Risk Management Performance of Alternative Distribution Functions

Transcription

1 Risk Management Performance of Alternative Distribution Functions January 2002 Turan G. Bali Assistant Professor of Finance Department of Economics & Finance Baruch College, Zicklin School of Business City University of New York 17 Lexington Avenue, Box New York, New York Phone: (646) Fax : (646) Turan_Bali@baruch.cuny.edu Panayiotis Theodossiou Professor of Finance School of Business Rutgers University 227 Penn Street Camden, New Jersey Phone: (856) Fax : (856) theodoss@camden.rutgers.edu Key words: value at risk, risk management, extreme value distributions, skewed fat-tailed distributions JEL classification: G12, C13, C22

2 2 Risk Management Performance of Alternative Distribution Functions ABSTRACT This paper compares the risk management performance of alternative distribution functions. The results indicate that the extreme value and skewed fat-tailed distributions, such as the skewed generalized t (SGT) and inverse hyperbolic sine (IHS) distributions, improve the standard VaR models that assume normality. This is because the SGT and IHS distributions put more emphasis on the tail areas of observed frequency distributions, and hence provide good predictions of catastrophic market risks during extraordinary periods. The empirical findings lead to a firm rejection of the hypothesis that one is obliged to use the distribution of extremes only (instead of the distribution of all returns) to obtain precise VaR measures. The results based on the actual and estimated VaR thresholds as well as the likelihood ratio tests point out that the maximum likely loss of financial institutions can be more accurately estimated using the generalized extreme value and skewed fat-tailed distributions. They perform surprisingly well in capturing both the rate of occurrence and the size of extreme observations in financial markets. In addition, this article proposes a conditional VaR approach that takes into account time-varying volatility, considers the non-normality of returns, and deals with extreme events. The results indicate that the actual VaR thresholds are time-varying to a degree not captured by the conditional normal density, but precisely estimated by the conditional SGT, IHS, and extreme value distributions.

3 3 Risk Management Performance of Alternative Distribution Functions I. Introduction The importance of sound risk management for financial institutions was emphasized by several high-profile risk management disasters in the early 1990s. 1 During the past decade, there has been an increased focus on the measurement of risk and the determination of capital requirements for financial institutions to meet catastrophic market risk. This increased focus has led to the development of various risk management techniques. The primary technique is Value at Risk (VaR), which determines the maximum expected loss on a portfolio of assets over a certain holding period at a given confidence level (probability). 2 The use of VaR techniques in risk management has exploded over the past few years. The two most popular VaR techniques are the variance-covariance analysis and historical simulation. The variance-covariance analysis relies on the assumption that financial market returns follow a multivariate normal distribution. This technique is easy to implement because the VaR can be computed from a simple quadratic formula with the variances and covariances of returns as the only inputs. Its major drawback is that financial market returns exhibit skewness and significant excess kurtosis (fat-tails and peakness), and as such they are not normal. Because of this, the size of actual losses is much higher than that predicted by the normal distribution. As a result, the variancecovariance analysis produces VaR thresholds that understate the true risk faced by financial institutions (tail bias). The variance-covariance approach is particularly weak where a VaR model should be strong in the prediction of large losses for regulatory purposes and risk control. Historical simulation does not rely on normality and as such it does not suffer from the tailbias problem. By applying the empirical distribution of all assets returns in the trading portfolio, the outcome will reflect the historical frequency of large losses over the specified data window. Unlike

4 4 the variance-covariance analysis, the historical approach can be used in a natural way to compute VaR for non-linear positions, such as derivative positions. The problem with historical simulation is that it is very sensitive to the particular data window. In particular, the inclusion of extraordinary periods, such the stock market booms and crashes, affects significantly the computation of VaR measures. This is because the empirical return distribution is very dense and smooth around the mean but discrete in the tails because of a few extremely large price movements. As a result, VaR measures based on historical simulation exhibit high variances. Moreover, at its lower end, the empirical return distribution drops sharply to zero and remains there, thus the probability of more severe losses than the past largest one is assigned the value of zero, which might be considered imprudent. 3 In light of the above, an alternative approach that approximates the tails of the distribution of returns asymptotically is more appropriate than imposing a symmetric thin-tailed functional form like the normal distribution. Although VaR models based on the normal distribution provide acceptable estimates of the maximum likely loss under normal market conditions, they fail to account for extremely volatile periods corresponding to financial crises. Longin (2000), McNeil and Frey (2000), and Bali (2001a) show that VaR measures based on the distribution of extreme returns (extreme value distributions), instead of the distribution of all returns, provide good predictions of catastrophic market risks during extraordinary periods. An important contribution of this paper is the application and assessment of several flexible probability distribution functions in computing VaR measures based on the distribution of all returns. These distributions are the inverse hyperbolic sine (IHS) of Johnson (1949), the exponential generalized beta of the second kind (EGB2) of McDonald and Xu (1995), the skewed generalized-t (SGT) of Theodossiou (1998), and the skewed generalized error (SGED) of Theodossiou (2001).

