Method of Dynamic VaR and CVaR Risk Measures Forecasting for Long Range Dependent Time Series on the Base of the Heteroscedastic Model

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1. Introduction

VaR (Value-at-Risk) and CVaR (Conditional Value-at-Risk) have become the standard measures of market risk management. Their popularity has led to a large number of publications on this topic in recent years. Definition, description of the properties and comparative analysis of these risk measures can be found, for example, in [1] [2] [3] . Various methods for their evaluation and forecasting that represents different approaches are proposed. Most of the methods that provide explicit formulas for CVaR estimation are described in [4] . Optimization approach for CVaR evaluation is given in [5] [6] . Non-parametric methods of estimation can be found, for example, in [7] [8] . A large number of works devoted to the method of VaR and CVaR estimating based on the stochastic time series model. The basic ideas of the approach can be found for example in [2] [9] [10] . A significant number of works show the practical application of the approach for estimating and forecasting of stock indices, see for example [10] [11] [12] [13] .

At the same time, during the global financial turmoil, the problem of constructing of new approaches for VaR and CVaR estimating and forecasting remains relevant. In this paper, we propose a new method for VaR and CVaR prediction for financial time series. The method takes into account the most statistically significant extreme values of data and the presence of the long-range dependence that is typical for financial time series [2] [14] . For the convenience of practical application, the method is formulated as an incremental algorithm. At each step, the system of tests is proposed to evaluate the quality of the obtained results.

The proposed algorithm is used for forecasting VaR and CVaR for the time series of daily log return Nikkey225 Stock Index. The analysis of the obtained forecast estimates confirms their high quality. The formatter will need to create these components, incorporating the applicable criteria that follow.

2. Key Definitions

The continuously distributed random variable $\left\{{X}_{t},t\in Z\right\}$ with finite mean defined on the probability space $\left(\Omega ,{\Psi}_{t},{{\rm P}}_{t}\right)$ is considered. Here ${\Psi}_{t}$ is the information set containing all available at the time $t$ information about the time series. Series $\left\{{X}_{t}^{2},t\in Z\right\}$ is assumed to be stationary. It is accepted that the time series has the property of the long-range dependence [15] : there is $0<\gamma <1$ and ${c}_{r}>0$ so that:

$\underset{k\to \infty}{\mathrm{lim}}\rho \left(k\right)/\left({c}_{r}{k}^{-\gamma}\right)=1,\text{}\rho \left(k\right)=\text{Corr}\left({X}_{t},{X}_{t+k}\right),\text{}k\in N\cup \left\{0\right\}$ (1)

For a fixed confidence level $\alpha $ dynamic risk measures $\text{VaR}$ and $\text{CVaR}$ are defined as [9] :

${\text{VaR}}_{\alpha}^{t}\left(t+h\right)=\mathrm{inf}\left\{x\in R|{{\rm P}}_{t}\left[{X}_{t+h}\le x\right]\ge \alpha \right\},$

${\text{CVaR}}_{\alpha}^{t}\left(t+h\right)={E}_{{\Psi}_{t}}\left[{X}_{t+h}|{X}_{t+h}\ge {\text{VaR}}_{\alpha}^{t}\left(t+h\right)\right],$

${E}_{{\Psi}_{t}}[\cdot ]$ denotes expectation with respect to ${\Psi}_{t}$ .

The aim of the study is to construct a model for ${\text{VaR}}_{\alpha}^{\stackrel{\u02dc}{t}}\left(\stackrel{\u02dc}{t}+h\right)$ and ${\text{CVaR}}_{\alpha}^{\stackrel{\u02dc}{t}}\left(\stackrel{\u02dc}{t}+h\right)$ where $\stackrel{\u02dc}{t}$ is an arbitrary moment of time. The forecasting values are determined by extrapolation of the values of this model (21 cm × 28.5 cm).

3. Forecast Methodology

In the article [16] , the most popular methods for dynamic $\text{VaR}$ and $\text{CVaR}$ estimating are analyzed, their classification is given and the recommendations for their use are proposed. In accordance with the formulated in the article the structural scheme of selection of dynamic risk measures estimation the approach based on a stochastic time series model is chosen.

