A Generalized Extreme Value Approach to Financial Risk Measurement

Transcription

1 TURAN G. BALI A Generalized Extreme Value Approach to Financial Risk Measurement This paper develops an unconditional and conditional extreme value approach to calculating value at risk (VaR), and shows that the maximum likely loss of financial institutions can be more accurately estimated using the statistical theory of extremes. The new approach is based on the distribution of extreme returns instead of the distribution of all returns and provides good predictions of catastrophic market risks. Both the in-sample and out-of-sample performance results indicate that the Box Cox generalized extreme value distribution introduced in the paper performs surprisingly well in capturing both the rate of occurrence and the extent of extreme events in financial markets. The new approach yields more precise VaR estimates than the normal and skewed t distributions. JEL codes: G12, C13, C22 Keywords: financial risk management, value at risk, extreme value theory, skewed fat-tailed distributions. RAPID GLOBALIZATION of financial and product markets, innovations in the design of derivative securities, and examples of spectacular losses associated with derivatives over the past decade have made financial institutions recognize the growing importance of risk management. Extraordinary events such as stock market crashes, bond market collapses, and foreign exchange crises are major concerns in risk management and financial regulation. Regulators are concerned with I thank the editor (Deborah Lucas) and an anonymous referee for their extremely helpful comments and suggestions. I also thank Geoffrey Booth, Peter Christoffersen, Haim Levy, Salih Neftci, Robert Schwartz, and Panayiotis Theodossiou for their useful comments and suggestions on earlier version of this article. I benefited from discussions with Linda Allen, Ozgur Demirtas, Armen Hovakimian, John Merrick, Lin Peng, and Liuren Wu on certain theoretical and empirical points. An earlier version of this paper was presented at the 2004 Econometric Society Meeting, Baruch College and the Graduate School and University Center of the City University of New York. I gratefully acknowledge the financial support from the Eugene Lang Research Foundation of the Zicklin School of Business, Baruch College, and the PSC-CUNY Research Foundation of the City University of New York. TURAN G. BALI is Professor of Finance, Department of Economics & Finance, Baruch College, Zicklin School of Business, City University of New York, 17 Lexington Avenue, Box , New York, NY ( Turan Bali@baruch.cuny.edu). Received May 6, 2005; and accepted in revised form November 6, Journal of Money, Credit and Banking, Vol. 39, No. 7 (October 2007) C 2007 The Ohio State University

2 1614 : MONEY, CREDIT AND BANKING the protection of the financial system against catastrophic events, which can be a source of systematic risk. In recent years, a central issue in risk management has been to determine capital requirement for financial institutions to meet catastrophic market risk. This increased focus on risk management has led to the development of various methods and tools to measure the risks financial institutions face. A primary tool for financial risk assessment is value at risk (VaR), which is defined as the maximum loss expected on a portfolio of assets over a certain holding period at a given confidence level (probability). The use of VaR techniques in risk management has exploded over the last few years. Financial institutions now routinely use VaR techniques in managing their trading risk, and non-financial firms have started adopting the technology for their risk-management purposes as well. Many implementations of VaR assume that asset returns are normally distributed. This assumption simplifies the computation of VaR considerably. However, it is inconsistent with the empirical evidence on asset returns, which suggests that the distribution of asset returns is skewed and fat-tailed. This implies that extreme events are much more likely to occur in practice than would be predicted by the symmetric thinner-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. Under these conditions, we think that an alternative approach that approximates the tail areas asymptotically may be more appropriate than imposing an explicit functional form like the normal or lognormal on the distribution. An important contribution of this paper is to develop a new approach to calculating VaR. The method introduced here is based on extreme value theory (EVT). The major challenge in implementing VaR analysis is the specification of a probability distribution for market variables used in the calculation of the VaR estimate. Focusing on extreme changes in risk factors, rather than the original time series generally, allows avoidance of the difficult choice of a probability distribution for risk factors. The traditional VaR models estimate the maximum likely loss of an institution under normal market conditions, which corresponds to the normal functioning of financial markets during ordinary periods. Therefore, the standard VaR measures provide inaccurate estimates of actual losses during highly volatile periods corresponding to financial crises. The new approach is based on the distribution of extreme returns instead of the distribution of all returns, and hence provides good predictions of catastrophic market risks during extraordinary periods. This paper shows that the statistical theory of extremes improves the traditional VaR models that assume normality because the new approach puts more emphasis on the tail areas of the observed frequency distributions and hence yields a more precise estimate for the thresholds in question. In order to overcome the drawbacks of the normal distribution, the skewed t distribution of Hansen (1994) that takes into account skewness and fat-tailedness of a conditional return distribution is used. Both the in-sample and out-of-sample performance results indicate that the extreme value distributions perform much better than the skewed t and normal distributions in capturing both the rate of occurrence and the (average) extent of extreme events

