A comparison of GARCH models for VaR estimation in three different markets.

Transcription

1 Uppsala University Department of Statistics Andreas Johansson, Victor Sowa Supervisor: Lars Forsberg A comparison of GARCH models for VaR estimation in three different markets..

2 Abstract In this paper the value at risk (VaR) forecasts are compared using three different GARCH models; ARCH(1), GARCH(1,1) and EGARCH(1,1). The implemented method is a one-day ahead out of sample forecast of the VaR. The forecasts are evaluated using the Kupiec test with a five percent significance level. The focus is on three different markets; commodities, equities and exchange rates. The goal of this thesis is to answer which of the models; ARCH(1), GARCH(1,1) and EGARCH(1,1) is best at forecasting the VaR for commodities, equities and exchange rates. Which assumed distribution for the conditional variance performs the best? Is the normal distribution or the Student-t a better option when forecasting VaR? The results shows that the ARCH(1) and EGARCH(1,1) model specifications are good options for forecasting the VaR for the chosen securities. These models give rise to a statistical significant forecast for the VaR with only one exception, the exchange rate between the SEK and USD. The worst performing model is the GARCH(1,1), which showed no significant results for any security. The normal distribution is the preferred conditional distribution assumption. In some securities the Student-t distribution shows a marginally better result, but the normal distribution is also a valid option in those cases. Keywords: ARCH, GARCH, EGARCH, Value at Risk, Volatility and Forecasting.

3 Contents 1. Theory Introduction Value at Risk (VaR) ARCH/GARCH models Earlier Results Data/Method Data Descriptive statistics Test of the data Moving out of sample Forecast Results/Discussion ARCH(1) results GARCH(1,1) results EGARCH(1,1) results Conclusion Summary Recommendations for investors Acknowledgments References Appendix Return series Parameter Estimation Comparison of ARCH(1) forecasts vs. the unconditional variance... 27

4 1. Theory 1.1 Introduction All investors try to maximize their returns on investments while minimizing risks; therefore the perceived risk is a key ingredient in all investment decisions. The investor will require a risk premium to invest in risky assets; hence the required returns are higher the more risk a specific investment contains 1. Each investor must choose his or her portfolio carefully since a large investment in a negatively returning security will generate a large loss and a too small investment in a good security will generate an opportunity cost. To manage risk is therefore crucial for maximizing the portfolios return while minimizing losses. A frequently used risk measure is the value at risk (VaR), which measures the likelihood that a portfolio will face its worst case outcome over a predetermined time period and at a predefined confidence level (Angelidis et al., 2004). One of the reasons why the VaR is such a prevalent method to estimate risk is due to the regulatory framework created by the Basel committee on Banking Supervision during the nineties. This regulatory framework forces the banks to calculate VaR for their portfolios, preventing them from taking on too much financial risk (Basel III, 2010). VaR is a relative simple concept, which is easy to implement and use for everyday investors. For the calculation of VaR an investor needs to estimate the securities volatility, i.e. risk. Empirical studies have concluded that financial instruments have heteroscedasticity in the variance. To address this observation the autoregressive conditional heteroscedasticity model (ARCH) and the general autoregressive conditional heteroscedasticity model (GARCH) were introduced by Engle (1982) and Bollerslev (1986). The family of GARCH models captures the changing volatility over time, since they are conditional upon heteroscedasticity (Orhan and Köksal, 2011). Since the initial introduction of the ARCH and GARCH over 100 new varieties of these have emerged (Bollerslev, 2010). However there is no definite answer to which of the models from the GARCH family that is the best at forecasting the volatility for all types of financial data. Due to the plethora of different GARCH models available the models that have been examined need to be restricted. This thesis will focus on three of the most commonly used, 1 True if investors are assumed to be rational and risk averse. 1

5 which are; the ARCH(1), GARCH(1,1) and the EGARCH(1,1) (Bollerslev et al., 2010). The three models will be used to estimate the VaR with a one-day ahead forecast horizon. Nine different securities representing three different markets (commodities, equities and currencies) in the time period June April 2013 are used to evaluate the VaR for the different models. The goal of this thesis is to answer: Which of the models; ARCH(1), GARCH(1,1) and EGARCH(1,1) is best at forecasting the VaR for commodities, equities and exchange rates? Which assumed distribution for the conditional variance performs the best? Is the normal distribution or the Student-t a better option when forecasting VaR? The outline of this paper is as follows. In Section 1 the theoretical framework is presented along with some earlier results. In Section 2 we describe the data and method that is used. In Section 3 the main results is discussed and Section 4 contains the concluding remarks. 1.2 Value at Risk (VaR) ( ) is defined as the upper limit of the left tail of the assumed distribution i.e. the limit that is to be violated in percent of the time (Lopez, 1999), this is called the nominal size. A violation is said to occur when the daily loss is larger than the VaR. In a perfectly specified model this violation should occur with percent probability (Figure 1). The observed probability of a violation is called the empirical size. A good estimate of VaR should result in an empirical size that is close to the nominal size. An underestimated VaR would trigger the investors to take on too much risk, while an overestimated VaR would make the investors too restrictive with their capital and miss potential profits, i.e. opportunity costs (Orhan and Köksal, 2011). 2

