Answer the questions in the space below. Written answers require no more than few compact sentences to show you understood and master the concept. Show your work to receive partial credit. Points are as indicated. Each subquestion (a,b, c, etc.) contributes to the total. Show your calculations to get partial credit. #. # 1 You are evaluating a project with the following after tax cash flows: Year After-tax Cash Flow ($mm) 0-100 (initial investment) 1 50 2 50 3 50 4 50 Assume: beta= 2 R free = 3% E(r M )= 10% 1 a (6 points) What is the NPV of the project? 1 b (3 points) The Internal Rate of Return (IRR) of this project is: 34.90% Compute the highest beta at which the project's NPV becomes zero. Page 1 of 12
2 Consider the following annual portfolio returns and the accompanying statistics: Year: 1 2 3 4 5 Return: 15.00% -10.00% -11.00% -7.50% 20.00% Arithmetic Mean = 1.30% Std.Dev= 14.95% Geometric Mean = 0.4455% 2 a (4 points) What is annualized holding period return earned if you held this portfolio over the last 5 years? 2 b (4 points) What return would you expect to earn next year if you assume the data above is a representative sample of past returns for this portfolio? 2 c (4 points) What is total, five-year holding period return you would have earned had you held this portfolio over the last five years? 2 d (4 points) What would be the value at the end of the five years of $100 invested in this portfolio? Page 2 of 12
Portfolio Expected Return FNCE 4030 Fall 2012 3 Consider the stocks below and the correlation matrix among the stocks: E(r) s Correlations A B C A 7% 10% A 1 0.75 0 B 9% 12% B 0.75 1 0 C 11% 15% C 0 0 1 3 a (6 points) Compute the expected return, variance and Std.Dev. of the equally weighted portfolio made of the three stocks. Identify on the StdDev-Mean chart the three stocks and the Equal Weight portfolio. Expected Return vs Standard Deviation 0.15 0.1 0.05 0 0% 5% 10% 15% 20% Std.Dev. Page 3 of 12
3 b (4 points) You build a portfolio only out of these three stocks A,B,C and you maximize Sharpe Ratio. What do you expect that optimal Sharpe Ratio to be at the very least? Assume risk free rate = 4% We add the risk free asset in the chart. We observe that the line connecting the risk free asset F to the 3 c (4 points) How would you improve the Sharpe Ratio of the equally weighted portfolio? Assume you can only use these three stocks and you cannot short-sell. Explain briefly. One way to improve the equally weighted portfolio comes from observing that stocks A and B are quite Page 4 of 12
Portfolio Expected Return FNCE 4030 Fall 2012 4 4 a 4 b There are only 3 assets traded: an equity instrument E, a debt instrument D, and the risk free asset F. Your utility function is in the form U = E(r) - 0.5 A s 2. The parameters are listed below. The family of risky portfolios is charted below for your convenience. (6 points) Identify the Max Sharpe portfolio if you can invest only in D and E. Estimate the weight of your investment in the Equity and in the Debt instrument. An estimate using a graphical approach is ok. (8 points) Draw the Capital Allocation Line (CAL) and given your risk aversion coefficient below, identify the optimal allocation portfolio on the chart. 4 c (6 points) Write your numerical estimate of how much you would invest in the three assets if you had $100. E(r) Std.Dev Utility Function - Risk Aversion Coefficient: E 14% 15% A 20.00 D 6% 8% F 2% - Correlation(D,E)= 25% 20% E(r) vs s - Family of risky portfolios 15% 10% Equity Asset 5% Risk Free Debt Asset Risky Portfolios 0% 0% 5% 10% 15% 20% 25% Portfolio Standard Deviation Page 5 of 12
(additional space to write your answer) Page 6 of 12
Portfolio Expected Return FNCE 4030 Fall 2012 5 Consider the stocks, the market portfolio and the risk-free rate below. Beta s(e) Market Risk Free A 0.8 10% R M 8% r free 3% B 1 12% s M 15% C 1.2 15% 5 a (6 points) Compute the expected return, variance and Std.Dev. of each stock. Identify the three stocks on the SML chart drawn below for your convenience. 15% E(r) vs b - Security Market Line 10% 5% Risk Free Market Portfolio SML 0% 0 0.5 1 1.5 2 beta Page 7 of 12
