a.) Assume that you manage a risky portfolio with an expected rate of return of
ID: 2383209 • Letter: A
Question
a.) Assume that you manage a risky portfolio with an expected rate of return of 22% and a standard deviation of 35%. The T-bill rate is 6%. Your client chooses to invest 75% of a portfolio in your fund and 25% in a T-bill money market fund.
What is the reward-to-volatility ratio (S) of your risky portfolio and your client’s portfolio?
Your reward to volatility ratio______
Your clients reward to volatility ratio______
b.)Consider historical data showing that the average annual rate of return on the S&P 500 portfolio over the past 85 years has averaged roughly 8% more than the Treasury bill return and that the S&P 500 standard deviation has been about 21% per year. Assume these values are representative of investors' expectations for future performance and that the current T-bill rate is 4%.
Calculate the expected return and variance of portfolios invested in T-bills and the S&P 500 index with weights as follows: (Leave no cells blank - be certain to enter "0" wherever required. Do not enter your answer as a percentage but in a decimal format. Round "Expected Return" to 4 decimal places and the "Variance" to 4 decimal places.)
WBills
WIndex
Expected Return
Variance
0.2
0.8
0.1040
0.0282
0.4
0.6
?
0.0
1.0
?
0.8
0.2
?
1.0
0.0
?
0.6
0.4
?
c.)Assume that you manage a risky portfolio with an expected rate of return of 21% and a standard deviation of 33%. The T-bill rate is 7%. Your client’s degree of risk aversion is A = 2.7, assuming a utility function U = E(r) - ½A².
-What proportion, y, of the total investment should be invested in your fund? (Do not round intermediate calculations. Round your answer to 2 decimal places. Omit the "%" sign in your response.)
-What is the expected value and standard deviation of the rate of return on your client’s optimized portfolio? (Do not round intermediate calculations. Round your answers to 2 decimal places. Omit the "%" sign in your response.)
Calculate the expected return and variance of portfolios invested in T-bills and the S&P 500 index with weights as follows: (Leave no cells blank - be certain to enter "0" wherever required. Do not enter your answer as a percentage but in a decimal format. Round "Expected Return" to 4 decimal places and the "Variance" to 4 decimal places.)
Explanation / Answer
Answer (a)
Expected rate of return on Risky portfolio E(r) = 22%
Standard deviation SD = 35%
T-Bill Rate = 6% (Risk-free rate)
Proportion of Client investment = 75% in risky portfolio + 25% in T-Bill Fund
Expected rate of return on client portfolio = 75% * 22% + 25% * 6%
= 16.5% + 1.5% = 18%
Variance of client's portfolio = (0.75*0.35)^2 25% of portfolio pertain to riskfree T-Bills
= 0.2625^2 = 0.06890625
Standard Deviation of Client's Portfolio = Square Root (0.06890625)
= 0.2625 or 26.25%
Reward to Volatility Ratio of Risky portfolio = (Expected Return - Riskfree return) / SD
= 0.22 - 0.06 / 0.35
= 0.16/0.35 = 0.45714
Reward to Volatility Ratio of Client's Portfolio = 0.18 - 0.06 / 0.2625
= 0.12/0.2625 = 0.45714
Answer (b)
Average Annual rate of return on S&P 500 Index = 8% more than T-Bill rate
T-Bill rate = 4% (Risk-free rate)
Standard Deviation = 21%
Expected return on Index = 8%+4% = 12%
calculations of the figures to fill the table are as follows
W Bills 0.4 and W Index 0.6 Expected Return =0.4*0.04+0.6*0.12 = 0.016+ 0.072 = 0.088
Variance = 0.6*0.21^2 = 0.0159
W Bills 0 W Index 1.0 Expected return = 1*.12 = 0.12 Variance = 0.12^2 = 0.0144
W B 0.8 WI 0.2 Expected return = 0.8*.04+0.2*.12 = 0.032+0.024 = 0.054
Variance = 0.2*0.21^2 = 0.0018
WB 0.6 WI 0.4 Expected return = 0.6*.04+0.4*0.12 = 0.024+0.048 = 0.072
Variance = 0.4*0.21^2 = 0.0071
The answers in tabular form are as follows
W Bills W Index Expected Return Variance
0.2 0.8 0.1040 0.0282 Example
0.4 0.6 0.088 0.0159
0.0 1.0 0.120 0.0144
0.8 0.2 0.054 0.0018
1.0 0.0 0.040 0.0
0.6 0.4 0.072 0.0071
Answer(c)
Expected rate of return of Risky portfolio r = 0.21 (21%)
Standard Deviation of Risky portfolio SD = 0.33 (33%)
T-Bill Rate rt = 0.07 (7%) (Risk-free rate)
Degree of Risk Aversion A = 2.7
Utility Function U = E(r) - (1/2)*(A*SD^2)
Substituting the above values
Utility Function U = 0.21 - (1/2)*(2.7*0.33^2)
= 0.21 - 0.5 * 0.29403
= 0.21 - 0.147
= 0.063
We find that U is less than the risk-free return of 0.07 (7%) of T-Bills
Hence to calculate the proportion the client can invest in the risky portfolio for creation of optimized portfolio
Assuming y is the proportion to be invested in risky portfolio
0.063 = 0.21*y + 0.07 *(y-1)
0.063 - 0.21y + 0.07y - 0.07
0.063+0.07 = 0.21y+0.07y
0.133 = 0.28y
y = 0.133/0.28 = 0.475
Hence 47.5% of the total investment should be invested in the risky portfolio
Based on the proportion of 47.5 in risky portfolio and 52.5 in T-Bills (in percentage terms)
Expected rate of return of optimized portfolio = 0.475*0.21+0.525*0.07
= 0.09975 + 0.03675
= 0.1365
Expected rate of return of optimized portfolio = 13.65%
Standard deviation of optimized portfolio = 0.475 * 0.33 = 0.15675 or 15.675%
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