\documentclass{article}
\input{15401-preamble}
\psetnum{6}
\date{2011/11/21}

\begin{document}
\begin{pset}
  \begin{problem}
    
    \begin{subproblem}
    \textbf{False}. Systematic risk inherent to all of the assets ---
    for example, due to changing economic conditions --- can't be
    eliminated by diversification.
    \end{subproblem}

    \begin{subproblem}
      \textbf{True}. If the stock with high standard deviation has a
      low or negative correlation with the other assets in the
      portfolio, it can contribute less to portfolio risk.
    \end{subproblem}
    
    \begin{subproblem}
      \textbf{False}. In a market with many assets, it might be the
      case that no individual asset lies on the efficient frontier --
      meaning that diversification can produce the same returns with
      lower risk.
    \end{subproblem}

    \begin{subproblem}
      \textbf{False}. If the return of the dominated asset has a low
      correlation with the other asset, their combination may have
      even lower risk.
    \end{subproblem}
  \end{problem}

  \begin{problem}

    \begin{subproblem}
      Yes, because the assets have a low correlation in returns, a
      portfolio containing some XYZ could have lower risk than one
      containing only ABC. For example, the 60\% ABC / 40\% XYZ
      portfolio discussed below has a lower standard deviation than
      either asset alone. 
    \end{subproblem}

    \begin{subproblem}
      If the returns are perfectly correlated, there is no longer any
      reason to invest in XYZ.
    \end{subproblem}

    \begin{subproblem}
      The expected return of the portfolio is
      \[ \Exp{R_p} = \omega_1 R_1 + \omega_2 R_2 = 0.6 \cdot 0.2 + 0.4
      \cdot 0.15 = 18\% \]

      The standard deviation is given by
      \begin{align*}
        \sigma_p^2 &= \omega_1^2 \sigma_1^2 + \omega_2^2 \sigma_2^2 + 2
        \omega_1 \omega_2 \sigma_1 \sigma_2 \rho_{1,2}\\
        &= 0.6^2 \cdot 0.2^2 + 0.4^2 \cdot 0.25^2 + 2 \cdot 0.2 \cdot
        0.4 \cdot 0.2 \cdot 0.25 \cdot 0.2\\
        &= 0.0292 \\
        \sigma_p &\approx 17.1\%
      \end{align*}
    \end{subproblem}

    \begin{subproblem}
      \begin{align*}
        \Exp{R_p} &= 19.5\% \\
         \omega_1 \cdot 0.2 + (1-\omega_1)
         \cdot 0.15 &= 0.195 \\
         \omega_1 &= 0.9
       \end{align*}

       The portfolio should have weights of 90\% ABC and 10\% XYZ. Its
       standard deviation is
       \begin{align*}
         \sigma_p^2 &= \omega_1^2 \sigma_1^2 + \omega_2^2 \sigma_2^2 + 2
         \omega_1 \omega_2 \sigma_1 \sigma_2 \rho_{1,2}\\
         &= 0.9^2 \cdot 0.2^2 + 0.1^2 \cdot 0.25^2 + 2 \cdot 0.2 \cdot
         0.4 \cdot 0.2 \cdot 0.25 \cdot 0.2\\
         &= 0.034825 \\
         \sigma_p &\approx 18.7\%
      \end{align*}
  \end{subproblem}

  \begin{subproblem}
    \begin{align*}
         0 &= \omega_1^2 \sigma_1^2 + \omega_2^2 \sigma_2^2 + 2
         \omega_1 \omega_2 \sigma_1 \sigma_2 \rho_{1,2}\\
         &= \omega_1^2 0.2^2 + (1-\omega_1)^2 0.25^2 - 2
         \omega_1 (1-\omega_1) 0.2 0.25 \\
         \omega_1 \approx 0.555
    \end{align*}

    The portfolio with 55.5\% ABC and 44.5\% XYZ would have zero
    standard deviation.
  \end{subproblem}
  \end{problem}

  \begin{problem}

    \begin{subproblem}
      The Sharpe ratio is maximized at the weights of 77\% ABC and
      23\% XYZ, so this is the tangency portfolio. This portfolio has
      expected return 18.85\% and standard deviation 17.5\%.
    \end{subproblem}

    \begin{subproblem}
      \begin{align*}
        19.5\% &= \omega R_t + (1-\omega) R_f \\
        &= \omega 0.1885 + (1-\omega) 0.05\\
        \omega &\approx 1.047
      \end{align*}

      We'd choose a portfolio weight of 1.047 for the tangency
      portfolio and -0.047 for the risk-free asset (i.e. we're selling
      the risk-free asset short to leverage our purchase of the
      tangency portfolio).

      Broken down, the portfolio has a weight of 0.806 for ABC, 0.241
      for XYZ, and -0.047 for the risk-free asset.

