Flexible Dynamic Conditional Correlation Multivariate GARCH models for Asset Allocation

Transcription

1 UNIVERSITA CA FOSCARI DI VENEZIA novembre 2005 Flexible Dynamic Conditional Correlation Multivariate GARCH models for Asset Allocation Monica Billio, Michele Gobbo, Masimiliano Caporin Nota di Lavoro

2 Flexible Dynamic Conditional Correlation Multivariate GARCH models for Asset Allocation Monica Billio Massimiliano Caporin Michele Gobbo September 2005 Abstract This paper introduces the Flexible Dynamic Conditional Correlation (FDCC) multivariate GARCH model which generalises the Dynamic Conditional Correlation (DCC) multivariate GARCH model proposed by Engle (2002). The FDCC model relax the assumption of common dynamics among all assets used in the DCC model. In fact, we cannot impose that the correlation dynamics of, say, European sectorial stock indexes are identical to the corresponding US ones. We thus extend the DCC model introducing a block-diagonal structure; in the FDCC the dynamics is constrained to be equal among groups of variables. We present an application to a sectorial asset allocation problem. Keywords: Multivariate GARCH, Dynamic Correlation, Volatility, Asset Allocation, Risk Management. Dipartimento di Scienze Economiche, Università Ca Foscari, Venezia, and School for Advanced Studies in Venice Foundation, billio@unive.it. Dipartimento di Scienze Economiche Marco Fanno, Università degli Studi di Padova and GRETA Associati, Venezia - corresponding author: Dipartimento di Scienze Economiche Marco Fanno, Università degli Studi di Padova, Via del Santo 33, Padova, Italy - massimiliano.caporin@unipd.it - phone GRETA Associati, Venezia - mgobbo@greta.it. 1

3 1 Introduction In today s global and highly volatile markets the efficient measurement and management of market risk has become a critical factor for the competitiveness and even survival of financial institutions. One of the inputs required by risk managers, seeking to hold efficient portfolios, is the correlation between the securities to be included in the portfolio. Until recently, correlation was assumed to be constant and stable over time. However, all empirical studies that attempted to verify this finding, have failed to confirm the validity of this assumption. In fact, most experienced practitioners would attest that correlations increase in periods of high volatility and that both the magnitude and persistence of correlation is affected by volatility. The asset allocation decision entails, inter alia, an assessment of the risks and returns of the various assets in the opportunity set. Optimal portfolio choice requires a forecast of the covariance matrix of the returns. Similarly, the calculation of the standard deviation of today s portfolio requires a covariance matrix of all the assets in the portfolio. For actual portfolios, with thousands of derivative and synthetic instruments, these functions require estimation and forecasting of very large covariance matrices. Over the past 20 years, a considerable literature has been developed and the dynamics of the covariance of assets has been explored, although the primary focus has been on univariate volatilities and not on correlations (or covariances). In fact, in the multivariate GARCH literature one of the most relevant problems is represented by the high number of parameters. In order to solve this difficulty, Bollerslev (1990) suggested to keep constant the correlations and put forward the Constant Conditional Correlation model 2

4 (CCC). Only recently, Engle (2002) proposes a new class of models that both preserve the ease of estimation of the Bollerslev s constant correlation model but allow the correlations to change over time. Engle adds to the CCC model a limited dynamic in the correlations, introducing a GARCH-type structure. However, the dynamics is constrained to be equal for all the correlations. In our view, this is an unnecessary restriction. In fact, we cannot impose that the correlations dynamic evolution of, say, European sectorial stock indexes, is identical to the corresponding US one. We thus extend the DCC model, introducing a block-diagonal structure that solves this problem. The correlation dynamic is constrained to be equal only among groups of variables. The suggested model provides a much more flexible parameterisation of correlation dynamics, by maintaining at the same time the parameter number at a feasible level; we call this new model Flexible Dynamic Conditional Correlation (FDCC). The block dynamic representation can be useful not only in asset allocation or asset pricing models (for instance extending Morelli, 2003) but also in other contexts. For example, to investigate whether the introduction of the euro in Europe has increased the correlation among national assets, or in more general terms, to analyse the interdependence and contagion issues, or the spillover effects as in Higgs and Worthington (2004); alternatively the block-structured correlation model could be used to evaluate trading strategies involving asset correlations such as in Chong (2004), or to analyse the relationship between exchange and interest rates, generalising the approach of Bautista (2003). The plan of the paper is as follows: in section 2 we introduce the Flex- 3

