Risk direction is going more and more of import after we have faced several times of fiscal crisis particularly the 1 we are enduring now for fiscal establishments and regulators. Before recent fiscal crisis, Value at Risk ( VaR ) is a simple and widely acceptable tool to step and manage hazard in the last 15 old ages since JP Morgan published its Riskmetrics to mensurate and pull off hazard, but late more and more analysts doubt its usefullness and efficiency during fiscal crisis.
In this survey we employ four widely used attacks to gauge VaR for three different fiscal assets at three different assurance degrees to prove their public presentation. The attacks we used in this survey are the Historical Simulation attack, Traveling mean attack, GARCH Normal attack and GARCH Student T attack, the three fiscal assets are S & A ; P 500, Brent oil and United States three month exchequer measure, the three assurance degrees are 95 % , 99 % and 99.9 % . There are two chief intents in this paper, the first is to prove the public presentation of four attacks to see which one is superior to others. The Second is to analyse the consequences and seek to reply the inquiry whether VaR can mensurate and pull off hazard efficaciously particularly during fiscal crisis clip period. The information we collected in this survey is day-to-day return of three fiscal assets from 1st Jun 1989 to 29th May 2009. The consequences of the survey show that GARCH pupil T attack is superior to other three attacks in most instances, it ‘s the lone attack did non underestimate hazard, even more in some instances it overestimated hazard. We come to a decision that VaR can mensurate and pull off hazard no affair in which clip period if we employ the proper attack for proper fiscal plus at proper assurance degree.
In visible radiation of recent fiscal crisis, hazard direction has drawn really high attending from regulators and fiscal establishments. Regulators and fiscal establishments are reexamining the tools to mensurate and pull off hazard and doing more rigorous steps to command hazard. Value at Risk ( VaR ) is a simple and widely used tool to step and manage hazard, it is popular in the last 15 old ages since JP Morgan published its Riskmetrics to mensurate and pull off hazard, but late more and more analysts doubt its usefullness and efficiency during fiscal crisis. In this survey we will seek to reply the above inquiry by proving the public presentation of VaR during different clip period.
The construct of hazard is refers to the volatility of unexpected results in finance. It includes concern hazard, strategic hazard and fiscal hazard. Business hazard and strategic hazard are hazards that relate to merchandise markets or economic and political environments of a company, fiscal hazard is the hazard associated with fiscal market activities. Financial hazard can be farther divided into several sub-categories: foremost is market hazard, it is the hazard due to alterations in market monetary values. Second is recognition hazard that counterparties are non able to carry through their contractual duties. Third is liquidness hazard, the hazard of inability to run into payment duties. Fourth is operation hazard, this hazard root from the internal staff or system failure or external events. Fifth is legal hazard, which is the hazard that due to improper minutess. This paper will concentrate merely on fiscal hazard, and more specifically, on how this type of hazard can be captured through four most normally used methods to gauge Value-at-Risk ( VaR ) based on three different features of fiscal assets and at different assurance degrees.
Historical Simulation Approach
Unlike other parametric VAR theoretical accounts, the historical simulation ( HS ) theoretical account [ one ] does non do specific premise about the distribution on the plus returns. Besides, it is a nonparametric attack. The VAR figure of the historical simulation is easy to understand, so it is more easy accepted by direction and the trading community. It is predict that the current places will play back the record of history. Besides, it is comparatively easy to implement. In most simple instance, historical simulation provides current weights to a clip serious of historical plus returns, that is, ( Jorion, 1995 )
There are several advantages of historical simulation. First, historical simulation is simple to implement which based on historical informations on hazard factors have been collected in-house for day-to-day taging to market. Second, historical simulation histories for fat dress suits which are present in the historical information. Third, historical simulation uses the pick of skyline for mensurating VaR. Besides, historical is intuitive. Users can travel back in clip and explicate the fortunes behind the VaR step. ( Best on 1998 )
On the other manus, the historical simulation attack has a figure of drawbacks. Due to the value of the portfolio alterations, the per centum value alterations in the portfolio no longer mention to the original portfolio value. One job of historical simulation attack is that utmost percentiles are hard to gauge exactly without a big sample of historical informations. Another job of historical simulation is that plus monetary values frequently exhibit swerving behaviour. A solution provided to cover with tendency job is to connote symmetricalness on the portfolio value distribution by taking the negative of the net incomes and losingss used in standard historical simulation, which doubles the informations used in calculating the percentiles and eliminates the tendency. ( Holt on 1998 )
The Variance-Covariance Approach
The variance-covariance attack is the simplest of the VaR methods in computation required. Normally, planetary Bankss used it to aggregate informations from a big figure of trading activities. Variance-Covariance attack is widely used by Bankss with relatively low degrees of trading sites and it besides the first VaR theoretical account to be provided in off-shelf computing machine bundles. Portfolio net incomes and loss are usually distributed in the variance-covariance attack which is based on premise that financial-asset returns. ( Colleen & A ; Marianne 1997 )
Define Rt to be the matrix of market returns at clip T and allow? t represent the variance-covariance theoretical account is that it has zero mean, which matches criterion market pattern. Based on this premise, Jackson ( 1997 ) points out that the appraisal mistake associated with ill determined mean estimations which may diminish the efficiency to gauge variance-covariance matrix. The return on a portfolio of foreign-exchange places can be expressed as a additive combination of exchange-rate returns due to we are non sing complex derived functions. In footings of the sensitiveness of portfolio, one hazard factor is explained the alteration of the portfolio value.
