5.2.2. Semi-Variance

Some groundbreaking articles are summarized in Table 5 below.


**Table 5.** Selected work on Portfolio Selection and Semi variance.

Roy (1952) proposed the concept of downside risk and defined it as a risk below the target value. Markowitz (1959) proposed a well-known mean-semi-variance model to estimate the weights

of portfolio. Hogan and Warren (1972) pointed out the advantage of using the mean-semi-variance criterion in portfolio selection over the mean-variance model. Stone (1973) gave two interrelated three-parameter risk measures, in which the semi-variance was a special case. Hogan and Warren (1974) compared the difference between mean-variance model and mean-semi-variance model. Porter (1974) analyzed the relationship between stochastic dominance and mean-semi-variance model. Jahankhani (1976) empirically verified the relationship between return and risk in the mean-variance model and mean-semi-variance asset pricing model. Bawa and Lindenberg (1977) extended the semi-variance to the generalized lower partial moment framework, developed a Capital Asset Pricing Model (CAPM) using a mean-lower partial moment framework and derive explicitly formulae for the equilibrium values of risky assets that hold for arbitrary probability distributions. Fishburn (1977) applied the downside risk to the utility function model. Bawa (1978) extended the downside risk to higher order and showed its usability. Nantell and Price (1979) calculated variance and semi-variance by means of the distribution of prior portfolio returns and found that asset market portfolio prices with semi-variance were higher than variance at a certain risk level. Choobineh and Branting (1986) provided a simple form of semi-variance approximation by using mean, variance, critical value and cumulative probability below the critical value. Lee and Rao (1988) proposed a new asset pricing model in the framework of mean lower partial moment, which used semi-variance and semi-deviation to measure risk. Lewis (1990) used semi-variance as a measure of risk, applied it to the capital market and utility theory, and explained its advantages and disadvantages. Chen et al. (1991) proposed a set of linear regression models to approximate the semi-variance of the total returns of items with independent distribution. Chow et al. (1992) pointed out that in the absence of prior knowledge about the parametric structure of asset return distribution and the form of investor preference function, variance may no longer be an appropriate risk measure. They used various risk-return measures independent of distribution to test the efficiency and decentralization effect of international portfolio investment and found that semi-variance could effectively and conveniently identify risks. Tse et al. (1993) put forward an optimal strategy for personal investment using downside risk and proposed a model for accurate calculation of failure probability under the assumption of Brown's motion process. Markowitz (1993) transformed the mean-semi-variance portfolio optimization problem into the mean-variance optimization problem and used the critical line algorithm to obtain the optimal solution. Josephy and Aczel (1993) proposed an unbiased, consistent and effective estimators for the semi-variance.

