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