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Open AccessArticle

Covariance Prediction in Large Portfolio Allocation

1
São Paulo School of Economics, FGV, São Paulo 01332-000, Brazil
2
Department of Statistics, University of Campinas, Campinas 13083-859, Brazil
3
UC3M-Santander Big Data Institute, Universidad Carlos III de Madrid, Getafe 28903, Spain
4
Department of Economics, Universidade Federal de Santa Catarina, Florianópolis 88040-970, Brazil
*
Author to whom correspondence should be addressed.
Econometrics 2019, 7(2), 19; https://doi.org/10.3390/econometrics7020019
Received: 12 November 2018 / Revised: 23 April 2019 / Accepted: 2 May 2019 / Published: 9 May 2019

Abstract

Many financial decisions, such as portfolio allocation, risk management, option pricing and hedge strategies, are based on forecasts of the conditional variances, covariances and correlations of financial returns. The paper shows an empirical comparison of several methods to predict one-step-ahead conditional covariance matrices. These matrices are used as inputs to obtain out-of-sample minimum variance portfolios based on stocks belonging to the S&P500 index from 2000 to 2017 and sub-periods. The analysis is done through several metrics, including standard deviation, turnover, net average return, information ratio and Sortino’s ratio. We find that no method is the best in all scenarios and the performance depends on the criterion, the period of analysis and the rebalancing strategy.
Keywords: Minimum variance portfolio; risk; shrinkage; S&P 500 Minimum variance portfolio; risk; shrinkage; S&P 500

1. Introduction

Forecasting returns, volatilities and conditional correlations has attracted the interest of researchers and practitioners in finance since these factors are crucial, for example, in portfolio allocation, risk management, option pricing and hedging strategies; see, for instance, Engle (2009), Hlouskova et al. (2009) and Boudt et al. (2013) for some references.
A well-known stylised fact in multivariate time series of financial returns is that not only conditional variances but also conditional covariances and correlations evolve over time. To describe this evolution, several methods have been proposed in the literature. In general, these methods involve different ways to circumvent the issue of dimensionality. The treatment of this problem is vital for the estimation of large portfolios (composed of hundreds or thousands of assets). As noted by Engle et al. (2017), when dealing with portfolios composed of a thousand time series, many multivariate GARCH models present unsatisfactory performance or computational problems in their estimation. For some multivariate GARCH models, estimation problems arise even for smaller dimensions; see, for instance, Laurent et al. (2012), Caporin and McAleer (2014), Caporin and Paruolo (2015) and de Almeida et al. (2018).
Our empirical application is based on an investor who adopts the minimum variance criterion in order to decide on portfolio allocations. A very large body of literature in portfolio optimization considers this particular policy; see, for instance, Clarke et al. (2011 2006) for extensive practitioner-oriented studies on the performance and composition of minimum variance portfolios. This policy can be seen as a particular case of the traditional mean-variance optimisation. The mean-variance problem, however, is known to be very sensitive to estimation of the mean returns (Frahm 2010; Jagannathan and Ma 2003).1 Very often, the estimation error in the mean returns degrades the overall portfolio performance and introduces an undesirable level of portfolio turnover. In fact, existing evidence suggests that the performance of optimal portfolios that do not rely on estimated mean returns is usually better, see DeMiguel et al. (2009).
To obtain the minimum variance portfolio, the key input is the estimate of the conditional covariance matrix. As far as we known, there are few works in the literature comparing the estimation of this matrix for large portfolios, with Creal et al. (2011), Hafner and Reznikova (2012), Engle et al. (2017), Nakagawa et al. (2018) and Moura and Santos (2018) being especially relevant. Given the myriad of models and methods in the literature to estimate the covariance matrix, empirical studies about the comparison of estimates in large portfolios are most welcome.
The paper is intended to assess the performance of several methods to predict one-step-ahead conditional covariance matrices in large portfolios. This is done empirically, by comparing the out-of-sample performance of minimum variance portfolios based on S&P500 stocks traded from 2 January 2000 to 30 November 2017, using measures such as average (AV), standard deviation (SD), information ratio (IR), Sortino’s ratio (SR) (Sortino and van der Meer 1991), turnover (TO) and average portfolio net of transaction cost (AV n e t ). Since not all stocks of the index were traded during the whole period, we consider portfolios of dimension N = 174 stocks. To assess the robustness of the results, we also the analyse three sub-periods: the pre-crisis period (January 2004 to December 2007), the subprime crisis period (January 2008 to June 2009), and the post-crisis period (July 2009 to November 2017).
We consider several attractive methods and models including recent proposals used by practitioners and academics to predict one-step-ahead conditional covariance matrices. They are selected mainly because they use different approaches to overcome the issue of dimensionality problem. Specifically, the paper compares the DCC model as used in Engle et al. (2017), the DECO model of Engle and Kelly (2012), the OGARCH model of Alexander and Chibumba (1996), the RiskMetrics 1994 and the RiskMetrics 2006 (Zumbach 2007) methods, the generalised principal volatility components analysis (GPVC) proposed by Li et al. (2016) as a generalisation of the procedure of Hu and Tsay (2014), and we also apply the robust version of the GPVC method proposed by Trucíos et al. (2019). DCC models are estimated using composite likelihood, as advocated in Pakel et al. (2014). In addition, the linear shrinkage (LS) and non-linear shrinkage (NLS) of Ledoit and Wolf (2004a) and Ledoit and Wolf (2012), respectively, are applied on all the previous methods. Therefore, compared to Engle et al. (2017), Hafner and Reznikova (2012) and Nakagawa et al. (2018), the set of competing methods is much bigger and the device of shrinkage is assessed in all the compared methods. We consider a total of 47 methods, including the equal-weighted portfolio strategy. This constitutes the main contribution of the paper.
The rest of the paper is organised as follows: Section 2 presents the methods and models used to predict the one-step-ahead volatility covariance matrix. It also presents the composite likelihood used to estimate the DCC model and the shrinkage method as presented in Pakel et al. (2014). The empirical application is given in Section 3. Section 4 concludes and the list of the estimation methods is in the Appendix A.

2. The Forecast Methods

Denote by r i , t , i = 1 , , N , t = 1 , , T the return of the i-th asset at time t, where N is the number of assets under consideration to construct the portfolio and T denotes the sample size. For simplicity, consider that E ( r i , t | F t 1 ) = 0 , where F t 1 denotes the information available at time ( t 1 ) . Let r t = ( r 1 , t , , r N , t ) ; the conditional covariance matrix is defined as H t = Cov ( r t | F t 1 ) with elements h i , j , t = Cov ( r i , t , r j , t | F t 1 ) . At time ( t 1 ) , we are interested in estimating H t in order to select a portfolio for the period ( t 1 , t ] . In the following we present some methods to estimate it.

2.1. The RiskMetrics Methods

One of the most popular methods used in risk analysis is the RiskMetrics method developed by the RiskMetrics Group at JP Morgan. We call this the RiskMetrics 1994 (RM1994) method. The main feature of the RiskMetrics method is that the predicted volatility is a linear function of the present and past squared returns. Although it has being widely used, it has some problems. In order to overcome some of these problems, the same group developed the RM2006 method. Like the RM1994 method, the RM2006 method is also data-oriented, in the sense that it was calibrated and tested to have good performance with the majority of the target empirical data, and was developed to take into account some of the stylised facts and weaknesses detected in the RM1994 method. We can summarize the main modifications in three types. In the first type, considering that the volatility has a long memory feature, the weights decay logarithmically instead of exponentially, as happens in the RM1994 method. The second is that the weights depend on the forecast horizon. The third is that the conditional distribution of the return is not multivariate Gaussian; the distribution is based on the estimated devolatilised residuals and it can be roughly defined as a Student-t distribution with scale correction. Finally, the return levels are modelled considering the lagged correlation between returns.

2.2. The CCC Model

The constant conditional correlation model (Bollerslev 1990) is one of the simplest MGARCH models to estimate, since basically the variances are modelled independently and the covariances are obtained using the conditional standard deviation and a constant conditional correlation matrix. The conditional covariance matrix H t evolves according to:
H t = D t R D t ,
D t = D i a g ( d 1 , t , , d N , t ) ,
R = D i a g ( H ) 1 / 2 H D i a g ( H ) 1 / 2 ,
H = Cov ( r t ) ,
with d i , t 2 = V a r ( r i , t | F t 1 ) (marginal univariate conditional variances). The advantage of the CCC model is its easy estimation, although, the main disadvantage is the strong assumption that conditional correlations are time-invariant. Engle (2002) extended this idea in a dynamic conditional correlation way, as detailed in the next section.

2.3. The DCC Model

In this section, we describe the scalar DCC model of Engle (2002) as used in Pakel et al. (2014) and Engle et al. (2017), and the composite likelihood. The non-linear shrinkage method, which is also used to estimate the DCC model, is presented in Section 2.8. In the DCC model, the marginal univariate conditional variances d i , t 2 = V a r ( r i , t | F t 1 ) are modelled first. Define the devolatilised residuals as s t = ( r 1 , t / d 1 , t , , r N , t / d N , t ) . We use the DCC model with correlation targeting as in Engle et al. (2017). The conditional covariance matrix H t evolves according to:
H t = D t R t D t ,
R t = D i a g ( Q t ) 1 / 2 Q t D i a g ( Q t ) 1 / 2 ,
Q t = ( 1 α β ) C + α s t 1 s t 1 + β Q t 1 ,
where D t is a diagonal matrix with the i-th element of the diagonal equal to d i , t 2 , C = C o r r ( r t ) = C o v ( s t ) is the unconditional correlation matrix, and R t = C o r r ( r t | F t 1 ) = C o v ( s t | F t 1 ) is the conditional correlation matrix at time t. The parameters α and β are non-negative with α + β < 1 . We have
r t | F t 1 W S ( 0 , H t ) ,
where W S ( 0 , H t ) means a multivariate distribution with mean zero and covariance matrix H t .
The model is usually estimated in three stages. In each stage, the estimation is conditional on the estimates found in previous stages. The stages are: (1) estimate D t usually assuming a GARCH(1,1) model for each t = 1 , , T , and evaluate the devolatilised residuals; (2) select an estimator of the correlation target matrix C using the devolatilised residuals; and (3) estimate the parameters α and β . We will comment on stage one in the application section and on stage 2 in Section 2.8. In the third stage, even with only two parameters, one may face estimation problems with a large number of assets because it is necessary to invert the conditional covariance matrix H t (for each t = 1 , , T ). One way to overcome this problem is through the use of the composite (log-)likelihood2 to compute it. This method was proposed in the 2008 version of Pakel et al. (2014). In the 2014 version, they showed that the estimators of α and β , given by maximizing the composite likelihood, are consistent although not efficient. They evaluate the composite likelihood by summing the likelihood of all contiguous pairs. Thus, there are only ( N 1 ) bivariate terms and for any contiguous pair it is only necessary to invert a matrix of order two. For instance, let r ( i ) = ( r i , 1 , , r i , T ) , i = 1 , , N , i.e., the series of returns of the ith asset, and denote by l i ( α , β ; r ( i ) , r ( i + 1 ) ) the likelihood of the pair ( r ( i ) , r ( i + 1 ) ) , i = 1 , , N 1 , assuming that each pair comes from a bivariate DCC model, defined similarly as the model given by Equations (5–7). Then, the composite likelihood is given by:
C L ( α , β ; r ( i ) , i = 1 , , N ) = i = 1 N 1 l i ( α , β ; r ( i ) , r ( i + 1 ) ) .
Engle et al. (2017) argue that the estimator of the conditional covariance matrix given by the DCC model using composite likelihood in stage three with the estimation of the unconditional correlation matrix using non-linear shrinkage in stage two is robust against model misspecification in large dimensions (N).

