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The Garch model and their The Garch model and their Applications to the VaRApplications to the VaR
The Garch model and their The Garch model and their Applications to the VaRApplications to the VaR
Ricardo A. TagliafichiRicardo A. Tagliafichi
The presence of the volatility in the assets returns
Selection of a Portfolio with models
as CAPM or APT
The estimation of Value at Risk of a Portfolio
The estimations of derivatives primes
The classic hypothesis
The capital markets are perfect, and has rates in a continuous form defined by: Rt=Ln(Pt)-Ln(Pt-1)
These returns are distributed identically and applying the Central Theorem of Limits the returns are n.i.d
These returns Rt, Rt-1, Rt-2, Rt-2,........, Rt-n,doesn't have any relationship among them, for this reason there is a presence of a Random Walk
The great questions as a result of the perfect markets and the
random walk
n =t (n/t) 0.5
s = 0
The periodic structure of the volatilityMerval Index
0
0,02
0,04
0,06
0,08
2 4 8 16 32 64 128 256
Difference between
and
n
2 0.54742
4 0.51488
8 0.52297
16 0.53161
32 0.52719
64 0.51825
128 0.51785
256 0.52206
)ln()ln()ln( 1
nx n
nn 1 xn n1
The memory of a process: The Hurst exponent
Is a number related with the probability that an event is autocorrelated
Hn ncSR )/(
)()()/( nLnHcLnSRLn n
The meaning of H
0.50 < H < 1 imply that the series is persistent, and a series is persistent when is characterized by a long memory of its process
0 < H < 0.50 mean that the series is antipersistent. The series reverses itself more often than a random series series would
The coefficient R/Sn
The construction of these coefficient doesn’t require any gaussian process, neither it requires any parametric processThe series is separated in a small periods, like beginning with a 10 periods, inside the total series, until arriving to periods that are as maximum half of the data analyzed
We call n the data analyzed in each sub period and Rn= max(Yt..Yn) - min (Yt..Yn) and . R/Sn = average of Rn/average of Sn where Sn is the volatility of this sub period
Some results of the coefficient H
0
1
2
3
4
2 3 4 5 6 7
Ln (n)
Ln
(R
/S)n
Index Dow Jones
Coeff.
H 0.628
S.E. of
H 0.011
R squared
0.974
Const. -0.617
Some results of the coefficient H
1
2
3
4
5
2 3 4 5 6 7 8
Ln (n)
Ln
(R
/S)n
Indice MervalCoeff.
H 0.589
S.E. of
H 0.006
R squared
0.987
Const. -0.184
The conclusions of the use of H
The series presents coefficients H over 0.50, that indicates the presence of persistence in the seriesUsing the properties of R/Sn coefficient we can observe the presence of cycles proved by the use of the FFT and its significant tests.
It is tempting to use de Hurst exponent to estimate de variance in annual terms, like the following: H
n n1
The market performanceAssets
Merval SidercaBono
Pro 2
Global
2027
Period 90-94 95-00 90-94 95-00 95-00 99-00
Obs. 1222 1500 1222 1500 1371 504
Mean 0.109 0.015 0.199 0.057 0.0511 -0.022
Volatility 3.566 2.322 4.314 3.107 1.295 1.1694
Skewness 0.739 -0.383 0.823 -0.32 -0.146 -0.559
Kurtosis 7.053 8.020 7.204 7.216 33.931 21.971
Maximum 24.40 12.08 26.02 17.98 14.46 9.39
Minimum -13.52 -14.76 -18.23 -21.3 -11.78 -9.946
The market performance.. are the returns n.i.d.?
The K-S test: P (Dn<n,0.99)= 0.95 is used to prove
that the series has n.i.d. shows the following results:
Asset Number of
Observations
Dn n,0.95
Merval Index 2722 0.0844 0.023534
Siderca 2722 0.0658 0.026534
Bono Pro2 1371 0.2179 0.036678
Bono Global 504 0.2266 0.060376
The independence of returns
The autocorrelation function is the relationship between the stock’s returns at different lags.
