We introduce a financial portfolio optimization framework that allows us to automatically select the relevant assets and estimate their weights by relying on a sorted $\ell_1$-Norm penalization, henceforth SLOPE. Our approach is able to group constituents with similar correlation properties, and with the same underlying risk factor exposures. We show that by varying the intensity of the penalty, SLOPE can span the entire set of optimal portfolios on the risk-diversification frontier, from minimum variance to the equally weighted. To solve the optimization problem, we develop a new efficient algorithm, based on the Alternating Direction Method of Multipliers. Our empirical analysis shows that SLOPE yields optimal portfolios with good out-of-sample risk and return performance properties, by reducing the overall turnover through more stable asset weight estimates. Moreover, using the automatic grouping property of SLOPE, new portfolio strategies, such as SLOPE-MV, can be developed to exploit the data-driven detected similarities across assets.
We examine machine learning and factor-based portfolio optimization. We find that factors based on autoencoder neural networks exhibit a weaker relationship with commonly used characteristic-sorted portfolios than popular dimensionality reduction techniques. Machine learning methods also lead to covariance and portfolio weight structures that diverge from simpler estimators. Minimum-variance portfolios using latent factors derived from autoencoders and sparse methods outperform simpler benchmarks in terms of risk minimization. These effects are amplified for investors with an increased sensiti…
This paper is devoted to study the optimal portfolio problem. Harry Markowitz's Ph.D. thesis prepared the ground for the mathematical theory of finance. In modern portfolio theory, we typically find asset returns that are modeled by a random variable with an elliptical distribution and the notion of portfolio risk is described by an appropriate risk measure. In this paper, we propose new stochastic models for the asset returns that are based on Jumps- Diffusion (J-D) distributions. This family of distributions are more compatible with stylized features of asset returns. On the other hand, in t…
In this paper, we propose $\ell_p$-norm regularized models to seek near-optimal sparse portfolios. These sparse solutions reduce the complexity of portfolio implementation and management. Theoretical results are established to guarantee the sparsity of the second-order KKT points of the $\ell_p$-norm regularized models. More interestingly, we present a theory that relates sparsity of the KKT points with Projected correlation and Projected Sharpe ratio. We also design an interior point algorithm to obtain an approximate second-order KKT solution of the $\ell_p$-norm models in polynomial time wi…
The optimization of the variance supplemented by a budget constraint and an asymmetric $\ell_1$ regularizer is carried out analytically by the replica method borrowed from the theory of disordered systems. The asymmetric regularizer allows us to penalize short and long positions differently, so the present treatment includes the no-short-constrained portfolio optimization problem as a special case. Results are presented for the out-of-sample and the in-sample estimator of the regularized variance, the relative estimation error, the density of the assets eliminated from the portfolio by the reg…
The optimization of a large random portfolio under the Expected Shortfall risk measure with an $\ell_2$ regularizer is carried out by analytical calculation. The regularizer reins in the large sample fluctuations and the concomitant divergent estimation error, and eliminates the phase transition where this error would otherwise blow up. In the data-dominated region, where the number $N$ of different assets in the portfolio is much less than the length $T$ of the available time series, the regularizer plays a negligible role even if its strength $η$ is large, while in the opposite limit, where …
The emergence of robust optimization has been driven primarily by the necessity to address the demerits of the Markowitz model. There has been a noteworthy debate regarding consideration of robust approaches as superior or at par with the Markowitz model, in terms of portfolio performance. In order to address this skepticism, we perform empirical analysis of three robust optimization models, namely the ones based on box, ellipsoidal and separable uncertainty sets. We conclude that robust approaches can be considered as a viable alternative to the Markowitz model, not only in simulated data but…
Jinan University（暨南大学）；College of Information Science and Technology, Jinan University（暨南大学信息科学技术学院）；Jinan University-University of Birmingham Joint Institute, Jinan University（暨南大学-伯明翰大学联合学院）
Expected Shortfall (ES), the average loss above a high quantile, is the current financial regulatory market risk measure. Its estimation and optimization are highly unstable against sample fluctuations and become impossible above a critical ratio $r=N/T$, where $N$ is the number of different assets in the portfolio, and $T$ is the length of the available time series. The critical ratio depends on the confidence level $α$, which means we have a line of critical points on the $α-r$ plane. The large fluctuations in the estimation of ES can be attenuated by the application of regularizers. In this…
The existing approaches to sparse wealth allocations (1) are limited to low-dimensional setup when the number of assets is less than the sample size; (2) lack theoretical analysis of sparse wealth allocations and their impact on portfolio exposure; (3) are suboptimal due to the bias induced by an $\ell_1$-penalty. We address these shortcomings and develop an approach to construct sparse portfolios in high dimensions. Our contribution is twofold: from the theoretical perspective, we establish the oracle bounds of sparse weight estimators and provide guidance regarding their distribution. From t…
We prove that the Omega measure, which considers all moments when assessing portfolio performance, is equivalent to the widely used Sharpe ratio under jointly elliptic distributions of returns. Portfolio optimization of the Sharpe ratio is then explored, with an active-set algorithm presented for markets prohibiting short sales. When asymmetric returns are considered we show that the Omega measure and Sharpe ratio lead to different optimal portfolios.
