Bootstrap Predictions for
VaR and ES using
Extreme Value Theory
The Impact of the Financial Crisis and Adapting to Basel III
Christiaan Hoiting
s1889494
MSc. Thesis EORAS: Econometrics University of Groningen
Supervisors:
Prof. Dr. P.A. BEKKER Ir. Drs. G.J. van WIGGEN Co-assessor:
University of Groningen
MSc. Thesis EORAS: Econometrics
Bootstrap Predictions for VaR and ES using Extreme Value
Theory
Author:
Christiaan Hoiting
s1889494
Supervisors:
Prof. Dr. P.A. Bekker (RuG)
Ir. Drs. G.J. van Wiggen (EY)
January 14, 2015
Abstract
In this research we have analysed the finite sample properties of different VaR and ES estimators. Using a GJR-Garch bootstrap method we extend the standard procedure to incorporate leverage effects in financial returns. Both stressed and tranquil financial times are taken into account in the empirical and Monte Carlo analysis. Results showed that the Hill estimator was best able to predict one-day 99%VaR estimates. The Generalized Pareto Distribution (GPD) method was best able to predict one-day 97.5% ES estimates. In terms of accuracy of the estimator we find that for the conditional volatility estimators the GPD method resulted in the least wide bootstrap prediction intervals. VaR could be estimated slightly more precise than ES in terms of interval width. However, the differences were negligible.
Contents
1 Introduction 1
2 Development of Financial Regulations 3
2.1 The Onset of Basel I . . . 3
2.2 Towards Basel 2 . . . 4
2.3 Basel 3 and 3.5 . . . 4
3 Theoretical Framework 7 3.1 Dynamics of Time Series . . . 7
3.1.1 ARCH Processes . . . 8
3.1.2 GARCH Processes . . . 8
3.2 Value-at-Risk and Expected Shortfall . . . 9
3.2.1 Value-at-Risk . . . 9
3.2.2 Expected Shortfall . . . 11
3.2.3 From Value-at-Risk to Expected Shortfall under Basel 3.5 . . . 12
3.3 Extreme Value Theory . . . 13
3.3.1 Generalised Pareto Distribution Method . . . 14
3.3.2 Hill Estimator . . . 18
3.4 Bootstrap Algorithms . . . 19
3.4.1 Historical Simulation . . . 20
3.4.2 GARCH Models . . . 20
3.4.3 Other Bootstrap Procedures and options . . . 23
4 Monte Carlo Analysis 25 4.1 Historical Simulation Approach . . . 25
4.2 GARCH Approach . . . 26
4.2.1 Stressed Simulations . . . 27
4.2.2 Tranquil Simulations . . . 29
4.2.3 Discussion of MC Results . . . 31
5 Empirical Data Analysis 31 5.1 Financial Data in General . . . 32
List of Tables
1 Monte Carlo Results Historical Simulation . . . 26
2 Monte Carlo Results GJR-Garch(1,1) VaR, high persistence . . . 28
3 Monte Carlo Results GJR-Garch(1,1) ES, high persistence . . . 28
4 Monte Carlo Results GJR-Garch(1,1) VaR, low persistence . . . 30
5 Monte Carlo Results GJR-Garch(1,1) ES, low persistence . . . 30
6 Descriptive Statistics . . . 32
7 Test Statistics Losses . . . 36
8 Residual Diagnostics . . . 37
9 Estimation Results Empirical Data . . . 42
List of Figures
1 Pillars Basel II . . . 42 Main Revisions of the Fundamental Review of the Trading Book . . . 5
3 Loss Distribution with VaR and ES for α = 70% . . . 10
4 Extreme Value Theory Models . . . 13
5 AEX Index from 01/01/2005-01/01/2014 . . . 32
6 Log Losses AEX from 01/01/2000-01/01/2014 . . . 33
7 QQ-plot Normal Distribution for AEX . . . 34
8 ACF Plot for AEX and Squared AEX . . . 35
9 GPD Fitted to the Residuals of the GJR-GARCH Model for AEX . . . 37
10 Probability and Quantile Plot of GPD Fit for AEX . . . 37
11 Backtest VaR AEX, sample size=1000 . . . 43
12 Backtest ES AEX, sample size=1000 . . . 44
13 Backtest VaR BRENT, sample size=250 . . . 45
Acknowledgments
This thesis is the last step in the pursuit of the MSc. degree in Econometrics, Actuarial Sciences and Operations Research at the University of Groningen. Here, Econometrics was the chosen specialisation. The thesis was written in combination with an internship at EY FSRisk in Amsterdam. I would like to thank all the colleagues for their tips and the discussions regarding the thesis. The internship was a fruitful experience.
Furthermore, I would like to thank my supervisor, Prof. Dr. Bekker, for his insightful suggestions. He has shown me the need for thorough and consistent notation which makes this thesis a better piece to read.
1
Introduction
Recent years have shown that financial markets are evidently instable and that excessively large losses can lead to failure of financial institutions. In order to reduce the number of bankruptcies, institutions are obliged to calculate risk metrics on which they base their capital requirements. Due to the financial crisis the need for more adequate metrics now has come to light. The Basel Committee of Banking Supervision (BCBS) is the institution proposing these appropriate metrics and now has proposed to move from using the Value-at-Risk (VaR) metric towards Expected Shortfall (ES). VaR is mathematically seen simply a quantile of a loss distribution and for risk management purposes usually the 99% quantile is used. It can answer the question how large the probability is that a loss exceeds a given amount in a fixed time period. For example, if one computes a one-day VaR of 1,000,000 Euro with probability 1%, one could say that there is a 1% chance that the losses on a particular portfolio could exceed this 1,000,000 Euro in the next day. In an informal way, one could also say that it is expected that in 1 out of 100 days the loss on a portfolio would be larger than 1,000,000 Euro. Closely related to VaR is ES. ES is namely the expectation of the losses larger than the VaR at a given quantile, i.e. E [L|L > VaR], where L are the losses. ES looks further into the tail of a loss distribution and is therefore able to give an estimate how large losses can become once a VaR is breached. However, the difficulty of estimating ES comes from the lack of many observations in the tail.
Risk managers would like to have their risk measurements as accurate as possible since an overly prudent estimation would lead to an overestimation of capital and therefore lower return on equity and an underestimation would lead to a lack of capital and could bring the bank in financial stress. Most financial institutions use the non-parametric, historical simulation approach as the method to compute their VaR estimates. Hoogerheide and van Dijk (2010) mention that this mainly has to do with time constraints for ’real time’ decision making. Simulation based analysis, with the required level of precision, usually takes too long for financial institutions in order to be used.
The Basel Committee has prescribed that financial institutions have to base their risk metric for capital requirements on both normal market conditions and stressed market conditions. In stressed market conditions it is usually the case that volatility increases and that more spikes follow each other in asset returns. Here, Extreme Value Theory (EVT) comes into play. The usefulness of EVT in financial market data is to capture and model the tail behaviour of a distribution even when few observations are available. Events in the tail are usually scarce but can have a large impact on the Profit and Loss (PnL) statements of a financial institution and therefore an appropriate method of tail estimation is needed.
order to gain a more stable risk metric. The purpose of this research is connected to this statement. The question I would like to answer is which estimation method of expected shortfall leads to more stable risk metric estimates and whether expected shortfall is really more stable than VaR when taking estimation risk into account. The answer to this question is pursued by using extreme value distributions and bootstrap methods for GARCH models (Bollerslev, 1986) to obtain confidence intervals of the risk metrics. Several tail estimators are used and compared for their adequacy.
Pascual et al. (2005) have developed a GARCH bootstrap procedure for the GARCH(1,1) model which I will extend to a bootstrap procedure for the Glosten, Jagannathan and Runkle (GJR)-GARCH(1,1) model originally developed by Glosten et al. (1993) in order to take into account the asymmetry in financial return data. History has namely shown that financial returns and the corresponding volatilities react more heavily to negative shocks than to positive shocks. Christoffersen and Gon¸calves (2005) show that using the GARCH bootstrap method for confidence intervals around VaR and ES estimates is valid even in the presence of conditional heteroskedasticity and non-normality. This is also a useful result when working with financial data.
