The Value at Risk : method historical and method GARCH

1. The Value-at-Risk
Lin Jibin and Verny Tania
Universit Paris 1 Panthon Sorbonne
Dissertation submitted to MOSEF, Faculty of Economics, Universit´e Paris 1
Panthon Sorbonne, as a partial fulfilment of the requirement for the Master 1 of
Economie Quantitative.
February 2015

2. Contents
List of Figures iii
1 Introduction 1
2 Value at Risk 4
2.1 Strengths and Weaknesses of VaR . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Overview of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.1 Sampling Period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.2 Analysis Horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.3 Sampling Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 VaR Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.1 Relative VaR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.2 Marginal VaR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.3 Incremented VaR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.4 Conditional VaR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 Overview of VaR Methodologies . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4.1 Historical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4.2 Parametric Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4.3 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.5 General Calculation Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.6 Strengths and Weaknesses of each Methodology . . . . . . . . . . . . . . . . 11
i

3. CONTENTS
3 Calculation of VaR 12
3.1 Historical VaR Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.1.1 Historical Method for Simulated Data . . . . . . . . . . . . . . . . . . 13
3.2 GARCH VaR Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2.1 Introduction of ARCH(1) Model . . . . . . . . . . . . . . . . . . . . . 16
3.2.2 Extension to a GARCH(p,q) Model . . . . . . . . . . . . . . . . . . . . 18
3.2.3 GARCH Simulation for Simulated Data . . . . . . . . . . . . . . . . . 19
4 S&P500 Data 24
4.1 Testing for Normality using a Q-Q Plot . . . . . . . . . . . . . . . . . . . . . 25
4.2 Historical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.3 GARCH Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.3.1 Parameters of the GARCH(p,q) Model . . . . . . . . . . . . . . . . . . 31
4.3.2 Application of GJR-GARCH on each column of the rolling window . 36
5 Conclusion 39
Bibliography 41
ii

4. List of Figures
3.1 Graph of Yield for 2000 Observations of Simulated Data . . . . . . . . . . . . 13
3.2 Table illustrating the VaR 1% and VaR 5% of the Rolling Window of Yields
for Simulated Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3 Historical Graph of VaR 1% and 5% . . . . . . . . . . . . . . . . . . . . . . . 15
3.4 Trend and Correlation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.5 White Noise and Information Criterion . . . . . . . . . . . . . . . . . . . . . 20
3.6 Table of Var 1% and VaR 5% of the Rolling Window of Returns . . . . . . . 21
3.7 Graph of Var 1% and Var 5% for the GARCH simulation . . . . . . . . . . . 22
3.8 Graph of yields, VaR 1% and VaR 5% for a Student Distribution . . . . . . . 23
4.1 Graph of Yield of S&P500 over 10 years . . . . . . . . . . . . . . . . . . . . . 24
4.2 Q-Q Plot Estimations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.3 Histogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.4 Test for Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.5 Graph of Q-Q Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.6 Graph of Var 1% and VaR 5% for the Rolling Window of Returns . . . . . . 29
4.7 Graph of VaR 1% and 5% for the historical method of S&P500 . . . . . . . . 30
4.8 Trend and Correlation Analysis for Yields . . . . . . . . . . . . . . . . . . . . 32
4.9 Table to determine the order of p and q . . . . . . . . . . . . . . . . . . . . . 33
4.10 Table of Different Criterions Information . . . . . . . . . . . . . . . . . . . . . 34
4.11 GJR-GARCH Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.12 Table of Var1% and 5% for the Rolling Window . . . . . . . . . . . . . . . . . 36
iii

5. LIST OF FIGURES
4.13 Graph of Var 1% and VaR 5% for the Rolling Window . . . . . . . . . . . . . 37
4.14 Graph of yields, VaR1% and VaR5% for the S&P500 . . . . . . . . . . . . . . 38
iv

