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36Journal of Siberian Federal University. Engineering & Technologies 1 (2012 5) 111-121
УДК 336.763
Extreme Value Theory
and Peaks Over Threshold Model
in the Russian Stock Market
Vladimir O. Andreeva*, Sergey E. Tinykovb, Oksana P. Ovchinnikovaa and Gennady P. Parahinc
а Oryol Regional Academy of State Service 5a Pobedy st., Oryol, 302028 Russia b Branch of Siberian Federal University, Zheleznogorsk 12a Kirov st., Zheleznogorsk, Russia c TelekomStroyService, Ltd. 61 Tsarev Brod st., Zarechensky, 302528 Russia 1
Received 6.02.2012, received in revised form 13.02.2012, accepted 20.02.2012
Traditional research methods adopts normal distributions as a pattern of the stock market behavior. This paper utilized POT model of extreme value theory, and GPD distribution which can give more accurate description on tail distribution of financial returns/losses. EVT and POT techniques are applied to a series of daily losses of the RTS index (RTSI) over a 15-year period (1995-2009), RTSI is total index of 50 largest Russian stocks. The focus is on the use of proposed methods to asses tail related risk providing a modeling tool for modern risk management.
Keywords: Extreme Value Theory, General Pareto Distribution, Peaks over Threshold, Tail Distribution, Value at Risk.
Introduction
The study of extreme events has attracted the special attention in connection with the global crisis of 2008-2009. The Russian stock market has been dramatic volatile over 15-year period (from 38 points on 05.10.1998 to around 2487 points on 19.05.2008 and back to about 498 points on 23.01.2009). In irregular financial market, it is necessary to set up models and systems to evaluate and control risks. In this paper we focus on the extreme behavior of financial series, unraveling the volatilities in the financial markets has always been an decipherable mystery. One of the purposes of this chapter is to test the validity of a popular risk management instrument: Value-at-Risk estimator in Russian equity market, which is a widely adopted technique in the developed countries for quantifying market risk. We have to deal with extreme events when a risk takes values from the tails of its probability distribution. In the field of market risk management it is a great concern the day by day determination of the Value-at-Risk (VaR) [1]. VaR is a high
* Corresponding author E-mail address: [email protected]
1 © Siberian Federal University. All rights reserved
quantile of the distribution of losses (for example the 95th percentile): VaRp=F'(p), where F is the loss cumulative distribution function and p the selected probability level. Traditional procedure calculating VaR based on normal distribution has limitations. VaR model reflects that it is to asses the possible maximum loss under regular market environments. Risk managers have become more concerned with events occurring under extreme market conditions [2,3]. This paper argues that extreme value theory (EVT) and POT (Peaks Over Threshold) model provide tools for estimating measures of tail risk under irregular volatility in market. We consider a fully parametric model, based on the GPD (Generalized Pareto Distribution), which can be easily estimated by maximum likelihood method [4,5].
Extreme value theory is a powerful and fairly robust framework to study the tail behavior of a distribution. There have been a number of extreme value studies in the finance literature in recent years: quantile estimation using the extreme value theory [6]; the estimation of the tails of loss severity distributions and the estimation of the quantile risk measures for financial time series using extreme value theory [7,8]; overview the extreme value theory as a risk management tool [9]; potentials and limitations of the extreme value theory [10,11]; an extensive overview of the extreme value theory for risk managers [12]; the estimation of tail-related risk measures for heteroskedastic financial time series [13]; comprehensive source of the extreme value theory to the finance and insurance literature
We use of Extreme Value The ory tomo del the tail returns and then show how our EVT estimates are incorporated into the risk measures. Two mam approaches are proposed in the literature [16]: the Block Maxima (BM) and the Peaks-over-Threshok1 models (POT). The group of models for threshold exceedances are more modern and powerful than tlie BM mrdels [06], we focus on ahis epproacm and its application to the losseo on the RTSI ttock index. We appty she parametric POT method r>ased on the Generalized Pareto distoibulioe (GPD) to deactibe tair behaviour. Begin ba assuming thmt market losses represent the eealizations a o°° a random variable X oves ¡en enough high threshold u. More particularly, if X has the cumulatike distoibution funciion Fka), we are; inlererted in the oumulative distribution function Fu(o0 of axceedantces of X over- a high threshold u, i.e. the consditional excess distribution function is defined as:
As to the sufficient large u, EVT provides us with a powerful key resula, whice states fou a large class of underlying distributions F(a) [2]:
I. Theoretical framework of the extreme value approach
[14,15].
