====== Week3-4 - Matrix Algebra ======

===== 3.8 - Matrix Albegra: Review part 1 =====

  * Here we note A a matrix nxm,
  * n: number of rows
  * m: number of columns

  * In R, to create a matrix: <code>matA = matrix(data=c(1,2,3,4,5,6),nrow=2,ncol=3)</code>
  * To create a vector: <code>xvec = c(1,2,3)</code>
  * To convert: <code>xvec.mat = as.matrix(xvec)</code>
  * To transpose a matrix or vector: <code>t(mat)
t(xvec)</code>
  * A matrix A is symetric if \(A=A^T\).

===== 3.9 - Matrix Albegra: Review part 2 =====

  * Matrix multiplication is not commutative: \(A \cdot B \neq B \cdot A\)
  * Matrix multiplication in R: <code> A %*% B</code>
  * nxn identity matrix in R is created with: <code>diag(n)</code>
  * In R to inverse a matrix we use: <code>invA = solve(A)</code>
  * For a 2x2 matrix we can compute the inverse as:
\[A^{-1} = \frac{1}{det(A)} \begin{bmatrix} a_{22} & -a_{12} \\ -a_{21} & a_{11} \end{bmatrix}, ~~ \text{assuming: } det(A) = a_{11}a_{22} - a_{12}a_{21} \neq 0\]

  * If we have a vector of 1 called one, then we can use: \(one^T x = \sum\limits_{i=1}^n x_i\)
  * In R we have the function crossprod(x,y) = \(x^Ty\)

===== 4.1 - Matrix Algebra: Portfolio Math =====

  * We consider 3 assets with return rates \(R_i\) normally distributed.
  * The portfolio is normally distributed too. We can compute its mean and variance already.
  * We then note: \(R = \begin{bmatrix} R_A \\ R_B \\ R_C \end{bmatrix}\), \(\mu = \begin{bmatrix} \mu_A \\ \mu_B \\ \mu_C \end{bmatrix}\), \(\mathbf{1} = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}\), \(x = \begin{bmatrix} x_A \\ x_B \\ x_C \end{bmatrix} \)
  * And finally \(\Sigma = \begin{bmatrix} \sigma_A^2 & \sigma_{AB} & \sigma_{AC} \\ \sigma_{AB} & \sigma_B^2 & \sigma_{BC} \\ \sigma_{AC} & \sigma_{BC} & \sigma_C^2 \end{bmatrix} \)

  * Portfolio weights sum to 1 so we have : \(\mathbf{1}^Tx = 1\)

==== Portfolio return ====

  * \(R_{p,x} = x^T R = R^T x\)

==== Portfolio expected return ====

  * \(\mu_{p,x} = x^T\mu = \mu^T x\)

==== Portfolio variance ====

  * \(\sigma_{p,x}^2 = x^T \Sigma x\)

==== Covariance Between 2 portfolio returns ====

  * 2 portfolio with weights x and y,
  * The covariance between the 2 is: \(Cov(R_{p,x},R_{p,y}) = x^T \Sigma y = y^T \Sigma x\)

===== 4.2 - Matrix Algebra: Bivariate Normal =====

  * We note: \(\mathbf{X} = \begin{bmatrix} X \\ Y \end{bmatrix}, ~ x = \begin{bmatrix} x \\ y \end{bmatrix}, ~ \mu = \begin{bmatrix} \mu_X \\ \mu_Y \end{bmatrix}, \Sigma = \begin{bmatrix} \sigma_X^2 & \sigma_{XY} \\ \sigma_{XY} & \sigma_Y^2 \end{bmatrix}\)

  * Then we can write the bivariate normal distribution as: 
\[f(x) = \frac{1}{2\pi det(\Sigma)^\frac 12} e^{- \frac 12 (x - \mu)^T \Sigma^{-1} (x-\mu)}\]

  * Where \(det(\Sigma) = \sigma_X^2 \sigma_Y^2 (1 - \rho_{XY}^2)\)

  * We use the shorthand notation: \(\mathbf{X} \sim N(\mu,\Sigma)\)