6. Chapter 1
Introduction
”Risk is like fire: if controlled it will help you, if uncontrolled it will rise up and destroy
you” Theodore Roosevelt
This citation is quite true as investing in risky assets may either bring you profits or
losses. A particular question that generally all the investors who have invested or are
looking forward to invest in a risky asset ask at certain period is, what is the most I can lose
on an investment? Financial markets were recently defined as a strange turbulent markets
and the management of risk was definitely required. In order to solve this situation, an
important factor called the calculation of Value-at-Risk was introduced. The concept of
value at risk also known as (VaR) is the most well-known risk measure and tries to give a
quite suitable answer to investors’ questions about the risks, at least within an appropriate
bound. It refers to the measurement of potential loss in one value of a financial assets or
portfolio over a certain period for a given confidence level.
During the 1950s, Harry Markowitz had derived and developed the Value at Risk and
was rewarded afterwards in 1990, the Nobel Prize in Economics for his fundamental dis-
covery according to the research and contributions in the area of portfolio theory. His
research and development resulted that the standard deviation and the expected returns
of portfolios could be approximate with a structure of the normal distribution and about
the risk involved. Due to his advance research and development, he motivated financial
managers to diversify their portfolio by performing different types of calculations such as
correlation, standard deviation and expected returns between financial assets within their
present position (Jorion 2007).
1

7. Moreover, during the 1970s and 1980s, many financial organizations started working
on models that were designed to measure and aggregate risk. The number of demand for
this type of model increased rapidly. Then the famous risk measure named Riskmetrics
was introduced by J.P. Morgan. His staff developed a measure risk which shows the max-
imum loss over the next trading day in one value and was called Value at Risk (Dowd
1998).
J.P Morgan provided Riskmetrics public in 1994 and therefore, many software providers
had implemented it in either new software or existing ones which lead various institutions
to use it (Dowd 1998). He named the service Riskmetrics and the use of the Value at Risk,
to describe the risk that arisen from data. This particular type of measure found a ready au-
dience with banks and many financial institutions. Riskmetrics was recently bought by a
company named MSCI Barra in 2010. Furthermore, there are different approaches used to
compute Value at Risk even though there are nowadays numerous variations within each
approach. The approaches are Historical Simulation, Parametric Simulation and Monte
Carlo Simulation. These different simulations will be discuss later in the project.
Even though, the application of VaR analysis and reporting now extends to non-financial
corporations, which has resulted in the adoption of related ”at risk” measures such as
Earnings-at-Risk (EaR), Earnings-Per-Share-at-Risk (EPSaR) and Cash-Flow-at-Risk (CFaR).
These different measures take into account special considerations of the corporate environ-
ment such as the use of accrual vs. mark-to-market accounting and hedge accounting for
qualifying transactions. Additionally, non-financial corporations focus longer-term impact
of risk on cash flows and earning (quarterly or annually) in the budgeting and planning
process.
However, the recent economic and financial crises show that their uses have major im-
perfections. Indeed, the financial crisis which occurred in 2007 is an irrefutable proof that
the management of risk in a macroeconomic framework is not yet developed. The risk
measures are part of the cause, among the latter; the Value-at-Risk is one of the most used
in the financial world measurement as described before. So it is interesting to ask whether;
the deficiency of the VaR calculations, does it comes from its methods of estimations for pe-
riods of high volatility? In other words, in recessionary periods and economic imbalance,
is it necessary to limit the use of VaR as a method of risk assessment?
An outline of the report is as follows: In chapter 2, we will give the explanation of the
2

8. Value at Risk, its strength and weakness and the different methodologies of obtaining it.
Chapter 3 describes the calculations of VaR with the Historical and GARCH Simulations
and then the application of it on Simulated Data. In chapter 4, we carried out the appli-
cation of these methods of simulations mentioned previously on the data of S&P500 and
make comparisons between both methods so as to see if they lead to the same result or
differ according to the particular data. Finally, we conclude our study in chapter 5.
3

9. Chapter 2
Value at Risk
VaR is defined as the predicted worst-case loss at a specific confidence level (e.g 90%, 95%, 99%)
over a certain period of time (say 1 day or 20 days for the purpose of regulatory capital re-
porting). An example is that J.P. Morgan takes a snapshot of its global trading positions to
estimates its daily earning at risk, which a VaR measure that Morgan defines as the 95%
confident worst case loss over the next 24 hours due to adverse market movements. It is
also defined as the following:
P(Rp ≤ V aRτ %) = 1 − τ%
where Rp = Rp − Rp−1 refers to the change in the value of portfolio and τ is the selected
confidence level.
The VaR of a portfolio depends essentially on three parameters:
• The distribution of the gains and losses of a portfolio or an asset.
• The level of confidence interval usually 95% or 99%. It is the probability that the
possible losses of the portfolio or the asset will not be over the Value at Risk by the
definition.
• Time horizon chosen. This parameter is fundamental as the longer is the time hori-
zon, the losses will be more important. An example is for a normal distribution of
yield. The VaR have to be multiplied by a day so as to obtain the VaR over the days.
Value at Risk of 50 million dollars at a 95% confident interval and a 1 day horizon means
4

