Show pageOld revisionsBacklinksBack to top This page is read only. You can view the source, but not change it. Ask your administrator if you think this is wrong. ====== 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: \(\mathbf{X} \sim N(\mu,\Sigma)\) public/courses/finance/computational_finance/matrix_algebra.txt Last modified: 2020/07/10 12:11by 127.0.0.1