- Email: [email protected]
Cross-market volatility index with Factor-DCC
Cross-market volatility index with Factor-DCC Sofiane Aboura, Julien Chevallier PII: DOI: Reference:
S1057-5219(14)00091-X doi: 10.1016/j.irfa.2014.06.003 FINANA 724
To appear in:
International Review of Financial Analysis
Received date: Revised date: Accepted date:
20 January 2014 19 May 2014 17 June 2014
Please cite this article as: Aboura, S. & Chevallier, J., Cross-market volatility index with Factor-DCC, International Review of Financial Analysis (2014), doi: 10.1016/j.irfa.2014.06.003
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Cross-market volatility index with Factor-DCC∗
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Sofiane Aboura† and Julien Chevallier‡
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Revised Version: May 2014
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This paper proposes a new empirical methodology for computing a cross-market volatility index – coined CMIX – based on the Factor DCC-model, implemented on volatility surprises. This approach solves both problems of treating high-dimensional data and estimating time-varying conditional correlations. We provide an application to a multi-asset market data composed of equities, bonds, foreign exchange rates and commodities during 1983-2013. This new methodology may be attractive to asset managers, since it provides a simple way to hedge multi-asset portfolios with derivatives contracts written on the CMIX.
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JEL Codes: C32; G01; F15 Keywords: Cross-Market Index; Factor-DCC; Volatility Surprise; Asset Management
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Acknowledgements: We are indebted to the Editor Brian Lucey and anonymous reviewers who enabled many improvements to this work. For helpful comments, we wish to thank Duc Khuong Nguyen, Lorne Switzer, Pascal Gantenbein, and seminar participants at the 2013 Paris Financial Management Conference (session ‘Asset Allocation and Valuation’). † DRM Finance, Universit´e Paris Dauphine, Place du Mar´echal de Lattre de Tassigny, 75775 Paris Cedex 16, France. Email: [email protected] ‡ Corresponding Author. IPAG Business School (IPAG Lab), 184 Boulevard Saint-Germain, 75006 Paris, France. Tel: +33 (0)1 49 40 73 86. Fax: +33 (0)1 49 40 72 55. Email: [email protected]
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Cross-market volatility index with Factor-DCC
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This paper proposes a new empirical methodology for computing a cross-market volatility index – coined CMIX – based on the Factor DCC-model, implemented on volatility surprises. This approach solves both problems of treating high-dimensional data and estimating time-varying conditional correlations. We provide an application to a multi-asset market data composed of equities, bonds, foreign exchange rates and commodities during 1983-2013. This new methodology may be attractive to asset managers, since it provides a simple way to hedge multi-asset portfolios with derivatives contracts written on the CMIX.
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JEL Codes: C32; G01; F15 Keywords: Cross-Market Index; Factor-DCC; Volatility Surprise; Asset Management
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In asset management, attention is usually focused only on predictable volatility, such as conditional volatility or implied volatility. According to Engle (1993), it is the difference that cannot be forecasted between squared residuals and the conditional variance which is worthy of interest. Such a quantity has been coined ‘volatility surprise’. Hamao, Masulis and Ng (1990) were the first to interpret this quantity as volatility surprise since it lags behind the conditional variance. Note that the volatility surprise component corresponds to an unexpected volatility in Engle (1993), while it corresponds to a foreign unexpected volatility in the context of Hamao et al. (1990). This new concept paved the way for numerous studies. In this paper, we use the ‘volatility surprise’ component in multivariate (instead of univariate) GARCH models. In addition, what is important in a ‘real-world’ context is the ability to track a representative index of a global portfolio of various assets. In practice, asset managers are however facing a problem of estimating high-dimensional matrices of assets. Two main problems arise. The first one lies in determining the correct correlation function that captures precisely the time-varying nature of market data. The second one deals with choosing the proper methodology to estimate loading factors of market data. This paper reconciliates both approaches. Indeed, we construct a new cross-market volatility index that captures the unpredicted risk. Second, we address the high-dimensional issue by resorting to the Factor-DCC framework by Zhang and Chan (2008). Indeed, numerical problems arise for classical multivariate GARCH models. For example, in the VECH model, with a four-asset matrix, the number of parameters grows to 210. A credible alternative would be to compute pairwise correlations by using the DCC model. However, the DCC estimation involves computing the correlation of too many pairs sampled n(n − 1)/2 times, which remains complicated to interpret. To cope with these limitations, Engle (2009) has suggested an approximate approach called the ‘MacGyver Method’ based on bivariate correlations. This estimation strategy uses separate log-likelihood functions that are maximised, and from which the estimators are averaged by using medians. In the same spirit, Engle, Shephard and Sheppard (2008) have proposed a new method of estimating large dimensional time-varying correlations that rely on the sum of partial quasi-likelihood functions instead of a standard full quasi-likelihood approach. In this paper, we follow the Factor-DCC methodology by Zhang and Chan (2008) in a multivariate framework. This model dramatically simplifies the estimation process, by estimating the correlation function on a small number of factors instead of multiple pairwise DCCs. Factor methods arise from the need for macroeconomists and central bankers to follow hundreds of time-series variables as proxies for the state of the economy. Thus, it appears necessary to gather as much information as possible from as many variables as possible. This methodology typically yields to models with a large number of variables and associated parameters to estimate. To that end, factor models have been developed to extract the information in datasets with many variables while, at the same time, keeping the model parsimonious (see Stock and Watson (2005, 2006) for a survey). Zhang and Chan (2008) have adapted these factor modeling techniques to the field of financial
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econometrics, with the inclusion of time-varying conditional correlations which are essential to capture the key trends in time series data. In their article, the authors use standard static Principal Components (PC) methods to extract two factors, by following the classical approach suggested by Stock and Watson (2002a,b). The use of PC methods ensures the identification of the model, since it normalizes all factors to have mean zero and variance one (Forni et al. (2000), Bai and Ng (2007)). Besides, the empirical implementation of the Factor-DCC model proceeds with PC estimates, due to their relative computational tractability. Alternative methods involve Markov chain Monte Carlo (MCMC) estimation of dynamic factors (see for instance Bekiros and Paccagnini (2014) for an application to large panel datasets with the use of MCMC). Besides their heavy computational burden, they require strong identification restrictions which may lead to factors with poor economic content. Hence our choice of the PC methods. To our best knowledge, no statistical tests have been implemented to determine the number of factors in a Factor-DCC model. In their empirical application to the Hong Kong stock market, Zhang and Chan (2008) allow an arbitrary choice of factors that enters the computation of the Factor-DCC model. To address this problem, we borrow a well known criterion from the factor modeling literature. The criterion adopted in our analysis is proposed by Alessi, Barigozzi, and Capasso (henceforth ABC, 2009), which is a refinement of the one by Bai and Ng (2002). ABC define a refined loss function, and evaluate it over a range of the constant and over random subsamples of the data. The estimated number of factors is then the number that is insensitive to neighboring values of the constant, and has no dependance on the subsamples. The choice of ABC’s criteria as opposed to various other methods – e.g. Connor and Korajczyk (1993), Hallin and Liska (2007), Onatski (2009) – is motivated by a Monte Carlo Study implemented on financial data comprising equities, commodities, credit spreads, interest rates, and currencies by Boon and Ielpo (2012). These authors demonstrate that ABC’s criterion is superior in accuracy (overall best in the Monte Carlo study, even when cross-section and serial correlation exist in the data), and precision (less sensitive to whether linear dependencies exist in the financial data, yielding the same estimation regardless of whether the criterion is applied to the data, or to the vector autoregressive residuals). When applying ABC’s criterion to our case study, we can restrict the number of factors to two, which is quite convenient to estimate the conditional correlation function. Our contribution is threefold. First, we update the concept of ‘volatility surprise’ by plugging it in the Factor-DCC model. Second, we apply the Factor-DCC methodology from a cross-market perspective. Indeed, our dataset includes equities, bonds, foreign exchange and commodities, and spans a long-term period of twenty years from 1983 to 2013. To our best knowledge, the Factor-DCC approach has never been implemented by mixing commodity markets with financial assets. Third, we show how to construct a multi-asset volatility index that we denote CrossMarket Volatility Index (CMIX). Contrary to private funds, the methodology used in this paper is of public use, and can be replicated on any underlying dataset tailored to investors’ needs. Besides, we provide in the paper some benchmark of comparison concerning the performance achieved by the CMIX vs. pre-existing volatility indices. The remainder of the article is structured as follows: Section 2 details the Factor-DCC; Section 3 the dataset; Section 4 the results; Section 5 the CMIX composition. Section 6 concludes.
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The Factor DCC model for volatility surprise
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In the following section, we define the concept of volatility surprise from which we extract factor loadings. Second, we briefly recall the use of PC methods to extract factors. Third, the framework of DCC and Factor-DCC models is exposed.
