Estimation of Extreme Risk Models:
GARCH-EVT and SV-EVT Approaches
Mengdie Wang
Student number: 11376864 Date of final version: August 2, 2017 Master’s programme: Econometrics
Specialisation: Financial Econometrics Supervisor: Prof. dr. H. P. Boswijk Second reader: Dr. S. A. Broda
Faculty of Economics and Business
Faculty of Economics and Business
Amsterdam School of Economics
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1 Introduction 1
2 Literature Review 3
3 Model and Method 6
3.1 Value-at-Risk . . . 6
3.2 Extreme Value Theory . . . 6
3.3 The GARCH-EVT Model . . . 9
3.4 The SV-EVT Model . . . 12
3.4.1 The SV Model . . . 12
3.4.2 MCMC Algorithm . . . 13
4 Data Description 17 5 Empirical Analysis 21 5.1 In-sample Evidence . . . 21
5.1.1 The GARCH-EVT model . . . 21
5.1.2 The SV-EVT model . . . 27
5.2 Out-of-sample Evidence . . . 31
5.2.1 Backtesting . . . 31
5.2.2 Test Statistics . . . 33
Introduction
Continued uncertainty about the economy and increased concerns about po-tential financial risks have required a more informed and efficient capital allocation process, since extreme events occur rarely but powerfully. The Asian financial crisis in 1997, the American banking collapse in 2008, the European debt crisis in 2010, the more frequent extreme price movements have captured the most attention on risk management in financial investment nowadays.
A widely used tool for measuring market risk is Value at Risk, which stems from a capital test the New York Stock Exchange (NYSE) firstly imposed on its member firms in 1922 (Holton 2002). It is believed that J.P. Morgan specifically presented the name "Value-at-Risk" in 1985 and its methodology RiskMetrics has been adopted by lots of financial institutions and corporations (Guldimann 2000). In 1988, the Basle Committee proposed the Basle Accord to regulate banks with a set of minimal capital requirements, which later in 1992 was enforced by law in G-10 countries as Basel I.
In general, Value at Risk (VaR) is defined as the potential future loss an asset could be subject to. In practice, institutions and companies would choose their own time period and probability to calculate VaR. In the Basel Accord, it is settled to focus on the 10-day 99% VaR to determine the minimum capital requirements. To make better predictions of VaR, one important step is to model the returns distri-bution more accurately. It is mentioned in previous papers that combining Extreme Value Theory with a time series model can estimate the tail part with greater accu-racy, which leads to the improved prediction of extreme losses. The GARCH model and the SV model are two well established and competing models in financial econo-metrics. While the GARCH model gains popularity in aforementioned literature, the SV model attracts little attention due to its complexity of computation. But they both have their own strengths under different conditions. In this thesis, we choose to grasp the nettle and employ the GARCH model, the SV model as the foundation model combined with EVT to compute VaR.
The focus is to compare the accuracy and efficiency of GARCH-EVT and SV-EVT models with other competing approaches, by computing one-day-ahead VaR
for 4 stock indices, namely, S&P 500, HSI, CAC40, N225. We choose financial data from several matured capital markets in the financial world: U.S., HongKong, France and Japan. First, we will fit the sample returns within suitable GARCH and SV models separately. Then by analyzing the standardized residuals within EVT, we can obtain the corresponding GARCH-EVT and SV-EVT models and then calcu-late their one-day-ahead VaR. To test whether the proposed models are successful, we separate the data into two groups to do in-sample and out-of-sample analysis, combined with the binomial test and the duration-based test. Since losses are of more interests than gains, we focus solely on the negative parts of returns.
The thesis is organized as follows. Chapter 2 gives a relevant literature review of VaR approaches and model constructions. Chapter 3 sets up the GARCH-EVT model and the SV-EVT model step by step and presents the estimation of VaR in detail. Chapter 4 describes the data used in the empirical analysis. Chapter 5 investigates the empirical findings from both in-sample and out-of-sample analysis. Chapter 6 draws the conclusion.
Literature Review
Among VaR approaches, there are three traditional calculation methods, namely, Historical Simulation, the Variance-Covariance Method and Monte Carlo Simula-tion.
Historical Simulation (HS) adopts historical data to simulate future returns for VaR calculation, based on the assumption that situations in the past indicate the trend in the near future. Since the methodology is easy to implement, which results in lower modeling risk but relatively higher precision, it is now frequently used by most institutions. However, there are researchers who evaluate HS to be inefficient since it ignores conditional information but concentrates on the average simulation. Pritsker (2000) states that HS is under-responsive to changes in conditional risk.
The Variance-Covariance Method computes VaR by fitting a normal distribu-tion to the financial data, which simplifies the problems. It is similar to the idea behind HS, by simply changing the actual data distribution into a fitted normal distribution. But in reality, most assets show an excess kurtosis which indicates the returns distribution having fatter tails than the normal distribution. Hence, whether this methodology provides accurate results is surely doubted.
Monte Carlo Simulation is a relatively accurate method compared to the other two. It develops the model for future stock returns and generates multiple trials through the model. Linsmeier and Pearson (1996) believe that this method is the most flexible approach of VaR computations. However, its methodology requires complicated computations, which leads to high costs for financial institutions.
How to model the return distribution is the main task in calculating VaR. As more and more researchers confirm, assets returns tend to display volatility clus-tering and a non-Gaussian distribution with fat tails. To better fit the distri-bution, many researchers turn to Extreme Value Theory (EVT), which was first developed in the application of cotton industry by Fisher and Leonard Tippett (1928). Then Gnedenko (1943) provided a more systematic development, as the Fisher–Tippett–Gnedenko theorem, stating that with the converged normalized maximum value of a sample there are three possibilities for the limit distribu-tion, named the Gumbel distribudistribu-tion, the Fréchet distribudistribu-tion, and the Weibull
distribution. This is called the "Annual Maxima Series" (AMS) approach, which is implemented by selecting block maxima series. Another method is known as the "Peak Over Threshold" (POT) method, deriving the peak values over a spe-cific threshold. Balkema and de Haan (1974) and Pickands (1975) propose this threshold-based method, which will be introduced in detail in Section 3.2, known as Pickands-Balkema-de Haan theorem. Scarrott and MacDonald (2012) review several estimation approaches for selecting the optimal threshold.
EVT has been widely used in hydrology and wind engineering since the early 20th century. Its application in financial industry, however, was first introduced systematically by Embrechts, Kluppelberg and Mikosh in 1997. Many researchers later apply EVT for VaR calculations, for example, Danielsson and de Vries (2000), Gencay and Selcuk (2004). Manganelli and Engle (2004) apply EVT into the quan-tile regression, and Yi, Feng and Huang (2014) combine the quanquan-tile autoregression method with the EVT approach, providing a robust estimator. Muller, Dacorogna and Pictet (1998) compare the EVT model with the GARCH model for foreign ex-change rates, showing the GARCH model to perform better.
The ARCH model has been popular in financial time series analysis since En-gle (1982) first proposed ARCH(1) to deal with volatility clustering by assuming autocorrelation in the volatility. The research and empirical findings on studying ARCH-type models are enormous. To have a better fit of daily returns, Bollerslev (1986) extended the ARCH model to the generalized ARCH (GARCH) model based on previous research. Nelson (1991) and Zakoian (1994) develop, respectively, the EGARCH model and the TGARCH model to capture the asymmetry in the volatil-ity part caused by leverage effects. Both models show superior performance in terms of asymmetry. The IGARCH(1,1) model is adopted to implement the RiskMetrics methodology proposed by J.P. Morgan (1996). Since then, ARCH-type models ap-pear to have widespread use in financial data analysis.