6 Figure 1 VaR for a security. VaR is the minimum amount that will be lost with a probability of τ. (τ = 0.05 in the figure). The probability function for VaR can be written as ( ( )) (1) where is the daily return. ( ) is defined as ( ) (2) where is the mean of the returns, is the assumed distributions critical value 2 with area and is the volatility obtained from the model estimate (Orhan and Köksal, 2011). The ( ) is the minimum that will be lost with the frequency of. For example, if $ is invested in a security with a VaR of 4 percent and a of 5 percent. Then the security would lose a minimum of $4000 five percent of the time. 2 For a normal distribution this would correspond to 1.65 for. 3

7 In this thesis VaR is evaluated using a likelihood ratio test developed by Kupiec (1995). This test allows the empirical size to be tested against the nominal size. The test statistic for this test is the likelihood ratio (( ) ( ) (( ) ( ) (3) where V is the total number of violations in the time period and hence is the empirical size. This can be found using the indicator function { ( ) ( ) (4) where receives a value of one if the daily loss is more than the VaR. The total number of violations are. In a perfectly specified model the probability for a violation is, creating the null hypothesis that, i.e. the empirical size is equal to the nominal size. The more deviates from the larger the Kupiec score (K) will become. K is known to belong to a chi-square distribution with 1 degree of freedom (Orhan and Köksal, 2011). The commonly used 5 percent significance level is used in this thesis, i.e. if the K value is larger than 3.84 the null hypothesis will be rejected. If the null hypothesis is rejected, the specific model is not a suitable specification to estimate the VaR for that security. There are some flaws in the Kupiec test. Firstly, the test does not take the sequence of violations into account. If the violations are not independent it may cause a researcher to underestimate the risk in times of economic uncertainty (Orhan and Köksal, 2011). Secondly, the Kupiec score is not affected by how large the violation is. This means that a 1 percent violation or a 4 percent violation will have the same weight (Lehar et al., 2002). In this thesis the volatility is estimated by using three different types of ARCH/GARCH models. 4

8 1.3 ARCH/GARCH models Different types of autoregressive conditional heteroscedasticity models (ARCH), first introduced by Engle (1982), are commonly used to estimate risk. The ARCH model expanded into the generalized autoregressive conditional heteroscedasticity (GARCH) model by Bollerslev (1986). These models capture the fluctuations in variance over time, which are present in most financial instruments. Another empirical observation is that the variance is usually higher during times of turmoil. Nelson (1991) created the exponential GARCH (EGARCH) model to capture this tendency. An EGARCH model allows positive and negative shocks 3 to have different effects on the estimated variance. The three different model specifications are presented below. The return of a security is defined as ( ) (5) where is the closing price of the security day t. The return is assumed to consist of two parts, a predictable and an unpredictable part ( ) (6) where is all available information at time t-1, which give rise to the expected return in time period t (Angelidis et al., 2004). The symbol is the unpredictable return. The conditional return is considered an autoregressive process ( ) (7) The assumption is that the market price includes all available information up to that point, as explained by the efficient market hypothesis (Lee, 2006: 103). 3 Positive and negative shocks are often referred to as positive and negative news in the literature. 5

9 The unpredictable part of the returns can be expressed as (8) where is the time varying volatility and is iid with mean zero and a unit variance, ( ). Research has shown that most securities have a leptokurtic unconditional distribution (fat tails and peaked). In this thesis the normal distribution will be compared to the Student-t distribution, which has a fatter tail. Observe that just because the unconditional variance is leptokurtic and skewed do not mean that the conditional variance need to have the same attributes. The ARCH(q) model is defined as (9) where,,, the series is stationary if. The ARCH model creates a process where today s variance depends on its own previous variance. This allows the model to capture the volatility clustering observed in financial markets. The parameter explains how fast the model reacts to news on the market. The one step ahead forecast for the ARCH(1) model is done by using the equation, (10) The GARCH(p,q) model adds a moving average term, making it similar to a regular ARMA(p,q) process. This allows a slower decay in variance from random shocks which is more coherent with observed data (Teräsvirta, 2009). The definition of the GARCH(p,q) model is (11) 6