5 ible Dynamic Conditional Correlation model (FDCC) as a generalisation of Engle s DCC model. In section 3 we present an empirical application of the FDCC model in an asset allocation framework; we analyse sectorial stock indexes and we empirically motivate the need for different correlation dynamics among groups of assets. Section 4 concludes. 2 Flexible Dynamic Conditional Correlation Engle and Sheppard (2001) and Engle (2002) generalised the Constant Conditional Correlation (CCC) model of Bollerslev (1990) developing the Dynamic Conditional Correlation model (DCC). In this paper we propose the Flexible DCC model (FDCC) as a generalisation of the Engle s DCC model. For a k-dimensional variable y t, the FDCC model has the following representation: y t = µ t + ε t E [ ε t I t 1] = 0 E [ ε t ε t I t 1] = H t H t = D t R t D t (1) R t = (Q t ) 1 Q t (Q t ) 1 (2) Q t = cc + aa η t η t + bb Q t 1 (3) where E [ I t 1 ] is the expectation conditional to the information set at time t 1, D t is a diagonal matrix of conditional standard deviations (D t = diag (σ 11,t, σ 22,t,...σ kk,t )), η t represents the standardised residuals (η t = D 1 t ε t ), equations (1) define the mean dynamics, the residuals conditional moments and the conditional variance structure, where represents the Hadamard product (i.e. element by element product), equations (2) and (3) describe the correlation dynamics where Q t = diag( q 11,t, q 22,t,... q kk,t ). Finally, 4

6 c, a and b are partitioned k-dimensional vectors with the following structure a = [ a 1 i m 1 a 2 i m 2... a w i m w ] (and similarly for b and c) where i h is an h-dimensional vector of ones and w is the number of blocks. The coefficient vectors thus contain w different coefficients, each of them possibly repeated m j times, with j = 1, 2,..., w. As a result of the block structure of the coefficient matrices, the dynamics is equal only for groups of variables and not for the whole correlation matrix. Furthermore, as in the CCC or DCC models, the conditional variances can be parameterised with any univariate GARCH model (see Engle (1982), Bollerslev (1986), and for a survey Bollerslev, Chou and Kroner (1992) and Bollerslev, Engle and Nelson (1994)); moreover, the simplest model structure allows the inclusion of spillover effects following the approach of Hwang and Satchell (2005). To avoid explosive patterns, the coefficients must satisfy a set of constraints, namely α i α j + β i β j < 1 for i, j = 1, 2,...w. In addition, the correlation matrix is positive definite by construction since, given a suitable starting point Q t is the sum of positive definite and semidefinite matrices. Finally, the Engle s DCC model can be obtained setting a = α i k, b = β i k, cc = (1 aa bb ) Γ. The main innovation with respect to the Engle s DCC model is the relaxation of the hypothesis of a common correlation dynamics among all the assets; we assume that dynamics are common among group of assets obtaining a feasible representation, competitive to other DCC-type models. Furthermore, the FDCC model is included in the generalised DCC extension 5

