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Given the utility function U = E(r) Ac€?o 0.5AA?A?2 and the fact that T-bill offer a risk-free rate of 4%, what is the minimum value for the risk-aversion coefficient A where an investor prefers the T-bills to an investment returning 10% with a standard deviation of 18%? (Hint: T-bills are risk-free)
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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%Assuming two risk-free rates for lending and borrowing in the market: r(f) and r(b). Suppose your utility function is described by U = E(r) - 0.5A xo² with A> 0, and you are combining the risk-free asset with the optimal risky asset to maximize the utility. Consider the following two situations: 1. Suppose r(b) = r(f): you form the optimal complete portfolio C1 by borrowing money at r(f) and invest y1 (i.e., y1 represents portfolio weight) in the optimal risky portfolio (P1). Your utility score under this situation is denoted as U1; II. Suppose r(b) >r(f): you form another optimal complete portfolio C2 by borrowing money at r(b) and invest y2 (1.e., y2 represents portfolio weight) in the optimal risky portfolio (P2). Your utility score under this situation is denoted as U2. For simplicity, let's assume the optimal risky portfolios under these two situations are the same (i.e., E(P1)=E(P2) and o(P1) = o(P2)). What are the relationship between y1 and y2, U1 and U2: O a. y1=y2 and U1=U2 O…On the basis of the utility formula below, which investment would you select if you were risk averse with A = 4? Investment Expected return E(r) Standard deviation σ 1 0.12 0.30 2 0.15 0.50 3 0.21 0.16 4 0.24 0.21
- a. Compute the expected rate of return on investment i given the following information: the market risk premium is 5%; Rf = 6%; βi = 1.2. b. Compute E(RM).Compute the expected rate of return on investment i given the followinginformation: Rf = 8%; E(RM) = 14%; βi = 1.0.b. Recalculate the required rate of return assuming βi is 1.8.25. a. Compute the expected rate of return on investment i given the followinginformation: the market risk premium is 5%; Rf = 6%; βi = 1.2.b. Compute E(RM)Supposing the return from an investment has the following probability distribution Return Probability R (%) 8 0.2 10 0.2 12 0.5 14 0.1 Required: What is the expected return of the investment? What is the risk as measured by the standard deviation of expected returns?
- 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 thereAPT An analyst has modeled the stock of Crisp Trucking using a two-factor APT model. The risk-free rate is 6%, the expected return on the first factor (r1) is 12%, and the expected return on the second factor (r2) is 8%. If bi1 = 0.7 and bi2 = 0.9, what is Crisp’s required return?Q. The market rate of return is 14%, beta is 1.5 and the required rate of return is 18.5%. What is risk-free rate of return?
- Suppose the CAPM is true. Consider two assets, X and Y, and the market M. Suppose cov(X,M) = .3, cov(Y,M) = .5. %3D (a) Is the expected return higher on X or Y? (b) Suppose var(M) = 1.5, what are the betas of X and Y? (c) Suppose the expected market return is 20% and the risk free rate is 5%, what is the expected returns of X and Y?. (d) Given your analysis in (a)-(c), what type of investor would prefer asset X to asset Y?6. On the basis of the utility formula below, which investment would you select if you were risk averse with A = 4? Standard Expected return E(t) deviation Investment 0.12 0.30 0.15 0.50 0.21 0.16 0.24 0.21Consider 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.