的数目，我国规定设立股份有限公司，应当有2人以上200人以下为发起人4、股份有限责任公司的全部资本划分为等额的股份，通过向社会公开发行的办法筹集资金，任何人在缴纳了股款之后，都可以成为公司股东，没有资格限制。

5、可转让性6、易于筹资2题：为什么我国《公司法》允许存在一人有限责任公司？一人有限责任公司与个人独资企业有何不同？答：1.就立法初衷而言，许可自然人投资设立一人有限责任公司的重要考虑是减少实质上的一人公司的设立，简化和明晰股权归属，减少纷争。

以往由于我国《公司法》禁止设立一人公司，使得投资人通过各种途径设立或形成的实质上的一人公司大量存在，挂名股东与真实股东之间的投资权益纠纷以及挂名股东与公司债权人之间的债务纠纷不断，令工商行政管理部门和司法机关无所适从。

在修订《公司法》的过程中，法律委员会、法制工作委员会会同国务院法制办、工商总局、国资委、人民银行和最高人民法院反复研究认为：从实际情况看，一个股东的出资额占公司资本的绝大多数而其他股东只占象征性的极少数，或者一个股东拉上自己的亲朋好友作挂名股东的有限责任公司，即实质上的一人公司，已是客观存在，也很难禁止。

根据我国的实际情况，并研究借鉴国外的通行做法，应当允许一个自然人投资设立有限责任公司。

《_金融数学-课后习题答案》

金融数学-课后习题答案本文档为金融数学课后习题的参考答案。

在解答问题时，我会尽量给出详细的步骤和推导过程，帮助读者更好地理解金融数学的概念和方法。

1. 第一章：时间价值1.1 问题一题目：如果我现在存入1000元，年利率是5%，请问5年后我能得到多少钱？解答：首先需要计算每年的复利，即每年利息和本金的总和。

根据复利计算公式：年末总金额 = 本金 * (1 + 年利率)^时间年数代入数据进行计算：年末总金额 = 1000 * (1 + 0.05)^5 = 1000 * 1.2762815625 ≈ 1281.28元因此，5年后你能得到大约1281.28元。

1.2 问题二题目：如果我希望在5年后拥有2000元，年利率是5%，请问我需要存入多少钱？解答：首先需要计算本金与利息的比例，然后根据比例计算需要的本金。

根据复利计算公式：年末总金额 = 本金 * (1 + 年利率)^时间年数可以将该式转化为：本金 = 年末总金额 / (1 + 年利率)^时间年数代入数据进行计算：本金 = 2000 / (1 + 0.05)^5 = 2000 / 1.2762815625 ≈ 1567.45元因此，你需要存入大约1567.45元。

2. 第二章：贴现与现值2.1 问题一题目：如果一笔未来支付3000元的现金流在5年后，年利率是6%，请问它的现值是多少？解答：为了计算现值，我们需要使用贴现率（年利率）和时间年数。

根据贴现计算公式：现值 = 未来支付金额 / (1 + 年利率)^时间年数代入数据进行计算：现值= 3000 / (1 + 0.06)^5 = 3000 / 1.33822557689 ≈ 2241.53元所以，该未来支付的现金流的现值大约为2241.53元。

2.2 问题二题目：如果我希望在5年后得到3000元的现金流，年利率是6%，请问我愿意支付多少现值？解答：为了计算现值，我们使用贴现率（年利率）和时间年数。

《金融学(第二版)》讲义大纲及课后习题答案详解第16章

CHAPTER 16CAPITAL STRUCTUREObjectives•To understand how a firm can create value through its financing decisions. •To show how to take account of a firm’s financing mix in evaluating investment decisions.Outline16.1 Internal versus External Financing16.2 Equity Financing16.3 Debt Financing16.4 The Irrelevance of Capital Structure in a Frictionless Environment16.5 Creating Value through Financing Decisions16.6 Reducing Costs16.7 Dealing with Conflicts of Interest16.8 Creating New Opportunities for Stakeholders16.9 Financing Decisions in Practice16.10 How to Evaluate Levered InvestmentsSummary•External financing subjects a corporation’s investment plans more directly to the discipline of the capital market than internal financing does.•Debt financing in its broadest sense includes loans and debt securities, such as bonds and mortgages, as well as other promises of future payment by the corporation, such as accounts payable, leases, and pensions.•In a frictionless financial environment, where there are no taxes or transaction costs, and contracts are costless to make and enforce, the wealth of shareholders is the same no matter what capital structure the firm adopts.•In the real world there are a number of frictions that can cause capital structure policy to have an effect on the wealth of shareholders. These include taxes, regulations, and conflicts of interest between the stakeholders of the firm. A firm’s management might therefore be able to create shareholder val ue through its capital structure decisions in one of three ways:•By reducing tax costs or the costs of burdensome regulations.•By reducing potential conflicts of interest among various stakeholders in the firm.•By providing stakeholders with financial assets not otherwise available tothem.•There are three alternative methods used in estimating the net present value of an investment project to take account of financial leverage: the adjusted present value method, the flows to equity method, and the weighted average cost of capital methodSolutions to Problems at End of ChapterDebt-Equity Mix1. Divido Corporation is an all-equity financed firm with a total market value of $100 million.The company holds $10 million in cash-equivalents and has $90 million in other assets.There are 1,000,000 shares of Divido common stock outstanding, each with a market price of $100.Divido Corporation has decided to issue $20 million of bonds and to repurchase $20 million worth of its stock.a.What will be the impact on the price of its shares and on the wealth of itsshareholders?Why?b.Assume that Divido’s EBIT has an equal probability of being $20 million,or $12 million, or $4million.Show the impact of the financial restructuring on the probability distribution of earnings per share in the absence oftaxes.Why does the fact that the equity becomes riskier not necessarily affect shareholder wealth?SOLUTION:a.In an M&M frictionless environment, where there are no taxes and contractsare costless to make and enforce, the wealth of shareholders is the same no matter what capital structure the firm adopts. In such an environment, neither the stock price nor shareholders’ wealth would be affected. In the real world Divido’s management might be ab le to create shareholder value by issuing debt and repurchasing shares in two ways:By reducing tax costsBy reducing the free cash flow available to management and exposing itself to greater market discipline.b.The formula for EPS without debt is:EPS all equity =EBIT1,000,000 sharesThe interest payments will be $1.2 million per year (.06 x $20 million)regardless of the realized value of EBIT. The number of shares outstanding after exchanging debt for equity will be 800,000. EPS with debt is therefore: EPS with debt = Net Earnings= EBIT – $1.2 million800,000 shares 800,000 sharesAlthough the shares of stock become riskier with debt financing, the expected earnings per share go up. In a frictionless financial environment, the net effect is to leave the price of the stock unaffected.Leasing2. Plentilease and Nolease are virtually identical corporations.The only difference between them is that Plentilease leases most of its plant and equipment whereas Nolease buys its plant and equipment and finances it by pare and contrast their market-value balance sheets.SOLUTION:Market-Value Balance Sheets of Nolease and Plentilease CorporationsThe main difference between the bonds and the lease as a form of debtfinancing is who bears the risk associated with the residual market value of the leased asset at the end of the term of the lease. Since Nolease Corporation has bought its equipment, it bears this risk. In Plentilease’s case, however, it is the lessor that bears this residual-value risk.Pension Liabilities3. Europens and Asiapens are virtually identical corporations.The only difference between them is that Europens has a completely unfunded pension plan, and Asiapens has a fully funded pension pare and contrast their market-value balance sheets.What difference does the funding status of the pension plan make to the stakeholders of these two corporations?SOLUTION:Balance Sheets of Asiapens and Europens CorporationsAsiapens has funded its pension plan by issuing bonds and investing the funds raised in a segregated pool of pension fund assets. These pension assets take the form of a diversified portfolio of stocks and bonds issued by othercompanies and serve as collateral for the pension benefits promised byAsiapens to its employees. In the case of Europens, there is no segregated poolof pension assets. The pension promises of Europens are backed by the assets of the company itself. Therefore, the employees of Asiapens are more secure about receiving their promised pension benefits, since the benefits arecollateralized by a more diversified portfolio of assets. In the case of bothcompanies, however, any unfunded pension liability reduces shareholdersequity.4. Comfort Shoe Company of England has decided to spin off its Tango Dance Shoe Division as a separate corporation in the United States.The assets of the Tango Dance Shoe Division have the same operating risk characteristics as those of Comfort.The capital structure of Comfort has been 40% debt and 60% equity in terms of marketing values, and is considered by management to be optimal.The required return on Comfort’s assets (if unlevered) is 16% per year, and the interest rate that the firm (and the division) must currently pay on their debt is 10% per year.Sales revenue for the Tango Shoe Division is expected to remain indefinitely at last year’s level of $10 million.Variable costs are 55% of sales.Annual depreciation is $1 million, which is exactly matched each year by new investments.The corporate tax rate is 40%.a.How much is the Tango Shoe Division worth in unlevered form?b.If the Tango Shoe Division is spun off with $5 million in debt, how muchwould it be worth?c.What rate of return will the shareholders of the Tango Shoe Divisionrequire?d.Show that the market value of the equity of the new firm would be justifiedby the earnings to the shareholders.SOLUTION:a.The unlevered free cash flow for the Tango Shoe Division would be (in$millions):Sales: $10.0Var. Cost: -5.5Depreciation - 1.0Taxable Income $ 3.5Taxes (@40%) -1.4After-Tax Income $2.1Depreciation 1.0Investment -1.0Free Cash Flow $2.1 millionUnlevered, Tango is worth: $2.1 million / 0.16 = $13.125 millionb.If Tango had $5 million of debt, its total value would be:Market Value with Debt = Market Value without Debt + PV of Interest Tax ShieldV L = V U + T x B= $13.125 + (.4 x 5) = $15.125 millionTango Equity = $15.125 - $5 = $10.125 millionc.Tango’s cost of equity capital would be .1778k e = k + (1-T)(k - r)D/E = .16 + (1-.4)(.16-.10)x 5/10.125 = .1778d.The value of the equity should be the present value of the expected net incomediscounted at the required rate of return on equity. The expected net income will be the unlevered cash flow less the after-tax cost of the interest of the debt: $2.1 - (.6) (.1 x $5) = $2.1 - $.3 = $1.8 million per yearS = $1.8 million / .1778 = $10.125 million5. Based on the above problem, Suppose that Foxtrot Dance Shoes makes custom designed dance shoes and is a competitor of Tango Dance Shoes. Foxtrot has similar risks and characteristics as Tango except that it is completely unlevered.Fearful that Tango Dance Shoes may try to take over Foxtrot in order to control their niche in the market, Foxtrot decides to lever the firm to buy back stock.a.If there are currently 500,000 shares outstanding, what is the value ofFoxtrot’s stock?b.How many shares can Foxtrot buy back and at what value if it is willing toborrow 30% of the value of the firm?c.What if it is willing to borrow 40% of the value of the firm?d.Should Foxtrot borrow more?SOLUTION:a.Current price per share: $13.125 million /.5 million shares = $26.25 per shareb.@30% debtAmount to borrow: 30% of 13.125 million = $3.9375 millionPV of Tax Shield = .4 x $3.9375 million = $1.575 millionValue of levered firm = $13.125 + $1.575 = $14.7 millionValue of equity in levered firm = $14.7 million $3.9375 million = $10.7625 millionTo compute the number of shares Foxtrot can repurchase, we need to know the price per share.If Foxtrot can repurchase shares at the existing price of $26.25 then the number of shares retired will be$3.9375 million/$26.25 per share = .15 million shares. This will leave .35million shares outstanding, and the price of each share will be $10.7625million/.35 million = $30.75.If the PV of the tax shield gets incorporated in the price of the shares before the repurchase, then the price of the shares will increase by $1.575 million/.5million = $3.15. So the price of the repurchased shares will be$26.25 + $3.15 = $29.40.Then the number of shares retired will be $3.9375 million/$29.40 per share = 133,929 shares. This will leave 366,071 shares outstanding each with a price of $29.40.c.@40% debtAmount to borrow: 40% of $13.125 million = $5.25 millionPV of Tax Shield = .4 x $5.25 million = $2.1 millionValue of levered firm = $13.125 + $2.1 = $15.225 millionValue of equity in levered firm = $15.225 million $5.25 million = $9.975 millionIf Foxtrot can repurchase shares at the existing price of $26.25 then the number of shares retired will be$5.25 million/$26.25 per share = .2 million shares. This will leave .3 million shares outstanding, and the price of each share will be $9.975 million/.3 million = $33.25.If the PV of the tax shield gets incorporated in the price of the shares before the repurchase, then the price of the shares will increase by $2.1 million/.5 million = $4.20. So the price of the repurchased shares will be$26.25 + $4.20 = $30.45.Then the number of shares retired will be $5.25million/$30.45 per share =172,414 shares. This will leave 327,586 shares outstanding each with a price of $30.45.d. Foxtrot’s management must trade off the tax savings due to additional debtfinancing against the costs of financial distress that rise with the degree of debt financing.6. Hanna-Charles Company needs to add a new fleet of vehicles for their sales force.The purchasing manager has been working with a local car dealership to get the best value for the company dollar.After some negotiations, a local dealer has offered Hanna-Charles two options:1) a three year lease on the fleet of cars or 2) 15% off the top to purchase outright. Option 2 would cost Hanna-Charles company about 5% less than the lease option in terms of present value.a.What are the advantages and disadvantages of leasing?b.Which option should the purchasing manager at Hanna-Charles pursueand why?SOLUTION:a.Advantages:The lessor bears all the residual-value riskTax BenefitsNo disposal concerns (or resale) when life of equipment is expended.Disadvantages:No ownership while maintaining maintenance responsibilityb.Lease or Buy:Hanna-Charles company should lease. Although they may spend more with the lease, they do not bear the residual-value risk.7. Havem and Needem companies are exactly the same differing only in their capital structures.Havem is an unlevered firm issuing only stocks whereas Needem issues stocks and bonds. Neither firm pays corporate taxes. Havem pays out all of its yearly earnings in the form of dividends and has 1 million shares outstanding.Its market capitalization rate is 11% and the firm is currently valued at$180 million.Needem is identical except that 40% of its value is in bonds and has 500,000 shares outstanding. Needem’s bonds are risk free and pay a coupon of 9% per year and are rolled over every year.a.What is the value of Needem’s shares?b.As an investor forecasting the upcoming year, you examine Havem andNeedem using three possible states of the economy that are all equallylikely: normal, bad, and exceptional.Assuming the earnings will be the same, one half, and one and a half respectively, produce a table that shows the earnings and the earnings per share for both Havem and Needem in all three scenarios.SOLUTION:a.Needem has $72 million in debt and $108 million in equity. Since there are500,000 shares, the value of each share is $216.b.Expected EBIT = $180 million x 11% = $19.8 million per yearInterest expense for Needem = $72 million x .09 = $6.48 million per yearb EBIT – Interest Expense8. Using the foregoing example, let us now assume that Havem and Needem must pay taxes at the rate of 40% annually.Given the same distribution of possible outcomes as previously:a.What are the possible after-tax cash flows for Havem and Needem?b.What are the values of the shares?c.If one was not risk averse, which company would that person invest in? SOLUTION:a.After-tax CF Havem = (1 -Tax Rate) EBIT = Net incomeNet income Needem = (1 -Tax Rate) (EBIT - Int. Pmt.)CF Needem = (1 -Tax Rate) (EBIT) + Tax Rate x Int. PmtCF Needem (bad) = (.60) $9.9 + (.40) x 6.48 = $8.532 millionCF Needem (normal) = (.60) $19.8 + (.40) x 6.48 = $14.472 millionCF Needem (except.) = (.60) $29.7 + (.40) x 6.48 = $20.412 millionb.The equity of Havem will be worth $11.88 million /.11 = $108 million or $108per shareThe total value of Needem’s debt + equity will be $108,000,000 + .4 x $72,000,000 = $136.8 million.Needem’s equity will be worth $136,800,000- $72,000,000 = $64.8 million.Since there are .5 million shares of Needem, each will be worth $129.60.c.Needem.9. The Griffey-Lang Food Company faces a difficult problem.In management’s effort to grow the business, they accrued a debt of $150 million while the value of the company is only $125 million.Management must come up with a plan to alleviate the situation in one year or face certain bankruptcy. Also upcoming are labor relations meetings with the union to discuss employee benefits and pension funds. Griffey-Lang at this time has three choices they can pursue: 1) Launch a new, relatively untested product that if successful (probability of .12) will allow G-L to increase the value of the company to $200 million, 2) Sell off two food production plants in an effort to reduce some of the debt and the value of the company thus making it even (.45 probability of success), or do nothing (probability of failure = 1.0).a.As a creditor, what would you like Griffey-Lang to do, and why?b.As an investor?c.As an employee?SOLUTION:a.As a Creditor:Option 2 best suits the creditor. Option 2 allows the creditor to regain some value through the sale of plant and equipment.b.As a shareholder:The shareholders have nothing to lose and everything to gain by taking a big chance with the new product.c.As an Employee:Selling off two production plants will eliminate jobs. Doing nothing means certain bankruptcy and may result in liquidation of the firm and the loss of all the jobs. For the employees, the best choice is option 1.。