3 TURAN G. BALI : 1615 in financial markets. In fact, the statistical theory of extremes appears to be a more natural and robust approach to VaR calculations. The paper extends the unconditional extreme value approaches developed by Longin (2000), McNeil and Frey (2000), and Bali (2003). Their methods are based on a sound statistical theory and offer a parametric form for the tail of a distribution, but they do not yield VaR measures that reflect the current mean-volatility background. Given the serial correlation and conditional heteroskedasticity of most financial data, we believe using an unconditional density function to be a major drawback of any kind of VaR-estimator. The paper evaluates the empirical performance of alternative distribution functions based on the unconditional and conditional coverage tests introduced by Kupiec (1995) and Christoffersen (1998), respectively. The VaR measures based on the unconditional extreme value, skewed t, and normal distributions do not account for systematic time-varying changes in the distribution of financial returns. The conditional coverage test results indicate that the actual VaR thresholds are time varying to a degree not captured by the unconditional density functions. The paper proposes a conditional extreme value approach to VaR by taking into account the dynamic behavior of the maximal and minimal returns. In addition to developing a generalized unconditional/conditional EVT approach to estimating VaR, this paper proposes an economic framework in which the loss-averse investors allocate financial assets based on the mean-var efficient frontier. There is a long literature about safety-first investors who minimize the chance of disaster (or the probability of failure). The portfolio choice of a safety-first investor is to maximize expected return subject to a downside risk constraint. Roy s (1952), Baumol s (1963), and Arzac and Bawa s (1977) safety-first investor uses a downside risk measure that is a function of VaR. Roy indicates that most investors are principally concerned with avoiding a possible disaster and that the principle of safety plays a crucial role in the decision-making process. Hence, we assume that the idea of a disaster exists and a risk averse safety-first investor will seek to reduce as far as is possible the chance of such a catastrophe occurring. In our framework, one can obtain a modified capital asset pricing model (CAPM) relation, where VaR of a portfolio is positively related to the portfolio s expected return. Our results indicate that VaR is not only a primary tool for risk management, but it can be useful for optimal asset allocation as well. The paper is organized as follows. Section 1 provides an economic framework for loss-averse investors. Section 2 presents the extreme value distributions. Section 3 introduces an unconditional extreme value approach to calculating VaR. Section 4 describes the data and provides the estimation results. Section 5 evaluates the insample performance of unconditional VaR models. Section 6 proposes a conditional extreme value approach to VaR, and compares its out-of-sample performance with the skewed t and normal distributions. Section 7 concludes the paper. 1. ECONOMIC FRAMEWORK The mean-variance theory of portfolio choice determines the optimum asset mix by maximizing (i) the expected risk premium per unit of risk in a mean-variance

4 1616 : MONEY, CREDIT AND BANKING framework or (ii) the expected value of a utility function approximated by the expected return and variance of the portfolio. In both cases, market risk of the portfolio is defined in terms of the variance (or standard deviation) of the portfolio s returns. Modeling portfolio risk with the traditional volatility measures assumes that investors are concerned only about the average variation (and covariation) of individual stock returns. This paper assumes a loss-averse investor who solves the optimal portfolio selection problem in a VaR framework. The loss-averse investor uses a mean-var approach to allocate financial assets by maximizing the expected value of a utility function approximated by the expected return and VaR of the portfolio. The focus on VaR as the appropriate measure of portfolio risk allows investors to treat losses and gains asymmetrically. We consider an investor who allocates her portfolio in order to maximize the expected utility of end-of-period wealth U(W). We assume that the distribution of returns on the investor s portfolio of risky assets is non-symmetrical and fat-tailed. The expected value of end-of-period wealth can be written as W = n i=1 w i R i + w f r f, where R i is unity plus the expected rate of return on the ith risky asset, r f is unity plus the rate of return on the riskless asset, w i is the fraction of wealth allocated to the ith risky asset, and w f is the fraction of wealth allocated to the riskless asset. Since our objective is to measure the effect of higher moments on the standard asset pricing models, we now approximate the expected utility by a Taylor series expansion around the expected wealth. For this purpose, the utility function is expressed in terms of the wealth distribution, so that E[U(W )] = U(W ) f (W ) dw, (1) where f (W) is the probability density function of the end-of-period wealth that depends on the multivariate distribution of returns and on the vector of weights w. Hence, the infinite-order Taylor series expansion of the utility function is U(W ) = U (k) ( W )(W W ) k, (2) k! k=0 where W = E(W ) denotes the expected end-of-period wealth. Under rather mild conditions (see Loistl 1976), the expected utility is given by: [ ] U (k) ( W )(W W ) k U (k) ( W ) E[U(W )] = E = E[(W W ) k ]. (3) k! k! k=0 Therefore, the expected utility depends on all central moments of the distribution of the end-of-period wealth. It should be noticed that the approximation of the expected utility by a Taylor series expansion is related to the investor s preference (or aversion) toward all moments of k=0

5 TURAN G. BALI : 1617 the distribution that are directly given by derivatives of the utility function. Scott and Horvath (1980) indicate that, under the assumptions of positive marginal utility, decreasing absolute risk aversion at all wealth levels together with strict consistency for moment preferences, one has 1 U (k) (W ) > 0 W if k is odd, U (k) (W ) < 0 W if k is even. Focusing on terms up to the fourth one, we obtain E[U(W )] = U( W ) + U (1) ( W )E[(W W )] U (2) ( W )E[(W W ) 2 ] + 1 3! U (3) ( W )E[(W W ) 3 ] + 1 4! U (4) ( W )E[(W W ) 4 ] + O(W 4 ), (4) where O(W 4 ) is the Taylor remainder. We define the expected return, variance, skewness, and kurtosis of the end-of-period return, R p,as 2 μ p = E[R p ] = W, σ 2 p = E[ (R p μ p ) 2] = E[(W W ) 2 ], s 3 p = E[ (R p μ p ) 3] = E[(W W ) 3 ], κ 4 p = E[ (R p μ p ) 4] = E[(W W ) 4 ]. Hence, the expected utility is simply approximated by the following preference function: E[U(W )] U( W ) U (2) ( W )σ 2 p + 1 3! U (3) ( W )s 3 p + 1 4! U (4) ( W )κ 4 p. (5) Under conditions established by Scott and Horvath (1980), the expected utility depends positively on the mean and skewness of R p, and negatively on the variance E[U(W )] E[U(W )] E[U(W )] E[U(W )] and kurtosis of R p, i.e., μ p > 0, < 0, > 0, and < 0. σp 2 s 3 p κ 4 p This indicates aversion to variance and kurtosis and preference for (positive) skewness. Based on the standard utility functions (such as CARA and CRRA), one can show E[U(W )] that an increase in VaR reduces the expected utility of wealth, VaR p < 0 because VaR increases with variance and kurtosis and decreases with positive skewness. 1. Further discussion on the conditions that yield such moment preferences or aversion can be found in Pratt and Zeckhauser (1987), Kimball (1993), and Dittmar (2002). 2. These definitions of skewness and kurtosis, as central higher moments, differ from the statistical definitions as standardized central higher moments E[((r p μ p )/σ p ) j ] for j = 3, 4.

