Problem Set I: Interest Rates and Present Value …sseriga.free.fr/course/uploads/Financial...

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Stockholm School of Economics in RigaFinancial Economics, Spring 2010

Jevgenijs Babaicevs

Problem Set I:Interest Rates and Present Value Calculations

Problem 1: The Power of Compounding and Interest Rate Quotesa) 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 EUR50,000 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) An account that pays 2.5% every six months for three years?

(ii) An account that pays 7.5% every 18 months for three years?

(iii) 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% APRcompounded daily. Assuming that there are 365 days in the year, compute the effective annualrate (EAR) for each investment choice.

e) Your bank account pays interest with an EAR of 5%. What is the APR quote for this accountbased 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 yearswhen interest is compounded (i) yearly; (ii) monthly; (iii) continuously?

Problem 2: The NPV Decision Rulea) 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 tenyears from now.

(i) What is the net present value (NPV) of the investment opportunity if the interest rate is6% per year? Should you undertake the opportunity?

(ii) Re-calculate the NPV if the interest rate is 2% per year? Should you undertake theopportunity?

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b) Your company has identified three potential investment projects. The projects and theirrespective cash flows are shown in the following table:

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 Perpetuitiesa) 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 budgetsLVL 30,000 per year forever for the party! If the interest rate is 8% per year, and if the firstparty 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 thedonation to account for the effect of inflation on the cost of the party in future years. AlthoughLVL 30,000 is adequate for the next year's party, the students estimate that the party's cost willrise by 4% per year thereafter. How much does Vanja need to donate now to satisfy theirrequest?

Problem 4: Valuing Annuitiesa) 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 1200. He has justmade 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 yearsleft on the mortgage)?

b) You work for a pharmaceutical company RigaFarm that has developed a new drug against theAH1N1 virus. The patent for the drug will last for 17 years. You expect that the drug's profitswill be LVL 2 million in its first year and then this amount will grow at a rate of 5% per yearfor the next 17 years. Once the patent expires, other pharmaceutical companies will be able toproduce the same drug and competition will likely drive profits to zero. What is the presentvalue of the new drug if the interest rate is 10% per year?

Project Cash Flow Today($)

Cash Flow in OneYear ($)

A -10 20

B 5 5

C 20 -10

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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 thepurchase price. SwedBank is offering you a 30-year mortgage that requires annual payments andhas an interest rate of 7% per year.

(i) What will your annual payment be if you sign up for this mortgage?

(ii) 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 toallow you to pay this amount each year, yet still borrow EUR 300,000, on condition thatat the end of the mortgage (in 30 years), you must make a balloon payment; that is, youmust repay the remaining balance on the mortgage. How much will this balloonpayment be?

b) Taavi is saving for his retirement. To live peacefully and comfortably, he decides to save up $2million by the time he is 65. Today is his 30th birthday, and he decides, starting today andcontinuing on every birthday up to and including his 65th birthday, that he will put the sameamount into his savings account. The interest rate is 5%.

(i) How much should Taavi set aside each year to make sure that he will have $2 million inthe account on his 65th birthday?

(ii) Instead assume that Taavi expects his income to increase over the lifetime; hence, forhim it will be more rational to save less now and more later. Instead of putting the sameamount aside each year, he decides to let the amount that he sets aside grow by 7% peryear. How much will he put into account today? Notice that Taavi is planning to makethe first contribution today!

c) Your spouse bought an annuity from BATVA Life Insurance Company for LVL 200,000 whenshe retired. In exchange for LVL 200,000, the company will pay her LVL 25,000 per year untilshe dies. The interest rate is 5%. How long must she live after the day she retired to come outahead; 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 at30% per year for the next five years. After that, as competitors from the South enter, earningsgrowth is expected to slow down to 2% per year and continue at that level forever. Veiko’scompany has just announced earnings of EEK 1 million. Assuming that all cash flows occur atthe end of the year, what is the present value of all future earnings if the interest rate is 8%?

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Additional Problems for Your own Study

Problem 6: Tricky but Interestinga) 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 onyour sixty-fifth birthday. After that point, you can make withdrawals as you like. You decidethat you will plan to live up to 100 and work until you turn 65. You estimate that to livecomfortably in retirement, you will need EUR 100,000 per year starting at the end of the firstyear of retirement and ending on your one hundredth birthday.

(i) If you plan to contribute the same amount to the plan at the end of every year that youwork, how much do you need to contribute each year to fund your retirement?

(ii) The setup of your retirement plan in a) is not very realistic because most retirementplans do not allow you to specify a fixed amount to contribute every year. Instead, youare 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 yearuntil you retire. Assuming that everything else stays the same as in a) above, whatpercentage of your income do you need to contribute to the plan every year in order tofund 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 yearthereafter. Assume that all revenue and maintenance costs occur at the end of the year. KHOintends to run the restaurant, which can be built and become operational immediately, as long asit continues to make a positive cash flow. If the restaurant costs $10 million to build, and theinterest rate is 6% per year, should KHO invest in the restaurant?

Answer: a) EUR 9,366.29 9.948%i C ii p b) Yes because 3,995,073.97 0NPV

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Jevgenijs Babaicevs

Problem Set II: The Law of One Price and Optimal Portfolio Choice

Problem 1: Arbitrage and the Law of One Pricea) 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:

(i) What is the no-arbitrage price of a security that pays cash flows of $100 in one year and$100 in two years?

(ii) What is the no-arbitrage price of a security that pays cash flows of $100 in one year and$500 in two years?

(iii) Suppose a security with cash flows of $50 in one year and $100 in two years is tradingfor 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 thesource of the arbitrage opportunity and how you would trade to exploit it.

Problem 2: The Price of RiskThe following table shows the no-arbitrage prices of securities A and B.

Security Market PriceToday ($)

Cash Flow in OneYear ($)

Cash Flow in TwoYears ($)

B1 94 100 0

B2 85 0 100

PayoffsCase 2 AssetPrice State 1 State 2

Asset 1 0.5 1 2

Asset 2 2.5 3 10

PayoffsCase 1 AssetPrice State 1 State 2

Asset 1 0.5 1 -0.5

Asset 2 3 5 -2.5

Cash Flows in One Year ($)Security Market PriceToday ($) Weak Economy Strong Economy

A 231 0 600

B 346 600 0

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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 thisportfolio?

c) Suppose security C has a payoff of $600 when the economy is weak and $1800 when theeconomy is strong.

