Abstract: Modern portfolio theory produces an optimal portfolio from estimates of expected returns and a covariance matrix. We present a method for portfolio optimization based on replacing expected returns with sorting criteria, that is, with information about the order of the expected returns but not their values. We give a simple and economically rational definition of optimal portfolios that extends Markowitz’ definition in a natural way; in particular, our construction allows full use of covariance information. We give efficient numerical algorithms for constructing optimal portfolios. This formulation is very general and is easily extended to more general cases: where assets are divided into multiple sectors or there are multiple sorting criteria available, and may be combined with transaction cost restrictions. Using both real and simulated data, we demonstrate dramatic improvement over simpler strategies.

Abstract: We consider the problem of portfolio selection within the classical Markowitz meanvariance framework, reformulated as a constrained least-squares regression problem. We propose to add to the objective function a penalty proportional to the sum of the absolute values of the portfolio weights. This penalty regularizes (stabilizes) the optimization problem, encourages sparse portfolios (i.e. portfolios with only few active positions), and allows to account for transaction costs. Our approach recovers as special cases the noshort- positions portfolios, but does allow for short positions in limited number. We implement this methodology on two benchmark data sets constructed by Fama and French. Using only a modest amount of training data, we construct portfolios whose out-of-sample performance, as measured by Sharpe ratio, is consistently and significantly better than that of the naive evenly-weighted portfolio which constitutes, as shown in recent literature, a very tough benchmark.

Abstract: In this paper, we give the axiomatic characterization of risk measures and discuss the treads of developments in this area. The main recently proposed risk measures are presented, and their properties and relations are discussed. The corresponding versions of dynamic risk measure are also briefly introduced.