Dynamic Factor Multivariate GARCH Model

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Dynamic factor multivariate GARCH model Andr´e A. P. Santos Department of Economics Universidade Federal de Santa Catarina Guilherme V. Moura Department of Economics Universidade Federal de Santa Catarina Hudson da Silva Torrent Universidade Federal do Rio Grande do Sul February 15, 2012 Abstract Factor models are well established as promising alternatives to obtain covariance matrices of portfolios containing a very large number of assets. In this paper, we consider a novel multivariate factor GARCH specification with a flexible modeling strategy for the common factors, for the individual assets, and for the factor loads. We apply the proposed model to obtain minimum variance portfolios of all stocks that belonged to the S&P100 during the sample period and show that it delivers less risky portfolios in comparison to benchmark models, including existing factor approaches. Keywords: dynamic conditional correlation (DCC), forecasting, Kalman filter, learning CAPM, performance evaluation, Sharpe ratio JEL classification: G11, G17, C58

1

Introduction

Obtaining covariance matrices for portfolios with a large number of assets remains a fundamental challenge in many areas of financial management, such as asset pricing, portfolio optimization and market risk management. Many of the the initial attempts to build models for conditional covariances, such as the VEC model of Bollerslev et al. (1988) and the BEKK model of Engle and Kroner (1995), among others, suffered from the so-called curse of dimensionality. In these specifications, the number of parameters increase very rapidly as the cross-section dimension grows, thus creating difficulties in the estimation process and entailing a large amount of estimation error in the resulting covariance matrices.

1

In this context, factor models emerge as promising alternatives to circumvent the problem of dimensionality and to alleviate the burden of the estimation process. The idea behind factor models is to assume that the co-movements of financial returns can depend on a small number of underlying variables, which are called factors. This dimensionality reduction allows for a great flexibility in the econometric specification and in the modeling strategy. In fact, alternative approaches for conditional covariance matrices based on factors models have been proposed in the literature. Generally, these models differ in their assumptions regarding the characteristics of the factors. For instance, Alexander and Chibumba (1996) and Alexander (2001) obtain common factors from statistical techniques such as principal components analysis (PCA) whereas Chan et al. (1999) use common factors extracted from asset returns. Engle et al. (1990), Alexander and Chibumba (1996), Alexander (2001), and Vrontos et al. (2003) assume that factors follow a GARCH process, whereas Aguilar and West (2000) and Han (2006) consider a stochastic volatility (SV) dynamics. Moreover, van der Weide (2002) extends previous studies by assuming that factors are not mutually orthogonal. In this paper, we put forward a novel approach to obtain conditional covariance matrices based on a flexible dynamic factor model. The model extends previous econometric specifications in at least two aspects. First, the proposed approach achieves great flexibility by allowing alternative econometric specifications for the common factors and for the individual assets in the portfolio. In particular, the model allows for a parsimonious multivariate specification for the covariances among factors based on a conditional correlation model and consider alternative univariate GARCH specifications to model the volatility of individual assets. Second, factor loads are assumed to be time-varying rather than constant. We treat factor loads as latent variables and consider richer dynamics based on recent developments in asset pricing theory (Adrian and Franzoni 2009). Finally, we discuss an estimation procedure to obtain conditional covariance matrices according to the proposed model. We apply the proposed model to obtain in-sample and out-of-sample one-step-ahead forecasts of the conditional covariance matrix of all assets that belong the S&P100 index during the sample period, and use the estimated matrices to compute short selling-constrained and unconstrained minimum variance portfolios. The performance of the proposed model is compared to that of alternative benchmark models, including existing factor approaches. The results indicate that the proposed model delivers less risky portfolios in comparison to the benchmark models. The paper is organized as follows. In Section 2 we describe the model specification and

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give details regarding estimation and related models. Section 3 discusses an application in the context of portfolio optimization and proposes a methodology to perform out-of-sample evaluation. Finally, Section 4 brings concluding remarks.