(i) Security C has the same payoffs as what portfolio of the securities A and B? What is theno-arbitrage price of security C?

(ii) What is the expected return of security C if both states are equally likely? What is itsrisk premium? What is the difference between the return of security C when theeconomy is strong and when it is weak?

(iii) If security C had a risk premium of 10%, what arbitrage opportunity would beavailable?

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, andthe correlation coefficient between the returns of these securities are shown in the following table:

Security Expected Return Standard Deviation Correlationcoefficient

Stock A 20% 20%

Stock B 15% 25%-0.4

a) Given the expected return and standard deviation of stock B, would anyone be interested ininvesting in it? Explain!

b) Toms, a prominent Latvian investor, invests 60% of his money in stock A and the rest in stockB. 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. Heasks you to solve for the portfolio weights analytically and calculate the expected return andstandard deviation of his rebalanced portfolio!

d) Additionally two more stocks (stock C and stock D) have been just introduced. Notice thatvolatilities of stock A and stock B have changed accordingly and the variance-covariance matrixis as follows:

4 4

10 1015 10

5 20 012 5 12

A B C DABCD

(i) 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 hisnew portfolio!

(iii) What are the betas of all four stocks relative the portfolio?

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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 +1.0. What is the optimalcombination of Security1 and Security 2 that should be held by the investor whose objective isto minimize risk (assume short-selling is not allowed)?

c) Suppose the correlation of returns on the two securities is -1.0. What fraction of the investor'snet worth should be held in Security 1 and in Security 2 in order to produce a zero riskportfolio?

d) What is the expected return on the portfolio in c)? How does this compare with the risk-freereturn 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 witheach other. What is the standard deviation of the returns on a portfolio that has equal holdings in100 stocks?

c) What is the standard deviation of a fully diversified portfolio of such stocks?

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Additional Problems for Your own StudyProblem 6: Dell vs. MicrosoftHistorical data on the key risk characteristics of Dell and Microsoft stocks are shown in the followingtable.

Stock Beta Standard Deviation Correlationcoefficient

Dell 2.21 62.7%

Microsoft 1.81 50.7%0.66

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 Microsoftand one-third in Treasury Bills?

c) What is the standard deviation if the portfolio is split evenly between Dell and Microsoft and isfinanced with 50% margin?

d) What is the approximate standard deviation of a portfolio composed of 100 stocks with betas of2.21 like Dell? How about 100 stocks like Microsoft?

Answer: a) 51.7%p b) 34.5%p

c) 103.4%p d) 33.15% 27.15%Dell Microsoftp p

Problem 7: Diversification Principle Once AgainAssume there are three states of the world and two different financial assets. Assets’ returns aredescribed in the following discrete probability distribution function.

ReturnsAssetsState 1 State 2 State 3

Asset 1 8% -2% 12%

Asset 2 -5% 14% 9%

Probability 0.5 0.3 0.2

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 betweenthe two returns?

d) Consider an equally-weighted portfolio of Asset 1 and Asset 2. Compute the mean return andstandard deviation of this portfolio!

Answer: a) 1 25.8% 3.5%E r E r b)1 2

2 228.36 75.25r r

c)1 2 1 2, ,27.1 0.587r r r r d) 4.65% 12.35%p pE r

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Jevgenijs Babaicevs

Problem Set III:The Capital Asset Pricing Model

Problem 1: The ABC's of the Capital Asset Pricing ModelAssume there are only three stocks traded on the stock market. All the necessary information regardingthese stocks is summarized in the table below.

Stock Expected Return Beta-Coefficient

Stock 1 12.0% 1.8Stock 2 7.0% 0.8Stock 3 1.2

a) Assume that a risk-free asset is also available. Explain what is meant by the Security MarketLine (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 therate 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 4 2.0 and stock 5 with 5 1.05 .Empirical evidence reveals that the average rate of return of stock 4 is 16.0%, and the averagerate 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 :

2 :

i F i M F

M Fi F i

M

SML r r r r

r rCML r r

You also have the following information: 15%, 6%, 15%.M F Mr r Answer the followingquestions, assuming that the CAPM holds.

a) Which equation would you use to determine the expected return on an individual security with astandard deviation of returns of 50% and a beta of 2; that is, 50%, 2 . What is theexpected return for such security?

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b) Which equation would you use to determine the expected return on a portfolio knowing that it isan efficient portfolio? If you were told that the standard deviation of returns on such portfolio isequal to M , what is the expected return on such portfolio?

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, andwhat type of risky assets equation (2) can be used for?

Problem 3: Capital BudgetingLembergs Ltd., a new conglomerate in Riga, has three operating divisions outlined in the followingtable.

Division Percentage of Firm ValueFood 50

Electronics 30Chemicals 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, ED

D ERimi Foods 0.8 0.3Sony Electronics 1.6 0.2Dow Chemicals 1.2 0.4

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.

b) Lembergs'D

D E-ratio is 0.4. If your estimates of divisional betas are correct, what is

Lembergs' equity beta?

c) Assume that the risk-free interest rate is 7% and the expected return on the market portfolio is15%. 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 youassumed that debt has a beta of 0.2?

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Problem 4: The CAPM in Practice - Part INote: 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 foryour future pension: (i) a Money-Market Fund which invests into 3-month Treasury bills with a returnof 2% per annum; (ii) an S&P 500 Index Fund, which is a good proxy for the market portfolio, deliversa premium of 8% and a standard deviation of 20% per annum; (iii) and Investment Unlimited EquityFund, managed by LittleDima&Co, the returns of which can be described by the following equation:

1t F Mt F tr r r r

where tr and Mtr are returns of the fund and market portfolios respectively; Fr is a return on the risk-free asset, is a constant, and t is the share of the fund’s returns not explained by the market. DonArnisimo has observed that the performance of the fund over the past years yields:

20.0 1.0 0.50R

a) Calculate the expected rate of return of the Investment Unlimited Equity Fund using the CAPMmodel?

b) Calculate the Sharpe Ratios for the S&P 500 Index Fund and Investment Unlimited Equity Fundrespectively.

c) Assume that CAPM holds! What is a composition of the optimal portfolio, i.e. how much youshould 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 anddiscovered that an estimate of is 0.02 (2% per annum) with a standard error of 0.002. Doesthis mistake matter for the composition of the optimal portfolio calculated in c)? If no, thenexplain why not; if yes - explain why so and calculate the composition of a new optimalportfolio (with the same expected return of 8%).

