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.     

On additivity of tail comonotonic risks

ABSTRACT

As perceived from daily experience together with numerous empirical studies, the multivariate risks demonstrate a strong coherence in the extremal dependence structure especially over the course of financial turmoil or industrial accidents and outbreaks. Under this motivating paradigm, we show the universal asymptotic additivity under upper tail comonotonicity, as the probability level approaching to 1, for Value-at-Risk and Conditional Tail Expectation for a portfolio of fixed number of risks, in which each marginal risk could be any one having a finite endpoint or belonging to one of the three max domains of attraction. Our obtained results do not require the tail equivalence assumption as needed in the existing literature. This resolves a lasting problem in quantitative risk management and covers most distributions commonly encountered in practice.     

How accurate is the square-root-of-time rule in scaling tail risk: A global study

The square-root-of-time rule (SRTR) is popular in assessing multi-period VaR; however, it makes several unrealistic assumptions. We examine and reconcile different stylized factors in returns that contribute to the SRTR scaling distortions. In complementing the use of the variance ratio test, we propose a new intuitive subsampling-based test for the overall validity of the SRTR. The results indicate that serial dependence and heavy-tailedness may severely bias the applicability of SRTR, while jumps or volatility clustering may be less relevant. To mitigate the first-order effect from time dependence, we suggest a simple modified-SRTR for scaling tail risks. By examining 47 markets globally, we find the SRTR to be lenient, in that it generally yields downward-biased 10-day and 30-day VaRs, particularly in Eastern Europe, Central-South America, and the Asia Pacific. Nevertheless, accommodating the dependence correction is a notable improvement over the traditional SRTR.     

A semi-parametric approach to risk management

Abstract

The benchmark theory of mathematical finance is the Black–Scholes–Merton (BSM) theory, based on Brownian motion as the driving noise process for stock prices. Here the distributions of financial returns of the stocks in a portfolio are multivariate normal. Risk management based on BSM underestimates tails. Hence estimation of tail behaviour is often based on extreme value theory (EVT). Here we discuss a semi-parametric replacement for the multivariate normal involving normal variance–mean mixtures. This allows a more accurate modelling of tails, together with various degrees of tail dependence, while (unlike EVT) the whole return distribution can be modelled. We use a parametric component, incorporating the mean vector μ and covariance matrix Σ, and a non-parametric component, which we can think of as a density on [0,∞), modelling the shape (in particular the tail decay) of the distribution. We work mainly within the family of elliptically contoured distributions, focusing particularly on normal variance mixtures with self-decomposable mixing distributions. We discuss efficient methods to estimate the parametric and non-parametric components of our model and provide an algorithm for simulating from such a model. We fit our model to several financial data series. Finally, we calculate value at risk (VaR) quantities for several portfolios and compare these VaRs to those obtained from simple multivariate normal and parametric mixture models.     

Risk bounds for factor models

Recent literature has investigated the risk aggregation of a portfolio \(X=(X_{i})_{1\leq i\leq n}\) under the sole assumption that the marginal distributions of the risks \(X_{i} \) are specified, but not their dependence structure. There exists a range of possible values for any risk measure of \(S=\sum_{i=1}^{n}X_{i}\), and the dependence uncertainty spread, as measured by the difference between the upper and the lower bound on these values, is typically very wide. Obtaining bounds that are more practically useful requires additional information on dependence.Here, we study a partially specified factor model in which each risk \(X_{i}\) has a known joint distribution with the common risk factor \(Z\), but we dispense with the conditional independence assumption that is typically made in fully specified factor models. We derive easy-to-compute bounds on risk measures such as Value-at-Risk (\(\mathrm{VaR}\)) and law-invariant convex risk measures (e.g. Tail Value-at-Risk (\(\mathrm{TVaR}\))) and demonstrate their asymptotic sharpness. We show that the dependence uncertainty spread is typically reduced substantially and that, contrary to the case in which only marginal information is used, it is not necessarily larger for \(\mathrm{VaR}\) than for \(\mathrm{TVaR}\).     

Insurance Portfolio Risk Retention

In this article, I introduce a statistic for managing a portfolio of insurance risks. This tool is based on changes in the risk profile when changes in a risk parameter, such as a deductible, coinsurance, or upper policy limit, are made. I refer to the new statistic as a risk measure relative marginal change and denote it as RM². By examining data from the Wisconsin Local Government Property Fund, I show how it can be used by an insurer to identify the “best” and “worst” risks in terms of opportunities for risk management. The RM² changes reflect the underlying dependence structure of risks; I use an elliptical copula framework to demonstrate the sensitivity of risk mitigation strategy to the dependence structure.     

