Comparative Analysis of the Markowitz Model and the Single-Index Model: An Empirical Study on N/T Ratios and Investment Constraints

Abstract

Two models have been extensively applied in the use of the modern portfolio theory. Markowitz model (MM), based on the mean–variance framework, prescribes precise theoretical implication of the efficient frontier under direct modeling of inter-asset covariance structure. Even though theoretically correct, however, MM does nevertheless continue to harbor serious shortcomings, viz., excessively high complication of computation and over-sensitiveness toward estimation error of the input data, particularly the estimation of the covariance matrix. Vehemently opposed to MM, on the other hand, is the Single-Index Model (IM) matrix, which incorporates the market index, a general factor, so drastically reducing the dimension of estimation of inter-asset correlation, and thus correspondingly increasing significantly the tractability and speed of computation. Unfortunately, however, the simplification does bring over-reliance toward the market factor, in effect excluding rigorous inter-asset correlations and thus potentially warping the shape of efficient frontier. Accordingly, both theoretically and practically, there will remain controversies over balancing accuracy and tractability between MM and IM. It undertakes systematic comparisons of the performance between the Markowitz model and the Single-Index Model under diverse conditions of varying sample sizes (T) of the data set, asset numbers (N), and constraint conditions of investment. It is discovered, under conditions of large-dimensional, small-sample, and no constraint, IM exhibits superior robustness, whereas under conditions of sufficiently large sample size and constraint, MM exhibits higher competitiveness. By filling the gap of theoretical and application values, the paper presents a concrete basis of model selection of application of portfolio optimization.