When the US stock market becomes extreme?

Aboura, Sofiane Article When the US stock market becomes extreme? Risks Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Aboura, Sofiane (2014) : When the US stock market becomes extreme?, Risks, ISSN 2227-9091, MDPI, Basel, Vol. 2, Iss. 2, pp. 211-225, https://doi.org/10.3390/risks2020211 This Version is available at: https://hdl.handle.net/10419/103611 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/3.0/ Risks 2014,2, 211-225; doi:10.3390/risks2020211 OPEN ACCESS risks ISSN 2227-9091 www.mdpi.com/journal/risks Article When the U.S. Stock Market Becomes Extreme? Sofiane Aboura Department of Finance, DRM-Finance, Université de Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 Paris CEDEX 16, France; E-Mail: [email protected]; Tel.: +33-1-4405-4565. Received: 5 March 2014; in revised form: 5 May 2014 / Accepted: 13 May 2014 / Published: 28 May 2014 Abstract: Over the last three decades, the world economy has been facing stock market crashes, currency crisis, the dot-com and real estate bubble burst, credit crunch and banking panics. As a response, extreme value theory (EVT) provides a set of ready-made approaches to risk management analysis. However, EVT is usually applied to standardized returns to offer more reliable results, but remains difficult to interpret in the real world. This paper proposes a quantile regression to transform standardized returns into theoretical raw returns making them economically interpretable. An empirical test is carried out on the S&P500 stock index from 1950 to 2013. The main results indicate that the U.S stock market becomes extreme from a price variation of ±1.5% and the largest one-day decline of the 2007–2008 period is likely, on average, to be exceeded one every 27 years. Keywords: extreme value theory; volatility; risk management JEL classifications: C4, G13, G32 1. Introduction The tail behavior of the financial series has been largely examined by many studies1with various applications. The key attraction of extreme value theory (EVT) is that it offers a set of ready-made approaches to risk management analysis. A general discussion of the application of EVT to risk management is proposed by [12,15,16]. However, if the oldest studies made use of raw returns 1Some of the main studies of univariate EVT in finance include [1–14]. Risks 2014,2212 (e.g., [9,10]etc.) in their EVT analysis, since then, the literature has applied EVT on independent and identically distributed data such as standardized returns (e.g., [13]). Therefore, many results stemming from EVT applications remain quiet difficult to interpret economically because of the use of these standardized returns and conclusion can be vague. Indeed, the practice is to extract standardized returns by filtering the raw returns with a GARCH family model; then run some standard tests to check for the approximate independent and identically distributed nature of the standardized returns and finally apply EVT on these data. However, these standardized returns have no economic meaning in the real world. In clear, they cannot be understood in terms of monetary units. One naive possibility would have been to apply an OLS procedure to study the relationship between a response variable (raw returns) and an explanatory variable (standardized returns), in order to numerically recompute all the raw returns corresponding to the standardized returns derived from the EVT analysis and forecasting. Recall that the least-squares regression describes how the mean of the response variable changes with the vector of covariates. But standard regression analysis has several limits such as not being robust to extreme observations whereas EVT is based on observations located in the tails. For that reason, this article proposes an empirical methodology to convert the results into monetary units such as stock index returns. This issue seems not to have been addressed previously, perhaps because the typical EVT paper is generally devoted to the sole statistical perspective used to describe the extreme behavior of financial returns, but not to explain the economic meaning. The method is a simple linear transformation of the standardized returns into raw returns using quantile regressions [17]. Quantile regression generalizes the standard regression model to conditional quantiles of the response variable. Hence, it can be be viewed as a natural extension of standard least squares estimation of conditional mean models to the estimation of a series of models for conditional quantile functions [18]. To our best knowledge, no such statistical test has been implemented so far to address this problem of economic interpretation of EVT results. This research is relevant because it gives a new approach for interpreting the EVT results. The main contribution consists in proposing an empirical methodology based on quantile regression for transforming EVT results expressed in the form of standardized returns into raw returns. An application is provided on the U.S. market from 1950 to 2013. The article is organized as follows. Section 2 opens with a brief review of the extreme value theory applications in finance. Section 3 presents an analysis of the data while Section 4 analyses the empirical results. Section 5 summarizes the main findings and concludes. 