The Fixed-weight Specification
Return covariance and discrepancy are changeless over the period, which is the premise of the fixed-weight attack. Hence, it is predict that future discrepancies and covariances are equal to the sample discrepancies and covariances calculated over the fixed-length informations history.
The indifferent and efficient calculator of the population variance-covariance matrix should utilize all informations which each observation is equal, if return discrepancies and discrepancies are changeless. One of the fixed-weight attack? the random-walk theoretical account which restricts the past information period to merely one observation ( i.e T=1 ) . The fixed-weight assumes that? T is a random-walk and that? T is based on much empirical work with plus returns, which suggests that comparatively old informations should be ignored ( Engel and Gizycki, 1998 )
Bollerslev ( 1996 ) described the generalized-autoregression conditional heteroscedasticy ( GARCH ) theoretical accounts which captures volatility bunch. These theoretical accounts apply both autoregression and traveling mean behaviour in discrepancy and covariance.
It is necessary to enforce limitations before prosecuting in appraisal as GARCH theoretical account is that the figure of hazard factors increases computation quickly becomes intractable.
Monte Carlo Simulation
The Monte Carlo simulation method is a parametric attack which can be priced utilizing full rating, bring forthing random motions in hazard factors from estimated parametric distributions. The Monte Carlo simulation attack returns in two stairss.
First, all hazard factors will be specified by the hazard director in a parametric stochastic procedure. Second, all the hazard factors simulate different monetary value waies. The portfolio of the Monte Carlo simulation method is marked to market utilizing full rating as in the historical simulation method, that is, V*k=V ( S*i, K ) , sing at each skyline. Hence, the Monte Carlo method is similar to the historical simulation attack, excepting the conjectural charges in monetary values? Si for plus I in equation which are created by random draws from a prespecified stochastic procedure alternatively of sampled from historical informations ( Sorvon 1995 ) .
The Monte Carlo methods interject an expressed statistical attack and use mathematical techniques to bring forth a big figure of possible portfolio-return results. The Monte Carlo attack takes into history the events that likely occur, but, in fact, they were non observed over the historical period. One of the chief advantages of Monte-Carlo methods is that it evaluates a richer set of events than contained within past history. In order to implement the Monte-Carlo method, a statistical attack of the plus returns must be selected. We use the Monte Carlo method into two statistical attacks: simple normal distribution and a mixture of normal distribution.
Monte- Carlo Methods Using Normally-Distributed Asset Returns
We consider use the premise that assets returns are usually distributed, which is the first execution of the Monte-Carlo attack. The discrepancy covariance matrix is estimated utilizing the fixed-weight variance-covariance attack. The VaR estimation is provided by the appropriate percentile and the resulting alterations in portfolio value. The consequences should be close to those obtained from the fixed-weight variance-covariance attack due to this method utilizing the same distributional premises as the variance-covariance method.
Monte-Carlo Methods Using a Mixture of Normal Distributions
A Monte-Carlo attack is proposed by Zangari ( 1996 ) which makes usage of a mixture of normal distributions. This attack is to double the fat-tailed nature of plus returns. The premise implies that an asset-return realisation is from two distributions: one with chance P and another 1 with chance ( 1-p ) . The parametric quantities of the mixture of normal distribution one estimated that both distributions have zero agencies.
Unfortunately, Hamilton ( 1991 ) proposed that this map does non hold a planetary upper limit. When one of the observation is precisely zero the likeliness is infinite, this belongings emerges. Although Hamilton has provided Bayesian solutions to this job, our theoretical account was to re-start the appraisal process with assorted get downing values. The standard Monte-Carlo theoretical account is used to get the VaR, when the parametric quantities have been estimated. In the assorted distribution, observations are simulated by pulling p observations from simulations of the first distribution and ( 1-P ) observations from 2nd distribution.
