Consider the two (excess return) index-model regression results for stocks A and B. The risk-free rate over the period was 6%, and the markets average return was 14%. Performance is measured using an index model regression on excess returns.

Stock A

Stock B

Index model regression estimates 1% + 1.2(rM rf ) 2% + 0.8(rM rf )

R-square

0.576 0.436

Residual standard deviation, (e) 10.3% 19.1%

Standard deviation of excess returns 21.6% 24.9%

a. Calculate the following statistics for each stock: i. Alpha ii. Information ratio iii. Sharpe ratio iv. Treynor measure

b. Which stock is the best choice under the following circumstances? i. This is the only risky asset to be held by the investor. ii. This stock will be mixed with the rest of the investors portfolio, currently composed solely of holdings in the market-index fund. iii. This is one of many stocks that the investor is analyzing to form an actively managed stock portfolio.

Consider a portfolio that offers an expected rate of return of 12% and a standard deviation of 18%. T-bills offer a risk-free 7% rate of return. What is the maximum level of risk aversion for which the risky portfolio is still preferred to T-bills?

Use this information for the following questions, Question 3-6. You manage a risky portfolio with an expected rate of return of 18% and a standard deviation of 28%. The T-bill rate is 8%.

Your client chooses to invest 70% of a portfolio in your fund and 30% in an essentially risk-free money market fund. What is the expected value and standard deviation of the rate of return on his portfolio?

Suppose that your risky portfolio includes the following investments in the given proportions: Stock A 25% Stock B 32% Stock C 43% What are the investment proportions of your clients overall portfolio, including the position in T-bills?

5) What is the reward-to-volatility (Sharpe) ratio (S) of your risky portfolio? Your clients? 6) Your clients degree of risk aversion is A = 3.5. a. What proportion, y, of the total investment should be invested in your fund? b. What is the expected value and standard deviation of the rate of return on your clients optimized portfolio?

In addition to risk-free securities, you are currently invested in the Tanglewood Fund, a broad based fund of stocks and other securities with an expected return of 10.39% and a volatility of 27.22%. Currently, the risk-free rate of interest is 3.11%. Your broker suggests that you add a venture capital fund to your current portfolio. The venture capital fund has an expected return of 21.55%, a volatility of 64.24%, and a correlation of 0.19 with the Tanglewood Fund. Calculate the required return and use it to decide whether you should add the venture capital fund to your portfolio. 7) You are currently only invested in the Natasha Fund (aside from risk-free securities). It has an expected return of 14% with a volatility of 20%. Currently, the risk-free rate of interest is 3.8%. Your broker suggests that you add Hannah Corporation to your portfolio. Hannah Corporation has an expected return of 20%, a volatility of 60%, and a correlation of 0 with the Natasha Fund. a. Is your broker right? b. You follow your brokers advice and make a substantial investment in Hannah stock so that, considering only your risky investments, 60% is in the Natasha Fund and 40% is in Hannah stock. When you tell your finance professor about your investment, he says that you made a mistake and should reduce your investment in Hannah. Is your finance professor right? c. You decide to follow your finance professors advice and reduce your exposure to Hannah. Now Hannah represents 15% of your risky portfolio, with the rest in the Natasha fund. Is this the correct amount of Hannah stock to hold? d. Calculate the Sharpe ratio of each of the three portfolios above and estimate the portfolio weight in Hannah stock maximizes the Sharpe ratio. 8) You are considering how to invest part of your retirement savings. You have decided to put $300,000 into three stocks: 60% of the money in GoldFinger (currently $23/share), 30% of the money in Moosehead (currently $71/share), and the remainder in Venture Associates (currently $4/share). If GoldFinger stock goes up to $40/share, Moosehead stock drops to $53/share, and Venture Associates stock rises to $14 per share, What is the new value of the portfolio? b. What return did the portfolio earn? c. If you dont buy or sell any shares after the price change, what are your new portfolio weights? 9) You own three stocks: 600 shares of Apple Computer, 10,000 shares of Cisco Systems, and 5000 shares of Colgate-Palmolive. The current share prices and expected returns of Apple, Cisco, and Colgate-Palmolive are, respectively, $547, $18, $95 and 12%, 10%, 8%. a. What are the portfolio weights of the three stocks in your portfolio? b. What is the expected return of your portfolio? c. Suppose the price of Apple stock goes up by $20, Cisco rises by $7, and Colgate-Palmolive falls by $14. What are the new portfolio weights? d. Assuming the stocks expected returns remain the same, what is the expected return of the portfolio at the new prices? 10) Consider a world that only consists of the three stocks shown in the following table:

a. Calculate the total value of all shares outstanding currently. b. What fraction of the total value outstanding does each stock make up? c. You hold the market portfolio, that is, you have picked portfolio weights equal to the answer to part b (that is, each stocks weight is equal to its contribution to the fraction of the total value of all stocks). What is the expected return of your portfolio? 11) There are two ways to calculate the expected return of a portfolio: either calculate the expected return using the value and dividend stream of the portfolio as a whole, or calculate the weighted average of the expected returns of the individual stocks that make up the portfolio. Which return is higher? 12) Using the data in the following table, estimate (a) the average return and volatility for each stock, (b) the covariance between the stocks, and (c) the correlation between these two stocks.

