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\begin{document}

\raggedright

\begin{small}
  \section*{NPV, Annuities, Perpetuties}
\begin{outline}
  \1 Discounted cash flow
$$PV = \sum_{t=1}^T \frac{C_t}{(1+r_t)^t}.$$
\1 Annual percentage rate (APR)
	\2 Effective annual rate (EAR) is $(1+r/n)^n - 1$.
	\2 Continuous compounding results in an EAR of $e^r$.
\1 Inflation: real CF after $t$ timesteps is $\frac{(\textrm{nominal CF})_t}{(1+i)^t}$.
\1 Annuities:
	$$PV = \frac{A}{1+r} + \ldots + \frac{A}{(1+r)^T} = \frac{A}{r} \left( 1 - \frac{1}{(1+r)^T} \right).$$
	\2 With growth rate $g$:
	$$PV = \begin{cases}\frac{A}{r-g} \left[ 1 - \left( \frac{1+g}{1+r} \right)^T \right], & r \neq g \\
											\frac{T}{1+r},& r = g \end{cases}.$$
	\2 Perpetuity: $PV = \frac{A}{r}.$. With growth $g$:
	$PV = \frac{A}{r-g}$.
      \end{outline}

      \section*{Fixed-Income}

      \begin{outline}
  	\1 Spot interest rate $r_t$: the (annualized) interest rate for maturity date $t$.  Average rate of interest between now and  $t$.
	\1 \textbf{Zero-coupon bond} with maturity date $t$ is a bond which pays \$A at $t$.
		\2 Can convert to spot interest rates: $P = \frac{A}{(1+r_t)^t}$, so we can extract $r_t$.
	\1 \textbf{Coupon bond}
        \2 Coupon bond with coupon payments $\{C_1, C_2,
        \ldots, C_T\}$ and a principal $A$ at maturity $T$
        should be priced as:
        $$P = \sum_{t=1}^T \frac{C_t}{(1+r_t)^t} + \frac{A}{(1+r_T)^T}.$$
        \2 Yield-to-maturity: $y$ for which
        $$P = \sum_{t=1}^T \frac{C_t}{(1+y)^t} + \frac{A}{(1+y)^T}.$$
        
        \1 Forward interest rates: rate to borrow from time $t-1$
        to $t_1$
        $$f_t = \frac{(1+r_t)^t}{(1+r_{t-1})^{t-1}} - 1.$$

	\1 \textbf{Duration}: PV-weighted average of maturity.
        Average time taken by a bond to repay investment
	$$D = \sum_{t=1}^T \frac{PV(CF_t)}{P} \times t = \frac{1}{P}
        \sum_{t=1}^T t  \frac{C_t}{(1+y)^t}.$$
	\2 Modified duration is first derivative of price wrt yield
	$$MD = - \frac{1}{P} \frac{\partial P}{\partial y} = \frac{D}{1+y}.$$

        \1 \textbf{Convexity} is the second derivative of price wrt yield:
	$$P \cdot V_M = \frac{1}{2P} \frac{\partial^2 P}{\partial y^2} =
        \frac{1}{(1+y)^2} \sum_{t=1}^T t  (t-1)  \frac{C_t}{(1+y)^t} $$
      \end{outline}

      \section*{Stocks}
      \begin{outline}
        \1 Dividend discount model: price is NPV of dividend stream
	$$P_0 = \sum_{t=1}^{\infty} \frac{D_t}{(1+r_t)^t}$$
        If fixed $D$ and $r$:
        $$ P_0 = \sum_{t=1}^{\infty} \frac{D}{(1+r)^t} = \frac{D}{r}.$$
        \2 Gordon growth model: with constant growth rate $g$, $D_t = D_0 (1+g)^t$:
        $$P_0 = \sum_{t=1}^{\infty} \frac{(1+g)^{t-1}}{(1+r)^t} D_1 =
        \frac{D_1}{r-g} = \frac{1+g}{r-g}D_0,$$

        \1 Terms
\2 Earnings ($E$ or $EPS$): total profit net of depreciation and taxes
\2 Payout ratio $p$: dividend/earnings = $DPS/EPS$
\2 Retained earnings: (earnings - dividends)
\2 Plowback ratio $b$: retained earnings/total earnings $= 1-p$
\2 Book value $BV$: cumulative retained earnings
\2 Return on book equity $ROE$: earnings/$BV$

	\1 PVGO: 	$P_0 = \frac{EPS_1}{r} + PVGO$
	\2 If $PVGO = 0$, $P/E = \frac{1}{r}$.  If $PVGO>0$, then
        $P/E$ ratio becomes higher, and $ROE$ must exceed cost of capital.
      \end{outline}

      \section*{Forwards/Futures}
      \begin{outline}
        \1 Forward: customized contract w/ specific counterparty;
        futures: exchange-traded through clearinghouse

        \1 Spot-future parity: $$F_{0,T} = P_0(1+r_f - \hat{y})^T$$
        $\hat{y} = y-c$ is the net convenience yield (convenience
        yield minus storage costs)

        \1 Contango: prices increase with maturity; $\hat{y}$
        negative. Compare backwardation.
      \end{outline}
    \end{small}
\end{document}