5 5 This paper shows that the SGT and IHS (henceforth referred as skewed fat-tailed distributions) produce precise VaR measures and compare favorably to the extreme value distributions. The latter is attributed to the fact that the SGT, IHS, and extreme value distributions provide an excellent fit to the tails of the empirical return distribution. 4 Value-at-risk measures are most often expressed as percentiles corresponding to the desired confidence level. The 99% confidence level is consistent with VaR standards, but this choice is really arbitrary. Therefore, we expand the analysis beyond the 99% level and consider the entire tail of the return distribution. We evaluate the empirical performance of alternative distribution functions in predicting the 0.5%, 1%, 1.5%, 2%, 2.5%, and 5% VaR thresholds. The results consistently point to the same conclusions and provide evidence that the extreme value and skewed fat-tailed distributions perform surprisingly well in modeling the asymptotic behavior of stock market returns. Moreover, the paper evaluates the relative performance of aforementioned distributions based on the unconditional and conditional coverage tests introduced by Kupiec (1995) and Christoffersen (1998), respectively. VaR measures based on the unconditional extreme value and skewed fat-tailed distributions do not account for systematic time-varying changes in the distribution of returns. The conditional coverage test results indicate that the actual thresholds are time-varying to a degree not captured by the unconditional density functions. This paper extends the unconditional VaR approach by taking into account the dynamic (time-series) behavior of financial return volatility. The dynamic behavior of returns is modeled using the autoregressive absolute GARCH process of Taylor (1986) and Schwert (1989). The paper is organized as follows. Section II contains a short discussion of the extreme value and flexible distributions. Section III provides the unconditional value-at-risk models based on

6 6 alternative distribution functions. Section IV describes the data. Section V presents the estimation results. Section VI compares the risk management performance of alternative VaR models. Section VII proposes a conditional VaR approach. Section VIII concludes the paper. II.A Extreme Value Distributions We investigate the fluctuations of the sample maxima (minima) of a sequence of i.i.d. nondegenerate random variables {X 1, X 2,, X n } with common cumulative distribution function (cdf) F(x), where M 1 = X 1, M 2 = max (X 1, X 2 ),, M n = max (X 1,, X n ), n 2. (1) Corresponding results for the minima can be obtained from those for maxima by using the identity: min (X 1,, X n ) = max ( X 1,, X n ). (2) The exact cdf of the maximum M n is easy to write: P(M n x) = P(X 1 x,, X n x) = F n (x), x R, n N. (3) According to this, extremes happen near the upper end of the support of the distribution, hence intuitively the asymptotic behavior of M n must be related to the cdf F(x) in its right tail. In this case this tail has finite support. We let x F = sup{x R: F(x) < 1} (4) denote the right endpoint of F(x). We immediately obtain, for all x < x F, P(M n x) = F n (x) 0, n, (5) and, in the case x F <, we have for x x F that P(M n x) = F n (x) = 1. (6) Thus M n P x F in probability as n, where x F. To obtain an asymptotic distribution theory, we need to look at the convergence in distribution of the centered and normalized maxima.

7 7 Here, the well-known Fisher-Tippett (1928) theorem has the following content: if there exist normalizing constants σ n > 0 and centering constants µ n R such that x= σ 1 ( M µ ) d H, n, (7) n n n for some non-degenerate distribution H, then H belongs to the type of one of the three so-called standard extreme value distributions: Frechet : H max, 1/ ξ exp( x ) if x > 0 ξ ( x) = (8) 0 otherwise 1 if x 0 Weibull : H max, ξ ( x) = (9) 1/ ξ exp( ( x) ) otherwise Gumbel: H max,0 (x) = exp[ exp( x)] - < x < + (10) Jenkinson (1955) proposes a generalized extreme value (GEV) distribution, which includes the three limit distributions in (8)-(10), distinguished by Gnedenko (1943): 1/ ξ M µ µ H max, ξ ( M ; µ, σ ) = exp 1 + ξ ; 1 + ξ M 0 (11) σ σ where ξ is a shape parameter. For ξ > 0, ξ < 0, and ξ = 0 we obtain the Frechet, Weibull and Gumbel families, respectively. The Frechet distribution is fat tailed as its tail is slowly decreasing; the Weibull distribution has no tail after a certain point there are no extremes; the Gumbel distribution is thin-tailed as its tail is rapidly decreasing. The shape parameter ξ, called the tail index, reflects the fatness of the distribution, whereas the parameters of scale, σ, and of location, µ, determine the average and standard deviation of the extremes along with ξ. An alternative approach to determine the type of asymptotic distribution for extremes can be based on the concept of generalized Pareto distribution (GPD). 5 Excesses over high thresholds can be modeled by the generalized Pareto distribution, which can be derived from the generalized

8 8 extreme value (GEV) distribution. The generalized Pareto distribution of the standardized maxima denoted by G max (x) is given by G max (x) = 1 + ln [H max (x)], where H max (x) is the GEV distribution: 1/ ξ M µ G max, ξ ( M ; µ, σ ) = ξ σ. (12) Notice that the generalized Pareto distribution presented in equation (12) nests the standard Pareto distribution, the uniform distribution on [-1,0], and the standard exponential distribution: 6 Pareto: G ( x) = 1 x max, ξ 1/ ξ Uniform: G ( x) = 1 ( x) max, ξ 1/ ξ for x 1, (13) for x [-1,0] (14) Exponential: G ( x) = exp( x) for x 0. (15) max,0 1 To determine whether the generalized Pareto or the generalized extreme value distribution yields a more accurate characterization of extreme movements in financial markets, Bali (2001b) proposes a more general extreme value distribution using the Box-Cox (1964) transformation: F max, ξ φ 1/ ξ M µ exp 1 + ξ 1 σ ( M; µσφ,, ) = + 1 φ (16) The Box-Cox-GEV distribution in equation (16) nests the generalized Pareto distribution of Pickands (1975) and the generalized extreme value distribution of Jenkinson (1955). More specifically, when φ equals one the Box-Cox-GEV reduces to the GEV distribution given in equation (11): when φ = 1 F max,ξ (x) H max,ξ (x). When φ equals zero, the Box-Cox-GEV converges to the generalized Pareto distribution presented in equation (12): when φ = 0 F max,ξ (x) G max,ξ (x). 7