6 1618 : MONEY, CREDIT AND BANKING These results indicate that the increase in the maximum likely loss reduces the expected utility of a loss-averse investor who takes into account higher-order moments of the return distribution. Hence, the investor uses a VaR-efficient portfolio to solve the following constrained optimization problem: max E t (R p,t+ ) w s.t.var t (w,α, ) = VaR b, (6) where the VaR-efficient allocation depends on the loss probability level α, the benchmark VaR level VaR b, limiting the authorized risk, the length of investment horizon, and the initial budget w allocated at time t among n financial assets. The VaR-efficient allocation w solves the first-order conditions: E t (R i,t+ ) = λ VaR t ( w w i,t,α, ) i,t ( VaR t w i,t,α, ) (7) = VaR b, where i = 1, 2,..., n, and λ is a Lagrange multiplier. By changing the VaR b level in equation (6), the expected return of the portfolio is maximized subject to a VaR constraint, which yields the efficient portfolio frontier that consists of maximum expected return for a given VaR. We now assume that based on the mean-var efficient frontier, the optimal risky portfolio is determined and a given budget is allocated between the optimal risky portfolio and a risk-free asset. Then let θ be the proportion between the optimal risky portfolio and the risk-free asset. That is, R p,t+ = θ n wi R i,t+ + (1 θ)r (8) i=1 is a portfolio that leads to expected return E t (R p,t+ ) for this θ, r is the risk-free interest rate, and w i is the proportion of security i in the optimal risky portfolio. The portfolio s expected return and VaR are given by: E t (R p,t+ ) = θ n wi E t(r i,t+ ) + (1 θ)r, (9) i=1 VaR p,t = θvar p,t (w,α, ) (1 θ)r. (10) Note that VaR p,t is the maximum loss the portfolio p is expected to suffer with probability α. We define the VaR of the risk-free asset to be equal to r as r > 0is the expected gain. In other words, the risk-free asset will suffer a loss of r. Hence, in equation (10) the risk-free rate reduces the portfolio s VaR with a proportional amount of (1 θ)r.

7 TURAN G. BALI : 1619 Substituting θ from equation (10) into equation (9) yields the relationship between the portfolio s expected return, E t (R p,t+ ) and its VaR: ( ) E p,t r E t (R p,t+ ) = C + VaR p,t + r VaR p,t, (11) where E p,t = n i=1 w i E t(r i,t+ ) and VaR p,t = VaR p,t (w, α, ) are the expected return and VaR of the optimal risky portfolio, and C = r( E p,t + VaR p,t VaR p,t + r ) is a constant. Equation (11) indicates that the portfolio s expected return E t (R p,t+ ) is positively related to its VaR denoted by VaR p, t. 3 It describes a theoretical relation between VaR and the expected return. For a long period of time, the standard way to model the risk-return relationship and to measure the market price of risk has been to use the CAPM of Sharpe (1964), Lintner (1965), and Black (1972). Equation (11) can be viewed as a modified CAPM relation, where downside risk of a portfolio (measured by VaR) is positively related to the portfolio s expected return. The results presented in this section provide evidence that VaR is not only a primary tool for financial risk management, but it can be used for optimal portfolio selection as well when investors care about higher moments of returns. Hence, an accurate measurement of VaR has important implications for both risk management and asset allocation. 2. EXTREME VALUE DISTRIBUTIONS Extreme movements in risk factors are measured here by large changes in asset prices denoted by X. Let us call f (x) the probability density function (PDF), and F(x) the cumulative distribution function (CDF) of X, which can take values between l and u. Let X 1, X 2,..., X n be a sequence of asset returns on days 1, 2,..., n. Extremes are defined as the maxima and minima of the n independent and identically distributed random variables X 1, X 2,..., X n. Let M n represent the highest daily changes (the maximum) and m n denote the lowest daily changes (the minimum) in market variables over n trading days: M n = max(x 1, X 2,...,X n ), (12) m n = min(x 1, X 2,...,X n ) = max( X 1, X 2,..., X n ). (13) To find a limit distribution for maxima, the maximum variable M n is transformed such that the limit distribution of the new variable is a non-degenerate one. Following the Fisher Tippett (1928) theorem, the variate M n is reduced with a location parameter, μ n, and a scale parameter, σ n, in such a way that x = (M n μ n )/σ n Hmax d (x). 3. As the expected return of a risky portfolio exceeds r, the slope coefficient in equation (11) is positive.

8 1620 : MONEY, CREDIT AND BANKING Assuming the existence of {μ n, σ n > 0}, Gnedenko (1943) obtains three types of non-degenerated distributions for the standardized maximum: Frechet: H max,ξ (x) = exp( x 1/ξ ), (14) Weibull: H max,ξ (x) = exp( ( x) 1/ξ ), (15) Gumbel: H max,0 (x) = exp[ exp( x)]. (16) Jenkinson (1955) proposes a generalized extreme value (GEV) distribution, which includes the three limit distributions, equations (14) (16), distinguished by Gnedenko (1943): { [ ( )] } M μ 1/ξ ( ) M μ H max,ξ (M; μ, σ) = exp 1 + ξ ; 1+ ξ 0, σ σ (17) 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 (i.e., the weight of the tails), whereas the parameters of scale σ and of location μ represent the dispersion and average of the extremes, respectively. One potential problem with equations (14) (17) is that financial returns are not i.i.d. As explained in Leadbetter, Lindgren, and Rootzen (1983), Resnick (1987), and Castillo (1988), when the data are dependent, it is possible that extreme value distributions cannot be described as in equations (14) (17). Not surprisingly, no general statement can be made without further assumptions on the exact nature of the dependence structure. However, the theory of extremes for the case of dependence, while not completely developed, has identified a number of empirically relevant cases for which inferences based on standard EVT remain valid. For example, this is true if the daily returns are stationary, and follow an MA(q), AR(p), or ARMA(p, q) model. Because these conditions describe our data reasonably well, we prefer to stick to the extreme value distributions described in equations (14) (17) as opposed to making assumptions on the dependence structure that would lead to other forms of extreme value distributions. 4 An alternative approach to determine the type of asymptotic distribution for extremes can be based on the concept of generalized Pareto distribution (GPD) introduced by Pickands (1975). Excesses over high thresholds can be modeled by the 4. To save space, we do not present the asymptotic distribution of sequences of dependent random variables. The full set of details is available from the author upon request. The interested reader may wish to consult Leadbetter, Lindgren, and Rootzen (1983), Resnick (1987), and Castillo (1988) as well.