Furthermore, in this research I would like to contribute to the literature using the recent extreme returns from the 2008 financial crisis. Tail modelling techniques can be analysed for their use in risk management. Especially for these stressed market conditions EVT might be a relevant issue. Even though there is a substantial literature regarding ES and VaR making use of EVT techniques it is argued that using the relatively recent data an updated view can be given regarding the use of extreme value theory in financial risk management. Kourouma et al. (2011) have taken a step using the 2008 data in combination with extreme value theory. They adopt the GJR-Garch model and apply EVT methods. Their results show that, as expected, the EVT in general performs better than historical simulation under normal and abnormal market condition. However, they do not implement bootstrapping techniques and there our research will be an addition. Focussing on different tail estimators with the bootstrap percentile intervals as a basis, this paper is different from the existing literature. The implementation of the GJR-GARCH bootstrap procedure also adds to the main literature. Furthermore, sample size can be quite important. The Basel Committee prescribes that estimations have to be done based on a limited number of days (250). Therefore, this will be taken into account in this research.
Based on all this information I set up the following research question:
”Which estimation method based on GARCH models leads to the most stable estimation of VaR and ES and is there a substantial difference in stability between the two?
The question will be answered using two sub-questions. Firstly, I look at the bootstrap confidence intervals. Secondly, I examine actual empirical data and perform formal backtests.
the research. Firstly, in Chapter 5 a Monte Carlo analysis with a data generating process is performed. Secondly, in Chapter 6 the data is introduced and the backtest and bootstrap results are displayed. Finally, the last chapter concludes and discusses further developments and research opportunities.
2
Development of Financial Regulations
2.1
The Onset of Basel I
In the aftermath of several financial crises the need of regulation in the financial sector was large. The industry needed a level playing field in which some standard practices had to be introduced regarding banks’ capital requirements.
The first Basel Capital Accord was released in 1988 and was called the ”International convergence of capital measurement and capital standards” document. The Basel committee at that time consisted of a group of central bank representatives of 12 different countries. The main proposal was first released as a consultative document in order to receive amendments from the financial sector and academia. Based on the comments from the industry the proposition was revised and released as a framework for the financial sector, which was not actually binding. Regulatory institutions in the different countries were free to choose whether they would actually implement the recommendations. The 1988 document was mainly focused on credit risk and the main objectives of the committee were to consolidate the stability of the banking sector and to reduce competition between international banks. This view led to specific capital requirements based on the types of assets which were in a bank’s possession, and the definition of Tier 1 and 2 capital.
After the release of the first consultative document, which was thus mainly based on credit risk, in 1991 the Basel Committee started a new initiative to incorporate market risk in the regulatory framework. They proposed amendments in which banks had to identify their ”trading book”. Balthazar (2006) identifies the trading book as the set of positions in financial instruments (including derivatives and other off-balance sheet items) held for the purpose of making short-term profits due to variation in prices or from brokering and/or market-making activities and the use of hedging for other positions in the trading book.
These new amendments also led to the definition of Tier 3 capital and the implementation of new capital requirement calculations; the Standardized and the Internal Models approach. The Standardized approach, as the name suggests, made use of the standard calculations of the Basel I framework. The Internal Models approach introduced the VaR calculations. This was the first time VaR models were officially proposed as an official risk measurement metric even though banks were already using the method, in an even more advanced way, for their own sake.
changing economic environment the need for a more elaborate framework was evident.
2.2
Towards Basel 2
The Basel 2 Capital Accord consists of three consultative documents from which the final one was published in 2004. The two main goals from the first accord, as mentioned above, remained and a third objective was added; the promotion of more stringent practice in the risk management activities of banks. Internal risk management practices, such as the collection of data and the development of models, became more and more important.
The Basel 2 framework stands on three pillars as shown in figure 1.
Figure 1: Pillars Basel II
As also mentioned in Sarma (2007) the first pillar focuses on the minimum regulatory standards and is a revised framework for capital adequacy standards based on the foundations of Basel 1. This pillar deals with credit, operational and market risk. The preferred risk metric for market risk in Basel 2 is VaR and therefore the first pillar is mostly connected to this research. The second pillar provides the banks a framework to incorporate risks which are not completely covered by the Basel accord. Moreover, it is a regulatory response to the first pillar such that regulators are able to supervise banks whether they meet the requirements and, if necessary, impose penalties.
Pillar three is focused on disclosure requirements of financial institutions in order to make the bank’s risk profile more transparent for investors.
2.3
Basel 3 and 3.5
and internal control regulation, market liquidity risk became of the keystones of the accord. Banks are now required to mitigate the risk of a bank run by keeping different levels of reserves for their different types of deposits and loans. For liquidity purposes, two new ratios were introduced. The ”Liquidity Coverage Ratio” (LCR) and the ”Net Stable Funding Ratio” (NSFR). The LCR is basically the short term liquidity ratio and the NSFR is aimed at a long term period of extended stress. we will not go into much details about these new capital ratios and other capital requirements introduced by Basel III, these are well described in many available sources including the well known Basel Committee working papers concerning capital and liquidity rules 1. The main focus here is on the revisions to the trading book
which is defined in the previous section. In this section we will briefly outline the main revision. Any particular details which are necessary for this research will be addressed in the corresponding sections.
In the second consultative document of ”The Fundamental Review of the Trading Book”, which is also called Basel 3.5, the BCBS (2013) argues that the revisions are aimed at contributing to a more resilient banking sector by strengthening capital standards for market risk. The capital requirements for trading book activities were not adequate to cover the large losses. In figure 2 the main revisions for the trading book are stated.
Figure 2: Main Revisions of the Fundamental Review of the Trading Book
The first thing discussed is the boundary between the trading and the banking book. In the current Basel Accord the criterion is quite subjective and banks are self-determining the placement of positions. In the new framework banks have to designate an instrument beforehand and are not allowed (only in extraordinary circumstances) to change trading book instruments to the banking book. ”Market conditions” alone are not a sufficient condition for the possibility of switching. Also note that if this switch leads to a lower capital charge, the difference will be disclosed as an additional Pillar 1 (IRC) charge. Furthermore, the supervision for instruments in the trading book will become much stricter, positions have to be marked-to-market daily in the PnL statement and policies have to be disclosed to the public and supervisors. The issue at hand here is the possibility of arbitrage which the committee is trying to reduce. Banks previously had the possibility to create capital benefits by switching instrument from the trading to the banking book due to lower capital requirements in the banking book.
I will briefly address the revisions in the treatment of credit. These revisions are mainly focused on the capital charges for securitisation and non-securitisation positions. According to the Basel Committee
some risks in securitisations are hard to measure and therefore the standardized charge is more appropri-ate and will comprise a credit spread risk component and a default risk component. Separation between these positions and introducing the spread and risk components are beneficial for capturing expected losses from default.
The incorporation of market liquidity risk is one of the more important revisions in the context of this research. The method to incorporate this risk is by using different liquidity horizons in the market risk metric. The liquidity horizons are applied to the risk factors in this case instead of to the instruments itself. Incorporation of market risks should be able to reduce the arbitrage possibilities between assigning instruments to the banking or the trading book.
The choice of market risk metric is of course most strongly connected to this paper. The Committee has proposed to use ES as the new risk instead of the VaR calculations. The ES calculations have to be calibrated to periods of financial stress. Briefly, the new metric entails a separation between a period of significant financial stress and a current regular operational period. The total ES calculation is then determined as the ratio of between the two particular ES calculations.
The relationship between the standardised and internal models-approach is addressed lastly. The committee wishes to enforce the connection between the two approach and obligates banks to regularly calculate capital charges for trading desks using the standardized method and disclose these results to the public to provide a clear comparison between bank performance. Furthermore, extending the adequacy of the standardized approaches would lead to a better substitute when internal models fail.
3
Theoretical Framework
In this chapter we will describe the details of the time series dynamics, the risk metrics investigated and the bootstrap procedures involved. The first thing to mention is the definition of the losses. For a given equity index, say the AEX, the index value at time t is denoted as Pt. The value at time t − 1 is then
Pt−1. In this research we will consider log-returns Rtand therefore note
Rt= ln P t Pt−1 , for t = 1, . . . , T, = ln Pt− ln Pt−1.
