12.10 Problems: Portfolio Theory with Matrix Algebra

Exercise 1.4 (Solve for global minimum variance portfolio) The global minimum variance portfolio $\mathbf{m}=(m_{1},m_{2},m_{3})^{\prime}$ for the $3$ asset case solves the constrained minimization problem:

$$\begin{align*} \min_{m_{1},m_{2},m_{3}}~\sigma_{p,m}^{2}&=m_{1}^{2}\sigma_{1}^{2}+m_{2}^{2}\sigma_{2}^{2}+m_{2}^{2}\sigma_{2}^{2}\\ &+2m_{1}m_{2}\sigma_{12}+2m_{1}m_{3}\sigma_{13}+2m_{2}m_{3}\sigma_{23}\\ &\textrm{s.t. }m_{1}+m_{2}+m_{3}=1. \end{align*}$$ The Lagrangian for this problem is: $$\begin{align*} L(m_{1},m_{2},m_{3},\lambda) & =m_{A}^{2}\sigma_{A}^{2}+m_{B}^{2}\sigma_{B}^{2}+m_{C}^{2}\sigma_{C}^{2}\\ & +2m_{A}m_{B}\sigma_{AB}+2m_{A}m_{C}\sigma_{AC}+2m_{B}m_{C}\sigma_{BC}\\ & +\lambda(m_{A}+m_{B}+m_{C}-1) \end{align*}$$

1. Write out the first order conditions

Exercise 1.5 (Solve for global minimum variance portfolio)

• Use 2 x 2 matrix inversion rule to explicitly solve for

Exercise 12.1 (Solve for tangency portfolio)

• Give explicit formula for tangency portfolio in 2 x 2 case