Modelling Value-at-Risk in Investment Banks: “Empirical Evidence of JP Morgan, Merrill Lynch and Bank of America”

The objective of paper is to assess the efficiency of financial model to capture increasing volatilities across asset class markets of the three investment banks. For which data will be collect to forecast the credit risk, and to know how well our standard tools forecast volatility, particularly during the turmoil that extend throughout the globe. Volatility prediction is a critical task in asset valuation and risk management for investors and financial intermediaries. The paper will focus on Value-at-Risk (VaR) which is a standard model that has been forecasted using both nonparametric and parametric approaches and then Backtesting procedure had been applied to achieve the both outcome. One is to detect the underlying credit risk which is associated with the market as well as portfolio risk, and other is to perceive model which provide more accurate forecasting.


Introduction
The global financial crises put practitioner and researcher to re-asses the efficiency of financial model.Increasing volatilities across asset class makes it important to know how well our standard tools forecast volatility, particularly amid episodes of turmoil that extend throughout the globe.Volatility prediction is a critical task in asset valuation and risk management for investors and financial intermediaries.The price of almost every derivative security is affected by swings in volatility.Therefore, the focus of this chapters is to detect the underlying credit risk which is associated with the market as well as portfolio risk.As market risk has sever effect on capital which banks maintain as a buffer for adversities, bad outcomes or hits, they have to maintain ample liquidity of capital to bear the hit or stress or adverse moment.The Value at Risk (VaR) approach is very useful to measures the potential loss in value of a risky asset or portfolio over a defined period within confidence interval.
To predict the efficient model to forecast value at risk (VaR) from the daily return data of historical price datafor daily predicted losses and for trading days we have calculated a total numbers of hit form the forecast and actual VaR on the daily basis from which we will get the total hit or points where the actual VaR exceed then forecast or loss actual losses exceed then forecast.This will help investment banks to predict more accurate losses as well to maintain enough capital for the losses.
The existing approach for VaR estimation may be classified into three approaches.First, the non-parametric Historical Simulation (HS) approach.Second, the fully parametric models approach based on an econometric model for volatility dynamics and the assumption of conditional normality describe the entire distribution of returns including possible volatility dynamic.Third, Extreme Value Theory approach parametrically models only the tail of the return distribution.So, here our focus would limit up to econometric models measuring VaR.

Econometric Modelling for Value-at-Risk
Value at Risk (VaR) provide the maximum losses not exceeding a given probability defined as the confidence level, over a given period of time.VaR methodology forecasting risk is applied by investment banks to measure the market risk of their asset portfolios (market value at risk), however VaR has a wide spectrum for quantitative risk management for many types of risks.
Here we present the various parametric and non-parametric models including various econometric models that are used for forecasting VaR mostly focuses on market side risk and it is one of the best approaches to capture the market risk through volatility.The VaR entails the estimation of quintile of the distribution returns.VaR is usually computed separately for the left and right tails of the returns distribution of the risk managers.The VaR of a long position (left tail of the distribution function) over a given time horizon t and probability p, while p is one minus the VaR confidence level, is defined as In the above equation F is the cumulative distribution function that describes the profit and loss distribution (P&L) of the risky financial position and  −1 denotes its inverse function.In order to estimate VaR for our research we will focus on some parametric and non-parametric models and then compare them to predict the best outcomes or most accurate models to estimate VaR.

