Introduction Stressed VaR MS&E 444: Project Shubhabrata Sengupta, Lewis Kaneshiro, Alireza Ebrahimi, Milad Sharif In this project we investigate various methods for computing Value-at-Risk (VaR). We use three indicesspx (S&P 500), CCMP (Nasdaq Composite Index) and CDX as our investment options. CDX is composed of 125 equally weighted credit default swaps on investment grade entities, distributed among 6 sub-indiceshigh Volatility, Consumer, Energy, Financial, Industrial, and Technology, Media and Tele-communications. We rst look at VaR calculation when a portfolio has a single asset only. We use various GARCH models, conditional EVT and quantile regression (CAViaR) to calculate VaR. We then construct a portfolio which comprises of all three assets with equal weightings. We use the GO-GARCH model to calculate VaR for this portfolio. For all our VaR calculations, we perform extensive backtesting. VaR calculation using GARCH models for single indices A standard GARCH model assumes normal innovation. As a sanity check for this assumption, we show below the plots for daily log returns for the three indices along with simulated returns for both normal and Students-t distributions. Daily negative log return for SPX (1990 2012) Daily negative log return for CCMP (1990 2012) 0.04 0.00 0.04 1990 03 12 1991 11 05 1993 06 30 1995 02 24 1996 10 19 1998 06 15 Day Simulated normal negative log return for SPX (1990 2012) 0.10 0.00 0.10 1990 03 12 1991 11 05 1993 06 30 1995 02 24 1996 10 19 1998 06 15 Day Simulated Students t negative log return for SPX (1990 2012) 0.4 0.2 0.0 0.2 0.10 0.00 0.10 1990 03 12 1994 08 10 1999 01 08 2003 06 08 2007 11 06 2012 04 05 Day Simulated normal negative log return for CCMP (1990 2012) 0.10 0.00 0.10 1990 03 12 1994 08 10 1999 01 08 2003 06 08 2007 11 06 2012 04 05 Day Simulated Students t negative log return for CCMP (1990 2012) 0.5 0.0 0.5 1990 03 12 1991 11 05 1993 06 30 1995 02 24 1996 10 19 1998 06 15 1990 03 12 1994 08 10 1999 01 08 2003 06 08 2007 11 06 2012 04 05 Day Day Daily negative log return for CDX (2003 2012) 0.2 0.0 0.1 2003 10 30 2005 07 08 2007 03 18 2008 11 24 2010 08 04 2012 04 13 Day Simulated normal negative log return for CDX (2003 2012) 0.10 0.00 2003 10 30 2005 07 08 2007 03 18 2008 11 24 2010 08 04 2012 04 13 Day Simulated Students t negative log return for CDX (2003 2012) 0.5 0.0 0.5 1.0 2003 10 30 2005 07 08 2007 03 18 2008 11 24 2010 08 04 2012 04 13 Day
In each case, we see that the simulated normal distribution does not model the frequency of the unexpectedly high shocks that the actual returns have. The simulated Students-t distribution is a better t for modeling the daily log returns for the indices. We also observe volatility clustering in all three daily returns. This means that GARCH models are a reasonable choice for VaR calculation. The following table shows important tests for normality. The Jarque-Bera test is denoted as JB. The Ljung-Box test for upto 10 lags is denoted as LB. The same Ljung-Box test when done for squared returns is denoted as LB2. The excess kurtosis, skewness and the Jarque-Bera test suggest departure from the normality assumption for all three indices. The Ljung-Box test rejects the null hypothesis that the daily returns are independently distributed. These suggest that various GARCH models with Students-t innovations may t our data well. Index Excess Kurtosis Skewness p value (JB) p value(lb1) p value(lb2) SPX 8.457104 0.2314239 < 2.2 10 16 5.783 10 8 < 2.2 10 16 CCMP 5.739155 0.07623512 < 2.2 10 16 0.05009 < 2.2 10 16 CDX 7.388752 0.4648892 < 2.2 10 16 5.23 10 10 < 2.2 10 16 We t four GARCH based models to the data. The models we consider are HS-GARCH, HS- GARCH-t, HS-APARCH-t and CONDEVT. HS-GARCH ts a GARCH(1,1) model with standard normal innovations. HS-GARCH-t ts a GARCH(1, 1) model with Students-t innovations. Hence the model can be written as follows, where ɛ t N(0, 1) for HS-GARCH and ɛ t Students-t for HS-GARCH-t. r t = µ + u t, u t = σ t ɛ t, σ 2 t = ω + α 1 σ 2 t 1 + β 1 u 2 t 1 The HS-APARCH-t models ts an APARCH(1, 1) model shown below. r t = µ + u t, u t = σ t ɛ t, σ δ t = ω + α 1 ( u t 1 + γ 1 u t 1 ) δ + β 1 σ δ t 1 The CONDEVT model ts a GARCH(1, 1) model assuming normal innovations just as in HS- GARCH. It does this via the QMLE method. It then ts the residuals to a Generalized Pareto distribution. In the following tables we show the number of violations on a yearly basis for all three indices. We trained each model on 1000 contiguous days of daily returns and then did a 1-day ahead VaR prediction. We then rolled the 1000-day training window forward by one day and computed VaR for the next day. This daily prediction was done for each day in our data-set starting at the 1001st day. Once we had the VaR, we calculated whether the actual return exceeded the calculated VaR. We present the number of such violations on a yearly basis in the tables in the next page. For SPX and CCMP we present data for 18 years. We only present last 4 years data for CDX since that is what was available. The key years to focus on are 1996, 2007, 2008 and 2009. As expected HS-GARCH has the worst VaR prediction among all the models tested since it has the highest number of violations. HS-CONVDEVT gives the best results for SPX in the year 2009 but performs worse than HS- GARCH-t and HS-APARCH-t for CCMP the same year. Both HS-GARCH and HS-APARCH-t seem to have the same level of predictive power when it comes to VaR across all three indices. Both these models result in the similar number of exceedances for all three indices. We also note that none of the models stay below the required number of violations during the stress period. Sometimes they exceed the allowable number of violations by 4 times as in 2007. However HS-GARCH-t and HS-APARCH-t have decent performance during 2008 and 2009.
