Beta Constrained Markowitz Minimum Variance Portfolio - Closed Form Solution

Question

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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.

Answer 1

Assuming that $\Sigma$ is invertible, then \begin{align} 2\omega' = \lambda_1\overrightarrow{1}'\Sigma^{-1}+\lambda_2\beta'\Sigma^{-1}. \end{align} We can then solve $\lambda_1$ and $\lambda_2$ from the system of equations \begin{align*} 2 &= \lambda_1\overrightarrow{1}'\Sigma^{-1}\overrightarrow{1}+\lambda_2\beta'\Sigma^{-1}\overrightarrow{1}\\ 2c &= \lambda_1\overrightarrow{1}'\Sigma^{-1}\beta+\lambda_2\beta'\Sigma^{-1}\beta. \end{align*} Consequently, $\omega$ can be obtained from the above equation.

Answer 2

This is an interesting problem. I don't think the problem is set up correctly quite yet. I rewrote it slightly to correspond to how it's generally written as a quadratic program.

The optimization problem you write down fixes betas to be a certain value. That could make sense but instead I wondered if we could simply minimize beta across the portfolio while minimizing correlations. In that case, the optimization problem becomes:

\begin{align} \min_w \quad& w^\prime \Sigma w + w^\prime\lvert\beta\rvert\\ s.t. \quad& w^\prime\vec 1 = 1 \\ \text{and} \quad & w > 0 \end{align}

I don't think a closed-form solution exists to this problem. But it's quite easy (and fast) to solve this with a quadratic optimizer such as provided by cvxopt. Here is some example code:

import numpy as np
import matplotlib.pyplot as plt
import cvxopt as opt
from cvxopt import blas, solvers
import pandas as pd

np.random.seed(123)

# Turn off progress printing 
solvers.options['show_progress'] = False

## NUMBER OF ASSETS
n_assets = 4

## NUMBER OF OBSERVATIONS
n_obs = 1000

# Generate returns
return_vec = np.random.randn(n_assets, n_obs)

betas = np.array([.1, .8, .2, .05])

def markowitz_beta(returns, betas):
    n = len(returns)
    returns = np.asmatrix(returns)

    # Convert to cvxopt matrices
    # minimize: w * mu*S * w + betas * x 
    S = opt.matrix(np.cov(returns))

    # Minimize betas, can add mean returns here if desired
    pbar = opt.matrix(np.abs(betas))

    # Create constraint matrices
    # Gx < h: Every item is positive
    G = opt.matrix(-np.eye(n))   # negative n x n identity matrix
    h = opt.matrix(np.zeros(n))
    # Ax = b sum of all items = 1
    A = opt.matrix(1.0, (1, n))
    b = opt.matrix(1.0)

    # CALCULATE THE OPTIMAL PORTFOLIO
    wt = solvers.qp(S, pbar, G, h, A, b)['x']
    return np.asarray(wt).ravel()

weights = markowitz_beta(return_vec, betas)
print weights
print betas.ravel()
print np.dot(weights, betas)


Output:

[  3.50677883e-01   3.34638307e-07   2.50326847e-01   3.98994936e-01]
[ 0.1   0.8   0.2   0.05]
0.105083172131

As you can see, the highest betas receive very low weights.

We also wrote a blog post on this (and the code is a version from there) which you can find here: https://www.quantopian.com/posts/the-efficient-frontier-markowitz-portfolio-optimization-using-cvxopt-repost-cloning-of-nb-now-enabled

Disclaimer: I work at Quantopian.