《金融学(第二版)》讲义大纲及课后习题答案详解十四章

CHAPTE R 14FORWARD AND FUTURE S PRICE SObjectives∙ To explain the economic role of futures markets∙To show what information can and cannot be inferred from forward and futures prices.Outline14.1 Distinctions Between Forward and Futures Contracts14.2 The Economic Function of Futures Markets14.3 The Role of Speculators14.4 Relation Between Commodity Spot and Futures Prices14.5 Extracting Information from Commodity Futures Prices14.6 Spot-Futures Price Parity for Gold14.7 Financial Futures14.8 The Implied Risk-Free Rate14.9 The Forward Price Is Not a Forecast of the Spot Price14.10 Forward-Spot Parity with Cash Payouts14.11 Implied Dividends14.12 The Foreign-Exchange Parity Relation14.13 The Role of Expectations in Determining Exchange RatesSummary∙ Futures contracts make it possible to separate the decision of whether to physically store a commodity from thedecision to have financial exposure to its price changes.∙ Speculators in futures markets improve the informational content of futures prices and make futures marketsmore liquid than they would otherwise be.∙ The futures price of wheat cannot exceed the spot price by more than the cost of carry:∙ The forward-spot price parity relation for gold is that the forward price equals the spot price times the cost ofcarry:This relation is maintained by the force of arbitrage . ∙One can infer the implied cost of carry and the implied storage costs from the observed spot and forward prices and the risk-free interest rate. ∙ The forward-spot parity relation for stocks is that the forward price equals the spot price times 1 plus the risk-free rate less the expected cash dividend.This relation can therefore be used to infer the implied dividend from the observed spot and forward prices and the risk-free interest rate.∙ The forward-spot price parity relation for the dollar/yen exchange rate involves two interest rates:where F is the forward price of the yen, S is the current spot price, r Y is the yen interest rate, and r $ is the dollarinterest rate.∙If the forward dollar/yen exchange rate is an unbiased forecast of the future spot exchange rate, then one can infer that forecast either from the forward rate or from the dollar-denominated and yen-denominated risk-free interest rates. F S C-≤F S r s =++()1F S r D=+-()1F r S r Y11+=+$Solutions to Problems at End of ChapterForward Contracts and Forward-Spot Parity.1. Suppose that you are planning a trip to E ngland. The trip is a year from now, and you have reserved a hotel room in London at a price of ₤ 50 per day. You do not have to pay for the room in advance. The exchange rate is currently $1.50 to the pound sterling.a.E xplain several possible ways that you could completely hedge the exchange rate risk in this situation.b.Suppose that r₤=.12 and r$=.08. Because S=$1.50, what must the forward price of the pound be?c.Show that if F is $0.10 higher than in your answer to part b, there would be an arbitrage opportunity. SOLUTION:a.Ways to hedge the exchange rate risk:Pay for the room in advanceBuy the pounds you will need in the forward market.Invest the present value of the rental payments in a pound-denominated riskless asset.b. F = S (1+r$)/(1+r£) = $1.50 x 1.08/1.12 = $1.4464 per poundc.If F is $1.55 then arbitrage profits can be made by borrowing dollars, investing in pounds and selling themforward at the inflated forward price. After paying off principle and interest on the dollars borrowed, you would have pure arbitrage profits left over. For example,Borrow $1.50,Convert it into 1 pound,Invest it in pound-denominated bonds to have 1.12 pounds a year from now,Sell 1.12 pounds forward at $1.55 per pound to have $1.736 a year from now,After 1 year, pay off the principle and interest on the loan ($1.50x 1.08 = $1.62).This series of transactions leaves you with $.116 a year from now with no initial outlay of your money.Forward-Spot Parity Relation with Known Cash Payouts2. Suppose that the Treasury yield curve is flat at an interest rate of 7% per year (compounded semiannually).a.What is the spot price of a 30-year Treasury bond with an 8% coupon rate assuming coupons are paidsemiannually?b.What is the forward price of the bond for delivery six months from now?c.Show that if the forward price is $1 lower than in your answer to part b, there should be an arbitrageopportunity.SOLUTION:b. The forward price for delivery six months from now is $1,124.089:F = S(1+r) - C = $1,124.724 x 1.035 - 40 =$1,124.089c. If the forward price is only $1,123.089, then arbitrage profits can be made by selling the bond short and buying itforward at the low forward price. It can be described as follows:Sell short a bond at $1,124.724; buy it forward at $1,123.089; invest the proceeds of the short sale to earn 3.5% for6 monthsAfter 6 months, take delivery of the bond and cover your short saleForward-Spot Parity Relation with Uncertain Dividends3. A stock has a spot price of $100; the riskless interest rate is 7% per year (compounded annually), and the expected dividend on the stock is $3, to be received a year from now.a.What should be the one-year futures price?b.If the futures price is $1 higher than your answer to part a, what might that imply about the expected dividend? SOLUTION:a.S = $100, r = .07, D = $3. F = S ( 1+r) - D = $104b.If F is $105, that might imply that D is really only $2.Storage Costs versus Dividend Yield4. Compare the forward-spot price-parity relation for gold to the one for stocks. Is it fair to say that stocks have a negative storage cost equal to the dividend yield?SOLUTIONOne could definitely say that stocks have a negative storage cost equal to the dividend.5. Suppose you are a distributor of canola seed and you observe the spot price of canola to be $7.45 per bushel while the futures price for delivery one month from today is $7.60. Assuming a $.10 per bushel carrying cost, what would you do to hedge your price uncertainty?SOLUTIONWe see that F> S+C. If you short the futures contract, you can sell your seed at $7.60 per bushel.6. Infer the spot price of an ounce of gold if you observe the price of one ounce of gold for forward delivery in three months is $435.00, the interest rate on a 91-day Treasury bill is 1% and the quarterly carrying cost as a percentage of the spot price is .2%.SOLUTIONDeduce from the futures price parity condition for gold that F = S0 (1 + r + s) so that S0 = $429.84.7. You are a dealer in kryptonite and are contemplating a trade in a forward contract. You observe that the current spot price per ounce of kryptonite is $180.00, the forward price for delivery of one ounce of kryptonite in one year is $205.20, and annual carrying costs of the metal are 4% of the current spot price.a.Can you infer the annual return on a riskless zero-coupon security implied by the Law of One Price?b.Can you describe a trading strategy that would generate arbitrage profits for you if the annual return on theriskless security is only 5%? What would your arbitrage profit be, per ounce of kryptonite?SOLUTIONa.By no-arbitrage, we require that the riskless rate r satisfy:F = S0 (1 + r + s)205.2 = 180 (1 +r +.04) = 187.2 + 180rr = 18/180 = .10 or 10%b.The implicit risk-free rate that you can earn by buying kryptonite, storing it, and selling it forward at $205.2 perounce is 10%. If the riskless borrowing rate is five percent, you should borrow at that rate and invest in hedged kryptonite. If you buy an ounce of kryptonite for $180, you will get $205.2 for it for sure a year from now. If you borrow the $180, you will have to pay principal and interest of $180 x 1.05 plus another .04 x $180 in storage costs.This totals $196.2, thus leaving you with $9 in arbitrage profits.8. Calculate the implicit cost of carrying an ounce of gold and the implied storage cost per ounce of gold if the current spot price of gold per ounce is $425.00, the forward price of an ounce of gold for delivery in 273 days is $460.00, the yield over 91 days on a zero-coupon Treasury bill is 2% and the term structure of interest rates is flat. SOLUTIONFirst, we solve it assuming a simple compounding method for the risk free interest rate. Over 273 days, the Risk free rate is 2%*3=6%. Therefore we have,F = S (1 + r + s )460 = 425 (1.06 + s)s = (460 - 450.5)/425 = 9.5/425 = .02235 for 273 daysThus the carrying costs are roughly 8.24% for 273 days or 10.98% per year.Second, we solve it assuming we need to compound the interest rates. The risk free rate over 273 days will be(1+2%)3-1=6.12%.plug in the above formulae we get s=.021145 for 273 days.Thus the carrying costs are roughly 8.23% for 273 days or 11.13% per year.9. The forward price for a share of stock to be delivered in 182 days is $410.00, whereas the current yield on a 91-day T-bill is 2%. If the term structure of interest rates is fiat, what spot price for the stock is implied by the Law of One Price?SOLUTIONF = $410; r = .02 per quarter.S = F/(1+r)2 = $394.0810. You observe that the one-year forward price of a share of stock in Kramer,Inc.,a New York tour-bus company and purveyor of fine clothing, is $45.00 while the spot price of a share is $41.00. If the riskless yield on a one-year zero-coupon government bond is 5%:a.What is the forward price implied by the Law of One Price?b.Can you devise a trading strategy to generate arbitrage profits? How much would you earn per share?SOLUTIONa.The no-arbitrage value of the forward price is F = $43.05.b.The observed forward price is excessive. Consider short-selling a forward contract and taking a long position ina portfolio consisting of one stock and the sale of a bond with face value of F. Future liabilities for this positionare zero, while the current cash inflow is $1.86.11. Infer the yield on a 273-day, zero-coupon Japanese government security if the spot price of a share of stock in Mifune and Associates is 4,750 yen whereas the forward price for delivery of a share in 273 days is 5,000 yen.SOLUTIONThe implied yield over the 273 day term is r = 5.26%.12. On your first day of trading in Vietnamese forward contracts, you observe that the share price of Giap Industries is currently 54, 000 dong while the one-year forward price is 60, 000 dong. If the yield on a one-year riskless security is fifteen percent, are arbitrage profits possible in this market? If not, explain why not. If so, devise an appropriate trading strategy.SOLUTIONArbitrage profits would seem to be possible, since the no-arbitrage forward price implied by these parameters isF = $62,100.The futures contract is underpriced, relative to this no-arbitrage value. Consider taking a long position in the forward contract and simultaneously selling a share of Giap stock and buying a riskless bond with a face value equal to the observed forward price. The liabilities from these joint positions are zero, while the current cash inflow is $1826.09.13. The share price of Schleifer and Associates, a financial consultancy in Moscow, is currently 10, 000 roubles whereas the forward price for delivery of a share in 182 days is 11,000 roubles. If the yield on a riskless zero-coupon security with term to maturity of 182 days is 15%, infer the expected dividend to be paid by Schleifer and Associates over the next six months.SOLUTIONThe implied dividend is 500 roubles.14. The spot rate of exchange of yen for Canadian dollars is currently 113 yen per dollar but the one-year forward rate is 110 yen per dollar. Determine the yield on a one-year zero-coupon Canadian government security if the corresponding yield on a Japanese government security is 2.21%.SOLUTIONThe implied Canadian rate over this term is approximately 5.00%.。