2

The model

In this section we introduce our conditional covariance model based on a factor model. Throughout the paper we assume that each one of the N individual asset returns, yit , is generated by K ≤ N factors,

yit = βit ft + εit

(1)

where ft is a mean zero vector of common factor innovations, εit ∼ N (0, hit ) is the ith measurement error, and βit is the ith row of the N × K matrix βt of factor loadings. We assume that the factors are i) conditionally orthogonal to the measurement errors, E [fit εjt |ℑt−1 ] = 0 ∀i ∈ {1, ..., K} ∀j ∈ {1, ..., N } and ii) are not mutually conditionally orthogonal, i.e. E [fit fjt |ℑt−1 ] ̸= 0 ∀i ̸= j, where ℑt−1 denotes the information set available up to t − 1. In addition, we assume that the measurement errors are conditionally orthogonal with timevarying conditional variances, i.e E [εit εjt |ℑt−1 ] = 0 ∀i ̸= j and E [εit εjt |ℑt−1 ] = hit ∀i = j. An important feature of the specification in (1) is that the factor loads are time-varying. Existing evidence suggest that allowing for time variation in factor loads lead to improvement in terms of pricing errors and forecasting accuracy; see, for instance, Jostova and Philipov (2005), Ang and Chen (2007), and Adrian and Franzoni (2009). In this paper, we consider that each βit evolves according to two alternative laws of motion. The first specification considers that factor loads are unobservable and follow a random walk (RW), βit = βit−1 + uit

(2)

where uit ∼ N (0, Σui ) is the 1 × K vector of errors in the law of motion of the factor loadings and εit and uit are independent. The second law of motion is based on the learning factor model of Adrian and Franzoni (2009). In the learning model, the factor load follows a meanreverting process and investors are unaware of its long-run level. Hence, they need to infer both the current level of the risk factor load and its long-run mean from the history of realized

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returns in a learning process. According to learning model, each βit evolves according to βit = (1 − ϕ)Bit + ϕβit−1 + uit

(3)

Bit = Bit−1

(4)

where Bit is the expected value of the long-run mean of βit at time t and uit ∼ N (0, Σui ) is the 1 × K vector of errors in the law of motion of the factor loadings in the learning model. We assume that εit , and uit are independent. The conditional covariance matrix, Ht , of the vector of returns in (1) is given by Ht = βt Ωt βt′ + Ξt

(5)

where Ωt is a symmetric positive definite conditional covariance matrix of the factors, and Ξt is a diagonal covariance matrix of the residuals from the factor model in (1), i.e. Ξt = diag(h1t , . . . , hN t ), where diag is the operator that transforms the N × 1 vector into a N × N diagonal matrix and hit is the conditional variance of the residuals of the i-th asset. Note that the positivity of the covariance matrix Ht in (5) is guaranteed as the two terms in the right-hand side are positive definite. To model Ωt , the conditional covariance matrix of the factors in (5), alternative specifications can be considered, including multivariate GARCH models (see Bauwens et al. 2006 and Silvennoinen and Ter¨asvirta 2009 for comprehensive reviews) and stochastic volatility models (Harvey et al. 1994; Aguilar and West 2000; Chib et al. 2009). In this paper, we consider the dynamic conditional correlation (DCC) model of Engle (2002), which is given by: Ωt = Dt Rt Dt where Dt = diag

(√

hf1t , . . . ,



(6)

) hfkt , hfkt is the conditional variance of the k-th factor, and

Rt is a symmetric positive definite conditional correlation matrix with elements ρij,t , where ρii,t = 1, i, j = 1, . . . , K. In the DCC model the conditional correlation ρij,t is given by qij,t ρij,t = √ qii,t qjj,t

(7)

where qij,t , i, j = 1, . . . , K, are collected into the K ×K matrix Qt , which is assumed to follow

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GARCH-type dynamics, ¯ + αzt−1 z ′ + βQt−1 Qt = (1 − α − β) Q t−1

(8)

√ where zft = (zf1t , . . . , zfkt ) with elements zfit = fit / hfit being the standardized factor ¯ is the K × K unconditional covariance matrix of zt and α and β are non-negative return, Q scalar parameters satisfying α + β < 1. We follow a similar modeling strategy of Cappiello et al. (2006) and consider alternative univariate GARCH-type specifications to model the conditional variance of the factors, hfkt , and the conditional variance of the residuals, hit . In particular, we consider the GARCH model of Bollerslev (1986), the asymmetric GJR-GARCH model of Glosten et al. (1993), the exponential GARCH (EGARCH) model of Nelson (1991), the threshold GARCH (TGARCH) model of Zakoian (1994), the asymmetric power GARCH (APARCH) model of Ding et al. (1993), the asymmetric GARCH (AGARCH) of Engle (1990), and the nonlinear asymmetric GARCH (NAGARCH) of Engle and Ng (1993). In all models, we use their simplest forms where the conditional variance only depends on one lag of past returns and past conditional variances. We describe in the Appendix the econometric specification of each of these models.

