1 

 

GFC-Robust Risk Management Strategies  
under the Basel Accord* 

 
 
 
 

Michael McAleer 
Econometric Institute 

Erasmus School of Economics 
Erasmus University Rotterdam  

and  
Tinbergen Institute 
The Netherlands 

and 
Institute of Economic Research 

Kyoto University 
Japan 

 
 

Juan-Ángel Jiménez-Martín 
Department of Quantitative Economics 

Complutense University of Madrid 
 
 

Teodosio Pérez-Amaral 
Department of Quantitative Economics 

Complutense University of Madrid 
 
 
 
 
 
 

Revised: October 2010  
 
 
 
 
 
 
 
 

* For financial support, the first author wishes to thank the Australian Research 
Council, National Science Council, Taiwan, and the Japan Society for the Promotion of 
Science. The second and third authors acknowledge the financial support of the 
Ministerio de Ciencia y Tecnología and Comunidad de Madrid, Spain. 



2 

 

Abstract 

A risk management strategy is proposed as being robust to the Global Financial Crisis 

(GFC) by selecting a Value-at-Risk (VaR) forecast that combines the forecasts of 

different VaR models. The robust forecast is based on the median of the point VaR 

forecasts of a set of conditional volatility models. This risk management strategy is 

GFC-robust in the sense that maintaining the same risk management strategies before, 

during and after a financial crisis would lead to comparatively low daily capital charges 

and violation penalties. The new method is illustrated by using the S&P500 index 

before, during and after the 2008-09 global financial crisis. We investigate the 

performance of a variety of single and combined VaR forecasts in terms of daily capital 

requirements and violation penalties under the Basel II Accord, as well as other criteria. 

The median VaR risk management strategy is GFC-robust as it provides stable results 

across different periods relative to other VaR forecasting models. The new strategy 

based on combined forecasts of single models is straightforward to incorporate into 

existing computer software packages that are used by banks and other financial 

institutions.  

 

Key words and phrases: Value-at-Risk (VaR), daily capital charges, robust forecasts, 
violation penalties, optimizing strategy, aggressive risk management strategy, 
conservative risk management strategy, Basel II Accord, global financial crisis. 
 
JEL Classifications: G32, G11, G17, C53, C22. 



3 

1. Introduction 

 

The Global Financial Crisis (GFC) of 2008-09 has left an indelible mark on economic 

and financial structures worldwide, and caused a generation of investors to wonder how 

things could have become so bad (see, for example, Borio (2008)). There have been 

many questions asked about whether appropriate regulations were in place, especially in 

the USA, to ensure the appropriate monitoring and encouragement of (possibly 

excessive) risk taking.  

 

The Basel II Accord1 was designed to monitor and encourage sensible risk taking, using 

appropriate models of risk to calculate Value-at-Risk (VaR) and subsequent daily 

capital charges. VaR is defined as an estimate of the probability and size of the potential 

loss to be expected over a given period, and is now a standard tool in risk management. 

It has become especially important following the 1995 amendment to the Basel Accord, 

whereby banks and other Authorized Deposit-taking Institutions (ADIs) were permitted 

(and encouraged) to use internal models to forecast daily VaR (see Jorion (2000) for a 

detailed discussion). The last decade has witnessed a growing academic and 

professional literature comparing alternative modelling approaches to determine how to 

measure VaR, especially for large portfolios of financial assets.  

 

The amendment to the initial Basel Accord was designed to encourage and reward 

institutions with superior risk management systems. A back-testing procedure, whereby 

actual returns are compared with the corresponding VaR forecasts, was introduced to 

assess the quality of the internal models used by ADIs. In cases where internal models 

led to a greater number of violations than could reasonably be expected, given the 

confidence level, the ADI is required to hold a higher level of capita (see Table 1 for the 

                                                            
1 When the Basel I Accord was concluded in 1988, no capital requirements were defined for market risk. 
However, regulators soon recognized the risks to a banking system if insufficient capital were held to 
absorb the large sudden losses from huge exposures in capital markets. During the mid-90’s, proposals 
were tabled for an amendment to the 1988 Accord, requiring additional capital over and above the 
minimum required for credit risk. Finally, a market risk capital adequacy framework was adopted in 1995 
for implementation in 1998. The 1995 Basel I Accord amendment provides a menu of approaches for 
determining market risk capital requirements, ranging from a simple to intermediate and advanced 
approaches. Under the advanced approach (that is, the internal model approach), banks are allowed to 
calculate the capital requirement for market risk using their internal models. The use of internal models 
was introduced in 1998 in the European Union. The 26 June 2004 Basel II framework, implemented in 
many countries in 2008 (though not yet in the USA), enhanced the requirements for market risk 
management by including, for example, oversight rules, disclosure, management of counterparty risk in 
trading portfolios. 



4 

penalties imposed under the Basel II Accord). Penalties imposed on ADIs affect 

profitability directly through higher capital charges, and indirectly through the 

imposition of a more stringent external model to forecast VaR.2 This is one reason why 

financial managers may prefer risk management strategies that are passive and 

conservative rather than active and aggressive (for more on this, see below). 

