Estimation of tail-related value-at-risk measures: range-based extreme value approach

Abstract

This study proposes a new approach for estimating value-at-risk (VaR). This approach combines quasi-maximum-likelihood fitting of asymmetric conditional autoregressive range (ACARR) models to estimate the current volatility and classical extreme value theory (EVT) to estimate the tail of the innovation distribution of the ACARR model. The proposed approach reflects two well-known phenomena found in most financial time series: stochastic volatility and the fat-tailedness of conditional distributions. This approach presents two main advantages over the McNeil and Frey approach. First, the ACARR model in this approach is an asymmetric model that treats the upward and downward movements of the asset price asymmetrically, whereas the generalized autoregressive conditional heteroskedasticity model in the McNeil and Frey approach is a symmetric model that ignores the asymmetric structure of the asset price. Second, the proposed method uses classical EVT to estimate the tail of the distribution of the residuals to avoid the threshold issue in the modern EVT model. Since the McNeil and Frey approach uses modern EVT, it may estimate the tail of the innovation distribution poorly. Back testing of historical time series data shows that our approach gives better VaR estimates than the McNeil and Frey approach.

Keywords:

• Risk management
• Value-at-risk (VaR)
• Asymmetric conditional autoregressive range (ACARR) model
• Extreme value theory (EVT)

JEL Classifications:

• C22—Time-series models
• C53—Forecasting and other model applications

Acknowledgements

We are grateful to the editors, Michael Dempster and Jim Gatheral, the anonymous referees, and the participants at the 4th Singapore Economic Review (SER) conference, 18th conference on Pacific Basin Finance, Economics, Accounting, and Management (PBFEAM), 18th conference of the Multinational Finance Society (MFS), 4th Financial Management Association (FMA) Asian conference, 19th conference on the Theories and Practices of Securities and Financial Markets (SFM), and a seminar at Academia Sinica, for their valuable comments and suggestions on earlier versions of this paper.

Notes

¹VaR is generally defined as the capital sufficient to cover, in most instances, losses from a portfolio over a holding period of a fixed number of days. The Basle accord proposes a 1% VaR over a 10-day holding period. According to the 1996 Capital Adequacy Directive by the Bank of International Settlement in Basle, the risk capital of a bank must be sufficient to cover losses on the bank’s trading portfolio over a 10-day holding period in 99% of occasions. For internal risk control purposes, most of the financial firms compute a 5% VaR over a one-day holding period.

²Three forms of extreme value distributions represent the GEV distribution: the Gumbel distribution, the Frechet distribution, and the Weibull distribution. See Section 3.2 for details.

³There are periods when the conditional distribution of financial returns appears light-tailed rather than heavy-tailed.

⁴See Section 5.1.2 for details about the McNeil and Frey (2000) approach.

⁵Their approach is vindicated by the very satisfying overall performance in various back testing experiments.

⁶There is a growing recognition of the fact that the range-based volatility models (e.g. the ACARR model) can provide sharper estimates and forecasts than the return-based volatility models (e.g. the GARCH model). Many insightful studies have provided powerful evidence to support this viewpoint including Parkinson (1980), Garman and Klass (1980), Wiggins (1991), Rogers and Satchell (1991), Kunitomo (1992), Gallant et al. (1999), Yang and Zhang (2000), Alizadeh et al. (2002), and more recently, Brandt and Jones (2006), Chou (2006), and Martens and van Dijk (2007).

⁷There are good reasons why the asset price should behave asymmetrically. For example, for investors the more relevant risk is generated by the downward movement rather than the upward movement of the asset price; the upward movement is important in generating investors’ expected returns. For more details of this and other related issues, see also Levy (1978), Engle et al. (1987), Nelson (1991), Duan (1995), Engle and Ng (1993), Campbell (1999), Barberis and Huang (2000), and Tsay (2002).

⁸Note that t is not a point in time, but a specified time interval, may be an exchange trading day, so the notions of highest, lowest, and opening prices apply. Thanks to the anonymous referees for pointing this out.

⁹See Bollerslev (1986) for a discussion of the parameters in the context of the GARCH model.

¹⁰See Chou (2006) for more details of this and other related issues.

¹¹See Gnedenko (1943), Gumbel (1958), Jenkinson (1955), and Kinnison (1985).

¹²Thanks to the anonymous referees for bringing this important clarification to our attention.

¹³See Gumbel (1958).

¹⁴According to McNeil and Frey (2000), the upper tail (the right tail) of the return distribution represents losses for an investor with a short position in futures, whereas the lower tail (the left tail) represents losses for an investor being long in futures (see also Section 5.1.2). Therefore, following McNeil and Frey (2000), we can assume that the upward price range is associated with a short position, and that the downward price range is associated with a long position. This is because the upper tail of the range distribution (i.e. the range (price) increases as measured by the upward price ranges ) represents losses for an investor with a short position in futures, whereas the lower tail (i.e. the range (price) decreases as measured by the downward price ranges ) represents losses for an investor being long in futures. Thanks so much to one anonymous referee for bringing this important point to our attention.

¹⁵See Embrechts et al. (1997) and McNeil and Frey (2000).

¹⁶We thank the anonymous referees for bringing this point to our attention.

¹⁷Like other standard numerical or statistical software, Matlab now also provides functions or routines for extreme value analysis.

¹⁸Note that according to Chou (2006), the model coefficients and measure the short-term impact effect and the long-term impact effect of volatility shocks respectively. By comparing the model coefficients, Chou reported that there is an asymmetric relationship of volatility shocks between the upward ranges and the downward ranges. As one anonymous referee has pointed out, this inference on volatility asymmetry between the two ranges is just too strong by using point estimation alone. We totally agree with the referee’s comment, and therefore, decide not to follow Chou’s argument in our current study. We will leave this rather interesting issue for future research. We thank the referee so much for bringing this important point to our attention.

¹⁹According to equations (5) and (8), the error terms in the ACARR model follow an iid distribution this is consistent with the basic EVT assumption that the random variables X ₁, X ₂, …, X_n also follow an iid distribution. Empirical evidence also supports the idea that pre-whitening of data by fitting of a dynamic model is a sensible prelude to EVT analysis in practice (see Embrechts et al. (1997) and McNeil and Frey (2000)). Thus, we extract the error terms from the adopted ACARR (1,1) model, and use (classical) EVT to model their tail distribution.

²⁰In the nonlinear context, it is not possible to construct an overall goodness-of-fit statistic.

²¹This model mimics many features of real financial return series.

²²In other words, the GDP is the natural model for the unknown excess distribution above sufficiently high thresholds.

²³This effectively gives us a random threshold at the (N + 1)th order statistic.

²⁴See equations (23) and (24).

²⁵See McNeil and Frey (2000) for the discussion on the choice of N = 100.

²⁶The estimations obtained above are now used to compute VaR for our approach and the McNeil and Frey (2000) approach. See equations (22), (30) and (32) for details.

²⁷The historical VaR is obtained based on the historical distribution.

²⁸To test for the statistical significance of any difference, we carry out a non-parametric Kolmogorov-Smirnov test. See Section 5.2.2 for details.

²⁹In 7 (3) out of 10 cases, the historical VaR is slightly underestimated (overestimated) by our approach.

³⁰The critical value of the Kolmogorov–Smirnov test statistic at the 5% significance level is 0.4670.