On some nonlinear dependence structure in portfolio design

Abstract

Constructing portfolios with high
returns and low risks is always in great
demand. Markowitz (1952) utilized correlation
coefficients between pairs of stocks to build
portfolios satisfying different levels of risk
tolerance. The correlation coefficient describes
the linear dependence structure between two
stocks, but cannot capture a lot of nonlinear
independence structures. Therefore, sometimes,
portfolio performances are not up to investors'
expectations. In this paper, based on the theory
of copula by Sklar (see [19]), we investigate
several new methods to detect nonlinear
dependence structures. These new methods
allow us to estimate the density of the portfolio
which leads to calculations of some popular risk
measurements like the value at risk (VaR) of
investment portfolios. As for applications,
making use of the listed stocks on the Ho Chi
Minh city Stock Exchange (HoSE), some
Markowitz optimal portfolios are constructed
together with their risk measurements.
Apparently, with nonlinear dependence
structures, the risk evaluations of some pairs of
stocks have noticeable twists. This, in turn, may
lead to changes of decisions from investors.