Problem Set I: Interest Rates and Present Value Calculations

Transcription

1 Stockholm School of Economics in Riga Financial Economics, Spring 2010 Jevgenijs Babaicevs Problem Set I: Interest Rates and Present Value Calculations Problem 1: The Power of Compounding and Interest Rate Quotes a) An amount of EUR 8,000 is invested at 5% per year. (i) What is the balance in the account after 3 years? After 13 years? (ii) How long does it take for the balance to reach EUR 32,000? b) How much should you have deposited in a bank account 5 years ago in order to have EUR 50,000 today, given that the interest rate has been 5% per year over the period? c) Which do you prefer: a bank account that pays 5% per year for three years or (i) (ii) (iii) An account that pays 2.5% every six months for three years? An account that pays 7.5% every 18 months for three years? An account that pays 0.5% per month for three years? d) You have found three investment choices for a one-year deposit: (i) 10% annual percentage rate (APR) compounded monthly; (ii) 10% APR compounded annually, and (iii) 9% APR compounded daily. Assuming that there are 365 days in the year, compute the effective annual rate (EAR) for each investment choice. e) Your bank account pays interest with an EAR of 5%. What is the APR quote for this account based on semiannual compounding? What is the APR with monthly compounding? f) An amount $1000 earns interest at 5% per year. What this amount has grown to after 10 years when interest is compounded (i) yearly; (ii) monthly; (iii) continuously? Problem 2: The NPV Decision Rule a) You are considering a unique investment opportunity. If you invest LVL 10,000 today, you will receive LVL 500 one year from now, LVL 1500 two years from now, and LVL 10,000 ten years from now. (i) (ii) What is the net present value (NPV) of the investment opportunity if the interest rate is 6% per year? Should you undertake the opportunity? Re-calculate the NPV if the interest rate is 2% per year? Should you undertake the opportunity? 1

2 b) Your company has identified three potential investment projects. The projects and their respective cash flows are shown in the following table: Project Cash Flow Today ($) Cash Flow in One Year ($) A B 5 5 C Assume that all cash flows are certain and the risk-free interest rate is 10% per annum. (i) What is the NPV of each project? (ii) If the firm can choose only one of these projects, which project should it undertake? (iii) If the firm can choose any two of these projects, which projects should it undertake? Problem 3: Valuing Perpetuities a) Vanja, a class of 1998, has just graduated from SSE Riga and wants to endow an annual graduation party at his alma mater. He wants the event to be a memorable one, so he budgets LVL 30,000 per year forever for the party! If the interest rate is 8% per year, and if the first party is in one year's time, how much does Vanja need to donate to endow the party? b) Before accepting the money, the SSE Riga Student Association has asked Vanja to increase the donation to account for the effect of inflation on the cost of the party in future years. Although LVL 30,000 is adequate for the next year's party, the students estimate that the party's cost will rise by 4% per year thereafter. How much does Vanja need to donate now to satisfy their request? Problem 4: Valuing Annuities a) When Armands purchased his new flat, he took out a 30-year annual-payment mortgage with an interest rate of 6% per year. The annual payment on the mortgage is EUR He has just made a payment and has decided to pay the mortgage off by repaying the outstanding balance. What is the payoff amount if Armands has lived in the house for 12 years (so there are 18 years left on the mortgage)? b) You work for a pharmaceutical company RigaFarm that has developed a new drug against the AH1N1 virus. The patent for the drug will last for 17 years. You expect that the drug's profits will be LVL 2 million in its first year and then this amount will grow at a rate of 5% per year for the next 17 years. Once the patent expires, other pharmaceutical companies will be able to produce the same drug and competition will likely drive profits to zero. What is the present value of the new drug if the interest rate is 10% per year? 2

3 Problem 5: Some Further Considerations... a) You are thinking of purchasing a house which costs EUR 350,000. You have EUR 50,000 in cash that you can use as a down payment on the house, but you need to borrow the rest of the purchase price. SwedBank is offering you a 30-year mortgage that requires annual payments and has an interest rate of 7% per year. (i) (ii) What will your annual payment be if you sign up for this mortgage? Suppose you would like to buy the house and take the mortgage described. Unfortunately, you can afford to pay only EUR 23,500 per year. SwedBank agrees to allow you to pay this amount each year, yet still borrow EUR 300,000, on condition that at the end of the mortgage (in 30 years), you must make a balloon payment; that is, you must repay the remaining balance on the mortgage. How much will this balloon payment be? b) Taavi is saving for his retirement. To live peacefully and comfortably, he decides to save up $2 million by the time he is 65. Today is his 30 th birthday, and he decides, starting today and continuing on every birthday up to and including his 65 th birthday, that he will put the same amount into his savings account. The interest rate is 5%. (i) (ii) How much should Taavi set aside each year to make sure that he will have $2 million in the account on his 65 th birthday? Instead assume that Taavi expects his income to increase over the lifetime; hence, for him it will be more rational to save less now and more later. Instead of putting the same amount aside each year, he decides to let the amount that he sets aside grow by 7% per year. How much will he put into account today? Notice that Taavi is planning to make the first contribution today! c) Your spouse bought an annuity from BATVA Life Insurance Company for LVL 200,000 when she retired. In exchange for LVL 200,000, the company will pay her LVL 25,000 per year until she dies. The interest rate is 5%. How long must she live after the day she retired to come out ahead; that is, to get more in value than what she paid in? d) Veiko is running a hot Estonian Music Company. Analysts predict that its earnings will grow at 30% per year for the next five years. After that, as competitors from the South enter, earnings growth is expected to slow down to 2% per year and continue at that level forever. Veiko s company has just announced earnings of EEK 1 million. Assuming that all cash flows occur at the end of the year, what is the present value of all future earnings if the interest rate is 8%? 3