《金融学（第二版）》讲义大纲及课后习题答案详解第八章

《⾦融学（第⼆版）》讲义⼤纲及课后习题答案详解第⼋章CHAPTER 8VALUATION OF KNOWN CASH FLOWS: BONDSObjectivesTo show how to value contracts and securities that promise a stream of cash flows that are known with certainty. To understand the shape of the yield curve .To understand how bond prices and yields change over time.Outline8.1 Using Present Value Formulas to Value Known Cash Flows8.2 The Basic Building Blocks: Pure Discount Bonds8.3 Coupon Bonds, Current Yield, and Yield to Maturity8.4 Reading Bond Listings8.5 Why Yields for the Same Maturity Differ8.6 The Behavior of Bond Prices over TimeSummaryA change in market interest rates causes a change in the opposite direction in the market values of all existing contracts promising fixed payments in the future.The market prices of $1 to be received at every possible date in the future are the basic building blocks for valuing all other streams of known cash flows. These prices are inferred from the observed market prices of traded bonds and then applied to other streams of known cash flows to value them.An equivalent valuation can be carried out by applying a discounted cash flow formula with a different discount rate for each future time period.Differences in the prices of fixed-income securities of a given maturity arise from differences in coupon rates, default risk, tax treatment, callability, convertibility, and other features.Over time the prices of bonds converge towards their face value. Before maturity, however, bond prices can fluctuate a great deal as a result of changes in market interest rates.Solutions to Problems at End of ChapterBond Valuation with a Flat Term Structure1. Suppose you want to know the price of a 10-year 7% coupon Treasury bond that pays interest annually.a. You have been told that the yield to maturity is 8%. What is the price?b. What is the price if coupons are paid semiannually, and the yield to maturity is 8% per year?c. Now you have been told that the yield to maturity is 7% per year. What is the price? Could you haveguessed the answer without calculating it? What if coupons are paid semiannually?SOLUTION:c. Price = 100. When the coupon rate and yield to maturity are the same, the bond sells at par value (i.e. the priceequals the face value of the bond).2. Assume six months ago the US Treasury yield curve was flat at a rate of 4% per year (with annual compounding) and you bought a 30-year US Treasury bond. Today it is flat at a rate of 5% per year. What rate of return did you earn on your initial investment:a.If the bond was a 4% coupon bond?b.If the bond was a zero coupon bond?c.How do your answer change if compounding is semiannual?SOLUTION:a and b.Step 3: Find rates of return:Rate of return = (coupon + change in price)/initial price4% coupon bond: r = (4 + 84.74 - 100)/100 = -0.1126 or -11.26%Zero-coupon bond: r = (0 + 23.71 - 30.83)/30.83 = -0.2309 or -23.09%. Note that the zero-coupon bond is more sensitive to yield changes than the 4% coupon bond.c.Step 2: Find prices of the bonds today:Step 3: Find rates of return:Rate of return = (coupon + change in price) / initial price4% coupon bond: r= (2 + 84.66 - 100)/100 = -0.1334 or -13.34%Zero coupon bond: r= (0 + 23.30 - 30.48)/30.48 = -0.2356 or -23.56%. Note that the zero-coupon bond is more sensitive to yield changes than the 4% coupon bond.Bond Valuation With a Non-Flat Term Structure3. Suppose you observe the following prices for zero-coupon bonds (pure discount bonds) that have no risk of default:a.What should be the price of a 2-year coupon bond that pays a 6% coupon rate, assuming couponpayments are made once a year starting one year from now?b.Find the missing entry in the table.c.What should be the yield to maturity of the 2-year coupon bond in Part a?d.Why are your answers to parts b and c of this question different?SOLUTION:a. Present value of first year's cash flow = 6 x .97 = 5.82Present value of second year's cash flow = 106 x .90 = 95.4Total present value = 101.22maturity.Coupon Stripping4. You would like to create a 2-year synthetic zero-coupon bond. Assume you are aware of the following information: 1-year zero- coupon bonds are trading for $0.93 per dollar of face value and 2-year 7% coupon bonds (annual payments) are selling at $985.30 (Face value = $1,000).a. What are the two cash flows from the 2-year coupon bond?b. Assume you can purchase the 2-year coupon bond and unbundle the two cash flows and sell them.i. How much will you receive from the sale of the first payment?ii. How much do you need to receive from the sale of the 2-year Treasury strip to break even? SOLUTION:a. $70 at the end of the first year and $1070 at the end of year 2.b. i. I would receive .93 x $70 = $65.10 from the sale of the first payment.ii. To break even, I would need to receive $985.30- $65.10 = $920.20 from the sale of the 2-year strip.The Law of One price and Bond Pricing5. Assume that all of the bonds listed in the following table are the same except for their pattern of promised cash flows over time. Prices are quoted per $1 of face value. Use the information in the table and the Law of One Price to infer the values of the missing entries. Assume that coupon payments are annual.SOLUTION:From Bond 1 and Bond 4, we can get the missing entries for the 2-year zero-coupon bond. We know from bond 1 that:1.0092 = 0.06/1.055 +1.06/(1.055)2. This is also equal to 0.06/(1+z1) + 1.06/(1+z2)2 where z1 and z2 are the yields to maturity on one-year zero-coupon and two-year zero-coupon bonds respectively. From bond 4 , we have z1, we can find z2.1.0092 – 0.06/1.0526 = 1.06/(1+z2)2, hence z2 = 5.51%.To get the price P per $1 face value of the 2-year zero-coupon bond, using the same reasoning:1.0092 – 0.06x0.95 = 1.06xP, hence P = 0.8983To find the entries for bond 3: first find the price, then the yield to maturity. To find the price, we can use z1 and z2 found earlier:PV of coupon payment in year 1: 0.07 x 0.95 = 0.0665PV of coupon + principal payments in year 2: 1.07 x 0.8983 =0.9612Bond Features and Bond Valuation6. What effect would adding the following features have on the market price of a similar bond which does not have this feature?a.10-year bond is callable by the company after 5 years (compare to a 10-year non-callable bond);b.bond is convertible into 10 shares of common stock at any time (compare to a non-convertible bond);c.10-year bond can be “put back” to the company after 3 years at par (puttable bond) (c ompare to a 10-year non-puttable bond)d.25-year bond has tax-exempt coupon paymentsSOLUTION:a.The callable bond would have a lower price than the non-callable bond to compensate the bondholders forgranting the issuer the right to call the bonds.b.The convertible bond would have a higher price because it gives the bondholders the right to convert their bondsinto shares of stock.c.The puttable bond would have a higher price because it gives the bondholders the right to sell their bonds backto the issuer at par.d.The bond with the tax-exempt coupon has a higher price because the bondholder is exempted from paying taxeson the coupons. (Coupons are usually considered and taxed as personal income).Inferring the Value of a Bond Guarantee7. Suppose that the yield curve on dollar bonds that are free of the risk of default is flat at 6% per year. A 2-year 10% coupon bond (with annual coupons and $1,000 face value) issued by Dafolto Corporation is rates B, and it is currently trading at a market price of $918. Aside from its risk of default, the Dafolto bond has no other financially significant features. How much should an investor be willing to pay for a guarantee against Dafolto’s defaulting on this bond?SOLUTION:value of a guarantee against default: 1073.3-918 = $155.3The implied Value of a Call Provision and Convertibility8. Suppose that the yield curve on bonds that are free of the risk of default is flat at 5% per year. A 20-year default-free coupon bond (with annual coupons and $1,000 face value) that becomes callable after 10 years is trading at par and has a coupon rate of 5.5%.a.What is the implied value of the call provision?b. A Safeco Corporation bond which is otherwise identical to the callable 5.5% coupon bond describedabove, is also convertible into 10 shares of Safeco stock at any time up to the bond’s maturity. If its yield to maturity is currently 3.5% per year, what is the implied value of the conversion feature? SOLUTION:is the implied value of the call provision: 1062.3 – 1000 = $62.3Note that the call provision decreases the value of the bond.This bond has the same features as the 5.5% default free callable bond described above, plus an additional feature: it is convertible into stocks. Hence the implied value of the conversion feature is the difference between the values of both bonds: 1284.2-1000 = $284.25. Note that the conversion feature increases the value of the bond.Changes in Interest Rates and Bond Prices9. All else being equal, if interest rates rise along the entire yield curve, you should expect that:i. Bond prices will fallii. Bond prices will riseiii. Prices on long-term bonds will fall more than prices on short-term bonds.iv. Prices on long-term bonds will rise more than prices on short-term bondsa. ii and iv are correctb. We can’t be certain that prices will changec. Only i is correctd. Only ii is correcte. i and iii are correctSOLUTION:The correct answer is e.Bond prices are inversely proportional to yields hence when yields increase, bond prices fall. Long-term bonds are more sensitive to yield changes than short-term bonds.。