9 TURAN G. BALI : 1621 GPD. For the standardized maxima, GPD is given by: [ ( )] M μ 1/ξ G max,ξ (M; μ, σ) = ξ. (18) σ Notice that the GPD presented in equation (18) nests the standard Pareto distribution, the uniform distribution on [ 1, 0], and the standard exponential distribution: 5 Pareto: G max,ξ (x) = 1 x 1/ξ for x 1, (19) Uniform: G max,ξ (x) = 1 ( x) 1/ξ for x [ 1, 0], (20) Exponential: G max,0 (x) = 1 exp( x) for x 0. (21) To determine whether the GPD or the GEV distribution yields a more accurate characterization of extreme movements in financial markets, we propose a more general extreme value distribution using the Box Cox (1964) transformation: [ { [ ( )] }] M μ 1/ξ λ F max,ξ (M; μ, σ, λ) = exp 1 + ξ 1 σ + 1. (22) The Box Cox GEV distribution in equation (22) nests the GPD of Pickands (1975) and the GEV distribution of Jenkinson (1955). More specifically, when λ equals one the Box Cox GEV reduces to the GEV distribution given in equation (17), i.e., when λ = 1, F max,ξ (x) H max,ξ (x). When λ equals zero, the Box Cox GEV converges to the GPD presented in equation (18). 6 We examine the parameter estimates of F max,ξ (x) to evaluate which of the two commonly used extreme value distributions is closer to the truth for stock returns and other daily financial time series. The results indicate that Box Cox GEV distribution provides a more accurate characterization of the actual data than the GPD and GEV distributions because the best fit is somewhere in between the two families. Two parametric approaches are commonly used to estimate the extreme value distributions: (i) the maximum likelihood method that yields parameter estimators that are unbiased, asymptotically normal, and of minimum variance, and (ii) the regression method that provides a graphical method for determining the type of asymptotic distribution. In this paper, the maximum likelihood method is used to determine the correct specification of the limit distribution for the maxima and minima. λ 5. In equations (19) (21), the shape parameter, ξ, determines the tail behavior of the distributions. For ξ > 0, the distribution has a polynomially decreasing tail (Pareto). For ξ = 0, the tail decreases exponentially (exponential). For ξ<0, the distribution is short tailed (uniform). 6. When λ = 0, the transformation is by L Hopital s rule: lim λ 0 F max,ξ (M; μ, σ, λ) = G max,ξ (M; μ, σ).

10 1622 : MONEY, CREDIT AND BANKING The Box Cox GEV distribution presented in equation (22) has a density function, ( 1 f max ( ; x) = σ { exp )[ 1 + ξ ( M μ σ [ ( 1 + ξ ( ) )] 1+ξ ξ which yields the following log-likelihood function: ( ))] } M μ 1/ξ λ, (23) ( ) 1 + ξ k ( ( )) Mi μ ln L f = kln σ k ln 1 + ξ ξ σ kλ k i=1 ( 1 + ξ i=1 σ ( )) Mi μ 1/ξ. (24) σ Differentiating the log-likelihood function in equation (24) with respect to μ, σ, ξ, and λ yields the first-order conditions of the maximization problem. Clearly, no explicit solution exists to these non-linear equations, and thus numerical procedures or search algorithms are called for. 3. AN UNCONDITIONAL EXTREME VALUE APPROACH TO ESTIMATING VaR The extreme value distributions will be fitted with daily stock return data. In continuous time diffusion models, (log)-stock price movements on non-dividend-paying stocks are described by the following stochastic differential equation, d ln P t = μ t dt + σ t dw t, (25) where W t is a standard Wiener process with zero mean and variance of dt, and μ t and σ t are the time-varying drift and diffusion parameters of the geometric Brownian motion. In discrete time, equation (25) yields a return process: ln P t+ ln P t = R t = μ t t + σ t z t, (26) where t is the length of time interval in which the discrete time data are recorded and W t = z t is the Wiener process with zero mean and variance of t. The critical step in calculating VaR is the choice of the threshold point that will define what variation in returns R t is to be considered extreme. We let this threshold be denoted by, with α the probability that a R t exceeding the threshold will occur.

11 TURAN G. BALI : 1623 The threshold is defined by: ( P(R t ) = P z >ϑ= μ ) t t = α, (27) σ t t where P( ) is the underlying probability distribution, assumed to be known. In traditional VaR models, a typical parameterization is ϑ = 2.33, and Normal = μ t t σ t t, (28) where P( ) is the cumulative normal distribution and α = 1%. Equation (28) gives the 1% threshold for the right tail of the distribution. The corresponding threshold for the left tail is Normal = μ t t 2.33σ t t. 7 The risk manager who has exposure to a risk factor R t needs to know how much capital to put aside to cover at least the fraction, 1 α, of daily losses during a year. In order to do this, the risk manager must first determine a threshold so that the event (R t ) has a probability α under P( ). The standard approach does this by using an explicit distribution that is in general the normal distribution. The alternative provided by EVT is to work with the extreme value distribution F( ) instead of P( ), and then determine the threshold level by going backward from α to by solving: F( ) = 1 α, (29) given the value of α. The use of extremal theory assumes that the tail behavior of the distribution of R t during periods of extreme market volatility can be better approximated by the asymptotic distribution of the maximum (or minimum) of the series. In applications of EVT to VaR calculations, one first picks a high cutoff level L so that all R t > L > 0 are defined to be in the positive tail of the distribution. This L has to be selected so that the threshold needed for VaR calculations, namely, is much farther into the tail, so that >L. Using this L, next we define the probabilities associated with R t : P(R t L) = Q(L), (30) P(R t L + e t ) = Q(L + e t ), (31) where e t > 0 is an exceedence of the threshold L at time t. Finally, let Q L (e t )begiven by: Q L (e t ) = Q(L + e t) Q(L). (32) 1 Q(L) 7. For a comprehensive survey on value-at-risk models, see Dowd (1998) and Jorion (2001). For testing and comparing value-at-risk models, see Lopez (1998) and Christoffersen, Hahn, and Inoue (2001).