9 9 II.B Skewed Fat-Tailed Distributions This section presents the probability density functions for the skewed generalized-t (SGT), the skewed generalized error (SGED), the inverse hyperbolic sine (IHS), and the exponential generalized beta of the second kind (EGB2). As shown in Hansen, McDonald, and Theodossiou (2001), all four densities accommodate diverse distributional characteristics when used for fitting data or as a basis for quasi-maximum likelihood estimation (QMLE) of regression models. The SGT probability density function is C 1 k SGT ( y; µσ,, n, k, λ ) = 1+ y µ + δσ (17) k k k σ (( n 2) k )( 1 + sign( y µ + δσ) λ) θ σ ( n 1) + k where ( 2 (( 2 ) ) ( )) 1 k θ 1, C = k n k B k n k, k ( ( 2) ) ( 1, ) 3,( 2) ( ) ( λ) θ = k n B k n k B k n k S, S ( λ) 1 3λ 4A λ = +, ( 2,( 1) ) ( 1, ) ( 3,( 2) ).5.5 A= B k n k B k n k B k n k, δ = 2λ AS( λ) 1, μ and σ is are the mean and standard deviation of the random variable y, n and k are positive kurtosis parameters, λ is a skewness parameter obeying the constraint λ <1, sign is the sign function, and B( ) is the beta function. In the above density, μ δσ is the mode and δ = (μ mode(y))/σ is Pearson s skewness. The SGT gives several well-known distributions as special cases. Specifically, it gives for λ=0 the generalized-t of McDonald and Newey (1988), for k=2 the skewed t of Hansen (1994), for n= the skewed generalized error distribution (also used in this paper), for n= and λ=0 the

10 10 generalized error distribution or power exponential distribution of Subbotin (1923) (used by Box and Tiao [1962] and Nelson [1991]), for n=, λ=0, k=1 the Laplace or double exponential distribution, for n=1, λ=0, k=2, the Cauchy distribution, for n=, λ=0, k=2 the normal distribution, and for n=, λ=0, k= the uniform distribution. The SGED probability density function is where C 1 SGED( y; µσ,, k, λ ) = exp y µ + δσ k σ k k ( 1+ sign( y µ + δσ) λ) θ σ ( 2θ ( 1 )), δ 2λ ( λ) 1 C = k Γ k = AS S ( λ) = 1+ 3λ 4A λ ( 1 k) ( 3 k) S( λ) θ =Γ Γ ( 2 ) ( 1 ) ( 3 ).5.5 A=Γ k Γ k Γ k k (18) μ, σ, λ, k, and sign are as defined previously. The inverse hyperbolic sine (IHS) probability density function is IHS y ( ; µσ,, k, λ) = k 2 ( ( ) y + ) 2 (( ) ( ) ) ( ) y µ δσ σ θ y µ δσ σ λ θ 2π θ µ δσ σ σ 2 k exp ln ln 2 (19) whereθ = 1 σ w, δ = µ w σw, 2.5k λ λ 2λ k 2 λ k.5 k.5 µ w =.5( e e ) e, σ w =.5( e + e + 2) ( e 1), μ w and σ w are the mean and standard deviation of w=sinh(λ+z/k), sinh is the hyperbolic sine function, z is a standardized normal variable, and µ and σ are the mean and standard deviation of y. 8 Note that negative (positive) values of λ generate negative (positive) skewness, and zero values no skewness. Smaller values of k result in more leptokurtic distributions.

11 11 The EGB2 probability density function is ( µσ ) EGB2 y;,, p, q = C e ( + ) p y µ δσ ( θσ) ( y µ + δσ) ( θσ) ( 1+ e ) p+ q, (20) where C ( B p q θσ ) δ ( ψ p ψ q ) θ θ ψ ( p) ψ ( q) = 1 (, ), = ( ) ( ), = 1 +, p and q are positive scaling constants, B( ) is the beta function, ψ ( z) = dln Γ ( z) dz and ψ ( ) ψ( ) z = d z dz are the psi function and its first derivative, and μ and σ are the mean and standard deviation of y. The EGB2 is symmetric for equal values of p and q, positively skewed for values of p > q, and negatively skewed for values of p < q. The EGB2 converges to the normal distribution for infinite values of p and q. III. Value at Risk Models with Alternative Distribution Functions The traditional VaR models assume that the probability distribution of log-price changes (log-returns) is normal, an assumption that is far from perfect. However, the distributions of logreturns of financial assets are usually skewed to the left, have fat-tails, and are peaked around the mode. Because of the fat tails, extreme outcomes happen much more frequently than would be predicted by the normal distribution. 9 This section presents alternative VaR models based on the extreme value as well as the aforementioned flexible distributions. 10 VaR calculations are performed in an environment where the stochastic process A t depends on a risk factor such as the interest rate, exchange rate, or equity return. The arbitrage-free price of a financial asset at time t, A t, is assumed to be a known function of R t and the parameters ϕ, i.e., A t = A(R t, t; ϕ). (21) The stochastic variation in A t during an infinitesimal interval dt can be given by Ito s Lemma: da t = A R dr t + A t dt + 1 A 2 2 RR t σ dt (22)

