The Garch model and their Applications to the VaR
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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 V alue a t R isk of a Portfolio. The estimations of derivatives primes.

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The presence of the volatility in the assets returns
The presence of the volatilityin 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 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'thave 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
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
The periodic structure of the volatility the random walkMerval Index

Difference between

and


The memory of a process the hurst exponent
The memory of a process: the random walkThe Hurst exponent

Is a number related with the probability that an event is autocorrelated


The meaning of h
The meaning of the random walkH

0.50 < H < 1imply 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 s n
The coefficient the random walkR/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

4 the random walk

3

2

1

0

2

3

4

5

6

7

Some results of the coefficient H

Index Dow Jones

Ln (R/S)n

Ln (n)



The conclusions of the use of h
The conclusions of the use of the random walkH

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:


The market performance
The market performance the random walk


The market performance1
The market performance the random walk

.. 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:


The independence of returns
The independence of returns the random walk

The autocorrelation function is the relationship between the stock’s returns at different lags.

The Ljung Box or Q-statistic at lag 10:


The test of hypothesis
The test of hypothesis the random walk

Ho: r0 .... r10 = 0 H1: some r1 ....rk ¹ 0


Hurst coefficient and ljung box q statistic
Hurst coefficient and the random walk Ljung Box Q-Statistic


Different crisis supported until government's change and the obtaining of the blinder from the MFI

Effect convertibility


Applying fractal an statistical analysis we can say
Applying Fractal an statistical analysis we can say.... obtaining of the blinder from the

  • The series of returns are notnid

  • Some rs ¹ 0

  • The st ¹ s1 t 0.5

  • 4) There values of kurtosis and skewness in the series denote the presence of Heteroscedasticity


The traditional econometrics assumed
The traditional econometrics assumed: obtaining of the blinder from the

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
The Arch model .... obtaining of the blinder from the

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:


Engle 1982
Engle 1982 obtaining of the blinder from the

ARCH (q)

Autoregressive Conditional Heterocedastic


Bollerslev 1986
Bollerslev 1986 obtaining of the blinder from the

GARCH (q,p)

Generalized Autoregressive Conditioned Heteroskedastic


A simple prediction of a volatility with arch model
A simple prediction of a volatility with Arch model obtaining of the blinder from the

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
If we regress the series on a constant…. obtaining of the blinder from the

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 e t 2 series
The ACF and the PAC of obtaining of the blinder from the et2series

The Ljung Box or Q-statistic at lag 10:


How to model the volatility
How to model the volatility obtaining of the blinder from the

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


The garch 1 1
The Garch (1,1) obtaining of the blinder from the

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
Some results of GARCH (1,1) applied to Merval Index obtaining of the blinder from the


The persistence of a garch 1 1
The persistence of a Garch (1,1) obtaining of the blinder from the

The autoregressive root that governs the persistence of the shocks of the volatility is the sum of a + b

Also a + b allows to predict the volatility for the future periods


The persistence and the evolution of a shock on e t in t t days
The persistence and the evolution of a shock on obtaining of the blinder from the et in (t+t) days


With a garch model it is assumed that the variance of returns can be a predictable process
With obtaining of the blinder from the a Garch model, it is assumed that the variance of returns can be a predictable process

If ...

for the future t periods ...


The news impact curve and the asymetric models
The news impact curve and the asymetric models obtaining of the blinder from the

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
The EGARCH (1,1) obtaining of the blinder from the

Nelson (1991)

This model differs from Garch (1,1) in this aspect:

Allows the bad news (et and g < 0) to have a bigger impact than the good news in the volatility prediction.


The tarch 1 1
The TARCH (1,1) obtaining of the blinder from the

Glosten Jaganathan and Runkle

and Zakoian (1990)

g is a positive estimator with weight when there are negative impacts


The presence of asymetry
The presence of asymetry. obtaining of the blinder from the

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


What is value at risk
What is obtaining of the blinder from the Value at Risk?

VaR measures the worst loss expected in a future time with a confidence level previously established

VaRforecasts the amount of predictable losses for the next period with a certain probability


Computing var
Computing obtaining of the blinder from the VaR

VaRmakes the sum of the worst loss of each asset over a horizonwithin 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 obtaining of the blinder from the VaR

s

t

days to be forecasted

market position

Volatility measure

VAR

Level of confidence

Report of potential loss


The success of var
The success of obtaining of the blinder from the 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 thevolatilityon which these forecast is based


The ewma to estimate the volatility
The EWMA obtaining of the blinder from the to estimate the volatility

EWMA, is used by Riskmetrics1 and this method established that the volatility is conditioned bay the past realizations

1 Riskmetrics is a trade mark of J.P.Morgan


The ewma and garch
The EWMA and GARCH obtaining of the blinder from the

Usingl = 0.94for 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
What happen after 1995 obtaining of the blinder from the

Today, the best model to compute the volatility of a global argentine bond is a Tarch(1,1)


Conclusions
Conclusions obtaining of the blinder from the

Using the ACF and PACF in one hand and using fractal geometry in the other hand we arrive to the following expressions:

rs ¹ 0 andsn ¹ st (n/t) 0.5

That allow the use of Garch models to forecast the volatility


Conclusions1
Conclusions obtaining of the blinder from the

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


Conclusions2
Conclusions obtaining of the blinder from the

If volatilityis 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 reportedresources are usefull, whengood news are presentresources are not needed


The future
The Future obtaining of the blinder from the

The use of derivatives for reducing de Var of a portfolio

To calculate the primes of derivatives Garch models will be use

Questions


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