Asset Allocation and return distribution: from mean-variance to mean-VaR

One of the most classic, but also most important issues in finance is that of asset allocation. If you have a sum of money that you want to invest, it makes common sense to divide this sum among the most advantageous investment alternatives, but a mathematically satisfactory solution to this problem is not trivial. The most intuitive approach seems to be to define a return metric and a risk metric to seek an optimal balance between the two; a similar logic leads directly to the consideration of maximizing a utility function that, in some sense, incorporates the portfolio's return and risk metrics. Sometimes these are only formal differences, but the two paths cross, as in the case of the meanvariance model, which takes an approach to the problem of asset allocation that has had great success finance, both in terms of theoretical approach and practicality. The emergence and diffusion of this model dates back to the 1950s and the work of Henry Markowitz. Although a long series of contributions has advanced the portfolio allocation problem in various ways, the guiding principles of the mean-variance model remain essentially the same. Although various criticisms have been made over the years, this model remains a fundamental reference point in both the financial industry and academia. One of the criticisms of the mean-variance model, however, is that the consideration of mean and variance as return-risk indicators is not considered entirely appropriate; variance, in particular, may be an inadequate measure of risk. In any case, the distribution of securities returns included in the basket of assets from which the optimal allocation of capital is sought plays a crucial role. For example, it can be shown that if these returns are described by a multivariate Normal distribution, the mean-variance approach should be used in every case, but this is not true for every distribution. Moreover, it is well known that financial returns in general do not fit the normality hypothesis well, and it is this that has sparked interest in finding alternatives to the model in question. In reality, although less well known, the solution of the mean variance is the one to refer to even for the more general class of Elliptical distributions. It is therefore of great interest to evaluate the adequacy of the latter when it comes to representing the reality of financial returns, which is a more delicate task than in the Normal case. Moreover, it is shown that, from the point of view of distribution, asymmetry is the most problematic feature with respect to the application of the mean-variance model. Therefore, it is particularly important to examine the presence of asymmetry in financial returns. The presence of asymmetry is generally advocated, but is itself not completely sure, especially for certain asset classes and certain time horizons. In any case, the need to overcome the mean-variance model has led to the development of alternative models, such as the multi-moment approach, which typically leads to considering the inclusion of third and fourth moments, but this leads to theoretical and computational problems. A more parsimonious alternative is to use the mean absolute deviation as a measure of risk. However, the more promising alternative seems to be the mean-CVaR model introduced by Rockafellar and Uryasev, which has been widely used in recent years. The proposed study aims to isolate the impact of the distributional properties of different asset baskets on portfolio optimization outcomes by highlighting the differences between the meanvariance model and alternatives. To achieve this goal, the paper is structured as follows: - A first theoretical part. A utilitarian justification is offered that differs from the traditional one. On this basis, the mean-variance model and the main alternative models are described. The importance of distributional hypotheses in general and asymmetry in particular with respect to the mean-variance model and alternatives will become clear. An in-depth analysis of the concept of asymmetry in the context of preferences will be offered and an overview of alternative distributional hypotheses to the Normal distribution will be provided, highlighting the role of Elliptical distributions. - In the second part, a review of the literature that has addressed the issue of the asymmetry of financial returns is presented. In doing so, some problematic nature of literature becomes apparent and solutions are proposed. Then, an empirical analysis is carried out for different asset classes. The same issue is addressed first in a univariate and then in a multivariate sense. In the multivariate case, we refer to the role of Elliptical distributions and therefore test the hypothesis of elliptical symmetry for different groups of assets. - In the third part, a direct comparison is made between the mean-variance and meanabsolute deviation and the mean-CVaR model; the two alternatives are chosen based on the considerations made in the first part, and the asset holdings underlying the optimizations are based on the data used in the second part. The comparison is based on a simulation strategy; in this phase, different degrees of asymmetry are assumed/adjusted to highlight the contribution of this asymmetry to the allocation differences. Before moving to the empirical part, a review of the literature that has addressed the same issue will be provided. This review will be critical and will highlight a number of problems that make the conclusions presented in some papers questionable. The strategy proposed here aims to overcome these problems.