Problem 1: Portfolio Choice with 2 Risky Assets (1 point)

Assume that all investors have some maximum standard deviation they are able to tolerate and are maximizing the expected return subject to not exceeding that standard deviation. There are investors with all kinds of maximum standard deviations. Furthermore, assume that investors can allocate their wealth across two assets. There is no borrowing. There are only 2 assets available to all investors. Asset 1: expected return 15%, standard deviation 20%. Asset 2: expected return 10%, standard deviation 30%, correlation with asset 1 return 0.2. Assets 1 and 2 are exactly the same except for differences in return properties stated above. Given that asset 1 is superior to asset 2 as it offers higher expected return with a lower standard deviation, will any investor allocate any fraction of their wealth to asset 2?

Problem 2: CAPM Concepts (2 points) Are statements below true or false? Explain your answer.

a) (0.5 point) Assume that CAPM holds. Given that stocks A and B, which are traded in the same market, have the same expected return, their betas must be the same.

b) (0.5 point) Stocks A and B, which are traded in the same market, have the same beta. Given that Correlation(A,Market)>Correlation(B,Market), it must be the case that Standard deviation(A)

c) (0.5 point) Assume that CAPM holds. In January 2015 stock A had some expected return. In February 2015 risk-free rate increased compared to January 2015, but the expected market return and stock A beta stayed the same. It must be the case that the expected return on stock 1

A increased in February 2015 compared to its expected return in January 2015. d) (0.5 point) If the stock market consists of only 1 stock, that stock must have a beta of 1. Problem 3: CAPM in Practice (3 points)

For this problem, you will need data in Excel file homework3 data 1.xlsx available on Black- board, which contains monthly returns on Microsoft, IBM, Oracle stocks and the aggregate stock market index. You do not know if CAPM holds.

a) (1 point) Compute expected returns (means) on Microsoft, IBM, Oracle stocks and the ag- gregate stock market index.

b) (1 point) Compute betas of Microsoft, IBM, Oracle stocks and the aggregate stock market index.

c) (1 points) One of the implications of CAPM is that stocks which have higher :s should

A(RM Rf)andRB =RF +B(RM Rf).Subtractingthesecondequationfromthefirst

Thus, as (RM RF ) is positive (nobody would hold the risky market portfolio, if it would on

Problem 4: Weighted Average Cost of Capital (2 points)

The firm has annual earnings of $1 million, which are assumed to be an annual perpetuity starting in year 1 (now is year 0). The firm is financed by 40% of debt and 60% of equity. The cost of equity capital is 10% and the cost of debt is 5%. Assume that there are no taxes.

a) (1 point) What is the firms weighted average cost of capital? b) (1 point) What is the net present value of the firm? Problem 5: Risk-free Rate and Expected Market Return (2 points) Assume that CAPM holds. Stocks A and B have the following characteristics:

also have higher expected returns. To see this, note that CAPM implies that RA = RF +

oneyields:RARB =RF +A(RM Rf)RF B(RM Rf)=(AB)(RM RF).

average pay less than the riskless asset), if (A B ) > 0, RA RB should also be positive. Consequently, one way to test CAPM in data is to check if stocks which have higher :s also have higher expected returns. Given your results from a) and b), is this implication of CAPM true in the sample of Microsoft, IBM, Oracle, and the aggregate stock market index?

2

Stock A 0.5

Expected return 5.25%

a) (1 point) What is the expected market return? b) (1 point) What is the risk-free rate?

Stock B 0.9 8.25%

Hint: Write down CAPM equations for both stocks and treat it as two linear equations with two unknowns.

Problem 6: Does CAPM hold? (1 point) Stock A has the expected return of 0.9%, standard deviation of 10%, and the correlation with

the market portfolio of 0.3. The risk-free rate is 1%. Does CAPM hold? Hint: Think about the signs (positive/negative), not exact values. Problem 7: Market Portfolio (1 point)

Suppose that the market consists of only 2 stocks. Stock 1 has the market capitalization (ag- gregate market value of its shares) of $50 million and of 0.85. Stock 2 has the market capitalization of $75 million. What is stock 2 ?

Problem 8: Portfolio Choice Using Numerical Methods (6 points)

For this problem, you will need data in Excel file homework3 data 2.xlsx available on Black- board, which contains monthly returns on Microsoft and Alcoa stocks. Assume that those are only 2 stocks on the market. You will need to use a statistical software. Microsoft Excel should satisfy your needs, but you can use anything else you prefer. Do all tasks below at monthly frequency. Use population statistics to compute return properties.

a) (0.5 point) Compute the expected return and standard deviation of Microsoft. b) (0.5 point) Compute the expected return and standard deviation of Alcoa. c) (0.5 point) Compute the covariance and correlation between Microsoft and Alcoa returns.

d) (1 point) Construct the grid of possible portfolio allocations. Assume that is the portfolio allocation to Microsoft. Assume that is between 0 and 1 and use the grid precision of 0.005 (0.5%). Thus, your grid is going to be 0,0.005, 0.01, 0.015,...,0.985, 0.99, 0.995, 1. For each portfolio allocation on the grid, compute the expected return and the standard deviation using 3

properties of individual assets from a)-c). Do not report the whole grid in the answer: report only the rows corresponding to the following values of : 0.1, 0.3, 0.7, and 0.95.

e) (0.5 point) Plot the opportunity set in the standard deviation-expected return domain. Use the grid constructed in part d) to answer the questions below:

f) (1 point) What is the minimum standard deviation you can achieve using those 2 assets? What is (approximately) the portfolio allocation to Microsoft to achieve that standard deviation?

g) (1 point) Suppose the maximum monthly standard deviation you can tolerate is 9%. What is your optimal portfolio allocation among Alcoa and Microsoft stocks?

h) (1 point) Suppose that the monthly risk-free rate is 0.1% and you can both invest and borrow at this rate. What is the weight of Microsoft in the aggregate market portfolio?

Problem 9: Debt-to-Equity Ratio (1 point)

The firm has the weighted average cost of capital of 15%. The cost of equity is 20% and the firm pays 8.5% interest rate on its debt to investors. Assume that there are no taxes. What is the firms debt-to-equity ratio?

Problem 10: Bounds on the Weighted Average Cost of Capital (2 points)

The firm is financed by 30% of debt and 70% of equity. The corporate tax rate is 21%. The firm pays 2% interest rate on its debt to investors. The risk-free rate in the economy is also 2% and the firm equity has beta of 2.5.

a) (1 point) What is the lower bound for the firms weighted average cost of capital? b) (1 point) What is the upper bound for the firms weighted average cost of capital? Problem 11: Tax Benefits of Debt (2 points)

Your firm will have annual earnings (before interest and taxes) of $1 million in year 1 (now is year 0). The earnings will grow at 5% annually forever. The tax rate is 35%. The firm has $400,000 of outstanding debt, on which it has to pay 5% annual interest rate, and its unlevered cost of capital is 12%. The tax rate for years 1-14 is 35%, but it will drop to 25% for year 15 and will stay at that value forever. The debt is assumed to stay constant. What is the value of the firm?

a) (1 point) What is the value of the firm? 4

b) (1 point) Your industry decides to hire a lawyer, which will negotiate with the government that the tax decrease will happen in year 10 instead of year 15 (that is only earnings in years 1-9 will be taxed at 35% rate). What is the maximum amount your firm will be willing to contribute in order to hire the lawyer?