 

Excessive conservatism can have a negative impact on the profitability of ADIs as 

higher capital charges are subsequently required. Therefore, ADIs should perhaps 

consider a strategy that allows an endogenous decision as to how often ADIs should 

violate, and hence incur violation penalties, in any financial year (for further details, see 

McAleer and da Veiga (2008a, 2008b), McAleer (2009), Caporin and McAleer (2010a), 

and McAleer et al. (2009)). Additionally, ADIs need not restrict themselves to using 

only a single risk model. McAleer et al. (2009) propose a risk management strategy that 

consists in choosing from among different combinations of alternative risk models to 

forecast VaR. They discuss a combination of forecasts that was characterized as an 

aggressive strategy, and another that was regarded as a conservative strategy.3  

 

Following such an approach, this paper suggests using a combination of VaR forecasts 

to obtain a crisis robust risk management strategy. The paper defines a crisis robust 

strategy as an optimal risk management strategy that remains unchanged regardless of 

whether it is used before, during or after a significant financial crisis. Parametric 

methods for forecasting VaR are typically fitted to historical returns assuming specific 

conditional distributions of returns, such as normality, Student-t, or generalized normal 

distribution. The VaR forecast depends on the parametric model, the conditional 

distribution and can be heavily affected by a few large observations. Some models 

provide many violations, but low daily capital charges. Additionally, these results can 

change drastically from tranquil to turbulent periods. In short, regardless of economic 

turbulence, is there a model to forecast VaR that provides a reasonable number of 

violations and daily capital charges? 

                                                            
2 In the 1995 amendment (p. 16), a similar capital requirement system was recommended, but the specific 
penalties were left to each national supervisor. We consider that the penalty structure contained in Table 1 
of this paper belongs only to Basel II, and was not part of Basel I or its 1995 amendment. 
 
3 This is a novel possibility. Technically, a combination of forecast models is also a forecast model. In 
principle, the adoption of a combination of forecast models by a bank is not forbidden by the Basel 
Accords, although it is subject to regulatory approval. 



5 

  

We estimate several univariate conditional volatility models to forecast VaR, assuming 

different returns distributions (specifically, Gaussian, Student-t and Generalized 

Normal). Additionally, we present 12 new strategies based on combinations of stardard 

model VaR forecasts, namely: lowerbound, upperbound (as defined in McAleer et al. 

(2009)), the average, and nine additional strategies based on the 10th,… 50th, … 90th 

percentiles. Models are compared over three different time periods to investigate 

whether we can find a risk management strategy that is robust over time (that is, crisis-

robust). We provide evidence that using the median of the point VaR forecasts of a set 

of univariate conditional volatility models is a robust risk measure. A risk management 

strategy based on the median forecast is found to be superior to alternative single and 

composite model alternatives.  

 

The remainder of the paper is organized as follows. In Section 2 we present the main 

ideas of the Basel II Accord Amendment as it relates to forecasting VaR and daily 

capital charges. Section 3 reviews some of the most well-known models of conditional 

volatility used to forecast VaR. In Section 4 the data used for estimation and forecasting 

are presented. Section 5 analyses the robust VaR forecasts before, during and after the 

2008-09 GFC. Section 6 presents some conclusions. 

 

2. Forecasting Value-at-Risk and Daily Capital Charges   

  

The Basel II Accord stipulates that daily capital charges (DCC) must be set at the higher 

of the previous day’s VaR or the average VaR over the last 60 business days, multiplied 

by a factor (3+k) for a violation penalty, wherein a violation involves the actual negative 

returns exceeding the VaR forecast negative returns for a given day:4 

  

   ______

60t t-1DCC = sup - 3+ k VaR ,  - VaR  (1) 

 

where  
 
                                                            
4 Our aim is to investigate the likely performance of the Basel II regulations. In this section we carry out 
our analysis applying the Basel II formulae to a period that includes the 2008-09 GFC, during which the 
Basel II Accord regulations were not fully implemented. 



6 

DCCt = daily capital charges, which is the higher of   60

______

t-1- 3 + k VaR  and  - VaR , 

 

tVaR  = Value-at-Risk for day t, 

 

tttt zYVaR ̂ˆ  , 

 

60

______

VaR  = mean VaR over the previous 60 working days, 
 

tŶ = estimated return at time t, 

 

tz = 1% critical value of the distribution of returns at time t,  

 

t̂ = estimated risk (or square root of volatility) at time t, 

 
0 k 1    is the Basel II violation penalty (see Table 1). 
 
 

[Insert Table 1 here] 

 

The multiplication factor5 (or penalty), k, depends on the central authority’s assessment 

of the ADI’s risk management practices and the results of a simple backtest. It is 

determined by the number of times actual losses exceed a particular day’s VaR forecast 

(Basel Committee on Banking Supervision (1996, 2006)). The minimum multiplication 

factor of 3 is intended to compensate for various errors that can arise in model 

implementation, such as simplifying assumptions, analytical approximations, small 

sample biases and numerical errors that tend to reduce the true risk coverage of the 

model (see Stahl (1997)). Increases in the multiplication factor are designed to increase 

the confidence level that is implied by the observed number of violations to the 99 per 

cent confidence level, as required by regulators (for a detailed discussion of VaR, as 

well as exogenous and endogenous violations, see McAleer (2009), Jiménez-Martin et 

al. (2009), and McAleer et al. (2009)). 