VaR 99% for SPX 1994 1995 1996 1997 1998 1999 2000 2001 2002 Trading 217 252 254 253 252 252 252 248 252 Expected Violations 2.17 2.52 2.54 2.53 2.52 2.52 2.52 2.48 2.52 HS-GARCH 8 2 10 8 8 3 6 4 4 HS-GARCH-t 5 2 7 5 7 2 2 2 2 HS-APARCH-t 4 2 9 5 6 2 2 2 1 HS-CONDEVT 4 2 6 5 4 0 2 2 2 VaR 99% for SPX 2003 2004 2005 2006 2007 2008 2009 2010 2011 Trading 252 252 252 251 251 253 252 252 252 Expected Violations 2.52 2.52 2.52 2.51 2.51 2.53 2.52 2.52 2.52 HS-GARCH 0 1 3 4 12 11 6 8 6 HS-GARCH-t 0 0 0 3 8 4 1 3 5 HS-APARCH-t 2 0 0 2 8 3 2 3 4 HS-CONDEVT 0 1 2 4 10 4 0 1 5 VaR 99% for CCMP 1994 1995 1996 1997 1998 1999 2000 2001 2002 Trading 217 252 254 253 252 252 252 248 252 Expected Violations 2.17 2.52 2.54 2.53 2.52 2.52 2.52 2.48 2.52 HS-GARCH 4 6 7 5 11 5 3 1 0 HS-GARCH-t 2 3 5 1 5 3 2 1 0 HS-APARCH-t 1 1 4 1 6 3 1 1 0 HS-CONDEVT 1 3 4 1 5 3 1 1 0 VaR 99% for CCMP 2003 2004 2005 2006 2007 2008 2009 2010 2011 Trading 252 252 252 251 251 253 252 252 252 Expected Violations 2.52 2.52 2.52 2.51 2.51 2.53 2.52 2.52 2.52 HS-GARCH 0 0 3 5 5 8 5 8 7 HS-GARCH-t 0 0 2 3 3 4 2 3 4 HS-APARCH-t 0 0 1 3 4 4 2 2 4 HS-CONDEVT 0 1 4 5 4 7 3 2 4 VaR 99% for CDX 2008 2009 2010 2011 Trading 230 259 261 260 Expected Violations 2.30 2.59 2.61 2.60 HS-GARCH 3 4 3 2 HS-GARCH-t 2 1 2 2 HS-APARCH-t 2 1 2 2 HS-CONDEVT 2 1 3 2
Backtesting results for GARCH models We rst do a two sided binomial test on the violations. In this test we assume that each violation is an independent coin toss with a chance of violation 0.01. Under this assumption we can use the R function binom.test to obtain the required p values. These are presented below. As we have seen before, the HS-GARCH model has the lowest p values showing the worst performance. None of the models accept the null hypothesis at 95% condence level in 2007 for SPX. HS-GARCH-t, HS-APARCH-t and HS-CONDEVT accept the null hypothesis at 95% for all years for CCMP and CDX. The result is particularly interesting for CDX. It is possible that the outcome would have been dierent if we had more data for this index. p value for VaR 99% violations for SPX 1994 1995 1996 1997 1998 1999 2000 2001 2002 HS-GARCH 0.00017 1.00 0.000028 0.00043 0.00042 0.74 0.0042 0.32 0.32 HS-GARCH-t 0.07 1.00 0.015 0.11 0.01 1.00 1.00 1.00 1.00 HS-APARCH-t 0.17 1.00 0.001 0.11 0.04 1.00 1.00 1.00 0.53 HS-CONDEVT 0.17 1.00 0.04 0.11 0.32 0.19 1.00 1.00 1.00 p value for VaR 99% violations for SPX 2003 2004 2005 2006 2007 2008 2009 2010 2011 HS-GARCH 0.19 0.53 0.74 0.32 0.000001 0.000006 0.0042 0.00042 0.0042 HS-GARCH-t 0.19 0.19 0.19 0.74 0.004 0.33 0.53 0.74 0.11 HS-APARCH-t 1.00 0.19 0.19 1.00 0.004 0.74 1.00 0.74 0.33 HS-CONDEVT 0.19 0.53 1.00 0.32 0.0002 0.33 0.19 0.53 0.11 p value for VaR 99% violations for CCMP 1994 1995 1996 1997 1998 1999 2000 2001 2002 HS-GARCH 0.17 0.00 0.00 0.11 0.00 0.11 0.74 0.52 0.19 HS-GARCH-t 1.00 0.74 0.11 0.53 0.11 0.74 1.00 0.53 0.19 HS-APARCH-t 0.73 0.53 0.33 0.53 0.04 0.74 0.53 0.53 0.19 HS-CONDEVT 0.73 0.74 0.33 0.53 0.11 0.74 0.53 0.53 0.19 p value for VaR 99% violations for CCMP 2003 2004 2005 2006 2007 2008 2009 2010 2011 HS-GARCH 0.19 0.19 0.74 0.11 0.11 0.00 0.11 0.00 0.00 HS-GARCH-t 0.19 0.19 1.00 0.74 0.74 0.33 1.00 0.74 0.33 HS-APARCH-t 0.19 0.19 0.53 0.74 0.32 0.33 1.00 1.00 0.33 HS-CONDEVT 0.19 0.53 0.33 0.11 0.32 0.01 0.74 1.00 0.32 p value for VaR 99% violations for CDX 2008 2009 2010 2011 HS-GARCH 0.50 0.33 0.75 1.00 HS-GARCH-t 1.00 0.53 1.00 1.00 HS-APARCH-t 1.00 0.53 1.00 1.00 HS-CONDEVT 1.00 0.53 0.75 1.00
