Copula Central Asymmetry of Equity Portfolios

Copula Central Asymmetry of Equity Portfolios

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{\it Keywords:} Dependence Asymmetry, Radial Symmetry, Reflection Symmetry, Financial Contagion

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Introduction

Asymmetric dependence is now considered an unavoidable characteristic of financial returns. Financial markets exhibit an increase in cross-sectional dependence during financial crises through different mechanisms collectively called financial contagion. This different probability of co-movements in the lower tail of the returns distribution generates an asymmetry in the dependence see, e.g., longin2001extreme; ang2002asymmetric; hong2007asymmetries. Market supervisors' financial stability considerations are highly influenced by dependence asymmetry and the systemic risk it entails. Risk managers should include this characteristic in their probabilistic models and raise their capital buffer requirements to avoid losses in distress periods. Asset managers should consider this feature by reallocating their assets on time to profit from arbitrage opportunities or avoid losses by making proper hedging decisions.

Even in tranquil times, asset managers should include the skewness, co-skewness, and odd moments that asymmetric dependence generates in their portfolio optimization. Albuquerque2012 develops an equilibrium model of this relationship. For portfolio optimization with higher moments, see jondeau2007financial and reference therein and expanding on the previous reference, I additionally recall that the large class of mixed risk-averse agents Caballe1996, tsetlin2009 prefer the risk with a higher moment for odd moments and the risk with a lower moment for even moments. If odd moments are sizeable, portfolio allocation of mixed risk-averse agents could differ dramatically from the one obtained by agents using the mean-variance analysis.

Historically, several statistics were proposed for the detection of asymmetric dependence. The pioneering works of longin2001extreme; ang2002asymmetric; hong2007asymmetries, used exceedance correlations. The positive (negative) exceedance correlation is the correlation conditional to the upper(lower) tail. A statistic for detecting asymmetric dependence is the difference between positive and negative exceedance correlation. Other similar statistics were differences between positive and negative exceedance covariances or exceedance betas hong2007asymmetries. See chen2015 and reference therein for more recent work in this line of research.

The approach has three main drawbacks: it is conditionally linear and cannot capture non-linear dependencies in the tails. It is bivariate, including information only for pairs of assets. The statistic depends on the choice of the threshold(s) to distinguish the tails from the center of the distribution.

A second approach, in the literature, is based on the difference between the upper and lower tail dependence functionsBormann2020. Those measures are usually considered more robust non-linear alternatives to exceedance covariances and correlations. The approach is still bivariate or applied pairwise, controlling the family-wise error. It depends on a tuning parameter to select extreme observations. Adaptation to the time-series context is not firmly justified theoretically.

Based on a distributional distance, I propose a non-parametric test of central symmetry applied to Copula functions. Central Symmetry is one possible generalization of univariate symmetry to higher dimensions. Elliptical distributions are a paradigmatic example of a centrally symmetric distribution. Under this symmetry, the upper and lower tails have the same probability. Violation of the multivariate distribution's central symmetry could then imply dependence asymmetry. Unfortunately, testing for the multivariate distribution's central symmetry requires knowledge or estimation of the center of symmetry, a highly nontrivial task in the multivariate setting. For this reason, I test copula central symmetry, where the center is known as the unit hypercube's center. In addition, the central asymmetry of copula functions is more directly related to dependence asymmetry and not contaminated by marginal asymmetry. This symmetry also goes under the names copula radial symmetry or copula reflection symmetry.\footnote{see pages 64-65 of joe2014dependence for a discussion on the different names.} Because I study the relationship between multivariate and copula symmetry, I adopt the copula central symmetry name in this paper. Non-parametric tests of copula central symmetry based on distributional distances were pioneered by aki1993nonparametric without even referring to copula functions. bouzebda2012test, dehgani2013measures, and genest2013tests introduced the test in the copula literature in the bivariate case. See billio2021 for a complete list of references of the different tests of this property and an extensive simulation study in the i.i.d. case, comparing the main procedures.

Testing non parametrically copula central symmetry is a multivariate non-linear approach that does not require any tuning parameter nor the estimation of a center of symmetry. It is independent of marginal symmetry and can be easily adapted to the context of weakly dependent data.

