This question is related to recent rule changes in the Quantopian Open.
I am trying to figure out a closed form solution to a beta constrained minimum variance portfolio problem but it doesn't seem particularly tractable. Has anyone else tried this? So far, I have set up the problem
$$\begin{align} \min_w \quad& w^\prime \Sigma w \\ s.t. \quad& w^\prime\vec 1 = 1 \\ \text{and} \quad & w^\prime \beta = c \end{align}$$
where
- $w$ is the vector of portfolio weights, our control
- $\Sigma$ is the total covariance matrix
- $\beta$ is the vector of CAPM-type market betas
- $c$ a constant that the portfolio beta should be equal to
Changing constraints to Lagrange multipliers the objective becomes
$$\min_w \quad w^\prime\Sigma w - \lambda_1(w^\prime\vec 1 - 1) - \lambda_2(w^\prime\beta - c)$$
the first order conditions are
$$\begin{align} 0 &= 2w^\prime\Sigma - \lambda_1\vec 1 - \lambda_2 \beta\\ 1 &= w^\prime\vec 1\\ c &= w^\prime\beta \end{align}$$
I cannot seem to get the equations to work out nicely, perhaps no closed form solution exists but I wanted to check here and see if anyone could get something reasonable before I go to numerical optimization.