Exemption Exam - Finance
Show all work clearly, if you desire partial credit for incomplete answers. If you need to
make any extra assumptions to answer any of the questions, do so. However, state the
additional assumptions clearly.
You plan to retire in 30 years, and you anticipate the need for an annual pre-tax retirement
income of $200,000 per year, in today's dollars.
a. Assuming an expected inflation rate of 4% over the next 30 years, what is your annual
retirement income needs in 2027 dollars?
b. Assume your life expectancy from today is 60 years. What is the present value of the
cost of your retirement, under the assumption that the discount rate is 7% for all future
c. Would the answer to (b) be very different if you assumed that you would live forever,
d. Assume that you may save tax-free in a personal retirement account, and that you will
only be taxed upon the money you withdraw annually upon retirement. You have two
investment choices for the account. You may invest in stocks with an expected return of
12% per year, or you may invest in bonds with an expected return of 7% per year.
Compare how much you will have to contribute to your retirement account each year for
the next 30 years to meet your income goal, under the two investment policies. Assume
inflation will be 4% indefinitely.
e. Is there a simple mathematical expression that approximates the relationship between
savings each year for the next 30 years, and the future annual retirement income required?
Suppose you have two stocks in your portfolio, Max and Min. The expected return of Max
iis 10% and the expected return of Min is 7%. The standard deviation of Max is 30% and
the standard deviation of Min is 20%. The correlation between the two securities is .3.
Suppose the riskless asset has an expected return of 5%.
a. What is the mean and the standard deviation of a portfolio composed of 50% Max and
b. Which has the highest Sharpe ratio, Max, Min or the 50/50 portfolio?
c. Suppose the correlation between Max and Min was -1. Could you identify an arbitrage
opportunity, and if so, what credit conditions would allow the investor to exploit this
opportunity? No need to calculate exact portfolio weights for the opportunity. Simply
explain the positions required.
Suppose the expected return to the market portfolio was 12%, and the standard deviation
of the market portfolio was 20%. Further, assume that all of the conditions in the previous
a. Assuming for a moment that market prices are efficient, and that the CAPM holds, what
is the CAPM beta of the Max company?
b Now consider what you might do if the price of the Max company were lower, such that
the firm had an expected return of 12%, however the beta remained the same. Construct
a portfolio (with positive and negative weights) of the market security, the riskless asset
and shares of Max such that the portfolio has an expected return greater than the riskless
rate, but zero beta.
Carbon Monoxide Unlevered Inc. (CUI) & Carbon Dioxide Levered Inc. (CLI) are two
environmentally conscious socially responsible companies manufacturing the same line
of pollution control equipment. The two firms are identical in every way except in their
capital structures: CUI is unlevered while CLI has debt of $ 30 mm yielding 10% p.a. Both
firms have expected EBIT of $ 20 mm p.a. Each firm has 2 million common shares
outstanding. CLI's shares trade at $ 20/share.
1) Suppose that corporate income is taxed @ 30%. There are no personal taxes.
1a) What should CUI's share trade at?
1b) Compute the WACC of each firm. Are the WACCs different? Why (not)?
2) Suppose that in addition to corporate taxes, ordinary income is taxed at 20%. Dividends
and capital gains are assumed to be tax exempt. At what price should CUI's shares trade
in the market?
3) We are back in the no personal tax scenario. Consider a project that costs $50 mm, having the same risk as that of the existing business of the two firms. The project yields an annual pre-tax cashflow of $ 15 mm p.a. What would the new share price of CLI be if it accepted this project? What would CUI's price be, if it took this project?
Professor: William N. Goetzmann
Question 1: The Basics
Assume an investor is considering an investment in a large number of different risky securities, all with different expected returns and standard deviations and correlations. Assume also, that the investor is free to create any portfolio of any of these securities.
1) Draw the investor's opportunity set in expected return and standard deviation space. Shade in all feasible portfolios of risky assets.
2) Label the minimum variance portfolio.
3) Draw the efficient frontier.
Now assume that a riskless asset exists and the investor can borrow and lend at the riskless rate.
4) Identify the risk and return of the riskless asset in the diagram.
5) Draw the efficient frontier, now that the riskless asset exists.
6) If you could invest in only two portfolios (the riskless asset is counted as a portfolio, by the way) which would they be?
Question 2: Risk and Return
Consider two corporations, called Risco and Safco. The annual expected return of shares of Risco stock is 15% per year, and the annual expected return of shares of Safco stock is 10% per year. The annual standard deviation of shares of Risco stock is 30% per year, and the annual standard deviation of shares of Safco stock is 20% per year. The correlation between Risco and Safco is .5.
1) What is the expected return of a portfolio composed of 50% Risco and 50% Safco?
2) What is the standard deviation of a portfolio composed of 50% Risco and 50% Safco?
3) Which of the three portfolios maximizes the probability of exceeding 0% return? Show or briefly explain why.
4) Does the 50/50 portfolio dominate either Risco stock or Safco stock as an investment? If so, why?
Question 3: Systematic Risk
Assume the CAPM holds, and that the risk-free rate is 5.5 % and the expected equity risk premium in 8.5.
1) If the expected return of Risco is 15%, what is its beta?
2) If the expected return of Safco is 10%, what is its beta?
3) Suppose you could borrow and lend at the riskless rate. What portfolio of Safco and the riskless security could you create that would have the same systematic risk (i.e. beta) as Risco?