 

In calculating the number of violations, ADIs are required to compare the forecasts of 

VaR with realised profit and loss figures for the previous 250 trading days. In 1995, the 

                                                            
5 The formula in equation (1) is contained in the 1995 amendment to Basel I, while Table 1 appears for 
the first time in the Basel II Accord in 2004.  



7 

1988 Basel Accord (Basel Committee on Banking Supervision (1988)) was amended to 

allow ADIs to use internal models to determine their VaR thresholds (Basel Committee 

on Banking Supervision (1995)). However, ADIs that propose using internal models are 

required to demonstrate that their models are sound. Movement from the green zone to 

the red zone arises through an excessive number of violations. Although this will lead to 

a higher value of k, and hence a higher penalty, violations will also tend to be associated 

with lower daily capital charges.6 

 

Value-at-Risk refers to the lower bound of a confidence interval for a (conditional) 

mean, that is, a “worst case scenario on a typical day”. If interest lies in modelling the 

random variable,  Yt
, it could be decomposed as follows: 

 

 1( | )t t t tY E Y F   . (2) 

 

This decomposition states that  Yt
 comprises a predictable component,  E(Y

t
| F

t1
) , which 

is the conditional mean, and a random component, 
t
. The variability of  Yt

, and hence 

its distribution, is determined by the variability of 
t
. If it is assumed that  t

 follows a 

conditional distribution, such that: 

 
),(~ 2

ttt D                                                         

 

where  t
 and   t

 are the conditional mean and standard deviation of  t
, respectively, 

these can be estimated using a variety of parametric, semi-parametric or non-parametric 

methods. The VaR threshold for  Yt
 can be calculated as: 

 

 1( | )t t t tVaR E Y F   , (3) 

 

where   is the critical value from the distribution of 
t

 to obtain the appropriate 

confidence level. It is possible for 
t
 to be replaced by alternative estimates of the 

                                                            
6 The number of violations in a given period is an important (though not the only) guide for regulators to 
approve a given VaR model. 



8 

conditional standard deviation in order to obtain an appropriate VaR (for useful reviews 

of theoretical results for conditional volatility models, see Li et al. (2002) and McAleer 

(2005), who discusses a variety of univariate and multivariate, conditional, stochastic 

and realized volatility models).  

 

Some recent empirical studies (see, for example, Berkowitz and O'Brien (2001), 

Gizycki and Hereford (1998), and Pérignon et al. (2008)) have indicated that some 

financial institutions overestimate their market risks in disclosures to the appropriate 

regulatory authorities, which can imply a costly restriction to the banks trading activity. 

ADIs may prefer to report high VaR numbers to avoid the possibility of regulatory 

intrusion. This conservative risk reporting suggests that efficiency gains may be 

feasible. In particular, as ADIs have effective tools for the measurement of market risk, 

while satisfying the qualitative requirements, ADIs could conceivably reduce daily 

capital charges by implementing a context-dependent market risk disclosure policy. For 

a discussion of alternative approaches to optimize VaR and daily capital charges, see 

McAleer (2008) and McAleer et al. (2009). 

 

The next section describes several volatility models that are widely used to forecast the 

1-day ahead conditional variances and VaR thresholds.  

 

3. Models for Forecasting VaR 

 

ADIs can use internal models to determine their VaR thresholds. There are alternative 

time series models for estimating conditional volatility. In what follows, we present 

several conditional volatility models to evaluate strategic market risk disclosure, namely 

GARCH, GJR and EGARCH, with normal, Student-t and Generalized normal 

distribution errors, where the parameters are estimated.  

 

These models are chosen as they are well known and widely used in the literature. For 

an extensive discussion of the theoretical properties of several of these models, see Ling 

and McAleer (2002a, 2002b, 2003a) and Caporin and McAleer (2010a). As an 

alternative to estimating the parameters, we also consider the exponential weighted 



9 

moving average (EWMA) method by Riskmetrics (1996) and Zumbauch, (2007) that 

calibrates the unknown parameters. We include a section on these models to present 

them in a unified framework and notation, and to make explicit the specific versions we 

are using. Apart from EWMA, the models are presented in increasing order of 

complexity.  

 

3.1 GARCH 

 

For a wide range of financial data series, time-varying conditional variances can be 

explained empirically through the autoregressive conditional heteroskedasticity 

(ARCH) model, which was proposed by Engle (1982). When the time-varying 

conditional variance has both autoregressive and moving average components, this 

leads to the generalized ARCH(p,q), or GARCH(p,q), model of Bollerslev (1986). It is 

very common to impose the widely estimated GARCH(1,1) specification in advance.  

