Almost marginal conditional stochastic dominance Michel M. Denuit a , Rachel J. Huang b,⇑,1 , Larry Y. Tzeng c,1 , Christine W. Wang c a Institut de statistique, biostatistique et sciences actuarielles (ISBA), Université Catholique de Louvain, Louvain-la-Neuve, Belgium b Graduate Institute of Finance, National Taiwan University of Science and Technology, Taiwan c Department of Finance, National Taiwan University, Taiwan article info Article history: Received 8 October 2012 Accepted 16 December 2013 Available online 2 January 2014 JEL classification: D81 Keywords: Marginal conditional stochastic dominance Almost stochastic dominance Asset allocation Optimal investment abstract Marginal Conditional Stochastic Dominance (MCSD) developed by Shalit and Yitzhaki (1994) gives the conditions under which all risk-averse individuals prefer to increase the share of one risky asset over another in a given portfolio. In this paper, we extend this concept to provide conditions under which most (and not all) risk-averse investors behave in this way. Instead of stochastic dominance rules, almost stochastic dominance is used to assess the superiority of one asset over another in a given portfolio. Switching from MCSD to Almost MCSD (AMCSD) helps to reconcile common practices in asset allocation and the decision rules supporting stochastic dominance relations. A financial application is further pro- vided to demonstrate that using AMCSD can indeed improve investment efficiency. Ó 2013 Elsevier B.V. All rights reserved. 1. Introduction The most common investment rule is certainly the mean–vari- ance (MV) rule. It is easy to compute, and in some cases even to ex- press analytically, which explains why the MV rule has become most widely accepted throughout the financial profession (see Liz- yayev and Ruszczyn ´ ski, 2012). On the other hand, Expected utility (EU) maximization lies at the heart of modern investment theory and practice. To be analytically consistent with EU maximization, the MV rule requires strong assumptions (such as quadratic utility functions or normally distributed returns), which seldom hold in practice. However, EU requires the specification of the investor’s utility function which appears extremely difficult. Stochastic dominance (SD) is an alternative approach which avoids all these shortcomings by considering the preferences shared by all the rational decision-makers. Therefore, it does not require a specific utility function nor a specific return distribution. Furthermore, it uses the whole probability distribution rather than the usual MV parameters of standard deviation and mean return. The second-degree stochastic dominance (SSD) rule is appropriate for the class of all risk-averse EU maximizers. It has the advantage that it requires no restrictions on probability distributions nor on investors’ utility functions outside of the requirement that inves- tors be risk-averse, EU maximizers. Given a portfolio of assets, marginal conditional stochastic dominance (MCSD) has been introduced by Yitzhaki and Olkin (1991) and Shalit and Yitzhaki (1994) as a condition under which all risk-averse EU maximizer individuals prefer to increase the share of one risky asset over that of another. Specifically, these authors consider risk-averse investors holding a given portfolio of risky assets and derive criteria expressed in terms of the joint probability distribution of the assets and of the underlying portfo- lio to ensure that the share of an asset is increased at the expense of another in the portfolio. This helps to detect inefficiency and to improve inefficient portfolios. MCSD has been successfully applied to solve asset allocation problems by several authors, including Clark et al. (2011), Clark and Kassimatis (2012, 2013), Shalit and Yitzhaki (2010). MCSD expresses the conditions under which all risk-averse investors holding a specific portfolio prefer one asset to another. Furthermore, MCSD has been shown to involve more than pairwise comparisons as developed by Shalit and Yitzhaki (2003). It is a less demanding concept and more adapted to empir- ical analysis than SSD because it considers only marginal changes of holding risky assets in a given portfolio. Despite their theoretical attractiveness, MV and SSD rules may create paradoxes in the sense that they fail to distinguish between some risky prospects, whereas it is obvious that the vast majority of investors would prefer one over the other. This is why Bali et al. (2009) considered almost stochastic dominance (ASD) as a viable 0378-4266/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jbankfin.2013.12.014 ⇑ Corresponding author. Tel.: +886 930553988. E-mail address: rachel@mail.ntust.edu.tw (R.J. Huang). 1 Research fellows at Risk and Insurance Research Center, National Chengchi University, Taiwan. Journal of Banking & Finance 41 (2014) 57–66 Contents lists available at ScienceDirect Journal of Banking & Finance journal homepage: www.elsevier.com/locate/jbf