Homework # 2

Finc 335

Due: Sept. 27, 2015

Consider the three stocks in the following table. P_t represents price per share at time t, and Q_t represents shares outstanding at time t (in millions).

	P0	Q0	P1	Q1
A	50	100	55	100
B	50	200	45	200
C	100	200	110	200

For the rst period (t to t ), compute rates of return on:

a price-weighted index of the three stocks;

a market value-weighted index of the three stocks;

an equally weighted index of the three stocks;

Suppose at t = 0, you started a portfolio consisting of $5,000 in stock A, $5,000 in stock B and $10,000 in stock C. Suppose also that you did not change your positions during the time period from t to t : Without further computation, based on your answers to part (a), (b) or (c), what is the rate of return of your portfolio for this period? Explain your answer.

Suppose instead at time t = 0, you want to invest $7,000 in the three stocks. You want your portfolio to mimic the return of the market value weighted index in the period from t to t : How many shares of each stock would you have in your portfolio and why?

Consider information regarding the following two stocks. The probability of each state ofthe economy is 1/3. No dividends are paid.

		Ending Price in
Stock	Initial Price	Boom	Normal	Bust
A	40	60	50	25
B	25	20	25	35

What is the expected value and the standard deviation of the rate of return of each stock?

Which stock is the better investment? Why?

Consider a risky portfolio. The end-of-year cash ow derived from the portfolio will be either $50,000 with probability 0.6 or $150,000 with probability 0.4. The alternative riskless investment in T-bill pays 5%.

If you require a risk premium of 10%, how much will you be willing to pay for the portfolio?

Suppose the portfolio can be purchased for the amount you found in (a). What will be the expected rate of return on the portfolio?

Now suppose you require a risk premium of 15%. What is the price you will be willing to pay now?

Comparing your answers to (a) and (c), what do you conclude about the relationship between the required risk premium on a portfolio and the price at which the portfolio will sell?

You manage a risky portfolio with an expected return of 17% and a standard deviation of27%. The T-bill rate is 5%.

Your client chooses to invest 70% of a portfolio in your fund and 30% in a T-bill money market fund. What is the expected value and standard deviation of the rate of return on your client s portfolio?

Suppose that your risky portfolio includes the following investments in given proportions:

Stock A: 27% Stock B: 33%

Stock C: 40%

What are the investment proportions in your client s overall portfolio including the position in T-bills?

What is the Sharpe (reward-to-volatility) ratio of your risky portfolio? Your client s?

Suppose your client decides to invest in your portfolio a proportion y of the total investment budget so that the overall portfolio will have an expected rate of return of 15%. What is the proportion y? What is the standard deviation of the rate of return on your client s portfolio?

Suppose Merck stock o?ers an expected rate of return of 12% and a standard deviation of25%. The risk-free rate is 5%.

Graph the Capital Allocation Lines (CAL) for Merck stock.

What is the Sharpe (reward-to-volatility) ratio of Merck stock.

If you have a portfolio of $15,000 with $5,000 invested in a risk free security and $10,000 in Merck stock, what is the expected rate of return of your portfolio? What is your portfolio s standard deviation? What is the reward-to-volatility ratio of your portfolio?

If your friend is more risk averse than you are and that she also has a portfolio of $15,000 invested in a risk free asset and Merck stock. Do you think she has more or less than $10,000 in Merck stock? Why?

Suppose Southwest Airlines stock o?ers an expected return of 10% and a standard deviation of 20%. Suppose American Airlines stock o?ers an expected return of 8% and a standard deviation of 16%. The correlation between American s stock and Southwest s stock is 0. The risk-free rate is 5%.

2

Graph Capital Allocation Lines (CAL) for Southwest and American.

If you can only invest in one of the airlines along with the risk-free security, which security will a risk averse investor choose? Why?

Derive and graph the CAL for a risky portfolio created by putting portfolio weights of 0.5 on Southwest and 0.5 on American.

Stocks A, B, and C have the same expected return and standard deviation. The followingtable shows the correlation coe cients between the returns on these stocks.

	Stock A	Stock B	Stock C
Stock A	1.0
Stock B	0.9	1.0
Stock C	0.1	-0.4	1.0

Given these correlations, consider the following four portfolios constructed from these stocks. Which portfolio has the lowest risk? Explain.