 

Consider the stationary AR(1)-GARCH(1,1) model for daily returns, ty :   

 

 t 1 2 t-1 t 2y = φ +φ y +ε , φ < 1  (4) 

 

for nt ,...,1 , where the shocks to returns are given by:  

 

 t t t t

2
t t -1 t-1

ε = η h , η ~ iid(0,1)

h =ω+αε + βh ,
 (5) 

 

and 0, 0, 0      are sufficient conditions to ensure that the conditional variance 

0th . The stationary AR(1)-GARCH(1,1) model can be modified to incorporate a non-

stationary ARMA(p,q) conditional mean and a stationary GARCH(r,s) conditional 

variance, as in Ling and McAleer (2003b). 

 

3.2 GJR 

 

In the symmetric GARCH model, the effects of positive shocks (or upward movements 

in daily returns) on the conditional variance, th , are assumed to be the same as the 



10 

negative shocks (or downward movements in daily returns). In order to accommodate 

asymmetric behaviour, Glosten, Jagannathan and Runkle (1992) proposed a model 

(hereafter GJR), for which GJR(1,1) is defined as follows:  

 

 2
t t-1 t -1 t-1h =ω+(α+ γI(η ))ε + βh ,  (6) 

 

where 0,0,0,0    are sufficient conditions for ,0th  and )( tI   is an 

indicator variable defined by: 

 

  
1, 0

0, 0
t

t
t

I






  
 (7) 

 

 as t  has the same sign as t . The indicator variable differentiates between positive 

and negative shocks, so that asymmetric effects in the data are captured by the 

coefficient  . For financial data, it is expected that 0  because negative shocks 

have a greater impact on risk than do positive shocks of similar magnitude. The 

asymmetric effect, ,  measures the contribution of shocks to both short run persistence, 

2  , and to long run persistence, 2    . Although GJR permits asymmetric 

effects of positive and negative shocks of equal magnitude on conditional volatility, the 

special case of leverage, whereby negative shocks increase volatility while positive 

shocks decrease volatility (see Black (1976) for an argument using the debt/equity 

ratio), cannot be accommodated (for further details on asymmetry versus leverage in the 

GJR model, see Caporin and McAleer (2010b)). 

 

3.3 EGARCH 

 

An alternative model to capture asymmetric behaviour in the conditional variance is the 

Exponential GARCH, or EGARCH(1,1), model of Nelson (1991), namely:  

 

 t -1 t-1
t t-1

t-1 t-1

ε ε
logh =ω+α + γ + βlogh , | β |< 1

h h
 (8) 

 



11 

where the parameters ,    and   have different interpretations from those in the 

GARCH(1,1) and GJR(1,1) models.  

 

EGARCH captures asymmetries differently from GJR. The parameters   and   in 

EGARCH(1,1) represent the magnitude (or size) and sign effects of the standardized 

residuals, respectively, on the conditional variance, whereas   and    represent the 

effects of positive and negative shocks, respectively, on the conditional variance in 

GJR(1,1). Unlike GJR, EGARCH can accommodate leverage, depending on the 

restrictions imposed on the size and sign parameters. 

 

As noted in McAleer et al. (2007), there are some important differences between 

EGARCH and the previous two models, as follows: (i) EGARCH is a model of the 

logarithm of the conditional variance, which implies that no restrictions on the 

parameters are required to ensure 0th ; (ii) moment conditions are required for the 

GARCH and GJR models as they are dependent on lagged unconditional shocks, 

whereas EGARCH does not require moment conditions to be established as it depends 

on lagged conditional shocks (or standardized residuals); (iii) Shephard (1996) observed 

that 1||   is likely to be a sufficient condition for consistency of QMLE for 

EGARCH(1,1); (iv) as the standardized residuals appear in equation (7), 1||   would 

seem to be a sufficient condition for the existence of moments; and (v) in addition to 

being a sufficient condition for consistency, 1||   is also likely to be sufficient for 

asymptotic normality of the QMLE of EGARCH(1,1).   

 

The three conditional volatility models given above are estimated under the following 

distributional assumptions on the conditional shocks: (1) normal, and (2) t, with 

estimated degrees of freedom. As the models that incorporate the t distributed errors are 

estimated by QMLE, the resulting estimators are consistent and asymptotically normal, 

so they can be used for estimation, inference and forecasting. 

 

3.4 Exponentially Weighted Moving Average (EWMA) 

 



12 

As an alternative to estimating the parameters of the appropriate conditional volatility 

models, Riskmetrics (1996) developed a model which estimates the conditional 

variances and covariances based on the exponentially weighted moving average 

(EWMA) method, which is, in effect, a restricted version of the ARCH( ) model. This 

approach forecasts the conditional variance at time t as a linear combination of the 

lagged conditional variance and the squared unconditional shock at time 1t  . The 

EWMA model calibrates the conditional variance as: 

 

 2
t t-1 t-1h = λh +(1- λ)ε  (9) 

 

where   is a decay parameter. Riskmetrics (1996) suggests that   should be set at 0.94 

for purposes of analysing daily data. As no parameters are estimated, there are no 

moment or log-moment conditions. 