金融学第二版讲义大纲及课后习题答案详解第十章

CHAPTER 10AN OVERVIEW OF RISK MANAGEMENTObjectives« To explore how risk affects finan cial decisi on-mak ing.« To provide a con ceptual framework for the man ageme nt of risk.«To explain how the financial system facilitates the efficient allocation of risk-bearing.Outline10.1 What Is Risk?10.2 Risk and Econo mic Decisi ons10.3 The Risk Ma nageme nt Process10.4 The Three Dime nsions of Risk Tran sfer10.5 Risk Tran sfer and Econo mic Efficie ncy10.6 In stituti ons for Risk Man ageme nt10.7 Portfolio Theory: Quan titative An alysis for Optimal Risk Man ageme nt10.8 Probability Distributions of ReturnsSummary* Risk is defined as uncertainty that matters to people. Risk management is the process of formulating the benefit- cost trade-offs of risk-reduction and deciding on a course of action to take. Portfolio theory is the quantitative analysis of those trade-offs to find an optimal course of action.* All risks are ultimately borne by people in their capacity as consumers, stakeholders of firms and other econo mic orga ni zati ons, or taxpayers.* The risk in ess of an asset or a tra nsacti on cannot be assessed in isolati on or in the abstract; it depe nds on the specific frame of refere nee. In on e con text, the purchase or sale of a particular asset may add to one ' s risk exposure; in another, the same transaction may be risk-reducing.* Speculators are in vestors who take positi ons that in crease their exposure to certa in risks in the hope of in creas ing their wealth. In con trast, hedgers take positi ons to reduce their exposures. The same pers on can be a speculator on some exposures and a hedger on others.* Many resource-allocation decisions, such as saving, investment, and financing decisions, are significantly in flue need by the prese nee of risk and therefore are partly risk-ma nageme nt decisi ons.* We disti nguish among five major categories of risk exposures for households: sick ness, disability, and death job loss; consumer-durable asset risk ; liability risk ; and financial asset risk .* Firms face several categories of risks: production risk , price risk of outputs , and price risk of in puts .* There are five steps in the risk-management process: risk identification, risk assessment, selection of riskman ageme nt tech ni ques, impleme ntati on, review.* There are four techniques of risk management: r isk avoidanee, loss prevention and control, risk retention, risk tra nsfer.* There are three dimensions of risk transfer: hedging , insuring , and diversifying .* Diversificati on improves welfare by spread ing risks among many people, so that the existi ng un certa inty matters less. * From society ' s perspective-n^ageme nt in stituti ons con tribute to econo mic efficie ncy in two importa nt ways. First, they shift risk away from those who are least willing or able to bear it to those who are most willing to bear it. Second, they cause a reallocation of resources to production and consumption in accordance with the new distribution of risk-bearing.By allowing people to reduce their exposure to the risk of undertaking certain bus in ess ven tures, they may en courage en trepre neurial behavior that can have a ben efit to society.* Over the cen turies, various econo mic orga ni zati ons and con tractual arra ngeme nts have evolved to facilitate a more efficient allocation of risk-bearing by expanding the scope of diversification and the types of risk that are shifted.* Among the factors limit ing the efficie nt allocati on of risks are tra nsacti ons costs and problems of adverse selecti on and moral hazard.Solutions to Problems at End of ChapterOn the Nature of Risk and Risk Management1. Suppose that you and a friend have decided to go to a movie together next Saturday. You will select any movie for which tickets are available when you get to the theater. Is this a risky situation for you? Explain. Now suppose that your friend has already purchased a ticket for a movie that is going to be released this Saturday. Why is this a risky situation? How would you deal with the risk?SOLUTION:No, the uncertainty doesn ' t represienncteriysokusdo not care which movie you see. However, if your friend has a ticket already, and if you wait till Saturday to buy yours, the show may be sold out. To eliminate the risk that you may not be able to sit with your friend and see the same movie, you might buy your ticket in advance.2. Suppose you are aware of the following investment opportunity: You could open a coffee shop around the corner from your home for $25,000. If business is strong, you could net $15,000 in after-tax cash flows each year over the next 5 years.a. If you knew for certain the business would be a success, would this be a risky investment?b. Now assume this is a risky venture and that there is a 50% chance it is a success and a 50% chance you gobankrupt within 2 years. You decide to go ahead and invest. If the business subsequently goes bankrupt, did you make the wrong decision based on the information you had at the time? Why or why not?SOLUTION:a. No, this investment would not be risky.b. No, you did not make a “ wrong ” decision. When you made your decision, you did not know for certain that thecompany would go bankrupt. You decided to invest for many reasons, including the possibility of making a lot of money.Given your tolerance for risk and the fact that you based our decision on the information available at the time, your decision was not wrong and may have been optimal at the time.3. Suppose you are a pension fund manager and you know today that you need to make a $100,000 payment in 3 months.a. What would be a risk-free investment for you?b. If you had to make that payment in 20 years instead, what would be a risk free investment?c. What do you conclude from your answers to Parts a and b of this question?SOLUTION:a. A risk-free investment for you would be a Treasury Bill (default risk free) which matures in exactly 3 months.b. A risk-free investment would be a zero coupon U.S. Treasury security maturing in 20 years and which would have thesame single payment of $100,000.c. Because risk is dependent upon circumstances, what is risk-free for one individual may be risky for another too. There canbe any number of risk-free investments depending upon circumstances. Your investment time horizon is critical tochoosing the best risk-free investment (so payments in can exactly match payments out so that you are left with no risk).4. Is it riskier to make a loan denominated in dollars or in yen?SOLUTION:It depends on the context. For people whose income and expenses are denominated in dollars (perhaps because they live in the U.S), denominating a loan in yen would be riskier than denominating it in dollars. But for someone whose income and expenses are denominated in yen, denominating the loan in yen would be less risky than in dollars.5. Which risk management technique has been chosen in each of the following situations?« Installing a smoke detector in your home« Investing savings in T-bills rather than in stocks« Deciding not to purchase collision insurance on your car« Purchasing a life insurance policy for yourselfSOLUTION:« Loss preve nti on and con trol.・Risk avoida nee« Risk rete nti on・Risk tran sfer6. You are considering a choice between investing $1,000 in a conventional one-year T-Bill offering an interest rate of 8% and a one-year Index 丄inked Inflation Plus T-Bill offering 3% plus the rate of inflation.a. Which is the safer investment?b. Which offers the higher expected return?c. What is the real return on the Index 丄inked Bond?SOLUTION:a. The inflation-indexed T-Bill offers a fixed real rate of return of 3% over the life of the investment. The realreturn on the conventional T- Bill ' s real return depends upon the expected rate of inflation over the life of thein vestme nt. The safer in vestme nt is the In flati on Plus T-Bill.b. The real rate of return on the conventional T-Bill depends upon the expected rate of inflation over the life of thein vestme nt. You do not know which expected retur n is higher unl ess you know what in flati on is expected to be.c. The real retur n on the in dex-l in ked T-Bill is 3%.Hedging and Insurance7. Suppose you are interested in financing your new home purchase. You have your choice of a myriad financing options. You could enter into any one of the following agreements: 8% fixed rate for 7 years, 8.5% fixed rate for 15 years, 9% fixed for 30 years. In addition, you could finance with a 30-year variable rate that begins at 5% and increases and decreases with the prime rate, or you could finance with a 30year variable rate that begins at 6% with ceilings of 2% per year to a maximum of 12% and no minimum.a. Suppose you believe that interest rates are on the rise. If you want to completely eliminate your risk of risinginterest rates for the longest period of time, which option should you choose?b. Would you consider that hedging or insuring? Why?c. What does you r risk management decision “ cost ” you in terms of quoted interest rates during the firstyear?SOLUTION:a. You would choose the 30-year fixed rate at 9%.b. That would be a hedge because you have elim in ated both the upside (decli ning rates) or dow nside ( rising rates).c. This costs me at least 4% since I could get a variable rate loa n at 5%.8. Referring to the information in problem 7, answer the following:a. Suppose you believe interest rates are going to fall, which option should you choose?b. What risk do you face in that transaction?c. How might you insure against that risk? What does that cost you (in terms of quoted interest rates?). SOLUTION:a. You would want one of the variable rate options, in particular the variable loan tied to the prime rate, currently equal to5%.b. You face the risk of rising rates.c. You could in sure aga inst that risk by purchas ing the opti on to have a 12% ceil ing on the rate (2% in crease per year.This option cost you 1% (the difference between 6% and 5%).9. Suppose you are thinking of investing in real estate. How might you achieve a diversified real estate investment?SOLUTION:« You could own several differe nt build ings in the same gen eral area.« You could own several differe nt build ings in differe nt geographic areas.« You could sell some of your equity own ership to other owners to lower your own in dividual exposure to decli ning market values.10. Suppose the following represents the historical returns for Microsoft and Lotus Development Corporation:Historical ReturnsYear MSFT LOTS110%9%215%12%3-12%-7%420%18%57%5%a. What is the mean return for Microsoft? For Lotus?b. What is the standard deviation of returns for Microsoft? For Lotus?c. Suppose the returns for Microsoft and Lotus have normally distributed returns with means and standarddeviations calculated above. For each stock, determine the range of returns within one expected standard deviation of the mean and within two standard deviations of the mean.SOLUTION:a. Mea n return Microsoft: 8.0%; Lotus: 7.4%b. If you use the formula for the sta ndard deviati on based on a sample of size n:You find that the standard deviations are: MSFT: 10.94%; Lotus: 8.357%.However, if you use the formula for the population standard deviation:You find that the standard deviations are: MSFT 12.23% and LOTS 9.34%.c. Range of returns within 1 standard deviation Microsoft: -2.94% to +18.94% Range of returns within 1 standarddeviation Lotus: -0.957% to + 15.76% Range of returns within 2 standard deviations Microsoft: -13.88% to+29.88% Range of returns within 1 standard deviation Lotus: -9.31% to + 24.11%。