4 Additional Problems for Your own Study Problem 6: Tricky but Interesting a) You have just turned 30 years old and you have accepted your first job - congratulations! Now you must decide how much money to put into your retirement plan which works as follows: Every dollar in the plan earns 7% per year. You cannot make withdrawals until you retire on your sixty-fifth birthday. After that point, you can make withdrawals as you like. You decide that you will plan to live up to 100 and work until you turn 65. You estimate that to live comfortably in retirement, you will need EUR 100,000 per year starting at the end of the first year of retirement and ending on your one hundredth birthday. (i) (ii) If you plan to contribute the same amount to the plan at the end of every year that you work, how much do you need to contribute each year to fund your retirement? The setup of your retirement plan in a) is not very realistic because most retirement plans do not allow you to specify a fixed amount to contribute every year. Instead, you are required to specify a fixed percentage of your salary that you want to contribute. Assume that your starting salary is EUR 75,000 per year and it will grow 2% per year until you retire. Assuming that everything else stays the same as in a) above, what percentage of your income do you need to contribute to the plan every year in order to fund the same retirement income? b) Kenneth Hornbill (KHO) is thinking of making an investment in a new restaurant in Helsinki. The restaurant will generate revenues of $1 million per year for as long as KHO maintains it. He expects that the maintenance cost will start at $50,000 per year and will grow at 5% per year thereafter. Assume that all revenue and maintenance costs occur at the end of the year. KHO intends to run the restaurant, which can be built and become operational immediately, as long as it continues to make a positive cash flow. If the restaurant costs $10 million to build, and the interest rate is 6% per year, should KHO invest in the restaurant? Answer: a) i C EUR 9, ii p 9.948% b) Yes because NPV 3,995,

5 Stockholm School of Economics in Riga Financial Economics, Spring 2010 Jevgenijs Babaicevs Problem Set II: The Law of One Price and Optimal Portfolio Choice Problem 1: Arbitrage and the Law of One Price a) Consider two securities that pay risk-free cash flows over the next two years and that have the current market prices shown in the following table: Security Market Price Today ($) Cash Flow in One Year ($) Cash Flow in Two Years ($) B B (i) (ii) (iii) What is the no-arbitrage price of a security that pays cash flows of $100 in one year and $100 in two years? What is the no-arbitrage price of a security that pays cash flows of $100 in one year and $500 in two years? Suppose a security with cash flows of $50 in one year and $100 in two years is trading for a price of $130. What arbitrage opportunity is available? b) There is an arbitrage opportunity in each of the following two cases. For each case, explain the source of the arbitrage opportunity and how you would trade to exploit it. Case 1 Asset Payoffs Price State 1 State 2 Asset Asset Case 2 Asset Payoffs Price State 1 State 2 Asset Asset Problem 2: The Price of Risk The following table shows the no-arbitrage prices of securities A and B. Security Market Price Cash Flows in One Year ($) Today ($) Weak Economy Strong Economy A B

6 a) What are the payoffs of a portfolio of one share of security A and one share of security B? b) What is the market price of this portfolio? What expected return will you earn from holding this portfolio? c) Suppose security C has a payoff of $600 when the economy is weak and $1800 when the economy is strong. (i) (ii) (iii) Security C has the same payoffs as what portfolio of the securities A and B? What is the no-arbitrage price of security C? What is the expected return of security C if both states are equally likely? What is its risk premium? What is the difference between the return of security C when the economy is strong and when it is weak? If security C had a risk premium of 10%, what arbitrage opportunity would be available? Problem 3: Don't Put All Your Eggs in One Basket: Diversify! Assume A and B are the only securities traded in the market. Expected returns, standard deviations, and the correlation coefficient between the returns of these securities are shown in the following table: Security Expected Return Standard Deviation Stock A 20% 20% Stock B 15% 25% Correlation coefficient -0.4 a) Given the expected return and standard deviation of stock B, would anyone be interested in investing in it? Explain! b) Toms, a prominent Latvian investor, invests 60% of his money in stock A and the rest in stock B. What is the expected return and standard deviation of his portfolio? c) Toms is not satisfied: He wants to form a portfolio (from stock A and B) with the lowest risk. He asks you to solve for the portfolio weights analytically and calculate the expected return and standard deviation of his rebalanced portfolio! d) Additionally two more stocks ( stock C and stock D) have been just introduced. Notice that volatilities of stock A and stock B have changed accordingly and the variance-covariance matrix is as follows: (i) A A B C D B C D Fill in the missing values and interpret numbers on the main diagonal! (ii) Madara suggest Toms constructing a portfolio consisting of 25% invested in stock A, 40% in stock B, 20% in stock C, and the rest in stock D. Calculate the variance of his new portfolio! (iii) What are the betas of all four stocks relative the portfolio? 2

