Tangency portfolio weights under a skew-normal model in small and large dimensions

In this paper, we investigate the distributional properties of the estimated tangency portfolio (TP) weights assuming that the asset returns follow a matrix variate closed skew-normal distribution. We establish a stochastic representation of the linear combination of the estimated TP weights that fully characterizes its distribution. Using the stochastic representation we derive the mean and variance of the estimated weights of TP which are of key importance in portfolio analysis. Furthermore, we provide the asymptotic distribution of the linear combination of the estimated TP weights under the high-dimensional asymptotic regime, i.e., the dimension of the portfolio <i>p</i> and the sample size <i>n</i> tend to infinity such that p/n→c∈(0,1). A good performance of the theoretical findings is documented in the simulation study. In an empirical study, we apply the theoretical results to real data of the stocks included in the S&amp;P 500 index.

本文针对资产收益率服从矩阵变量闭合斜正态分布（matrix variate closed skew-normal distribution）的情形，探究了估计得到的切线投资组合（tangency portfolio, TP）权重的分布特性。我们推导得到估计切线投资组合权重线性组合的随机表示形式，该形式可完整刻画其分布特征。借助该随机表示，我们推导出切线投资组合估计权重的均值与方差——二者均为投资组合分析中的核心关键指标。此外，我们在高维渐近框架下给出了估计切线投资组合权重线性组合的渐近分布，其中投资组合维度$p$与样本量$n$均趋于无穷，且满足$p/n→c∈(0,1)$。仿真研究证实了本文理论结果具备优异的实证表现。在实证研究中，我们将本文理论结果应用于标普500（S&P 500）指数成分股的真实交易数据。