\documentclass[twocolumn]{article} \usepackage{fancyhdr} \usepackage{amsfonts, amsmath, amssymb, latexsym} \usepackage{geometry} \usepackage{enumerate} \usepackage{multicol} \usepackage{outlines} \usepackage{verbatim} \usepackage{times} \setlength{\parindent}{0pt} \setlength{\parskip}{0pt plus 0.5ex} %\setlength{\parskip}{1ex plus 0.5ex minus 0.2ex} \usepackage{fullpage} \makeatletter \renewcommand{\section}{\@startsection{section}{1}{0mm}% {-1ex plus -.5ex minus -.2ex}% {0.5ex plus .2ex}%x {\normalfont\large\bfseries}} \renewcommand{\subsection}{\@startsection{subsection}{2}{0mm}% {-1explus -.5ex minus -.2ex}% {0.5ex plus .2ex}% {\normalfont\normalsize\bfseries}} \renewcommand{\subsubsection}{\@startsection{subsubsection}{3}{0mm}% {-1ex plus -.5ex minus -.2ex}% {1ex plus .2ex}% {\normalfont\small\bfseries}} \makeatother % Don't print section numbers \setcounter{secnumdepth}{0} \thispagestyle{empty} \newcommand{\ps}{\\ \vspace{0.05in}} \newcommand{\T}{\texttt} \newcommand{\R}{\textrm} \newcommand{\B}[1]{[\textrm{#1}]} \newcommand{\boe}{\begin{outline}[enumerate]} \newcommand{\eo}{\end{outline}} \newenvironment{sect}[1]{\section{#1}\begin{outline}[enumerate]}{\end{outline}} \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 $g$: $$PV = \begin{cases}\frac{A}{r-g} \left[ 1 - \left( \frac{1+g}{1+r} \right)^T \right], & r \neq g \\ \frac{AT}{1+r},& r = g \end{cases}.$$ \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} \section*{Options} \begin{outline} \1 Call option: right to buy an asset (price $S$) for a given strike price $K$ on or before the expiration date $T$ \2 Value of call option at expiration = max(0, market price - exercise price) \1 Put option: right to sell an asset for a given price on or before the expiration date \2 Value of put option at expiration = max(0, exercise price - market price) \1 European: exercise only on expiration date; American: can exercise before \1 Put-call parity: two equivalent scenarios \2 Own a call option $C$ on a stock for exercise price $K$, and invest enough elsewhere (bonds, etc.) to have $K$ at the exercise date. \2 Own a put option $P$ on a stock for exercise price $K$ and own a share of the stock at price $S$ \3 The total value for both is max(stock price, $K$). \2 So: $S+P = PV(K) + C$ \1 Binomial model \2 Assume stock either goes up to $uS$ with probability $p$ or down to $dS$ \2 Option value is either $C_u$ or $C_d$ depending on which case \2 Can replicate with $a$ shares stock + $b$ risk free bond portfolio \2 $(uS)a + (1+r)b = C_u$ ; $dS_a + br = C_d$ \[ C_0 = \frac{1}{1+r}\left( \frac{1+r-d}{u-d}C_u + \frac{u-(1+r)}{u-d}C_d\right) \] \1 Black-Scholes formula: $\B{Value of call option} = (\B{delta} \times \B{share price}) - \B{bank loan} = [N(d_1) \times S] - [N(d_2) \times PV(K)],$ where \2 $$d_1 = \frac{\log[S/PV(K)]}{\sigma\sqrt{t}} + \frac{\sigma\sqrt{t}}{2}, \qquad d_2 = d_1 - \sigma \sqrt{t};$$ \2 $N(d)$ is the cumulative normal probability distribution ($d$ is unitless; it is measured in standard deviations) \2 $K$ is the exercise price of the option; $PV(K)$ is calculated by discounting at the risk-free interest rate. Thus, $PV(K) = KR^{-T}$ \2 $t$ is in units of years. \2 $S$ is the price of the stock now \2 $R = 1+r_f$. \2 $\sigma$ is the volatility per year, i.e. the standard deviation per period of continuously compounded rate of return on stock \end{outline} \section{Portfolio Theory} \begin{outline} \1 $Cov[x,y] = \rho_{xy} \sigma_x \sigma_y$ \1 Two-asset portfolio, weights $\omega_1$ and $\omega_2$ \2 $E[R_P] = \omega_1 \mu_1 + \omega_2 \mu_2$ \2 $\sigma_p = \omega_1^2 \sigma_1^2 + \omega_2^2 \sigma_2^2 + 2 \omega_1 \omega_2 \sigma_1 \sigma_2 \rho_{1,2}$ \3 can be less than both $\sigma_1$ and $\sigma_2$ \1 Diversication eliminates ideosyncratic risk; systematic risk remains \1 If there's a risk-free asset... \2 Sharpe ratio measures risk/reward tradeoff $$ S - \frac{E[R] - R_f}{\sigma} $$ \2 Market portfolio is the tangency portfolio, which maximizes Sharpe ratio. \2 Combinations of the tangency portfolio and risk-free asset make up the Capital Market Line \3 Optimal portfolios are on the CML. None are above it. \end{outline} \section{CAPM} \begin{outline} \1 Beta measures systematic risk $$\beta_A = \frac{Cov[A]}{Cov[M]} = \frac{\rho_{AM} \sigma_A}{\sigma_M}$$ \1 All expected returns lie on the Securities Market Line $$ E[R] = R_f + \beta(E[R_M] - Rf) $$ \1 Investors care about beta, not standard deviation, because non-systematic risks are diversifiable \end{outline} \section{Capital Budgeting} \begin{outline} \1 NPV rule \end{outline} \end{small} \end{document}