2.1

Estimation

The model is estimated in a multi-step procedure. First, the time-varying factor loadings in (1) are estimated via maximum likelihood (ML). Second, the conditional covariance matrix of the factors in (6) is obtained by fitting a DCC model to the time series of factor returns assuming Gaussian innovations. The DCC parameters are estimated using the composite likelihood (CL) method proposed by Engle et al. (2008). Finally, we consider alternative univariate GARCH specifications to obtain conditional variances of the residuals of the factor model. For each residual series, we pick the specification that minimizes the Akaike Information Criterion (AIC). Next we detail the estimation procedure.

Estimation of the Time Varying Factor Loadings Given the factors ft , the systems of equations (1)-(2) and (1)-(3) form a linear and Gaus-

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sian state space model for each asset i: yit = Ht xit + εit xit = F xit−1 + Ruit

(9) (10)

where Ht contains the factors, and xit contains the time varying factor loadings, which are treated as unobserved states. The transition matrix F ensures that the states evolve according to (2) and (3)-(4) in the RW and learning models, respectively1 . The error term uit is normally distributed with mean zero and covariance matrix Σui , and R is a selection vector whose entries are either 0 or 1. This model can be estimated via ML using the Kalman filter to build the likelihood function, which is then maximized in order to obtain parameter estimates. The vector of unobserved states xit , which in the case of the learning model includes not only βit but also Bit , can be estimated conditional on the past and current observations yi1 , . . . , yit via Kalman filter. Define xit|t−1 as the expectation of xit given yi1 , . . . , yit−1 with mean square error (MSE) matrix Pt|t−1 . For given values of xit|t−1 and Pt|t−1 , when observation yit is available, the prediction error can be calculated vit = yit − ft′ xit|t−1 . Thus, after observing yit , a more accurate inference of xit|t and Pt|t can be made: xit|t = xit|t−1 + Pt|t−1 ft′ ∆−1 t vt , Pt|t = Pt|t−1 − Pt|t−1 ft′ ∆−1 t ft Pt|t−1 , where ∆t = ft′ Pt|t−1 ft + σv is the prediction error covariance matrix. An estimate of the state vector in period t + 1 conditional on y1 , . . . , yt , is given by the prediction step xt+1|t = F xt|t , Pt+1|t = F Pt|t F ′ + RΣui R′ .

(11)

For a given time series y1 , . . . , yT , the Kalman filter computations are carried out recursively for t = 1, . . . , T . Because of the nonstationary transition equation (2), the initialization was implemented using the exact initial Kalman filter proposed by Koopman (1997). The parameters in the covariance matrix Σui are treated as unknown coefficients which are collected in the parameter vector ψ. Estimation of ψ is based on the numerical maximization 1

Note that the vector of states in the learning model contains not only βit but also Bit , and thus, Ht must include some columns of zeros.

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of the loglikelihood function that is constructed via the prediction error decomposition and given by2 T T 1∑ 1∑ ′ NT l(ψ) = − log 2π − log |Ft | − vt logFt−1 vt . 2 2 2 t=1

(12)

t=1

Estimation of the Conditional Covariance Matrix of the Factors To obtain the conditional covariance matrix of the factors, Ωt , we use a DCC specification in (6). The estimation of the DCC model can be conveniently divided into volatility part and correlation part. The volatility part refers to the estimation of the univariate conditional volatilities of the factors using a GARCH-type specification. The parameters of the univariate volatility models are estimated by quasi maximum likelihood (QML) assuming Gaussian likelihoods3 . The correlation part refers to the estimation of the conditional correlation matrix in (7) and (8). In order to estimate the parameters of the correlation part, we consider the CL method proposed by Engle et al. (2008). As Engle et al. point out, the CL estimator delivers more accurate parameter estimates in comparison to the two-step procedure proposed by Engle and Sheppard (2001) and Sheppard (2003), specially in high dimensional problems.