Problem 5: The CAPM in Practice - Part IINote: 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 stocksand large stocks. If you regress the excess return on a small stock, indexed by i, on the excess marketreturn, you find the following relation:

0.03 1.3it Ft Mt Ft itR R R R

Every small stock behaves in this way, but different small stocks have different it which areuncorrelated 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:

0.01 0.9jt Ft Mt Ft jtR R R R

Again, every large stock behaves in this way, but different large stocks have different jt which areuncorrelated with each other.

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a) What fraction of the stock market value in Finlandia is accounted for by large stocks and whatfraction by small stocks?

b) Show that if you have an infinite number of stocks available, and you can freely trade in smalland large stocks, then there is an arbitrage opportunity.

c) Does the Capital Asset Pricing Model hold in Finlandia? (Hint: Your answer should be aqualitative, verbal explanation in one sentence!)

d) Now suppose that it of different small stocks have a correlation with each other, and jt ofdifferent large stocks have also correlation with each other. How does this affect youranswer in part b)?

Additional Problem for Your own Study

Problem 6: The Arbitrage Pricing TheoryImagine that there are only two pervasive macroeconomic factors. Securities X, Y and Z havesensitivities to these factors summarizes in the following table.

Securities 1 2

X 1.75 0.25

Y -1.00 2.00

Z 2.00 1.00

Assume that the expected risk premium is 4% on factor 1 and 8% on factor 2 (Treasury Bills obviouslyoffer zero risk premium).

a) What is the risk premium on each of the three securities according to the Arbitrage PricingTheory (APT)?

b) Suppose you buy $80 of security X and $60 of security Y and sell $40 of security Z. What is thesensitivity 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 isthe 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 Yoffered a premium of 14%, and security Z offered a premium of 16%. Suggest an investmentstrategy that has zero sensitivity to each factor and that has a negative risk premium.

Answer: Let ir be the risk premium on security i, and j - the portfolio's sensitivity to the factor j.

a) 9.0% 12.0% 16.0%X Y Zr r r b) 1 20 1.0 8%pr

c) 1 21.0 0 4%pr d) 2.0 0.5 1.5 1% 0X Y Z pw w w r

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Jevgenijs Babaicevs

Problem Set IV:Valuation of Options - Basics

Problem 1: Position DiagramsFor each of the following cases below, draw a position diagram (i.e. show the gross value at the date ofmaturity) to illustrate how options can be used. Be very specific to state the individual contractsconstituting 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 thestock.

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 ofbonds and options with the identically the same return profile as a short stock. Illustrate using aposition diagram!

e) You are discussing with your colleague whether to issue a loan to a small business with a highdebt/equity ratio. To better understand all the risks involved, explain:

(i) Why equity can be viewed as a call option on a firm? Express the position of anequityholder in terms of call options.

(ii) How can debt be viewed as an option on a firm? Express the position of a debtholder interms of put options.

f) This summer you are offered an internship opportunity at the options-trading department at theinvestment bank CHASE. Prior to this, you have decided to train yourself up and draw positiondiagrams for each of the following strategies. Assume that all options are identical but theexercise prices.

(i) Buy 1 call at 1E and 1 put at 2E 1 2E E .

(ii) Long 1 call at 1E and short 1 call at 2E 1 2E E .

(iii) Long 1 call at 1E and 1 call at 2E , short 1 call at 3E and 1 call at 4E 1 3 4 2E E E E .

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Problem 2: The Put-Call Parity Conditiona) 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 compoundedinterest rate is 10% per annum.

(i) Do these figures represent equilibrium in the market?

(ii) Using the figures above, demonstrate which transactions should be made in order toexploit the disequilibrium situation!

b) A European put is sold for LVL 1. It has an expiration date in two month’s time and the riskfree interest rate is 2% during this period. The underlying equity does not pay a dividend and itscurrent 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? Pleasenotice 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 PirateBank’s stock is selling for $8. The call’s exercise price is $100. The continuously compoundedrisk-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 buyone. How can you achieve this?

(ii) Suppose that puts are traded. What should a 26-week put with an exercise price of $100sell for?

Problem 3: Understanding OptionsThe corporate charter of Lidosta Derivatives contains a paragraph stating that the firm will beliquidated after exactly one year from today and the market value of total assets will be distributed tothe firm’s debtholders and shareholders. Lidosta Derivatives is financed by: (i) A seasoned equityoffering of 35 million LVL, and (ii) two zero-coupon loans each with a principal of 100 million LVLthat will be repaid after exactly one year. The total market value today of equity and debt is 200 millionLVL. Loan-AA is senior, i.e. it has priority at liquidation over Loan-BB that is subordinate. Loan-AAhas a simple interest rate of 8.5% that is equal to the market interest rate. The simple risk free interestrate is 8%.

a) Illustrate in a single position diagram the possible values of the three sources of finance afterexactly one year. Be sure to state what you measure on the two axes, and indicate precisely thecritical 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) consistingof two call options with different exercise prices. Calculate the premiums today of the two calloptions.

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Problem 4: Introduction to the Binomial Option Pricing ModelAssume that there are only two possible states of the world. In State 1, the stock price rises by 50% andin State 2, it drops by 25%. The current stock price is $100, the exercise price of the call option writtenon 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.

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

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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 6th 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 10th 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?

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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?

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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?

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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 $20.30 2008 $1.55 $0.30 $12.30 2007 $1.70 $0.25 $14.20 2006 $0.90 $0.25 $9.80 2005 $0.85 $0.25 $8.50 2004 $1.02 $0.25 $10.20

The average payout ratio in the industry is about 25%. You also notice that on the ex-dividend 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%.

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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 $500 2.0 22% B $600 1.5 20% C $500 1.0 12%

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?

