13th AFIR Colloquium 2003
The estimation of Market
VaR using Garch models and
a heavy tail distributions
The dynamic VaR and The Static VaR
The Garch Models
The Heavy tails
distributions
13th AFIR Colloquium 2003
The market VaR
The principal
components
•The volatility
•The probability
distributions of returns
•The probability defined
for the maximum loss to
be accepted
13th AFIR Colloquium 2003
Why we need a credible VaR
1) Because when we calculate a VaR position we
need to make a reserve outside the portfolio
2) Because the traders must believe in this VaR
and constraint the portfolio in order to comply
with the limits as a result of VaR estimation
3) Because when we make a reserve we reduce
the dividends, and add additional costs for
this frozen funds
13th AFIR Colloquium 2003
The first component of VaR:
The volatility
How it is presented the volatility in the market?
1) The volatility don’t follows the law of t0.5
2) The volatility is presented in clusters. There
are moments of great volatility followed by
moments of tranquility
3) The volatility series is a predictable process
0
04/02/2003
01/02/2003
10/02/2002
07/02/2002
04/02/2002
01/02/2002
10/02/2001
07/02/2001
04/02/2001
01/02/2001
10/02/2000
07/02/2000
04/02/2000
01/02/2000
10/02/1999
07/02/1999
04/02/1999
01/02/1999
10/02/1998
07/02/1998
04/02/1998
01/02/1998
13th AFIR Colloquium 2003
Daily Volatilities of MSCI General Index
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
01/07/2003
11/07/2002
09/07/2002
07/07/2002
05/07/2002
03/07/2002
01/07/2002
11/07/2001
09/07/2001
07/07/2001
05/07/2001
03/07/2001
01/07/2001
11/07/2000
09/07/2000
07/07/2000
05/07/2000
03/07/2000
13th AFIR Colloquium 2003
Daily Volatility of Perez Companc (ARG)
20
15
10
5
13th AFIR Colloquium 2003
How to forecast the volatility
Regress the series returns on a constant and the
model is:
Rt  c   t
The constant is the mean of the series and the
residuals, t are the volatility or the difference
between the value observed and the constant or
the mean of the series.
13th AFIR Colloquium 2003
The presence of Arch in the
model
First step: Test the hypothesis
Ho : r1  r2  r3  . . . .  rk = 0
H1 : some rk  0
r
2
Use the statistic: Q  n (n  2) 
 k
i 1 n  i
k
2
i
13th AFIR Colloquium 2003
Second step: The Arch LM Test
Ho : There are absence of Arch
H1 : There are presence of Arch
Estimate the following auto regression model:
ˆ   0   
2
t
2
1 t 1
 
2
2 t 2
 
2
k t k
And calculate the Observations * R2 = TR2
This coefficient TR2  2k
13th AFIR Colloquium 2003
Some results of different series
Q(8)
Prob
T*R2
k
Prob
Dow Jones
164.71
0.00
100.13
3
0.000
Bovespa
42.12
0.00
22.71
1
0.000
MSCI
67.37
0.00
27.65
2
0.000
T.bond 5 y
83.20
0.00
18.32
3
0.000
IDP
95.15
0.00
137.32
4
0.000
Merval
300.09
0.00
97.39
2
0.000
Series
13th AFIR Colloquium 2003
With the presence of Arch the
forecast volatility may be done by
nonlinear models
1) Garch models
2) RiskMetrics™ or EWMA
3) Asymmetric Garch models
RiskMetrics is a trade mark of J.P.Morgan
13th AFIR Colloquium 2003
The Garch model
p
     
2
t
i 1
2
i t 1
q
   i
i 1
2
t i
 vt
If the volatility for tomorrow is a result of:
“a little of the error of my prediction of
today plus a little of the prediction for
today”
Then we are in presence of a Garch(1,1)
13th AFIR Colloquium 2003
The beauty of Garch (1,1) model
The square error of an heteroscedasticity process
seems an ARMA (1,1). The autoregressive root that
governs the persistence of the shocks of volatility is
the sum of (  )
      
2
t
2
t 1
2
t 1
       (  
2
t
2
t 1
2
t 2
  )
2
t 2
k

(
1


)
2
j 1 2
k 2
t 
     t  j    t 1
1 
j 1
k
13th AFIR Colloquium 2003
Now we can estimate the volatility
For the day t
t
1  (   )
t 2
E  
    )  t
1  (   )
 )
2
t
For t days or the volatility between t and t+t
 )
E t 1  t ,t
t 1
t




1  (   )
1  (   ) 2


t
 
t  1)     )
1  (   ) 
1  (   )  1  (   )

