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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?

Hypothetical drug-development program requires $200 million in out-of pocket

costs over a 10-year period during which no revenues are generated, and with

only a 5% probability of success. However, if the drug development is successful, it

is plausible to assume that it could generate a net income of $2 billion per year over

a 10-year period of exclusivity from years 11–20. The present value of this income

stream in year 10 is $12.3 billion (using a 10% cost of capital).

(a) Compute the expected return and standard deviation (over a 10-year period) of

this investment

costs over a 10-year period during which no revenues are generated, and with

only a 5% probability of success. However, if the drug development is successful, it

is plausible to assume that it could generate a net income of $2 billion per year over

a 10-year period of exclusivity from years 11–20. The present value of this income

stream in year 10 is $12.3 billion (using a 10% cost of capital).

(a) Compute the expected return and standard deviation (over a 10-year period) of

this investment

(a) Explain the advantages of using Value Added Statements (VAS) for interdivision for comparisons in decentralized firm.

Given three securities:

Expected Standard Correlations of Returns

Return Deviation Stock 1 Stock 2 Stock 3

Stock 1 0.15 0.20 1.00 0.20 0.30

Stock 2 0.20 0.30 1.00 0.80

Stock 3 0.08 0.10 1.00

(a) Find the expected return and standard deviation of a portfolio with 25% in stock

1, 50% in stock 2, and 25% in stock 3.

(b) For the portfolio in part (a), find the covariance of its return with the return of an

equally weighted portfolio of stocks 1 and 2.

(c) Suppose the risk-free rate is 5%. Someone claims you that an equally weighted

portfolio of Stocks 1 to 3 is the tangency portfolio of these three stocks. Do you

believe his claim? Justify your answer.

Expected Standard Correlations of Returns

Return Deviation Stock 1 Stock 2 Stock 3

Stock 1 0.15 0.20 1.00 0.20 0.30

Stock 2 0.20 0.30 1.00 0.80

Stock 3 0.08 0.10 1.00

(a) Find the expected return and standard deviation of a portfolio with 25% in stock

1, 50% in stock 2, and 25% in stock 3.

(b) For the portfolio in part (a), find the covariance of its return with the return of an

equally weighted portfolio of stocks 1 and 2.

(c) Suppose the risk-free rate is 5%. Someone claims you that an equally weighted

portfolio of Stocks 1 to 3 is the tangency portfolio of these three stocks. Do you

believe his claim? Justify your answer.

Hypothetical drug-development program requires $200 million in out-of pocket

costs over a 10-year period during which no revenues are generated, and with

only a 5% probability of success. However, if the drug development is successful, it

is plausible to assume that it could generate a net income of $2 billion per year over

a 10-year period of exclusivity from years 11–20. The present value of this income

stream in year 10 is $12.3 billion (using a 10% cost of capital).

(a) Compute the expected return and standard deviation (over a 10-year period) of

this investment.

(b) Suppose we setup a megafund to fund 150 drug-development programs, with the

same amounts of investment and payoff as the one in part (a). Assuming that the

success or failure of each program is independent of each other, what is the probability

for the megafund to score three or more successes in the 150 drug-development

programs.

(c) Compute the expected return and standard deviation (over a 10-year period) of the

megafund.

costs over a 10-year period during which no revenues are generated, and with

only a 5% probability of success. However, if the drug development is successful, it

is plausible to assume that it could generate a net income of $2 billion per year over

a 10-year period of exclusivity from years 11–20. The present value of this income

stream in year 10 is $12.3 billion (using a 10% cost of capital).

(a) Compute the expected return and standard deviation (over a 10-year period) of

this investment.

(b) Suppose we setup a megafund to fund 150 drug-development programs, with the

same amounts of investment and payoff as the one in part (a). Assuming that the

success or failure of each program is independent of each other, what is the probability

for the megafund to score three or more successes in the 150 drug-development

programs.

(c) Compute the expected return and standard deviation (over a 10-year period) of the

megafund.

• A coupon bond with a coupon rate of 8% and a face value of $1,000. Coupons

are paid out annually and the bond has 1 year to maturity. The current coupon

has just been paid out. The current price of the bond is $1018.772.

• A zero coupon bond with a face value of $1,000 and 2 years to maturity. The

bond trades at $907.029.

• An annuity that pays $50 every year for the next 3 years. The next payment will

be a year from now and the last payment will be 3 years from now. The annuity

is currently worth $136.967.