Many researchers combine EVT with GARCH models to estimate VaR, and the combined models usually have better results than the single model. McNeil and Frey (2000) propose a two-stage method as the GARCH-EVT model for 5 index returns. Bystrom (2004) also applies the GARCH-EVT model to the Swedish and Ameri-can stock markets, but extended to compare different methods of EVT-AMS and POT, which in general perform similarly. Ozun, Cifter and Yilmazer (2010) com-bine different GARCH models with EVT to calculate VaR for the Istanbul stock market, which indicates the filtered EVT model performs better than parametric VaR models. The GARCH-EVT model can also be applied in managing portfolio risks. Chebbi and Hedhli (2014) associate the GARCH-EVT model with a time-varying copula to analyze 4 stock markets, namely, Tunisian, American, French and Moroccan. Koliai (2016) presents a GARCH-EVT model with the R-vine model to manage portfolio risks, considering three sets of financial data: equity indices, exchange rates and commodities.
Another popular investigated method in modeling the distribution is the Stochas-tic Volatility (SV) model. Unlike the models mentioned above, the SV model con-siders a volatility shock separately but still keeps the conditional variance positive as the EGARCH model does. The SV model is served as a natural alternative to ARCH-type models in dealing with time-varying volatility. It was pioneered by Clark (1973), who introduced a stochastic concept into financial data analysis and compared the fitted performance of finite-variance distribution with the normal dis-tribution. Compared to the GARCH model, the SV model are better-behaved in representing the properties of financial data observed in reality. Geweke (1994) and Fridman and Harris (1998) find that the SV model provides more flexibility than ARCH-type models. Since then, the SV model has been employed in option pric-ing intensively. Hull and White (1987) concentrate on a European call option with stochastic volatility. Melino and Turnbull (1990) investigate a stochastic volatility model for exchange rates, which turn out to be a much better fit to the empirical distribution. There are different types of SV models when taking into account of the distribution type of the error part. Early studies, like Jacquier, Polson and Rossi (1994), would consider it as a Gaussian distribution. Chib, Nardari and Shephard (1998) assume it to be a Student’s t-distribution, which is applied in our paper as well. There are other fat-tailed distributions like Skew-Exponential Power Distribu-tion, as in Andersen (1996), and Skew-Generalized Error DistribuDistribu-tion, as in Nunzio, Diego and Davide (2004).
Though the SV model has a better performance especially in high-frequency data, its empirical applications are limited by the difficulties in the estimation. While the GARCH model is easily estimated by maximum likelihood, the SV model is hard to perform via likelihood due to its unobserved variance, which requires com-plicated numerical techniques. The commonly used approaches for the SV model are Method of Moments, Maximum Likelihood and Markov Chain Monte Carlo. Ander-sen, Chung and Sørensen (1999) perform the efficient method of moments (EMM) for the SV model. Melino and Turnbull (1990) adopt the generalized method of mo-ments (GMM) in exchange rates analysis. Harvey, Ruiz and Shephard (1994) pro-pose a quasi-maximum likelihood (QML) estimation method to capture the common movements in volatility. As sampling methods develop, Jacquier, Poison and Rossi (1994) first employ a Markov Chain Monte Carlo (MCMC) approach to estimate the SV model. Chib, Nardani and Shephard (2002) apply the MCMC method in estimating the SV model under a Student’s t distribution assumption.
Since the GARCH-EVT model outperforms the single GARCH model in most conditions, it is worth to investigate whether the SV model combined with EVT will produce more satisfying outcomes, see Zhou et al. (2012). Many researchers have focused on comparing the performance of the GARCH and SV models, i.e. Carnero, Pena and Ruiz (2004), Gerlach and Tuyl (2006). It is also interesting to investigate how GARCH-EVT and SV-EVT differ from each other.
Model and Method
3.1
Value-at-Risk
Value-at-Risk is defined as the maximum loss of an asset given a time period and probability. Denote the value of the asset as V , the time period as [t, t + `], the probability as p. Its mathematical expression is as follows:
p = Pr[Vt+`− Vt ≥ −VaR(`)] (3.1) = 1 − Pr[Vt+`− Vt< −VaR(`)]
From the equation above, we can tell VaR focus on the upper tail of the loss function. For simplicity, in the empirical study later, we shall use the negative logreturns of the asset. With probability p, the potential loss of the asset is less than VaR over the horizon `.
Let F`(·) denote the cumulative distribution function (cdf) of P&L (Vt+`− Vt) and q = 1 − p.
−VaR(`) = xq = inf{x|F`(x) ≥ q} = F`−1(1 − q) (3.2) We have xq defined as the qth quantile of F`(x). Hence, the main challenge in calculating VaR is to have a reliable estimate of the tail part of the cdf of the loss function. In this paper, we set up two models to compute Value at Risk, both based on EVT.
3.2
Extreme Value Theory
Extreme Value Theory concentrates on assessing the probability of rare events, which corresponds to our main interests – the tail-related measures of extreme losses. In practice, there are two popular approaches in applying EVT, namely, "An-nual Maxima Series" and "Peak Over Threshold". They rely on different references to determine the extreme values. Figure 1 below shows directly the differences be-tween AMS and POT. AMS selects the maximum value given a specific period or
block, like x4, x6, x11 in each period, which is also named as block maxima. As for POT, it focuses on the observations exceeding the chosen threshold u, i.e. x5, x6, x7, x10, x11 in the figure.
Figure 3.1: AMS (left) and POT (right)
Since the financial returns tend to display volatility clustering, AMS might not fit the scenario where extreme values are likely to occur successively and applying AMS might lose useful information. Hence we adopt POT method in the following analysis. The POT method specifies the observations above the chosen threshold as extreme values and focuses on the "exceedance" part rather than the whole data set.
Assume that X1, X2, · · · , Xn are independent and identically distributed ran-dom variables (i.i.d.), which follow the cdf F (x) = P r(Xi ≤ x). Denote u as the threshold. We have the excess residuals over u as yi = Xi−u for X > u, i = 1, · · · , k. The probability distribution of y is given as follows:
Fu(y) = Pr{X − u ≤ y|X > u} = Pr{X − u ≤ y, X > u} Pr{X > u} = p{u < X ≤ y + u} p{X > u} = F (y + u) − F (u) 1 − F (u)
For X > u, we have x = y + u and the equation below is derived as:
F (x) = Fu(y)[1 − F (u)] + F (u) (3.3) In order to obtain the estimate of F (x), we shall have the value of Fu(y) and F (u). For F (u), we can estimate it as (n − k)/n, where n is the total number of observations and k is the number of exceedances. For Fu(y), based on Pickands-Balkema-de Haan theorem, as u → ∞, the conditional excess distribution Fu(y)
converges to Generalized Pareto Distribution (GPD): Gξ,β(y) = 1 − (1 + ξyβ)−1ξ if ξ 6= 0 1 − exp(−βy) if ξ = 0 (3.4)
where β > 0, y > 0 for ξ ≥ 0, 0 ≤ y ≤ −β/ξ for ξ < 0. The parameter ξ is known as the shape parameter and β is known as the scale parameter. According to different values of ξ, GPD is divided into three groups:
• ξ = 0: Exponentially-decaying-tailed distribution, i.e. the Normal Distribu-tion, the Gamma Distribution.
• ξ > 0: Heavy-tailed distribution, i.e. the Pareto Distribution, the Student’s t-Distribution.
• ξ < 0: Short-tailed distribution, i.e. the Uniform Distribution, the Beta Dis-tribution.
As for financial data, we generally choose ξ > 0 for analysis since it tends to display heavy tails.
