PROBLEM 1:
1. Question #1: Capital Allocation:
Consider the following capital market: a risk-free asset yielding 1.00% per year and a mutual fund consisting of 65% stocks and 35% bonds. The expected return on stocks is 11.75% per year and the expected return on bonds is 4.25% per year. The standard deviation of stock returns is 36.00% and the standard deviation of bond returns 11.50%. The stock, bond and risk-free returns are all uncorrelated.
1. What is the expected return on the mutual fund?
2. What is the standard deviation of returns for the mutual fund?
Now, assume the correlation between stock and bond returns is 0.40 and the correlations between stock and risk-free returns and between the bond and risk-free returns are 0 (by construction, correlations with the risk-free asset are always zero).
3. What is the standard deviation of returns for the mutual fund? Is it higher or lower than the standard deviation found in part 2? Why?
Now, assume that the standard deviation of the mutual fund portfolio is exactly 26.50% per year and a potential customer has a risk-aversion coefficient of 2.50.
4. What correlation between the stock and bond returns is consistent with this portfolio standard deviation?
5. What is the optimal allocation to the risky mutual fund (the fund with exactly 26.50% standard deviation) for this investor?
6. What is the expected return on the complete portfolio?
7. What is the standard deviation of the complete portfolio?
8. What is the Sharpe ratio of the complete portfolio?
Question 2
1. Question #2: Markowitz Optimization:
Open the associated Excel file named QPS2 Data Fall I 2015 Problem 2 in My Course Content: Problem Set Spreadsheets. The data file includes 60 months of returns for 11 exchange traded funds; their names and ticker symbols follow:
Ticker Name of Exchange Traded Fund
1 SPY SPDR S&P 500 ETF
2 MDY SPDR S&P MidCap 400 ETF
3 IWM iShares Russell 2000 ETF
4 QQQ Power Shares QQQ ETF
5 EFA iShares MSCI EAFE ETF
6 VWO Vanguard FTSE Emerging Markets Stock Index ETF
7 VNQ Vanguard REIT Index ETF
8 BND Vanguard Total Bond Market ETF
9 PFF iShares US Preferred Stock ETF
10 GLD SPDR Gold Shares ETF
11 JNK SPDR Barclays High Yield Bond ETF
All students will do problem 2 using 9 of the above ETFs; all students will include the first 8 ETFs listed above: SPY, MDY, IWM, QQQ, EFA, VWO, VNQ, and BND. All students will include one of the last 3 ETFs: PFF, GLD, and JNK as instructed on your version.
Version C: Include JNK and exclude PFF and GLD.
Use the data on your 9 ETFs to answer the following questions:
1. What is the average return for each of the nine indexes?
2. Show the covariance matrix of returns. Briefly describe how you constructed the covariance matrix.
Consider the simple case where short sales are allowed, but short positions must be greater than or equal to -50% and long positions must be less than or equal to 50%. Use Excel Solver to find the Minimum Variance Portfolio (MVP).
3. What is the expected portfolio return for the MVP portfolio?
4. What is the portfolio standard deviation for the MVP portfolio?
5. What is the portfolio composition (i.e., what are the weights for the nine ETFs)?
Consider the simple case where short sales are allowed, but short positions must be greater than or equal to -30% and long positions must be less than or equal to 30%. Use Excel Solver to find the Maximum return portfolio with a standard deviation of exactly 4.50%.
6. What is the expected portfolio return for this portfolio?
7. What is the portfolio composition (i.e., what are the weights for the nine ETFs)?
Consider the more realistic case where short sales are NOT allowed and no more than 30% of the portfolio and no less than 4% is invested in any ETF. Use Excel Solver to find the Minimum Variance Portfolio (MVP).
8. What is the expected portfolio return for the MVP portfolio?
9. What is the portfolio standard deviation for the MVP portfolio?
10. What is the portfolio composition (i.e., what are the weights for the nine ETFs)?
Consider the simple case where short sales are NOT allowed and no more than 25% and no less than 4% of the portfolio is invested in any ETF. Use Excel Solver to find the Market Portfolio if the risk-free rate is 0.15%/month (1.80%/year).
11. What is the expected portfolio return for this portfolio?
12. What is the portfolio standard deviation for this portfolio?
13. What is the portfolio composition (i.e., what are the weights for the nine ETFs)?
14. What is the maximum Sharpe ratio?
PROBLEM 2
Question #1: Capital Allocation:
Consider the following capital market: a risk-free asset yielding 1.00% per year and a mutual fund consisting of 65% stocks and 35% bonds. The expected return on stocks is 11.75% per year and the expected return on bonds is 4.25% per year. The standard deviation of stock returns is 36.00% and the standard deviation of bond returns 11.50%. The stock, bond and risk-free returns are all uncorrelated.
Risk Free Rate = 1%
Mutual Fund:
Weights Return Std. Dev.
Stock 65.00% 11.75% 36.00%
Bonds 35.00% 4.25% 11.50%
1 Expected Return = Weighted Average of returns
9.13%
2 Standard Deviation of Mutual Fund =
23.74%
"Now, assume the correlation between stock and bond returns is 0.40 and the correlations between stock and risk-free returns and between the bond and risk-free returns are 0 (by construction, correlations with the risk-free asset are always zero). 3. What is the standard deviation of returns for the mutual fund? Is it higher or lower than the standard deviation found in part 2? Why?
3 Correlation = 0.4
Standard Deviation
25.28%
"Now, assume that the standard deviation of the mutual fund portfolio is exactly 26.50% per year and a potential customer has a risk-aversion coefficient of 2.50.
4. What correlation between the stock and bond returns is consistent with this portfolio standard deviation?
5. What is the optimal allocation to the risky mutual fund (the fund with exactly 26.50% standard deviation) for this investor?
6. What is the expected return on the complete portfolio?
7. What is the standard deviation of the complete portfolio?
8. What is the Sharpe ratio of the complete portfolio?
Standard Deviation 26.50%
Correlation 0.7352
Portfolio
Weights Return Std. Dev.
Mutual Fund 1 0.09125 0.265
Risk Free 0 0.01 0
Total 1
4 Standard Deviation 26.50%
Correlation 0.7352
5 Portfolio Return = 9.13%
7 Standard Deviation 26.50%
8 Sharp Ratio = 0.3066
6 Expected Return on Complete Portfolio = 9.13%
Expected Standard Deviation of the Portfolio = 26.50%
Sharp Ratio of Portfolio = 0.3066
PROBLEM 3
1. What is the average return for each of the nine indexes?
Index SPY MDY IWM QQQ EFA VWO VNQ BND JNK
Avg. Return 1.28% 1.31% 1.33% 1.64% 0.67% 0.10% 1.07% 0.22% 0.52%
2. Show the covariance matrix of returns. Briefly describe how you constructed the covariance matrix.