13) Use the data in Problem 5, consider a portfolio that maintains a 50% weight on stock A and a 50% weight on stock B. a. What is the return each year of this portfolio? b. Based on your results from part a, compute the average return and volatility of the portfolio. c. Show that (i) the average return of the portfolio is equal to the average of the average returns of the two stocks, and (ii) the volatility of the portfolio equals the same result as from the calculation in Eq. 11.9. d. Explain why the portfolio has a lower volatility than the average volatility of the two stocks.

14) Suppose two stocks have a correlation of 1. If the first stock has an above average return this year, what is the probability that the second stock will have an above average return? Because the correlation is perfect, they move together (always), so the probability is 1.

15) Arbor Systems and Gencore stocks both have a volatility of 36%. Compute the volatility of a portfolio with 50% invested in each stock if the correlation between the stocks is (a) +1.0, (b) 0.50, (c) 0, (d) 0.50, and (e) 1.0. In which of the cases is the volatility lower than that of the original stocks?

16) Wesley Publishings stock has a volatility of 40%, while Addison Printings stock has a volatility of 20%. If the correlation between these stocks is 15%, what is the volatility of the following portfolios of Addison and Wesley: (a) 100% Addison, (b) 75% Addison and 25% Wesley, and (c) 50% Addison and 50% Wesley.

17) Suppose Avon and Nova stocks have volatilities of 55% and 26%, respectively, and they are perfectly negatively correlated. What portfolio of these two stocks has zero risk?

18) Suppose Tex stock has a volatility of 44%, and Mex stock has a volatility of 22%. If Tex and Mex are uncorrelated, a. What portfolio of the two stocks has the same volatility as Mex alone? b. What portfolio of the two stocks has the smallest possible volatility?

19) Suppose the average stock has a volatility of 49%, and the correlation between pairs of stocks is 18%. Estimate the volatility of an equally weighted portfolio with (a) 1 stock, (b) 30 stocks, (c) 1000 stocks. 20) What is the volatility (standard deviation) of an equally weighted portfolio of stocks within an industry in which the stocks have a volatility of 40% and a correlation of 40% as the portfolio becomes arbitrarily large? 21) Consider an equally weighted portfolio of stocks in which each stock has a volatility of 40%, and the correlation between each pair of stocks is 27%. a. What is the volatility of the portfolio as the number of stocks becomes arbitrarily large? b. What is the average correlation of each stock with this large portfolio?

22) Stock A has a volatility of 58% and a correlation of 27% with your current portfolio. Stock B has a volatility of 97% and a correlation of 28% with your current portfolio. You currently hold both stocks. Which will increase the volatility of your portfolio: (i) selling a small amount of stock B and investing the proceeds in stock A, or (ii) selling a small amount of stock A and investing the proceeds in stock B? 23) You currently hold a portfolio of three stocks, Delta, Gamma, and Omega. Delta has a volatility of 44%, Gamma has a volatility of 47%, and Omega has a volatility of 48%. Suppose you invest 30% of your money in Delta, and 35% each in Gamma and Omega. a. What is the highest possible volatility of your portfolio? b. If your portfolio has the volatility in (a), what can you conclude about the correlation between Delta and Omega? 24) Suppose Ford Motor stock has an expected return of 15% and a volatility of 38%, and Molson-Coors Brewing has an expected return of 12% and a volatility of 28%. If the two stocks are uncorrelated, a. What is the expected return and volatility of a portfolio consisting of 70% Ford Motor stock and 30% of Molson-Coors Brewing stock? b. Given your answer to part (a), is investing all of your money in Molson-Coors stock an efficient portfolio of these two stocks? c. Is investing all of your money in Ford Motor an efficient portfolio of these two stocks? 25) Suppose Intels stock has an expected return of 26% and a volatility of 50%, while Coca-Colas has an expected return of 6% and volatility of 25%. If these two stocks were perfectly negatively correlated (i.e., their correlation coefficient is 1), a. Calculate the portfolio weights that remove all risk. b. If there are no arbitrage opportunities, what is the risk-free rate of interest in this economy? a. If the two stocks are perfectly correlated negatively, they fluctuate due to the same risks, but in opposite directions. Because Intel is twice as volatile as Coke, we will need to hold twice as much Coke stock as Intel in order to offset Intels risk. That is, our portfolio should be 2/3 Coke and 1/3 Intel. For Problems 2628, suppose Johnson & Johnson and Walgreens Boots Alliance have expected returns and volatilities shown below, with a correlation of 21%.