An important issue to discuss here is how to choose the optimal threshold in empirical analysis. The choice of threshold is a trade-off between bias and variation. If the threshold is too high with few exceedances, then the analysis is likely to yield unsubstantiated conclusions with large variations. On the other hand, if the threshold is too low with too many exceendances, the analysis would be unlikely to follow GPD unless the raw data is GPD. McNeil and Frey (2000) compare the GPD approach with the empirical distribution by simulations. They find out as long as k is sufficiently large (k > 50 in their cases), the GPD method is robust regardless of the threshold value. The choice of u in GPD method is not as critical as the other methods. In the thesis, we will make use of the Mean Excess plot, a widely used tool in risk and extreme value fields, to determine the threshold value. The plot consists of the Mean Excess Function,
f (u) = Pn i=1(Xi− u) Pn i=1I(Xi>u) (3.5) where I is an indicator function as
I = 1 if Xi > u 0 if Xi ≤ u
The threshold value can be determined by looking at the Mean Excess plot. The interpretation will be introduced in detail in Section 5.
y = x − u, the tail estimator is as follows: ˆ F (x) = Gξ, ˆˆβ(x − u)(1 − n − k n ) + n − k n = " 1 − (1 + ˆ ξ(x − u) ˆ β ) −1 ˆ ξ # k n + n − k n = 1 − k n " 1 + ξ(x − u)ˆ ˆ β #−1ˆ ξ (3.6)
where ˆξ and ˆβ can be estimated by maximum likelihood. The probability density function (pdf) of GPD is:
fξ,β(y) = 1 β(1 + ξ y β) −1ξ−1 with ξ > 0 (3.7) Derive the log-likelihood function:
`(ξ, β) = −k ln(β) − (1 + 1 ξ) k X i=1 ln(1 + ξyi β ) (3.8)
Solving the first-order conditions with respect to ξ and β, ∂` ∂β = − k β + (1 + 1 ξ) k X i=1 yi β(β + ξyi) = 0 (3.9) ∂` ∂ξ = 1 ξ2 k X i=1 ln(1 + ξyi β ) − (1 + 1 ξ) k X i=1 yi β(β + ξyi) = 0 (3.10)
we can obtain the value of ˆξ and ˆβ.
For q > F (u), transform the tail estimator above to get:
ˆ xq = u + ˆ β ˆ ξ (1 − q k/n ) − ˆξ− 1 (3.11)
3.3
The GARCH-EVT Model
To utilize EVT, an important assumption is for the data to be i.i.d., for which the asset returns aren’t qualified. To solve the problem, McNeil and Frey (2000) propose a two stage approach to set up the GARCH-EVT model. The approach is summarized as follows:
• Fit a suitable GARCH model to the returns by quasi-maximum likelihood, assuming the standardized Student’s t-distribution.
• Consider the standardized residuals to be a realization of a white noise process. Apply EVT to model the tail part and estimate xq for a given probability q.
Assume the daily returns rt can be represented by the AR(p) model: rt = φ0 + φ1rt−1+ · · · + φprt−p+ at = φ0 + p X i=1 φirt−i+ σtt (3.12)
with φ0 as the constant, φi as parameters, rt−i as the lagged returns, σt as the conditional variance of at, t as the standardized residual with t = at/σt. To be specific, t
iid
∼ N (0, 1) is a strict white noise. To fit σt into a GARCH model, we first introduce the representations of several GARCH-type models.
The most common model–GARCH(m,s) is given by:
σt2 = ω + m X i=1 αia2t−i+ s X j=1 βjσ2t−j (3.13) where α0 > 0, αi ≥ 0, βj ≥ 0, and Pmax(m,s) i=1 (αi+ βj) < 1.
Since financial returns tend to display leverage effects, which is the negative cor-relation between returns and its volatility, we introduce two asymmetric GARCH models to address the problem.
The exponential GARCH(m,s) (EGARCH) model of Nelson (1991) is repre-sented as follows:
log σt2 = ω + m X
i=1
[αit−i+ γi(|t−i| − E(|t−i|))] + s X
j=1
βjlog(σ2t−j) (3.14)
The threshold GARCH(m,s) (TGARCH) model of Zakoian (1994) is represented as follows:
σt = ω + m X
i=1
αi(|at−i| − γiat−i) + s X
j=1
βjσt−j (3.15)
The last two models are extensions of the GARCH model and designed to deal with asymmetry problem, i.e. leverage effects.
Applying the data of returns into different models and choose the optimal num-ber of lags, which will be explained explicitly in Chapter 5. To better fit the models, we use the standardized Student’s t-distribution instead of the normal distribution, since returns are fat-tailed rather than normally distributed. The Standardized Student’s t-distribution is given by:
f (t, ν) = Γ[ν+12 ] pπ(ν − 2)Γ(ν 2) 1 + 2 t ν − 2 −ν+12 (3.16)
with degrees of freedom ν > 0 determining the thickness of the tail, Γ(z) = R∞
0 x
By applying quasi-maximum likelihood, we first derive the log-likelihood func-tion under the assumpfunc-tion that the random errors follow the standardized Student’s t-distribution. `t(θ) = log Γ( ν + 1 2 ) − 1 2log(π(ν − 2)) log Γ( ν 2) − ν + 1 2 log(1 + 2 t (ν − 2)) − 1 2log σ 2 t `(θ) = m X t=1 `t(θ)
where θ represents the unknown parameters in GARCH-type models, θ = (ω, α, γ, β). Solving the first-order conditions with respect to ω, αi, γi, βj, we end up obtaining a specified GARCH-type model and its standardized residual t.
Denote rt = µt+ σtt, with µt= φ0+Ppi=1φirt−i.
Next, we make forecast of conditional mean µt+1 and variance σt+1 with the estimated parameters from QML above. The conditional mean ˆµt+1 is given by:
ˆ µt+1 = φˆ0+ p X i=1 ˆ φirt+1−i (3.17)
The conditional variance ˆσ2
t+1 for GARCH, EGARCH, TGARCH respectively, is:
ˆ σt+12 = ω +ˆ m X i=1 ˆ αiaˆ2t−i+1+ s X j=1 ˆ βjσˆt−j+12 ˆ σt+12 = exp " ˆ ω + m X i=1
[ ˆαiˆt−i+1+ ˆγi(|ˆt−i+1| − E(|ˆt−i+1|))] + s X j=1 ˆ βjlog(ˆσt−j+12 ) # ˆ σt+1 = ω +ˆ m X i=1 ˆ αi(|ˆat−i+1| − ˆγiˆat−i+1) + s X j=1 ˆ βjσˆt−j+12
Hence, we can obtain the standardized residuals as a white noise process ˆt = ˆ
at/ˆσt. By EVT method explained in Section 3.2, we can have an estimate of condi-tional VaR as follows:
VaRq = µˆt+1+ ˆσt+1xˆq (3.18) with ˆxq obtained from (3.11).
3.4
The SV-EVT Model
3.4.1 The SV ModelTo introduce the Stochastic Volatility model, we first derive the mean-corrected returns yt as: yt = rt− 1 T T X i=1 ri (3.19)
Introduce ht to satisfy equation σt = exp(ht/2). A basic SV model is set up as follows: yt= exp(h2t)t ht+1= µ + φ(ht− µ) + ηt (3.20) with t = 0, · · · , T , where t iid ∼ N (0, 1), ηt iid ∼ N (0, σ2 η).
We construct htas an AR(1) process, where φ stands for how quickly the volatil-ity approaches its mean µ. Therefore, for series ht+1 being stationary, we have |φ| < 1. The parameter ht represents the amount of volatility at time t and φ de-termines the autocorrelation condition between the logged volatility, known as the persistence.
By stationarity, we can obtain the unconditional mean and variance for ht, E[ht] = µ + φ(E[ht−1] − µ) + E[ηt]
= µ (3.21) V ar[ht] = E[(ht− µ)2] = φ2E[(ht−1− µ)2] + E[ηt2] = φ2V ar(ht−1) + V ar(ηt) = σ 2 η 1 − φ2 (3.22) Therefore, we have ht ∼ N (µ, ση2/(1 − φ2)).
The standard SV model assumes the error part to follow a normal distribution with zero mean and unit variance. However, as we have discussed before, returns are usually fat-tailed instead of normally distributed. Hence, we add the scale variable√λt into the basic SV model, in which case
√
λtt follows the standardized Student’s t-distribution with unknown degrees of freedom ν, to better fit the models and compare the performances with GARCH-type models, where
λt ∼ IG( ν 2,
ν 2)
with IG stands for inverse gamma with pdf as: IG(λ|a, b) = b a Γ(a)λ −(a+1) exp(−b λ) (3.23)
When λt= 1, it implies a basic SV model with normal distribution error part. To address the leverage effect, we introduce the correlation coefficient ρ between the error term t and the log-volatility ηt. Then the transformed SV model we are applying in the analysis is as follows:
yt= exp(h2t) √ λtt ht+1 = µ + φ(ht− µ) + ηt (3.24) t ηt ! ∼ N " 0 0 ! , 1 ρση ρση σ2η !# (3.25)
When ρ < 0, there exists leverage effects, that is the returns decrease as the volatility increases. When ρ = 0, it implies a basic SV model without leverage effects.