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The volatility surprise
Following Engle (1993), we make use in this study of the volatility component that cannot be forecasted, called ‘volatility surprise’. Let us consider the mean equation of a standard GARCH (1,1) specification:
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where rt is the asset price returns, ǫt is the innovations, and µ is the mean. The purpose of the time-varying conditional σt is to capture as much of the conditional variance in the residual ǫt as follows:
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More specifically, Engle (1993) defines the so-called ’volatility surprise’, ς, as the difference between the squared residuals ǫ2 and the conditional variance σ 2 . For scaling purposes, we normalize this quantity by the conditional variance σ 2 . The ‘normalized volatility surprise’, ς˜, is given by:
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! 2 ǫt − σt2 ς˜t = σt2
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The residuals are extracted from the mean equation of the conditional variance which is chosen among appropriate GARCH specifications using an AIC criterion. In addition, the conditional mean is zero.
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Extracting factors based on Principal Component Analysis
P Given a n-dimensional random variable ~x = (x1 , . . . , xn )′ with covariance matrix ~ x , principal component analysis consists in using a few linear combinations of xi in order to explain the structure P of ~ x (Tsay (2010))1 . Let w ~ i = (wi1 , . . . , win )′ be a n-dimensional real-valued vector, with i = 1, . . . , n. The linear combination of the random vector ~x writes: 1
Note that PCA can also be used for the correlation matrix ρ ~x of ~x.
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– yi and yj are uncorrelated for i 6= j; – the variances of yi are as large as possible.
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where ~x are the returns of the n variables, and yi is the return of the portfolio which assigns the weight wij to the jth variable. As stated above, the main insight behind PCA consists in finding the linear combinations w ~ i , such that:
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Therefore, the first principal component of ~x is the linear combination y1 = w ~ 1′ ~x which maxi′ mizes V ar(y1 ) subject to w ~ 1w ~ 1 = 1. The second principal component of ~x is the linear combination ′ y2 = w ~ 2 ~x which maximizes V ar(y2 ) subject to w ~ 2′ w ~ 2 = 1 and Cov(y1 , y2 ) = 0. The ith principal component of ~x is the linear combination yi = w ~ i′ ~x which maximizes V ar(yi ) subject to w ~ i′ w ~i = 1 and Cov(yi , yj ) = 0 for j = 1, . . . , i − 1. The interested reader can refer to Tsay (2010) for the resolution of this problem, based on the P spectral decomposition of the covariance matrix ~ x . The result is that the proportion of the total variance in ~x explained by the ith principal component is the ratio between the ith eigenvalue and P the sum of all eigenvalues of ~ x . In empirical applications, it is also useful to compute the cumulative proportion of total variance explained by the first i principal components. By following this procedure, a small number of factors (but with a large corresponding cumulative proportion) can be selected.
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The DCC model
Let ς˜t denote a n × 1 vector of volatility surprises at time t, which is assumed to be conditionally normal with mean zero and covariance n × n matrix Ht : ς˜t |Ωt−1 ∼ N (0, Ht )
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where Ωt−1 represents the information set at time t − 1. The conditional covariance matrix Ht can be decomposed as follows (Engle (2009)): Ht = Dt Rt Dt
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!p p p Rt is the n × n time-varying correlation matrix. Dt = diag h1,t , ..., hi,t , ..., hn,t is the n × n diagonal matrix of time-varying standard deviations extracted from univariate GARCH models p with hi,t = σi,t on the ith diagonal. The dynamic conditional correlation structure in matrix form is given by: Rt = Q∗−1 Qt Q∗−1 t t
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The Factor DCC model
Factor models have been at the center of asset pricing theory. The idea is to specify a number of factors that summarize all the existing dependence between asset returns from which a matrix of variance-covariance can be estimated. These factors impact on the correlation function over time. Ding (1994) and Alexander (2000, 2001) set up the orthogonal GARCH (O-GARCH) model by applying principal component analysis techniques to extract m principal components. More recently, Zhang and Chan (2008) resorted to independent component analysis to construct Factor GARCH models in the generalized orthogonal GARCH (GO-GARCH) framework, where the orthogonality constraint of the factor loading matrix is relaxed. This model is well suited to high-dimensional data. It can be estimated in two steps. In the first step, the factors are estimated given some statistical criteria. In the second step, a GARCH model is estimated for each factor. The innovation
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ǫt are assumed to be generated from the latent uncorrelated factors yt = (y1,t , ..., ym,t ) by linear transformation such as: ǫt = Ayt Each factor yi,t is described as a GARCH(1,1) process as:
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From now, the conditional covariance matrix of series ς˜ can be computed. But, in the case the loading factors might still have some conditional correlation, Zhang and Chan (2008) propose to model the remaining time-varying conditional correlation between factors with the DCC model. This approach is named the Factor-DCC model. Actually, the Factor-DCC model is simply constructed as an extension of the Factor GARCH model in order to model the conditional correlation between factors. The conditional covariance matrix between factors is given by:
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where Rt denotes the conditional correlation matrix of yt expressed in the form of the equations !p (7) and (9). Dt = diag hyi,t is a diagonal matrix composed of the square root of the diagonal elements of the conditional covariance matrix Σt of factors yi,t with i = 1, 2, ..., m, with m < n. The associated quasi-log-likelihood function is given by Zhang and Chan (2008). Among the models proposed, we choose to implement the specification of the conditional-decorrelation DCC (CD-DCC) since it gives the best performance when using financial data. In this paper, it is essential to catch the idea that the inputs to the Factor-DCC model are the volatility surprises ς˜ computed from eq.(3), instead of returns in the classical approach. By doing so, we propose a new method to create a cross-market volatility index.
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Descriptive statistics are shown in Table 1. The interested reader can check (by means of one-sided t-tests) that the ‘volatility surprise’ components calculated from eq.(3) are mean-zero, as they are re-used as an input to the Factor-DCC model. The data is retrieved from Thomson Financial Datastream. Thirty time series have been gathered in daily frequency over a period of thirty years, from January 25, 1983 to January 25, 2013. The total number of observations is equal to 7,828. The data can be decomposed into subgroups:
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1. Equities: F T SE100 is the FTSE 100 Price Index; SP 500 is the S&P 500 Price Index; DJIA is the Dow Jones Industrial Average Price Index.
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2. Bonds: Bonds is the US Benchmark 10-Year DS Government Index - Clean Price Index.
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3. Foreign Exchange Rates: EU DOLLR is the Euro to USD (WMR & DS) Exchange Rate; U KEU RSP is the UK GBP to Euro (WMR & DS) Exchange Rate; U KDOLLR is the UK
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6. Precious Metals Individual Price Series: Gold is the S&P GSCI Gold Spot Price Index; Silver is the Silver Fix LBM Cash in Cents/Troy ounce; P latinum is the London Platinum Free Market in USD/Troy ounce.
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7. Industrial Metals Individual Price Series: Copper is the S&P GSCI Copper Spot Price Index; Aluminum is the LME Aluminum 99.7% 3-Month in USD/MT; N ickel is the LME Nickel 3-Month in USD/MT.
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8. Agricultural Individual Price Series: Cof f ee is the S&P GSCI Coffee Spot Price Index; W heat is the S&P GSCI Wheat (CBOT) Spot Price Index; Soya Oil is the Soya Oil Crude Decatur Cents/Lb; Corn is the Corn No.2 Yellow in Cents/Bushel; Sugar is the Raw Sugar ISA Daily Price in Cents/Lb; Cocoa Butter is the Cocoa Butter African US Del in USD/MT; Cotton is the Cotton 1.1/16Str Low-Midl, Memph in Cents/Lb; Soybean is the Soyabeans No.1 Yellow in Cents/Bushel.
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5. Commodity Disaggregated Indices: Industrial M etals is the S&P GSCI Industrial Metals Spot Price Index; Energy is the S&P GSCI Energy Spot Price Index; Agricultural is the S&P GSCI Agricultural Spot Price Index; Livestock is the S&P GSCI Livestock Spot Price Index; Grains is the S&P GSCI Grains Spot Price Index; P reciousM etals is the S&P GSCI Precious Metals Spot Price Index.
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4. Commodity Aggregated Indices: SP GSCI is the S&P GSCI Commodity Spot Price Index; CRB is the CRB BLS Spot Price Index.
9. Energy Individual Price Series: Brent is the Crude Oil Brent Current Month FOB in USD/BBL.
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GBP to USD (WMR) Exchange Rate.
Individual commodity price series have been converted to USD when necessary using the appropriate bilateral foreign exchange rates from Thomson Financial Datastream. All time series are found to be stationary when transformed to volatility surprises (as explained in section 2.1). Standard unit root tests (ADF, PP, KPSS) are not provided to save space, and can be accessed upon request to the authors. Next, we present the results obtained when performing the estimation of the Factor-DCC model with these 30 time series of ‘volatility surprises’.