Robust backtesting for GARCH models The previous test does not capture time series dependence among violations. To do so, we consider four dierent testsmarkov, Ljung-Box, Geometric and Weibull. In the following paragraphs, we denote a sequence of violations by {I t }, where I t = 1 if the loss exceeded VaR at time t and 0 otherwise. If I t is a rst order Markov process, the one-step ahead transition probabilities P (I t+1 I t ) are given by (1 π 01 )π 11 (1 π 11 )π 01, where π ij is the transition probability P (I t+1 = j I t = i). Under the null hypothesis, the violations have a constant conditional mean that implies the two linear restrictions, π 01 = π 11 = p. A likelihood ratio test of restrictions can be computed from the following likelihood function, where T ij denotes the number of observations with a j following an i and T i is the number of i. L(I; π 01, π 11 ) = (1 π 01 ) (T 0 T 01 ) π T 01 01 (1 π 11) (T 1 T 11 ) π T 11 11 To construct the Ljung-Box test LB(k), we check whether the rst k auto-correlations are 0 for the sequence {I t p} where p = 0.01 for 99% VaR. Both the Geometric and the Weibull tests take as input, D i, the duration between two violations. Under the null hypothesis, the hazard function of the durations, λ(d i ), should be at and equal to p = 0.01 for 99% VaR. This is shown in the equation below. λ(d i = d) = (1 p) d 1 p 1 d 2 j=0 (1 p)j p = p Under the alternative hypothesis, the violation sequence, and hence the duration, display dependence or clustering. The only continuous distribution without duration dependence is the exponential. Thus under the null hypothesis, the distribution of the durations should be given by the following equation. f(d; p) = pe pd A more powerful test is when the distributions follow Weibull distribution as shown below. f(d; a, b) = a b bd b 1 exp ( ad)b In the following page, we present the results of these tests as p values. To get a sucient number of violations we consider a time period of four years. We note some interesting trends in the data. While a particular test (say Ljung-Box) is pretty consistent across various models and various time periods, there are inconsistencies between tests. For example, we see that we have high p values for the Ljung-Box test, showing that the {I t } series is independently distributed. Some clustering is noted in the '08-'11 period. However both the Geometric and Weibull tests show low p values for time periods where Ljung-Box tests show that the I t is independently distributed. A case in point is '95-'99 for SPX. However Ljung-Box, Geometric and Weibull seem to be consistent where clustering exists in {I t }. We notice inconsistencies for the Kupiec test as well, where it shows very low p values for '00-'03. This period shows high p values for the other tests. Lastly, we were unable to t a Weibull distribution to the violations for CCMP. This is a problem when the number of violations is too low.