26) Calculate (a) the expected return and (b) the volatility (standard deviation) of a portfolio that is equally invested in Johnson & Johnsons and Walgreens stock. 27) For the portfolio in question 26 if the correlation between Johnson & Johnsons and Walgreens stock were to increase, a. Would the expected return of the portfolio rise or fall? b. Would the volatility of the portfolio rise or fall?

28) Calculate (a) the expected return and (b) the volatility (standard deviation) of a portfolio that consists of a long position of $8500 in Johnson & Johnson and a short position of $1500 in Walgreens. 29) Using the same data as for question 26, calculate the expected return and the volatility (standard deviation) of a portfolio consisting of Johnson & Johnsons and Walgreens stocks using a wide range of portfolio weights. Plot the expected return as a function of the portfolio volatility. Using your graph, identify the range of Johnson & Johnsons portfolio weights that yield efficient combinations of the two stocks, rounded to the nearest percentage point. Hint: The set of efficient portfolios is approximately those portfolios with no more than 60% invested in J&J (this is the portfolio with the lowest possible volatility).

30) A hedge fund has created a portfolio using just two stocks. It has shorted $36,000,000 worth of Oracle stock and has purchased $95,000,000 of Intel stock. The correlation between Oracles and Intels returns is 0.65. The expected returns and standard deviations of the two stocks are given in the table below:

a. What is the expected return of the hedge funds portfolio? b. What is the standard deviation of the hedge funds portfolio? 31) Fred holds a portfolio with a 21% volatility. He decides to short sell a small amount of stock with a 48% volatility and use the proceeds to invest more in his portfolio. If this transaction reduces the risk of his portfolio, what is the minimum possible correlation between the stock he shorted and his original portfolio? Answer: For a small transaction size, short selling A and investing in P changes risk according to SD(Rp) SD(Ra)Corr(Ra,Rp). We gain the risk of the portfolio and lose the risk A has in common with the portfolio. For this to be negative, we must have SD(Rp)/SD(Ra) < Corr(Ra,Rp) or Corr > 21%/48% = 43.75%. 31) Suppose Targets stock has an expected return of 21% and a volatility of 43%, Hersheys stock has an expected return of 11% and a volatility of 22%, and these two stocks are uncorrelated. a. What is the expected return and volatility of an equally weighted portfolio of the two stocks? Consider a new stock with an expected return of 16% and a volatility of 30%. Suppose this new stock is uncorrelated with Targets and Hersheys stock. b. Is holding this stock alone attractive compared to holding the portfolio in (a)? c. Can you improve upon your portfolio in (a) by adding this new stock to your portfolio? Explain. 32) You have $8600 to invest. You decide to invest $17,000 in Google and short sell $8400 worth of Yahoo! Googles expected return is 14% with a volatility of 26% and Yahoo!s expected return is 12% with a volatility of 26%. The stocks have a correlation of 0.92. What is the expected return and volatility of the portfolio? 33) You expect HGH stock to have a 20% return next year and a 45% volatility. You have $25,000 to invest, but plan to invest a total of $50,000 in HGH, raising the additional $25,000 by shorting either KBH or LWI stock. Both KBH and LWI have an expected return of 15% and a volatility of 35%. If KBH has a correlation of +0.9 with HGH, and LWI has a correlation of 0.90 with HGH, which stock should you short? 34) Suppose you have $100,000 in cash, and you decide to borrow another $25,000 at a 6% interest rate to invest in the stock market. You invest the entire $125,000 in a portfolio J with a 24% expected return and a 26% volatility. a. What is the expected return and volatility (standard deviation) of your investment? b. What is your realized return if J goes up 13% over the year? c. What return do you realize if J falls by 26% over the year? 35) You have $55,000 to invest. You choose to put $105,000 into the market by borrowing $50,000. a. If the risk-free interest rate is 3% and the market expected return is 11%, what is the expected return of your investment? b. If the market volatility is 16%, what is the volatility of your investment? a. E[r] = 3% + 105,000/55,000 ( (11% 3%) = 18.27 % b. Vol = 105,000/55,000 ( 16% = 30.55% 36) You currently have $100,000 invested in a portfolio that has an expected return of 12% and a volatility of 8%. Suppose the risk-free rate is 5%, and there is another portfolio that has an expected return of 20% and a volatility of 12%. a. What portfolio has a higher expected return than your portfolio but with the same volatility? b. What portfolio has a lower volatility than your portfolio but with the same expected return? 37) Assume the risk-free rate is 4%. You are a financial advisor, and must choose one of the funds below to recommend to each of your clients. Whichever fund you recommend, your clients will then combine it with risk-free borrowing and lending depending on their desired level of risk.