10 where,,,,. The process will be stationary if. If the stationarity condition is fulfilled the conditional variance will converge towards the unconditional variance model reacts to news on the market while heteroscedasticity is over time. If the. The parameter again explains how fast the ( ) states how persistent the conditional parameter is large, effects from economic news in the market will have a tendency to linger. The GARCH(1,1) is the most used model specification, often used as a benchmark model within this area. The one step ahead forecast for the GARCH(1,1) model is done by using the equation, (12) The EGARCH(p,q) model captures the asymmetric effect on variance from positive and negative news (Nelson, 1991). From empirical data the market volatility seem to react differently depending on the sign of the shocks, negative shocks usually results in periods of higher volatility compared to positive news (Nelson, 1991). By including a third parameter the EGARCH allows the model to react differently depending on the different type of news. The EGARCH model is defined as ( ) ( ( )) ( ) (13) where and ( ) will depend on the assumed distribution, for a normal distribution ( ). If ( ( )) the market is returning less than expected, clearly a negative shock. If the estimation shows that it implies that the model is symmetric. However, if the estimation shows that it will imply that negative news cause a higher future volatility than a positive, hence the model is asymmetric. The EGARCH model differs from the ARCH and GARCH models because the logarithm of the variance is what is being estimated. By taking the logarithm of the conditional variance it ensures a positive value. The logarithm also relaxes the parameters constraint; they no longer need to be positive. and are still expected to have positive values. It is troublesome for inference and forecasting if they are negative. The however is expected to have a negative value, 7

11 which means that a negative shock in the market will increase the future volatility ( MathWorks, 2013). The EGARCH model is stationary if The one step ahead forecast for the EGARCH(1,1) model is done by using the equation, ( ) ( ( )) ( ). (14) 1.4 Earlier Results Research in this area has been extensive. There is no definitive conclusion on which model in the GARCH family that is best suited to forecast the volatility for specific types of securities. Most research has shown that a leptokurtic conditional distribution is better at describing the VaR (Angelidis et al., 2004; Orhan and Köksal, 2011; Köksal, 2009; Aloui and Mabrouk, 2010). Usually the normal distribution is compared against Student-t or the generalized error distribution (GED). There is a big difference in which specific GARCH model that performs the best. Orhan & Köksal (2011) showed that the ARCH(1) model was the best one for estimating risk for currencies compared to 15 other GARCH models. Köksal (2009) and Hansen & Lunde (2005) showed that more sophisticated models outperformed the GARCH(1,1) model when forecasting the ISE-100 index and the DM-$ exchange. A majority of research have concluded that models, which take information asymmetry into consideration, outperform symmetric models (Hansen and Lunde, 2005; Aloui and Mabrouk, 2010). When compared to other families of volatility models the GARCH models outperforms others such as Black and Scholes models, stochastic volatility models, regime switching models and grey theorem (Lehar et al., 2002; Kung and Yu, 2008; Luo et al., 2010; Pederzoli, 2006). Ultimately, the GARCH models are still the preferred choice when forecasting VaR. More recent research has shown that a better estimate of the variance is to use the realized variance instead of the squared returns (Andersen and Bollerslev, 1998; Hansen et al., 2012). This is however outside the scope of this paper and we will use the squared returns as an estimate for the daily variance. 8

12 CURRENCIES EQUITIES COMMODITIES 2. Data/Method All calculations are done in Matlab R2012b Simulink version and the MFE toolbox ( MFE Toolbox, n.d.). 2.1 Data The data that have been used were downloaded from Thomson Reuters Datastream ( Datastream, n.d.). Financial instruments are chosen to cover different parts of the market (Table 1). The datasets are divided into three groups; commodities, equities and currencies. Equities have been selected to represent some of Sweden s largest companies. Security Corn No2 Yellow Crude Oil-WTI Gold Bullion Hennes & Mauritz VOLVO Ericsson SEK to EUR SEK to GBP SEK to USD Description Corn future Is a grade of crude oil used as a benchmark in oil pricing. Exchange traded commodity backed by physical gold (USD) Is a multinational retail-clothing company. Manufacturer of cars, trucks, buses and construction equipment. Company that provides telecommunication equipment. Exchange rate between SEK and EUR Exchange rate between SEK and GBP Exchange rate between SEK and USD Table 1 - The datasets used in this paper to compare the different conditional volatility models. The data is from the start of June 2009 until the start of April Only trading days are included in the samples. Each model is estimated using the adjusted closing price for the last 1000 trading days for each security 4. The data used in the estimates are from the start of June 2009 up to the start of April 2013 for each dataset. 4 First 500 observations used for the starting estimate, followed by the 500 one-step-ahead forecasts. 9

13 2.2 Descriptive statistics n Mean std Min Max Kurtosis Skewness Corn No2 Yellow Crude Oil-WTI Gold Bullion Hennes & Mauritz VOLVO Ericsson SEK to EUR SEK to GBP SEK to USD Table 2 - Descriptive statistics. All calculations are done for the last 1000 observations for each dataset, capturing the first 500 observations used for the first model estimation plus the 500 rolling forecasts. (Everything is calculated using percentage points 1 = 1%). Figure 2 Two examples of the returns. Each plot uses the last 1000 observations for each dataset, capturing the first 500 observations used for the first model estimation plus the 500 rolling forecasts. The graph for each security is available in Appendix 8.1. Each dataset has a mean return very close to zero and all the time series shows signs of heteroscedasticity, as expected (Figure 2). Among the commodities and equities there are large differences between the minimum and maximum values. This is in contrast with the relative small maximum and minimum values in the exchange rates (Table 2). The standard deviations are similar between the different securities though, which suggest that extreme values are more common among commodity and equity prices than exchange rates. All securities have a positive excess kurtosis suggesting that they have leptokurtic distributions. Most of the securities also have skewed distributions, some are positive and some are negative. This is a bit surprising when earlier work has shown that securities have a negative skewness in returns (Orhan and Köksal, 2011). 10

