The Garch model and their
Applications to the VaR
Ricardo 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
rs = 0
sn = st (n/t)
0.5
The periodic structure of the volatility
Merval Index
n
ln( s n )  ln( s 1 )
x
ln( n)
2
0.54742
4
0.51488
8
0.52297
0,06
16
0.53161
0,04
32
0.52719
0,02
64
0.51825
128
0.51785
256
0.52206
Difference between
s n  s1 n
and
sn s1 n
0,08
0
2
4
8
16
32
64
128 256
x
The memory of a process:
The Hurst exponent
Is a number related with the probability that
an event is autocorrelated
(R / S )n  c n
H
Ln( R / S ) n  Ln(c)  H Ln(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 process
The 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
Index Dow Jones
4
Coeff.
H
0.628
S.E. of
H
0.011
R
square 0.974
d
Ln (R/S)n
3
2
1
0
2
3
4
5
Ln (n)
6
7
Const. -0.617
Some results of the coefficient H
Indice Merval
Coeff.
H
0.589
S.E. of
H
0.006
R
square 0.987
d
5
Ln (R/S)n
4
3
2
1
2
3
4
5
Ln (n)
6
7
8
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 series
Using 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
s n  s1 n
The market performance
Period
Bono
Merval
Siderca
Pro 2
90-94 95-00 90-94 95-00 95-00
Obs.
1222
Mean
0.109 0.015 0.199 0.057 0.0511 -0.022
Volatility
3.566 2.322 4.314 3.107
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
Assets
1500
1222
1500
1371
1.295
Maximum 24.40 12.08 26.02 17.98 14.46
Global
2027
99-00
504
1.1694
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<en,0.99)= 0.95 is used to prove
that the series has n.i.d. shows the following results:
Asset
Number of
Observations
Dn
en,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
Q  n(n  2)
i 1
r
2
i
ni
 
2
10
The test of hypothesis
Ho: r0 .... r10  0
H1: some r1 ....rk  0
Hurst coefficient and
Ljung Box Q-Statistic
Series
Dow Jones
Hurst
Q – Statistic
Coefficient
for k = 10
33.205
0.628
Merval
52.999
0.589
Siderca
51.157
0.787
Pro 2 in dollars
46.384
0.782
Different crisis supported until
government's change and the
obtaining of the blinder from
the MFI
Effect
convertibility
Graph 1 - Daily returns Merval Index
30,0
25,0
20,0
15,0
10,0
5,0
0,0
-5,0
-10,0
-15,0
-20,0
08/31/2000
02/29/2000
09/02/1999
03/04/1999
09/10/1998
03/09/1998
09/16/1997
03/21/1997
10/01/1996
04/09/1996
10/13/1995
04/24/1995
11/01/1994
05/10/1994
11/17/1993
05/27/1993
12/01/1992
06/11/1992
12/16/1991
06/27/1991
12/28/1990
07/05/1990
01/10/1990
11/29/2000
06/30/2000
01/19/2000
08/18/1999
03/12/1999
10/13/1998
05/05/1998
12/01/1997
07/04/1997
02/03/1997
09/05/1996
04/03/1996
11/03/1995
06/07/1995
01/06/1995
08/09/1994
03/09/1994
10/12/1993
05/12/1993
12/10/1992
07/14/1992
02/11/1992
09/12/1991
04/12/1991
11/08/1990
06/08/1990
01/08/1990
Graphic 2 - Daily volatilities of Merval Index
30,000
25,000
20,000
15,000
10,000
5,000
0,000
Applying Fractal an statistical
analysis we can say....
1) The series of returns are not nid
2) Some rs  0
3) The st  s1 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:
e t  Yt  Yˆt
Engle 1982
e    e  e  .  e
2
t
2
1 t 1
2
2 t 2
2
q t q
 ht
ARCH (q)
Autoregressive Conditional Heterocedastic
Bollerslev 1986
q
e     ie
2
t
i 1|
p
2
t 1
   jh
j 1
2
t j
GARCH (q,p)
Generalized Autoregressive
Conditioned Heteroskedastic
 vt
A simple prediction of a volatility
with Arch model
s    1 ( Rt 1  R )
2
t
2
Where:

s2t = variance at day t
Rt-1- R = deviation from the mean at
day t-1
If we regress the series on a
constant….
Rt  c  e t
c = constant or a mean of the series
et = deviation at time t
...if series et is a black noise then
there is a presence of ARCH
The ACF and the PAC of et
2 series
The Ljung Box or Q-statistic at lag 10:
MERVAL
01/90
11/94
12/94
12/00
359.48 479.52
SIDERCA
GLOBAL
2017
01/90
11/94
12/94
12/00
11/98
12/00
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)
16
14
12
10
8
6
4
10/13/2000
07/06/2000
03/21/2000
12/02/1999
08/25/1999
05/12/1999
02/01/1999
10/21/1998
07/08/1998
03/23/1998
12/11/1997
09/04/1997
05/26/1997
02/13/1997
11/05/1996
07/30/1996
04/18/1996
01/10/1996
09/28/1995
06/22/1995
03/10/1995
12/01/1994
2
0
The Garch (1,1)
s    e
2
t
2
1 t 1
 1s
2
t 1
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
0.088
(0.0019) (0.030)
0.203
(0.035)
0.503
(0.138)

0.141
0.137
(0.0090) (0.018)
0.152
(0.012)
0.122
(0.020)

0.847
0.862
(0.0090) (0.016)
0.814
(0.015)
0.760
(0.040)
0.779
0.757
P(Q8) 0.516
0.774
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 et in (t + t) days
1.2
1
0.8
0.6
0.4
0.2
0,986
0,975
0,95
0,9
23
21
19
17
15
13
11
9
7
5
3
1
0
0,8
With a Garch model, it is assumed that
the variance of returns
can be a predictable process
If ... s     e
2
t
2
1 t 1
 1s
2
t 1
for the future t periods ...
 
1  (   )
s 
(t  1)  (   )

1  (   ) 
1  (   )
t 1
2
1,t
 1  (   ) 2
  1 | (   ) s 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)
Nelson (1991)
log( s )     log( s
2
t
2
t 1
e t 1
e t 1
) 

s t 1
s t 1
This model differs from Garch (1,1) in this
aspect:
Allows the bad news (et and  < 0) to have a
bigger impact than the good news in the
volatility prediction.
The TARCH (1,1)
Glosten Jaganathan and Runkle
and Zakoian (1990)
s    s
2
t
2
t 1
 e
2
t 1
  d t 1 e
2
t 1
where d t 1  1 si e t 1  0 y
d t 1  0 in other case
 is a positive estimator with weight
when there are negative impacts
NEWS IMPACT CURVE
USING Garch and Asymetric Models
35
30
25
20
15
10
5
0
-10
Garch (1,1)
-5
0
Tarch (1,1)
5
10
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 et/st
Number of
re(t),e(t-k) not
null in the
first –10
values 
Merval Index
01/90
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
s
market
position
t
Volatility measure
days to be
forecasted
VAR
Level of confidence
Report of potential loss
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
k


j 2
s t  (1   )  s t ,t  j 
j 1


1 Riskmetrics
2
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:
rs  0 and sn  st (n/t) 0.5
That allow the use of Garch models
to forecast the volatility
Conclusions
With the right model of Garch we can
forecast the volatility for different
purposes in this case for the VaR
There are different patterns between
the returns previous 1995 (Mexican
crisis) and after it
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