Week3-4 - Matrix Algebra

• 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 \(A=A^T\).

• 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) = \(x^Ty\)

• 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 : \(\mathbf{1}^Tx = 1\)

• \(R_{p,x} = x^T R = R^T x\)

• \(\mu_{p,x} = x^T\mu = \mu^T x\)

• \(\sigma_{p,x}^2 = x^T \Sigma x\)

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

• 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: \(\mathbf{X} \sim N(\mu,\Sigma)\)