GARCH model
According to Engle (1982) and Bollerslev (1986) the VaR is based upon the assumption that the standard deviation in returns does not change over time (homoskedasticity), Engle argues that we get much better estimates by using models that explicitly allow the standard deviation to change of time.Suggesting two variants -Autoregressive Conditional Heteroskedasticity (ARCH) and Generalized Autoregressive Conditional Heteroskedasticity (GARCH)that provide better forecasts of variance and, by extension, better measures of Value at Risk.Engle (1982) Ng (1993) provide the ability to forecast the volatility on the basis of conditional mean and variance in the financial market with the proper selection of the financial assets to structure an investment portfolio.Merton (1980) shows that the expected returns on the market are related to the accurate volatility forecast.There is a relationship between risk and return when dealing the fair valuation of assets, Glosten, Jagannathan &Runkle (1993) pointed out in certain period of time, there is a higher yield from riskier asset, depended on particular strategy.
The aim of applying GARCH model isparticularly to capture the volatility clustering in assets returns.As Madelbrot (1963) shows periods of time where the returns of the assets do not show high variation and certain period in which the variation of returns related to their mean are high.The econometric challenge is to specify how the information is used to forecast the mean and variance of the return, conditional on the past information.While many specifications have been considered for the mean return and have been used in efforts to forecast future returns, virtually no methods were available for the variance before the introduction of ARCH models.Nelson (1991) proposed the extended version of GARCH which unlike the ARCH and GARCH allows for the symmetry in the responsiveness to shocks, does not impose the non-negative constraints on parameters and reduces the effect of outliers on the estimation results.EGARCH has been commonly used to examine interest rate, inflation rate of future markets, exchange rate and in the analysis of stock returns (Tsay, 2005).
The GARCH model that has been described is typically called the GARCH (p,q) model.The (1,1) in parentheses is a standard notation in which the first number refers to how many autoregressive lags, or ARCH terms, appear in the equation, while the second number refers to how many moving average lags are specified, which here is often called the number of GARCH terms.The GARCH models explicitly model the conditional volatility as a function of past conditional volatilities returns.

Data Methodology
Computing VaR forecast to calculate expected shortfall or expected value of the loss given that an event outside a given probability level has occurred out-of-sample for the period of 2001-to-2011 for banks which survived the crises, and forecast VaR for the period of 2006-to-2008 banks who were failed during the sub-prime from the data of daily historical price.The historical return series data have been taken for three banks (i.e.JP Morgan Chase & Co., Merril Lynch, and Bank of America) as per the methodology and these three banks have identical operations but different attributes.The JPM is one of the largest investment banks and does not offer general banking services but have been more stable during pre and post crises period, while ML Banks who also offered identical services but collapsed during the 2007-08 turmoil and lastly BofA, offering investment banking services as well general banking (i.e.accepting deposits and loans of customer's) also survived in the crises.To study offorecasted value-at-risk predicted volatility and capital requirement to overcome from the expected shortfall and number of Hit's when bank's losses exceeded then their daily return.We have taken bank's daily return data of historical price from the period of 1991-2011 except ML data has been analysed for the limited period of 2001-2008 due to nonavailability of historical price data.
Further, the VaR has been computed using both non-parametric and parametric approaches.The non-parametric includes Historical Simulation (HS) approach while parametric approach includes GARCH, Exponential-GARCH (E-GARCH), and Threshold ARCH (TARCH) and another standard approach which combines both nonparametric and GARCH model well known as Filtered Historical Simulation (FHS).
The important assumptions for applying VaR is confidence level  on given time horizon at losses with amount that will exceed with a probability of 1 −  on given time horizon.In our research the VaR computed on both 95% and 99% confidence level,  and 1 −  represent significance level at 0.05% or 0.01% with the assumption that return is normally distributed with mean zero and standard deviation one.
Journal of Global Economy, Volume 14 No 2, June, 2018 So, VaR formula with the confidence level  and 1 −  represent significance level Where: L is the lower threshold of loss with the probability of losing more than L on a particular time horizon is 1 − .

Descriptive Analysis
The return series has been generated from the daily stock price data from Bloomberg, to convert daily data into return series the normal log has been computed with the standard formula i.e. ln   / −1 taking from natural logarithm current information divided with previous information.The Figure 5.1.Shows the descriptive statistics.As the descriptive statistics shows that a mean of BOA -0.00077, JPM -0.000101 and ML-0.000108.hence mean value is non-zero so we reject the our Null Hypothesis rejecting assumption of mean equal to zero, with median, and std.deviation of all three banks with the Jarque-Bear statistic shows that the null hypothesis of normality is rejected at any level of significance, as evidence by high excess kurtosis and negative skewness due to conditional volatility.The unconditional distributions are non-normal and have a long left tail relative to a symmetric distribution cause of higher volatility near future due to negative shock.
Further, another table display the descriptive results compute garch/e-garch and tgarch with the help of the statistical package in eview's the results have been shown in the table 5.2.
From the Table : 2. We could conclude that the volatility term in the mean equation is statistically significant as indicating that rather than being constant the mean return is dependent on the level of volatility.
The GARCH (p,q)-model has been computed using one lag i.e. p lags of the squared error.To

RETURN
checking hetreoscadisticity the modified white test has been conducted.White test statistic is computed as the number of observations times of r-squared.It is asymptotically distributed with degrees of freedom.The table: 3. shown below of Fstatistics shows that almost it is non-significant as the F-test value is almost nonsignificant so our alternate hypothesis rejected that assumes presence of hetroscadisticy.

