VaR Estimation by Historical Simulation

In this method, historical financial data produce future behavior information. Therefore, it is possible to use financial data history to obtain significant predictions of future performance.

Historical simulation is a relatively simple method that is easy to implement and has the advantage of not needing to assume that returns follow a normal distribution.

In general, the historical method depends on observed data, so if the sample size is not large enough, there is a greater chance that the VaR estimate lacks precision.

Various articles, such as ^{Hendricks (1996)} and ^{Barone-Adesi et al. (2000)}, provide a detailed discussion of the historical simulation approach.

VaR Estimation by Monte Carlo Simulation

Through this method, recently addressed by ^{Hong, Hu and Liu (2014)}, an approximation of the expected return performance of a portfolio or financial instrument is obtained by means of simulations that generate random trajectories of the returns of the portfolio or financial instrument, considering certain initial assumptions about volatilities and correlations of risk factors.

Parametric VaR

The parametric method is used under the assumption that the observed data follow some rules or models with unknown parameters. We use these data to obtain parameter estimates and apply the rule or model established to calculate the VaR, as ^{Mentel (2013)} pointed out. The parametric method offers two approaches: unconditional and conditional.

Unconditional Approach

In this method, the assumption is that the financial returns in each period of time are independent and identically distributed random variables (IID) that follow a multivariate normal distribution. However, various investigations show that multivariate Gaussian distribution cannot explain some properties of the empirical distribution of financial data. For example, ^{Fama (1965)}, ^{Hull and White (1998)} point out that changes in several market variables (stock prices, prices of zero coupon bonds, exchange rates, commodity prices, etc.) are leptokurtic, that is, there are too many values close to the average and too many in the extreme tails. These changes increase the probability of very large and very small movements in the value of market variables and decrease the probability of moderate movements. Therefore, there is not enough evidence to support the Gaussian hypothesis, so we propose a distribution with different characteristics.

Conditional Approach

It admits that time series of financial returns depend on past information. Traditionally, the Autoregressive Moving Average Model (ARMA) describes a series dependence, where we obtain a stationary series. However, ARMA models assume that the variance is constant, and given that generally speaking the volatility of financial time series is not constant, these models are not suitable to use with them.

The most popular of several studies aimed at finding models that describe the volatility variable in time, a common characteristic of financial returns, is ^{Engle’s Autoregressive Conditional Heteroscedasticity model (ARCH, 1982)}, where the variance conditioned on past information is not constant and depends on the square of past innovations. Subsequently, ^{Bollerslev (1986)} generalized arch models by proposing the Generalized Conditional Autoregressive Heteroscedasticity (GARCH) models. Here, the conditional variance depends not only on the squares of perturbations, as in arch models, but also on the conditional variances of previous periods. Both models may be combined with the ARMA model, obtaining the ARMA-ARCH and ARMA-GARCH models.

In the standard GARCH model, widely used today, we assume that observed data fits a Gaussian distribution. For many financial return series, however, the Gaussian distribution is inadequate, since it does not consider leptokurtosis.

Strengths and Weaknesses of VaR

VaR is a popular risk measure, because it can quantify the market risk of a financial institution by calculating a unique numerical value. This value is the maximum possible loss of a portfolio or financial instrument in a given period of time, for a certain level of confidence (^{Barone-Adesi, Giannopoulos and Vosper, 2000}). At the same time, one of its disadvantages is that, by definition, it ignores losses whose probability of occurrence is less than that chosen as the level of confidence in the estimation. For these reasons, Bali affirms that standard VaR provides an inadequate estimation of losses during periods of high volatility such as those corresponding to financial crises (^{Bali, 2007}).

On the other hand, ^{Artzner, Delbaen, Eber and Heath (1999)} indicate that one of its main disadvantages is that it is not a coherent measure of risk and non-convex. A measure of risk is coherent if it satisfies the axioms of monotonicity, subadditivity, positive homogeneity and invariance under translations.

