Solution to Problem Set 4 ECN 134 Finance Economics Prof Farshid Mojaver Risk and Return 1 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 mean 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 0 06 1 a 06 and 10 Find P X 0 P Z 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 11 and 16 Find out what the rate of return is for the worst five years of the S P That is what was the rate of return for the worst 10 of the last 48 a 11 years We must solve for a 10 P X a P Z 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 a 11 1 28 16 a 09 This tells us approximately that in the worst 5 years of the last 48 years the S P lost 9 or more 2 When we specify utility by U E r 0 5A 2 the utility level for T bills is 0 07 The utility level for the risky portfolio is U 0 12 0 5A 0 18 2 0 12 0 0162A In 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 09 A 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 0 00 0 05 0 10 0 15 0 20 0 25 2 0 0000 0 0025 0 0100 0 0225 0 0400 0 0625 E r 0 05000 0 05375 0 06500 0 08375 0 11000 0 14375 The bold line in the following graph labeled Q4 for Question 4 depicts the indifference curve E r U Q4 A 4 5 U Q3 A 3 U Q5 A 0 4 U Q6 A 0 4 Repeating the analysis in Problem 3 utility is now U E r 0 5A E r 2 0 0 04 The 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 0 00 0 05 0 10 0 15 0 20 0 25 2 0 0000 0 0025 0 0100 0 0225 0 0400 0 0625 E r 0 0400 0 0450 0 0600 0 0850 0 1200 0 1650 The 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 5 The portfolio expected return and variance are computed as follows rPortfolio 1 WBills 2 rBills 3 WIndex 4 rIndex 1 2 3 4 0 0 0 2 0 4 0 6 0 8 1 0 5 5 5 5 5 5 1 0 0 8 0 6 0 4 0 2 0 0 13 5 13 5 13 5 13 5 13 5 13 5 13 5 11 8 0 135 0 118 10 1 0 101 8 4 0 084 6 7 0 067 5 0 0 050 Portfolio 3 20 2 Portfolio 20 0 20 16 0 16 12 0 12 8 0 08 4 0 04 0 0 00 0 0400 0 0256 0 0144 0 0064 0 0016 0 0000 6 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 WBills WIndex 0 0 0 2 0 4 0 6 0 8 1 0 1 0 0 8 0 6 0 4 0 2 0 0 rPortfolio Portfolio 2Portfolio U A 3 U A 5 0 135 0 118 0 101 0 20 0 16 0 12 0 084 0 067 0 050 0 08 0 04 0 00 0 0400 0 0256 0 0144 0 0064 0 0016 0 0000 0 0750 0 0796 0 0794 0 0744 0 0646 0 0500 0 0350 0 0540 0 0650 0 0680 0 0630 0 0500 The 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 Client …