Asymptotics for large claims reinsurance in a time-dependent renewal risk model

Abstract

We study the asymptotic tail behaviour of reinsured amounts of the LCR and ECOMOR treaties under a time-dependent renewal risk model, in which a dependence structure is introduced between each claim size and the interarrival time before it. Assuming that the claim size distribution has a subexponential tail, we derive some precise asymptotic results for both treaties.     

Bounds for Functions of Dependent Risks

The problem of finding the best-possible lower bound on the distribution of a non-decreasing function of n dependent risks is solved when n=2 and a lower bound on the copula of the portfolio is provided. The problem gets much more complicated in arbitrary dimensions. When no information on the structure of dependence of the random vector is available, we provide a bound on the distribution function of the sum of risks which we prove to be better than the one generally used in the literature.     

Asymptotic Analysis of the Loss Given Default in the Presence of Multivariate Regular Variation

Consider a portfolio of n obligors subject to possible default. We propose a new structural model for the loss given default, which takes into account the severity of default. Then we study the tail behavior of the loss given default under the assumption that the losses of the n obligors jointly follow a multivariate regular variation structure. This structure provides an ideal framework for modeling both heavy tails and asymptotic dependence. Multivariate models involving Archimedean copulas and mixtures are revisited. As applications, we derive asymptotic estimates for the value at risk and conditional tail expectation of the loss given default and compare them with the traditional empirical estimates.     

Insurance premiums and default risk in mutual insurance

Abstract

Two types of default risk are discussed in the article: The traditional “probability of ruin” (insurer being unable to meet his obligations) and a “perceived probability of ruin” (the probability of the insured being affected by ruin). The explicit relationship between these probabilities on the actuarial loading factors of a mutual insurer were developed. The explicit mathematical formulae obtained for these complex relationships were followed also by numerical results. A second concept presented in the paper is related to the idea of actuarially fair premiums. It is shown that the premium must also be a function of the payments of the other insured as well as their claim distributions, reflecting thereby the simultaneity and mutual dependence of the insured.     

Multivariate extremes for models with constant conditional correlations

Models with constant conditional correlations are versatile tools for describing the behavior of multivariate time series of financial returns. Mathematically speaking, they are solutions of a special class of stochastic recurrence equations (SRE). The extremal behavior of general solutions of SRE has been studied in detail by Kesten [Kesten, H., 1973. Random difference equations and renewal theory for products of random matrices. Acta Mathematica 131, 207–248] and Perfekt [Perfekt, R., 1997. Extreme value theory for a class of Markov chains with values in ^d. Advances in Applied Probability 29, 138–164]. The central concept to understanding the joint extremal behavior of such multivariate time series is the multivariate regular variation spectral measure. In this paper, we propose an estimator for the spectral measure associated with solutions of SRE and prove its consistency. Our estimator is the tail empirical measure of the multivariate time series. Successful use of the estimator depends on a good choice of k, the number of upper order statistics contributing to the empirical measure. We introduce a new criteria for the choice of k based on a scaling property of the spectral measure. We investigate the performance of our estimation technique on exchange rate time series from HFDF96 data set. The estimated spectral measure is used to calculate probabilities of joint extreme returns and probabilities of large movements in an exchange rate conditional on the occurrence of extreme returns in another exchange rate. We find a high level of dependence between the extreme movements of most of the currencies in the EU. We also investigate the changes in the level of dependence between the extreme returns of pairs of currencies as the sampling frequency decreases. When at least one return is extreme, a strong dependence between the components is present already at the 4-hour level for most of the European currencies.     