2.2

Forecasting

One-step-ahead forecasts of the conditional covariance matrices based the model can be obtained as: ′ Ht|t−1 = βt|t−1 Ωt|t−1 βt|t−1 + Ξt|t−1 ,

(13)

where βt|t−1 , Ωt|t−1 , and Ξt|t−1 are, respectively, one-step-ahead forecasts of the factor loads computed according to (2), one-step-ahead forecasts of the conditional conditional covariance matrix of the factors computed according to (6), and one-step-ahead forecasts of the conditional residual variances computed according to a GARCH-type model and collected into the 2

For details about parameters and state estimation using the Kalman filter, see Durbin and Koopman (2001). See Appendix B in Adrian and Franzoni (2009) for a full derivation of the Kalman filter for the learning-CAPM. 3 Reviews of estimation issues, such as the choice of initial values, numerical algorithms, accuracy, as well as asymptotic properties are given by Berkes et al. (2003), Robinson and Zaffaroni (2006), Francq and Zakoian (2009), and Zivot (2009). It is important to note that even when the normality assumption is inappropriate, the QML estimator based on maximizing the Gaussian loglikelihood is consistent and asymptotically Normal provided that the conditional mean and variance functions of the GARCH model are correctly specified; see Bollerslev and Wooldridge (1992).

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diagonal matrix Ξt|t−1 .

2.3

Benchmark models

We consider four alternative benchmark specifications for the conditional covariance matrix of returns. The first benchmark model is a version of the present specification in (5) with time-invariant factor loads, i.e. Ht = βΩt β ′ + Ξt ,

(14)

where β is the ordinary least squares (OLS) estimate of the regression model yit = βi ft + εit . The second benchmark is the orthogonal GARCH (OGARCH) model of Alexander and Chibumba (1996) and Alexander (2001), Ht = W Λt W ′ ,

(15)

where W is a N ×K matrix of eigenvectors of the first K ≤ N orthogonal factors obtained via principal components analysis (PCA) and Λt is a diagonal covariance matrix of the conditional variances of the principal components, i.e. Λt = diag (hP C1t , . . . , hP Ckt ) where hP Ct follows a GARCH model. The third benchmark is the factor model proposed by Chan, Karceski, and Lakonishok (1999) (CKL), Ht = βt Γt βt′ + Υt ,

(16)

where the covariance matrix Γt is modeled as a 252-day rolling window sample covariance t−1 ∑ 1 matrix of the factors, i.e. Γt = 252 fi fi′ , and Υt is a diagonal matrix of of residual i=t−252

variances from the rolling 252-day regression of the asset returns on the factors. Finally, the fourth benchmark model is the Risk Metric (RM) model, which consists of an exponentially-weighted moving average scheme to model conditional covariances. In this approach, the conditional covariance matrix is given by ′ Ht = (1 − λ)Yt−1 Yt−1 + λHt−1 ,

with the recommended value for the model parameter for daily returns being λ = 0.94.

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(17)

3

Application to portfolio optimization

To evaluate the performance of the proposed model in comparison to benchmark models we consider the minimum variance portfolio (MVP) problem. We examine the properties of MVP under two alternative weighting schemes: unconstrained and short-sales constrained. In the unconstrained case, the MVP can be formulated as: min wt Ht|t−1 wt wt

(18)

subject to wt′ ι = 1

where ι is a N × 1 vector of ones. The solution to the unconstrained MVP problem in (18) is given by:

wt =

−1 Ht|t−1 ι −1 ι′ Ht|t−1 ι

.

(19)

In the short-sales constrained case, we add in (18) a restriction to avoid negative weights, i.e. wt ≥ 0. Previous research has shown that imposing such constraints may substantially improve performance, mostly by reducing portfolio turnover, see Jagannathan and Ma (2003), among others. In the constrained case, optimal MVP weights are obtained using numerical methods.

3.1

Data and implementation details

To evaluate the performance of the model vs. benchmark models, we use a data set composed of daily observations of all stocks that belonged to the S&P100 index from January 4, 2000 until December 31, 2010. This yields a total of 78 stocks with 2766 observations. Returns are computed as the differences in log prices. Data was obtained from the EcoWin/Reuters database. The first 1766 observations are used to estimate the parameters of all models and to obtain in-sample forecasts, whereas the last 1000 observations are used to obtain out-ofsample forecasts. These forecasts are nonadaptative, i.e. the parameters estimated in the in-sample period were kept fixed in the out-of-sample period. Table 1 reports descriptive statistics for the in-sample and out-of-sample periods. We observe a lower average return and a higher average standard deviation in the out-of-sample period in comparison to the in-