3.4.2 MCMC Algorithm
The SV model above is a hierarchical model made up of several layers: the mean-corrected return y = (y1, · · · , yT), the latent logarithmic volatility h = (h1, · · · , hT), the scale variable λ = (λ1, · · · , λT) and the hyperparameter θ = (µ, φ, ρ, ση, ν).
Based on previous papers, we decide to introduce the Markov Chain Monte Carlo method and the BUGS tool to implement the SV model.
The basic idea of MCMC is to draw variables from a posterior distribution by repeatedly sampling a Markov process until the current process converges to the posterior density.
By definition, a Markov process is a sequence {Xt} if its conditional distribution function satisfies P (Xh|Xs, s ≤ t) = P (Xh|Xt), h > t > s. In words, a Markov process has the property that with h > t > s, the value of Xs does not affect the value of Xt under the condition of Xt, which is called "memoryless". To start MCMC, it is vital to construct a transition probability function of the Markov process in order to generate the Markov chain. In our case, the Markov chain is {θ(m+1), λ(m+1), h(m+1)|(θ(m), λ(m), h(m)}M
m=1. Starting from some chosen initial values (θ(0), λ(0), h(0)), we set up the transition kernel as
P (x, A) = P (θ(m+1), λ(m+1), h(m+1) ∈ A|(θ(m), λ(m), h(m) ∈ x) (3.26) With sufficiently large simulations, the value of the Markov chain gradually con-verges to the stationary posterior distribution.
To obtain the posterior distribution, we need Bayes’s Theorem. p(Y |X) = p(X|Y )p(Y )
p(X)
By transforming Bayes’s Theorem, we can have the relationship between the poste-rior distribution, the likelihood function and the pposte-rior distribution.
p(Y |X) ∝ p(X|Y ) × p(Y ) posterior ∝ likelihood × prior
We will implement the Bayesian analysis of the SV model via WinBugs. In con-trast to the tailored programming in R or C++, this software simplifies the process of estimation since it has no requirement of computing the precise formulas for the density functions. Any changes, like prior distributions, are easily accomplished in practice. However, it may suffer from a slow convergence and inefficiency.
In order to obtain the posterior distribution p(h, θ, λ|y), we need to make use of the likelihood function p(y|h, θ, λ) and the prior distribution p(h, θ, λ).
Derive the likelihood function as follows:
p(y|h, θ, λ) = p(y1, y2, · · · , yT|h, θ, λ) = T Y t=1 p(yt|ht, θ, λ)
Since h is a Markov Chain, it satisfies a "two-sided" Markov property, which is that h−t in the conditioning part of distribution can be replaced by ht−1 and ht+1. That is,
p(ht|h−t, θ, λ, y) = p(ht|ht−1, ht+1, θ, λ, y)
The dependence property simplifies the computation by reducing the influence area within the immediate past and future. Hence, the joint density of volatilities and parameters, also as the prior distribution, can be written as:
p(h, θ, λ) = p(h1, h2, · · · , hT, θ, λ) = p(h1|θ, λ) T Y t=1 p(ht|ht−1, θ, λ)p(θ, λ) ∝ p(ht|ht−1, θ, λ) × p(ht+1|ht, θ, λ) × p(yt|ht)
The joint posterior distribution is computed as: p(h, θ, λ|y) ∝ p(y|h, θ, λ) × p(h, θ, λ) ∝ T Y t=1 p(yt|ht, θ, λ) × p(h1|θ, λ) T Y t=1 p(ht|ht−1, θ, λ)p(θ, λ)
We assume the prior distribution of the hyperparameter are independent and will use the prior distribution proposed by Kim (1998).
To implement the Bayesian inference via WinBugs, Meyer and Yu (2000) pro-posed the reformulated SV model as follows:
yt|ht+1, ht, µ, φ, ση2, ρ, λt ∼ N ρ ση exp(ht 2)[ht+1− µ − φ(ht− µ)], λtexp(ht)(1 − ρ 2) ht+1|ht, µ, φ, ση2, λt ∼ N (µ + φ(ht−1− µ), λtση2) h1|µ, φ, ση2, λ0 ∼ N (µ, λ0ση2 1 − φ2) λt|ν ∼ IG( ν 2, ν 2)
Once we have obtained the posterior distributions, we combine Gibbs Sampling proposed by Geman and Geman (1984) to complete the MCMC procedure. Gibbs Sampling generates the sample from the posterior distributions by iteratively draw-ing samples from each of the conditional posterior distributions. Followdraw-ing Nakajima and Omori (2009), we use the method below:
Step 1: Initialize the values of volatility h(0), the latent variable λ(0) and the pa-rameter vector θ(0).
Step 2: Draw a random sample from their respective conditional distributions and repeat the iterations M times. For time t = i, draw
• θ(i) from p(θ|h(i−1), λ(i−1), y∗) • λ(i) from p(λ t|h(i−1), θ(i−1), y∗) • h(i) from p(h t|h (i−1) t−1 , θ(i−1), λ(i−1), y ∗).
In practice, for a sufficiently large M , we usually discard the first m ran-dom draws since the Markov chains in previous iterations do not converge and the marginal distributions are not stationary. The first m samples are called "burn-in" sample.
Take htfor example, we can compute its point estimate and variance as follows:
ˆ ht = 1 M − m M X j=m+1 h(j)t (3.27) ˆ σt = 1 M − m M X j=m+1 exp(h(j)t ) (3.28)
During the sampling procedure, for each iteration we need to repeat T times, which is quite time consuming in the process. BUGS applies certain sophisticated sampling approaches to draw samples. If the density is not log-concave, a Metropolis-Hastings algorithm is utilized. This method generates the random draws from an approximate distribution compared to the original density. It is applicable especially in conditions when the original density is infeasible in practice but an approximate distribution is easy to compute. The approximate distribution is called the jumping distribution and must be symmetric. If the rejection of a draw from the jumping distribution is required, we need to re-sample the new chain from a different distri-bution, trying to exploit the information of rejection.
Once the whole Gibbs Sampling is finished, we will obtain the estimates of pa-rameters and ht. For ht, we are looking for the filtered one instead of smoothed one, that is an estimate based on information up to time t, denoted as ˆht|t−1. Then we make forecast under the SV model to generate the realizations of rn+1 and hn+1.
• Draw ht+1 from a normal distribution with mean µ + φ(ht− µ) and variance ση2.
• Compute ˆσt+1 with σt+1= exp(ht+1/2).
• Compute the standardized residuals sequence t: t = rt√− Et−1(rt) λtσt|t−1 (3.29) • Compute ˆµt+1 with µt+1 = T1 PT i=1rt
Similarly as in the GARCH-EVT model, by EVT method explained in Section 3.2, we can have an estimate of conditional VaR as follows:
VaRq = µˆt+1+ q
ˆ
λt+1σˆtxˆq (3.30) with ˆxq obtained from Equation 3.11.
Data Description
The selected data set includes 4 specific indices collected daily from 1/31/2000 to 1/31/2017, namely S&P500 of U.S., HSI of Hong Kong, CAC40 of France and N225 of Japan, obtained at the closing time from Yahoo Finance. The chosen indices provide a complete representation of the financial condition of the most developed financial markets in the world, consisting of a relatively large number of traded stocks. Due to the time difference, there might be few deviations when the closing prices are considered. Hence we analyze the 4 stock indices separately.
Before setting up the GARCH-EVT and SV-EVT models, we first do the pre-liminary data transformation. To ensure the stationarity of return data, the daily returns rt are defined as:
rit = log Pit Pi,t−1
∗ 100 (4.1)
where Pit stands for the daily closing price of the index i on day t.
