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Two firms operating under oligopoly are faced with two choices, to charge a high price or a low price. If one firm charges a low price while the other a high price, the firm that charges a low price attracts customers and earns a profit of $600,000 while the firm charging a high price loses customers and earns only $100,000. If both firms charge a high price they earn $400,000 each while if both charge a low price, they earn $200,000 each.

a) What profits are the firms likely to earn in the absence of cooperation?

b) If the firms cooperated, what profits would each firm earn?

a) What profits are the firms likely to earn in the absence of cooperation?

b) If the firms cooperated, what profits would each firm earn?

1. The profits from projects A and B have variances of $500, 000 and $45, 000 respectively. Which of the two projects is more risky if the expected values of the profits are $5000 and $500, respectively?

2. A project has expected risky cash flows of $90, 000 in perpetuity while the certainty equivalent cash flows are $60, 000. The risk free rate is 10%. What is the risk adjusted rate of return on the risky cash flows?

3. Should an investor invest in this project if the initial cost is $650, 000?

4. What would be the internal rate of return if the cost of the project was $600, 000?

2. A project has expected risky cash flows of $90, 000 in perpetuity while the certainty equivalent cash flows are $60, 000. The risk free rate is 10%. What is the risk adjusted rate of return on the risky cash flows?

3. Should an investor invest in this project if the initial cost is $650, 000?

4. What would be the internal rate of return if the cost of the project was $600, 000?

A firm is considering employing one of the two machines A and B over a period of 4 years at the end of which the salvage value of each is zero. The cost of machine A is $10, 000 while that of machine B is $11, 000. The probability distributions of the returns for each machine are given in the table below.

PROBABILTY MACHINE A($) MACHINE B($)

0.25 6000 7000

0.50 5000 5000

0.25 4000 3000

The risk free discount rate is 10% while the risk premium applied as follows.

STANDARD DEVIATION($) RISK PREMIUM

0 – 999 0%

1000 – 1999 10%

2000 – 2999 10%

3000 – 3999 20%

Which of the two machines should be installed?

PROBABILTY MACHINE A($) MACHINE B($)

0.25 6000 7000

0.50 5000 5000

0.25 4000 3000

The risk free discount rate is 10% while the risk premium applied as follows.

STANDARD DEVIATION($) RISK PREMIUM

0 – 999 0%

1000 – 1999 10%

2000 – 2999 10%

3000 – 3999 20%

Which of the two machines should be installed?

The owner of a firm expects to make a profit of $100 for each of the two years and be able to sell the firm at the end of the second year at $800. The owner of the firm believes the appropriate discount rate for the firm is 15%. What is the value of the firm?

A firm owned by an investor who invested $500, 000 earns a profit of $150, 000 a year. The best investment alternative for the investor is a portfolio of stocks and bonds earning a return of 12%. What is the economic profit?

Which would be most helpful when considering a large expenditure that might require repeating payments?

1 careful consideration of short-term goals

2 recording income and spending over the past year

3 creating a budget to consider future income and spending

4 learning more about different kinds of accounts to manage money

1 careful consideration of short-term goals

2 recording income and spending over the past year

3 creating a budget to consider future income and spending

4 learning more about different kinds of accounts to manage money

The current price of a non-dividend paying stock is $2600. You collect the following

additional information on the stock: E[R] = 9%, and σ[R] = 20%. The term structure

of interest rates is flat with r = 2%. For this question, you will be considering options

with a maturity T of one year and a strike price X = $2650

a) What is the expected return of the following trading strategy: (i) buy two call

options on the stock, (ii) sell two put options on the stock, (iii) buy $5300/(1+2%)

of 1-year zero coupon bonds?

additional information on the stock: E[R] = 9%, and σ[R] = 20%. The term structure

of interest rates is flat with r = 2%. For this question, you will be considering options

with a maturity T of one year and a strike price X = $2650

a) What is the expected return of the following trading strategy: (i) buy two call

options on the stock, (ii) sell two put options on the stock, (iii) buy $5300/(1+2%)

of 1-year zero coupon bonds?

A spread is a combination of option positions that involves four strike prices (see figure below, which shows the gross payoffs). Assume options are European with the same maturity and a > 0.

(a) Find the option positions and strike prices necessary to obtain the the spread. Show that your portfolio replicates the spread payoff and assume that the underlying options are all call options.