p value for LB(1) test for SPX p value for LB(5) test for SPX '95-'99 '00-'03 '03-'07 '08-'11 '95-'99 '00-'03 '03-'07 '08-'11 GARCH 0.15 0.18 0.80 0.22 0.41 0.47 0.01 0.35 GARCH-t 0.38 0.80 0.92 0.61 0.71 1.00 1.00 0.38 APARCH-t 0.44 0.82 0.90 0.61 0.20 1.00 1.00 0.38 CONDEVT 0.17 0.85 0.82 0.63 0.47 1.00 0.00 0.00 p value for Kupiec test for SPX p value for Markov test for SPX '95-'99 '00-'03 '03-'07 '08-'11 '95-'99 '00-'03 '03-'07 '08-'11 GARCH 0.00 0.04 0.50 0.00 0.00 0.08 0.74 0.00 GARCH-t 0.03 0.50 0.01 0.08 0.01 0.75 0.03 0.17 APARCH-t 0.00 0.31 0.03 0.08 0.00 0.57 0.09 0.17 CONDEVT 0.47 0.17 0.30 0.15 0.08 0.37 0.56 0.28 p value for Geometric test for SPX p value for Weibull test for SPX '95-'99 '00-'03 '03-'07 '08-'11 '95-'99 '00-'03 '03-'07 '08-'11 GARCH 0.00 0.64 0.02 0.06 0.05 0.06 0.66 0.01 GARCH-t 0.02 0.48 0.96 0.06 0.00 0.05 0.13 APARCH-t 0.00 0.54 0.91 0.03 0.01 0.02 0.13 0.04 CONDEVT 0.00 0.03 0.66 0.0 0.00 0.07 0.48 0.00 p value for LB(1) test for CCMP p value for LB(5) test for CCMP '95-'99 '00-'03 '03-'07 '08-'11 '95-'99 '00-'03 '03-'07 '08-'11 GARCH 0.92 0.92 0.68 0.36 0.02 1.0 0.4 0.0 GARCH-t 0.63 0.95 0.80 0.68 0.00 1.0 1.0 0.1 APARCH-t 0.67 0.97 0.80 0.7 0.14 1.0 1.0 0.04 CONDEVT 0.65 0.97 0.65 0.61 0.54 0.0 0.41 0.18 p value for Kupiec test for CCMP p value for Markov test for CCMP '95-'99 '00-'03 '03-'07 '08-'11 '95-'99 '00-'03 '03-'07 '08-'11 GARCH 0.0 0.03 0.37 0.0 0.0 0.03 0.57 0.0 GARCH-t 0.24 0.0 0.5 0.38 0.4 0.0 0.74 0.57 APARCH-t 0.52 0.0 0.5 0.56 0.68 0.0 0.74 0.73 CONDEVT 0.36 0.0 0.24 0.08 0.54 0.0 0.41 0.18 p value for Geometric test for CCMP p value for Weibull test for CCMP '95-'99 '00-'03 '03-'07 '08-'11 '95-'99 '00-'03 '03-'07 '08-'11 GARCH 0.00 0.50 0.11 0.00 GARCH-t 0.10 0.22 0.97 0.37 APARCH-t 0.26 0.06 0.97 0.55 CONDEVT 0.16 0.06 0.30 0.08
We show eight plotstwo for each GARCH model, for index SPX. The plots from year 2008 are on the left and those from 2009 are on the right. Going from top down, the models are HS-GARCH, HS-GARCH-t, HS-APARCH-t and HS-CONDEVT. The blue line shows the 95% VaR and the red line shows the 99% VaR. Visual inspection shows that the VaR prediction from APARCH-t is most responsive to the daily log returns. Year 2008 negative log return and estimated VaR Year 2009 negative log return and estimated VaR 0.15 0.15 0.06 0.04 0.02 0.00 0.02 0.04 0.06 Year 2008 negative log return and estimated VaR Year 2009 negative log return and estimated VaR 0.15 0.15 0.05 0.00 0.05 Year 2008 negative log return and estimated VaR Year 2009 negative log return and estimated VaR 0.15 0.15 Year 2008 negative log return and estimated VaR Year 2009 negative log return and estimated VaR 0.15 0.15
We show eight plotstwo for each GARCH model, for index CCMP. The plots from year 2008 are on the left and those from 2009 are on the right. Going from top down, the models are HS- GARCH, HS-GARCH-t, HS-APARCH-t and HS-CONDEVT. The blue line shows the 95% VaR and the red line shows the 99% VaR. Visual inspection shows that the VaR prediction from APARCH-t is most responsive to the daily log returns. Year 2008 negative log return and estimated VaR (CCMP) Year 2009 negative log return and estimated VaR (CCMP) 0.15 0.15 0.06 0.04 0.02 0.00 0.02 0.04 0.06 Year 2008 negative log return and estimated VaR (CCMP) Year 2009 negative log return and estimated VaR (CCMP) 0.15 0.15 0.05 0.00 0.05 Year 2008 negative log return and estimated VaR (CCMP) Year 2009 negative log return and estimated VaR (CCMP) 0.15 0.15 Year 2008 negative log return and estimated VaR (CCMP) Year 2009 negative log return and estimated VaR (CCMP) 0.15 0.15
We show eight plotstwo for each GARCH model, for index CDX. The plots from year 2008 are on the left and those from 2009 are on the right. Going from top down, the models are HS-GARCH, HS-GARCH-t, HS-APARCH-t and HS-CONDEVT. The blue line shows the 95% VaR and the red line shows the 99% VaR. Visual inspection shows that the VaR prediction from APARCH-t is most responsive to the daily log returns. This can be seen near the sudden changes in returns where the corresponding change in VaR predicted from APARCH-t is the lowest of all models. Year 2008 negative log return and estimated VaR (CDX) Year 2009 negative log return and estimated VaR (CDX) 0.2 0.1 0.0 0.1 0.2 0.3 0.05 0.00 0.05 0.10 0.15 0.20 0 50 100 150 200 Year 2008 negative log return and estimated VaR (CDX) Year 2009 negative log return and estimated VaR (CDX) 0.2 0.1 0.0 0.1 0.2 0.3 0.05 0.00 0.05 0.10 0.15 0.20 0 50 100 150 200 Year 2008 negative log return and estimated VaR (CDX) Year 2009 negative log return and estimated VaR (CDX) 0.2 0.1 0.0 0.1 0.2 0.3 0.05 0.00 0.05 0.10 0.15 0 50 100 150 200 Year 2008 negative log return and estimated VaR (CDX) Year 2009 negative log return and estimated VaR (CDX) 0.2 0.1 0.0 0.1 0.2 0.3 0.05 0.00 0.05 0.10 0.15 0.20 0 50 100 150 200
