While the SD rules developed within the expected utility paradigm are employed to construct the various efficient sets as well as to predict some investors’ choices between a pair of portfolios, and are also employed to examine whether one can rationalize the holding of the market portfolio, it is difficult to find all the SD efficient diversification strategies which is the main drawback of the SD procedure. The reason for this difficulty is that one needs to conduct an infinite number of pairwise comparisons. On the other hand, there are some attempts to mitigate this disadvantage of the SD rules. Specifically, Shalit & Yitzhaki (1994) introduce the concept of MCSD so that it can be applied more readily to financial markets. If using this criterion for a given portfolio, pairwise dominating and dominated assets are determined, by marginally increasing the dominating asset at the expense of the dominated asset in the given portfolio the resulting portfolio dominates other possible portfolios constructed from available assets by SSD. It means that the risk averse investors expected utility does increase as a result of such marginal changes; that is, the risk averse investors expected utility of the resulting portfolio by marginally increasing the dominating asset and marginally decreasing the dominated asset in a given portfolio is larger than the expected utility of the given portfolio. Therefore, the concept of MCSD considers marginal changes in the proportions of two assets in a given portfolio and states the conditions under which all risk averse investors, when presented with a given portfolio, prefer to increase the share of one risky asset over that of another. However, MCSD only provides the conditions for dominance in the case of two assets, given a portfolio, and thus, it is possible to find some marginal changes in the shares of three or more assets in the given portfolio such that the resulting portfolio dominates the given portfolio by SSD, even if there is no dominance by MCSD (or SSD). Thus, Shalit & Yitzhaki (2003) consider marginal changes in the proportions of three or more assets and extend this analysis to allow for changes in many assets of a portfolio in order to find no portfolios which dominate the given portfolio by SSD. Therefore, the use of this criterion allows one to determine whether a given portfolio is SSD efficient relative to other possible portfolios created from available assets. Hence, the concept of MCSD deals with SSD efficiency analysis with diversification.

However, based on several factors which should be considered in evaluating the quality of various competing investment decision making rules, it would seem that MCSD cannot determine whether a given portfolio dominates the other possible portfolios or some portfolio dominates the given portfolio, cannot determine whether some investors have partial information by finding such information satisfying the conditions of the corresponding criterion, and cannot predict whether some investors would choose a given portfolio over the other possible portfolios. On the other hand, based on the effectiveness of each decision rule in terms of the relative size of the obtained efficient set, the higher order SD rules allow one to determine a definite ranking of portfolios because the more assumptions are imposed on preferences, the smaller the obtained efficient set. Based on the underlying assumptions needed to justify the employment of each decision rule in addition to the characteristics of the decision and the effectiveness of each decision rule, the PSD and MSD rules allow one to partially characterize the set of investors who choose a portfolio over the other one because one has a deal of freedom in specifying partial information. Based on the characteristics of the decision one faces an issue that should be addressed in addition to the effectiveness of each decision rule and the underlying assumptions, the RSSD rule allows one to predict risk loving investors’ choices between a pair of two portfolios. It would seem that these decision rules might be able to overcome the several problems of the concept of MCSD; however, there are no MCSD versions which are derived from the higher order SD framework, the PSD and MSD framework, and the RSSD framework.

In order to be able to deal with various SD efficiency analyses with diversification and to overcome the problems of the concept of MCSD, we extend the concept of MCSD to the case of:
(a) The higher order SD rules. Because the more information on preferences, generally one have the smaller the derived efficient set, the use of the extension of MCSD to higher order SD allows one to determine whether a given portfolio dominates the other possible portfolios by the higher order SD rule in some situations where the original MCSD (or SSD) fails.
(b) The PSD and MSD rules. Because there is no reason why risk loving cannot prevail in the negative or positive (but not in all) domains, the extension of MCSD to the PSD and MSD rules allows one to partially characterize the set of investors who have S-shape or reverse S-shape preferences by finding such information satisfying the conditions of this extension.
(c) The RSSD rule. Because it is not clear whether there is no dominance by RSSD although there is no dominance by SSD in addition to the fact that the quantitative aspects, that is, the relative size (vs ranking) of efficient sets under various criteria and specifications, constitute interesting empirical questions, the answers to which depend on the particular set of data used, the extension of the RSSD rule allows one to predict risk loving investors’ choices among a given portfolio and the other possible portfolios in some situation where the original MCSD (or SSD) fails.

