Correlation, hierarchies, and networks in financial markets

We discuss some methods to quantitatively investigate the properties of correlation matrices. Correlation matrices play an important role in portfolio optimization and in several other quantitative descriptions of asset price dynamics in financial markets. Specifically, we discuss how to define and obtain hierarchical trees, correlation based trees and networks from a correlation matrix. The hierarchical clustering and other procedures performed on the correlation matrix to detect statistically reliable aspects of the correlation matrix are seen as filtering procedures of the correlation matrix. We also discuss a method to associate a hierarchically nested factor model to a hierarchical tree obtained from a correlation matrix. The information retained in filtering procedures and its stability with respect to statistical fluctuations is quantified by using the Kullback-Leibler distance.Comment: 37 pages, 9 figures, 3 table

Comparing quantile residual life functions by confidence bands

A quantile residual life function is the quantile of the remaining life of a surviving subject, as it varies with time. In this article we present a nonparametric method for constructing confidence bands for the difference of two quantile residual life functions. These bands provide evidence for two random variables ordering with respect to a quantile residual life order introduced in Franco-Pereira et al. (2010). A simulation study has been carried out in order to evaluate and illustrate the performance and the consistency of this new methodology. We also present applications to real data examples.Quantile residual life, Confidence bands

The percentile residual life up to time t0: ordering and aging properties

Motivated by practical issues, a new stochastic order for random variables is introduced by comparing all their percentile residual life functions until a certain instant. Some interpretations of these stochastic orders are given, and various properties of them are derived. The relationships to other stochastic orders are studied, and also an application in Reliability Theory is described. Finally, we present some characterization results of the decreasing percentile residual life up to time t0 aging notion.Aging notion, Hazard rate, Mean residual life, Percentile residual life, Reliability, Stochastic ordering

Characterization of bathtub distributions via percentile residual life functions

In reliability theory and survival analysis, many set of data are generated by distributions with bathtub shaped hazard rate functions. Launer (1993) established several relations between the behaviour of the hazard rate function and the percentile residual life function. In particular, necessary conditions were given for a special type of bathtub distributions in terms of percentile residual life functions. The purpose of this paper is to complete the study initiated by Launer (1993) and to characterize (necessary and sufficient conditions) all types of bathtub distributions.Percentile residual life, Bathtub hazard rate, Aging notions,

Economic sector identification in a set of stocks traded at the New York Stock Exchange: a comparative analysis

We review some methods recently used in the literature to detect the existence of a certain degree of common behavior of stock returns belonging to the same economic sector. Specifically, we discuss methods based on random matrix theory and hierarchical clustering techniques. We apply these methods to a set of stocks traded at the New York Stock Exchange. The investigated time series are recorded at a daily time horizon. All the considered methods are able to detect economic information and the presence of clusters characterized by the economic sector of stocks. However, different methodologies provide different information about the considered set. Our comparative analysis suggests that the application of just a single method could not be able to extract all the economic information present in the correlation coefficient matrix of a set of stocks.Comment: 13 pages, 8 figures, 2 Table