Stock Price Monte Carlo
The job of non-linearity is solved by a batch of estimates, utilizing a 2nd order Taylor series of enlargement. This attack brings two chief jobs. First, the Taylor series is non able to cover all non-linearities good plenty, particularly the stock monetary value in the comparatively big motions, which merge in a hazard direction puting. Second, the normal distribution of portfolio returns is lost, which makes the delta theoretical account computationally efficient and easy to implement. Harmonizing to compare three estimates with regard to truth and computational clip to a full rating manner, Pritsker ( 1997 ) finds that in 25 % of the estimates a Monte Carlo simulation utilizing the 2nd order Taylor series, usually underestimated the true VaR by an norm of 10 % . It assumes that no bound on computational clip, the full rating theoretical account implemented considers all non-linear relationships. This theoretical account implements a VaR calculation focused on a Monte Carlo simulation, remaining within the Black-Scholes model of changeless volatility and stock monetary value motion.
In this survey, it takes an analytical attack to turn out the consequences by presuming an independent world. Arbnor & A ; Bjerke ( 1994 ) points out that the feature of this attack is its cyclic nature. This characteristic can Begins and ends with facts and these facts can take to the start of a new rhythm. When it applies for this survey, it means to choose a good theoretical account to depict the nonsubjective world or prove a theoretical account whether it is nice tantrum for describe the nonsubjective world. In add-on, the attack shows quantitative character and involves some complicated mathematics calculation apply to the theoretical account.
We use a big sum of empirical informations apply the approached we employed to gauge VaR, it means that the consequences come from a batch of historical informations trial and analysis. This manner of survey shows we are taking a quantitative attack. Harmonizing to Arbor & A ; Bjerkne ( 1994 ) , quantitative attack show much clear about the variables and cover a great sum of historical informations comparison to qualitative attack. This attack besides assumes that the theoretical constructs can be measured. A batch of empirical informations is collected to be tested to mensurate the attacks whether can gauge VaR exactly.
Deductive attack begins with a general construct, given regulation or bing theory and so traveling on to a more specific decision. Woolfolk ( Woolfolk, 2001, p. 286 ) describes this attack as “ pulling decisions by using regulations or rules ; logically traveling from a general regulation or rule to a specific solution ” .
In this survey, we test the public presentation of four normally use VaR attacks based on three different underlying assets with different features at different assurance degree. The intent is to analyze the theoretical accounts non make a new theoretical account to gauge VaR.
The concluding decision might beef up some attacks for some specific implicit in assets and might weaken some attacks for other attacks on other implicit in assets at different assurance degree.
All the empirical information is used in this survey can be checked in public beginnings, and a certain sum of old surveies sing to the attacks used in this survey to gauge VaR. One can look into the consequences whenever they want to see whether the consequences are dependable by look intoing whether the consistent consequences are the same as this survey shows, if non the same, means this survey is non dependable.
It is really of import to demo cogency when justify an attack or a theoretical account. This means if the consequences is non able tell the truth of the world, the consequences which is employed by the attack or theoretical account is non cogency, the consequence is non meaningful. In other words, the grade of cogency is depends on how closer we get a true image of the world of a given state of affairs.
To better show cogency, it is critical to cognize the relation between the theory and informations. If the information is accommodating to the theories in a continuously manner, that means it hold a strong cogency of the theory or the theoretical account which employed for the survey. This is confirmed by the survey of Holme & A ; Solvang ( 1991 ) . In this paper, different attacks are chosen to gauge VaR base on three different assets and empirical clip series informations at three assurance degrees. The cogency will be enhanced if the information tantrum the attacks or theoretical account continuously.
Appraisal of VaR
In this survey, four attacks are employed to gauge VaR for three different underlying assets. The ideal state of affairs is that appraisal value of VaR is fit for the future value of returns. But the existent state of affairs is that the attack might overrate or undervalue VaR comparison to the existent returns. For illustration of bank industry, if VaR is overestimated, it means that Bankss hold inordinate capital to cover losingss under the ordinance of Basel II agreement. While in instance of VaR is underestimated, it might take to failure to cover unexpected losingss. This is why some American bank went to bankruptcy during the recent fiscal crisis.
The four attacks are Historical simulation attack, Traveling mean attack, GARCH normal attack and GARCH pupil T attack. The implicit in assets are being analyzed are Brent oil, S & A ; P 500 and United States three month exchequer measure.
When utilizing the parametric attacks to gauge VaR, we do surmise that whether the returns of implicit in assets are fit for our premise of distribution, such as normal distribution for traveling mean attack and GARCH normal attack. Harmonizing to Jorion ( 2007 ) that economic clip series are seldom usually distributed. So the public presentation of these parametric attacks will be showed less efficiency if the implicit in assets are off from normal distribution.