The Ljung Box or Q-statistic at lag 10:
10
1
210
2
)2(i
i
innnQ
The test of hypothesis
Ho: some
Hurst coefficient and Ljung Box Q-Statistic
Series Q – Statistic
for k = 10
Hurst Coefficient
Dow Jones 33.205 0.628
Merval 52.999 0.589
Siderca 51.157 0.787
Pro 2 in dollars 46.384 0.782
Graph 1 - Daily returns Merval Index
-20,0
-15,0
-10,0
-5,0
0,0
5,0
10,0
15,0
20,0
25,0
30,0
01/10/1990
07/05/1990
12/28/1990
06/27/1991
12/16/1991
06/11/1992
12/01/1992
05/27/1993
11/17/1993
05/10/1994
11/01/1994
04/24/1995
10/13/1995
04/09/1996
10/01/1996
03/21/1997
09/16/1997
03/09/1998
09/10/1998
03/04/1999
09/02/1999
02/29/2000
08/31/2000Effect convertibility
Different crisis supported until government's change and the obtaining of the blinder from the MFI
Graphic 2 - Daily volatilities of Merval Index
0,000
5,000
10,000
15,000
20,000
25,000
30,000
01/0
8/19
90
06/0
8/19
90
11/0
8/19
90
04/1
2/19
91
09/1
2/19
91
02/1
1/19
92
07/1
4/19
92
12/1
0/19
92
05/1
2/19
93
10/1
2/19
93
03/0
9/19
94
08/0
9/19
94
01/0
6/19
95
06/0
7/19
95
11/0
3/19
95
04/0
3/19
96
09/0
5/19
96
02/0
3/19
97
07/0
4/19
97
12/0
1/19
97
05/0
5/19
98
10/1
3/19
98
03/1
2/19
99
08/1
8/19
99
01/1
9/20
00
06/3
0/20
00
11/2
9/20
00
Applying Fractal an statistical analysis we can say....
1) The series of returns are not nid
2) Some s 0
3) The t t 0.5
4) There values of kurtosis and skewness in the series denote the presence of Heteroscedasticity
The traditional econometrics assumed:
The variance of the errors is a constant
The owner of a bond or a stock should be interested in the prediction of a
volatility during the period in that he will be a possessor of the asset
The Arch model ....
We can estimate the best model to predict a variable, like a regression model or an ARIMA model
In each model we obtain a residual series like:
ttt YY ˆ
Engle 1982
tqtqttt h 22
222
112 .
ARCH (q)
Autoregressive Conditional Heterocedastic
Bollerslev 1986
t
p
jjtj
q
itit vh
1
2
|1
21
2
GARCH (q,p)
Generalized Autoregressive Conditioned Heteroskedastic
A simple prediction of a volatility with Arch model
211
2 )( RRtt Where:
2t = variance at day t
Rt-1- R = deviation from the mean at day t-1
If we regress the series on a constant….
tt cR
c = constant or a mean of the seriest = deviation at time t
...if series t is a black noise then
there is a presence of ARCH
The ACF and the PAC of t2
series
The Ljung Box or Q-statistic at lag 10:
MERVAL SIDERCAGLOBAL
2017
01/90
11/94
12/94
12/00
01/90
11/94
12/94
12/00
11/98
12/00
359.48 479.52 477.93 392.65 151.35
How to model the volatility
With the presence of a black noise and....
Analyzing the ACF and PACF using the same considerations for an ARMA process ....
We can identify a model to predict the volatility
Volatility of Merval Index modelling whith Garch (1,1)
0
2
4
6
8
10
12
14
16
12/01/1994
03/10/1995
06/22/1995
09/28/1995
01/10/1996
04/18/1996
07/30/1996
11/05/1996
02/13/1997
05/26/1997
09/04/1997
12/11/1997
03/23/1998
07/08/1998
10/21/1998
02/01/1999
05/12/1999
08/25/1999
12/02/1999
03/21/2000
07/06/2000
10/13/2000
The Garch (1,1)
211
211
2 ttt
This model was used during 1990-1995 with a great success, previous to the “tequila effect” or Mexican crisis
Some results of GARCH (1,1) applied to Merval Index
90-00 90-94 95-00 98-00
0.125
(0.0019)
0.088
(0.030)
0.203
(0.035)
0.503
(0.138)
0.141
(0.0090)
0.137
(0.018)
0.152
(0.012)
0.122
(0.020)
0.847
(0.0090)
0.862
(0.016)
0.814
(0.015)
0.760
(0.040)
P(Q8) 0.516 0.774 0.779 0.757
The persistence of a Garch (1,1)
The autoregressive root that governs the persistence of the shocks of the
volatility is the sum of +
Also + allows to predict the volatility for the future periods
The persistence and the evolution of a shock on t in (t + days
0
0.2
0.4
0.6
0.8
1
1.21 3 5 7 9
11 13 15 17 19 21 23
0,986 0,975 0,95 0,9 0,8
With a Garch model, it is assumed that the variance of returns
can be a predictable process
If ...2
112
112
ttt
for the future t periods ...