10. that 1 day in 20, one could expect to lose more that 50 million dollars due to market move-
ment. Then the amount of the VaR coupled with confident level and time period param-
eters, allow to anticipate that losses exceeding 50 million dollars would occur 5% of the
time; that is losses less then 50 million dollars would occur 95% of the time.
5

11. 2.1 Strengths and Weaknesses of VaR
2.1 Strengths and Weaknesses of VaR
6

12. 2.2 Overview of Parameters
2.2 Overview of Parameters
2.2.1 Sampling Period
Sampling Period
• Determines how much data gets used in calculating risk measures.
• Rolling window period which is the most common.
• Fixed dates or multiple date.
2.2.2 Analysis Horizon
Analysis Horizon
• Its use is to scale out the returns.
• Exemple: 5 days will give a 1 week VaR number.
• Does not scale the statistic directly but rather the sample return.
2.2.3 Sampling Frequency
7

13. 2.3 VaR Measures
2.3 VaR Measures
There are in fact three related Value at Risk measures:
2.3.1 Relative VaR
Relative Value at Risk measure the risk of underperformance relative to a pre-defined
benchmark, such S&P500 Index. It is relevant to many institutions investors, including
investment managers and mutual funds as their performance is usually compared to a
target benchmark.
2.3.2 Marginal VaR
Marginal VaR measures how much risk a position adds to a portfolio. Precisely, it measure
how much portfolio VaR would change if the position were removed entirely, that is Value
at Risk with position minus Value at Risk without position.
2.3.3 Incremented VaR
Incremented Value at Risk is strictly related to marginal VaR. Marginal VaR measures the
difference in portfolio risk brought about by removing an entire position as we said previ-
ously whereas incremented VaR measures the impact of small changes in position weight-
ing. It is also very useful for identifying best candidates for gradual risk reduction that is
which position to partially hedge.
2.3.4 Conditional VaR
The conditional value at risk is also called as Expected Shortfall. It refers to the average loss
that one can experience beyond the VaR threshold, so its provides additional information
as to how severe the loss can be.
8

14. 2.4 Overview of VaR Methodologies
2.4 Overview of VaR Methodologies
Markets risk models are designed to measure potential losses due to adverse changes in the
prices of financial instruments. There are various approaches to forecast market risk and
no single method is best for every simulation. Inspired by modern portfolio theory, Value
at Risk models forecast risk by analyzing historical movements of the market variables.
Generally, there are three known methods to calculate the VaR.
2.4.1 Historical Simulation
Historical method is a better and simplest methodology of estimating the value at risk
for many types of portfolios. In this approach, the VaR is being calculated by creating a
theoretical time series of returns on that portfolio and is obtained by running the portfolio
through actual historical data and computing the changes that would have occurred in
each period.
2.4.2 Parametric Simulation
It estimates Value at Risk with equation that specifies parameters such as correlation,
volatility, delta and gamma.
2.4.3 Monte Carlo Simulation
Monte Carlo simulation refers to a type of value at risk method which is a much more
difficult analytical tool. It simulates random processes with has an effect on the prices of
financial instruments. In order for this method to be efficient, a large amount of simula-
tions are required. It is based on the geometric Brownian motion which has been widely
used to simulate stock prices. It estimates VaR by simulating random scenarios and reval-
uating positions in the portfolio.
9

15. 2.5 General Calculation Steps
2.5 General Calculation Steps
10

16. 2.6 Strengths and Weaknesses of each Methodology
2.6 Strengths and Weaknesses of each Methodology
In the following chapters, we has decided to perform the calculations of the Value at Risk
by making use of the historical method and the parametric method particulary the GARCH
method on both Simulated data and the S&P500 data.
11

