124 hello tack;e all the questions

12. You can form a portfolio of two assets, A and B, whose returns have the following characteristics: Stock | E[R] Standard Deviation Correlation A 0.10 0.20 0.5 B 0.15 0.40 If you demand an expected return of 12%, what are the portfolio weights? What is the portfolio's standard deviation? 13. Your have decided to invest all your wealth in two mutual funds: A and B. Their returns are characterized as follows: . the mean returns are FA = 20% and FB = 15% . the covariance matrix is TA TB TA 0.3600 0.0840 TB 0.0840 0.1225 21 @ 2001, Andrew W. Lo and Jiang Wang 1.6 Risk & Portfolio Choice 1 QUESTIONS You want your total portfolio to yield a return of 18%. What proportion of your wealth should you invest in fund A and B? What is the standard deviation of the return on your portfolio? 14. In addition to the fund A and B in the previous question, now you decide to include fund C to your portfolio. Its expected return is fc = 10%. The covariance matrix of the three funds is TA TB TC TA 0.3600 0.0840 0.1050 TB 0.0840 0.1225 0.0700 TC 0.1050 0.0700 0.0625 Your portfolio now consists of fund A, B and C. You would like to have an expected return of 16% on your portfolio and a minimum risk (measured by standard deviation of the return). What portfolio should you hold? What is the return standard deviation of your portfolio? (Hint: You would need to use Excel Solver or some other optimization software to solve the optimal portfolio.) 15. You can only invest in two securities: ABC and XYZ. The correlation between the returns of ABC and XYZ is 0.2. Expected returns and standard deviations are as follows: Security | E[R] (R) ABC 20% 20% XYZ 15% 25% a) It seems that ABC dominates XYZ in that it has a higher expected return and lower standard deviation. Would anyone ever invest in XYZ? Why? b) What is the expected return and standard deviation of a portfo- lio that invests 60% in ABC and 40% in XYZ? c) Suppose instead that you want your portfolio to have an expected return of 19.5%. What portfolio weights do you select now? What is the standard deviation of this portfolio? 16. You have the same data as the previous question. In addition, you have a risk-free security with a guaranteed return of 5%. The tangency portfolio has an expected return of ?? and standard deviation of ??. (a) What weights are placed on ABC and XYZ in the tangency port- folio?1. Suppose that a consumer with utility U(x1, 12) is given an initial endowment 71, 72 so that his total budget set is p171, 1272 (a) Derive a formula for the slutsky equation with endowments (b) Suppose that pi, p2 are such that consumption is exactly equal to the initial en- dowment. Show that on = #23 in this special case. 2. You are given the following partial information about a consumer's purchases. he consumes only two goods.: Year 1 Year 2 Quantity Price Quantity Price Good 1 100 100 Good 1 120 100 Good 2 100 100 Good 2 ? 80 Over what range of quantitites of good 2 consumed in year 2 would you conclude: (a) That his behaviour is inconsistent (i.e. it contradicts the weak axiom of revealed preference) (b) That the consumer's consumption bundle in year 1 is revealed preferred to that in year 2? (c) That the conumer's consumption bundle in year 2 is revealed preferred to that in year 1? (d) That there is insufficient information to justify (a),(b),(c) (e) *That good 1 is an inferior good (at some price) for this consumer? Assume that the weak axiom is satisfied (f) "That good 2 is an inferior good (at some price) for this consumer? Assume that the weak axiom is satisfied. 3. Recall that for demand functions to come from a utility maximization function: . They must be homogeneous of degree 0. That is r(tp, tp2, tm) = x(p1, p2, m) . They must satisfy the budget constraint p . I = m . The slutsky matrix must be symmetric. (2 + . The slutsky matrix is equal to the matrix of hicksian demand functions: ah = Dm (a) Suppose that we have 2 goods and consider the demand functions x(p,m) defined by: (p, m) = P2 m" P1 + P2 PI 12(p, m) = Bpi m P1 + P2 P2 Determine a and B (b) "Suppose that we have 2 goods and the hicksian demand function for good one hi(p, u) : hi(p, u) = 2u P1 1. Given that oh = ", show that for p2 > 0, ha(p, u) = In ( ) u + 9(pz, u) where g is an arbitrary function. 