12 1624 : MONEY, CREDIT AND BANKING We thus obtain Q L, the conditional distribution of how extreme a R t is, given that it already qualifies as an extreme. As shown in Pickands (1975), the distance between G max,ξ and Q L will converge to zero if L is a high threshold (or as L increases). Hence, we assume that the Box Cox GEV distribution proposed in the paper is very close to the true distribution for a high threshold level. To calculate VaR using the EVT approach, assume that the observations (L + e t1,...,l + e tn ) represent maximal changes for the random variable R t at time t i so that the e ti represent the exceedences. Further, let L be a high enough level so that the Box Cox GEV is a good approximation for the true distribution Q L. Then, the probabilities associated with these extreme observations are given by: [ { [ ( )] }] eti μ 1/ξ λ Q L (R t L < e ti ) = exp 1 + ξ 1 σ + 1. (33) Once the parameters μ, σ, ξ, and λ of the Box Cox GEV distribution are estimated using the maximum likelihood method, we can approximate the tails at each L + e ti using the approximation given in equation (32): [ { [ ( 1 Q(L + e ti ) = k 1 eti μ exp 1 + ξ N σ λ λ )] }] 1/ξ λ, (34) where k and N are the number of extremes and the number of total data points, respectively. The ratio, k/n, is an estimate of 1 Q(L), the unconditional probability that an observation will exceed L. The critical value that corresponds to various levels of α in equation (29) can be found from this estimate by first letting [ { [ ( α = k μ 1 exp 1 + ξ N σ λ )] }] 1/ξ λ and then solving for the unknown value. The solution yields: Box Cox GEV = μ + (35) ( ) [ σ ( 1λ ( ξ ln 1 αnλ )) ξ 1]. (36) k One can also assume that the GPD and GEV distributions are good approximations for the true distribution Q L. That is, Q L will be very close to H max,ξ and G max,ξ if L is a high threshold. Notice that the threshold Box Cox GEV in equation (36) reduces to GPD (when λ = 0) and to GEV (when λ = 1):

13 GPD = μ + ( σ ξ TURAN G. BALI : 1625 ) [ ( ) αn ξ 1], (37) k GEV = μ + ( ) [ ( ( σ ln 1 αn ξ k )) ξ 1]. (38) There is substantial empirical evidence that the distribution of returns on equities and other financial assets is typically skewed, peaked around the mode, and has fat tails. 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 and to account for skewness and kurtosis in the data, we use the skewed fat-tailed distribution of Hansen (1994) that takes into account the non-normality of returns and deals with relatively infrequent events. One of our objectives is to evaluate the risk measurement performance of two broad classes of distributions: (i) extreme value distributions that focus on the tail observations and (ii) skewed fat-tailed and normal distributions that use all data points in the empirical return distribution. The skewed t of Hansen (1994) is considered in this paper as an alternative to extreme value distributions. Hansen (1994) introduced a generalization of the Student t distribution where asymmetries may occur, while maintaining the assumption of a zero mean and unit variance. The skewed t density that provides a flexible tool for modeling the empirical distribution of stock market returns exhibiting skewness and fat tails is given by equation (39). bc f (z t ; μ, σ, v,ρ) = ( bc ( v v 2 ( ) ) bzt + a 2 v ρ ( ) ) bzt + a 2 v ρ if z t < a/b if z t a/b, where z t = R t μ is the standardized market return, the constants a, b, and c are given σ by ( ) v + 1 ( ) Ɣ v 2 a = 4ρc, b 2 = 1 + 3ρ 2 a 2 2, c = v 1 ( v ). π(v 2)Ɣ 2 Hansen shows that this density is defined for 2 < v < and 1 <ρ<1. This density has a single mode at a/b, which is of opposite sign with the parameter ρ. (39)

14 1626 : MONEY, CREDIT AND BANKING Thus, if ρ>0, the mode of the density is to the left of zero and the variable is skewed to the right, and vice versa when ρ<0. Furthermore, if ρ = 0, Hansen s distribution reduces to the traditional standardized t distribution. If ρ = 0 and v =, it reduces to a normal density. 8 An alternative approach to calculating VaR is based on the lower tail of the skewed t distribution. Specifically, we estimate the parameters of the skewed t density (μ, σ, v, ρ) using daily returns and then find a specific percentile of the estimated distribution. Assuming that R t f v,ρ (z) follows a skewed t (ST) density, parametric VaR is the solution to: ST (α) f v,ρ (z) dz = α, (40) where ST (α) is the VaR threshold based on the skewed t density with a loss probability of α. Equation (40) indicates that VaR can be calculated by integrating the area under the PDF of the skewed t distribution. 4. DATA AND ESTIMATION RESULTS The data consist of daily index levels of the Dow Jones Industrial Average (DJIA). The time period of investigation for the Dow 30 equity index extends from May 26, 1896 through December 29, 2000, giving a total of 28,758 daily observations. Appendix A provides descriptive and correlogram statistics of the daily percentage returns on DJIA. The unconditional mean of the daily returns is about 0.019% with a standard deviation of 1.09%. The maximum and minimum values are about 14.27% and 27.96%, respectively. We also report the skewness, excess kurtosis, and the Ljung Box statistics for testing the null hypotheses of independent and identically distributed normal variates. The skewness statistic for daily returns is negative and statistically significant at the 1% level. The excess kurtosis statistic is positive and significant at the 1% level, implying that the distribution of equity returns has much thicker tails than the normal distribution. The fat-tail property is more dominant than skewness in the sample. The first-order autocorrelation coefficient is found to be positive and large enough to reject the first-order zero correlation null hypothesis at the 1% level. Panel B of Appendix A 8. The parameters of skewed t density are estimated by maximizing the log-likelihood function of R t with respect to the parameters μ, σ, v, and ρ: ( ) v + 1 LogL = n ln b + n ln Ɣ n ( v ) 2 2 ln π n ln Ɣ(v 2) n ln Ɣ 2 ( ) v + 1 n ( ) n ln σ ln 1 + d2 t, 2 (v 2) t=1 where d t = bzt + a (1 ρs) and s is a sign dummy taking the value of 1 if bz t + a < 0 and s = 1 otherwise.