2.4. The DECO Model

Engle and Kelly (2012) propose a dynamic equicorrelation (DECO) model as a trade-off between a model which imposes many restrictions in the covariance matrix and a less structured model. They contend that imposing too much structure can lead to an efficient estimation when the restrictions are correct, but can suffer from breakdown in the presence of misspecification. On the other hand, the lack of restrictions may lead to the issue of dimensionality. Considering this trade-off, they propose a model where the cross-correlations between any pair of returns are equal on the same day, but it can vary over time. In addition, as in the CCC and DCC models, the DECO model also assumes that the marginals are modelled by a univariate volatility model. Using the same notation, we have d i , t 2 = V a r ( r i , t | F t 1 ) , and the covariance matrix is written as H t = D t R t D t as in Equation (5). The equicorrelation matrix is given by:
R t = ( 1 ρ t ) I N + ρ t J N ,
where ρ t is the equicorrelation, I N denotes the N-dimensional identity matrix and J N is the N × N matrix of ones. According to Engle and Kelly (2012), R t 1 exist if and only if ρ t 1 and ρ t 1 / ( N 1 ) , and R t is positive definite if and only if ρ t ( 1 / ( N 1 ) , 1 ) . The evaluation of the likelihood is easy because we have closed forms for R t 1 and det ( R t ) , given by:
R t 1 = 1 1 ρ t I N ρ t ( 1 ρ t ) ( 1 + [ N 1 ] ρ t ) J N ,
and
det ( R t ) = ( 1 ρ t ) N 1 1 + ( N 1 ) ρ t ,
respectively. This description of the DECO model corresponds to a single block. The DECO model can also be used considering many blocks, as described in Engle and Kelly (2012).

2.5. The OGARCH Model

Alexander and Chibumba (1996) propose the Orthogonal GARCH (OGARCH) model, a dimension reduction technique to model the conditional covariance matrix. The model intends to simplify the problem of modelling an N-dimensional system into modelling a system of K-dimension orthogonal components where those components are obtained through principal component analysis ( K N ). Since the components are orthogonal, the conditional covariance matrix of the whole system can be obtained as:
H t = A D t A + V ϵ ,
where A is an N × k matrix whose columns are the normalised eigenvectors associated with the unconditional covariance matrix, D t is a diagonal matrix whose elements are the conditional variances of the k principal orthogonal components associated with the k largest eigenvalues, and V ϵ is the covariance matrix of the errors that can be ignored. The conditional variances of each component can be modelled by a GARCH-type model.
Alexander and Chibumba (1996) and Alexander (2002) emphasise the importance of using a number of components k much smaller than N. However, Bauwens et al. (2006) and Becker et al. (2015) suggest using k = N to avoid problems related with the inverse of H t . The OGARCH model with k = N is a particular case of the GO-GARCH model (Van der Weide 2002).

2.6. The Generalised Principal Volatility Components Model

The generalised principal volatility components (GPVC) procedure is a dimension reduction technique recently proposed by Li et al. (2016), which decomposes a series into two groups of volatility components. The first group corresponds to a small number of components with volatility evolving over time while the second one corresponds to components whose volatility is constant over time. The GPVC procedure considers an orthogonal matrix M = [ A : B ] and decomposes an N-dimensional vector y t = ( y 1 t , , y N t ) with E ( y t | F t 1 ) = 0 into:
y t = M M y t = ( A A + B B ) y t = A f t + f f l t ,
with f t = A y t and f f l t = B B y t . The matrix M is obtained through the decomposition G M = Λ M , where Λ is a diagonal matrix with elements given by the eigenvalues in decreasing order and M is the associated matrix of normalised eigenvectors. The columns of matrices A and B are the eigenvectors associated with the non-zero and zero eigenvalues, respectively, which are obtained from the eigenvalue decomposition of the matrix G . In practice, G is given by:
G = k = 1 g t = 1 T ω ( y t ) E 2 y t y t Σ I ( y t k y t ) ,
where g is a positive integer that gives the maximum lag order considered, ω ( · ) is a weight function, Σ is the unconditional covariance matrix and · is the L 1 norm. Then, after some calculations, the conditional covariance matrix can be obtained by:
H t = A H t f A + A A Σ B B + B B Σ ,
where H t f is the conditional covariance matrix of the volatility components with volatility evolving over time and the remaining are terms as defined previously3. The matrix G is estimated as:
G ^ = k = 1 g τ = 1 T ω ( y τ ) 1 T k t = k + 1 T y t y t Σ ^ I ( y t k y τ ) 2 .
The estimated version of Equation (16) is obtained by replacing the true values with the estimated ones.

2.7. The Robust GPVC Model

Trucíos et al. (2019) show the non-robustness of the GPVC procedure of Li et al. (2016) and propose an alternative procedure to obtain volatility components that is robust to outliers. This procedure is based on a robust estimator of the unconditional covariance matrix, a weighted estimator of E y t y t Σ I ( y t k y t ) , and robustified filters. The matrix (17) is replaced by a less sensitive matrix, defined as:
G ^ R = k = 1 g τ = 1 T ω ( y τ ) t = k + 1 T δ ( d t 2 ) ( y t y t Σ ^ R ) I ( y t k y τ ) 2 ,
where d t 2 is the robust squared Mahalanobis distance given by d t 2 = ( y t μ ^ R ) Σ ^ t 1 ( y t μ ^ R ) with Σ ^ t = 0.01 ρ ( y t 1 y t 1 ) + 0.99 Σ ^ t 1 , Σ ^ 1 = Σ ^ R and μ ^ R , Σ ^ R being robust estimates of the unconditional mean and covariance matrix. Trucíos et al. (2019) use the minimum covariance determinant (MCD) estimator of Rousseeuw (1984), implemented by the algorithm of Hubert et al. (2012). The robust filters, ρ ( · ) and δ ( · ) are given by ρ ( x t ) = x t if d t 2 c , ρ ( x t ) = Σ ^ R if d t 2 > c ; δ ( x ) = 1 if x c , δ ( x ) = 1 / x if x > c and δ ( · ) = δ ( · ) / | | δ ( · ) | | , where · is the L 1 norm. For details, see Trucíos et al. (2019).
To avoid returns corresponding to periods with high volatility being considered as possible outliers, the robust procedure incorporates in the squared Mahalanobis distance a covariance matrix evolving over time, which can be seen as a robust RM1994 method with λ = 0.99 .
Finally, the conditional covariance matrix H t is obtained as in Equation (16).

2.8. Linear and Non-Linear Shrinkage

Besides the estimation of the covariance matrix ( H t ), in some of the aforementioned models, we have to estimate the unconditional covariance or correlation matrix; for instance, the matrix C in Equation (7) of the DCC model. Generally, the estimation of the unconditional correlation (covariance) matrix is done using the sample correlation (covariance) matrix. However, this is inefficient in the large dimensional case because we could end up with a number of parameters with the same order of magnitude as the dataset, or even larger (see, for instance, the simulation study in the Appendix of Engle et al. (2017)). In general, comparing the eigenvalues of the true correlation matrix with the eigenvalues of the sample correlation matrix, there is a tendency to underestimate the smaller eigenvalues and overestimate the larger ones. A natural way to reduce this bias is to increase the smaller eigenvalues and decrease the larger sample eigenvalues and then reconstruct the estimate of the correlation matrix. This is the main idea behind the shrinkage method. Engle et al. (2017) analyse the use of three types the shrinkage: linear shrinkage of Ledoit and Wolf (2004b) with shrinkage target given by (a multiple of) the identity matrix; linear shrinkage of Ledoit and Wolf (2004a) with shrinkage target given by the equicorrelation matrix; and the non-linear shrinkage of Ledoit and Wolf (2012) for the estimation of the unconditional correlation matrix in Equation (7). Using simulation, they conclude that the three types of shrinkage have better performance than the use of the sample correlation matrix in the estimation of H t , and the best performance is obtained from the non-linear shrinkage. They conclude that the application of non-linear shrinkage improves the estimation, and the improvement generally increases for a larger number of assets. In the application, they also apply the non-linear shrinkage to the estimated one-step-ahead conditional covariance matrix, which is not done in the simulation study. In the empirical application, they construct portfolios of global minimum variance with portfolio sizes 100 , 500 and 1000 and updated monthly. As in the simulation study, they construct portfolios with H t modelled by DCC and CCC models and the RiskMetrics 2006 method. However, besides applying the linear and non-linear shrinkage to the target correlation matrix, they also apply the shrinkages to the one-step-ahead prediction of the volatility matrix. The best performance is achieved by the DCC model with the non-linear shrinkage applied only to the estimation of the intercept matrix, followed by the non-linear shrinkage applied both to the intercept matrix and to the one-step-ahead prediction matrix. We use the linear shrinkage towards the equicorrelation matrix, because in Engle et al. (2017) it presented slightly better performance than the shrinkage towards the identity matrix, although the estimator does not belong to the class of rotation-equivariant estimators.
For a light introduction to the main idea behind shrinkage, suppose we want to estimate the covariance matrix Σ and we have an estimate C ^ based on a sample of size T. For instance, C ^ could be the sample covariance matrix and Σ , the population matrix (unconditional covariance matrix). This is the case of the estimation of the DCC, where Σ is the intercept matrix. When the ratio N / T , called concentration ratio, becomes large, we have in-sample overfitting due to the excessive number of parameters, introducing a bias in the estimation of the eigenvalues. One way to correct this problem is through the shrinkage method.
For the linear shrinkage towards the equicorrelation matrix, denote by c ^ i j the element of the estimate C ^ . The mean of the estimated correlations is given by:
r ¯ = 2 ( N 1 ) N i = 1 N 1 j = i + 1 N c ^ i , j c ^ i , i c ^ j , j ,
such that for the target matrix F we have f i , i = c ^ i , i and f i , j = r ¯ c ^ i , i c ^ j , j . The shrinkage estimate is given by:
Σ ^ S h r i n k = δ F + ( 1 δ ) C ^ ,
where the shrinkage intensity, δ , is such that it minimizes the expected quadratic loss as in Ledoit and Wolf (2004a). For the shrinkage intensity δ , define the quadratic loss function
L ( δ ) = | | δ F + ( 1 δ ) C ^ Σ | | 2 .
Ledoit and Wolf (2004a) propose to use the shrinkage intensity, which minimizes the risk function R ( δ ) = E ( L ( δ ) ) . The formulae and the derivation of the estimated shrinkage intensity can be found in the Appendix B of Ledoit and Wolf (2004a).
Regarding the non-linear shrinkage, let C ^ having dimension ( N × N ) , ( λ ^ 1 , , λ ^ N ) , sorted in descending order, be the set of eigenvalues, and ( u ^ 1 , , u ^ N ) the corresponding eigenvectors, such that:
C ^ = i = 1 N λ ^ i u ^ i u ^ i .
For an investor holding a portfolio with weights ω , the estimated variance is given by ω C ^ ω . The non-linear shrinkage of Ledoit and Wolf (2004b) is a transformation from ( λ ^ 1 , , λ ^ N ) to λ ˜ = ( λ ˜ 1 , , λ ˜ N ) , such that substituting λ ^ i for λ ˜ i in Equation (21) gives a consistent estimator of the out-of-sample variance ω Σ ω . Denote by λ = ( λ 1 , , λ N ) the set of eigenvalues of Σ in descending order. Ledoit and Wolf (2004b) define QuEST functions ( q 1 ( λ ) , , q N ( λ ) , such that λ ˜ minimizes the Euclidean distance between the QuEST functions and the sample eigenvalues, i.e., given by:
λ ˜ = arg min λ [ 0 , ) N i = 1 N [ q i ( λ ) λ ^ i ] 2 .
A definition of the QuEST functions and a rigorous exposition of non-linear shrinkage can be found in Ledoit and Wolf (2012), while a lighter presentation can be found in the Supplementary Material of Engle et al. (2017).