《金融学（第二版）》讲义大纲及课后习题答案详解十二章

《⾦融学（第⼆版）》讲义⼤纲及课后习题答案详解⼗⼆章CHAPTER 12CHOOSING AN INVESTMENT PORTFOLIOObjectivesTo understand the process of personal investing in theory and in practice.To build a quantitative model of the tradeoff between risk and reward.Outline12.1 The Process of Personal Portfolio Selection12.2 The Trade-off between Expected Return and Risk12.3 Efficient Diversification with Many Risky AssetsSummaryThere is no single portfolio selection strategy that is best for all people.Stage in the life cycle is an imp ortant determinant of the optimal composition of a person’s optimal portfolio of assets and liabilities.Time horizons are important in portfolio selection. We distinguish among three time horizons: the planning horizon, the decision horizon, and the trading horizon.In making portfolio selection decisions, people can in general achieve a higher expected rate of return only by exposing themselves to greater risk.One can sometimes reduce risk without lowering expected return by diversifying more completely either withina given asset class or across asset classes.The power of diversification to reduce the riskiness of an investor’s portfolio depends on the correlations among the assets that make up the portfolio. In practice, the vast majority of assets are positively correlated with each other because they are all affected by common economic factors. Consequently, one’s ability to reduce risk through diversification among risky assets without lowering expected return is limited.Although in principle people have thousands of assets to choose from, in practice they make their choices from a menu of a few final products offered by financial intermediaries such as bank accounts, stock and bond mutual funds, and real estate. In designing and producing the menu of assets to offer to their customers theseintermediaries make use of the latest advances in financial technology.Solutions to Problems at End of Chapter1. Suppose that your 58-year-old father works for the Ruffy Stuffed Toy Company and has contributed regularly to his company-matched savings plan for the past 15 years. Ruffy contributes $0.50 for every $1.00 your father puts into the savings plan, up to the first 6% of his salary. Participants in the savings plan can allocate their contributions among four different investment choices: a fixed-income bond fund, a “blend” option that invests in large companies, small companies, and the fixed-income bond fund, a growth-income mutual fund whose investments do not include other toy companies, and a fund whose sole investment is stock in the Ruffy Stuffed Toy Company. Over Thanksgiving vacation, Dad realizes that you have been majoring in finance and decides to reap some early returns on that tuition money he’s been investing in your education. He shows you the most recent quarterly statement for his savings plan, and you see that 98% of its current value is in the fourth investment option, that of the Ruffy Company stock..a.Assume that your Dad is a typical risk-averse person who is considering retirement in five years. Whenyou ask him why he has made the allocation in this way, he responds that the company stock has continually performed quite well, except for a few declines that were caused by problems in a division that the company has long since sold off. Inaddition, he says, many of his friends at work have done the same. What advice would you give your dad about adjustments to his plan allocations? Why?b.If you consider the fact that your dad works for Ruffy in addition to his 98% allocation to the Ruffy stockfund, does this make his situation more risky, less risky, or does it make no difference? Why? SOLUTION:a.Dad has exposed himself to risk by concentrating almost all of his plan money in the Ruffy Stock fund. This is analogous to taking 100% of the money a family has put aside for investment and investing it in a single stock.First, Dad needs to be shown that just because the company stock has continually performed quite well is no guarantee that it will do so indefinitely. The company may have sold off the divisions which produced price declines in the past, but future problems are unpredictable, and so is the movement of the stock price. “Past performance is no guarantee of future results” is the lesson.Second, Dad needs to hear about diversification. He needs to be counseled that he can reduce his risk by allocating his money among several of the options available to him. Indeed, he can reduce his risk considerably merely by moving all of his money into the “blend” fund because it is diversifi ed by design: it has a fixed-income component, a large companies component, and a small companies component. Diversification isachieved not only via the three differing objectives of these components, but also via the numerous stocks that comprise each of the three components.Finally, Dad’s age and his retirement plans need to be considered. People nearing retirement age typically begin to shift the value of their portfolios into safer investments. “Safer” normally connotes less variability, so that the risk of a large decline in the value of a portfolio is reduced. This decline could come at any time, and it would be very unfortunate if it were to happen the day before Dad retires. In this example, the safest option would be the fixed-income bond fund because of its diversified composition and interest-bearing design, but there is still risk exposure to inflation and the level of interest rates. Note that the tax-deferred nature of the savings plan encourages allocation to something that produces interest or dividends. As it stands now, Dad is very exposed to a large decline in the value of his savings plan because it is dependent on the value of one stock.Individual equities over time have proven to produce the most variable of returns, so Dad should definitely move some, probably at least half, of his money out of the Ruffy stock fund. In fact, a good recommendation given his retirement horizon of five years would be to re-align the portfolio so that it has 50% in the fixed- income fund and the remaining 50% split between the Ruffy stock fund (since Dad insists) and the “blend” fund.Or, maybe 40% fixed-income, 25% Ruffy, 15% growth-income fund, and 20% “blend” fund. This latterallocation has the advantage of introducing another income-producing component that can be shielded by the tax-deferred status of the plan.b.The fact that Dad is employed by the Ruffy Company makes his situation more risky. Let’s say that the companyhits a period of slowed business activities. If the stock price declines, so will th e value of Dad’s savings plan. If the company encounters enough trouble, it may consider layoffs. Dad’s job may be in jeopardy. At the same time that his savings plan may be declining in value, Dad may also need to look for a job or go onunemployment. Thus, Dad is exposed on two fronts to the same risk. He has invested both his human capital and his wealth almost exclusively in one company.2. Refer to Table 12.1.a.Perform the calculations to verify that the expected returns of each of the portfolios (F, G, H, J, S) in thetable (column 4) are correct.b.Do the same for the standard deviations in column 5 of the table.c.Assume that you have $1million to invest. Allocate the money as indicated in the table for each of the fiveportfolios and calculate the expected dollar return of each of the portfolios.d.Which of the portfolios would someone who is extremely risk tolerant be most likely to select? SOLUTION:d.An extremely risk tolerant person would select portfolio S, which has the largest standard deviation but also thelargest expected return.3. A mutual fund company offers a safe money market fund whose current rate is4.50% (.045). The same company also offers an equity fund with an aggressive growth objective which historically has exhibited an expected return of 20% (.20) and a standard deviation of .25.a.Derive the equation for the risk-reward trade-off line.b.How much extra expected return would be available to an investor for each unit of extra risk that shebears?c.What allocation should be placed in the money market fund if an investor desires an expected return of15% (.15)?SOLUTION:a.E[r] = .045 + .62b.0.62c.32.3% [.15 = w*(.045) + (1-w)*(.020) ]4. If the risk-reward trade-off line for a riskless asset and a risky asset results in a negative slope, what does that imply about the risky asset vis-a-vis the riskless asset?SOLUTION:A trade-off line wit h a negative slope indicates that the investor is “rewarded” with less expected return for taking on additional risk via allocation to the risky asset.5. Suppose that you have the opportunity to buy stock in AT&T and Microsoft.a.stocks is 0? .5? 1? -1? What do you notice about the change in the allocations between AT&T andMicrosoft as their correlation moves from -1 to 0? to .5? to +1? Why might this be?b.What is the variance of each of the minimum-variance portfolios in part a?c.What is the optimal combination of these two securities in a portfolio for each value of the correlation,assuming the existence of a money market fund that currently pays 4.5% (.045)? Do you notice any relation between these weights and the weights for the minimum variance portfolios?d.What is the variance of each of the optimal portfolios?e.What is the expected return of each of the optimal portfolios?f.Derive the risk-reward trade-off line for the optimal portfolio when the correlation is .5. How much extraexpected return can you anticipate if you take on an extra unit of risk?SOLUTION:a.Minimum risk portfolios if correlation is:-1: 62.5% AT&T, 37.5% Microsoft0: 73.5% AT&T, 26.5% Microsoft.5: 92.1% AT&T, 7.9% Microsoft1: 250% AT&T, short sell 150% MicrosoftAs the correlation moves from -1 to +1, the allocation to AT&T increases. When two stocks have negativec orrelation, standard deviation can be reduced dramatically by mixing them in a portfolio. It is to the investors’benefit to weight more heavily the stock with the higher expected return since this will produce a high portfolio expected return while the standard deviation of the portfolio is decreased. This is why the highest allocation to Microsoft is observed for a correlation of -1, and the allocation to Microsoft decreases as the correlationbecomes positive and moves to +1. With correlation of +1, the returns of the two stocks will move closely together, so you want to weight most heavily the stock with the lower individual standard deviation.b. Variances of each of the minimum variance portfolios:62.5% AT&T, 37.5% Microsoft Var = 073.5% AT&T, 26.5% Microsoft Var = .016592.1% AT&T, 7.9% Microsoft Var = .0222250% AT&T, short 150% Microsoft Var = 0c. Optimal portfolios if correlation is:-1: 62.5% AT&T, 37.5% Microsoft0: 48.1% AT&T, 51.9% Microsoft.5: 11.4% AT&T, 88.6% Microsoft1: 250% AT&T, short 150% Microsoftd. Variances of the optimal portfolios:62.5% AT&T, 37.5% Microsoft Var = 048.1% AT&T, 51.9% Microsoft Var = .022011.4% AT&T, 88.6% Microsoft Var = .0531250% AT&T, short 150% Microsoft Var = 0e. Expected returns of the optimal portfolios:62.5% AT&T, 37.5% Microsoft E[r] = 14.13%48.1% AT&T, 51.9% Microsoft E[r] = 15.71%11.4% AT&T, 88.6% Microsoft E[r] = 19.75%250% AT&T, short 150% Microsoft E[r] = -6.5%f.Risk-reward trade-off line for optimal portfolio with correlation = .5:E[r] = .045 + .66/doc/31dbf23b580216fc700afd59.html ing the optimal portfolio of AT&T and Microsoft stock when the correlation of their price movements is 0.5, along with the results in part f of question 12-5, determine:a.the expected return and standard deviation of a portfolio which invests 100% in a money market fundreturning a current rate of 4.5%. Where is this point on the risk-reward trade-off line?b.the expected return and standard deviation of a portfolio which invests 90% in the money market fundand 10% in the portfolio of AT&T and Microsoft stock.c.the expected return and standard deviation of a portfolio which invests 25% in the money market fundand 75% in the portfolio of AT&T and Microsoft stock.d.the expected return and standard deviation of a portfolio which invests 0% in the money market fundand 100% in the portfolio of AT&T and Microsoft stock. What point is this?SOLUTION:a.E[r] = 4.5%, standard deviation = 0. This point is the intercept of the y (expected return) axis by the risk-rewardtrade-off line.b.E[r] = 6.03%, standard deviation = .0231c.E[r] = 15.9%, standard deviation = .173d.E[r] = 19.75%, standard deviation = .2306. This point is the tangency between the risk-reward line from 12-5part f and the risky asset risk-reward curve (frontier) for AT&T and Microsoft.7. Again using the optimal portfolio of AT&T and Microsoft stock when the correlation of their price movements is 0.5, take $ 10,000 and determine the allocations among the riskless asset, AT&T stock, and Microsoft stock for:a. a portfolio which invests 75% in a money market fund and 25% in the portfolio of AT&T and Microsoftstock. What is this portfolio’s expected return?b. a portfolio which invests 25% in a money market fund and 75% in the portfolio of AT&T and Microsoftstock. What is this portfolio’s expect ed return?c. a portfolio which invests nothing in a money market fund and 100% in the portfolio of AT&T andMicrosoft stock. What is this portfolio’s expected return?SOLUTION:a.$7,500 in the money-market fund, $285 in AT&T (11.4% of $2500), $2215 in Microsoft. E[r] = 8.31%, $831.b.$2,500 in the money-market fund, $855 in AT&T (11.4% of $7500), $6645 in Microsoft. E[r] = 15.94%, $1,594.c.$1140 in AT&T, $8860 in Microsoft. E[r] = 19.75%, $1,975.8. What strategy is implied by moving further out to the right on a risk-reward trade-off line beyond the tangency point between the line and the risky asset risk-reward curve? What type of an investor would be most likely to embark on this strategy? Why?SOLUTION:This strategy calls for borrowing additional funds and investing them in the optimal portfolio of AT&T and Microsoft stock. A risk-tolerant, aggressive investor would embark on this strategy. This person would be assuming the risk of the stock portfolio with no risk-free component; the money at risk is not onl y from this person’s own wealth but also represents a sum that isowed to some creditor (such as a margin account extended by the investor’s broker).9. Determine the correlation between price movements of stock A and B using the forecasts of their rate of return and the assessments of the possible states of the world in the following table. The standard deviations for stock A and stock B are0.065 and 0.1392, respectively. Before doing the calculation, form an expectation of whether that correlation will be closer to1 or -1 by merely inspecting the numbers.SOLUTION:Expectation: correlation will be closer to +1.E[r A] = .05*(-.02) + .15*(-.01) + .60*(.15) + .20*(.15) = .1175, or, 11.75%E[r B] = .05*(-.20) + .15*(-.10) + .60*(.15) + .20*(.30) = .1250, or, 12.50%Covariance = .05*(-.02-.1175)*(-.20-.125) + .15*(-.01-.1175)*(-.10-.125) +.60*(.15-.1175)*(.15-.125) + .20*(.15-.1175)*(.30-.125) =.008163Correlation = .008163/(.065)*(.1392) = .90210.Analyze the “expert’s” answers to the following questions:a.Question:I have approx. 1/3 of my investments in stocks, and the rest in a money market. What do you suggestas a somewhat “safer” place to invest another 1/3? I like to keep 1/3 accessible for emergencies.Expert’s answer:Well, you could try 1 or 2 year Treasury bonds. You’d get a little bit more yie ld with no risk.b.Question:Where would you invest if you were to start today?Expert’s answer:That depends on your age and short-term goals. If you are very young – say under 40 –and don’tneed the money you’re investing for a home or college tuition or such, you would put it in a stockfund. Even if the market tanks, you have time to recoup. And, so far, nothing has beaten stocks overa period of 10 years or more. But if you are going to need money fairly soon, for a home or for yourretirement, you need to play it safer.SOLUTION:a.You are not getting a little bit more yield with no risk. The real value of the bond payoff is subject to inflationrisk. In addition, if you ever need to sell the Treasury bonds before expiration, you are subject to the fluctuation of selling price caused by interest risk.b.The expert is right in pointing out that your investment decision depends on your age and short-term goals. In addition, the investment decision also depends on other characteristics of the investor, such as the special character of the labor income (whether it is highly correlated with the stock market or not), and risk tolerance.Also, the fact that over any period of 10 years or more the stock beats everything else cannot be used to predict the future.。