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Jevgenijs Babaicevs

Problem Set V:The Binomial Option Pricing Model

Problem 1: The Binomial Option Pricing Model on a Non-Dividend Paying StockThe current stock price of Slava Corporation is $50. In each of the next three years, this stock price caneither go up by 20% or go down by 10%. According to the corporate charter of Slava Corporation, thecompany does not pay any dividends. A one-year simple risk-free interest rate is 5%.

a) Using the Binomial Option Pricing Model, calculate the price of the call option on Slava stockwith an exercise price of $40. Use (i) a risk-neutral valuation and (ii) a dynamic replicationapproach respectively!

Problem 2: The Binomial Option Pricing Model on a Dividend Paying StockIn a two period multiplicative binomial model the current stock price of Alfs Limited is EUR 100 andmay either be EUR 117.52 or EUR 88.57 after six months. The annual continuously compoundedinterest rate is 6%.

a) Calculate the value today of an European call with one year to maturity and an exercise price ofEUR 90 given that the stock does not pay any dividends!

b) Assume now instead that a dividend of EUR 5 is paid after six months. Determine the valuetoday of the call in part a) but with a difference that it is an American call instead.

c) Determine the exact composition of the replicating portfolio for the American call in part b) ateach of the nodes in the binomial tree.

d) If the underlying stock does not pay a dividend, calculate the value today of an American putwith an exercise price of EUR 110 and one year to maturity. Explain your result!

Problem 3: Option Pricing with Path DependencyNote: This is a slightly modified version of Problem 4 from Re-Exam 2008.

Consider a leading Latvian real estate company Looting Ltd. with two subsidiaries in Tullinn andBlinius. The stock price of the company is currently traded at $100. Every six-month the stock priceincreases by 40% if the real estate market is on the upturn or decreases by 25% if the market is on thedownturn. A six-month simple interest rate is 4%.

a) You are offered to buy an American call option on the stock of Looting Ltd. with an exerciseprice of $100 and a one year to maturity. Besides that you know that this option is path-dependent and after the first downturn, the exercise price drops to $80.

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(i) Use a multiplicative binomial model to calculate the price of this option.

(ii) Calculate a composition of the replicating portfolio at all nodes of the binomial tree.

b) Kenneth offers you to convert your path-dependent American option into an Asian with thesame exercise price and time to maturity as in a) and which can be exercised at the end of theyear only. Note: The payoff of an Asian option is the arithmetic average price of the underlyingasset throughout the holding period less the exercise price!

(i) Will you keep your path-dependent American call or convert it into an Asian if it doesnot cost you anything to convert and you could do it immediately after the purchase.

(ii) Explain why the vega of an Asian option is lower than the vega of an European with thesame parameters!

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Jevgenijs Babaicevs

Problem Set VI:The Black-Merton-Scholes Option Pricing Model

Problem 1: Introduction to the BMS Modela) 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 aprice of $120 and has a standard deviation of 40% per year. The continuously compounded risk-free interest rate is 6.18% per year.

(i) Use the Black-Merton-Scholes formula to calculate the price of the call.

(ii) 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 yourcompany 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. Youmay decide to sell the shares of CS that you will receive in the market at any time. However, aspart of the offer, CS also agrees that at any time during the next year, it will buy the shares backfrom you for LVL 25 per share if you want. Assume the current continuously compounded risk-free interest rate is 6.18% a year, and the volatility of CS stock is 30% per annum. Assumefurther 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 TimeThe current stock price of Paalzow Oil (PO) is SEK 200. The standard deviation is 22.3% per year, andthe continuously compounded interest rate is 21% a year. A one-year call option on PO has an exerciseprice 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 theone-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 stockprice rises, and (iii) next period if the stock price falls. Show at each point how you wouldreplicate a call option with a levered investment in the company’s stock.

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Problem 3: Collateralized Debt ObligationsNote: This is a slightly modified version of Problem 5 from Re-Exam 2008.

Diana P. has just informed you that your summer internship is going to be at SweetBank this year. Thebank is planning to offer three types of claims, which mature in one year, on the value of its underlyingportfolio 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, withtranche 1 1T being most senior and tranche 3 3T being most junior. The claims of the tranches arepaid in the order of their seniority: 1T pays the value of the underlying loan portfolio up to a maximumof 1D , and is paid first; 2T pays the excess value of the loan portfolio up to a maximum of 2D , after 1Thas been paid; 3T is a claim to the residual value of the loan portfolio after the claims of the first twotranches are satisfied.

a) Using real options language, draw separate position diagrams for the three tranches as afunction of the value of the underlying loan portfolio V . Do not forget to denote the criticalvalues on the axes!

b) Explain the character (type) of the options associated with each of the claims! Express the valueof each of the tranches as a function of V and C E - the price of a European call optionwritten 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 ofSweetBank. As part of your compensation package you receive several units of one of thetranches. After taking over as CFO your first task is to decide how many loans N to includein the underlying portfolio in such a way that the market value of your compensation package ismaximized! You know that each of the loans is equally weighted in the portfolio. The return oneach of the loans has an annual variance of 2 and any two returns have a correlationcoefficient of 0 1 .

(i) You are given several units of tranche 1T . How many loans will you decide to includein the underlying portfolio?

(ii) You are given several units of tranche 3T . How many loans will you decide to includein 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 3T andyou have optimally chosen the number of loans to include in the portfolio. Moreover, thematurity of the structured products has been extended from one year to two years. Enthused bythis change, you do some research and discover that at the annual frequency the returns on someloans exhibit momentum (Loans-AA), while others exhibit mean-reversion (Loans-BB). Theannual variance of returns of both types of loans is still 2 .

(i) Which type of loans will you choose to include in the underlying portfolio?

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Additional Problem for Your own Study

Problem 4: BMS Model and Land-OwnersAlexandre owns a one-year call option on 1 acre of real estate in Paris. The exercise price is $2 millionand the current, appraised market value of the land is $1.7 million. The land is currently used as aparking lot, generating just enough money to cover real estate taxes. The annual standard deviation is15% 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 realestate 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) 71,167.285C USD b) ' 28,640.445C USD


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 semi-annually 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.

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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?

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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 Aaa-rated 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?

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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 12.75-year maturity zero-coupon bond selling at a yield to maturity of 8% (effective annual yield) has convexity of 150.3 and modified duration of 11.81 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: 231.2. 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?


