Measuring large comovements in financial markets

A general, copula-based framework for measuring the dependence among financial time series is presented. Particular emphasis is placed on multivariate conditional Spearman's rho (MCS), a new measure of multivariate conditional dependence that describes the association between large or extreme negative returns—so-called tail dependence. We demonstrate that MCS has a number of advantages over conventional measures of tail dependence, both in theory and in practical applications. In the analysis of univariate financial series, data are filtered to remove temporal dependence as a matter of routine. We show that standard filtering procedures may strongly influence the conclusions drawn concerning tail dependence. We give empirical applications to two large data sets of high-frequency asset returns. Our results have immediate implications for portfolio risk management, derivative pricing and portfolio selection. In this context we address portfolio tail diversification and tail hedging. Amongst other aspects, it is shown that the proposed modeling framework improves the estimation of portfolio risk measures such as the value at risk.     

Multivariate geometric expectiles

A generalization of expectiles for d-dimensional multivariate distribution functions is introduced. The resulting geometric expectiles are unique solutions to a convex risk minimization problem and are given by d-dimensional vectors. They are well behaved under common data transformations and the corresponding sample version is shown to be a consistent estimator. We exemplify their usage as risk measures in a number of multivariate settings, highlighting the influence of varying margins and dependence structures.     

A note on Mossin’s theorem for deductible insurance given random initial wealth

Mossin’s theorem for deductible insurance given random initial wealth is re-examined. For a fair premium, it is shown that a necessary and sufficient condition, in the spirit of the Generalized Mossin Theorem for coinsurance, is impossible using the notion of expectation dependence. Next, it is established that for a fair premium, full insurance will be optimal for a risk-averse individual if the random loss and the random initial wealth are negative quadrant dependent, improving upon an extant result in the literature. In view of a set of examples given in this paper, such a sufficient condition cannot be obtained using the notion of expectation dependence. Finally, for an unfair premium, it is shown that partial insurance will always be optimal, irrespective of the risk preference of the individual as well as the dependence structure between the random loss and the random initial wealth.     

Combining random forest and copula functions: A heuristic approach for selecting assets from a financial crisis perspective

In this paper we propose a heuristic strategy aimed at selecting and analysing a set of financial assets, focusing attention on their multivariate tail dependence structure. The selection, obtained through an algorithmic procedure based on data mining tools, assumes the existence of a reference asset we are specifically interested to. The procedure allows one to opt for two alternatives: to prefer those assets exhibiting either a minimum lower tail dependence or a maximum upper tail dependence. The former could be a recommendable opportunity in a financial crisis period. For the selected assets, the tail dependence coefficients are estimated by means of a proper multivariate copula function. Copyright © 2010 John Wiley & Sons, Ltd.     

Risk management under extreme events

This article presents two applications of extreme value theory (EVT) to financial markets: computation of value at risk (VaR) and cross-section dependence of extreme returns (i.e., tail dependence). We use a sample comprised of the United States, Europe, Asia, and Latin America. Our main findings are the following. First, on average, EVT gives the most accurate estimate of VaR. Second, tail dependence of paired returns decreases substantially when both heteroscedasticity and serial correlation are filtered out by a multivariate GARCH model. Both findings are in agreement with previous research in this area for other financial markets.     

Multivariate Tweedie lifetimes: the impact of dependence

Systematic longevity risk is increasingly relevant for public pension schemes and insurance companies that provide life benefits. In view of this, mortality models should incorporate dependence between lives. However, the independent lifetime assumption is still heavily relied upon in the risk management of life insurance and annuity portfolios. This paper applies a multivariate Tweedie distribution to incorporate dependence, which it induces through a common shock component. Model parameter estimation is developed based on the method of moments and generalized to allow for truncated observations. The estimation procedure is explicitly developed for various important distributions belonging to the Tweedie family, and finally assessed using simulation.     

Extreme dependence in investor attention and stock returns – consequences for forecasting stock returns and measuring systemic risk

We characterize co-movements in investor attention by modeling multivariate internet search volume data. Using a variety of copula models that can capture both asymmetric and skewed dependence, we find empirical evidence of strong non-linear and asymmetric dependence in the attention investors give to companies. Modeling three years of daily stock returns and search volumes from Google Trends for 29 bank names, we find a striking similarity between the dependence structure inherent in stock returns and the dependence in the corresponding time series of search queries. We then document the existence of significant asymmetric and skewed tail dependence in the joint distribution of stock returns and investor attention. Finally, stock returns and internet search volumes appear to evolve concurrently in real time with neither one leading the other. Our findings have important implications, e.g. for the analysis of banks' interconnectedness based on equity data and the pricing of investor attention in the cross-section of stock returns.     

