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1) Suppose you are willing to buy a European call option on XYZ ...

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1) Suppose you are willing to buy a European call option on XYZ ...

    Take Home Midterm Exam

    FIN 4504, Summer A 2005

    May 26, 2005, Thursday

    Instructor Cem Demiroglu

    Name:

Section I. Open Ended Short Essay Questions (5 points each)

1) Explain the “limited liability rule”.

    Limited liability is the legal protection given stockholders whereby they are responsible for the debts and obligations of a corporation only to the extent of their capital contributions.

2) Why do firms repurchase their own stock?

    1. To take advantage of potential undervaluation

    2. To distribute excess cash (a substitute for dividends)

    3. To increase leverage

    4. To fend off takeovers

    5. To counter the dilution effects of stock options

    3) What are the three alternative mechanisms to price and allocate IPOs: What are the strengths and weaknesses of each mechanism?

See Lecture Notes 2.

    4) What are some of the factors that firms consider when choosing their IPO underwriters?

    1. How the underwriter values the firm relative to other underwriters 2. All-star analyst coverage

    3. Underwriter reputation, etc.

Section II. Problems (10 points each)

    1) Suppose you are willing to buy a European call option on XYZ stock which currently (at time 0) trades at $50 per share in the NYSE. The exercise price of the call option is $55 and the option expires in 3-months. Consider two different scenarios about the distribution of the price of XYZ at the option expiration date (time T):

Scenario #1:

    $53 with probability ? and $57 with probability ?

Scenario #2:

    $50 with probability ? and $60 with probability ?

    a) Calculate the expected value of the stock and the expected payoff of the call option under each scenario.

Scenario #1:

    Expected value of the stock = ? * $53 + ? * $57 = $55

    Expected payoff of the call option = ? * $0 + ? * ($57-$55) = $1

Scenario #2:

    Expected value of the stock = ? * $50 + ? * $60 = $5

    Expected payoff of the call option = ? * $0 + ? * ($60-$55) = $2.5

    b) Based on your answer in part (a) under which scenario should the call premium (c) be higher? Why? (Assumption: At time 0 the option holder and the option seller both know the distribution of XYZ’s price at time T)

    The call premium should be higher under Scenario #2 because the expected payoff of the option is higher under this scenario. The higher the expected payoff of a financial instrument the higher is its price.

    c) Based on your answer in part (b) explain how the riskiness of the underlying stock affects the premium paid on a call option.

    As the volatility of the underlying stock increases the value of the call option increases.

    2) Consider the following information to answer the questions below. MSFT currently trades at $20 per share in the market. The call premium for an MSFT European call option with an exercise price equal to $18 and expiration date T is $4. On the other hand, the put premium for an MSFT European put option with an exercise price equal to $18 and expiration date T is $1. The periodic interest rate from time 0 to time T is 3%.

    a) Prepare the payoff table, the payoff diagram and the profit diagram for a portfolio which consists of the following: long one share of MSFT, short one MSFT call option, long one MSFT put option.

    See the put-call parity section under Lecture Note 9 for the payoff tables and diagrams.

    b) Are there any arbitrage opportunities with these prices? If yes, explain what trades you should do to take advantage of the arbitrage opportunity and how much profit you make. Be specific and tell me what you are buying (or selling) and at what price.

Check whether the put-call parity is violated:

S + p c = PV(X) 0

S + p c = $20 + 1 - $4 = $17 0

PV(X) = $18 / (1.03) = $17.48

    Yes, put call parity is violated. To take advantage of the arbitrage opportunity Borrow $17.48 at 3%

    Buy the stock, long put, short call.

You make a sure profit of 48 cents.

    3) Firm A currently has $1,000 in cash. It does not have any fixed assets. Therefore, if Firm A were liquidated today its value would be equal to the amount of cash it holds ($1,000).

The face value of Firm A’s debt is $1,000 and this debt will mature in six month. There

    are two projects available to the Firm A. Both projects require an investment outlay of $800. Project 1 pays off $600 with probability 0.5 and $1,000 with probability 0.5. Project 2 pays off $700 with probability 0.50 and $900 with probability 0.50. The pay offs of both projects will be due just before the maturity of the firm’s debt.