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sample period. This difference in sample moments is due to the fact that the out-of-sample period ranges from Jan. 2007 to Dec. 2010 and thus include the financial crisis observed during that period. It is worth pointing out three technical remarks regarding the implementation of the benchmark models. First, when implementing the CKL model we use a rolling estimation window of 252 observations. Therefore, we do not include the first 252 observations when evaluating the performance of all models. Second, the Risk Metrics approach does not involve any unknown coefficients as we set λ = 0.94. Finally, when implementing the OGARCH model we consider alternative number of factors. In particular, we implement by varying the number of factors from 1 up to N . To facilitate the exposition and discussion of results, we report the results for the OGARCH only for the best performing specification.

Table 1: Descriptive statistics The Table reports average in-sample and out-of-sample descriptive statistics for the data set composed of all stocks that belonged to the S&P100 index (78 stocks).

Number of obs. Mean Std. Deviation Median Kurtosis Skewness

3.2

In-sample 01/01/2000 - 09/01/2007 1766 0.018 2.111 0.007 19.666 -0.600

Out-of-sample 09/01/2007 - 31/12/2010 1000 -0.012 2.541 0.014 14.454 -0.214

Choice of the factor model

In order to implement the factor model we consider an extension of the Fama and French’s (1993) 3-factor model proposed by Carhart (1997), yit = αit + β1it (Rm − Rf ) + β2it SM Bt + β3it HM Lt + β4it P R1Y Rt + εit

(20)

where Rm − Rf is the excess return of the value-weight return on all NYSE, AMEX, and NASDAQ stocks minus the one-month Treasury bill rate and SM B, HM L and P R1Y R are returns on value-weighted, zero-investment, factor-mimicking portfolios for size, book-tomarket equity, and one-year momentum in stock returns, respectively.4 See Fama and French 4

All factors were obtained from the web page of Kenneth French (http://mba.tuck.dartmouth.edu/pages/ faculty/ken.french/data_library.html).

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(1993) and Carhart (1997) for details regarding the construction of these factor-mimicking portfolios. Each parameter in (20) is time-varying and evolves according to (2) and (3).5

3.3

Methodology for evaluating performance

We examine the portfolios’ performance in terms of the variance of returns (ˆ σ 2 ), Sharpe ratio (SR) and turnover. These statistics are computed as follows:

σ ˆ2 =

T −1 1 ∑ ′ (wt Rt+1 − µ ˆ)2 T −1 t=1

T −1 µ ˆ 1 ∑ ′ SR = , where µ ˆ= wt Rt+1 σ ˆ T −1 t=1

Turnover =

1 T −1

T −1 ∑ N ∑

) ( wj,t+1 − wj,t+ ,

t=1 j=1

where wj,t+ is the portfolio weight in asset j at time t + 1 but before rebalancing and wj,t+1 is the desired portfolio weight in asset j at time t + 1. As pointed out by DeMiguel et al. (2009), turnover as defined above can be interpreted as the average fraction of wealth traded in each period. To test the statistical significance of the difference between the variances and Sharpe ratios of the returns for two given portfolios, we follow DeMiguel et al. (2009) and use the stationary bootstrap of Politis and Romano (1994) with B=1,000 bootstrap resamples and expected block length b=5.6 The resulting bootstrap p-values are obtained using the methodology suggested in Ledoit and Wolf (2008, Remark 3.2).

3.4

Results

Table 2 reports the in-sample and out-of-sample portfolio variance, the Sharpe ratio and the portfolio turnover of the short-sales constrained and unconstrained minimum variance portfolio policies obtained with covariance matrices generated by the proposed model with time-varying factor loads based on the RW model (FlexFGARCH-RW) and on the learning model (FlexFGARCH-learning), and by the benchmark models. The benchmark models are 5

In unreported results, we consider a version of the factor model in (20) with time-invariant alphas. The results are very similar to those reported here and are available upon request. 6 We performed extensive robustness checks regarding the choice of the block length, using a range of values for b between 5 and 250. Regardless of the block length, the test results for the variances and Sharpe ratios are similar to those reported here.