The t copula with multiple parameters of degrees of freedom: bivariate characteristics and application to risk management

The t copula is often used in risk management as it allows for modeling the tail dependence between risks and it is simple to simulate and calibrate. However, the use of a standard t copula is often criticized due to its restriction of having a single parameter for the degrees of freedom (dof) that may limit its capability to model the tail dependence structure in a multivariate case. To overcome this problem, the grouped t copula was proposed recently, where risks are grouped a priori in such a way that each group has a standard t copula with its specific dof parameter. In this paper we propose the use of a generalized grouped t copula, where each group consists of one risk factor only, so that a priori grouping is not required. The copula characteristics in the bivariate case are studied. We explain simulation and calibration procedures, including a simulation study on the finite sample properties of the maximum likelihood estimators and Kendall's tau approximation. This new copula is significantly different from the standard t copula in terms of risk measures such as tail dependence, value at risk and expected shortfall.     

Stock–bond dependence and flight to/from quality

This paper investigates the stock–bond dependence structure using a dependence-switching copula model. The model allows stock–bond dependence to switch between positive dependence regimes (contagions or crashes of the two markets during downturns or booms in both markets during upturns) and negative dependence regimes (flight-to-quality from stock markets to bond markets or flight-from-quality from bond markets to stock markets). Using data from four developed markets including the US, Canada, Germany, and France for the period between January 1985 and August 2022, we find that the within-country stock–bond (extreme) dependence could be both positive and negative. In the positive dependence regimes, the stock–bond dependence is asymmetric with stronger left tail dependence than the right tail dependence, giving evidence of a higher likelihood of joint stock–bond market crashes or contagions during market downturns than the collective stock–bond market booms. Under the negative dependence regimes, we find both flight-from-quality and flight-to-quality, with flight-to-quality being more dominant in the North American markets while flight-from-quality is more prominent in the European markets. Further, the dependence switches between positive and negative regimes over time. Moreover, the dependence is mainly in the positive regimes before 2000 while mostly in the negative regimes after that, indicating contagions mostly before 2000 and flights afterwards. Further, the dependence switches between positive and negative regimes around financial crises and the COVID-19 pandemic. These results greatly enrich the findings in the existing literature on the co-movements of stock–bond markets and are important for risk management and asset pricing.     

Tail dependence and indicators of systemic risk for large US depositories

In this study, we investigate the extreme loss tail dependence between stock returns of large US depository institutions. We find that stock returns exhibit strong loss dependence even in their limiting joint extremes. Motivated by this result, we derive extremal dependence-based systemic risk indicators. The proposed systemic risk indicators reflect downturns in the US financial industry very well. We also develop a set of firm-level average extremal dependence measures. We show that these firm-level measures could have been used to identify the firms that were more vulnerable to the 2007–2008 financial crisis. Additionally, we explore the performance of selected systemic risk indicators in predicting the crisis performance of large US depository institutions and find that the average stock return correlations are also good predictors of crisis period returns. Finally, we identify factors predictive of extremal dependence for the US depository institutions in a panel regression setting. Strength of extremal dependence increases with asset size and similarity of financial fundamentals. On the other hand, strength of extremal dependence decreases with capitalization, liquidity, funding stability and asset quality. We believe the proposed indicators have the potential to inform the prudential supervision of systemic risk.     

Multivariate lifetime distributions for the exponential dispersion family

We consider a general form of a multivariate lifetime model in which dependence is induced via a common shock component. The univariate marginal distributions come from the well-known and widely applied exponential dispersion family that includes the normal, compound-Poisson, gamma and negative binomial distributions. Any combination of truncation or censoring, either left or right, is considered, for which all moments are derived. This allows for the model to be calibrated to any affine transformation of lifetime data.     

Economic evaluation of asymmetric and price range information in gold and general financial markets

Gold is widely perceived as a good diversification or safe haven tool for general financial markets, especially in market turmoil. To fully understand the potential, this study constructs an asymmetric multivariate range-based volatility model to investigate the dependence and volatility structures of gold, stock, and bond markets and further to compare the difference between the financial crisis and post-financial crisis periods. We find a striking explanatory ability to volatility structures provided by the price range information and significant evidence of asymmetric dependence across gold, stock, and bond markets. We implement an asset-allocation strategy incorporating asymmetric dependence and price range information to explore their economic importance. The out-of-sample results show that between 35 and 517 basis points and between 90 and 1111 basis points are earned annually when acknowledging asymmetric dependence and price range information, respectively. These economic benefits are inversely related to the level of investors’ risk aversion and are particularly significant in the period of the global financial crisis.     

Non-linear Dependence in Stock Returns: Does Trading Frequency Matter?

The UK has a quote-driven pure dealer market structure that is very different from order driven markets such as the NYSE and Japanese markets. This paper investigates non-linear dependence in stock returns for an exhaustive sample of UK stocks for a 21 year period. The results are analysed on the basis of trading frequency. It is found that non-linear dependence is highly significant in all cases for both individual stocks and stock portfolios formed on the basis of trading frequency. The non-linear dependence is primarily over a one day interval, although statistically significant non-linear dependence exists consistently even up to five trading days. Most of the non-linear dependence is in the form of ARCH-type conditional heteroskedasticity. However, statistically significant non-linearity in addition to an EGARCH(1,1) dependence also appears to be present. This additional non-linearity is greater for individual stocks than for portfolios and greater for smaller, less-liquid portfolios. Non-linear dependence does not appear to be caused by non-stationarity in underlying economic fundamentals or by non-linearity in the conditional mean. However, low dimensional chaos is not generally supported. The limited evidence on chaotic behaviour is stronger for portfolios with long price adjustment delays across component stocks. The main results are consistent with US studies on stock indices, suggesting that the process generating non-linear dependence is not dependent on market microstructure characteristics.     

