Mean semi-deviation from a target and robust portfolio choice under distribution and mean return ambiguity

Abstract

a b s t r a c tWe consider the problem of optimal portfolio choice using the lower partial moments risk measure for a market consisting of n risky assets and a riskless asset. For when the mean return vector and variance/covariance matrix of the risky assets are specified without specifying a return distribution, we derive distributionally robust portfolio rules. We then address potential uncertainty (ambiguity) in the mean return vector as well, in addition to distribution ambiguity, and derive a closed-form portfolio rule for when the uncertainty in the return vector is modelled via an ellipsoidal uncertainty set. Our result also indicates a choice criterion for the radius of ambiguity of the ellipsoid. Using the adjustable robustness paradigm we extend the single-period results to multiple periods, and derive closed-form dynamic portfolio policies which mimic closely the single-period policy.

New Safe Approximation of Ambiguous Probabilistic Constraints for Financial Optimization Problem

“…Proof. We already know that (20)- (21) represent the RC of (4), with the perturbation set being (16). Now let us prove that if the independent random variables , = 1, .…”

“…. , , satisfy hypotheses P.1-2 and , , are feasible for (20)- (21), then is feasible for (4) with probability of at least 1 − exp{− 2 /2}, = min{Ω, / √ }. We can pass from (4) to the following inequality…”

“…Huang et al [20] considered the relative robust conditional value-at-risk portfolio selection problem, where the underlying probability distribution of portfolio return belonged to a certain set. Pınar and Paç [21] derived the distributionally robust portfolio rules, addressed potential uncertainty in the mean return vector, and derived a closed-form portfolio rule when the uncertainty in the return vector was modeled via an ellipsoidal uncertainty set. Liu et al [22] adopted an uncertainty set in distributionally robust portfolio selection problems, where the considered uncertainties are in terms of both the distribution and the first two order moments.…”

Portfolio selection under uncertainty: a new methodology for computing relative‐robust solutions

“…Initial contributions, under the absolute-robust design, focused on the formulation of the robust counterparts of classic portfolio optimization problems or the development of deterministic algorithms in order to solve them (El Ghaoui and Lebret, 1997;Ben-Tal and Nemirovski, 1998;Goldfarb and Iyengar, 2003;Halldórsson and Tütüncü, 2003). More recent contributions explored the close relationship between the structure of the uncertainty set and the risk measure selected (Natarajan et al, 2009;Ben-Tal et al, 2010;Paç and Pınar, 2014), analyzed the effects of the uncertainty sets' structure and scale (Lu, 2006(Lu, , 2011Roy, 2010;Gregory et al, 2011;Kaläı et al, 2012;Fabretti et al, 2014), and compared the characteristics of absolute-robust portfolios to classic portfolios (Kim et al, 2013a(Kim et al, , 2013b(Kim et al, , 2014c.…”

“…There are several deterministic methods available in the literature for solving similar minimax optimization problems (Yaman et al., 2007; Pınar and Paç, 2014; Pınar, 2016; Paç and Pınar, 2018). In this work, a different path was taken.…”