\documentclass{article}
\input{15401-preamble}
\psetnum{3}
\date{2011/10/12}

\begin{document}
\begin{pset}
  
  \begin{problem}
      \begin{align*}
        P_A &= \sum_{t=1}^\infty \frac{\$2.20}{1.095^t} &=
        \frac{\$2.20}{0.095} &\approx \$23.16\\
        P_B &= \sum_{t=1}^\infty \frac{\$1.40 \cdot
          (1.03)^{t-1}}{1.095^t} &= \frac{\$1.40}{0.095 - 0.03}
        &\approx \$21.54\\
        P_C &= \frac{\$1.00}{1.095} + \frac{\$1.50}{1.095^2} +
        \frac{\$2.00}{1.095^3} + \frac{\$2.50}{1.095^4} +
        \sum_{t=5}^\infty \frac{\$2.50}{1.095^t} \\&\approx 0.913 +
        1.251 + 1.523 + 1.739 + \frac{1}{1.095^4} \frac{\$2.50}{0.095}
        && \approx \$23.73
      \end{align*}    
  \end{problem}

  \begin{problem}
    
    \begin{subproblem}
      The firm has a payout ratio of 0.4 and a plowback ratio of
      0.6. If it maintains its ROE forever, it will grow at a rate of
      \[g = b \cdot ROE = 0.6 \cdot 0.1 = 6\%\]. Therefore, its stock
      is valued at
      \begin{align*}
        P_0 = \frac{D_1}{r-g} = \frac{1+g}{r - g}D_0 = \frac{1.06}{0.08
          - 0.06} \$0.40 = \$21.20
      \end{align*}

      Its forward PE ratio is
      \[ \nicefrac{P_0}{EPS_1} = \frac{P_0}{(1+g) EPS_0} =
      \frac{\$21.20}{1.06 \cdot \$1} = 20 \]
      which is higher than $\nicefrac1r$ because the firm's PVGO is positive.
    \end{subproblem}

    \begin{subproblem}
      The firm maintains a ROE of 10\% for ten years --- we'll
      interpret that to mean that the ROE falls to 6\% and the payout
      ratio increases to 1 \emph{after} year 10. Therefore, its book
      value, earnings, and dividend will increase as above for years 1
      -- 10.
      
      At the present time, the book value of the firm is \$10 per
      share (earnings / ROE). The book value will grow by 6\% for the
      first ten years, so after year ten the book value will be $\$10
      \cdot1.06^{10} \approx \$17.91$ per share. Thereafter, it will
      not grow, because a payout ratio of 1 implies no further
      retained earnings. Because the ROE drops to $6\%$, earnings (and
      thus dividends) will fall to $D_{11} = 0.06 \cdot \$17.91
      \approx \$1.075$ per share.

      \begin{align*}
        P_0 &= \sum_{i=1}^{10} \frac{D_i}{1.08^i} + \sum_{i=11}^\infty
        \frac{D_{11}}{1.08^i} \\
        &= \frac{(1+g) D_0}{r-g} \left[ 1-
          \left(\frac{1+g}{1+r}\right)^{10} \right] + \frac{1}{1.08^{10}}
        \frac{D_{11}}{0.08} \\
        &= \frac{1.06 \cdot \$0.40}{0.02} \left[1 -
          \left(\frac{1.06}{1.08}\right)^{10} \right] + \frac{1}{1.08^{10}}
        \frac{\$1.075}{0.08}& \\
        &\approx \$3.614 + \$6.221 &\approx \$9.84
      \end{align*}
    \end{subproblem}

    \begin{subproblem}
      Without retention (i.e. a payout ratio of 1), the stock will pay
      a dividend of \$1 per share for years 1--10. Thereafter, with the
      reduced ROE, it will pay out a dividend of \$0.6 per
      share. Thus, its price would be
      \begin{align*}
        P &= \sum_{i=1}^{10} \frac{\$1}{1.08^i} + \sum_{i=11}^{\infty}
        \frac{\$0.6}{1.08^i} \\
        &= \frac{\$1}{0.08}\left[1 -
          \left(\frac{1}{1.08}\right)^{10}\right] +
        \frac{1}{1.08^{10}} \frac{\$0.60}{0.08} \\
        &\approx \$6.710 + \$3.474 \approx \$9.72
      \end{align*}