For risk metric purposes it is usually more useful to consider a loss distribution. This simply turns the distribution around such that the tail of interest is the right tail. Denote the losses from now on as Lt.
Hence we can write,
Lt= −Rt.
These losses then have underlying distribution FL.
3.1
Dynamics of Time Series
It is known that financial time series often show behaviour which is not independently and identically distributed (iid). Volatility tends to change over time and increased volatility appears often in clusters. This brings us to the choice of a GARCH volatility model for this research in which one can take account of the above mentioned stylized facts. As a basis we will first outline the general autoregressive processes on which the conditional heteroskedasticity models are based. Subsequently, we will define the dynamics of the losses which are assumed. For the random losses defined above we will namely assume that Lt
can follow two different processes. Both the GARCH(1,1) and GJR-GARCH(1,1) are considered. The definitions given in this section are based on Hayashi (2000) and McNeil et al. (2005).
However, as a starting point we can write the loss process without conditional heteroskedasticity, also known as an ARMA(p,q) model, as:
Lt= φ0+ p X i=1 φiLt−i+ t+ q X j=1 θjt−j, ∀t ∈ Z,
where {t} is a white noise process. In financial time series we often drop the assumption that variance
3.1.1 ARCH Processes
Here we build on the ARMA(p,q) process defined above and assume that the {t} are no longer white
noise. We also assume that conditional portfolio returns have mean zero and that we can write the loss process as Lt= σtZt, σt2= α0+ r X i=1 αiL2t−i, where α0> 0 and αi ≥ 0, i = 1, . . . , p.
Furthermore, Zt∼ SW N (0, 1) is a strict white noise process with zero mean and unit variance which
is called the innovation sequence. Regarding the distribution of the innovations we consider two options in this paper; the normal distribution and the (scaled) student-t distribution.
3.1.2 GARCH Processes
The GARCH processes, defined in this section, are the time series dynamics models used in the research. The extension to the ARCH model is developed by Bollerslev (1986) under the name ’Generalized ARCH’. Basically it entails adding lagged values of the conditional variance to the above mentioned ARCH process. The GARCH process can be written as
Lt= σtZt, (1) σ2 t = α0+ r X i=1 αiL2t−i+ s X j=1 βjσt−j2 , (2) where α0> 0, αi≥ 0, i = 1, . . . , p, and βj≥ 0, j = 1, . . . , q.
Again (Zt)t∈Z∼ SW N (0, 1). One can also regroup the parameters of the GARCH model and write
θ= (α0, α1, . . . , αj, β1, . . . , βj) 0
.
The most often used representation of the GARCH(r,s) model, which will also be the main represen-tation used in this research, is the GARCH(1,1) model. This can simply be shown as
Lt= σtZt,
σt2= α0+ α1L2t−1+ βσ2t−1.
Then, θ can be written as θ = (α0, α1, β)0.
As mentioned above, usually it is assumed that Zt, the innovation sequence, has a standard normal
numerical analysis differences between the two innovation distributions will be investigated.
Next to the GARCH(r,s) model I will also look into the GJR-GARCH model introduced by Glosten et al. (1993). In this model there is a distinction between the impact of a negative Lt−1 and a positive
Lt−1, i.e. positive and negative losses, a so called leverage effect. This is intuitively interesting as it
sounds reasonable that, for example, the stock market reacts more strongly to a negative return than to a positive one in terms of volatility. The variance process of the GJR-GARCH can be written as
σ2t = α0+ r X i=1 (αi+ δiILt−i<0)L 2 t−i+ s X j=1 βjσt−j2 ,
where It−1is an indicator function giving a value of zero or one when
ILt−i<0= 0 if Lt−1≥ 0 1 if Lt−1< 0 .
For the GARCH(r,s) models all the parameters will be estimated by maximum likelihood methods. Using this estimation method requires to determine beforehand what kind of distribution the innovation sequence has. One can also use quasi maximum likelihood methods when one has assumed erroneously that the innovation sequence follows a normal distribution. The model adopted in this research will be the most straightforward GJR-GARCH(1,1) model. In a similar way as before we can write the variance process as
σt2= α0+ (α1+ δILt−1<0)L
2
t−1+ βσ2t−1,
and the parameter vector as θ0 = (α0, α1, δ, β)0.
The next section will elaborate on the VaR and ES estimates. One thing to mention at this point is that we will focus on one-day VaR and ES forecasts. In order to take the current level of volatility into account, this implies that we also need a one-day forecast for σ. The conditional variance at time t + 1, σt+1, is known when the parameters of the GARCH model are known. However, this is not the case and
therefore the ML estimates of θ are used to obtain ˆσt+1.
3.2
Value-at-Risk and Expected Shortfall
In the introduction the simple explanation of VaR and ES are stated. In this section we will describe the mathematical definitions of the two risk metrics and describe how they are usually estimated in the literature and in the industry. For mathematical definitions used, we refer to McNeil et al. (2005) and Hull (2012).
3.2.1 Value-at-Risk
actually become. Formally, VaR can be denoted as follows. When we have a portfolio of assets with losses Ltand a fixed time horizon we can write
FL(l) = P (L ≤ l)
as the underlying distribution function of the losses. Then given some confidence level α ∈ (0, 1),
VaRα= inf{l ∈ R : P (L > l) ≤ 1 − α} = inf{l ∈ R : FL(l) ≥ α}.
VaR is the quantile of the loss distribution, i.e. the inverse of the cumulative distribution function and can be written as
VaRα= qα(FL) = FL−1(α),
where qα(FL) is the quantile function and FL−1(α) the inverse of the distribution function. Using the
assumption that the losses follow a (GJR-)GARCH process as in Equation (1) we can also write the one day VaR forecast for the losses Ltat time t as
VaRα,t= σt+1qα(FZ),
where qα is the quantile of the innovation sequence Zt which has an underlying distribution FZ and
where σt+1is the conditional variance from the GARCH process. Using this type of VaR calculation, we
can take into account the current volatility level.
For illustration purposes, in Figure 3 we have plotted a probability density function of a standard normal distribution where the vertical line denotes the VaR level at a (fictional) 70% confidence level. If
-3 -2 -1 0 1 2 3 0.0 0.1 0.2 0.3 0.4 Loss Probabilit y Densit y E(L) V aR ES α = 0.7
Figure 3: Loss Distribution with VaR and ES for α = 70%
we suppose that the innovations Zt are standard normally distributed the VaR is denoted as
where Φ represents the cumulative distribution function of the standard normal distribution and the corresponding inverse, Φ−1, is the quantile function. Similarly, we can write for t-distributed innovations
at time t
VaR(tν)
α,t = σt+1t−1ν (α), (3)
where t−1
ν (α) is the quantile function of the t-distribution with ν degrees of freedom.
Christoffersen and Gon¸calves (2005) have, for comparative reasons, analysed VaR and ES estimates using the normal distribution. In this research, we will rather focus on the Student’s t-distribution which is more flexible with regards to the tail. The reason for using the Student’s t-distribution is the ease of estimation and therefore its usefulness for risk management purposes.
3.2.2 Expected Shortfall
Expected Shortfall is based on the extended VaR model ’conditional VaR’, developed by Artzner et al. (1997). It is the expected loss when the VaR is exceeded and can be written as ESα= E [L|L ≥ V aRα].
Again denoting a random loss Lt from underlying distribution FL, ES with confidence level α ∈ (0, 1)
and u ≥ α formally is defined as
ESα= 1 1 − α Z 1 α qu(FL)du,
where qu(FL) = FL−1(u) is the quantile function of FL for confidence level u ≥ α which implies that the
relationship between VaR and ES can be denoted as
ESα= 1 1 − α Z 1 α VaRudu.
Again assuming the GARCH dynamics from (1), we can use the ES estimate of the innovations and multiply it with the corresponding volatility to obtain
ESα,t= σt+1E [Zt|Zt> V aRα] .
Expected Shortfall looks further into the tail of the loss distribution by basically averaging VaR at several confidence levels. This means that ES is always larger than VaR and it is the reason why the Basel Committee has suggested to compute ES at a lower confidence interval than VaR. In Figure 3 the ES is displayed at the vertical dotted line in the yellow shaded part.