In this respect, a recent powerful tie-break bootstrap Seo2024, valid also for time series and useful only in the copula setup, allows statistical power with a large number of series and a small number of observations. The basic idea at the core of the new bootstrap was developed in the context of a randomization test beare2020 for copulas. billio2022 investigated the outstanding performance of the randomization procedure of beare2020 for testing copula central symmetry in the i.i.d. case. Unfortunately, the randomization test was not easily portable to time series. In this paper, I show comparable performance improvements, in the case of time series, using the tie-break bootstrap of Seo2024 in testing copula central symmetry.

In the empirical application, I use the benchmark dataset of US portfolio returns available from Kenneth French. This dataset was used in ang2002asymmetric, Hong2006,Bormann2020, and several other papers in the dependence asymmetry literature. Differently from the previous studies, the power of our test allows me to test symmetry separately for each year in the sample, investigating the time-varying dimension of dependence asymmetry using non-overlapping windows.

The paper is structured as follows. Section (ref) defines central symmetry, discusses the relationships between the different measures of dependence asymmetry. Section (ref) introduces the test, the tie-break bootstrap, and studies the statistical power of the procedure by simulation. Section (ref) apply the test of copula central symmetry to US equity portfolios based on different characteristics. Section (ref) concludes by summarizing our findings and outlining future research directions.

Central Symmetry

In this section, I introduce several multivariate symmetry concepts for random vectors and relate them to properties important in the financial context or used in dependence asymmetry measurement. I summarize these theoretical results in figure (ref).

I denote a random quantity by capital letters and deterministic quantities by lowercase letters. ${\buildrel d \over =}$ denotes the equality in distribution.\\ I start by considering univariate symmetry.

In particular, if I introduce the cumulative distribution function (CDF) of $X$, $F_{X}\left(x\right) = \mathbb{P}\left(X\leq x\right)$ and the survival function (SF), $\bar{F}_{X}\left(x\right) = \mathbb{P}\left(X> x\right)=1-F_{X}\left(x\right) $, then, if the CDF is continous\footnote{ Otherwise (ref) holds only at the points of continuity of the CDF}

and symmetry is valid if and only if the equality \[F_{X}\left(x\right)=F_{Y}\left(x\right)=\bar{F}_{X}\left(-x\right),\] holds for every $x\in\mathbb{R}$. Several extensions of the symmetry property to a random vector are possible. In the following, bold symbols represent vectors, for example $\mathbf{R}=\left(R_1,\ldots,R_N\right)^{\prime}$, and inequalities with bold symbols represent joint inequalities, for example $\left\{\mathbf{R}\leq \mathbf{r} \right\}= \left\{\bigcap^{N}_{i=1} R_i \leq r_i\right\}$.

The straightforward generalization of univariate symmetry is marginal symmetry :

Under this multivariate symmetry, I require that all the marginals are symmetric. This symmetry disregards the dependence among the components.

Another possible generalization of univariate symmetry, taking dependence into account, is central symmetry:

Let me define $\mathcal{N}= \left\{1,\ldots,N\right\}$ and the multi-index $\mathcal{I}=\left\{i_1,\ldots,i_k\right\}\subseteq \mathcal{N}$ where $\left\vert\mathcal{I}\right\vert=K\leq N$. The subvector of $\mathbf{X}$ with components in $\mathcal{I}$ is $\mathbf{X}_{\mathcal{I}} = \left(X_{i_1},\ldots,X_{i_k}\right)^{\prime}$. The marginal CDF of $\mathbf{X}_{\mathcal{I}} $ is $F_{\mathbf{X}_{\mathcal{I}}} \left(\mathbf{x}_{\mathcal{I}}\right) = \mathbb{P}\left(\mathbf{X}_{\mathcal{I}}\leq \mathbf{x}_{\mathcal{I}}\right) $. The multivariate SF $\bar{F}_{\mathbf{X}}\left( \mathbf{x}\right)=\mathbb{P}\left(\mathbf{X}> \mathbf{x} \right)$ and CDF $F_{\mathbf{X}}\left(\mathbf{x}\right)=\mathbb{P}\left(\mathbf{X}\leq \mathbf{x}\right)$, satisfy the following relationship

In analogy with the univariate case, central symmetry holds if and only if, the following property of continuous multivariate CDF and SF holds for every $\mathbf{x}\in \mathbb{R}^N$

The next result shows the consequences of the central symmetry hypothesis, impacting financial markets.

