VALUE AT RISK BASED ON ARMA-GARCH AND GARCH-EVT: EMPIRICAL EVIDENCE FROM INSURANCE COMPANY STOCK RETURN

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1 VALUE AT RISK BASED ON ARMA-GARCH AND GARCH-EVT: EMPIRICAL EVIDENCE FROM INSURANCE COMPANY STOCK RETURN Ely Kurniawati 1), Heri Kuswanto 2) and Setiawan 3) 1, 2, 3) Master s Program in Statistics, Institut Teknologi Sepuluh Nopember Jl. Arief Rahman Hakim, Surabaya Indonesia 1) allykoer@gmail.com, 2) heri_k@statistika.its.ac.id, 3) setiawan@statistika.its.ac.id; ABSTRACTS Value at Risk (VaR) is one of the most well known methods in quantifying and controlling risk of a portfolio. It measures the maximum value of risk in a time frame and in an assigned confidence level. Variance Covariance method is the most widely used method due to its simplicity in calculation. However it requires normal distribution assumption to the data. The assumption of normality is invalid when the data series have a fat tailed indicated by high occurrence of extreme event. By using normal distribution assumption, the risk of high quantile is underestimate. This paper will evaluate VaR estimation using ARMA- GARCH and GARCH-EVT to accommodate volatility and extreme event in financial data. The result concludes that GARCH-EVT approach can capture both volatility clustering and extreme value. The different setting of threshold does not give significant effect in estimating VaR. Keywords: return, Value at Risk, EVT, GARCH, Dynamic VaR. INTRODUCTION Value at Risk (VaR) is one of the most well known methods in quantifying and controlling risk of a portfolio. It measures the maximum value of risk in a time frame and in an assigned confidence level (Franke, Hardl, & Hafnere, 2011). There are three common methods in estimating VaR, i.e. Variance Covariance Method (VCM), Historical Simulation (HS), and Monte Carlo Simulation (MCS). The VCM is the most widely used method due to its simplicity in calculation. However it requires normal distribution assumption to the data. The assumption of normality is not valid when the data series have a fat tailed which is indicated by high occurrence of extreme event. By using normal distribution assumption, the risk of high quantile is underestimate (Bensalah, 2000). In the recent studies, ARMA-GARCH approach is applied in dealing with clustered volatility occurred in financial data. This approach is more complicated compared with the VCM, since it needs to model time series data to estimate mean parameter, and to model the presence of heteroscedastic in variance using GARCH. However, ARMA-GARCH model is less sensitive to capture the existence of extreme value. Studies performed by Gencay et.al (2003) used Extreme Value Theory (EVT) approach to model VaR of extreme events wihtout considering the volatility in the data. To capture the presence of volatility and extreme event, this study will use dynamic VaR proposed by McNeil dan Frey (2000). It combines GARCH model to estimate the current volatility and EVT to estimate the tail of innovation distribution A-1-1

2 of the GARCH model. EVT method in this study is using Peak Over Threshold (POT). This approach has beed used by Singh, Allen and Powell (2011) to estimate VaR and found that it outperformed other methods. This study will be applied into stock return data of PT. Lippo General Insurance (LPGI) during period of January 2006 until March VaR of 1-day ahead forecast will use sliding window method with window size =1000. To find out the more accurate method in estimating VaR, both approach will be compared by using backtesting method. METHODOLOGY Value at Risk using ARMA-GARCH Approach Value at Risk estimation using ARMA-GARCH follow 2 steps. The first is modelling ARMA to time series data to estimate the mean value, and the second is modelling GARCH to model the presence of heteroscedasticity in ARMA model to estimate the variance. ARMA(p,q) model is a combination of AR(p) and MA(q) to model stationary time series data. AR(p) model explain that t- th observation is influenced by p -th observation before. While MA(q) model explain that t th observation is influenced by q th error before. The common form of ARMA (p,q) is as follow (Wei, 2006): φ ( B)( Y µ ) = θ ( B) ε, (1) p t q t p dan q : order of AR and MA model j B : backward shift operator which defined BYt = Yt j, φ p ( B ) : AR(p) component which can be write, p φp( B) = 1 φ1b φpb, θ q ( B ) : MA(q) component which can be write, q θq( B) = 1 θ1b θqb, µ : Mean of { Y t } series, and ε t 2 : White noise error with mean 0 and varians σ ε. A GARCH (generalized autoregressive conditionally heteroscedastic) model uses values of the past squared observations and past variances to model the variance at time t. GARCH(p,q) model can be written as follow (Wei, 2006): p dan q : order of GARCH model h t : variance at time t ht 1 : variance at time t 1 ε : error at time t 1 t 1 h = α + αε + βh, (2) 2 t 0 1 t 1 1 t 1 After modelling ARMA and GARCH, VaR calculated as follow : VaR 1 where F ( α ) is q-th quantile ( q 1 α ) 1 ( ) ˆ F ( ) α = µ + α ˆ σ (3) t t t = of standard normal distribution function, ˆt µ is estimated using ARMA model and ˆt σ is estimated using GARCH (Gencay & Selcuk, 2004). A-1-2