7 Problem 4: A Simple One... Key characteristics of the two securities are summarized in the following table: Security Expected Return Standard Deviation Security 1 10% 5% Security 2 16% 8% a) Which security should an investor choose if she wants to (i) maximize expected returns, (ii) minimize risk (assume the investor cannot form a portfolio)? b) Suppose the correlation of returns on the two securities is What is the optimal combination of Security1 and Security 2 that should be held by the investor whose objective is to minimize risk (assume short-selling is not allowed)? c) Suppose the correlation of returns on the two securities is What fraction of the investor's net worth should be held in Security 1 and in Security 2 in order to produce a zero risk portfolio? d) What is the expected return on the portfolio in c)? How does this compare with the risk-free return on Treasury Bills of 10%? Would the investor want to invest in Treasury Bills? Problem 5: How Well Diversified is Your Portfolio? a) How many variance terms and how many covariance terms do you need to calculate the risk of a 100-share portfolio? b) Suppose all stocks have a standard deviation of 30% and a correlation coefficient of 0.4 with each other. What is the standard deviation of the returns on a portfolio that has equal holdings in 100 stocks? c) What is the standard deviation of a fully diversified portfolio of such stocks? 3

8 Problem 6: Dell vs. Microsoft Additional Problems for Your own Study Historical data on the key risk characteristics of Dell and Microsoft stocks are shown in the following table. Stock Beta Standard Deviation Dell % Microsoft % Correlation coefficient Assume the standard deviation of the return on the market portfolio to be 15%. a) What is the standard deviation of a portfolio invested half in Dell and half in Microsoft? b) What is the standard deviation of a portfolio invested one-third in Dell, one-third in Microsoft and one-third in Treasury Bills? c) What is the standard deviation if the portfolio is split evenly between Dell and Microsoft and is financed with 50% margin? d) What is the approximate standard deviation of a portfolio composed of 100 stocks with betas of 2.21 like Dell? How about 100 stocks like Microsoft? Answer: a) p 51.7% b) p 34.5% 0.66 Dell Microsoft c) p 103.4% d) p 33.15% p 27.15% Problem 7: Diversification Principle Once Again Assume there are three states of the world and two different financial assets. Assets returns are described in the following discrete probability distribution function. Assets Returns State 1 State 2 State 3 Asset 1 8% -2% 12% Asset 2-5% 14% 9% Probability a) What is the mean return on Asset 1 and Asset 2? b) What is the variance of the return on Asset 1 and Asset 2? c) What is the covariance of the returns of two assets? What is the correlation coefficient between the two returns? d) Consider an equally-weighted portfolio of Asset 1 and Asset 2. Compute the mean return and standard deviation of this portfolio! Answer: a) E r E r c) 5.8% 3.5% b) r1, r2 r1, r2 2 2 r r d) E r 4.65% 12.35% p p 4

9 Stockholm School of Economics in Riga Financial Economics, Spring 2010 Jevgenijs Babaicevs Problem Set III: The Capital Asset Pricing Model Problem 1: The ABC's of the Capital Asset Pricing Model Assume there are only three stocks traded on the stock market. All the necessary information regarding these stocks is summarized in the table below. Stock Expected Return Beta-Coefficient Stock % 1.8 Stock 2 7.0% 0.8 Stock a) Assume that a risk-free asset is also available. Explain what is meant by the Security Market Line (SML) in the context of CAPM? b) Construct a SML from the information above and interpret the values of its coefficients. Calculate the expected rate of return on stock 3 according to the CAPM model? How big is the rate of return on the risk-free asset? What is the expected rate of return on the market portfolio? c) Two more stocks have been just introduced, stock 4 with and stock 5 with Empirical evidence reveals that the average rate of return of stock 4 is 16.0%, and the average rate of return of stock 5 is 7%. What inference can you draw from this information? Explain! Problem 2: SML or CML? Consider the following two equations of the Security Market Line (SML) and the Capital Market Line (CML): 1 SML : r r r r rm r F 2 CML : ri rf i M i F i M F You also have the following information: rm 15%, rf 6%, M 15%. Answer the following questions, assuming that the CAPM holds. a) Which equation would you use to determine the expected return on an individual security with a standard deviation of returns of 50% and a beta of 2; that is, 50%, 2. What is the expected return for such security? 1

10 b) Which equation would you use to determine the expected return on a portfolio knowing that it is an efficient portfolio? If you were told that the standard deviation of returns on such portfolio is equal to, what is the expected return on such portfolio? M c) What is a beta of the portfolio in b)? d) Given your answers above, explain what type of risky assets equation (1) can be used for, and what type of risky assets equation (2) can be used for? Problem 3: Capital Budgeting Lembergs Ltd., a new conglomerate in Riga, has three operating divisions outlined in the following table. Division Percentage of Firm Value Food 50 Electronics 30 Chemicals 20 You want to estimate the cost of capital for each division. As part of your project in Market Research, you identified three principal competitors shown in the table below. Competitor Estimated Equity Beta, E D D E Rimi Foods Sony Electronics Dow Chemicals Answer the following questions, assuming that betas are estimated accurately and the CAPM holds. a) Assuming that the debt of these firms is risk-free, estimate the asset beta for each of Lembergs' divisions. D b) Lembergs' -ratio is 0.4. If your estimates of divisional betas are correct, what is D E Lembergs' equity beta? c) Assume that the risk-free interest rate is 7% and the expected return on the market portfolio is 15%. Estimate the cost of capital for each of Lembergs' divisions. d) How much would your estimates of each of Lembergs division's cost of capital change if you assumed that debt has a beta of 0.2? 2