POT model and Generalized Pareto distribution
Fu (x) = P(X - u < x | X > u)
u
F ( x+u )—F (u ) 1-F (u )
(1)
F (x) ® G^p (x), u ->oo,
where Generalized Pareto Distribution is defined by:
1 i - exp(-x I p\if % = 0
{ [0,oo), ecjm <¡¡>0 where x si
ecjm £ < 0
GPD subsumes three other distributions under its parameterization [2]. So, when tail index <2=0, we obtain a Type 1 (exponentially declining) distribution. If <<0, we have a Type 2 (power declining). For <>0, we obtain a Type 3 (constant declining) diltribution. Given these threetypes of distribution, one of our tasks in this paper well be to uncover which ly pe best describes the extremes of stock returns on the emerging Russianmarket.
Identification of GPD parameters
Let (XhX2,...,Xk(u)) be a sequence of iid random variables from an unknown Uistribution (F X>u. Shape <and scale ?>0 parameSets are then defined on he threshold u [16]. These GPD perametero can be determined by maximum likolihood (ML) meOhode. The [og likelihood eunction of the GPD for 160 is:
XUt, t) = -£(u)(leg/t)-rC-^C/n)^leg(l-t f /t, (3)
i=c
where xt satisfies the constnaints specified for x,-. If X= 0, i)re log likeehood function is:
L(J3) = -k(u) lnGt-tT xf. (4)
i=i
ML estimates are thenfound by maximizingthe log-likelihood function using numeral optimization methods. We can get these 0 and p estimates through solving simultaneous equations (4^0):
1 ko x- 1 k^ X.
-^TT1 = tTI 10 gd + ) - T + 1 ) krj- = 0
k T i=1 P T i=1 P + T
dL(k,P)_ k(u) 1 | f £x, = 0 dp p s tk P2kPTxi
Tail evaluation formula
Assuming that u is sufficiently high, by combining expressions (2) and (3) the distributionrunction F(x) for exceedances can be written as:
*) = (1 -F(b ))Fb (x) + F+(b) (5)
= )()+Ue-b)-^) .
k+
Fu(x) is a GPD and F(u) is given by [n-k(u)]/n; n is Che total number of obsesvations, k(u) tlie number of observations at>ove the threshold u, < and g? are the parameters of the GPD.
Estimating VaR
For a given probability p>FSu) and threshold uk the value-at-risk (VaR) is calculated by inverting the tail estimation formula (5g:
VaRp = u + £{[^(1 - -!}• (6)
- 113 -
Choosing threshold value
Choice of the threshold u is the important issue to deal with: u too high results in too few exceedances and consequently high variance estimators. On the other hand, u too small provides biased estimators and the approximation tv a GPD could not be Oeasible. It is possible to choose an asymptotically optimal yares°old by a quaatification of a bias versus vrriance Orade-off.
Mean excess function
One suggestion which is of immediate use in practice is baeed on the linearity ob the menn excess function e(u) for the GPD. From [2] we know that Oor a random valuetS with a GPD distribution function Glfi the mean exces: function is:
e(u) = E(X -u |X >u) =£+kkL, (7)
P + uk> 0,k< 1-
It suggests a graphical approach for ohoosing u: choose u >0 such that e(a) es approximately linear for a3u. Using ptos to compare resulting estimates acroes a variety of u-aalues, due to She usual presence of multiple choice of the threshold, is recommended.