Suppose that the time series $\left\{{X}_{t},t\in Z\right\}$ is a trajectory of stochastic process, that is:

${X}_{t}={\mu}_{t}+{\epsilon}_{t}={\mu}_{t}+{\sigma}_{t}{Z}_{t}$ (2)

where conditional mean ${\mu}_{t}$ and variation ${\sigma}_{t}$ are defined on the information space ${\Psi}_{t}$ , $\left\{{Z}_{t}\right\}\stackrel{iid}{{\displaystyle ~}}{F}_{t}\left(0,1\right)$ (independent, identically distributed random variables with a conditional distribution function ${F}_{t}\left(0,1\right)$ ). Let $Z$ is a random variable with the same distribution as any random variable from $\left\{{Z}_{t}\right\}$ . Then [2] [9] [10] :

$\begin{array}{l}{\text{VaR}}_{\alpha ,k}^{t}={\mu}_{t+k}+{F}^{-1}\left(\alpha \right){\sigma}_{t+k}={\mu}_{t+k}+{\text{VaR}}_{\alpha}\left(Z\right){\sigma}_{t+k},\\ {\text{CVaR}}_{\alpha ,k}^{t}={\mu}_{t+k}+{\text{CVaR}}_{\alpha}\left(Z\right){\sigma}_{t+k}.\end{array}$ (3)

It is necessary to construct the forecast model for ${\sigma}_{t}$ to determine its $P$ days forecast and to estimate $\text{VaR}$ and $\text{CVaR}$ for a random variable $Z$ . Then the forecasting values for dynamic risk measures can be found under the following formulas:

$\begin{array}{l}{\text{VaR}}_{\alpha}^{t+P}={\mu}_{t+P}+{\text{VaR}}_{\alpha}\left(Z\right){\sigma}_{t+P},\\ {\text{CVaR}}_{\alpha}^{t+p}={\mu}_{t+P}+{\text{CVaR}}_{\alpha}\left(Z\right){\sigma}_{t+P}\mathrm{..}\end{array}$ (4)

Hereinafter it is assumed that the trend, that defines ${\mu}_{t}$ , is absent (or removed from the data) [2] . Please do not revise any of the current designations.

4. An Algorithm for Constructing the Dynamic Risk Measures VaR and CVaR Forecast Taking into Account the Long-Range Dependence Presence

For the convenience of the practical application the proposed method for $\text{VaR}$ and $\text{CVaR}$ forecasting is formulated as an incremental algorithm.

Step 1. For the time series a time series of variances (TSV) is constructed. General analysis of the studied time series and the TSV is carried out, the dependence of time series members (and their squares) from their previous values; the volatility and normality are analyzed.

Step 2. The TSV is tested on the long-range dependence. The Hurst parameter is estimated using five standard methods: the aggregated variance method, the method of absolute values of the aggregated series, the periodogram method, the method of residuals of regression, the R/S method [17] . Average value ${\stackrel{\u2322}{H}}_{mn}$ is chosen as the Hurst parameter estimation.

Step 3. The model for ${\sigma}_{t}$ forecasting is estimated using the $\text{FIGARCH}$ model and taking into account the long-range dependence of the WFD. The actualization of the model by reducing it to the model AR (∞) is performed. The method of smoothing of the autocorrelation function (ACF) proposed by the authors in [18] (the new method) is used. The least square method is used to determine the autoregression coefficients ${\left({a}_{1},\cdots ,{a}_{N},\cdots \right)}^{\prime}$ . So the problem is reduced to the infinite system of Yule-Walker equations [18] :

$\underset{j=0}{\overset{\infty}{\sum}}{\rho}_{\left|i-j\right|}{a}_{j}}={\rho}_{i+1},\text{}i=0,\cdots ,\infty .$ (5)

The regression equation for ACF based on the definition of the long-range dependence (1) is used to get estimates for ${\rho}_{i}:\rho \left(k\right)={\alpha}_{1}H\left(2H-1\right){k}^{2H-2}+{\alpha}_{2}+{\epsilon}_{k}$ , ${\epsilon}_{k}-iid$ , ${k}_{0}\le k\le N$ . With the help of the optimization procedure [17] the Hurst parameter estimate and the estimates $\stackrel{\u2322}{\rho}\left(k\right)$ are corrected.