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Table 1: Descriptive statistics: ‘Volatility surprise’ Kurtosis
Standardized Skewness
Standardized Kurtosis
-0.04 -0.10 -0.11
284.92 258.26 279.02
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279.46 844.44 112.43
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187.64 216.32 181.24
5.21 11.04 4.80
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0.26 1.08 0.22
0.04 0.09 0.17 0.00 0.08 0.05 0.09 0.16 0.17 0.13 -0.04 -0.04 0.41 -0.05 -0.03 0.07 -0.55 -0.36 -0.01 -0.10 0.06 0.03 0.11
187.59 243.23 215.32 183.11 177.74 161.22 180.35 263.45 279.12 218.20 216.51 216.51 239.66 191.47 220.32 251.30 311.04 149.93 208.10 585.85 249.32 263.95 252.38
7.38 18.33 11.43 5.89 4.66 3.44 5.01 12.12 12.20 9.93 7.71 7.71 14.57 5.55 9.64 10.16 87.90 87.82 9.25 69.72 10.11 22.80 13.81
123.23 710.68 263.96 74.83 38.60 21.28 44.44 255.34 242.96 208.25 110.04 110.04 461.96 56.14 187.44 203.06 775.70 775.10 211.33 563.23 170.39 981.63 342.77
0.03 0.07 0.05 0.03 0.02 0.02 0.02 0.04 0.04 0.04 0.03 0.03 0.06 0.02 0.04 0.04 0.02 0.05 0.04 0.11 0.04 0.08 0.05
0.65 2.92 1.22 0.40 0.21 0.13 0.24 0.96 0.87 0.95 0.50 0.50 1.92 0.29 0.85 0.80 2.49 5.16 1.01 0.96 0.68 3.71 1.35
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Equities FTSE100 SP500 DJIA Bonds 10-Year U.S. Govt. Benchmark Bond Index Exchange Rates EUDOLLR UKEURSP UKDOLLR Commodities SP GSCI CRB Industrial Metals Energy Agricultural Livestock Grains Precious Metals Gold Copper Coffee Wheat Brent Soya Oil Corn Sugar Cocoa Butter Cotton Soybean Silver Platinum Aluminum Nickel
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Note: The number of daily observations is equal to 7,828 from January 25, 1983 to January 25, 2013. The standardized skewness and kurtosis are the skewness and kurtosis of the ‘volatility surprise’ component standardized by their estimated standard deviation. The source of the series is Thomson Financial Datastream.
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For the Factor-DCC implementation, we need to determine a priori a methodology to find the number of factors, which is not provided in the paper by Zhang and Chan (2008). For that reason, the choice of the number of factors is either subject to an arbitrary choice, or it can be determined by using an appropriate methodology. In this paper, we propose to use the formal statistical approach by Alessi et al. (2009) to determine the number of factors. Their criterion modifies the criterion by Bai and Ng (2002) by multiplying the penalty function by a positive real number, which allows to tune its penalizing power by analogy with the method developed by Hallin and Liˇska (2007). The criterion by Alessi et al. (2009) offers the advantage of achieving better performance than that by Bai and Ng (2002), especially in the case of large idiosyncratic disturbances. Therefore, we choose to implement the methodology by Alessi et al. (2009), which is computationally easy to implement by iteratively applying the criterion of Bai and Ng (2002). ∗T Let rc;N denote the number of factors pointed out by the Alessi et al. (2009) criterion with T the time and N the cross-section dimensions of the dataset. The test statistic use the two information criteria: NT N +T T∗ log , c ∈ R+ IC1,N (k) = log [V (k)] + c k (16) NT N +T
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We can divide the Factor-DCC model’s econometric methodology by Zhang and Chan (2008) into two steps. In the first step, we extract the common factors from a volatility surprise dataset of equities, bonds, foreign exchange rates and commodities by using the Principal Components approach of Stock and Watson (2002a,b). Based on a statistical criterion, we restrict the number of factors to the first two Principal Components extracted from the dataset of daily time-series. In the second step, the CD-DCC, which represents the time-varying conditional correlation equation of the model, has been estimated by following thoroughly Zhang and Chan’s (2008) approach.
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T T In Figure 1, the number of factors rc,N is represented by the solid black line. rc,N is greater T than one, and smaller than two. Hence, we round up rc,N , and conclude in favor of the presence of two factors. The stability of the estimated number of factors with respect to sample size is
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ABC estimated number of factors (volatility surprise) r*T c;N
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considering subsamples of sizes (nj , τj ) with j = 0, . . . , J such that n0 = 0 < n1 < n2 < . . . < nJ = N and τ0 = 0 < τ1 < τ2 < . . . < τJ = T . The empirical variance, represented by dotted lines in Figure 1, confirms our diagnostic. Hence, we present below the results obtained for the Principal Component Analysis with a two-factor model. Figure 2 shows the factor loadings. It shows the contribution of each individual series to each factor. Visually, we observe that the contributions to the factors differ largely across variables. Precious metals, energy products and equities account for the largest weights in both factors (see Table 1 for the list of variables). Table 2 lists the fraction of total variance in the dataset explained by the two factors, along with the volatility time-series in the dataset that each of them is most strongly correlated with. The cumulative variance of factors is given by the sum of their eigenvalues. The total variance is the sum of the variances of the individual volatility series in the dataset, measured as the sum of
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their eigenvalues. Factor 1 accounts for 21% of the variability contained in the dataset, while Factor 2 accounts for 18% of the total variation. This confirms that the Factor-DCC model captures a significant fraction (approximately 40%) of the total variation of the 30 volatility series contained in our dataset. Factor 1 is characterized by a mix of precious metals and energy contributions. Factor 2 captures mainly influences from gold and energy products. The over-weight in precious metals can be understood as a means to hedge away volatility risk. On the contrary, energy products
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are known for their pro-cyclical – even ‘bubble-like’ – behavior. Taken together, these results are informative of investors’ behaviour in presence of (volatility) risk: gold serves as a refuge for value (Baudry et al. (2011)), while energy (mainly petroleum) products can be seen as speculative investment vehicles (Buyuksahin et al. (2011)). When looking at the composition of the CMIX, these individual volatility series contributions to the index are not surprising.
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Table 2: Share of variance explained by factors Factor 1 (21.00% of total variance) R2 Precious Metals 0.77 Gold 0.75 S&P GSCI 0.28 Oil Brent 0.21 GSCI Energy 0.17 Factor 2 (17.51% of total variance) S&P GSCI 0.32 Oil Brent 0.31 GSCI Energy 0.26 Gold 0.22 Precious Metals 0.22 Note: This table presents the R-squared of univariate regressions of the factors extracted from the dataset on all individual volatility series. For each factor, a list of five variables with which the factors
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Table 3 displays the estimates of the Factor-DCC parameters, which appear standard compared to GARCH-family models. In addition, the average conditional correlation between the two factors is equal to -0.01, which means that the two factors are almost decorrelated as is the objective of the CD-DCC model. Recall that, in the conditional-decorrelation DCC model, the factors are conditionally as uncorrelated as possible, which should describe the multivariate financial time series best. For simplicity, the four parameters DCC(p), DCC(q), ARCH(p), and GARCH(q) are set equal to 1. Figure 3 shows the conditional correlation between the first and second factors estimated by PCA. We can see that the conditional correlation estimated by the CD-DCC model is timevarying with a zero-average correlation. This result is conform to the objective of the conditionaldecorrelation GARCH model. We remark that some during very brief time periods the conditional correlation exhibits various signs, and some clustering effects as is usual for volatility series. To cite a few, we identify the 1987 stock market crash, the 1989 junk bond market collapse, the 1991 first Gulf war, the 1997-98 LTCM and currency crises, the 2000 dot-com bubble burst, the 2001-02
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Table 3: Factor-DCC estimates for volatility surprise Model CD-DCC Parameters α ˆ DCC 0.0234*** (0.0112) βˆDCC 0.9109*** (0.1741) Diagnostics Log-L -45188.6 BIC -9.0297 ρij -0.01
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Enron bankruptcy, or the 2007-08 sub-primes crisis. Next, we detail how to obtain the cross-market volatility index (CMIX) based on this new formal statistical approach.
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Note: Numbers in parentheses are the standard errors. Statistical significance is indicated at the 1%(***), 5%(**), and 10%(*) levels. In the Factor-DCC model, α ˆ DCC and βˆDCC are conditional-decorrelation (CD) GARCH parameters, in which the factors are conditionally as uncorrelated as possible. Log L is the Log-Likelihood, and BIC is the Approximate Bayesian Information criterion. ρij is the average conditional correlation between the estimated variances of two factors.
