Value-at-Risk via mixture distributions reconsidered

Applied Mathematics and Computation 215 (2009) 2103–2119

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Value-at-Risk via mixture distributions reconsidered Markus Haas Department of Statistics, University of Munich, Akademiestrasse 1, D-80799 Munich, Germany

a r t i c l e

i n f o

Keywords: Fat-tailed distributions Forecasting Gaussian mixture Markov-switching Nonlinear time series Stock Markets Value-at-Risk

a b s t r a c t Value-at-Risk (VaR) has evolved as one of the most prominent measures of downside risk in financial markets. Zhang and Cheng [M.-H. Zhang, Q.-S. Cheng, An Approach to VaR for capital markets with Gaussian mixture, Applied Mathematics and Computation 168 (2005) 1079–1085] proposed an approach to VaR for daily returns based on Gaussian mixtures, which have become rather popular in empirical economics and finance since the seminal paper of Hamilton [J.D. Hamilton, A new approach to the economic analysis of nonstationary time series and the business cycle, Econometrica 57 (2) (1989) 357–384]. However, they do not conduct tests to assess the accuracy of the mixture-implied VaR measures. Recently, Guidolin and Timmermann [M. Guidolin, A. Timmermann, Term structure of risk under alternative econometric specifications, Journal of Econometrics, 131 (2006) 285–308] showed that Markov mixture models do well in measuring VaR at a monthly frequency, but the results may not hold for daily returns due to their more pronounced non-Gaussian features. This paper provides an extensive application of various Markov mixture models to VaR for daily returns of major European stock markets, including out-of-sample backtesting. To accommodate the properties of daily returns, we consider both Gaussian and Student’s t mixtures, and we compare the performance of both uni- and multivariate models under different parameter updating schemes. We find that a univariate mixture of two Student’s t distributions performs best overall. However, by the example of the recent turmoil in financial markets, we also highlight a weak point of the approach. Ó 2009 Elsevier Inc. All rights reserved.

1. Introduction Both in industry and in academia, Value-at-Risk (VaR) is a widely employed measure to characterize the downside risk of a financial position (see, e.g. [28,2]). VaR can be defined in terms of the conditional quantile of the portfolio return distribution for a given horizon (typically a day or a week) and a given (typically small) shortfall probability. To be more precise, consider a time series of portfolio returns, r t , and an associated series of ex-ante VaR measures with shortfall probability n; VaRt ðnÞ. The VaRt ðnÞ implied by a model M is defined by

FM t1 ðVaR t ðnÞÞ ¼ n; FM t1

ð1Þ

where is the (conditional) cumulative distribution function (cdf) derived from model M using the information up to time t  1. Thus, economically, VaRt ðnÞ is defined so that, over the next period, the probability that the portfolio suffers a loss larger than the VaR is 100  n%. It should perhaps be noted that, as a risk measure, VaR is not without its drawbacks. In particular, ‘‘it is concerned only with the number of losses that exceed the VaR and not the magnitude of these losses” E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.08.005

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[11, p. 85]; for an axiomatic analysis of risk measures, see [6]. Alternative risk measures that avoid these shortcomings, such as expected shortfall (ES), have been proposed; however, as VaR is the industry standard, accurate VaR measures are certainly of great importance, as well as a prerequisite for accurate ES calculations. To measure VaR accurately, specification of the distribution function in (1) is crucial. In particular, it has been documented extensively that, at frequencies higher than a month, the distribution of the returns of most financial assets, such as stocks or exchange rates, is characterized by pronounced departures from Gaussianity. Most notably, the empirical return distributions tend to be leptokurtic, that is, they are more peaked and fatter tailed than the Gaussian distribution with the same variance, and there can be non-negligible deviations from symmetry as well (for an overview of the extensive literature, see [22]; and the references therein). Moreover, returns are not independently distributed but, as already observed by Mandelbrot [33], characterized by pronounced volatility clustering, meaning that ‘‘large [price] changes tend to be followed by large changes-of either sign-and small changes tend to be followed by small changes”. In this paper, it is investigated whether Markov-switching (MS) models provide an appropriate framework for modeling the VaR of daily stock returns. Since the pioneering works of Hamilton [23] and Turner et al. [41], these processes have become rather popular in empirical finance since they possess several properties that make their application in this area very attractive: In addition to being able to reproduce the above-mentioned unconditional and conditional properties of returns (cf. [40]), they often lead to an economically plausible description of the return generating process (see Section 2 for discussion). Moreover, the EM algorithm of Dempster et al. [13] provides a simple tool for numerically stable and reliable parameter estimation even in multivariate situations, which compares favorably with other popular return models such as GARCH (e.g., [32,39]) and stochastic volatility (e.g. [7,9]). In a recent paper, Guidolin and Timmermann [19] have shown that Gaussian Markov mixtures do very well in assessing the risk of stock and bond returns on a monthly basis. This is a promising result; however, the non-Gaussian features are much more pronounced in returns at higher frequencies such as daily. Using daily data and an independent mixture model, Zhang and Cheng [45] demonstrate that, at a descriptive level, a Gaussian mixture approach is able to provide substantial insight into the markets’ risk structure, but they do not conduct out-of-sample tests. To examine the suitability of the MS framework in this context, we calculate the daily Value-at-Risk for major European stock markets using MS models with two and three regimes allowing for both Gaussian and Student’s t component densities, the latter being introduced in order to account for the oftentimes rather acute outlier activity in these series. Moreover, we compare Value-at-Risk measures based on both expanding and moving window parameter updating schemes, as well as from uni- and multivariate specifications. A mixture of two Student’s t distributions appears to be the most reliable specification, although the period of the recent financial crisis constitutes an acid test for all models. The organization of the paper is as follows. The next section introduces Markov mixture models for asset returns. Section 3 discusses issues of model specification and in-sample parameter estimates. Section 4 presents an application to the calculation and backtesting of out-of-sample Value-at-Risk measures for major European stock markets, and Section 5 concludes.

2. Markov-switching models for asset returns A particularly appealing approach to modeling the distribution of financial asset returns that has become rather prominent in recent years is the use of mixture distributions. A p-dimensional random vector x is said to have a k-component finite mixture distribution if its density is given by Table 1 Likelihood-based goodness-of-fit measures for the in-sample period (1990–1999): univariate models. France

Gaussian models k¼1 k¼2 k¼3 k¼4 Student’s t models k¼1 k ¼ 2; m1 ¼ m2 k ¼ 2; m1 –m2 k¼3 k¼4

Germany

United Kingdom

Portfolio

K

log L

BIC

log L

BIC

log L

BIC

log L

BIC

2 6 12 20

3880.7 3734.2 3706.1 3699.7

7777.0 7515.3 7506.2 7556.0

3998.2 3694.3 3626.6 3617.8

8012.1 7435.5 7347.1 7392.1

3665.1 3531.8 3506.9 3500.7

7345.9 7110.6 7107.7 7157.9

3453.3 3218.8 3170.2 3161.1

6922.3 6484.5 6434.3 6478.6

3 7 8 13 21

3797.4 3729.9 3729.9 3705.1 3699.0

7618.3 7514.6 7522.3 7511.9 7562.3

3816.8 3656.4 3656.1 3617.7 3607.2

7657.1 7367.6 7374.9 7337.0 7378.7

3592.1 3523.8 3522.0 3503.7 3498.8

7207.7 7102.3 7106.6 7109.1 7161.8

3319.7 3208.8 3205.5 3167.4 3158.7

6662.9 6472.3 6473.6 6436.5 6481.6

Reported are likelihood-based goodness-of-fit measures for univariate Markov-switching models fitted to the three stock market indices and an equally weighted portfolio of these. The left-most column specifies the form of the component densities and the number of components in (2), where ‘‘Gaussian” and ‘‘Student’s t” refer to densities (3) and (4), respectively, and k is the number of regimes. K denotes the number of parameters of a model, log L is the value of the maximized log-likelihood, and BIC is the Bayesian information criterion of Schwarz [38], i.e., BIC ¼ 2  log L þ K log T, where T is the sample size. Smaller values of BIC are preferred, and boldface entries indicate the best model according to BIC.

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f ðx; p1 ; . . . ; pk1 ; h1 ; . . . ; hk Þ ¼

k X

pj fj ðx; hj Þ;

ð2Þ

j¼1

P where pj > 0; j ¼ 1; . . . ; k; j pj ¼ 1, are the mixing weights, and fj ð; hj Þ; j ¼ 1; . . . ; k, are the component densities, each of which is characterized by a parameter vector hj . In most applications, the component densities are taken to be Gaussian, i.e.,

fj ðx; hj Þ ¼ f ðx; lj ; Rj Þ ¼

1 ð2pÞ

p=2

  1 0 pffiffiffiffiffiffiffi exp  ðx  lj Þ R1 j ðx  lj Þ ; 2 jRj j

ð3Þ

j ¼ 1; . . . ; k, where the lj s and Rj s are the component means and covariance matrices, respectively, giving rise to a finite mixture of normal distributions. Modeling the return density via (2) and (3) is very attractive for two reasons. Firstly, the class of finite normal mixtures is known to exhibit an enormous flexibility and is capable of capturing the skewness and excess

Table 2 Likelihood-based goodness-of-fit measures for the in-sample period (1990–1999): multivariate models. Gaussian

K log L BIC

Student’s t

k¼1

k¼2

k¼3

k¼4

k¼1

k ¼ 2; m1 ¼ m2

k ¼ 2; m1 –m2

k¼3

k¼4

9 10229 20530

20 9788.6 19734

33 9666.7 19592

48 9619.6 19615

10 9879.1 19836

21 9689.2 19543

22 9686.9 19546

34 9629.7 19525

49 9594.9 19573

Reported are likelihood-based goodness-of-fit measures for multivariate Markov-switching models fitted to the three stock market indices. See the footnote of Table 1 for further explanations.