15 TURAN G. BALI : 1627 reports the Ljung Box Q-statistics and their p-values for the cumulative effect of up to twelfth-order autocorrelation. The Q-statistic at lag j is a test statistic for the null hypothesis that there is no autocorrelation up to order j. The Ljung Box Q-statistic for daily raw returns and squared returns indicates the presence of intertemporal dependencies in the first and second moments of the return distribution. The Q- statistics reject the zero correlation null hypothesis, indicating that the distribution of the next squared return depends not only on the current return but on several previous returns. The descriptive and correlogram statistics of daily raw returns and squared returns shown in Panels A B of Appendix A indicate that stock market returns are not i.i.d. As shown by Leadbetter, Lindgren, and Rootzen (1983), Resnick (1987), and Castillo (1988), even if the underlying series exhibits m-dependence structure, or follows a moving average of order q, autoregressive of order p, or ARMA(p, q) process, the limit distribution of extremes belongs to the domain of attraction of standard EVT distributions. More specifically, as long as the underlying series (in our case, DJIA daily returns) is stationary and follows an MA(q), AR(p), or ARMA(p, q) process then the conventional procedures developed for i.i.d. data perform adequately on data with dependence structures. However, to make sure that our results are not driven by a misspecified conditional extreme value distribution, we follow Diebold, Schuermann, and Stroughair (1998) and generate the maxima and minima from daily returns, standardized by an asymmetric GARCH model. We first fit a conditional mean-volatility model to the raw daily returns, standardize the data by the estimated conditional mean and volatility, and then repeat our analysis based on these standardized residuals. There is also substantial empirical evidence that the distribution of index returns is leptokurtic, i.e., the unconditional return distribution shows high peaks and fat tails. This implies that extreme events are much more likely to occur in practice than would be predicted by the thin-tailed normal distribution. To overcome the drawbacks of the normal distribution, we use the skewed t distribution with an asymmetric GARCH model of Glosten, Jagannathan, and Runkle (GJR-GARCH, 1993), that takes into account time varying volatility characterized by persistence, asymmetric volatility response to past positive and negative shocks, and considers the conditional nonnormality of returns. The extremal analyses are carried out on returns which have been standardized by the following skewed t AR(1) GJR-GARCH(1,1) model first: R t = α 0 + α 1 R t 1 + z t σ t t 1 = μ t t 1 + z t σ t t 1, (41) σ 2 t t 1 = β 0 + β 1 σ 2 t 1 z2 t 1 + β 2σ 2 t 1 + δs t 1 σ 2 t 1 z2 t 1, (42) S t 1 = 1ifσ t 1z t 1 < 0, and S t 1 = 0 otherwise, where R t is the return for period t, z t is a random variable drawn from the skewed t density, μ t t 1 = α 0 + α 1 R t 1 is the conditional mean, and σ 2 t t 1 is the conditional

16 1628 : MONEY, CREDIT AND BANKING variance of returns based on the information set up to time t 1, and SR t = (R t μ t t 1 )/σ t t 1 is a series of identically and independently distributed (i.i.d.) standardized returns. The parameter δ allows for an asymmetric volatility response to past positive and negative information shocks. Panel C of Appendix A presents the correlogram statistics of standardized returns, SR t, and squared standardized returns. Intertemporal dependencies diminished after the standardization operation with the skewed t AR(1) GJR-GARCH(1,1) model. However, we still observe higher-order dependencies in the standardized residuals. The Q-statistics reject the zero correlation null hypothesis for the squared standardized returns after lag 3, indicating higher-order dependencies in volatility. 9 In this paper, extreme values are defined as excesses over high thresholds (see Embrechts, Kluppelberg, and Mikosch 1997). We set the threshold as 2.5%, 5%, and 10% of the right and left tails of the standardized return distribution. It is well known in the extreme value literature that the maxima (and minima) of a random variable over fixed time interval has a GEV distribution and all exceedences will have a GPD. At an earlier stage of the study, we obtained the extremes over n trading days (n = 23 days or 1 month) and alternatively using the mean excess function approach described in Embrechts, Kluppelberg, and Mikosch. Since the qualitative results turn out to be very similar, we choose not to present them. They are available upon request. Panels A, B, and C of Table 1 present the maximum likelihood estimates of the Box Cox GEV, GPD, and GEV distributions based on the 2.5%, 5%, and 10% tails of the empirical distribution. The results are clear-cut and allow one to determine unambiguously the type of extreme value distribution: for both the largest falls and rises of stock 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. 10 The LR test between the GPD and the Box Cox GEV distributions also indicates a firm rejection of the GPD in all cases. As shown in Table 1, λ max is estimated to be , , and for the 2.5%, 5%, and 10% tails of the return distribution, respectively. The corresponding figures for λ min are found to be , , and for the minimal returns on DJIA. The LR test results indicate that both the GPD with λ = 0 and the GEV with λ = 1 are strongly rejected in favor of the Box Cox GEV distribution with 0 <λ<1. This implies that neither the Frechet nor the Pareto distribution yields an accurate characterization of extreme movements for daily stock returns. 9. Bollerslev, Engle, and Nelson (1994) and Bali and Theodossiou (2007) show that the standardized residuals from a GARCH process with time-varying mean and volatility are not i.i.d. At an earlier stage of the study, we use 11 different specifications of GARCH and obtain similar results on the correlogram statistics of standardized and squared standardized residuals. Which GARCH specification used makes a little difference. 10. The likelihood ratio (LR) statistic is calculated as LR = 2 [log-l log-l], where log-l is the value of the log likelihood under the null hypothesis, and log-l is the log likelihood under the alternative. This statistic is distributed as chi-square with one degree of freedom.