14 2.3 Test of the data The data is plotted and compared to the normal distribution. As expected the returns seem to have a leptokurtic distribution (Figure 3). No conclusion about the overall sign of the skewness can be concluded by looking at the graphs, some show positive skewness and some have negative skewness. Figure 3 Example of the unconditional distributions in the different datasets. Only two securities are presented here but the others can be provided upon request (lewkiz@yahoo.com). The ticks on the x-axis represent one standard deviation each. The returns are modeled as in Eq. 6, and then the residuals are tested for autocorrelation. There is no significant autocorrelation present in the residuals 5. The squared-residuals showed significant autocorrelation on the earlier lags, which suggests that the variance could be modeled using an autoregressive process. By using the Jarque Bera test with a significance level of five percent it showed that the assumption of normality was rejected in each of the time series. The Engle test with 5 lags rejects the null hypothesis in all samples, suggesting that there is an ARCH effect present in the samples. To summarize, a GARCH model should be able to capture the structure of these time series. 5 There are some significant higher level lags but those are ignored when they do not have any economic effect. 11

15 2.4 Moving out of sample Forecast Figure 4 Flowchart of the method used to compare the VaR between the models. Each model, with the two different conditional distribution assumptions is estimated using a sample size of 500 observations, the estimation window. Each model is estimated 500 times each, moving the estimation window one step forward each time. Each model estimate is used to do a one-step-ahead forecast of the VaR. The forecasted VaR is then compared to the actual observed value in the next time period. The violations, larger losses than the predicted VaR, are summarized and the Kupiec score is calculated to compare the different models. 12

16 A common way to test the predictive power of a model is to do a sequence of out of sample forecasts (Figure 5). The estimation window, in the initial stage, consists of the first 500 days of our sample. With those 500 days the 501st day s variance is forecasted and compared to the observed variance for that day. The observed value for the 501st observation is then included while the first observation in the initial estimation window is dropped, making the estimation window again consist of 500 observations but now from the second to the 501st day and the 502nd is forecasted. This is done until the entire forecast window of 500 observations has been forecasted (Figure 4). Figure 5 A rolling forecast is estimated for each dataset. 500 observations are used for the model estimate, which is then used to predict the next day s variance. Estimated VaR is then compared with the actual observed returns for the next period. Then the rolling forecast is moved one step forward and the procedure is repeated. A total of 500 predictions are calculated for each model and the number of violations, R t+1 < VaR, is summarized. Each model is estimated using the three different GARCH models. The estimated model is then used to make an out of sample prediction, which is compared to the observed value (Hansen and Lunde, 2005). 3. Results/Discussion 3.1 ARCH(1) results Each distribution is calculated by using a one-step-ahead out of sample forecast that is compared to the observed value. The expected VaR for next time period can then be evaluated by comparing the nominal size against the empirical size. The Kupiec test shows that the null hypothesis,, cannot be rejected with a 5 percent significance for the ARCH(1) models with one exception, the exchange rate between 13

17 CURRENCIES EQUITIES COMMODITIES SEK and USD (Table 3). Comparing this with the Student-t distribution where the null hypothesis is rejected in all except three securities. This implies that the normal distribution is a more accurate conditional distribution assumption than the Student-t when predicting VaR for these securities. This is inconsistent with earlier results that have shown that the Student-t is overall better at forecasting VaR, since it captures the leptokurtic distribution in returns 6. Corn No2 Yellow Crude Oil-WTI Gold Bullion NORMAL STUDENTS NORMAL STUDENTS NORMAL STUDENTS V/N 3.8% 2.4% 4.6% 3.0% 6.2% 3.2% Kupiec Hennes & Mauritz VOLVO Ericsson NORMAL STUDENTS NORMAL STUDENTS NORMAL STUDENTS V/N 4.8% 2.6% 6.2% 4.6% 5.0% 2.4% Kupiec SEK to EUR SEK to GBP SEK to USD NORMAL STUDENTS NORMAL STUDENTS NORMAL STUDENTS V/N 5.2% 4.0% 3.4% 3.2% 2.6% 2.6% Kupiec Table 3 - ARCH(1) out of sample forecast results. The VaR is estimated using a nominal size of five percent, τ = This means that the null hypothesis, H 0 : V/N = τ, is rejected at a five percent significance level if the Kupiec score is larger than 3.8 (Acceptance of the null hypothesis is green and rejection is red). The ARCH(1) model seem to result in a rigid one step ahead forecast, it only seem to react during periods of large changes in variance (Figure 6). All the ARCH(1) models have a relative small α 1 parameter value, so only large shocks will have an effect on the forecast (Appendix 8.2). It results in a forecast very close to the unconditional variance (Appendix 8.3). 6 Just because the unconditional distribution shows a leptokurtic distribution does not mean that the conditional distribution need to have the same distribution though. 14