Firm A’s management has three options:

    a) immediately liquidate the firm and pay $1,000 to debt holders

    b) Invest in Project 1

    c) Invest in Project 2

    Which option is preferred by equity holders? Which one is preferred by debt holders? Please show your work.

No project:

    Payoff to debtholders = $1,000

    Payoff to equityholders = $0

Project #1:

    Expected payoff to debtholders = ? * $800 + ? * $1000 = $900

    Expected payoff to equityholders = ? * $0 + ? * $200 = $100

Project #2:

    Expected payoff to debtholders = ? * $900 + ? * $1000 = $950

    Expected payoff to equityholders = ? * $0 + ? * $100 = $50

    Equityholders prefer Project #1. Debtholders prefer no investment (option a).

    4) Ceredian Corp decided to hire an investment banker to help take it public. After talking with some regular clients, the underwriter decides that the true value of a firm’s

    share will be $7 or $13 with equal probability. The underwriter decided to sell 1 million shares, but also needs to compute the appropriate offer price. There is a group of uninformed investors willing to submit bids for 1 million shares of this IPO as long as their expected profit is non-negative. These uninformed investors know the probability distribution of the true share price as stated above, but do not know the true value of Ceredian as informed investors do. The informed investors are willing to order 1 million shares of the IPO if the offer price is lower than the true price. If the IPO is oversubscribed, shares are allocated on a pro-rata basis.

    a) Calculate the expected profits to each agent (e.g. underwriter, informed investors and uninformed investors) when the offer price is set equal to the expected true price of the firm. Explain the direction and the magnitude of wealth transfers between the agents. Explain why expected true price of the firm cannot be the equilibrium offer price.

Expected true price = ? * $7 + ? * $13 = $10

    Expected profit informed = ? * 500,000 * $3 = $750,000

    Expected profit uninformed = ? * 1,000,000 * -$3 + ? * 500,000 * $3 = -$750,000 Expected profit firm/underwriter = $0 (selling shares at fair price)

    There is a wealth transfer from uninformed investors to informed. If the uninformed keep losing money like this they will not come to the IPO market. However, if uninformed leave the market, firms can sell IPO shares only when the issue is hot. All cold issues will fail. This obviously can’t be an equilibrium. Firms should underprice their offers to make sure that uninformed investors break even on average and hence stay in the market.

    b) Compute the equilibrium offer price that the underwriter should set so that both cold and hot offers go through (e.g. succesfully completed).

? * 1,000,000 ($7 - OP) + ? * 500,000 ($13 OP) = 0

OP = $9

    c) What is the expected profit (in dollars) to each agent if the offer price is equal to the value computed in part (b)? Are there any wealth transfers between agents?

There is no wealth transfer from/to uninformed investors.

    Issuers lose 1,000,000 * ($10 - $9) = $1,000,000 because of underpricing. This wealth is transferred to informed investors.

    5) John purchases 2,000 shares of GOOG stock at $10 a share by putting up the minimum amount cash and borrowing the remaining from his broker. The initial margin is 50% and the maintenance margin is 40%. Assume that the interest rate on the broker’s loan is 2%

    per month.

    a) How much would to the stock price have to fall (in percentage) at the end of first month before John will get a margin call?

Initial equity = 50% * 2,000 * $10 = $10,000

Amount borrowed = Value of stock Initial equity = $20,000 - $10,000 = $10,000

    At the end of the month accumulated loan interest is $10,000 * 2% = $200

Let’s call the price where margin call comes as P. Then at price = P,

[2,000*P Loan Interest Charge] / 2,000*P = 0.40

[2,000*P - $10,000 - $200] / 2,000*P = 0.40

P = $8.5 How much a fall is this? 15%!

