\documentclass{article}
\input{15401-preamble}
\psetnum{4}
\date{2011/10/31}

\begin{document}
\begin{pset}

  \begin{problem}

    \begin{subproblem}
      The coffee producer will sell three futures contracts: one
      for 150,000 lbs coffee to be delivered at the end of this month,
      and one for each of the following months. The prices of the
      future contracts are (for 150,000 pounds):
      \begin{align*}
        F_{0,1} &= S_0 \left(1+r-\hat{y}\right)^{\nicefrac{1}{12}} &=
        150,000 \cdot \$2.90 (1+0.005-0.04)^{\nicefrac{1}{12}} &\approx
        \$433,710 \\
        F_{0,2} &= S_0 \left(1+r-\hat{y}\right)^{\nicefrac{2}{12}} &=
        150,000 \cdot \$2.90 (1+0.005-0.04)^{\nicefrac{2}{12}} &\approx
        \$432,425 \\
        F_{0,3} &= S_0 \left(1+r-\hat{y}\right)^{\nicefrac{3}{12}} &=
        150,000 \cdot \$2.90 (1+0.005-0.04)^{\nicefrac{3}{12}} &\approx
        \$431,143
      \end{align*}

      The producer will receive revenue of \$433,710, \$433,425, and
      \$431,143 in the next three months respectively.
    \end{subproblem}

    \begin{subproblem}
      Forwards are individualized contracts with a specific
      counterparty, while futures are standardized contracts traded
      through a clearinghouse. This means that a forward contract
      could be set up with any desired asset or maturity, assuming a
      counterparty can be found. However, such contracts could be
      quite illiquid, making it difficult to obtain the necessary
      forward at a reasonable price; this is unlikely to be the case
      with futures.

      Forwards have a significant counterparty risk: the buyer must
      account for the possibility that the seller will default, and
      vice versa. This is unlikely to be the case for futures, as the
      contracts are guaranteed by the clearinghouse. The
      clearinghouse's default risk is low because it requires each
      party to post collateral on a margin account. This can be seen
      as a disadvantage for futures; with forwards, no money is needed
      until the settlement date.
    \end{subproblem}
  \end{problem}

  \begin{problem}
    
    \begin{subproblem}

      \begin{align*}
  %      F_t = S_0 (1+r_t)^{\nicefrac{t}{12}} \\
        \text{1 month:    } \qquad  & 10741.60 = 10740.71
        (1+r_1)^{\nicefrac{1}{12}} & \Rightarrow r_1 = 0.10\% \\
        \text{3 months:    } \qquad & 10749.56 = 10740.71
        (1+r_1)^{\nicefrac{3}{12}} & \Rightarrow r_3 = 0.33\% \\
        \text{6 months:    } \qquad & 10780.91 = 10740.71
        (1+r_1)^{\nicefrac{6}{12}} & \Rightarrow r_6 = 0.75\% \\
        \text{12 months:    } \qquad & 10839.52 = 10740.71
        (1+r_1)^{\nicefrac{12}{12}} & \Rightarrow r_{12} = 0.92\%
      \end{align*}
    \end{subproblem}

    \begin{subproblem}
      To obtain today's price for the three months of copper
      production in Q1 2012, sell 80,000 contracts of each of the Jan-12, Feb-12
      and Nov-11 copper futures. Revenue will be

      \begin{align*}
        \text{Jan-12  }\qquad & 80,000 \cdot \$10769.74  = \$861,579,200 \\
        \text{Feb-12  }\qquad & 80,000 \cdot \$10780.91  = \$862,472,800 \\
        \text{Mar-12  }\qquad & 80,000 \cdot \$10790.75  = \$863,260,000
      \end{align*}
    \end{subproblem}
    
    \begin{subproblem}
      If copper prices increase to \$20,000, the Jan-12 futures
      position will have a loss of $80,000 \cdot (\$20,000 - 10769.74)
      = \$738,420,800$.