Historical Simulation (HS)
The first and the most commonly used method is referred to calculating value-atrisk was called historical simulation or historical VaR.A contemporaneous description of historical simulation is provided by Linsmeier and Pearson (1996).The main idea behind the HS is assumption that a historical distribution of returns will remain the same over the next periods (i.e.assumption of that price changes behaviour repeats itself over time).As a result, the VaR based on HS is simply the empirical quintile of the distribution associated with the desired likelihood level.
Where quantile have been calculated with confidence at 95% (i.e. at the significant of 5%) to furcate for the given period from the daily return data.1. Quantile has been computed based on first quantile of the return data using excel.

Filtered Historical Simulation
To overcome with the shortcoming of the non-parametric approach of historical simulation the another approach introduce by Hull and White (1988) and Barone-Adesi et al (1999) combines with the Historical simulation and the GARCH model known as Filtered Historical Simulation (FHS).The importance of the model is that not making any distributional assumption for the standardized return during forecasting variance in the volatility model.The FHS model is assumed to be more superior then historical simulation.FHS combine the GARCH-model for return series and use Historical Simulation to infer the distribution of the residuals through applying the quantiles 1 of the standardized residuals (i.e.residual divide by conditional standard deviation) and the conditional mean forecasts from a volatility model.
The Figure-4.Representing the computed FHS, calculated with confidence at 95% (i.e. at the significant of 5%) to furcate for the given period from the daily return data of forecasted value-at-risk (VaR) for the Bank of America, JP Morgan Chase & Co., and Merrill Lynch

GARCH
GARCH extends ARCH process to include past squared return and past variance in the model.So, in the Generalized Autoregressive Conditional Heteroskedasticity (GARCH)-model the conditional variance is defined as a linear function of lagged conditional variance, and squared past returns with the assumption of iid and zero-mean.Where,   conditional mean, and   is conditional variance process.The model have following assumption:   ~ ((  ) = 0. (  ) = 1 .  2 = (Ω −1 ) Where,   2 = (  ΙΩ −1 ) =   conditional mean with given the information set of available at time t-1, Ω −1 , {}   = 0 is the innovation process with conditional variance(  ΙΩ −1 ) =   2 , f (.) is the density function of {}   = 0 and g is the functional form of conditional volatility.To computing VaR in GARCH model assumed normal distribution with the following equation of VaR.
+1,  =  +1 +    +1 Where,  +1 and  +1 are the conditional forecasts of the mean and the standared deviation at time t+1, given the information at time t.In the AR (1)-GARCH (1,1) model for value of p and q result of a specification search in terms of AIC an BIC criteria.
This model is fitted to data series using a pseudo maximum likelihood estimation assuming normal distributed innovation to estimate parameters  and standardized residuals( ).
The Figure-5.Representing the computed GARCH, calculated with confidence at 95% (i.e. at the significant of 5%) to furcate for the given period from the daily return data for forecasted value-at-risk (VaR) for the Bank of America, JP Morgan Chase & Co., and Merrill Lynch

T-GARCH
The appropriate method of measuring is Threshold GARCH proposed by Zakoian (1994.The threshold GARCH model in the contest of conditionally heteroscedastic time series have been found appropriate to analysing asymmetric volatilities.While the effect of current volatility on the future volatility decreases to zero at an exponential rate for standard-threshold-GARCH (TGARCH) processes, here we introduce a class of T-GARCH processes exhibiting persistent (in volatilities) properties when current volatility constantly remains for long time in future volatilities for all-step ahead forecasts Nelson (1990).
A general class of Threshold-GARCH (T GARCH) model can be derived as: Where, ℎ  volatility depends on whether the past squared value that are positive or negative asymmetric GARCH via 'threshold'.  is a sequence of iid random variable with mean zero and variance unity.
The time series {  } governed by (1,1) will be referred to as T-GARCH (p,q).
The Figure-6.Representing the computed GARCH, calculated with confidence at 95% (i.e. at the significant of 5%) to furcate for the given period from the daily return data for forecasted value-at-risk (VaR) for the Bank of America, JP Morgan Chase & Co., and Merrill Lynch