4) Again, suppose that the CAPM were violated and that Risco plotted above the security market line by 4%. How might you take advantage of this situation, without incurring systematic risk, or incurring any net expense? For this question, you may assume that both securities can be purchased and shorted with no transactions costs, and that you could borrow and lend at the riskless rate. Write out the exact portfolio you would create.
Question 4: Leverage
For this question alone, assume that 40 % of the capital structure of Risco is equity, and 60% is debt. Further assume that Safco has no debt in its capital structure.
1) Is the systematic risk of the assets in Risco greater or less than the systematic risk of the assets in Safco?
2) Consider the effect of changing the capital structure of Safco to a 40% equity, 60% debt financed company. What would be the expected return of the equity of Safco if it had the same capital structure as Risco?
Question 5: Efficient Markets
Currently, Safco stock is trading at $50 per share and Risco stock is trading at $10 per share. Suppose the board of directors of Safco, meeting behind closed doors on Monday afternoon, discussed whether their company should acquire Risco. While nothing was firmly resolved, a majority of the directors said they would vote in favor of the acquisition at the next board meeting, on Wednesday of that week, at which time the exact price to be offered for Risco shares would be decided. One of the directors of Safco "leaked" this information to the newspapers that evening after the discussion. You read a tiny notice about the possibility of the merger on page six of the Tuesday morning paper.
1) If you buy Risco shares in anticipation of the price going up, are you trading on insider information?
2) If you buy Risco shares in anticipation of the price going up, are you a believer in strong form efficiency?
3) If you buy Risco shares in anticipation of the price going up, are you a believer in semi-strong form efficiency?
4) Suppose the market is semi-strong form efficient. On Wednesday, Safco directors decide to make a bid to acquire Risco for $12 per share. Do you expect the price of Risco shares to change significantly on that day? Why or why not?
Extra Credit: Valuation
This question is only to be done if you have completed answers to all the other questions. You will get NO CREDIT if you have not done the rest of the exam.
For this question, assume that a single-stage discounted cash flow model under certainty fairly represents the relationship between current price, earnings, discount rates and growth for each firm. That is:
where P is the price of the firm, E is the earnings, R is the discount rate, and g is the growth rate in future earnings.
1) Assume that the P/E ratio of Risco is 10 and the P/E ratio of Safco is 10. Assume that the average analysts' forecast the growth rate in earnings for Risco is 6% and 4 % for Safco. What are the implied discount rates (R) for Safco and for Risco?
2) If Safco and Risco both had earnings of 24 million per year, and the CAPM holds, what price would you pay for each company?
Questions For Qualifying Exam For Financial Management I.
1. The expected return of asset A is 10%/yr. The expected return of asset B is 15%/year. The annual standard deviation of A's returns are 12% and the annual standard deviation of B's returns are 20%. The correlation between the returns of A and B is .5.
A. What is the annual standard deviation of a portfolio composed of 25% A and 75% B?
B. Draw the efficient frontier composed of mixtures of A & B, in mean-standard deviation space. To do so, you must estimate the location of the minimum variance portfolio. Please label it in the picture.
C. Now draw the efficient frontier under two extreme assumptions: first , that the correlation between A and B = 1, and second, that the correlation between A & B = -1.
D. Now, suppose that there is a riskless asset that returns 5%/ yr. Show the new efficient frontier, under the assumption that the correlation coefficient between A & B is .5, and that an investor can borrow and lend at the riskless rate.
E. Can you identify the portfolio of A & B that maximizes the investor probability of exceeding a return in excess of the riskless rate? Assume you may NOT invest in the riskless asset.
2. The Capital Asset Pricing Model
A. Assume that the assumptions underlying the Capital Asset Pricing Model are valid, and the model itself is true. Investor J has a personal wealth of $100,000, The world's wealth is $35,000,000,000,000. Company A has 12,000,000 shares outstanding of common stock, and has no debt. The current share price is $50. What is the dollar value of A's stock held by investor J?
B. Assume the CAPM is true, and hold exactly. The expected return of asset A is 12%, the expected return of the market portfolio is 10% and the beta of A is 1.4. What is the riskless rate?
C. Assume the CAPM is true and hold exactly, and the riskless rate is the number calculated in B. Is it possible to have an asset in this economy with a beta of 1.2 and an expected return of 12%?
D. This is an APT question. Suppose you observed the market conditions specified in C, and that you were a risk-neutral investor. What actions would you take to earn positive expected returns, without increasing the beta of your portfolio?
E. Suppose you took the actions you specified in D. Would this be true arbitrage? Why or why not?
F. Assume the CAPM is true, and hold exactly, and the riskless rate is the number calculated in B. Company A has a beta of 1.4, and a standard deviation of 25%/ yr.. Company B has a beta of 1.2 and a standard deviation of 30%/ year. The expected return of the market portfolio is 10%. The correlation between A and B is .3. Calculate the expected return of A, the expected return of B and the expected return of a portfolio comprised of 50% A and 50% B.
3. Capital Structure
A. The beverage industry as a whole has an average levered equity beta of 1.07, and an average ratio of debt to total capital of 20%. Estimate the beta of an all-equity financed company within the beverage industry.
B. Company C in the beverage industry has a ratio of debt to total capitalization of 50%. Assuming that its assets are typical of the industry, and that it's debt has a beta of zero, what is the beta of its publicly traded stock?