21
2,1 )(|1
)(1)(1
)(1)()1(
)(1 t
tt
t t
The news impact curve and the asymetric models
After 1995, the impact of bad news in the assets prices, introduced the concept of the asymetric models, due to the effect of the great negative impact.
The aim of these models is to predict the effect of the catastrophes or the impact of bad news
The EGARCH (1,1)
elson (1991)
1
1
1
121
2 )log()log(
t
t
t
ttt
This model differs from Garch (1,1) in this aspect:
Allows the bad news (t and < 0) to have a bigger impact than the good news in the volatility prediction.
The TARCH (1,1)
caseotherind
ysidwhere
d
t
tt
ttttt
0
01
1
11
211
21
21
2
Glosten Jaganathan and Runkle
and Zakoian (1990)
is a positive estimator with weight when there are negative impacts
NEWS IMPACT CURVEUSING Garch and Asymetric Models
0
5
10
15
20
25
30
35
-10 -5 0 5 10
Garch (1,1) Tarch (1,1) Egarch (1,1)
The presence of asymetry.
To detect the presence of asymetry we use the cross correlation function between the squared residuals of the model and the standarized residuals calculated as t/t
Number of
rt-k) not
null in the
first –10 values
Merval Index01/90
11/94
12/94
12/00
11/99
12/00
0 5 4
What is Value at Risk?
VaR measures the worst loss expected in a future time with a confidence level previously established
VaR forecasts the amount of predictable losses for the next period with a certain probability
Computing VaR
VaR makes the sum of the worst loss of each asset over a horizon within an interval of confidence previously established
“ .. Now we can know the risk of our portfolio, by asset and by the individual manage … “
The vice president of pension funds of Chrysler
The steps to calculate VaR
market position
tVolatility measure days to be
forecasted
Report of potential loss
VAR
Level of confidence
The success of VaR
Is a result of the method used to estimate the risk
The certainty of the report depends from the type of model used to compute the volatility on which these forecast is based
The EWMA to estimate the volatility
EWMA, is used by Riskmetrics1 and this method established that the volatility is conditioned bay the past realizations
2
1
2,)1(
k
jjtt
jt
1 Riskmetrics is a trade mark of J.P.Morgan
The EWMA and GARCH
Using 0.94 for EWMA models like was established by the manuals of J. P. Morgan for all assets of the portfolio is the same as using a Garch (1,1) as follows:
What happen after 1995
Today, the best model to compute the volatility of a global argentine bond is a Tarch(1,1)
Limits for the estimation of VaR with Tarch(1,1) y Riskmetrics for Global
Argentine Bond 1999-2000
-15
-10
-5
0
5
10
15
Conclusions
Using the ACF and PACF in one hand and using fractal geometry in the other hand we arrive to the following expressions:
s 0 and n t (n/t) 0.5
That allow the use of Garch models to forecast the volatility
Conclusions
There are different patterns between the returns previous 1995 (Mexican crisis) and after it
With the right model of Garch we can forecast the volatility for different purposes in this case for the VaR
Conclusions
If volatility is corrected estimated the result will be a trustable report
Each series have its own personality, each series have its own model to predict volatility
In other words.. When bad news are reported resources are usefull, when good news are present resources are not needed
The Future
The use of derivatives for reducing de Var of a portfolio
To calculate the primes of derivatives Garch models will be use
Questions