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The Cross-Market Index for Volatility Surprise
Several computational steps are required to go from the Factor loadings to the composition of the Cross-Market Volatility Index (CMIX):
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– This absolute value |Si,n | is divided by the sum of each factor loading’s absolute values P30 n=1 |Si,n |, and multiplied by 100.
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– For each factor loading Fi , i = {1, 2}, the absolute value of each series’ Si,n contribution, n = 1, . . . , 30 is taken.
– The contribution of each series to the index is computed as the average percentage between P |+|S | |S the two factor loadings’ contributions ωn = 1,n 2 2,n , with 30 n=1 ωn = 1.
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These calculations provide for each series its contribution in percentage to the CMIX.
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Figure 4 reveals the composition of the CMIX in a pie chart. The average contribution of each class of asset is equal to 5%, with a minimum of 0.8% (DJIA) and a maximum of 10% (Gold). Interestingly, we notice that the composition of the CMIX seems over-weighted in precious metals (around 30%). This is consistent with the view of precious metals as a safe-haven during bear financial markets followed by recessions. The CMIX reflects the ‘right’ level of risk among various asset classes. Indeed, the overweight of precious metals is a consequence of the large swings usually observed in their volatilities. As an example, during the 2008-2011 economic slowdown, the Gold ounce rised from 800 USD to 1900 USD, and declined thereafter. Due to our computational methodology, the CMIX absorbs this information and restitutes it in real-time by showing such an impressive percentage allocated to precious metals.
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Stemming from this composition in percentage, we can create the time series of the Cross Market Index CM IX. It is computed as an average of the 30 assets composing the index, P30 weighted by their respective contribution: CM IXt = n=1 ωn Pn,t . Equivalently, CM IX = P P30 t=7828 n=1 ωn Pn,t . To obtain the graph of the CMIX, we report the contribution (in %) of each volatility surprise series to the index, as detailed in the pie chart below.
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Figure 4: Composition of the Cross-Market Index for volatility surprise
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We investigate the benefits for asset managers to use the Cross-Market Volatility Index. The first use of the CMIX index could be for hedging strategies of portfolios consisting in traditional assets. In that case, derivatives contracts such as options or variance swaps would be appropriate instruments. The second use could be to consider the cross-market volatility index as an alternative class of asset, in itself. In that latter case, we propose an illustration. Indeed, this section reports the Sharpe ratios resulting from a mean-variance portfolio optimization problem2 involving the CMIX vs. other pre-existing volatility indices. The purpose of this exercise consists in assessing the relative performance of the newly created CMIX compared to its likely competitors currently used by market practitioners.
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The CMIX index appears in Figure 5 in comparison with the MSCI World Minimum Volatility index and the VIX (accessed from Thomson Financial Datastream). The CMIX appears with a base 100 on January 25, 1983. Its scale is given by the left Y-axis (along with the VIX multiplied by ten for comparability). The scale of the MSCI World volatility index is given by the right Y-axis. As it can be seen from the graph, the obvious advantage of relying on our methodology is that it allows to compute volatility indices with quite a long historical. On the contrary, the MSCI Volatility index is available only from December 31, 1998 onwards, while the VIX is available since 1990. Compared to the VIX – i.e. the CBOE measure of the implied volatility for S&P 500 index options – the CMIX exhibits similar volatility peaks. However, given that the composition of our volatility index and the VIX widely differ, both indices start to behave differently from 2009 onwards. On the one hand, the VIX returned to lower volatility values past Q4:2008. On the other hand, the CMIX picked up other influences (from commodity markets for instance), which might explain its increasing slope in the current period. Compared to the MSCI World Volatility index– which aims to reflect the performance characteristics of a minimum variance strategy in the large and mid cap equity universe across 24 countries – the CMIX appears much more stable. Indeed, there exists a stark difference between the two volatility indices during 2001-08: the CMIX has a positive slope until the summer 2008, while the MSCI World index is characterized by a brutal increase (and subsequent drop).
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Table 4: Asset performance VIX MSCI Volatility Index CMIX
Portfolio Returns (%) [2.86 - 3.28] [1.04 - 1.19] [1.39 - 1.56]
Portfolio Risks (%) [9.92 - 47.17] [2.07 - 23.86] [5.43 - 28.67]
Sharpe Ratio 0.16 0.15 0.15
Information Ratio 0.18 0.17 0.17
We include the risk-free rate by using the 30-day U.S. Treasury Bill, and consider three com-
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peting portfolios invested either in the VIX, the MSCI World Minimum Volatility – Price Index, or the CMIX. The purpose of our stylized exercise is to compare the relative performance of these volatility indices for asset managers. This section uses data from the creation of the MSCI Volatility Index on December 31, 1998 to January 25, 2013. For the VIX, in the first two columns of Table 4, the range of monthly returns is between 2.86% and 3.28%. These rather high expected returns come at the price of risk levels in the range of 9.92% to 47.17%. One may cautiously attempt to interpret these results. The CBOE Volatility Index (VIX) is a central measure of market expectations of near-term volatility conveyed by S&P 500 stock index option prices. Since its creation in 2003 (and handback calculated to 1990 by the CBOE), the VIX has been widely considered as a barometer of investor sentiment and market volatility. Hence, we find that it can bring appealing expected returns to investors, while remaining a risky strategy. The Sharpe ratio (Sharpe (1966)) is a well-known measure of return-to-risk which plays a central role in portfolio analysis. More precisely, a portfolio that maximizes the Sharpe ratio is also the tangency portfolio on the efficient frontier from the mutual fund theorem. As investors should be driven by Sharpe ratios, they should be both interested in expected returns and in the
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Figure 5: Comparison of the CMIX with the re-scaled VIX (VIX × 10) and the MSCI World Volatility Indices
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This paper proposes a new empirical methodology for computing a cross-market volatility index – coined CMIX – based on the Factor DCC-model. This approach solves both problems of treating high-dimensional data and estimating time-varying conditional correlations. We provide an application to a multi-asset market data composed of equities, bonds, foreign exchange rates and commodities during 1983-2013. This new methodology may be attractve to asset managers as it provides a simple way to hedge multi-asset portfolios with derivatives contracts written on the CMIX. Future research will use this index for option pricing purposes to gauge its hedging capability.
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risk associated to these returns. The maximum Sharpe ratio for the VIX is given in the third column of Table 4. Its current value (0.18) is lower than the ones found in previous literature.3 A related ratio is the information ratio, which uses relative returns (see Goodwin (1988)). It is given in the last column of Table 4, and confirms the previous result. For the MSCI World Minimum Volatility Index, we obtain lower expected returns (lower bound 1.04%, upper bound 1.19%). These are accompanied by more reasonable risk levels (lower bound 2.07%), although investing in volatility indices can be overall quite risky (upper bound 23.86%) – even for an index coined ‘world minimum volatility’.4 Recall that the MSCI World Minimum Volatility Index is based on the composition of its MSCI parent index, and designed to achieve the lowest volatility for a given set of constraints. The Sharpe ratio coefficient (0.15) is lower than that of the VIX. For the CMIX, constructed following the methodology in this paper including 30 assets, we obtain a reasonable performance with expected returns in the range of 1.39%-1.56%. Investing in the CMIX turns out to be slightly more exposed in terms of risks (lower bound 5.43%, upper bound 28.67%) than in the case of the MSCI World Minimum Volatility index. These results can be partially explained by the overweight of the CMIX in Precious Metals and Gold. The bottom line is that investors could certainly consider including the CMIX as a candidate in their investment universe (along with the pre-existing MSCI volatility index). Or, they can tailor another volatility index that suits them best by replicating our methodology. Due to its reliance on S&P 500 option prices, an investment strategy based on the VIX is characterized by (nearly twice) higher levels of risks.
