Pierpaolo Uberti

In this paper, we investigate the information content of the lower part of the spectrum of financial correlation matrices to evaluate market synchronization and the loss of diversification opportunities. In a financial context, classical applications of Principal Component Analysis and Random Matrix Theory identify the largest eigenvalues as indicators of dominant market factors and synchronization patterns. We complement this perspective by providing a structural interpretation of the smallest eigenvalues in terms of effective market dimensionality. By combining numerical nullity and the Marchenko-Pastur lower bound, we establish a quantitative link between the lower spectrum of the correlation matrix and its numerically rank deficient structure. The paper presents a methodological proposal and validates its effectiveness through comprehensive real data experiments in both descriptive and predictive settings.

This paper presents a generalization of principal component analysis aimed at reducing the risk of any given benchmark portfolio. Taking the benchmark investment as the starting point, we propose a method to iteratively construct an orthogonal basis of the return space. These orthogonal vectors, called pseudo-principal portfolios after suitable normalization, are combined with the benchmark through an allocation rule to achieve risk reduction. From a theoretical perspective, we connect this construction to the mean–variance framework and derive geometric properties with meaningful financial implications. From an empirical perspective, we provide in-sample and out-of-sample experiments on different datasets of real financial data to support the effectiveness of the strategy, which combines the original benchmark with the pseudo-principal portfolios to lower risk, measured in terms of return volatility.

In this research, starting from a widely accepted definition of risk, we support the idea that risk reduction is a more realistic objective than risk minimization, which represents a theoretical utopia. Furthermore, significant risk reduction can be achieved without relying on risk measurement and risk minimization. To this end, we propose a generalization of the numerical rank and the condition number of a matrix—specifically the return matrix in this application. This generalization considers the entire matrix spectrum instead of focusing only on the smallest eigenvalue, as the condition number does. The approach directly provides an order among a finite number of risky scenarios. Risk reduction is obtained by identifying the riskiest scenarios and reducing investment exposures corresponding to them. The validity of this theoretical proposal is supported by a comprehensive experiment performed on real data. The capacity of the proposed approach to effectively reduce risk is proven by measuring the variability of out-of-sample returns for benchmark portfolios—constructed by minimizing standard risk measures—compared to the strategy of reducing exposure in high-risk scenarios. Finally, preventing large losses with limited active management—thereby controlling the impact of transaction costs—not only reduces risk but also preserves the average return and, consequently, the portfolio’s Sharpe ratio.

This paper introduces a novel and effective methodology for constructing portfolio diversification measures derived from any reference risk measure. The central contribution lies in leveraging the extensive theoretical developments in risk measurement to systematically inform and enhance the design of diversification metrics. The link between risk and diversification is exploited through an optimization framework in which the objective function is defined as a weighted Euclidean distance dependent on risk. We prove that the resulting objective function satisfies key axiomatic properties typically required to diversification measures, and that the corresponding optimization problem admits a unique solution that is inherently related to the intuitive concept of geometric diversification, thereby providing theoretical support for it. The key economic interpretation relies on determining the point in the allocation space that is equally distant – under a risk-sensitive metric – from the vertices of the simplex, i.e. the fully concentrated portfolios. An important economic insight of our approach is its applicability within a general long-short investment framework–a significant advancement, given that most classical diversification measures are restricted to long-only portfolios. Finally, to support the robustness of our findings, we present a comprehensive empirical analysis across multiple real-world financial datasets, highlighting meaningful comparisons between our proposed measure and several widely used diversification metrics.

In well-defined experimental settings, we evaluate the out-of-sample performance of two asset allocation paradigms: minimum risk and maximum diversification. Specifically, for each given risk measure, we compare the optimal minimum risk allocation with the allocation obtained by maximizing a portfolio diversification measure induced by the same risk measure. The experiment is performed in an out-of-sample, long-only framework, accounting for proportional transaction costs and different lengths of both the estimation window and the holding period. The strategies are compared in terms of numerical stability, return, the Sharpe ratio, and risk, as measured through the same risk measures used for calculating the optimal allocation: variance of returns, mean absolute deviation, value at risk, and expected shortfall at significance levels of 1% and 5%. We show that the maximum diversification strategies are highly competitive, if not generally superior, to the risk minimization allocations. This result supports well-known empirical findings of naive investment strategies that are difficult to beat in practice. Risk minimization strategies require highly accurate forecasts of future returns to perform well. Moreover, these strategies exhibit extreme numerical instability, where even infinitesimal variations in the inputs can dramatically alter the optimal allocation. Therefore, implementation costs are high, significantly impairing performance. In contrast, maximum diversification strategies are less sensitive to minor changes in the input parameters, providing stable allocations that are less affected by transaction costs. Furthermore, these strategies do not require accurate predictions of future returns and are effective in controlling investment risk.

Despite the growing attention in the last years on the topic of systemic risk, a widely accepted definition of systemic crisis is missing. We use a theoretical scheme to subjectively define a systemic event. This permits the analysis of a financial crisis as a standard binary classification problem, providing an intuitive and useful framework to compare systemic risk measures defined in very different fields. Then we focus the empirical analysis on the comparison of the performance of correlation-based systemic risk measures using the standard tools for the evaluation of binary classifiers as the receiver operating characteristic (ROC) curve and the area under the curve (AUC). We show that the binary classification framework is useful but unable to capture some significant differences among the measures under comparison. The experimental approach, developed on real financial data, is divided in an in-sample exercise, able to evaluate the descriptive power of the different systemic risk measures, and an out-of-sample application to evaluate the capacity of the measures in preventing and predicting systemic events. The forecasting ability of a measure can be fundamental for policy makers and investors respectively to stabilize market fluctuations and to reduce the losses.