11 Problem 4: The CAPM in Practice - Part I Note: This is a slightly modified problem from Re-Exam 2008! Don Arnisimo, a young specialist in the field of pension economics, offers you three funds to invest for your future pension: (i) a Money-Market Fund which invests into 3-month Treasury bills with a return of 2% per annum; (ii) an S&P 500 Index Fund, which is a good proxy for the market portfolio, delivers a premium of 8% and a standard deviation of 20% per annum; (iii) and Investment Unlimited Equity Fund, managed by LittleDima&Co, the returns of which can be described by the following equation: is a return on the risk- where r t and free asset, is a constant, and r r r r 1 t F Mt F t r Mt are returns of the fund and market portfolios respectively; r F t is the share of the fund s returns not explained by the market. Don Arnisimo has observed that the performance of the fund over the past years yields: R 0.50 a) Calculate the expected rate of return of the Investment Unlimited Equity Fund using the CAPM model? b) Calculate the Sharpe Ratios for the S&P 500 Index Fund and Investment Unlimited Equity Fund respectively. c) Assume that CAPM holds! What is a composition of the optimal portfolio, i.e. how much you should invest in each of the assets, to achieve an expected return of 8% per annum? d) Don Arnisimo found a mistake in his calculations! He has re-estimated equation 1 and discovered that an estimate of is 0.02 (2% per annum) with a standard error of Does this mistake matter for the composition of the optimal portfolio calculated in c)? If no, then explain why not; if yes - explain why so and calculate the composition of a new optimal portfolio (with the same expected return of 8%). Problem 5: The CAPM in Practice - Part II Note: This is a slightly modified problem from Re-Exam 2009! Suppose that the local stock market in Finlandia is made up of only two kinds of stocks, small stocks and large stocks. If you regress the excess return on a small stock, indexed by i, on the excess market return, you find the following relation: R R R R it Ft Mt Ft it Every small stock behaves in this way, but different small stocks have different it which are uncorrelated with each other. Similarly, if you regress the excess return on a large stock, indexed by j, on the excess market return, you find the following relation: R R R R jt Ft Mt Ft jt Again, every large stock behaves in this way, but different large stocks have different uncorrelated with each other. jt which are 3

12 a) What fraction of the stock market value in Finlandia is accounted for by large stocks and what fraction by small stocks? b) Show that if you have an infinite number of stocks available, and you can freely trade in small and large stocks, then there is an arbitrage opportunity. c) Does the Capital Asset Pricing Model hold in Finlandia? ( Hint: Your answer should be a qualitative, verbal explanation in one sentence!) d) Now suppose that it of different small stocks have a correlation with each other, and jt of different large stocks have also correlation with each other. How does this affect your answer in part b)? Additional Problem for Your own Study Problem 6: The Arbitrage Pricing Theory Imagine that there are only two pervasive macroeconomic factors. Securities X, Y and Z have sensitivities to these factors summarizes in the following table. Securities 1 2 X Y Z Assume that the expected risk premium is 4% on factor 1 and 8% on factor 2 (Treasury Bills obviously offer zero risk premium). a) What is the risk premium on each of the three securities according to the Arbitrage Pricing Theory (APT)? b) Suppose you buy $80 of security X and $60 of security Y and sell $40 of security Z. What is the sensitivity of your portfolio to each of the two factors? What is the expected risk premium? c) Suppose you buy $160 of security X and $20 of security Y and sell $80 of security Z. What is the sensitivity of your portfolio to each of the two factors? What is the expected risk premium? d) Suppose that the APT did not hold and that security X offered a risk premium of 8%, security Y offered a premium of 14%, and security Z offered a premium of 16%. Suggest an investment strategy that has zero sensitivity to each factor and that has a negative risk premium. Answer: Let r i be the risk premium on security i, and j - the portfolio's sensitivity to the factor j. a) r 9.0% r 12.0% r 16.0% b) r p 8% X Y Z % d) w 2.0 w 0.5 w 1.5 r 1% 0 c) 1 2 r p X Y Z p 4

13 Stockholm School of Economics in Riga Financial Economics, Spring 2010 Jevgenijs Babaicevs Problem Set IV: Valuation of Options - Basics Problem 1: Position Diagrams For each of the following cases below, draw a position diagram (i.e. show the gross value at the date of maturity) to illustrate how options can be used. Be very specific to state the individual contracts constituting the portfolio. a) You want to create an insurance against declining stock prices when you already own the stock. Can you achieve the same result with other financial instruments? b) You want to create a hedge against all changes of the stock price when you already own the stock. c) You want to gain from all changes in the stock price without owning the underlying stock. d) Suppose that short selling of the stock is not allowed. Describe how to construct a portfolio of bonds and options with the identically the same return profile as a short stock. Illustrate using a position diagram! e) You are discussing with your colleague whether to issue a loan to a small business with a high debt/equity ratio. To better understand all the risks involved, explain: (i) (ii) Why equity can be viewed as a call option on a firm? Express the position of an equityholder in terms of call options. How can debt be viewed as an option on a firm? Express the position of a debtholder in terms of put options. f) This summer you are offered an internship opportunity at the options-trading department at the investment bank CHASE. Prior to this, you have decided to train yourself up and draw position diagrams for each of the following strategies. Assume that all options are identical but the exercise prices. (i) Buy 1 call at E 1 and 1 put at 2 E E E. 1 2 (ii) Long 1 call at E 1 and short 1 call at 2 E E E. 1 2 (iii) Long 1 call at E 1 and 1 call at E 2, short 1 call at E3 and 1 call at 4 E E E E E