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[4] Bai, J., Ng, S. 2002, Determining the number of factors in approximate factor models. Econometrica 70(1), 191-221
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[3] Alexander, C., 2001, Market models: A guide to financial data analysis, Wiley, Chichester, UK
[5] Bai, J., Ng, S. 2007, Determining the number of primitive shocks in factor models. Journal of Business and Economic Statistics 25(1), 52-60 [6] Baudry, P., Collard, F., Portier, F. 2011, Gold rush fever in business cycles. Journal of Monetary Economics 58, 84-97
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[2] Alexander, C., 2000, Orthogonal methods for generating large positive semi-definite covariance matrices, ISMA Center Discussion Papers in Finance 2000-06, University of Reading, UK
[7] Bekiros, S., Paccagnini, A. 2014. Bayesian forecasting with small and medium scale factoraugmented vector autoregressive DSGE models. Computational Statistics and Data Analysis 71, 298-323
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[8] Boon, LN., Ielpo, F. 2012, Determining the maximum number of uncorrelated strategies in a global portfolio. SSRN Working Paper, Social Science Research Network, USA [9] Buyuksahin, B., Harris, J.H. 2011, Do Speculators Drive Crude Oil Futures Prices? Energy Journal 32, 167-202
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[1] Alessi, L., Barigozzi, M., Capasso, M. 2010, Improved penalization for determining the number of factors in approximate factor models. Statistics and Probability Letters 80(23-24):1806-1813
[10] Connor, G., Korajczyk, RA. 1993, A test for the number of factors in an approximate factor model. Journal of Finance 48(4), 1263-1291
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[11] Dash, S., Moran, M.T. 2005, VIX as a Companion for Hedge Fund Portfolios. Journal of Alternative Investments 8, 75-80
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[14] Engle, R., 2009, Anticipating correlations: A new paradigm for risk management, Princeton University Press, USA, 154 pages
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Novel methodology to create volatility index. Update on the concept of ‘volatility surprise’. Methodology reproducible by asset managers. Cross Market Volatility Index computed from Factor DCC model.
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Submitted to the International Review of Financial Analysis
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For the paper entitled “Cross-market volatility index with Factor-DCC”
S1057-5219(14)00091-X doi: 10.1016/j.irfa.2014.06.003 FINANA 724
To appear in:
International Review of Financial Analysis
Received date: Revised date: Accepted date:
20 January 2014 19 May 2014 17 June 2014
Please cite this article as: Aboura, S. & Chevallier, J., Cross-market volatility index with Factor-DCC, International Review of Financial Analysis (2014), doi: 10.1016/j.irfa.2014.06.003
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Cross-market volatility index with Factor-DCC∗
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Sofiane Aboura† and Julien Chevallier‡
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Revised Version: May 2014
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This paper proposes a new empirical methodology for computing a cross-market volatility index – coined CMIX – based on the Factor DCC-model, implemented on volatility surprises. This approach solves both problems of treating high-dimensional data and estimating time-varying conditional correlations. We provide an application to a multi-asset market data composed of equities, bonds, foreign exchange rates and commodities during 1983-2013. This new methodology may be attractive to asset managers, since it provides a simple way to hedge multi-asset portfolios with derivatives contracts written on the CMIX.
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JEL Codes: C32; G01; F15 Keywords: Cross-Market Index; Factor-DCC; Volatility Surprise; Asset Management
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Acknowledgements: We are indebted to the Editor Brian Lucey and anonymous reviewers who enabled many improvements to this work. For helpful comments, we wish to thank Duc Khuong Nguyen, Lorne Switzer, Pascal Gantenbein, and seminar participants at the 2013 Paris Financial Management Conference (session ‘Asset Allocation and Valuation’). † DRM Finance, Universit´e Paris Dauphine, Place du Mar´echal de Lattre de Tassigny, 75775 Paris Cedex 16, France. Email: [email protected] ‡ Corresponding Author. IPAG Business School (IPAG Lab), 184 Boulevard Saint-Germain, 75006 Paris, France. Tel: +33 (0)1 49 40 73 86. Fax: +33 (0)1 49 40 72 55. Email: [email protected]
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Cross-market volatility index with Factor-DCC
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Abstract
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This paper proposes a new empirical methodology for computing a cross-market volatility index – coined CMIX – based on the Factor DCC-model, implemented on volatility surprises. This approach solves both problems of treating high-dimensional data and estimating time-varying conditional correlations. We provide an application to a multi-asset market data composed of equities, bonds, foreign exchange rates and commodities during 1983-2013. This new methodology may be attractive to asset managers, since it provides a simple way to hedge multi-asset portfolios with derivatives contracts written on the CMIX.
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JEL Codes: C32; G01; F15 Keywords: Cross-Market Index; Factor-DCC; Volatility Surprise; Asset Management
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In asset management, attention is usually focused only on predictable volatility, such as conditional volatility or implied volatility. According to Engle (1993), it is the difference that cannot be forecasted between squared residuals and the conditional variance which is worthy of interest. Such a quantity has been coined ‘volatility surprise’. Hamao, Masulis and Ng (1990) were the first to interpret this quantity as volatility surprise since it lags behind the conditional variance. Note that the volatility surprise component corresponds to an unexpected volatility in Engle (1993), while it corresponds to a foreign unexpected volatility in the context of Hamao et al. (1990). This new concept paved the way for numerous studies. In this paper, we use the ‘volatility surprise’ component in multivariate (instead of univariate) GARCH models. In addition, what is important in a ‘real-world’ context is the ability to track a representative index of a global portfolio of various assets. In practice, asset managers are however facing a problem of estimating high-dimensional matrices of assets. Two main problems arise. The first one lies in determining the correct correlation function that captures precisely the time-varying nature of market data. The second one deals with choosing the proper methodology to estimate loading factors of market data. This paper reconciliates both approaches. Indeed, we construct a new cross-market volatility index that captures the unpredicted risk. Second, we address the high-dimensional issue by resorting to the Factor-DCC framework by Zhang and Chan (2008). Indeed, numerical problems arise for classical multivariate GARCH models. For example, in the VECH model, with a four-asset matrix, the number of parameters grows to 210. A credible alternative would be to compute pairwise correlations by using the DCC model. However, the DCC estimation involves computing the correlation of too many pairs sampled n(n − 1)/2 times, which remains complicated to interpret. To cope with these limitations, Engle (2009) has suggested an approximate approach called the ‘MacGyver Method’ based on bivariate correlations. This estimation strategy uses separate log-likelihood functions that are maximised, and from which the estimators are averaged by using medians. In the same spirit, Engle, Shephard and Sheppard (2008) have proposed a new method of estimating large dimensional time-varying correlations that rely on the sum of partial quasi-likelihood functions instead of a standard full quasi-likelihood approach. In this paper, we follow the Factor-DCC methodology by Zhang and Chan (2008) in a multivariate framework. This model dramatically simplifies the estimation process, by estimating the correlation function on a small number of factors instead of multiple pairwise DCCs. Factor methods arise from the need for macroeconomists and central bankers to follow hundreds of time-series variables as proxies for the state of the economy. Thus, it appears necessary to gather as much information as possible from as many variables as possible. This methodology typically yields to models with a large number of variables and associated parameters to estimate. To that end, factor models have been developed to extract the information in datasets with many variables while, at the same time, keeping the model parsimonious (see Stock and Watson (2005, 2006) for a survey). Zhang and Chan (2008) have adapted these factor modeling techniques to the field of financial
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econometrics, with the inclusion of time-varying conditional correlations which are essential to capture the key trends in time series data. In their article, the authors use standard static Principal Components (PC) methods to extract two factors, by following the classical approach suggested by Stock and Watson (2002a,b). The use of PC methods ensures the identification of the model, since it normalizes all factors to have mean zero and variance one (Forni et al. (2000), Bai and Ng (2007)). Besides, the empirical implementation of the Factor-DCC model proceeds with PC estimates, due to their relative computational tractability. Alternative methods involve Markov chain Monte Carlo (MCMC) estimation of dynamic factors (see for instance Bekiros and Paccagnini (2014) for an application to large panel datasets with the use of MCMC). Besides their heavy computational burden, they require strong identification restrictions which may lead to factors with poor economic content. Hence our choice of the PC methods. To our best knowledge, no statistical tests have been implemented to determine the number of factors in a Factor-DCC model. In their empirical application to the Hong Kong stock market, Zhang and Chan (2008) allow an arbitrary choice of factors that enters the computation of the Factor-DCC model. To address this problem, we borrow a well known criterion from the factor modeling literature. The criterion adopted in our analysis is proposed by Alessi, Barigozzi, and Capasso (henceforth ABC, 2009), which is a refinement of the one by Bai and Ng (2002). ABC define a refined loss function, and evaluate it over a range of the constant and over random subsamples of the data. The estimated number of factors is then the number that is insensitive to neighboring values of the constant, and has no dependance on the subsamples. The choice of ABC’s criteria as opposed to various other methods – e.g. Connor and Korajczyk (1993), Hallin and Liska (2007), Onatski (2009) – is motivated by a Monte Carlo Study implemented on financial data comprising equities, commodities, credit spreads, interest rates, and currencies by Boon and Ielpo (2012). These authors demonstrate that ABC’s criterion is superior in accuracy (overall best in the Monte Carlo study, even when cross-section and serial correlation exist in the data), and precision (less sensitive to whether linear dependencies exist in the financial data, yielding the same estimation regardless of whether the criterion is applied to the data, or to the vector autoregressive residuals). When applying ABC’s criterion to our case study, we can restrict the number of factors to two, which is quite convenient to estimate the conditional correlation function. Our contribution is threefold. First, we update the concept of ‘volatility surprise’ by plugging it in the Factor-DCC model. Second, we apply the Factor-DCC methodology from a cross-market perspective. Indeed, our dataset includes equities, bonds, foreign exchange and commodities, and spans a long-term period of twenty years from 1983 to 2013. To our best knowledge, the Factor-DCC approach has never been implemented by mixing commodity markets with financial assets. Third, we show how to construct a multi-asset volatility index that we denote CrossMarket Volatility Index (CMIX). Contrary to private funds, the methodology used in this paper is of public use, and can be replicated on any underlying dataset tailored to investors’ needs. Besides, we provide in the paper some benchmark of comparison concerning the performance achieved by the CMIX vs. pre-existing volatility indices. The remainder of the article is structured as follows: Section 2 details the Factor-DCC; Section 3 the dataset; Section 4 the results; Section 5 the CMIX composition. Section 6 concludes.