We derive a closed form portfolio optimization rule for an investor who is diffident about mean return and volatility estimates, and has a CRRA utility. The novelty is that confidence is here represented using ellipsoidal uncertainty sets for the drift, given a volatility realization. This specification affords a simple and concise analysis, as the optimal portfolio allocation policy is shaped by a rescaled market Sharpe ratio, computed under the worst case volatility. The result is based on a max-min Hamilton-Jacobi-Bellman-Isaacs PDE, which extends the classical Merton problem and reverts to…
We introduce performance-based regularization (PBR), a new approach to addressing estimation risk in data-driven optimization, to mean-CVaR portfolio optimization. We assume the available log-return data is iid, and detail the approach for two cases: nonparametric and parametric (the log-return distribution belongs in the elliptical family). The nonparametric PBR method penalizes portfolios with large variability in mean and CVaR estimations. The parametric PBR method solves the empirical Markowitz problem instead of the empirical mean-CVaR problem, as the solutions of the Markowitz and mean-C…
The paper Zhao et al. (2015) shows that mean-CVaR-skewness portfolio optimization problems based on asymetric Laplace (AL) distributions can be transformed into quadratic optimization problems under which closed form solutions can be found. In this note, we show that such result also holds for mean-risk-skewness portfolio optimization problems when the underlying distribution is a larger class of normal mean-variance mixture (NMVM) models than the class of AL distributions. We then study the value at risk (VaR) and conditional value at risk (CVaR) risk measures on portfolios of returns with NM…
This paper develops a method to derive optimal portfolios and risk premia explicitly in a general diffusion model for an investor with power utility and a long horizon. The market has several risky assets and is potentially incomplete. Investment opportunities are driven by, and partially correlated with, state variables which follow an autonomous diffusion. The framework nests models of stochastic interest rates, return predictability, stochastic volatility and correlation risk. In models with several assets and a single state variable, long-run portfolios and risk premia admit explicit formu…
Portfolio optimization approaches inevitably rely on multivariate modeling of markets and the economy. In this paper, we address three sources of error related to the modeling of these complex systems: 1. oversimplifying hypothesis; 2. uncertainties resulting from parameters' sampling error; 3. intrinsic non-stationarity of these systems. For what concerns point 1. we propose a L0-norm sparse elliptical modeling and show that sparsification is effective. The effects of points 2. and 3. are quantifified by studying the models' likelihood in- and out-of-sample for parameters estimated over train…
A large portfolio of independent returns is optimized under the variance risk measure with a ban on short positions. The no-short selling constraint acts as an asymmetric $\ell_1$ regularizer, setting some of the portfolio weights to zero and keeping the out of sample estimator for the variance bounded, avoiding the divergence present in the non-regularized case. However, the susceptibility, i.e. the sensitivity of the optimal portfolio weights to changes in the returns, diverges at a critical value $r=2$. This means that a ban on short positions does not prevent the phase transition in the op…