      If copper prices fall to \$3000, the Jan-12 futures position
      will realize a gain of $80,000 \cdot (\$10769.74 - \$3000) =
      \$621,579,200$.
    \end{subproblem}
  \end{problem}

  \begin{problem}

    \begin{subproblem}
      \begin{align*}        
%        F_t = S_0 (1+r_t-\hat{y}_t)^{\nicefrac{t}{12}} \\
        \text{1 month:    } \qquad & 10807.08 = 10740.71
        (1+0.001-\hat{y}_1)^{\nicefrac{1}{12}} & \Rightarrow \hat{y}_1
        \approx -7.57\% \\
        \text{6 months:    } \qquad & 11186.06 = 10740.71
        (1+0.001-\hat{y}_1)^{\nicefrac{6}{12}} & \Rightarrow \hat{y}_6
        \approx -7.71\% \\
        \text{12 months:    } \qquad & 11186.06 = 10740.71
        (1+0.001-\hat{y}_1)^{\nicefrac{12}{12}} & \Rightarrow \hat{y}_{12}
        \approx -8.80\% 
      \end{align*}

    \end{subproblem}

    \begin{subproblem}
      A strategy for speculating on copper is as follows: borrow
      $\$955,256,800$ from the market. Use this money to purchase
      80,000 tons of copper at the spot price (cost: $80,000 \cdot
      10740.71 = \$859,256,800$) and pay the storage fee (cost:
      $80,000 \cdot \$1,200 = \$96,000,000$). After one year, sell the
      copper and repay the loan. Using the implied interest rate of
      $0.92\%$ from above, the cost of doing so is $\$955,256,800
      \cdot 1.0093 = \$964,140,688.24$.

      By contrast, if we instead purchased 80,000 tons of Aug-12
      copper futures, the cost would be $80,000 \cdot \$11,784.73 =
      \$942,778,400$. Purchasing the futures is cheaper after taking
      into account the storage fee and interest rate.
    \end{subproblem}
  \end{problem}

  \begin{problem}

    \begin{subproblem}
      Consider first purchasing a single futures contract while
      holding the bonds.  At the end of the year, the value of the
      zero-coupon bonds will be $\$337,500 \cdot 1.06 =
      \$357,750$. The current price of the index future will be $(1 +
      0.06 - 0.03)^1 \cdot \$1350 = \$1390.50$. Therefore:

      \begin{itemize}
      \item if the index finishes at 1,200, the value of the portfolio
        will be $\$357,750 + 250 \cdot (1200-1390.5) = \$310,125$.
      \item if the index finishes at 1,400, the value of the portfolio
        will be $\$357,750 + 250 \cdot (1400-1390.5) = \$360,125$.
      \end{itemize}

      If instead we had purchased 250 units of the index (and no
      bonds), the value of the portfolio would be the value of the
      index plus the dividend yield
      \begin{itemize}
      \item if the index finishes at 1,200: $\$250 \cdot 1,200 +
        \$337,500 \cdot 3\% = \$310,125$
      \item if the index finishes at 1,400: $\$250 \cdot 1,400 +
        \$337,500 \cdot 3\% = \$360,125$
      \end{itemize}

      The two portfolios behave the same way.
    \end{subproblem}

    \begin{subproblem}
      We could implement this change by purchasing \$20,000,000 in
      S\&P 500 futures. This requires purchasing
      $\frac{\$20,000,000}{250*\$1350} = 59$ contracts.

      After one year, if the index finishes at 1400, the portfolio
      consists of
      
      \begin{itemize}
      \item cash from maturing bond: $\$50,000,000 \cdot 1.06 =
        \$53,000,000$
      \item cash from dividends: $\$50,000,000 \cdot 0.03 =
        \$1,500,000$
      \item cash from settlement of S\&P index futures: $59 \cdot 250 \cdot (1400-1390.50) =
        \$140,125$.
      \item S\&P index (held): $\$50,000,000 \cdot
        \frac{\$1400}{\$1350} = \$51,851,852$
      \end{itemize}

      The total value of the portfolio is now $\$106,491,977$. 51.3\% of
      the portfolio is in cash, and 48.7\% in the S\&P 500.
    \end{subproblem}
  \end{problem}
\end{pset}
\end{document}