14 Problem 2: The Put-Call Parity Condition a) Assume that a call option is traded for $10 with an exercise price of $110. The price of the underlying security is $120. The expiration is one year and the continuously compounded interest rate is 10% per annum. (i) (ii) Do these figures represent equilibrium in the market? Using the figures above, demonstrate which transactions should be made in order to exploit the disequilibrium situation! b) A European put is sold for LVL 1. It has an expiration date in two month s time and the risk free interest rate is 2% during this period. The underlying equity does not pay a dividend and its current price is LVL 160. The exercise price of the option is LVL 165. (i) Given these numbers, is it possible to exploit a risk-free arbitrage opportunity? Please notice that there are NO call options traded on this share! c) The common stock of Pirate Bank is selling for $90. A 26-week call option written on Pirate Bank s stock is selling for $8. The call s exercise price is $100. The continuously compounded risk-free interest rate is 10% per annum. (i) Suppose that puts on Pirate Bank s stock are not traded but you are very eager to buy one. How can you achieve this? (ii) Suppose that puts are traded. What should a 26-week put with an exercise price of $100 sell for? Problem 3: Understanding Options The corporate charter of Lidosta Derivatives contains a paragraph stating that the firm will be liquidated after exactly one year from today and the market value of total assets will be distributed to the firm s debtholders and shareholders. Lidosta Derivatives is financed by: (i) A seasoned equity offering of 35 million LVL, and (ii) two zero-coupon loans each with a principal of 100 million LVL that will be repaid after exactly one year. The total market value today of equity and debt is 200 million LVL. Loan-AA is senior, i.e. it has priority at liquidation over Loan-BB that is subordinate. Loan-AA has a simple interest rate of 8.5% that is equal to the market interest rate. The simple risk free interest rate is 8%. a) Illustrate in a single position diagram the possible values of the three sources of finance after exactly one year. Be sure to state what you measure on the two axes, and indicate precisely the critical values on both axes. b) Loan-AA may be interpreted as a synthetic portfolio of one risk free asset and one put option. Indicate the exercise price of the put option, who is the buyer and the seller of the put option, and calculate the premium of the put option today. c) Loan-BB may be interpreted as a synthetic portfolio (positive vertical price spread) consisting of two call options with different exercise prices. Calculate the premiums today of the two call options. 2

15 Problem 4: Introduction to the Binomial Option Pricing Model Assume that there are only two possible states of the world. In State 1, the stock price rises by 50% and in State 2, it drops by 25%. The current stock price is $100, the exercise price of the call option written on this stock is $105, and the simple risk-free rate for the period until expiration is 5%. a) Define and calculate the hedge ratio in this specific case? b) How much should you borrow to hedge the call? c) Use the one period binomial option pricing model to determine the value of the call. 3

16 Stockholm School of Economics in Riga Financial Economics, Spring 2010 Tālis Putniņš Problem Set IX: Valuing common stocks, and payout policy Exercise 1: Valuing common stocks Computer stocks currently trade at a required return on equity of 16%. Krāslavas Kompji, a large computer company will pay a year-end dividend of $2 per share. a) If the stock is selling at $50 per share, what must be the market s expectation of the growth rate of dividends? b) If dividend growth forecasts for Krāslavas Kompji are revised downward to 5% per year, what will happen to the price of Krāslavas Kompji stock? What (qualitatively) will happen to the company s P/E ratio? Exercise 2: Valuing common stocks You believe that next year Brāļu Corp. will have earnings per share equal to 6 on its common stock. Thereafter you expect earnings to grow at a rate of 8% p.a. in perpetuity. The payout ratio is 1/3. You require a return of 12% on your investment. a) How much should you be prepared to pay for the stock? b) Compute the return on equity (ROE) and present value of growth opportunities (PVGO). c) Assume that ROE has changed to 10%. How much should you be prepared to pay for the stock? Compute PVGO. d) Assume that ROE has changed to 15%. How much should you be prepared to pay for the stock? Compute PVGO. Exercise 3: Valuing common stocks Čiptek is an established computer chip manufacturer with profitable existing projects and new products in development. The company earned $1 per share last year and just paid a $0.50 dividend. The required return on equity in the computer chip 1

17 manufacturing industry is 15%. Investors believe the company will maintain the payout ratio of 50% and that the ROE of 20% will persist. a) What is the market price of Čiptek stock? b) Suppose you discover that Čiptek s competitor has developed a new chip that will eliminate Čiptek s technological advantage. The new product will reduce Čiptek s ROE in 2 year s time to 15% and at this time Čiptek will have to reduce its plowback ratio to 40% due to falling demand. What is your estimate of Čiptek s intrinsic value per share? c) No one in the market will become aware of the change to Čiptek s competitive status until the end of year 2. What will be rate of return on Čiptek stock in each of the 3 years from now (years 1, 2, and 3)? Exercise 4: Valuing common stocks Bauskas Corp. s cash flows from operations before interest and taxes were $2m in the year just ended and management expects that they will grow by 5% per year forever. To make this happen the firm will have to invest 20% of pretax cash flow each year. The tax rate is 35%. Depreciation was $200,000 in the year just ended and is expected to grow at the same rate as operating cash flows. The company s weighted average cost of capital is 12%, the firm currently has $2m worth of debt outstanding and 1m shares on issue. Use the free cash flow approach to find the intrinsic value of a share. Exercise 5: Valuing common stocks Lodiņu Corp. currently reinvests all earnings (pays no dividends) and is expected to continue doing so for the next 5 years. Its latest EPS was $10. The firm s ROE for the next 5 years is 20% per year. In the 6 th year from now ROE on new investments is expected to fall to 15% and the company is expected to start paying out 40% of earnings as dividends. The required return on equity is 15%. a) What is Lodiņu Corp. s intrinsic value per share? b) Assuming the current market price equals intrinsic value what will happen to its stock price over the next year? Next two years? Between years 6 and 7? c) What is the dividend yield expected to be in the 10 th year? d) Suppose Lodiņu Corp. only pays out 20% of earnings starting year 6. What effect will this have on your estimate of intrinsic value? 2