Hill plot
Let X1>X2>.>Xn tie the order statistics of positive random variables iid. The Hill estimator of the tail index £ using k+1 ordor statirtics is defined by [17]:
$ = Xjn -). (8)
Hill plot is a good instrument to find the optimal threshold [18]. Over a specifie range of exceedances, the Hill plot may be under the statiovary series, and the turning point is a good choice of optimal threshold. We use Che following intuitive tdeas:
(1) The sequence of the turning point is less than ~n/10 [19].
(2) The Hill estimator in the turning point has a relative large deviation from the fitted stationary straight line.
(3) The turning point is the last sequence of point that satisfies the two conditions stated above.
Empirical results
We consider a exfreme value approach, working on the serie a of daily log losses (negative returns) of the Russian RTSI Index over a period of fifteen years (1995-20009). The Russian Trading System Index (RTSI) comprises o° 50 of tine largest stocks capturing 85%of the total market capiaalizarion oa the Russian Trading System enchange. The data used in this paper are obtained from RTS web site [21]. The empiric study uses the series of log daily losses of the RTSI Index, containing 3 447 trading days (closing prices). Fig.1 shows the plot of daily dynamics of RTSI index values, and log daily losses.
Table 1 shows the summary statistics for the series of log daily changes. This table shows that kurtosis value is 9.7024 and skewness value is 0.3752. Relative value of Normal distribution is 3 and 0, respectively. So we can see empirical distribution of log daily losses and normal distribution is not compatible.
a) daily dynamics of index values
b) log daily losse s
Fig. 1. RTSI Index - sampleperiod 01.09.1995 - 30.06.2009 (closing values)
In addition to this, Jarqua-Bera statistic shows that law of log daily losses is obviously differeni from normal distribution. The JB test statistics is defined as [10]:
jB _ n [ STD2 + (Kurtosis-3)2 | UL> ~ 6 L 6 f 24 J •
The JB statistic has approximately a chi-squared cCistrilbution, with two degrees of freedom. The Jarqua-Bera test deprnds can skewness and kurtosis statistics. If tlif JIB test statistic equals zero, it means that the diitribution has zero skewnesr and kurireis is sbout equal 3, and so it can be concluded that the normality assemption holds. Skewness values far from zero ant1 kurtosis values far from 3 lead to an increase in JB values. The test returnr the logical value h = 1 ff it rejects the null hypothesis at the /><0.05 significance level, and h =0 if it cawnot. We have for data oS Table h: JB value=6532.8, p~0, h=1. It means that we can reject the hypothesis that the distribution of daily losses is normal.
Table 1. Summary statistics for daily losses in RTS
mean min max
-0.0007 -0.2020 0.2120
std skewness kurtosis
0.0289 0.3752 9.7024
variance JB test n
-0.0008 6532.8 h=1,p <0.001 3447
In Fig.2 we represent the Hill graph, which plots the Hill estimator of versus the k upper order statistics (and threshold u, respectively). We select the last area to k~0.1*3447~350, where the Hill estimator is more stable.
The mean excess function (7) allows to establish the behavior of the distribution tails [23]: we choose threshold u looking at the linear shape (with positive slope) of the graph (Fig.3). Considering Hill plot and the mean excess function, we choose u=0.0334 (the number of observation exceeding threshold u is equal k=294).
The results of ML estimation of the GPD parameters (on chosen threshold u=0.0334) are £=0.1492 and p = 0.0206:
Maximum Likelihood (ML) estimates of £,fi: out =
par_ests: [0.1492 0.0206] funval: -803.6979 par_ses: [0.0688 0.0018] threshold: 0.0334 data: [1x294 double] p_less_thresh: 0.9675
QQ-plot graph makes us able to evaluate the goodness of fit of the empirical series to a parametric GPD model (Fig.4) [24]. Notice that a concave departure from the straight line in the QQ-plot (Frg.4a) is an indication of heavy tailed distribution, whereas a convex departure is an inrrcation of a thin tail.
