11.9 Problems: Introduction to Portfolio Theory

Exercise 4.4 Annual estimates of the GWN model parameters for Boeing and Microsoft are given in the table below:
Boeing Microsoft
\(\mu\) 0.149 0.331
\(\sigma^{2}\) 0.069 0.136
\(\rho\) -0.008 -0.008
  1. Create the following portfolios.
    • Combinations of Boeing and Microsoft (with \(x_{\text{{boeing}}}=-1,-0.9,...,2\) and \(x_{\text{{msft}}}=1-x_{\text{{boeing}}}\))
    • Combinations of Boeing and T-bills (with \(x_{\text{{boeing}}}=0,0.1,...,2\))
    • Combinations of Microsoft and T-bills (with \(x_{\text{{msft}}}=0,0.1,...,2\))
    • Use an annual risk-free rate of \(3\%\) per year for the T-bill. For each portfolio, compute \(E(R_{p})\), var\((R_{p})\) and SD\((R_{p})\) using the appropriate formulas. For each portfolio, plot \(E(R_{p})\) vs. SD\((R_{p})\) and put these values on the same graph. Compute the Sharpe’s slope for Boeing and Microsoft. Which asset has the highest slope value?
  2. Compute the global minimum variance portfolio using the analytical formula (11.10).
    • Make a bar chart showing the weights of Boeing and Microsoft in global minimum variance portfolio.
    • Compute \(E(R_{p})\), var\((R_{p})\) and SD\((R_{p})\) for the tangency portfolio.
    • Compute Sharpe’s slope for the tangency portfolio.
    • Indicate the location of the tangency portfolio on the graph you created previously in question 1.
  3. Using a risk-free rate of \(3\%\) per year for the T-bill, compute the tangency portfolio using the analytic formula (11.16).
    • Make a bar chart showing the weights of Boeing and Microsoft in the tangency portfolio.
    • Compute \(E(R_{p})\), var\((R_{p})\) and SD\((R_{p})\) for the tangency portfolio.
    • Compute the Sharpe’s slope for the tangency portfolio.
    • Indicate the location of the tangency portfolio on the graph you created previously in question 1.
  4. Consider a portfolio that has \(10\%\) in the tangency portfolio and \(90\%\) in T-bills.
    • In this portfolio, what is the percent invested in Boeing and what is the percent invested in Microsoft? Give a bar chart showing the percents invested in T-bills, Boeing and Microsoft.
    • Compute \(E(R_{p})\), var\((R_{p})\) and SD\((R_{p})\) for this portfolio.
    • Compute Sharpe’s slope for this portfolio.
    • Indicate the location of the tangency portfolio on the graph you created previously in question 1.
  5. Find the efficicient portfolio (combination of T-bills and tangency portfolio) that has the same risk (SD) as Microsoft.
    • In this portfolio, what is the percent invested in Boeing and what is the percent invested in Microsoft? Give a bar chart showing the percent invested in T-bills, Boeing and Microsoft.
    • Compute \(E(R_{p})\), var\((R_{p})\) and SD\((R_{p})\) for this portfolio.
    • Compute Sharpe’s slope for this portfolio.
    • Indicate the location of the tangency portfolio on the graph you created previoulsy in question 1.
Exercise 4.5 (Real world portfolios revisited)

  • Re-do the real-world portfolio analysis using forward looking estimates: For each 5-year period use the previous 5-year period to estimate GWN model parameters and then do asset allocation.

  • compare results to those in the section.