7 proposed by Engle (2002), who suggested the following structure Q t = [ii A B] Γ + A η t 1 η t 1 + B Q t 1 (4) where A, B and C are square matrices and Γ is the unconditional correlation matrix of η t. Unfortunately, the model of equation (4) is quickly intractable since the number of parameter grows rapidly and creates the same problems of a standard large dimension BEKK or Vech GARCH model (see Engle and Kroner (1995)). Differently, the FDCC model generalises the approach of Franses and Hafner (2003), who suggest [ ] k Q t = 1 α i β Γ + αα η t 1 η t 1 + βq t 1 (5) i=1 where β ia a scalar and α a k-dimensional vector. The flexibility of the FDCC model has a cost, since we loose the variance targeting property. The unconditional correlation is not included in the FDCC model, as a consequence we have some additional parameters to estimate. However, the correlation targeting property could be included in the model with an additional set of parameter constraints, namely α i α j + β i β j + c i c j = 1 for i, j = 1,...w. The FDCC model can easily estimated in the same way of the DCC model, as described in the Engle s (2002) paper. The approach is based on a two-step procedure: the first step focuses on univariate estimation, while the second focuses on correlations. Unfortunately, standard likelihood ratio tests cannot be used in comparing CCC and DCC to our FDCC model since they are not nested representations. Therefore, the comparison can be made only in terms of information criteria. 6

8 3 An empirical application: sectorial asset allocation Now, we consider an empirical application focused on a dataset of daily data from the Italian Stock market and we tackle a sectorial asset allocation problem. There are three major sectors that compose the Italian Mibtel general index: Industrials, Services and Finance. Each of these three sub indexes is further divided into several sub-sectors. The composition is summarised in Table 1. INSERT TABLE 1 HERE All time series were downloaded from DataStream and are expressed in euro. They run from January 1991 to September 2003, yielding more than 3000 daily observations. The returns are calculated, as usual, through a log difference transformation. If we consider a dynamic analysis of the sector indexes, the volatility is clearly far from being constant and the GARCH models can be useful to capture these features. Given the characteristics of the series, we fitted an asymmetric GARCH specification (in particular a GJR-GARCH, see Glosten, Jagannathan and Runkle, 1993) able to capture both the excess of kurtosis and the asymmetric effects. The estimation results are not reported here but can be requested to the authors. Our main interest is the correlations among the univariate GARCH standardised residuals. A preliminary rolling window correlation analysis evidences that almost all the correlations vary through time. Furthermore, these correlations also present different patterns between groups; that is, the correlation among indices belonging to the Finance macro sector are similar 7

9 but different from the correlations between indices belonging to the Service macro-sector. Figures 1 reports an example. INSERT FIGURE 1 To describe these patterns some dynamic correlation models are estimated: we considered the CCC-GARCH proposed by Bollerslev (1990), the DCC-GARCH proposed by Engle (2002) and we compare them with the suggested FDCC model. Table 2 reports the CCC model estimates while Table 3 reports the DCC parameter estimates over the whole set of indices and the Finance, Service and Industry macro-sectors singularly considered. It is evident that parameters describing the correlation dynamics are different among the sectors. Table 4 reports the FDCC parameter estimates. INSERT TABLES 2, 3 AND 4 HERE The advantages of considering a FDCC model are clearly evidenced by a standard likelihood ratio test which rejects the null hypothesis of the DCC model (common parameters in all blocks and correlation targeting restriction - the LR statistics is around 975). Moreover, all the parameters are highly significant (by considering quasi maximum likelihood standard errors). The comparison among the correlation models cannot be restricted to a pure statistical analysis but should be combined with some empirical evidence. For this reason, we considered a simulated exercise. Within a Markovitz approach and with a restricted sample (the last 2 years data) we estimate mean variance portfolios with CCC, DCC and FDCC time varying correlations structures. Portfolio weights are computed assuming no risk-free asset, with or without positivity restrictions (short selling), no transaction 8