3. Empirical Application

3.1. Data and Methods

In this section, we implement the procedures described in Section 2 and use the predicted one-step-ahead conditional covariance matrix to construct the minimum variance portfolio (MVP) of the stocks used in the composition of the S&P 500 index, traded from 2 January 2000 to 30 November 2017. Because not all stocks of the index were traded during the whole period, we ended up with N = 174 stocks.
To evaluate the out-of-sample portfolio performance, we consider a rolling window scheme. The out-of-sample portfolio performance is evaluated in four different periods, namely: pre-crisis period (January 2004 to December 2007, 1008 days), subprime crisis period (January 2008 to June 2009, 378 days), post-crisis period (July 2009 to November 2017, 2218 days), and full period (January 2004 to November 2017, 3503 days). In each window, the one-step-ahead covariance matrix is estimated and the MVP values with and without short-sale constraints are obtained. The weights in the MVP portfolio are rebalanced with both daily and monthly frequencies. In the latter case, we follow Engle et al. (2017), that is, we obtain the portfolio returns daily but update the weights monthly (following the common convention we use 21 consecutive trading days as a month). Monthly updating is common in practice to reduce transaction costs.
The procedures described in Section 2 are combined with the linear and non-linear shrinkage estimator described in SubSection 2.8. The linear and non-linear shrinkage are applied at the beginning and/or at the end of the estimation procedure. A detailed description of each combination of the estimation procedures is given in the Appendix A. In addition, for the sake of comparison, we also implement the naive equal-weighted portfolio. In the line of Engle et al. (2017), Gambacciani and Paolella (2017), Trucíos et al. (2018) among others, we consider the following annualised out-of-sample performance measures. Denote by R p = { r p , 1 , , r p , k } the observed out-of-sample returns from a given method where k in the length of the out-of-sample period. The measures considered in this paper: the annualised average portfolio return (AV), standard deviation portfolio return (SD), information ratio (IR), Sortino’s ratio (SR) and average turnover (TO) are computed as follows:
  • AV: equal to 252 × R ¯ p , where R ¯ p is the average of the elements of R p .
  • SD: equal to 252 × S p , where S p is the standard deviation of the elements of R p .
  • IR: AV/SD.
  • SR: AV/ 252 × S 2 , where S 2 is the mean of r p , i , i = 1 , , k , with r p , i = r p , i 2 if r p , i less than the minimal acceptable return, which is taken to zero, and zero otherwise.
  • TO: k 1 t = 2 k j = 1 N | ω j , t ω j , t 1 | where ω j , t is the portfolio weight at time t for the j-th asset, and k is the number of the out-of-sample portfolio returns.
As pointed out by Kirby and Ostdiek (2012), Santos and Ferreira (2017), Olivares-Nadal and DeMiguel (2018), among others, transaction costs (c) can have an impact on the portfolio’s performance. In order to take into account those costs, we also compute the portfolio returns net of transaction cost. For a given c, the portfolio return net of transaction costs at time t is given by r p , t n e t = ( 1 c × t u r n o v e r t ) ( 1 + r p , t ) 1 and then the annualised average portfolio return net of transaction costs is AV n e t = 252 × R ¯ p n e t where R ¯ p n e t is the average of the portfolio return net of transaction costs r p , 1 n e t , , r p , k n e t . We consider c = 20 b p (intermediate) and c = 50 b p (high level) transaction costs where a basis point (bp) is a unit of measure commonly used in finance and is equivalent to 0.01 % . The annualised average portfolio return net of transation costs considering c = 20 b p and c = 50 b p are denoted by AV 20 b p n e t and AV 50 b p n e t , respectively.

3.2. Results

Table 1, Table 2, Table 3, Table 4, Table 5, Table 6, Table 7 and Table 8 report annualised out-of-sample performance measures for MVP with performance for the pre-crisis, crisis, post-crisis and full periods. Table 1, Table 2, Table 3 and Table 4 report the results for daily rebalanced portfolios whereas Table 5, Table 6, Table 7 and Table 8 report the results for monthly rebalanced portfolios. We also have results for MVP with no short-sale constraints. However, in this paper we focus on the results for MVP with short-sale constraints and give a short summary of the main findings for the case without short-sale constraints. A detailed analysis of the case without short-sale constraints is given in the Supplementary Material.
In Table 1, Table 2, Table 3, Table 4, Table 5, Table 6, Table 7 and Table 8 we report (in parentheses) the rank of the methods according to the SD criterion in the second column. Moreover, for each criterion, the best five methods are highlighted in shadowed cells. The equal-weighted portfolio strategy is represented by 1 / N .
Taking into account the fact that portfolios are chosen in order to have the minimum variance, the analysis is first done according to the SD criterion. For portfolios rebalanced daily or monthly, the largest SD is reported by the equal-weight portfolio strategy. For portfolios rebalanced daily (Table 1, Table 2, Table 3 and Table 4), the five smallest SDs are obtained by the DCC based-methods, except in the crisis period, in which case the five smallest SDs are spread among the DCC, OGARCH and GPVC based-methods. In the crisis-period, the smallest SD is obtained by the GPVC procedure with the non-linear shrinkage applied to the one-step-ahead conditional covariance matrix. For portfolios rebalanced monthly (Table 5, Table 6, Table 7 and Table 8), the smallest SDs are obtained by the RM2006-LS4, NLS-DCC, NLS-GPVC and RM2006-LS procedures for the full, pre-crisis, crisis and post-crisis periods, respectively.
The best performance in terms of the AV criterion differs depending on the period and rebalance strategy. For instance, for daily rebalancing the best performance in the full period is achieved by the RPVC followed by the RPVC with non-linear shrinkage applied to the one-step-ahead conditional covariance matrix. However, for the pre-crisis, crises and post-crisis periods, the best performance is achieved by the OGARCH with non-linear shrinkage applied to the unconditional covariance matrix (NLS-OGARCH), RPVC with linear shrinkage applied to the one-step-ahead conditional covariance matrix (RPVC-LS) and RiskMetrics method with linear shrinkage applied to the one-step-ahead conditional covariance matrix (RM1994-LS), respectively. For monthly rebalancing, the best performances in the full, pre-crisis, crisis and post-crisis periods are achieved by the RPVC, OGARCH-NLS, GPVC-LS and equal-weight portfolio strategy, respectively.
In terms of average turnover, the five smallest average turnovers are in the OGARCH and GPVC groups, with the best performance being achieved by the OGARCH with non-linear shrinkage applied to the one-step-ahead conditional covariance matrix in almost all cases. The only two exceptions are observed in the crisis period, in which case the best performance is achieved by the GPVC procedure with non-linear shrinkage applied to the one-step-ahead conditional covariance matrix. Additionally, note that regardless of whether portfolio is rebalanced daily or monthly, the average turnover reported by all dimension reduction techniques is smaller than reported by the non-dimension reduction procedures.
As for the annualised average portfolio returns taking into account transaction costs, the procedures with the five largest values of AV 20 b p n e t and AV 50 b p n e t are the same procedures with the largest AV, except in some cases in the pre-crisis period, where one of five largest AV 50 b p n e t is obtained by the NLS-OGARCH-NLS procedure.
For each period, the five best methods in terms of information criteria are the same (except in Table 8, where four methods are the same). We omit the analysis in the crisis period because these criteria values are negative. Overall, for daily rebalancing, RiskMetrics based methods are among the best in the full and post-crisis periods, RPVC and RPVC-NLS are among the best in the full and pre-crisis periods, and NLS-OGARCH and LS-OGARCH are among the best in the pre-crisis period. For monthly rebalancing, some OGARCH-based methods are among the best in the pre-crisis and full periods, some CCC-based methods are among the best in the post-crisis and full periods, RM1994-LS is among the best for the post-crisis period, and RPVC is among the best for the full period.
The analysis of Table 1, Table 2, Table 3, Table 4, Table 5, Table 6, Table 7 and Table 8 reveals that none of the methods is the best in all scenarios and the performance depends on the criterion, the period and the rebalancing strategy. In this sense, the analysis will focus on the full period (Table 1 and Table 5) in order to account for periods with different volatility levels. When portfolios are rebalanced on a daily basis, we find that DCC-based methods are the best in terms of SD; RM2006-LS, RM2006-NL, RPVC and RPVC-NLS are the best in terms of {AV, AV 20 b p n e t , AV 50 b p n e t } and {IR, SR}, and some OGARCH-based are the best regarding TO. For monthly rebalanced portfolios, the best methods in terms of SD are DCC, LS-DCC, NLS-DCC, RM2006 and RM2006-LS, whereas the best performances in terms of {AV, AV 20 b p n e t , AV 50 b p n e t } and {IR, SR} are given by (RPVC, RPVC-NLS), (OGARCH-NLS, NLS-OGARCH-NLS) and CCC. In addition, the equal-weighted strategy is the second best in terms of AV, but the worst regarding SD, IR and SR criteria.
To show when the shrinkage method improves performance in terms of SD, the analysis is again focused on the full period (Table 1 and Table 5). For daily and monthly portfolio rebalancing: shrinkage always improves the performance of the RM2004 and GPVC methods (except LS-GPVC for monthly rebalancing) whereas it always worsens the DCC method; linear shrinkage at the end improves RM2006; just linear/non-linear shrinkage at the beginning improves DECO; OGARCH-NLS and NLS-OGARCH-NLS improves OGARCH; LS-CCC improves CCC (as well as NLS-DCC for daily rebalancing). Additionally, for daily rebalancing, shrinkage always improves the performance of RPVC (except LS-GPVC), whereas for monthly rebalancing, linear shrinkage applied at the beginning and/or end improves RPVC. Nakagawa et al. (2018) also reports that in some cases the use of non-linear shrinkage on the unconditional covariance matrix of the devolatilised returns in the DCC model increases the standard deviation of the out-of-sample portfolio returns.
We now discuss the effect of shrinkage in terms of AV 50 b p n e t . For daily rebalancing, shrinkage improves the performance of the RM2006 and DECO methods, and worsens the performance of the DCC and RPVC methods. In addition, CCC-NLS is better than CCC, RM1994-NLS is better than RM1994, and LS-GPVC is better than GPVC. For monthly rebalancing, shrinkage does not improve the performance of the CCC, DCC, GPVC and RPVC methods. In addition, RM2006-LS is better than RM2006, RM1994-NLS is better than RM1994, DECO-NLS and NLS-DECO-NLS are better than DECO, and OGARCH-NLS and NLS-OGARCH-NLS are better than OGARCH.
Finally, we list next the main findings when short-selling is allowed for optimisation of the portfolio variance. A detailed analysis of these cases is given in the Supplementary Material. First, none of the methods is the best in all scenarios and the performance depends on the criterion, the sample period and the portfolio rebalancing scheme. Second, the analysis of the full period reveals that for daily rebalancing, DCC methods are the best regarding SD and are among the best in terms of IR and SR. RM1994-LS and RM2006-LS are the best according to AV, AV 20 b p n e t , AV 50 b p n e t , IR and SR. For monthly rebalancing, DCC-LS and LS-DCC-LS are among the best in terms of SD, RM2006-NLS is the best in terms of SD and is among the best regarding IR and SR. RM 1994 and RM1994-LS are the first and second best in terms of AV, AV 20 b p n e t , AV 50 b p n e t but are among the worst in terms of SD. Third, the analysis of the turnover and average net returns in the no short-sale constraints case must be carefully done. This is because since no limits are imposed on the weights of the portfolio, large turnover values can be obtained and consequently we can have a large loss (average return) but huge net gain (average net portfolio return taking into account transaction costs). Fourth, in many cases shrinkage improves the performance of the methods in terms of SD, and this improvement can be substantial. Fifth, the top-five models in terms of SD are the same in both restricted and unrestricted minimum variance portfolios for daily rebalancing, except in the crisis period.

