\documentclass{article}
\input{15401-preamble}
\psetnum{1}
\date{2011/09/19}

\begin{document}
\begin{pset}
  \begin{problem}
    First, note that the money market account's interest rate is 0.2\%
    per month. Then:
    \begin{align*}
      V_{\text{year 6}} &= \$60000 \cdot 1.002^{72} - \$45,000 \cdot
      1.002^{72-30} + \sum_{k=1}^{6} 4000 \cdot 1.002^{72-12k} \\
      &\approx \$69283.08 - \$48939.19 + \$25503.92 \\
      &\approx \$45747.81
    \end{align*}
  \end{problem}
  
  \begin{problem}
    
    \begin{subproblem}
      \begin{align*}
        \$300,000 &= \frac{M}{0.005}
                     \left(1-\frac{1}{1.005^{360}}\right) \\
                  &\approx 166.79 M \\
        M         &\approx \frac{\$300000}{166.79}
                  &\approx \$1798.65
      \end{align*}
    \end{subproblem}

    \begin{subproblem}
      \begin{align*}
        \$300,000 &= \frac{\$1600}{0.005}
                     \left(1-\frac{1}{1.005^{360}}\right) +
                     \frac{X}{1.005^{360}} \\
                  &\approx \$266866.58 + \frac{X}{6.023} \\
        X &\approx 6.023 \cdot (\$300000 - \$266866.58) \\
        &\approx \$199548.50
      \end{align*}
    \end{subproblem}
  \end{problem}

  \begin{problem}
    
    \begin{subproblem}
      Consider the present value of the revenue and operating cost
      streams as two separate perpetuties:
      \begin{align*}
        PV_R &= \frac{\$2000000}{0.10} &= \$20,000,000 \\
        PV_E &= \frac{\$1000000}{0.10 - 0.05} &= \$20,000,000 \\
        PV &= PV_R - PV_E &= 0
      \end{align*}
    \end{subproblem}

    \begin{subproblem}
      Now consider the same streams as annuities:
      \begin{align*}
        PV_R &= \frac{\$2000000}{0.10} \left(1 - \frac{1}{(1.1)^{15}}\right)
          &\approx \$15,212,159 \\
        PV_E &= \frac{\$1000000}{0.10 - 0.05} \left(1 -
          \left(\frac{1.05}{1.1}\right)^{15} \right)\cdot &\approx
        \$10,046,422 \\
        PV &= PV_R - PV_E &\approx \$5,165,736
      \end{align*}
    \end{subproblem}
  \end{problem}

  \begin{problem}
    
    \begin{subproblem}
      
      \begin{align*}
        \$8239.05 &= \frac{M}{\nicefrac{0.142}{12}} \left(
          1-\frac{1}{\left(1 + \nicefrac{0.142}{12} \right)^{48}}
        \right) \\
        M &= \$225.97
      \end{align*}
    \end{subproblem}

    \begin{subproblem}
      \begin{align*}
        X = \sum_{i=1}^{48} \frac{M}{\left(1 +
            \nicefrac{0.08}{12}\right)^i}
        = \$9256.23
      \end{align*}
     ...and \$9,256.23 is higher than the price of the car. 
    \end{subproblem}

    \begin{subproblem}
      This article is nonsense. Paying cash will always be a better
      choice as long as the loan interest rate is higher than the
      savings interest rate. While it's true that the ``average amount
      on which he'd be paying interest [is] only about half the total
      amount borrowed'', it's incorrect to say that the investment
      ``would earn interest on his full deposited principal, plus
      continually compounded interest''. This doesn't take into
      account that the monthly loan payments need to be subtracted
      from the investment balance --- and the loan payments will be
      higher than the interest as long as the loan interest rate is
      higher than the CD interest rate.
    \end{subproblem}
  \end{problem}
\end{pset}
\end{document}
