Sample Tangency Portfolio, Representativeness and Ambiguity: Impact of the Law of Small Numbers

We provide a model for understanding the impact of the sample size neglect when an investor, hoping for the tangency portfolio uses the sample estimator of the covariance matrix for this purpose. By assuming a wrong hypothesis, we are looking for a family of covariance matrices such as their difference in terms of the utility function with the sample one is a decreasing function of the latter under a wrong hypothesis regarding the market structure of returns. This approach allows us to characterize the ambiguity of investor reliance on the Sharpe model (the most, the less 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, than for the relative ambiguous which is better than the one obtained from the less ambiguous profiles. However, they are all better than the sample covariance matrix. We show how the relative profile denotes actually an equilibrium state between the two extreme cases, and may be viewed as a multi-criteria maxmin approach. We show that ambiguity comes actually from the finite sample property of the investment universe and follows a power law distribution. We also derive an analytical expression of the risk aversion coming from the sample size neglect.