12 12 where A R = AR ( t, t) R t and A RR = 2 AR (, t) R t 2 t are the delta and gamma of the asset, respectively. To compute VaR, one imposes a model on the stochastic differential dr t. For most risk factors, researchers choose the stochastic differential equation of the form: dr t = µ t dt + σ t dw t. (23) Assuming that t denotes the length of time interval, the discrete time approximation of the stochastic process in (23) is written as: R = µ t+ σ t z, (24) t t t where z is standard normal with mean zero and variance one. The critical step in calculating VaR measures is the estimation of the threshold point defining what variation in returns R t is considered to be extreme. Let Φ be the probability that R t will exceed the threshold ϑ. That is, ϑ µ t t Pr( Rt > ϑ) = Pr z > a = =Φ σ t t (25) where Pr( ) is the underlying probability distribution. In the traditional VaR model, a = 2.326, and ϑ Normal = µ t t σ t t (26) where Pr( ) is the cumulative normal distribution and Φ is 1%. 11 The VaR at time t is obtained from equation (22) by letting A t = A RR = 0: VaR (A, Φ, t) = A R ϑ Normal. (27) The risk manager who has exposure to a risk factor R t, which changes by discrete increments of R t, needs to know how much capital to put aside to cover at least the fraction 1 Φ of daily losses during a year. For this purpose, the risk manager must first determine a threshold ϑ so that the event ( R t ϑ) has a probability Φ under Pr( ). The standard approach does this by using an explicit

13 13 distribution that is in general the normal distribution. The alternative approach is to use a cumulative probability distribution F(ϑ) based on one of the extreme value and flexible probability distributions then solve for ϑ to obtain the threshold, i.e., ( ) ϑ = F 1 1 Φ. (28) As shown in Bali (2001b), the Box-Cox-GEV distribution yields the following VaR threshold: ξ σ 1 ΦNφ ϑbox Cox GEV = µ + ln 1 1 (29) ξ φ n where n and N are the number of extremes and the number of total data points, respectively. Once the location (µ), scale (σ), shape (ξ), and φ parameters of the Box-Cox-GEV distribution are estimated one can find the VaR threshold, ϑ Box-Cox-GEV, based on the choice of confidence level (Φ). With the GPD (φ = 0) and GEV (φ = 1) distributions, the VaR threshold in equation (29) reduces to: ξ σ ΦN ϑgpd = µ + 1 (30) ξ n ξ σ ΦN ϑgev = µ + ln 1 1. (31) ξ n As will be discussed in the paper, there is substantial empirical evidence that the distribution of stock returns is typically skewed to the left and leptokurtic, that is, the unconditional return distribution shows high peaks, fat tails, and more outliers on the left tail. This implies that extreme events are much more likely to occur in practice than would be predicted by the thin-tailed normal distribution. This also suggests that the normality assumption can produce VaR numbers that are inappropriate measures of the true risk faced by financial institutions. In order to overcome the drawbacks of the normal distribution, we use the flexible distributions that take into account the

14 14 non-normality of returns, and deals with events that are relatively infrequent. The VaR threshold is computed using ϑ = µ t + aσ t, (32) flexible where a is the cut-off for the standardized cdf associated with probability 1 Φ, i.e., F(a)=1 Φ, and µ and σ are the mean and standard deviation parameters of the corresponding flexible distribution. IV. Data The data set consists of daily (percentage) log-returns (log-price changes) for the Dow Jones Industrial Average (DJIA) and S&P500 composite indices. The time period investigation for the DJIA is May 26, 1896 to December 29, 2000 (28,758 observations) and for the S&P500 is January 4, 1950 to December 29, 2000 (12,832 observations). 12 Table 1 shows that the unconditional mean of daily log-returns for the DJIA and S&P500 are % and % with a standard deviation of 1.09% and 0.87%, respectively. The maximum and minimum values are 14.27% and 27.96% for the DJIA, and 8.71% and 22.90% for the S&P500. The table also reports the skewness and excess kurtosis statistics for testing the distributional assumption of normality. The skewness statistics for daily returns are negative and statistically significant at the 1% level. The excess kurtosis statistics are considerably high and significant at the 1% level, implying that the distribution of equity returns has much ticker tails than the normal distribution. The fat-tail property is more dominant than skewness in the sample. The maximal and minimal returns are obtained from the original daily data described above. Following the extreme value theory, we define the extremes as excesses over high thresholds [see Embrechts et al. (1997, pp )]. Specifically, the extreme changes are defined as the 5 percent of the right and left tails of the empirical distribution. Panel B of Table 1 shows the means, standard

15 15 deviations, maximum and minimum values of the extremes. In addition to the 5% tails, the extremes are obtained from the 2.5% and 10% tails of the empirical distribution. The qualitative results are found to be robust across different threshold levels. To save space we choose not to present the empirical findings based on the 2.5% and 10% tails. 13 V.A Empirical Results for the Extreme Value Distributions Table 2 presents the regression method estimates of the Box-Cox-GEV, generalized Pareto, and generalized extreme value distributions. The empirical results are clear-cut and allow one to determine unambiguously the type of extreme value distribution: for both the largest falls and rises of equity returns, the asymptotic distribution belongs to the domain of attraction of the Box-Cox- GEV distribution. A likelihood ratio (LR) test between the GEV and Box-Cox-GEV distributions leads to a firm rejection of the GEV distribution for the DJIA and S&P500 stock indices. 14 The LR test between the GPD and Box-Cox-GEV distributions also indicates a firm rejection of the GPD distribution in all cases except for the DJIA maximal returns. 15 In estimating the parameters of the Box-Cox-GEV distribution, a one dimensional grid search method and a nonlinear least square estimation technique are used. 16 Since one of our goals in this section is to determine whether the asymptotic distribution of extremes belongs to the domain of attraction of GEV or GPD, the value of φ is expected to be between zero and one. The parameter φ is estimated by scanning this range in increments of 0.1. When a minimum of the sum of squares is found, greater precision is desired, and the area to the right and left of the current optimum is searched in increments of As shown in Table 2, φ max is estimated to be 0.02 and 0.51 for the DJIA and S&P500, respectively, whereas the corresponding figures for φ min are found to be 0.66 and 0.94 for the minimal returns. The LR test results indicate that both the generalized Pareto