The graphs of daily price series and daily returns series are shown in Figure 4.1. The graphs on the left side show time-varying trends and fluctuations for each index. Most bottom points are around year 2008 due to the global financial crises. From the graphs on the right it is clear that high volatility is followed by high volatility, while low volatility by low volatility. As we have suggested before, the daily returns tend to display volatility clustering.
Table 4.1 represents the descriptive statistics of daily returns. We have 4,275, 4,238, 4,349 and 4,186 observations for S&P 500, HSI, CAC40 and N225 respectively, with constant mean close to 0. However, the positive/negative of mean values indi-cate the corresponding markets displaying general upward/downward trends during the sample period. In general, U.S. and HongKong still show a rising trend, while France and Japan suffer from a decreasing economy. With the negative skewness value and high kurtosis value, it is evident that each return displays a non-normality distribution, which is also confirmed by Jarque-Box statistics and its p-value.
To determine whether the distribution is fat-tailed, we apply a more direct method–the Q-Q (quantile-quantile) plot besides the descriptive statistics. In our
Table 4.1: Descriptive Statistics of Daily Returns
S&P500 HSI CAC40 N225 No. of Observations 4,275 4,238 4,349 4,186 Mean 0.011510 0.009630 -0.004035 -0.000617 Median 0.048804 0.018154 0.027210 0.030308 Min. -9.469512 -13.582025 -9.471537 -12.111020 Max. 10.957197 13.406811 10.594590 13.234592 Stdev 1.240987 1.502420 1.485144 1.567581 Skewness -0.187983 -0.054520 -0.027705 -0.356024 Kurtosis 11.268283 11.026265 7.750063 9.568611 Jarque-Bera 12,218 11,393 4,095.3 7,624.5 *** *** *** *** Note: The table includes descriptive statistics of S&P500, HSI, CAC40 and N225 over period from 1/31/2000 to 1/31/2017. Stdev stands for the standard deviation. Jarque-Bera statistics test the null hypothesis that the returns follow normal distribution, with *** stands for significance at 0.01%.
case, we plot the quantiles of the selected distribution against the Normal Distribu-tion to compare these two distribuDistribu-tions. As seen from Figure 4.2, we can tell that all returns show an evident concave departure from the normal distribution, which indicate the fat-tailedness.
Figure 4.2: Q-Q Plot of Returns against the Normal Distribution
From the correlogram in Figure 4.3, there are significant negative correlations between current squared returns and the lagged level returns. Therefore, there exists an evident asymmetry in each returns, which is the leverage effect we are looking for.
Figure 4.3: Cross Correlogram Plots of Returns and Squared Returns
In general, returns display the following properties: extreme movements, volatil-ity clustering, the leverage effect and non-normal distributions with fat tails, as we have proposed before.
Empirical Analysis
In the empirical analysis, we intend to do in-sample and out-of-sample analysis for each model. The whole data set is divided into two parts: in-sample data from 1/31/2000 to 1/31/2015 and out-of-sample data from 2/1/2015 to 1/31/2017. In-sample analysis focuses on constructing the model using the In-sample data, while out-of-sample analysis will apply backtesting to do VaR forecasts. After implementing several tests, we are able to compare the out-of-sample results of the GARCH-EVT and SV-EVT models based on their performances of accuracy and efficiency. Since VaR concentrates on the loss part of returns, we will select the extreme values from the returns’ negative part.
5.1
In-sample Evidence
5.1.1 The GARCH-EVT modelFirst, find an appropriate AR-EGARCH/TGARCH model for the returns re-spectively. Based on the partial autocorrelation functions (PACF) as in Figure 5.1, we have an initial guess of p in AR(p) for each return. For instance, we fit AR(2) for S&P500 and estimate its parameters. However, since the initial choice might not be the most suitable one, we have to try adjacent choices for p to test the model’s validity. Not only the parameter should be significant, but also the information cri-terion AIC and BIC should be the minimum and the log likelihood value should be the maximum. The procedure is similar in deciding m and s in GARCH(m,s). At last, we have chosen AR(2)-EGARCH(2,1) for S&P500, AR(0)-EGARCH(2,1) for HSI, AR(2)-EGARCH(2,1) for CAC40 and AR(0)-TGARCH(1,1) for N225. Results are listed in Table 5.1. Except the constant term in CAC40 and N225, other pa-rameters are all found to be significant, which proves the specified models are valid. The significance of γ suggests that asymmetry indeed exists in returns data.
After obtaining the appropriate AR-EGARCH/TGARCH model, we can calcu-late the corresponding conditional mean ˆµt and variance ˆσ2t.
For the next step to set up EVT part, we need the standardized residuals t,
Figure 5.1: ACF and PACF Plot of Returns
which should be i.i.d..
t =
rt− ˆµt ˆ σt
(5.1) Table 5.2 lists the descriptive statistics of standardized residuals t. Like daily returns, the standardized residuals are not normally distributed as shown by negative skewness value, high kurtosis value and high Jarque-Bera statistics. However, the non-normality is not as obvious as in the return data. From the histograms of standardized residuals in Figure 5.2, we can tell that S&P and N225 represent more fat-tailedness in the negative part than HSI and CAC40. Whether there are ARCH terms left is tested within the ARCH LM tests. With the high p-value, it is evident that we have eliminated ARCH terms. The results from Table 5.2 and Figure 5.2 strongly suggest that the obtained standardized residuals are close to i.i.d. and have fat-tailedness, which are valid to be analyzed in the EVT part.
Table 5.1: Estimation Results of AR-GARCH model
Parameters S&P500 HSI CAC40 N225 Mean Equation φ0 0.041795 0.035988 0.014083 0.018760 ** * (0.2437) (0.3421) φ1 -0.060066 -0.051246 ** ** φ2 -0.037162 -0.025110786 * * Variance Equation ω -0.005621 0.006651 0.003791 0.039334 * ** * *** α1 -0.284341 -0.194019 -0.233171 0.086738 *** *** *** *** α2 0.144804 0.134641 0.098555 *** *** *** β1 0.982695 0.987087 0.982609 0.904775 *** *** *** *** γ1 -0.178124 -0.177241 -0.130776 0.580593 ** ** *** *** γ2 0.302117 0.303715 0.237232 *** *** ***
Note: The models are AR(2)-EGARCH(2,1) for S&P500, EGARCH(2,1) for HSI, AR(2)-EGARCH(2,1) for CAC40 and AR(0)-TGARCH(1,1) for N225. Specific p-value is shown if it is insignificant, while
*** stands for significance at 0.01%. ** stands for 0.1%.
* stands for 1%.
Table 5.2: Descriptive Statistics of Standardized Residuals
Standardized Residuals S&P500 HSI CAC40 N225 No. of Observations 3773 3745 3835 3694 Mean -0.037841 -0.012427 -0.019685 -0.015903 Stdev 0.999127 1.009635 1.003600 1.002411 Skewness -0.441111 -0.153977 -0.294550 -0.351522 Kurtosis 4.194681 3.776364 3.835004 4.459438 Jarque-Bera 347.67 109.35 167.44 405.06 *** *** *** *** ARCH LM test 0.88858 0.05700 5.183e-02 5.283 (Lag=5) (0.7799) (0.9947) (0.9954) (0.0884) Note: The table includes descriptive statistics of the standardized resid-uals of S&P500, HSI, CAC40 and N225 over period from 1/31/2000 to 1/31/2017. Stdev stands for the standard deviation. Jarque-Bera statistics test the null hypothesis that the returns follow normal distribution, with *** stands for significance at 0.01%.
Then, we apply EVT into the standardized residuals. An important step is to obtain the threshold value since we are adopting the POT method. As we have proposed in Section 3.2, the Mean Excess plot is adopted in this part to determine the threshold value as in Figure 5.3 by using the reversed standardized residuals. Embrechts et al. (1997) and Davison and Smith (1990) have shown that an empirical estimate of the Mean Excess Function (Mean Excess Plot) is positively linear for high values of the threshold, which indicates the underlying distribution to be in the domain of a GPD distribution. Observing the plot can be subjective, but we can narrow down the ranges of threshold values by ensuring a straight line with positive slope above a certain threshold. Then we can fit GPD at a range of candidate threshold values and try to find the optimal threshold u by keeping the standard error as small as possible.
