\documentclass{article}
\input{15401-preamble}
\psetnum{7}
\date{2011/12/07}

\begin{document}
\begin{pset}
  \begin{problem}
    
    \begin{subproblem}
      \textbf{True.} A negative beta means that the investment's risk
      is not just uncorrelated but \emph{anti}correlated with the
      market. Holding this investment would reduce the risk of a
      market portfolio even more than a risk-free asset, so it will
      have a market price that yields a lower return than the
      risk-free rate.
    \end{subproblem}

    \begin{subproblem}
      \textbf{False.} The expected \emph{excess} return on the
      investment (over a risk-free asset) is twice as high as the
      expected excess return of the market portfolio.
    \end{subproblem}
    
    \begin{subproblem}
      \textbf{False.} Investors demand higher expected rates of return
      on stocks with a higher beta. It's possible for a stock to have
      more variable returns, but lower beta because its returns are
      uncorrelated with the market (as in question 2).
    \end{subproblem}

    \begin{subproblem}
      \textbf{False.} The CAPM predicts that a security with a beta of
      zero will offer the same expected return as the risk-free rate.
    \end{subproblem}

    \begin{subproblem}
      \textbf{False.} Assuming T-bills are risk-free, they have a beta
      of zero, and the market portfolio has beta 1, so the investor's
      beta will be $\nicefrac23$.
    \end{subproblem}
    
  \end{problem}    

  \begin{problem}

    \begin{subproblem}
      \[ \beta = \frac{\Cov{R_A, R_M}}{\Var{R_M}} = \frac{\phi_{AM}
        \sigma_A \sigma_M}{\sigma_M^2} = \frac{\phi_{AM}
        \sigma_A}{\sigma_M} = \frac{0.8 \cdot 0.3}{0.25} = 0.96\]
    \end{subproblem}

    \begin{subproblem}
      \begin{align*}
        \beta &= \frac{\phi_{BM} \sigma_B}{\sigma_M} \\
        \phi_{BM} &= \beta \frac{\sigma_M}{\sigma_B} \\
        &= 0.8 \frac{0.25}{0.4} = 0.5
      \end{align*}
    \end{subproblem}

    \begin{subproblem}
      \begin{align*}
        \Exp{R_A} &= \beta_A (\Exp{R_M} - R_F) + R_F &= 0.96\cdot 7\% + 2\%
        &= 7.76\%\\
        \Exp{R_B} &= \beta_B (\Exp{R_M} - R_F) + R_F &= 0.8\cdot 6\% + 2\%
        &= 6.8\%
      \end{align*}
    \end{subproblem}

    \begin{subproblem}
      Stock A has a lower standard deviation but a higher beta
      compared to B. That is, stock B's price is subject to larger
      fluctuations but they are less correlated with the market
      price. Thus, B's stock, taken individually, is riskier, but more
      of its risk is diversifiable.
    \end{subproblem}
  \end{problem}

  \begin{problem}
    
    \begin{subproblem}
      The expected excess return is 12\% - 4\% = 8\%.
    \end{subproblem}

    \begin{subproblem}
      We can't say anything about the beta based on the realized
      return, as the CAPM is about expected return --- except inasmuch
      as realized return can be a predictor for expected return.
    \end{subproblem}

    \begin{subproblem}
      The stock has an expected excess return (over the risk-free
      rate) of 10\%, so by the CAPM its beta must be $\nicefrac{10\%}{8\%}
      = 1.25$.
    \end{subproblem}
  \end{problem}

  \begin{problem}
    First, note that the total portfolio value is \$1,388,000, so the
    portfolio weights of the three stocks are
    \begin{align*}
      \omega_A &= \frac{10000\cdot\$25}{\$1,388,000} &\approx 0.1801\\
      \omega_B &= \frac{15000\cdot\$38}{\$1,388,000} &\approx 0.4107\\
      \omega_C &= \frac{8000\cdot\$71}{\$1,388,000} &\approx 0.4092
    \end{align*}

    Now, let's compute the beta of the individual stocks
    \begin{align*}
      \beta_A &= \frac{\phi_{AM}\sigma_A}{\sigma_M} &= \frac{0.83
        \cdot 0.65}{0.21} &\approx 2.569\\
      \beta_B &= \frac{\phi_{BM}\sigma_B}{\sigma_M} &= \frac{0.55
        \cdot 0.41}{0.21} &\approx 1.269\\
      \beta_C &= \frac{\phi_{CM}\sigma_C}{\sigma_M} &= \frac{0.47
        \cdot 0.55}{0.21} &\approx 1.231
    \end{align*}

    The beta of the portfolio, therefore, is
    \begin{align*}
      \beta_P &= \omega_A \beta_A + \omega_B \beta_C + \omega_C \beta_C
      \\
      &\approx 0.1801 \cdot 2.569 + 0.4107 \cdot 1.269 + 0.4092 \cdot
      1.231
      &\approx 1.488
    \end{align*}

    So by the CAPM its expected return is
    \[\Exp{R_P} = \beta_P (11\% - 6\%) + 6\% \approx 13.44\%\]
  \end{problem}
  
\end{pset}
\end{document}