3 Value at Risk using GARCH-EVT Approach McNeil and Frey (2000) proposed a dynamic VaR forecasting method using EVT. The method is implemented in following two steps. Step (1) GARCH(1,1) model is fitted to data which will produce residual for step (2) and also 1 day ahead prediction of µ t + 1 and σ t + 1. Step (2) EVT POT with k = 50 and k = 100 is applied to the residual from step (1), and estimate the POT VaR quantile as follow : VaR t ( α ) ˆ ξ ˆ σ 1 q = z( k+ 1) + 1 ˆ ξ k / n where, z ( k + 1) = threshold ˆ σ = scale estimator ˆ ξ = shape estimator n = sample k = observation exceed threshold Dynamic VaR is calculated using the result from step (1) and step (2) as follow : RESULTS AND DISCUSSION VaR q t+ 1 t+ 1 t (4) = ˆ µ + ˆ σ VaR ( α) (5) Financial data is having volatility clustering and extreme value as well (Tsay, 2005). This study tries to examine whether the statement is true. Empirically, using LPGI stock return, it can be seen from the following Pic 1 that stock return is playing around zero value, which is stationary. Besides that, the fluctuation in this plot also confirm the presence of volatility clustering. This volatility clustering tend to result a heteroscedastic problem, which later will be overcame using ARMA-GARCH and GARCH EVT approach in calculating VaR LReturn Time Pic 1. Clustered Volatility of Stock Return The following Table 1 is showing descriptive statistics of closing price and stock return. We may see that return fluctuated from -67,79% to 59,25%. This range indicate high variation. From the value of kurtosis (12,59) shows that there is positif excess kurtosis which indicate fat tailness in the distribution of data. A-1-3

4 Table 1. Descriptive Statistics Price and Stock Return Table 2 shows VaR estimation of each approach. GARCH EVT is showing positive value regarding loss which is caused by the use of negative return in analysis to capture the extreme value above assigned threshold ( k = 50 and k = 100). GARCH-EVT give a higher estimate in each quantile compared to ARMA-GARCH. Table 2. VaR Estimate of LPGI Stock Return Picture 2 below shows plot of VaR estimate of LPGI stock return. Loss is shown by those value plotted below zero line, and gain is plotted above zero. This plot is also reveal that EVT-GARCH (red and purple line) can capture volatility and extreme value as well. ARMA- GARCH approach (green line) is able to capture the volatility but still under GARCH-EVT in accomodating extreme event. Different with the result of 0,5% and 1% quantile, ARMA- GARCH seems to have a similar performance with GARCH-EVT to accommodate extreme value using quantile 5%. The different threshold value ( k = 50 and k = 100 ) in EVT-POT is resulting almost similar estimation of VaR, shown by red line and purple line. A-1-4

5 Pic 2. Loss (bottom) and Gain (top) for LPGI Stock Return using ARMA-GARCH (green line), EVT; k = 50 (purple line) and EVT; k = 100 (red line) at Quantile 0,5%(a), 1%(b) and 5%(c) A violation based backtesting method (McNeil and Frey, 2000) is used in this study to find out the more accurate approach. For the next day predicted quantile r ˆq and actual return r t + 1, a violation is said to occur if r ˆ 1 > r, i.e. the actual lost is greater than the forecasted VaR. If t+ q A-1-5

6 q is the quantile for VaR, the estimated number of violation is given by (1 q) of Total Predictions (Trials). Backtesting result is presented in Table 3. It can be said that for quantile 0,5% and 1%, GARCH-EVT approach is giving more accurate 1-day VaR forecast. While for quantile 5%, ARMA-GARCH is slightly more accurate compare to GARCH-EVT. Table 3. Backtesting Result of VaR Forecast CONCLUSIONS AND RECOMMENDATIONS Based on the result of all research process, this research concluded that GARCH EVT approach can capture both volatility clustering and extreme value as well. The different setting of threshold does not give significant effect in estimating VaR. Some recommendations for future research: since this study is only using univariate VaR, it is needed to analysis VaR by involving other influenced variable. For example is the influence of competitor, or the possibility of macroeconomic changing. REFERENCES Bensalah, Y. (2000). Steps in Applying Extreme Value Theory to Finance : A Review. Bank of Canada Working Paper Franke, J., Hardl, W. K., & Hafnere, C. M. (2011). Statistics of Financial Markets. Berlin: Springer. Gencay, R., & Selcuk, F. (2004). Extreme value theory and Value-at-Risk: Relative performance in emerging markets. International Journal of Forecasting, Gencay, R., Selcuk, F., & Ulugulyagci, A. (2003). High volatility, thick tail and extreme value theory in value at risk estimation. Insurance: Mathematics & Economics, A-1-6

7 McNeil, A. J., & Frey, R. (2000). Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach. Journal of Empirical Finance, Setiawan, S. (2013). Prospek dan Daya Saing Sektor Perasuransian Indonesia Di Tengah Tantangan Integrasi Jasa Keuangan ASEAN. In K. Tulisan, Serial Analisis Kebijakan Fiskal: Penguatan Hubungan Ekonomi dan Keuangan Internasional dalam Mendukung Pembangunan Nasional. Jakarta: Nagamedia. Singh, A. K., Allen, D. E., & Powell, R. J. (2011). Value at Risk Estimation Using Extreme Value Theory. 19th International Congress on Modelling and Simulation. Perth: Tsay, R. S. (2005). Analysis of Financial Time Series - 2nd Ed. Canada: Wiley-Interscience. Wei, W. W.S. (2006). Time Series Analysis, Univariate and Multivariate Methods. Pearson. A-1-7