Optimal leverage from non-ergodicity

In modern portfolio theory, the balancing of expected returns on investments against uncertainties in those returns is aided by the use of utility functions. The Kelly criterion offers another approach, rooted in information theory, that always implies logarithmic utility. The two approaches seem incompatible, too loosely or too tightly constraining investors' risk preferences, from their respective perspectives. The conflict can be understood on the basis that the multiplicative models used in both approaches are non-ergodic which leads to ensemble-average returns differing from time-average returns in single realizations. The classic treatments, from the very beginning of probability theory, use ensemble-averages, whereas the Kelly-result is obtained by considering time-averages. Maximizing the time-average growth rates for an investment defines an optimal leverage, whereas growth rates derived from ensemble-average returns depend linearly on leverage. The latter measure can thus incentivize investors to maximize leverage, which is detrimental to time-average growth and overall market stability. The Sharpe ratio is insensitive to leverage. Its relation to optimal leverage is discussed. A better understanding of the significance of time-irreversibility and non-ergodicity and the resulting bounds on leverage may help policy makers in reshaping financial risk controls.Comment: 17 pages, 3 figures. Updated figures and extended discussion of ergodicit

Theory and Applications of Robust Optimization

In this paper we survey the primary research, both theoretical and applied, in the area of Robust Optimization (RO). Our focus is on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying prominent theoretical results of RO, we also present some recent results linking RO to adaptable models for multi-stage decision-making problems. Finally, we highlight applications of RO across a wide spectrum of domains, including finance, statistics, learning, and various areas of engineering.Comment: 50 page

Noise Dressing of Financial Correlation Matrices

We show that results from the theory of random matrices are potentially of great interest to understand the statistical structure of the empirical correlation matrices appearing in the study of price fluctuations. The central result of the present study is the remarkable agreement between the theoretical prediction (based on the assumption that the correlation matrix is random) and empirical data concerning the density of eigenvalues associated to the time series of the different stocks of the S&P500 (or other major markets). In particular the present study raises serious doubts on the blind use of empirical correlation matrices for risk management.Comment: Latex (Revtex) 3 pp + 2 postscript figures (in-text

An Evolutionary Optimization Approach to Risk Parity Portfolio Selection

In this paper we present an evolutionary optimization approach to solve the risk parity portfolio selection problem. While there exist convex optimization approaches to solve this problem when long-only portfolios are considered, the optimization problem becomes non-trivial in the long-short case. To solve this problem, we propose a genetic algorithm as well as a local search heuristic. This algorithmic framework is able to compute solutions successfully. Numerical results using real-world data substantiate the practicability of the approach presented in this paper

Rapid X-ray Variability of Seyfert 1 Galaxies

The rapid and seemingly random fluctuations in X-ray luminosity of Seyfert galaxies provided early support for the standard model in which Seyferts are powered by a supermassive black hole fed from an accretion disc. However, since EXOSAT there has been little opportunity to advance our understanding of the most rapid X-ray variability. Observations with XMM-Newton have changed this. We discuss some recent results obtained from XMM-Newton observations of Seyfert 1 galaxies. Particular attention will be given to the remarkable similarity found between the timing properties of Seyferts and black hole X-ray binaries, including the power spectrum and the cross spectrum (time delays and coherence), and their implications for the physical processes at work in Seyferts.Comment: To appear in From X-ray Binaries to Quasars: Black Hole Accretion on All Mass Scales, ed. T. J. Maccarone, R. P. Fender, and L. C. Ho (Dordrecht: Kluwer

The scaling of X-ray variability with luminosity in Ultra-luminous X-ray sources

We investigated the relationship between the X-ray variability amplitude and X-ray luminosity for a sample of 14 bright Ultra-luminous X-ray sources (ULXs) with XMM-Newton/EPIC data, and compare it with the well established similar relationship for Active Galactic Nuclei (AGN). We computed the normalised excess variance in the 2-10 keV light curves of these objects and their 2-10 keV band intrinsic luminosity. We also determined model "variability-luminosity" relationships for AGN, under several assumptions regarding their power-spectral shape. We compared these model predictions at low luminosities with the ULX data. The variability amplitude of the ULXs is significantly smaller than that expected from a simple extrapolation of the AGN "variability-luminosity" relationship at low luminosities. We also find evidence for an anti-correlation between the variability amplitude and L(2-10 keV) for ULXs. The shape of this relationship is consistent with the AGN data but only if the ULXs data are shifted by four orders of magnitudes in luminosity. Most (but not all) of the ULXs could be "scaled-down" version of AGN if we assume that: i) their black hole mass and accretion rate are of the order of ~(2.5-30)x 10E+03 Msolar and ~ 1-80 % of the Eddington limit, and ii) their Power Spectral Density has a doubly broken power-law shape. This PDS shape and accretion rate is consistent with Galactic black hole systems operating in their so-called "low-hard" and "very-high" states.Comment: 10 pages, 5 figures, 2 tables, accepted for publication in A&

Statistics of Atmospheric Correlations

For a large class of quantum systems the statistical properties of their spectrum show remarkable agreement with random matrix predictions. Recent advances show that the scope of random matrix theory is much wider. In this work, we show that the random matrix approach can be beneficially applied to a completely different classical domain, namely, to the empirical correlation matrices obtained from the analysis of the basic atmospheric parameters that characterise the state of atmosphere. We show that the spectrum of atmospheric correlation matrices satisfy the random matrix prescription. In particular, the eigenmodes of the atmospheric empirical correlation matrices that have physical significance are marked by deviations from the eigenvector distribution.Comment: 8 pages, 9 figs, revtex; To appear in Phys. Rev.

A Closed-Form Solution of the Multi-Period Portfolio Choice Problem for a Quadratic Utility Function

In the present paper, we derive a closed-form solution of the multi-period portfolio choice problem for a quadratic utility function with and without a riskless asset. All results are derived under weak conditions on the asset returns. No assumption on the correlation structure between different time points is needed and no assumption on the distribution is imposed. All expressions are presented in terms of the conditional mean vectors and the conditional covariance matrices. If the multivariate process of the asset returns is independent it is shown that in the case without a riskless asset the solution is presented as a sequence of optimal portfolio weights obtained by solving the single-period Markowitz optimization problem. The process dynamics are included only in the shape parameter of the utility function. If a riskless asset is present then the multi-period optimal portfolio weights are proportional to the single-period solutions multiplied by time-varying constants which are depending on the process dynamics. Remarkably, in the case of a portfolio selection with the tangency portfolio the multi-period solution coincides with the sequence of the simple-period solutions. Finally, we compare the suggested strategies with existing multi-period portfolio allocation methods for real data.Comment: 38 pages, 9 figures, 3 tables, changes: VAR(1)-CCC-GARCH(1,1) process dynamics and the analysis of increasing horizon are included in the simulation study, under revision in Annals of Operations Researc

Superconductivity and Antiferromagnetism: Hybridization Impurities in a Two-Band Spin-Gapped Electron System

We present the exact solution of a one-dimensional model of a spin-gapped correlated electron system with hybridization impurities exhibiting both magnetic and mixed-valence properties. The host supports superconducting fluctuations, with a spin gap. The localized electrons create a band of antiferromagnetic spin excitations inside the gap for concentrations x of the impurities below some critical value x_c. When x = x_c the spin gap closes and a ferrimagnetic phase appears. This is the first example of an exactly solvable model with coexisting superconducting and antiferromagnetic fluctuations which in addition supports a quantum phase transition to a (compensated) ferrimagnetic phase. We discuss the possible relevance of our results for experimental systems, in particular the U-based heavy-fermion materials.Comment: 4 page