Historical Simulation Approach
Estimating VaR by utilizing the historical simulation attack is non a complicated mathematically computations but it requires a batch of historical informations. As presented in Chapter 2, the right window size is critical because if the empirical information is excessively short, it might hold a extremely changing VaR, while a longer window length would bring forth a better appraisal but the older empirical informations might be low relevancy of future returns.
The first undertaking of this attack is to choose an empirical window length to calculate the future returns. We will choose a traveling window of the old 2000 observations which approximately eight calendar old ages. The window length chosen are based on entire sample size which is more than 5000 observations for three underlying assets and assurance degree 95 % , 99 % and 99.9 % are used in this survey. This window length should bring forth better public presentation of this attack at higher assurance degree.
We use PERCENTILE map in Excel to cipher the n per centum percentile of the value of a clip series informations. The value by percentile map is normally non an exact value in informations set, and Excel will cipher a coveted value between two closest values by making a additive insertion. The consequences will be showed in subsequently chapter.
Traveling Average Approach
The first undertaking of this attack is the same as historical simulation attack is taking a window size. In this survey we choose 45 yearss which is 9 calendar hebdomads to cipher the standard divergence.
It is easy to utilize STDEV map to cipher standard divergence base on traveling 45 yearss window size in Excel, so utilize the consequence apply to the parametric VaR expression to acquire the value of VaR. The consequences will be showed in subsequently chapter.
GARCH Normal Approach
Like traveling mean attack we need to cipher standard divergence foremost to cipher the concluding VaR, but before making that, we have to gauge the parametric quantities, and foremost. The parametric quantities are estimated by utilizing maximal likelihood appraisal. It is a challenge occupation because the old literatures have been studied in this field merely described the MLE map but did non demo how to implement it.
In this survey, we estimate the parametric quantities by in EVIEWS. Then the inquiry is how to make up one’s mind the traveling window size. As MLE map besides assumes that the returns are usually distributed, so the smaller the window size, the larger the hazard that those values off from normal distribution. First estimation the value with window size of 1000, 2000, 3000, 4000, 5000 severally, we found that the value of window size 3000 is most close to the value shows by Jorion ( 2007 ) base on the similar fiscal assets, so we take 3000 as the window size to gauge these three parametric quantities value. The small different between our estimated value comparison to the value advised by Jorion ( 2007 ) is because we use different clip period and the implicit in plus is non precisely the same, and we believe the consequences is dependable. The value and consequences showed by EVIEWS are showed in appendix 1.
After we got the value of parametric quantities, and, we input the value to the expression of GARCH ( 1,1 ) theoretical account to cipher the volatility, the VaR is estimated base on the volatility value consequently and the consequences will be shown in the consequences and analysis chapter.
GARCH pupil T Approach
Under this attack, we estimate the parametric quantities, and foremost, it is the same occupation as GARCH normal attack, we did it by EVIEWS and the consequences are showed in appendix 1.
When we got the parametric quantities value, we employ it to gauge the value of volatility, so we apply it to cipher VaR with the critical value under pupil T distribution ( Jorion, 2007 ) . All these computations are done by Excel and the consequences of VaR will be showed in subsequently chapter.
The lopsidedness indicates a distribution looks compare with a normal distribution. For normal distribution, the form of distribution is symmetrically distributed around its mean. In the instance of normal distribution, its skew is 0. As mentioned before, the fiscal assets is non precisely usually distributed, they might hold a positive or negative lopsidedness. Below graph show the negative skew and positive skew.
Negative skew means that it has a longer left tail, most of the distribution is concentrated on the right once more the mean. While Positive skew is the positive instance of negative skew. Skewness is to be concerned because traveling mean attack and GARCH attack are assumes the implicit in assets are under usually distribution, if the implicit in assets are to a great extent skewed, their appraisal of VaR will less accurate. The two attacks might undervalue or overrate the VaR value harmonizing to the skew of the implicit in assets return ( Lee, Lee & A ; Lee 2000 ) .
When kurtosis is 0 and symmetrically distributed, it is the instance of normal distribution. A high kurtosis of underlying assets indicates that there are more utmost values on this implicit in plus distribution than those of normal distribution. The positive extra kurtosis is called leptokurtic and the negative extra kurtosis is called platkurtic. The VaR values will be underestimated when it is negative kurtosis.
The Source of Data
In this survey, we use day-to-day empirical informations of three fiscal assets, they are clip series informations. The clip period is selected from 1st Jun 1989 to 29th May 2009 for all three assets. The first two 1000s observations ( from 1989 to 1997 ) are used as historical informations for prognosis the hereafter, so the remainder of informations are classify into two period, period 1 is from 1997 to 2009 ( more than 3000 observations ) stand foring for the clip period including normal clip and fiscal crisis clip, while period 2 is from 2008 to 2009 ( about 355 observations ) stand foring for fiscal crisis clip period. Divide this two period is for the intent of this survey.