10 costs and two weeks revision (10 days). The models are thus re-estimated every 10 days and one-step-ahead correlation and variance forecasts are computed and stored. With the one-step-ahead variance-covariance matrix forecasts, a mean-variance problem is then considered. Additional assumptions refer to the index returns on which we base the portfolio weights computation and on the portfolio required return. For the first assumption we consider the last two months returns to get a closer matching with market movements. In the portfolio weights computation we consider several cases of portfolio returns: the last two months return of an equally weighted portfolio; the last two months return of the general Mibtel index; the global optimal portfolio; a 20% annual return. Table 5 reports the annualised optimal portfolio standard deviations of the final estimation (i.e. the last two months of the sample): INSERT TABLE 5 Moving from the constant correlation assumption to the Engle s DCC model the optimal portfolio variance marginally decreases; this result is stable over the whole sample considered. Differently, the variance reduction implied by our Flexible DCC is more evident. Table 6 reports the estimated portfolio weights for global optimal portfolios in the final estimation. INSERT TABLE 6 In the final step we assume that parameters are stable for a two-week horizon and we computed a 10-day sequence of one-step-ahead forecasts of the variance-covariance matrix. These forecasts have then been used to compute portfolio returns for the constrained and unconstrained cases and several 9

11 assumptions on portfolio returns. As a general result we can state that, under the same assumption for portfolio returns, the FDCC model provides the lowest portfolio variance and the highest portfolio return. Figure 2 reports the returns evolution over one year of a particular case: portfolio weights are the global optimal portfolio weights. INSERT FIGURE 2 4 Conclusions We develop an extension of the Dynamic Conditional Correlation model proposed by Engle (2002). The Flexible DCC preserves the ease of estimation of the Engle s model but also allows the correlations to have different dynamics. In fact, in the DCC model, the dynamics is constrained to be equal for all the correlations. From a practical point of view, this is an unnecessary restriction, therefore we extend the DCC model introducing a constant and a block-diagonal structure. Bearing in mind the idea of sectorial or geographical portfolio diversification, the dynamics is thus constrained to be equal only among groups of variables. In this respect, the FDCC model could be useful for extending and improving current research areas involving the portfolio diversification topic, figuring out the relationship between block-type structures and factor analysis starting from Hui (2005); alternatively, a different research area may consider the relations between macroeconomic dynamic and financial markets as in Ewing et al. (2003). After introducing the model, we consider an empirical application comparing the CCC, DCC and Flexible DCC models. The analysis focuses on Italian sectorial stock indexes and on a sectorial asset allocation problem. 10

12 The estimates of the three models confirm, for the period of analysis, the presence of dynamics in the correlations and also shows the presence of dissimilarities in these dynamics among macro sectors. Furthermore, a simulated portfolio allocation exercise (under a Markovitz approach) shows that the FDCC model provide the lowest optimal portfolio variance and the highest portfolio returns. Acknowledgements. The first and second author acknowledge financial support from the Italian MIUR project Econometric Models for the Analysis of Financial Markets: The Integration Process in the Area of the Euro. The authors wish to thank Michael McAleer, Domenico Sartore, and the participant to the SCO2003 conference, the Forecasting Financial Market 2003 conference and the Second European Deloitte Conference in Risk Management for helpful comments and suggestions. References [1] Bautista, C.C., 2003, Interest rate - exchange rate dynamics in the Philippines: a DCC analysis, Applied Economics Letters, 10, [2] Bollerslev, T., 1986, Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics 31, [3] Bollerslev T., 1990, Modelling the coherence in short-run nominal exchange rates: a multivariate generalized ARCH approach, Review of Economic and Statistics, 72,

13 [4] Bollerslev T., R. Y. Chou & K. F. Kroner (1992) ARCH modeling in finance: A review of the theory and empirical evidence. Journal of Econometrics 52, [5] Bollerslev T., R. F. Engle & D. B. Nelson (1994) ARCH models. In R.F. Engle & D. McFadden, Handbook of econometrics, Vol. 4, pp Elsevier, Amsterdam. [6] Chong, J., 2004, Options trading profits from correlation forecasts, Applied Financial Economics, 14, [7] Engle, R.F., 1982, Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation, Econometrica 50, [8] Engle R.F., 2002, Dynamic conditional correlation - a simple class of multivariate GARCH models, Journal of Business and Economic Statistics, 20(3), [9] Engle R.F. and K. Sheppard, 2001, Theoretical and empirical properties of dynamic conditional correlation multivariate GARCH, University of California, San Diego, Discussion paper [10] Ewing, B.T., S.M. Forbes and J.E. Payne, The effects of macroeconomic shocks on sector-specific returns, Applied Economics, 35, [11] Franses, P.H. and C.M. Hafner, 2003, A Generalised Dynamic Conditional Correlation Model for Many Asset Returns, Econometric Institute Report EI , Erasmus University Rotterdam 12