4. Conclusions

The main conclusion of the paper is that none of the methods is the best in all scenarios and the performance depends on the criterion, the sample period, the portfolio rebalancing scheme and whether or not short-selling constraints are included in the portfolio optimisation process.
When short-selling constraints are included in the portfolio optimisation process, the main results can be summarised as follows. First, none of the methods is the best in all scenarios and the performance depends on the criterion, the sample period and the portfolio rebalancing scheme. Second, when considering the SD criterion, the five smallest SDs are obtained by the DCC based-methods, except in the crisis period, in which case, the five smallest SDs are spread among the DCC, OGARCH and GPVC based-methods. In the crisis-period, the smallest SDs are obtained by the GPVC procedure with the non-linear shrinkage applied to the one-step-ahead conditional covariance matrix. For portfolios rebalanced monthly, the smallest SDs are obtained by the RM2006-LS, NLS-DCC, NLS-GPVC and RM2006-LS procedures for the full, pre-crisis, crisis and post-crisis periods, respectively. Third, unlike Engle et al. (2017) and Nakagawa et al. (2018), we do not find that applying non-linear shrinkage to the unconditional correlation matrix of the devolatilised returns improves the performance of the portfolio in terms of SD when the DCC model is used, and this also happens when applied in other methods. It is important to point out that Engle et al. (2017) use portfolio of 1000 assets, Nakagawa et al. (2018) use portfolios of 100, 500 and 1000 assets and we use a portfolio with 174 assets.
When short-selling is allowed for optimisation of the portfolio variance, the main conclusions are: none of the methods is the best in all scenarios and the performance depends on the criterion, the sample period and the portfolio rebalancing scheme; in many cases shrinkage improves the performance of the methods in terms of SD and this improvement can be substantial; for daily rebalancing the top-five models in terms of SD are the same of those when short-selling constraints are imposed, except in the crisis period cases. Finally, focusing on the analysis of the full period cases we can say that overall the DCC and Riskmetrics-based methods are the best; and the analysis of the turnover and average net returns in the no short-selling constraints case should be carefully done.

Supplementary Materials

The following are available online at https://www.mdpi.com/2225-1146/7/2/19/s1, File: Covariance Prediction in Large Portfolio Allocation: Supplementary Material.

Author Contributions

This paper has been a collaborative effort, with all authors contributing equally to this work. This includes conceptualization and investigation of the main ideas in the manuscript, methodology proposals, and formal analysis, as well as all aspects of the writing process.

Funding

The first three authors acknowledge financial support from the São Paulo Research Foundation (FAPESP), grants 2016/18599-4, 2018/03012-3, 2013/00506-1 and 2018/04654-9. The fourth author is grateful to the National Council for Scientific and Technological Development (CNPq) for grant 303688/2016-5. The third author is also grateful to CNPq for grant 313035/2017-2.

Acknowledgments

The first three authors acknowledge support of the Centre for Applied Research on Econometrics, Finance and Statistics (CAREFS) and the Centre of Quantitative Studies in Economics and Finance (CEQEF). The authors are also grateful to two anonymous referees and the academic editor for providing useful comments and suggestions on earlier version of the paper.

Conflicts of Interest

The authors declare no conflict of interest.

Appendix A. Estimation Methods

Here we present the detailed list of the estimation methods implemented in the paper. The marginal variances in the CCC, DCC and DECO models were modelled by the GJR-(1,1) model (Glosten et al. 1993) and the parameters were estimated by quasi-maximum likelihood assuming a Student-t distribution. The volatility components in the GPVC and RPVC procedures were modelled by the GJR(1,1)-cDCC(1,1) model and its robust version proposed by Boudt et al. (2013) and Laurent et al. (2016), respectively. The univariate variances in the OGARCH model were also modelled by the GJR-(1,1).
In the GPVC and RPVC procedures, the number of selected volatility components was estimated using criteria of Ahn and Horenstein (2013), Bai and Ng (2002) and Kaiser-Guttman Guttman (1954), and using the ratio estimator proposed by Lam and Yao (2012). Following these criteria and the suggestions in Trucíos et al. (2019), we use one volatility component in the GPVC procedure and four volatility components in the RPVC procedure.
The CCC, DCC, DECO, RM1994 and RM2006 procedures were implemented using the MFE Matlab Toolbox of Kevin Sheppard. The OGARCH, GPVC and RPVC procedures were implemented in R (R Core Team 2017) using the R packages rugarch of Ghalanos (2017), Rcpp of Eddelbuettel and François (2011) and covRobust of Wang et al. (2017). For the shrinkage procedures, we used the R packages RiskPortfolios (Ardia et al. 2018) and nlshrink (Ramprasad 2016) for the linear and non-linear shrinkage, respectively, coupled with the MATLAB toolbox QuEST (Ledoit and Wolf 2017) for the non-linear shrinkage and the MATLAB function covCor5. Whenever a program presented other options, we used the default options.
CCC based-methods
  • CCC: Estimated by quasi-maximum likelihood.
  • LS-CCC: Estimated as in CCC, but with the unconditional covariance matrix (Equation (4)) estimated using linear shrinkage.
  • NLS-CCC: Estimated as in LS-CCC, but replacing linear by the non-linear shrinkage.
  • CCC-LS: Estimated as in CCC, with the application of the linear shrinkage to the one-step-ahead conditional covariance matrix H T + 1 .
  • CCC-NLS: Estimated as in CCC-LS, but replacing linear by non-linear shrinkage.
  • LS-CCC-LS: Estimated as in LS-CCC, with the application of non-linear shrinkage to the one-step-ahead conditional covariance matrix H T + 1 .
  • NLS-CCC-NLS: Estimated as in NLS-CCC, with the application of non-linear shrinkage to the one-step-ahead conditional covariance matrix H T + 1 .
DCC based-methods
  • DCC: Estimated by composite likelihood (Pakel et al. 2014) using consecutive pairs.
  • LS-DCC: Estimated as in DCC, but with the unconditional covariance matrix of the devolatilised returns ( C in Equation (7)) estimated using linear shrinkage.
  • NLS-DCC: Estimated as in LS-DCC, but replacing linear by non-linear shrinkage.
  • DCC-LS: Estimated as in DCC, with the application of linear shrinkage to the one-step-ahead conditional covariance matrix H T + 1 .
  • DCC-NLS: Estimated as in DCC-LS, but replacing linear by non-linear shrinkage.
  • LS-DCC-LS: Estimated as in LS-DCC, with the application of linear shrinkage to the one-step-ahead conditional covariance matrix H T + 1 .
  • NLS-DCC-NLS: Estimated as in NLS-DCC, with the application of non-linear shrinkage to the one-step-ahead conditional covariance matrix H T + 1 .
DECO based-methods
  • DECO: Estimated using a single block.
  • LS-DECO: Estimated as in DECO, but the unconditional covariance matrix of the devolatilised returns is estimated using linear shrinkage.
  • NLS-DECO: Estimated as in LS-DECO, but replacing linear by non-linear shrinkage.
  • DECO-NLS: Estimated as in DECO-LS, but non-linear shrinkage is applied to the one-step-ahead conditional covariance matrix H T + 1 .
  • NLS-DECO-NLS: Estimated as in NLS-DECO model, but with non-linear shrinkage applied to the H T + 1 and linear shrinkage towards the equicorrelation matrix
Because in the DECO model the estimated unconditional covariance matrix and H T + 1 are already equicorrelated there is no sense in using linear shrinkage towards the equicorrelation matrix, since it has no effect.
RiskMetrics based-methods
  • RM1994: RM1994 method.
  • RM1994-LS: Estimated as in RM1994 with linear shrinkage applied to the one-step-ahead conditional covariance matrix H T + 1 .
  • RM1994-NLS: Estimated as in RM1994-LS but replacing linear by non-linear shrinkage.
  • RM20066: RM2006 method (Zumbach 2007).
  • RM2006-LS: Estimated as in RM2006 with linear shrinkage applied to the one-step-ahead conditional covariance matrix H T + 1 .
  • RM2006-NLS: Estimated as in RM2006-LS but replacing linear by non-linear shrinkage.
OGARCH based-methods
  • OGARCH: The OGARCH model considers k = N components.
  • LS-OGARCH: Estimated as in OGARCH, but the unconditional covariance matrix used in the spectral decomposition is estimated using linear shrinkage.
  • NLS-OGARCH: Estimated as in LS-OGARCH, but replacing linear by non-linear shrinkage.
  • OGARCH-LS: Estimated as in OGARCH with the linear shrinkage applied to the one-step-ahead conditional covariance matrix H T + 1 .
  • OGARCH-NLS: Estimated as in OGARCH-LS, but replacing linear by non-linear shrinkage.
  • LS-OGARCH-LS: Estimated as in LS-OGARCH, but linear shrinkage is applied to the predicted one-step-ahead conditional covariance matrix H T + 1 .
  • NLS-OGARCH-NLS: Estimated as in NLS-OGARCH, but non-linear shrinkage is applied to the predicted one-step-ahead conditional covariance matrix H T + 1 .
GPVC based-methods
  • GPVC: The GPVC procedure considers k = 1 volatility component, as explained later. We use g = 5 as in Li et al. (2016).
  • LS-GPVC: Estimated as in the GPVC model with the unconditional covariance matrix Σ ^ in Equation (17) estimated using linear shrinkage.
  • NLS-GPVC: Estimated as in LS-GPVC, but replacing linear by non-linear shrinkage.
  • GPVC-LS: Estimated as in GPVC with linear shrinkage applied to the one-step-ahead conditional covariance matrix H T + 1 .
  • GPVC-NLS: Estimated as in GPVC-LS, but replacing linear by non-linear shrinkage.
  • LS-GPVC-LS: Estimated as in LS-GPVC with linear shrinkage applied to the predicted one-step-ahead conditional covariance matrix H T + 1 .
  • NLS-GPVC-NLS: Estimated as in NLS-GPVC with non-linear shrinkage applied to the predicted one-step-ahead conditional covariance matrix H T + 1 .
RPVC based-methods
  • RPVC: The RPVC procedure considers k = 4 volatility components, as explained later. We use g = 5 as in Li et al. (2016) and c as in Trucíos et al. (2019).
  • LS-RPVC: Estimated as in RPVC, but linear shrinkage is applied to the robust unconditional covariance matrix Σ ^ R used in Equation (18).
  • NLS-RPVC: Estimated as in LS-RPVC, but replacing linear by non-linear shrinkage.
  • RPVC-LS: Estimated as in RPVC with linear shrinkage applied to the one-step-ahead conditional covariance matrix H T + 1 .
  • RPVC-NLS: Estimated as in RPVC-LS, but replacing linear by non-linear shrinkage.
  • LS-RPVC-LS: Estimated as in LS-RPVC with the linear shrinkage applied to the predicted one-step-ahead conditional covariance matrix H T + 1 .
  • NLS-RPVC-NLS: Estimated as in NLS-RPVC with non-linear shrinkage applied to the predicted one-step-ahead conditional covariance matrix H T + 1 .