VaR calculation using CAViaR for single indices The CAViaR method proposed by Engle and Manganelli models the VaR directly by tting a model to it. The initial VaR is calculated as a quantile of the historical data. This is then used as a starting value to an autoregressive model that takes VaRs as input. This is in contrast to the GARCH based models that model the return and use it to calculate VaR. A general CAViaR specication is given by the following equation where β i are parameters to be estimated and l i are lagged observables. VaR t = β 0 + q β i VaR t i + i=1 r β j l t j We t two specic realizations of the above model. The rst is Symmetric Absolute Value shown below. r t is the return at time t. j=1 VaR t = β 1 + β 2 VaR t 1 + β 3 r t 1 The second CAViaR model we t is called the Indirect GARCH (1, 1) model which is expressed by the following equation. VaR t = β 1 + β 2 VaR 2 t 1 + β 3 r 2 t 1 We estimate the parameters of each model by using the rst 2982 daily returns. This is done by choosing the set of parameters (β i ) that minimize the number of violations over that period. The initial value of VaR needed to start evaluating each model, is calculated by looking at the 0.01% quantile for the rst 300 returns of the historical period of 2982 daily returns. Once we nd the parameters we forecast the VaR for a year into the future. Assuming there are 250 trading days in a year, a set of parameters estimated from 2982 historical daily returns will forecast VaR for upcoming 250 trading days. We then slide the estimation window forward by 250 trading days and repeat the process. The following table shows the predicted number of violations per year along with expected number of violations for both the models for SPX and CCMP. We could not run the model for CDX since we did not have sucient data to train either model. We note that both models are quite eective in predicting VaR outside of the stress period of 2007-2009. Both models perform equally worse during the stress period. We also note that both HS-APARCH-t and CONDEVT perform better than CAViaR models. However the CAViaR models predict an entire years VaR after estimating the parameters while GARCH models only do 1-day ahead prediction, so they have the advantage of a more updated history. It is possible to do 1-day ahead VaR prediction with CAViaR models as well but the required computational horsepower prevented us from trying it out. 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 Trading 252 252 252 252 251 251 253 252 252 252 Expected Violations 2.52 2.52 2.52 2.52 2.51 2.51 2.53 2.52 2.52 2.52 Symmetric Absolute Value VaR 99% SPX 4 1 0 0 0 3 9 10 4 7 CCMP 0 2 1 1 3 3 7 4 7 4 Indirect GARCH VaR 99% SPX 4 1 0 1 1 7 11 4 5 6 CCMP 3 1 1 0 3 2 8 4 3 4
We show plots for years 2008 on the left and that of 2009 on the right for index SPX. The VaR for the top row is calculated by the Symmetric Absolute Value model and that for the bottom calculated by the Indirect GARCH model. We notice that visually the two models don't look very dierent however the VaR curve is much more smooth than those t by the GARCH models since they are predicted for almost 250 trading days, instead of just 1 day. Year 2008 negative log return and estimated VaR Year 2009 negative log return and estimated VaR 0.06 0.04 0.02 0.00 0.02 0.04 0.06 Year 2008 negative log return and estimated VaR Year 2009 negative log return and estimated VaR 0.05 0.00 0.05 We show the same four plots for index CCMP. Yet again we note the smoothness of these ts compared to the corresponding ts with a GARCH model. Year 2008 negative log return and estimated VaR Year 2009 negative log return and estimated VaR 0.06 0.04 0.02 0.00 0.02 0.04 0.06 Year 2008 negative log return and estimated VaR Year 2009 negative log return and estimated VaR 0.05 0.00 0.05