18 CURRENCIES EQUITIES COMMODITIES Figure 6 Two examples of the ARCH(1) out of sample forecast with the normal distribution assumption. The red dots represent the forecasted values and the blue line is the observed variance 7. Note that the last 500 observations are used for the one-step-ahead forecast for each dataset. 3.2 GARCH(1,1) results In comparison with the ARCH(1) model the GARCH(1,1) seems to be a worse option when forecasting VaR. The null hypothesis is rejected in all Kupiec tests. This implies that the GARCH(1,1) is bad at forecasting the VaR for the selected securities (Table 4). Corn No2 Yellow Crude Oil-WTI Gold Bullion NORMAL STUDENTS NORMAL STUDENTS NORMAL STUDENTS V/N 12.4% 13.2% 8.8% 9.0% 16.2% 13.6% Kupiec HENNES & MAURITZ VOLVO ERICSSON NORMAL STUDENTS NORMAL STUDENTS NORMAL STUDENTS V/N 11.4% 8.4% 16.4% 15.6% 11.6% 13.6% Kupiec SEK to EUR SEK to GBP SEK to USD NORMAL STUDENTS NORMAL STUDENTS NORMAL STUDENTS V/N 12.0% 11.2% 14% 13.0% 16.4% 16.0% Kupiec Table 4 - GARCH(1,1) out of sample forecast results. The VaR is estimated using a nominal size of five percent, τ = This means that the null hypothesis, H 0 : V/N = τ, is rejected at a five percent significance level if the Kupiec score is larger than 3.8 (Acceptance of the null hypothesis is green and rejection is red). 7 The other print outs are available upon request (lewkiz@yahoo.com). 15

19 Looking closer at the empirical size,, the GARCH(1,1) model underestimates the potential variance in the next period (Table 4). The graph of the predicted values show that the forecasted variance is very close to today s value, meaning that whenever the variance deviates from today s variance the forecast fails (Figure 7). Figure 7 - Two examples of the GARCH(1,1) out of sample forecast with the normal distribution assumption. The red dots represent the forecasted values and the blue line is the observed variance 8. Notice that all the predicted values are shifted one step to the right of the observed variance. In contrast to the ARCH(1) model there is no clear conclusion on which distribution is the best in forecasting the VaR in a GARCH(1,1) model. 3.3 EGARCH(1,1) results The graphed forecasts indicates that the EGARCH(1,1) is very good at predicting the next periods variance (Figure 8). This is confirmed by the Kupiec test where the null hypothesis is only rejected in 5 out of 18 models (Table 5). 8 The other print outs are available upon request (lewkiz@yahoo.com). 16

20 CURRENCIES EQUITIES COMMODITIES Figure 8 - Two examples of the EGARCH(1,1) out of sample forecast with the normal distribution assumption. The red dots represent the forecasted values and the blue line is the observed variance 9. There are some mixed results but overall it seems like the normal distribution once again is the better option in predicting the VaR. Under the normal distribution assumption the null hypothesis, empirical size = nominal size, is only rejected in the exchange rate between SEK and USD (Table 5). Corn No2 Yellow Crude Oil-WTI Gold Bullion NORMAL STUDENTS NORMAL STUDENTS NORMAL STUDENTS V/N 3.6% 2.8% 5.4% 4.2% 6.8% 5.4% Kupiec HENNES & MAURITZ VOLVO ERICSSON NORMAL STUDENTS NORMAL STUDENTS NORMAL STUDENTS V/N 4.8% 4.2% 6.4% 4.0% 4.2% 2.6% Kupiec SEK to EUR SEK to GBP SEK to USD NORMAL STUDENTS NORMAL STUDENTS NORMAL STUDENTS V/N 4.4% 2.4% 6.2% 4.6% 3.0% 1.8% Kupiec Table 5 - EGARCH(1,1) out of sample forecast results. The VaR is estimated using a nominal size of five percent, τ = This means that the null hypothesis, H 0 : V/N =, is rejected at a five percent significance level if the Kupiec score is larger than 3.8 (Acceptance of the null hypothesis is green and rejection is red). 9 The other print outs are available upon request (lewkiz@yahoo.com). 17