    b) Suppose that the price of GOOG shares suddenly fell to $7.5 at the end of the month. How much money would John has to add to his account to maintain ownership of all his GOOG stock if the share price falls by the amount in part (a). (Assume that he needs to bring his margin to back to maintenance margin)

Total value of the GOOG stocks = 2,000 * $7.5 = $15,000

Minimum acceptable equity position = $15,000 * (0.40) = $6,000

Existing equity position = $15,000 - $10,000 - $200 = $4,800

Required new cash inflow = $6,000 - $4,800 = $1,200

c) Let’s suppose that the share price falls by the amount in part (b) and John does not

    have any money to add to his account. Therefore, he instructs his broker to sell some of his GOOG shares and use the proceeds to pay off a portion of the loan. How many shares does John have to sell to comply with the maintenance margin?

Number of shares sold = X

Total value of GOOG shares after the sale= $7.5 * (2,000 X)

Total loan & interest after partial payment with stock sale proceeds

    = $10,200 - $7.5 X

Total equity position = $7.5 * (2000 X) [$10,200 - $7.5 X] = $4,800

We try to satisfy the following equality:

$4,800 / [$7.5 (2,000 X)] = 0.4 ? X = 400 shares

6) Suppose that you are considering investing in two stocks. After analyzing the two

    stocks you think that there are two possible states for the economy over the next year:

    “Good” and “Bad”. Each state is equally likely (probability 0.5). The returns on the two

    securities in each state are as follows:

     State____ Return Stock A Return Stock B

     Good 15% 8%

     Bad 5% 10%

    a) What is the expected return and standard deviation of each stock?

E(r) = (0.5)*(15%) + (0.5)*(5%) = 10% a

    E(r) = (0.5)*(10%) + (0.5)*(8%) = 9% b

    22Var(r) = (0.5)*(15%-10%) + (0.5)*(5%-10%) = 0.0025 a

    1/21/2Stdev (r) = [Var(r)] = (0.0025) = 5% aa

    22Var(r) = (0.5)*(10%-9%) + (0.5)*(8%-9%) = 0.0001 b

    1/21/2Stdev (r) = [Var(r)] = (0.000625) = 1% bb

    b) What is the covariance and correlation between the two stocks?

Cov (r,r) = (0.5)* (15%-10%)*(8%-9%) + (0.5)*(5%-10%)*(10%-9%) = -0.0005 ab

    Correlation (r,r) = Cov (r,r) / [Stdev (r) * Stdev (r) ] ababab

     = -0.0005 / [(0.05)*(0.01)]

     = -1

    c) Draw a picture to illustrate the tradeoff between risk and return that is available by

    investing in these two stocks.

E(r) = w * E(r) + (1-w) * E(r) paaab

     22Var(r) =w*Var (r)+(1-w)*Var (r)+2*w*(1-w)*Corr (r,r)*Stdev(r)*Stdev paaabaaaba

    (r) b

E(r) = w * E(r) + (1-w) * E(r) ? E(r) = w * (10%) + (1-w) * (9%) paaabpaa

     ? E(r) = 9% + 1% w pa

     22Var (r) = w * Var (r) + (1-w) * Var (r) - 2 * w* (1-w) * Stdev(r) * Stdev (r) paaaba aab

     2 Simplify, Var (r) = [ w Stdev (r) - (1-w) Stdev (r) ] paaab

     22 Var (r) = [w * 5% - (1-w) 1%] = [ - 1 % + 6% w]paaa

     ? Stdev (r) = - 1 % + 6% * wpa

w = [ Stdev (r) + 1%) ] / 6% ap

E(r) = 9% + 1% * [ Stdev (r) + 1 %) ] / 6% pp

E(r) = 9.16% + 0.17 * Stdev (r) pp

This is the equation of a line, right? Let’s draw the line:

    E(r)

     Slope = 0.17

     9.16 %

    Stdev (r)

    d) Suppose that a risk free investment of 6% is also available. Does this present a profit opportunity for you? Why or why not?

    Nobody will buy a risk-free asset with 6% return in this world. Because by combining two risky assets we can create a risk-free asset with a higher return (9.16%).

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