Backtesting CAViaR models for single indices If our model ts the data well then P (r t < VaR t ) = θ where θ = 99% for 99% VaR. This is equivalent to requiring that the sequence of associated indicator functions {I t } be i.i.d. Hence a property that any VaR estimate should satisfy is that of providing a lter to transform a possibly serially correlated and heteroscedastic time series into a serially independent sequence of indicator variables. Various tests have been proposed to detect the presence the serial correlation in the sequence of indicator functions 1 ; this is only a necessary but not sucient condition to assess the performance of a quantile model. To get around this problem, Engle and Manganelli, propose two new tests called the in-sample and out-of-sample dynamic quantile tests (DQ tests). Please refer to CAViaR: Conditional Autoregressive Value at Risk by Regression Quantiles for details and proofs of their results. In the following table we present both in-sample and out-of-sample DQ test results for both indices. We note that the in-sample tests reject the null hypothesis for the years 2008-2010 for both Symmetric Absolute Value and Indirect GARCH. However we see that the in-sample tests reject the null hypothesis for the years 2004-2010 for the index CCMP even though the market was not under stress in 2004-2006. The out-of-sample accept the null hypothesis for all the years for both indices. This means that the predicted indicator function is not serially correlated. Symmetric Absolute Value VaR 99% DQ in-sample p-value 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 SPX 0.39 0.4 0.4 0.4 0.36 0.39 0.013 0.00065 0.00041 0.31 CCMP 0.61 0.027 0.000026 0.000015 0.000015 0.000016 0.000027 0.000016 0.033 0.026 Symmetric Absolute Value VaR 99% DQ out-of-sample p-value SPX 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 CCMP 0.89 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.99 Indirect GARCH VaR 99% DQ in-sample p-value 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 SPX 0.21 0.56 0.6 0.43 0.58 0.6 0.4 0.00043 0.00046 0.011 CCMP 0.015 0.022 0.0082 0.037 0.018 0.027 0.027 0.027 0.032 0.027 Indirect VaR 99% DQ out-of-sample p-value SPX 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.99 CCMP 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1 We have seen some of these in the section on robust backtesting.
Portfolio VaR with multivariate GARCH To estimate VaR of a portfolio of indices, we use the multivariate GARCH model. A multivariate GARCH model can be written as the following equation, where r t is the n-dimensional vector of returns, µ t is the n-dimensional vector of conditional mean returns, ɛ t is an i.i.d vector white noise process with identity covariance matrix and H t is the conditional covariance matrix of r t. r t = µ t + H t ɛ t Various methods have been developed to estimate the conditional variance H t. The VEC- GARCH model of Bollerslev, Engle and Wooldridge (1988), models the conditional variance as a function of all lagged conditional covariances and lagged cross-product of returns. This can be written as the following equation where vech operator stacks all the columns of a lower triangular matrix. vech(h t ) = c + q A j vech(r t j r t j) + j=1 p vech(h t j ) The rst problem with this model is the sheer number of parameters to estimate. c is a n(n+1)/2 parameter vector and A j, B j are n 2 (n+1) 2 /4 parameter matrices. The second problem is that there exist only sucient, rather restrictive, conditions for H t to be positive denite for all t. Various simplications to this model have been suggested to ameliorate both these shortcomings. Bollerslov, Engle and Wooldridge (1988) proposed a simplied version where A j and B j are diagonal matrices. This greatly reduces the number of parameters to estimate but is too restrictive since no interaction between dierential conditional variances and covariances are allowed. The Baba, Engle, Kraft and Kroner (BEKK) model introduced by Engle and Kroner (1995) relaxes some of these constraints but ensures that the conditional covariance matrices are postive denite. In the BEKK model, the conditional covariance matrix is expressed as the following product, where C is a lower triangular matrix. The form of C ensures that H t is positive denite. j=1 H t = CC + q j=1 k=1 K A kjr t j r t ja kj + q j=1 k=1 K B kjh t j B kj A dierent approach to reducing the number of parameters to estimate is taken by Orthogonal GARCH (O-GARCH) and Generalized Orthogonal GARCH (GO-GARCH). Both these models assume that r t can be expressed as a linear combination of uncorrelated factors z t, where each component of z t is modeled as GARCH process. The relationship between the two can be expressed by the following equation, where the orthogonal matrix W does not vary with time. r t = Wz t The GO-GARCH model relaxes the orthogonality constraint imposed on W by O-GARCH, making it more exible. An example of a GO-GARCH(1, 1) model is shown in the following set of equations, where h it are diagonal elements of matrix H t and z it are components of vector z t.