21 The parameter estimates are unstable for some of the EGARCH(1,1) models (Appendix 8.2). It is probably due to the logarithmic transformation and the problems that arise when there are periods with zero returns. Sometimes the estimation procedure hit the parameter restrictions, which also contributes to the unstable parameter estimations. 4. Conclusion 4.1 Summary The results shows that the best models to forecast the VaR for these securities are the ARCH(1) and the EGARCH(1,1) model. These models are good options for each security with one exception, the exchange rate SEK to USD. The good results of the EGARCH model confirms earlier results, models that take asymmetry into consideration outperforms symmetric models. Even if the empirical size is very similar to the ARCH(1) model, the EGARCH(1,1) seem to be better at following the currently present variance (Figure 8). This suggest that the EGARCH(1,1) is better at adjusting for the heteroscedasticity in variance in comparison to the ARCH(1) model. The worst performing model is the GARCH(1,1), which showed poor results for all securities. This is a bit surprising when the GARCH(1,1) model is often used as a benchmark in this area of research. Orhan & Köksal (2011) showed that the ARCH(1) model outperformed other GARCH models when looking at exchange rates, so even if it is uncommon that the ARCH(1) outperforms the GARCH(1,1) model it is still plausible. The conditional normal distribution shows better results than the Student-t assumption with few exceptions. This is in contrast with earlier results that mostly have shown that the Student-t is a better option. 18

22 4.2 Recommendations for investors The best performing model for each security is summarized in Table 6. Overall the ARCH(1) and EGARCH(1,1) models are very close in their empirical size. The ARCH(1) model is easier to estimate than the EGARCH(1,1), so for a quick estimate the ARCH(1) model with conditional normal distribution is a good choice 10. The EGARCH(1,1) seem to be better at adjusting for current volatility in comparison to the ARCH(1) model (Figure 8). It means that the EGARCH(1,1) is quicker to adjust when structural breaks happens. Our final recommendation is therefore to use the EGARCH(1,1) model with the appropriate distribution assumption. Model Distribution V/N Corn No2 Yellow ARCH(1) Normal 3.8% ARCH(1) Normal 4.6% Crude Oil-WTI EGARCH(1,1) Normal 5.4% Gold Bullion EGARCH(1,1) Student-t 5.4% Hennes & Mauritz ARCH(1) EGARCH(1,1) Normal Normal 4.8% 4.8% VOLVO ARCH(1) Student-t 4.6% Ericsson ARCH(1) Normal 5.0% SEK to EUR ARCH(1) Normal 5.2% SEK to GBP EGARCH(1,1) Student-t 4.6% SEK to USD EGARCH(1,1) Normal 3.0%* Table 6 The best performing models for VaR estimation for each security. Note that there are very small differences between the ARCH(1) and the EGARCH(1,1) empirical size. *) It is significantly different from the nominal size of 5 percent. 5. Acknowledgments Oscar Andersson, Johan Sowa and Rebecca Tingvall. 10 In these samples the ARCH(1) model makes forecasts very close to the unconditional variance. This suggests that the average variance for the last trading days should give a good estimate for the next period s variance (Appendix 8.3). This implies that an even easier way to forecast the VaR is to use the unconditional variance instead of the ARCH(1) estimate. 19

23 6. References Aloui, C., Mabrouk, S., Value-at-risk estimations of energy commodities via longmemory, asymmetry and fat-tailed GARCH models. Energy Policy 38, Andersen, T.G., Bollerslev, T., Answering the skeptics: Yes, standard volatility models do provide accurate forecasts. International Economic Review 39, 885. Angelidis, T., Benos, A., Degiannakis, S., The use of GARCH models in VaR estimation. STAT METHODOL 1, Badescu, A.M., Kulperger, R.J., GARCH option pricing: A semiparametric approach. Insurance Mathematics and Economics 43, Basel III: A global regulatory framework for more resilient banks and banking systems, Bank for international settlements, Basel, Switzerland. Bollerslev, T., Generalized autoregressive conditional heteroscedasticity. Journal of econometrics 31, Bollerslev, T., Glossary to ARCH (GARCH). Bollerslev, T., Russell, J., Watson, M., Volatility and Time Series Econometrics. Oxford University Press. Datastream [WWW Document], n.d. URL 01/Web.Public/Login.aspx?brandname=datastream&version= &protocol=0 (accessed ). Engle, R.F., Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation. Econometrica 50, Hansen, P.R., Huang, Z., Shek, H. howan, Realized GARCH: a joint model for returns and realized measures of volatility. Journal of Applied Econometrics 27, Hansen, P.R., Lunde, A., A forecast comparison of volatility models: does anything beat a GARCH(1,1)? Journal of Applied Econometrics 20, Kung, L.-M., Yu, S.-W., Prediction of index futures returns and the analysis of financial spillovers A comparison between GARCH and the grey theorem. European Journal of Operational Research 186, Kupiec, P.H., Techniques for verifying the accuracy of risk measurement models. Board of Governors of the Federal Reserve System (U.S.), Finance and Economics Discussion Series: 95-24,