金融数学引论答案第二版

金融数学引论答案第二版【篇一：北大版金融数学引论第二章答案】>第二章习题答案1．某家庭从子女出生时开始累积大学教育费用5万元。

如果它们前十年每年底存款1000元，后十年每年底存款1000+x 元，年利率7%。

计算x 。

解：s = 1000s?7%+xs?7%20p10p20px = 50000 ? 1000s?7% = 651.72s?p7%102．价值10,000元的新车。

购买者计划分期付款方式：每月底还250元，期限4年。

月结算名利率18%。

计算首次付款金额。

试用x和y 表示d 。

解： a?p2n= a?pn+ a?p (1 ? d)则nny ? xd = 1 ? ( x ) n5．已知：a?p7= 5.58238, a?= 7.88687, a?= 10.82760。

计算i。

11p18p解：a?p = a?p + a?p v718711解得=i = 6.0%10?p +a∞?p6.证明： 11?v10s。

解：p v = 100a?+ 100a20?8p3% p3% = 2189.7168．某人现年40岁，现在开始每年初在退休金帐号上存入1000元，共计25年。

然后，从65岁开始每年初领取一定的退休金，共计15年。

设前25年的年利率为8%，后15年的年利率7%。

计算每年的退休金。

解：设每年退休金为x，选择65岁年初为比较日=解得x = 8101.658。

《金融学(第二版)》讲义大纲及课后习题答案详解第七章

《金融学〔第二版〕》讲义大纲及课后习题答案详解第七章CHAPTER 7PRINCIPLES OF ASSET VALUATIONObjectives? Understand why asset valuation is important in finance.? Explain the Law of One Price as the principle underlying all asset-valuation procedures. ? Explain the meaning and role of valuation models.? Explain how information gets reflected in security prices.Outline7.1 The Relation Between an Asset’s Value and Its Price 7.2 Value Maximization and Financial Decisions 7.3 The Law of One Price and Arbitrage7.4 Arbitrage and the Prices of Financial Assets 7.5 Exchange Rates and Triangular Arbitrage 7.6 Interest Rates and the Law of One Price 7.7 Valuation Using Comparables 7.8 Valuation Models7.9 Accounting Measures of Value7.10 How Information Gets Reflected in Security Prices 7.11 The Efficient Markets HypothesisSummary? In finance the measure of an asset’s value is the price it would fetch if it were sold in a competitive market. Theability to accurately value assets is at the heart of the discipline of finance because many personal and corporate financial decisions can be made by selecting the alternative that maximizes value.? The Law of One Price states that in a competitive market, if two assets are equivalent they will tend to have thesame price. The law is enforced by a process called arbitrage, the purchase and immediate sale of equivalent assets in order to earn a sure profit from a difference in their prices.? Even if arbitrage cannot be carried out in practice to enforce the Law of One Price, unknown asset values canstill be inferred from the prices of comparable assets whose prices are known.? The quantitative method used to infer an asset’s value from information about the prices of comparable assets iscalled a valuation model. The best valuation model to employ varies with the information available and the intended use of the estimated value. ? The book value of an asset or a liability as reported in a firm’s financial statements often differs from its currentmarket value.? In making most financial decisions, it is a good idea to start by assuming that for assets that are bought and soldin competitive markets, price is a pretty accurate reflection of fundamental value. This assumption is generally warranted precisely because there are many well-informed professionals looking for mispriced assets who profit by eliminating discrepancies between the market prices and the fundamental values of assets. The proposition that an asset’s current price fully reflects all publicly-available information about future economic fundamentals affecting the asset’s value is known as the Efficient Markets Hypothesis.? The prices of traded assets reflect information about the fundamental economic determinants of their value.Analysts are constantly searching for assets whose prices are different from their fundamental value in order to buy/sell these “bargains.〞 In deciding the best strategy for the purchase/sale of a “bargain,〞 theanalyst has to evaluate the accuracy of her information. The market price of an asset reflects the weighted average of all analysts opinions with heavier weights for analysts who control large amounts of money and for those analysts who have better than average information.Instructor’s ManualChapter 7 Page 106Solutions to Problems at End of ChapterLaw of One Price and Arbitrage1. IBX stock is trading for $35 on the NYSE and $33 on the Tokyo Stock Exchange. Assume that the costs of buying and selling the stock are negligible. a. How could you make an arbitrage profit?b. Over time what would you expect to happen to the stock prices in New York and Tokyo?c. Now assume that the cost of buying or selling shares of IBX is 1% per transaction. How does this affectyour answer?SOLUTION:a. Buy IBX stock in Tokyo and simultaneously sell them in NY. Your arbitrage profit is $2 per share.b. The prices would converge.c. Instead of the prices becoming exactly equal, there can remain a 2% discrepancy between them, roughly $.70 inthis case.2. Suppose you live in the state of Taxachusetts which has a 16% sales tax on liquor. A neighboring state called Taxfree has no tax on liquor. The price of a case of beer is $25 in Taxfree and it is $29 in Taxachusetts.a. Is this a violation of the Law of One Price?b. Are liquor stores in Taxachusetts near the border with Taxfree going to prosper?SOLUTION:a. This is not a violation of the Law of One Price because it is due to a tax imposed in one state but not in the other.Illegal arbitrage will probably occur, with lawbreakers buying large quantities of liquor in Taxfree and selling it in Taxachusetts without paying the tax.b. It is likely that liquor stores will locate in Taxfree near the border with Taxachusetts. Residents of both stateswill buy their liquor in the stores located in Taxfree, and liquor stores in Taxachusetts will go out of business.Triangular Arbitrage3. Suppose the price of gold is 155 marks per ounce.a. If the dollar price of gold is $100 per ounce, what should you expect the dollar price of a mark to be?b. If it actually only costs $0.60 to purchase one mark, how could one make arbitrage profits?SOLUTION:a. $100 buys the same amount of gold (1 ounce) as 155 DM, so 1 DM should cost 100/155 or $.645.b. The marks are “cheaper〞 than they should be, so the arbitrage transaction requires you to buy marks at thecheap price, use them to purchase gold, and sell the gold for dollars. Example:1. Start with $1 million, which you borrow for only enough time to carry out the arbitrage transaction.2. Use the million dollars to buy 1,666,667 marks (1,000,000 / 0.60)3. Buy 10,752.69 ounces of gold (1,666,667 / 155)4. Sell the gold for $1,075,269 (10752.69 x 100)Your risk-free arbitrage profit is $75,269.4. You observe that the dollar price of the Italian lira is $0.0006 and the dollar price of the yen is $0.01. What must be the exchange rate between lira and yen for there to be no arbitrage opportunity?SOLUTION:.0006$/lira?.06Yen/lira.01$/YenInstructor’s ManualChapter 7 Page 1075. Fill in the missing exchange rates in the following table: US dollar British pound German mark Yen US dollar $1 $1.50 $.5 $.01 British pound ￡0.67 German mark DM2.0 Japanese ￥100 Yen SOLUTION: US dollar British pound German mark Japanese Yen US dollar $1 $1.50 $.5 $.01 British pound ￡0.67 1 = .67 / 2 = .67 / 100 German mark DM2.0 = 2 / .67 1 = 2 / 100 Japanese ￥100 = 100 / .67 = 100 / 2 1 Yen US dollar British pound German mark Japanese Yen US dollar $1 $1.50 $.5 $.01 British pound ￡0.67 ￡1 ￡.33 ￡.0067 German mark DM2.0 DM3.0 DM1.0 DM.02 Japanese ￥100 ￥150 ￥50 ￥1 Yen Valuation Using Comparables6. Suppose you own a home that you purchased four years ago for $475,000. The tax assessor’s office has just informed you that they are increasing the taxable value of your home to $525,000. a. How might you gather information to help you appeal the new assessment?b. Suppose the house next door is comparable to yours except that it has one fewer bedroom. It just sold for$490,000. How might you use that information to argue your case? What inference must you make about the value of an additional bedroom?SOLUTION:a. You should retrieve as much information as you can about recent sales of comparable homes. If you canconvince the assessor’s office that your home is comparable (and the market value of the recent sales is less than $525,000) you should have a good case. You can gather the information about home sales from a real estate broker.b. The difference between your house’s assessed value and the actual market value of the home next door is$35,000 ($525,000 - $490,000). If you can convince the tax assessor’s office that the value of a bedroom is less than $35,000, then the assessor must agree that your home is worth less than $525,000. For example, if comparable sales figures show that one additional bedroom (all else reasonably equivalent) is worth only $10,000, then you should be able to argue that your home is worth $500,000 rather than $525,000.7. The P/E ratio of ITT Corporation is currently 6 while the P/E ratio of the S&P 500 is 10. What might account for the difference? SOLUTION: There are several possible reasons:? ITT may be riskier than the S&P500 either because it is in a relatively risky industry or has a relatively higherdebt ratio.? ITT’s reported earnings may be higher than they are expected to be in the future, or they may be inflated due tospecial accounting methods used by ITT.Instructor’s ManualChapter 7 Page 1088. Suppose you are chief financial officer of a private toy company. The chief executive officer has asked you to come up with an estimate for the company’s price per share. Your company’s earnings per share were $2.00 in the year just ended. You know that you should look at public company comparables, however, they seem to fall into two camps. Those with P/E ratios of 8x earnings and those with P/E ratios of 14x earnings. You are perplexed at the difference until you notice that on average, the lower P/E companies have higher leverage than the higher P/E group. The 8x P/E group has a debt/equity ratio of 2:1. The 14x P/E group has a debt/equityratio of 1:1. If your toy company has a debt/equity ratio of 1.5:1, what might you tell the CEO about your company’s equity value per share? SOLUTION:It would be reasonable to apply a P/E of 11x earnings (= (8 + 14) / 2) because your leverage is midway between the two groups. Hence, your company’s price per share would be: 11x $2.00 = $22.00 per share.9. Assume that you have operated your business for 15 years. Sales for the most recent fiscal year were $12,000,000. Net income for the most recent fiscal year was $1,000,000. Your book value is $10,500,000. A similar company recently sold for the following statistics: Multiple of Sales: 0.8x Multiple of Net Income 12x Multiple of Book Value 0.9xa. What is an appropriate range of value for your company?b. If you know that your company has future investment opportunities that are far more profitable than thecompany above, what does that say about your company’s likely valuation? SOLUTION:a. Multiple of Sales: .8x = $12 million x .8 Multiple of Net Income 12x = $1 million x 12 Multiple of Book Value .9x = $10.5 million x .9 An appropriate range might be 9 to 12 millionb. Higher end of the range = $9.6 million = $12 million = $9.45 millionEfficient Markets Hypothesis10. The price of Fuddy Co. stock recently jumped when the sudden unexpected death of its CEO was announced. What might account for such a market reaction?SOLUTION:Investors may believe that the company’s future prospects look better(i.e., either higher earnings or less risky) without the deceased CEO.11. Your analysis leads you to believe that the price of Outel’s stock should be $25 per share. Its current market price is $30.a. If you do not believe that you have access to special information about the company, what do you do?b. If you are an analyst with much better than average information, what do you do?SOLUTION:a. If you believe that the market for Outel stock is an informationally efficient one then the $30 market price(which is a weighted average of the valuations of all analysts) is the best estimate of the stock’s true value. You should question whether your own analysis is correct.b. You sell the stock because you think you have superior information. Real Interest Rate Parity12. Assume that the world-wide risk-free real rate of interest is 3% per year. Inflation in Switzerland is 2% per year and in the United States it is 5% per year. Assuming there is no uncertainty about inflation, what are the implied nominal interest rates denominated in Swiss francs and in US dollars?SOLUTION: Switzerland: (1.03 x 1.02) =1.0506 hence nominal interest rate = 5.06% US: (1.03 x 1.05) = 1 .0815 hence nominal interest rate = 8.15%Instructor’s ManualChapter 7 Page 109Integrative Problem13. Suppose an aunt has passed away and bequeathed to you and your siblings (one brother, one sister) a variety of assets. The original cost of these assets follows:ITEM COST WHEN PURCHASEDJewelry $500 by Grandmother 75 years ago House 1,200,000 10 years ago Stocks and Bonds 1,000,000 3 years ago Vintage (used) Car 200,000 2 months ago Furniture 15,000 various dates during last 40 yearsBecause you are taking a course in finance, your siblings put you in charge of dividing the assets fairly among the three of you. Before you start, your brother approaches you and says: “I’d really like the car for myself, so when you divide up the assets, just give me the car and deduct the $200,000 from my share.〞Hearing that, your sister says: “That sounds fair, because I really like the jewelry and you can assign that to me and deduct the $500 from my share.〞You have always loved your aunt’s house and its furnishings, so you would like to keep the house and the furniture.a. How do you respond to your brother and sister’s requests? Justify your responses.b. How would you go about determining appropriate values for each asset?SOLUTION:a. Because the market price of the car is close to the what your brother is willing to give up for it, your brother’srequest is reasonable. It is, however, quite possible (even likely), that the antique jewelry is worth much more today than what your relative’s grandmother paid for it in the past. Assigning only its acquisition cost to your sister’s share is quite likely a gross miscalculation. If she wants the jewelry, she should be “charged〞 an amount equal to today’s market value. It does not matter that your sister does not want to sell the jewelry for a profit, because the jewelry has VALUE even if you do not sell it. Fairness is all about equal VALUE.b. You would probably have to hire a professional appraiser for the furniture and the jewelry. You can look up thevalue of the stocks and bonds in a financial newspaper. You can estimate the value of the house by inquiring for how much similar houses in the same neighborhood have recently been sold. The car was purchased only twomonths ago, so it is probably reasonable to assume that the current market price is very close to what your distant relative paid for the car. Instructor’s ManualChapter 7 Page 110。