E-GARCH
Exponential GARCH-model consider the best model and basically computed logarithm of conditional variance and address that a negative return affect more than positive.As it variance calculate in logarithm and predicted more accurate as squared of value become positive.In the GARCH model, it's assumed that only the magnitude of unanticipated excess returns determines  2 .Otherwise it would be argued that not only the magnitude but also the direction of the returns affects volatility i.e. negative shocks (event/news) tend to impact volatility more than positive shocks.The limitation is that the how does a shock linger in the volatility estimate.This two limitation are the main factor for developing E-GARCH model: Where, , , , ,   are coefficients, and   comes from a generalized error distribution.
Using E-GARCH we can drive better estimate for the volatility for assets return then classic GARCH model.
The Figure -7.Representing the computed GARCH, calculated with confidence at 95% (i.e. at the significant of 5%) to furcate for the given period from the daily return data for forecasted value-at-risk (VaR) for the Bank of America, JP Morgan Chase & Co., and Merrill Lynch   huge volatility in the crises period.However, the VaR of Bank of America is seems to be more stable as the gaph's shows less volatility captured during the defined period.
The violation occurs when a realized return is greater than the predicted VaR. the violation ration is defined as the total number of violation, divided by the total number of one-period forecast.If the violation ration at the pth-quantile is greater than α percent, then it's implies excessive underestimation of the realized return.If the violation ration remain less than α percent at the pth-quantile, there is excessive overestimation of the realized return by the underlying model.For example, if the violation ration is 4% at the 95 th quantile, the realized return is only 4% of the time greater than what the model predicts.Sometimes it argued by the experts or researcher that if violation is too low i.e. not-significant that mean bank's maintain enough capital to cover their expected losses but missed out the profit opportunities, as there is trade-off between risk and return, but it's important that banks should maintain ample capital bad event.
The Table : 1 and Table : 2 shows violation at 95% and 99% confidence level at long tail (buy side) with the number of violation and percentage of violation under different model.We will check the efficiency and relevance of the model when comparing predicted VaR with actual returns and further back-testing.

Statistical Parametric Backtesting
The purpose of statistical backtesting methodology are adopted by banks for their internal risk measurement comparison of daily profits and losses with model-generated risk measures to assess the quality of and accuracy for their risk measurement systems.This whole process is called backtesting, and known a technique for evaluating banks risk measurement with a selection of more accurate model.At present bank select different model for the purpose of backtesting with an aim of selecting more accurate and inaccurate models.
The out-of-sample problem has been reviewed by many expert; i.e.Kupiec (1995), Berkowitz andO'Brien (2002), Engle and Manganelli (2004), to name but a few.These references have assumed the correct specification of the VaR model in forecast evaluations.Therefore, prior to the forecasting stage, the risk manager has to decide, using the available information, i.e. all the sample, which econometric model is most adequate for the conditional VaR process.This preliminary stage involves model selection and validation, hence the importance of quantile specification tests associated to VaR.

The Backtesting Framework 1
The underlying idea of applying backtesting are basically adopting strategy for internal risk measurement through best models.Originally the backtesting consist a periodic comparison of the bank's daily value-at-risk measures with the daily return series.Initially to measure the market base and operation risk side of credit risk of the portfolio of the banks.The comparison of risk measure with the daily return series and marked the number of times when risk measure were larger than the daily return series of the bank called Hit's.
Then these outcome of hit's actually compared with the assumed level of coverage to assess the performance of bank's risk model.For comparison of those hits a large number of statistical tests could be applied.As the VaR framework provide the risk assessment for an estimate of the amount that could be lost in a day due to change in market movement over a given holding period with a pre-determined confidence level.The standard procedure of applying Backtesting is to compare observed percentage of outcomes of covered risk with pre-determine i.e. 95% or 99% level of confidence.This implies that banks 95 th or 99 th percentile risk measures truly covered 95% or 99% of the bank's actual outcome.
The another approach of applying backtesting with specifying the appropriate risk measures and actual outcomes arises due to sensitivity of a static portfolio with adverse volatility in price, or price shocks.In the VaR and its backtesting procedure end-of-day return series are consider as an input for the risk measurement model.And this series to be applied for the possible change in the value of static portfolio due to high volatility in the price of return series during the holding period.
The Basel framework of backtesting had suggested the use of risk measures calibrated to a one-day holding period.As it is appropriate to employ one day actual outcome for the purpose of benchmark in backtesting.While other suggested that the actual trading outcomes experience by the banks to be consider most important also, relevant for risk management purpose, which should be benchmarked against the reality.
To be more precisely the backtesting procedure is viewed purely a statistical test of the quality of the value-at-risk measures.As suggested that it should employ a daily results that allows for an "uncontaminated" test.Backtesting using actual outcomes of profit and losses become also important as it can uncover cases where the risk measurement technique are accurately capturing daily volatility in spite of being measuring with integrity.
The framework adopted by Basel Committee for backtesting purpose is most straightforward procedure for comparing the risk measures with the daily outcomes.Which is a simple method of calculating the number of hit's (i.e.exceptions) that is and not covered by daily outcome by the risk measures.