Table 3 Gaussian Markov mixture parameter estimates for the three index return series over the in-sample period (1990–1999). k¼1

k¼2

k¼3

1 

0.510   0:047 0:049 0:044 ð0:036Þ ð0:038Þ ð0:033Þ 0 1 0:801 0:750 1:164 B C B ð0:063Þ ð0:066Þ ð0:051Þ C B 0:617 1:447 0:643 C B C B ð0:098Þ ð0:058Þ C B C @ 0:685 0:526 1:031 A ð0:059Þ 0.410   0:076 0:088 0:058 ð0:031Þ ð0:024Þ ð0:028Þ 0 1 0:708 0:228 0:355 B ð0:041Þ ð0:024Þ ð0:031Þ C B C B 0:428 0:402 0:205 C B C B C ð0:031Þ ð0:020Þ B C @ 0:536 0:411 0:620 A ð0:033Þ 0.080   0:152 0:254 0:008 ð0:175Þ ð0:192Þ ð0:154Þ 0 1 5:136 4:113 3:009 B ð0:624Þ ð0:600Þ ð0:436Þ C B C B 0:716 6:427 2:758 C B C B ð0:800Þ ð0:463Þ C B C @ 0:671 0:550 3:917 A ð0:465Þ 0 1 0:204 0:960 0:010 B ð0:011Þ ð0:007Þ ð0:056Þ C B C B 0:013 0:979 0:024 C B C B ð0:006Þ ð0:006Þ ð0:027Þ C B C @ 0:027 0:011 0:772 A ð0:009Þ ð0:006Þ ð0:053Þ 0.965

b2 p b 02 l

 0:042 0:040 0:045 ð0:023Þ ð0:024Þ ð0:021Þ 0 1 1:306 0:842 0:774 B C B ð0:037Þ ð0:032Þ ð0:029Þ C B 0:616 1:434 0:638 C B C B ð0:041Þ ð0:028Þ C B C @ 0:646 0:508 1:099 A ð0:031Þ 0 –

b2 b 2 =R R

–

p^ 3 l ^03

0 –

0.792   0:069 0:094 0:048 ð0:022Þ ð0:021Þ ð0:021Þ 0 1 0:843 0:406 0:493 B C B ð0:032Þ ð0:026Þ ð0:025Þ C B 0:515 0:737 0:346 C B C B ð0:037Þ ð0:024Þ C B C @ 0:622 0:467 0:745 A ð0:030Þ 0.208   0:057 0:162 0:036 ð0:082Þ ð0:094Þ ð0:073Þ 0 1 3:027 2:450 1:827 B ð0:256Þ ð0:239Þ ð0:178Þ C B C B 0:705 3:992 1:730 C B C B C ð0:332Þ ð0:186Þ B C @ 0:674 0:556 2:425 A ð0:195Þ 0 –

b3 b 3 =R R

–

–

b P

1

0

^ d

–

0.865

b1 p

b l 01

b1 b 1 =R R

0:972 B ð0:007Þ B @ 0:028 ð0:007Þ

1 0:107 ð0:026Þ C C 0:893 A ð0:026Þ

The table reports, for k ¼ 1; 2, and 3 regimes, the parameter estimates for multivariate models based on the Gaussian distribution (3), with standard errors b j ; j ¼ 1; . . . ; k, is the b j =R given in parentheses. The order of the assets is alphabetically, i.e., the return vector at time t is r t ¼ ðr France;t ; rGermany;t ; r UK;t Þ0 . R b j are reported in the lower-triangular parts of the covariance/correlation matrix of regime j, where the elements of the estimated correlation matrix R b and b b j ; j ¼ 1; . . . ; k, are the stationary probabilities of the Markov chain, as implied by the estimated transition matrix, P, d is the respective matrices. p b implied persistence of the Markov chain, given by the second largest eigenvalue of P.

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kurtosis detected in financial return data (cf. [34,40]). Secondly, normal mixtures often provide an economically appealing disaggregation of the stochastic mechanism generating asset returns. For example, the presence of different mixture components can be attributed to the distribution of returns depending on an unobserved state (or regime) of the market. In particular, expected returns as well as variances and correlations will be different in bull and bear market regimes (cf. [3,4,37,44,45,18,20]). Alternatively, a mixture of normals could arise from different groups of actors in the market, with the groups possessing different sets of information or possibly processing market information differently. Kon [30], for example, argues that returns on individual stocks may be drawn from ‘‘a noninformation distribution, a firm-specific information distribution, and a market-wide information distribution-hence, a mixture of three normal distributions”, and Vigfusson [42], Ahrens and Reitz [1], and Dean and Faff [12] identify different mixture components associated with different types of traders, such as chartists and fundamentalists. However, in applications to daily returns, use of Gaussian components may not be appropriate. Namely, although a normal mixture can accommodate a considerable degree of excess kurtosis, its tails will eventually decay in a Gaussian manner, i.e., squared exponentially, whereas the tails of financial returns are often found to be more realistically described by a power-law behavior due to the regular occurrence of rather large realizations. This feature, which is of particular importance for risk management, can be captured by replacing the Gaussian with a Student’s t distribution, i.e., by replacing (3) in (2) with



fj ðx; hj Þ ¼ f ðx; lj ; Rj ; mj Þ ¼

C



(

mj þp 2

pffiffiffiffiffiffiffi 1 þ

)ðmj þpÞ=2 ðx  lj Þ0 R1 j ðx  lj Þ

mj

Cðmj =2Þðpmj Þp=2 jRj j

ð4Þ

;

j ¼ 1; . . . ; k, where mj > 0 is the degrees of freedom parameter characterizing the tail of the jth component. The interpretation of the other parameters in (4) is similar to (3) with the exception that, for mj > 2, the covariance matrix of component j is mj ðmj  2Þ1 Rj rather than just Rj . In general, as mj is the tail index of component j, moments of (2) and (4) will be finite only for orders lower than minfm1 ; . . . ; mk g. Further properties of the multivariate t distribution are described, for example, in

Table 4 Student’s t Markov mixture parameter estimates for the three index return series over the in-sample period (1990–1999).

p^ 1 ^01 l

k¼1

k¼2

k¼3

1 

0.505   0:074 0:089 0:054 ð0:031Þ ð0:033Þ ð0:029Þ 0 1 0:586 0:542 0:962 B ð0:049Þ ð0:041Þ ð0:037Þ C B C B 0:535 1:066 0:439 C B C B ð0:061Þ ð0:036Þ C B C @ 0:662 0:471 0:813 A ð0:042Þ 0.364   0:050 0:066 0:054 ð0:029Þ ð0:023Þ ð0:028Þ 0 1 0:580 0:181 0:296 B ð0:037Þ ð0:020Þ ð0:025Þ C B C B 0:406 0:343 0:173 C B C B C ð0:023Þ ð0:018Þ B C @ 0:535 0:405 0:530 A ð0:032Þ 0.131   0:069 0:140 0:004 ð0:103Þ ð0:120Þ ð0:094Þ 0 1 2:761 2:601 1:825 B ð0:298Þ ð0:303Þ ð0:220Þ C B C B 0:813 3:703 1:848 C B C B ð0:384Þ ð0:235Þ C B C @ 0:739 0:647 2:206 A ð0:234Þ 10:55 ð1:272Þ 0 1 0:988 0:010 0:021 B ð0:004Þ ð0:004Þ ð0:010Þ C B C B 0:007 0:988 0:004 C B C B ð0:003Þ ð0:005Þ ð0:006Þ C B C @ 0:005 0:975 A 0:002 ð0:003Þ ð0:003Þ ð0:011Þ 0.982

b2 p b 02 l

 0:058 0:068 0:053 ð0:020Þ ð0:020Þ ð0:019Þ 0 1 0:845 0:490 0:504 B ð0:031Þ ð0:024Þ ð0:023Þ C B C B 0:575 0:861 0:390 C B C B ð0:033Þ ð0:021Þ C B C @ 0:643 0:493 0:728 A ð0:027Þ 0 –

b2 b 2 =R R

–

b3 p b 03 l

0 –

0.581   0:059 0:083 0:060 ð0:024Þ ð0:020Þ ð0:022Þ 0 1 0:662 0:214 0:356 B ð0:031Þ ð0:020Þ ð0:023Þ C B C B 0:395 0:442 0:187 C B C B ð0:026Þ ð0:018Þ C B C @ 0:583 0:374 0:564 A ð0:027Þ 0.419   0:039 0:020 0:033 ð0:041Þ ð0:047Þ ð0:037Þ 0 1 1:375 1:147 0:893 B ð0:081Þ ð0:081Þ ð0:060Þ C B C B 0:717 1:858 0:868 C B C B C ð0:114Þ ð0:066Þ B C @ 0:704 0:589 1:170 A ð0:067Þ 0 –

b3 b 3 =R R

–

–

b m b P

5:813 ð0:369Þ 1

7:919 ð1:010Þ 0 0:992 B ð0:003Þ B @ 0:008 ð0:003Þ

b d

–

0.982

b1 b 1 =R R

1 0:010 ð0:004Þ C C 0:990 A ð0:004Þ

The table reports, for k ¼ 1; 2, and 3 regimes, the parameter estimates for multivariate models based on the Student’s t distribution (4), with standard errors given in parentheses. See the footnote of Table 3 for further explanations.

