Value at Risk with Stable Distributions

Transcription

1 Value at Risk with Stable Distributions Tecnológico de Monterrey, Guadalajara Ramona Serrano B

2 Introduction The core activity of financial institutions is risk management. Calculate capital reserves given a comfortable level of risk is essential to ensure viability and good reputations for financial institutions. Therefore, it is necessary to have an accurate model and a proper measurement that describes the risks. A commonly used methodology for estimation of market risks is the Value at Risk (VaR). It became a key measure of market risk since the Basle Committee established in 1996 that commercial banks should cover losses on their trading portfolios over a 10-day horizon, 99% of the time.

3 Objectives The first objective of the present investigation is to consider the family of Stable Distributions in VaR modeling as an alternative in the Mexican market. The stable non-gaussian approach is given by its attractive properties that are almost the same as the normal one. The second objective of this work is to consider the GST distribution as an alternative distribution for VaR modelling of data sets involving asymmetric and heavy tails with diverse levels of peakedness around location. Finally, we propose a performance comparison among the estimation of the Value at Risk obtained with the Gaussian, Skew Generalized t and the Stable distributional assumption for the financial returns.

4 Literature Review Over the past decades financial theory was developed under the assumption that assets returns follow a normal distribution. The Gaussian hypothesis was not seriously questioned until the seminal papers of Mandelbrot (1963) and Fama (1965) were published. The excess kurtosis found in Mandelbrot's and Fama's investigations led them to reject the normal assumption and to propose the stable Paretian distribution as a statistical model for asset returns. In later years, the stable Paretian assumption was supported by numerous empirical investigations (Bawa, 1979; Janicki and Weron, 1994; Panorska, Mittnik and Rachev, 1995; Nolan, 1997 and 2003; and Rachev and Mittnik, 2000).

5 A VaR measure is the highest possible loss over a certain period of time at a given confidence level, without abrupt changes in the market conditions. More formally, VaR is defined as the upper limit of the one-side upper interval: Value at Risk where 1-q is the confidence level and X is the relative change (return) in the portfolio value over the time horizon τ. It means, the VaR values are obtained from the probability distribution of portfolio value returns: The VaR methodologies mainly differ in ways of constructing the probability density function

6 Stable distributions capture the leptokurtic nature of empirical financial data. A random variable X is said to follow stable distribution if for any a>0 and b>0, there exist constants c>0 and d R such that Stable Distribution where X₁ y X₂ are independent copies of equality in distribution. X and = d denotes the In general, stable distributions do not have closed form expressions for probability density function (PDF) and cumulative distribution function (CDF). A stable random variable X is commonly described by its characteristic function (CF), which is defined by

7 Characteristic Function where 0 <α 2 is the index of stability or characteristic exponent, 1 β 1 is the skewness parameter, γ 0 is the scale parameter, and δ R is the location parameter. This parameterization call 1-parameterization is convenient for theorical purposes. The 0-parameterization is recommended for numerical work and statistical inference.

8 Generalized Skewed t distribution (GST) Generalized skewed t (GST) distribution is the generalization of student's t distribution, which allows skewness. The probability density function for the standard GST distribution is defined as follows: where f z;, 2, 1 1. bc bc The constants a, b and c are given by The parameter controls the tail thickness and controls the skewness. bz a 1 bz a 1 a 4 c 2 1, b a 2, c z a b z a b

9 Data description In this paper, we pick up five members of the Mexican Stock Exchange Index (IPC), which are from five different industrials. Grupo Financiero Banorte, S.A.B DE C.V., serie O (GFNORTE) Cemex, S.A.B. DE C.V., serie CPO (CEMEX) Empresas ICA, S.A.B. DE C.V., (ICA) Grupo Mexico, S.A.B. DE C.V., serie B (GMEXICO) Wal-Mart de México, S.A. DE C.V., (WALMEX) These five stocks represent the five major industrials constitute into IPC index. We observe the price of these five stocks (based on daily closing prices) from the time period May 2, 2002 to December 31, 2015, about 3446 observations for each stock. The reference currency used is the Mexican peso, as it is the currency of listing companies.

10 The figure 1 represents the series of stock prices Stocks daily closing prices Daily Prices /2002 1/2003 1/2004 1/2005 1/2006 1/2007 1/2008 1/2009 1/2010 1/2011 1/2012 1/2013 1/2014 1/2015 CEMEX WALMEX GFNORTE GMEXICO ICA

11 The series plot of the 5 stocks returns are shown in figure 2. RCEMEX RFNORTE Daily Returns (%) RGMEXICO RWALMEX RICA

12 Descriptive statistics of daily returns are shown in Table 1 Descriptive statistics of daily returns Series CEMEX GFNORTE GMEXICO ICA WALMEX Sector Materials Finance Services Metals & Mining Industrials Consumer Staples Mean Variance Skewness Kurtosis Jarque-Bera Probability (JB) We can see that the skewness of the data is not zero, meaning that the data are asymmetric. The kurtosis of the data suggests that the data are fat-tailed. In addition, the Jarque-Bera statistic is also very large and statistically significant, rejecting the assumption of normality.