The information we collect is historical day-to-day monetary value of three assets: S & A ; P 500 index, Brent oil and US three month exchequer measure. It is easy to acquire the informations from public beginning. S & A ; P 500 informations that we can acquire it from Yahoo finance ( hypertext transfer protocol: //finance.yahoo.com ) , Brent oil informations can acquire from Energy Information Administration ( hypertext transfer protocol: //www.eia.doe.gov ) , US three month exchequer measure data we get from Federal Reserve Bank of St. Louis ( hypertext transfer protocol: //research.stlouisfed.org ) .
The S & A ; P 500 is a valued leaden index which consist 500 large-cap companies of United States, it can be viewed as a well diversified portfolio, so their volatility is non every bit high as Brent oil and US three month exchequer measure. Figure 3.6a shows the day-to-day returns of S & A ; P 500. From table 3.5 we see the value of mean day-to-day volatility of S & A ; P 500 is 1.38 % and one-year is 21.74 % , it indicates that public presentation of monetary value alteration is pretty stable. The lopsidedness of S & A ; P 500 is -0.15 and the mean day-to-day ln monetary value alteration is 0.94 % , it shows the distribution curve of S & A ; P 500 is a nice lucifer with normal distribution without sing the kurtosis.
The kurtosis is 7.33, it shows that the distribution is narrower than the normal distribution and has a fatter dress suit. This value of kurtosis is between assets Brent oil and US three month exchequer measure, plus consideration of lopsidedness value, it can be concluded that Brent oil is the plus that most fit normal distribution, so S & A ; P 500 and US three month exchequer measure. This can be viewed by comparing the histogram of figure 3.6b, 3.7b and 3.8c. Therefore it can anticipate that S & A ; P 500 will to bring forth a public presentation between Brent oil and US three month exchequer measure by traveling mean attack and GARCH normal attack. The value of high kurtosis make this plus performs less effectual by parametric attacks which assume assets are follow normal distribution.
The Brent oil monetary values is the 2nd volatile among the three underlying assets, this can be seen from the tabular array 3.5 comparison with other two assets. The day-to-day volatility of Brent oil is 2.71 % and one-year is 43.02 % shows that it much more volatile than S & A ; P 500.
The Kurtosis is 4.46 indicates that it is a small spot narrower and has a little fatter dress suits than normal distribution, while the lopsidedness is -0.18, it shows that the distribution is comparatively high symmetric. Combine with the value of kurtosis and lopsidedness, it indicates that Brent oil is the plus that most fit for normal distribution, and this can be seen of figure 3.7b. Therefore plus Brent oil suppose to execute better in parametric attack than other two assets. The day-to-day ln monetary value alteration is 1.94 % , it shows this plus are high volatility plus, it might execute less effectual by nonparametric attack because the nonparametric attack is non good at assets with high volatility.
The US three month exchequer measures is the most volatility plus of all three assets we employed in this survey, its day-to-day volatility is 8.74 % and mean ln monetary value alteration is 2.32 % , while the value of lopsidedness is -3.27 and a really high kurtosis of 360.84, above values suggest the return distribution of this plus is hapless fit normal distribution, it shows this characteristic in figure 3.8c.
From the figure 3.8b, US three month exchequer measure was non a high volatility plus before 2007, but since 2007, its volatility aggressively goes up and go really high volatility which can be seem from figure 3.8a. If the clip period chosen is before 2007, its volatility should be less than Brent oil. But this period is non fit the intent of this survey, our survey is focus on the clip period including recent fiscal crisis. Base on above features of this plus, it can be expected that this plus will execute the worst no affair nonparametric or parametric attacks.
For clip series informations, it is of import to look into whether it has autocorrelation or others call consecutive correlativity. Autocorrelation in clip series informations means the informations correlative with itself over clip and can be measured by a one lagged Durbin-Watson trial in a arrested development. The being of autocorrelation indicates that the employed attack is hapless tantrum to the clip series informations that the monetary value of today can non be described as a additive map of monetary value of yesterday. Tsay ( 2002 ) states that in instance of autocorrelation for clip series informations, there are other factors besides the historical monetary values that affect today ‘s monetary value, the consequences will be less effectual if we use this attacks forecast the hereafter monetary values.
The void hypothesis will be rejected if the value of DW is non in the parts of the interval which is the value between DL and DU, from table 3.9 shows that all these three assets are within the interval, therefore the void hypothesis will non be rejected and there is no grounds of demoing autocorrelation for these three clip series informations.