Conditional Value at Risk

An alternative coherent risk measure is CVaR, introduced by Artzner, Delbaen, Eber and Heath, defined here as the expected loss that exceeds the VaR. In general, CVaR provides the conditional expectations of loss above the VaR. Some other names for CVaR are Expected Shortfall, Mean Excess Loss, Beyond VaR, Tail VaR, Conditional Loss and Expected Tail Loss.

CVaR provides better adjustment and consistency in the estimation of risk with respect to the VaR, since it complements p-value information provided by the VaR, becoming very useful with asymmetric distributions (^{Artzner, Delbaen, Eber and Heath, 1997}).

^{Rockafellar and Uryasev (2002)} show that the CVaR is a coherent measure of risk, as well as the significant advantages of this methodology with respect to the traditional VaR. On the other hand, ^{Kibzun and Kuznetsov (2003)} mention that under normal conditions, CVaR is a convex function with respect to the positions taken, which allows the construction of an efficient optimization algorithm. ^{Peña (2002)} indicates that despite the advantages of VaR over CVaR, both are complementary measures; that is, if the objective of VaR is to control market risk under normal conditions, the objective of CVaR is to control market risks in extreme conditions. In addition, it is possible to establish a relationship between VaR and CVaR if a specific parametric distribution is assumed. For the specific case of normality, the VaR and CVaR are similar.

Therefore, we can consider the use of CVaR as a complement to VaR. So, if the objective of the VaR is to control market risk under normal conditions, the objective of the CVaR is to control market risks in extreme conditions.

Currently, both risk measures are widely used both in the field of research and in empirical studies and applications.

Portfolio Optimization

Regardless of whether we apply VaR or CVaR to measure the risk of a portfolio or financial instrument, one of the natural objectives of risk management is portfolio optimization. In Markowitz Portfolio Theory, the concepts of correlation and covariance are key elements of the model. In addition, it introduced the concept of diversification, that is, the addition of certain assets to an investment portfolio, to decrease risk.

The objective of ^{Markowitz (1952)} was to find the optimal allocation for each investment instrument, minimizing variance (considered as the risk measure) subject to a condition on the expected return; that is, the investor wants to obtain the highest return and is adverse to risk. For Markowitz, a portfolio is considered efficient if it provides the maximum possible return for a given level of risk, or equivalently, if it provides the lowest possible risk for a specific level of profitability.

Thus, we can say that an optimal portfolio is a combination of financial instruments that represent a risk-return ratio that maximizes investor satisfaction. Here we assume that individual assets follow Gaussian distributions and there is a correlation matrix for the dependence between the assets.

Variance may be replaced by other risk measures such as VaR and CVaR. Since VaR is not a consistent measure, though, it is difficult to find the optimal portfolio for minimizing it. ^{Gaivoronski and Pflug (2005}) propose an approach to VaR using a soft measure called SVaR (local minimizer) that filters irregularities locally. The approach, however, is too complex.

On the other hand, CVaR is a coherent and convex measure, so a unique global minimizer may be found. ^{Rockafellar and Uryasev (2002)} point out a simple way to optimize a portfolio by minimizing CVaR, which enables transforming the problem into a classic linear programming problem. Recently, various studies that extend this model have come to light, including ^{Andersson et al. (2001)}; ^{Charpentier and Oulidi (2008)}; ^{Glasserman et al. (2002)}; and ^{Krokhmal et al. (2001)}.

All these authors, however, use the historical approach to calculate portfolio allocation. Therefore, if there is not enough historical data, risk may be severely underestimated. Another drawback is that they do not address the dependence of the historical data series.

As already mentioned, the financial data series is not perfectly independent, since there is a correlation between the returns on different actions and different types of risks. Therefore, it is essential to describe such risk management correlations and establish the necessity of knowing about joint distribution of financial data.