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Appendix B of Zellner [43], and McLachlan and Peel [34, Chapter 7] and Peel and McLachlan [35] show how mixtures of t distributions can be estimated via the EM algorithm. Several aspects of univariate t mixtures are also discussed in Giacomini et al. [16]. Mixtures of normals and t distributions could also be constructed (see, e.g., [36]); however, in this paper, we shall restrict attention to the case where all components belong to the same parametric family, i.e., either (3) or (4). To complete the formulation of the mixture model (2), we have to specify the stochastic mechanism generating the market regimes, i.e., the evolution of the mixing weights pj ; j ¼ 1; . . . ; k. To allow for predictability of regimes and in particular the pronounced volatility clustering of returns, we adopt the Markov-switching technique that has become very popular in empirical finance since the seminal work of Hamilton [23]. We can write the Markov mixture model with k regimes for a time series fxt g as 1=2

xt ¼ lDt þ RDt zt ;

ð5Þ

where fzt g is an iid sequence from either (3) or (4) with zero mean and unit scale (i.e., the identity matrix), and fDt g is a Markov chain with finite state space S ¼ f1; . . . ; kg and primitive (irreducible and aperiodic) transition matrix P,

0

p11 B . B P ¼ @ .. p1k

1    pk1 .. C C  . A;    pkk

ð6Þ

where pij ¼ pðDt ¼ jjDt1 ¼ iÞ; i; j ¼ 1; . . . ; k. It is assumed that fzt g and fDt g are independent. Denote the stationary distribution of the Markov chain by p ¼ ðp1 ; . . . ; pk Þ0 . Then the stationary distribution of the process (5) and (6) is given by (2) with either (3) or (4). The moment structure of Markov-switching models is investigated in Timmermann [40], and parameter estimation via the EM algorithm is discussed in Hamilton [24], Hamilton [25] and McLachlan and Peel [34, Chapter 13].

3. Application to stock returns We consider three index return series of major European stock markets, namely, daily returns of the MSCI indices for France, Germany and the United Kingdom (UK) from January 1990 to April 2009 (T ¼ 4993 observations). All returns are measured in Euro. Continuously compounded percentage returns are used, i.e., the return of index i at time t is

Equally weighted portfolio index level 400

index level

300 200 100 0 1990

1992

1994

1996

1998

2000 time

2002

2004

2006

2008

2010

2006

2008

2010

Equally weighted portfolio index returns

index returns

10 5 0 −5 −10 1990

1992

1994

1996

1998

2000 time

2002

2004

Fig. 1. Index level (upper panel) and continuously compounded index returns (lower panel) of an equally weighted portfolio of the MSCI indices for France, Germany, and the United Kingdom from January 1990 to April 2009.

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rit ¼ 100  logðIit =Ii;t1 Þ, where Iit denotes the level of index i at time t, and the return vector at time t is defined as r t ¼ ðr 1t ; r2t ; r3t Þ0 , where r 1t ; r 2t , and r 3t are the time-t returns of the indices for France, Germany, and the UK, respectively. In addition to the three country indices, we include the returns of an equally weighted portfolio of the three indices, calculated as the simple average of the individual indices’ returns. In this regard, it should be noted that, due to our use of continuously compounded returns, the linear relation between the individual returns and the portfolio return is only an approximation; however, for daily returns, this approximation is usually rather accurate and standard practice, see Fama [14, Chapter 1], and Campbell et al. [8, Chapter 1], for discussion. In Section 4.2, we calculate and compare Value-at-Risk measures for both the individual indices and the portfolio as implied by both univariate and multivariate models. To this end, we first estimate the models over the (approximately) first ten years of data, i.e., the first 2500 observations (ranging from January 1990 to the end of August 1999), while the remaining observations are retained for the computation and backtesting of out-of-sample Value-at-Risk measures. 3.1. Model specification To keep the analysis in Section 4.2 manageable, we have to select an upper bound for the number of regimes, i.e., the value of k in (2). It is well-known that standard test theory breaks down in the context of mixture models, so that likelihood ratio tests cannot be used to determine the appropriate number of mixture components. More rigorous approaches have been devised, but they are computationally rather expensive [34, Chapter 6,15, Chapter 4]. An alternative is to rely on model selection criteria such as the Bayesian information criterion (BIC) of Schwarz [38], given by BIC ¼ 2  log L þ K log T, where log L is the value of the maximized log-likelihood achieved by a given model, K is the number of parameters and T is the sample size, see, e.g., Guidolin and Timmermann [19,21] for applications to Markov-switching models. The literature on mixture models provides some theoretical and practical support for the appropriateness and good performance of the BIC in this framework (see, e.g. [34, Chapter 6,15, Chapter 4], and the references therein). Moreover, in models based on Student’s

Table 5 Evaluation of Value-at-Risk measures for the French market, September 1999 to December 2007: Gaussian models. 100  n

0.25

Model

Coverage ð100  b nÞ

Gaussian, moving window k¼1 uni., k ¼ 2 uni., k ¼ 3 multi., k ¼ 2 multi., k ¼ 3

0.5

1

2.5

5

10

1.575*** 0.324 0.463* 0.602*** 0.648***

1.899*** 0.602 0.880** 1.019*** 1.112***

2.409*** 1.297 1.621*** 1.945*** 1.806***

3.983*** 3.057 3.566*** 3.891*** 4.122***

6.346*** 5.697 6.716*** 6.531*** 7.179***

9.680 9.680 11.58** 11.35** 11.90***

Gaussian, expanding window 1.621*** k¼1 0.324 uni., k ¼ 2 0.371 uni., k ¼ 3 0.509** multi., k ¼ 2 0.648*** multi., k ¼ 3

1.992*** 0.741 0.741 1.019*** 1.204***

2.640*** 1.251 1.390* 1.853*** 1.945***

4.169*** 3.057 3.289** 3.937*** 4.122***

6.670*** 5.790 6.577*** 6.855*** 7.689***

10.51 9.819 11.35** 10.98 11.63**

Independence ðb d¼b q 00 þ b q 11  1Þ Gaussian, moving window 0.074** k¼1 0.003 uni., k ¼ 2 0.005 uni., k ¼ 3 0.071* multi., k ¼ 2 0.007 multi., k ¼ 3

0.055** 0.071* 0.009 0.036 0.031

0.054** 0.023 0.042 0.029 0.008

0.055** 0.031 0.017 0.003 0.004

0.073*** 0.009 0.005 0.002 0.020

0.057** 0.017 0.041** 0.019 0.016

Gaussian, expanding window 0.071** k¼1 0.003 uni., k ¼ 2 0.004 uni., k ¼ 3 0.086** multi., k ¼ 2 0.007 multi., k ¼ 3

0.051* 0.007 0.007 0.036 0.012

0.045* 0.025 0.020 0.007 0.029

0.049** 0.031 0.010 0.008 0.008

0.070*** 0.011 0.003 0.030 0.008

0.045** 0.015 0.036* 0.009 0.005

Shown are the results for correct coverage and independence of out-of-sample Value-at-Risk (VaR) measures implied by Gaussian Markov-switching models for the French market, where ‘‘uni.” and ‘‘multi.” refers to uni- and multivariate models, respectively. For the coverage test (8), we report the empirical percentage shortfall probabilities, 100  b n ¼ x=T, observed for a nominal VaR level n; n ¼ 0:0025; 0:005; 0:01; 0:025; 0:05; 0:1, where x is the empirical shortfall frequency, and T is the number of forecasts evaluated. For the independence test (10), we report the degree of Markov persistence q 11  1. associated with the estimated transition matrix Q in (9), i.e., b d¼b q 00 þ b * Indicates significance at the 10% level, as obtained from the likelihood ratio tests (8) and (10) for coverage and independence, respectively. ** Indicates significance at the 5% level, as obtained from the likelihood ratio tests (8) and (10) for coverage and independence, respectively. *** Indicates significance at the 1% level, as obtained from the likelihood ratio tests (8) and (10) for coverage and independence, respectively.

