Lab for Lecture 5: Portfolio Risk and Return
Portfolio math, diversification, and the investment opportunity set
1 Combining Stocks and Bonds in a Portfolio
1.1 Data
Use the asset_class_returns.xlsx dataset to obtain annual returns on the S&P 500 (sp500) and 10-year Treasury bonds (tbond10). You will also need the T-bill rate (tbill) to calculate Sharpe ratios.
| Column Name | Data |
|---|---|
| sp500 | Annual returns on S&P 500 (includes dividends) |
| tbond10 | Annual returns on US T. Bonds (10-year) |
| tbill | Average 3-month T.Bill rate per year |
1.2 Analysis
Using this data:
- Calculate the mean return, standard deviation, variance, and Sharpe ratio for each asset class (stocks and bonds) individually.
- Calculate the covariance and correlation between stock and bond returns.
- Construct a 60-40 portfolio (60% stocks, 40% bonds):
- Estimate the portfolio’s expected return and standard deviation in two ways:
- Using statistics of the portfolio returns in the past
- Using statistics of the assets in the portfolio
- Using statistics of the portfolio returns in the past
- Calculate the Sharpe ratio of this portfolio and compare it with the Sharpe ratios of the individual assets.
- Estimate the portfolio’s expected return and standard deviation in two ways:
- Construct the investment opportunity set:
- Create 11 portfolios with stock weights ranging from 0% to 100% in increments of 10%.
- Calculate the expected return and standard deviation for each portfolio.
- Plot this investment opportunity set with standard deviation on the x-axis and expected return on the y-axis.
- Explore the impact of correlation on diversification:
- Recalculate the investment opportunity set assuming the correlation between stocks and bonds is 0, -1, and +1 (keeping the individual means and standard deviations the same).
- Plot all four investment opportunity sets on the same graph to compare.
1.3 Questions
- How does the Sharpe ratio of the 60-40 portfolio compare to the Sharpe ratios of stocks and bonds individually? What does this tell you about diversification?
- Looking at the investment opportunity set, is there a portfolio that dominates others (higher return for the same risk, or lower risk for the same return)?
- What is the minimum variance portfolio (approximately)? Why might an extremely risk-averse investor not want to hold 100% bonds?
- How does the shape of the investment opportunity set change as you vary the assumed correlation? What happens when correlation is -1?
- The historical correlation between stocks and bonds has not been constant over time. If you expected future correlation to be higher than historical correlation, how would that affect your portfolio allocation?