金融数学附答案修订版

金融数学附答案修订版 IBMT standardization office【IBMT5AB-IBMT08-IBMT2C-ZZT18】1、给定股票价格的二项模型，在下述情况下卖出看涨期权 S 0 S u S d X r τ 股数50 60 40 55 0.55 1/2 1000（1）求看涨期权的公平市场价格。

（2）假设以公平市场价格+0.10美元卖出1000股期权，需要买入多少股股票进行套期保值，无风险利润是多少？答案：（1）d u d r S S S e S q --=τ0=56.0406040505.005.0=--⨯⨯e （2）83.2>73.2，τr e S V -∆+∆='0083.2> τr e S -∆+∆'0 406005--=--=∆d u S S D U =25.0股 104025.00'-=⨯-=∆-=∆d S D 753.9975.0105.005.0'-=⨯-=∆⨯-e 美元则投资者卖空1000份看涨期权，卖空250股股票，借入9753美元所以无风险利润为1.85835.005.0=⨯e 美元2、假定 S 0 = 100，u=1.1，d=0.9，执行价格X=105，利率r=0.05，p=0.85，期权到期时间t=3，请用连锁法则方法求出在t=0时该期权的价格。

(答案见课本46页)3、一只股票当前价格为30元，六个月期国债的年利率为3%，一投资者购买一份执行价格为35元的六个月后到期的美式看涨期权，假设六个月内股票不派发红利。

波动率σ为0.318.问题：（1）、他要支付多少的期权费？【参考N（0.506)=0.7123；N（0.731）=0.7673 】{提示：考虑判断在不派发红利情况下，利用美式看涨期权和欧式看涨期权的关系}解析：在不派发红利情况下，美式看涨期权等同于欧式看涨期权！所以利用Ｂ—Ｓ公式，就可轻易解出来这个题！同学们注意啦，N（d1）=N（-0.506），N（d2）=N（-0.731）。

《金融学(第二版)》讲义大纲及课后习题答案详解第九章

CHAPTER 9VALUATION OF COMMON STOCKSObjectives∙To explain the theory and application of the discounted cash flow valuation method as applied to the equity of a firm.Outline9.1 Reading Stock Listings9.2 The Discounted Dividend Model9.3 Earnings and Investment Opportunities9.4 A Reconsideration of the Price/Earnings Multiple Approach9.5 Does Dividend Policy Affect the Value of a Share?Summary∙The discounted cash flow (DCF) method of valuing assets consists of discounting expected future cash flows ata risk-adjusted discount rate.∙The discounted dividend model (DDM) for valuing shares of stock starts from the observation that an investor in common stock expects a rate of return (consisting of cash dividends and price appreciation) that is equal to the market capitalization rate. The resulting formula shows that the current price of a share is the present value of all expected future dividends.∙In the constant growth rate DDM, the growth rate of dividends is also the expected rate of price appreciation.∙Growth per se does not add value to a share’s current price. What adds value is the opportunity to invest in projects that yield a rate of return in excess of the market capitalization rate.∙In a “frictionless” financial envir onment, where there are no taxes and no transaction costs, the wealth of shareholders is the same no matter what dividend policy the firm adopts.∙In the real world there are a number of frictions that can cause dividend policy to have an effect on the wealth of shareholders. These include taxes, regulations, the costs of external finance, and the information content of dividends.Solutions to Problems at End of Chapter1.The DDM Corporation has just paid a cash dividend (D0) of $2 per share. It has consistently increased its cash dividends in the past by 5% per year, and you expect it to continue to do so. You estimate that the market capitalization rate for this stock should be 13% per year.a.What is your estimate of the intrinsic value of a share (derived using the DDM model)?b.Suppose that the actual price of a share is $20. By how much would you have to adjust each of thefollowing model parameters to “justify” this observed price:i.The growth rate of dividendsii.The market capitalization rateSOLUTION:a.P0 = D0(1+g)/(k-g) = 2(1+0.05)/(0.13-0.05) = $26.25b.If the actual price of the share is $20, then some of our input parameters might need some adjustments:i.Assuming all other parameters are left as given, then solving for g =(.13 x 20 – 2)/(2+20) = 0.0273= 2.73%ii.Similarly, solving for k = 2(1.05)/20 + 0.05 = 0.155 = 15.5%2.The Rusty Clipper Fishing Corporation is expected to pay a cash dividend of $5 per share this year. You estimate that the market capitalization rate for this stock should be 10% per year. If its current price is $25 per share, what can you infer about its expected growth rate of dividends?SOLUTION:D1 = $5; k = 10%; P0= $25Hence g = 0.1 - 5/25 = -0.1 = -10%3. The Constant Growth Corporation (CGC) has expected earnings per share (E1) of $5. It has a history of paying cash dividends equal to 20% of earnings. The market capitalization rate for CGC’s stock is 15% per year, and the expected ROE on the firm’s future investments is 17% per year? U sing the constant growth rate discounted dividend model,a. What is the expected growth rate of dividends?b. What is the model’s estimate of the present value of the stock?c. If the model is right, what is the expected price of a share a year from now?d.Suppose that the current price of a share is $50.By how much would you have to adjust each of the following model parameters to “justify” this observed price:i.The expected ROE on the firm’s future investments.ii.The market capitalization rateiii.The dividend payout ratio.SOLUTION:a. g = earnings retention ratio x ROE = .8 x .17 = .136 = 13.6%b. P0 = D1/(k-g)D1 = .2 x $5 = $1 per shareP0 = $1/(.15 -.136) = $1/.014 = $71.43c. The stock price grows at the same rate as dividends, i.e., 13.6% per year:P1 = P0 x (1 + g) = $71.43 x 1.136 = $81.14d.If the market is efficient then the $50 price represents the best estimate of the stock’s true value. To “justify” thisprice, one of the input parameters in the model needs to be adjusted:i.Assuming all other parameters are correct, if we were to adjust for the ROE:50 = D1/(k-g), where g = 0.8 x ROESolving for g, then for ROE: g = .15 – 1/50 = 0.13 = 13%,hence ROE is equal to 13/0.8 = 16.25%ii. If we were to adjust the market capitalization k then:k = 1/50 + .136 = .156 = 15.6%iii.Dividend payout ratio x E1 = 50 x (.15-.136) = 0.7,hence Dividend payout ratio = 0.7/5=0.14 = 14%4. The stock of Slogro Corporation is currently selling for $10 per share. Earnings per share in the coming year are expected to be $2 per share. The company has a policy of paying out 60% of its earnings each year in dividends. The rest is retained and invested in projects that earn a 20% rate of return per year. This situation is expected to continue forever.a. Assuming the current market price of the stock reflects its intrinsic value as computed using the constantgrowth rate DDM, what rate of return do Slogro’s investors require?b. By how much does its value exceed what it would be if all earnings were paid as dividends and nothingwere reinvested?c. If Slogro were to cut its dividend payout ratio to 25%, what would happen to its stock price? What ifSlogro eliminated the dividend altogether?d. Suppose that Slogro wishes to maintain its current 60% dividend payout policy but that it also wishes toinvest an amount each year equal to that year’s total earnings. All the money would be invested inprojects earning 20% per year. One way that Slogro could do so would be to issue an amount of new stock each year equal to one-half that year’s earnings. What do you think would be the effect of this policy on the current stock price?SOLUTION:a. P0 = $10, E1 = $2, b = .4, ROE = .2k = D1/P0 + gD1 = .6 x $2 = $1.20g = b x ROE = .4 x .2 = .08Therefore, k = $1.20/$10 + .08 = .12 + .08 = .2 or 20%b. If all earnings were paid as dividends its price would be:P0 = $2/.2 = $10Thus, its price is the same whether it reinvests or not. This is because k = ROE.c. Since k = ROE, the stock price would be unaffected by cutting the dividend and investing the additionalearnings.d.Again, this should have no impact on the stock’s price since the NPV of the investments would be zero (the IRRof those projects (20%) is equal to the investors’ required rate of return, hence the firm’s c ost of capital).5. The Corporation currently pays no cash dividends, and it is not expected to for the next 5 years. Its sales have been growing at 25% per year.a.Can you apply the constant growth rate DDM to estimate its intrinsic value? Explain.b.It is expected to pay its first cash dividend $1 per share 5 years from now. It its market capitalization rateis 20% and its dividends are expected to grow by 10% per year, what would you estimate its intrinsic value to be?c.If its current market price is $100 per share, what would you infer the expected growth rate of its futuredividends to be?SOLUTION:a.Yes, we can apply the DDM model even if the company doesn’t pay dividends for the first 5 years. Thecompany will eventually have to pay dividends in the future.b.P4 = D5/(k-g) = 1/(.2-.1) = $10P0 = 10/1.24 = $4.82c.If P0 = $100 then P4 = 100 x 1.24 = 207.36 and g = 0.2 – 1/207.36 = 19.518%6. The Digital Growth Corp. pays no cash dividends currently and is not expected to for the next 5 years. Its latest EPS was $10, all of which was reinvested in the company. The firm’s expected ROE for the next 5 years is 20% per year, and during this time it is expected to continue to reinvest all of its earnings. Starting 6 years from now, the firm’s ROE on new investments is expected to fall to 15%, and the company is expected to start paying out 40% of its earnings in cash dividends, which it will continue to do forever after. DG’s market capitalization rate is 15% per year.a. What is your estimate of DG’s int rinsic value per share?b. Assuming its current market price is equal to its intrinsic value, what do you expect to happen to its priceover the next year? The year after?c. What effect would it have on your estimate of DG’s intrinsic value if you expecte d DG to pay out only20% of earnings starting in year 6?56P0 = P5/(1+k)5 = $180.82/1.155 = $89.90b.The price should rise by 15% per year until year 5 after which it will grow at the dividends’ growth rate g (=9%).c. Since ROE =k, the dividend payout ratio will have no effect on current price.7. The 2Stage Co. just paid a dividend of $1 per share. The dividend is expected to grow at a rate of 25% per year for the next 3 years and then to level off to 5% per year forever. You think the appropriate market capitalization rate is 20% per year.a. What is your estimate of the intrinsic value of a share of the stock?b. If the market price of a share is equal to this intrinsic value, what is the expected dividend yield?c. What do you expect its price to be one year from now? Is the implied capital gain consistent with yourestimate of the dividend yield and the market capitalization rate?P 3 = D 4/(k – g) = 2.05078/(.20 -.05) = $13.67P 0 = D 1/(1+k) + D 2/(1+k)2 + (D 3 + P 3)/(1+k)3 = $1.25/1.2 + $1.5625/1.22 + ($1.953 + $13.67)/1.23 = $11.17 b. If the market price of a share is equal to this intrinsic value, the expected dividend yield is D 1/P 0, which is1.25/11.17 = .1119 or 11.2%c. Its price one year from now = P 1 = D 2/(1+k) + (D 3 + P 3)/(1+k)2 = $1.5625/1.2 + ($1.953 + $13.67)/1.22 =$12.15.The implied capital gain is $12.15 - $11.17 = $.98, which is 8.8% of the price P 0. Thus the dividend yield plusthe capital gain rate add up to 20%, which is k.8. The Bearded ladies’ Stock guide offers the following method for selecting stocks:Compute the stock’s PEG ratio by dividing its P/E mu ltiple by its growth rate of earnings. Select only those stocks whose PEG ratio is in the lowest quartile.a. If the stock is fairly priced according to the constant-growth-rate DDM, what should be its PEG ratio asa function of the following three variables: the stock’s market capitalization rate (k), the expectedprofitability of its future investments (ROE), and its plowback ratio (b)? (Assume the P/E ratio used in computing PEG is the ratio of the stock’s current price to its expected earnings per share, P 0/E 1)b. Assume the CAPM and the DDM are valid. The risk free rate is .04 and the risk premium on the marketportfolio is .06. What should be the relationship between the PEG for a stock whose ROE is .10 and a stock whose ROE is .15, assuming the two stocks have the same beta (equal to 1) and plowback ratio (equal to .6)?c. What do you think of the Bearded Ladies’ method?SOLUTION:a. If the DDM holds we know that P 0 = D 1 / (k-g), furthermore, we know that g = b x ROE and D 1=(1-b) E 1b(ROE)(k -b(ROE))b (ROE)b E k -b(ROE)E )b (g E P PEG -=⨯-==1111110b. The values of k for each of the stocks will be: k = .04 + .06 beta = .04 + .06 =.1PEG 1= (1-.6) / (.6 x .10 x (.1 - .6 x .10)) = 166.67PEG 2= (1-.6) / (.6 x .15 x (.1 - .6 x .15)) = 444.44PEG 1 < PEG 2c. As we can see in part b, the PEG rule would lead us to choose the stock with the lowest ROE.In general, if the stock market is informally efficient, then any stock will offer an expected rate of return that is commensurate with the stock’s perceived market risk, regardless of the stock’s PEG.Using the Internet for Stock Pricing9. Pick a company whose stock is traded on the NYSE. Use one of the stock valuation models discussed in this chapter together with information that you can find by searching the Internet to compute an intrinsic value for the s tock. Compare your estimate of intrinsic value with the stock’s actual price. Would you be willing to make an investment decision on the basis of your research? Why or why not?SOLUTION:One simple model that we can use to value a company is to find the average P/E multiple of the industry in which the company operates and multiply it by the expected earnings per share of that company. The difference between this intrinsic value and the actual market value of the stock can be explained by the difference between our assumptions regarding the company’s future investment opportunities and the market’s expectations. For example, if the market value of the stock is higher than the intrinsic value found, then this difference reflects the investors’ belief that the company will have a greater-than-average future investments opportunities with a rate of return greater than the market capitalization rate for this particular industry. If markets are efficient, then this market value is supposed to be the “real” value o f the company, and represents the view of the majority of investors, hence I would be reluctant to follow my own findings of the intrinsic value as a basis for an investment decision.Dividend Policy10. Divido Corporation is an all-equity financed firm with a total market value of $100 million. The company holds $10 million in cash-equivalents and has $90 million in other assets. There are 1,000,000 shares of Divido common stock outstanding, each with a market price of $100. What would be the impact on Di vido’s stock price and on the wealth of its shareholders of each of the following decisions? Consider each decision separately.a. The company pays a cash dividend of $10 per share.b. The company repurchases 100,000 shares.c. The company pays a 10% stock dividend.d. The company has a 2-for-1 stock split.e. The company invests $10 million in an expansion that has an expected IRR equal to the firm’s cost ofcapital.SOLUTION:a.The stock price falls by $10, but shareholder wealth remains the same in a frictionless world becauseshareholders receive $10 in cash on each share they own. In the real world, shareholder’s wealth may decline because personal taxes may have to be paid on the cash dividend.b.The stock price is unchanged and so is shareholder wealth. Some of the shareholders who sold their shares mayhave to pay taxes on their capital gains in the real world.c.The number of shares outstanding rises to 1,100,000, and the stock price falls to $90.909 (=$100MM/$1.1MM)per share. Shareholder wealth is unchanged: instead of having one share at $100, now the shareholder will have1.1 shares at $90.909/share (1.1 x 90.909 = 100)d.The number of shares outstanding rises to 2,000,000, and the stock price falls to $50 per share. Theoretically,shareholder wealth is unchanged.e.The composition of the firm’s assets changes. Cash falls by $10 million and other assets go up by the sameamount. There is no change in either the stock price or in shareholder wealth.11. It has been found empirically, that on average the total market value of their stock rises when firms announce a stock split. What hypotheses might you offer to explain this phenomenon?SOLUTION:Theoretically, when a firm announces a stock-split, the number of shares doubles (if 2-to-1 stock split) and the market value per share drops by half. Empirically, we have observed a small increase in market value of the stock after the announcement of a stock-split. This can be explained by the informational content of the split. Outside investors may interpret this stock dividend as a positive sign that the company is doing well, hence increasing the price of the stock. Another possible interpretation is that since the price per share is now lower after the split, it can become more affordable for some investors.12. Suppose that a company has had an extraordinarily profitable year, and it announces that it will use most of its net cash inflow to buy back shares of its stock in the market. Would you expect the price of its stock to rise or fall when the announcement is made? Explain.SOLUTION:Theoretically, the price of the stock should not change after a stock repurchase. But the announcement could send a positive signal to investors that the company has been doing very well and has enough cash to buy back shares as a form of dividends. This might increase the price of the stock after the announcement.。