Statistical Backtesting methods
An accurate VaR model must qualify unconditional coverage test.In a good VaR model the number of violation should be as per the selected confidence level.Which always may not be true.In mathematical terms, VaR to be consider for a portfolio's value-at-risk is defined to be the 'α' quantile of the portfolio's profit and loss distribution: Where,  −1 ( •|Ωt) to be consider the quantile function of the profit and loss distribution which varies over time with a market conditions and the portfolio's composition, as embodied in Ω  change.This can be drawn with the help of example as if 5% VaR, i.e.VaR at (0.05) is Rs 100,000 then it could be said that at 5% time we should expect to observe a loss on this portfolio in excess of Rs100,00.Since bank's adhere to risk-based capital requirements by using their own internal risk models to determine their 5% or 1% value-at-risk i.e.VaR at (0.05) or (0.01).Therefore, it is important to have a means of examining whether or not reported VaR represents an accurate measure of a bank's actual level of risk.

Backtesting implementation process: Methodology
The following is the standard procedure conducting backtesting:

The Exception of Hit's (Violations)
The table: 1 summarize the violation (i.e.exceptions) are occurred at the 95 th and 99 th percent confidence level under different model's applied for predicting VaR on daily basis for the data from 2001-to-2011 for the BOFA and JP Morgan Chase while Merrill Lynch the VaR has been forecasted for the period of 2006-to-2008 due to limited data availability.The VaR predicted mostly to considering the period of highly vulnerable for banks and most of banks failed during the period of crises which includes Merrill Lynch, which later on merged with Bank of America.So, the study of ML data includes pre-merger data i.e. till 2008.
As per the table the violation define the relative performance of each model is calculated in terms of violation ratio.The violation for each banks under different model have been calculated with the percentage of violation occurred (the percentage of violation have been as calculated as per the period in which total points of portfolio loss and divided by the number of days VaR estimated).These empirical results are prepared for the purpose of backtesting which actually explain the best model for bank's for more accurate forecasting.In the table: 2. Root mean squared error and mean absolute error have been computed through Conditional forecasting and not a high realized return frequency mode which is consider true.Eview's forecasting of VaR is based on r-squared conditional frequency with an assumption that is true.The selection of model is based on least error in the term of root mean squared error and mean absolute error, as the table suggested for BoA, the T-GARCH have least error in both case to be consider as more accurate.In the same method for the JP Morgan GARCH to be consider good model and for ML, T-GARCH to be consider appropriate model but subject to backtesting procedure as merely a least error can't predict the validity of accurate model.Result of forecasting VaR of daily return

Test of unconditional coverage
The unconditional coverage test is suggested one of the good, and suitable method for the purpose of backtesting of a value-at-risk (VaR) model to record the failure, which gives us the numbers that VaR is exceeded in a given observation.
Unconditional coverage refers to the fact that the fraction of overshooting's (ex post loss exceeds ex ante forecasted VaR) observed should be as per determined confidence level of the VaR.Failure of unconditional coverage means that the calculated VaR does not measure the risk accurately, while passing the unconditional coverage test mean model is more accurate to capture the underlying risk.