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the proposed model with time invariant factor loads (FlexFGARCH-OLS), the OGARCH model, the Risk Metrics model, and the CKL model. The table also reports bootstrap pvalues for the differences between portfolios variance and Sharpe ratio with respect to those obtained with the CKL model. The results in terms of portfolio variance indicate that the FlexFGARCH-RW model delivers the lowest portfolio variance, for both in- and out-of-sample periods and for both constrained and unconstrained policies. For instance, in the out-of-sample period the FlexFGARCHRW model achieves a portfolio variance of 0.403 for the unconstrained policy, which is substantially (and statistically) lower than the portfolio variance delivered by the OGARCH, RM and CKL models (2.216, 2.777, and 1.112, respectively). We also observe that the portfolio variance obtained with the FlexFGARCH-RW model is lower in comparison to the FlexFGARCH-learning and FlexFGARCH-OLS models. The results for the Sharpe ratios indicate that the FlexFGARCH-RW model deliver a better risk-adjusted performance in comparison to the benchmark models. For instance, the FlexFGARCH-RW model delivered a SR of 0.022 for the unconstrained policy during the outof-sample period, whereas the CKL model delivered a SR of 0.012. However, the differences in SR are not statistically significant. In terms of portfolio turnover, we observe that the best performance was achieved by the OGARCH model in all cases. We also find that the the FlexFGARCH-RW model yields higher turnover in comparison to the FlexFGARCHlearning and FlexFGARCH-OLS models. This result suggest that RW factor loads leads to optimal portfolios with lower variance but also with higher turnover. Finally, we observe that the portfolio turnover turnover associated to the constrained policies tends to be lower in comparison to that of unconstrained policies. This result is in line with previous empirical literature such as DeMiguel et al. (2009). To further illustrate the results, we depict in Figure 1 the boxplot of MVP returns for the in- and out-of-sample periods and for the unconstrained and constrained policies. The figure shows that the dispersion of portfolio returns obtained with the FlexFGARCH-RW model is substantially lower in comparison to the remaining models in all cases, which corroborates the results in Table 2. We also observe that the dispersion of portfolio returns during the out-of-sample period is substantially higher in comparison to the in-sample period, which reflects the impact of the worldwide financial crises of 2007-2008. Alternative re-balancing frequencies The results discussed above are based on the assumption that investors adjust their port-

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Unconstrained MVP, out of sample

Unconstrained MVP, in sample FlexFGARCH−RW FLexFGARCH−Learning FlexFGARCH−OLS OGARCH Risk Metrics CKL −10

−5

0

5

10

−10

−5

0

5

10

Constrained MVP, out of sample

Constrained MVP, in sample FlexFGARCH−RW FLexFGARCH−Learning FlexFGARCH−OLS OGARCH Risk Metrics CKL −10

−5

0

5

10

−10

−5

0

5

10

Figure 1: Boxplot of portfolio returns folio on a daily basis. The transaction costs incurred with such frequent trading can possibly deteriorate the net portfolio performance. Obviously this effect can be reduced by adjusting the portfolio less frequently, such as on a weekly or monthly basis, which in fact is done in practice by many institutional investors. A drawback of re-balancing the portfolio less frequently is that the portfolio weights become outdated, which may harm its performance. We examine the performance of the MVP under alternative re-balancing frequencies. Tables 3 and 4 show the results for weekly and monthly re-balancing frequencies, respectively. As expected, we find that lowering the re-balancing frequency results in a substantial reduction in portfolio turnover. We observe that the FlexFGARCH-RW model delivers the lowest portfolio variance in the majority of the cases. Summarizing the results in Tables 2 to 4, we find that lowering the re-balancing frequency does not lead to better risk-adjusted performance in terms of SR. In fact, the best overall out-of-sample performance in terms of portfolio variance and SR is achieved when the

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FlexFGARCH-RW model is used to obtained unconstrained MVP under daily re-balancing.

4

Concluding remarks

Factor models are currently established as an alternative to alleviate the problem of dimensionality and the burden of the estimation process when modeling covariance matrices of portfolios containing a large number of assets. In this paper, we put forward a novel, flexible approach to obtain conditional covariance matrices based on a factor model that extends previous econometric specifications. The proposed approach achieves great flexibility by allowing a parsimonious specification for the common factors and alternative specifications the individual assets in the portfolio. Moreover, we treat factor loads as time-varying latent variables and consider richer dynamics based on recent developments in asset pricing theory. We apply the proposed model to obtain in-sample and out-of-sample one-step-ahead forecasts of the conditional covariance matrix of all assets that belong the S&P100 index during the sample period, and use the estimated matrices to compute short selling-constrained and unconstrained minimum variance portfolios. The performance of the proposed model is compared to that of alternative benchmark models, including existing factor approaches. The results indicate that the proposed model delivers less risky portfolios in comparison to the benchmark models.

