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- Assuming your utility function U = E(r) - Ao². Consider the investments shown in the table. If your risk aversion coefficient is -2, which investment would you choose? Investment E[r] #1 #2 #3 #4 #2 #3 #4 #1 12% 15% 15% 24% σ 40% 40% 30% 40%2. Suppose that you have a riskfree asset and N risky assets for investment. The rate of return on the riskfree asset is r,, while the (Nx1) vector of the rate of return on the N risky assets is r, which is multivariate normal, i.e., r N(u, E). Your utility function for a portfolio that consists of the riskfree asset and the N risky asset is u(r,)=r,-=o, 2 Suppose that the sum of investment proportions on the riskfree and risky assets is one. Answer the following question. A. What is your optimal investment proportion in the risky assets? How is your investment on the riskfree asset affected by different values of 2? B. Suppose that there is only one risky asset i. Show the effects of the Sharpe ratio (4,/0, ) on the investment proportion in the risky asset.Think about whether a risk-free asset should earn a risk-premium beyond the risk-free rate. Thinking about that should give you an idea of the beta for a risk-free asset. Or, look again at the CAPM equation: E(Ri)=Rf+βi[E(RM)−Rf] Given this equation, what beta sets the E(R) of the risk free asset equal to the risk-free rate? A) zero B) 0.5 C) 1.0 D) its random
- We believe that the single factor model can predict any individual asset’s realized rate of return well. Both Portfolio A and Portfolio B are well-diversified: ri = E(ri) + βiF + Ei, where E(ei) = 0 and Cov(F, i) = 0 A B β 1.2 0.8 E(r) 0.1 0.08 (1) What is the rate of return of the risk-free asset? (2) What is the expected rate of return of the well-diversified portfolio C with βC = 1.6, which also exists in the market? (3) A fund constructs a well-diversified portfolio D. Studies show that βD = 0.6. The expected rate of return of D is 0.06. Is there an arbitrage opportunity? If so, construct a trading strategy to earn profits with no risk. If not, why?Consider an economy with a (net) risk-free return r1 = 0:1 and a market portfolio with normally distributed return, with ErM = 0:2 and 2M = 0:02. Suppose investor A has CARA preferences, with risk aversion coe¢ cient equal to 1 and an endowment of 10. a) Write down the maximization problem for the investor. b) Determine the amount invested in the risky portfolio and in the risk-free asset. c) Suppose another investor (B) has a coe¢ cient of absolute risk aversion equal to 2 (and the same endowment 10). Compute his optimal portfolio and compare it to that of investor A. Explain the di§erent results for investors A and B. d) Finally, consider Investor C with mean-variance preferences Ec V ar(c) (and endowment 10). Compute his optimal portfolio and compare it to that of investors A and B (as obtained in questions b and c). Compare your result with those obtained for investors A and B.Consider a two-factor Arbitrage Pricing Theory (APT) model, T; = a; + b1,ifı + b2.i f2 + €i, with the following information Asset i Asset 1 0.07 0.50 0.25 Asset 2 0.15 1.10 0.75 Asset 3 0.20 b1,3 Hi b1i b2i 1.0 and the risk-free rate rp is 0.025. (a) Find the value of b13 to preclude arbitrage opportunity. (b) is an Asset 4 with 4 = 0.13, b14 = 0.8, and b2.4 = 0.4. Explain how you would exploit an arbitrage opportunity if there
- Consider a one-period model with N = {1,2,3} and two risky assets S1, S2 whose with current prices are 71 = 42 and 12 = 23 respectively. At time one, the price of S1 is believed to be either 44, 43, or 40 while the price of S2 is believed to be either 27, 22, or 20. Suppose that the risk-free interest rate is 3%. What is the unique risk-neutral probability measure Q in this situation? Select one: О а. Q(1) — 0.35, Q(2) — 0.27, Q(3) — 0.38 O b. Q(1) = 0.63, Q(2) = 0.34, Q(3) = 0.03 Ос. Q(1) — 0.25, Q(2) — 0.62, Q(3) — 0.13 O d. Q(1) = 0.35, Q(2) = 0.62, Q(3) = 0.03Consider a world with only two risky assets, A and B, and a risk-free asset. The two riskyassets are in equal supply in the market, i.e., the market portfolio M = 0.5A + 0.5B. Itis known that R¯M = 11%, σA = 20%, σB = 40% and ρAB = 0.75. The risk-free rate is2%. Assume CAPM holds.(a) What is the beta for each stock?(b) What are the values for R¯A and R¯B?Consider the following graph. According to Markowitz’ portfolio theory, which point on the graph represents optimal portfolio? C A B D
- Suppose you observe the following situation:Security Beta Expected ReturnDiamond Co 1.3 0.2Spade Co 0.8 0.14 (a) According to the above information, could we figure out the market return and risk-free rate? Explain your answer. (b) Discuss the possibility of including zero beta or negative beta assets in your portfolio. Explain the pros and cons of including these types of assets.Suppose you want to establish a bullish spread strategy. The are two call options. The first one has X1=$50 and C1=$5. The second one has X2=$42 and C2=$6. When the underlying asset price is S(t)=$45, what is the profit from the strategy? What is the maximum profit of the strategy? What is the minimum payoff of the strategy?Write out the equation for the Capital Market Line (CML), and draw it on the graph. Interpret the plotted CML. Now add a set of indifference curves and illustrate how an investors optimal portfolio is some combination of the risky portfolio and the risk-free asset. What is the composition of the risky portfolio?