Lab for Lecture 8: Optimal Asset Allocation
The complete portfolio and constrained optimization
1 Finding the Optimal Complete Portfolio
1.1 Data
Use the asset_class_returns.xlsx dataset with the same four asset classes as in Lecture 7: S&P 500 (sp500), 10-year Treasury bonds (tbond10), gold (gold), and real estate (real_estate), plus the T-bill rate (tbill).
| 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
Building on the tangency portfolio from Lecture 7:
- Calculate the expected return, standard deviation, and Sharpe ratio of the tangency portfolio.
- Compare the Sharpe ratio of the tangency portfolio to the Sharpe ratios of the individual assets.
- Calculate the optimal complete portfolio for different levels of risk aversion:
- Using the optimal capital allocation formula, calculate the weight in the tangency portfolio (versus T-bills) for investors with risk aversion coefficients \(A\) = 1, 2, 3, 4, 5, and 6.
- For each value of \(A\), calculate the final weight in each of the four risky assets and in T-bills.
- Report the expected return and standard deviation of each complete portfolio.
- Solve the problem with constrained optimization (using Solver in Excel):
- Re-solve for the tangency portfolio weights with the constraint that all weights must be non-negative (no short selling allowed) and no asset in the portfolio can have a weight larger than 30%.
- Report the new optimal weights and compare the Sharpe ratio of this constrained tangency portfolio to the unconstrained version.
- Calculate the optimal complete portfolio for \(A\) = 4 using the constrained tangency portfolio.
1.3 Questions
- How does the Sharpe ratio of the tangency portfolio compare to the best individual asset’s Sharpe ratio? What does this tell you about the value of diversification?
- How does the allocation to T-bills change as risk aversion increases? At what level of risk aversion does an investor hold more than 50% in T-bills?
- Compare the final asset weights for an investor with \(A = 2\) versus \(A = 5\). How do the portfolios differ in terms of their composition and risk-return characteristics?
- How much does the Sharpe ratio decline when you impose the optimization constraints?
- The optimal allocation depends on your risk aversion coefficient. How confident are you in your own estimate of \(A\)? How sensitive are the results to small changes in this parameter?
- If you were advising a client who is 30 years from retirement versus one who is 5 years from retirement, how might you adjust the risk aversion coefficient, and what would be the impact on their optimal allocation?
- What are some practical considerations (beyond what we’ve modeled) that might lead you to deviate from the mathematically optimal allocation?