5 b (6 points) Now assume that the expected return of the three stocks is given below. If the simple CAPM is valid, which of the three following situations (independently) is possible. Explain in few words. E[r] Beta A 10.40% 0.8 B 8.00% 0.5 C 11.60% 1.2 5 c (6 points) Now assume that the expected return of the three stocks is still that given in (b) and that the the market expected return and risk free rate are not known. If the simple CAPM is valid and stocks A and B are priced correctly, what is R M and R free? Page 8 of 12
6 Assume there are two independent economic factors F 1 and F 2. Also assume R free = 6.0% and that the following are well-diversified portfolios: Portfolio b 1 b 2 Expected Return ( E[r] ) A 1.5 1 20% B 2 0.75 20% 6 a (6 points) What is the expected return-beta relationship in this economy? 6 b (4 points) Now suppose the risk premia are actually 1.2 times what you just calculated. How would you revise the fair Expected Returns for the two portfolios in the table above? Page 9 of 12
7 Assume there are two independent economic factors F 1 and F 2 and two corresponding factor portfolios with expected returns below. Also assume: R free = 3.0% Portfolio P 1 P 2 E(r) 9% 15% Portfolio A has factor loadings: b 1 = 0.5 b 2 = 0.75 7 a (3 points) What is A's risk premium? 7 b (3 points) Now suppose there is a second portfolio B with same factor loadings and expected return equal to 13.00% What investment strategy do you recommend? Explain briefly. 8 a (4 points) Assume that the expected return of the market is = 11.0% and its Std.Dev. Is = 20.0% You are a money manager and a potential client has invested her wealth in the risk-free asset and the rest in the market portfolio with the weights below. Assume the client has the standard utility function U = E(r) - 0.5 A s 2 and is acting optimally. Calculate the client s risk aversion level. The risk free rate is 3% Weight in market= 0.8 Weight in risk-free= 0.2 Page 10 of 12
9 9 a You have calculated the following statistics for the historic risk premia of a portfolio of investment-grade corporate bonds and a portfolio of large stocks. Use the data to answer parts (a) through (g). Parts (e), (f) and (g) appear on the next page. You may also need some of the Normal Distribution values on the formula sheet. Arithm. Mean Stdev Skew Kurt Corp Bonds 6% 8% 0.52-0.75 Large Stocks 12% 24% -0.85 1.5 (4 points) Assume Normality. Calculate the probability of a realized loss next year for Corporate Bonds. Calculate the probability of a realized loss next year Large Stocks. Which portfolio has the greater probability of losing money? 9 b (4 points) Assume Normality. Calculate the 68.26% confidence interval of returns for Bonds and Stocks. 9 c (4 points) Assume Normality. Calculate the probability of losing in Corporate bonds next year more than 8% In other words, for Corp. Bonds, what is the probability of r < -8% 9 d (4 points) Now consider all of the return statistics for Corporate Bonds. Is the probability calculation from Part (b) to big or too small or does the consideration of the additional statistics give mixed signals? EXPLAIN. Page 11 of 12
9 e (4 points) Calculate the 5% VaR for Large Stocks assuming that the distribution of returns is Normal. Briefly define the meaning of the 5% VaR. 9 f (3 points) If you calculated the 5% VaR for Large Stocks using the numerical sorting method covered in the project (often called a Back Simulation or the Historic Approach ), would you expect a larger or smaller VaR than the value from part (e)? Why? 9 g (5 points) Assume that the returns in the table are Excess Returns, in other words, the risk free rate was already removed. Compute the Sharpe ratios for both Corporate Bonds and Large Stocks and, after considering all statistics, are Sharpe Ratios a good comparison of these asset classes? EXPLAIN. Page 12 of 12