 

4. Data  

 

The data used for estimation and forecasting are the closing daily prices for Standard 

and Poor’s Composite 500 Index (S&P500), which were obtained from the Ecowin 

Financial Database for the period 3 January 2000 to 16 March 2010. Although it is 

unlikely that an ADI’s typical market risk portfolio only tracks the S&P500 index, it is 

used as an illustration of the broad movements of profits and losses of the equity 

portfolios of ADIs.  

 

If tP  denotes the market price, the returns at time t ( )tR  are defined as: 

  1log / t t tR P P . (10) 

 

 

[Insert Figures 1-2 and Table 2 here] 

 

Figure 1 shows the S&P500 returns, for which the descriptive statistics are given in 

Table 2. The extremely high positive and negative returns are evident from September 



13 

2008 onward, and have continued well into 2009. The mean is close to zero, and the 

range is between -9.5% and +11%. The Jarque-Bera Lagrange multiplier test rejects the 

null hypothesis of normally distributed returns. As the series displays high kurtosis, this 

would seem to indicate the existence of extreme observations, as can be seen in the 

histogram, which is not surprising for daily financial returns data. 

 

Several measures of volatility are available in the literature. In order to gain some 

intuition, we adopt the measure proposed in Franses and van Dijk (1999), where the true 

volatility of returns is defined as: 

 

   2

1|  t t t tV R E R F , (11) 

 

where 1tF  is the information set at time t-1.  

 

Figure 2 shows the S&P500 volatility, as the square root of Vt in equation (11). The 

series exhibit clustering that needs to be captured by an appropriate time series model. 

The volatility of the series appears to be high during the early 2000s, followed by a 

quiet period from 2003 to the beginning of 2007. Volatility increases dramatically after 

August 2008, due in large part to the worsening global credit environment. This 

increase in volatility is even higher in October 2008. In less than four weeks in October 

2008, the S&P500 index plummeted by 27.1%. In less than three weeks in November 

2008, starting the morning after the US elections, the S&P500 index plunged a further 

25.2%. Overall, from late August 2008, US stocks fell by a scarcely unbelievable 42.2% 

to reach a low on 20 November 2008.  

 

An examination of daily movements in the S&P500 index from 2000 suggests that large 

changes by historical standards are 4% in either direction. From January 2000 to August 

2008, there was a 0.31% chance of observing an increase of 4% or more in one day, and 

a 0.18% chance of seeing a reduction of 4% or more in one day. Therefore, 99.5% of 

movements in the S&P500 index during this period had daily swings of less than 4%. 

Prior to September 2008, the S&P500 index had only 7 days with massive 4% gains, but 

since September 2008, there have been a further 12 such days. On the downside, before 



14 

the current stock market meltdown, the S&P500 index had only 4 days with huge 4% or 

more losses whereas, during the recent panic, there were a further 17 such days.  

 

This comparison is between more than 99 months and less than 6 months. During this 

short time span of financial panic, the 4% or more gain days chances increased 80 times 

while the chances of 4% or more loss days increased 32 times. Such movements in the 

S&P500 index are unprecedented. 

 

5. Robust Forecasting of VaR and Evaluation Framework 

 

As seen in McAleer et al. (2010), the global financial crisis has affected the best risk 

management strategies by changing the optimal model for minimizing daily capital 

charges. The objective here is to provide a robust risk management strategy, namely one 

that does not change over time, even in the context of a GFC. This robust risk 

management strategy also has to lead to daily capital charges that are not excessive, and 

violation frequencies that are compatible with the Basel II Accord.  

 

ADIs need not restrict themselves to using only a single risk model. We propose a risk 

management strategy that consists in choosing a forecast from among different 

combinations of alternative risk models to forecast VaR. McAleer et al. (2010) 

developed a risk management strategy that used combinations of several models for 

forecasting VaR. It was found that an aggressive risk management strategy (namely, 

choosing the supremum of VaR forecasts, or upperbound) yielded the lowest mean 

capital charges and largest number of violations. On the other hand, a conservative risk 

management strategy (namely, by choosing the infinum, or lowerbound) had far fewer 

violations, and correspondingly higher mean daily capital charges.  

 

In this paper, we forecast VaR using combinations of the forecasts of individual VaR 

models, namely the rth percentile of the VaR forecasts of a set of univariate conditional 

volatility models. Alternative single models with different error distributions and 

several combinations are compared over three different time periods to investigate 

which, if any, of the risk management strategies may be robust.  

 



15 

We conduct an exercise to analyze the performance of existing VaR forecasting models, 

as permitted under the Basel II framework, when applied to the S&P500 index. 

Additionally, we analyze twelve new strategies based on combinations of the previous 

standard single-model forecasts of VaR, namely: lowerbound (0th percentile), 

upperbound (100th percentile), average, and nine additional strategies based on the 10th 

through to the 90th percentiles. It is intended to determine whether we can select a 

robust VaR forecast irrespective of the time period, and to provide reasonable daily 

capital charges and number of violation penalties. 