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t regimes, we restrict the degrees of freedom parameters mj in (4) to be equal across the regimes for models with k P 3, as estimation with regime-specific mj turns out to be extremely slow for k > 2, and the improvement in likelihood-based fit is negligible. For k ¼ 2, we consider both cases m1 ¼ m2 and m1 – m2 . Tables 1 and 2 report, for the in-sample period from January 1990 to August 1999, the values of the maximized log-likelihood and the BIC for models with k 6 4 for univariate and multivariate models, respectively. Considering the results for univariate models in Table 1, we observe that, according to BIC, a Gaussian mixture with k ¼ 3 is preferred for France and the equally weighted portfolio, whereas Student’s t mixtures with two and three regimes are preferred for the UK and Germany, respectively, with m1 ¼ m2 imposed in the former case. Among the multivariate models, as reported in Table 2, a Student’s t mixture with three regimes performs best. Moreover, in no case is a model with k ¼ 4 preferred over a model with k ¼ 3, which leads us to restrict the analysis to specifications with k 6 3. 3.2. Parameter estimates Before we turn to out-of-sample VaR backtesting, we briefly discuss several parameter estimates for the in-sample period in order to illustrate the typical pattern of the regime-specific means, variances, and correlations, and to highlight the impact of the component densities, i.e. (3) versus (4), on the inferred structure of the regimes and transition probabilities. We restrict attention to the multivariate models; the results for the univariate specifications are qualitatively similar. Estimation results are reported in Tables 3 and 4 for models with Gaussian and Student’s t distributions, respectively, where

Table 6 Evaluation of Value-at-Risk measures for the French market, September 1999 to December 2007: Student’s t models. 100  n

0.25

Model

Coverage ð100  ^ nÞ

0.5

1

2.5

5

10

Student’s t, moving window uni., k ¼ 1 uni., k ¼ 2; m1 ¼ m2 uni., k ¼ 2; m1 –m2 uni., k ¼ 3 multi., k ¼ 1 multi., k ¼ 2; m1 ¼ m2 multi., k ¼ 2; m1 –m2 multi., k ¼ 3

0.463* 0.278 0.278 0.371 0.695*** 0.509** 0.463* 0.463*

1.065*** 0.463 0.509 0.695 1.297*** 0.926** 0.834** 0.741

1.853*** 0.973 0.926 1.390* 1.945*** 1.806*** 1.760*** 1.390*

3.520*** 2.779 2.733 3.520*** 3.891*** 3.844*** 3.844*** 3.335**

7.133*** 5.697 5.604 6.577*** 7.133*** 6.577*** 6.531*** 6.207**

11.63** 10.05 9.912 11.90*** 11.72*** 11.53** 11.58** 11.26*

Student’s t, expanding window uni., k ¼ 1 uni., k ¼ 2; m1 ¼ m2 uni., k ¼ 2; m1 –m2 uni., k ¼ 3 multi., k ¼ 1 multi., k ¼ 2; m1 ¼ m2 multi., k ¼ 2; m1 –m2 multi., k ¼ 3

0.463* 0.278 0.278 0.371 0.602*** 0.417 0.417 0.417

1.158*** 0.463 0.463 0.509 1.297*** 0.926** 0.741 0.695

1.945*** 1.158 1.204 1.251 1.992*** 1.667*** 1.714*** 1.390*

3.937*** 2.872 2.872 3.242** 3.983*** 3.613*** 3.613*** 3.474***

7.318*** 5.743 5.743 6.577*** 7.550*** 6.484*** 6.438*** 6.855***

12.00*** 10.24 10.28 11.30** 12.14*** 11.26* 11.26* 11.58**

^00 þ q ^11  1Þ Independence ð^ d¼q Student’s t, moving window uni., k ¼ 1 uni., k ¼ 2; m1 ¼ m2 uni., k ¼ 2; m1 –m2 uni., k ¼ 3 multi., k ¼ 1 multi., k ¼ 2; m1 ¼ m2 multi., k ¼ 2; m1 –m2 multi., k ¼ 3

0.096** 0.003 0.003 0.004 0.060* 0.086** 0.096** 0.005

0.077** 0.005 0.005 0.007 0.095*** 0.041 0.048 0.007

0.058** 0.038 0.041 0.054* 0.077*** 0.086*** 0.062** 0.054*

0.045* 0.006 0.007 0.018 0.059** 0.048* 0.048* 0.037

0.070*** 0.009 0.011 0.003 0.070*** 0.035 0.036 0.037

0.044** 0.009 0.012 0.034 0.051** 0.033 0.036 0.008

Student’s t, expanding window uni., k ¼ 1 uni., k ¼ 2; m1 ¼ m2 uni., k ¼ 2; m1 –m2 uni., k ¼ 3 multi., k ¼ 1 multi., k ¼ 2; m1 ¼ m2 multi., k ¼ 2; m1 –m2 multi., k ¼ 3

0.096** 0.003 0.003 0.004 0.071* 0.107** 0.107** 0.004

0.069** 0.005 0.005 0.005 0.095*** 0.041 0.055 0.007

0.053* 0.029 0.027 0.025 0.075** 0.068** 0.093*** 0.087***

0.045* 0.004 0.004 0.011 0.043* 0.029 0.042* 0.047*

0.064*** 0.001 0.001 0.003 0.071*** 0.060** 0.062*** 0.013

0.039* 0.018 0.014 -0.035* 0.040* 0.049** 0.049** -0.004

Shown are the results for correct coverage and independence of out-of-sample Value-at-Risk (VaR) measures implied by Student’s t Markov-switching models for the French market. See the footnote of Table 5 for further explanations.

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the regimes have been ordered with respect to a declining stationary regime probability, i.e., p1 > p2 > p3 . For purpose of comparison, we also report the results for single-regime models, where k ¼ 1. For the models with k ¼ 2 regimes, the first regime is, for Gaussian as well as Student’s t components, characterized by a higher expected return, lower variances and lower correlations, implying that it can be interpreted as the bull market regime, whereas the second regime represents the bear market. However, there are also some differences between the results in Tables 3 and 4. Namely, in the Gaussian model, the bear market regime is less persistent (the estimated ‘‘staying probability” p22 in (6) is lower) and has a lower stationary probability p2 than in the Student’s t model. This can be related to the observation that, in the former model, the bear market is more ‘‘extreme” than in the latter. In particular, with the exception of the UK, expected bear market returns are negative in the Gaussian model, whereas they are positive in the t model, and the ratios of bear market variances to bull market variances are 3.6, 5.4, and 3.3 in the Gaussian model for the French, German, and UK market, respectively, whereas they are only 2.1, 4.2, and 2.1 in the t model. On the other hand, the differences between the correlations are more distinctive in the model based on Student’s t regimes. These observations can be explained by the pronounced nonnormality ^ ¼ 7:9. In particular, as observed by Klaof the regime densities implied by the estimated degrees of freedom parameter m assen [29] and Ardia [5], a Gaussian regime-switching model will detect regime switches more frequently due to an untypical observation in an otherwise low- or high-volatility period, whereas Student’s t components, as a result of their higher peaks and fatter tails, can better accommodate such realizations within a given regime, leading to regimes that are more persistent. Moreover, it can be shown, e.g. [40] that, in normal mixture models, excess kurtosis is exclusively generated by the ratio of the component variances. On the other hand, in Student’s t mixtures, there is an additional source of leptokurtosis, namely conditional (within-regime) kurtosis, so that, to achieve the same degree of overall kurtosis, the differences in volatility across regimes can be more moderate. Similarly, by using a t distribution, extreme observations are given less weight in calculating the parameter estimates [35], resulting in the aforementioned differences in the bear market expected returns. When k ¼ 3 regimes are considered, both models identify a regime (regime 3) with very high volatility and a stationary probability of approximately 10%. In this regime, expected returns are negative with the exception of the UK in the Student’s model, but these regimes’ means are also characterized by large standard errors due to the scarcity and high volatility of observations generated by this regime. Expected returns are positive in the other two regimes. Regime 1 mimics the ‘‘unconditional” variances and correlations as represented by the single-regime models ðk ¼ 1Þ (particularly so in the Gaussian model), whereas regime 2 is a low-volatility regime with low correlations. Moreover, as expected, the estimated degrees of freedom parameter is somewhat larger in the k ¼ 3 than in the k ¼ 2 specification.