Kupiec (1995) unconditional coverage test
A likelihood ration test proposed by kupiec in 1995.To examine that the failure rate is statistically equal to the expected one.Let  be the expected failure rate i.e. (=1-p), where p is the confidence level for the VaR.Once the total number of such trials denoted by T, then the number of failure denoted with N can be modelled with a binomial distribution with the probability of occurrence equal to .
In such a case the correct Null and Alternative Hypothesis could be drawn as: ≠ , The appropriate likelihood ratio statistic is: is define the number of days over a T period in which portfolio loss was larger than the VaR forecasted,  +1 simply the sequence of VaR violation with the following assumption: The appropriate likelihood ratio statistic is given below: Where,  →  2 (1) under  0 of good specification.This backtesting procedure is two side and rejected mode if it generates too many or low violation, but on the base of risk expert, can accept model that generate dependent exceptions.
According to the  2 2 (Chi-squared) distribution table provided the critical value at the 95% and 99% confidence level are 3.841 at 5% significance level and 6.635 at 1% significance level.The kupice unconditional coverage test provide the value given in the table 3. The "green" marked showing these model are provided accurate forecast estimate as the value of this model is near to the critical value of  2 distribution.Further, they also put for testing to select the more accurate model.

Loss Function
The underlying idea of using loss function is provided by Lopez (1998) propose another way to control for the magnitude of the exceedance in the violation.The basic approach of loss function is reflect with a negative orientation.They provide higher score when failure take place.Only those model which minimises the loss is preferred over the other models.VaR model selected on the base of comparison, and least loss model is selected with the assumption of it provide accurate estimate.
The following quadratic loss function suggested by Lopez with the magnitudes of violation.
Where,( +1 −  +1 ) is a magnitude of difference between value-at risk and return series.Thus a score of one is imposed at exception occurs, with this numerical score increase with the magnitude of the exception.

HS FHS GARCH E-GARCH T-GARCH
the accurate model while for JP Morgan and Bank of America, T-GARCH consider to be more accurate model as per loss function results suggested.

The Basel committee's traffic light test
The broad framework of rules has been outlined regarding back tests of different risk models, the size of risk capital requirement depends of the outcome of the model backtest.If risk become larger so will the capital requirement (Basel Committee, 1996).The traffic light test begins with same method of computing violation (i.e.exception) from the computed VaR over the trading day period.
According to Jorion (2001) the expected number of violation is define by T*(1-p), where T is the number of days and p selected confidence level.Usually Basel Committee have accepted four exception over 250 trading days with 99% of VaR and fall it in the "green zone" of the traffic light test (i.e. as in traffic light green is signal of general rule of acceptance).However, if the exceptions fall between the ranges of 5-9, they fall into the second category in the yellow zone.If exception found to be more than nine they fall into the third category i.e. red zone and need have concerned by experts or regulators.
However, the direct approach of applying the traffic light test is to check the violation ratio, and compare it with the critical value of chi-square distribution.If violation ratio to be close to the critical value it would be consider the model is accepted and fall in green zone.While if the violation ratio are little away from the critical value then the lower probability of accepting inaccurate model and if they are more far then critical value then there is high probability of accepting inaccurate models.The red zone clearly indicate the problem in model.Here traffic light test assume that model computing with the low exception to be efficient and if the violation ratio are more less than critical value it also has problem.In such a case usually banks have enough capital or risk is overestimated due to which bank's not able to invest their surplus or losing its profitable opportunity.

Backtesting Conclusion & Outcome
The pioneer exercising of Model validation in the form of backtesting provides very important results for the accuracy of VaR models at the end user of VaR forecasting.The emphasized should be given to the VaR quantitative parameters for backtesting.For the selection of more accurate model the experts should avoid high confidence level for backtesting as it found that the selection of high confidence level decrease the effectiveness of the statistical test as mention by Jorion (2007).
As per our results the Merrill Lynch Bank passed almost all test including showing very low exception.Which researcher found the bias, and that is only due to the difference in sample size, as the sample size are too low at comparison of the other banks.If this violation proportion take with the other banks sample size then it would be under more threat of banks performance.
The most appropriate and simple model for the purpose of backtesting is proposed by kupiec's POF-Test.The underlying issue with the model is if statistically too many or too few exceptions are observed, the model is rejected.As mention in the research in table: 3, of kupiecun conditional coverage test.The good model are those which value's lies near the critical value of the  2 distribution and model which generate few and too exception are rejected.The other suitable model for the purpose of bank's risk T-GARCH and Filtered Historical Simulation (FHS) model found to be the good model for all banks, almost predicted more accurate results as per loss function given.FHS has good property that become best model for forecasting VaR for all three banks.According to the model all banks having hit's which could become the case of distress in entire banking if banks underestimate VaR forecast and do not maintain enough capital to cover daily loss.
Presently, the VaR is become the most appropriate model for estimation of risks by not only banks but other financial institution and regulators also.However, the problem of implementing VaR aspect and interpretation are different for this agent.i.e., regulators view to accurate model are those who generate few exception but banks and financial institution preferred model which generate violation (exception) which results are near to the critical value of pre-estimate.
Figure 5.2.1A.Historical Simulation: Value-at-risk (VaR) forecasted for the period of 2001 to 2011 from the daily Historical Price of Bank of America.Source: Bloomberg daily data have been taken for the period of 1991-2011 for the purpose of VaR forecast to calculated expected shortfall or expected value of the loss given that an event outside a given probability level has occurred for the period of2001-to-2011.