Figure 5.3: Mean Excess Plot of Standardized Residuals
For instance, from the mean excess plot of S&P 500, we will narrow down the candidate threshold between 1.5 - 2.0. And within the range, we choose the optimal threshold value based on their performances. For S&P 500, 1.62 is the optimal one since we can have the significant parameter estimates and its standard error smallest. Based on MEP, we have chosen the threshold value u for other returns, 1.55 for HSI, 1.62 for CAC40 and 1.62 for N225. Specifically for N225, to obtain enough exceedances above the threshold, we set the threshold value relatively lower than MEP would suggest. To find the optimal estimate of ξ and β in GPD, we apply MLE as we have discussed in (3.9) and (3.10). The estimated parameters are listed in Table 5.3 as well.
Table 5.3: Estimation Results of AR-GARCH-EVT model
S&P 500 HSI CAC40 N225 In-Sample Observations n 3773 3745 3835 3694 Threshold u 1.62 1.55 1.62 1.62 No. of Exceedances k 240 250 240 197 Percentage of Exceedances k/n 6.4% 6.7% 6.3% 5.3% GPD Parameters ξ 0.0286 0.0384 0.1374 0.0588 SE 0.0669 0.0719 0.0740 0.0601 β 0.5989 0.5218 0.4432 0.5726 SE 0.0557 0.0499 0.0433 0.0533 Note: Since we have transformed the negative returns into positive to sim-plify the calculations, the threshold value u here is positive number.
To quantify the accuracy of the estimated parameters, we have computed Stan-dard Errors (SE) to make comparisons. From Table 5.3, it is indicated that the relative precision of ξ is always lower than β for all returns. A larger shape param-eter β would lead to a more spread-out distribution, while a larger scale paramparam-eter ξ would lead to a heavier distribution tail. S&P 500 here obtains the largest shape parameter and CAC40 gets the largest scale parameter, which indicates that the highest VaR estimates in high probability conditions.
It is assumed that the conditional excess distribution converges to GPD as we have used in EVT analysis with ξ > 0. To confirm the assumptions, we plot the exceedances over threshold into a GPD as in Figure 5.4. It is evident that the empirical distribution is consistent with the corresponding GPD, which proves the model’s validity.
Figure 5.4: GPD plot fitted to exceedances over threshold
Take S&P 500 with 95% probability VaR for instance, ˆ x0.95 = u + ˆ β ˆ ξ (1 − q k/n ) − ˆξ− 1 = 1.62 + 0.59886668 0.02863172 (1 − 0.95 6.4% ) −0.02863172− 1 = 1.765569
After verifying the fitness of the model, we can obtain VaR estimates as follows. Take S&P 500 with 95% probability VaR for instance as well, then for t equal to the final day of the sample,
VaRt0.95 = − [µt+1+ ˆσt+1× (−ˆx0.95)]
= −h0.08507848 +√0.9729299 × (−1.765569)i = 2.2263350
We compute VaR estimates under 90%, 95% and 99% for all returns. The results can be found on Table 5.7.
5.1.2 The SV-EVT model
Similarly as in the GARCH-EVT model, we take the first step to fit an ap-propriate SV model for each return. Based on previous papers, we set the prior distribution in MCMC as: µ ∼ N (−1, 1) (5.2) φ + 1 2 ∼ Beta(20, 1.5) (5.3) ρ ∼ U (−1, 1) (5.4) 1 σ2 η ∼ Gamma(2.5, 0.025) (5.5) ν ∼ Gamma(16, 0.8) (5.6)
The sampler is initialized as µ = 0, φ = 0.95, 1/σ2
η = 100, ρ = −0.4, ν = 15.
We take iterations 15,000 times and set the "burn-in" sample as 5,000 times. Table 5.4 reports the estimation results of the SV model, including the posterior mean, the standard deviation, the MC error and the 95% interval. The Monte Carlo error defines the accuracy of approximations, which is calculated as:
M C = Standard Deviation√ No. of Iterations
the volatility clustering, while the ν value showing the validity of applying the standardized Student’s t-distribution. According to the negative value of ρ, it is verified that the leverage effect exists in sample data. We can expect the error to decrease with the square root of the number of trials as shown in MC error.
Table 5.4: Estimation Results of the SV model
Parameters Mean Stdev MC error 95% Interval S&P 500 µ -0.095 0.081 6.61e-04 [-0.229,0.038] ν 14.246 3.126 0.026 [10.510,29.511] φ 0.990 0.001 8.16e-06 [0.989,0.992] ρ -0.896 0.039 3.18e-04 [-0.962,-0.826] ι 50.294 8.790 0.072 [34.129,64.210] HSI µ 0.388 0.135 1.12e-03 [0.169,0.608] ν 16.364 5.435 0.044 [8.091,24.670] φ 0.994 0.001 8.16e-06 [0.992,0.996] ρ -0.573 0.057 4.65e-04 [-0.675,-0.482] ι 90.376 20.704 0.169 [63.97,134.1] CAC40 µ 0.280 0.817 6.67e-03 [0.149,0.416] ν 19.349 3.651 0.030 [13.95,25.47] φ 0.990 0.001 8.16e-06 [0.989,0.992] ρ -0.895 0.032 2.613e-04 [-0.949,-0.843] ι 43.036 4.205 0.034 [35.990,49.931] N225 µ 0.544 0.085 6.94e-04 [0.405,0.684] ν 16.905 4.463 0.036 [10.520,25.160] φ 0.985 0.002 1.63e-05 [0.981,0.988] ρ -0.549 0,051 4.16e-04 [-0.650,-0.474] ι 35.702 4.568 0.037 [28.94,43.751] Note: ι = 1/ση2.
For the next step to set up the EVT part, we obtain the standardized residuals t as in (3.29) and apply the EVT algorithm.
The descriptive statistics of standardized residuals are in Table 5.5. Unlike stan-dardized residuals obtained from the GARCH model, residuals from the SV model don’t display evident excess skewness and kurtosis. The Jarque-Bera tests indicate that the residuals do not follow a normal distribution. But from the histograms of standardized residuals in Figure 5.5, except CAC40 and N225, others do not display a fat tail in the negative part. This gives rise to a question whether it is still worth to apply EVT to its tail-part since there doesn’t exist significant fat-tailedness. However, we will still set up the SV-EVT model to value its performance but also calculate VaR by the SV model to compare their results.
Table 5.5: Descriptive Statistics of Standardized Residuals
Standardized Residuals S&P500 HSI CAC40 N225 No. of Observations 3773 3745 3835 3694 Mean 0.001815 0.007396 -0.001279 0.006964 Stdev 0.979775 0.982506 0.999844 0.957325 Skewness -0.234211 -0.099275 -0.168093 -0.155779 Kurtosis 3.239618 3.187503 2.976139 2.799874 Jarque-Bera 43.678 11.743 18.154 21.024 *** ** ** ***
Note: The table includes descriptive statistics of the standardized resid-uals of S&P500, HSI, CAC40 and N225 over period from 1/31/2000 to 1/31/2017. Stdev stands for the standard deviation. Jarque-Bera statistics test the null hypothesis that the returns follow normal distribution, with *** stands for significance at 0.01%, ** stands for significance at 0.1%.
Figure 5.5: Histogram of Standardized Residuals
From results in Table 5.6, it is hard to fit the conditional excess distribution into a GPD. Therefor we choose at around 100 exceedances to fit the distribution and it turns out to have a negative ξ value in S&P500 and HSI, which is against our assumptions in EVT set-up. Residuals display a rather short tail instead of a fat tail. And SE is higher compared to that of the GARCH-EVT model, implying the relative lower precisions.