《金融数学》(第二版)习题参考答案(修订版)

δ0.5 = a' (0.5) / a (0.5) =（0.08*0.5+0.03）/(1+2.5%)=0.06829
∫ 1.15
A(3) = 100⋅ exp⎜⎝⎛⎜
3
t2
0
/ 100dt⎞⎠⎟⎟+
X
= 109.42 +
X
∫ A(6) = (109.42 + X )⋅exp⎛⎜⎜⎝
6t2
3
/ 100dt⎞⎠⎟⎟
第三部分是自55年开始支付5次每次支付2x故有050545500001051053278516由已知115045869228001001由过去法第2期后未偿还本金金额为10001018692288453852次支付的本金金额为001784689由将来法第4期后未偿还本金金额为次支付的利息金额为0015138212517第69期还款额中本金金额为3606929269rvrv故由已知292292094473rv和70期偿还的本金金额比为944186970同样解得01196518由已知前10次付款等于应付利息故十年末的未偿还贷款余额仍为1000第11至20次付款等于应付利息的两倍即本金偿还值等于应付利息值有11101110111011101211101010201000809209209209209210004343885后10期每期付款等于x故206474008104343885519分别用将来法计算两种偿还方式在第5次付款之后的未偿还本金有10005100050051000800kvia487914
200a ×(1+ 0.5%)123 = 33873 123 0.5%
200a ×(1+ 0.5%)70 = 26005 123 0.5%

习题答案(第二版)

6、计算分析题参考答案第二章货币时间价值1. ① P0=20 X [ ( P/A, 10% 9) +1]=20 X( 5.759+1 ) =135.18 万元② P4=25 X( P/A ，10% 10)=25 X 6.145=153.63 万元P0=153.63 X (P/S,10%,3)=153.63 X 0.751=115.38 万元应选择第二种方案2. A(F/A,15%,15)=21000 X( P/A,15%,4 ) (1+15%)+21000 X( P/A,15%,4 )(P/F,15%,1)r=i/m=10% - 2=5%t=m • n=5X 2=1010=1 000 X 0.614 = 614(元)0 12 3 45、PV0=A0- 1/(1+i) +A1 • 1/(1+i) +A2 • 1/(1+i) +A3 - 1/(1+i) +A4 - 1/(1+i) =1000 X PVIF5%,0+2000X PVIF5%,1+1000X PVIF5%,2+3000 X PVIF5%,3+4 000 X PVIF5% 4 =1 000 X 1.000+2 000 X 0.952+1000 X 0.907+3 000 X 0.864+4 000 X 0.823 = 9695 元V 0 =10000 X 丄=100000 元10%解得A=2544.87元 3. 6000 X( P/A,8%,10 ) =40260v 42000, 故不能取得借款 4、如果是每年计息一次，则 n=5, i=10%, FV5=1 000，那么: PV =FV5 • PVIFi,n=1 000-PVIF10% 5=1 000 X 0.621=621(元) 如果每半年计息一次，则m=2则：PV=FV10- PVIF5%7.( 1) 16*(1-28%)( P/A,10%,31)(1+10%) = 120.11 万元 (2) 4.6+10.1055 X( 1-28%) (P/A,10%,30)=113.19 万元所以选择第一种方案 8. (1) 8 年后复利终值为：F=P(F/P,7%,8)=1718.19(元) i 8% 每3个月复利一次；F =(1+-)4n =1000冥(1 +」)804 4 =4874(元)(8)P=1000勺1+10%)10=385.5(元)P=1000<(1+ 1|%)-2 泮0=0.4546(元)因为：实际利率i =(1 +r)M-1M12% 12所以：i =(1+ --- ) -1 = 12.68%12 6% 52i =(1 +——)52 -1 =6.18% 521000(1+i)10 =2060 i=7.49% i 10设本金为 P, 2P=P(1+?) =2 , i=14.35%第三章风险与收益甲股票的期望收益率=(10%+ 6%+ 8% /3 = 8% 乙股票的期望收益率=(7%+ 13%+ 10% /3 = 10%(10%-8%)2 +(6%-8%)2 + (8%-8%)2------ =2%乙股票收益率的标准差=J (7% -10%)2+(13%-10%)2+(10%-10%)2= 3%(3 )甲、乙两只股票收益率的变化系数:甲股票收益率的变化系数=2%/8%= 0.25乙股票收益率的变化系数=3%/10%= 0.3(4)甲、乙两只股票的 3值:由于市场达到均衡，则期望收益率=必要收益率根据资本资产定价模型: 甲股票：8沧 5%^ 3X( 12%^5% ,贝3 = 0.4286乙股票：10%= 5%^ 3 X( 12沧 5%),贝U 3 = 0.7143(5 )甲、乙股票的收益率与市场组合收益率的相关系数:根据p j = r X 旦，贝U— b j(6)组合的3系数、组合的风险收益率和组合的必要收益率:组合的 3 系数=30%X 0.4286 + 70%X 0.7143 = 0.63 组合的风险收益率= 0.63 X( 12%- 5% = 4.4% 组合的必要收益率=5%+ 4.4% = 9.4 。

《金融数学》 第二版 习题参考答案 修订版

《金融学(第二版)》讲义大纲及课后习题答案详解 第15章

金融数学课后习题答案 2

《金融数学》(第二版)练习题(修订版)

金融数学引论答案第二版

金融学第二版课后习题答案

《_金融数学-课后习题答案》

《金融学(第二版)》讲义大纲及课后习题答案详解 第16章

《金融学(第二版)》讲义大纲及课后习题答案详解 十四章

《金融学（第二版）》讲义大纲及课后习题答案详解第八章

金融学第二版讲义大纲及课后习题答案详解第十章

《金融学（第二版）》讲义大纲及课后习题答案详解十二章

金融数学引论答案第二版

《金融学(第二版)》讲义大纲及课后习题答案详解 第七章

金融数学附答案修订版

《金融学(第二版)》讲义大纲及课后习题答案详解 第九章

《金融数学》(第二版)习题参考答案(修订版)

习题答案(第二版)

《金融数学》第二版习题参考答案修订版

《金融学(第二版)》讲义大纲及课后习题答案详解第15章

《金融学(第二版)》讲义大纲及课后习题答案详解第16章

《金融学(第二版)》讲义大纲及课后习题答案详解十四章

《金融学(第二版)》讲义大纲及课后习题答案详解第七章

《金融学(第二版)》讲义大纲及课后习题答案详解第九章