24 Köksal, B., A Comparison of Conditional Volatility Estimators for the ISE National 100 Index Returns. Journal of Economic and social research 11, 1. Lee, C.F., Encyclopedia of finance. Springer, New York. Lehar, A., Scheicher, M., Schittenkopf, C., GARCH vs. stochastic volatility: Option pricing and risk management. Journal of Banking and Finance 26, Lopez, J.A., Methods for Evaluating Value-at-Risk Estimates. Economic Review ( ) 3. Luo, C., Seco, L.A., Wang, H., Wu, D.D., Risk modeling in crude oil market: a comparison of Markov switching and GARCH models. Kybernetes 39, MathWorks [WWW Document], MathWorks. URL (accessed ). MFE Toolbox [WWW Document], n.d. Kevin Sheppard. URL (accessed ). Nelson, D.B., Conditional Heteroscedasticity in Asset Returns: A New Approach. Econometrica 59, Orhan, M., Köksal, B., A comparison of GARCH models for VaR estimation. eswa 2012, Pederzoli, C., Stochastic volatility and GARCH: a comparison based on UK stock data. European Journal of Finance 12, Teräsvirta, T., An Introduction to Univariate GARCH Models, in: Mikosch, T., Kreiß, J.-P., Davis, R.A., Andersen, T.G. (Eds.), Handbook of Financial Time Series. Springer Berlin Heidelberg, pp

25 7. Appendix 7.1 Return series Figure 9 - Time series data of the returns for the first six securities examined in this thesis. Each plot uses the last 1000 observations for each dataset, capturing the first 500 observations used for the first model estimation plus the 500 rolling forecasts. 22

26 Figure 10 Time series data of the returns for the last three securities used in this thesis. Each plot uses the last 1000 observations for each dataset, capturing the first 500 observations used for the first model estimation plus the 500 rolling forecasts. All securities show returns centered around zero and there are signs of heteroscedasticity in each of the time series (Figure 10). 7.2 Parameter Estimation Each model is estimated 500 times, one time per one-step-ahead forecast. To check the parameter values the average of all 500 estimates are presented in Table 7, Table 8 and Table 9. In the ARCH(1) model the autoregressive parameter is relative small, meaning that the last period shock only has a small impact on the next day s variance (Table 7). For example in the dataset SEK to GBP the is only 0.01, meaning that there is almost no effect on the next days variance due to today s shock. 23

27 ARCH - 'Normal' ARCH - 'Student-t' Security α 0 α 1 Security α 0 α 1 Corn Oil Gold H&M VOLVO Ericsson SEK to EUR SEK to GBP SEK to USD 4.26 (0.32) 3.35 (0.27) 1.22 (0.11) 1.98 (0.11) 4.53 (0.37) 3.83 (0.60) (0.05) 0.72 (0.05) (0.07) (0.07) 0.13 (0.04) 0.05 (0.04) (0.01) 0.04 (0.04) Corn Oil Gold H&M VOLVO Ericsson SEK to EUR SEK to GBP SEK to USD 4.61 (0.28) 3.42 (0.14) 1.34 (0.18) 2.26 (0.25) 4.62 (0.40) 3.78 (0.56) (0.05) 0.72 (0.04) (0.05) 0.08 (0.05) 0.24 (0.06) 0.13 (0.04) 0.11 (0.04) (0.04) Table 7 Average parameter values for the 500 estimations of the rolling forecast for each ARCH(1) model. Standard deviation is within parenthesis. The GARCH(1,1) autoregressive parameter is still very small but the moving average term is very large (Table 8). The sum of and β decides how fast a shock returns to the long run variance. This suggests that a shock has a long lasting effect. Also when the β is large, the last days shock has a very strong influence on the next day s variance, meaning that the last shock will have a huge impact on the next day s variance 11. This is very clear when looking at the forecast series in the GARCH(1,1) model. The prediction is usually very close to today s variance (Figure 7). 11 In the forecast. 24