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1
See Wied et al. (2013) for a test for the presence of structural breaks in minimum variance portfolios
2
From now on we just call the log-likelihood likelihood.
3
Note that when Σ = I , H t = A H t f A + B B Σ = A H t f A + Σ f f l as presented in Li et al. (2016).
4
The acronyms are described in the Appendix A.
5
6
This method was implemented using the MFE Matlab Toolbox of Kevin Sheppard with the default options. An R implementation of the same procedure can be found in Trucios (2017).
Table 1. Annualised performance measures: AV, SD, IR, SR and TO stand for the average, standard deviation, information ratio, Sortino’s ratio and turnover of the out-of-sample MVP returns. AV 20 b p n e t  and AV 50 b p n e t stand for the average out-of-sample MVP return net of transaction costs considering 20 and 50 basis-points, respectively. Period January 2004 to November 2017. The shaded cells denote the top five for each criterion. Weights are rebalanced on a daily basis considering short-selling constraints.
Table 1. Annualised performance measures: AV, SD, IR, SR and TO stand for the average, standard deviation, information ratio, Sortino’s ratio and turnover of the out-of-sample MVP returns. AV 20 b p n e t  and AV 50 b p n e t stand for the average out-of-sample MVP return net of transaction costs considering 20 and 50 basis-points, respectively. Period January 2004 to November 2017. The shaded cells denote the top five for each criterion. Weights are rebalanced on a daily basis considering short-selling constraints.
AVSDIRSRTOAV 20 bp net AV 50 bp net
1/N8.30220.058 (47)0.4140.570---
CCC7.70611.839 (12)0.6510.8900.2977.5097.279
CCC LS7.00411.881 (14)0.5900.8070.3076.8156.578
CCC NLS7.87611.932 (17)0.6600.9050.2777.6857.470
LS CCC7.50611.816 (11)0.6350.8680.3027.3117.078
NLS CCC7.34511.809 (10)0.6220.8480.2987.1536.923
LS CCC LS6.62811.918 (16)0.5560.7590.3056.4396.205
NLS CCC NLS7.52211.910 (15)0.6320.8650.3037.3277.091
DCC7.73711.613 (2)0.6660.9080.3087.5327.296
DCC LS6.94111.689 (5)0.5940.8100.3146.7496.508
DCC NLS7.71111.695 (6)0.6590.9050.2857.5137.292
LS DCC7.70711.613 (1)0.6640.9040.3087.5027.266
NLS DCC7.62911.616 (3)0.6570.8940.3077.4247.188
LS DCC LS6.90711.688 (4)0.5910.8060.3146.7156.474
NLS DCC NLS7.64511.699 (7)0.6530.8960.2837.4477.227
RM20068.64911.809 (9)0.7320.9950.2718.4468.234
RM2006 LS8.74611.724 (8)0.7461.0170.2828.5648.343
RM2006 NLS8.73411.865 (13)0.7361.0110.2688.5378.327
RM19948.50212.220 (22)0.6960.9470.2838.2898.069
RM1994 LS8.39112.012 (18)0.6990.9530.2778.1967.979
RM1994 NLS8.76312.151 (19)0.7210.9900.2258.5818.405
DECO5.98012.258 (25)0.4880.6600.2975.7975.568
DECO NLS6.10312.485 (41)0.4890.6690.3605.8845.604
LS DECO5.98012.257 (24)0.4880.6600.2975.7975.568
NLS DECO5.98112.257 (23)0.4880.6600.2975.7985.569
NLS DECO NLS6.10312.485 (42)0.4890.6690.3605.8845.604
OGARCH8.36312.341 (27)0.6780.9360.0958.2718.196
OGARCH LS7.05212.544 (43)0.5620.7730.1036.9746.893
OGARCH NLS8.12612.154 (20)0.6690.9280.0728.0527.996
LS OGARCH7.95112.477 (39)0.6370.8770.0957.8607.786
NLS OGARCH8.36512.341 (27)0.6780.9360.0958.2738.198
LS OGARCH LS6.88012.710 (44)0.5410.7430.1016.8026.723
NLS OGARCH NLS8.12612.154 (20)0.6690.9280.0728.0517.996
GPVC7.82512.467 (38)0.6280.8610.1327.7007.598
GPVC LS7.43812.274 (26)0.6060.8340.1067.3417.259
GPVC NLS6.72712.369 (31)0.5440.7490.1136.6216.533
LS GPVC7.99412.452 (36)0.6420.8910.1177.8727.781
NLS GPVC7.67212.433 (33)0.6170.8450.1307.5477.447
LS GPVC LS7.47012.429 (32)0.6010.8260.1617.3597.238
NLS GPVC NLS6.72512.365 (30)0.5440.7490.1136.6196.533
RPVC9.65712.785 (45)0.7551.0470.2229.4799.310
RPCV LS7.98912.439 (34)0.6420.8890.1807.8617.724
RPVC NLS9.18612.485 (40)0.7361.0260.1849.0358.893
LS RPVC8.54312.347 (29)0.6920.9530.2018.3878.235
NLS RPVC8.06413.142 (46)0.6140.8500.1917.9047.755
LS RPCV LS7.49312.439 (35)0.6020.8280.1677.3787.252
NLS RPVC NLS7.65812.460 (37)0.6150.8500.1727.5097.376
Table 2. Annualised performance measures: AV, SD, IR, SR and TO stand for the average, standard deviation, information ratio, Sortino’s ratio and turnover of the out-of-sample MVP returns. AV 20 b p n e t  and AV 50 b p n e t stand for the average out-of-sample MVP return net of transaction costs considering 20 and 50 basis-points, respectively. Period January 2004 to December 2007. The shaded cells denote the top five for each criterion. Weights are rebalanced on a daily basis considering short-selling constraints.
Table 2. Annualised performance measures: AV, SD, IR, SR and TO stand for the average, standard deviation, information ratio, Sortino’s ratio and turnover of the out-of-sample MVP returns. AV 20 b p n e t  and AV 50 b p n e t stand for the average out-of-sample MVP return net of transaction costs considering 20 and 50 basis-points, respectively. Period January 2004 to December 2007. The shaded cells denote the top five for each criterion. Weights are rebalanced on a daily basis considering short-selling constraints.
AVSDIRSRTOAV 20 bp net AV 50 bp net
1/N12.73212.755 (47)0.9981.418---
CCC11.4258.381 (6)1.3631.9630.25611.13710.934
CCC LS9.8188.495 (14)1.1561.6550.2649.5699.362
CCC NLS11.1578.404 (11)1.3281.9070.24710.86310.668
LS CCC11.3058.394 (9)1.3471.9400.25811.03010.826
NLS CCC11.4618.399 (10)1.3651.9660.25111.19510.997
LS CCC LS9.6328.628 (17)1.1161.5960.2589.3869.183
NLS CCC NLS11.1728.426 (12)1.3261.9100.26310.90110.692
DCC11.1448.203 (3)1.3591.9470.26310.84310.636
DCC LS9.4508.394 (8)1.1261.6050.2689.2018.992
DCC NLS10.9198.234 (5)1.3261.8980.25310.60910.410
LS DCC11.1038.199 (2)1.3541.9410.26310.80210.596
NLS DCC11.0358.196 (1)1.3461.9290.26210.73310.527
LS DCC LS9.4238.391 (7)1.1231.6010.2689.1748.965
NLS DCC NLS10.8298.226 (4)1.3161.8840.25210.51910.321
RM200611.9838.553 (15)1.4012.0450.25811.63011.426
RM2006 LS10.9888.435 (13)1.3031.8870.26810.72810.516
RM2006 NLS9.8528.686 (19)1.1341.6190.2599.5209.318
RM19949.4969.148 (29)1.0381.5030.2829.1218.902
RM1994 LS8.4988.866 (23)0.9591.3740.2758.1827.967
RM1994 NLS10.0809.112 (28)1.1061.5840.2209.7429.571
DECO9.2829.062 (25)1.0241.4570.2539.0408.840
DECO NLS8.9989.197 (32)0.9781.3880.3028.7258.487
LS DECO9.2809.063 (26)1.0241.4560.2539.0398.838
NLS DECO9.2719.064 (27)1.0231.4550.2549.0308.829
NLS DECO NLS8.9989.197 (33)0.9781.3880.3028.7258.487
OGARCH13.3569.188 (31)1.4542.0970.08313.16513.100
OGARCH LS11.56510.105 (45)1.1441.6020.08811.43511.367
OGARCH NLS12.8059.203 (34)1.3911.9980.07112.63812.582
LS OGARCH13.0689.257 (36)1.4122.0300.08112.88512.821
NLS OGARCH13.3629.188 (30)1.4542.0980.08313.17213.106
LS OGARCH LS11.30510.326 (46)1.0951.5280.08211.17511.110
NLS OGARCH NLS12.8049.203 (34)1.3911.9970.07112.63712.582
GPVC11.4979.268 (37)1.2411.7570.10911.24611.163
GPVC LS11.0249.282 (39)1.1881.6800.08210.83510.772
GPVC NLS11.2109.320 (43)1.2031.6900.09910.99310.918
LS GPVC12.2139.294 (41)1.3141.8680.09411.95311.881
NLS GPVC11.2749.348 (44)1.2061.7030.10811.02010.938
LS GPVC LS10.3259.288 (40)1.1121.5590.12910.15310.052
NLS GPVC NLS11.1659.318 (42)1.1981.6830.09710.94910.876
RPVC12.9668.680 (18)1.4942.1690.19312.64212.492
RPCV LS10.4239.000 (24)1.1581.6460.15210.21810.100
RPVC NLS12.2338.697 (20)1.4072.0180.17111.95111.818