2. Methodology 2.1. Tail Distribution A theorem from [19,20] shows that when the threshold uis sufficiently high, the distribution function Fuof the excess beyond this threshold can be approximated by the Generalized Pareto Distribution (GPD): Fu(x)≈Gξ,β(x).(1) This limit distribution has a general form given by: Risks 2014,2213 Gξ,β(x) =    1−(1 + ξx/β)−1/ξ,for ξ6= 0 1−exp(−x/β),for ξ= 0 (2) where β≥0and where x≥0when ξ≥0and where 0≤x≤ −β/ξ when ξ < 0.βis a scaling parameter and ξis the tail index. The tail index is an indication of the tail heaviness, the larger ξ, the heavier the tail. This distribution encompasses other type of distributions. If ξ > 0(ξ < 0) then it is a reformulated version of the ordinary Pareto distribution and if ξ= 0, it corresponds to the exponential distribution. 2.2. Threshold Selection Threshold selection is subject to a trade off between finding a high threshold where the tail estimate has a low bias with a high variance or finding a low threshold where the tail estimate has a high bias with a low variance. Low (high) bias refers to a (non-reliable) reliable estimate for the tail index. There are two approaches for threshold selection. The first approach is visual inspection and the second one is automatic detection. The first approach of visual selection [16,21,22] denotes a plausible threshold choice based on the results of the two given plot methods, namely, the mean residual life plot and the threshold plot. The mean residual life plot is the first visual method: "u, 1 nu u X i=1 (xi−u)#(3) with nurepresenting the number of exceedances over the threshold u. The threshold detection begins with a slope is positive. The threshold plot is the second visual inspection method that consists in fitting the GPD over a range of thresholds allowing observation of both the parameter estimate stability and variance: hb ξ, Gξ,β(x) = 1 −(1 + ξx/β)−1/ξi.(4) The second approach, of optimal threshold, corresponds to the application of an automated method which aims at minimizing the Asymptotic Mean Squared Error (AMSE)2. For the optimal threshold selection method, let’s consider an ordered sample of size nu,Xnu≤... ≤X1with Xnubeing the nth u upper order statistic. The [24] estimator is defined by: "nu,b ξH nu=1 nu nu X i=1 (logXn−i+1 −logXn−nu)#.(5) It has been popular for the optimal threshold to be estimated such that the bias and variance of the estimated Hill tail index vanish at the same rate where the mean squared error is asymptotically minimized. Usually, AMSE is obtained throughout a sub-sample bootstrap procedure. This paper 2See [15], Section 4.7 ii. More general discussion on AMSE includes [23]. Risks 2014,2214 follows [15] who develop a criterion for which the AMSE of the Hill estimator is minimal for the optimal number of observations in the tail. In clear, this approach computes the optimal sample fraction needed to apply the tail index estimator. 2.3. Three Extreme Risk Management Measures Every day risk management practices require evaluating the potential risk of loss. Three risk management measures can be used to adress this evaluation problem. Most of the recent literature [12,25–29] confirms the superiority of the in and out-the-sample performance of the risk management models when combining a heavy-tailed GARCH filter with an extreme value theory-based approach. For that reason, these three risk management measures are based on returns standardized by GARCH models. First, the value-at-risk (VaR) is a common risk measure on any portfolio of financial assets with a given probability and time horizon. For example, a one-day 1% VaR of 1 million euros means that there is a 0.01 probability that the portfolio will fall in value by more than 1 million euros over the next day period. However, there is no need to fully identify the probability distribution because only the extreme quantiles are of interest. Therefore, in our setting, the value-at-risk is computed by inverting the tail estimation formula based on the loss distribution: V aRq=u+β ξnu n(1 −q)−ξ−1.(6) Nevertheless, VaR models have been criticized for their partial inadequacy. First, VaR can be misleading during volatile periods [30,31]. Second, the VaR of a portfolio can be higher than the sum of the respective VaRs of each individual asset in the portfolio [32]. Third, VaR disregards any loss beyond the VaR level. Second, a complementary measure known as the expected shortfall (ES) is usually used, for example, in margin requirements. ES particularly focuses on the loss beyond the VaR level. It accounts for the size of tail losses since it evaluates the expected loss size given that VaR is exceeded: ESq=V aRq+E(X−V aRq|X > V aRq)(7) Third, the return level (RL) [33] highlights the effect of extrapolation, which is useful for forecasting, even if scarcity produces large variance estimates. Let’s consider xmas