Table 7 Evaluation of Value-at-Risk measures for the German market, September 1999 to December 2007: Gaussian models. 100  n

0.25

Model

Coverage ð100  ^ nÞ

Gaussian, moving window k¼1 uni., k ¼ 2 uni., k ¼ 3 multi., k ¼ 2 multi., k ¼ 3

0.5

1

2.5

5

10

1.621*** 0.371 0.185 0.417 0.648***

2.362*** 0.741 0.509 0.880** 0.834**

3.011*** 1.343 1.158 1.853*** 1.806***

4.539*** 3.150* 3.103* 4.261*** 4.122***

6.531*** 6.160** 6.623*** 7.133*** 7.272***

9.958 10.89 11.53** 11.86*** 12.32***

Gaussian, expanding window 1.621*** k¼1 0.417 uni., k ¼ 2 0.185 uni., k ¼ 3 0.371 multi., k ¼ 2 0.509** multi., k ¼ 3

2.362*** 0.787* 0.602 0.926** 0.834**

3.150*** 1.436* 1.158 1.899*** 1.945***

4.771*** 3.381** 3.011 4.215*** 4.215***

6.901*** 6.484*** 6.762*** 7.272*** 7.457***

10.19 11.21* 11.58** 12.41*** 12.46***

0.074*** 0.030 0.019 0.014 0.011

Independence ðb d¼b q 00 þ b q 11  1Þ Gaussian, moving window 0.071** k¼1 0.004 uni., k ¼ 2 0.002 uni., k ¼ 3 0.004 multi., k ¼ 2 0.065* multi., k ¼ 3

0.076*** 0.118*** 0.005 0.009 0.048

0.128*** 0.056* 0.012 0.007 0.034

0.081*** 0.028 0.014 0.001 0.004

Gaussian, expanding window 0.071** k¼1 0.004 uni., k ¼ 2 0.002 uni., k ¼ 3 0.004 multi., k ¼ 2 0.005 multi., k ¼ 3

0.076*** 0.051 0.006 0.009 0.008

0.119*** 0.051* 0.012 0.005 0.020

0.103*** 0.022 0.001 0.010 0.013

0.056** 0.015 0.021 0.030 0.013

0.106*** 0.026 0.001 0.003 0.001 0.074*** 0.013 0.009 0.003 0.023

Shown are the results for correct coverage and independence of out-of-sample Value-at-Risk (VaR) measures implied by Gaussian Markov-switching models for the German market. See the footnote of Table 5 for further explanations.

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4. Application to Value-at-Risk Now we turn to the statistical evaluation of the one-step-ahead out-of-sample Value-at-Risk measures implied by the different models. We first describe the methodology in Section 4.1 and then present the empirical results in Section 4.2. 4.1. Backtesting Value-at-Risk (VaR) measures For a nominal VaR shortfall probability n and a correctly specified VaR model, we expect 100  n% of the observed return values not to exceed the respective VaR forecast. To test the models’ suitability for calculating accurate ex-ante VaR measures, we say that a violation or hit occurs at time t if r t < VaRt and define the binary sequence

 It ¼

1;

if r t < VaRt ;

0;

if r t P VaRt :

ð7Þ

P Then the empirical shortfall probability is b n ¼ x=T, where x ¼ Tt¼1 It is the number of observed violations, and T is the numb ber of forecasts evaluated. If n is significantly higher (less) than n, then the model under study tends to underestimate (overestimate) the risk of the financial position. To formally test whether a model correctly estimates the VaR, that is, whether the empirical shortfall probability, b n, is statistically indistinguishable from the nominal shortfall probability, n, we use the likelihood ratio test statistic (see, e.g. [31])

Table 8 Evaluation of Value-at-Risk measures for the German market, September 1999 to December 2007: Student’s t models. 100  n

0.25

Model

Coverage ð100  ^ nÞ

0.5

1

2.5

5

10

Student’s t, moving window uni., k ¼ 1 uni., k ¼ 2; m1 ¼ m2 uni., k ¼ 2; m1 –m2 uni., k ¼ 3 multi., k ¼ 1 multi., k ¼ 2; m1 ¼ m2 multi., k ¼ 2; m1 –m2 multi., k ¼ 3

0.278 0.139 0.139 0.139 0.787*** 0.324 0.278 0.278

0.787* 0.556 0.556 0.417 1.528*** 0.695 0.741 0.556

1.945*** 1.158 1.112 0.973 2.686*** 1.436* 1.436* 1.065

4.308*** 3.103* 3.057 2.918 4.863*** 3.705*** 3.659*** 3.474***

7.596*** 6.438*** 6.207** 6.531*** 7.596*** 6.716*** 6.670*** 6.346***

13.25*** 11.35** 11.30** 11.90*** 12.51*** 12.23*** 12.23*** 11.35**

Student’s t, expanding window uni., k ¼ 1 uni., k ¼ 2; m1 ¼ m2 uni., k ¼ 2; m1 –m2 uni., k ¼ 3 multi., k ¼ 1 multi., k ¼ 2; m1 ¼ m2 multi., k ¼ 2; m1 –m2 multi., k ¼ 3

0.371 0.232 0.232 0.093* 0.787*** 0.232 0.371 0.185

0.880** 0.556 0.556 0.371 1.436*** 0.741 0.695 0.509

2.177*** 1.251 1.251 0.926 2.640*** 1.482** 1.482** 1.065

4.539*** 3.428*** 3.428*** 3.057 5.141*** 3.335** 3.474*** 3.428***

7.874*** 6.346*** 6.346*** 6.670*** 7.828*** 6.484*** 6.670*** 6.531***

13.90*** 11.72*** 11.72*** 12.23*** 13.39*** 12.27*** 12.32*** 11.86***

^00 þ q ^11  1Þ Independence ð^ d¼q Student’s t, moving window uni., k ¼ 1 uni., k ¼ 2; m1 ¼ m2 uni., k ¼ 2, m1 –m2 uni., k ¼ 3 multi., k ¼ 1 multi., k ¼ 2; m1 ¼ m2 multi., k ¼ 2; m1 –m2 multi., k ¼ 3

0.003 0.001 0.001 0.001 0.111*** 0.003 0.003 0.003

0.111*** 0.078* 0.078* 0.004 0.077** 0.060* 0.055 0.006

0.053* 0.069** 0.073** 0.010 0.114*** 0.051* 0.051* 0.033

0.090*** 0.030 0.031 0.035 0.089*** 0.026 0.028 0.047*

0.070*** 0.046** 0.053** 0.009 0.070*** 0.068*** 0.070*** 0.041*

0.097*** 0.033 0.034 0.002 0.086*** 0.020 0.020 0.015

Student’s t, expanding window uni., k ¼ 1 uni., k ¼ 2; m1 ¼ m2 uni., k ¼ 2; m1 –m2 uni., k ¼ 3 multi., k ¼ 1 multi., k ¼ 2; m1 ¼ m2 multi., k ¼ 2; m1 –m2 multi., k ¼ 3

0.122** 0.002 0.002 0.001 0.111*** 0.002 0.004 0.002

0.097** 0.078* 0.078* 0.004 0.051* 0.118*** 0.060* 0.005

0.065** 0.062** 0.062** 0.009 0.099*** 0.048* 0.048* 0.033

0.091*** 0.020 0.020 0.031 0.098*** 0.037 0.033 0.034

0.068*** 0.034 0.034 0.019 0.069*** 0.076*** 0.077*** 0.014

0.079*** 0.019 0.015 0.014 0.077*** 0.019 0.022 0.025

Shown are the results for correct coverage and independence of out-of-sample Value-at-Risk (VaR) measures implied by Student’s t Markov-switching models for the German market. See the footnote of Table 5 for further explanations.

2112

M. Haas / Applied Mathematics and Computation 215 (2009) 2103–2119 asy LRT1 ¼ 2fx logðn=^nÞ þ ðT  xÞ log½ð1  nÞ=ð1  ^nÞg  v2 ð1Þ:

ð8Þ

While (8) constitutes a test for an unconditionally correctly specified VaR model, it is also relevant to assess whether the violations are independent. For example, a model that fails to account appropriately for the volatility clustering may be correct unconditionally, but in any given period will have uncorrect probability of violation conditionally, leading to violation clustering. Thus, we also apply a test for the independence of the series fIt g defined in (7), as proposed by Christoffersen [10]. This tests the null of independence against the alternative of a first-order Markov structure of fIt g, with transition matrix

Q¼



q00

1  q11

1  q00

q11

 ;

ð9Þ

where qii ¼ pðIt ¼ ijIt1 ¼ iÞ; i ¼ 0; 1. Let T ij be the number of observations of It assuming value i followed by j, i; j ¼ 0; 1. Then ^00 ¼ T 00 =ðT 00 þ T 01 Þ, and b q 11 ¼ T 11 =ðT 10 þ T 11 Þ. Under the null hypothesis, we have q00 ¼ 1  q11 , the conwe estimate q ^ ^00 ¼ ðT 00 þ T 10 Þ=ðT 01 þ T 11 þ T 00 þ T 10 Þ, and the likelihood ratio test statistic for the null hypothesis strained estimator is q d ¼ q00 þ q11  1 ¼ 0 is asy

LRT2 ¼ 2ðlog L0  log L1 Þ  v2 ð1Þ;