LS RPVC11.6358.577 (16)1.3571.9440.17511.35411.218
NLS RPVC10.8788.829 (22)1.2321.7600.17110.57910.447
LS RPCV LS10.3049.271 (38)1.1111.5580.13910.12510.016
NLS RPVC NLS10.6288.760 (21)1.2131.7230.15810.33610.215
Table 3. Annualised performance measures: AV, SD, IR, SR and TO stand for the average, standard deviation, information ratio, Sortino’s ratio and turnover of the out-of-sample MVP returns. AV 20 b p n e t  and AV 50 b p n e t stand for the average out-of-sample MVP return net of transaction costs considering 20 and 50 basis-points, respectively. Period January 2008 to June 2009. The shaded cells denote the top five for each criterion. Weights are rebalanced on a daily basis considering short-selling constraints.
Table 3. Annualised performance measures: AV, SD, IR, SR and TO stand for the average, standard deviation, information ratio, Sortino’s ratio and turnover of the out-of-sample MVP returns. AV 20 b p n e t  and AV 50 b p n e t stand for the average out-of-sample MVP return net of transaction costs considering 20 and 50 basis-points, respectively. Period January 2008 to June 2009. The shaded cells denote the top five for each criterion. Weights are rebalanced on a daily basis considering short-selling constraints.
AVSDIRSRTOAV 20 bp net AV 50 bp net
1/N−30.66843.046 (47)−0.713−0.960---
CCC−25.40722.009 (20)−1.154−1.4640.362−25.564−25.799
CCC LS−25.52222.003 (19)−1.160−1.4710.365−25.680−25.917
CCC NLS−23.68222.613 (27)−1.047−1.3440.300−23.820−24.026
LS CCC−26.28821.934 (13)−1.199−1.5160.369−26.448−26.686
NLS CCC−27.14421.965 (16)−1.236−1.5580.365−27.301−27.537
LS CCC LS−27.05221.967 (17)−1.232−1.5530.368−27.211−27.449
NLS CCC NLS−25.37222.346 (25)−1.135−1.4460.326−25.521−25.743
DCC−26.52021.580 (5)−1.229−1.5540.389−26.683−26.928
DCC LS−26.70221.596 (7)−1.236−1.5630.391−26.866−27.112
DCC NLS−24.63621.926 (12)−1.124−1.4460.312−24.777−24.989
LS DCC−26.63921.582 (6)−1.234−1.5610.390−26.802−27.047
NLS DCC−27.02021.596 (7)−1.251−1.5810.392−27.184−27.429
LS DCC LS−26.83321.599 (9)−1.242−1.5700.392−26.997−27.243
NLS DCC NLS−24.89921.952 (14)−1.134−1.4600.311−25.039−25.249
RM2006−22.72821.862 (11)−1.040−1.3260.281−22.858−23.054
RM2006 LS−22.91221.815 (10)−1.050−1.3380.279−23.041−23.235
RM2006 NLS−21.26721.958 (15)−0.969−1.2640.216−21.372−21.529
RM1994−20.79322.108 (22)−0.941−1.2050.260−20.914−21.096
RM1994 LS−21.23422.053 (21)−0.963−1.2320.259−21.355−21.537
RM1994 NLS−20.97422.161 (23)−0.946−1.2360.178−21.060−21.188
DECO−31.85922.706 (33)−1.403−1.7420.408−32.030−32.288
DECO NLS−29.18722.618 (28)−1.291−1.6330.386−29.358−29.615
LS DECO−31.85422.706 (32)−1.403−1.7420.408−32.026−32.284
NLS DECO−31.82922.702 (31)−1.402−1.7410.408−32.001−32.258
NLS DECO NLS−29.18822.618 (29)−1.291−1.6330.386−29.359−29.615
OGARCH−21.67123.390 (36)−0.927−1.2180.107−21.722−21.799
OGARCH LS−21.74523.360 (35)−0.931−1.2230.108−21.796−21.873
OGARCH NLS−20.11821.541 (3)−0.934−1.2230.071−20.153−20.205
LS OGARCH−23.67724.009 (45)−0.986−1.2910.109−23.728−23.804
NLS OGARCH−21.67123.390 (36)−0.927−1.2180.107−21.722−21.799
LS OGARCH LS−23.57123.957 (41)−0.984−1.2880.109−23.622−23.699
NLS OGARCH NLS−20.11821.541 (3)−0.934−1.2230.071−20.153−20.205
GPVC−19.78922.287 (24)−0.888−1.1510.105−19.831−19.894
GPVC LS−16.84122.700 (30)−0.742−0.9730.113−16.890−16.964
GPVC NLS−23.69221.444 (1)−1.105−1.4340.050−23.711−23.740
LS GPVC−18.38022.823 (34)−0.805−1.0790.112−18.429−18.503
NLS GPVC−20.57421.983 (18)−0.936−1.2070.102−20.614−20.674
LS GPVC LS−21.13723.982 (43)−0.881−1.1440.193−21.208−21.315
NLS GPVC NLS−23.71621.451 (2)−1.106−1.4350.050−23.735−23.764
RPVC−17.36923.870 (40)−0.728−0.9620.188−17.446−17.561
RPCV LS−15.91123.839 (39)−0.667−0.8880.189−15.990−16.109
RPVC NLS−22.22922.432 (26)−0.991−1.2960.114−22.277−22.350
LS RPVC−21.00423.672 (38)−0.887−1.1530.195−21.076−21.183
NLS RPVC−25.11927.169 (46)−0.925−1.2310.156−25.192−25.302
LS RPCV LS−21.16423.982 (44)−0.883−1.1450.193−21.235−21.342
NLS RPVC NLS−25.49223.964 (42)−1.064−1.3890.115−25.543−25.620
Table 4. Annualised performance measures: AV, SD, IR, SR and TO stand for the average, standard deviation, information ratio, Sortino’s ratio and turnover of the out-of-sample MVP returns. AV 20 b p n e t  and AV 50 b p n e t stand for the average out-of-sample MVP return net of transaction costs considering 20 and 50 basis-points, respectively. Period July 2009 to November 2017. The shaded cells denote the top five for each criterion. Weights are rebalanced on a daily basis considering short-selling constraints.
Table 4. Annualised performance measures: AV, SD, IR, SR and TO stand for the average, standard deviation, information ratio, Sortino’s ratio and turnover of the out-of-sample MVP returns. AV 20 b p n e t  and AV 50 b p n e t stand for the average out-of-sample MVP return net of transaction costs considering 20 and 50 basis-points, respectively. Period July 2009 to November 2017. The shaded cells denote the top five for each criterion. Weights are rebalanced on a daily basis considering short-selling constraints.
AVSDIRSRTOAV 20 bp net AV 50 bp net
1/N13.13016.057 (47)0.8181.148---
CCC11.83010.561 (15)1.1201.6060.30611.66911.427
CCC LS11.45510.599 (18)1.0811.5540.31811.28811.037
CCC NLS11.93210.502 (10)1.1361.6280.28811.78111.554
LS CCC11.71310.540 (13)1.1111.5950.31011.55011.304
NLS CCC11.52510.512 (11)1.0961.5710.30811.36211.119
LS CCC LS11.19310.629 (19)1.0531.5140.31511.02710.778
NLS CCC NLS11.64010.552 (14)1.1031.5830.31711.47311.222
DCC12.21310.366 (1)1.1781.6810.31512.04711.797
DCC LS11.73610.429 (8)1.1251.6120.32211.56611.311
DCC NLS11.94210.383 (5)1.1501.6440.29511.78611.553
LS DCC12.20410.366 (1)1.1771.6790.31412.03811.788
NLS DCC12.17510.367 (3)1.1741.6740.31412.00911.761
LS DCC LS11.71510.427 (7)1.1241.6090.32211.54511.290
NLS DCC NLS11.92210.383 (4)1.1481.6410.29311.76811.536
RM200612.64810.498 (9)1.2051.6860.27512.50212.283
RM2006 LS13.31410.403 (6)1.2801.8120.28913.16012.930
RM2006 NLS13.54210.518 (12)1.2881.8200.28113.39313.169
RM199413.24310.941 (35)1.2101.6910.28713.09112.863
RM1994 LS13.61310.686 (25)1.2741.7990.28113.46313.239
RM1994 NLS13.43010.808 (32)1.2431.7470.23513.30513.117
DECO11.14410.806 (29)1.0311.4780.29810.98610.749
DECO NLS11.00711.214 (39)0.9821.4100.38310.80510.501
LS DECO11.14510.806 (31)1.0311.4780.29810.98710.750
NLS DECO11.14510.806 (29)1.0311.4780.29810.98710.750
NLS DECO NLS11.00711.214 (39)0.9821.4100.38310.80510.501
OGARCH11.33310.671 (22)1.0621.5080.09811.28011.201
OGARCH LS10.03010.684 (24)0.9391.3340.1099.9729.885
OGARCH NLS10.92711.000 (36)0.9931.4220.07210.88910.833
LS OGARCH11.14510.658 (20)1.0461.4850.09911.09211.012
NLS OGARCH11.33310.671 (22)1.0621.5080.09811.28011.201
LS OGARCH LS10.19410.669 (21)0.9561.3600.10810.13610.050
NLS OGARCH NLS10.92611.000 (37)0.9931.4210.07210.88910.833
GPVC10.99211.289 (41)0.9741.3770.14810.91310.795
GPVC LS10.05210.781 (27)0.9321.3240.1169.9919.898
GPVC NLS10.00811.374 (44)0.8801.2510.1329.9399.835
LS GPVC10.68111.061 (38)0.9661.3660.12810.61210.508
NLS GPVC10.98511.300 (42)0.9721.3750.14610.90710.790
LS GPVC LS11.20310.569 (16)1.0601.5320.17011.11410.981
NLS GPVC NLS10.03011.364 (43)0.8831.2560.1319.9629.858
RPVC12.89211.524 (46)1.1191.5920.24112.76612.578
RPCV LS11.08410.772 (26)1.0291.4660.19210.98510.836