Using the criterion (a) for a given portfolio, pairwise dominating and dominated assets can be determined, and thus, the use of the MCSD version which is derived from the higher order SD framework allows one to determine whether there is a combination of changes in the proportions of two assets so that the resulting portfolio dominates a given portfolio by the higher order SD rule whenever we can assume the underlying assumptions. As pointed out by Tehranian (1980), the empirical studies in the area of SD have encountered practical problems because of the lack of a definite ranking of portfolios in many instances. This extension might be able to determine a definite ranking among a given portfolio and the other possible portfolios because additional information on the derivatives can be utilized in deriving higher order SD rules in addition to the fact that the more information, generally one have the smaller the derived efficient set. Thus, the switch from MCSD to the case of higher order SD can be expected to avoid such practical problems by finding marginal changes in the shares of two assets in a given portfolio such that the resulting portfolio dominates the given portfolio by the higher order SD rule.

Using the criterion (b), we can investigate whether by altering the proportions of two assets in a given portfolio the resulting portfolio dominates the given portfolio by PSD or MSD. In other words, the use of this extension allows one to determine whether some investors have S-shape or reverse S-shape preferences by finding such information satisfying the conditions of the MCSD versions which are derived from the PSD and MSD framework. As pointed out by Post and Levy (2005), the traditional mean-variance capital asset pricing model fares poorly in explaining observed cross-sectional average stock returns, while the maintained utility assumptions of mean-variance capital asset pricing model are a possible explanation for a violation of the central prediction of this model. Specifically, there is compelling evidence that many decision-makers are risk seeking over a range, and the switch from MCSD to the case of PSD and MSD can be expected to deal with such difficulties by partially characterizing the set of investors who have S-shape or reverse S-shape preferences satisfying the conditions of these criteria.

Using the criterion (c) for a given portfolio, we can predict risk seeking investors’ choices with diversification whenever investors’ utility functions are unknown except for the fact that they are risk-seeking. If there is dominance by this criterion, knowing only risk-seeking the risk-seeking investors would choose the resulting portfolio by marginally increasing the dominating asset and marginally decreasing the dominated asset in a given portfolio over the given portfolio or would choose a pair of dominating and dominated assets via such marginal changes. This is because the risk seeking investors expected utility of the resulting portfolio is larger than the expected utility of the given portfolio or because the risk seeking investors expected utility does increase as a result of the marginal changes. As pointed out by Fong (2013), we might be able to rationalize some investors’ preference for lottery-type stocks if we assume that investors are locally risk seeking using RSSD tests. Because it is not clear whether there is no dominance by RSSD although there is no dominance by SSD, and vice-versa. Thus, if we can assume risk loving, this extension also might be able to rationalize risk-seeking investor's behavior and choices in some situation where the original MCSD fails, and the switch from MCSD to the case of RSSD can be expected to help understand such a phenomenon.

This paper is an attempt to extend the concept of MCSD to the cases of higher order SD, PSD, MSD, and RSSD in order to deal with these SD efficiency analyses with diversification so that the concept of these SD rules can be applied more readily to financial markets. On the other hand, this extension only considers marginal changes in the proportions of two assets; thus, it is possible to find some combination of changes in the proportions of three or more assets in a given portfolio so that the resulting portfolio dominates the given portfolio. Thus, we also consider marginal changes in the proportions of multiple assets in a given portfolio and attempt to extend our analysis to allow for changes in the shares of multiple assets to find no portfolios which dominate the given portfolio. Therefore, the extension of MCSD to each SD rule can determine whether a given portfolio is SD efficient or not, relative to all other possible portfolios constructed from available assets. We note that the MCSD versions which are derived from these SD frameworks are more confining concepts than the original SD rule because they only consider marginal changes in the given portfolio weights.