r t = Wz t z it F t 1 N(0, h it ) h it = ω i + α i z 2 it + β i h i,t 1 The above equations imply the following relations. r t F t 1 N(0, V t ) V t = WH t W We follow a two-step strategy to build our GO-GARCH models. Each component of the one-day ahead prediction of the covariance matrix of asset returns at time t, is expressed as the following equation, where σi S, σs j are one-day ahead prediction of volatilities and rij L is the one-day ahead prediction of correlation of asset volatilities. σ t ij = σ S i σ S j r L ij L and S signify the time window used to estimate the corresponding quantities. We choose S {0.25L, 0.5L, 0.75L} where L = 1000 most recent trading days. More specically if S = 0.5L, at time t, we use the last 1000 trading days to estimate r ij and use the last 500 trading days to estimate σ i. By doing this, we are trying to capture the long term correlation between two assets and the short term volatilities in the individual assets. We then use σ ij to calculate the one-day ahead VaR. The number of yearly violations for a equally weighted portfolio consisting of SPX, CCMP and CDX is shown below. Once again we note that all the GO-GARCH models perform equally well after 2008 and perform equally badly in 2008. This suggests that combining the short-term volatilities with long-term correlations did not make much of a dierence. It is possible that choosing a much shorter window to estimate volatilities might work better. 2008 2009 2010 2011 Trading 253 252 252 252 Expected Violations 2.53 2.52 2.52 2.52 S = L 8 1 0 1 S = 0.75L 7 1 0 1 S = 0.5L 5 1 0 2 S = 0.25L 7 1 1 5
The following plots show the VaR estimates by the GO-GARCH model, with predictions for the year 2008 on the left and that of the year 2009 on the right. Going from top to bottom we go from S = 0.75L to S = 0.25L. Each graph shows the VaR predicted when S = L. We notice that the VaR predicted when S = L is lower than that predicted when S < L. Multi time (3/4 length) GO GARCH VaR at 99% (2008 detail) Multi time (3/4 length) GO GARCH VaR at 99% (2009 detail) Multi 3/4 GO GARCH VaR at 99% Full GO GARCH VaR at 99% Portfolio s (EQ weight) Multi 3/4 GO GARCH VaR at 99% Full GO GARCH VaR at 99% Portfolio s (EQ weight) 2008 detail 2009 detail Multi time (1/2 length) GO GARCH VaR at 99% (2008 detail) Multi time (1/2 length) GO GARCH VaR at 99% (2009 detail) Multi 1/2 GO GARCH VaR at 99% Full GO GARCH VaR at 99% Portfolio s (EQ weight) Multi 1/2 GO GARCH VaR at 99% Full GO GARCH VaR at 99% Portfolio s (EQ weight) 2008 detail 2009 detail Multi time (1/4 length) GO GARCH VaR at 99% (2008 detail) Multi time (1/4 length) GO GARCH VaR at 99% (2009 detail) Multi 1/4 GO GARCH VaR at 99% Full GO GARCH VaR at 99% Portfolio s (EQ weight) Multi 1/4 GO GARCH VaR at 99% Full GO GARCH VaR at 99% Portfolio s (EQ weight) 2008 detail 2009 detail
Stressed VaR Limitations of VaR have been highlighted by the recent nancial turmoil. Financial industry and regulators now regard stress tests as no less important than VaR methods for assessing a bank's risk exposure. A new emphasis on stress testing also derives from the amended Basel II framework which requires banks to compute a valid stressed VaR number. This framework says that the banks must calculate a Stressed VaR measure intended to replicate a VaR calculation that would be generated on the banks current portfolio if the relevant market factors were experiencing a period of stress. The period of stress should be approved by a supervisor and should reect the losses experienced in the 2007/2008 period. However the Basel framework does not specify a model to calculate the VaR and leaves it up to the banks to choose an appropriate model to capture material risks they face. In our view, this leaves a big gap in the regulatory framework since there has been little research in creating risk models that capture rare stress events. A survey of stress testing practices conducted by the Basel Committee in 2005 showed that most stress tests are designed around a series of scenarios based either on historical events, hypothetical events, or some combination of the two. Such methods have been criticizes by Berkowitz (2000a). Without using a risk model the probability of each scenario is unknown, making its importance dicult to evaluate. There is also the possibility that many extreme yet plausible scenarios are not even considered. The pressing technical issue now facing nancial institutions that intend to comply with the amended Basel II framework is to understand how to calculate a valid stressed VaR number. An over-simplistic interpretation of this specication might be to increase the assumed volatilities of the securities in a portfolio. This would have the eect of lengthening the tails of the Gaussian (normal) loss distributions that underlie all standard VaR calculations. We can see the eect of stress period on volatilities in the following two plots. These plots show the volatility for SPX, CDX and CCMP during the stress periods of 2008 and 2009 as estimated by GO-GARCH. We see that the spike in volatility in CDX dwarfs the spike in volatility in SPX and CCMP underscoring the severity of problems in the credit markets. Time Varying Variance of SPX, CCMP, and CDX (2008) Time Varying Variance of SPX, CCMP, and CDX (2009) Variance 0.000 0.005 0.010 0.015 SPX CCMP CDX Variance 0.000 0.005 0.010 0.015 SPX CCMP CDX Date 2008 Date 2009
