Introduction

Matrices are used to represent the risk and returns of the varioius portfolio.

Key Terms and Applications

1. Linear questions --> the gamma and vega hedges for an options portfolio
2. Covariance Matrix --> Variance of a linear portfolio
3. Cholesky decomposition --> Simulating returns on correlated assets, Monte Carlo VaR
4. Eigenvalue, Eigenvector --> Positive Definite of Matrix
5. Principal Component Analysis

Brief

Orthogonal

When dot product of tow non-zero vector is 0. the vectors are called orthogonal.

Singular matrices

Det = 0

then, No inverse matrix exist.

then, we call it a singular matrix.

That means there is a linear relationship between the rows (or columns).

Determinant

$ \begin{vmatrix} a & b\\ c & d \end{vmatrix} = ad-bc$

$\begin{vmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{vmatrix} = (1) \begin{vmatrix} 5 & 6\\ 8 & 9 \end{vmatrix} - (2) \begin{vmatrix} 4 & 6\\ 7 & 9 \end{vmatrix} + (3) \begin{vmatrix} 4 & 5\\ 7 & 8 \end{vmatrix}$

$det(A)= a_{i1}C_{i1} + .... a_{in}C_{in}$

$det(A) = a_{1j}C_{1j} + .... a_{1j}C_{1j}$

where, $C_{ij} = (-1)^{i+j}M_{ij}$

$C_{ij}$ is cofactor , $M_{ij}$ is a minor (the determinant from det(A) by deleting the ith row and the jth column from A)

Quadratic

$A$ is a square matrix with dimension n. $x$ is an n-dimensional vector.

$x'Ax$ is a quadratic form.

$$\begin{pmatrix} x_1 & x_2 \end{pmatrix} \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = 1x_1^2 + (2+3)x_1x_2 + 4x_2^2 $$

We can change A to symetric matrix B.

$B = \frac{1}{2}(A+A') = \begin{pmatrix} 1 & 2.5 \\ 2.5 & 4 \end{pmatrix}$

$$\begin{pmatrix} x_1 & x_2 \end{pmatrix} \begin{pmatrix} 1 & 2.5 \\ 2.5 & 4 \end{pmatrix}\begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = 1x_1^2 + (2.5+2.5)x_1x_2 + 4x_2^2 $$

References:

[1] Alexander, Carol. Market Risk Analysis. Vol. I. Chichester, England: Wiley, 2008. Print.