In the section above was described how to estimate VaR for Gaussian and t-distributed innovations. The same methodology holds for ES in these two distributions. For the Gaussian loss distribution ES is given by
ES(N )α,t = σt+1
φ(Φ−1(α))
1 − α ,
can write ES(tν) α,t = σt+1 gν(t−1ν (α)) 1 − α ν + (t−1 ν (α))2 ν − 1 , (4)
with gν being the density function.
3.2.3 From Value-at-Risk to Expected Shortfall under Basel 3.5
As mentioned above, the Basel Committee has proposed a change in the use of risk metric to measure market risks. Expected Shortfall will have to replace the VaR. Even though VaR is easy to implement and practical to use, there are quite some issues when using the method. These issues are based on the coherence of risk measures. Over the years academia and professionals from the industry have shown that VaR is not a ’coherent’ risk measure. Especially Artzner et al. (1999) have shown that there are certain theoretical issues when VaR is used and has there set up a number of axioms to define the appropriateness of a risk measure.
A risk measure is coherent when it satisfies a numbers of axioms which will be shortly discussed here. The main flaw of VaR is that it is not subadditive. This implies that for a portfolio with random asset losses L(i)(where (i) denotes the particular asset) there is a possibility that
ρL(1)+ L(2)≥ ρL(1)+ ρL(2),
where ρ(·) is a specific risk measure function such as a VaR estimate. In words this means that in a portfolio of assets the total VaR of your portfolio might be larger than the sum of the VaR’s of the individual assets. This is counter-intuitive because when building a portfolio one usually tries to diversify and hedge price movements.
In total, four coherence axioms are defined for a risk measure to be appropriate. Subaddivity is the first one and explained above. The second characteristic is closely related to subadditivity. It is defined as ’positive homogeneity’ and implies that in a portfolio without diversification the following should hold:
ρ(λL) = λρ(L), λ > 0.
Hence, if one multiplies the portfolio by λ > 0 the risk level increases proportionally by a factor λ. The last two axioms Artzner et al. (1999) state are ’translation invariance’, which implies that for every fixed c ∈ R it holds that ρ(L + c) = ρ(L) + c, and ’monotonicity’, which implies that for losses L(1) ≤ L(2)
almost surely one can write ρ(L(1)) ≤ ρ(L(2)).
At the moment the VaR is measured at a 99% confidence level. When ES will be introduced the confidence level will be lowered to 97.5%. The committee argues that lowering the confidence level will lead to a more stable output at a relatively comparable level of risk. Furthermore, the new risk metric should be able to capture the risk in periods of financial stress. Therefore, the committee has proposed to calculate ES under stressed and normal periods and compute a ratio of the two in order to set the regulatory capital base. For the specifics of the computations the Fundamental Review of the Trading Book (BCBS, 2013) can be consulted. Due to this proposal, in this research the performance of VaR and ES under both normal and stressed market conditions will be analysed.
3.3
Extreme Value Theory
Extreme value theory comes into play when one is dealing with financial time series data showing evidence of having a fat tail. This fat tail can be described as a power decay of the density function (Gen¸cay et al., 2003). Extremes are usually modelled in two different ways; the block-maximum method and the threshold exceedance method. In the block maximum method the returns are divided into several blocks (weeks, months, years). From each block the minimum and maximum are then analysed in order to model the tail. Bystr¨om (2004) shows that in the limit, i.e. when n → ∞, this distribution is either a Gumbel, Fr´echet or Weibull distribution2, depending on the decay of the tails. The drawback of this
method is the fact that it is quite wasteful of the (already rather scarce) data. Therefore, in this research the threshold exceedance method is utilized. In this method one basically has to choose an appropriate threshold u and use the values of exceeding this threshold to model the tail of the distribution. In the next section will be described how to appropriately define the threshold.
A simple example of the difference between the two approaches is shown in Figure 4. For the left
0 1 2 3 4 5 0 1 2 3 4 5 6 7 x2 x6 x8 x12 x15 0 1 2 3 4 5 0 1 2 3 4 5 6 7 x1 x2 x3 x8 x12 x14
Block Maximum vs. Exceedance over Threshold Method
Figure 4: Extreme Value Theory models, the left panel displays the block maximum method, the right panel the threshold method with u = 4.5
panel of Figure 4, the block maximum method, losses are divided in 5 blocks and for each particular block the maximum observation is obtained to include in the set of observations to model the tail. Hence, x2,
x6, x8, x12and x15are the observations to model the tail. For the right panel, the Peak-over-Threshold
method, one chooses a threshold u and all values exceeding the threshold are considered as extreme values. One can observe the wastefulness of the block maximum method in this example as it results in fewer observations for the extreme value distribution. Gilli and K¨ellezi (2006) also implement the block maximum method in order to compare the performance of the two methods. The wastefulness of data in the block maximum method leads to a disappointing result in risk metric estimation.
Two different Peak over Threshold estimators will be analysed for tail modelling. First we will explain the method based on the Generalised Pareto Distribution. Subsequently, the Hill method is explained. 3.3.1 Generalised Pareto Distribution Method
For this section we apply the Extreme Value Theory models to the innovations Ztin Equation (1). This
is also known as conditional EVT since for estimation first a GARCH model is fitted to data and the tail estimations are applied to the standardized residuals ˆZt = ˆLt/ˆσt. McNeil and Frey (2000) argue that
obtaining GARCH filtered residuals is beneficial for extreme value theory analysis. Moreover, Gilli and K¨ellezi (2006) argue that the choice of a conditional or unconditional method depends on the forecasting horizon. For a long horizon unconditional methods are more appropriate and for short horizons, days or weeks, the opposite holds true. With regards to the performance of conditional and unconditional extreme value theory methods in normal and stressed market conditions, Bystr¨om (2004), contributes to the literature. Using data from the Swedish stock exchange before and during the financial crisis, he finds that conditional methods are also more able to adapt to normal market conditions compared to the more conservative unconditional methods. Combining both arguments and using the fact that we will focus on a one-day forecast under stressed and normal market conditions, we will apply the conditional approach.
Again, we assume that the innovation distribution has an unknown distribution function FZand that
the random innovations are iid. Let k observations exceed the before-mentioned threshold u. In general, for an excess distribution over a threshold u of the set of random variables {Z1, . . . , ZT} one can write
for the random exceedances
Xj = Zj− u > 0 , j = 1, . . . , k.
The innovations exceeding the threshold u follow the excess distribution function
Fu(x) = P (Z − u ≤ x|Z > u) =
FZ(x + u) − FZ(u)
1 − FZ(u)
, (5)
where x = z − u are realisations of the random excesses and 0 ≤ x ≤ ZF − u. Furthermore, ZF is
the right endpoint of the innovation distribution FZ. Estimation of FZ is usually not the problem since
converges to the Generalised Pareto Distribution. Formally, Fu(x) d −→ Gξ,λ(x), where, Gξ,λ(x) = 1 − (1 + ξx/λ)−1/ξ, if ξ 6= 0, 1 − exp(−x/λ), if ξ = 0,
in which ξ is a shape parameter and λ the scale parameter. Here, λ > 0 and 0 ≤ x ≤ (ZF − u) when
ξ ≥ 0 and 0 ≤ x ≤ −λ/ξ when ξ < 0.
The parameters of the GPD, ξ and λ, can be estimated by maximum likelihood estimation. Maximum likelihood estimation of the parameters of the GPD is performed as
max ln L(ξ, λ; x1, . . . , xk) = max k
X
j=1
ln gξ,λ(xj),
where the density function of the GPD is given by
gξ,λ(x) = 1 λ 1 + ξx λ −(1/ξ+1) .
Then, since ξ can be either positive, negative or 0 one distinguishes between ξ 6= 0 and ξ = 0. For ξ 6= 0 we can write max ln L(ξ, λ; x1, . . . , xk) = max k X j=1 ln gξ,λ(xj), = −k ln λ − (1 + 1/ξ) k X j=1 ln1 + ξxj λ
subject to λ > 0, 1 + ξxj/λ > 0 and where x1, . . . , xj are observed realisations of excesses. In the case
ξ = 0 the maximum likelihood problem can be solved as
max ln L(0, λ; x1, . . . , xk) = max k X j=1 ln g0,λ(xj) = −k ln λ −1 λ k X i=1 xj.