16 16 distribution with φ = 0 and the generalized extreme value distribution with φ = 1 are strongly rejected in favor of the Box-Cox-GEV distribution with 0 < φ < 1. The tail index ξ for the GPD and GEV distributions is found to be positive and statistically different from zero. This implies a rejection of the thin-tailed (ξ = 0) Gumbel and exponential distributions with rapidly decreasing tails against the fat-tailed (ξ > 0) Frechet and Pareto distributions with slowly decreasing tails, and a fortiori a rejection of the short-tailed (ξ < 0) Weibull and uniform distributions. 17,18 The asymptotic t-statistics of the estimated shape parameters (ξ) clearly indicate the non-normality of extremes. Another notable point in Table 2 is that the estimated shape parameters for the minimal returns (ξ min ) turn out to be greater than those for the maximal returns (ξ max ). More specifically, the estimated ξ max values are in the range of 0.26 to 0.42 for the DJIA and 0.17 to 0.33 for the S&P500, while for the minimal returns the estimates of ξ min vary from 0.28 to 0.45 for the DJIA and from 0.39 to 0.46 for the S&P500. Since the higher ξ the fatter the distribution of extremes, the minimal returns have thicker tails than the maximal returns. A comparison of the estimated scale parameters (σ) of the Box-Cox-GEV, GPD, and GEV distributions indicates that both σ max and σ min are overestimated by the generalized Pareto and underestimated by the generalized extreme value distribution. For the DJIA (Panel A of Table 2), the scale parameters (σ max, σ min ) are found to be (0.656%, 0.625%) for the Box-Cox-GEV, (0.657%, 0.727%) for the GPD, and (0.409%, 0.489%) for the GEV distribution. The qualitative results turn out to be the same for the S&P500: σ max and σ min are 0.440% and 0.303% for the Box-Cox-GEV, 0.503% and 0.383% for the GPD, and 0.316% and 0.283% for the GEV distribution. Since the volatility of extremes depends on the scale (σ) and shape (ξ) parameters of the asymptotic extremal

17 17 distributions, there is no clear evidence whether the maximal returns are more volatile than the minimal returns or vice versa. V.B Empirical Results for the Flexible Distributions Table 3 presents the estimated parameters of the skewed generalized t (SGT), skewed generalized error (SGED), inverse hyperbolic sign (IHS), exponential generalized beta of the second kind (EGB2), and normal distributions for the DJIA and the S&P500 log-returns. The estimates are obtained using the maximum likelihood method and the iterative algorithm described in Theodossiou (1998). The first two columns of the table present the estimates for the mean and standard deviation of log-returns for the DJIA and S&P500. As expected, these estimates are quite similar across distributions and do not differ much from the simple arithmetic means and standard deviations of log-returns presented in Table 1. The third and fifth column present the estimates for the kurtosis parameters k and n (SGT only). In the case of SGT, the values of k and n are, respectively, 1.76 and 3.39 for the DJIA and 1.60 and 5.21 for the S&P500. In both cases, the values are quite different from those of the normal distribution of k = 2 and n =. Both pairs of values indicate that the DJIA and S&P500 log-returns are characterized by excess kurtosis. Note that in the SGT model, the parameter k controls mainly the peakness of the distribution around the mode while the parameter n controls mainly the tails of the distribution, i.e., adjusting the tails to the extreme values. The parameter n has the degrees of freedom interpretation as in the Student-t distribution. The fourth column presents the skewness parameter λ which is negative and statistically significant for both DJIA and S&P500 log-returns.

18 18 Column 6 and 7 present the standardized measures for skewness S k = E(y-µ) 3 /σ 3 and kurtosis K u = E(y-µ) 4 /σ 4 based on the parameter estimates for k, n, and λ. 19 The standardized skewness values, S k, for both series are negative indicating that the distributions of log-returns are skewed to the left. The standardized kurtosis for the DJIA is not defined because the parameter n = 3.39 < 4. Note that the moments of the SGT exist up to the value of n. In the case of the S&P500, the standardized kurtosis is The above results provide strong support to the hypothesis that both the DJIA and S&P500 log-returns are not normal. The normality hypothesis is also rejected by the LR statistics for testing the null hypothesis of normality against that of SGT. Note that the LR statistics, presented in column 9, are quite large and statistically significant at the 1% percent level. To test the overall fit of the SGT, we also use the Kolmogorov-Smirnov statistic (KS). The KS statistics, presented in the last column of Table 3, are small and statistically insignificant at both the 1% and 5% levels providing support to the null hypothesis that the data are SGT and IHS distributed. The SGED estimates for the kurtosis parameter k are close to one (k = for the DJIA and for the S&P500) and they are considerably lower than those of SGT. This is because the SGED s kurtosis is controlled by parameter k only, thus, to account for the excess kurtosis in the data the parameter k has to be smaller than that of SGT. 20 The skewness parameter λ is negative and statistically significant for both the DJIA and the S&P500 data. The standardized skewness and kurtosis parameters are smaller than those of SGT. SGED s nesting property allows us to test the null hypothesis that the data follow the SGED against the alternative hypothesis that they follow the SGT. The log-likelihood ratios for testing the latter hypothesis are for the DJIA and for the S&P500. These ratios, which follow chi-square distribution with one degree of freedom, are large and statistically significant at the 1% level, thus suggesting that the SGT provides a better fit

