A Parametric Sharpe Ratio Optimization Approach for Fuzzy Portfolio Selection Problem

Abstract

When facing to make a portfolio decision, investors may care more about every portfolio's performance on a return and risk tradeoff. In this paper, a new low partial moment measurement that only punishes the loss risk is defined for selection variables based on L-S integral. Furthermore, a new performance measure for portfolio evaluation is proposed to generalize the Sharpe ratio in the fuzzy context. With the optimal performance criterion, a new parametric Sharpe ratio portfolio optimization model is developed wherein uncertain returns are presented as parametric interval-valued fuzzy variables. To make the proposed model easy to solve, we transform the fractional programming into an equivalent form and solve it with domain decomposition method (DDM). Finally, we apply the proposed performance measure into a portfolio selection problem, compare the computational results in different cases, and analyze the influence of different parameters on the optimal portfolio.

The Performance Measurement of Generalized Sharpe Ratio and Economic Performance Measure: A Hedge Funds Example

“…Thus, we can conclude that the EPN(NP), the GSR(NP), and the GSR(P-A) produce more similar rankings than the SR. Only the GSR(P-A) approximates well for the three parametric estimation methods. Our findings are consistent with [7,8,16,17,18] Notes: The SR stands for the Sharpe ratio. The GSR(P-A) is the parametric estimate of the generalized Sharpe ratio (GSR) proposed by Alexander [1].…”

“…Another parametric method to estimate the GSR was developed by Alexander [1] and is called GSR(P-A). She used the Taylor expansion of the expected utility to get the certainty equivalent of the portfolio investment and obtain the maximum expected utility by using the approximating method by [17]. The formula is as follows:…”

“…For instance, Henriksson and Merton [11] argued that applying the Sharpe ratio had been restricted and caused performance valuation problems because of non-linear payoffs. Furthermore, the Sharpe ratio generates the wrong valuation when the return distribution has deviated from normality [17]. The study of [17] also proved that flattening returns can increase the Sharpe ratio for a long time.…”

“…Furthermore, the Sharpe ratio generates the wrong valuation when the return distribution has deviated from normality [17]. The study of [17] also proved that flattening returns can increase the Sharpe ratio for a long time. As a result, the Sharpe ratio cannot perform well for portfolio performance measures.…”

Uncertain Portfolio Selection with Borrowing Constraint and Background Risk

“…Kar et al [27] considered the Sharpe ratio and the value-at-risk ratio and then proposed a biobjective fuzzy portfolio selection model. In addition, papers by Xia et al [28], Huang [29], Deng and Li [30], Li et al [31], Guo et al [32], Liu and Li [33], and Zhou et al [34] have also developed the theory of fuzzy portfolio selection. However, further studies show that paradoxes appear when fuzzy set theory is employed to address belief degree (Liu [22]).…”