ð10Þ

^ ^ ^T0000 þT 10 ð1  q ^00 ÞT 01 þT 11 , and L1 ¼ q ^T0000 ð1  q ^00 ÞT 01 q ^11 ÞT 10 . ^T1111 ð1  q where L0 ¼ q It should be noted, however, that some authors argue that unconditional correctness of VaR measures is actually more important than the absence of violation clustering. As put forward by Jorion [27], ‘‘less variable capital charges could be achieved at the expense of more bunching. Less variability is economically beneficial. [. . .] Whether bunching is intrinsically bad, however, is not so obvious.” See Jorion [27] for an extensive discussion of this issue. 4.2. Empirical results We calculate one-step-ahead out-of-sample VaR measures and consider the VaR levels n ¼ 0:0025; 0:005; 0:01; 0:025; 0:05, and 0.1, as defined in (1). For each model, the parameter estimates are updated every week (5 trading days)

Table 9 Evaluation of Value-at-Risk measures for the UK market, September 1999 to December 2007: Gaussian models. 100  n

0.25

Model

Coverage ð100  ^ nÞ

Gaussian, moving window k¼1 uni., k ¼ 2 uni., k ¼ 3 multi., k ¼ 2 multi., k ¼ 3

0.5

1

2.5

5

10

1.158*** 0.463* 0.371 0.648*** 0.602***

1.806*** 0.787* 0.787* 1.065*** 0.880**

2.177*** 1.390* 1.390* 1.528** 1.436*

3.613*** 2.779 3.150* 3.150* 3.150*

5.651 5.095 5.049 5.558 6.068**

9.495 10.01 10.51 10.61 11.72***

Gaussian, expanding window 1.204*** k¼1 0.417 uni., k ¼ 2 0.324 uni., k ¼ 3 0.556** multi., k ¼ 2 0.741*** multi., k ¼ 3

1.853*** 0.695 0.741 1.112*** 1.204***

2.316*** 1.390* 1.297 1.806*** 1.853***

3.844*** 2.872 2.918 3.242** 3.613***

5.790 5.188 4.817 5.419 6.207**

9.912 9.727 9.958 10.65 11.67**

Independence ðb d¼b q 00 þ b q 11  1Þ Gaussian, moving window 0.029 k¼1 0.096** uni., k ¼ 2 0.004 uni., k ¼ 3 0.065* multi., k ¼ 2 0.006 multi., k ¼ 3

0.060** 0.051 0.008 0.033 0.009

0.086*** 0.020 0.014 0.015 0.015

0.069*** 0.023 0.028 0.002 0.043*

0.053** 0.013 0.005 0.003 0.057**

0.084*** 0.038* 0.015 0.033 0.028

Gaussian, expanding window 0.027 k¼1 0.004 uni., k ¼ 2 0.003 uni., k ¼ 3 0.006 multi., k ¼ 2 0.007 multi., k ¼ 3

0.058** 0.060* 0.007 0.031 0.027

0.079*** 0.020 0.013 0.008 0.007

0.060** 0.004 0.019 0.011 0.042*

0.066*** 0.002 0.030 0.003 0.045*

0.071*** 0.035 0.013 0.022 0.034

Shown are the results for correct coverage and independence of out-of-sample Value-at-Risk (VaR) measures implied by Gaussian Markov-switching models for the UK market. See the footnote of Table 5 for further explanations.

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employing both a moving and an expanding window of data, where in case of the first updating scheme we use the most recent 2500 observations in the sample. In this manner, we obtain, for each model and return series, 2493 one-step-ahead out-of-sample VaR measures. The inclusion of both a moving and an expanding estimation window was suggested by an anonymous referee since, particularly with multi-regime models, it is not a priori clear which of the two choices is more appropriate. On the one hand, as suggested by the referee, it may be unnecessary or even misguided to use a moving window in the context of Markov-switching models, since these processes by construction are capable of accommodating shifts in the marginal densities. In this perspective, dropping old observations may result in a loss of valuable information in particular if, for example, periods of financial turmoil are abandoned which may, however, incorporate useful but rare insight into the structure of the high-volatility regimes. On the other hand, one may suspect that even the within-regime densities are subject to slow evolution over time and/or abrupt structural breaks. In this case, a moving window should be preferred, provided that the window is long enough to include both low- and high-volatility periods. Our backtesting sample spans the period from September 1999 to April 2009, and thus covers the recent financial crisis which seriously affected the stock markets since the beginning of 2008. To illustrate, Fig. 1 plots, in its upper and lower panel, respectively, the index level and returns of the equally weighted portfolio of the three country indices included in our study. The figure shows that the sharp decline of the markets in 2008 was accompanied by a burst of volatility that appears rather extreme even in view of the former high-volatility periods covered by the sample under study. In view of these latest events, we first perform the backtesting for the time span September 1999 to

Table 10 Evaluation of Value-at-Risk measures for the UK market, September 1999 to December 2007: Student’s t models. 100  n

0.25

Model

Coverage ð100  ^ nÞ

0.5

1

2.5

5

10

Student’s t, moving window uni., k ¼ 1 uni., k ¼ 2; m1 ¼ m2 uni., k ¼ 2; m1 –m2 uni., k ¼ 3 multi., k ¼ 1 multi., k ¼ 2; m1 ¼ m2 multi., k ¼ 2; m1 –m2 multi., k ¼ 3

0.417 0.371 0.371 0.324 0.463* 0.463* 0.463* 0.509**

0.834** 0.695 0.741 0.695 0.787* 0.787* 0.787* 0.787*

1.714*** 1.343 1.297 1.297 1.760*** 1.343 1.390* 1.297

3.566*** 2.872 2.918 3.103* 3.520*** 2.964 2.918 2.594

6.021** 4.771 4.817 5.280 6.160** 5.604 5.604 5.373

10.93 10.47 10.47 11.07 11.02 10.70 10.75 10.65

Student’s t, expanding window uni., k ¼ 1 uni., k ¼ 2; m1 ¼ m2 uni., k ¼ 2; m1 –m2 uni., k ¼ 3 multi., k ¼ 1 multi., k ¼ 2; m1 ¼ m2 multi., k ¼ 2; m1 –m2 multi., k ¼ 3

0.417 0.324 0.324 0.278 0.417 0.556** 0.602*** 0.648***

0.834** 0.602 0.556 0.695 0.834** 0.741 0.648 0.787*

1.806*** 1.204 1.158 1.251 1.806*** 1.343 1.251 1.390*

3.752*** 2.501 2.547 2.733 3.798*** 2.918 2.964 3.057

6.531*** 4.724 4.863 4.910 6.577*** 5.419 5.512 5.975**

11.16* 10.01 10.14 10.56 11.26* 10.65 10.61 10.79

q 11  1Þ Independence ðb d¼b q 00 þ b Student’s t, moving window uni., k ¼ 1 uni., k ¼ 2; m1 ¼ m2 uni., k ¼ 2, m1 –m2 uni., k ¼ 3 multi., k ¼ 1 multi., k ¼ 2, m1 ¼ m2 multi., k ¼ 2; m1 –m2 multi., k ¼ 3

0.107** 0.004 0.004 0.003 0.096** 0.005 0.005 0.005

0.048 0.060* 0.055 0.007 0.051 0.051 0.051 0.008

0.093*** 0.021 0.023 0.013 0.089*** 0.056* 0.054* 0.059*

0.071*** 0.020 0.019 0.030 0.073*** 0.066** 0.068** 0.065**

0.042* 0.031 0.030 0.018 0.055** 0.072*** 0.081*** 0.062**

0.072*** 0.021 0.021 0.021 0.070*** 0.040* 0.044* 0.046**

Student’s t, expanding window uni., k ¼ 1 uni., k ¼ 2; m1 ¼ m2 uni., k ¼ 2; m1 –m2 uni., k ¼ 3 multi., k ¼ 1 multi., k ¼ 2; m1 ¼ m2 multi., k ¼ 2; m1 –m2 multi., k ¼ 3

0.107** 0.003 0.003 0.003 0.107** 0.078* 0.071* 0.007

0.048 0.071* 0.006 0.007 0.048 0.055 0.065* 0.008

0.060** 0.027 0.029 0.013 0.060** 0.021 0.062** 0.054*

0.064** 0.012 0.011 0.007 0.062** 0.068** 0.082*** 0.062**

0.059** 0.022 0.029 0.028 0.058** 0.060** 0.057** 0.060**

0.070*** 0.043* 0.040* 0.019 0.068*** 0.056** 0.057** 0.038*

Shown are the results for correct coverage and independence of out-of-sample Value-at-Risk (VaR) measures implied by Student’s t Markov-switching models for the UK market. See the footnote of Table 5 for further explanations.