All these securities are risk-free. Note that there is no direct borrowing and lending

here, so if you want to borrow (lend) you need to sell (buy) an appropriate bond.

If you want to borrow (lend) you need to sell (buy) an appropriate bond.

Bank of Montreal offers a forward rate over year 3, f3, of 3%. That rate is good

for a loan or deposit of $10,000. Can you make money and eat a free lunch at

Bank of Montreal’s expense? If so, how?

are paid out annually and the bond has 1 year to maturity. The current coupon

has just been paid out. The current price of the bond is $1018.772.

• A zero coupon bond with a face value of $1,000 and 2 years to maturity. The

bond trades at $907.029.

• An annuity that pays $50 every year for the next 3 years. The next payment will

be a year from now and the last payment will be 3 years from now. The annuity

is currently worth $136.967.

All these securities are risk-free. Note that there is no direct borrowing and lending

here, so if you want to borrow (lend) you need to sell (buy) an appropriate bond.

If you want to borrow (lend) you need to sell (buy) an appropriate bond.

Bank of Montreal offers a forward rate over year 3, f3, of 3%. That rate is good

for a loan or deposit of $10,000. Can you make money and eat a free lunch at

Bank of Montreal’s expense? If so, how?

• A coupon bond with a coupon rate of 8% and a face value of $1,000. Coupons

are paid out annually and the bond has 1 year to maturity. The current coupon

has just been paid out. The current price of the bond is $1018.772.

• A zero coupon bond with a face value of $1,000 and 2 years to maturity. The

bond trades at $907.029.

• An annuity that pays $50 every year for the next 3 years. The next payment will

be a year from now and the last payment will be 3 years from now. The annuity

is currently worth $136.967.

All these securities are risk-free. Note that there is no direct borrowing and lending

here, so if you want to borrow (lend) you need to sell (buy) an appropriate bond.

(c) Expectations theory of interest rates. He wonders

what the price of the zero coupon bond will be in a year. Compute the expected price for him.

(d) RBC offers a forward rate over year 2, f2, of 4%. That rate is good for a loan or

deposit of $10,000. Can you make money and eat a free lunch at RBC’s expense? If so, how?

are paid out annually and the bond has 1 year to maturity. The current coupon

has just been paid out. The current price of the bond is $1018.772.

• A zero coupon bond with a face value of $1,000 and 2 years to maturity. The

bond trades at $907.029.

• An annuity that pays $50 every year for the next 3 years. The next payment will

be a year from now and the last payment will be 3 years from now. The annuity

is currently worth $136.967.

All these securities are risk-free. Note that there is no direct borrowing and lending

here, so if you want to borrow (lend) you need to sell (buy) an appropriate bond.

(c) Expectations theory of interest rates. He wonders

what the price of the zero coupon bond will be in a year. Compute the expected price for him.

(d) RBC offers a forward rate over year 2, f2, of 4%. That rate is good for a loan or

deposit of $10,000. Can you make money and eat a free lunch at RBC’s expense? If so, how?

Today is his 24th birthday. He plans to retire at 65 years old and he expects to live for another 20 years afterwards. He wants an income of $30,000 per year during his retirement years, to be paid annually on his birthday (starting from his 65th birthday). He plans to save some amount at each birthday from the age 25 to 64. He thinks about saving a constant amount for the first 10 years and then increases his saving at 3% each year until the last one before his retirement. The bank provides two types of accounts. One account pays 6.9%/year compounded quarterly. The other account pays 7%/year compounded annually?

(a) What is the balance of your brother’s account right after he makes his deposit in

his saving account on his 50th birthday?

(a) What is the balance of your brother’s account right after he makes his deposit in

his saving account on his 50th birthday?

are paid out annually and the bond has 1 year to maturity. The current coupon

has just been paid out. The current price of the bond is $1018.772.

• A zero coupon bond with a face value of $1,000 and 2 years to maturity. The

bond trades at $907.029.

• An annuity that pays $50 every year for the next 3 years. The next payment will

be a year from now and the last payment will be 3 years from now. The annuity

is currently worth $136.967.

All these securities are risk-free. Note that there is no direct borrowing and lending

here, so if you want to borrow (lend) you need to sell (buy) an appropriate bond.

(c) Expectations theory of interest rates. He wonders

what the price of the zero coupon bond will be in a year. Compute the expected price for him.

(d) RBC offers a forward rate over year 2, f2, of 4%. That rate is good for a loan or

deposit of $10,000. Can you make money and eat a free lunch at RBC’s expense? If so, how?