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Table 2: Minimum variance portfolio performance The Table reports the average daily portfolio variance, the Sharpe ratio and the portfolio turnover of the short-sales constrained and unconstrained minimum variance portfolio policies obtained with covariance matrices generated by the FlexFGARCH-RW model and FlexFGARCH-learning models. The benchmark models are the FlexFGARCHOLS, the OGARCH model, the RiskMetrics model, and the CKL model. Bootstrap p-values for the differences between portfolios variance and Sharpe ratio with respect to those obtained with the CKL model appear in parenthesis below each coefficient.

Variance

Sharpe ratio Constrained

Turnover

0.286 (0.000) 0.602 (0.008) 0.594 (0.036) 0.746 (0.000) 0.546 (0.666) 0.537 (1.000)

0.028 (0.808) −0.005 (0.020) 0.006 (0.168) 0.018 (0.670) 0.002 (0.124) 0.023 (1.000) Constrained

0.333

0.657 (0.000) 1.343 (0.006) 1.128 (0.534) 2.222 (0.000) 1.170 (0.628) 1.149 (1.000)

−0.009 (0.882) −0.004 (0.448) −0.002 (0.148) −0.004 (0.628) −0.007 (0.638) −0.013 (1.000)

In-sample FlexFGARCH-RW FlexFGARCH-learning FlexFGARCH-OLS OGARCH RiskMetrics CKL Out-of-sample FlexFGARCH-RW FlexFGARCH-learning FlexFGARCH-OLS OGARCH RiskMetrics CKL

0.089 0.095 0.028 0.266 0.040

0.476 0.082 0.094 0.030 0.247 0.036

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Variance

Sharpe ratio Turnover Unconstrained

0.231 (0.000) 0.527 (0.668) 0.651 (0.000) 0.745 (0.000) 1.366 (0.000) 0.515 (1.000)

0.026 (0.960) 0.000 (0.106) 0.004 (0.358) 0.019 (0.894) 0.004 (0.480) 0.023 (1.000) Unconstrained

0.403 (0.000) 1.158 (0.634) 1.198 (0.250) 2.216 (0.000) 2.777 (0.000) 1.112 (1.000)

0.022 (0.676) 0.003 (0.738) 0.025 (0.374) −0.005 (0.646) 0.006 (0.862) 0.012 (1.000)

0.452 0.136 0.138 0.034 1.426 0.068

0.585 0.118 0.151 0.031 1.633 0.073

Table 3: Minimum variance portfolio performance with weekly re-balancing The Table reports the average daily portfolio variance, the Sharpe ratio and the portfolio turnover of the short-sales constrained and unconstrained minimum variance portfolio policies obtained with covariance matrices generated by the FlexFGARCH-RW and FlexFGARCH-learning models. The benchmark models are the FlexFGARCH-OLS, the OGARCH model, the RiskMetrics model, and the CKL model. Bootstrap p-values for the differences between portfolios variance and Sharpe ratio with respect to those obtained with the CKL model appear in parenthesis below each coefficient.

Variance

Sharpe ratio Constrained

Turnover

0.592 (0.172) 0.633 (0.000) 0.593 (0.12) 0.745 (0.000) 0.560 (0.42) 0.544 (1.000)

0.004 (0.185) 0.003 (0.077) 0.003 (0.079) 0.018 (0.501) 0.014 (0.369) 0.025 (1.000) Constrained

0.129

1.219 (0.324) 1.383 (0.000) 1.138 (0.807) 2.220 (0.000) 1.237 (0.144) 1.149 (1.000)

0.013 (0.132) −0.002 (0.394) 0.000 (0.118) −0.004 (0.678) −0.008 (0.766) −0.012 (1.000)

In-sample FlexFGARCH-RW FlexFGARCH-learning FlexFGARCH-OLS OGARCH RiskMetrics CKL Out-of-sample FlexFGARCH-RW FlexFGARCH-learning FlexFGARCH-OLS OGARCH RiskMetrics CKL

0.040 0.042 0.014 0.131 0.021

0.167 0.038 0.043 0.015 0.130 0.020

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Variance

Sharpe ratio Turnover Unconstrained

0.438 (0.000) 0.568 (0.156) 0.651 (0.000) 0.743 (0.000) 1.338 (0.000) 0.525 (1.000)