The proposition implies that under central symmetry, I can neglect skewness, co-skewness, and higher-order odd moments in portfolio optimization and that every portfolio built from a centrally symmetric $\mathbf{R}$ has a symmetric distribution. Proposition (ref) has the following important corollary as a subcase of iii):

This result was already stated in the bivariate case in nelsen1993some.

The literature on financial contagion focuses on dependence asymmetry in the tails conditioning on exceedances. Following ang2002asymmetric and hong2007asymmetries the positive (negative) exceedance at level $a$ of two random variable $R_1,R_2$ , given their standardization $\tilde{R}_i = \dfrac{R_1-\mu_i}{\sigma_i}$, $i=1,2$ is the event $\left\{\tilde{R}_1> a \right\}\cup \left\{\tilde{R}_2> a \right\} $, $ \left(\left\{\tilde{R}_1<- a \right\}\cup \left\{\tilde{R}_2< - a \right\} \right)$.The means and standard deviations conditional to the exceedance at level $a$ for $i=1,2$ are

the covariances at exceedance at level $a$ are:

Finally, correlations at exceedance at level $a$ are defined by the following equations

The following proposition studies the effect of central symmetry on exceedance covariance and correlations

The proposition holds for each pair of components in case $\mathbf{R}$ is $N$-dimensional and centrally symmetric.\\

It is helpful to differentiate the role of marginal random variables from the contributions of their dependence structure in a different way. This task can be accomplished using notions from copula theory. I provide only the results and definitions needed in the following, but the interested reader could refer to nelsen2007introduction, joe2014dependence and durante2015principles.

Let $F_{X_i}\left(x_i\right)$, be the marginal cumulative distribution function (CDFs) of $X_i$, the $i$-th component of $\mathbf{X}$. The component-wise probability integral transform (PIT) applied to $\mathbf{X}$ leads to a random vector $\mathbf{U}$ with components distributed uniformly on the unit interval:

Following 1959fonctions, if the marginals are continous, the joint CDF of $\mathbf{X}$ equivalent to:

$C_{\mathbf{U}}$ is the copula corresponding to $F_{\mathbf{X}}$ and represents the joint CDF of $\mathbf{U}$. An equivalent strategy applies to the marginal survival functions of $\mathbf{X}$, $\bar{F}_{X_i}\left(x_i\right)$, $ i\in \mathcal{N}$. At each $\mathbf{x}\in \mathbb{R}^N$, I can write the joint survival function of $\mathbf{X}$:

Then, an alternative expression for the joint CDF of $\mathbf{Y}$ is: \[ F_{\mathbf{Y}}\left(\mathbf{x}\right) =\bar{F}_{X}\left(-\mathbf{x}\right)=\bar{C}_{\mathbf{U}}\left(\bar{F}_{X_1}\left(-x_1\right),\ldots,\bar{F}_{X_N}\left(-x_N\right)\right)\]

and

\[\left(F_{Y_1}\left(Y_1\right),\ldots,F_{Y_N}\left(Y_N\right)\right)=\left(\bar{F}_{X_1}\left(-\left(-X_1\right)\right),\ldots,\bar{F}_{X_N}\left(-\left(-X_N\right)\right) \right)=\mathbf{1}_N- \mathbf{U}, \] where $\mathbf{1}_N$ be a $N$-dimensional vector with all components equal to one. The survival copula $\bar{C}_{\mathbf{U}}$ represents the joint CDF (not the joint SF) of $\mathbf{1}_N-\mathbf{U}$. The following definition of copula central symmetry follows:\\

To understand the relationship between central symmetry, marginal symmetry, and copula central symmetry, I define the symmetrization and the antisymmetrization of $C_{\mathbf{U}}$

I remark that $C_{\mathbf{U}^{S}}$, being a convex combination of copula functions is a copula. It is the CDF of

The following Theorem disentangles the contribution coming from marginal and dependence asymmetry.

