Revisiting the safe haven role of Gold across time and frequencies during the COVID-19 pandemic

Abstract

This research empirically evaluates the potential diversification benefits of Gold during the COVID-19 pandemic period, when including it in equity-based asset allocation strategies. This study proposes minimum VaR portfolios, with monthly rebalance and different wavelet scales (short-run, mid-run and long-run), doing both an in-sample and out-of-sample analysis. We find much more unstable weights as the frequency of the decomposition becomes lower, and strong evidence of the outperformance of the mid-run decompositions over the rest of active management strategies and the passive management of buy and hold the variety of single equity indices. Thus, we may shed some light on the role of Gold as a safe haven when properly filtering aggregated data.

1. Introduction

At the beginning of the year 2020, an ounce of Gold -equivalent to 28.3495 g- was trading at $1,530. Although the first cases of the new Coronavirus were already diagnosed in the Chinese city of Wuhan, it was not yet considered a global sanitary crisis, so the Gold price remained stable during the early weeks of the year. However, in February 2020, when the disease struck Europe, this precious metal began an upward trend that was only slightly halted in mid-March by the global collapse of the economies as a result of the first home confinements and the clear contagion effect between markets of very different natures and geographies (Banerjee, 2021, Davidovic, 2021, Guo et al., 2021, So et al., 2021). Since the end of March, Gold underwent an exponential recovery and began a bullish rally that reached an all-time peak in early August 2020, hitting $2,035 per ounce (Sanderson, 2020). The major driver behind the strong purchasing of Gold during the year 2020 lies in its store of value nature. Besides, from the economic crisis caused by the pandemic, countercyclical and expansive fiscal policies were applied, leading to the idea of future inflationary pressures in the long-term horizon. Thus, precious metals are also influenced by the US dollar patterns. Over 2020 the greenback US dollar was weak, then investors rushed to buy Gold. In this regard, the large international hedge funds – Elliott and Caxton, among others – echoed the increasing decline of currencies and bet on radically rising their weightings in Gold as a safe haven for their institutional portfolios, see Fletcher and Sanderson (2020).1

The pandemic caused by the COVID-19 outbreak led to a sharp increase in stock market volatility, drastically reducing market liquidity.2 As reported by leading journals such as the Financial Times the instability experienced during the first waves of the pandemic is evidenced in the collapse of the Standard & Poor’s 500 index, which jumped from an all-time high on February 19, 2020 to a 34% drop in just one month (Elder, 2020). In this context of global uncertainty and fear, market participants shifted from risky alternatives -e.g., emerging currencies- to others traditionally considered as safe havens -e.g., precious metals-. Gold is considered as one of the most relevant stores of wealth and investors turn to it in the face of large market downturns. The diversification benefits it provides to equity portfolios and its long-term profitability have led to an overall acceptance among a wide variety of market participants (Ratner and Klein, 2008, Dyhrberg, 2016, Beckmann et al., 2019, AlKhazali et al., 2020). The key factor of a diversification strategy is the dependence structure among the different securities that integrate such portfolio. The inverse correlation among Gold and stock markets enables investors to reduce their overall risk exposure, as potential losses in equities may be offset by gains in the precious metal (Grauer and Shen, 2000, Driessen and Laeven, 2007). This paper aims to shed some light on the performance of Gold by examining the dynamic structure of dependence of such precious metal and different international equity markets across time and frequencies. Based on a careful and state-of-the-art methodology, we provide empirical evidence to help answer a recurring question for many academics and practitioners: is Gold truly a safe haven in bearish and turbulent markets?

A variety of market forces affect economic relations between Gold and equity markets (Corbet et al., 2020, Ji et al., 2020). These economic shocks are localized in time and exhibit oscillations of different frequency (Al-Yahyaee et al., 2020). Some economic indicators and their own interrelationships lag behind, ahead of or coincide with other variables (Dai et al., 2020). Additionally, different market players in the financial system understand the mechanics on shorter and longer timescales (Reboredo and Rivera-Castro, 2014, Baruník et al., 2016, Liu et al., 2017, Beckmann et al., 2019). Thus, the “short-run“ and ”long-run“ terms are considered to be fundamental in modelling the complex relationships between these financial variables. In this regard, wavelets decompose time series data at different scales and can reveal relationships not evident in the aggregate data (Maghyereh et al., 2019, Mishra et al., 2019, Paul et al., 2019, Maghyereh and Abdoh, 2020, Rehman, 2020, Rehman and Kang, 2020, Živkov et al., 2020, Živkov et al., 2020, Berger and Czudaj, 2020, Bhatia et al., 2020, Dai et al., 2020, Lim, 2020, Mensi et al., 2020, Sun et al., 2021, Ghosh et al., 2021, Karim et al., 2021, Khraief et al., 2021).

Based on the properties of wavelet coefficients, in this paper we would contribute to the previous literature deriving scale-based estimators of volatility and correlation and test whether there are significant differences in the contribution of Gold to portfolio diversification for different decomposition frequencies. Specifically, our methodology consists of examining the time-varying behavior of equity-based portfolios when Gold is included. We use a variety of monthly rebalance strategies based on a minimum Value-at-Risk (VaR) optimization approach, which allows an evaluation of the overall Gold performance and contributions over the COVID-19 period. This research goes further the existing literature in minimum VaR portfolio optimization, by adapting the seminal study of Consigli (2002) to the context of skewed distributions and out-of-sample forecasting rebalance. We implement novel techniques in modelling the conditional dependences between pairs of asset returns by fitting the dynamic correlation matrix via an Asymmetric Dynamic Conditional Correlation (ADCC) model. We fit the different univariate volatility processess based on a variety of Generalized Autoregresive Conditional Heteroscedasticity (GARCH) models.