Which fund would you recommend without knowing your clients risk preference? 38) Assume all investors want to hold a portfolio that, for a given level of volatility, has the maximum possible expected return. Explain why, when a risk-free asset exists, all investors will choose to hold the same portfolio of risky stocks. 39) The Optima Mutual Fund has an expected return of 19.5% and a volatility of 20.4%. Optima claims that no other portfolio offers a higher Sharpe ratio. Suppose this claim is true, and the risk-free interest rate is 5.5%. a. What is Optimas Sharpe Ratio? b. If eBays stock has a volatility of 37.8% and an expected return of 9.3%, what must be its correlation with the Optima Fund? c. If the SubOptima Fund has a correlation of 85% with the Optima Fund, what is the Sharpe ratio of the SubOptima Fund? 40) When the CAPM correctly prices risk, the market portfolio is an efficient portfolio. Explain why. All investors will want to maximize their Sharpe ratios by picking efficient portfolios. When a riskless asset exists this means that all investors will pick the same efficient portfolio, and because the sum of all investors portfolios is the market portfolio this efficient portfolio must be the market portfolio. 41) A big pharmaceutical company, DRIg, has just announced a potential cure for cancer. The stock price increased from $5 to $100 in one day. A friend calls to tell you that he owns DRIg. You proudly reply that you do too. Since you have been friends for some time, you know that he holds the market, as do you, and so you both are invested in this stock. Both of you care only about expected return and volatility. The risk-free rate is 3%, quoted as an APR based on a 365-day year. DRIg made up 0.2% of the market portfolio before the news announcement. a. On the announcement, your overall wealth went up by 1% (assume all other price changes canceled out so that without DRIg, the market return would have been zero). How is your wealth invested? b. Your friends wealth went up by 2%. How is he invested? 42) Your investment portfolio consists of $18,000 invested in only one stockMicrosoft. Suppose the risk-free rate is 6%, Microsoft stock has an expected return of 13% and a volatility of 44%, and the market portfolio has an expected return of 12% and a volatility of 19%. Under the CAPM assumptions, a. What alternative investment has the lowest possible volatility while having the same expected return as Microsoft? What is the volatility of this investment? b. What investment has the highest possible expected return while having the same volatility as Microsoft? What is the expected return of this investment? Answers a. Under the CAPM assumptions, the market is efficient; that is, a leveraged position in the market has the highest expected return of any portfolio for a given volatility and the lowest volatility for a given expected return. By holding a leveraged position in the market portfolio, you can achieve an expected return of E[R] = 6% + X (12% 6%). Setting this equal to 13% gives 13 E[R] = 6% + X (12% 6%) Solving it gives X = 1.1666 So the portfolio with the lowest volatility that has the same return as Microsoft has 18,000 ( 1.16667 = 21,000 in the market portfolio, and borrows 21,000 18,000 = 3,000; that is, $3,000 in the risk-free asset. SD (Rp) = 1.16667 ( 19 = 22.17% Note that this is considerably lower than Microsofts volatility. b. A leveraged portion in the market has volatility equal to SD (Rp) = X ( 19% Setting this equal to the volatility of Microsoft gives 44% = X ( 19%, X = 2.3157 So the portfolio with the highest expected return that has the same volatility as Microsoft has 18,000 ( 2.3157 = 41,684 in the market portfolio, and borrows 41,684 18,000 = 23,684, that is 23,684 in the risk-free asset. E[Rp] = 0.06 + 2.3157(12% 6%) = 19.89% Note that this is considerably higher than Microsofts expected return. 43) Suppose you group all the stocks in the world into two mutually exclusive portfolios (each stock is in only one portfolio): growth stocks and value stocks. Suppose the two portfolios have equal size (in terms of total value), a correlation of 0.5, and the following characteristics:

The risk-free rate is 3%. a. What is the expected return and volatility of the market portfolio (which is a 5050 combination of the two portfolios)? b. Does the CAPM hold in this economy? (Hint: Is the market portfolio efficient?) 44) Suppose the risk-free return is 3.5% and the market portfolio has an expected return of 11.2% and a volatility of 17.9%. Merck & Co. (Ticker: MRK) stock has a 21.5% volatility and a correlation with the market of 0.045. a. What is Mercks beta with respect to the market? b. Under the CAPM assumptions, what is its expected return? 45) Consider a portfolio consisting of the following three stocks: The volatility of the market portfolio is 10% and it has an expected return of 8%. The risk-free rate is 3%. a. Compute the beta and expected return of each stock. b. Using your answer from part (a), calculate the expected return of the portfolio. c. What is the beta of the portfolio? d. Using your answer from part (c), calculate the expected return of the portfolio and verify that it matches your answer to part (b).