Forecasting stock-market tail risk and connectedness in advanced economies over a century: The role of gold-to-silver and gold-to-platinum price ratios

We examine the predictive value of risk perceptions as measured in terms of the gold-to-silver and gold-to-platinum price ratios for stock-market tail risks and their connectedness in eight major industrialized economies using monthly data for the period 1916:02–2020:10 and 1968:01–2020:10, where we use four variants of the popular Conditional Autoregressive Value at Risk (CAViaR) framework to estimate the tail risks for both 1% and 5% VaRs. Our findings for the short sample period show that the gold-to-silver price ratio resembles the gold-to-platinum price ratios in that it is a useful proxy for global risk. Our findings for the long sample period show, despite some heterogeneity across economies, that the gold-to-silver price ratio often helps to out-of-sample forecast for both 1% and 5% stock market tail risks, particularly when a forecaster suffers a higher loss from underestimation of tail risks than from a corresponding overestimation of the same absolute size. We also find that using the gold-to-silver price ratio for forecasting the total connectedness of stock markets is beneficial for an investor who suffers a higher loss from an underestimation of total connectedness (i.e., an investor who otherwise would overestimate the benefits from portfolio diversification) than from a comparable overestimation.     

Re-examination of international bond market dependence: Evidence from a pair copula approach

The finance literature provides substantial evidence on the dependence between international bond markets across developed and emerging countries. Early works in this area were based on linear models and multivariate GARCH models. However, based on the limitations of these models this paper re-examines the non-linearity, multivariate and tail dependence structure between government bond markets of the US, UK, Japan, Germany, Canada, France, Italy, Australia and the Eurozone, from January 1970 to February 2019 using ARMA-GARCH based pair- copula models. We find that the bond markets in our sample tend to have both upper tail dependence in terms of positive shocks and lower tail dependence in terms of negative shocks. The estimated C-vine shows Eurozone has the highest average dependency. The D-vine, with optimal chain dependency structure shows the best order of connectedness to be the UK, the USA, Italy, Japan, Eurozone, France, Canada, Germany and Australia. The R-vine copula results underline the complex dynamics of bond market relations existing between the selected economies. The estimated R-vine shows Eurozone, Germany and Australia are the most inter-connected nodes. The multivariate distribution structure (interdependency) of bond markets for all countries were modelled with the C-vine, D-vine and R-vine copulas. In this application, the R-vine copula allows for detailed modelling of all bond markets and hence provides a more accurate goodness of fit and mean square error for the interdependency between all markets. In light of the changing volatility in bond markets, we conduct additional tests using time-varying copulas and find that the dependence structure among the bond markets examined is time-varying with the dynamic dependence parameter plots revealing that the nature of the dependence structure is intense during crisis periods.     

Modelling Co-movements and Tail Dependency in the International Stock Market via Copulae

This paper examines international equity market co-movements using time-varying copulae. We examine distributions from the class of Symmetric Generalized Hyperbolic (SGH) distributions for modelling univariate marginals of equity index returns. We show based on the goodness-of-fit testing that the SGH class outperforms the normal distribution, and that the Student-t assumption on marginals leads to the best performance, and thus, can be used to fit multivariate copula for the joint distribution of equity index returns. We show in our study that the Student-t copula is not only superior to the Gaussian copula, where the dependence structure relates to the multivariate normal distribution, but also outperforms some alternative mixture copula models which allow to reflect asymmetric dependencies in the tails of the distribution. The Student-t copula with Student-t marginals allows to model realistically simultaneous co-movements and to capture tail dependency in the equity index returns. From the point of view of risk management, it is a good candidate for modelling the returns arising in an international equity index portfolio where the extreme losses are known to have a tendency to occur simultaneously. We apply copulae to the estimation of the Value-at-Risk and the Expected Shortfall, and show that the Student-t copula with Student-t marginals is superior to the alternative copula models investigated, as well the Riskmetics approach.     

Multiresolution approximation for volatility processes

Abstract

We present an application of wavelet techniques to non-stationary time series with the aim of detecting the dependence structure which is typically found to characterize intraday stock index financial returns. It is particularly important to identify what components truly belong to the underlying volatility process, compared with those features appearing instead as a result of the presence of disturbance processes. The latter may yield misleading inference results when standard financial time series models are adopted. There is no universal agreement on whether long memory really affects financial series, or instead whether it could be that non-stationarity, once detected and accounted for, may allow for more power in detecting the dependence structure and thus suggest more reliable models. Wavelets are still a novel tool in the domain of applications in finance; thus, one goal is to try to show their potential use for signal decomposition and approximation of time-frequency signals. This might suggest a better interpretation of multi-scaling and aggregation effects in high-frequency returns. We show, by using special dictionaries of functions and ad hoc algorithms, that a pre-processing procedure for stock index returns leads to a more accurate identification of dependent and non-stationary features, whose detection results are improved compared with those obtained by other traditional Fourier-based methods. This allows generalized autoregressive conditional heteroscedastic models to be more effective for statistical estimation purposes.     

