Abstract

Modern portfolio theory indicates that portfolio optimization can be carried out based on the mean-variance model, where returns and risk are represented as the average and variance of the historical data of the stock’s returns, respectively. Several studies have been carried out to find better risk proxies, as variance was not that accurate. On the other hand, fewer papers are devoted to better model/characterize returns. In the present paper, we explore the use of the reliability measure P(Y<X) to choose between portfolios with returns given by the distributions X and Y. Thus, instead of comparing the expected values of X and Y, we will explore the metric P(Y<X) as a proxy parameter for return. The dependence between such distributions shall be modelled by copulas. At first, we derive some general results which allows us to split the value of P(Y<X) as the sum of independent and dependent parts, in general, for copula-dependent assets. Then, to further develop our mathematical framework, we chose Frank copula to model the dependency between assets. In the process, we derive a new polynomial representation for Frank copulas. To perform a study case, we considered assets whose returns’ distributions follow Dagum distributions or their transformations. We carried out a parametric analysis, indicating the relative effect of the dependency of return distributions over the reliability index P(Y<X). Finally, we illustrate our methodology by performing a comparison between stock returns, which could be used to build portfolios based on the value of the the reliability index P(Y<X).

Highlights

  • Portfolio management and optimization consist of selecting a set of assets, and their respective portfolio participation weights, which best satisfy the investor’s ideal risk–return relationship [1]

  • We illustrate our methodology by performing a comparison between stock returns, which could be used to build portfolios based on the value of the the reliability index P(Y < X )

  • When modelling the reliability of a given component whose strength is described by a random variable X subjected to stresses modelled by a random variable Y, the component fails if the stress applied to it exceeds the strength, while the component works whenever

Read more

Summary

Introduction

Portfolio management and optimization consist of selecting a set of assets, and their respective portfolio participation weights, which best satisfy the investor’s ideal risk–. Several authors covered the following topics: Evolutionary computation in the discovery of rules in algorithmic trading for shares [6]; swarm intelligence research for portfolio optimization [7]; portfolio optimization problem with the Markowitz mean-variance structure [3]; 20 years of portfolio optimization based on operational research [8] Despite these contributions, fewer papers were devoted to better model/characterize returns. This paper presents, at first, some general mathematical formulations and theorems which will latter support an empirical exploratory analysis of the portfolio optimization problem Such analysis considered dependence as being modelled by Frank copulas and that asset prices follow distributions from the Dagum family. The codes developed and used in the present paper are presented at the end of the paper

Reliability
Copulas and Reliability Measures
Revisiting Frank Copula
Reliability Calculations
Numerical Calculations
Parametric Study
Applications to Real Stock Data
Modelling Stock Return Data as Log–Dagum Random Variables
Reliability Analyses
Methods
Full Text
Published version (Free)

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call