Previously, among the assumptions about the joint distribution function was multivariate normal distribution. With this distribution, dependency is described by the correlation matrix, and obtaining the risk measure is simple (^{Markowitz, 1952}). However, this assumption does not work well for financial data.

In addition to normal distribution, another approach will be used in this paper to describe financial series: NIG distribution, which is a particular case of the family of GH distributions that will be described in detail in the next section.

Descriptive Statistics

Skewness and kurtosis were calculated for each equity series in order to validate distributions with higher skewness values, either positive or negative. So we can expect that the empirical data do not correspond to a normal distribution.

The following table presents some statistics for the series considered:

By analyzing the excess of kurtosis, different behavior from the normal distribution is noted, so the presence of heavy tails is expected.

Normality Test

When we use the Anderson-Darling and Shapiro-Francia normality tests, the null hypothesis of normality may be rejected. In this case, the proposed NIG distribution becomes a candidate to fit the empirical data. We establish both normality tests as follows:

For the null hypothesis test, the p-value of each data series was set using both ^{Anderson-Darling (1954)} and ^{Shapiro-Francia (1972)} proofs, with a significance level of 0.05, so, if p - value ≥ 0.05 the null hypothesis is not rejected.

Shapiro-Francia Test

The normality test developed by ^{Shapiro and Francia (1972)} is an approximate and simplified version of the Shapiro Wilk test to prove the normality of a larger series of data. The test parameter is obtained by calculating the slope of the regression line by simple least squares, i.e.,

Anderson Darling Test

The ^{Anderson-Darling (1954)} criteria test the hypothesis that a data series comes from a population that adheres to a continuous Cumulative Distribution Function (CDF). We can express the test as follows:

Normal Inverse Gaussian Distribution

As mentioned before, multiple studies prove that NIG distribution fits the financial series. ^{Barndorff-Nielsen (1997)} defined this kind of distribution:

where

and

where K₁ is the modified Bessel function of third order and index 1. Also, α, β, μ and δ are parameters, satisfying 0 ≤ |β| ≤ α, μ ∈ R and 0 < δ.

Parameters α and β determine the shape, and μ and δ scale the distribution. Parameter α denotes the flatness of the density function, i.e. a high value of α means a greater concentration of the probability around μ. Parameter β defines a kind of skewness of the distribution. When β = 0, the NIG distribution is symmetric around the mean. A negative value represents a heavier left tail. Parameter δ describes the distribution scale, and parameter μ is responsible for the distribution density shift.

Goodness of Fit Tests

By simulating a vector with the obtained parameters, we test the similarity of both distributions with Kolmogorov-Smirnov and Anderson-Darling criteria in which the p-values correspond with the acceptance region of the null hypothesis. So according to the statistical tests, it can be stated that NIG distribution is capable of modeling returns even during a period of economic crisis.

VaR and CVaR Estimates

From definition 1 we can obtain the VaR knowing the cumulative distribution function of the asset’s returns:

where Fx-1 is the inverse of the cumulative distribution function of the returns of the asset in a period. In other words, the VaR is the quantile q of F_x. Therefore, calculation of the VaR consists mainly of the estimate of the lower quantiles of the cumulative distribution function of the assets’ returns.

Assuming that the distribution of asset returns is normal, described by mean parameters μ and standard deviation σ, the VaR calculation consists of finding the q^th percentile of the standard normal distribution z_q:

where X* = z_qσ + μ, and Φ(z) is the standard normal density function, N(z) is the cumulative normal distribution function, X is the asset performance, g(x) is the normal distribution function of the yields with mean μ and standard deviation σ, and X* is the minimum yield at confidence level 1 - q.

The CVaR of X at level 1-q is defined as the expected loss, given that this loss exceeds the VaR, that is:

The CVaR complements the information provided by the VaR. As mentioned earlier, it is very useful when we have asymmetric distributions, as well as being an excellent risk management tool, applicable in distributions with peaks.