Conditional tail risk measures for the skewed generalised hyperbolic family

https://doi.org/10.1016/j.insmatheco.2019.02.008Get rights and content

Abstract

This paper deals with the estimation of loss severity distributions arising from historical data on univariate and multivariate losses. We present an innovative theoretical framework where a closed-form expression for the tail conditional expectation (TCE) is derived for the skewed generalised hyperbolic (GH) family of distributions. The skewed GH family is especially suitable for equity losses because it allows to capture the asymmetry in the distribution of losses that tends to have a heavy right tail. As opposed to the widely used Value-at-Risk, TCE is a coherent risk measure, which takes into account the expected loss in the tail of the distribution. Our theoretical TCE results are verified for different distributions from the skewed GH family including its special cases: Student-t, variance gamma, normal inverse gaussian and hyperbolic distributions. The GH family and its special cases turn out to provide excellent fit to univariate and multivariate data on equity losses. The TCE risk measure computed for the skewed family of GH distributions provides a conservative estimator of risk, addressing the main challenge faced by financial companies on how to reliably quantify the risk arising from the loss distribution. We extend our analysis to the multivariate framework when modelling portfolios of losses, allowing the multivariate GH distribution to capture the combination of correlated risks and demonstrate how the TCE of the portfolio can be decomposed into individual components, representing individual risks in the aggregate (portfolio) loss.

Introduction

This paper addresses one of the main challenges faced by financial companies on how to evaluate the risk arising from the distribution of losses. Indeed, the quantification of exposure in a portfolio of losses presents a key challenge for practitioners. Developing a standardised framework for measuring risks and estimating loss severity from historical data was initially addressed by Morgan/Reuters (1996) who developed a standardised Value-at-Risk (VaR) measure that has been widely used by practitioners in financial institutions with significant trading volumes. Despite its widespread use, the VaR fails to satisfy the coherency principle. Specifically, it is unable to quantify the expected loss in the tail of the distribution and exploit the benefits of portfolio diversification. Computation of the VaR requires knowledge about the distribution of the underlying data. Practitioners have widely adopted the assumption of normality as it leads to straightforward calculation of the VaR. However, as the literature shows, financial returns exhibit heavy tails and excess kurtosis (see, e.g., McNeil et al. (2015) and Ignatieva and Platen (2010)), while financial losses exhibit extreme tails and are clearly severely right-skewed (see, e.g., McNeil (1997), Embrechts et al. (2002), Bernardi et al. (2012), Eling (2012) and Miljkovic and Grün (2016)). Although the normal distribution is the most popular distribution used for modelling in economics and finance, it is not appropriate for modelling financial risks or portfolio losses (see, e.g., Lane (2000) and Vernic (2006)).

In order to address the shortcomings of VaR we describe the risk of the underlying portfolio distribution using the Tail Conditional Expectation (TCE) risk measure, in combination with the skewed generalised hyperbolic (GH) family of distributions. To the best of our knowledge, very few contributions provide a tractable model in which the risk measure of the sum of positive random variables has a closed-form expression, as our model does. The TCE used in risk quantification describes the expected amount of risk that could be experienced given that risk factors exceed a particular threshold value, which is different from the VaR, which merely defines the minimum loss that would incur in an unfavourable scenario. At the same time, the skewed GH family allows to model heavy tails of the distribution, and is more flexible compared to its symmetric counterpart, the so-called symmetric generalised hyperbolic distribution (SGH) family discussed in Ignatieva and Landsman (2015) in the sense that it allows us to capture the asymmetry in the distribution of underlying variables. This is particularly important for financial companies that aim to capture the distribution of financial losses, which tends to be heavily skewed. Furthermore, the motivation for consideration of the skewed distributions is that they are becoming popular in the literature, see, e.g., Adcock (2010) and Eling (2012) who consider skewed normal and skewed Student-t distribution to model financial and insurance data, respectively. Some previous results on elliptical distributions in the context of the loss assessment and calculation of capital requirement can be found in Landsman and Valdez (2003) and Valdez et al. (2009). Landsman and Valdez (2005) discuss TCE for exponential dispersion models, while non-Gaussian distributions used in context of risk diversification have been studied in Desmoulins-Lebeault and Kharoubi-Rakotomalala (2012). Some additional properties on the additivity of the VaR for heavy-tailed distributions are addressed in Embrechts et al. (2009). Miljkovic and Grün (2016) model loss data using mixture distributions. Further risk applications related to loss severity were addressed by Peters et al., 2010, Peters et al., 2011 and Chavez-Demoulin et al. (2006), but the authors do not provide closed-form expressions for risk measures. Problems related to the use of risk measures in capturing portfolio risk and decomposition of an aggregate loss into its individual components have also been studied.

