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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Ricardo A. Tagliafichi
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 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
rs = 0
sn = st (n/t) 0.5
Is a number related with the probability that an event is autocorrelated
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 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
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:
.. are the returns n.i.d.?
The K-S test: P (Dn
The autocorrelation function is the relationship between the stock’s returns at different lags.
The Ljung Box or Q-statistic at lag 10:
Ho: r0 .... r10 = 0 H1: some r1 ....rk ¹ 0
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
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:
Autoregressive Conditional Heterocedastic
Generalized Autoregressive Conditioned Heteroskedastic
s2t = variance at day t
Rt-1- R = deviation from the mean at day t-1
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 Ljung Box or Q-statistic at lag 10:
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
This model was used during 1990-1995 with a great success, previous to the “tequila effect” or Mexican crisis
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
for the future t periods ...
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
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.
Glosten Jaganathan and Runkle
and Zakoian (1990)
g is a positive estimator with weight when there are negative impacts
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
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
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
days to be forecasted
Level of confidence
Report of potential loss
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
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
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:
Today, the best model to compute the volatility of a global argentine bond is a Tarch(1,1)
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
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
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 use of derivatives for reducing de Var of a portfolio
To calculate the primes of derivatives Garch models will be use