How fat are the tails of equity market indices?

• Author: Stoyan Stoyanov,Frank J. Fabozzi,Lixia Loh
• DOI: http://doi.org/10.1002/ijfe.1577
• Date: 01 July 2017
• Published date: 01 July 2017

Received: 8 December 2015 Revised: 27 October 2016 Accepted: 12 February 2017

DOI: 10.1002/ijfe.1577

RESEARCH ARTICLE

How fat are the tails of equity market indices?

Stoyan Stoyanov1Lixia Loh2Frank J. Fabozzi3

1Stony Brook University,Harr iman Hall

109, Stony Brook, 11974-3775, NY,USA

2Consultant, 294 Lavender Street, Singapore

338807, Singapore

3EDHEC Business School, 393, Promenade

des Anglais - BP3116 06202 Nice Cedex 3,

France

Correspondence

Frank J. Fabozzi, 858 Tower ViewCircle,

New Hope, PA 18938, USA.

Email: frank.fabozzi@edhec.edu

Abstract

Using a generalized autoregressive conditional heteroskedasticity model to explain

away the volatility clustering of volatility effect and extreme value theory to anal-

yse the residuals' left and right tails, we study the tail thickness of 22 developed and

19 emerging equity market indices. In-sample and out-of-sample tests indicate that

exponential tails of the residuals cannot be strongly rejected. We study the disper-

sion of extremes of developed and emerging markets, and we report a statistically

significant tail asymmetry in both types of markets and a significant change in both

tail risk and tail asymmetry of emerging markets after the financial crisis of 2008.

KEYWORDS

conditional value-at-risk, extreme value theory, fat tails, GARCH, value-at-risk, tail risk

1INTRODUCTION

The properties of asset return distributions have been studied

extensively since the fundamental workof Mandelbrot (1963)

and Fama (1965). Empirical research has established the fol-

lowing four stylizedfacts about asset returns: (a) clustering of

volatility,(b) auto-regressive behaviour, (c) skewness, and (d)

fat tails (see Stoyanov, Rachev, Racheva-Iotova, and Fabozzi

(2011) for a discussion and literature reviewand Kim, Rachev,

Bianchi, Mitov, and Fabozzi (2011) for an empirical study).

The degree to which these properties are present in return

data depends heavily on the frequency of returns and asset

class. The higher the frequency is, the more pronounced is

the clustering of volatility and fat tailedness. With respect to

asset class, returns of large-cap stocks are known to exhibit

lighter tails, reflecting better liquidity and lower probability

of default.

Although the functional form of return distributions is

unknown, there have been significant efforts to determine the

functional form of the tail decay. This is theoretically possi-

ble because of a fundamental result in extreme value theory

(EVT) that asserts that the extreme tail can be approximately

described through a simple parametric family of distribu-

tions known as the generalized Pareto distribution (GPD). In

finance, the rate of tail decay has important implications for

portfolio risk measurement and stress testing where a distri-

butional model is combined with a risk measure sensitive to

tail events, as well as option pricing where the so-called smile

effect can be explained by the presence of fat tails.

Since the application of EVT in finance by Parkinson

(1980) and Longin (1996), the theory has played an increas-

ing role in the estimation of the frequency of extreme events

and risk forecasting using value-at-risk (VaR) or conditional

value-at-risk (CVaR) as risk measures. Studies on predic-

tive performance of various VaR methods have found the

EVT-based method to be particularly accurate (Bekiros &

Georgoutsos, 2005; Danielsson & de Vries, 1997; Fernandez,

2005; McNeil & Frey, 2000; Pownall & Koedij, 1999;

Tolikas, Koulakiotis, & Brown, 2007). In studies of the dis-

tribution of extreme stock returns, the principal finding is that

market index returns exhibit Pareto-liketails, see for example,

Jondeau and Rockinger (2003), Gettinby, Sinclair, Power,

and Brown (2004), Longin (2005), and Tolikas and Gettinby

(2009). However, in these studies, EVT is applied uncondi-

tionally without explaining away the clustering of volatility

which is known to cause fat tails in the unconditional return

distribution.

This paper contributes to the literature in a number of

ways. First, we consider a sample of 22 developed and 19

emerging market indices which is a much larger number of

Int J Fin Econ. 2017;22:181–200. wileyonlinelibrary.com/journal/ijfe Copyright © 2017 John Wiley & Sons, Ltd. 181

182 STOYANOV ET AL.

countries investigated than in previous studies. Second, we

explain away the clustering of volatility effect and study the

tail thickness of both tails of the residual to assess the relative

impact of the clustering of volatility effect on tail thickness.