Sovereign Risk Contagion in East Asia: A Mixture of Time-Varying Copulas Approach

This study analyzes sovereign risk contagion between four East Asian economies (China, Hong Kong, Japan, and Korea) and its structural changes through the Global Financial Crisis (GFC) and the European Debt Crisis (EDC) by applying the mixture of time-varying copulas to those economies’ credit default swap (CDS) spreads.

This article first finds a strong contagion from the US and PIIGS economies to the East Asian sovereign CDS markets and intraregional contagion within the East Asian markets. Second, the impact of contagion is different according to whether it is measured by the linear (Gaussian) or the upper tail dependence. Third, Japan plays an important role in increasing the linear dependence whereas China and Korea are crucial in terms of the upper tail dependence. Lastly, the GFC has structurally increased the linear dependence but not the upper tail dependence between the East Asian sovereign CDS markets.     

Multivariate models for operational risk

Böcker and Klüppelberg [Risk Mag., 2005, December, 90–93] presented a simple approximation of OpVaR of a single operational risk cell. The present paper derives approximations of similar quality and simplicity for the multivariate problem. Our approach is based on the modelling of the dependence structure of different cells via the new concept of a Lévy copula.     

Worst case portfolio vectors and diversification effects

We consider the problem of identifying the worst case dependence structure of a portfolio X ₁,…,X _n of d-dimensional risks, which yields the largest risk of the joint portfolio. Based on a recent characterization result of law invariant convex risk measures, the worst case portfolio structure is identified as a μ-comonotone risk vector for some worst case scenario measure μ. It turns out that typically there will be a diversification effect even in worst case situations. The only exceptions arise when risks are measured by translated max correlation risk measures. We determine the worst case portfolio structure and the worst case diversification effect in several classes of examples as, e.g. in elliptical, Euclidean spherical, and Archimedean type distribution classes.     

Testing the Gaussian copula hypothesis for financial assets dependences

Abstract

Using one of the key properties of copulas that they remain invariant under an arbitrary monotonic change of variable, we investigate the null hypothesis that the dependence between financial assets can be modelled by the Gaussian copula. We find that most pairs of currencies and pairs of major stocks are compatible with the Gaussian copula hypothesis, while this hypothesis can be rejected for the dependence between pairs of commodities (metals). Notwithstanding the apparent qualification of the Gaussian copula hypothesis for most of the currencies and the stocks, a non-Gaussian copula, such as the Student copula, cannot be rejected if it has sufficiently many ‘degrees of freedom’. As a consequence, it may be very dangerous to embrace blindly the Gaussian copula hypothesis, especially when the coefficient of correlation between the pairs of assets is too high, such that the tail dependence neglected by the Gaussian copula can became large, leading to the ignoring of extreme events which may occur in unison.     

Collective efficacy and natural hazards: differing roles of social cohesion and task-specific efficacy in shaping risk and coping beliefs

Abstract

Previous research in non-disaster contexts has shown that the concept of collective efficacy, which is a group’s sense of its ability to achieve a specific objective, assists understanding of community readiness and households’ decisions to take preparedness actions. Collective efficacy expands the concept of social capital, which refers to social resources such as trust, norms and networks, by addressing how likely communities are to activate these resources for specific tasks. This paper empirically investigates the effect of three distinct collective efficacy components on risk perception, fear and self-efficacy regarding natural hazards in Austria. The three components have differing impacts on risk and coping beliefs: (1) Social cohesion decreases risk perception and fear but has no effect on self-efficacy; (2) Efficacy belief in social support increases self-efficacy; (3) Efficacy belief in citizen groups increases risk perception and fear. The combination of efficacy belief in social support and citizen groups seems to be most promising for stimulating protective action, as they together promote both risk and coping appraisal. However, overreliance on social support may have the undesirable effect of creating a false sense of safety among disaster-prone households. The findings demonstrate that collective efficacy provides a meaningful perspective from which to examine risk and coping beliefs but caution against treating it as an umbrella concept, given the differing effects of its components. Future studies are needed to investigate the impact of collective efficacy on other key explanatory factors of protective action, such as response efficacy or non-protective responses.