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In the following section, we define the concept of volatility surprise from which we extract factor loadings. Second, we briefly recall the use of PC methods to extract factors. Third, the framework of DCC and Factor-DCC models is exposed.
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where rt is the asset price returns, ǫt is the innovations, and µ is the mean. The purpose of the time-varying conditional σt is to capture as much of the conditional variance in the residual ǫt as follows:
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More specifically, Engle (1993) defines the so-called ’volatility surprise’, ς, as the difference between the squared residuals ǫ2 and the conditional variance σ 2 . For scaling purposes, we normalize this quantity by the conditional variance σ 2 . The ‘normalized volatility surprise’, ς˜, is given by:
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Extracting factors based on Principal Component Analysis
P Given a n-dimensional random variable ~x = (x1 , . . . , xn )′ with covariance matrix ~ x , principal component analysis consists in using a few linear combinations of xi in order to explain the structure P of ~ x (Tsay (2010))1 . Let w ~ i = (wi1 , . . . , win )′ be a n-dimensional real-valued vector, with i = 1, . . . , n. The linear combination of the random vector ~x writes: 1
Note that PCA can also be used for the correlation matrix ρ ~x of ~x.
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– yi and yj are uncorrelated for i 6= j; – the variances of yi are as large as possible.
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where ~x are the returns of the n variables, and yi is the return of the portfolio which assigns the weight wij to the jth variable. As stated above, the main insight behind PCA consists in finding the linear combinations w ~ i , such that:
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Therefore, the first principal component of ~x is the linear combination y1 = w ~ 1′ ~x which maxi′ mizes V ar(y1 ) subject to w ~ 1w ~ 1 = 1. The second principal component of ~x is the linear combination ′ y2 = w ~ 2 ~x which maximizes V ar(y2 ) subject to w ~ 2′ w ~ 2 = 1 and Cov(y1 , y2 ) = 0. The ith principal component of ~x is the linear combination yi = w ~ i′ ~x which maximizes V ar(yi ) subject to w ~ i′ w ~i = 1 and Cov(yi , yj ) = 0 for j = 1, . . . , i − 1. The interested reader can refer to Tsay (2010) for the resolution of this problem, based on the P spectral decomposition of the covariance matrix ~ x . The result is that the proportion of the total variance in ~x explained by the ith principal component is the ratio between the ith eigenvalue and P the sum of all eigenvalues of ~ x . In empirical applications, it is also useful to compute the cumulative proportion of total variance explained by the first i principal components. By following this procedure, a small number of factors (but with a large corresponding cumulative proportion) can be selected.
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The DCC model
Let ς˜t denote a n × 1 vector of volatility surprises at time t, which is assumed to be conditionally normal with mean zero and covariance n × n matrix Ht : ς˜t |Ωt−1 ∼ N (0, Ht )
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where Ωt−1 represents the information set at time t − 1. The conditional covariance matrix Ht can be decomposed as follows (Engle (2009)): Ht = Dt Rt Dt
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where the unconditional covariance matrix Q is composed of the n × n vector of standardized ς˜ residuals ei,t = hi,t computed from the first stage procedure for which ei,t ֒→ N (0, Rt ). A and B i,t √ √ are n × n diagonal matrices where A = diag( a) and B = diag( b). The scalar version of the DCC model can be written as:
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The Factor DCC model
Factor models have been at the center of asset pricing theory. The idea is to specify a number of factors that summarize all the existing dependence between asset returns from which a matrix of variance-covariance can be estimated. These factors impact on the correlation function over time. Ding (1994) and Alexander (2000, 2001) set up the orthogonal GARCH (O-GARCH) model by applying principal component analysis techniques to extract m principal components. More recently, Zhang and Chan (2008) resorted to independent component analysis to construct Factor GARCH models in the generalized orthogonal GARCH (GO-GARCH) framework, where the orthogonality constraint of the factor loading matrix is relaxed. This model is well suited to high-dimensional data. It can be estimated in two steps. In the first step, the factors are estimated given some statistical criteria. In the second step, a GARCH model is estimated for each factor. The innovation
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ǫt are assumed to be generated from the latent uncorrelated factors yt = (y1,t , ..., ym,t ) by linear transformation such as: ǫt = Ayt Each factor yi,t is described as a GARCH(1,1) process as:
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From now, the conditional covariance matrix of series ς˜ can be computed. But, in the case the loading factors might still have some conditional correlation, Zhang and Chan (2008) propose to model the remaining time-varying conditional correlation between factors with the DCC model. This approach is named the Factor-DCC model. Actually, the Factor-DCC model is simply constructed as an extension of the Factor GARCH model in order to model the conditional correlation between factors. The conditional covariance matrix between factors is given by:
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where Rt denotes the conditional correlation matrix of yt expressed in the form of the equations !p (7) and (9). Dt = diag hyi,t is a diagonal matrix composed of the square root of the diagonal elements of the conditional covariance matrix Σt of factors yi,t with i = 1, 2, ..., m, with m < n. The associated quasi-log-likelihood function is given by Zhang and Chan (2008). Among the models proposed, we choose to implement the specification of the conditional-decorrelation DCC (CD-DCC) since it gives the best performance when using financial data. In this paper, it is essential to catch the idea that the inputs to the Factor-DCC model are the volatility surprises ς˜ computed from eq.(3), instead of returns in the classical approach. By doing so, we propose a new method to create a cross-market volatility index.
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Descriptive statistics are shown in Table 1. The interested reader can check (by means of one-sided t-tests) that the ‘volatility surprise’ components calculated from eq.(3) are mean-zero, as they are re-used as an input to the Factor-DCC model. The data is retrieved from Thomson Financial Datastream. Thirty time series have been gathered in daily frequency over a period of thirty years, from January 25, 1983 to January 25, 2013. The total number of observations is equal to 7,828. The data can be decomposed into subgroups:
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1. Equities: F T SE100 is the FTSE 100 Price Index; SP 500 is the S&P 500 Price Index; DJIA is the Dow Jones Industrial Average Price Index.
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2. Bonds: Bonds is the US Benchmark 10-Year DS Government Index - Clean Price Index.
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3. Foreign Exchange Rates: EU DOLLR is the Euro to USD (WMR & DS) Exchange Rate; U KEU RSP is the UK GBP to Euro (WMR & DS) Exchange Rate; U KDOLLR is the UK
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6. Precious Metals Individual Price Series: Gold is the S&P GSCI Gold Spot Price Index; Silver is the Silver Fix LBM Cash in Cents/Troy ounce; P latinum is the London Platinum Free Market in USD/Troy ounce.
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7. Industrial Metals Individual Price Series: Copper is the S&P GSCI Copper Spot Price Index; Aluminum is the LME Aluminum 99.7% 3-Month in USD/MT; N ickel is the LME Nickel 3-Month in USD/MT.
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8. Agricultural Individual Price Series: Cof f ee is the S&P GSCI Coffee Spot Price Index; W heat is the S&P GSCI Wheat (CBOT) Spot Price Index; Soya Oil is the Soya Oil Crude Decatur Cents/Lb; Corn is the Corn No.2 Yellow in Cents/Bushel; Sugar is the Raw Sugar ISA Daily Price in Cents/Lb; Cocoa Butter is the Cocoa Butter African US Del in USD/MT; Cotton is the Cotton 1.1/16Str Low-Midl, Memph in Cents/Lb; Soybean is the Soyabeans No.1 Yellow in Cents/Bushel.
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5. Commodity Disaggregated Indices: Industrial M etals is the S&P GSCI Industrial Metals Spot Price Index; Energy is the S&P GSCI Energy Spot Price Index; Agricultural is the S&P GSCI Agricultural Spot Price Index; Livestock is the S&P GSCI Livestock Spot Price Index; Grains is the S&P GSCI Grains Spot Price Index; P reciousM etals is the S&P GSCI Precious Metals Spot Price Index.
