Lab for Lecture 6: Optimal Capital Allocation

The Capital Allocation Line, risk aversion, and optimal portfolio choice

1 Allocating Capital Between a Risky Asset and the Risk-Free Asset

1.1 Data

Use the asset_class_returns.xlsx dataset to obtain annual returns on the S&P 500 (sp500) and the T-bill rate (tbill).

Column Name Data
sp500 Annual returns on S&P 500 (includes dividends)
tbill Average 3-month T.Bill rate per year

1.2 Analysis

Using this data:

  • Calculate the mean return and standard deviation of the S&P 500, and the average T-bill rate over the sample period.
  • Construct the Capital Allocation Line (CAL):
    • Create 11 portfolios combining the market (S&P 500) and T-bills, with market weights ranging from 0% to 100% in increments of 10%.
    • Calculate the expected return and standard deviation for each portfolio.
    • Plot the CAL with standard deviation on the x-axis and expected return on the y-axis.
  • Verify that the slope of the CAL equals the Sharpe ratio of the market portfolio.
  • Solve for the weight in the market portfolio if you want to achieve:
    • A portfolio standard deviation of 10%
    • A portfolio expected return of 8%
  • Calculate the optimal capital allocation:
    • Calculate the optimal weight in the market portfolio for investors with risk aversion coefficients \(A\) = 1, 2, 3, 4, 5, and 6.
    • For each value of \(A\), calculate the resulting portfolio’s expected return and standard deviation.

1.3 Questions

  • Why is the CAL a straight line, while the investment opportunity set from Lecture 5 was curved?
  • What does the slope of the CAL represent economically?
  • If you wanted a portfolio with a standard deviation of 25% (higher than the market’s), what would you need to do? What weight would you need in the market?
  • How does the optimal allocation to the market change as risk aversion increases? Does this relationship make intuitive sense?
  • An investor with \(A = 2\) ends up with a very different portfolio than one with \(A = 5\). What is the difference in their allocations, and what does this imply about how risk aversion affects wealth accumulation over time?
  • Think about your own risk tolerance. What level of risk aversion coefficient do you think best describes you? What would your optimal allocation be?
  • Why might an investor’s risk aversion coefficient change over their lifetime? How would this affect their optimal capital allocation?
  • The optimal allocation formula assumes investors have quadratic utility. What are the limitations of this approach?