17. Chapter 3
Calculation of VaR
3.1 Historical VaR Methodology
Historical value at risk is given by applying a three step process:
• Select a sample of actual daily risk factor or price changes over a given period
• Apply those daily changes to the current value of the risk factors or prices, and
revalue the current portfolio as many times as the number of datapoints (i.e. days)
in the historical sample
• Rank revalued profit/loss and identify the VaR that isolates the pre-defined confi-
dent level of the distribution in the left tail.
Moreover, to determine the rank of the profit/loss data, we make use of the following
formula:
V aR(A%, 1k) = N ∗ (100% − A%) (3.1)
where where N is the number of historical data, A refers to the VaR either 95% or 99%
in general. An example is if N = 500, and we take A to be 95%, we apply the previous
formula and obtained as result, a VaR at 95% which will be the 5th value of the data.
12

18. 3.1 Historical VaR Methodology
3.1.1 Historical Method for Simulated Data
In order to carry out our study, we have taken 2000 observations as simulated data with
a student distribution at 4 degree of liberty. We then performed our calculations with the
powerful program known as SAS and obtained the following results.
Figure 3.1: Graph of Yield for 2000 Observations of Simulated Data
Conclusion: The graphical representation 3.1 shows the returns of the 2000 observa-
tions for a student distribution of 4 degree of liberty.
Now, for the followings, we will carried out a rolling window of the returns with an
overlap of 50(i.e. from 1 to 100, then 50 to 150, 100 to etc...) on the 2000 observations.
We then obtained 39 columns, the goal of this specific type of methodology allows higher
accuracy. This latter is widely used in the labor market especially in the financial market.
The following table refers to the VaR 1% and VaR 5% obtained for the rolling window
13

19. 3.1 Historical VaR Methodology
which is given by applying the same steps of the process of calculating the value at risk for
a historical simulation described previously.
Figure 3.2: Table illustrating the VaR 1% and VaR 5% of the Rolling Window of Yields for
Simulated Data
Conclusion: We notice that the VaR at 1% is less than the VaR at 5%.
14

20. 3.1 Historical VaR Methodology
Figure 3.3: Historical Graph of VaR 1% and 5%
Conclusion: We note that the figure 3.1 shows that the VaR 5% is relatively stable
whereas the VaR 1% has been slumped from the 5th observations until the 9th observations
and afterwards it recovered faster to its initial level. As it refers to simulated data, we
cannot deduce a relevant consequence, so, we will determine this with the data of S&P500
lately.
15

21. 3.2 GARCH VaR Methodology
3.2 GARCH VaR Methodology
In order to understand the GARCH model, we must first give a brief explanation of the
ARCH model which the source of the GARCH model.
3.2.1 Introduction of ARCH(1) Model
The ARCH models stands for Autoregressive Conditional Heteroscedasticity. The AR rep-
resents the facts that these models are autoregressive models in squared returns. The con-
ditional part comes from the fact that in the models, next period’s volatility is conditional
on information this period. Finally Heteroscedasticity means that non constant volatility.
In a linear regression where yt = α + βxt + t , when the variance of the residuals, t is
constant, so we call that homoscedastic and make use of the ordinary least square to esti-
mate both α and β. But if on the other side, the variance of the residuals is not constant, we
then call that heteroscedastic and we can make use of weighted least squares to estimate
the regression coefficients. Assume that the return on an assets is given as follows:
rt − µt = bt,
= ht t,
= σt t.
where t represents the sequence of N(0, 1) i.i.d. random variables. If the variance was
constant, then it could be calculated as below:
σ2
= h =
T
i=1
b2
t
T
But as it depends on time, we would need another model.Then the first model was pro-
posed by Engle(1982). It consists of modeling the variance as a process of the moving
average of order q. An example is for an ARCH(q), we have:
σ2
t = ht = α0 +
q
i=1
αib2
t−1 + vt
16

