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\documentclass[a4paper]{article} | |
\usepackage{multicol, amsmath, amsfonts, xcolor} | |
\usepackage[landscape, margin=0.2in]{geometry} | |
\title{15.415 Cheat Sheet} | |
\author{Zaz Brown} | |
\date{Dec 2021} | |
\newcommand{\dd}{\mathrm d} | |
\newcommand{\ddd}[2]{\frac{\dd^2 #1}{\dd #2^2}} | |
\newcommand{\pd}[2]{\frac{\partial #1}{\partial #2}} | |
\newcommand{\topd}[1]{ \stackrel{\partial {#1}}\longrightarrow } | |
\newcommand{\pdd}[2]{\frac{\partial^2 #1}{\partial #2^2}} | |
\newcommand{\sep}{ \textbf{;} \ } | |
\newcommand{\var}{ \mathrm{Var} } | |
\newcommand{\cov}{ \mathrm{Cov} } | |
\newcommand{\PV}{ \mathrm{PV} } % present value | |
\newcommand{\CF}{ \mathrm{CF} } % cash flow | |
\newcommand{\io}{{ \boldsymbol\iota }} | |
\newcommand{\w}{{ \mathbf w }} | |
\newcommand{\pmat}[1]{ \begin{pmatrix} #1 \end{pmatrix} } | |
\setlength{\parindent}{0pt} | |
\begin{document} | |
\begin{multicols*}{3} | |
\textbf{2021-12-15 15.415 Foundations of Modern Finance, Zaz Brown} \\ | |
\\ | |
Yield-to-maturity of a bond, y: | |
$$ | |
B = \sum_{t=1}^T \frac{C_t}{(1+y)^t} + \frac{P}{(1+y)^T} | |
$$ | |
Where $B$ is bond price. | |
\\ | |
Macaulay duration is the weighted average term to maturity: | |
$$ | |
D = \sum_{t=1}^T \frac{\PV(\CF_t)}{B} t = \frac 1 B \sum_{t=1}^T \frac{\CF_t}{(1+y)^t} t | |
$$ | |
Modified duration: | |
$$ | |
MD = \frac{D}{1+y} | |
$$ | |
Convexity of bond measures the curvature of the bond price as a function of yield: | |
$$ | |
CX = \frac{1}{2 B} \ddd{B}{y} | |
$$ | |
\textbf{OTHER, GENERAL INFO:} \\ | |
\\ | |
\textbf{Linear Algebra} \\ | |
\begin{align*} | |
\begin{bmatrix} | |
a & b \\ | |
c & d | |
\end{bmatrix}^{-1} | |
= \frac{1}{ad-bc} \begin{bmatrix} | |
d & -b \\ | |
-c & a | |
\end{bmatrix}, \ \ | |
\det = ad-bc, \\ | |
\lambda = \frac 1 2 \left(a + d \pm \sqrt{a^2 + d^2 + 4 b c - 2 a d}\right) | |
\end{align*} | |
\textbf{Eigenvalues} $(A-\lambda I)\mathbf v = \mathbf 0 \iff \mid A-\lambda I \mid = 0$ \\ | |
$\det(A) = \prod \boldsymbol\lambda, \ \ \mathrm{tr}(A) = \sum \boldsymbol\lambda$ \\ | |
If all upper-left determinants $> 0$, all $\lambda > 0$ \\ | |
\textbf{Lagrange: Maximize $f$ while $g = c$:} \\ | |
\indent \ \ \ \ \textbf{3. Identify critical poits by solving:} | |
\small $$\nabla(f(\mathbf v) - \lambda(g(\mathbf v)-c)) = 0$$ | |
\scriptsize | |
\indent \ \ \ \ \ \ \ \ \textit{\textbf{Alternatively, when $\mathbf v=[x,y]$, solve:}} | |
\small $$\pd f x \Big/ \pd g x = \pd f y \Big/ \pd g y, \ g=c$$ \\ | |
\scriptsize | |
\indent \ \ \ \ \textbf{2. Determine if max or min using eigenvalues of} $\mathbf H(\mathbf v) | |
{\color{gray}\ \sim \nabla^2(\mathbf v)}$ \\ | |
\indent \ \ \ \ \ \ \ \ \textit{\textbf{Hack: compare with nearby points}} \\ | |
\indent \ \ \ \ \ \ \ \ All positive $\implies$ local minimum \\ | |
\indent \ \ \ \ \ \ \ \ All negative $\implies$ local maximum \\ | |
\indent \ \ \ \ \ \ \ \ Some zero $\implies$ flat directions \\ | |
\indent \ \ \ \ \ \ \ \ Mixed +/- $\implies$ saddle point \\ | |
\indent \ \ \ \ \textbf{1. Check endpoints.} \\ | |
\\ | |
\textbf{Series:} | |
$$ | |
\sum_{k=0}^n r^n = \frac{1-r^{n+1}}{1-r}, \ \ | |
\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} = \ln(2), \ \ | |
\sum_{n=1}^\infty \frac{1}{n^p} = \infty \iff p \leq 1 | |
$$ | |
\normalsize | |
\textbf{Probability Basics:} ($f_X = $ PDF of $X$) | |
$$ | |
f_X \dd x = \frac{f_X}{|\dd y / \dd x|} \dd y, \ \ | |
\mathrm{Skew}(X) = \mathbb E \left[ \left(\frac{X-\mu}{\sigma}\right)^3 \right], \ \ | |
\mathrm{Kurt}(X) = \mathbb E \left[ \left(\frac{X-\mu}{\sigma}\right)^4 \right] - 3 | |
$$ | |
\begin{tabular}{ r c c c l } | |
\hline | |
$X$ & \textbf{PDF:} $f_X(x)$ & $\mathbb E[X]$ & $\var(X)$ \\ \hline | |
\textbf{Uniform distribution:} & $[[1 \geq x \geq 0]]$ & $\frac 1 2$ & $\frac 1 {12}$ \\ | |
\textbf{Normal distribution:} & $\frac{1}{\sqrt{2\pi}\sigma} \exp(-\frac{(x-\mu)^2}{2\sigma^2})$ & $\mu$ & $\sigma^2$ \\ | |
\textbf{LogNormal distribution:} & $\frac{1}{\sqrt{2\pi}\sigma x} \exp(-\frac{(\ln x-\mu)^2}{2\sigma^2})$ & $\exp(\mu+\sigma^2/2)$ & \scriptsize$(\exp(\sigma^2)-1)\exp(2\mu+\sigma^2)$\normalsize \\ | |
\textbf{Cauchi distribution:} & $\frac{A}{\pi A^2 + x^2}$ & undef. & undef. \\ | |
\\ \hline | |
$X$ & \textbf{PMF:} $f_X(k)$ & $\mathbb E[X]$ & $\var(X)$ \\ \hline | |
\textbf{Bernoulli distribution:} & $p[[k=1]] + q[[k=0]$ & $p$ & $pq$ & $q=1-p$ \\ | |
\textbf{Binomial distribution:} & ${n \choose k}p^k q^{n-k}$ & $np$ & $npq$ & sum of $n$ IID Bernouilli\\ | |
\textbf{Poisson distribution:} & $\frac{e^{-\lambda}\lambda^k}{k!}$ & $\lambda$ & $\lambda$ & $\lambda = \#/t$ \\ | |
\textbf{Geometric distribution:} & $(1-p)^{k-1} p$ & $1/p$ & $\frac{1-p}{p^2}$ \\ | |
\end{tabular} | |
\scriptsize | |
\end{multicols*} | |
\end{document} |
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15.455 Quantitative Finance cheat sheet PDF final version.