General convex order on risk aggregation

Using a general notion of convex order, we derive general lower bounds for risk measures of aggregated positions under dependence uncertainty, and this in arbitrary dimensions and for heterogeneous models. We also prove sharpness of the bounds obtained when each marginal distribution has a decreasing density. The main result answers a long-standing open question and yields an insight in optimal dependence structures. A numerical algorithm provides bounds for quantities of interest in risk management. Furthermore, our numerical results suggest that the bounds obtained in this paper are generally sharp for a broader class of models.     

A multivariate jump-driven financial asset model

We discuss a Lévy multivariate model for financial assets which incorporates jumps, skewness, kurtosis and stochastic volatility. We use it to describe the behaviour of a series of stocks or indexes and to study a multi-firm, value-based default model. Starting from an independent Brownian world, we introduce jumps and other deviations from normality, including non-Gaussian dependence. We use a stochastic time-change technique and provide the details for a Gamma change. The main feature of the model is the fact that—opposite to other, non-jointly Gaussian settings—its risk-neutral dependence can be calibrated from univariate derivative prices, providing a surprisingly good fit.     

Nonparametric Tests for Positive Quadrant Dependence

We consider distributional free inference to test for positivequadrant dependence, that is, for the probability that two variablesare simultaneously small (or large) being at least as greatas it would be were they dependent. Tests for its generalizationto higher dimensions, namely positive orthant dependence, arealso analyzed. We propose two types of testing procedures. Thefirst procedure is based on the specification of the dependenceconcepts in terms of distribution functions, while the secondprocedure exploits the copula representation. For each specification,a distance test and an intersection-union test for inequalityconstraints are developed for time-dependent data. An empiricalillustration is given for U.S. insurance claim data, where wediscuss practical implications for the design of reinsurancetreaties. Another application concerns detection of positivequadrant dependence between the HFR and CSFB/Tremont marketneutral hedge fund indices and the S&P 500 index.     

Nonlife actuarial models,theory, methods and evaluation by Yiu-Kuen Tse

We consider two different portfolios of proportional reinsurance of the same pool of risks. This contribution is concerned with Gaussian-like risks, which means that for large values the survival function of such risks is, up to a multiplier, the same as that of a standard Gaussian risk. We establish the tail asymptotic behavior of the total loss of each of the reinsurance portfolios and determine also the relation between randomly scaled Gaussian-like portfolios and unscaled ones. Further, we show that jointly two portfolios of Gaussian-like risks exhibit asymptotic independence and their weak tail dependence coefficient is nonnegative.     

A new time-varying optimal copula model identifying the dependence across markets

This paper proposes a new time-varying optimal copula (TVOC) model to identify and capture the optimal dependence structure of bivariate time series at every time point. In the TVOC model, half-rotated copulas are constructed to measure the nonlinear and asymmetric negative dependence, and the distribution-free test for independence is introduced to verify the dependent relationship and reduce the computational time. The TVOC model is then employed to research the dependence structure between security and commodity markets. We find evidence that the dependence structures across different markets vary over time and that emergencies are usually the major cause of sudden changes in the dependence structure. We also show that the TVOC model captures the dynamic characteristics of the direction and intensity of the dependence as well as the dynamic characteristics of the types of dependence structure. In particular, the half-rotated copulas can accurately describe the asymmetric negative extreme dependence across different markets.     

Multivariate Lévy processes with dependent jump intensity

In this work we propose a new and general approach to build dependence in multivariate Lévy processes. We fully characterize a multivariate Lévy process whose margins are able to approximate any Lévy type. Dependence is generated by one or more common sources of jump intensity separately in jumps of any sign and size and a parsimonious method to determine the intensities of these common factors is proposed. Such a new approach allows the calibration of any smooth transition between independence and a large amount of linear dependence and provides greater flexibility in calibrating nonlinear dependence than in other comparable Lévy models in the literature. The model is analytically tractable and a straightforward multivariate simulation procedure is available. An empirical analysis shows an accurate multivariate fit of stock returns in terms of linear and nonlinear dependence. A numerical illustration of multi-asset option pricing emphasizes the importance of the proposed new approach for modeling dependence.     

Can analysts predict rallies better than crashes?

We use the copula approach to study the structure of dependence between sell-side analysts’ consensus recommendations and subsequent security returns, with a focus on asymmetric tail dependence. We match monthly vintages of I/B/E/S recommendations for the period January–December 2011 with excess security returns during six months following recommendation issue. Using a mixed Gaussian–symmetrized Joe–Clayton copula model we find evidence to suggest that analysts can identify stocks that will substantially outperform, but not underperform relative to the market, and that their predictive ability is conditional on recommendation changes.