Value at Risk (VaR)

VaR methodology is commonly used for measuring market risk; it is a suitable measurement since regulators accept this quantity as a basis for setting capital requirements for market risk exposure.

A VaR measure is the maximum loss on a period of time (τ) at a specific confidence level (1 - q). More formally, VaR is defined as:

where X represents the portfolio’s returns.

This definition can be rewritten in terms of the probability distribution of portfolio value returns as follows:

where FX-1· is the inverse cumulative distribution function of portfolio returns in one period.

VaR estimates can be obtained via parametric approach, historical simulation and Monte Carlo simulation.

Probability density functions

The development of more robust approaches for VaR estimates is crucial. In this paper, we apply probability and time series theory to improve estimations of appropriate underlying distributions, to capture fat tails and volatility of conditional return distributions; and as a result, improve the estimation of VaR.

In this work, we introduce three families of distributions: Hansen’s skew t distribution (Hansen-GST), NIG and stable distribution, which can capture the kurtosis and skewness of financial returns (^{Rachev & Han, 2000}; ^{Mittnik et al. 2000}).

Hansen’s skew t distribution (Hansen-GST). ^{Hansen (1994)} proposed a different parametric approach to modeling the conditional density of the normalized error. His suggestion is to select a distribution which depends upon a low-dimensional parameter vector, and then let this parameter vector vary as a function of the conditional variables.

Definition. Hansen-GST distribution is a simple skewed generalization of t-Student density. The probability density function is defined as follows:

where 2 < η < ∞, - 1 < λ < 1. The constants a, b and c are given by a=4λcη-2η-1, b=1+3λ2-a2 and c=Γη+12πη-2Γη2.

The parameter η controls the tail thickness and λ controls the skewness. When η → ∞, it is reduced to skewed normal distribution. When λ = 0, it is reduced to Student’s t distribution. Figure 1 gives the probability density function of Hansen-GST with different parameters. We can see that the smaller η is, the fatter the tail is.

Figure 1 Hansen’s skew t distribution with λ=0.5 and η=4, 10, 20 

If a random variable Z follows a standard Hansen-GST distribution with parameter η and λ, we write it as Z + GST(η,λ).

Normal Inverse Gaussian distribution (NIG). The NIG distribution is a specific case of the generalized hyperbolic distribution with λ = −1/2.

Definition. A random variable X follows a NIG distribution with parameters λ = -1/2, X ≥ 0 and Ψ ≥ 0 if the probability density function is

where α, δ ≥ 0 and μ, β ∈ ℝ such that β ∈ [-α,α] and k_λ is the third order modified Bessel function. The parameter β represents the asymmetry, a the heaviness of the tails, λ captures the form of the distribution, μ is a location parameter and δ measures the dispersion (^{Corlu et al., 2016}).

Stable distribution. The stable distribution is capable of capturing skewness and heavy tails and having many intriguing mathematical properties (^{Devroye & James, 2014}; ^{Fofack, & Nolan, 1999}; ^{Zolotarev, 1989}). The class was characterized by Paul Lévy in his study of sums of independent identically distributed terms in the 1920’s.

Stable distributions do not have mathematical expressions for their probability density (PDF) and cumulative distribution (CDF) functions, instead they are described by their characteristic function (CF).

Definition. A characteristic function of a stable random variable X is defined as follows:

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.

If a stable random variable X follows a stable distribution, we write it as X ~ S (α, β, γ, δ). See Figure 2 for plots of stable densities.

Figure 2 Stable densities S (α, β, γ, δ) 

Since we cannot get closed form of PDF and CDF for stable distributions (except that α = 2), we calculate them numerically (^{Mittnik, Doganoglu, & Chenyao, 1999}; Nolan, 1997).

Goodness of fit. We use the Kolmogorov-Smirnov (KS) and Anderson-Darling (AD) statistics to compare the goodness-of-fit of the distributions of interest. The KS test statistic computes the difference between the fitted cumulative distribution function F(x) and the empirical cumulative distribution function F^x as follows: D=sup⁡Fx-F^x.