Using $\stackrel{\u2322}{\rho}\left(k\right)$ instead of ${\rho}_{k}$ the reduced system of normal Equations (5) is constructed and using the Holetskogo method the vector of assessments ${\stackrel{^}{\stackrel{\to}{a}}}_{N}={\left({\stackrel{^}{a}}_{1},\cdots ,{\stackrel{^}{a}}_{N}\right)}^{\prime}$ is found. As it is shown in [19] the solution of the reduced system converges to the exact solution.

The lag of the reduced AR model $M\le N$ is determined using the information criterions: AIK (Akaike information criterion), HQC (Hannan-Quinn information criterion), SBIC (Bayesian information criterion) [14] . The lag value is chosen on the basis of minimum deviation.

The quality of the obtained AR model is checked. The variance ratio test [20] is used to test if the residuals of the model are iid (independent and identically distributed). The resulting model is used to obtain ${\stackrel{^}{\sigma}}_{t}$ .

Step 4. The residuals of the model (2) are analyzed. Using ${\stackrel{^}{\sigma}}_{t}$ (step 3) the implementations of a random variable ${Z}_{t}:{\stackrel{^}{Z}}_{t}={X}_{t}/{\stackrel{^}{\sigma}}_{t}$ are built. ${Z}_{t}$ are analyzed on iid (the variance ratio test) and other properties. In accordance with the results using the classification scheme given in [21] , the method to get ${\text{VaR}}_{\alpha}\left(Z\right)$ and ${\text{CVaR}}_{\alpha}\left(Z\right)$ estimates is chosen. The estimates ${\stackrel{^}{\text{VaR}}}_{\alpha}\left(Z\right)$ , ${\stackrel{^}{\text{CVaR}}}_{\alpha}\left(Z\right)$ are obtained.

Step 5. With the results of steps 3 and 4 the model for dynamic risk measures estimating (3) is ready. After building the dynamic risk measures estimations $\stackrel{^}{{\text{VaR}}_{\alpha}^{t}}$ и $\stackrel{^}{{\text{CVaR}}_{\alpha}^{t}}$ their quality is analyzed using the Kupiec test, the Kristoffersen test and the $V$ test [10] [16] .

Step 6. The built dynamic risk measures model is used to get the forecast. Using the model from step 3 the $P$ -step forecast for ${\sigma}_{t}$ is built by the formulas:

${\stackrel{^}{\sigma}}_{l+p}^{2}={\displaystyle \underset{i=p}{\overset{M+p}{\sum}}{\stackrel{^}{a}}_{i}{\stackrel{^}{\sigma}}_{l-i+1}^{2}},\text{}l=N+1,\cdots ,p=1,\cdots ,P.$ (6)

Using the estimates ${\stackrel{^}{\text{VaR}}}_{\alpha}\left(Z\right)$ , ${\stackrel{^}{\text{CVaR}}}_{\alpha}\left(Z\right)$ (step 4) the $P$ -step forecast for dynamic risk measures ${\stackrel{^}{\text{VaR}}}_{\alpha}^{t+P}$ and ${\stackrel{^}{\text{CVaR}}}_{\alpha}^{t+P}$ (4) is obtained. Then the

index of the time series is increased by $P$ and the procedure is repeated as many times as necessary. Thus in each cycle of the algorithm application the model is updated to take into account new data.

Step 7. Using the back testing procedure, the quality of the predicted values ${\stackrel{^}{\text{VaR}}}_{\alpha}^{t+P}$ and ${\stackrel{^}{\text{CVaR}}}_{\alpha}^{t+P}$ (step 6) is checked, the prediction errors $ME$ , $MAE$ , $MSE$ are calculated. For $\stackrel{^}{\text{CVaR}}$ estimates the BPoE-test [22] is used.

