Sample tangency portfolio, representativeness and ambiguity: impact of the law of small numbers

ABSTRACT We provide a model for understanding the impact of sample-size neglect for cases where an investor uses the sample estimator of the covariance matrix in order to obtain a tangency portfolio. By assuming an erroneous hypothesis, we look for a family of covariance matrices such that their deviation from the sample, in terms of the utility function, is a decreasing function of the latter under an erroneous hypothesis regarding the market structure of returns. This approach allows us to characterize the ambiguity of investor in relation to the Sharpe model (the most, the least and the relative ambiguous investors), and to compute a covariance matrix characterizing each ambiguity profile. We show that the expected loss is better for the most ambiguous investors than it is for the relative ambiguous - and the expected loss for the relative ambiguous investors is, in turn, better than that obtained for the least ambiguous profiles. All are better than the sample covariance matrix. We show how the relative profile actually denotes a state of equilibrium between the two extreme cases, and may be viewed as a multicriteria maxmin approach. We show that the ambiguity actually derives from the finit; sample property of the investment universe and follows a power-law distribution. We also derive an analytical expression for the risk aversion arising from sample-size neglect.