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:
    matA = matrix(data=c(1,2,3,4,5,6),nrow=2,ncol=3)
  • To create a vector:
    xvec = c(1,2,3)
  • To convert:
    xvec.mat = as.matrix(xvec)
  • To transpose a matrix or vector:
    t(mat)
    t(xvec)
  • 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:
     A %*% B
  • nxn identity matrix in R is created with:
    diag(n)
  • In R to inverse a matrix we use:
    invA = solve(A)
  • For a 2×2 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: \(\mathbf{X} \sim N(\mu,\Sigma)\)