17 TURAN G. BALI : 1629 TABLE 1 MAXIMUM LIKELIHOOD ESTIMATES OF THE EXTREME VALUE DISTRIBUTIONS FOR DJIA Panel A. 2.5% Tails of the Empirical Distribution Maxima μ max σ max ξ max λ max Log-L LR Box Cox GEV , (130.03) (13.867) (6.4518) (2.4507) GPD , (129.90) (13.832) (6.4309) GEV , (113.81) (13.540) (8.5252) Minima μ min σ min ξ min λ min Log-L LR Box Cox GEV , ( ) (14.821) (8.2170) (7.1173) GPD , ( ) (14.753) (6.9671) GEV , ( ) (14.266) (8.6766) Panel B. 5% Tails of the Empirical Distribution Maxima μ max σ max ξ max λ max Log-L LR Box Cox GEV , (176.10) (19.044) (9.4771) (2.4781) GPD , (175.38) (19.038) (9.4499) GEV , , (155.13) (18.785) (12.400) Minima μ min σ min ξ min λ min Log-L LR Box Cox GEV , ( ) (19.753) (11.081) (7.0983) GPD , ( ) (20.210) (9.3828) GEV , , ( ) (19.644) (12.012) Panel C. 10% Tails of the Empirical Distribution Maxima μ max σ max ξ max λ max Log-L LR Box Cox GEV , (206.47) (27.397) (13.650) (4.3672) GPD , (203.63) (27.508) (12.921) GEV , , (179.57) (27.208) (17.118) Box Cox GEV , ( ) (27.449) (15.061) (8.9845) (Continued)

18 1630 : MONEY, CREDIT AND BANKING TABLE 1 CONTINUED Panel C. 10% Tails of the Empirical Distribution Minima μ min σ min ξ min λ min Log-L LR GPD , ( ) (28.015) (13.099) GEV , , ( ) (27.370) (17.047) NOTE: This table shows the maximum likelihood estimates of the location (μ), scale (σ ), shape (ξ), and Box Cox GEV (λ) parameters based on the 2.5%, 5%, and 10% tails of the empirical distribution. Asymptotic t-statistics are given in parentheses. The maximized log-likelihood values (log-l) are reported. The likelihood ratio test results between the Box Cox GEV and GPD, and between the Box Cox GEV and GEV distributions are shown in the last column. 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 fattailed (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. 11 The asymptotic t-statistics of the estimated shape parameters (ξ) clearly indicate the non-normality of extremes. Another notable point in Table 1 is that the estimated shape parameters for the minimal returns (ξ min ) are generally greater than those for the maximal returns (ξ max ). More specifically, the estimated ξ max values are in the range of 0.25 to 0.44 for the 2.5% tails, 0.26 to 0.45 for the 5% tails, and 0.26 to 0.42 for the 10% tails, while for the minimal returns the estimates of ξ min vary from 0.35 to 0.44 for the 2.5% tails, from 0.32 to 0.42 for the 5% tails, and from 0.29 to 0.42 for the 10% tails. Since the higher ξ the fatter the distribution of extremes, the minimal returns have thicker tails than the maximal returns. One may think that the estimation results presented above cannot be generalized, as the best fit for stock returns may not be the best fit for interest rates and foreign currency. At an earlier stage of the study, we use daily observations for the U.S. 3- month Treasury bill and federal funds interest rates (January 8, 1954 to December 31, 2000), the logarithmic daily percentage returns on the S&P 500 and Dow Jones Industrial Average (January 4, 1965 to December 31, 2000), and the logarithmic daily changes in the U.S. Dollar/Deutsche Mark and the U.S. Dollar/French Franc exchange rates (January 4, 1971 to December 31, 2000). Extreme values are defined as excesses over high thresholds. We set the threshold as almost 2 standard deviations around the sample mean, which corresponds to almost 2.5% of the right and left tails of the empirical distribution. Appendix B provides the maximum likelihood estimates of the Box Cox GEV distribution. λ max is estimated to be in the range of 0.16 to 0.83 for the maximal changes in risk factors, whereas λ min estimates are found to be between 0.15 and Although not presented in the paper, the maximized log-likelihood values of the Gumbel and exponential distributions are found to be much lower than those of the GEV and GPD distributions for both the maximal and minimal returns on DJIA.