Solving is done in a numerical optimization program and estimated parameters can then be used in modelling of the tail.
Modelling the Tail
it as
FZ(z) = Fu(x) [1 − FZ(u)] + FZ(u)
To obtain an estimate of FZ(z) we make use of the Pickands-Balkema-de Haan theorem and plug in the
estimate of the GPD for Fu(x) when u is large enough. Furthermore, FZ(u) = P (Z ≤ u) can simply be
estimated as
b FZ(u) =
n − k n ,
where k is again the number of exceedances over a certain threshold u and n is the total number of losses in a particular sample. Hence we can write,
b FZ(z) = bFu(x) h 1 − bFZ(u) i + bFZ(u), =1 − (1 + ˆξ(GP D)(z − u)/ˆλ(GP D))−1/ ˆξ(GP D) 1 − n − k n +n − k n , =1 − (1 + ˆξ(GP D)(z − u)/ˆλ(GP D))−1/ ˆξ(GP D)(k/n) + (1 − k/n) , = 1 − k n 1 + ˆξ(GP D)(z − u)/ˆλ(GP D)−1/ ˆ ξ(GP D) ,
where ˆξ(GP D) and ˆλ(GP D) are the ML estimates for ξ and λ, respectively.
b
FZ can be inverted in order to obtain quantiles for VaR estimates. For the GPD distribution the
VaR is equal to d VaR(GP D)α = u + ˆ λ(GP D) ˆ ξ(GP D) n k(1 − α) − ˆξ(GP D) − 1 . In Appendix A.1 the derivation for the VaR estimate of the GPD is given.
Recall that for the loss dynamics in equation (1) we can use the estimate of the tail based on the GPD and estimate the VaR of Ltat time t
d
VaRα,t= ˆσt+1VaRd
(GP D)
α . (6)
This holds due to the fact that we have applied EVT methods to the standardized residuals from the fitted (GJR-)Garch.
Finding the Threshold
The threshold u has been mentioned above. However, an important issue in the threshold method is the choice of u. The k observations which exceed the threshold can, for example, be examined by means of a mean excess plot. To obtain a mean excess plot first the mean excess function is needed and can be written as
For the Generalized Pareto Distribution it is known that the mean excess function is given by
e(u) = λ + ξu
1 − ξ , for ξ < 1,
where one can use the fact that for a threshold υ > u it also has to hold that Fυ(l) = Gξ,λ(υ−u) which
leads to the mean excess function
e(υ) = λ + ξ(υ − u) 1 − ξ , = ξυ 1 − ξ + λ − ξu 1 − ξ . (7)
However, the parameters are unknown and therefore one often uses the empirical mean excess function. The sample mean excess function is defined as:
ek(u) =
Pk
i=1(Zi− u)IZi>u
Pk
i=1IZi>u
,
where the indicator function IZi>u takes a value of one when the realized innovation is larger than the
threshold u and zero otherwise. In a mean excess plot we can plot the value of ek(u) against several
threshold values u and check for linearity. The threshold u from which point onwards the curve becomes linear is then the optimal threshold to use for ML estimation of the GPD parameters. Furthermore, the threshold for which the curve becomes linear is also an indication that the standardised residuals follow a Generalised Pareto distribution. A linear upward trend in a mean excess plot indicates a positive shape parameter ξ. A negative downward trend indicates the opposite. A mean excess plot can be used for each particular sample to determine a threshold. However, in a bootstrap procedure multiple samples are generated. It is therefore infeasible to determine the threshold visually for each sample. For GPD estimation McNeil et al. (2005) have taken an approach in which they consider the 10% largest values from a sample as extreme values.
It may seem that the definition of the mean excess function is rather redundant in this paper given the fact that we will use a fixed number of excesses for each bootstrap sample. However, keep in mind that this definition is useful for the derivation of the ES estimate. Namely, note that
ESα= VaRα+ E(Z − VaRα|Z > VaRα).
Then, using (7) and replacing υ by VaR(GP D)α for the expectation of the excesses over a higher threshold
we find the GPD estimate of the Expected Shortfall,
Then, in a similar fashion as we did for (6) we can write c ESα,t= ˆσt+1ESc (GP D) α . 3.3.2 Hill Estimator
The second tail estimation procedure we will apply is the Hill estimator. The Hill estimator can be used to model the tail of heavy-tailed distributions. The difference with the GPD estimator is that the Hill estimator is specifically applicable to heavy tailed loss distributions, i.e. for which ξ > 0, where the GPD method is adequate for both short and heavy tailed loss distributions. Hence, for the Hill method it is assumed that the distribution has a heavy tail of the following form:
FZ(z) = 1 − Q(z)z−1/ξ. (8)
This holds for all Zj > u and Q(z) is a slowly varying function3. The tail of the underlying distribution
is in the maximum domain of attraction of a Fr´echet distribution. For a large enough z, Q(Z) can be approximated by a constant C due to the slowly varying function properties. Using the threshold u and
b
FZ(u) = (n − k)/n we can rewrite (8) as
C =1 − ˆFZ(u) u1/ξ, = k nu 1/ξ,
from which it follows that for the Hill tail estimator it holds that
FZ(z) = 1 − k nu 1/ξz−1/ξ, = 1 − k n z u −1/ξ .
To obtain an estimate of FZ(z) we need the estimate of ξ. The Hill estimator of ξ is computed as
ˆ ξ(h) = 1 k k X j=1 ln Z(j)− ln(u),
where Z(j) denotes the jth-order statistic of Zj > u for which it holds that Z(1) < Z(2) < · · · < Z(k).
Then bFZ(z) is obtained as b FZ(z) = 1 − k nu 1/ ˆξ(h) z−1/ ˆξ(h), = 1 − k n z u −1/ ˆξ(h) .
For comparative reasons it is rewritten in the GPD form as b F (z) = 1 − k n 1 + ˆ ξ(h)(z − u) ˆ ξ(h)u !−1/ ˆξ(h) .
However, for the computation of the inverse of the tail distribution function, which is needed for com-putation of the risk metric, we will focus on the first notation and write the VaR as
VaR(h)α = u hn k(1 − α) i−ξ(h) , and the ES as ESα(h)= V aR(h)α 1 − ˆξ(h),
such that for the VaR and ES estimates for Lt at time t we write
d
VaRα,t= ˆσt+1VaR(h)α and EScα,t= ˆσt+1ES(h)α .
Again, refer to Appendix A.2 for the derivation.
3.4
Bootstrap Algorithms
Hartz et al. (2006) argue that the use of the bootstrap in VaR estimation is beneficial due to the fact that the sampling distribution of a point forecast is usually unknown and a bootstrap resampling algorithm approximates this distribution while being computationally relatively simple. The bootstrap idea originates from Efron and Tibshirani (1993) and basically entails resampling n observations from a dataset of the same size n to generate a bootstrap distribution for a certain statistic, for example VaR or ES. Bootstrap procedures are also very useful in estimating bootstrap percentile intervals. As mentioned in Qi (2008) the bootstrap method is widely used to build percentile intervals around a certain risk metric such as VaR or ES. It avoids the need to estimate parameters for the statistic’s limiting distribution which might be unknown and difficult to obtain. The method we use for estimation of VaR and ES statistics suffers from parameter estimation risk. The bootstrap method takes the uncertainty in parametric estimation models into account in the construction of bootstrap intervals.
3.4.1 Historical Simulation
Historical Simulation (HS) is relatively the most easy method to estimate VaR. It is based on quantiles of the observed, ordered losses and can be computed as
d VaR(HS)α = qα {Lt} T t=1 ,
where qα(·) is the quantile function of a set of observations {Lt} T
t=1for confidence level α. The estimation
method for ES can be based on a nonparametric method in which the sample average of losses larger than the VaR is computed. This will also be one of the estimators in this research, comparable to the HS method when using VaR. As described by Christoffersen and Gon¸calves (2005), the HS estimator of ES can be written as c ES(HS)α = 1 Pn t=1ILt≥VaR(HS)α n X t=1 LtIL t≥VaR(HS)α ! ,
where ILt≥V aRαis again an indicator function. This type of estimator will be used for the HS method. For
the conditional (GARCH type) volatility and EVT estimators of VaR and ES we will use the definitions from the previous sections.
