A direct approach to risk approximation for vast portfolios under gross-exposure constraint using high-frequency data

It is well known that the traditional estimated risk for the Markowitz mean-variance optimization had been demonstrated to seriously depart from its theoretic optimal risk due to accumulation of input estimation errors. Fan et al. (in J. Am. Stat. Assoc. 107:592–606, <CitationRef CitationID="CR4">2012a</CitationRef>) addressed the problem by introducing the gross-exposure constrained mean-variance portfolio selection. In this paper, we present a direct approach to estimate the risk for vast portfolios using asynchronous and noisy high-frequency data. This approach alleviates accumulation of the estimation error of tens of hundreds of integrated volatilities (or co-volatilities), and on the other hand it has the advantage of smoothing away the microstructure noise in the spatial direction. Based on the simple approach, together with the “pre-averaging” technique, we obtain a sharper bound of the risk approximation error than that in Fan et al. (in J. Am. Stat. Assoc. 107:412–428, <CitationRef CitationID="CR5">2012b</CitationRef>). This bound is locally dependent on the allocation plan satisfying the gross-exposure constraint. The bound does not require exponential tail of the distribution of the microstructure noise. Finite fourth moment suffices. Our work also demonstrates that the mean squared error of the risk estimator can be decreased by choosing an optimal tuning parameter depending on the allocation plan. This is more pronounced for the moderately high-frequency data. Our theoretical results are further confirmed by simulations. Copyright Sociedad de Estadística e Investigación Operativa 2013