 

5.1 Evaluating Crisis-Robust Risk Management Strategies 

 

In Table 3 the performance of the different VaR forecasting models is evaluated using 

several standard criteria that are relevance for the risk manager, namely: daily capital 

charges (DCC), number of violations (NoV), the failure rate (that is, the ratio of NoV to 

number of days) (FailRa), accumulated losses 7  (AcLoss), and the value of the 

asymmetric linear tick loss function 8  (AlTick) that allows a comparison of model 

performance. As Giacomini and Komunjer (2005) argue, the tick loss function is the 

implicit loss function whenever the target is a forecast of a particular α-quantile. This 

loss function takes into account the magnitude of the implicit cost associated with VaR 

forecasting errors. As VaR estimates are used frequently to assist in strategic financial 

decision-making and managing market risk, VaR forecasting errors can imply financial 

distress, such as incorrectly estimating daily capital charges subject to regulatory 

control.   

 

The performance criteria are calculated for each model and error distribution, and for 

each of the three sub-samples, before, during, and after the 2008-09 GFC, where before 

is from 2 January 2008 to 11 August 2008, during is from 12 August 2008 to 9 March 

2009, and after is from 10 March 2009 to 16 March 2010. Table 3 shows the 

                                                            
7 López (1999) suggested measuring the accuracy of the VaR forecast on the basis of the distance 
between the observed returns and the forecasted VaR values if a violation occurs:  

      




01 1 11 1
1

0

R VaR if R and R VaRt t tt t t t
t

otherwise
 , a preferred VaR model is the one that minimizes 

the total loss value, 1 1t
T

tt    . 

8 The tick loss function of order α defined as     1 01 1 1L e e et t t      , where 1 1 1e R VaRt t t    . 



16 

performance statistics for comparing the VaR forecasts for each sub-period. We also 

include 12 new forecasting strategies based on combinations of individual model VaR 

forecasts, namelt : lowerbound, upperbound, average, and nine additional strategies 

based on the 10th through to the  90th percentiles, including the median. 

 

No risk model is found always to be superior to its competitors as there is no strategy 

that optimizes every evaluation statistic for the three sub-periods. Nonetheless, the 50th 

percentile strategy (namely, the median - shaded row of Table 3) is found to be robust, 

as it produces adequate VaR forecasts that exhibit stable results across different periods 

relative to the other risk models. Furthermore, we compute for each sub-sample and 

each statistic the values of the 1st, 2nd and 3rd quartiles. In Table 4, the median strategy 

statistics fall in almost all cases in the 2nd quartile. The EGARCH-t model performance 

is close to the median strategy, except that the former has average daily capital charges, 

(AvDCC) slightly higher for the three periods. In general, the median strategy provides 

a robust VaR forecast, regardless of whether there is a GFC. 

 

[Insert Tables 3-4 here] 

 

As can be seen in Table 3, before the GFC, AvDCC based on the Riskmetrics VaR 

forecast is lower (9.03%) than that based on the median (9.77%). During the GFC, 

AvDCC based on the median strategy increased drastically (20.57%), although not as 

much as when the Riskmetrics model is used (22.51%). Finally, after the GFC, AvDCC 

decreased, becoming smaller for the median strategy (10.95%) than for the Riskmetrics 

model (11.19%).  

 

Figure 3 shows the evolution of the evaluation statistics for the strategies based on the 

10th through to the 90th percentiles in the three sub-samples. GFC-robustness, in our 

context, implies a trade-off between the number of violations and daily cost of capital 

charges. In general, when the chosen percentile increases, it is found that DCC 

decreases and NoV increases, all the more so when the selected percentile is greater 

than the median.  

 

Figure 4 shows VaR forecasts using four models (Riskmetrics, median, upperbound and 

lowerbound) and S&P500 returns, where the vertical axis represents returns, and the 



17 

horizontal axis represents the period from 2 January 2008 to 16 March 2010. The 

S&P500 returns are given as the upper blue line that fluctuates around zero. The upper 

red line represents the supremum of the VaR calculated for the individual volatility 

models, which reflects an aggressive risk management strategy, whereas the lower 

green line represents the infimum of the VaR calculated for the individual volatility 

models, which reflects a conservative risk management strategy. These two lines 

correspond to a combination of alternative risk models. The brown line is the median of 

the VaR, calculated for the individual models of volatility. As a benchmark, the black 

line denotes the Riskmetrics model.  

 

[Insert Figures 3-4 here] 

 

As can be seen in Figure 4, VaR forecasts obtained from the different models of 

volatility have fluctuated, as expected, during the first few months of 2008. It has been 

relatively low, at below 5%, and relatively stable between April and August 2008.  

Around September 2008, VaR started increasing until it peaked in October 2008, 

between 10% and 15%, which is almost a four-fold increase in VaR in a matter of one 

and a half months. In the last two months of 2008, VaR decreased to values between 5% 

and 8%, which is still twice as large as it had been just a few months earlier. Therefore, 

volatility has increased substantially during the GFC, and has remained relatively high 

thereafter.  

