ECN 134Risk and ReturnRisk and ReturnSolution to Problem Set 4ECN 134Finance Economics Prof. Farshid MojaverRisk and Return1. Let rx, ry, rz be returns of portfolios X,Y, and Z.(i) P(rx <0) = P(z<(0-5)/20) = P(z< -0.25) is greater than P(ry <0) = P(z<(0-7)/20) = P(z< -0.35), which in turn is greater than P(rz <0) = P(z<(0-5)/10) = P(z< -0.5).You can determine the rank order of the probabilities by the rank order of the z scores without looking at the table for the normal distributions since P(z<-0.25) > P(z<-0.35) > P(z > -0.5).(ii) Similarly, P(rx <5) = P(z<0); P(ry <5) = P(z<-0.1); P(rz <5) = P(z<0) and P(z<0) >P(z<-0.1).(iii) P(rx <10) = P(z<0.25); P(ry <10) = P(z<0.15); P(rz <10) = P(z<0.5). Clearly, P(z<0.5)> P(z<0.25) > P(z<0.15).(iv) No, since X has lower mean than Y with the same risk.(You can draw a graph in the (risk=μ, mean=σ) plane.)(v) Yes, if he is risk-lover.(vi) No, Z offers the same expected return but lower risk.2. E(r) = [0.2 × (−25%)] + [0.3 × 10%] + [0.5 × 24%] =10%3. E(rX) = [0.2 × (−20%)] + [0.5 × 18%] + [0.3 × 50%] =20% E(rY) = [0.2 × (−15%)] + [0.5 × 20%] + [0.3 × 10%] =10%4. X 2 = [0.2 × (– 20 – 20)2] + [0.5 × (18 – 20)2] + [0.3 × (50 – 20)2] = 592 X = 24.33% Y 2 = [0.2 × (– 15 – 10)2] + [0.5 × (20 – 10)2] + [0.3 × (10 – 10)2] = 175 X = 13.23%5. E(r) = (0.9 × 20%) + (0.1 × 10%) =19%Risk and Return 1. a. μ = .06 and σ = .10 : Find P (X < 0) = P(Z < 0−. 06.10) = P (Z < −.6) .Turning to the normal tables we find that the probability associated with the critical value of −.6 is 27.43%. That is 27.43% of the time, and thus approximately 13 years , the return on corporate bonds was less than 0%. b. μ = .11 and σ = .16 : Find out what the rate of return is for the worst fiveyears of the S&P. That is: what was the rate of return for the worst 10% of the last 48 years. We must solve for a: .10 = P (X < a) = P(Z<a−. 11.16). Turning to the normal tables we find that the critical value leading to a .10 outcome is −1.28. Thus find a such that −1.28 =a−. 11.16 => a =-.09This tells us (approximately) that in the worst 5 years of the last 48 years the S&P lost9% or more.2. When we specify utility by U = E(r) – 0.5Aσ2, the utility level for T-bills is: 0.07The utility level for the risky portfolio is: U = 0.12 – 0.5A(0.18)2 = 0.12 – 0.0162AIn order for the risky portfolio to be preferred to bills, the following inequality must hold:0.12 – 0.0162A > 0.07  A < 0.05/0.0162 = 3.09A must be less than 3.09 for the risky portfolio to be preferred to bills.3. Points on the curve are derived by solving for E(r) in the following equation:U = 0.05 = E(r) – 0.5A = E(r) – 1.5The values of E(r), given the values of , are therefore: 2E(r)0.000.00000.050000.05 0.0025 0.053750.10 0.0100 0.065000.15 0.0225 0.083750.20 0.0400 0.110000.25 0.0625 0.14375The bold line in the following graph (labeled Q4, for Question 4) depicts the indifference curve.4. Repeating the analysis in Problem 3, utility is now:U = E(r) – 0.5A = E(r) – 2.0 = 0.04The equal-utility combinations of expected return and standard deviation are presented in the table below. The indifference curve is the upward sloping line in the graph above, labeled Q4 (for Question 4). 2E(r)0.000.00000.04000.05 0.0025 0.04500.10 0.0100 0.06000.15 0.0225 0.08500.20 0.0400 0.12000.25 0.0625 0.1650The indifference curve in Problem 4 differs from that in Problem 2 in both slope and intercept. When A increases from 3 to 4, the increased risk aversion results in a greater slope for the indifference curve since more expected return is needed in order to compensate for additional . The lower level of utility assumed for Problem 4 (0.04 rather than 0.05) shifts the vertical intercept down by 1%.E(r) 5 4U(Q3,A=3)U(Q4,A=4)U(Q5,A=0)U(Q6,A<0)5. The portfolio expected return and variance are computed as follows:(1)WBills(2)rBills(3)WIndex(4)rIndexrPortfolio(1)(2)+(3)(4)Portfolio(3)  20% 2 Portfolio0.05%1.013.5%13.5% = 0.13520% = 0.200.04000.25%0.813.5%11.8% = 0.11816% = 0.160.02560.45%0.613.5%10.1% = 0.10112% = 0.120.01440.65%0.413.5%8.4% = 0.0848% = 0.080.00640.85%0.213.5%6.7% = 0.0674% = 0.040.00161.05%0.013.5%5.0% = 0.0500% = 0.000.00006. Computing utility from U = E(r) – 0.5  A = E(r) – 1.5 , we arrive at the values in the column labeled U(A = 3) in the following table:WBillsWIndexrPortfolioPortfolio2PortfolioU(A = 3) U(A = 5)0.01.00.1350.200.04000.07500.03500.20.80.1180.160.02560.07960.05400.40.60.1010.120.01440.07940.06500.60.40.0840.080.00640.07440.06800.80.20.0670.040.00160.06460.06301.00.00.0500.000.00000.05000.0500The column labeled U(A = 3) implies that investors with A = 3 prefer a portfolio that is invested 80% in the market index and 20% in T-bills to any of the other portfolios in the table.7. The column labeled U(A = 5) in the table above is computed from:U = E(r) – 0.5A = E(r) – 2.5The more risk averse investors prefer the portfolio that is invested 40% in the market index, rather than the 80% market weight preferred by investors with A = 3.8. Expected return = (0.7  18%) + (0.3  8%) = 15%Standard deviation = 0.7  28% = 19.6%9. Your reward-to-variability ratio: S=18−828= 0 .3571Client's reward-to-variability ratio: S=15−819 . 6=0 .357110.ClientP051015202530010203040E(r)%CAL (Slope = 0.3571)11. a. y*=E(rP)−rfAσP2=0 . 18−0. 083 .5×0 .282=0 . 100 . 2744=0 . 3644Therefore, the client’s optimal proportions are: 36.44% invested in the risky portfolio and 63.56% invested in T-bills.b. E(rC) = 8 + 10y* = 8 + (0.3644  10) = 11.644%C = 0.3644  28 =