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: \(\mathbf{X} \sim N(\mu,\Sigma)\)