After we get estimates £,p, use them in (5), get the formula for of tail evaluation:
k(n) f - 4.4334 -—
p = F(f) = 1 --n-(1 + 4.1402*-) 41402
n 4.4246
k(n) = 204, n=3447, k(u)/n=8.53%
Employ the result in (6), get the VaR formula on GPD model:
4.4246 3447 0I402
VaR = 4.4334 +-{[-(I - p)]-41402 - 1}.
p 4.1402U 204 ( }
In Table 2 we report 95%, 99%, 99.5%, 99,9% Value-a0-Risk estimates of three different VaR estimation methods. The perfotmrnce of the dif/etrntVaR estimation methods can be evaluated by comparing the estimates with the acfual losses otnserfed, in particurar by computiag (oud testing) the number of VaR violrtions. VaR approochet bastd on are rstumption or normal distribucion are;
Order Statistics
a) Hill estimator versus k upper order statistics (k<350)
Hillplot
Order Statistics
b) Hill estimator versus k upper order statistics (k<350) (plot zooming) Fig;. 2. Hill estimator vessus k upper order sSrtistics (probability level p=0.95)
Fig. 3. Mean excess function
Ordered Data
a) QQ-plot:empirical vs exponentialdistribution
Ordered Data
b) QQ-plot: empirical vs GPD distribution ( 4=0.1492, p=0.0206,u=0.0334) Fig. 4. QQ-plot versus GPD diseribution and exponential distribution
Table 2. VaR estimation for daily RTSI losses:one day horizon
VaR approach p=0.950 p=0.975 p=0.990 p=0.995 p=0.999
Normalmodel 0.0394 0.0451 0.0520 0.0565 0.0663
Historical simulation 0.0452 0.0607 0.0849 0.1083 0.1771
GPD model 0.0499 0.0611 0.0856 0.1062 0.1620
definitely to underestimate high percentiles, while estimates based on historical simulation face with the problem of out of sample performence. The extreme value approach on GPD model seems appeopriate and easy to implemenf
Conclusions
Since last century, volatility of international financial system is getting severe. A stable financial system is so desirable. Therefore, risk management has aroused growing attention. As a measurement of market risk, VaR has been widely used in risk management. However, derivation between VaR estimation of normal hypothesis and abnormal distribution of practical benefit rate of financial always cause the bigger error in estimation. Aiming at this problem, through GPD model which fits tail distribution of financial products more accurately, this paper recalculates VaR by POT method. Compared with traditional method of risk study, this paper has made some progress in research approach and philosophy and more applicable in practice, which has been demonstrated by example of the Russian market analysis.
We use software systems: EVIM [25,26], MATLAB [27] and LOGOS-EVT, developed by authors of this paper.
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Применение метода превышений порога и теории экстремальных значений для моделирования рисков на российском фондовом рынке
В.О. Андреев3, С.Е. Тиняков6, О.П. Овчинникова8, Г.П. Парахинв
а Орловская региональная академия государственной службы Россия 302028, Орел, ул. Победы, 5а б Железногорский филиал СФУ Россия, Железногорск, ул. Кирова, 12а в ТелекомСтройСервис Россия 302528, Орловская обл., п. Зареченский, ул. Царев Брод, 61
При моделировании поведения фондового рынка обычно используется нормальное распределение Гаусса. Однако, метод превышений порога (РОТ) теории экстремальных значений (ЕУТ), обобщенное распределение Парето (GPD) позволяют более точно описывать финансовые доходности (потери) от операций с ценными бумагами, особенно на хвостах вероятностных распределений. В данной работе описывается применение предложенных методов к моделированию и анализу потерь на основе индекса РТС, представляющего собой усредненную цену акций 50 крупнейших российских компаний, за 15-летний период (1995 - 2009 г.). Особое внимание уделено использованию предложенных методов для оценки экстремального рыночного риска на хвостах распределений, что позволяет получить современный инструмент моделирования для системы управления рисками.
Ключевые слова: теория экстремальных значений, обобщенное распределение Парето, метод превышений порогового значения, вероятностное распределение на хвосте, стоимость под риском.