The HS bootstrap is proposed by Christoffersen and Gon¸calves (2005). It is relatively easy to imple-ment since one does not have to take into account parameter estimation or any distributional assumptions. To generate B bootstrap samples for VaR and ES one uses the original sample of losses Ltand resamples
with replacement to obtain a sample with estimators of Lt; L∗t : t = 1, . . . , T . Then, for each bootstrap
sample i = 1, . . . , B one can compute the historical VaR and ES as
VaR∗(HS)α,T+1= qα {L∗t} T t=1 , ES∗(HS)α,T+1=Pn 1 t=1IL∗ t≥V aR∗α n X t=1 L∗ tIL∗ t≥V aR∗α ! ,
where qαis the α-quantile of the distribution of ordered losses. Regarding this non-parametric bootstrap
method one has to note that serial dependency of returns is not taken into account. However, since the historical simulation method is in practice the most used method by financial institutions we will make use of this bootstrap method for comparative reasons. Once one has generated the B bootstrap samples for the VaR and ES estimates the p% bootstrap percentile intervals can be calculated as
q(1−p)/2 n VaR∗(HS)α,T+1 oB i=1 , q1−(1−p)/2 n VaR∗(HS)α,T+1 oB i=1 , (9) q(1−p)/2 n ES∗(HS)α,T+1oB i=1 , q1−(1−p)/2 n ES∗(HS)α,T+1oB i=1 . (10) 3.4.2 GARCH Models
bootstrap to standardized residuals. First I will describe the bootstrap procedure developed by Pascual et al. (2005) for comparative purposes. Subsequently, the extension to the GJR-GARCH(1,1) will be given.
GARCH(1,1)
The bootstrap procedure is performed taking the following steps. First, the GARCH(1,1) parameters ˆ
θ=αˆ0, ˆα1, ˆβ1
0
are estimated using the maximum likelihood method. From the estimated models we use ˆσt to compute ˆZt= Lt/ˆσt for t = 1, . . . , T and
ˆ σ12= ˆ α0 1 − ˆα1− ˆβ1 , ˆ σt2= ˆα0+ ˆα1L2t−1+ ˆβ ˆσ2t−1, for t = 2, . . . , T. Here, ˆσ2
1 is the estimated marginal variance and ˆσt2 are the conditional variances. To implement the
bootstrap we resample with replacement from the ˆZtto create B bootstrap samples. We then create an
empirical distribution ˆFB of the residuals. For each bootstrap sample we recreate the time series
ˆ
σ∗2t = ˆα0+ ˆα1L∗2t−1+ ˆβ ˆσ∗2t−1,
ˆ
L∗t = Zt∗σˆ∗t, for t = 1, . . . , T.
Note that for each bootstrap sample we start with ˆσ∗2
1 = ˆσ12. Again, using maximum likelihood estimation
we compute the bootstrap parameters of the GARCH model ˆθ∗ =αˆ∗ 0, ˆα∗1, ˆβ∗
0
. These parameters are then implemented in the 1-step ahead forecast estimation for ˆLT+1 and ˆσ2T+1 which are computed by
ˆ σT∗2+1= ˆα∗0+ ˆα∗1∗2T + ˆβ ∗σˆ∗2 T , (11) L∗ T+1= ZT+1∗ σˆ∗T+1. (12)
Note that the Z∗
T+1 are then random draws with replacement from the original sample, L∗2T = L2T and
the conditional variance for time T based on past observations is given by
ˆ σ∗2T = ˆ α∗ 0 1 − ˆα∗ 1− ˆβ∗ + α ∗ 1 T −2 X j=0 ˆ β∗j L2T −j−1− ˆ α∗ 0 1 − ˆα∗ 1− ˆβ∗ .
Even though this bootstrap procedure enables us to compute future returns, we only focus on the pre-dicted volatilities for our VaR and ES calculations. The bootstrap procedure Nieto and Ruiz (2010) have proposed would give smaller prediction intervals in small samples or under a non-consistent conditional variance. However, using a reasonable sample size the intervals will be similar and therefore the single bootstrap procedure will be implemented in this research. We focus only on one day ahead forecasts of the risk metrics. Therefore, based on the estimates for ˆσ∗2
boot-strap VaR and ES estimates and compute bootboot-strap prediction intervals in a similar way as in equation (9).
Pascual et al. (2005) have also examined a bootstrap procedure in which they only resampled from the residuals ˆZt and used the original parameter vector ˆθ
0
= (ˆα0, ˆα1, ˆβ) to obtain forecasts ˆyT+1 and
ˆ
σT+1. This is computationally significantly faster since GARCH parameters have to be estimated only
once. However, in forecasting volatility, the coverage rates of bootstrap confidence intervals are evidently closer to the nominal coverage rate of 95% when the parameters of the GARCH model are estimated a second time. Hence, for the extension to the GJR-GARCH(1,1) model, I will also apply the two step bootstrap method since this provides the most accurate result of the two.
GJR-GARCH(1,1)
In this section we extend the originally proposed bootstrap procedure for the GARCH(1,1) model. The bootstrap procedure for the GJR-GARCH model is a slight adaptation from the original one proposed above which is able to capture asymmetry of returns and which therefore should be able to more ade-quately capture the characteristics of the time series in order to forecast the volatility. Recall that the GJR-GARCH(1,1) model is defined as
Lt= σtZt,
σ2t = α0+ (α1+ δ1ILt−1<0)L
2
t−1+ βσ2t−1.
Performing similar steps as in the previous section one can first estimate the parameter vector of the original data by means of quasi maximum likelihood. Consecutively, the standardized residuals ˆZt =
Lt/ˆσt are calculated and for the conditional variance one can write
ˆ σ21= ˆ α0 1 − ˆα1− ˆδ/2 − ˆβ , ˆ σ2t = ˆα0+ (ˆα1+ ˆδ1ILt−1<0)L 2 t−1+ ˆβ ˆσ2t−1, for t = 1, . . . , T,
where for the unconditional variance, ˆδ is divided by 2 due to the assumption of symmetry in the innovation sequence. If the innovation sequence is symmetric it can be assumed that approximately half of the returns will be negative and half will be positive. Using the bootstrap principle by sampling from the standardized residuals we recreate B bootstrap time series
ˆ σ∗2t = ˆα0+ ( ˆα1+ ˆδ1ILt−1<0)L ∗2 t−1+ ˆβ ˆσ∗2t−1, L∗ t = Zt∗σˆ∗t, for t = 1, . . . , T.
such that ˆ σT∗2+1= ˆα0∗+ ( ˆα∗1+ ˆδ∗1ILT<0)L ∗2 T + ˆβ ∗ˆσ∗2 T , ˆ L∗ T+1= ZT∗+1ˆσT∗+1.
The main difference between the GARCH(1,1) bootstrap and the GJR-GARCH(1,1) bootstrap is the way the variance process is written in terms of past observations. The leverage effect in the GJR leads to the following definition for ˆσ∗2T :
ˆ σ∗2 T = ˆ α∗ 0 1 − ˆα∗ 1− ˆδ∗/2 − ˆβ∗ + α ∗ 1 T −2 X j=0 ˆ β∗jL2 T −j−1 −αˆ∗1+ ˆδ∗/2 αˆ∗ 0 1 − ˆα∗ 1− ˆδ∗/2 − ˆβ∗ T −2 X j=0 ˆ β∗j+ δ∗ T −2 X j=0 ˆ β∗jL2T −j−1ILT −j−1<0.
This expression can be found by iterative substitution and applying geometric series properties. The formal derivation can be found in Appendix B. Again, based on the predictions the VaR and ES estimates can be computed and bootstrap prediction intervals can be found.
3.4.3 Other Bootstrap Procedures and options
Several attempts have been made in using the bootstrap originated by Efron and Tibshirani (1993) aimed at confidence intervals around parameters of ARCH and GARCH processes. First Ruiz and Pascual (2002) adopted a general framework for bootstrapping the parameters of GARCH models to fit financial time series. In their paper they argued that the standard bootstrapping techniques in which one simply resampled the direct losses was not adequate when the data suffered from heteroskedasticity and excess kurtosis. Therefore, they proposed to resample from the residuals.