Tālis Putniņš

Problem Set VIII: The term structure of interest rates; futures, swaps and risk management

Exercise 1: Term structure of interest rates What is the relation between forward rates and the market’s expectation of future short term rates? Explain in the context of both the expectations and liquidity preference theories of the term structure of interest rates. Exercise 2: Term structure of interest rates You are given the following information about five government bonds with face value of 100:

Bond Years to maturity Coupon payment Price (PV) A 1 7.0 100 B 2 8.0 102 C 3 6.7 95 D 4 7.0 93 E 5 12.0 109

a) Estimate the spot rates for the next five years. b) Draw the term structure of the interest rates, with the spot rates on the y-axis and the time to maturity on the x-axis. c) Discuss the possible explanations for the shape of the term structure. d) Calculate the one-year futures rates, 1f2, 2f3, 3f4, and 4f5. e) Calculate the two-year futures rates, 1f3 and 2f4 (two-year spot rates in one and two year’s time).

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Exercise 3: Term structure of interest rates You have the following information about two bonds:

Time to maturity Coupon payment Bond 1 4 years 5.0% Bond 2 4 years 10.0%

Interest payments are annual and the first coupon will be paid after exactly one year. The face value of each bond is 100. Furthermore, assume that the spot rates are:

Maturity Spot rate 1 5.00% 2 4.75% 3 4.50% 4 4.25%

a) Calculate the price for both bonds. b) Calculate the yield for both bonds. c) Calculate the one-year forward rates up to 4 years forward. Exercise 4: Term structure of interest rates Two government bonds have face value EUR 100 and a time to maturity of 6 years. The first bond has an annual coupon of 6% and a yield of 12%. The second bond has an annual coupon of 10% and a yield of 8%. Using this information, determine the 6-year zero coupon rate. Exercise 5: Term structure of interest rates Assume a downward sloping term structure of zero-coupon rates:

T Zero rate 1 7.00 2 6.70 3 6.30 4 5.80 5 4.60

a) Look at the rates. Why must some of the rates be wrong? b) How can you make an arbitrage profit from the situation?

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Exercise 6: Term structure of interest rates The term structure for zero coupon bonds is currently:

T (years) YTM on a zero1 4% 2 5% 3 6%

Next year at this time you expect it to be:

T (years) YTM on a zero1 5% 2 6% 3 7%

a) What do you expect the rate of return to be over the coming year on a 3-year zero coupon bond? b) Under the expectations hypothesis, what yields to maturity on 1-year and 2-year zeros (the 1-year and 2-year spot rates) does the market expect to see in 1 year’s time? Is the market’s expectation of the return on the 3-year bond greater or less than yours? Exercise 7: Term structure of interest rates The yield to maturity (YTM) on 1-year zero coupon bonds is 5% and the YTM on 2-year zeros is 6%. The 12% annual coupon bonds with 2 years to maturity have YTM of 5.8%. What arbitrage opportunity is available for an investment banking firm? What is the profit on this activity? Exercise 8: Term structure of interest rates The one-year spot rate is r1 = 6%, and the one-year forward rates are: 1f2 = 6.4%; 2f3 = 7.1%; 3f4 = 7.3%; 4f5 = 8.2%. a) What are the spot rates r1, r2, r3, r4 and r5? b) If the expectations hypothesis holds, what can you say about expected future interest rates? c) Suppose your company will receive $100m at t = 4, but will make a $107m payment at t = 5. Show how the company can lock in the interest rate at which it will invest at t = 4 (not using futures contracts). Will the $100m invested at this lock-in rate be sufficient to cover the $107m liability? d) What is the YTM of a 5 year government bond that pays a 5% annual coupon?

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Exercise 9: Futures, swaps and risk management a) Why is there no futures market for cement?

b) Why might individuals purchase a futures contract rather than the underlying asset?

Exercise 10: Futures, swaps and risk management You enter into the long side of a March maturity S&P 500 futures contract with multiplier $250 and futures price of 1,477.20. The margin requirement is 10%. a) How much must you deposit with your broker? b) If the March futures price were to increase to 1,500 what percentage return would you earn on your net investment? c) If the March futures price falls by 1% what is your percentage return? Exercise 11: Futures, swaps and risk management You are a corporate treasurer of Užavas Corp. and will purchase $1m of bonds for the sinking fund in 3 months. You believe rates will soon fall and you would like to purchase the bonds in advance of requirements. Unfortunately to obtain approval to do so from the board of directors will take up to 2 months. How can you hedge adverse movements in bond yields and prices (qualitative answer is sufficient)? Will you be short or long? Exercise 12: Futures, swaps and risk management You manage a $13.5m portfolio currently all invested in equities, and believe the market is on the verge of a big but short lived downturn. You would move your portfolio temporarily into T-bills, but do not want to incur the transactions costs of liquidating and reestablishing your equity position. Instead you decide to temporarily hedge your equity holdings with S&P 500 index futures contracts. a) Should you be long or short the contracts? Why? b) If your equity holdings are invested in a market index fund how many contracts should you enter? The S&P 500 index is now at 1,350 and the contract multiplier is $250. c) How does your answer to (b) change if the beta of your portfolio is 0.6?

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Exercise 13: Futures, swaps and risk management The following table shows gold futures prices for varying contract lengths. Being an investment good, assume gold has negligible storage costs and convenience yield. Calculate the annually compounded interest rate faced by traders of gold futures for each of the contract lengths below. The spot price of gold is $664.30 per ounce.

Exercise 14: Futures, swaps and risk management The S&P 500 portfolio pays a dividend yield of 1% annually. Its current value is 1,300. The T-bill rate is 4%. Suppose the S&P 500 futures price for delivery in 1 year is 1,330. Construct an arbitrage strategy to exploit this mispricing and show that your profits will equal the mispricing. Exercise 15: Futures, swaps and risk management The spot price of a British pound is currently $2.00. The risk-free interest rate on 1-year government bonds is 4% in the US and 6% in the UK. a) What must be the forward price of the pound for delivery in 1 year? b) Suppose you call your broker and he quotes you a forward price of F0 = $2.03/£. How can you make risk free arbitrage profits? Calculate the profits per contract. Exercise 16: Futures, swaps and risk management You are the financial manager of Cēsu Corp. and plan to issue $10m of 10-year bonds in 3 months. At current yields the bonds would have a modified duration of 8 years. The T-note futures contract is selling at F0 = 100 and has modified duration of 6 years. How can you use the futures contract to hedge the risk surrounding the yield at which you will be able to sell the bonds (quantitative solution required)? Both the bond and the contract are at par value.