The line (ref) represents the contribution to central asymmetry coming from the marginals embedded in the symmetrization of the dependence structure. The line (ref) adds the contributions from the dependence structure. Point 3 of Theorem (ref) is the generalization to the multivariate case of Theorem 3.2 in nelsen1993some. The decomposition is new even in the bivariate case and shows a compensation between marginal asymmetry and dependence asymmetry if they have opposite signs. The different sign of those contributions in financial markets is empirically documented and theoretically motivated in Albuquerque2012. In particular, the latter paper shows positive skewness and negative co-skewness. The Theorem and the empirical findings in Albuquerque2012 imply that negative dependence asymmetry, usually associated with financial contagion, could be masked by positive marginal symmetry if we use measures based on the multivariate distribution as exceedance correlations and covariances. In this work, instead, I focus on the asymmetry coming from the dependence structure and the null hypothesis will be of copula central symmetry. Copula central asymmetry has consequences for other measures of asymmetric dependence. Those measures are based on the difference between the upper and lower tail dependence functionsBormann2020.

The lower tail dependence function of $\mathbf{1}_N- \mathbf{U}$ is equal to the upper dependence function of $\mathbf{U}$ joe2014dependence leading to the following result that I state as a Lemma:

Analogously more refined upper and lower tail orders Hua2011 and upper and lower directional joe2010 and directional tail-weighted dependence measures Li2024 are equal under central symmetry. Copula central Symmetry is not impacted by marginal symmetry as central symmetry and implies the nullity of measures based on upper and lower tail dependence. In addition, as remarked in the introduction, it is a genuinely non-linear and multivariate property. On theoretical grounds, testing copula central symmetry appears to be the best procedure to detect dependence asymmetry.

Testing Time Series for Copula Central Symmetry

In this section, I introduce the test statistic and the tie-break bootstrap for testing the null hypothesis of copula central symmetry \[\mathcal{H}_0 : \mathbf{U} {\buildrel d \over =} \mathbf{1}_N- \mathbf{U}.\]

or equivalently using the copula and the survival copula

\[\mathcal{H}_0 : C_{\mathbf{U}}\left(\mathbf{u}\right) =\bar{C}_{\mathbf{U}}\left(\mathbf{u}\right)\]

An informative test statistic for the null hypothesis $\mathcal{H}_0$, could be a functional of $\Delta_{\mathbf{U}}\left(\mathbf{u}\right)$. A non-parametric consistent estimation of $\Delta_{\mathbf{U}}$ can use the Empirical Copula. Let us consider an independent sample of size $T$ from $N$-dimensional random vector $\mathbf{X}$, $\left\{\left\{X_{ti}\right\}^{T}_{t=1}\right\}^{N}_{i=1}\equiv\left\{\mathbf{X}_{t}\right\}^{T}_{t=1}$ be . I denote the set $A$ indicator as $\mathbb{I}\left(A\right)$. In addition, I define the normalized ranks $U_{T,ti}=\dfrac{1}{T+1}\sum^T_{s=1}\mathbb{I}\left(X_{s i}\leq X_{t i}\right)$, $t=1,\ldots,T$ and $i\in\mathcal{N}$ . With those definitions, the empirical copula and the empirical survival copula are:

Then, the antisymmetrization of the empirical copula is

An application of the functional delta method van1996weak on results for central limit theorem (CTL) of the multivariate empirical process for strongly mixing data with mixing coefficient $\alpha_n= o\left(n^{-a}\right)$ for some $a>0$, that can be found in rio1999theorie, allow bucher2013consistent to obtain the weak convergence result for the empirical copula process under the following non-restrictive assumptions on copula derivatives

Under assumption A (ref), for strongly mixing data with mixing coefficient $\alpha_n= o\left(n^{-a}\right)$ for some $a>0$, the empirical copula process $\mathbb{C}_T=\sqrt{T}\left(C_{T}\left(\mathbf{u}\right)-C_{\mathbf{U}}\left(\mathbf{u}\right)\right)$ converge weakly, in the Hoffman-Jorgensen sense, in $\ell^{\infty}\left(\left[0,1\right]^N\right)$ the space of bounded function on the $N$-dimensional unit hypercube:

where $\mathbb{B}_{_{\mathbf{U}}}$ is a d-dimensional Brownian sheet with covariance function

In the following proposition, I derive the weak convergence results for the empirical survival copula process for strongly mixing data, under assumption A (ref).