22. 3.2 GARCH VaR Methodology
and for an ARCH(1), it is given as follows:
σ2
t = ht = α0 + α1b2
t−1 + vt
where α0 > 0 and αi ≥ 0 to ensure positive variance and α1 < 1 for stationarity. In
order that the variance is non negative, α0, αi have to be non negative too as well as if
q
i=1
αi < 1, then the process is stationary. Consequently in this particular case, two vari-
ances could be calculated, a conditional variance α0 +
q
i=1
α1b2
t−1 and finally an uncondi-
tional variance given as:
σ2
=
α0
1 −
q
i=1
α1b2
t−1
In this case of non-stationarity, only the conditional variance could be calculated. We
also notice that in the case of an ARCH(1), the conditional distribution of bt is as follows:
bt | bt − 1 ∼ N(0, α0 + α1b2
t−1)α
which can be used to calculate the VaR.
An ARCH(1) is the same as an AR(1) model on squared residuals, b2
t . To see this, we
define the difference between the squared residual return and our conditional expectation
of the squared residual return, as:
vt = b2
t − E[b2
t | It],
= b2
t − σ2
t
where It − 1 refers to the information at time t − 1. We note that vt is a zero mean, uncor-
related series. So, the ARCH(1) equation is obtained as:
σ2
t = α0 + α1b2
t−1,
b2
t − vt = α0 + α1b2
t−1,
b2
t = α0 + α1b2
t−1 + vt,
17

23. 3.2 GARCH VaR Methodology
which is an AR(1) process on squared residuals.
3.2.2 Extension to a GARCH(p,q) Model
In an ARCH(q) model, the next period’s variance will depend only on last period’s squared
residual so a crisis that caused a large residual would not have the sort of persistence
that we observe after current crisis. This has led to an extension of the ARCH model to a
GARCH which stands for Generalized ARCH model and was first developed by Bollerslev
(1986). The GARCH(p,q) is written as:
σ2
t = ht,
= α0 +
q
i=1
αib2
t−1 +
p
i=1
βiσ2
t−1 + vt
and the GARCH(1,1) is given as follows:
σ2
t = α0 + α1b2
t−1 + β1σ2
t−1 + vt
where α0, α1, β1 > 0 and α1 + β1 < 1. Therefore, if the condition of stationarity is satisfied,
then the process will give an infinite variance and the unconditional variance is defined as:
σ2
=
α
(1 − α1 − β1)
The GARCH models can used in many applications and moreover some of the different
types of GARCH are the Integrated GARCH (IGARCH), Exponential GARCH (EGARCH),
GARCH in Mean (GARCH-M)and Quadratic GARCH (QGARCH) and the GJR-GARCH.
18

24. 3.2 GARCH VaR Methodology
3.2.3 GARCH Simulation for Simulated Data
Figure 3.4: Trend and Correlation Analysis
Conclusion: According to the figure 3.4 , we notice the presence of autocorrelation. We
will then verify by the portmanteau test if autocorrelation really exists.
19

25. 3.2 GARCH VaR Methodology
Figure 3.5: White Noise and Information Criterion
Conclusion: We note that the 24 lags are all not significant with a critical value of 5%
and the Pvalue( too high in terms of critical value at 5% or 10%) confirm this conclusion.
Moreover, the parameters of p and q are 1 and 1 respectively.
20

26. 3.2 GARCH VaR Methodology
Figure 3.6: Table of Var 1% and VaR 5% of the Rolling Window of Returns
21

27. 3.2 GARCH VaR Methodology
Figure 3.7: Graph of Var 1% and Var 5% for the GARCH simulation
Conclusion: According to the figures, we notice that the VaR at 5% are all more neg-
atives than the VaR at 1% which is normal as by the definition of the VaR, it refers to the
maximal potential loss that a portfolio can have for a specific probability, here the critical
value of 5% possible errors, we have a potential maximal loss which is greater for each
point at 1% critical value (i.e. this latter has a low maximal potential loss for a low proba-
bility). We also deduce that the VaR 5% is more unstable that the other one.
We then merge the two graphs of the different VaR mentioned for both historical and
GARCH method with the returns.
22

28. 3.2 GARCH VaR Methodology
Figure 3.8: Graph of yields, VaR 1% and VaR 5% for a Student Distribution
Conclusion: As a result for the figure 3.8 , we observe that the value at risk for both
methods are coincident which are totally true due to the simulated data taken. Further-
more, the value at risk for both methods do not exceed the returns which means that it
allow to advocate the maximal potential losses, thus everything is under control.
23

29. Chapter 4
S&P500 Data
We will now consider the data of the S&P500 which can be found and downloaded from
the site of Yahoo France over a period of 10 years starting from the 15th December 2004 till
15 December 2014.
Figure 4.1: Graph of Yield of S&P500 over 10 years
The figure 4.1 represents simply the daily yields of the data S&P500 for a period of
10 years. It shows that during the period after the financial crisis, we have an explosive
24