With the deepening of risk research, the downside risk has attracted more and more attention (Rom and Ferguson 1994). Kaplan and Alldredge (1997) used a specific risk-based index, which could maintain a certain level of risk in different periods of time, to make a series of trade-offs between risk and return and studied its properties and performance in the case of semi-variance. Hamza and Janssen (1998) took transaction cost into consideration and applied the mean-semi-variance model to the portfolio selection problem, introduced a series of binary variables and separable constraints, and finally solved the portfolio optimization problem using separable techniques. Grootveld and Hallerbach (1999) analyzed the similarities and differences of using variance and downside risk as risk measures from empirical data and theory. Costa and Nabholz (2002) considered different computational forms of mean and semi-variance with errors and formulated robust mean-semi-variance portfolio selection problems based on linear matrix inequality optimization problems. Estrada (2004) noted that semi-variance was supported by theoretical facts and practical considerations and was a feasible measure of risk, and that the mean-semi-variance behavior criterion was perfectly consistent with the expected utility and the average compound return utility. Ballestero (2005) defined semi-variance as a weighted sum of squares deviating from the objective value of return on assets and applied it to portfolio selection. Jin et al. (2006) proved that no matter the market conditions and the distribution of stock returns, the effective strategy of mean-semi-variance in a single period could always be realized. They also established the realizability of the mean-semi-variance model under the condition of no arbitrage and extended it to the general downside risk measurement problem. Sira (2006) described the significant differences in portfolio outcomes using variance and semi-variance to measure risk and emphasized that using semi-variance as a risk measure could lead to more robust and effective boundaries. Chabaane et al. (2006) used a group of hedge funds with significant deviations from normal to consider the portfolio problem by maximizing expected return under the constraints of standard deviation, semi-variance, VaR and CVaR. However, if the asset return data do not follow the normal distribution, the mean-semi-variance model may produce inefficient portfolios. Consequently, Eldomiaty (2007) proposed the mean-semi-deviation model to measure the average loss rate. Huang (2008) proposed two fuzzy mean-semi-variance models and proved the properties of semi-variance in the case of fuzzy variables. Sayilgan and Mut (2010) regarded the portfolio problem as a multi-objective optimization, used the semi-variance and the lower partial moment as the risk measurement, and took genetic algorithm to solve the multi-objective optimization to achieve Pareto efficient portfolio. Cumova and Nawrocki (2011) transformed the exogenous asymmetric matrix into a symmetric matrix and proved that there was indeed a closed form of solution. On this basis, the critical line algorithm could be used to solve the mean semi-variance problem. Assuming investment capital and net cash flow as fuzzy variables, Zhang et al. (2011) proposed the reliability return index and the reliability risk index by using the expected value of credibility and the lower semi-variance of the fuzzy variables and gave the comprehensive risk return index for selecting the optimal investment strategy. Zhang et al. (2012) proposed a probabilistic mean-semi-variance entropy model to deal with multi-period portfolio selection under fuzzy returns. Metaxiotis and Liagkouras (2012) used a multi-objective evolutionary algorithm to solve the constrained mean-semi-variance portfolio optimization problem. Alimi et al. (2012) used fuzzy programming technology to solve multi-objective fuzzy mean semi-variance portfolio optimization model. Brito et al. (2016) proposed a flexible approach to portfolio selection using skewness/semi-variance bio-objective optimization framework, which allowed investors to analyze the effective balance between biases and semi-variables. Salah et al. (2016) noted that estimating portfolio risk by conditional variance or conditional semi-variance could obtain information about the future development of different asset returns and help investors to obtain more effective portfolio. Chen et al. (2017) considered that stock returns limited by expert estimates were described as uncertain variables and then verified three properties of semi-variances of uncertain variables. Based on the concept of semi-variances of uncertain variables, the mean-semi-variance models of two types uncertain portfolio selection were proposed.

The semi-variance is also used in the multi-period case. Bi et al. (2013) discussed the continuous time mean-semi-variance portfolio selection problem with probability distorted by nonlinear transformation, provided 'necessary and sufficient' conditions for the existence of feasibility and optimal strategy, and gave the general form of the solution when the optimal solution existed. Zhang (2015) considered the multi-period portfolio selection problem in a fuzzy investment environment, in which the return and risk of assets were characterized by probability mean and semi-variance, respectively. At the same time, based on the possibility theory, a new multi-period possible portfolio selection model was proposed, which includes risk control, transaction cost, borrowing constraints, threshold constraints and cardinality constraints. Liu and Zhang (2015) considered the multi-period fuzzy portfolio optimization problem with the shortest trading lot. Based on the possibility theory, a mean-semi-variance portfolio selection model was proposed to maximize the final wealth and minimize the cumulative risk within the entire investment level. Najafi and Mushakhian (2015) proposed a multi-stage stochastic mean-semi-variance CVaR model to deal with portfolio optimization problems. The parameters of semi-variance and CVaR were controlled at a certain confidence level. Huang et al. (2016) took the correlation between items and time sequence into account to propose a new mean-variance and mean-semi-variance model. Chen et al. (2018) took securities returns as uncertain variables to establish a multi-period mean-semi-variance portfolio optimization model with realistic constraints: transaction costs, cardinality and boundary constraints. Furthermore, if the security return was zigzag uncertain variable, they gave the equivalent deterministic form of mean-semi-variance model and proposed a modified imperialist competitive algorithm to solve the corresponding optimization problems.