13th AFIR Colloquium 2003
Risk Metrics™
The analysts have fruitfully applied the Garch
methodology in assets pricing models and in the
volatility forecast. Risk Metrics use a special
Garch model when use the decay factor l  0.94.
The behavior of this model is similar to:
Garch (1,1) with   1l),   l and   0
™] Risk Metrics is a trade mark of J. P. Morgan
13th AFIR Colloquium 2003
The limitations of Garch (1,1)
1) Garch models only are sensitive to the
magnitude of the excess of returns and not to the
sign of this excess of return.
2) The non negative constraints on , , and 
which are imposed to ensure that 2t remains
positive
3) The conditional moments, may explode when
the process itself is strictly stationary and
ergodic.
13th AFIR Colloquium 2003
The solutions for the limitations
of Garch (1,1)
The asymmetric models
Egarch
(p,q)
Tarch
(1,1)
 )

q
Ln       j Ln 
2
t

2
t
j 1
   
where
2
t  j1
2
t 1
)
  t i
 t i 

   i
i
 


i 1 
t i
t i 
p
 
2
t 1
d t 1  
d t 1  1if  t 1  0,
and d t 1  0 otherwise
2
t 1
13th AFIR Colloquium 2003
How to detect the asymmetry
and select the correct model
Log
likelihood
1
2
T
   1  log( 2 )  log  '  / T )
2
A.I.C.
 2 / n  2k / n
S.C.
 2  / n  [k log( n)] / n
The asymmetry test
13th AFIR Colloquium 2003
The cross correlation for the
asymmetry test
rxy (l ) 
n l
 (x
C xy (l )
C xy (l )
C xx (0) C yy (0)
Where:
t
 x )( yt i  y ) / n
for l  0,1,2,.....
(y
 y )( xt l  x ) / n
for l  0,1,2,.....
t 1
n l
t 1
t
13th AFIR Colloquium 2003
The asymmetry test
We must do a cross correlation between the
squared residuals of the Garch model and
the standardized residuals of the same (t/t)
The result of this cross correlation will be a
white noise if the model is symmetric or in
other words the Garch model is correctly
specified, and a black noise is the model is
asymmetric.
13th AFIR Colloquium 2003
The results applied to Tbond 5 y.
Garch (1.1)
Tarch(1,1)
Egarch(1,1)
C
-0.100257
-0.145632
-0.1811300
.
0.019 0.05
0.009 0.22
-0.003 0.77
.
0.129 0.00
0.031 0.04
0.014 0.20
.
0.879 0.00
0.909 0.00
0.998 0.00
0.151 0.00
-0.111 0.00

13th AFIR Colloquium 2003
The results applied to Tbond 5 y.
Garch
(1,1)
Tarch
(1,1)
Egarch
(1,1)
-1359.12
-1351.34
-1343.76
AIC
3.61
3.59
3.57
SC
3.63
3.62
3.60
Log
likelihood
13th AFIR Colloquium 2003
The tests to confirm the use of
an asymmetry model for
Treasury 5 years
The cross correlogram
Limits to
accept a
white noise
1 2 3 4 5
13th AFIR Colloquium 2003
The second component of VaR
The probability distribution
It was demonstrated that the returns
don’t follows a normal distribution, for
that reason I include the Heavy tails
distributions
What probability distribution follows
the returns?
13th AFIR Colloquium 2003
The heavy tails distributions
found in returns series
The
Logistic
Distribution
z
f ( x) 
 (1  z ) 2
para    x  
  scale factor  3 
ze
 x 





1
F ( x) 
1  e z
13th AFIR Colloquium 2003
The heavy tails distributions
found in returns series
The
Weibull
Distribution
x
 
 1   

f ( x)   x e


, F ( x)  1  e
x
 
 

c ln( a)  log( b)
ln(ln( 4))

y 
c 1
ln( b)  ln(  )
ln(ln( 4))
c
 0.262167b
ln(ln( 4 ))
3
13th AFIR Colloquium 2003
The EVD
This distribution depends of three parameters:
 = mode; y = location and  = shape
1
 
 

z
P(Y  y)  Fy ( y)  F0y ( z )  exp  11   z 
 

1  z  0
Where z = (y – ) / y
1
f ( z )  (1  z ) 
0
0
>0
1
1


exp  1  z ) 


Gumbel Distribution
Frechet Distribution
Weibull Distribution
 0
13th AFIR Colloquium 2003
The PWM for estimate EVD parameters
1 n
mˆ r (  ,y ,  )   X iU ir
n i 1
Where U is a plotting position that follows a free
distribution and k takes the probability as:
pk,n = [(n-k)+0.5]/n.
1  y
mr 
 