Exercise 17: Futures, swaps and risk management The US yield curve is flat at 4% and the Euro yield curve is flat at 3%. The current exchange rate is $1.50 per Euro. What will be the swap rate on an agreement to exchange currency over a 3-year period? The swap will call for an exchange of 1 million Euros for a given number of dollars each year.

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Exercise 18: Futures, swaps and risk management A $100 million interest rate swap (including principal exchange) has a remaining life of 10 months. Under the terms of the swap 6-month LIBOR is exchanged for 12% per annum (compounded semi-annually). LIBOR on all maturities is currently 10% continuously compounded. The 6-month LIBOR rate was 9.6% p.a. (continuously compounded) 2 months ago. What is the current price of the swap to the party paying floating? What is its value to the party paying fixed? Exercise 19: Futures, swaps and risk management A currency swap has a remaining life of 15 months. It involves exchanging interest at 10% on £20 million for interest at 6% on $30 million once a year. The term structure of interest rates in both the UK and US is currently flat, with 4% in the US and 7% in the UK. All interest rates are quoted with annual compounding. The current exchange rate ($ per £) is 1.85. What is the value of the swap to the party paying £?


Tālis Putniņš

Problem Set X: Capital structure policy

Exercise 1: Capital structure Companies A and B differ only in their capital structure. A is financed with 30% debt and 70% equity; B is financed with 10% debt and 90% equity. The debt is risk free. Assume perfect capital markets. a) Rosencrantz owns 1% of the common stock of A. What other investment package would produce identical cash flows for Rosencrantz? b) Guildenstern owns 2% of the common stock of B. What other investment package would produce identical cash flows for Guildenstern? c) Show that neither Rosencrantz nor Guildenstern would invest in the common stock of B if the total value of company A were less than that of B. Exercise 2: Capital structure Brengulis Corp., operating in a perfect capital market, pays no taxes and is financed entirely by common stock. The stock has a beta of 0.8, a price-earnings ratio of 12.5, and is priced to offer an 8% expected return. Brengulis Corp. now decides to repurchase half its common stock and substitute it with an equal amount of debt. The debt yields a risk free 5%. Calculate the following: a) The beta of common stock after the refinancing. b) The required return on the stock after refinancing. c) The required return on the company (i.e., stock and debt combined) after the refinancing. Assume that the operating profit is expected to remain constant in perpetuity. Calculate: d) The percentage increase in expected earnings per share. e) The new price-earnings multiple.

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Exercise 3: Capital structure A manufacturing firm with no debt outstanding and a market value of $100 million is considering borrowing $40 million and buying back stock. The interest rate on debt is 9% and the firm faces a tax rate of 35%. a) Estimate the annual interest tax savings each year from the debt. b) Estimate the present value of the interest tax savings assuming the debt change is permanent. c) Estimate the present value of the interest tax savings assuming the debt will be taken on for 10 years only. d) What will happen to the present value of interest tax savings if interest rates drop tomorrow to 7% but the debt itself is fixed rate debt? Exercise 4: Capital structure The following are the book and market value balance sheets of Lāčplēšu Corp.: Book values Net working capital $20 $60 Equity Long-term assets $80 $40 Debt Market values Net working capital $20 $120 Equity Long-term assets $140 $40 Debt Assume that Modigliani and Miller’s theory with taxes holds. There is no growth and the $40 of debt is expected to be permanent. The corporate tax rate is 40%. a) How much of the firm’s value is accounted for by the debt-generated tax shield? b) How much better off will Lāčplēšu Corp.’s shareholders be if the firm borrows $20 more and uses it to repurchase stock? c) Now instead suppose that Congress passes a law, which eliminates the deductibility of interest for tax purposes after a grace period of 5 years. What will be the new value of the firm, other things equal? (Assume an 8% borrowing rate.)

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Exercise 5: Capital structure Tērvetes Corp, an unlevered firm, has EBIT = $2 million per year. Tērvetes’ tax rate is 40% and market value is V = E = $12 million. The stock has a beta of 1, the risk-free rate is 9% and E(Rm) – Rf = 6%. Management is considering the use of debt; debt would be issued and used to buy back stock, leaving the firm’s assets unchanged. The default-free interest rate on debt is 12%. Because interest is tax deductible, the value of the firm would tend to increase as debt is added to the capital structure, but there would be an offset in the form of increasing expected bankruptcy costs. The firm’s analysts have estimated that the present value of any bankruptcy costs is $8 million and that the probability of bankruptcy will increase with leverage according to the following schedule.

Value of debt Probability of default $2.50m 0.00% $5.00m 8.00% $7.50m 20.5% $8.00m 30.0% $9.00m 45.0% $10.0m 52.5% $12.5m 70.0%

a) What is the current cost of equity and WACC (no debt added yet)? b) What is the optimal capital structure when bankruptcy costs are considered? c) What is the value of the firm at the optimal capital structure? Exercise 6: Capital structure Your boss, the CFO of Valmiermuiža Corp., is fond of saying that what kills a business is the debt burden it chooses to carry. Recently, however, one of the directors questioned the wisdom of this low-debt policy by pointing out the tax savings from debt financing. Your boss instructs you to illustrate the tax savings of debt by comparing the market values of two hypothetical firms that are very similar except in debt policy. You draw up the following parameters for the hypothetical firms:

Gaišais Corp. Tumšais Corp. EBIT $200m $200m Debt $200m $0 Growth rate (of EBIT) 0% 0% Payout ratio 100% 100% Interest rate, rd 10% n/a Corporate tax rate, tc 34% 34% Personal tax rate on interest, ti 40% n/a Personal tax rate on dividends, te 20% 20%