the return level that is exceeded on average once every mobservations. Let ζube the probability of exceeding the threshold u. Return level is expressed in annual scale so that the N-year return level is the level expected to be exceeded once every Nyears. From a risk management perspective, RL is a waiting time period before observing a minimum daily loss, while VaR is the minimum daily loss that can occur with a given probability over a certain horizon. Both measures are threfore closed and complementary. n250 is the average number of trading days per year with m=N×n250. N-year return level is an average waiting time period. For example, the N-year return level is the level expected to be exceeded once every N years. It comes for the N-year return level: Risks 2014,2215 xm=u+β ξh(N×n250ζu)ξ−1i(8) 3. Data Analysis 3.1. Data Description Let’s define the market log-returns as {Rt}t=1,...,T with T= 15,950 raw daily log-returns computed from closing price of the U.S. S&P 500 stock index. The time period begins on 3 January 1950 and ends on 28 May 2013. When applying EVT analysis, parameters estimates along with standard errors can be quite unreliable. This underlies the importance of having long time-series with an adaptive econometric filter analysis to obtain identically distributed innovations. Indeed, the presence of autocorrelation and hetereskedasticity in log-returns series requires implementing filters for avoiding any type of bias in the analysis since the data need to be independent and identically distributed before applying extreme value theory. Therefore, this study makes exclusively use of filtered data. 3.2. Data Filtering Process We examine all the possible specifications within five lags. We test 25 specifications of ARMA(p,q) + GARCH(1,1) models with p= 1, ..., 5and q= 1, ..., 5. We select the more parsimonious model. Four criteria are used for comparison: the Schwarz criterion, the autocorrelogram of residuals and squared residuals and the ARCH effect test. We take care of the trade off between parsimony and maximizing criteria. We find that the ARMA(1,1) + GARCH(1,1) model produces the best fit. We then test an alternative model, ARMA(1,1) + TGARCH(1,1), that allows for leverage effects by considering the contribution of the negative residuals in the ARCH effect. Finally, the ARMA(1,1) + TGARCH(1,1) specification has the best fit considering the four criteria. The ARMA (1,1) component takes into account the mean reversion nature of the returns, while the TGARCH specification allows for leverage effects by considering the contribution of the negative residuals in the ARCH effect. Table 1 displays the descriptive statistics showing that except the AR(1) term, all the parameters are statistically significant. The AR(1) term was not removed because withdrawing φ1did not clearly improved all the criteria. The ARMA(1,1) + TGARCH (1,1) specification is given by: Rt=µ+φ1Rt−1+θ1t−1+t(9) where the innovations tbeing functions of Ztand σt: t=Ztσt.(10) The standardized returns Ztare independent and identically distributed with Zt∼FZ(0,1). The purpose of the time-varying σtis to capture as much of the conditional variance in the residual tin order to leave Ztapproximately independent and identically distributed: σ2 t=ω+α(Zt−1σt−1)2+γ(Zt−1σt−1)2IZt−1σt−1<0+βσ2 t−1.(11) Risks 2014,2216 Table 1. Descriptive statistics. The Table presents the descriptive statistics of the S&P 500 stock index standardized daily log-returns (Z)from 3 January 1950 to 31 May 2013. Sample: 15,950 observations. Z Z Mean −0.0032 Q5 (residual) 2.9403 (p-value) (0.401) Median 0.0202 Q5 (squared residual) 7.2983 * (p-value) (0.063) Maximum 7.0191 Q10 (residual) 11.504 (p-value) (0.175) Minimum −13.1417 Q10 (squared residual) 11.546 (p-value) (0.173) Std. Dev. 1.0001 Q20 (residual) 20.090 (p-value) (0.328) Skewness −0.5062 Q20 (squared residual) 16.385 (Z-statistic, p-value) (−16.2613, 2.2×10−16) (p-value) (0.566) Kurtosis 7.8954 µ0.0002 *** (Z-statistic, p-value) (36.6956, 2.2×10−16) (Z-statistic) (4.7722) Jarque–Bera 16607.86 *** φ1−0.0812 (p-value) (0.0000) (Z-statistic) (−1.0679) Engle LM (1) 6.0183 ** θ10.1853 ** (p-value) (0.0142) (Z-statistic) (2.4722) Engle LM (2) 6.3745 ** ω9.37e-07 *** (p-value) (0.0413) (Z-statistic) (15.5038) Engle LM (5) 7.2949 * α0.0306 *** (p-value) (0.1996) (Z-statistic) (13.0564) Engle LM (10) 11.4924 γ0.0884 *** (p-value) (0.3205) (Z-statistic) (28.0765) q1% −2.5373 β0.9154 *** (Z-statistic) (415.2990) q5% −1.6240 Log-likelihood 54439.85 q95% 1.5658 Akaike criterion −6.8220 q99% 2.3426 Number 15950 *, ** and *** denotes parameter statistically significant at the 90%, 95% and 99% confidence level. q1%, q5%, q95% and q99% represent the empirical quantile measures at respectively 1%, 5%, 95% and 99%. The results for the maximum likelihood estimation of this model are displayed in Table 1. This model provides very good fit according to the selected criteria; all the parameters of the TGARCH(1,1) specification are statistically significant. We therefore extract the maxima and minima from the return shocks {Zt}t=1,...,T corresponding