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December 2007 (corresponding to the first 2159 observations of the backtesting sample); subsequently, we do the tests for the entire out-of-sample period. By following this procedure, it is possible to assess whether a model’s (potential) failure is chronic or can be attributed mainly to a clustering of violations due to the outbreak of unprecedented volatility during the turmoil. 4.2.1. Results for the period September 1999 to December 2007 The empirical results for the period from September 1999 to December 2007 are summarized in Tables 5, 7, 9, and 11 for the Gaussian models and in Tables 6, 8, 10, and 12 for the Student’s t models for France, Germany, the UK, and the equally weighted portfolio, respectively. For each VaR level, n, we show, for the coverage test (8), the empirical percentage shortfall probability 100  b n of the respective models. For the independence test (10), we report the degree of Markov persistence q 11  1. associated with the estimated transition matrix Q in (9), i.e., b d¼b q 00 þ b We first observe that the single-regime models, both Gaussian and Student’s t, are not adequate for all markets. They have a strong tendency to underestimate the risk, and, not surprisingly, independence of violations, as tested by (10), is also rejected in almost all cases, reflecting the fact that these models cannot capture the volatility dynamics. Considering the multiregime specifications, the most striking result is the superiority of VaR measures calculated on the basis of univariate as compared to multivariate models. In fact, both for multivariate Gaussian as well as for multivariate Student’s t specifications and for all series, correct coverage is rejected for most VaR levels, n, the only exception (to some extent) being the Student’s t models for the UK market. This may seem surprising at first view, given that the multivariate models incorporate additional information about the correlation structure of the returns, which may help identifying the current regime. A possible explanation is that the coherence between the regime variables of the individual markets is less than perfect, which may be assessed by comparing the conditional regime probabilities (employed as conditional mixing weights in (2) for calculating outof-sample VaR measures) implied by uni- and multivariate models. These are shown in Fig. 2 for the two-regime Student’s t models with m1 ¼ m2 and a moving estimation window. For the French market, there are remarkable differences between the regime forecasts of the uni- and multivariate models (top and bottom panel, respectively) until the middle of 2002, where the univariate model on average attaches a higher weight to the first regime. Subsequently, the models’ inferences are more coherent. However, the fact that the multivariate model more often ascribes a higher weight to the high-volatility regime at the beginning of the sample implies that, compared to the univariate process, more of the less extreme observations are used to determine its parameters; this results, for example, in lower volatility estimates in this regime and leads to lower VaR

Table 11 Evaluation of Value-at-Risk measures for equally weighted portfolio, September 1999 to December 2007: Gaussian models. 100  n

0.25

Model

Coverage ð100  b nÞ

Gaussian, moving window k¼1 uni., k ¼ 2 uni., k ¼ 3 multi., k ¼ 2 multi., k ¼ 3

0.5

1

2.5

5

10

1.575*** 0.509** 0.417 0.648*** 0.787***

1.945*** 0.880** 0.695 1.204*** 1.112***

3.103*** 1.343 1.019 2.270*** 2.084***

4.771*** 3.242** 3.103* 4.400*** 4.215***

7.226*** 6.114** 6.392*** 7.318*** 7.596***

10.42 10.05 11.39** 12.18*** 12.69***

Gaussian, expanding window 1.714*** k¼1 0.509** uni., k ¼ 2 0.371 uni., k ¼ 3 0.602*** multi., k ¼ 2 0.973*** multi., k ¼ 3

2.223*** 0.926** 0.602 1.297*** 1.297***

3.381*** 1.528** 1.390* 2.409*** 2.131***

5.234*** 3.613*** 3.335** 4.817*** 4.632***

7.642*** 6.577*** 6.484*** 7.828*** 8.337***

11.21* 10.89 11.44** 12.74*** 12.69***

Independence ð^ d¼b q 00 þ b q 11  1Þ Gaussian, moving window 0.074** k¼1 0.086** uni., k ¼ 2 0.004 uni., k ¼ 3 0.065* multi., k ¼ 2 0.008 multi., k ¼ 3

0.053* 0.044 0.007 0.027 0.031

0.060** 0.021 0.010 0.002 0.024

0.072*** 0.026 0.001 0.002 0.013

0.060** 0.001 0.006 0.003 0.004

0.077*** 0.032 0.019 0.017 0.005

Gaussian, expanding window 0.065** k¼1 0.086** uni., k ¼ 2 0.004 uni., k ¼ 3 0.071* multi., k ¼ 2 0.010 multi., k ¼ 3

0.041 0.041 0.006 0.023 0.013

0.050** 0.015 0.020 0.005 0.000

0.057** 0.002 0.006 0.010 0.007

0.062*** 0.005 0.016 0.008 0.006

0.065*** 0.021 0.015 0.012 0.009

Shown are the results for correct coverage and independence of out-of-sample Value-at-Risk (VaR) measures implied by Gaussian Markov-switching models for an equally weighted portfolio of the French, German, and UK markets. See the footnote of Table 5 for further explanations.

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measures even if the conditional probability of the bearish regime is close to unity. Note that the differences in performance between the uni- and multivariate two-component models are likewise more pronounced for France than for the two other country indices. Comparing the univariate multi-regime specifications, we find that the two-regime Student’s t processes perform best overall, both with m1 ¼ m2 and m1 –m2 . When combined with a moving estimation window, the hypothesis of correct coverage of VaR measures implied by these models cannot be rejected with the exception of the higher VaR levels ðn P 0:025Þ for the German market (Table 8). Their performance with an expanding window is comparable yet slightly worse for Germany and the equally weighted portfolio (Table 12). The performance of the Student’s t models with k ¼ 3 regimes is less consistent; although their results mimic those of the two-regime specifications for Germany and the UK (Table 10), they have a strong tendency to underestimate the risk at the higher VaR levels for France (Table 6) and the equally weighted portfolio. A similar observation holds for the Gaussian two- and three-regime models. For example, with k ¼ 2, the Gaussian models do well for France (Table 5) and, with a moving and expanding estimation window, respectively, for Germany (Table 7) and the UK (Table 9), but less so for the other series. With regard to the Markov test (10), we note that even for the models that perform best according to correct coverage, independence of the hit sequence (7) is occasionally rejected. When compared to the single-regime models, however, there is a considerable improvement. Moreover, in view of the moderate degree of clustering, it seems reasonable to accept Jorion’s argument referred to at the end of Section 4.1, namely, that such mild bunching should be accepted as a trade-off for unbiasedness. We also note that the univariate t model with three regimes performs somewhat better than the

Table 12 Evaluation of Value-at-Risk measures for equally weighted portfolio, September 1999 to December 2007: Student’s t models. 100  n

0.25

Model

Coverage ð100  b nÞ

0.5

1

2.5

5

10

Student’s t, moving window uni., k ¼ 1 uni., k ¼ 2; m1 ¼ m2 uni., k ¼ 2; m1 –m2 uni., k ¼ 3 multi., k ¼ 1 multi., k ¼ 2; m1 ¼ m2 multi., k ¼ 2; m1 –m2 multi., k ¼ 3

0.463* 0.371 0.324 0.371 0.926*** 0.602*** 0.509** 0.556**

1.158*** 0.648 0.695 0.695 1.482*** 0.880** 0.973*** 0.741

1.806*** 1.343 1.251 1.158 2.362*** 1.853*** 1.760*** 1.343

4.585*** 2.872 2.964 2.964 4.910*** 3.659*** 3.520*** 3.196**

8.059*** 5.651 5.604 6.346*** 7.967*** 7.133*** 7.179*** 6.577***

13.48*** 10.51 10.65 12.04*** 13.20*** 12.32*** 12.23*** 12.14***

Student’s t, expanding window uni., k ¼ 1 uni., k ¼ 2; m1 ¼ m2 uni., k ¼ 2; m1 –m2 uni., k ¼ 3 multi., k ¼ 1 multi., k ¼ 2; m1 ¼ m2 multi., k ¼ 2; m1 –m2 multi., k ¼ 3

0.602*** 0.371 0.324 0.371 0.926*** 0.509** 0.602*** 0.463*

1.251*** 0.695 0.648 0.602 1.482*** 0.926** 0.880** 0.787*

1.992*** 1.204 1.158 1.158 2.547*** 1.760*** 1.806*** 1.390*

5.049*** 2.918 2.872 3.289** 5.327*** 3.937*** 3.844*** 3.613***

8.569*** 5.929* 5.836* 6.438*** 8.522*** 7.179*** 7.133*** 7.226***

14.36*** 10.89 10.98 12.04*** 13.80*** 12.65*** 12.65*** 12.41***

0.079*** 0.037 0.034 0.002 0.077*** 0.041* 0.045* 0.042*

Independence ðb d¼b q 00 þ b q 11  1Þ Student’s t, moving window uni., k ¼ 1 uni., k ¼ 2; m1 ¼ m2 uni., k ¼ 2; m1 –m2 uni., k ¼ 3 multi., k ¼ 1 multi., k ¼ 2; m1 ¼ m2 multi., k ¼ 2; m1 –m2 multi., k ¼ 3

0.096** 0.122** 0.140** 0.004 0.041 0.071* 0.086** 0.006

0.069** 0.065* 0.060* 0.007 0.048* 0.044 0.038 0.007

0.060** 0.021 0.025 0.029 0.076*** 0.032 0.036 0.056*

Student’s t, expanding window uni., k ¼ 1 uni., k ¼ 2; m1 ¼ m2 uni., k ¼ 2; m1 –m2 uni., k ¼ 3 multi., k ¼ 1 multi., k ¼ 2; m1 ¼ m2 multi., k ¼ 2; m1 –m2 multi., k ¼ 3

0.071* 0.122** 0.140** 0.004 0.041 0.086** 0.071* 0.005

0.062** 0.060* 0.065* 0.006 0.048* 0.041 0.044 0.008

0.051* 0.027 0.029 0.029 0.048* 0.036 0.034 0.054*

0.063** 0.035 0.037 0.005 0.054** 0.045* 0.060** 0.016

0.050** 0.010 0.011 0.005 0.052** 0.049** 0.048** 0.028

0.067*** 0.040* 0.036 0.003 0.058*** 0.040* 0.042* 0.031

0.054** 0.012 0.005 0.015 0.055** 0.055** 0.056** 0.019

0.058*** 0.035 0.033 0.010 0.050** 0.031 0.031 0.024

Shown are the results for correct coverage and independence of out-of-sample Value-at-Risk (VaR) measures implied by Student’s t Markov-switching models for an equally weighted portfolio of the French, German, and UK markets. See the footnote of Table 5 for further explanations.