19 19 than the SGED. Moreover, the KS statistic rejects the null hypothesis that the DJIA and S&P500 log-returns follow the SGED. The superiority of the SGT over the SGED can be attributed to the fact that it provides a better fit to the tails of the distribution because of the parameter n. The IHS and EGB2 distributions are not linked directly to each other or the SGT, but the EGB2 is linked with the normal distribution. Specifically, as the parameters p and q approach infinity the EGB2 converges to the normal distribution. The possible values for kurtosis are limited to the range [3, 9] for EGB2; see Hansen et al. (2001). Like in the case of SGT and SGED, the results for IHS and EGB2 indicate that the DJIA and S&P500 log-returns exhibit skewness and significant excess kurtosis. The LR statistic for testing the null hypothesis of normal distribution against the alternative hypothesis of EGB2 rejects normality. The LR ratio for IHS does not exist because the normal distribution is not nested within IHS. The KS statistics indicate that the hypothesis that the data follow IHS cannot be rejected but the EGB2 hypothesis is rejected. The latter may be attributed to the inability of EGB2 to model kurtosis values outside the range 3 to 9. VI. VaR Calculations with Alternative Distribution Functions Table 4 presents the estimated thresholds for the extreme value, normal, and flexible distributions. The DJIA results show that the extreme tails yield threshold points, ϑ Box-Cox-GEV, that are up to 40% higher than the normal thresholds, ϑ Normal. Moreover, the ϑ Box-Cox-GEV, ϑ GPD, and ϑ GEV thresholds for the extreme negative increments are much greater than those for the extreme positive increments. The multiplication factors (ϑ Box-Cox-GEV /ϑ Normal ) for the extreme tails (Φ=0.5% and 1%) of the DJIA are in the range of 1.22 to 1.40 for the minimal returns and 1.12 to 1.28 for the maximal returns. The VaR thresholds for the Box-Cox-GEV, GPD, and GEV distributions indicate that the

20 20 two tails are asymmetric. The multiplication factors for the S&P500 extreme tails (Φ=0.5% and 1%) are in the range of 1.13 to 1.23 for the maxima and 1.09 to 1.22 for the minima. 21 The VaR measures for the flexible distributions are similar to those of the extreme value distributions. Specifically, the average ratio of VaR thresholds for the maximal (minimal) DJIA returns are ϑ Box-Cox-GEV /ϑ SGT = (1.0207), ϑ Box-Cox-GEV /ϑ SGED = (1.0028), ϑ Box-Cox- GEV/ϑ EGB2 = (1.0497), and ϑ Box-Cox-GEV /ϑ IHS = (1.0096). The corresponding figures for the S&P500 are ϑ Box-Cox-GEV /ϑ SGT = (0.9803), ϑ Box-Cox-GEV /ϑ SGED = (0.9609), ϑ Box-Cox- GEV/ϑ EGB2 = (0.9753), and ϑ Box-Cox-GEV /ϑ IHS = (0.9679). These findings imply that the tails of the empirical distribution approximated by the flexible distributions are similar to those of the extreme value distributions. The above results indicate that the tail areas obtained from the extreme value and flexible distributions are quite different and potentially more useful than those of the normal distribution (i.e., standard approach). Table 5 presents the risk management performance statistics for the extreme value, normal, and flexible distributions. The results show that the normal VaR thresholds for both the DJIA and S&P500 at the various tails are quite inadequate. Given that the DJIA data includes 28,758 daily returns, one would expect 144, 288, 431, 575, 719, and 1,438 returns to fall respectively into the 0.5%, 1%, 1.5%, 2%, 2.5%, and 5% negative and positive tails. The number of returns for the normal VaR thresholds falling into the negative (positive) 0.5% tail are 377 (295), 1% tail are 480 (390), 1.5% tail are 585 (479), 2% tail are 672 (552), 2.5% tail are 747 (616), and 5% tail are 1,100 (975). Based on these results, the normal distribution underestimates the actual VaR thresholds at the 0.5%, 1%, and 1.5% tails and overestimates the VaR thresholds for most of the remaining tails. The results for the S&P500 (see Panel B of Table 5) are quite similar to those of

21 21 the DJIA. The normal VaR estimates have a mean absolute percentage error (MA%E) of 42.53% for the DJIA and 34.53% for the S&P500. The Box-Cox-GEV distribution has the best overall performance, although the GPD, GEV, SGT, and IHS appear to be very similar in performance to the Box-Cox-GEV in both the DJIA and S&P500 samples. All five distributions estimate the actual VaR thresholds very well. Their mean absolute percentage errors in the DJIA samples are 2.04% for the Box-Cox-GEV distribution, 2.22% for the GPD, 5.14% for the GEV, 4.92% for the SGT, and 3.81% for the IHS. In the S&P500 sample, these errors are 3.12% for the Box-Cox-GEV distribution, 4.95% for the GPD, 4.29% for the GEV, 4.26% for the SGT, and 6.59% for the IHS. The remaining two flexible distributions, SGED and EGB2, perform better than the normal distribution, however, their MA%E are quite larger than those of the other distributions. The results indicate that the extreme value and flexible distributions are superior to the normal distribution in calculating value at risk. Given the obvious importance of VaR estimates to financial institutions and their regulators, evaluating the accuracy of the distribution functions underlying them is a necessary exercise. The evaluation of VaR estimates is based on hypothesis testing using the binomial distribution. This latter test is currently embodied in the MRA. 22 Under the hypothesis-testing method, the null hypothesis is that the VaR estimates exhibit the property characteristics of accurate VaR measures. If the null hypothesis is rejected, the VaR estimates do not exhibit the specified property, and the underlying distribution function can be said to be inaccurate. Otherwise, the VaR model is acceptably accurate. Under the MRA, banks will report their VaR estimates to their regulators, who observe when actual portfolio losses exceed these estimates. As discussed by Kupiec (1995), assuming that the VaR measures are accurate, such exceptions can be modeled as independent draws form a