18 Exercise 6: Valuing common stocks The market consensus is that Līvu Corp. has ROE = 9%, a beta of 1.25 and plans to indefinitely maintain its plowback ratio of 2/3. This year s earnings were $3 per share and the annual dividend was just paid. The consensus estimate of the coming year s market return is 14% and T-bills currently offer a yield of 6%. a) Find the price at which a share in Līvu Corp. should sell. b) Calculate the forward P/E ratio. c) Calculate the present value of growth opportunities. d) Suppose your research convinces you that Līvu Corp. will shortly announce that it will immediately reduce its plowback ratio to 1/3. Find the intrinsic value of the stock. Why is the intrinsic value different from the current market price? Exercise 7: Valuing common stocks A stock has just paid a dividend of $0.50 per share. The dividend is expected to grow at a rate of 6% p.a. for the next 20 years, after which it will level off. If the discount rate of shares of similar risk is 9% p.a., what is the value of the shares? Exercise 8: Payout policy Užavas Corp. has 1 million shares outstanding with a total market value of $20 million. The firm is expected to pay $1 million of dividends next year, and thereafter the amount paid is expected to grow by 5% per year in perpetuity. Thus the expected dividend is $1.05 in year 2, $1.105 in year 3 and so on. The company has heard that that the value of a share depends on the flow of dividends, and therefore it announces that next year s dividend will be increased to $2 million and that the extra cash will be raised by an issue of shares immediately after the stock goes ex-dividend (i.e., the new shares are not entitled to the $2 dividend). After that the total amount paid out each year will be as previously forecast, i.e., $1.05 in year 2, $1.105 in year 3 and so on. a) At what price will the new shares be issued in year 1? b) How many shares will the firm need to issue? c) What will be the expected dividend payments on these new shares, and what, therefore, will be paid out to the old shareholders after year 1? d) Show that the present value of the cash flows to the current shareholders remains $20 million. e) Now assume the new shares are issued in year 1 at $10 a share. Who gains and who loses? Is dividend policy still irrelevant? Why or why not? 3

19 Exercise 9: Payout policy Adherent of the dividends-are-good school sometimes point to the fact that stocks with high dividend yields tend to have above-average P/E multiples. Is this evidence convincing? Discuss. Exercise 10: Payout policy Suppose there are just three types of investors with the following tax rates: Individuals Corporations Institutions Dividends 50% 5% 0% Capital Gains 15% 35% 0% Individuals invest a total of $80 billion in stock and corporations invest $10 billion. The remaining stock is held by institutions. All three groups simply seek to maximize their after-tax income. These investors can choose from three types of stock offering the following pretax payouts: Low payout Medium payout High payout Dividends $5 $5 $30 Capital Gains $15 $5 $0 These payouts are expected to exist in perpetuity. The low-payout stocks have a total market value of $100 billion; the medium-payout stocks have a value of $50 billion; and the high-payout stocks have a value of $120 billion. a) Who are the marginal investors that determine the prices of the stocks? b) Suppose that this marginal group of investors requires a 12% after-tax return. What are the prices of the low-, medium-, and high-payout stocks? c) Calculate the after-tax returns of the three types of stock for each investor group. d) What are the dollar amounts of the three types of stock held by each investor group? 4

20 Exercise 11: Payout policy Your company has been under pressure from some investors to increase dividends. The following table provides historic information about your company s dividend policy. Year EPS Dividends per share Price 2009 $2.05 $0.30 $ $1.55 $0.30 $ $1.70 $0.25 $ $0.90 $0.25 $ $0.85 $0.25 $ $1.02 $0.25 $10.20 The average payout ratio in the industry is about 25%. You also notice that on the exdividend date the company s stock price tends to drop in the range of 60% to 80% of the amount of the dividend. a) Calculate the average dividend payout ratio and dividend yield for the company for the last few years. Does the firm s dividend policy deviate from the industry average significantly? b) If the tax on capital gains is 15%, what is the range of the marginal investor s tax rate on dividends? c) How would a pension fund that pays no tax on capital gains and no tax on dividends make profits on the ex-dividend day? Are there any risks involved in your suggested strategy? Exercise 12: Payout policy A corporation operates two identical projects, each yielding annual cash flows of $50 million. The corporation has 1000 million shares on issue and currently pays all its cash flows out as dividends. If the corporation decides to increase its dividend by 50 cents per share, and the cost of issuing new shares is 0.5%, calculate the impact on shareholder wealth of the corporation s strategy. Assume a discount rate of 10%. 5

21 Exercise 13: Payout policy Piebalgas Corp. must decide how much to pay out as dividends to its shareholders. It expects to have a net income of $1,000 (after depreciation of $500), and has the following investment opportunities: Project Initial investment Beta IRR A $ % B $ % C $ % The current T-bond rate is 9% and the expected premium on the stock market index is 6%. The firm has revenues of $5,000, which it expects to grow at 8%. Working capital will be maintained at 25% of revenues. How much should the firm return to shareholders as a dividend? 6

