Many financial optimization problems involve future values of security prices, interest rates and exchange rates which are not known in advance, but can only be forecast or estimated. Several methodologies have therefore been proposed to handle the uncertainty in financial optimization problems. One such methodology is Robust Statistics, which addresses the problem of making estimates of the uncertain parameters that are insensitive to small variations. A different way to achieve robustness is provided by Robust Optimization, which looks for solutions that will achieve good objective function values for the realization of the uncertain parameters in given uncertainty sets. Robust Optimization thus offers a vehicle to incorporate an estimation of uncertain parameters into the decision making process. This is true, for example, in portfolio asset allocation. Starting with the robust counterparts of the classical mean-variance and minimum-variance portfolio optimization problems, in this paper we review several mathematical models, and related algorithmic approaches, that have recently been proposed to address uncertainty in portfolio asset allocation, focusing on Robust Optimization methodology. We also give an overview of some of the computational results that have been obtained with the described approaches. In addition we analyze the relationship between the concepts of robustness and convex risk measures.
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In many risk-management applications, the multivariate normal distribution for asset returns does not reflect reality. In any case, the validity of such a claim may also depend on the investment horizon involved. Some evidence can be found in McNeil et al. (2005) where, by applying tests of normality to an arbitrary subgroup of 10 of the stocks in the Dow Jones index from 1993 to 2000, it can be seen that the daily, weekly and monthly return data fail the multivariate tests of normality, whereas return data over a quarter year are close to a normal distribution.
A way to define G is related to probabilistic guarantees in the likelihood that the actual realization of the uncertain coefficients will lie in the ellipsoidal uncertainty set S v . The definition of matrix G can be based on the data used to produce the estimates of the regression coefficients of the factor model (Fabozzi et al. 2007).
This kind of analysis has already been reviewed in Sect. 2.2.
In Artzner et al. (1999), a mapping ρ:Φ→ℝ is called a coherent measure of risk if it satisfies the four axioms of translation invariance, positive homogeneity, monotonicity and subadditivity, where subadditivity states that, ∀Y,Z∈Φ, ρ(Y+Z)≤ρ(Y)+ρ(Z); convexity is a consequence of these axioms.
A generalization of this theorem also exists in the case where Ω is an infinite set.
The goal is to form a portfolio in which the expected return is maximized, while some index of risk is minimized.
In presenting the soft robustness approach, we refer to D. Brown’s Ph.D. dissertation (2006), where emphasis is put on the concept of penalty functions.
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This is an updated version of the paper that appeared in 4OR, 8(2), 113–139 (2010).
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Scutellà, M.G., Recchia, R. Robust portfolio asset allocation and risk measures. Ann Oper Res 204, 145–169 (2013). https://doi.org/10.1007/s10479-012-1266-3
- Portfolio asset allocation
- Risk measures
- Mathematical models
- Algorithmic approaches