13 th afir colloquium 2003

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

The market VaR

The principal components

•The volatility

•The probability distributions of returns

•The probability defined for the maximum loss to be accepted


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

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


Daily Volatilities of MSCI General Index

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

01/02/1998

04/02/1998

07/02/1998

10/02/1998

01/02/1999

04/02/1999

07/02/1999

10/02/1999

01/02/2000

04/02/2000

07/02/2000

10/02/2000

01/02/2001

04/02/2001

07/02/2001

10/02/2001

01/02/2002

04/02/2002

07/02/2002

10/02/2002

01/02/2003

04/02/2003


Daily Volatility of Perez Companc (ARG)

0

5

10

15

20

03/07/2000

05/07/2000

07/07/2000

09/07/2000

11/07/2000

01/07/2001

03/07/2001

05/07/2001

07/07/2001

09/07/2001

11/07/2001

01/07/2002

03/07/2002

05/07/2002

07/07/2002

09/07/2002

11/07/2002

01/07/2003

How to forecast the volatility


Regress the series returns on a constant and the model is:

tt cR 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.

The presence of Arch in the model


First step: Test the hypothesis

Ho : k = 0

H1 : some k 0

Use the statistic: 2

1

2

)2( k

k

i

i

innnQ


Second step: The Arch LM Test

Ho : There are absence of Arch

H1 : There are presence of Arch

Estimate the following auto regression model:22

222

1102ˆ ktkttt

And calculate the Observations * R2 = TR2

This coefficient TR2 k

Some results of different series


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

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

The Garch model


q

ititi

p

itit v

1

2

1

21

2

“a little of the error of my prediction of today plus a little of the prediction for today”

If the volatility for tomorrow is a result of:

Then we are in presence of a Garch(1,1)

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 (

21

1

212

22

22

21

2

21

21

2

1

)1(

)(

tk

k

jjt

jk

t

tttt

ttt

Now we can estimate the volatility


For the day

22

)(1

)(1tE

For days or the volatility between t and t+

21

,1 )(1

)(1

)(1

)(11

)(1 tttE

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 . The behavior of this model is similar to:

Garch (1,1) with and

™] Risk Metrics is a trade mark of J. P. Morgan

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 2

t remains positive3) The conditional moments, may explode when the process itself is strictly stationary and ergodic.

The solutions for the limitations of Garch (1,1)

The asymmetric models


p

i it

iti

it

iti

q

jjtjt LnLn

11

21

2

Egarch

(p,q)

otherwisedand

ifdwhere

d

t

tt

ttttt

0

,01

1

11

211

21

21

2

Tarch

(1,1)

How to detect the asymmetry and select the correct model


The asymmetry test

TT

/'log)2log(12

nkn /2/2

nnkn /)]log([/2

Log likelihood

A.I.C.

S.C.

1

2

The cross correlation for the asymmetry test


)0()0(

)()(

yyxx

xyxy

CC

lClr

,.....2,1,0/))((

,.....2,1,0/))((

)(

1

1

lfornxxyy

lfornyyxx

lCln

tltt

ln

titt

xy

Where:

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.

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

The results applied to Tbond 5 y.


Garch

(1,1)

Tarch

(1,1)

Egarch

(1,1)Log likelihood -1359.12 -1351.34 -1343.76

AIC 3.61 3.59 3.57

SC 3.63 3.62 3.60

The tests to confirm the use of an asymmetry model for

Treasury 5 years


The cross correlogram

1 2 3 4 5

Limits to accept a white noise

The second component of VaRThe 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?

The heavy tails distributions found in returns series


z

x

exF

ez

factorscale

xparaz

zxf

1

1)(

3

)1()(

2

The

Logistic

Distribution

The heavy tails distributions found in returns series


The

Weibull

Distribution

xx

exFexxf 1)(,)( 1

bc

by

c

bac

262167.0))3

4ln(ln(

))4ln(ln(

)ln()ln(

))4ln(ln(

1

)log()ln(

The EVD13th AFIR Colloquium 2003

This distribution depends of three parameters:= mode; = location and = shape

0111exp)()()(1

0

zzzFyFyYP z

Gumbel Distribution Frechet Distribution

Weibull Distribution

Where z = (y – ) /

01exp)1()(11

1

zzzf

The PWM for estimate EVD parameters


0,1

)1(

11

1

1

rr

mr

n

i

riir UX

nm

1

1),,(ˆ

Where U is a plotting position that follows a free distribution and k takes the probability as:

pk,n = [(n-k)+0.5]/n.

The EVD 13th AFIR Colloquium 2003

Weibull distribution with different values of

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-4

-3.6

-3.1

-2.7

-2.2

-1.8

-1.3

-0.9

-0.4

0.05 0.5

0.95 1.4

1.85 2.3

2.75 3.2

3.65 4.1

The EVD <013th AFIR Colloquium 2003

0

0.1

0.2

0.3

0.4

0.5

0.6

-4

-3.6

-3.1

-2.7

-2.2

-1.8

-1.3

-0.9

-0.4 0.0

0.5

0.9

1.4

1.8

2.3

2.7

3.2

3.6

4.1

Frechet distribution with different values of

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:

2)(

)1(

xf

ppzKupiec

0.010 -2.326 -3.7331

0.020 -2.054 -2.8915

0.025 -1.960 -2.6712

0.030 -1.881 -2.5071

An example of returns: Tbond 5 y.


Tbond 5y. daily returns

TheGoodnessof fit test

K/S

0.0479

AD

2.4008

An example of returns: Bovespa


Bovespa daily returns

TheGoodnessof fit test

K/S

0.0198

AD

0.5067

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

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

What type of distributions we found


Series ObsFirst

Dist Fit

Second

Dist Fit

Test Goodness of Fit

KS

AD

1st 2nd 1st 2nd

Tbond 755 Logistic EVD 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

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

Simulation with daily returns of Tbond 5y


Simulation daily returns

D.Jones


The Logistic Distribution

The EVD Distribution

Some results13th AFIR Colloquium 2003

Asset Model Outliers% of outlies / Observations

Tbond 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

The Market VaR of Tbond


Backtesting 1% Daily VaR Returns of Tbond with Egarch (1,1) and Normal Distribution for 5 yearsTBond

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1% VaR with Egarch (1,1) 1% VaR Simulated with Logistic DistNegative Daily Returns

The Market VaR of DJI


Backtesting 1% VaR Daily Returns of Dow Jones Indexwith Egarch (1,1) and Normal probability distribution

-8

-7

-6

-5

-4

-3

-2

-1

0

Negative Daily Returns 1% VaR with Egarch (1,1) and normal dist.

1% VaR simulation with Logistic 1% VaR simulation with Weibull dist.

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

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

13 th afir colloquium 2003

Documents

static var

estimation of market

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accepted13th afir colloquium

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