14 [12] Glosten L., R. Jagannathan and D. Runkle, 1993, Relationship between the expected value and the volatility of the nominal excess returns on stocks, Journal of Finance, 48, [13] Higgs H. and A.C. Worthington, 2004, Transmission of returns and volatility in art markets: a multivariate GARCH analysis, 11, [14] Hui, T., 2005, Portfolio diversification: a factor analysis approach, Applied Financial Economics, 15, [15] Hwang, S. and S.E. Satchell, 2005, GARCH models with cross-sectional volatility: GARCHX models, Applied Financial Economics, 15, [16] Morelli, D., 2003, Capital asset pricing model on UK securities using ARCH, Applied Financial Economics, 13,

15 MIBTEL (General) Table 1: Italian indexes composition. INDUSTRIAL SERVICE FINANCE FOOD CARS PAPER CHEMICALS CONSTRUCTION ELECRONICS PLANTS MACHINE INDUSTRIALS MISC MINERALS METALS TEXTILE CLOTHING DISTRIBUTION MEDIA PUBLIC UTILITY SERVICES TRANSPORT TOURISM INSURANCE BANKS FINANCE HOLDINGS FINANCE MISC REAL ESTATE FINANCE SERVICES

16 FOOD CARS PAPER CHEMICALS CONSTRUCTION ELECRONICS PLANTS & MACHINE INDUSTRIALS MISC MIN TEX DISTRIBUTION MEDIA PUB. UTIL. SERV. TRANS & TOURISM INSURANCE BANKS FINANCE HOLDINGS FINANCE MISC. REAL ESTATE FINANCE SERVICES Table 2: CCC correlation estimates.

17 Parameters Estimates Standard deviations z-statistics FULL SET α β Log Likelihood: INDUSTRIALS α β Log Likelihood: SERVICE α β Log Likelihood: FINANCE α β Log Likelihood: Table 3: DCC estimates full sample full set and macro sectors Parameters Estimates Standard deviations z-statistics c c c a a a b b b Log-Likelihood Table 4: FDCC estimates full sample all sectors

18 1.00 Food/Paper Cars/Minerals Metals Chemicals/Finance Holding Figure 1 Correlation dynamics.

19 CCC DCC FDCC CCC DCC FDCC FOOD CARS PAPER CHEMICALS CONSTRUCTION ELECRONICS PLANTS MACHINE INDUSTRIALS MISC MINERALS METALS TEXTILE CLOTHING DISTRIBUTION MEDIA PUBLIC UTILITY SERVICES TRANSPORT TOURISM INSURANCE BANKS FINANCE HOLDINGS FINANCE MISC REAL ESTATE FINANCE SERVICES Table 6 Portfolio allocation in the Markovitz approach in a constrained and a non constrained problem (without risk free asset) global optimal portfolios Type Portfolio Return Correlation Model CCC DCC FDCC Equally Weighted Unconstrained Mibtel Global Optimal % Equally Weighted Constrained Mibtel Global Optimal Max index return Table 5 annualised optimal portfolio variances based on last two months of the sample; last two months annualised Mibtel standard deviation is 8.214; optimal portfolio variances depend on the required portfolio return which is in turn set equal to: last two month return of an equally weighted portfolio; last two month return of the Mibtel index; global optimal portfolio; 20% annual return (unconstrained only); among the 20 sectorial indices, the maximum last two moths return.

20 10 CCC DCC FDCC /10/02 25/11/02 04/01/03 13/02/03 25/03/03 04/05/03 13/06/03 23/07/03 01/09/03 Figure 2: portfolio returns - unconstrained Markovitz approach with objective return set to 20% (annual return)