0.017 (0.657) 0.011 (0.329) 0.002 (0.261) 0.019 (0.705) 0.019 (0.823) 0.026 (1.000) Unconstrained

0.955 (0.014) 1.223 (0.422) 1.184 (0.514) 2.214 (0.000) 2.767 (0.000) 1.132 (1.000)

0.000 (0.697) 0.005 (0.863) 0.027 (0.232) −0.005 (0.703) 0.003 (0.909) 0.009 (1.000)

0.174 0.061 0.062 0.016 0.740 0.037

0.215 0.053 0.069 0.016 0.861 0.039

Table 4: Minimum variance portfolio performance with monthly re-balancing The Table reports the average daily portfolio variance. the Sharpe ratio and the portfolio turnover of the short-sales constrained and unconstrained minimum variance portfolio policies obtained with covariance matrices generated by the FlexFGARCH-RW and FlexFGARCH-learning models. The benchmark models are the FlexFGARCH-OLS, the OGARCH model, the RiskMetrics model, and the CKL model. Bootstrap p-values for the differences between portfolios variance and Sharpe ratio with respect to those obtained with the CKL model appear in parenthesis below each coefficient.

Variance

Sharpe ratio Constrained

Turnover

0.852 (0.000) 0.648 (0.000) 0.590 (0.302) 0.744 (0.000) 0.577 (0.556) 0.561 (1.000)

0.013 (0.477) 0.004 (0.097) 0.001 (0.061) 0.018 (0.499) 0.022 (0.817) 0.025 (1.000) Constrained

0.041

1.472 (0.000) 1.405 (0.000) 1.145 (0.497) 2.220 (0.000) 1.364 (0.028) 1.166 (1.000)

−0.014 (0.859) −0.001 (0.316) 0.001 (0.11) −0.004 (0.674) −0.020 (0.487) −0.012 (1.000)

In-sample FlexFGARCH-RW FlexFGARCH-learning FlexFGARCH-OLS OGARCH RiskMetrics CKL Out-of-sample FlexFGARCH-RW FlexFGARCH-learning FlexFGARCH-OLS OGARCH RiskMetrics CKL

0.016 0.018 0.006 0.058 0.012

0.050 0.016 0.019 0.008 0.055 0.012

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Variance

Sharpe ratio Turnover Unconstrained

0.614 (0.294) 0.581 (0.188) 0.661 (0.000) 0.741 (0.000) 1.343 (0.000) 0.542 (1.000)

0.016 (0.511) 0.012 (0.271) −0.002 (0.134) 0.018 (0.134) 0.036 (0.702) 0.026 (1.000) Unconstrained

1.075 (0.257) 1.244 (0.46) 1.212 (0.554) 2.219 (0.000) 3.373 (0.000) 1.161 (1.000)

−0.004 (0.555) 0.006 (0.815) 0.023 (0.404) −0.004 (0.673) −0.012 (0.523) 0.010 (1.000)

0.059 0.025 0.027 0.008 0.332 0.021

0.069 0.022 0.031 0.008 0.388 0.023

Appendix: Univariate volatility models In this Appendix we describe the univariate GARCH specifications we use to model the conditional variances of the factors and the conditional variance of the residuals of the factor model.

GARCH: 2 σt2 = ω + αϵ2t−1 + βσt−1

Glosten–Jagannathan–Runkle GARCH (GJR-GARCH): 2 σt2 = ω + αϵ2t−1 + γI[ϵt−1 < 0]ϵ2t−1 + βσt−1

Exponential GARCH (EGARCH): |ϵt−1 | 2 ln(σt2 ) = ω + α √ + γ √ϵt−1 + βσt−1 2 2 σt−1

σt−1

Threshold GARCH (TGARCH): σt = ω + α|ϵt−1 | + γI[ϵt−1 < 0]|ϵt−1 | + βσt−1

Asymmetric power GARCH (APARCH): λ σtλ = ω + α (|ϵt−1 | + γϵt−1 )λ + βσt−1

Asymmetric GARCH (AGARCH): 2 σt2 = ω + α(ϵt−1 + γ)2 + βσt−1

Nonlinear asymmetric GARCH (NAGARCH): √ 2 )2 + βσ 2 σt2 = ω + α(ϵt−1 + γ σt−1 t−1

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