Problem Set 6 Answer Key ECN 134 Financial Economics Part A Risk Aversion Prof Farshid Mojaver Consider a portfolio that offers an expected rate of return of 12 and a standard deviation of 18 T bills offer a risk free 7 rate of return What is the maximum level of risk aversion for which the risky portfolio is still preferred to bills Answer 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 1 When an investment advisor attempts to determine an investor s risk tolerance which factor would they be least likely to assess A the investor s prior investing experience B the investor s degree of financial security C the investor s tendency to make risky or conservative choices D the level of return the investor prefers E the investor s feeling about loss Assume an investor with the following utility function U E r 3 2 s2 2 To maximize her expected utility she would choose the asset with an expected rate of return of and a standard deviation of respectively A 12 20 B 10 15 C 10 10 D 8 10 E none of the above U 0 10 3 2 0 10 2 8 5 highest utility of choices U E r A 2 s2 where A 4 0 3 Based on the utility function above which investment would you select A 1 B 2 C 3 D 4 E cannot tell from the information given U c 0 21 4 2 0 16 2 15 88 highest utility of choices 4 Which investment would you select if you were risk neutral A 1 B 2 C 3 D 4 E cannot tell from the information given If you are risk neutral your only concern is with return not risk 5 An investor invests 30 percent of his wealth in a risky asset with an expected rate of return of 0 13 and a variance of 0 03 and 70 percent in a T bill that pays 6 percent His portfolio s expected return and standard deviation are and respectively A 0 114 0 128 B 0 087 0 063 C 0 295 0 125 D 0 081 0 052 E none of the above 1 2 E rP 0 3 13 0 7 6 8 1 sP 0 3 0 03 5 19 6 What percentages of your money must be invested in the risky asset and the risk free asset respectively to form a portfolio with an expected return of 0 09 A 85 and 15 B 75 and 25 C 67 and 33 D 57 and 43 E cannot be determined 9 w1 12 1 w1 5 9 12 w1 5 5 w1 4 7 w1 w1 0 57 1 w1 0 43 0 57 12 0 43 5 8 99 Your client Bo Regard holds a complete portfolio that consists of a portfolio of risky assets P and T Bills The information below refers to these assets 7 What is the equation of Bo s Capital Allocation Line A E rC 7 2 3 6 Standard Deviation of C B E rC 3 6 1 167 Standard Deviation of C C E rC 3 6 12 0 Standard Deviation of C D E rC 0 2 1 167 Standard Deviation of C E E rC 3 6 0 857 Standard Deviation of C The intercept is the risk free rate 3 60 and the slope is 12 00 3 60 7 20 1 167 8 What are the proportions of Stocks A B and C respectively in Bo s complete portfolio A 40 25 35 B 8 5 7 C 32 20 28 D 16 10 14 E 20 12 5 17 5 Proportion in A 8 40 32 proportion in B 8 25 20 proportion in C 8 35 28 Part B Optimal Risky Portfolio 1 a Even though it seems that gold is dominated by stocks gold might still be an attractive asset to hold as a part of a portfolio If the correlation between gold and stocks is sufficiently low gold will be held as a component in a portfolio specifically the optimal tangency portfolio b If the correlation between gold and stocks equals 1 then no one would hold gold The optimal CAL would be comprised of bills and stocks only Since the set of risk return combinations of stocks and gold would plot as a straight line with a negative slope see the following graph these combinations would be dominated by the stock portfolio Of course this situation could not persist If no one desired gold its price would fall and its expected rate of return would increase until it became sufficiently attractive to include in a portfolio 2 5 2 0 1 8 S t o c k s 1 5 1 0 G o l d 5 r f 0 0 0 0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 S t a n d a r d D e v i a t i o n 2 The probability distribution is Probability 0 7 0 3 Rate of Return 100 50 Mean 0 7 100 0 3 50 55 Variance 0 7 100 55 2 0 3 50 55 2 4725 Standard deviation 47251 2 68 74 3 P 30 y y y 0 75 E rP 12 0 75 30 12 25 5 4 Since we do not have any information about expected returns we focus exclusively on reducing variability Stocks A and C have equal standard deviations but the correlation of Stock B with Stock C 0 10 is less than that of Stock A with Stock B 0 90 Therefore a portfolio comprised of Stocks B and C will have lower total risk than a portfolio comprised of Stocks A and B 5 Rearranging the table converting rows to columns and computing serial correlation results in the following table Nominal Rates Small company stocks 1920s 3 72 Large company stocks 18 36 Long term government bonds 3 98 Intermed term government bonds 3 77 Treasury bills Inflation 3 56 1 00 1 25 4 60 3 91 0 30 2 04 20 63 9 11 3 59 1 70 0 37 5 36 19 01 19 41 0 25 1 11 1 87 2 22 1960s 13 72 7 84 1 14 3 41 3 89 2 52 1970s 8 75 5 90 6 63 6 11 6 29 7 36 1980s 12 46 17 60 11 50 12 01 9 00 5 10 1990s 13 84 18 20 8 60 7 74 5 02 2 93 Serial Correlation 0 46 0 22 0 60 0 59 0 63 0 23 1930s 7 28 1940s 1950s For example to compute serial …