28 GARCH - 'Normal' GARCH - 'Student t' Security α 0 α 1 β Security α 0 α 1 β Corn Oil Gold H&M VOLVO Ericsson SEK to EUR SEK to GBP SEK to USD 0.70 (0.90) 0.99 (0.26) 0.02 (0.007) 0.54 (0.64) 0.13 (0.07) 1.22 (1.8) 0.02 (0.01) 0.07 (0.08) (0.06) 0.04 (0.01) 0.11 (0.04) (0.01) 0.81 (0.21) 0.60 (0.08) 0.94 (0.01) 0.66 (0.31) (0.43) 0.80 (0.07) 0.83 (0.20) 0.92 Corn Oil Gold H&M VOLVO Ericsson SEK to EUR SEK to GBP SEK to USD (0.79) 0.58 (0.28) 0.02 (0.005) 0.29 (0.26) 0.09 (0.05) 0.06 (0.11) 0.02 (0.008) 0.07 (0.08) (0.01) (0.005) (0.01) 0.89 (0.17) 0.74 (0.10) 0.94 (0.01) 0.77 (0.13) (0.04) 0.81 (0.06) 0.82 (0.18) 0.92 Table 8 - Average parameter values for the 500 estimations of the rolling forecast for each GARCH(1,1) model. Standard deviation is within parenthesis. The EGARCH(1,1) parameter estimation showed large differences between different securities. There are also some positive leverage effects, implying that there are some securities where positive news results in higher variance (Table 9). This is in conflict with previous research that has shown that negative news usually induces larger variance. EGARCH - 'Normal' EGARCH - 'Student t' Security α 0 α 1 β γ Security α 0 α 1 β γ Corn Oil Gold H&M VOLVO Ericsson SEK to EUR SEK to GBP SEK to USD 0.85 (0.95) 0.16 (0.26) 0.03 (0.55) 0.25 (0.23) 0.18 (0.53) 0.86 (0.70) (0.72) (0.85) (0.16) 0.06 (0.06) 0.08 (0.06) (0.10) 0.28 (0.08) 0.09 (0.04) 0.12 (0.07) 0.21 (0.06) 0.04 (0.05) (0.67) 0.88 (0.21) 0.35 (0.67) 0.72 (0.31) 0.86 (0.42) 0.37 (0.48) 0.82 (0.26) (0.68) 0.82 (0.38) (0.11) (0.15) 0.03 (0.08) (0.04) 0.04 (0.04) 0.01 (0.04) 0.03 (0.04) 0.06 (0.06) Corn Oil Gold H&M VOLVO Ericsson SEK to EUR SEK to GBP SEK to USD 0.50 (0.76) 0.15 (0.25) 0.06 (0.57) 0.07 (0.09) 0.17 (0.54) 0.04 (0.18) (0.71) (0.88) (0.17) (0.08) 0.02 (0.11) (0.07) 0.18 (0.05) 0.05 (0.05) (0.52) 0.88 (021) 0.48 (0.53) 0.91 (0.13) 0.86 (0.43) 0.96 (0.17) 0.86 (0.25) 0.04 (0.68) 0.75 (0.47) Table 9 - Average parameter values for the 500 estimations of the rolling forecast for each EGARCH(1,1) model (0.09) (0.18) (0.05) (0.04) 0.07 (0.06)

29 The average parameter values seem to be very plausible. Problems arise when each single parameter-estimation is examined. The value of the estimated parameter fluctuates depending on the exact sample used (Figure 11). This fluctuation in parameter values are present in most samples but it is clearly the worst in the EGARCH(1,1) models (Table 9). There are some parameter-value-spikes present in most EGARCH model estimates. This could be due to a couple of reasons. Firstly, the EGARCH uses a logarithmic transformation, which means that periods with zero returns results in undefined values. Secondly, the optimization code hit the hard coded parameter restrictions during some sample estimations, creating artificial boundaries on the parameter values. Finally, there might be some problems with the starting values used in the optimization process, resulting in miss defined maximum likelihood estimates. All models are approximations of reality. A good model should be stable and give rise to similar results no matter which sample that is used. These fluctuations suggest that the EGARCH model might be problematic to use when forecasting the VaR for these securities. This is outside the scope of this paper but could be interesting to investigate in future research. Figure 11 Example of the two worst performing parameter estimation. Both are EGARCH(1,1) models under normal conditional distribution assumption (shows similar results with the Student-t). The other EGARCH models only have a couple of spikes around a baseline. Each line represents the parameter value during each step-ahead forecast of the model. 26

30 CURRENCIES EQUITIES COMMODITIES 7.3 Comparison of ARCH(1) forecasts vs. the unconditional variance The ARCH(1) model makes forecasts very close to the unconditional variance. To test if this is true; the same estimate for the VaR was done using the unconditional variance as the forecast of the next period s variance. As shown by Table 10 the unconditional variance shows similar results as the ARCH(1) model. This is not a surprise when the alpha values are very small in the different ARCH(1) models (Appendix 8.2), meaning that the unconditional variance will have a large impact on the forecasted values. Corn No2 Yellow Crude Oil-WTI Gold Bullion ARCH(1) Avg. Var. ARCH(1) Avg. Var. ARCH(1) Avg. Var. V/N 3.8% 3.6% 4.6% 4.4% 6.2% 6.0% Kupiec Hennes & Mauritz VOLVO Ericsson ARCH(1) Avg. Var. ARCH(1) Avg. Var. ARCH(1) Avg. Var. V/N 4.8% 4.6% 6.2% 6.2% 5.0% 5.4% Kupiec SEK to EUR SEK to GBP SEK to USD ARCH(1) Avg. Var. ARCH(1) Avg. Var. ARCH(1) Avg. Var. V/N 5.2% 4.2% 3.4% 3.4% 2.6% 2.4% Kupiec Table 10 A comparison between the ARCH(1) forecast and the unconditional variance (Average Variance) for the forecast. Both are using the conditional normal distribution assumption and a moving rolling forecast of 500 observations. Kupiec test with 5% significance level is used to evaluate the VaR with a 5% nominal size. H 0 : V/N = τ, green acceptance, red rejection. 27