We also plot the conditional variance of the GARCH and SV models in Figure 5.5. The red line is the GARCH model while the black line is the SV model. It is evident that the figures have many overlapping lines, which represents that the conditional variance for each return share similar traces. Around 2008, every return show extremely high variance. For S&P 500 and HSI, the GARCH model has more wild fluctuations. On the contrary, the SV models of CAC40 and N225 fluctuates
Table 5.6: Estimation Results of the SV-EVT model
S&P 500 HSI CAC40 N225 In-Sample Observations n 3773 3745 3835 3694 Threshold u 2.0 2.0 2.0 1.9 No. of Exceedances k 102 95 101 99 Percentage of Exceedances k/n 2.70% 2.54% 2.63% 2.68% GPD Parameters ξ -0.0866 -0.1380 0.0219 0.0727 SE 0.0886 0.0921 0.1361 0.1115 β 0.4758 0.4500 0.3731 0.3062 SE 0.0631 0.0617 0.0628 0.0459 Note: Since we have transformed the negative returns into positive to sim-plify the calculations, the threshold value u here is positive number.
more.
Figure 5.6: Conditional Variance of the GARCH and SV models
From Table 5.7, we can tell S&P500 displays a distinct property, with the lowest VaR estimates of HS among all other methods while for other returns, HS computes the highest estimates. It is because the log returns of S&P 500 has a shorter tail compared with other three returns. It is indicated that Historical Simulation is likely to induce overestimation and underestimation since it depends on the sample distribution in the past. For the GARCH and GARCH-EVT, SV and SV-EVT estimates, they are quite close to each other for each return series. Comparing VaR estimates of the GARCH-EVT and SV-EVT models above, when using 90% and 95% confidence levels, S&P500 and N225 have higher VaR value while HSI and CAC40 have more conservative estimates. With a 99% level, the GARCH-EVT model offers
Table 5.7: VaR Estimates on Day n + 1
Models S&P 500 HSI CAC40 N225 q = 90% Historical Simulation 1.387192 1.668510 1.679971 1.739759 GARCH 1.733854 1.419123 1.547177 1.535804 GARCH-EVT 1.682396 1.443038 1.627505 1.429320 SV 1.537216 1.249034 1.126879 1.551890 SV-EVT 1.804412 1.175957 1.180748 1.960530 q = 95% Historical Simulation 1.989810 2.403513 2.415879 2.394210 GARCH 2.332552 1.853753 2.040025 2.026439 GARCH-EVT 2.226335 1.841090 1.981470 1.875689 SV 2.014279 1.637480 1.467802 2.028744 SV-EVT 2.289313 1.499449 1.494563 2.149506 q = 99% Historical Simulation 3.517525 4.299062 4.332298 4.104841 GARCH 3.660916 2.719354 3.070138 3.066818 GARCH-EVT 3.527744 2.807188 2.945841 2.985024 SV 3.011068 2.423203 2.152819 3.004442 SV-EVT 3.309019 2.141281 2.125856 2.688747
a much larger value than SV-EVT estimates. It is suggested that the GARCH-EVT model might overestimate the risks. To make more reasonable comparisons, we then apply backtesting method to further evaluate their performances since large estimates don’t ensure a better risk measurement.
5.2
Out-of-sample Evidence
5.2.1 BacktestingIn order to test whether the VaR estimation is indeed credible, we will apply backtesting against the out-of-sample period. Denote n as the number of in-sample observations, m as the number of out-of-sample observations. For backtesting, we use n as the time window. Along with the parameter estimates, we can compute each VaRt
q during the out-of-sample period by using previous n returns. That is, VaRt
qwith t in the set n, · · · , m−1 is calculated with returns rt−n+1, · · · , rt. This im-plementation is rolled forward for each day, which effectively captures time-varying characteristics. Since the backtesting period is relatively long, using MEP to de-termine the threshold value is not feasible. Therefore, we turn to set k/n at the percentage equivalent in the previous cases, in order to simplify the procedures.
We apply the violation rate to determine the performances of each model, which is a widely used tool to check the accuracy of VaR calculations. The violation rate
(VR) is defined as follows:
V R = Pm
i=1I[−rt+1>VaRtq]
m (5.7)
where I is an indicator function as
I = 1 if −rt+1>VaRtq 0 if −rt+1≤VaRtq (5.8)
Note that VaRtq is a positive number here.
For a correctly specified model, the violation rate should be equal to the hy-pothesis probability p. Therefore, if the violation rate is larger than p, it is indicated that the model has excessively underestimated the returns. If the violation rate is smaller than p, the model has excessively overestimated the returns. One way to compare the performances is to rank the distances between the empirical violation rate to p. The closest one outperforms others.
Since some standardized residuals do not show the fat-tailedness property, it might be helpful to apply the GARCH model and the SV model separately to compute VaR estimates and compare their performances. Table 5.8 lists the out-of-sample VaR violation rates of Historical Simulation, the GARCH model, the GARCH-EVT model, the SV model and the SV-EVT model under different α value. We also rank the violation rates according to its distance with expected probability. Consider, for example, S&P 500 under p = 10%. The GARCH-EVT model displays the best violation rate of 8.37% which underestimates the assumed confi-dence level by 1.63%. The second best model is GARCH, which amounts to 1.83% gap. The SV model, Historical Simulation and the SV-EVT model are found to be the least performing models. When p = 5%, the SV-EVT model and the GARCH-EVT model outperform other models, when Historical Simulation is still the worst performing model. When comes to extreme condition α = 1%, we have the best forecast model SV and GARCH model while the worst is Historical Simulation.
For S&P 500 and HSI, the performance of the SV model is better than the SV-EVT model especially when reaching the 99% confidence level. It is because for these two returns, the standardized residuals are not fat-tailed in the negative part and applying EVT will not optimize the VaR computations. On the other hand, for CAC40 and N225, the SV-EVT model outperforms the SV model. Comparing the GARCH model and the GARCH-EVT model, the advantage of adding EVT present gradually as the confidence interval increases. And in most cases, the GARCH-EVT model outperforms the GARCH model with a relatively closer violation rate com-pared to the expected rate. The overall ranking shows that the GARCH-EVT model forecasts most accurately among all the methods and markets.
5.2.2 Test Statistics
Binomial Test
Another method to compare the performances is to test whether the violation rates are statistically significant. Since Irt+1>VaRt
q ∼ Bernoulli(1 − q), we will have
Pm
i=1Irt+1>VaRtq follows a binomial distribution, that is
Pm
i=1Irt+1>VaRtq ∼ B(m, 1−q).
Based on the binomial distribution, we set up the statistic as: V R − (1 − q)
q q(1−q)
T
∼ N (0, 1) (5.9)
where T = m − n is the number of out-of-sample days.
It is a one-sided statistic with the null hypothesis V R = 1 − q. For the alternate hypothesis the model is underestimated, that is V R > (1 − q), we have the rejection region as V R−(1−q)q
q(1−q) T
> N (0, 1)0.05. On the other hand, for the alternate hypothesis the model is overestimated, that is V R < (1 − q), we have the rejection region as V R−(1−q)
q
q(1−q) T
< −N (0, 1)0.05.
Table 5.8 also lists the binomial test results of backtesting under 90%, 95% and 99% probability. We set the p-value at 0.05 as the significance level for consistency. Consider S&P 500 as an example. Under 90% probability, the GARCH-EVT model and the GARCH model do not reject the null hypothesis which prove their validity. The SV model and the SV-EVT models here tend to overestimate VaR and therefore lead to low violation rates compared to the expected rate. Though, almost all models do not pass the test when the confidence level is 95%, they perform well in higher confidence interval except HS. Under the confidence level 90% and 99%, models for HSI behave disappointingly in the binomial test by rejecting the null hypothesis. Models tend to overestimate in low confidence level and underestimate in high confidence level.
In general, the null hypothesis is rejected 5 times under HS, 4 times under the GARCH model, 4 times under the GARCH-EVT model, 7 times under the SV model and 7 times under the SV-EVT model. Especially under high confidence level, the SV model and the SV-EVT model do not have satisfying performance. They tend to compute overestimated or underestimated violation rates in different conditions. Under 95%, the GARCH-EVT model performs the best with binomial test passing for all returns.