We look at the dynamics of tail thickness across different

market conditions as well as its variation cross-sectionally

across markets. Finally, we test hypotheses both in-sample

and out-of-sample by carrying out VaR back-testing calcula-

tions at three different quantile levels forthe lef t and the right

tail (at 1%, 2.5%, and 5% tail probabilities); in previous stud-

ies, the tests are only in-sample. To this end, we employ the

standard VaR-based tests as well as a new CVaR test which

has not been employed in earlier studies.

Our findings confirm that the GARCH-EVT model is very

successful at describing high quantiles in both tails of the

return distribution for the 41 markets we investigate.We find

that having explained the clustering of volatility, an expo-

nential tail decay is sufficient to describe both tails of the

return distribution both in-sample and out-of-sample. As a

consequence, the Pareto-type tail decay reported in previous

studies may be a consequence primarily of the clustering of

volatility effect. The immediate conclusion that distributions

in the maximum domain of attraction (MDA) of the Gumbel

distribution present natural starting points in parametric mod-

eling of the residual in GARCH-type processes should be

considered with care. The Gaussian distribution is in that

domain of attraction, and it is known to be unrealistic even

if combined with a GARCH-type process. Our results sug-

gest that realistic models in the Gumbel MDA would be those

with relatively fast convergence to the limit (i.e., distributions

whose tail beyond the 2.5% quantile is very close to exponen-

tial). The Gaussian distribution is notorious for the very slow

convergence rate to the Gumbel limit.

As a by-product in the empirical comparisons, we conclude

that the commonly used VaR-basedtests for the out-of-sample

performance of risk models in the practice of model valida-

tion appear rather weak. At the 1% tail probability level which

is commonly used for regulation-related testing, these tests

cannot distinguish between models with tail index ∈[0,.2].

This presents a potential problem in model validation in that

the neighborhood of statistically acceptable models is quite

wide. Our results suggest that the VaR-basedtests be extended

with a CVaR test which is more informative because of the

better sensitivity of CVaR to tail thickness (see Stoyanov,

Rachev, and Fabozzi, 2013).

Finally, having explained away the clustering of volatility,

our results indicate that there is no significant difference in

tail thickness by country. This suggests that the presence of

such difference reported in previous studies (LeBaron and

Samanta, 2005) based on the unconditional return distribution

may be explained by the clustering of volatility. Assuming an

exponential tail decay of the residual, we measure the lower

and upper tail behaviour of the developed and the emerging

markets in the sample in terms of the average dispersion of

extremes (the average of the corresponding scale parameters

of the GPD). We find statistically significant tail asymmetry

in both types of markets and also a significant change after

the financial crisis of 2008. In the bull market before the cri-

sis, emerging markets are characterized by a better upside

potential which comes at the cost of higher volatility, while

after the crisis, the better upside potential comes at a cost of

a worse downside than developed markets. The dispersion of

the lower and the upper extremes of the residuals of the devel-

oped markets were relatively unchanged before and after the

crisis suggesting that the clustering of volatility is the pri-

mary source of the increased tail risk of the developed markets

during the crisis.

From a portfolio risk management perspective, the conclu-

sion that the dynamics in volatility is the most significant

factor in the dynamics of the left tail has important impli-

cations and emphasizes the importance of volatility manage-

ment in managing tail risk. For example, dynamic portfolios

attempting to achieve constant volatility would be an inter-

esting tool to manage the dynamics of tail risk regardless of

the risk measure used. However, as far as the question of

international diversification of tail risk is concerned, volatil-

ity management may be insufficient especially if emerging

markets are included in the portfolio because of the presence

of non-linearity in tail dependence (Christoffersen, Errunza,

Jacobs, and Langlois, 2012).

The paper is organized in the following way. Section 2

discusses the peak-over-threshold (POT) method, the condi-

tional EVT risk model, and the statistical tests used in the

paper. Sections 3 and 4 describe the data and the results,

respectively. Section 5 concludes.

2STATISTICAL METHODOLOGY

In finance, EVT has been traditionally applied to estimate

probabilities of extreme losses or loss thresholds such that

losses beyond it occur with a predefined small probability. In

fact, EVT provides a model for the extreme tail of the distri-

bution which turns out to have a relatively simple structure

described through the GPD. In this section, we first briefly

describe the POT method and the conditional EVT method

for risk estimation. Then, we proceed with the back-testing

methodology for the out-of-sample tests.

2.1 The peak-over-threshold method

The approach for testing for extreme values in this paper is

based on the POT method. Suppose that we have selected

a high loss threshold u, and we are interested in the condi-

tional probability distribution of the excess losses beyond u.