13 Kolmogorov- Smirnov goodness-offit test To compare the goodness-of-fit of the stable Paretian distribution, we consider the Kolmogorov-Smirnov goodness-of-fit test: D sup F x Fˆ x The Table 2 reports the observed value of D and the p-value. Estimated parameters of the stable densities are presented in Table 3. x Series CEMEX GFNORTE GMEXICO ICA WALMEX D p-value Series CEMEX GFNORTE GMEXICO ICA WALMEX α β γ δ

14 TS-GARCH model We use stable distribution as the innovation in GARCH model to describe the asymmetry and fat-tail property. In addition, since stable distributions do not have the second absolute expectation, we use the TS-GARCH. Individual stock return is modeled as (1) where R t is the series of individual stock return at time t, μ t and σ t are, respectively, the conditional mean and conditional standard deviation of R t, and z t are i.i.d. standardized stable paretian random variables, z t S(α,β,1,0;0).

15 Table 4 present the maximum likelihood estimates for the GARCH model based on the stable Paretian distribution. Series CEMEX GFNORTE GMEXICO ICA WALMEX Stable estimation and GST GARCH model a a b We consider the GST distribution as the innovation in GARCH model to describe the asymmetry and fat-tail property. The model is set up as follows. Individual stock return is modeled as (2) R t z t 2 t a t t 0 t t a c b t 1 1 t 1

16 Estimated parameters of the GST densities are presented in Table 5. GST and normal GARCH estimation Tabla 6 present the maximum likelihood estimates for the GARCH model based on the Hansen's Skew t Distribution. Series CEMEX GFNORTE GMEXICO ICA WALMEX a e a C b Tabla 7 present the maximum likelihood estimates for the GARCH model based on the Normal Distribution. Series CEMEX GFNORTE GMEXICO ICA WALMEX a a b

17 VaR Estimates We compute the VaR estimates at the confidence level 1-q using the following algorithm: Use MLE to estimate the parameters for the GARCH models in (1) and (2). Use MLE to estimate the parameters for the stable, GST and normal distribution in (1) and (2), respectively. Forecast and by en (1) and (2). t 1 t 1 Calculate the observed z i, i=1,,t from (1) and (2), using the parameter we get in step 1. Simulate S realizations of 2. z t 1 Calculate the simulated return., using the parameter we get in step Measure VaR as the negative of the q-th quantile of the simulated return's distribution.

18 The 99% and 95% are reported in Tables 8 and 9, respectively. Series CEMEX GFNORTE GMEXICO ICA WALMEX Normal GST α-estable Tabla 8. VaR al 99% VaR Estimates Series CEMEX GFNORTE GMEXICO ICA WALMEX Normal GST α-estable Tabla 9. VaR al 95% The bounds of admissible VaR exceedings and exceedings frequencies, for testing at level of significance 5% are provided in Table 10.

19 Evaluation of performance of heteroscedastic VaR models We give the backtesting results in Tables 11 and 12. We indicate by the bold font the numbers, which are outside of acceptable ranges. Series CEMEX GFNORTE GMEXICO ICA WALMEX Normal GST α-estable Table 11 Backtesting del VaR al 99% From Table 11 we can see that the normal model for the 99% VaR estimates for the CEMEX and ICA series give numbers of exceedings above the admissible interval, which implies that the conditional normal VaR measurements significantly underestimates the potential losses. Conditional GST model for the 99% VaR estimates showed that numbers of exceedings are within permissible range. Similarly, conditional stable model provided admissible number of exceptions, except the WALMEX series.

20 Evaluation of performance of heteroscedastic VaR models Table 12. Backtesting del VaR al 95% From Table 12 we can see that the 95% conditional normal and GST VaR estimates, are within the permissible range. However, stable modeling is not satisfactory, all series give numbers of exceedings under the permissible range.

21 Conclusions In this paper, we have considered the family of Stable Distributions in VaR modeling as an alternative in the Mexican market. The statisticals results suggest that the conditional stable VaR model provides more accurate and conservative VaR measurements for the 99% VaR estimation, i.e., our model improve the performance VaR measurements at high confidence level with the stable distributional assumption. Secondly, we have considered the GST distribution as an alternative distribution for VaR modelling of data sets involving asymmetric and heavy tails. The empirical results provide evidence that the conditional GST VaR model support dominant properties for the 99% VaR estimation than the normal model, i.e., the model based on the GST assumption outperforms the normal modeling for high values of the VaR confidence level. Finally, we have proposed a performance comparison among the estimation of the Value at Risk obtained with the Gaussian, Skew Generalized t and the Stable distributional assumption for the financial returns. Results illustrate that the normal modeling significantly underestimates 99% VaR, in contrast is acceptable for 95% VaR estimation. The heteroscedastic conditional stable VaR approach considered in this paper, provides superior fit in modeling VaR at high confidence level. However, additional research is needed. Future work will be employed multivariate dependent stable distribution to describe and examine portfolios, or used copula to describe the correlations between the stock returns considering the stable distribution as the marginal distribution.

22 Thanks!!!