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4. Commodity Aggregated Indices: SP GSCI is the S&P GSCI Commodity Spot Price Index; CRB is the CRB BLS Spot Price Index.
9. Energy Individual Price Series: Brent is the Crude Oil Brent Current Month FOB in USD/BBL.
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GBP to USD (WMR) Exchange Rate.
Individual commodity price series have been converted to USD when necessary using the appropriate bilateral foreign exchange rates from Thomson Financial Datastream. All time series are found to be stationary when transformed to volatility surprises (as explained in section 2.1). Standard unit root tests (ADF, PP, KPSS) are not provided to save space, and can be accessed upon request to the authors. Next, we present the results obtained when performing the estimation of the Factor-DCC model with these 30 time series of ‘volatility surprises’.
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Table 1: Descriptive statistics: ‘Volatility surprise’ Kurtosis
Standardized Skewness
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0.04 0.09 0.17 0.00 0.08 0.05 0.09 0.16 0.17 0.13 -0.04 -0.04 0.41 -0.05 -0.03 0.07 -0.55 -0.36 -0.01 -0.10 0.06 0.03 0.11
187.59 243.23 215.32 183.11 177.74 161.22 180.35 263.45 279.12 218.20 216.51 216.51 239.66 191.47 220.32 251.30 311.04 149.93 208.10 585.85 249.32 263.95 252.38
7.38 18.33 11.43 5.89 4.66 3.44 5.01 12.12 12.20 9.93 7.71 7.71 14.57 5.55 9.64 10.16 87.90 87.82 9.25 69.72 10.11 22.80 13.81
123.23 710.68 263.96 74.83 38.60 21.28 44.44 255.34 242.96 208.25 110.04 110.04 461.96 56.14 187.44 203.06 775.70 775.10 211.33 563.23 170.39 981.63 342.77
0.03 0.07 0.05 0.03 0.02 0.02 0.02 0.04 0.04 0.04 0.03 0.03 0.06 0.02 0.04 0.04 0.02 0.05 0.04 0.11 0.04 0.08 0.05
0.65 2.92 1.22 0.40 0.21 0.13 0.24 0.96 0.87 0.95 0.50 0.50 1.92 0.29 0.85 0.80 2.49 5.16 1.01 0.96 0.68 3.71 1.35
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Equities FTSE100 SP500 DJIA Bonds 10-Year U.S. Govt. Benchmark Bond Index Exchange Rates EUDOLLR UKEURSP UKDOLLR Commodities SP GSCI CRB Industrial Metals Energy Agricultural Livestock Grains Precious Metals Gold Copper Coffee Wheat Brent Soya Oil Corn Sugar Cocoa Butter Cotton Soybean Silver Platinum Aluminum Nickel
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Note: The number of daily observations is equal to 7,828 from January 25, 1983 to January 25, 2013. The standardized skewness and kurtosis are the skewness and kurtosis of the ‘volatility surprise’ component standardized by their estimated standard deviation. The source of the series is Thomson Financial Datastream.
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For the Factor-DCC implementation, we need to determine a priori a methodology to find the number of factors, which is not provided in the paper by Zhang and Chan (2008). For that reason, the choice of the number of factors is either subject to an arbitrary choice, or it can be determined by using an appropriate methodology. In this paper, we propose to use the formal statistical approach by Alessi et al. (2009) to determine the number of factors. Their criterion modifies the criterion by Bai and Ng (2002) by multiplying the penalty function by a positive real number, which allows to tune its penalizing power by analogy with the method developed by Hallin and Liˇska (2007). The criterion by Alessi et al. (2009) offers the advantage of achieving better performance than that by Bai and Ng (2002), especially in the case of large idiosyncratic disturbances. Therefore, we choose to implement the methodology by Alessi et al. (2009), which is computationally easy to implement by iteratively applying the criterion of Bai and Ng (2002). ∗T Let rc;N denote the number of factors pointed out by the Alessi et al. (2009) criterion with T the time and N the cross-section dimensions of the dataset. The test statistic use the two information criteria: NT N +T T∗ log , c ∈ R+ IC1,N (k) = log [V (k)] + c k (16) NT N +T
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We can divide the Factor-DCC model’s econometric methodology by Zhang and Chan (2008) into two steps. In the first step, we extract the common factors from a volatility surprise dataset of equities, bonds, foreign exchange rates and commodities by using the Principal Components approach of Stock and Watson (2002a,b). Based on a statistical criterion, we restrict the number of factors to the first two Principal Components extracted from the dataset of daily time-series. In the second step, the CD-DCC, which represents the time-varying conditional correlation equation of the model, has been estimated by following thoroughly Zhang and Chan’s (2008) approach.
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T T In Figure 1, the number of factors rc,N is represented by the solid black line. rc,N is greater T than one, and smaller than two. Hence, we round up rc,N , and conclude in favor of the presence of two factors. The stability of the estimated number of factors with respect to sample size is
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considering subsamples of sizes (nj , τj ) with j = 0, . . . , J such that n0 = 0 < n1 < n2 < . . . < nJ = N and τ0 = 0 < τ1 < τ2 < . . . < τJ = T . The empirical variance, represented by dotted lines in Figure 1, confirms our diagnostic. Hence, we present below the results obtained for the Principal Component Analysis with a two-factor model. Figure 2 shows the factor loadings. It shows the contribution of each individual series to each factor. Visually, we observe that the contributions to the factors differ largely across variables. Precious metals, energy products and equities account for the largest weights in both factors (see Table 1 for the list of variables). Table 2 lists the fraction of total variance in the dataset explained by the two factors, along with the volatility time-series in the dataset that each of them is most strongly correlated with. The cumulative variance of factors is given by the sum of their eigenvalues. The total variance is the sum of the variances of the individual volatility series in the dataset, measured as the sum of
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their eigenvalues. Factor 1 accounts for 21% of the variability contained in the dataset, while Factor 2 accounts for 18% of the total variation. This confirms that the Factor-DCC model captures a significant fraction (approximately 40%) of the total variation of the 30 volatility series contained in our dataset. Factor 1 is characterized by a mix of precious metals and energy contributions. Factor 2 captures mainly influences from gold and energy products. The over-weight in precious metals can be understood as a means to hedge away volatility risk. On the contrary, energy products
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are known for their pro-cyclical – even ‘bubble-like’ – behavior. Taken together, these results are informative of investors’ behaviour in presence of (volatility) risk: gold serves as a refuge for value (Baudry et al. (2011)), while energy (mainly petroleum) products can be seen as speculative investment vehicles (Buyuksahin et al. (2011)). When looking at the composition of the CMIX, these individual volatility series contributions to the index are not surprising.
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Table 2: Share of variance explained by factors Factor 1 (21.00% of total variance) R2 Precious Metals 0.77 Gold 0.75 S&P GSCI 0.28 Oil Brent 0.21 GSCI Energy 0.17 Factor 2 (17.51% of total variance) S&P GSCI 0.32 Oil Brent 0.31 GSCI Energy 0.26 Gold 0.22 Precious Metals 0.22 Note: This table presents the R-squared of univariate regressions of the factors extracted from the dataset on all individual volatility series. For each factor, a list of five variables with which the factors
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2nd step: DCC estimates
Table 3 displays the estimates of the Factor-DCC parameters, which appear standard compared to GARCH-family models. In addition, the average conditional correlation between the two factors is equal to -0.01, which means that the two factors are almost decorrelated as is the objective of the CD-DCC model. Recall that, in the conditional-decorrelation DCC model, the factors are conditionally as uncorrelated as possible, which should describe the multivariate financial time series best. For simplicity, the four parameters DCC(p), DCC(q), ARCH(p), and GARCH(q) are set equal to 1. Figure 3 shows the conditional correlation between the first and second factors estimated by PCA. We can see that the conditional correlation estimated by the CD-DCC model is timevarying with a zero-average correlation. This result is conform to the objective of the conditionaldecorrelation GARCH model. We remark that some during very brief time periods the conditional correlation exhibits various signs, and some clustering effects as is usual for volatility series. To cite a few, we identify the 1987 stock market crash, the 1989 junk bond market collapse, the 1991 first Gulf war, the 1997-98 LTCM and currency crises, the 2000 dot-com bubble burst, the 2001-02
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Table 3: Factor-DCC estimates for volatility surprise Model CD-DCC Parameters α ˆ DCC 0.0234*** (0.0112) βˆDCC 0.9109*** (0.1741) Diagnostics Log-L -45188.6 BIC -9.0297 ρij -0.01
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Enron bankruptcy, or the 2007-08 sub-primes crisis. Next, we detail how to obtain the cross-market volatility index (CMIX) based on this new formal statistical approach.