      The PVGO is the difference in price caused by retained earnings,
      i.e. $\$9.84 - \$9.72 = \$0.12$.
    \end{subproblem}
  \end{problem}

  \begin{problem}

    We'll start by computing the starting book value, EPS, dividend,
    and amount reinvested for each of the first five years:
    
    \begin{tabular}{r|ccccc}
Year & 1 & 2 & 3 & 4 & 5\\
\hline
Book value/share &  \$100.00  &  \$116.00  &  \$132.70  &  \$146.64  &  \$156.90 \\
ROE & 20\% & 18\% & 15\% & 10\% & 8\%\\
EPS &  \$20.00  &  \$20.88  &  \$19.91  &  \$14.66  &  \$12.55 \\
Payout Ratio & 20\% & 20\% & 30\% & 30\% & 60\%\\
Dividend &  \$4.00  &  \$4.18  &  \$5.97  &  \$4.40  &  \$7.53 \\
Reinvested &  \$16.00  &  \$16.70  &  \$13.93  &  \$10.26  &  \$5.02 \\
    \end{tabular}

    \begin{subproblem}
      \begin{align*}
        P_0 &= \frac{D_1}{(1+r)} &+ \frac{D_2}{(1+r)^2} &+ \frac{D_3}{(1+r)^3}
        &+ \frac{D_4}{(1+r)^4} &+ \frac{1}{(1+r)^4} \cdot \frac{D_5}{0.07 - 0.08
        \cdot 0.6} \\
      &\approx \frac{\$4.00}{1.07} &+ \frac{\$4.18}{1.07^2} &+ \frac{\$5.97}{1.07^3}
      &+ \frac{\$4.40}{1.07^4} &+ \frac{1}{1.07^4} \cdot \frac{\$7.53}{0.07 - 0.08
        \cdot 0.4} \\
      &\approx 3.74 &+ 3.65 &+ 4.87 &+ 3.36 &+ 151.20 &\approx \$166.82
      \end{align*}
    \end{subproblem}

    \begin{subproblem}
      If the firm pays out its entire earnings each year, the price
      would be
      \[ \frac{\$20}{1.07} + \frac{\$18}{1.07^2} + \frac{\$15}{1.07^3}
      + \frac{\$10}{1.07^4} + \frac{1}{1.07^5} \frac{\$8}{0.07}
      \approx \$135.77 \]
      so the PVGO is
      \[ PVGO = \$166.82 - \$135.77 \approx \$31.05\]
    \end{subproblem}

    \begin{subproblem}
      \[\nicefrac{P}{E} = \frac{\$156.93}{\$20} = 8.34 \]
    \end{subproblem}

    \begin{subproblem}
      In year 5, the firm has EPS of \$12.55, and a growth rate $g =
      8\% \cdot 40\% = 3.2\%$. The EPS in year 11, therefore, is
      \[ EPS_{11} = \$12.55 \cdot 1.08^{(11-5)} \approx 15.16 \]
      and so the dividends will be
      \[ D_{11} = 0.6 \cdot EPS_{11} \approx \$9.10 \]
      The ex-dividend stock price will be
      \[ P_{10} = \frac{D_{11}}{r - g} = \frac{\$9.10}{0.07 - 0.032}
      \approx \$239.38 \]
      The forward P/E ratio is
      \[P/E = \frac{P_{10}}{EPS_{11}} \approx \frac{\$239.38}{\$15.16}
      \approx 15.79 \]
      The PVGO is
      \[ PVGO = P_{10} - P_{10}^* = \frac{D_{11}}{r - g} -
      \frac{EPS_{11}}{r} \approx \$239.38 - \frac{\$15.16}{0.07}
      \approx \$239.38 - \$216.57 \approx \$22.81\]
    \end{subproblem}
  \end{problem}

  \begin{problem}
    Consider the possibilities for a 45\% drop in price given the
     the valuation formula
    \[ P_0 = \frac{D_1}{r-g} \]
    where $D_1$ is the expected dividend for the next year, $r$ is the
    discount rate (including risk premium), and $g$ is the (constant)
    dividend growth rate:

    One possible explanation for 2008--2009's 45\% drop in price is
    that the expected dividend for the next year and all future years
    dropped. However, they would have to drop by 45\% to completely
    explain the price drop. Although dividends were lower in 2009 than
    2008, they fell by only about 20\%. So this clearly isn't the
    whole story.

    A more likely explanation is that the expected rate of dividend
    growth changed as a result of updated information. Consider the
    amount of change $\Delta g$ required to change the price by 45\%:

    \begin{align*}
      P (1-.45) &= \frac{D_1}{r - g - \Delta g} \\
      \frac{D_1}{r - g} \cdot 0.55 &= \frac{D_1}{r - g - \Delta g} \\
       \Delta g &= - \frac{.45}{.55} (r - g)
     \end{align*}

     We can estimate $r-g$ from previous data, given that $r-g =
     \frac{D_{x+1}}{P_x}$ in year $x$. For the previous three years,
     $r-g$ was approximately $2\% \pm 0.1\%$. So a change of
     \[ \Delta g = - \frac{.45}{.55} (r - g) = -1.6\% \]
     in the dividend growth rate would give a 45\% drop in the stock
     price. A 1.6\% drop in the growth rate seems like a fairly
     reasonable expectation given the events of 2008.

     A combination of these two factors --- a drop in the next year's
     expected dividend rate and a slowdown of future dividend growth
     --- could provide a better explanation. For example, if dividends
     were expected to fall by 20\% (as they did), a drop of only 0.6\%
     in the dividend growth rate would explain the 45\% decrease in
     stock prices.
  \end{problem}
\end{pset}
\end{document}