M4 L2A 10 Portfolio Variance Using Factor Model V4
The formula that we calculated for portfolio variance is equivalent to the matrix notation formula that we saw a few videos ago.
Var(rp)=XT(BFBT+S)X \textrm{Var}(r_p) = \mathbf{X}^T(\mathbf{B}\mathbf{F}\mathbf{B}^T + \mathbf{S}) \mathbf{X} Var(rp)=XT(BFBT+S)X
Where
F=(Var(f1)Cov(f1,f2)Cov(f2,f1)Var(f2)) \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} F=(Var(f1)Cov(f2,f1)Cov(f1,f2)Var(f2))
B=(βi,1,βi,2βj,1,βj,2) \mathbf{B} = \begin{pmatrix}\beta_{i,1}, \beta_{i,2}\\ \beta_{j,1}, \beta_{j,2}\end{pmatrix} B=(βi,1,βi,2βj,1,βj,2)
BT=(βi,1,βj,1βi,2,βj,2) \mathbf{B}^T = \begin{pmatrix}\beta_{i,1}, \beta_{j,1}\\ \beta_{i,2}, \beta_{j,2}\end{pmatrix} BT=(βi,1,βj,1βi,2,βj,2)
S=(Var(si)00Var(sj)) \mathbf{S} = \begin{pmatrix}\textrm{Var}(s_i) & 0\\ 0 & \textrm{Var}(s_j)\end{pmatrix} S=(Var(si)00Var(sj))
X=(xixj) \mathbf{X} = \begin{pmatrix}x_{i}\\ x_{j}\end{pmatrix} X=(xixj)
XT=(xixj) \mathbf{X}^T = \begin{pmatrix}x_{i} & x_{j}\end{pmatrix} XT=(xixj)
Var(rp)=(xixj)((βi,1,βi,2βj,1,βj,2)(Var(f1)Cov(f1,f2)Cov(f2,f1)Var(f2))(βi,1,βj,1βi,2,βj,2)+(Var(si)00Var(sj)))(xixj) \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} Var(rp)=(xixj)((βi,1,βi,2βj,1,βj,2)(Var(f1)Cov(f2,f1)Cov(f1,f2)Var(f2))(βi,1,βj,1βi,2,βj,2)+(Var(si)00Var(sj)))(xixj)
Next Concept