      This portfolio has a standard deviation of $17.5\% \cdot 1.047 =
      18.32\%$.
    \end{subproblem}

    \begin{subproblem}
      This portfolio is strictly preferable to the 90\% ABC / 10\% XYZ
      portfolio above. It offers the same return (19.5\%), with a
      lower standard deviation.

      The difference in the two portfolios is that this one has the
      same proportions as the tangency portfolio, but uses leverage
      with the risk-free asset to increase the proportion of risk-free
      assets and get a higher return. The previous portfolio has to
      increase the weight of ABC to get a higher return, which means
      it loses out on some of the benefit of diversification to reduce
      risk.
    \end{subproblem}

    
  \end{problem}

  \begin{problem}
    
    \begin{subproblem}
      The monthly returns are attached. The sample mean and standard
      deviations are:

      \begin{center}
      \begin{tabular}{c|c|c|c|}
        & \bf AAPL & \bf BRK/A & \bf BHP \\
        \hline
        \bf mean	& 0.010489364	& 0.000545504	& 0.031195384 \\
        \bf stddev	& 0.107696294	& 0.060310459	&  0.111027939 \\
      \end{tabular}
    \end{center}
    \-\\

      The correlation matrix is

      \begin{center}
      \begin{tabular}{c|c|c|c}
        	& \bf AAPL	& \bf BRK/A	& \bf BHP\\
                \hline
\bf AAPL	& 1	& 0.182381136	& 0.50051187\\
\bf BRK/A	& 0.182381136	& 1	& 0.210546788\\
\bf BHP	& 0.50051187	& 0.210546788	& 1\\
\end{tabular}
\end{center}

\-\\
      
      so the covariance matrix is
      
      \begin{center}      
        \begin{tabular}{c|c|c|c}
        	& \bf AAPL	& \bf BRK/A	& \bf BHP\\
                \hline
\bf AAPL	& 0.011598492	& 0.001184604	& 0.005984769\\
\hline
\bf BRK/A	& 0.001184604	& 0.003637351	& 0.001409852\\
\hline
\bf BHP	& 0.005984769	& 0.001409852	& 0.012327203\\
      \end{tabular}
    \end{center}
  \end{subproblem}

    \begin{subproblem}

      \begin{center}
      \begin{tabular}{|c|c|c|c|c|}
        \hline
        \bf mean	& \bf stddev	& \bf weight AAPL	& \bf weight BRK/A	&
        \bf weight BHP\\
        \hline
4.50\%	& 15.59\%	& -23.50\%	& -29.16\%	& 152.66\%\\
4.00\%	& 13.91\%	& -18.85\%	& -15.99\%	& 134.84\%\\
3.50\%	& 12.27\%	& -14.20\%	& -2.82\%	& 117.02\%\\
3.00\%	& 10.68\%	& -9.55\%	& 10.35\%	& 99.20\%\\
2.50\%	& 9.18\%	& -4.90\%	& 23.52\%	& 81.38\%\\
2.00\%	& 7.81\%	& -0.24\%	& 36.69\%	& 63.55\%\\
1.50\%	& 6.66\%	& 4.41\%	& 49.87\%	& 45.73\%\\
1.00\%	& 5.86\%	& 9.06\%	& 63.04\%	& 27.90\%\\
0.50\%	& 5.55\%	& 13.71\%	& 76.20\%	& 10.09\%\\
0.00\%	& 5.83\%	& 18.36\%	& 89.38\%	& -7.74\%\\
\hline
      \end{tabular}
    \end{center}
  \end{subproblem}

    \begin{subproblem}
      The efficient frontier is plotted below. BHP lies nearly on the
      efficient frontier by itself, with a mean of 3.1\% and standard
      deviation of 11.1\%. Compare the 3\% mean data point on the
      efficient frontier, which adds a small short position in AAPL and a
      long position in BRK/A to reduce diversifiable risk.

      Note in particular that the (0.5\%, 5.55\%) point has both a
      higher return and lower risk than BRK/A. This is possible
      because the other two assets, despite having higher standard
      deviations, have a low correlation with BRK/A.
      
      \includegraphics[scale=0.7]{ps6-4-efficientfrontier}
    \end{subproblem}

    \begin{subproblem}
      The tangent portfolio uses short positions in AAPL and BRK/A and
      a long position in BHP, with portfolio weights are
      $-42.87\%$ AAPL, $-83.99\%$ BRK/A, and 226.86\% BHP. This gives
      a higher mean (6.58\%) and standard deviation (22.8\%) than we
      have seen previously. Not surprisingly, combinations of the
      risk-free asset and this tangent portfolio dominate the points
      we saw earlier on the efficient frontier.
      
      \includegraphics[scale=0.7]{ps6-4-cml}
    \end{subproblem}
  \end{problem}

  
\end{pset}
\end{document}