Portfolio management with higher moments: the cardinality impact

Abstract

This paper extends the study of the cardinality impact on portfolio performance, from the traditional mean‐variance framework to more general frameworks that include higher moments. For each framework, we propose a biobjective model that allows the investor to explicitly analyze the efficient trade‐off between expected utility and cardinality. We applied the proposed methodology to data from the Portuguese Stock Index (PSI20 index). The empirical results show that, in‐sample, the certainty equivalent and the Sharpe ratio increase with the cardinality level in all frameworks. The results also suggest that there are no performance gains, in‐sample, in terms of certainty equivalent, when higher moments are considered. Out of sample, the turnover increases up to a certain cardinality level, then decreases. For certain cardinality levels, there are gains in terms of out‐of‐sample certainty equivalent and Sharpe ratio, when skewness and kurtosis are considered. Finally, we check the robustness of these results in a large dataset from the EUROSTOXX50 index.

On the Gains of Using High Frequency Data in Portfolio Selection

“…Following the same notation as in Brito et al (2017aBrito et al ( , 2017b, suppose that the investor has a certain wealth to invest in a set of stocks. In this setting, the portfolio at time is defined by a × 1 vector, , of weights representing the proportions of the total wealth invested into the stocks.…”

“…In Problem (1), the investor's expected utility, [ ( , +1 )], needs to be estimated. In this paper, and following Brito et al (2017b), we consider the approximations for the investor's expected utility based on the second, third and fourth order Taylor expansions around the expected return of the portfolio, ( , +1 ). The expansions are truncated at the fourth order because there are no theoretical grounds, in terms of the investor's preferences, to include higher polynomial terms (for further details see Dittmar, 2002;Kimball, 1993;Martellini and Ziemann, 2010).…”

“…where represents the relative risk aversion coefficient (a higher value of implies more risk aversion). A CRRA-utility allows the incorporation of preferences toward higher moments in a parsimonious manner (Brandt et al, 2009;Brito et al, 2017aBrito et al, , 2017b. According to the previous section, three different frameworks are considered, denoted by MV, MVS and MVSK, where the approximated expected utility is given by Equation ( 2), Equation ( 6) and Equation ( 11), respectively.…”

“…Consequently, it seems that investors take into account the non-normal features of the returns distribution, showing preference for positive skewness (see, e.g., the seminal work of Arditti, 1967) and disliking high kurtosis (see, e.g., the empirical work of Maringer and Parpas, 2009). On the other hand, several empirical studies suggest that there are performance gains when higher moments (namely skewness and kurtosis) are considered in portfolio selection (see, e.g., Amaya et al, 2015;de Athayde and Flores, 2004;Brito et al, 2017aBrito et al, , 2017bHarvey et al, 2010;Maringer and Parpas, 2009). For many years, GARCH (see Bollerslev, 1986;Engle, 1982;Nelson, 1991) and stochastic volatility models (see Taylor, 1986) have been widely used in the financial services industry.…”

“…This paper contributes empirically to answer this question. Motivated by the works of Brandt et al (2009) and Brito et al (2017aBrito et al ( , 2017b, the portfolio selection problem is defined in a Constant Relative Risk Aversion (CRRA) world, where the returns distribution is characterized not only by the first two moments but also by the skewness and kurtosis. Therefore, three different frameworks are considered: MV (mean-variance), MVS (mean-variance-skewness) and MVSK (meanvariance-skewness-kurtosis).…”