Spectral measures of risk for international futures markets: A comparison of extreme value and Lévy models

Abstract

This paper investigates Lévy spectral risk measures (SRM) as a coherent alternative to generalized Pareto spectral risk measures. Specifically, using futures data from major indexes, we consider using SRM for conditional distributions belonging to the generalized hyperbolic family of Lévy processes, and compare and contrast the results with those obtained from the traditional unconditional extreme value approach. Compared with Lévy models, the extreme value model provides poor estimates of quantiles outside the fixed tails, which in turn yield poor estimates of the spectral risk measure itself. The superiority of the Lévy models is increasingly apparent as investors become increasingly risk averse.

Introduction

A measure of risk that generalizes both value at risk (VaR) and expected short-fall (ES) is the spectral risk measure (SRM). The SRM has the advantage of being independent of any particular extreme event. The measure provides the weighted average of possible losses with coverage spread over the entire spectrum. The more popular risk measure VaR has the advantage of simplicity, but comes with inherent weaknesses. Specifically, VaR does not satisfy the subadditivity requirement for a risk measure (which implies that it does not satisfy the coherence requirement). VaR fixes tail events corresponding to a given confidence level. It considers the conditional likelihood of tail events while ignoring the actual size of extreme catastrophic events. Thus VaR gives a partial snapshot of potential losses but fails to take into account the actual size of extreme losses after the cutoff point.

To overcome this weakness and to ensure that the subadditivity (and hence coherence) requirement is met, in this paper, we propose using the SRM in preference to the more commonly applied expected short-fall (ES) measure (where ES estimates the potential loss by averaging all the possible losses in the tail of the distribution, and, as such, overcomes the weakness of VaR—namely that it provides just a snapshot of loss at a particular confidence level). A number of authors, including Acerbi (2004), Cotter and Dowd (2006), and Sorwar and Dowd (2010), have studied the SRM because it meets the requirement that catastrophic tail events and usual nontail events should have different weights and that the weight of the catastrophic tail events should be allowed to vary according to how averse an investor is towards the risk. This generalization includes an extra parameter and a recognized risk aversion function. This extra flexibility, however, comes at a greater computational cost. The computational issues are discussed by Cotter and Dowd (2006) in the context of an extreme value (EV) approach. Specifically, Cotter and Dowd evaluate the integrals associated with the calculation of VaR, ES, and SRM.

In this study, we focus on estimating the SRM using both Lévy and extreme value (EV) models. To the best of our knowledge, Lévy models have not previously been used to estimate the SRM. We also discuss the computational challenges that arise in this process. EV models have been widely used to model extreme behavior in many applications, such as extremities of weather, reserves, and financial outcomes. Conditional Lévy models have also recently been applied in modelling extreme behavior. The main difference between the two is that Lévy models use all the data to estimate the model parameters, whereas an EV model uses only the data remaining in the left tail of the distribution after the cutoff point.

Our approach follows a procedure of fixing the tail as applied in the EV calibration, followed by calculating the SRM. We then calibrate the Lévy models using all available data and proceed to calculate the SRM with those models. By comparing the resulting estimates, we can identify the extent to which using only a fraction of the data on the tail affects the performance of a risk measure that is not tail-based.

The Lévy approach, although mathematically elegant, comes with a major drawback: with few exceptions, there are no closed-form formulas for risk measures, so even a relatively straightforward VaR estimation is difficult to implement. The risk measure SRM is a compounded version of VaR, and its implementation must therefore be even more protracted.

The motivation for this study comes from the fact that both VaR(α) and ES(α) are restricted to the tail at the extreme end of the density distribution (with α values as high as 0.95 or even 0.99). VaR(α) is a point estimate and ES(α) is the value of the integral $\frac{1}{1-\alpha }{\int }_{\alpha }^1\mathit{VaR}\left(u\right)\mathit{du}$. However, SRM with an exponential risk aversion function is an integral of the form ${\int }_0^1\frac{{Re}^{-R\left(1-u\right)}}{1-e^{-R}}\mathit{VaR}\left(u\right)\mathit{du}$, which is not restricted to such a tail but also embraces the data outside it. Thus, we are motivated to examine how EV/SRM compares with Lévy/SRM. To highlight the potential pitfall of applying an EV/SRM estimation, we consider SRM for both EV and Lévy models, using futures data from the S&P500, FTSE, DAX, HangSeng, and Nikkei 225 indexes.

The paper is structured as follows. Section 2 briefly describes the Lévy and EV frameworks, while Section 3 provides the initial data analysis. In Section 4 we discuss conceptual matters regarding estimation and bootstrapping of SRM. In Section 5, we discuss goodness of fit under Lévy and EV models. Section 6 describes the empirical findings, and the final section concludes the paper.