Portfolio optimization using Mean Absolute Deviation (MAD) and Conditional Value-at-Risk (CVaR)

Abstract

This paper investigates the efficiency of traditional portfolio optimization models when the returns of financial assets are highly volatile, e.g., in financial crises periods. We also develop alternative optimization models that combine the mean absolute deviation (MAD) and the conditional value at risk (CVaR), attempting to mitigate inefficient, low return and/or high-risk, portfolios. Three methodologies for estimating the probability of the asset's historical returns are also compared. By using historical data on the Brazilian stock market between 2004 and 2013, we analyze the efficiency of the proposed approaches. Our results show that the traditional models provide portfolios with higher returns, but our propose model are able to generate lower risk portfolios, which might be more attractive in volatile markets. In addition, we find that models that do not use equiprobable scenarios produce better results in terms of return and risk.

Fuzzy Portfolio Optimization of Power Generation Assets

“…It can be seen from Table 1 as well that for the given value of the portfolio return, the risk level in FSMAD Model 1 is slightly smaller. It means that the return value for each technology expressed by the expected value from probability theory (see model (5)) is almost the same as in the case of the trapezoidal membership function from model (8).…”

“…Moreover, the MAD model is more compatible with the fundamental principle of rational decision-making. Despite the fact that the mean-variance Markowitz (1952) model is a solid, well-known method, which laid the foundation for portfolio theory, it has been proven that MAD produces similar portfolio returns [5,27], which makes us confident in MAD's optimization ability. Considering this alternative definition of risk MAD, as proposed by Konno and Yamazaki [2], the portfolio rate of return, R p , is given as:…”

“…Assuming that each rate of return, R i , is a possibilistic variable, the simple conversion of model (5) presented above is replacing the expected value of the rate of return for asset i, E(R i ), by an adequate fuzzy mean value. This fuzzy mean value depends on the form of membership function assumed.…”

“…Reconsidering the semi-mean-absolute deviation portfolio selection model (5) and by replacing (13) with Equations (11) and (12), respectively, the following problem results:…”

“…Application of the MAD model (based on the linear semi-mean-absolute deviation risk function) enabled Mansini and Speranza [4] to introduce new specifications derived from market structure and from operative constraints into the portfolio selection model of the Milan stock exchange. De Silva et al [5] applied the MAD as well as the CVaR approach in order to avoid inefficient, low return, and/or high-risk portfolios on the Brazilian stock exchange. Furthermore, Liu [6] used the concept of the mean-absolute deviation function proposed by [2] when the asset returns from financial markets are represented by interval data.…”