Figure 5
Figure 5.2.1B.Historical Simulation: Value-at-risk (VaR) forecasted for the period of 2001 to 2011 from the daily Historical Price of JP Morgan Chase & Co.Source: Bloomberg daily data have been taken for the period of 1991-2011 for the purpose of VaR forecast to calculated expected shortfall or expected value of the loss given that an event outside a given probability level has occurred for the period of 2001-to-2011.

Figure 5 .
Figure 5.2.1C.Historical Simulation: Value-at-risk (VaR) forecasted for the period of 2006 to 2008 from the daily Historical Price of Merrill Lynch Bank.Source: Bloomberg daily data have been taken for the period of 2001-2008 for the purpose of VaR forecast to calculated expected shortfall or expected value of the loss given that an event outside a given probability level has occurred for the period of 2006-to-2008.

Figure 5
Figure 5.2.2A.Filtered Historical Simulation: Value-at-risk (VaR) forecasted for the period of 2001 to 2011 from the daily Historical Price of Bank of America.Source: Bloomberg daily data have been taken for the period of 1991-2011 for the purpose of VaR forecast to calculated expected shortfall or expected value of the loss given that an event outside a given probability level has occurred for the period of 2001-to-2011.

Figure 5
Figure 5.2.2B.Filtered Historical Simulation: Value-at-risk (VaR) forecasted for the period of 2001 to 2011 from the daily Historical Price of JP Morgan Chase & Co.Source: Bloomberg daily data have been taken for the period of 1991-2011 for the purpose of VaR forecast to calculated expected shortfall or expected value of the loss given that an event outside a given probability level has occurred for the period of2001-to-2011.

Figure 5
Figure 5.2.2C.Filtered Historical Simulation: Value-at-risk (VaR) forecasted for the period of 2006 to 2008 from the daily Historical Price of Merrill Lynch Bank.Source: Bloomberg daily data have been taken for the period of 2001-2008 for the purpose of VaR forecast to calculated expected shortfall or expected value of the loss given that an event outside a given probability level has occurred for the period of 2006-to-2008.

Figure 5 .Figure 5
Figure 5.2.3A.GARCH: Value-at-risk (VaR) forecasted for the period of 2001 to 2011 from the daily Historical Price of Bank of America.Source: Bloomberg daily data have been taken for the period of 1991-2011 for the purpose of VaR forecast to calculated expected shortfall or expected value of the loss given that an event outside a given probability level has occurred for the period of2001-to-2011.

Figure 5
Figure 5.2.3C.GARCH: Value-at-risk (VaR) forecasted for the period of 2006 to 2008 from the daily Historical Price of Merrill Lynch Bank.Source: Bloomberg daily data have been taken for the period of 2001-2008 for the purpose of VaR forecast to calculated expected shortfall or expected value of the loss given that an event outside a given probability level has occurred for the period of 2006-to-2008.

Figure 5
Figure 5.2.4A.T-GARCH: Value-at-risk (VaR) forecasted for the period of 2001 to 2011 from the daily Historical Price of Bank of America.Source: Bloomberg daily data have been taken for the period of 1991-2011 for the purpose of VaR forecast to calculated expected shortfall or expected value of the loss given that an event outside a given probability level has occurred for the period of 2001-to-2011.

Figure 5 .
Figure 5.2.4B.T-GARCH: Value-at-risk (VaR) forecasted for the period of 2001 to 2011 from the daily Historical Price of JP Morgan Chase & Co.Source: Bloomberg daily data have been taken for the period of 1991-2011 for the purpose of VaR forecast to calculated expected shortfall or expected value of the loss given that an event outside a given probability level has occurred for the period of 2001-to-2011.