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24 Stockholm School of Economics in Riga Financial Economics, Spring 2010 Jevgenijs Babaicevs Problem Set VI: The Black-Merton-Scholes Option Pricing Model Problem 1: Introduction to the BMS Model a) Peteris is interested in purchasing a European call on a new hot high-tech stock Moodle Inc. The call has a strike price of $100 and expires in 90 days. Currently, the stock is trading at a price of $120 and has a standard deviation of 40% per year. The continuously compounded riskfree interest rate is 6.18% per year. (i) (ii) Use the Black-Merton-Scholes formula to calculate the price of the call. Compute the price of the put with the same strike and expiration date. b) Assume that you own a small IT company. You have just received an offer to buy your company from a large, publicly traded firm, Cisco Systems (CS). Under the terms of the offer, you will receive 1 million shares of CS. CS stock currently trades at LVL 25 per share. You may decide to sell the shares of CS that you will receive in the market at any time. However, as part of the offer, CS also agrees that at any time during the next year, it will buy the shares back from you for LVL 25 per share if you want. Assume the current continuously compounded riskfree interest rate is 6.18% a year, and the volatility of CS stock is 30% per annum. Assume further that no dividends are paid. (i) Calculate the value of the offer and explain why it is worth more than LVL 25 million? Problem 2: Option Pricing in a Discrete and Continuous Time The current stock price of Paalzow Oil (PO) is SEK 200. The standard deviation is 22.3% per year, and the continuously compounded interest rate is 21% a year. A one-year call option on PO has an exercise price of SEK 180. a) Use the Black-Merton-Scholes model to value the call option on PO. b) Calculate the up-step and down-step that you would use if you valued the PO option with the one-period Binomial Option Pricing Model. Then value the option using this model! c) Re-value the option by using the two-period Binomial Option Pricing Model. d) Use your answer to part c) to calculate the option delta (i) today, (ii) next period if the stock price rises, and (iii) next period if the stock price falls. Show at each point how you would replicate a call option with a levered investment in the company s stock. 1

25 Problem 3: Collateralized Debt Obligations Note: This is a slightly modified version of Problem 5 from Re-Exam Diana P. has just informed you that your summer internship is going to be at SweetBank this year. The bank is planning to offer three types of claims, which mature in one year, on the value of its underlying portfolio consisting of commercial loans issued to residents of the Independent Republic of Junkbondia. The three claims differ in the seniority of their claims to the value of the underlying portfolio, with tranche 1 1 T being most senior and tranche 3 T being most junior. The claims of the tranches are 3 paid in the order of their seniority: T 1 pays the value of the underlying loan portfolio up to a maximum of D 1, and is paid first; T 2 pays the excess value of the loan portfolio up to a maximum of D 2, after T1 has been paid; T 3 is a claim to the residual value of the loan portfolio after the claims of the first two tranches are satisfied. a) Using real options language, draw separate position diagrams for the three tranches as a function of the value of the underlying loan portfolio V. Do not forget to denote the critical values on the axes! b) Explain the character (type) of the options associated with each of the claims! Express the value C E - the price of a European call option of each of the tranches as a function of V and written on the value of the underlying loan portfolio with an exercise price of E. c) Your summer internship has slowly turned to your full employment and you are now a CFO of SweetBank. As part of your compensation package you receive several units of one of the tranches. After taking over as CFO your first task is to decide how many loans N to include in the underlying portfolio in such a way that the market value of your compensation package is maximized! You know that each of the loans is equally weighted in the portfolio. The return on 2 each of the loans has an annual variance of and any two returns have a correlation coefficient of 0 1. (i) You are given several units of tranche T 1. How many loans will you decide to include in the underlying portfolio? (ii) You are given several units of tranche T 3. How many loans will you decide to include in the underlying portfolio? In both cases, show your reasoning explicitly! d) Suppose the board of SweetBank has decided to compensate you with units of tranche T 3 and you have optimally chosen the number of loans to include in the portfolio. Moreover, the maturity of the structured products has been extended from one year to two years. Enthused by this change, you do some research and discover that at the annual frequency the returns on some loans exhibit momentum (L oans-aa), while others exhibit mean-reversion (L oans-bb). The 2 annual variance of returns of both types of loans is still. (i) Which type of loans will you choose to include in the underlying portfolio? 2

26 Additional Problem for Your own Study Problem 4: BMS Model and Land-Owners Alexandre owns a one-year call option on 1 acre of real estate in Paris. The exercise price is $2 million and the current, appraised market value of the land is $1.7 million. The land is currently used as a parking lot, generating just enough money to cover real estate taxes. The annual standard deviation is 15% and the continuously compounded interest rate is 12%. a) Use the Black-Merton-Scholes formula to calculate the price of a call belonging to Alexandre? b) Assume now that the land is occupied by a warehouse generating rents of $150,000 after real estate taxes and all other expenses. The value of the land plus warehouse is again $1.7 million. Alexandre has a European call option. What is the value of the call now? Answer: a) C USD 71, b) C ' USD 28,

27 Stockholm School of Economics in Riga Financial Economics, Spring 2010 Tālis Putniņš Problem Set VII: Bond pricing, duration, convexity and immunization Exercise 1: Bond pricing Suppose five-year government bonds are selling on a yield of 4% p.a. and have a coupon of 6% p.a. a) Calculate the price of the bonds assuming they are issued by a European government and the coupon payments are made annually. b) Calculate the price of the bonds assuming they are issued by the US Treasury so the coupon payments are made semi-annually and the given yield refers to a semiannually compounded rate. Exercise 2: Bond pricing Two Treasury bonds have face values of $100,000 and pay coupons at the rate of 10%, semi-annually. Bond P has four years to maturity and bond Q has eight years to maturity. a) If the yield on the bonds is 7.5% p.a., what are the prices of the two bonds? b) If the yield rises to 12% p.a., what are the prices of the two bonds? c) What do the prices illustrate about the relations between price, yield, coupon rate and maturity? Exercise 3: Bond pricing Which security has a higher effective annual interest rate? a) A 3-month Treasury note selling at $97,645 with par value $100,000. b) A coupon bond selling at par and paying a 10% coupon semi-annually. 1