Several measures were proposed to test for copula central symmetry ( see billio2021 and references therein). I focus on a Cram\'er–von Mises statistic under the random measure generated by the empirical copula:

The 2-dimensional version of this statistic was introduced in bouzebda2012test and investigated further in dehgani2013measures and genest2013tests. This measure is one of the best performing for $d > 2$ in the i.i.d. case billio2021 and studied in the high dimensional case, in the context of a randomization test, in billio2022. I derive the asymptotic convergence in the time series setting in the following proposition.

Tie Break Dependent Bootstrap

In this subsection, I describe the tie-break bootstrap procedure for time series introduced in Seo2024 and derive the asymptotic behavior of the bootstrapped version of our test statistic.

Seo2024 proposes a bootstrap procedure improving finite sample performance by breaking ties induced by the block bootstrap on the bootstrapped normalized ranks. The tie-break bootstrap procedure for time series consists of the following steps:

The procedure is valid under the following assumption on the data generating process and choice of $l_{T}$.

Theorem 3.1 Seo2024 derive the following result under A (ref) and A (ref)

where the result is valid in $\ell^{\infty}\left(\left[0,1\right]^d\right)$ and $\overset{\mathbb{P}}{\underset{ \mathbf{X}}{\,\leadsto}\,}$ represents weak convergence conditional to the data in probability. The following proposition derives the analogous result for the Tie Break survival empirical copula process converge weakly conditional to the data in probability in $\ell^{\infty}\left(\left[0,1\right]^d\right)$ to the following limit

The validity of the approximate P-value in equation (ref) comes from the following proposition whose proof is analogous to the proof of proposition (ref).

Statistical Power

This section uses simulations to study the finite sample properties of the proposed multivariate copula central symmetry test. In all the experiments performed, the number of bootstrap or randomization replicates is M = 250, and the estimated rejection probabilities are computed using 1000 Monte Carlo independent replicates. Each table in the section presents a different number of observations $T\in\{50,100,250,500\}$, dimension of the random vector $N\in\{2,6,10,25,50\}$. To study the power of the tests based on $S_{n}$ in the time series large dimensional context, I use a recent asymmetric generalization of a score-driven copula factor model introduced in ohpatton with factors distributed according to a Skew-t distribution with common asymmetry parameter $\gamma\in\left[-1,1\right]$. I vary $\gamma$ in steps of $0.1$ and choose a number of groups equal to $2$ for $N\leq 10$ and $5$ in the other cases. The rest of the parameter values are from the simulation study in ohpatton. Here and in the empirical application section the block length follows the heuristic proposed in bucher2013consistent The results are reported in figure (ref).

The figure shows good statistical power of the test even in the case of small asymmetry and large dimensions if I consider more than 250 observations. ohpatton in their supplementary material report a value of $\gamma$ between -0.4 and -0.2 for their main specifications. I report a satisfactory power level in comparable cases even with 25 series. On the contrary, considering 50 series $\gamma=-0.2$, the power level is below the significance level, even if it is satisfactory for higher values of $\gamma$.

Empirical Evidence in US Stock Market

In this section, I study the copula central asymmetry of equity portfolio returns composed of US stocks. I use a benchmark dataset publicly available thanks to Kenneth French in the data library section of his website\footnote{\url{http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html}}. At first, I use the daily returns of five groups composed of 10 portfolios chosen by industry, deciles of market value, deciles of book-to-market ratio (value), and deciles of past performance(momentum)\footnote{A detailed description of the dataset is available on the Kenneth French website}. I consider data from 1 January 1927 to 31 December 2023 and test for central symmetry of the joint copula of the ten portfolios.

The statistical power outlined in the subsection (ref) allows me to test for copula central symmetry each year. The results are reported in figure (ref).