The same spike in volatility can be observed in the plots for NKY and GSCI indices shown below. This is especially true for NKY. We see the volatilities decaying in 2009. In both cases the volatilities were estimated by the GO-GARCH model. Time Varying Variance of SPX, nky, and gsci (2008) Time Varying Variance of SPX, nky, and gsci (2009) Variance 0.000 0.001 0.002 0.003 0.004 0.005 SPX nky gsci Variance 0.000 0.001 0.002 0.003 0.004 0.005 SPX nky gsci Date 2008 Date 2009 However, in order to calculate stressed VaR accurately, it is also necessary to stress the correlation matrix used in all VaR methodologies. It has been observed that during times of extreme volatility, correlations are dramatically perturbed relative to their historical values. In general, most correlations tend to increase during market crises, asymptotically approaching 1.0 during periods of complete meltdown. In the following two plots we show pairwise daily correlation estimated by the GO-GARCH model during 2008 (left) and 2009 (right). We see that the correlation between SPX and CCMP do not change much and both are very highly correlated. We see rapid changes in correlation between CDX and CCMP and CDX and SPX in the latter half of 2008. We see that at least in 2008, it is hard to discern a stress period from the pairwise correlations of these indices since they don't approach 1.0. We do however see rapid uctuations in correlations during the latter half of 2008. This shows that it is possible to be in a stress period even when the correlations do not approach 1.0. Time Varying Correlation of SPX, CCMP, and CDX (2008) Time Varying Correlation of SPX, CCMP, and CDX (2009) Correlation 1.0 0.5 0.0 0.5 1.0 CCMP & SPX CDX & SPX CDX & CCMP Correlation 1.0 0.5 0.0 0.5 1.0 CCMP & SPX CDX & SPX CDX & CCMP Date 2008 Date 2009
In the following plots we show the pair-wise correlation for NKY, GSCI and SPX. We see that the correlations vary wildly during the stress period but does not reach 1.0 Time Varying Correlation of SPX, NKY, and GSCI (2008) Time Varying Correlation of SPX, NKY, and GSCI (2009) Correlation 1.0 0.5 0.0 0.5 1.0 nky & SPX gsci & SPX gsci & nky Correlation 1.0 0.5 0.0 0.5 1.0 nky & SPX gsci & SPX gsci & nky Date 2008 Date 2009 Additionally, we wanted to check if the parameters of one of our models showed a specic pattern during the stress period. In the plots below, we show how parameters ω, µ (top left and right), α 1 and β 1 (bottom left and right) for a GARCH(1, 1) model vary from 1995 to 2012. We used normal innovations for the GARCH model and used 250 days of trading data to t estimate the parameters. Once again we do not notice a discernible change in the parameters during the stress period, thus underscoring the diculty in adjusting model parameters to compute a stressed VaR. mu 0.0015 0.0010 0.0005 0.0000 0.0005 1995 2000 2005 2010 omega 0e+00 1e 05 2e 05 3e 05 4e 05 5e 05 1995 2000 2005 2010 alpha1 0.00 0.05 0.10 0.15 betha1 0.5 0.6 0.7 0.8 0.9 1.0 1995 2000 2005 2010 1995 2000 2005 2010 Kupiec (1998) suggested a conditional stress approach, where the risk factor distributions are conditional on an extreme value realization of one or more of the risk factors. Conditional on a large move of at least one factor, the conditional factor covariance matrix exhibits much higher correlations among the remaining factors. However the conditional correlations remain unchanged.
Kupiec showed that the conditional stress test performs very well on portfolios constructed during the Asian currency crisis. However we did not have time to extend his methods to the 2008-2009 meltdown in our study. An alternative approach to conditional correlation is to stress the unconditional correlation matrix of the risk factors. Unfortunately, this approach is not as straightforward as the conditional correlation approach or stretching the tails of the loss distributions. The VaR calculation requires a positive denite correlation matrix and stressing this matrix may violate this property. In addition, as we have shown in the plots above, it is unclear what a stressed correlation matrix should look like since we don't observe a high spike in the pairwise correlations. An alternate approach might employ fat-tailed distributions to model the extreme loss events more accurately. Examples of these extreme value theory (EVT) distributions are the Gumbel, Generalized Pareto, Weibull, Fréchet, and the Tukey distributions. We tried this approach in our HS-CONDEVT model where we t a GARCH(1, 1) through QMLE and then t a Pareto distribution to the residuals. As we have discussed before, HS-CONDEVT did not result in fewer violations than other GARCH models like HS-APARCH-t Conclusion and Future directions Thus we see that stressed VaR is a very nascent eld where there has been far too little academic research. Models like HS-APARCH-t show some promise but still they have far too many violations during the stress period than a 99% exceedance limit will allow. GO-GARCH models work reasonably well for multi-asset portfolios but it remains to be seen if perturbing the volatilities obtained from a stress period and feeding them to GO-GARCH model will result in meaningful VaR models. Additionally we would like to evaluate portfolios with options which have non-linear payo and examine if GARCH models show the same performance for such portfolios. Acknowledgment We would like to thank Prof. David Donoho for his suggestion to use separate periods for estimating volatility and correlation in the GO-GARCH model. We also thank Prof Kay Geisecke, Gerald Teng and Andrew Abrahams for their help throughout the quarter. References Berkowitz, J. (2000a): A coherent framework for stress-testing, Journal of Risk, vol 2, pp 1-11. Kupiec, P. (1998): Stress testing in a value-at-risk framework, Journal of Derivatives, pp 7-24. Berkowitz, J., Christoersen, P. and Pelletier D. (2009): Evaluating Value-at-Risk Models with Desk-Level Data, Management Science, pp 1-15. Engle, R. and Manganelli, S. (2004) CAViaR: Conditional Autoregressive Value at Risk by Regreqion Quantiles, Journal of Business and Economic Statistics, pp 367-381.