The AD test statistic computes the weighted average of the squared differences as follows: A2=Fx-F^x2, where the weights are chosen in such way that the discrepancies in the tails are emphasized.

GARCH models

Volatility plays an important role in financial models of pricing and hedging. For this reason finding the conditional distributions on which their estimates of this are more efficient is crucial. The traditional GARCH-normal model fails to capture the non-normal characteristics of financial returns. In this paper, we use three flexible distributions to describe the volatility of the stock returns characterized by leptokurtosis and skewness.

Empirical studies support that the GARCH (1, 1) model works well for financial data (^{Bali, & Theodossiou, 2007}; ^{Liu, & Brorsen, 1995}; ^{Panorska, Mittnik, & Rachev, 1995}). It is important to mention that ^{Starica (2003)} shows that the GARCH (1,1) model has a poor volatility forecasting over long horizons, although at the same time he states that its model provides a precise estimation in a sample size of 2000 observations approximately.

GARCH (1, 1) model with stable distribution (stable-GARCH). Following ^{Panorska, Mittnik, & Rachev (1995)} and ^{Naka & Oral (2013)} we used the GARCH model proposed by Taylor (1986) and ^{Schwert (1989)}. Since stable distributions do not have the second absolute expectation (except when α = 2), the conditional variance is expressed in terms of the conditional standard deviation as follows:

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 random variables, z_t ~ S(α, β, 1, 0) with 1 < α < 2 .

To estimate the parameters in (6), we use the Maximum Likelihood Estimation (MLE) method³, where the PDFs of z_t were approximated using the computer program STABLE.⁴

Hansen-GST GARCH model. We consider the Hansen-GST distribution as the innovation in the GARCH model to describe the asymmetry and fat-tail property. The model is set up as follows.

Individual stock return is modeled as

To estimate the parameters in (7), we use the MLE method, where the pdfs of z_t are approximated using the approach proposed by ^{Hansen (1994)}.

NIG and Gaussian GARCH model. We consider the NIG and the Gaussian distributions as the innovation in the GARCH model. Individual stock return is modeled as (7) with c = 0. To estimate the parameters of the NIG distribution, we use MLE.

Estimating VaR

VaR estimation is realized by Monte Carlo simulation (Embrechts, McNeil & Frey, 2005; Glasserman, 2003) considering a heteroscedastic VaR model on the basis of the stable distribution, Hansen-GST, NIG and normal distribution. We simulated 10,000 realizations of the series of individual stock returns at time t and measure VaR as the negative of the q-th quantile of the simulated return’s distribution.

Backtesting

Since financial institutions have the freedom to specify their own model to compute their VaR, the procedure to backtesting becomes extremely important for regulators to assess the quality of the models.

We evaluate and compare the performance for distributions-based heteroscedastic VaR models based on the ^{Kupiec Unconditional Coverage (UC) test (1995)} and the ^{Christoffersen Conditional Coverage (CC) test (1998)}.

The Kupiec Unconditional coverage test (UC). Suppose we use the most recent k historical data to forecast the current VaR, define the indicator for VaR violations as follows:

where H are the historical returns and s = 1, ..., k.

The Kupiec likelihood ratio test (^{Kupiec, 1995}) tests the null hypothesis:

i.e. it tests whether the expected proportion of violations is equal to α.

The likelihood ratio test statistic is given by:

where N is the sample size, n is the number of violations and p = n/N is the percentage of violations. This test follows an asymptotic chi-square distribution with one degree of freedom.

The Christoffersen Conditional Coverage test (CC). The Conditional Coverage test (^{Christoffersen, 1998}) requires a correct unconditional coverage and furthermore, it ensures that the result series is i.i.d. This statistic test follows an asymptotic chi-square distribution with two degrees of freedom and is given by:

where n_ij is the number of observations with value i followed by j, and π_ij = Pr(I_s (α) = j|I_s-1 (α) i).

