Orthant-based variance decomposition in investment portfolios

A traditional and useful approach in portfolio theory is to consider the total risk of one security partitioned into two components: market risk and specific risk. In this paper, we propose a new variance decomposition based on a four-orthant partitioning of a bivariate normal distribution representing the returns on two stock portfolios. Four Euclidian quadrants around the central mean point are considered with their correspondent truncated distributions. We can consider stock pairs as events in which both stocks rise together, both decline or one rises and the other declines. The question that arises is what the contribution is of each quadrant to the overall mean return. And, what is the contribution of each quadrant to the total variance? We consider the mixture of four truncated bivariate normal distributions, where the weighting coefficients coincide with the quadrant probabilities. Through the law of total variance and the first and second moments of each truncated distribution, the requested decomposition formulas are deduced. These results are validated with straightforward simulations. The equations obtained here show higher variance concentration when considering diagonal quadrants, more than could be expected when compared to the subset probability mass. These results show that pair trading and low variance strategies could be better interpreted with this variance decomposition. Finally, a comparison with principal component theory is carried out showing that greater variance concentration can be found within this orthant scheme.