 

Figure 5 includes daily capital charges based on the median VaR for the previous 60 

days, which are the lower lines. The red line corresponds to the aggressive risk 

management strategy based on the supremum of the daily VaR forecasts of the 

alternative models of volatility, the green line corresponds to the conservative risk 

management strategy based on the infinum of the VaR forecasts, and the brown line 

denotes the robust strategy based on the median of the VaR forecasts. As a benchmark, 

we include DCC based on the Riskmetrics model. Table 3 shows that the Riskmetrics 

AvDCC and FailRa are higher than the median AvDCC, except for before the GFC. 

This can be seen in Figure 5 when comparing the DCC Riskmetrics and DCC median 

lines. The DCC median line is close to the upperbound DCC line virtually throughout 

the sample, but FailRa for the median strategy is always much lower and is falling 

within the Basel Accord limits than when the upperbound strategy is considered.  



18 

 

 [Insert Figure 5 here] 

 

6. Conclusion 

 

In this paper we proposed robust risk forecasts that use combinations of several 

conditional volatility models for forecasting VaR. Different strategies for combining 

models were compared over three different time periods, using S&P500 to investigate 

whether we can determine a GFC-robust risk management strategy. Backtesting 

provided evidence that a risk management strategy based on VaR forecast 

corresponding to the 50th percentile (median) of the VaR forecasts of a set of univariate 

conditional volatility models is robust, in that it yields reasonable daily capital charges, 

number of violations that do not jeopardize institutions that might use it, and more 

importantly, is invariant before, during and after the 2008-09 GFC.  

 

It is worth noting that, as in McAleer et al. (2010), the VaR model minimizing DCC 

before, during and after the GFC changed frequently. Although the median model is not 

derived as the best model for minimizing DCC and the number of violation penalties, it 

is nevertheless a model that balances daily capital charges and violation penalties in 

minimizing DCC.  

 

The idea of combining different VaR forecasting models is entirely within the spirit of 

the Basel II Accord, although its use would require approval by the regulatory 

authorities, as for any forecasting model. This approach is not computationally 

demanding, even though several models have to be specified and estimated over time. 

Further research is needed to compute the standard errors of the forecasts of the 

combination models, including the median forecast. 

 



19 

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20 

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22 

 

Figure 1. Daily Returns on S&P500 Index 

 3 January 2000 – 16 March 2010 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2. Daily Volatility in S&P500 Returns 

3 January 2000 – 16 March 2010 

 

 

 

-12%

-8%

-4%

0%

4%

8%

12%

00 01 02 03 04 05 06 07 08 09

0%

2%

4%

6%

8%

10%

12%

00 01 02 03 04 05 06 07 08 09



23 

Figure 3. Criteria for Comparing Percentile Strategies 

 

BEFORE CRISIS 

 

 

 

 

 

 

 

 

 

 

DURING CRISIS 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

AFTER CRISIS 

 

 

 

 

 

 

 

 



24 

 

Figure 4. VaR for S&P500 Returns 

2 January 2008 – 16 March 2010 

 

 

 

-20

-15

-10

-5

0

5

10

15

2008M07 2009M01 2009M07 2010M01

S&P Returns VaR Upperbound
VaR RSKM VaR Median
VaR Lowerbound

Before During After

Note: The upper blue line represents daily returns for S&P500 index. The 
upper red line represents the supremum, the lower green line the infinum, the 
bold brown line the median of the VaR forecasts of the alternative models (see 
Section 3), and the black line is the Riskmetrics VaR forecast. 

 Peak_value Date Trough_value date 

S&P Returns 1305.323 11/8/2008 676.5302 9/3/2009



25 

 

Figure 5.  DCC and S&P500 Returns 
 

 

 

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

-40

-30

-20

-10

0

10

20

2008M07 2009M01 2009M07 2010M01

S&P Returns DCC Lowerbound
DCC Median DCC RSKM
DCC Upperbound

Before During After



26 

 

 

Table 1: Basel Accord Penalty Zones 

 

Zone Number of Violations k 

Green 0 to 4 0.00 

Yellow 5 0.40 

 6 0.50 

 7 0.65 

 8 0.75 

 9 0.85 

Red 10+ 1.00 

Note: The number of violations is given for 250 business days. The 
penalty structure under the Basel II Accord is specified for the 
number of violations and not their magnitude, either individually or 
cumulatively.   

 



27 

Table 2. Descriptive Statistics for S&P500 Returns (%) 

3 January 2000 – 15 March 2010 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0

200

400

600

800

1,000

1,200

-10 -8 -6 -4 -2 0 2 4 6 8 10

Observations 2660

Mean      -0.009
Median   0.008
Maximum  10.96
Minimum -9.470
Std. Dev.   1.368
Skewness  -0.113
Kurtosis   11.06

Jarque-Bera  7209.7
Probability  0.0000



28 

 Table 3. Comparing Alternative Models of Volatility 
 BEFORE CRISIS  DURING CRISIS AFTER CRISIS 

Model AvDCC NoV FailRa AcLoss AlTick  AvDCC NoV FailRa AcLoss AlTick   AvDCC NoV FailRa AcLoss AlTick 