The best is the value equal to the figure of observations times the result of one minus the selected assurance degree, the parts shows the acceptable interval for the exclusions of VaR. The nearer of value of exclusions near to best value, the higher public presentation the attack do. If the exclusions are over the parts a batch, it indicates that the attack underestimation hazard a batch in future, on the other side, it overestimates risk a batch in future.
Data Result & A ; Analysis
Backtesting Results of Christoffersen
The backtesting consequences of Christoffersen based on three underlying assets and four attacks are shown below. The consequences will be analyzed and discussed harmonizing to the different assets, different attacks and different clip period at different assurance degrees. The drumhead consequences of Christoffersen for the four attacks are shown in Appendix 2. We choose two period informations to prove the VaR public presentation. Period 1 is from Apr 1997 to May 2009 ( more than 3000 observations ) , stand foring the clip period that includes the normal economic clip and the fiscal crisis clip. Period 2 represents the recent fiscal crisis clip period from 2008 to 2009 ( 355observations ) . Period 2 was selected in order to happen out how its VaR performs compared to that of period 1. One might inquire if period 2 makes sense by utilizing about 355 observations at a assurance degree of 99.9 % . It is true that 355 observations is non a good sample size for a 99.9 % assurance degree, but our intent here is to see whether the attacks underestimate hazard in fiscal crisis clip, because this can be indicated by the exclusions figures over the parts, we are non concentrating on whether the attacks overestimate hazard at this assurance degree 99.9 % because the observations is excessively little.
Historical Simulation Approach
Sing the plus S & A ; P 500, it produces bad consequences in period 1 and awful consequences in period 2. This attack can non gauge the peril of this plus decently. At a assurance degree 99 % and 99.9 % , the figure of exclusions are twice compared to the part ‘s maximal bound in period 1, while in period 2, the figure reveals that it is worse. It indicates that this attack produce a hapless consequence to gauge VaR in the above two periods with regard to the plus of S & A ; P 500. We should detect that because the above two periods include the recent fiscal crisis clip period, it affects the consequences of this attack given that it is non good at foretelling hazard during utmost clip periods. For illustration, at 99.9 % assurance degree, if the clip does non include the fiscal crisis clip period ( period 1 subtraction period 2 ) , its consequence is within the parts, that means it work at 99.9 % assurance.
For Brent oil, it besides produces bad consequence, but it shows better than S & A ; P 500, nevertheless it still shows that this attack underestimation hazard for this plus. The figure of exclusions is somewhat over the parts at 95 % and 99 % assurance and within the parts at 99.9 % in period 1. In period 2, it works ill at 95 % and 99 % assurance degree while at a assurance of 99.9 % , no clear consequences can be drawn because the figure of exclusions within the parts and the sample size are excessively little. If the period is does non include period 2, the consequences will be within the parts at 99 % and 99.9 % assurance degree, which means this attack can bring forth an acceptable consequence for this implicit in plus during the normal clip period before the recent fiscal crisis clip period at 99 % and 99.9 % assurance degree.
With regard to US three month exchequer measure, this attack performs the worst. This means the attack is non appropriate for gauging the hazard for this implicit in plus. The figures are unacceptable both in the two periods at all assurance degrees. The ground is that this plus is the most volatile 1 among the three underlying assets with a day-to-day volatility 8.74 % and with a really high kurtosis. Because historical simulation attack weight all the returns every bit, it takes clip to respond to utmost fluctuations of the returns. Another ground for this bad consequences might be the pick of window length, 2000 empirical returns might takes excessively much old information which is non relevant for gauging future return for this extremely volatile plus, particularly to gauge hazard since 2007.
The consequences indicate that this attack produces really hapless public presentation at 95 % assurance degree and hapless public presentation at 99 % assurance degree for all three assets, particularly for the plus of US three month exchequer measure no affair the clip period, its figure shows it underestimate hazard a batch. At 99.9 % assurance degree, this attack works to some extent for plus of Brent oil, and it besides works for S & A ; P 500 in the normal period ( the period non including period 2 ) . Therefore in this trial, historical simulation attack performs really hapless.
This attack assumes that the hereafter is indistinguishable to the past, but with the fact that more and more uncertainness is impacting the hereafter, the volatility in future seems non indistinguishable to the yesteryear, that is why this attack produces really bad consequences for the three assets. Overall for all the attacks in this survey, the public presentation of historical simulation attack is the worst one compared to the other three. However this does non intend that this attack is non valid for gauging VaR. It can be used with regard to higher assurance degree and proper assets, like the plus Brent oil in high assurance degree and non with assets like the US three month exchequer measure with really high volatiltiy.
Traveling Average Approach
This attack assumes the return of underlying assets follow a normal distribution which is non realistic because the returns of most fiscal assets including the present 1 has been shown by the lopsidedness and kurtosis figure in chapter three. Because the three assets are non truly usually distributed, so the public presentation of this attack is affected by the grade to which the underlying assets ‘ distribution expression like that of a normal distribution.
