# how to construct a diversified portfolio based on correlation

I have a porfolio of indexes and I built up a python model based on spearman correlation (I used a spearman and not a pearson because, after running some test on outliers and normality checks, I have some outliers and normality checks failed) to eliminate all the indexes with correlation coefficient higher than 0.8 expecting to reduce the risk and improve sharpe ratio. I have no target variable obviously, so the reason for eliminating feature is not to run a machine learning algorithm for prediction but just to improve the performance of the portfolio. It is a portfolio construction problem as I assume for now, equal weights, no optimisation at this point.

My first question is: is this a good approach to diversify? or there is a better way? I thought of PCA but I do not want to lose interpretability Second question, are indexes with negative correlations to be removed as well? looking at the definition of a portfolio volatility, the risk should reduce with neg correlation however, returns have opposite effect on the trend and they should cancel each other out.

• Check out: hudsonthames.org/…
– JPN
Jun 29 '20 at 13:43
• its a good reference.So correlation between asset/index makes the rows of my covariance matrix more linearly dependent thus the condition number of such matrix increases and the inversion of covariance generates instability. so removing the highly correlated ones may help, but, here I'm not solving an optimisation problem as I wrote. I just want to know if removing highly correlated index is considered correct in particular wrt to negative corr. During a portfolio construction, is there a metric which can tell me if my reduced portfolio is better than the larger original one? the sharpe ratio? Jun 29 '20 at 15:47

min $$\mathbf{w^{T} C w}$$
subject to constraints that weights sum to 1 and are non-negative, where $$\mathbf{C}$$ is the correlation matrix of multivariate asset returns. If you also regularize the portfolio weights with an L2-norm by adding $$\| \mathbf{w} \|^2$$ to the objective function, this should also alleviate multicollinearity by assigning equal weights within each correlated group of assets and improve out-of-sample performance by reducing overfitting