RSKM  9.03  4  2.5%  1.60  6.28    22.51  6  4.0%  6.21  16.27  11.19  5  1.9%  1.62  10.88 
GARCH  9.08  6  3.8%  1.89  6.42    21.39  7  4.7%  7.40  16.95  10.76  6  2.3%  1.74  10.73 
GJR  9.00  3  1.9%  1.03  5.75    20.11  4  2.7%  5.16  15.53  10.71  8  3.0%  2.75  11.10 

EGARCH  8.87  4  2.5%  1.13  5.82    19.92  10  6.7%  12.10  20.49  9.92  10  3.8%  3.81  11.52 
GARCH_t  11.16  1  0.6%  0.21  6.03    24.52  2  1.3%  2.85  15.51  13.67  1  0.4%  0.13  11.52 
GJR_t  10.80  1  0.6%  0.57  6.26    24.27  2  1.3%  2.88  15.57  12.21  3  1.1%  0.56  10.47 

EGARCH_t  10.75  1  0.6%  0.48  6.19    20.95  2  1.3%  4.51  15.23  11.14  4  1.5%  0.80  9.99 
GARCH_g  9.81  2  1.3%  0.79  5.90    22.11  5  3.4%  4.41  15.35  11.94  2  0.8%  0.73  10.75 
GJR_g  9.82  1  0.6%  0.80  5.96    21.97  3  2.0%  3.96  15.37  11.08  4  1.5%  1.34  10.39 

EGARCH_g  9.75  1  0.6%  0.72  5.89    19.58  6  4.0%  7.10  16.68  10.20  6  2.3%  2.22  10.56 
Inf  11.78  1  0.6%  0.21  6.42    25.26  2  1.3%  2.64  15.78  13.99  1  0.4%  0.07  11.56 
Sup  8.45  6  3.8%  2.06  6.28    20.01  11  7.4%  12.28  20.52  9.71  10  3.8%  4.31  11.89 
Mean  9.77  1  0.6%  0.69  5.82    20.73  3  2.0%  4.57  15.21  11.11  3  1.1%  0.99  10.21 

10th  Per.  11.43  1  0.6%  0.34  6.38    24.56  2  1.3%  2.87  15.62  13.34  2  0.8%  0.13  11.10 
20th Per.  10.81  1  0.6%  0.51  6.21    23.21  2  1.3%  3.49  15.49  12.39  2  0.8%  0.40  10.60 
30th Per.  10.37  1  0.6%  0.56  6.03    22.07  2  1.3%  3.96  15.34  11.85  2  0.8%  0.62  10.40 
40th Per.  10.06  1  0.6%  0.65  5.94    21.23  3  2.0%  4.40  15.32  11.38  3  1.1%  0.85  10.27 
50th Per. 
(Median) 

9.71  1  0.6%  0.76  5.86 
 
  20.57  3  2.0%  4.81  15.37      10.95  4  1.5%  1.07  10.14 

60th Per.  9.39  2  1.3%  0.87  5.80    22.29  5  3.4%  5.21  15.41  10.66  5  1.9%  1.51  10.29 
70th Per.  9.06  3  1.9%  1.05  5.80    21.86  7  4.7%  6.00  15.84  10.68  8  3.0%  2.05  10.53 
80th Per.  8.65  4  2.5%  1.45  5.95    21.32  8  5.4%  7.66  17.04  10.32  9  3.4%  2.90  11.06 
90th Per.  8.28  4  2.5%  1.81  6.11  ‐ 20.97  10  6.7%  10.53  19.19  10.01  10  3.8%  3.75  11.54 

Note: Percentage of Days Minimizing Daily Capital Charges (%DmDCC), Average of Daily Capital Charges (AvDCC), Number of Violations (NoV), Failure 
Rate (FailRa), Accumulated Loses (AcLoss), and Asymmetric Linear Tick loss function value (AlTick) for alternative models of volatility. 



29 

 

Table 4. Quartiles for Evaluation Statistics 
 

  Before        During      After   

Quartiles AvDCC NoV FailRa AcLoss Altick   DCC NoV FailRa AcLoss Altick.  DCC NoV FailRa AcLoss Altick. 

1  9.06  1.0  0.6%  0.65  5.88      20.34  2.5  1.7%  4.19  15.37    10.74  3.5  1.3%  0.77  10.43 

2  9.75  1.0  0.6%  0.79  5.96      21.39  4.0  2.7%  4.81  15.53    11.08  4.0  1.5%  1.34  10.73 

3   10.28  3.5  2.2%  1.08  6.23      22.31  6.0  4.0%  6.65  16.48    11.56  6.0  2.3%  1.98  10.99 

Models                                     

RSKM  1  4  4  4  4      4  4  4  3  3    3  3  3  3  3 

EGARCH_t  4  1  1  1  3      2  1  1  2  1    3  2  2  2  1 

Median  2  1  1  2  1      2  2  2  2  1    2  2  2  2  1 

Note: The top part of the table gives the value of the rth quartile for the evaluation statistics in Table 3 when considering all the GARCH-family models and 
Riskmetrics. The bottom part of the table is the quartile where the evaluation statistics fell for the RSKM, EGARCH_t and Median models. For example, for 
the RSKM model, an entry of ‘one’ in the first column means that the corresponding AvDCC fell in the first quartile.