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a) What is the total annual income to equity holders in Tumšais Corp. after both corporate and personal taxes? b) What is the market value of Tumšais Corp. if the required return on equity is 15%? c) What is the total annual income to bond holders in Gaišais Corp. after both corporate and personal taxes? d) What is the total annual income to equity holders in Gaišais Corp. after both corporate and personal taxes? e) What is the total combined annual income to bond holders and equity holders in Gaišais Corp. after both corporate and personal taxes? f) What is the present value of the tax advantage to Gaišais Corp.? g) What is the total market value of the firm and market value of the equity for Gaišais Corp.? h) What is the WACC of Gaišais Corp.? Is it higher or lower than the WACC of Tumšais Corp.? Exercise 7: Capital structure The total market value of a firm with $500,000 of debt is $1,700,000. Earnings before interest and taxes (EBIT) are expected to be constant in perpetuity. The interest rate on debt is 10%. The company is in the 34% tax bracket. If the company were 100% equity financed, the equity holders would require a 20% return. a) What would the value of the firm be if it were financed entirely with equity? b) What is the net income to the stockholders of this levered firm? Exercise 8: Capital structure Three companies, A, B and C, operate in the same line of business. A has a long-term target debt/assets ratio of 35%, whereas B has chosen a 50% debt-assets ratio. C has chosen to remain 100% equity financed. Company A has an excellent debt rating paying in effect only the risk free rate for its debt. Company B pays a premium of 0.5% p.a. over the risk-free rate. The beta of company C’s equity is 1.4. However, the equity betas for A and B are unknown. The corporate tax rate is 29% for all companies. The risk-free rate is 4.5% p.a. and the market risk premium is 5% p.a.. a) Estimate the cost of equity for companies A, B and C. b) Estimate the WACC for companies A, B and C.

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Exercise 9: Capital structure Circular File’s market value balance sheet: Net working capital $20 $5 Common Stock Fixed assets $10 $25 Bonds outstanding (Face value $50) Total assets $30 $30 Total value Who gains and who loses from the following maneuvers? a) Circular scrapes up $5 in cash and pays a cash dividend. b) Circular halts operations, sells its fixed assets, and converts net working capital into $20 cash. Unfortunately the fixed assets fetch only $6 on the second hand market. The $26 in cash are invested in T-bills. c) Circular encounters an acceptable investment opportunity, NPV = $0, requiring an investment of $10. The firm borrows to finance the project. The new debt has the same security, seniority, etc. as the old. d) Suppose the new project has NPV = $2 and is financed by an issue of preferred stock. e) The lenders agree to extend the maturity of their loan from one year to two in order to give Circular a chance to recover. Exercise 10: Capital structure Recently, your boss, the CFO of Līvu Corp. came across the notion that adjusting the firm’s capital structure might significantly increase its market value. Your boss has asked you to do a preliminary study on this issue. Līvu Corp. currently has a very conservative leverage policy with D/E ratio of only 10%. Regressions of the past 5 years’ data result in an estimated equity beta of 1.05. The marginal tax rate is 34%. The long-term risk-free rate is 8% and the market risk premium is 5.5%. An investment banker recently estimated the pretax cost of debt at various levels of the D/E ratio and the estimates are as follows.

D/E Pretax cost of debt10% 10.30% 20% 10.70% 30% 11.30% 40% 12.00% 50% 12.60% 60% 13.40% 70% 15.00% 80% 16.80% 90% 19.00% 100% 23.00%

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From Līvu Corp.’s most recent financial statement you find that EBIT = $120 million, depreciation = $10.5 million, capital expenditure = $15 million and the growth rate of cash flows to the firm is 6% p.a. Based on recent evidence in Schaefer and Strebulaev (2007), your boss instructs you to use the approximation that the beta of the firm’s debt is zero. a) What is the weighted cost of capital (WACC) today when the D/E ratio is 10%? b) What is the total market value of Līvu Corp. today when the D/E ratio is 10%? c) What is the unlevered beta of the firm today? d) Construct a table showing the relation between the D/E ratio, beta of equity and the cost of equity. e) Construct a table showing the relation between the D/E ratio, WACC and market value of the firm. f) What is the optimal capital structure for the firm? What is the minimum WACC and maximum firm value? g) If the firm decides to move to the optimal capital structure, what will be the gain to shareholders? Will debt holders be harmed by the move?


Tālis Putniņš

Problem Set XI: Capital budgeting/valuation with leverage;

and mergers and acquisitions

Exercise 1: Capital budgeting and valuation Walter Corp (WC) has the opportunity to invest $1 million now (t = 0) and expects after-tax returns of $600,000 at t = 1 and $700,000 at t = 2. The project will last for two years only. The appropriate cost of capital is 12% with all equity financing, the borrowing rate is 8% and WC will borrow $300,000 against the project. The debt must be repaid in two equal instalments. The corporate tax rate is 30%. Calculate the project’s APV. Exercise 2: Capital budgeting and valuation In year 1 Kimmel Corp. will earn $2000 before interest and taxes. The market expects these earnings to grow at 3% per year. The firm will make no net investments or changes to net working capital. The corporate tax rate equals 40% and the firm has $5000 debt, which is effectively risk free. Kimmel plans to keep a constant D/E ratio, so that on average the debt will grow at a rate of 3% per year. The risk free rate is 5% and expected return on the market is 11%. The asset beta for this industry is 1.11. a) If Kimmel were an all-equity (unlevered) firm, what would its market value be? b) Assuming the debt is fairly priced, what how much interest will Kimmel pay in year 1? c) Even though Kimmel’s debt is risk free, the future growth of the debt is uncertain because it depends on the value of the firm. What is the present value of the tax shields? d) Using the APV method, what is Kimmel’s total levered market value? What is Kimmel’s market value of equity? e) What is Kimmel’s WACC? f) From WACC calculate Kimmel’s cost of equity?

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g) Assuming that proceeds from increases in debt are paid out to equity holders, use the FCF to equity method to find the market value of equity. How does that compare to your answer in (d)? Exercise 3: Capital budgeting and valuation Consider a perpetual project that requires an initial investment of $1 million and produces a pre-tax cash inflow of $95,000 per year in perpetuity. The cost of capital with all-equity financing is 10% and the project allows the firm to firm to borrow at 7%. The corporate tax rate is 35%. Use APV to calculate the project’s value under the following two scenarios: a) The project will be partly financed with $400,000 of debt which is fixed in perpetuity. b) The initial borrowing of $400,000 will be increased or reduced in proportion to changes in the future market value of the equity to maintain a target D/E ratio. Explain the difference between your answers to (a) and (b). Exercise 4: Capital budgeting and valuation Now suppose that the project described in the previous exercise will be undertaken by a university. Funds for the project will be withdrawn from the university’s endowment, which is invested in a widely diversified portfolio of stocks and bonds. However, the university can also borrow at 7%. The university is tax exempt. The university treasurer proposes to finance the project by issuing $400,000 of perpetual bonds at 7% and by selling $600,000 worth of common stocks from the endowment. The expected return on common stocks is 10%. He therefore proposes to evaluate the project by discounting at a weighted average cost of capital, calculated as:

