Systems of Linear Equations

\begin{equation} T v = v' \end{equation}

every system of linear equations is decomposed into this. Classically, there’s either a unique solution, no solution, infinite solutions—

problems with zero

“zero” is really hard to define. For instance:

\begin{equation} 6.23423 \times 10^{192} - 1 \times 10^{7} = 6.23423 \times 10^{192} \end{equation}

so in this case \(10^{7}\) literally behaves like zero. (small numbers have the opposite problem)

so, we use elementary row operations to make sure that enormous numbers are essentially standardized—if a row has huge numbers, we may want to scale it down to smaller numbers to make them nice.

see linear temporal logic

An exact solution for a dynamic system with quadratic costs and linear differential equation describing the dynamics.

For some non-linear function, we can use its first Jacobian to create a linear system. Then, we can use that system to write the first order Taylor:

\begin{equation} y’ = \nabla F(crit)y \end{equation}

where \(crit\) are critical points.

Phase Portrait stability

• if all \(Re[\lambda] < 0\) of \(\qty(\nabla F)(p)\) then \(p\) is considered stable—that is, points initially near \(p\) will exponentially approach \(p\)

• if at least one \(Re[\lambda] > 0\) of \(\qty(\nabla F)(p)\) then \(p\) is considered unstable—that is, points initially near \(p\) will go somewhere else

CAPM, a Review

Note that we will be using the Sharpe-Linter version of CAPM:

\begin{equation} E[R_{i}-R_{f}] = \beta_{im} E[(R_{m}-R_{f})] \end{equation}

\begin{equation} \beta_{im} := \frac{Cov[(R_{i}-R_{f}),(R_{m}-R_{f})]}{Var[R_{m}-R_{f}]} \end{equation}

Recall that we declare \(R_{f}\) (the risk-free rate) to be non-stochastic.

Let us begin. We will create a generic function to analyze some given stock.

Data Import

We will first import our utilities

import pandas as pd
import numpy as np

Let’s load the data from our market (NYSE) as well as our 10 year t-bill data.