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The Cross-Market Index for Volatility Surprise
Several computational steps are required to go from the Factor loadings to the composition of the Cross-Market Volatility Index (CMIX):
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– For each factor loading Fi , i = {1, 2}, the absolute value of each series’ Si,n contribution, n = 1, . . . , 30 is taken.
– The contribution of each series to the index is computed as the average percentage between P |+|S | |S the two factor loadings’ contributions ωn = 1,n 2 2,n , with 30 n=1 ωn = 1.
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Figure 4 reveals the composition of the CMIX in a pie chart. The average contribution of each class of asset is equal to 5%, with a minimum of 0.8% (DJIA) and a maximum of 10% (Gold). Interestingly, we notice that the composition of the CMIX seems over-weighted in precious metals (around 30%). This is consistent with the view of precious metals as a safe-haven during bear financial markets followed by recessions. The CMIX reflects the ‘right’ level of risk among various asset classes. Indeed, the overweight of precious metals is a consequence of the large swings usually observed in their volatilities. As an example, during the 2008-2011 economic slowdown, the Gold ounce rised from 800 USD to 1900 USD, and declined thereafter. Due to our computational methodology, the CMIX absorbs this information and restitutes it in real-time by showing such an impressive percentage allocated to precious metals.
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Stemming from this composition in percentage, we can create the time series of the Cross Market Index CM IX. It is computed as an average of the 30 assets composing the index, P30 weighted by their respective contribution: CM IXt = n=1 ωn Pn,t . Equivalently, CM IX = P P30 t=7828 n=1 ωn Pn,t . To obtain the graph of the CMIX, we report the contribution (in %) of each volatility surprise series to the index, as detailed in the pie chart below.
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We investigate the benefits for asset managers to use the Cross-Market Volatility Index. The first use of the CMIX index could be for hedging strategies of portfolios consisting in traditional assets. In that case, derivatives contracts such as options or variance swaps would be appropriate instruments. The second use could be to consider the cross-market volatility index as an alternative class of asset, in itself. In that latter case, we propose an illustration. Indeed, this section reports the Sharpe ratios resulting from a mean-variance portfolio optimization problem2 involving the CMIX vs. other pre-existing volatility indices. The purpose of this exercise consists in assessing the relative performance of the newly created CMIX compared to its likely competitors currently used by market practitioners.
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The CMIX index appears in Figure 5 in comparison with the MSCI World Minimum Volatility index and the VIX (accessed from Thomson Financial Datastream). The CMIX appears with a base 100 on January 25, 1983. Its scale is given by the left Y-axis (along with the VIX multiplied by ten for comparability). The scale of the MSCI World volatility index is given by the right Y-axis. As it can be seen from the graph, the obvious advantage of relying on our methodology is that it allows to compute volatility indices with quite a long historical. On the contrary, the MSCI Volatility index is available only from December 31, 1998 onwards, while the VIX is available since 1990. Compared to the VIX – i.e. the CBOE measure of the implied volatility for S&P 500 index options – the CMIX exhibits similar volatility peaks. However, given that the composition of our volatility index and the VIX widely differ, both indices start to behave differently from 2009 onwards. On the one hand, the VIX returned to lower volatility values past Q4:2008. On the other hand, the CMIX picked up other influences (from commodity markets for instance), which might explain its increasing slope in the current period. Compared to the MSCI World Volatility index– which aims to reflect the performance characteristics of a minimum variance strategy in the large and mid cap equity universe across 24 countries – the CMIX appears much more stable. Indeed, there exists a stark difference between the two volatility indices during 2001-08: the CMIX has a positive slope until the summer 2008, while the MSCI World index is characterized by a brutal increase (and subsequent drop).
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Table 4: Asset performance VIX MSCI Volatility Index CMIX
Portfolio Returns (%) [2.86 - 3.28] [1.04 - 1.19] [1.39 - 1.56]
Portfolio Risks (%) [9.92 - 47.17] [2.07 - 23.86] [5.43 - 28.67]
Sharpe Ratio 0.16 0.15 0.15
Information Ratio 0.18 0.17 0.17
We include the risk-free rate by using the 30-day U.S. Treasury Bill, and consider three com-
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Basics of portfolio theory can be found in Grinold and Kahn (2000).
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peting portfolios invested either in the VIX, the MSCI World Minimum Volatility – Price Index, or the CMIX. The purpose of our stylized exercise is to compare the relative performance of these volatility indices for asset managers. This section uses data from the creation of the MSCI Volatility Index on December 31, 1998 to January 25, 2013. For the VIX, in the first two columns of Table 4, the range of monthly returns is between 2.86% and 3.28%. These rather high expected returns come at the price of risk levels in the range of 9.92% to 47.17%. One may cautiously attempt to interpret these results. The CBOE Volatility Index (VIX) is a central measure of market expectations of near-term volatility conveyed by S&P 500 stock index option prices. Since its creation in 2003 (and handback calculated to 1990 by the CBOE), the VIX has been widely considered as a barometer of investor sentiment and market volatility. Hence, we find that it can bring appealing expected returns to investors, while remaining a risky strategy. The Sharpe ratio (Sharpe (1966)) is a well-known measure of return-to-risk which plays a central role in portfolio analysis. More precisely, a portfolio that maximizes the Sharpe ratio is also the tangency portfolio on the efficient frontier from the mutual fund theorem. As investors should be driven by Sharpe ratios, they should be both interested in expected returns and in the
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Figure 5: Comparison of the CMIX with the re-scaled VIX (VIX × 10) and the MSCI World Volatility Indices
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This paper proposes a new empirical methodology for computing a cross-market volatility index – coined CMIX – based on the Factor DCC-model. This approach solves both problems of treating high-dimensional data and estimating time-varying conditional correlations. We provide an application to a multi-asset market data composed of equities, bonds, foreign exchange rates and commodities during 1983-2013. This new methodology may be attractve to asset managers as it provides a simple way to hedge multi-asset portfolios with derivatives contracts written on the CMIX. Future research will use this index for option pricing purposes to gauge its hedging capability.
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risk associated to these returns. The maximum Sharpe ratio for the VIX is given in the third column of Table 4. Its current value (0.18) is lower than the ones found in previous literature.3 A related ratio is the information ratio, which uses relative returns (see Goodwin (1988)). It is given in the last column of Table 4, and confirms the previous result. For the MSCI World Minimum Volatility Index, we obtain lower expected returns (lower bound 1.04%, upper bound 1.19%). These are accompanied by more reasonable risk levels (lower bound 2.07%), although investing in volatility indices can be overall quite risky (upper bound 23.86%) – even for an index coined ‘world minimum volatility’.4 Recall that the MSCI World Minimum Volatility Index is based on the composition of its MSCI parent index, and designed to achieve the lowest volatility for a given set of constraints. The Sharpe ratio coefficient (0.15) is lower than that of the VIX. For the CMIX, constructed following the methodology in this paper including 30 assets, we obtain a reasonable performance with expected returns in the range of 1.39%-1.56%. Investing in the CMIX turns out to be slightly more exposed in terms of risks (lower bound 5.43%, upper bound 28.67%) than in the case of the MSCI World Minimum Volatility index. These results can be partially explained by the overweight of the CMIX in Precious Metals and Gold. The bottom line is that investors could certainly consider including the CMIX as a candidate in their investment universe (along with the pre-existing MSCI volatility index). Or, they can tailor another volatility index that suits them best by replicating our methodology. Due to its reliance on S&P 500 option prices, an investment strategy based on the VIX is characterized by (nearly twice) higher levels of risks.
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[4] Bai, J., Ng, S. 2002, Determining the number of factors in approximate factor models. Econometrica 70(1), 191-221
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[3] Alexander, C., 2001, Market models: A guide to financial data analysis, Wiley, Chichester, UK
[5] Bai, J., Ng, S. 2007, Determining the number of primitive shocks in factor models. Journal of Business and Economic Statistics 25(1), 52-60 [6] Baudry, P., Collard, F., Portier, F. 2011, Gold rush fever in business cycles. Journal of Monetary Economics 58, 84-97
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[2] Alexander, C., 2000, Orthogonal methods for generating large positive semi-definite covariance matrices, ISMA Center Discussion Papers in Finance 2000-06, University of Reading, UK
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[1] Alessi, L., Barigozzi, M., Capasso, M. 2010, Improved penalization for determining the number of factors in approximate factor models. Statistics and Probability Letters 80(23-24):1806-1813
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[11] Dash, S., Moran, M.T. 2005, VIX as a Companion for Hedge Fund Portfolios. Journal of Alternative Investments 8, 75-80
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Novel methodology to create volatility index. Update on the concept of ‘volatility surprise’. Methodology reproducible by asset managers. Cross Market Volatility Index computed from Factor DCC model.
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Submitted to the International Review of Financial Analysis
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For the paper entitled “Cross-market volatility index with Factor-DCC”