30. 4.1 Testing for Normality using a Q-Q Plot
volatility as well as with the sovereign debt crisis, it has slightly been more eventful.
4.1 Testing for Normality using a Q-Q Plot
Figure 4.2: Q-Q Plot Estimations
Conclusion: The average of the yield is given as −0, 000199 over the specific period. The
standard deviation being 0, 01287079 (more than 100 times the e means), we can consider
that the volatility of the yield is high. Then the coefficient of asymmetry is 0, 33535453,
therefore a symmetry exists on the right hand side et means that the average is greater that
the median. The kurtosis which measures the size of the tails in the distribution, gives a
value of 11, 0977807 which means that the distribution tends to have fatter tails than those
25

31. 4.1 Testing for Normality using a Q-Q Plot
provided by the normal distribution assumption. So, the returns distribution do not follow
a normal distribution. Consequently, we will now work with a student distribution of 4
degree of liberty.
Figure 4.3: Histogram
Conclusion: We notice through the graph 4.3 that the extreme yields are outside the
density of the normal distribution. Thus, this shows clearly our argument.
26

32. 4.1 Testing for Normality using a Q-Q Plot
Figure 4.4: Test for Normal Distribution
27

33. 4.1 Testing for Normality using a Q-Q Plot
Conclusion: The table 4.4 represents different test carried out by the program SAS.
Here, the kolmogorov-Sminov which refers to a test that decides if a sample comes from a
population with a specific distribution, demonstrate that the normality hypothesis is being
rejected (with a Pvalue = 1%). It is exactly the similar case for the others two tests such as
Cramer-Von Mises and Anderson-Darling (with both a Pvlaue= 0, 5%).
Figure 4.5: Graph of Q-Q Plot
Conclusion: The graphical representation 4.5 would have been a normal distribution
if its curve is on a right. As previous results, we conclude that the yield of the S&P500 is a
non-normal distribution.
Now, for the followings, we will carried out a rolling window of returns as described
previously with an overlap of 50 on the 2522 observations (10 years). We then obtained 50
columns, the goal of this specific type of methodology allows higher accuracy. This latter
is widely used in the labor market especially in the financial market.
28

34. 4.2 Historical Method
4.2 Historical Method
The following table of the different VaR 1% and VaR 5% obtained for the rolling window
is given by applying the same steps of the process of calculating the value at risk for a
historical simulation described previously in Chapter 3 (Section 1).
Figure 4.6: Graph of Var 1% and VaR 5% for the Rolling Window of Returns
Conclusion: We note that the value at risk at 5% still exceeds the value at risk at 1%.
Afterwards, we will do the parametric method of GARCH where we find similar results.
29

35. 4.2 Historical Method
Figure 4.7: Graph of VaR 1% and 5% for the historical method of S&P500
Conclusion: We notice that the volatility linked to VaR will differ according to the
period. In fact, we see that the volatility of the value at risk has passed through a jump
after the 13th units on the x-axis (represents the year 2007) as well as at the 28th units on
the x-axis (i.e. 2010). This is due to the financial crisis taken in 2007 where the prices of
different indexes are unstable. We also note that for the second period, the effect is slightly
delayed which is surely true as it refers to the forecast of the data that we have.
30

36. 4.3 GARCH Method
4.3 GARCH Method
4.3.1 Parameters of the GARCH(p,q) Model
The first step is to determine the order of the model where:
• p represent the number of lags of the MA part and q is defined by the number of lags
of the AR part of the process of the yield according to the portfolio.
• To be able to identify how much lags can we obtain using the program SAS, we need
to create a procedure called ARIMA which will let us identify estimate and then
forecast the model.
31

37. 4.3 GARCH Method
Figure 4.8: Trend and Correlation Analysis for Yields
Conclusion: According to the figure 4.8 , we notice the presence of autocorrelation. We
will then verify by the portmanteau test if autocorrelation really exists.
32