Equally invested in stocks A and B

Equally invested in stocks A and C

Equally invested in stocks B and C (d) Totally invested in stock C

Homework # 2 Finc 335 Due: Sept. 27, 2015 1. Consider the three stocks in the following table. Pt represents price per share at time t, and Qt represents shares outstanding at time t (in millions). A B C P0 50 50 100 Q0 100 200 200 P1 55 45 110 Q1 100 200 200 For the ...rst period (t = 0 to t = 1), compute rates of return on: (a) a price-weighted index of the three stocks; (b) a market value-weighted index of the three stocks; (c) an equally weighted index of the three stocks; (d) Suppose at t = 0, you started a portfolio consisting of $5,000 in stock A, $5,000 in stock B and $10,000 in stock C. Suppose also that you did not change your positions during the time period from t = 0 to t = 1: Without further computation, based on your answers to part (a), (b) or (c), what is the rate of return of your portfolio for this period? Explain your answer. (e) Suppose instead at time t = 0, you want to invest $7,000 in the three stocks. You want your portfolio to mimic the return of the market value weighted index in the period from t = 0 to t = 1: How many shares of each stock would you have in your portfolio and why? 2. Consider information regarding the following two stocks. The probability of each state of the economy is 1/3. No dividends are paid. Stock A B Initial Price 40 25 Ending Price in Boom Normal Bust 60 50 25 20 25 35 (a) What is the expected value and the standard deviation of the rate of return of each stock? (b) Which stock is the better investment? Why? 3. Consider a risky portfolio. The end-of-year cash ow derived from the portfolio will be either $50,000 with probability 0.6 or $150,000 with probability 0.4. The alternative riskless investment in T-bill pays 5%. (a) If you require a risk premium of 10%, how much will you be willing to pay for the portfolio? (b) Suppose the portfolio can be purchased for the amount you found in (a). What will be the expected rate of return on the portfolio? (c) Now suppose you require a risk premium of 15%. What is the price you will be willing to pay now? (d) Comparing your answers to (a) and (c), what do you conclude about the relationship between the required risk premium on a portfolio and the price at which the portfolio will sell? 4. You manage a risky portfolio with an expected return of 17% and a standard deviation of 27%. The T-bill rate is 5%. (a) Your client chooses to invest 70% of a portfolio in your fund and 30% in a T-bill money market fund. What is the expected value and standard deviation of the rate of return on your client's portfolio? (b) Suppose that your risky portfolio includes the following investments in given proportions: Stock A: 27% Stock B: 33% Stock C: 40% What are the investment proportions in your client's overall portfolio including the position in T-bills? (c) What is the Sharpe (reward-to-volatility) ratio of your risky portfolio? Your client's? (d) Suppose your client decides to invest in your portfolio a proportion y of the total investment budget so that the overall portfolio will have an expected rate of return of 15%. What is the proportion y? What is the standard deviation of the rate of return on your client's portfolio? 5. Suppose Merck stock oers an expected rate of return of 12% and a standard deviation of 25%. The risk-free rate is 5%. (a) Graph the Capital Allocation Lines (CAL) for Merck stock. (b) What is the Sharpe (reward-to-volatility) ratio of Merck stock. (c) If you have a portfolio of $15,000 with $5,000 invested in a risk free security and $10,000 in Merck stock, what is the expected rate of return of your portfolio? What is your portfolio's standard deviation? What is the reward-to-volatility ratio of your portfolio? (d) If your friend is more risk averse than you are and that she also has a portfolio of $15,000 invested in a risk free asset and Merck stock. Do you think she has more or less than $10,000 in Merck stock? Why? 6. Suppose Southwest Airlines' stock oers an expected return of 10% and a standard deviation of 20%. Suppose American Airlines' stock oers an expected return of 8% and a standard deviation of 16%. The correlation between American's stock and Southwest's stock is 0. The risk-free rate is 5%. 2 (a) Graph Capital Allocation Lines (CAL) for Southwest and American. (b) If you can only invest in one of the airlines along with the risk-free security, which security will a risk averse investor choose? Why? (c) Derive and graph the CAL for a risky portfolio created by putting portfolio weights of 0.5 on Southwest and 0.5 on American. 7. Stocks A, B, and C have the same expected return and standard deviation. The following table shows the correlation coe cients between the returns on these stocks. Stock A Stock B Stock C Stock A 1.0 0.9 0.1 Stock B Stock C 1.0 -0.4 1.0 Given these correlations, consider the following four portfolios constructed from these stocks. Which portfolio has the lowest risk? Explain. (a) Equally invested in stocks A and B (b) Equally invested in stocks A and C (c) Equally invested in stocks B and C (d) Totally invested in stock C 3