Schematic description of the proposed method is shown in Figure 1. Please take note of the following items when proofreading spelling and grammar.

5. Numerical Testing of the Algorithm

To demonstrate the proposed algorithm a forecast for dynamic risk measures $\left(\alpha =0.9\right)$ for the time series of log returns on a daily basis is built. Data are collected from the oldest and the most well-known index of Asian markets $\text{Nikkey225StockIndex}$ (the time series $N225\_RED$ )―a composite index of the 225 largest companies publicly traded in $\text{Tokyo}\text{}\text{StockExchange}$ for the period from 2005 to 2015. $N225\_RED$ has a relatively low homogeneous volatility. The aim of this study is to forecast risk measures at a regular market behavior, so data without three time intervals with high volatility of the global financial system (01.07.2008-01.07.2009, 01.01.2011-01.07.2011, 01.02.2013-01.12.2013) are considered. Historical data of $\text{Nikkey225StockIndex}$ are not available online, but upon request.

Table 1 demonstrates the descriptive statistics for the time-series $\left({X}_{t}\right)$ and the squared series $\left({X}_{t}^{2}\right)$ .

Figure 1. Schematic description of the proposed method of dynamic risk measures VaR and CVaR forecasting.

Table 1. Basic descriptive statistics of the N225_RED.

Skewness is about 0 and kurtosis is about 3, so the distributions are close to normal. Ljung-Box test [2] results for $m=7$ confirm the dependence of data (and squared data) on their previous values (the values of Q-statistic are larger than critical value 12.017).

Consider the half of the general sample-843 values. The estimates of the Hurst parameter are ${\stackrel{\u2322}{H}}_{mn}=0.7387$ and ${\stackrel{\u2322}{H}}_{opt}=0.7281$ (step 2). These values confirm the long-range dependence of the time series.

Simulate ${\sigma}_{t}$ (step 3) using the method $\text{SACF}$ (the designation $\_\text{SACF}$ ) and for comparison the standard methodology (the designation $\_st$ ). The standard methodology uses the $AR\left(M\right)$ model with the coefficients found by the maximum likelihood method (MLH). The lag of the reduced $AR$ model is $M=55$ . The results of the variance ratio test (0.99 < 1.96 for the $\text{SACF}$ method and 0.69 < 1.96 for the standard method) confirm that the residuals of the models are iid.

For both models ${Z}_{t}$ are found (step 4) and their analysis is carried out. The results of the variance ratio test (0.98 < 1.96 for the method $\text{SACF}$ and 0.97 < 1.96 for the standard method) show that the residuals of the model (2) are iid.

Estimates ${\stackrel{^}{\text{VaR}}}_{0.9}\left(Z\right)$ , ${\stackrel{^}{\text{CVaR}}}_{0.9}\left(Z\right)$ are obtained using the following methods [16] : the historical simulation method $\left(\text{hist}\right)$ , the explicit formulas under the assumption of a normal distribution with the maximum likelihood estimates of the parameters $\left(\text{paramdistr}\right)$ ,the explicit formulas using $GEV$ and $GPD$ functions with the maximum likelihood estimates of the parameters ( $GEV\_\text{quant}$ and $GPD\_\text{quant}$ respectively), the empirical $POT$ method $\left(POT\_\text{emp}\right)$ . Table 2 demonstrates the results of the estimating.

Using the results of steps 3, 4 estimates (3) of the dynamic ${\stackrel{^}{\text{VaR}}}_{0.9}^{t}$ and ${\stackrel{^}{\text{CVaR}}}_{0.9}^{t}$ (step 5) are obtained. Figure 2 demonstrates the simulated dynamic ${\stackrel{^}{\text{VaR}}}_{0.9}^{t}\_\text{SACF}$ and ${\stackrel{^}{\text{CVaR}}}_{0.9}^{t}\_\text{SACF}$ (first 836 values) where the explicit formulas under the assumption of a normal distribution $\left(\text{paramdistr}\right)$ were used for risk measures model residuals estimating.