r 1

 1   )
 > 1,   0
1 
 
 (1  r ) 
The EVD >0
13th AFIR Colloquium 2003
Weibull distribution with different values of 
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.50
0.99
0.24
  0.75
4.1
3.65
3.2
2.75
2.3
1.85
1.4
0.95
0.5
0.05
-0.4
-0.9
-1.3
-1.8
-2.2
-2.7
-3.1
-3.6
-4
0.10
The EVD <0
13th AFIR Colloquium 2003
Frechet distribution with different values of 
0.6
0.5
0.9
0.10
0.4
0.3
0.5
0.2
0.1
4.1
3.6
3.2
2.7
2.3
1.8
1.4
0.9
0.5
0.0
-0.4
-0.9
-1.3
-1.8
-2.2
-2.7
-3.1
-3.6
-4
0
13th AFIR Colloquium 2003
The Kupiec
solution
Kupiec demonstrate that on base a normal distribution
that it is possible to extend the tails of the distribution
in form that contemplate the probability of a
catastrophe. The value that takes the abscissa named z
of a standardized normal distribution extended by
Kupiec is:
0.010 -2.326 -3.7331
z Kupiec 
p(1  p)
2
f ( x)
0.020
-2.054 -2.8915
0.025
-1.960 -2.6712
0.030
-1.881 -2.5071
13th AFIR Colloquium 2003
An example of returns: Tbond 5 y.
The
Goodness
of fit test
K/S
0.0479
AD
2.4008
Tbond 5y. daily returns
13th AFIR Colloquium 2003
An example of returns: Bovespa
The
Goodness
of fit test
K/S
0.0198
AD
0.5067
Bovespa daily returns
13th AFIR Colloquium 2003
The Goodness of fit tests
1) Kolmogorov Smirnov
The Kolmogorov Smirnov test is a test
that is independent of any Gaussian
distribution, and have the benefit that not
need a great number of observations.
There is one critical value that depends
on the number of observations and the
level of confidence
13th AFIR Colloquium 2003
The Goodness of fit tests
2) Anderson Darling
The Anderson Darling test is a
refinement of KS test, specially studied
for heavy tails distributions. There are
several critical values for each
distribution fitted and depends from the
number of observations and the level of
confidence
13th AFIR Colloquium 2003
What type of distributions we found
Test Goodness of Fit
First
Dist Fit
Second
Dist Fit
Logistic
EVD
1st 2nd 1st 2nd
0.06 0.14 2.40 32.4
D.Jones 1056 Logistic
Weibull
0.01 0.07 0.17 10.8
Bovespa 1036 Logistic
Weibull
0.02 0.08 0.49 10.6
Merval
773
Logistic
EVD
0.05 0.13 3.08 30.5
IDP
445
Logistic
Weibull
0.11 0.18 8.25 23.1
MSCI
1395 Logistic
Weibull
0.02 0.05 1.80 8.10
Series
Tbond
Obs
755
KS
AD
13th AFIR Colloquium 2003
Simulations
After we define the best probability
distribution for the series returns we can
simulate 20.000 trials using two methods
1)Montecarlo
2)Latin Hypercube
The objective is found the value of the
first percentile to determine the worst
loss possible
13th AFIR Colloquium 2003
Simulation with daily returns of
Tbond 5y
13th AFIR Colloquium 2003
Simulation
daily returns
D.Jones
The Logistic
Distribution
The EVD Distribution
13th AFIR Colloquium 2003
Some results
Model
Outliers
% of outlies /
Observations
Egarch(1,1)
6
1.0
Dow Jones Egarch(1,1)
10
1.0
Bovespa
Tarch (1,1)
10
1.0
Merval
Egarch(1,1)
13
1.1
IDP
Egarch(1,1)
3
0.9
MSCI
Garch (1,1)
10
0.8
Asset
Tbond
13th AFIR Colloquium 2003
The Market
VaR of Tbond
Backtesting 1% Daily VaR Returns of Tbond with
Egarch (1,1) and Normal Distribution for 5 yearsTBond
0
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
-11
1% VaR with Egarch (1,1)
Negative Daily Returns
1% VaR Simulated with Logistic Dist
13th AFIR Colloquium 2003
The Market
VaR of DJI
Backtesting 1% VaR Daily Returns of Dow Jones Index
with Egarch (1,1) and Normal probability distribution
0
-1
-2
-3
-4
-5
-6
-7
-8
Negative Daily Returns
1% VaR with Egarch (1,1) and normal dist.
1% VaR simulation with Logistic
1% VaR simulation with Weibull dist.
13th AFIR Colloquium 2003
Conclusions
The asymmetric Garch models, like Tarch
or Egarch model, not only fulfill with the
movements of the volatility, as we can
observe with the back testing presented,
also it is not necessary to use the heavy tails
distributions, because the negative impact
or the negative returns are included by the
model form and is the form of a dynamic
VaR
13th AFIR Colloquium 2003
Conclusions
The static VaR estimated with the heavy tails
distribution don’t follows the volatility
movements and create reserves in excess.
The time series history, complies with the
requirements of Basel
II, to make the
volatility forecast. It is easy to teach this model
to the traders, but not for the actuaries. The
traders and the shareholders only import the
recent past