Figure 5 .
Figure 5.2.4C.T-GARCH: Value-at-risk (VaR) forecasted for the period of 2006 to 2008 from the daily Historical Price of Merrill Lynch Bank.Source: Bloomberg daily data have been taken for the period of 2001-2008 for the purpose of VaR forecast to calculated expected shortfall or expected value of the loss given that an event outside a given probability level has occurred for the period of 2006-to-2008.

Figure 5 .
Figure 5.2.5A.E-GARCH: Value-at-risk (VaR) forecasted for the period of 2001 to 2011 from the daily Historical Price of Bank of America.Source: Bloomberg daily data have been taken for the period of 1991-2011 for the purpose of VaR forecast to calculated expected shortfall or expected value of the loss given that an event outside a given probability level has occurred for the period of 2001-to-2011.

Figure 5 .
Figure 5.2.5BE-GARCH: Value-at-risk (VaR) forecasted for the period of 2001 to 2011 from the daily Historical Price of JP Morgan Chase & Co.Source: Bloomberg daily data have been taken for the period of 1991-2011 for the purpose of VaR forecast to calculated expected shortfall or expected value of the loss given that an event outside a given probability level has occurred for the period of 2001-to-2011.

Figure 5 .
Figure 5.2.5C.E-GARCH: Value-at-risk (VaR) forecasted for the period of 2006 to 2008 from the daily Historical Price of Merrill Lynch Bank.Source: Bloomberg daily data have been taken for the period of 2001-2008 for the purpose of VaR forecast to calculated expected shortfall or expected value of the loss given that an event outside a given probability level has occurred for the period of 2006-to-2008 5.5 Summary: Violation of VaR Form the computed Value-at-risk now we have to calculate the total number of Hit's made during the forecasted periods which shows the probability of losses exceeded then predicted as shown the table below.Which has been calculated from 2001-to-2011for BOA & JPM, while 2006-to-2008 for ML Bank's daily return timeseries data for the 2767 days VaR forecasted and total number of violation has been calculated at those point where daily return exceeded from VaR forecasted.The high volatility is captured by VaR during the period of global turmoil which proved that bad event has more weight in volatility then good news, all model capture

Bank 44.7999 15.3549 1.582671 0.564155 14.8667 0.00415 18.31787 0.0042 14.8667 0.17152 JP Morgan & Chase 7.56911 18.4689 7.555492 1.771521 88.8954 4.8189 96.05138 3.8949 91.2342 8.33194 Bank of America 14.4323 22.5943 0.331478 0.389631 103.643 3.08284 98.53298 1.7715 103.643 3.08284 Table 6.3. Results of Unconditonal coverage test at 95% and 99% confidence level Banks Note 1. Green Box passing the kupiec as per calculation and test suggested by kupiec.
According to the Lopez, (1998) loss function, a mode is preferred over other which minimizes the total loss i.e.   = ∑    = .The table 4. have been showing computed value of loss function as define by Lopez, under both 95% and 99% confidence level.The loss function are applied only those model which passed the kupice unconditional coverage test (i.e.those model not pass under   not qualify for the loss function for further testing purpose).The previous table of Kupiec test Shows that in 95% confidence level only FHS model qualified for loss function, while at 99% confidence level all model qualified except Historical Simulation (HS).After applying loss function to select more accurate model is one which have least value or least loss model.Here, for Merrill Lynch FHS consider

Table : 6
.1: of computing number of violation (exception) we can draw traffic light test graph below:-As per the traffic light approach at 99% FHS, GARCH, E-GARCH and T-GARCH has to be consider good model to provide more accurate forecasting purpose.However, at 95% confidence level FHS-model to be consider inaccurate, and other model's e.g.GARCH for Merrill Lynch and T-GARCH for Bank of America to be more accurate model while for JP Morgan E-GARCH should be consider more appropriate model for forecasting.

Table : 6.5: Summary of Traffic Light Test with 95% and 99% Confidence falling within Three Zone. Total No of Hit's occurred under different Model
Percentage of violation 6.649801 2.022391 5.240332 1.120347 1.409469 0.686664 1.481749 0.758945 1.409469 0.686664

Note: Hit's are calculated based on violation occurred under different models, for 2767 days for JPM & BOFA, and 782 days for ML Banks.
assessment seems to be Basel suggested traffic light test which suggest good model only those which generate few exception so banks have enough capital to mitigate the risk.