19 TURAN G. BALI : 1631 for the minimal changes. The LR test results indicate that both the GPD with λ = 0 and the GEV with λ = 1 are strongly rejected in favor of the Box Cox GEV distribution with 0 <λ<1. This implies that neither the GEV nor the GPD yields an accurate characterization of extreme movements in financial markets quite generally. For the Box Cox GEV distribution, the estimated tail indices for the maximal changes (ξ max ) are in the range of 0.17 to 0.44, whereas the estimates of ξ min range from 0.23 to The results in Appendix B indicate that the Box Cox GEV distribution provides more accurate characterization of interest rates, stock market indices, and exchange rates than the GPD and GEV distributions RESULTS FROM THE UNCONDITIONAL EXTREME VALUE, SKEWED t, AND NORMAL DISTRIBUTIONS Table 2 compares the in-sample performance of alternative distribution functions for the DJIA. The top numbers at each row are the VaR thresholds estimated by the unconditional extreme value, skewed t and normal distributions. The results show that the extreme tails yield threshold points, Box Cox GEV, that are up to 42% higher than the normal thresholds, Normal. The thresholds for the Box Cox GEV, GPD, and GEV distributions clearly indicate that the two tails are asymmetric, and the extreme drops have tail areas farther away from normal compared to the extreme rises. There is strong evidence that the VaR thresholds are higher for the extreme negative returns than for the extreme positive returns. The above results indicate that the tail areas obtained from the extremal theory are quite different from and potentially more useful than the tails obtained from the skewed t and normal distributions. The second row in Table 2 presents the estimated counts or the number of observations falling in various tails of the extreme value, skewed t, and normal distributions. The results show that the normal VaR thresholds at various tails are quite inadequate. Given that the DJIA data include 28,758 daily returns from May 26, 1896 to December 29, 2000, one would expect 144, 288, 431, 575, and 719 returns to fall, respectively, into the 0.5%, 1%, 1.5%, 2%, and 2.5% negative and positive tails. The number of returns for the normal VaR thresholds falling into the negative (positive) 0.5% tail are 384 (290), 1% tail are 487 (378), 1.5% tail are 602 (479), 2% tail are 685 (540), and 2.5% tail are 763 (608). These results imply that for the minimal returns the normal distribution underestimates the actual VaR thresholds at all tails considered in the paper. For the maximal returns, the normal distribution underestimates the actual thresholds for the 0.5%, 1%, and 1.5% tails and overestimates for the 2% and 2.5% tails. The normal VaR estimates have a mean absolute percentage error (MA%E) of 33.06% for the right tail and 60% for the left tail of the return distribution. 12. The empirical results in the following sections are based on the daily returns on DJIA that cover the period from May 26, 1896 to December 29, 2000 because we need large number of observations to increase the power of unconditional and conditional coverage tests and the DJIA is the longest time series we could obtain.

20 1632 : MONEY, CREDIT AND BANKING TABLE 2 IN-SAMPLE PERFORMANCE OF THE UNCONDITIONAL DISTRIBUTION FUNCTIONS 2.5% tails of the distribution 5% tails of the distribution 10% tails of the distribution Entire distribution Maxima Box-Cox GEV GPD GEV Box-Cox GEV GPD GEV Box-Cox GEV GPD GEV Skewed t Normal α = 0.5% % % % % % % % % % % % 140 (0.10) 140 (0.10) 154 (0.71) 142 (0.02) 142 (0.02) 159 (1.56) 142 (0.02) 140 (0.10) 154 (0.71) 185 (10.88) 290 (115.2) α = 1% % % % % % % % % % % % 286 (0.01) 286 (0.01) 285 (0.02) 287 (0.00) 287 (0.00) 308 (1.43) 286 (0.01) 285 (0.02) 311 (1.88) 291 (0.04) 378 (26.13) α = 1.5% % % % % % % % % % % % 429 (0.01) 429 (0.01) 404 (1.80) 424 (0.13) 425 (0.09) 444 (0.37) 424 (0.13) 423 (0.17) 463 (2.30) 392 (3.76) 479 (5.16) α = 2% % % % % % % % % % % % 587 (0.25) 587 (0.25) 558 (0.53) 578 (0.01) 578 (0.01) 579 (0.03) 576 (0.00) 576 (0.00) 609 (1.99) 500 (10.48) 540 (2.24) α = 2.5% % % % % % % % % % % % 729 (0.14) 729 (0.14) 770 (3.63) 707 (0.20) 707 (0.20) 686 (1.57) 704 (0.32) 707 (0.20) 746 (1.03) 604 (19.92) 608 (18.52) Average MA%E 1.48% 1.48% 4.86% 1.11% 1.06% 5.13% 1.19% 1.50% 6.40% 13.52% 33.06% Minima Box-Cox GEV GPD GEV Box-Cox GEV GPD GEV Box-Cox GEV GPD GEV Skewed t Normal α = 0.5% % % % % % % % % % % % 158 (1.36) 158 (1.36) 162 (2.23) 157 (1.18) 154 (0.71) 167 (3.58) 155 (0.86) 151 (0.86) 160 (1.77) 214 (29.94) 384 (275.9) α = 1% % % % % % % % % % % % 295 (0.19) 304 (0.93) 295 (0.19) 304 (0.93) 304 (0.93) 316 (2.75) 304 (0.53) 298 (0.38) 321 (3.78) 338 (8.45) 487 (115.6) α = 1.5% % % % % % % % % % % % 419 (0.36) 428 (0.03) 410 (1.09) 431 (0.00) 432 (0.00) 436 (0.05) 432 (0.00) 432 (0.00) 462 (2.16) 442 (0.26) 602 (61.05) α = 2% % % % % % % % % % % % 565 (0.18) 558 (0.53) 542 (1.99) 571 (0.03) 592 (0.50) 572 (0.02) 585 (0.17) 587 (0.25) 622 (3.79) 564 (0.22) 685 (20.18) α = 2.5% % % % % % % % % % % % 744 (0.88) 692 (1.05) 771 (3.77) 717 (0.01) 736 (0.41) 702 (0.41) 735 (0.36) 736 (0.41) 765 (2.96) 690 (1.21) 763 (2.71) Average MA%E 4.03% 4.54% 6.55% 3.11% 3.61% 5.95% 3.48% 2.60% 8.87% 14.89% 60.00% NOTE: This table presents statistics on the in-sample risk measurement performance of the unconditional extreme value, skewed t and normal distributions. Given that the DJIA data include 28,758 daily returns from May 26, 1896 to December 29, 2000, the actual number of returns to fall in the 0.5%, 1%, 1.5%, 2%, and 2.5% left (minima) and right (maxima) tails are 144, 288, 431, 575, and 719, respectively. The top numbers at each row give the estimated VaR thresholds for each distribution. The numbers below the thresholds give the estimated counts with their unconditional coverage test statistics (LRuc) in parentheses. Average MA%E is the average mean absolute percentage error based on the actual and estimated counts., denote significance at the 5% and 1% levels, respectively.