(b) Same as (a) but assuming underlying options are all put options

(c) Consider: (i) buy a call with a strike price of X, and (ii) buy a put with a strike price of X+3a. Assuming no arbitrage, show mathematically that initial investment (i.e. the amount of money you need to pay at t = 0) to create the strategy is higher than initial investment to create the spread.

ST on horizontal axis and payoff (a) on vertical axis with (X, X+a, X+2a, X+3a) on horizontal axis at each point of 3-sided trapezoid (X has 0 payoff, X+a has 'a' payoff, X+2a has 'a' payoff, and X+3a has 0 payoff)

(a) Find the option positions and strike prices necessary to obtain the the spread. Show that your portfolio replicates the spread payoff and assume that the underlying options are all call options.

(b) Same as (a) but assuming underlying options are all put options

(c) Consider: (i) buy a call with a strike price of X, and (ii) buy a put with a strike price of X+3a. Assuming no arbitrage, show mathematically that initial investment (i.e. the amount of money you need to pay at t = 0) to create the strategy is higher than initial investment to create the spread.

ST on horizontal axis and payoff (a) on vertical axis with (X, X+a, X+2a, X+3a) on horizontal axis at each point of 3-sided trapezoid (X has 0 payoff, X+a has 'a' payoff, X+2a has 'a' payoff, and X+3a has 0 payoff)

Assume that the CAPM holds. You're currently managing two

well-diversified portfolios with the following characteristics:

Portfolio β

A 0.8

B 1.3

The risk-free rate rate Rf is 2%, the expected market risk premium E[Rm −Rf ] is 7%,

and the standard deviation of the market return σm is 20%.

(a) One of your clients would like to invest in a portfolio with weights equal to 40%,

and 60% in portfolios A and B, respectively. What is the expected return of

the portfolio? What is the amount of systematic risk in the portfolio (i.e., the

standard deviation of returns that is explained by the market factor)?

(b) Another client wants you to build a portfolio that generates an expected return

of 12%. Show how can you offer such a portfolio using portfolios A and B. What

is the CAPM beta of that portfolio?

well-diversified portfolios with the following characteristics:

Portfolio β

A 0.8

B 1.3

The risk-free rate rate Rf is 2%, the expected market risk premium E[Rm −Rf ] is 7%,

and the standard deviation of the market return σm is 20%.

(a) One of your clients would like to invest in a portfolio with weights equal to 40%,

and 60% in portfolios A and B, respectively. What is the expected return of

the portfolio? What is the amount of systematic risk in the portfolio (i.e., the

standard deviation of returns that is explained by the market factor)?

(b) Another client wants you to build a portfolio that generates an expected return

of 12%. Show how can you offer such a portfolio using portfolios A and B. What

is the CAPM beta of that portfolio?

Current price of a non-dividend paying stock is $2600. E[R] = 9%, and σ[R] = 20%. The term structure of interest rates is flat with r = 2%. For this question, consider options

with a maturity T of one year and a strike price X = $2650.

(a) Assuming that the assumptions underlying the Black-Scholes formula holds, compute

the no-arbitrage prices of a European call option and a European put option

on the stock.

(b) What is the expected return of the following trading strategy: (i) buy two call

options on the stock, (ii) sell two put options on the stock, (iii) buy $5300/(1+2%)

of 1-year zero coupon bonds?

(c) You observe the following market prices for the call and the put options priced in

(a): C = 210, and P = 203 . Assume that trading in option markets entails a cost

of $z per transaction, i.e., trading in the stock and the risk-free bond bears no

transaction costs but each time a an option is bought or sold, a cost z is incurred.

What minimum value for z guarantees the absence of arbitrage?

with a maturity T of one year and a strike price X = $2650.

(a) Assuming that the assumptions underlying the Black-Scholes formula holds, compute

the no-arbitrage prices of a European call option and a European put option

on the stock.

(b) What is the expected return of the following trading strategy: (i) buy two call

options on the stock, (ii) sell two put options on the stock, (iii) buy $5300/(1+2%)

of 1-year zero coupon bonds?

(c) You observe the following market prices for the call and the put options priced in

(a): C = 210, and P = 203 . Assume that trading in option markets entails a cost

of $z per transaction, i.e., trading in the stock and the risk-free bond bears no

transaction costs but each time a an option is bought or sold, a cost z is incurred.

What minimum value for z guarantees the absence of arbitrage?