Figure 2. Simulated and predicted values (last 5 values) of dynamic risk measures with new method.

Table 2. The estimates of the statics ${\stackrel{^}{\text{VaR}}}_{0.9}\left(Z\right)$ , ${\stackrel{^}{\text{CVaR}}}_{0.9}\left(Z\right)$ .

Conduct the analysis of quality (step 5) for ${\stackrel{^}{\text{VaR}}}_{0.9}^{t}$ estimates using the Kupiec test ( $p-$ values of statistics $\text{LRpof}$ ), the Kristoffersen test ( $p-$ values of statistics $\text{LRind}$ ) and their combination ( $p-$ values of statistics $\text{LRcc}$ ). The obtained estimates are reliable if $p-$ values exceed the given significance level (0.1 in our case). To analyze the ${\stackrel{^}{\text{CVaR}}}_{0.9}^{t}$ estimates the $V$ test with statistics ${V}_{1}$ , ${V}_{2}$ , $V$ is used. If the estimates are good the statistics, ${V}_{1}$ , ${V}_{2}$ , $V$ are close to zero. Table 3 presents the results of the analysis of the dynamic risk measures estimates for the $\text{SACF}$ method and Table 4 for the standard method.

The analysis of the results shows that the method $\text{paramdistr}$ (on the assumption of the normal distribution of residuals) gives the best ${\stackrel{^}{\text{VaR}}}_{0.9}^{t}$ estimates for both methods: $p-$ values of statistics are essentially more than 0.1. This is consistent with the results of the basic analysis (Table 1) and is confirmed by the results of the Jarque-Bera test [2] conducted for ${Z}_{t}$ (5.624 < 5.649 and 4.38 < 5.649 for both proposed and standard methods, respectively). At the same time, all estimates obtained with $POT\_\text{emp}$ method show the poor quality. The popular historical simulated method $\left(\text{hist}\right)$ gives quality estimates only with the $\text{SACF}$ method. In addition all values of statistics for the $\text{SACF}$ method are greater than the appropriate values for the standard method. V-test shows good results for both $\text{SACF}$ and standard methods.

The built models are used for dynamic risk measures forecasting. Forecasting procedure is performed on the window length equal to the half of the general sample power (843 values). 5-day $\left(P=5\right)$ forecast is built (see Figure 2). Thus, it is assumed that the parameters of the model are adequate for a period 5 (or more) days, the estimates of static risk measures at the forecast horizon are unchanged.

The forecasting procedure (steps 2 - 6) is repeated 168 times, and each time 5 new values (the accumulation window) are added. Table 5 presents minimum ${H}_{\mathrm{min}}$ , maximum ${H}_{\mathrm{max}}$ and average ${H}_{mn}$ values of the Hurst parameter obtained for the windows.

Table 5 shows, the values of the Hurst parameter confirm the long-range dependence for all windows (all values are essentially greater than 0.5). The minimum range of values (about 7%) shows stability of this characteristic.

Table 3. The results of the analysis of the dynamic risk measures estimates (SACF method).

Table 4. The results of the analysis of the dynamic risk measures estimations (standard method).

Table 5. Hurst parameter estimates.

Figure 3 demonstrates the results of variance forecasting using (6) with SACF and standard methods.

Visual comparison of the predicted and real values shows that the proposed new method better describes the dynamic behavior of the time series. Extreme values obtained with the new method are much closer to real values. The new method also exhibits less lag in extreme values determination. This can be explained by the fact that the new method uses the ACF prediction and takes into account the property of the long-range dependence. It should also be noted that the optimization procedure in the determination of the Hurst parameter has significantly improved the forecast stability.

Minimum, maximum and average values of the static risk measures for different windows are shown in Table 6 (the $\text{SACF}$ method) and Table 7 (the standard method).

Table 6 and Table 7 can be used to compare the quality of static risk measures estimations obtained by $\text{SACF}$ and standard methods. The range of val-

Figure 3. Real values of variance for real data and forecast estimates obtained by $\text{SACF}$ and standard methods.