22 22 binomial distribution with a probability of occurrence equal to (say) 1 percent. Accurate VaR estimates should exhibit the property that their unconditional coverage κ = q/n equals 1 percent, where q is the number of exceptions in N trading days. Since the probability of observing q exceptions in a sample of size N under the null hypothesis is N = 99 q q N q Pr( ) , (33) q the appropriate likelihood ratio statistic for testing whether κ = 0.01 is q N q q N q LR uc = 2[ln( κ ( 1 κ) ) ln( )]. (34) Note that the LR uc unconditional coverage test is uniformly the most powerful for a given sample size and has an asymptotic χ 2 (1) distribution. 23 Table 6 presents the likelihood ratio test results from testing the null hypothesis that the reported VaR estimates are acceptably accurate. According to the LR statistics for the binomial method, the standard approach that assumes normality of asset returns is strongly rejected for the 0.5%, 1%, 1.5%, 2.5%, and 5% VaRs for both the maximal and minimal returns. The extreme value distributions (Box-Cox-GEV, GPD, GEV) produce acceptably accurate VaR estimates for all tails and for all data sets considered in the paper. As shown in Panel B of Table 6, the SGT and IHS distributions cannot be rejected for all VaR tails and for both the maximal and minimal returns on S&P 500. In general, the flexible distributions produce acceptably accurate VaR measures except for few cases for the minimal returns on S&P 500: the 2% VaR estimates of EGB2, and the 2%, 2.5%, and 5% VaR estimates of SGED are found to be inaccurate at the 5% level of significance. The relative performance of the flexible distributions turns out to be slightly lower for the Dow Jones Industrial Average. Panel A of Table 6 indicates that the VaR estimates of SGED and EGB2 be inaccurate for the 0.5%, 1%, 2.5%, and 5% VaR tails. However, the VaR measures obtained

23 23 from the SGT and IHS distributions are found to be very precise for the Dow 30 index, except for the 0.5% tail of the minimal returns. As discussed by Christoffersen (1998), VaR estimates can be viewed as interval forecasts of the lower tail of the return distribution. Interval forecasts can be evaluated conditionally and unconditionally, that is, with or without reference to the information available at each point in time. The LR uc given in equation (34) is an unconditional test statistic because it simply counts exceedences (or violations) over the entire period. However, in the presence of volatility clustering or volatility persistence, the conditional accuracy of VaR estimates becomes an important issue. The VaR models that ignore mean-volatility dynamics may have correct unconditional coverage, but at any given time, they will have incorrect conditional coverage. In such cases, the LR uc test is of limited use since it will classify inaccurate VaR estimates as acceptably accurate. Moreover, as indicated by Kupiec (1995), Christoffersen (1998), and Berkowitz (2001), the unconditional coverage tests have low power against alternative hypotheses if the sample size is small. This problem does not exist here since our daily data sets cover a long period of time (28,758 observations for DJIA and 12,832 observations for S&P500). The conditional coverage test developed by Christoffersen (1998) determines whether the VaR estimates exhibit both correct unconditional coverage and serial independence. In other words, if a VaR model produces acceptably accurate conditional thresholds then not only the exceedences implied by the model occur x % (say 1 %) of the time, but they are also independent and identically distributed over time. Given a set of VaR estimates, the indicator variable is constructed as 1 if exceedence occurs I t = 0 if no exceedence occurs (35)

24 24 and should follow an iid Bernoulli sequence with the targeted exceedence rate (say 1 %). The LR cc test is a joint test of two properties: correct unconditional coverage and serial independence, LR cc = LR uc + LR ind, (36) which is asymptotically distributed as the Chi-squared with two degrees of freedom, χ 2 (2). The LR ind is the likelihood ratio test statistic for the null hypothesis of serial independence against the alternative of first-order Markov dependence. We calculated the LR ind and LR cc statistics for all the distribution functions considered in the paper. 24 We also computed the first-order serial correlation coefficient corr(i t,i t-1 ), a diagnostic suggested by Christoffersen and Diebold (2000), and test its statistical significance. The LR cc and corr(i t,i t-1 ) statistics indicate a strong rejection of the null hypothesis, implying that given an exceedence (or violation) on one day there is a high probability of a violation the next day. VII. A Conditional VaR Approach with Flexible Distributions These findings in the previous section suggest that the actual thresholds are time-varying to a degree not captured by the unconditional VaR models. The earlier results are based on a sound statistical theory, but do not yield VaR measures reflecting the current volatility background. In light of the fact that conditional heteroskedasticity and serial correlation are present in most financial time series, the unconditional VaR estimates cannot provide an accurate characterization of the actual thresholds. 25 We extend the unconditional VaR approach by taking into account the dynamic behavior of financial return volatility in extreme values. In order to improve the existing VaR methods we use the absolute GARCH (ABS-GARCH) process of Taylor (1986) and Schwert (1989) that takes into account time-varying volatility characterized by persistence, considers the conditional non-normality of returns, and deals with extreme events.

