The $$\alpha$$ α -Tail Distance with an Application to Portfolio Optimization Under Different Market Conditions

It is very important to find some new distance measurement methods to estimate the correlation of the return of stocks because that the traditional distance measurement methods do not consider the influence of the market conditions. In this paper, a new distance measurement which called the $$\alpha$$ α -tail distance is introduced to measure the correlations of stock’s returns under the different market conditions. We give some properties of the $$\alpha$$ α -tail distance and provide some details on how to determine the parametric $$\alpha$$ α in the $$\alpha$$ α -tail distance via the market condition evaluation indices. A mean variance model with variable cardinality constraints based on the hierarchical clustering is given as an application of the $$\alpha$$ α -tail distance. Moreover, the numerical example verifies that the $$\alpha$$ α -tail distance is more suitable for measuring the correlation between stock’s returns than other traditional distance measurements under the different market conditions.