RPVC NLS13.32711.476 (45)1.1611.6820.20213.22113.062
LS RPVC12.33110.816 (33)1.1401.6360.21412.21912.051
NLS RPVC12.63010.801 (28)1.1691.6770.20712.52112.358
LS RPCV LS11.25610.596 (17)1.0621.5350.17711.16411.026
NLS RPVC NLS12.14510.837 (34)1.1211.6190.18912.04711.898
Table 5. Annualised performance measures: AV, SD, IR, SR and TO stand for the average, standard deviation, information ratio, Sortino’s ratio and turnover of the out-of-sample MVP returns. AV 20 b p n e t  and AV 50 b p n e t stand for the average out-of-sample MVP return net of transaction costs considering 20 and 50 basis-points, respectively. Period January 2004 to November 2017. The shaded cells denote the top five for each criterion. Weights are rebalanced on a monthly basis considering short-selling constraints.
Table 5. Annualised performance measures: AV, SD, IR, SR and TO stand for the average, standard deviation, information ratio, Sortino’s ratio and turnover of the out-of-sample MVP returns. AV 20 b p n e t  and AV 50 b p n e t stand for the average out-of-sample MVP return net of transaction costs considering 20 and 50 basis-points, respectively. Period January 2004 to November 2017. The shaded cells denote the top five for each criterion. Weights are rebalanced on a monthly basis considering short-selling constraints.
AVSDIRSRTOAV 20 bp net AV 50 bp net
1/N8.30220.058 (47)0.4140.570---
CCC7.94612.262 (10)0.6480.9020.3197.9387.925
CCC LS6.83212.263 (11)0.5570.7750.3296.8236.810
CCC NLS7.72512.388 (16)0.6240.8670.2967.7177.704
LS CCC7.73112.261 (9)0.6310.8780.3237.7237.709
NLS CCC7.58812.278 (14)0.6180.8590.3197.5797.566
LS CCC LS6.47112.359 (15)0.5240.7280.3256.4626.449
NLS CCC NLS7.75812.439 (20)0.6240.8690.3217.7497.736
DCC7.42512.182 (3)0.6100.8450.3257.4167.403
DCC LS6.56712.200 (6)0.5380.7470.3346.5586.544
DCC NLS6.90112.247 (8)0.5630.7800.3026.8926.879
LS DCC7.38612.184 (4)0.6060.8400.3257.3777.364
NLS DCC7.29612.193 (5)0.5980.8290.3257.2877.274
LS DCC LS6.51812.203 (7)0.5340.7410.3346.5096.495
NLS DCC NLS6.78112.266 (12)0.5530.7640.3006.7726.760
RM20067.35012.012 (2)0.6120.8430.2877.3427.329
RM2006 LS7.44211.870 (1)0.6270.8670.2947.4347.421
RM2006 NLS7.10112.274 (13)0.5790.7980.2967.0937.081
RM19947.77712.644 (29)0.6150.8480.2967.7697.756
RM1994 LS7.15712.391 (17)0.5780.7960.2927.1497.136
RM1994 NLS7.90612.606 (27)0.6270.8650.2547.8997.888
DECO5.63112.899 (43)0.4370.6080.3175.6225.609
DECO NLS5.64113.162 (44)0.4290.5990.3865.6305.614
LS DECO5.63112.899 (42)0.4370.6080.3175.6225.609
NLS DECO5.63112.899 (41)0.4370.6080.3175.6225.609
NLS DECO NLS5.64013.162 (45)0.4290.5990.3865.6305.614
OGARCH7.81912.556 (24)0.6230.8590.1017.8167.812
OGARCH LS6.84812.687 (32)0.5400.7440.1136.8456.840
OGARCH NLS7.98512.451 (22)0.6410.8910.0787.9847.981
LS OGARCH7.58112.716 (37)0.5960.8210.1037.5797.575
NLS OGARCH7.82112.555 (23)0.6230.8590.1017.8187.814
LS OGARCH LS7.02912.893 (40)0.5450.7510.1117.0267.021
NLS OGARCH NLS7.99312.451 (21)0.6420.8910.0787.9917.988
GPVC7.28212.707 (34)0.5730.7890.1557.2777.271
GPVC LS7.22512.435 (19)0.5810.8010.1207.2227.218
GPVC NLS6.56012.672 (31)0.5180.7120.1326.5576.552
LS GPVC7.20012.713 (36)0.5660.7830.1387.1967.190
NLS GPVC7.22312.697 (33)0.5690.7820.1537.2197.212
LS GPVC LS6.52112.568 (25)0.5190.7180.1726.5166.509
NLS GPVC NLS6.56812.665 (30)0.5190.7130.1306.5656.559
RPVC8.45312.712 (35)0.6650.9200.2488.4468.436
RPCV LS7.35512.415 (18)0.5920.8220.1937.3507.342
RPVC NLS8.01112.816 (39)0.6250.8630.2018.0057.997
LS RPVC7.00012.615 (28)0.5550.7650.2276.9946.985
NLS RPVC6.48813.243 (46)0.4900.6760.2036.4826.474
LS RPCV LS6.53512.588 (26)0.5190.7180.1806.5306.523
NLS RPVC NLS6.87412.741 (38)0.5400.7430.1826.8696.862
Table 6. Annualised performance measures: AV, SD, IR, SR and TO stand for the average, standard deviation, information ratio, Sortino’s ratio and turnover of the out-of-sample MVP returns. AV 20 b p n e t  and AV 50 b p n e t stand for the average out-of-sample MVP return net of transaction costs considering 20 and 50 basis-points, respectively. Period January 2004 to December 2007. The shaded cells denote the top five for each criterion. Weights are rebalanced on a monthly basis considering short-selling constraints.
Table 6. Annualised performance measures: AV, SD, IR, SR and TO stand for the average, standard deviation, information ratio, Sortino’s ratio and turnover of the out-of-sample MVP returns. AV 20 b p n e t  and AV 50 b p n e t stand for the average out-of-sample MVP return net of transaction costs considering 20 and 50 basis-points, respectively. Period January 2004 to December 2007. The shaded cells denote the top five for each criterion. Weights are rebalanced on a monthly basis considering short-selling constraints.
AVSDIRSRTOAV 20 bp net AV 50 bp net
1/N12.73212.755 (47)0.9981.418---
CCC10.6368.697 (7)1.2231.7500.26510.62910.618
CCC LS7.6058.868 (19)0.8581.2080.2847.5977.585
CCC NLS10.3568.732 (8)1.1861.6940.25410.34910.338
LS CCC10.1588.738 (9)1.1631.6590.26910.15010.139
NLS CCC10.1868.758 (11)1.1631.6590.26310.17910.168
LS CCC LS7.3199.045 (23)0.8091.1370.2817.3127.300
NLS CCC NLS9.8028.809 (16)1.1131.5810.2779.7959.784
DCC10.9398.612 (3)1.2701.8250.27110.93210.920
DCC LS7.6918.796 (15)0.8741.2330.2867.6837.671
DCC NLS10.7638.661 (6)1.2431.7820.26010.75610.745
LS DCC10.9238.608 (2)1.2691.8230.27110.91510.904
NLS DCC10.8898.599 (1)1.2661.8190.26910.88210.871
LS DCC LS7.6728.795 (14)0.8721.2300.2847.6647.653
NLS DCC NLS10.7258.649 (5)1.2401.7780.25810.71810.707
RM200610.3788.765 (13)1.1841.7060.29210.36910.357
RM2006 LS9.2958.629 (4)1.0771.5400.3009.2879.275
RM2006 NLS9.5788.884 (20)1.0781.5270.3139.5699.556
RM19948.1129.545 (37)0.8501.2090.3238.1038.089
RM1994 LS6.8139.279 (24)0.7341.0330.3176.8046.791
RM1994 NLS9.9129.282 (25)1.0681.5200.2659.9049.892
DECO6.8839.577 (39)0.7191.0090.2776.8756.864
DECO NLS6.2579.784 (43)0.6400.8870.3406.2476.233
LS DECO6.8829.577 (39)0.7191.0080.2776.8756.863
NLS DECO6.8739.577 (41)0.7181.0070.2776.8656.854
NLS DECO NLS6.2579.784 (44)0.6400.8870.3406.2476.233
OGARCH12.6829.305 (26)1.3631.9580.08812.68012.676
OGARCH LS11.22910.166 (45)1.1051.5560.09711.22611.222
OGARCH NLS12.8789.376 (29)1.3741.9710.06312.87712.874
LS OGARCH12.5889.346 (28)1.3471.9280.08812.58612.582
NLS OGARCH12.6829.305 (26)1.3631.9580.08812.68012.676
LS OGARCH LS11.41410.359 (46)1.1021.5480.09011.41111.408
NLS OGARCH NLS12.8789.376 (29)1.3741.9710.06312.87712.874
GPVC11.0149.504 (36)1.1591.6360.14511.01011.004
GPVC LS11.0649.438 (31)1.1721.6570.10511.06111.057
GPVC NLS10.6379.478 (33)1.1221.5690.13410.63410.628
LS GPVC11.2359.595 (42)1.1711.6520.12011.23211.226
NLS GPVC10.9399.576 (38)1.1421.6110.14510.93510.929
LS GPVC LS9.1839.503 (35)0.9661.3450.1399.1799.174
NLS GPVC NLS10.6569.473 (32)1.1251.5720.13210.65210.647
RPVC11.5588.741 (10)1.3221.8960.21611.55211.544
RPCV LS10.1729.038 (22)1.1261.5940.17410.16810.161
RPVC NLS11.0238.761 (12)1.2581.7910.19311.01811.010
LS RPVC9.8598.845 (18)1.1151.5660.2029.8549.846
NLS RPVC9.8028.925 (21)1.0981.5580.1939.7979.789
LS RPCV LS9.1889.490 (34)0.9681.3460.1519.1849.178
NLS RPVC NLS9.9958.828 (17)1.1321.6000.1839.9909.982
