The Garch model and their Applications to the VaR

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# The Garch model and their Applications to the VaR - PowerPoint PPT Presentation

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 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 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

rs = 0

sn = st (n/t) 0.5

The memory of a process: The Hurst exponent

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

The meaning of H

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 coefficientR/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

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Some results of the coefficient H

Index Dow Jones

Ln (R/S)n

Ln (n)

The conclusions of the use ofH

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

.. 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 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

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

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....
• 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 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:

Engle 1982

ARCH (q)

Autoregressive Conditional Heterocedastic

Bollerslev 1986

GARCH (q,p)

Generalized Autoregressive Conditioned Heteroskedastic

A simple prediction of a volatility with Arch model

Where:

s2t = variance at day t

Rt-1- R = deviation from the mean at day t-1

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If we regress the series on a constant….

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 et2series

The Ljung Box or Q-statistic at lag 10:

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

The Garch (1,1)

This model was used during 1990-1995 with a great success, previous to the “tequila effect” or Mexican crisis

The persistence of a Garch (1,1)

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

With 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

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)

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)

Glosten Jaganathan and Runkle

and Zakoian (1990)

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

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

What is 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

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 VaR

s

t

days to be forecasted

market position

Volatility measure

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 thevolatilityon 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

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

The EWMA and GARCH

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

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

Conclusions

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

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 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 use of derivatives for reducing de Var of a portfolio

To calculate the primes of derivatives Garch models will be use

Questions