In conclusion it is noted that up to this point an attempt has been concerned with the preferences of investors and considering ways of determining a definite ranking, of predicting some investors’ choices, and of partially characterizing the set of investors who choose a portfolio, among a given portfolio and the other possible portfolios. However, Hanoch and Levy (1969) consider the choice of an individual decision-maker among alternative risky ventures and note that this choice may be regarded as a two-step procedure. Firstly, the decision-maker chooses an efficient set among all available portfolios, independently of the decision-maker’s tastes or preferences. Secondly, the decision-maker applies the decision-maker’s individual preferences to this set, in order to choose the desired portfolio. They deal with the analysis of the first step; that is, they deal with optimal selection rules which minimize the efficient set, by discarding any portfolio that is inefficient, in the sense that it is inferior to a member of the efficient set, from point of view of each and every individual, when all individuals’ utility functions are assumed to be of a given class of admissible functions. Based on this two-step procedure, our attempt is more confining concept than such a portfolio theory because rather than build an optimal portfolio we only consider determining whether a given portfolio of available assets belongs to some SD efficient sets given the limited knowledge of the investor’s preferences, although generally portfolio theory deals with the choice of investments that maximize an investor’s expected utility. Furthermore, Hadar and Russell (1969) note that in many theoretical as well as practical situations one is frequently confronted with the necessity (or at least desirability) of making a prediction about a decision maker’s preference (choice) between given pairs of uncertain alternatives without having any knowledge of the decision maker’s utility function. They propose two rules which are more powerful than the moment method such as Markowitz (1952) and specify conditions which will permit us to make predictions about preferences. They also note that the single most important property of these rules is that they are not only sufficient, but also necessary for the respective class of utility functions; that is, by means of the SD conditions we can not only predict preference, but we can also make a statement about the characteristics of the uncertain prospects. Based on the single most important property of their SD rules, our attempt is also and more confining concept than their theorems because rather than make a statement about the characteristics of the uncertain prospects we only consider determining a definite ranking, predicting some investors’ choices, and partially characterizing the set of investors who choose a portfolio, among a given portfolio and the other possible portfolios. In fact, they note that a result is obtained not only for differences between expected utilities, but also for differences between the expectations of any monotonically increasing function. Thus, Hadar and Russell (1969), Hadar and Russell (1974), and Thistle (1993) note that if there is dominance by FSD, then all the odd moments of the dominating prospect are larger than those of the dominated prospect. Therefore, if a prospect is preferred to the other one for all monotonic utility functions, then we can immediately say that the former prospect dominates the latter prospect by FSD, and thus, all the odd moments of the chosen prospect are larger than the respective moments of the other prospect. While from the above discussions, an attempt in this paper has several deficiencies, all of the preceding results might be able to be useful in determining a definite ranking, predicting some investors’ choices, and partially characterizing the set of investors who choose a portfolio when a group of investors will be unanimous in choosing among a given portfolio and the other possible portfolios.

Risk-seeking stochastic dominance (RSSD) is appropriate for all risk-seeking or risk loving investors (i.e., for all investors with increasing and convex utility functions) while second order stochastic dominance (SSD) is valid for all risk averse investors. Thus, the use of RSSD helps predict risk-seeking investors’ behavior and choices in some situation where SSD fails. However, a simple technique helping to predict risk-seeking investors' choices with diversification is not available in the RSSD framework because one needs to conduct an infinite number of pairwise comparisons. In this paper, we consider only marginal changes in the shares of two assets in a given portfolio and extend marginal conditional stochastic dominance (MCSD) to the case of RSSD. Therefore, the use of this criterion allows one to determine whether a given portfolio is RSSD efficient relative to all other possible portfolios constructed from available assets and to predict some risk-seeking investors' choices with diversification in some situation where the original MCSD (or SSD) fails. In addition, we extend this analysis to allow for changes in available assets of a portfolio so that it is possible to find no alternative portfolios that are pairwise preferred by all risk-seeking investors.

Finance; Stochastic dominance; Portfolio diversification; Behavioral finance; Security markets.

Higher order stochastic dominance (SD) rules assume additional information on the derivatives of the utility function, and the more information we assume on the utility function, generally one have the smaller the derived efficient set. Therefore, higher order SD rules can be employed for determining a definitive ranking result in some situation where second order stochastic dominance (SSD) fails; however, the SD procedure suffers from the lack of a search algorithm that builds efficient combinations of assets because it involves infinite pairwise comparisons. On the other hand, the concept of marginal conditional stochastic dominance (MCSD) mitigates this disadvantage of the SD rules, which deals with SSD efficiency analysis with diversification. In this paper, we extend MCSD to higher order SD, which investigates whether a portfolio is higher order SD efficient relative to the infinite possible portfolios constructed from multiple assets. This extension is a more confining concept than higher order SD because it considers only marginal changes in holding risky assets in a portfolio. In addition, MCSD provides the conditions for dominance in the case of two assets, given a portfolio, and thus, we extend this analysis to allow for changes in many securities of a portfolio in order to find no portfolio which dominates a given portfolio by higher order SD. The switch from MCSD to this extension can also determine a definite ranking for each set of portfolios because the more information we assume on the utility function, generally one have the smaller the derived efficient set.

JEL classification: G11.

Keywords: Finance; Stochastic dominance; Portfolio diversification; Efficiency analysis.