Table 7. Annualised performance measures: AV, SD, IR, SR and TO stand for the average, standard deviation, information ratio, Sortino’s ratio and turnover of the out-of-sample MVP returns. AV 20 b p n e t and AV 50 b p n e t stand for the average out-of-sample MVP return net of transaction costs considering 20 and 50 basis-points, respectively. Period January 2008 to June 2009. The shaded cells denote the top five for each criterion. Weights are rebalanced on a monthly basis considering short-selling constraints.
Table 7. Annualised performance measures: AV, SD, IR, SR and TO stand for the average, standard deviation, information ratio, Sortino’s ratio and turnover of the out-of-sample MVP returns. AV 20 b p n e t and AV 50 b p n e t stand for the average out-of-sample MVP return net of transaction costs considering 20 and 50 basis-points, respectively. Period January 2008 to June 2009. The shaded cells denote the top five for each criterion. Weights are rebalanced on a monthly basis considering short-selling constraints.
AVSDIRSRTOAV 20 bp net AV 50 bp net
1/N−30.66843.046 (47)−0.713−0.960---
CCC−25.34422.796 (13)−1.112−1.4600.381−25.355−25.371
CCC LS−25.39022.796 (14)−1.114−1.4620.383−25.401−25.418
CCC NLS−23.54023.614 (33)−0.997−1.3180.318−23.550−23.566
LS CCC−25.29522.748 (12)−1.112−1.4630.385−25.306−25.323
NLS CCC−26.76022.916 (21)−1.168−1.5320.373−26.771−26.787
LS CCC LS−26.81822.925 (22)−1.170−1.5350.375−26.828−26.844
NLS CCC NLS−25.22223.536 (26)−1.072−1.4140.335−25.233−25.248
DCC−26.14622.840 (16)−1.145−1.5080.419−26.158−26.175
DCC LS−26.24822.850 (17)−1.149−1.5130.419−26.259−26.276
DCC NLS−23.98223.354 (24)−1.027−1.3590.333−23.993−24.008
LS DCC−26.38422.858 (18)−1.154−1.5200.419−26.395−26.412
NLS DCC−27.05622.905 (20)−1.181−1.5540.419−27.067−27.084
LS DCC LS−26.49522.865 (19)−1.159−1.5260.421−26.506−26.523
NLS DCC NLS−24.84923.458 (25)−1.059−1.4010.331−24.859−24.875
RM2006−22.35622.084 (4)−1.012−1.3270.356−22.367−22.383
RM2006 LS−23.04522.006 (2)−1.047−1.3700.356−23.055−23.071
RM2006 NLS−21.10923.116 (23)−0.913−1.2060.274−21.116−21.126
RM1994−22.68522.716 (11)−0.999−1.3070.337−22.695−22.711
RM1994 LS−23.38822.619 (10)−1.034−1.3500.335−23.398−23.413
RM1994 NLS−21.73923.572 (28)−0.922−1.2150.235−21.745−21.755
DECO−28.18424.101 (42)−1.169−1.5500.404−28.197−28.215
DECO NLS−27.58823.858 (35)−1.156−1.5330.367−27.599−27.617
LS DECO−28.18224.100 (41)−1.169−1.5500.404−28.195−28.213
NLS DECO−28.16624.098 (40)−1.169−1.5490.404−28.178−28.197
NLS DECO NLS−27.59123.859 (36)−1.156−1.5330.367−27.602−27.620
OGARCH−20.67723.592 (30)−0.877−1.1450.124−20.680−20.683
OGARCH LS−20.85523.577 (29)−0.885−1.1550.126−20.857−20.860
OGARCH NLS−19.60822.343 (7)−0.878−1.1560.063−19.610−19.613
LS OGARCH−20.51624.433 (44)−0.840−1.0980.130−20.518−20.522
NLS OGARCH−20.67723.592 (30)−0.877−1.1450.124−20.680−20.683
LS OGARCH LS−20.56424.390 (43)−0.843−1.1030.132−20.567−20.570
NLS OGARCH NLS−19.60822.343 (7)−0.878−1.1560.061−19.610−19.613
GPVC−14.45422.017 (3)−0.657−0.8680.138−14.457−14.462
GPVC LS−14.10022.418 (9)−0.629−0.8310.136−14.103−14.107
GPVC NLS−20.43622.235 (5)−0.919−1.2090.048−20.438−20.440
LS GPVC−15.36122.807 (15)−0.674−0.9020.165−15.364−15.368
NLS GPVC−14.82921.853 (1)−0.679−0.8920.134−14.832−14.837
LS GPVC LS−17.99124.031 (38)−0.749−0.9950.226−17.996−18.004
NLS GPVC NLS−20.47122.244 (6)−0.920−1.2100.048−20.472−20.474
RPVC−15.07623.561 (27)−0.640−0.8490.203−15.080−15.086
RPCV LS−14.84123.612 (32)−0.629−0.8370.201−14.844−14.850
RPVC NLS−23.34123.711 (34)−0.984−1.2890.134−23.344−23.349
LS RPVC−18.34023.935 (37)−0.766−1.0170.226−18.345−18.353
NLS RPVC−26.86226.877 (46)−0.999−1.3310.178−26.868−26.876
LS RPCV LS−17.99124.031 (38)−0.749−0.9950.226−17.996−18.004
NLS RPVC NLS−25.37924.937 (45)−1.018−1.3380.140−25.383−25.388
Table 8. Annualised performance measures: AV, SD, IR, SR and TO stand for the average, standard deviation, information ratio, Sortino’s ratio and turnover of the out-of-sample MVP returns. AV 20 b p n e t  and AV 50 b p n e t stand for the average out-of-sample MVP return net of transaction costs considering 20 and 50 basis-points, respectively. Period July 2009 to November 2017. The shaded cells denote the top five for each criterion. Weights are rebalanced on a monthly basis considering short-selling constraints.
Table 8. Annualised performance measures: AV, SD, IR, SR and TO stand for the average, standard deviation, information ratio, Sortino’s ratio and turnover of the out-of-sample MVP returns. AV 20 b p n e t  and AV 50 b p n e t stand for the average out-of-sample MVP return net of transaction costs considering 20 and 50 basis-points, respectively. Period July 2009 to November 2017. The shaded cells denote the top five for each criterion. Weights are rebalanced on a monthly basis considering short-selling constraints.
AVSDIRSRTOAV 20 bp net AV 50 bp net
1/N13.13016.057 (47)0.8181.148---
CCC12.59210.935 (21)1.1521.6660.33312.58312.569
CCC LS12.20010.873 (17)1.1221.6280.34212.19112.178
CCC NLS12.03810.852 (15)1.1091.6000.31012.02912.017
LS CCC12.45510.936 (22)1.1391.6480.34012.44612.432
NLS CCC12.46610.894 (18)1.1441.6560.33312.45712.443
LS CCC LS11.99210.932 (20)1.0971.5920.33611.98311.970
NLS CCC NLS12.65610.945 (23)1.1561.6790.34012.64612.633
DCC11.72910.801 (11)1.0861.5540.33611.72011.706
DCC LS11.87310.761 (7)1.1031.5900.34011.86411.850
DCC NLS10.56010.716 (3)0.9851.4020.31510.55110.538
LS DCC11.71410.800 (10)1.0851.5510.33311.70511.691
NLS DCC11.70110.801 (12)1.0831.5490.33311.69211.678
LS DCC LS11.84510.761 (6)1.1011.5850.34011.83511.821
NLS DCC NLS10.53410.714 (2)0.9831.3970.31210.52510.512
RM200611.19710.716 (4)1.0451.4760.27111.18911.177
RM2006 LS11.98710.531 (1)1.1381.6270.27911.97811.966
RM2006 NLS10.94310.776 (8)1.0161.4420.29110.93510.923
RM199413.04011.344 (37)1.1501.6300.27513.03213.021
RM1994 LS12.75811.017 (27)1.1581.6530.27312.75012.738
RM1994 NLS12.22911.068 (28)1.1051.5660.25212.22212.211
DECO11.05411.293 (34)0.9791.4150.32111.04511.032
DECO NLS11.26211.791 (45)0.9551.3920.40911.25111.235
LS DECO11.05411.293 (34)0.9791.4150.32111.04511.032
NLS DECO11.05511.293 (36)0.9791.4150.32111.04611.033
NLS DECO NLS11.26211.791 (45)0.9551.3920.40911.25111.235
OGARCH10.57610.959 (25)0.9651.3770.10310.57310.569
OGARCH LS9.69410.852 (16)0.8931.2810.1209.6919.686
OGARCH NLS10.56811.197 (33)0.9441.3480.08810.56610.563
LS OGARCH10.20010.921 (19)0.9341.3320.10310.19710.192
NLS OGARCH10.58010.958 (24)0.9661.3770.10310.57710.573
LS OGARCH LS9.85310.847 (14)0.9081.3040.1159.8509.845
NLS OGARCH NLS10.58111.197 (32)0.9451.3500.08810.57910.576
GPVC9.37411.733 (43)0.7991.1210.1619.3709.363
GPVC LS9.19411.126 (31)0.8261.1680.1249.1919.186
GPVC NLS9.42511.598 (41)0.8131.1460.1459.4219.416
LS GPVC9.29511.436 (38)0.8131.1430.1419.2919.285
NLS GPVC9.37911.742 (44)0.7991.1220.1599.3759.368
LS GPVC LS9.61710.737 (5)0.8961.2830.1809.6129.605
NLS GPVC NLS9.43511.585 (40)0.8151.1490.1459.4329.426
RPVC11.16311.486 (39)0.9721.3760.26811.15511.144
RPCV LS9.96510.803 (13)0.9221.3190.2019.9599.951
RPVC NLS12.15811.600 (42)1.0481.4970.21812.15212.143
LS RPVC10.15011.125 (30)0.9121.2910.23910.14310.133
NLS RPVC10.84711.091 (29)0.9781.3880.21210.84110.833
LS RPCV LS9.63710.782 (9)0.8941.2790.1879.6329.624
NLS RPVC NLS11.13010.964 (26)1.0151.4510.18911.12511.118
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