2. Suppose that when pi = p2, hi(p, u) = u, h2(p, u) = 0. Determine hz(p, u).4. Consider the utility function In(r) + y. (a) Optional: Solve for r(p,m), y(p,m), u(p.m), e(p,u), a" (p,m), y" (p,m). Follow the solutions from PS2 if necessary. (b) Suppose that there are 5 people in the economy each with endowments m', i = 1, 2, 3, 4, 5. 1. Suppose that m' > py Vi. Construct the aggregate demand function for r and y. What properties do the individual demand functions have that simplifty this problem? 2. Now suppose that m' py for i = 3,4,5. Construct the aggregate demand for 71,12 5. Suppose that an agent is strictly risk-averse who has an initial wealth of w but who runs a risk of a loss of D dollars. The probability of loss is *. It is possible, however, for the decision maker to buy insurance. One unit of insurance costs q dollars and pays 1 dollar if the loss occurs. Thus, if a units of insurance are bought, the wealth of the individual will be w -aq if there is no loss and w - aq - D + a if the loss occurs. The utility maximizer thus solves: 420 max(1 - a)u(w - aq) + nu(w - aq - D + a) (a) Assume that at the optimum a > 0. Find the FOC for the problem. (b) Suppose that insurance is actuarially fair in the sense of it being equal to the expected cost of insurance. Find a" (c) "Optional: Suppose that the agent is insuring his car and once a week he goes to white castle for burgers. There is no drive through and his car must be pushed in order for him to restart it which requires him some effort e. If he turns off the car (and thus expends effort) he has a probability a(e) of his car being stolen while in the restaurant. if he does not spend the effort he has a probability of 7 (0) of his car being stolen where , (0) > (e). 19 1. Suppose that he is able to get the amount of insurance as in part (b) above. Show that he will always choose to leave his car running. 2. Let u(x) = In(z). If the agent lets his car run while in white castle his utility is: [1 - #(0)] In(w - aq) + *(0) In(w - aq - D+ c) If the agent turns off his car his utility is: [1 - m(e)] In(w - aq) + m(e) In(w - aq - D + a) -e Show that for the agent to choose to turn off his car, a SB py Vi. Construct the aggregate demand function for r and y. What properties do the individual demand functions have that simplifty this problem? 2. Now suppose that m' py for i = 3,4,5. Construct the aggregate demand for 71,12 5. Suppose that an agent is strictly risk-averse who has an initial wealth of w but who runs a risk of a loss of D dollars. The probability of loss is *. It is possible, however, for the decision maker to buy insurance. One unit of insurance costs q dollars and pays 1 dollar if the loss occurs. Thus, if a units of insurance are bought, the wealth of the individual will be w -aq if there is no loss and w - aq - D + a if the loss occurs. The utility maximizer thus solves: 420 max(1 - a)u(w - aq) + nu(w - aq - D + a) (a) Assume that at the optimum a > 0. Find the FOC for the problem. (b) Suppose that insurance is actuarially fair in the sense of it being equal to the expected cost of insurance. Find a" (c) "Optional: Suppose that the agent is insuring his car and once a week he goes to white castle for burgers. There is no drive through and his car must be pushed in order for him to restart it which requires him some effort e. If he turns off the car (and thus expends effort) he has a probability a(e) of his car being stolen while in the restaurant. if he does not spend the effort he has a probability of 7 (0) of his car being stolen where , (0) > (e). 19 1. Suppose that he is able to get the amount of insurance as in part (b) above. Show that he will always choose to leave his car running. 2. Let u(x) = In(z). If the agent lets his car run while in white castle his utility is: [1 - #(0)] In(w - aq) + *(0) In(w - aq - D+ c) If the agent turns off his car his utility is: [1 - m(e)] In(w - aq) + m(e) In(w - aq - D + a) -e Show that for the agent to choose to turn off his car, a SB