For plus S & A ; P 500, this attack does non execute good both in period 1 and period 2. For all three assurance degrees, it produces the worst consequence for this plus among the three assets. The ground for this may be because of the window size significance that a 45 yearss window size is more fit for the assets Brent oil and US three month exchequer measure because they have high volatility compared to plus S & A ; P 500. It might hold provided better consequences if another window size were used.
For Brent oil, this attack produces a really good consequence in period 2, and it is works good at a assurance degree of 95 % in period 1. It indicates that this attack can bring forth a better public presentation at a low assurance degree for assets like Brent oil and can besides work during the utmost clip period for this plus. The consequences are good because the feature of this plus ‘s returns ressemble that of the normal distribution.
Sing the plus US three month exchequer measure, this attack merely performs good at a assurance degree of 95 % in both periods. It indicates once more that this attack can bring forth a better consequence at a low assurance degree for an plus like US three month exchequer measure with a really high lopsidedness and kurtosis. Compared to the low assurance degree, it shows a bad public presentation at a high assurance degree no affair the clip period. Another ground why this attack can non bring forth a good consequence for this plus is the constellating issue. We can see the day-to-day return graph in chapter three reveals that the US three exchequer measure have a earnestly constellating utmost value during the recent fiscal crisis, and traveling mean attack does non take the bunch phenomenon into consideration.
Overall for all the attacks in this survey, this attack produces a good public presentation at low assurance degree and is more appropriate for assets like the Brent oil that whose return distribution ressembles the normal distribution though it performs ill at higher assurance degree. Compared to the GARCH attacks, it shows less efficiency to gauge VaR.
GARCH normal Approach
The GARCH normal attack besides works under the premise of normal distribution, so it is the same as the traveling mean attack where the consequences might be less efficient if the distribution of assets perverts from the normal distribution. Contrary to the traveling mean attack, the GARCH theoretical account can take into consideration the bunch phenomenon. That will enable the GARCH theoretical account to bring forth better consequences than the moving mean theoretical account.
The above two tabular arraies show that this attack performs really good in period 1 at a assurance degree of 95 % with the figure of its exclusions being more or less like the best mark figure for three assets. Compared to the low assurance degree, it produces bad consequences at higher assurance degree in both period 1 and period 2 with the exclusion of the plus Brent oil. The consequences once more demonstrated that parametric attacks under the premise of normal distribution produce better consequences if the distribution of the returns of the implicit in plus ressembles the normal distribution.
Though GARCH theoretical account can cover with constellating jobs, but the consequences of GARCH normal attack is non mostly superior to the traveling mean attack. Goorbergh & A ; Vlaar claim that the feature of volatility bunch is the most of import feature when ciphering VaR. However from the consequences utilizing the traveling mean attack and GARCH normal attacks which are tested by Christoffersen, it did non do a large difference. We agree that the volatility bunch is an of import feature when gauging VaR, but it is non the most of import 1. The most of import should be the distribution of the implicit in assets. Because both the moving mean attack and the GARCH normal attack are under the premise of normal distribution, if the implicit in assets are non really like the normal distribution, it will bring forth more or less acceptable consequences. The consequences of these two attacks prove above statement.
Another factor with regard to this attack is the parametric quantities estimation utilizing the MLE. The value of parametric quantities, and can impact the consequences of VaR appraisal. In this survey we take 3000 observations to gauge the parametric quantities utilizing EVIEWS, which might non exactly represents the existent value of the parametric quantities, and. Besides, because the maximal likeliness map is under the premise of the normal distribution hence, it might do this attack less efficient.
Overall for all four attacks, the GARCH normal green goodss better consequences than the historical simulation attack and the moving mean attack no affair whether in period 1 or period 2 at 95 % and 99 % assurance degree. But at higher assurance degree, it performs ill. Compared to the historical attack and moving mean attack, GARCH normal does a better occupation in covering with the volatility constellating phenomenon by utilizing an advanced manner to gauge volatility and do the appraisal exactly. But under the premise of normal distribution, its public presentation is non powerful, it still underestimate hazard at higher assurance degree and for the utmost clip period. To better trade with this job, GARCH with pupil Ts attack does a really good occupation.
GARCH with a pupil t-distribution
Unlike the above two parametric attacks, this attack is under the premise of student-t distribution which assumes that the underlying assets have a heavy tail. This is more realistic for fiscal assets and enables the GARCH theoretical account to bring forth better consequences. In add-on, this attack can take into history the volatility and bunch phenomenon.