For each group of portfolios, I perform one test for each year, i.e., 97 tests. Since the number of tests is large, I control the whole testing procedure's false discovery rate (FDR) of each group of portfolios using the approach of FDR. I report results controlling for the FDR at the nominal levels 1% (red), 5% (orange), and 10% (yellow). The amount of copula central asymmetry is time-varying for all five groups of portfolios and is less present in the second part of the sample. In particular, the 1929 crisis led to a strongly significant asymmetric event for all five groups, while the 2008 subprime crisis is relevant in only two of the four groups, and even in those cases, the magnitude of the statistics in the two years is not comparable. Asymmetry is more robust if we choose different sizes and weaker if we consider assets with different book-to-market or momentum. Strong asymmetry in size is consistent with the bivariate tests in ang2002asymmetric and Hong2006. Weaker asymmetry in book-to-market is consistent with the bivariate analysis in Hong2006 results, while we have less asymmetry in momentum. In particular, asymmetry almost disappears for the latter two characteristics in the second part of the sample. Using industry classification leads to less asymmetry than size but more asymmetry than book-to-market or momentum. In addition, asymmetry, in this case, is present during market downturns, consistently with the financial contagion narrative. This is particularly evident in the second part of the sample in which the Black Monday 1987, the subprime 2008, and the COVID 2020 are all labeled by the test as asymmetric years. I further explore this relationship between asymmetries and financial crisis with a detailed analysis based on industry classification at different levels of granularity in figure (ref). Increasing sector granularity seems to capture more asymmetry. The exception is the finest level of granularity of 49 industries, but, in this case, the analysis of statistical power of the subsection (ref) shows that the test is not entirely reliable with a comparable number of series. In addition, increasing granularity reduces differences in the first and the second part of the sample. For example, the peak of the statistic for 1929 becomes more and more comparable with the peak of 2008 if I increase the number of series considered. With 30 industries, the peaks are significant at the same confidence level and are comparable in magnitude. Those findings can be justified by the possibility that asymmetry develops in different parts of the economy during different times because the importance and riskiness of various industries change over time. Increasing the granularity of the industrial classification then appears more robust in capturing asymmetry.

Conclusion

The detection of dependence asymmetry is relevant for portfolio optimization, market asymmetry, and financial stability. I focus on copula central asymmetry and explain, by new theoretical results, its relationship with other properties used by previous authors to measure dependence asymmetry. The results also imply that the detection of dependence asymmetry by some of these other measures can be obfuscated by marginal symmetry of the opposite sign. Testing copula central symmetry represents a multivariate non-linear approach that does not require tuning parameters or estimation of a center of symmetry. By construction, it is independent of marginal symmetry. I adapt to the time series context, a previously developed test based on distributional distance, using a novel, robust bootstrap framework introduced in Seo2024. A simulation study with a state-of-the-art copula factor model DGP shows reliable statistical power with only 250 observations and a number of series less or equal to 25. The power of the test allows a yearly application of the procedure using daily returns data spanning almost a century for groups of portfolios based on different characteristics. I find that dependence asymmetry is a time-varying property and has been less relevant recently. Asymmetry is more present in portfolios based on size and less in portfolios based on book-to-market and momentum. In portfolios based on industry classification, asymmetry appears to be linked to market downturns in line with the financial contagion narrative. Increasing the granularity of the classification considered leads to a more robust detection of asymmetry due to the time-varying riskiness of different parts of the economy. The detection of dependence asymmetry analyzed in this paper could be extended in several directions. First, the test statistic cannot tell us the sign of the asymmetry, which would be essential given that a large class of utility functions implies a preference for positive asymmetry. Given a signed measure, it is possible to devise a portfolio optimization strategy that includes the measure. The measure introduced in Krupskii2016 has this characteristic but showed lower statistical performance using more than two seriesbillio2021 in the i.i.d. case. The importance of this line of research for financial stability should aim instead to a different methodological advance. Given the time-varying nature of asymmetry, a structural break in this characteristic could be used as an early warning indicator for financial turmoil. The suggested change point detection analysis requires the extension of the theorems of this paper to the sequential empirical copula process Bucher2014.