Nieto and Ruiz (2010) have also proposed an extension to the bootstrap model of Pascual et al. (2005) in which they perform a second bootstrap step. They estimate ˆσ∗2
T+1 in the same way as in equation
(11). However, quantiles are then obtained directly by resampling again from the empirical residuals ˆZt.
Bayesian Approach
Connected to the confidence interval estimation in the frequentist way is the Bayesian estimation ap-proach. In the Bayesian world, one beliefs that the model parameters have no true value, but come from an underlying distribution. Using sampling techniques, in the Bayesian methods one automatically generates a sample of parameter estimates which then can be used for risk metric computation. Little literature discussing this method for VaR and ES estimation is available. Using the bootstrap method of Pascual et al. (2005), Aussenegg and Miazhynskaia (2006) contribute to the literature using Bayesian inference. Particularly in comparison to this research in which bootstrap prediction intervals for the risk metrics are analysed, the Bayesian approach can give an interesting comparison. They compare the confidence intervals from the bootstrap method, thereby incorporating the parameter uncertainty, with a Bayesian GARCH framework. They argue that using Bayesian inference is beneficial to the calculations of VaR as it gives lower uncertainty in the VaR predictions. However, looking at backtests they do not find any differences in the proportions of failure. The idea behind using the Bayesian framework is that the risk manager or specialist can add his or her prior beliefs regarding the parameters used in the risk metric and thereby influencing the outcome for the VaR or expected shortfall estimate. From a regulatory view this might not be an optimal way of estimating the metrics since risk managers could then influence the outcomes and the level of regulatory capital they need to attract. The only way the Bayesian methods might be appropriate is when risk managers are obliged to use diffuse, non-informative, prior beliefs or when regulatory instances are able to set up proper priors. Aussenegg and Miazhynskaia (2006) have tested these subjections and found that varying the prior informativity actually had no significant influence on the results.
Another point of discussion using the Bayesian approach is that it does not automatically gives point forecasts which are actually needed for backtesting4. The reasoning behind the approach is that the underlying parameters have no true value but are also random variables. Therefore, one can only estimate the distribution of VaR and ES for a given point of time t. From this point one has to use another technique to come up with the appropriate point forecast which is further discussed by Ardia (2008). The authors conclude that the way to incorporate Bayesian methods for market risk management remains a question for further research.
A third and final objection against the use of Bayesian methods in financial risk management is the fact that even for one point estimate, one has to compute a complete parameter distribution. If one considers only one risk factor this might not be a significant problem. Banks, on the other hand, which are in possession of large portfolios might be restricted in their choice of computational method due to time constraints. Even though Hoogerheide and van Dijk (2010) found a Bayesian method using a mixture of t-distributions which is computationally fast and could therefore be appropriate in real time decision making the fact that one has to estimate a complete parameter distribution seems tedious. Due
4Backtesting is the technique in which actually observed losses are compared with VaR and ES estimates and ”breaks”
to the objections stated in this section I have chosen not to proceed with Bayesian estimation methods of GARCH models.
4
Monte Carlo Analysis
In this chapter the Monte Carlo simulations are performed in order to test the adequacy of the boot-strap confidence intervals for our ES and VaR estimates. Firstly, an analysis is done for the Historical Simulation procedure. Secondly, the GARCH bootstrap confidence intervals are analysed.
4.1
Historical Simulation Approach
As a simple starting point we consider iid randomly generated data and apply the historical simula-tion approach. For the historical simulasimula-tion estimasimula-tion method we make use of different sample sizes and different underlying distributions. In this research sample sizes of 250, 500, 1000 observations are considered. A sample size of 250 observations may seem rather small. However, the Basel Committee prescribes the use of a minimum of 250 daily observations and therefore it is useful to test the perfor-mance of bootstrap intervals for this, relatively, small sample. For the underlying distributions we use both the normal and the scaled student t-distribution. Both are scaled to have mean 0 and variance 202/252. This variance entails a variance of 20% per year and the total setup is quite similar as the one
in Christoffersen and Gon¸calves (2005). However, the difference is that they do not take into account the Basel prescribed minimum number of observations and the corresponding confidence levels for VaR and ES. The reason we compute the VaR and ES estimates using randomly generated iid data and use the historical simulation technique is to look at the accuracy of estimation of ES and VaR at different sample sizes. Since most banks use the historical simulation technique this is quite an interesting question.
The true VaR and ES values are found in a rather empirical way. We have drawn 10 million obser-vations from the particular underlying distributions and computed the quantile for the VaR and average of the quantiles above a certain threshold for the ES estimate. The true values are reported in Table 1. For this Monte Carlo analysis we generate 1000 samples and for each sample 1000 bootstrap samples. Note that this is quite intensive, especially for the later case in which we have to compute the GARCH bootstrap prediction intervals.
In this section we particularly focus on the performance of the 99% VaR and the 97.5%ES since these are the confidence levels the Basel Committee prescribes. Furthermore, since the data generated is iid only the historical simulation based bootstrap confidence intervals can be used. GARCH models are not defined in this case.
Table 1: Monte Carlo Results Historical Simulation
Properties Bootstrap Interval Properties Distribution T Metric True Value Mean Bias RMSE Coverage Rate Min. Max. Width t(8), σ = 202/252 250 VaR 3.125 3.109 -0.016 0.417 0.904 2.502 4.1557 0.529 ES 3.117 3.154 0.036 0.386 0.824 2.604 3.752 0.530 500 VaR 3.082 -0.044 0.274 0.848 2.644 3.596 0.305 ES 3.204 0.088 0.288 0.876 2.786 3.653 0.305 1000 VaR 3.110 -0.015 0.204 0.876 2.796 3.434 0.204 ES 3.194 0.077 0.208 0.866 2.894 3.512 0.205 N (0, 202/252) 250 VaR 2.950 2.938 -0.012 0.316 0.896 2.462 3.568 0.375 ES 2.922 2.936 0.013 0.273 0.816 2.534 3.326 0.379 500 VaR 2.877 -0.073 0.216 0.826 2.564 3.235 0.228 ES 2.945 0.023 0.195 0.854 2.657 3.237 0.230 1000 VaR 2.912 -0.038 0.156 0.846 2.681 3.137 0.154 ES 2.942 0.0120 0.132 0.876 2.739 3.147 0.156
VaR estimations are at 99% confidence level, ES estimations at 97.5%.
Looking at the results in Table 1 we can see that for the Student’s t-distribution the bias of the VaR is, as expected, smallest for the samples with 1000 observations. This also holds for the RMSE. However, for the ES estimates, the bias is smallest for the sample with 250 observations. This implies that the ES estimation is most accurate at the smallest sample size. The interesting aspect of the table is that the Historical Simulation approach underestimates the VaR and overestimates the ES slightly. From a prudential point of view it could then be said that the use of ES would lead to a more safe capital level. Looking at the coverage rates we see that coverage rates are under the nominal coverage level of 95% in all cases. Regarding sample size it can then be argued that the use of a 1000 observation sample leads to coverage rates closest to nominal for ES estimates. For the VaR estimations the smallest sample size again gives the highest nominal coverage rate. Note that the results in this table are produced using a Data Generating Process (DGP) which generates iid data. This is in compliance with the assumption when using the Historical Simulation approach that the data is iid. However, as mentioned before, empirical data often does not exhibit iid properties. Therefore, in the next section, we continue using the more realistic DGP in which autocorrelation and conditional heteroskedasticity are present.
4.2
GARCH Approach
In this section a DGP of a GJR-GARCH type, in which we distinguish between two different sets of true parameters, is used. The differences in the true parameter values implicate a different level of persistence of the volatility of the previous day. The lower β and α0,the less volatility is dependent on the previous
As mentioned above, we estimate both the Hill and GPD estimator for VaR and ES. The reason for this is that McNeil and Frey (2000) find that using a GPD estimator is more stable with respect to the choice of the threshold compared to the Hill quantile estimator for high quantiles, say α > 0.99. However, no notion is made regarding the lower (96.5%) quantiles which are actually particularly of interest in this research. Moreover, the GPD estimator requires estimation of a second parameter. This ’scale’ parameter leads to extra estimation uncertainty which might not be desirable in a regulatory metric. A last note regarding their paper is the fact that they choose to make their calculations using a sample of 1000 observations. With respect to the Basel regulations, which require risk metric calculations based on a minimum of 250 observations, it is useful to check stability using a smaller dataset.