to standardized returns using a time-varying volatility model. Our discussion is hereafter restricted to the standardized (raw) return maxima +Z(+R) and minima Risks 2014,2217 −Z(−R). Figure 1 shows the evolution of the S&P 500 stock index (1) prices; (2) volatility; (3) raw returns and (4) standardized returns. Figure 1. S&P 500 stock index. The Figure displays graphics from S&P 500 stock index from 3 January 1950 to 28 May 2013. From upper left to lower right corner: The stock index prices. The ARMA(1,1)-TGARCH historical volatility. The raw returns. The next four plots for visual threshold selections from the S&P 500 stock price standardized log-returns. (+Z) stands for upper tail and (−Z) stands for lower tail. The mean residual life plots are approximately linear with positively sloped, which indicates Pareto behavior overall for the −Zlower tail. The threshold plots display the maximum likelihood estimates for the tail index ξagainst a range of thresholds with 95% confidence limits. The stability in the parameter estimates is checked when the parameter estimates are approximately constant above the threshold range. The last two plots display for the lower tail −Z, the 100-year return level for the fitted GPD with 95% confidence intervals and tail plot based on a generalized Pareto model fitted to losses over the −Z-threshold where the three vertical lines (from left to right) locate the 99%, 99.90% and 99.99% expected shortfall level. Risks 2014,2218 3.3. Descriptive Statistics S&P 500 standardized returns have excess skewness and kurtosis. In all cases, the Jarque–Bera statistics induces a strong rejection of the normality hypothesis. When considering various percentiles from 1% to 99% as a comparison with those implied by the Normal distribution, it shows a clear departure. The Q-statistics, distributed as a χ2 5,χ2 10 and χ2 20 with 95% critical value, is 11.07, 18.31 and 31.41. The correlogram for the residuals exhibits no more dependence since the Q-statistics for the series are lower than the critical values. The Engle’s Lagrange multiplier test statistic measures the ARCH effect in the residuals; it shows no more evidence of remaining ARCH effects at lag five at the 95% confidence level. 3.4. Quantile Regression The transformation for recomputing “theoretical raw returns” from standardized returns is very useful for understanding the economic meaning of the statistical inference drawn from EVT results. Indeed, many articles applying EVT to standardized returns leave the reader with few economic interpretations of the results extracted from the filtered series. Therefore, it is important to transform the standardized returns computed from EVT into theoretical raw returns. As recent literature, to our knowledge, does not propose any solution, this article applies a linear transformation based on a semi-parametric technique that extends the ordinary least squares regression model to conditional quantiles. Indeed, standard ordinary least squares modelize the relationship between one or more covariates X and the conditional mean of the response variable Y given X. The quantile regression [17,18] permits a more complete description of the conditional distribution since it generalises the regression model to conditional quantiles of the response variable, such as the 95th or 99th percentile. It appears to be the appropriate approach for the estimation of conditional quantiles of a response variable Y, given a vector of covariates X. Hence, it can be used to measure the effect of covariates, not only in the center of a distribution, but also in the upper and lower tails. Quantile regression is useful when the rate of change in the conditional quantile depends on the quantile. In fact, it can be used with heterogeneous data for which the tails and the center of the conditional distribution vary differently with the covariates. Moreover, it is viewed as a natural extension of OLS estimation with a better consideration brought to extreme observations. It is more robust then OLS because there is no need to do any distributional assumption about the error term, contrary to the normality assumption in the standard regression model. In clear, quantile regression is, by construction, particularly robust to extreme values ([34], p. 7). The linear conditional quantile function can be estimated by solving the following minimization problem: ˆγnu(q) = argminγ(q)EnςqY−X0γ(q)o (12) The check function which weights positive and negative values asymmetrically for any quantile 0< q < 1is ςq(v) = v(q−I(v < 0)), where I(.)denotes the indicator function. The vector of coefficients γ(q) = (γ1q, γ2q, ..., γpq)0corresponds to pexplanatory variables X= (c, X1, X2...Xp)0that take the value of the intercept cand the standardized returns Z.Yis the variable of interest corresponding the the raw returns R. Risks 2014,2225 38. Christie, A. The stochastic behavior of common stock variances—Value, leverage, and interest rate effects. J. Financ. Econ. Theory 1982,10, 407–432. c 2014 by the author; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).