Research into deriving TCE-based allocations has experienced a rapid growth in the literature. TCE-based allocation for the multivariate normal family was developed in Panjer (2002) and extended to a larger class of elliptical distributions in Landsman and Valdez (2003). Multivariate skew-Elliptical and skew-Normal distributions were considered in Cai and Tan (2005) and Vernic (2006), respectively. Cai and Li (2005) have extended the literature on the TCE-based allocations for the phase-type distributions while Furman and Landsman (2007) have considered multivariate Gamma distribution. Further, Goovaerts et al. (2005) and Chiragiev and Landsman (2007) have investigated multivariate Pareto distribution. Guillén et al. (2013) emphasise the importance of a simple closed-form risk measure expression for loss aggregation and risk allocation. The authors derive the results for a set of distributions that generalise the Beta distribution. TCE for capital allocation was also discussed in Asimit et al. (2011) and risk decomposition was analysed by Pérignon and Smith (2010) and Engsted et al. (2012). Hendriks and Landsman (2017) have provided the TCE-capital allocation for the new multivariate Generalised Pareto family of distributions. Several research works have considered the capital allocation based on the TCE measure for the Sarmanov’s class of distributions: Vernic (2017) provides the allocation for the case of exponential marginals; Hashorva and Ratovomirija (2015), Ratovomirija et al. (2017) and Willmot and Woo (2015) consider capital allocation for mixed Erlang distributed risks joined by the Sarmanov distribution; Cossette et al. (2013) consider the FGM copula with mixed Erlang marginals, which is a special case of the Sarmanov distribution. A sequential approach to capital allocation has been proposed in Targino et al. (2015).

In the theoretical section of this paper we derive closed-form expressions for the TCE risk measure for the family of skewed GH distributions and obtain a TCE-based portfolio allocation of aggregate losses. In the empirical section of the paper we assess the ability of the GH family to fit the equity losses arising from stocks Amazon, Goldman Sachs, IBM, Google and Apple as constituents of the S&P 500 index.

When assessing the quality of fit to the univariate losses data arising from individual stocks using various goodness-of-fit statistics we find that none of the distributions from the GH family could be rejected, suggesting that all distributions provide good fit to univariate losses. We observe that the univariate TCE is comparable for all distributions from the GH family at lower quantiles. For the extreme quantiles, the VG and the HYP distributions result in consistently lower TCEs while the Student-t distribution leads to significantly larger risk estimates. When assessing the quality of the multivariate fit to the data on Amazon, Goldman Sachs, IBM, Google and Apple losses, we fit five-dimensional skewed GH distributions to the data. We observe that the best fit is attained by the multivariate Student-t distribution, followed by the GH distribution. We also find that these best performing distributions result in larger TCE estimates, i.e. they are more conservative estimators of risk. We show how TCE-based allocations in the portfolio of losses can be used to decompose an aggregate loss into the individual components, which are assumed to be correlated, and report the average percentage TCE-based allocations for different distributions. This appears to be useful from a portfolio manager’s point of view, when quantifying the capital requirement needed for each individual investment, and allocating the contribution of each individual risk to an aggregate risk.

The remainder of the paper is organised as follows. Section 2 introduces a family of skewed generalised hyperbolic (GH) distributions, starting from the most general multivariate case, and then considering its univariate, special, and limiting cases. Section 3 introduces the concept of the tail conditional expectation (TCE), and proceeds with developing innovative theoretical results for the closed-form TCE formulae for the univariate and multivariate distributions from the GH family. It also discusses portfolio risk decomposition, where individual risks are aggregated in form of the TCE. The empirical analysis presented in Section 4 demonstrates how our theoretical methodology is applied to the quantification of risk of an equity portfolio, whereby five stocks are used to estimate univariate TCEs arising from Amazon, Goldman Sachs, IBM, Google and Apple losses, as well as an aggregate TCE constructed for the five-dimensional portfolio of losses. The section proceeds with a discussion of TCE-based allocations to each of its constituents. Section 5 concludes and provides final remarks.

Section snippets

The class of generalised hyperbolic distributions

In this section we introduce a framework for a sufficiently rich class of distributions, which allows us to mix the normal distribution with different stochastic means and variances, resulting in a class of normal mean–variance mixture distributions. In particular, the skewed generalised hyperbolic (GH) distributions can be represented as a normal mean–variance mixture where the mixture variable follows the Generalised Inverse Gaussian (GIG) distribution. In the following, we introduce

Tail condition expectation

In this section, we review the concept of the tail conditional expectation (TCE). This risk measure corresponds to the expression of the tail Value-at-Risk (TVaR) or the expected short-fall (ES), assuming a continuous random variable representing financial loss, refer to McNeil et al. (2015).2 We develop TCE formulae for the univariate and

Empirical analysis

In this section we demonstrate how the methodology outlined above can be applied to quantifying risk of the portfolio. We discuss the fit of the univariate and multivariate GH family of distributions to the data on individual stocks, which are constituents of the S&P 500 index. We consider a sample of five stocks (Amazon, Goldman Sachs, IBM, Google, and Apple) covering a time frame from the 1st of January 2015 to the 1st of January 2017. The total number of computed return observations is 502.

Conclusion

This paper develops closed-form representation for the TCE risk measure and TCE-based portfolio allocation for skewed generalised hyperbolic distributions, which include a general, more flexible GH distribution (that has more flexibility due to an additional parameter) as well as its special cases (Student-t, VG, NIG, and HYP distributions). TCE is computed for the univariate loss as well as an aggregate loss arising from the portfolio of losses. As opposed to a widely used VaR measure, TCE is

Acknowledgment

This research was supported by the Zimmerman Foundation for the study of Banking and Finance .

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