%8.8

000,000,1

000,60010.0

000,000,1

000,40007.0

V

Er

V

DrWACC ED

What is right or wrong about the treasurer’s approach? Should the university invest in the project? Would the project’s value to the university change if the treasurer financed the project entirely by selling common stocks from the endowment? Exercise 5: Capital budgeting and valuation Gulbenes Corp. is assessing its capital structure. In January 2010 its equity had a market value of $24.27 billion, its debt had a market value of $2.8 billion and an AAA rating. Its equity beta is 1.47 and it faces a corporate tax rate of 40%. The Treasury bond yield is 6.5% and corporate AAA bonds are trading at a spread of

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0.30% over the Treasury rate. The expected stock market premium is 5.5%. Gulbenes Corp. has a policy of rebalancing its capital structure to maintain a target D/E ratio and its bonds are frequently refinanced at market rates. a) Estimate the current cost of capital for Gulbenes Corp. b) If Gulbenes Corp. moves to a debt-to-equity ratio of 30% it is estimated its bonds will have a BBB rating. BBB bonds have a spread of 2% over the Treasury rate. Estimate the cost of capital if Gulbenes Corp. moves to this new debt-to-equity ratio. c) Assuming a constant growth rate of 6% in the firm’s value, how much will firm value change if management decides to move to the debt-to-equity ratio of 30%? What will the effect be on the stock price? Exercise 6: Capital budgeting and valuation Lāčplēsis Corp. is assessing whether it should increase its leverage and you have been hired as a consultant. The firm has $527 million in market value of debt and $1.76 billion in market value of equity. The firm has EBIT of $131 million and faces a corporate tax rate of 36%. The company’s bonds are rated BBB and the cost of debt is 8%. At this rating the firm has a probability of default of 2.30% and the present value of bankruptcy costs is 30% of the firm’s value without its tax shields. The T-bond rate is 6%. Lāčplēsis Corp. has a policy of holding the level of its debt fixed in perpetuity, and the debt is frequently refinanced at market rates. a) Estimate the unlevered value of the firm. b) Estimate the levered value of the firm at a debt-to-equity ratio of 50%. At that level of leverage the firm’s bond rating would be CCC with a default probability of 46.61%. c) Would you recommend Lāčplēsis Corp. moves to the higher debt-to-equity ratio? Exercise 7: Mergers and acquisitions Aldaris Corp. currently has EPS = $2.00, but management are keen to report EPS = $2.67. Management therefore decides to acquire Cēsu Corp. even though there are no economic gains form the merger. Aldaris Corp. issues just enough of its own shares to ensure its $2.67 EPS objective. You are given the following data on the companies:

Aldaris Corp Cēsu Corp Merged firm EPS $2.00 $2.50 $2.67 Price per share $40 $25 ? P/E ratio 20 10 ? Number of shares 100,000 200,000 ? Total earnings $200,000 $500,000 ? Total market value $4m $5m ?

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a) Complete the table above for the merged firm. b) How many shares of Aldaris Corp. are exchanged for each share of Cēsu Corp.? c) What is the cost of the merger to Aldaris Corp.? d) What is the change in the total market value of the Cēsu Corp. shares that were outstanding before the merger? Exercise 8: Mergers and acquisitions Consider two small firms, Jubilejas Corp. and Senču Corp., that operate independently and have the following financial characteristics:

Jubilejas Corp. Senču Corp. Revenue $8,000 $4,000 - COGS $6,000 $2,400 EBIT $2,000 $1,600 Expected growth rate 4% 6% Cost of capital 9% 10%

Both firms are in steady state, with capital spending offset by depreciation. No working capital is required, and both firms face a tax rate of 50% a) Find the value of the combined (merged) firm if no synergies are realized. b) Now assume that combining the firms will create economies of scale in the form of shared distribution and advertising costs, which will reduce the cost of goods sold (COGS) from 70% of combined revenues ($8,400/$12,000) to 65% of combined revenues. This synergy is as risky as the combined enterprise. Estimate the value of the synergy. c) Now consider an alternative synergy. Suppose that as a result of the merger the combined firm is able to enter new markets and is expected to increase the future growth in the revenue of the combined entity to 6%. Also assume that the cost of goods sold is expected to remain at 70% of revenues. Estimate the value of the revenue enhancement synergy assuming it is as risky as the combined enterprise. Exercise 9: Mergers and acquisitions Inčukalna Corp. and Lodiņa Corp. have agreed to merge in response to increased competition in the industry. The merger is of strategic significance in that the combined group will have a stronger global business presence. The feasibility of the merger, however, hinges on potential revenue enhancements that would come from increased pricing power. The CFOs of both firms have agreed that incremental revenues equal to 6% of existing annual revenue could be achieved due to cross-selling of products to each customer base. To achieve this, $40 million would have to

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be spent initially in restructuring the company (no ongoing expenses) and the restructuring expense could be used to offset the company’s tax liability. The sum of Inčukalna Corp’s and Lodiņa Corp’s revenues (not including the revenue enhancement) was expected to grow from $300 million currently at a nominal rate of 5% p.a. The merged company is expected to have a cost of debt of 8%, cost of equity of 13% and a WACC of 9.5%. The tax rate is 30%. The risk-free rate is 5.5%. What is the expected value of the synergy? Exercise 10: Mergers and acquisitions The fixed exchange ratio in an acquisition of Līvu Corp. by Latgales Corp. is 1.4:1. The buyer and seller agree to a floating collar, which has a low trigger of $20 and a high trigger of $40, i.e.: the exchange ratio is 1.4 to 1, unless the buyer’s stock price falls below $20, in which case the exchange ratio is equal to $28 divided by the buyer’s share price; if the buyer’s share price is greater than $40, the exchange ratio is equal to $56 divided by the buyer’s share price. Calculate and graph the number of shares issued for one share of the seller’s stock, assuming the following share prices for the buyer: $0.01, $10.00, $15.00, $20.00, $30.00, $40.00 and $50.00. Also calculate and graph the value of the bid at these prices.

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