For the Monte Carlo simulations including the bootstrap we generate a total of 1000 samples with each 1000 bootstrap subsamples in order to compute the confidence intervals. The reason we reduce the number of simulations is due to the fact that a Monte Carlo simulation together with the bootstrap for GARCH models is computationally intensive. It makes use of an optimisation process which occasionally fails to converge to an optimal set of parameters when the covariance matrix of parameters is nearly singular or when the estimated parameters do not satisfy the stationarity conditions. When this occurs, a new sample has to be generated. Effectively, this results in a computational process which is larger than the necessary amount of bootstrap samples. Next to the bootstrap confidence intervals we also compute point statistics for VaR and ES. These are all based on 10.000 MC simulations since the computational time is significantly shorter. All computations for the GARCH Monte Carlo analysis and the forthcoming empirical data analysis have been executed on the University of Groningen’s Millipede Cluster. This computing cluster provides nodes with each 24 processing cores. Using this cluster reduces the computational time significantly compared to a normal computer which usually only has 4 cores. 4.2.1 Stressed Simulations
In Table 2 and 3 one can see the results of the DGP with α1 = 0.1, β = 0.8, δ = 0.15 and α0 =
202/252 × (1 − α
1− β − δ/2). The innovations are simulated from a Student’s t-distribution with 8
degrees of freedom. This DGP resembles the stressed financial times in which volatility is high and in which there is high persistence of conditional volatility. The left columns of the tables show the point estimate properties of the different estimators. For each Monte Carlo sample we have computed the true VaR and ES using the true parameters which are given in the corresponding tables. Based on these parameters the true σT+1 is computed and the VaR and ES estimates of the innovation sequence are
computed as in Equation (3) and (4) using a t-distributed innovation sequence with eight degrees of freedom. It is not feasible to compute one true VaR or ES for all MC samples together since we have to take the conditional volatilities into account. For each sample the bias is computed and the average of all samples is shown.
with previous research by Christoffersen and Gon¸calves (2005)5and McNeil and Frey (2000).
For the bootstrap properties first the coverage rate is stated. We are computing 95% confidence intervals and therefore we are expecting coverage rates close to the nominal 95%. The bootstrap confi-dence intervals are said to be adequate when it covers the true VaR or ES in 95% of the MC simulations. Furthermore, in the right columns of Table 2 and 3 the average minimum, maximum and interval width are also shown to give an idea of the properties of the accuracy of the estimators.
Table 2: Monte Carlo Results GJR-Garch(1,1) VaR, high persistence
DGP with α1= 0.1, β = 0.8, δ = 0.15 and α0= 202/252 × (1 − α1− β − δ/2)
Point Estimate Properties Bootstrap Interval Properties Est. Method Mean Bias RMSE Coverage Rate Min. Max. Width n=250 Historical Simulation 3.227638 0.628624 2.224225 0.465 2.241207 4.729298 1.073719 GPD 2.552481 -0.04653 0.486668 0.974 1.600798 4.79311 1.306115 Hill 3.107737 0.508722 0.964825 0.951 1.851126 6.074403 1.717192 t-dist. 2.56312 -0.03589 0.403344 0.978 1.669554 4.872263 1.309793 n=500 Historical Simulation 3.230342 0.615986 2.481956 0.339 2.478716 4.246833 0.7985 GPD 2.600978 -0.01338 0.380686 0.956 1.848839 4.271833 1.084912 Hill 3.144825 0.530468 0.879062 0.826 2.140859 5.270119 1.389722 t-dist. 2.59948 -0.01488 0.299432 0.951 1.903868 4.260057 1.059299 n=1000 Historical Simulation 3.310004 0.703541 2.316374 0.226 2.679823 4.195151 0.675935 GPD 2.611528 0.005065 0.26288 0.95 2.047506 4.243132 0.998375 Hill 3.134224 0.527761 0.813082 0.593 2.385932 5.145502 1.246125 t-dist. 2.603263 -0.0032 0.239698 0.947 2.086058 4.227475 0.977473 VaR estimations are at 99% confidence level.
Table 3: Monte Carlo Results GJR-Garch(1,1) ES, high persistence
DGP with α1= 0.1, β = 0.8, δ = 0.15 and α0= 202/252 × (1 − α1− β − δ/2)
Point Estimate Properties Bootstrap Interval Properties Est. Method Mean Bias RMSE Coverage Rate Min. Max. Width n=250 Historical Simulation 3.280925 0.616005 2.210312 0.408 2.36908 4.312227 0.819592 GPD 2.603252 -0.06167 0.569914 0.972 1.606285 5.059897 1.372313 Hill 3.366254 0.701334 1.269751 0.938 1.93358 6.85104 1.943898 t-dist. 2.628305 -0.03661 0.433086 0.977 1.69701 5.024375 1.326014 n=500 Historical Simulation 3.464618 0.783968 2.597312 0.284 2.719903 4.447404 0.762993 GPD 2.650116 -0.03053 0.4001 0.958 1.863601 4.401076 1.103947 Hill 3.389285 0.708635 1.145298 0.784 2.242756 5.794238 1.527858 t-dist. 2.665992 -0.01466 0.328267 0.952 1.935987 4.376257 1.068558 n=1000 Historical Simulation 3.529378 0.856822 2.470529 0.181 2.913294 4.287724 0.599621 GPD 2.660133 -0.01242 0.275124 0.951 2.070039 4.33262 1.002252 Hill 3.365998 0.693442 1.041522 0.51 2.512337 5.574622 1.341857 t-dist. 2.669611 -0.00295 0.258965 0.95 2.126326 4.340969 0.984734 ES estimations are at 97.5% confidence level.
Regarding the point estimates we observe that both the Historical Simulation estimator and the Hill estimator have a large bias compared to the GPD and t-dist. method for both VaR and ES estimates. Both also overestimate the true value while the other two underestimate it. Furthermore, the RMSE
5Christoffersen and Gon¸calves (2005) actually find that using the largest 2% of losses is optimal with regards to
of the GPD and t-dist. is significantly smaller than for the HS and Hill estimator. One interesting observation is the comparison in performance between the GPD method and the Student’s t-distribution method. Since the t-dist. is the true underlying distribution of the error terms, it is natural to expect that this method would perform most adequately. However, the GPD is not doing truly worse. This is also seen when looking at the the bootstrap intervals. First of all, the HS estimator has a coverage rate well under nominal which deteriorates as the sample size gets larger. Apparently, in a larger sample the dynamics in the GARCH process become such that the HS estimator is not able to find the true VaR. The same holds for the Hill estimator. The coverage rate deteriorates as the sample size gets larger. The opposite can be observed for the GPD and the t-dist. estimators. Coverage rates are above nominal for all sample sizes and for both VaR and ES estimators. Hence, the equivalence in performance of the two again becomes visible. Concerning the difference in performance between VaR and ES estimation we can observe that with regards to the bias of the estimators there is no clear consensus whether the VaR or ES bias is larger. This differs for the different estimation methods. However, looking at the coverage rates one can observe that the coverage rate are always lower for ES estimates. This would imply that VaR estimates cover the true VaR more often and hence are more accurate. This is also confirmed when looking at the average width of the intervals as a proportion of the true VaR. Width of VaR estimates are smaller in most samples. A conclusion that can be drawn from the results in Table 2 and 3 is that for stressed financial times the GPD estimation method performs extremely well. Coverage rates are at or above nominal levels and the bias is low.
4.2.2 Tranquil Simulations
Furthermore, in Table 4 and 5 the results are shown for the low persistence case. For this analysis we have chosen true α1 = 0.2, β = 0.5, δ = 0.15 and α0 = 102/252 × (1 − α1 − β − δ/2). This value
results in less extreme outcomes and less volatility clustering with an average volatility of 10% per year. The interesting part of this choice is that we can compare the performance of the different estimation methods under stressed and ’relatively’ calm periods.
Note in the results table that the average of the different estimators is between 1.5 and 2, compared to a average of around 2.5 and 3 for the above mentioned simulations corresponding to stressed financial times. Hence, the first thing that is observed is that the risk metrics are lower which is the result we want from the simulations of tranquil financial periods.
