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Beta Constrained Markowitz Minimum Variance Portfolio

There was an interesting question on StackExchange recently (https://quant.stackexchange.com/questions/18160/beta-constrained-markowitz-minimum-variance-portfolio-closed-form-solution/19186#19186) asking how we could use Markowitz to find a portfolio that not only minimizes correlations but also minimizes beta (in response to the Quantopian open).

I took a quick stab at it and posted an example here. For more information, see the full NB at https://www.quantopian.com/posts/the-efficient-frontier-markowitz-portfolio-optimization-using-cvxopt-repost-cloning-of-nb-now-enabled.

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2 responses

You should generate the market return time series and stock residual time series separately and correlate them by using the betas. This creates a simulated world where CAPM is the model that generated the data. As the code is written, the stocks are independent and your estimate of covariance is pure statistical noise. Also, when taking the covariance matrix S = opt.matrix(np.cov(returns)) you should not be using the full matrix of stock returns, that defeats the purpose of a factor model. The covariance matrix should be calculated as $\Sigma = \beta\sigma^2\beta^\prime + \text{COV}[\epsilon]$ where $\beta$ is the beta vector, $\sigma^2$ is the variance of the market, and $\epsilon$ is the matrix of residuals.

Hopefully this makes sense.

Wayne,

thanks for your response. I completely agree -- would be a more realistic example if there actually was any correlation between the returns. Before I used this code to generate correlated returns:

def gen_cov(n_assets=5, var=1, covar=0, var_vec=None, noise=0):  
    X = np.random.randn(n_assets, n_assets) * noise + covar  
    if var_vec is None:  
        np.fill_diagonal(X, var)  
    else:  
        np.fill_diagonal(X, var_vec)  
    return X

def gen_returns(mean=0., cov=None, size=100, var=1, covar=0, n_assets=5):  
    if cov is None:  
        cov = gen_cov(var=var, covar=covar, n_assets=n_assets)  
    if np.isscalar(mean):  
        mean = np.ones((cov.shape[0],), np.float32) * mean  
    return pd.DataFrame(np.random.multivariate_normal(mean, cov, size=size))  

I posted the code and some more simulation studies here: https://www.quantopian.com/posts/monte-carlo-simulations-of-correlated-portfolios-the-quest-for-uncorrelated-return-streams