28 Exercise 4: Bond pricing Consider a bond that pays a coupon rate of 10% p.a. semi-annually when the market interest rate is only 4% per half year. The bond has 3 years to maturity. a) Find the bond s price today and in 6 months from now after the next coupon is paid (assuming interest rates do not change). b) What is the total (6 month) rate of return on the bond if you buy it today and sell it in 6 months? c) Repeat (b), but instead of interest rates remaining unchanged, in 6 months time they have fallen to 3% per half year. Exercise 5: Bond pricing A $1000 face value bond with a coupon rate of 7% makes semi-annual coupon payments on January 15 and July 15 of each year. The Wall Street Journal reports the asked price for the bond on 30 January at 100:02. What is the invoice price of the bond? The coupon period has 182 days. Exercise 6: Bond pricing Brengulis Corp. issues two bonds with 20-year maturities, face values of $1000 and annual coupons. Both bonds are callable at $1050. The first bond is issued at a deep discount with a coupon rate of 4% and a price of $580 to yield 8.4%. The second bond is issued at par value with a coupon rate of 8.75%. a) What is the yield to maturity on the par bond? Why is it higher than the yield of the discount bond? b) If you expect rates to fall substantially in the next 2 years, which bond would you prefer to hold? c) In what sense does the discount bond offer implicit call protection? Exercise 7: Bond pricing A 10-year bond of a firm in severe financial distress pays annual coupons with a coupon rate of 14% and sells for $900. The firm is currently renegotiating the debt, and it appears that the lenders will allow the firm to reduce the coupon payments on the bond to one-half of the originally contracted amount. The firm can handle these lower payments. What is the stated and expected yield to maturity on the bonds? 2

29 Exercise 8: Duration, convexity and immunization Consider the following three bonds: Bond Time to maturity Coupon Bond 1 1 year 10 % Bond 2 3 years 0 % Bond 3 4 years 20 % The yield curve is flat and the yield to maturity is 9%. The face value is 100 for all three bonds. a) Compute the duration for all three bonds. b) Suppose that you purchase one Bond 1 and one Bond 3. Compute the duration of this portfolio. c) How can an investor adjust the portfolio in b) to get a duration of 2 years? Exercise 9: Duration, convexity and immunization Rank the effective durations of the following pairs of bonds: a) Bond A is an 8% coupon bond with 20 years to maturity selling at par value. Bond B is an 8% coupon bond with 20 years to maturity selling below par value. b) Bond A is a 20-year non-callable 8% coupon bond selling at par value. Bond B is a 20-year callable coupon bond with a coupon of 9%, also selling at par. c) Bond A is a 3-year 6% coupon bond making annual coupon payments priced at a yield of 4%. Bond B is a 3-year 6% coupon bond making semiannual coupon payments, also priced at a yield of 4%. d) Bond A is a 3-year 6% coupon bond making annual coupon payments priced at a yield of 4%. Bond B is a 3-year 8% coupon bond making annual coupon payments priced at a yield of 4%. e) Bond A is Baa-rated with an 8% coupon and 20 years to maturity. Bond B is Aaarated with an 8% coupon and 20 years to maturity. Exercise 10: Duration, convexity and immunization An insurance company must make payments to a customer of $10 million in 1 year and $4 million in 5 years. The yield curve is flat at 10%. a) If it wants of fully fund and immunize the obligation to this customer with a single issue of a zero-coupon bond, what maturity bond must it purchase? 3

30 b) What must be the face value and market value of that zero-coupon bond? Exercise 11: Duration, convexity and immunization Pension funds pay lifetime annuities to recipients. If a firm will remain in business indefinitely, the pension fund obligation will resemble a perpetuity. Suppose, therefore, that you are managing a pension fund with obligations to make perpetual payments of $2 million per year to beneficiaries. The yield to maturity on all bonds is 16%. a) If the duration of 5-year maturity bonds with annual coupons of 12% is 4 years and the duration of 20-year maturity bonds with 6% annual coupons is 11 years, how much of each of these coupon bonds (in market value) should you hold to fully fund and immunize your obligation? b) What will be the par value of your holdings in the 20-year bond? Exercise 12: Duration, convexity and immunization A year maturity zero-coupon bond selling at a yield to maturity of 8% (effective annual yield) has convexity of and modified duration of years. A 30-year maturity 6% annual coupon bond also selling at a yield to maturity of 8% has nearly identical duration (11.79 years) but considerably higher convexity: a) Suppose the yield to maturity on both bonds increases to 9%. What will be the actual percentage capital loss on each of the bonds? What percentage capital loss would be predicted by the duration-with-convexity rule? b) Repeat (a), but this time assume the yield to maturity decreases to 7%. c) Compare the performance of the two bonds in the two scenarios, one involving an increase in rates, the other involving a decrease. Based on the comparative investment performance, explain the attraction of convexity. d) In view of your answer to (c), do you think it is possible for two bonds with equal duration but different convexity to be priced initially at the same yield to maturity if the yields on both bonds always increased or decreased by equal amounts, as in this example? Would anyone be willing to buy the bond with lower convexity under these circumstances? 4