15. Portfolio Variance using Factor Model

Portfolio Variance in Non-Matrix and Matrix Notation

The formula that we calculated for portfolio variance is equivalent to the matrix notation formula that we saw a few videos ago.

Portfolio Variance (Matrix Notation)

\textrm{Var}(r_p) = \mathbf{X}^T(\mathbf{B}\mathbf{F}\mathbf{B}^T + \mathbf{S}) \mathbf{X}

Where

\mathbf{F} = \begin{pmatrix}\textrm{Var}(f_1) & \textrm{Cov}(f_1,f_2) \\ \textrm{Cov}(f_2,f_1) & \textrm{Var}(f_2) \end{pmatrix}

\mathbf{B} = \begin{pmatrix}\beta_{i,1}, \beta_{i,2}\\ \beta_{j,1}, \beta_{j,2}\end{pmatrix}

\mathbf{B}^T = \begin{pmatrix}\beta_{i,1}, \beta_{j,1}\\ \beta_{i,2}, \beta_{j,2}\end{pmatrix}

\mathbf{S} = \begin{pmatrix}\textrm{Var}(s_i) & 0\\ 0 & \textrm{Var}(s_j)\end{pmatrix}

\mathbf{X} = \begin{pmatrix}x_{i}\\ x_{j}\end{pmatrix}

\mathbf{X}^T = \begin{pmatrix}x_{i} & x_{j}\end{pmatrix}

Portfolio Variance Matrix Notation (putting the pieces together)

\textrm{Var}(r_p) = \begin{pmatrix}x_{i} & x_{j}\end{pmatrix}\left ( \begin{pmatrix}\beta_{i,1}, \beta_{i,2}\\ \beta_{j,1}, \beta_{j,2}\end{pmatrix}\begin{pmatrix}\textrm{Var}(f_1) & \textrm{Cov}(f_1,f_2) \\ \textrm{Cov}(f_2,f_1) & \textrm{Var}(f_2) \end{pmatrix}\begin{pmatrix}\beta_{i,1}, \beta_{j,1}\\ \beta_{i,2}, \beta_{j,2}\end{pmatrix}+\begin{pmatrix}\textrm{Var}(s_i) & 0\\ 0 & \textrm{Var}(s_j)\end{pmatrix}\right )\begin{pmatrix}x_{i}\\ x_{j}\end{pmatrix}