25 25 Following the introduction of ARCH models by Engle (1982) and their generalization by Bollerslev (1986), there have been numerous refinements of this approach to modeling conditional variance. The symmetric and asymmetric GARCH models, that parameterize the current conditional variance as a function of the last period s squared shocks and the last period s variance, produce dramatic increases in variance for very large shocks. The exponential GARCH (EGARCH) model of Nelson (1991), that defines the current conditional variance as an exponential function of the lagged shocks, lagged absolute shocks, and lagged log-variance, also overestimates the actual variance during highly volatile periods. Friedman and Laibson (1989) argue that large shocks constitute extraordinary events and propose to truncate their influence on the conditional variance. Taylor (1986) and Schwert (1989) suggest a less drastic approach. They propose an ARCH model that specifies the conditional standard deviation as a moving average of lagged absolute residuals. Since the GARCH and EGARCH models may overestimate the actual VaR thresholds, we choose to use the ABS-GARCH process to be able to estimate extreme return volatility more accurately. The following AR(1) ABS-GARCH(1,1) conditional mean-volatility specification is utilized with the generalized error distribution to estimate time-varying conditional VaR thresholds: R t = + α1rt 1 α 0 + z σ (37) t t σ t t 1 = β0 + β1 zt 1 σ t 1 + β2σ t 1 (38) v v exp[( 1/ 2) zt / Π f v ( zt Ω t 1) = (39) [( v+ 1) / v] Π2 Γ(1/ v) where 2 Π = Γ(1/ v) Γ(3/ v) ( 2 / v) 1/ 2, z t = R µ t σ t t 1 t t 1 is drawn from the conditional GED density f v (z t, Ω t-1 ), and can be viewed as an unexpected shock to the stock market, and v is the degrees of freedom or

26 26 tail-thickness parameter. 26 µ t t-1 = α 0 + α 1 R t-1 is the conditional mean, σ tt 1 is the conditional standard deviation, and Ω t-1 is the information set at time t The conditional value-at-risk measure with coverage probability, Ф, at time t is defined as the conditional quintile,ψ t t-1 (Ф), where Pr(R t Ψ t t-1 (Ф) Ω t-1 ) = Ф. (40) The conditionality of the VaR measure is crucial because of the dynamic behavior of asset returns. When the innovations are assumed to be Gaussian, the conditional VaR threshold is Ψ t t-1 (Ф) = µ t t-1 + ρσ t t-1 (41) where µ t t-1 and σ t t-1 are the estimated conditional mean and volatility of daily log-returns R t, and ρ is the critical value for the normal distribution, e.g., ρ = for Ф = 1%. When estimating the conditional VaR thresholds for the flexible distributions, ρ is calculated by integrating the area under the unconditional probability density function of standardized returns, i.e., ( ) R µ σ t tt 1 tt 1. An implicit assumption in the context of risk management is that the return series standardized by the conditional mean and conditional standardized deviation volatility are i.i.d. We should note that in our case the transformation of conditional volatility is based on the ABS- GARCH process given in equation (38). If (R t µ t t-1 )/σ t t-1 is not i.i.d for any transformation of volatility, then the conditional volatility alone is not sufficient for characterization of conditional VaR threshold. As discussed in Christoffersen, Hahn, and Inoue (2001), if (R t µ t t-1 )/σ t t-1 is i.i.d then the conditional quantile is some linear function of volatility, where the relevant coefficients of such a linear function [e.g., ρ in eq. (41)] is determined by the common distribution of the standardized returns.

27 27 We compare the risk management performance of alternative distribution functions for the standardized residuals. The estimated and actual counts as well as the likelihood ratio test statistics are presented in Table The relative performance of distribution functions is not affected when conditional heteroskedasticity and serial correlation are eliminated from the original data. The extreme value and flexible distributions outperform the normal distribution in estimating value at risk and provide the best predictions of catastrophic market risks for all quantiles considered in the paper. The SGT and IHS distributions produce very similar VaR thresholds for the standardized returns and perform better than the SGED and EGB2 distributions. The results provide strong evidence that it is not true as stated by Longin (2000) and Bali (2001a) that one is obliged to use the distribution of extremes only to be able to obtain precise VaR measures. The same accuracy level can be obtained from the distribution of all returns if the flexible distributions, which take into account skewness, leptokurtosis and volatility clustering in the financial data, are used in estimating conditional VaR thresholds. VIII. Conclusions This paper provides strong evidence that the skewed fat-tailed distributions perform as well as the extreme value distributions in modeling the asymptotic behavior of stock market returns. They yield a more precise and robust approach to risk management and value-at-risk calculations than the symmetric thin-tailed normal distribution. A notable point is that the SGT and IHS distributions produce very similar VaR thresholds, and perform better than the SGED and EGB2 distributions in approximating the extreme tails of the return distribution. It is also important to note that the Box-Cox-GEV distribution gives a more accurate characterization of the actual data than the GPD and GEV distributions because the truth is somewhere in between.

28 28 The statistical results indicate that given the conditional heteroskedasticity and serial correlation of most financial data, using an unconditional density function is a major drawback of any kind of VaR-estimator. Therefore, the unconditional VaR approach is extended to take into account the dynamic behavior of the mean and volatility of equity returns in extreme values. A conditional VaR approach is proposed based on the standardized residuals modeled with an autoregressive absolute GARCH process. The conditional coverage test results indicate that the actual thresholds are time varying to a degree not captured by the conditional normal distribution. The conditional skewed fat-tailed distributions, however, perform considerably well in modeling the time-varying conditional VaR thresholds. Based on the likelihood ratio tests, the normal distribution is strongly rejected in favor of the extreme value and skewed fat-tailed distributions.

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