Duration-based Test
Besides the binomial statistic above, we apply the duration-based test to check whether the violation process follows the independence hypothesis based on Christof-fersen and Pelletier (2004). It is also known as the conditional coverage test, which serves as a stronger test for adequacy of the proposed models. While the binomial
test focuses on the amount of the violations, the duration-based test investigates the frequency of the violations to see if the violations are clustered or not. The null hypothesis is that the duration of days between exceedances has no memory.
Table 5.9: Duration Based Test Results of Backtesing
Models S&P 500 HSI CAC40 N225 q = 90% Historical Simulation -113.8324 -145.4277 -157.8339 -154.0998 *** (0.8792) *** (0.1767) GARCH -117.3055 -130.9906 -142.0634 -147.6972 (0.8515) (0.0561) (0.7076) (0.6456) GARCH-EVT -120.0796 -125.2955 -131.5661 -154.5628 (0.8766) *** (0.5874) (0.3429) SV -108.2387 -145.4277 -176.7487 -151.4017 *** (0.8792) *** (0.1168) SV-EVT -94.56189 -107.3384 -155.8922 -121.0504 *** (0.1628) *** *** q = 95% Historical Simulation -58.24763 -78.01915 -83.05951 -105.3612 *** *** (0.0531) *** GARCH -38.87893 -99.04188 -81.5408 -102.0189 *** (0.3178) (0.6284) (0.3532) GARCH-EVT -43.86192 -99.04188 -91.23366 -113.6311 (0.1028) (0.3178) (0.6674) (0.4484) SV -64.16721 -96.02072 -122.6049 -115.2774 *** (0.3087) *** *** SV-EVT -64.16721 -89.45664 -108.7406 -102.8145 *** (0.1677) *** *** q = 99% Historical Simulation -4.097367 -6.977166 -7.242223 -27.93755 *** (0.5039) *** (0.9015) GARCH -7.155604 -48.95291 -36.93058 -40.59632 (0.7226) (0.8185) (0.5920) (0.3988) GARCH-EVT -7.155604 -40.9352 -36.93058 -40.59632 (0.7226) (0.7962) (0.5920) (0.3988) SV -21.4468 -40.55975 -59.87219 -59.47028 (0.0523) (0.3659) (0.1721) (0.2159) SV-EVT -21.4468 -44.29229 -56.41724 -76.75796 (0.0523) (0.2246) (0.2468) (0.2093) Note: *** stands for the null hypothesis being rejected under the signif-icance level 5%. In other words, the violation rates are not independent.
Table 5.9 lists the results of duration based tests of HS, the GARCH model, the GARCH-EVT model, the SV model and the SV-EVT model under 10%, 5% and 1% probability. We also set the p-value at 0.05 as the significance level for consis-tency. Consider S&P 500 as an example. Like the results from binomial test, the GARCH-EVT model and the GARCH model behave satisfactory under the confi-dence level 90% and the GARCH-EVT model is the only one passing the test under the confidence level 95%. Except HS, all the models do not reject the null hypothesis
under the confidence level 99%. The performance of the SV-EVT model is similar to that of CAC40 and N225, rejecting in low confidence level and accepting in high confidence level. Unlike the binomial test, models for HSI show satisfactory results here with only two situation rejecting.
It is indicated that almost all the models perform better under higher confi-dence level than low conficonfi-dence level. Altogether, the GARCH-EVT model and the GARCH model perform best by rejecting only once, followed by the SV model and the SV-EVT model. HS rejects the test under 7 cases and therefore its results of violation rates are not independent.
In terms of test results, the GARCH-EVT model performs the best especially under the confidence level 95%, while the SV-EVT model does not behave satisfac-tory in tests. Combined with the rankings of violation rate, it is concluded that we have the GARCH-EVT model better than the other models under chosen conditions in this thesis.
Table 5.8: Out-of-sample VaR Violation Rates and Results of Bi-nomial Tests
Models S&P 500 HSI CAC40 N225 q = 90% Historical Simulation 6.00%(4) 7.71%(2) 9.34%(1) 9.55%(1) *** *** (0.3086) (0.3705) GARCH 8.17%(2) 7.71%(2) 7.98%(4) 8.94%(5) (0.0855) *** (0.0631) (0.2173) GARCH-EVT 8.37%(1) 7.30%(3) 7.20%(5) 9.55%(1) (0.1112) *** *** (0.3705) SV 6.18%(3) 8.72%(1) 11.28%(3) 9.35%(3) *** (0.1721) (0.7948) (0.3153) SV-EVT 5.18%(5) 5.88%(5) 9.34%(1) 7.11%(5) *** *** (0.3086) *** q = 95% Historical Simulation 2.39%(4) 3.65%(5) 3.89%(3) 5.89%(2) *** *** (0.1243) (0.8186) GARCH 3.19%(2) 5.27%(2) 3.89%(3) 5.49%(1) *** (0.6099) (0.1243) (0.6902) GARCH-EVT 3.59%(1) 5.27%(2) 4.47%(1) 6.30%(3) (0.0730) (0.6098) (0.2924) (0.9072) SV 3.19%(2) 5.07%(1) 6.80%(4) 6.71%(4) *** (0.5288) *** *** SV-EVT 3.18%(3) 4.67%(4) 5.84%(2) 5.89%(2) *** (0.3666) (0.8079) (0.7591) q = 99% Historical Simulation 0.40%(5) 0.40%(1) 0.39%(3) 1.42%(1) *** (0.0924) (0.0820) (0.8270) GARCH 0.80%(3) 2.23%(5) 1.56%(1) 1.83%(2) (0.3236) *** (0.8976) *** GARCH-EVT 0.79%(4) 1.83%(2) 1.56%(1) 1.83%(2) (0.3236) *** (0.8976) *** SV 0.99%(1) 1.83%(2) 2.72%(4) 2.85%(4) (0.4964) *** *** *** SV-EVT 0.99%(1) 2.03%(4) 2.72%(4) 3.86%(5) (0.4964) *** *** *** Note: *** stands for the null hypothesis of the binomial test being rejected under the significance level 5%. In other words, the violation rates are either overestimated or underestimated.
Conclusion
The thesis aims to contribute to the VaR analysis literature by comparing the GARCH-EVT and SV-EVT models. For our analysis, we augment the widely used GARCH model and the SV model, with leverage effects and the Student’s t-distribution. We carefully analyze the tail part with Extreme Value Theory and set up the GARCH-EVT and the SV-EVT models. This is because the financial re-turns tend to display volatility clustering, fat tails and asymmetry properties. EVT helps to capture the statistical characteristics of extreme values.
The main challenges in the thesis are the computation of the EVT parameters and estimation of the SV model. We choose the Peak Over Threshold method to analyze the tail part of the returns distributions and Markov Chain Monte Carlo approach to estimate the SV model. Combining the R and WinBugs software, we are able to set up the GARCH-EVT model, the SV-EVT models and successfully compute VaR estimates.
The thesis is roughly divided into two parts, theoretical model construction and empirical analysis. The theoretical part introduces the literature review and the detailed constructions of proposed models and how to compute Value at Risk esti-mates. In latter part we implement the models over 4 stock indices and perform an in-sample and out-of-sample analysis.
In order to determine whether these methods are valid, we implement two tests– binomial test and duration-based test. The binomial test focus on the amount of the violations to decide whether overestimation or underestimation has occurred. The duration-based test examines the frequency of the violations to distinguish if it is clustered. The GARCH-EVT model outperforms other models by passing the most tests among all conditions. The SV-EVT model does not behave satisfactory with statistically overestimating results.
In conclusion, the GARCH model combined with the EVT can be used to com-pute VaR effectively for risk measurements. It helps the financial institutions and regulators to better distinguish the risky position in markets under uncertain ex-treme movements. Further research can investigate how to combine this method to compute VaR with diversified financial portfolios instead of single stock index.
Though the SV-EVT model does not perform well in the thesis, it is still interesting to discover how the SV model can be improved to compute VaR. But whether the complicated estimation process is worth its results is an answer to be found.
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