Data and descriptive statistics

For the empirical analysis, five assets with a different trading volume from five different industries listed on the Mexican Stock Exchange (BMV) have been chosen. These assets correspond to the following companies: Fomento Economico Mexicano, S.A.B. de C.V. (FEMSA) is a company that through its subsidiaries produces, distributes and markets non-alcoholic beverages throughout Latin America as part of the Coca-Cola system. The Company owns and operates convenience stores in Mexico and Colombia and holds a stake in Heineken; Grupo Carso (GCARSO) one of the most important conglomerates in Latin America, the company controls and operates companies in the industrial, commercial, infrastructure and construction sectors; Grupo Mexico S.A.B. de C.V. (GMEXICO) holds concessions to operate the Pacifico-Norte and Chihuahua-Pacifico Railroad lines. Grupo Mexico, through subsidiaries, operates open-pit copper mines, underground mines, a coal mine, copper smelters, a rod mill facility, and a precious metals refinery; Grupo Financiero Banorte S.A.B. de C.V. (GFNORTE) is a financial institution in Mexico. The Company offers banking services, premium banking, whole- sale banking, leasing and factoring, warehousing, insurance, pensions and retirement savings; and Grupo Televisa, S.A.B., (TELEVISA), operates media and entertainment businesses in the Spanish speaking world. The Company has interests in television production and broadcasting, programming, direct-to-home satellite services, publishing and publishing distribution, cable television, radio production, show business, feature films and Internet portals.

We estimate the VaR of these five stocks (based on daily closing prices) over the period January 2, 2002 to December 31, 2009, about 2018 observations for each stock. The reference currency used is the Mexican peso and the asset returns are logarithmic returns. The series plot of the 5 stocks returns are shown in Figure 3.

Figure 3 Daily returns (%) 

Descriptive statistics of daily returns are presented in Table 1. Based on the skewness and kurtosis we observed that the data are asymmetric and have thicker tails than the normal distribution. In addition, the Jarque-Bera statistic is large and statistically significant, thereby implying that the assumption of normality is rejected.

Goodness of fit

We use the KS and AD statistics to compare the goodness-of-fit of the distributions of interest. Table 2 presents the hypothesis test KS and AD to verify if the time series of the daily returns follow the proposed probability distribution, the p-value is reported in brackets. The null hypothesis is that the observed data originates from the hypothesized distribution and is rejected if the p-value is lower than α = 1%. It can be seen that each time series is adjusted to some probability distribution under a significance level of 1%.

To estimate the different parameters of the probability density functions that have been proposed in this work, we use the MLE method. Table 3 shows these estimators, standard errors are in brackets.

Estimating GARCH models

Tables 4 and 5 present the maximum likelihood estimation for the GARCH models assuming the alternative distribution functions, standard errors are in brackets.

We can observe from Tables 4 and 5 that the sum of the parameters are less than one, ensuring the conditions for stationarity, in addition, for the Hansen-GST GARCH model the asymmetric coefficients are significant, indicating the presence of asymmetric leverage volatility effects. Finally, the Akaike Information Criterion (AIC) indicates that the GARCH stable is the model that better captures the dynamics of the returns series; it is followed by the NIG GARCH model, next to the Hansen-GST and the normal GARCH models.

VaR estimates

VaR estimation is realized by Monte Carlo simulation based on 10,000 realizations of the series of individual stock. Table 6 provides the one-day ahead estimates of VaR for each returns, at 95% and 99% confidence levels.

As observed in Table 6, except for GMEXICO, the stable-VaR estimates at 95% and 99% confidence levels are higher than the normal, NIG and Hansen-GST VaR estimates, i.e., the stable VaR measurements provide more conservative5⁵ VaR estimates of potential losses. On the contrary, the normal distribution produces the lowest VaR estimates at 99% among the four VaR models.