Lab for Lecture 7: Tangency Portfolios
Finding the optimal risky portfolio with multiple asset classes
1 Constructing the Optimal Risky Portfolio
1.1 Data
Use the asset_class_returns.xlsx dataset to obtain annual returns on four asset classes: S&P 500 (sp500), 10-year Treasury bonds (tbond10), gold (gold), and real estate (real_estate). You will also need the T-bill rate (tbill) to calculate excess returns.
| Column Name | Data |
|---|---|
| sp500 | Annual returns on S&P 500 (includes dividends) |
| tbond10 | Annual returns on US T. Bonds (10-year) |
| gold | Annual percentage change in gold prices |
| real_estate | Average annual price appreciation in residential real estate |
| tbill | Average 3-month T.Bill rate per year |
1.2 Analysis
Using this data:
- Calculate the mean return, standard deviation, and Sharpe ratio for each of the four risky asset classes.
- Calculate the variance-covariance matrix and correlation matrix for the four asset classes.
- Find the optimal risky portfolio (tangency portfolio) using the formula in the lecture notes.
1.3 Questions
- Which asset classes receive the largest weights in the tangency portfolio? Does this surprise you given their individual Sharpe ratios?
- Gold and real estate often have lower Sharpe ratios than stocks. Why might they still receive meaningful weights in the optimal portfolio?
- Looking at the correlation matrix, which pairs of assets have the lowest correlations? How do these correlations relate to the weights in the tangency portfolio?
- The optimal weights are based on historical data. What are the risks of using historical estimates to construct portfolios for the future?
- If one asset’s estimated mean return increased by 2%, how do you think that would affect the optimal weights? Would the change be proportional?
- Why do we call this the “tangency” portfolio? What is it tangent to?
- In practice, many investors face constraints (e.g., no short selling). How might such constraints affect the composition and Sharpe ratio of the optimal portfolio?