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**Problem Set**

This problem set is due on Monday, 23 February 2015, 23:55 (GMT). This problem set is an individual assignment, and you should NOT consult or collaborate with your classmates in its completion. However, feel free to email me if you have any clarification questions. Please answer the problems in Excel and submit the Excel spreadsheet. A template Excel file “Finance for Executives – PS#2” that contains the relevant data is provided for your convenience. Feel free to modify it as you see fit to answer the questions. You may also use a financial calculator to calculate the answers. If you do so, then type your answers into the Excel file.

1. Consider portfolios with positions in the US and Brazilian equity markets. The (annual) expected return and standard deviation of returns (in US dollars) for the 2 markets are as follows: US Brazil E[r] 10% 15% SD[r] 20% 30% The correlation between the returns is 0.2. In addition, assume that the (annual) risk-free (T-bill) rate is 5%.

a. Calculate the expected returns and standard deviations of the following portfolios: (i) 50% in the risk-free asset, 50% in the US (ii) 50% in the risk-free asset, 50% in Brazil (iii) 50% in the US, 50% in Brazil (Note: You already did this calculation for PS#1.) (iv) 50% in the risk-free asset, 50% in the portfolio in (iii) above

b. Calculate the Sharpe ratios of (i) the US market (ii) the Brazilian market (iii) the portfolio in Q.1a(iii) (iv) the portfolio in Q.1a(iv)

c. Find the weights (T-bill, US, Brazil) for a portfolio with the same expected return as Brazil (15%), using only a combination of the risk-free rate and the portfolio in Q.1a(iii)? What is the standard deviation of this portfolio? What is the correlation of this portfolio with the portfolio in Q.1a(iii)?

2. Assume the risk-free rate is 5% (rf = 5%), the expected return on the market portfolio is 10% (E[rM] = 10%) and the standard deviation of the return on the market portfolio is 20% (σM = 20%). (All numbers are annual.) Assume the CAPM holds.

a. What are the expected returns on securities with the following betas: (i) β = 1.4 (ii) β = 0.6 (iii) β = -0.2

b. What are the betas of securities with the following expect returns: (i) 10% (ii) 5% (iii) -1%

c. What are the portfolio weights (in the risk-free asset and the market portfolio) for efficient portfolios (portfolios on the efficient frontier/CML) with expected returns of (i) 6% (ii) 10% (iii) 20%

d. What are the portfolio weights (in the risk-free asset and the market portfolio) for efficient portfolios (portfolios on the efficient frontier/CML) with standard deviations of (i) 8% (ii) 20% (iii) 25%

e. Can securities or portfolios with the following characteristics exist in equilibrium, assuming the CAPM holds (yes or no): (i) expected return 0%, standard deviation 40% (ii) expected return 8%, standard deviation 9% (iii) expected return 15%, standard deviation 50%

f. A stock with a beta of 1 (β = 1.0) has a current price of $40/share. (i) Assuming it pays no dividends, what is the expected price in 1 year? (ii) If it is expected to pay a dividend of $4/share at the end of the year, what is the expected price in 1 year (after the payment of the dividend)?

g. For a moment (but just a moment) assume that the CAPM may not hold. A non-dividend paying stock has a current price of $50/share and an expected price in 1 year of $57/share (based on your personal analysis of the company’s prospects). (i) If the stock has a beta of 1 (β = 1.0), what is its alpha (α)? (ii) What is the alpha (α) if the beta is 2 (β = 2.0)?

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3. XYZ Inc. has expected earnings over the next year of $2/share (E[E1] = 2). The company is expected to maintain an earnings retention rate of 40% (b = 0.4), i.e., 60% of earnings are expected to be paid out as dividends every year. The company has a beta of 2, the risk-free rate is 4% (rf = 4%), and the market risk premium is also 4% (E[rM]-rf = 4%).

a. If the growth rate in earnings is expected to be 4% in perpetuity

i. What is the value of the stock?

ii. What is the expected price a year from now?

iii. What is the expected holding period return over the next year?

iv. What ROE justifies (is consistent with) this growth rate?

b. If the ROE is expected to be 15% in perpetuity

i. What is the implied growth rate?

ii. What is the value of the stock?

c. If the current price of the stock is $18/share

i. What is the implied growth rate?

ii. What is the implied ROE?

4. XYZ Inc. is expected to pay no dividends for the next 5 years. However, at the end of the sixth year (at time 6), the company is expected to pay a dividend of $1/share. Dividends are expected to grow at 10% per year for the following 9 years (through the end of the 15th year, i.e., time 15), then to grow at 5% every year thereafter (forever). Assume the appropriate discount rate (required return) is 10%.

a. What is the expected value of the stock at time 15?

b. What is the expected value of the stock at time 5?

c. What is the value of the stock today?

5. Consider a 2-year, risk-free bond with a coupon rate of 6% (annual coupons) and a face amount of $1,000. If the yield on the bond is 6% (i.e., P = $1,000),

a. What is the Macaulay duration of this bond?

b. If the yield increases to 7% immediately, what does the duration approximation predict will be the percentage change in the bond price?

c. If the yield increases to 7% immediately, what is the actual percentage change in the bond price?

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