38. 4.3 GARCH Method
Figure 4.9: Table to determine the order of p and q
Conclusion: The portmanteau test shows that there is effectively autocorrelation with
a critical value of 5% for the lags 1 and 2, then 16, 18, etc... and no autocorrelation if the
critical value is 10%. In other words, we can reject the nullity hypothesis of the coefficients
that are correlated at 10% critical value. The returns are autocorrelated at 5% but not at
10% critical value. So, we will take the 10% threshold in order to continue our studies.
furthermore, according to the minimum information criterion, we obtained the GARCH
parameters as p = 2 and q = 1. We additionally applied different types of GARCH such as
GARCH(2, 1), IGARCH(2, 1), GARCH-M(2, 1), QGARCH(2, 1) and GJR- GARCH(2, 1) on
the S&P500 and obtained the following:
33

39. 4.3 GARCH Method
Figure 4.10: Table of Different Criterions Information
Conclusion: The figure 4.10 described the different information criterions obtained by
the application of different GARCH models mentioned previously on the S&P500. Ac-
cording to the table, we note that the GJR-GARCH is the most suitable due to a minimum
information criterions and a maximum log likelihood. We would then estimate the first
two order of our sample by the GJR-GARCH(2, 1).
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40. 4.3 GARCH Method
Figure 4.11: GJR-GARCH Estimation
Conclusion: Through the figure 4.10, the first two moements of the GJR-GARCH(2, 1)
are an expectation as −0, 001047 and a standard deviation as 0, 000140. We will then take
a student distribution with (1/0, 1360 ≈ 7, 35 = 7) to calculate the value at risk.
35

41. 4.3 GARCH Method
4.3.2 Application of GJR-GARCH on each column of the rolling window
Figure 4.12: Table of Var1% and 5% for the Rolling Window
Conclusion: Once more, we can see on the table 4.12 that the VaR at 1% is still higher of
the VaR at 5%.
36

42. 4.3 GARCH Method
Figure 4.13: Graph of Var 1% and VaR 5% for the Rolling Window
Conclusion: We note that the value at risk for the GARCH simulation is very close of
the one for the historical method.
Let us model this on a graphical representation so as to clearly see the difference of the
historical method and the GARCH method for the SP&500:
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43. 4.3 GARCH Method
Figure 4.14: Graph of yields, VaR1% and VaR5% for the S&P500
Conclusion: According to the figure 4.14, we note effectively that the curves are coin-
cident for both methods of calculations. We can ask ourselves that errors might have been
occurred in the analysis of the data due to the similar results. We also observe that the VaR
is over the returns (especially around the 39th till 44th, i.e. end of 2011 and beginning of
2012). Now, we can say that during the crisis period or the economic shock, the volatility
of different indexes explode, consequently the measures of risks such as the value at risk
do not give relevant results.
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44. Chapter 5
Conclusion
After our research study on value at risk on a specific sample of the S&P500, we can ob-
serve that the methods of calculations used resulted in similar outcomes, thus we can use
either one. Nevertheless, during periods of crisis such as economic or financial shocks, the
VaR and other measures of risk do not necessarily allow reliable estimations of the risk
incurred. Indeed once we cross the level of confidence, we do not know the amount of
losses that may incurred and economic agents have a tendency to underestimate the risk
of financial securities or portfolios.
This is why, two portfolios with the same VaR may generate extremely different losses
for which the VaR does not give any information. Moreover, the area of extreme values is
where the underlying assumptions of the VaR calculations are the most vulnerable: the cal-
culation of VaR through the variance-covariance matrix relies on the correlation between
assets. The stability of these correlations are not always verified especially beyond the con-
fidence interval and this is one reason why financial institutions have to have to comple-
ment VaR measures by carrying out and reporting on stress tests to measure extreme-event
losses.
Despite their limitations, value at risk models are widely used and this may further
increase especially as their usage has been validated by Governments (who nevertheless
imposed quite strict conditions). On the other hand, we notice a generalization of this
concept which in addition of being used by banks to evaluate the risk of their trading
activities, fits gradually in the management of funds and corporate treasury.
Finally, we continuously try to optimize risk measurement tools and several types of
39

45. risk measures derived from VaR such as conditional VaR, incremented VaR and expected
shortfall have been developed. The question to ask is whether these measures of risks can
allow to significantly reduce risks of losses during periods of economic instability?
40

46. Bibliography
Dowd, K. (1998), ‘Beyond value at risk: the new science of risk management’, Chichester.
John Wiley and Sons Ltd. .
Jorion, P. (2007), ‘Value at risk’, The New Benchmark For Managing Financial Risk 3.
41