I would like to find a Derivation of the efficient frontier set for the markowitz problem: enter image description here


1 Answer 1


To solve this constraint minimization problem, first form the Lagrangian Function \begin{align} L(w,\lambda_1,\lambda_2)=w'\Sigma w + \lambda_1(w'\boldsymbol{\mu}-m) + \lambda_2 (w'\boldsymbol{1}-1). \end{align}

The first order conditions for a minimum are then given by \begin{align} \frac{\delta L(w,\lambda_1,\lambda_2)}{\delta w}&=2 \Sigma w + \lambda_1 \boldsymbol{\mu} + \lambda_2 \boldsymbol{1}=\boldsymbol{0} \\ \frac{\delta L(w,\lambda_1,\lambda_2)}{\lambda_1}&=w'\boldsymbol{\mu}-m=0 \\ \frac{\delta L(w,\lambda_1,\lambda_2)}{\lambda_2}&=w'\boldsymbol{1}-1=0. \end{align}

This system of linear equations using matrix algebra can be represented as \begin{align} \begin{bmatrix} 2\Sigma & \boldsymbol{\mu} & \boldsymbol{1} \\ \boldsymbol{\mu}' & 0 & 0 \\ \boldsymbol{1}' & 0 & 0 \end{bmatrix} \begin{bmatrix} w \\ \lambda_1 \\ \lambda_2 \end{bmatrix}= \begin{bmatrix} \boldsymbol{0} \\ m \\ 1 \end{bmatrix}, \end{align} or \begin{align} \boldsymbol{A}\boldsymbol{z}=\boldsymbol{b}, \end{align} where

\begin{align} \boldsymbol{A}:=\begin{bmatrix} 2\Sigma & \boldsymbol{\mu} & \boldsymbol{1} \\ \boldsymbol{\mu}' & 0 & 0 \\ \boldsymbol{1}' & 0 & 0 \end{bmatrix}, \boldsymbol{z}:= \begin{bmatrix} w \\ \lambda_1 \\ \lambda_2 \end{bmatrix} \boldsymbol{b}:= \begin{bmatrix} \boldsymbol{0} \\ m \\ 1 \end{bmatrix}. \end{align} The solution for $\boldsymbol{z}$ is then given by (A has full rank and is thus invertible)

\begin{align} \boldsymbol{z}=\boldsymbol{A}^{-1} \boldsymbol{b} \end{align}

The first element of $\boldsymbol{z}$ gives you the set of efficient portfolios varying m.

  • $\begingroup$ You might add that one can analytically block invert $A$. This means that if all you really want is the first element of $z$, then you can save yourself some calculations. $\endgroup$
    – John
    Commented Mar 22, 2017 at 20:29

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.