It can be seen from the above two tabular arraies that GARCH with the student-t attack produces better consequences for all the assets at all assurance degrees in both periods. All the figure of exclusions is less than the maximal value for the parts connoting that this attack does non underestimate hereafter hazard for all three assurance degrees and underlying assets no affair what clip period is used. This shows this attack produces powerful consequences to gauge VaR among the four attacks. Similar to the GARCH normal attack, the appraisal of the parametric quantities, and have an consequence on the truth of the VaR appraisal. However from the figure of exclusions shown in the above two tabular arraies, it can detect that this attack can capture the hazard wholly and may sometimes overrate the peril.
In period 1, the figure of exclusions at 95 % and 99 % assurance degree is less than the minimal value of the parts. This is rejected by Christoffersen trial because it mostly overestimate hazard. It is excessively conservative to gauge VaR at these two assurance degree in period 1. While at 99.9 % assurance degree, it performs rather good as the figure of exclusions is more or less like the best expected value. Therefore this attack is excessively conservative in period 1 at 95 % and 99 % assurance degree, and is rather acceptable at 99.9 assurance degree.
In period 2, it performs better than in period 1 while demoing sensible public presentation at 95 % and 99.9 % assurance degree for all three assets connoting it does rather good during the fiscal crisis clip period. It is nevertheless excessively conservative at 99 % assurance degree for the Brent oil and US three month exchequer measure during the utmost clip period.
The difference between this attack and the GARCH normal attack is merely the premise of assets ‘ distribution, but it produces wholly different consequences. This once more demonstrates that the distribution is the most of import characteristic that affect VaR appraisal. Because these three underlying assets are in world non usually distributed, the traveling mean attack and GARCH normal produce hapless consequences. The three assets might non precisely follow pupil T distribution, but they have heavier dress suits than normal distribution and the dress suits are non every bit heavy as pupil T distribution, hence, GARCH pupil T attack overestimation hazard in period 1 at 95 % and 99 % assurance degree in period 1.
Overall for the four attacks, this attack produces the best consequence when gauging VaR. Besides this attack shows overestimated hazard at lower assurance degree in period 1 significance that the attack is excessively conservative when gauging VaR at lower assurance degree. Such a feature is non welcomed by houses because they do non desire to maintain so much capital modesty to forestall hazard which is non really bing. On the other manus, it performs really good at higher assurance degree no affair the clip period, particularly during the fiscal crisis clip period.
We want to one time once more emphasize the cogency of the findings because it is truly of import for any survey. We stress that the attacks we applied and the consequences we got are extremely valid in this survey. We will look at two facets to demo the cogency of this survey – 1 is the surface cogency and another one is external cogency.
Sing to the surface cogency, the consequences we concluded might be conflicting with old surveies, or old surveies have non been done under the same conditions before because we are utilizing four attacks to gauge VaR for three underlying assets with different characteristic, at three different assurance degrees and the clip periods are really near to now ( clip period 1 is from 1st Jun 1989 to 29th May 2009, clip period 2 is from 1st Jan 2008 to 29th May 2009 ) . In add-on, the attacks are the parametric and nonparametric technique with the inclusion of the premise of normal distribution and pupil T distribution for the parametric attack.
The old surveies shows that the historical simulation attack is a simple attack but can still bring forth comparatively good consequences, though it performed ill in this sturdy because the clip period and the assets we chose were different. Including recent fiscal clip period and a extremely volatile plus like the US three month exchequer measure affect the consequences of the historical simulation attack. In add-on, there are some surveies demoing that the traveling mean attack outperform GARCH normal attack or GARCH normal attack outperform traveling mean attack. In this survey, we found that the GARCH normal performs better than the moving mean attack as the figure of exclusions indicate. For the implicit in assets, it is of import to cognize their features, for illustration, for the Brent oil, both the moving mean attack and the GARCH normal can make a good occupation in period 1 at 95 % assurance degree. But for the plus US three month exchequer measure, the consequences provide a large different between the parametric and the nonparametric attack.
For the internal cogency, although there are some divergencies from the consequences we expected compared to the existent consequences, the consequences are rather good in general. We expected that the historical simulation attack should hold a really hapless public presentation for the plus US three month exchequer measure which is extremely volatile. The Brent oil performs the best of the three assets utilizing the parametric attacks under the premise of the normal distribution and the GARCH student-t attack produces superior consequences than the other attacks used in this survey.
The general grade of cogency in this survey can be seen as comparatively high, even though some consequences shows somewhat different as we expected, but the general consequences are rather good to reflect this fact. The nonparametric attack is non utile when including utmost clip period though it might execute good at high assurance degree in normal clip period. The parametric attack does non execute really good under the premise of normal distribution because the normal distribution is non realistic, while under the student-t distribution, it performs good and non merely can capture hazard wholly, but besides overestimate hazard.