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Conditional probability of high−volatility regime: univariate model for France 1

probability

0.8 0.6 0.4 0.2 0 1999

2000

2001

2002

2003

2004

2005

2006

2007

2008

2009

2010

time Conditional probability of high−volatility regime: univariate model for Germany 1

probability

0.8 0.6 0.4 0.2 0 1999

2000

2001

2002

2003

2004

2005

2006

2007

2008

2009

2010

time Conditional probability of high−volatility regime: univariate model for the UK 1

probability

0.8 0.6 0.4 0.2 0 1999

2000

2001

2002

2003

2004

2005

2006

2007

2008

2009

2010

time Conditional probability of high−volatility regime: multivariate model 1

probability

0.8 0.6 0.4 0.2 0 1999

2000

2001

2002

2003

2004

2005

2006

2007

2008

2009

2010

time Fig. 2. For the out-of-sample period and a two-regime Student’s t model with m1 ¼ m2 and moving window parameter updating scheme, the figure shows the conditional probabilities of the second (high-volatility) regime implied by both the univariate models and the multivariate model (bottom panel).

two-regime model with respect to the independence test; an explanation for this observation will be given in the next subsection.

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M. Haas / Applied Mathematics and Computation 215 (2009) 2103–2119 Table 13 Evaluation of Value-at-Risk measures, September 1999 to April 2009: Univariate Student’s t models with moving estimation window. 100  n

0.25

Model

Coverage ð100  b nÞ

0.5

1

2.5

5

10

France uni., k ¼ 1 uni., k ¼ 2; m1 ¼ m2 uni., k ¼ 2; m1 –m2 uni., k ¼ 3

0.682*** 0.401 0.401 0.401

1.364*** 0.722 0.762* 0.762*

2.286*** 1.203 1.163 1.524**

4.332*** 3.329** 3.289** 3.771***

8.022*** 6.458*** 6.378*** 6.980***

13.08*** 10.83 10.67 12.23***

Germany uni., k ¼ 1 uni., k ¼ 2; m1 ¼ m2 uni., k ¼ 2; m1 –m2 uni., k ¼ 3

0.401 0.201 0.201 0.201

0.963*** 0.602 0.602 0.562

2.447*** 1.324 1.284 1.083

4.813*** 3.490*** 3.450*** 3.169**

8.022*** 6.980*** 6.779*** 6.899***

14.08*** 11.91*** 11.87*** 12.23***

United Kingdom uni., k ¼ 1 uni., k ¼ 2; m1 ¼ m2 uni., k ¼ 2; m1 –m2 uni., k ¼ 3

0.842*** 0.802*** 0.762*** 0.481**

1.284*** 1.123*** 1.123*** 0.882**

2.246*** 1.765*** 1.725*** 1.484**

4.412*** 3.369*** 3.369*** 3.329**

7.702*** 5.576 5.616 5.856*

13.04*** 11.55** 11.47** 11.55**

Equally weighted portfolio uni., k ¼ 1 uni., k ¼ 2; m1 ¼ m2 uni., k ¼ 2; m1 –m2 uni., k ¼ 3

0.722*** 0.521** 0.481** 0.481**

1.484*** 0.842** 0.882** 0.842**

2.367*** 1.645*** 1.564*** 1.284

5.174*** 3.450*** 3.530*** 3.410***

9.105*** 6.498*** 6.458*** 6.699***

14.80*** 11.11* 11.19* 12.27***

Shown are, for the period from September 1999 to April 2009, the results for correct coverage of out-of-sample Value-at-Risk (VaR) measures implied by Student’s t Markov-switching models with a moving estimation window. See the footnote of Table 5 for further explanations.

4.2.2. Results for the entire out-of-sample period Now we turn to the VaR measures for the entire backtesting sample, including the crisis period. For the univariate Student’s t models with a moving estimation window, we report the results in Table 13. As may have been expected, the performance of the models worsens dramatically. Although the two-component models still measure the VaR satisfactorily for the lower VaR levels ðn 6 0:01Þ in case of the French and German markets, they fail in almost any other instance. The reason for this failure can be seen in the fact that the conditional variance of these processes is essentially bounded by the volatility parameter of their high-volatility component, which incorporates the information of previous high-volatility periods. Thus, as rolling parameter estimates evolve slowly, the models will have difficulties to react promptly if volatility rises to a historically new level. This is illustrated in Fig. 3 using the example of the equally weighted portfolio of the three stock markets. For the two-component process with m1 ¼ m2 and moving estimation window, the upper panel of Fig. 3 depicts the out-ofsample return series along with the ex ante VaR measures at the 1% level. Fig. 3 shows that there is a clustering of violations due to the outbreak of extraordinarily high volatility at the end of September 2008. For example, in the period from September 29 to October 31, there are 7 violations of the 1% VaR, corresponding to a 28% hit rate in this period, which clearly deteriorates the overall performance. In view of the fact that the three-component models tend to identify a ‘‘very high-volatility regime” (regime 3 in Tables 3 and 4), we may expect them to do better in such turbulent periods, and the lower panel of Fig. 3, which mimics the upper panel for the three-regime model, shows that this is indeed the case. For this model, we only register 4 violations in the period in question, which is still a hit rate of 16%, however. A similar comparison between the models with two and three components could be made for the high-volatility period beginning in the middle of 2002, which explains the better performance of the three-component models in the independence tests noted at the end of the preceding subsection. 5. Discussion The results reported in Section 4.2 show that, most of the time, a Markov mixture of two Student’s t distributions provides a reliable model for calculating Value-at-Risk measures for national stock market indices. However, during the period of the recent financial crisis, all models failed, where most of the violations leading to this failure were concentrated in a single month, namely October 2008. The reason for the breakdown of the models can be seen in the fact that the conditional volatility of the class of Markov-switching models studied in this paper is essentially bounded by that of the highest-volatility regime. Since rolling parameter estimates adjust only sluggishly, this makes these models unable to cope with a burst of previously unseen volatility. During such periods, three-regime specifications turned out to be much more capable of detecting sudden jumps in the risk level, but their performance was less satisfying in ‘‘normal times”. In view of the recent financial turmoil, it would therefore be interesting to evaluate whether models reacting immediately and more flexibly to shocks, such as GARCH or Markov-switching GARCH [26,17], would have been more appropriate to meet the associated burst of volatility.

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Returns and 1% Value−at−Risk (VaR) implied by a two−regime model 12 10

index returns and 1% VaR

8 6 4 2 0 −2 −4 −6 −8 −10 1999

2000

2001

2002

2003

2004 2005 time

2006

2007

2008

2009

2010

Returns and 1% Value−at−Risk (VaR) implied by a three−regime model 8

index returns and 1% VaR

6 4 2 0 −2 −4 −6 −8 −10 1999

2000

2001

2002

2003

2004 2005 time

2006

2007

2008

2009

Fig. 3. The top panel shows the returns of the equally weighted portfolio of the three MSCI country indices along with ex ante 1% VaR measures implied by the two-regime Student’s t model with m1 ¼ m2 and moving estimation window. The bottom panel repeats this, but for the three-regime Student’s t mixture.

Acknowledgement The author is grateful to an anonymous referee for exceptionally detailed and constructive comments that led to a greatly improved version of the paper. The research for this paper was supported by the German Research Foundation (DFG). References [1] R. Ahrens, S. Reitz, Heterogeneous expectations in the foreign exchange market: evidence from daily DM/US dollar exchange rates, Journal of Evolutionary Economics 15 (2005) 65–82. [2] C. Alexander, Market Risk Analysis, Value-at-Rsik Models, vol. 4, John Wiley & Sons, Chichester, 2008. [3] A. Ang, G. Bekaert, International asset allocation with regime shifts, Review of Financial Studies 15 (2002) 1137–1187. [4] A. Ang, J. Chen, Asymmetric correlations of equity portfolios, Journal of Financial Economics 63 (2002) 443–494. [5] D. Ardia, Bayesian estimation of a Markov-switching threshold asymmetric GARCH model with Student-t innovations, Econometrics Journal 12 (2009) 105–126.

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