Table 6. ${\stackrel{^}{\text{VaR}}}_{0.9}\left(Z\right)$ , ${\stackrel{^}{\text{CVaR}}}_{0.9}\left(Z\right)$ estimates for different windows (SACF method).

Table 7. ${\stackrel{^}{\text{VaR}}}_{0.9}\left(Z\right)$ , ${\stackrel{^}{\text{CVaR}}}_{0.9}\left(Z\right)$ estimates for different windows(standardmethod).

ues (max-min) for the $\text{SACF}$ method is less than the range of values for the standard method due to the fact that the proposed method explicitly uses the smoothing procedure of ACF and as a result the distribution function is more stable.

The obtained results are used to get the time series of dynamic risk measures estimates (3). As an example Figure 4 shows the forecast estimates for ${\stackrel{^}{\text{VaR}}}_{0.9}^{t}$ and ${\stackrel{^}{\text{CVaR}}}_{0.9}^{t}$ , obtained with the use of the paramdistr method.

The prediction errors of ${\stackrel{^}{\text{VaR}}}_{0.9}^{t}$ and ${\stackrel{^}{\text{CVaR}}}_{0.9}^{t}$ for both methods (for different methods of static risk measures estimating) are shown in Table 8 and Table 9.

Table 8 and Table 9 show that the prediction errors obtained for the $\text{SACF}$ method is less than the prediction errors obtained for the standard method. This proves the advantage of the proposed method. In addition Table 8 shows that methods $\text{hist}$ and $\text{paramdistr}$ gives the best estimates. This once again confirms the previously accepted hypothesis of data normal distribution.

The quality of built ${\stackrel{^}{\text{CVaR}}}_{0.9}^{t}$ forecast estimates is analyzed with BPoE test. Table 10 shows the BPoE values obtained for the initial data $\left(\text{real}\right)$ and for the

Figure 4. Forecast estimates for ${\stackrel{^}{\text{VaR}}}_{0.9}^{t}$ and ${\stackrel{^}{\text{CVaR}}}_{0.9}^{t}$ for 168 windows.

Table 8. The prediction errors of ${\stackrel{^}{\text{VaR}}}_{0.9}^{t}$ and ${\stackrel{^}{\text{CVaR}}}_{0.9}^{t}$ (SACF method).

Table 9. The prediction errors of ${\stackrel{^}{\text{VaR}}}_{0.9}^{t}$ and ${\stackrel{^}{\text{CVaR}}}_{0.9}^{t}$ (standard method).

Table 10. Results of BPoЕ-test for ${\stackrel{^}{\text{CVaR}}}_{0.9}^{t}$ .

forecast estimates with the use of the new method $\left(\text{SACFmeth}\right)$ and the standard method $\left(st\text{meth}\right)$ . These values are compared with the chosen level of risk measures $\alpha =0.9$ . The results show the high quality of the forecast estimates ${\stackrel{^}{\text{CVaR}}}_{0.9}^{t}$ obtained by the new method.

Table 10 may be used to liken the methods used for forecasting by comparing the value of confidence level $\alpha $ for predicted and real risk measures values. The standard method based on $\text{paramdistr}$ demonstrates the best results. At the same time, the proposed method shows the best results with the historical simulation method $\left(\text{hist}\right)$ . Table 10 shows that the deviation of $\alpha $ for the new method $\left(0.84\%\right)$ is substantially less than the deviation for the standard method $\left(1.8\%\right)$ .

6. Conclusion

In the article, a multi-step procedure for constructing the dynamic risk measures VaR and CVaR forecast is proposed. The procedure is designed for volatile series with the long-range dependence and is based on the heteroscedastic time series model. The optimization procedure for constructing and forecasting of ACF is used to find the model parameters. For the convenience of practical application, the prediction procedure is formulated as an algorithm. To test the proposed algorithm, the risk measures forecast for the time series of daily log return $\text{Nikkey225StockIndex}$ is built. Different tests carried out at different stages of the algorithm confirm the good quality of the obtained estimates.

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