A number of prior studies apply similar econometric methodologies to the one proposed in the current research. On the one hand, Maghyereh et al. (2019) analyze the potential dynamic connection between Gold and Islamic stocks at different timescales, remarking Gold shows hedging properties in certain investment horizons. On the other hand, Ghosh et al. (2021) uses a wavelet-based time-varying dynamic approach to investigate interdependencies between the global financial market and some selected energy markets. Interesting results about the dynamic time-varying connectedness between the global financial market and the energy markets studied in this research may have relevant implications for portfolio managers. Lastly, Bhatia et al. (2020) investigate the time-varying relation among different precious metals and the major developed (G7) and emerging (BRICS) equity markets, by implementing a hybrid wavelet-DCC-GARCH model over the period 2000–2007. Current paper closely follows this latest research but going further in both methodology and economic-financial implications, conducting an in-sample and out-of-sample study to avoid overfitting and spurious findings. Besides, unlike Bhatia et al. (2020), who construct minimum variance portfolios without any rebalancing or financial evaluation, we minimize a measure of special relevance for investors and regulators, the VaR under the assumption that multivariate distributions are asymmetric student-t defined. Additionally, we conduct an ex-post performance assessment both in terms of overall risk-return and downside risk measures.

Other authors as Akhtaruzzaman et al., 2021b, Banerjee and Pradhan, 2021 conduct studies very similar to ours in terms of subject matter. One the one hand, Akhtaruzzaman, Boubaker, Lucey, et al. (2021) study the properties of gold as a safe haven during the COVID-19 pandemic in a high-frequency correlation analysis via a DCC GARCH and the spillover models of Diebold and Yilmaz (2012). In addition, they construct minimum variance portfolios on an hourly basis with the basic approach of Kroner and Ng (1998). We extend their study in that we implement a much more sophisticated methodology for rebalancing portfolios, both at the level of decomposing returns with wavelets and at the level of the optimization problem, in which we minimize a VaR measure that considers the markedly asymmetric and leptokurtic nature of most financial series. Moreover, unlike them, we conduct a monthly rebalancing, which obviously versus their hourly recompositing of portfolio weights, is much more optimal due to the possibility that the potential diversification benefits induced by gold are limited by the transaction costs of such high frequencies. Additionally, we also extend the pandemic period considered by the authors by introducing the new waves and the 2020 Christmas period. On the other hand, Banerjee and Pradhan (2021), like Akhtaruzzaman, Boubaker, Lucey, et al. (2021), use an intraday study to assess the safe haven properties of precious metals (besides gold, they add silver and platinum) in a bifurcated pre- and during the COVID-19 pandemic period. They model correlations with a Copula VAR ADCC GARCH approach. We differ from them mainly in that while they only go as far as a study of dependencies, we implement them to shed some light on the true benefit of diversification with gold via an active management portfolio construction and rebalancing strategy. Lastly, we use a different methodology to model correlations, which then allows us to distinguish between strategies according to the investment planning horizons of each market player based on wavelet frequencies.

To conduct our time-varying portfolio analysis on a monthly basis, we follow a four-stage procedure. The first step of the research consists in getting the wavelets of the time series of Gold price and international stock market returns in frequency domain by applying the maximal overlap discrete wavelet transform (MODWT) over the entire sample period (January 2018 to December 2020). The previous results extracted from the wavelet approach suggest the use of an ADCC-GARCH specification to model the relationship between the return series of Gold and stock market within timescales in terms of dynamicity and persistence. Thus, at second stage we implement a two-stage ADCC model, fitting the marginals of the univariate MSCI indices and Gold, and subsequently the dependence structure among indices-Gold pairs over an in-sample calibration period (January 2018 to December 2019). Third, we implement a re-fitting procedure with one-day ahead forecasting over the out-of-sample pandemic period (January-December 2020). Fourth, we construct and rebalance portfolios on the basis of a minimum skewed student’s t VaR strategy and assess them from the view of ex-post risk and performance in the range January-December 2020. A robustness analysis is conducted as well, which includes a bifurcated analysis of pre- and during the COVID-19 pandemic. A data set with an equivalent length to the ones used for the main analysis (from January 2017 to December 2018) is selected for the first estimation of the models (calibration), and 2019 is reserved for the out-of-sample experiment.

A preliminary analysis of the out-of-sample period shows the temporal behavior of both the weights and the dependence structure among asset pairs. The fluctuations observed in the time evolution and across frequencies of the dynamic correlations support the relevance of conditional autoregressive models and wavelet decompositions in portfolio management. Overall, going from short- to the long-run increasingly unstable weights are found. Our results highlight the lack of stability of portfolios based on wavelet decomposition techniques -which may have some implications in terms of transaction costs- but reveal that understanding and fitting such variability via autoregressive models, we will be able to maximize the diversification benefits suggested by the new market information. The greatest diversification, together with a constant swapping of positions between Gold and MSCI indices is found in the mid-run. Furthermore, our cross-currency correlation analysis suggests that North American dollar-denominated investments are better diversifiers for undecomposed returns, while local currency ones are more suitable in the case of wavelet decomposition of returns. Regarding the weakness of the correlations and hence, the diversification benefits induced by gold across periods, we find that dependences among undecomposed returns are often weaker in the pre-pandemic period both from the view of the unified dollar and the national currency returns, whereas the dynamic dependences are weaker during the pandemic one when examining dollar returns decomposed at high frequencies, suggesting that the modelling of short-term decompositions for dollar returns is relevant and necessary in turbulent market scenarios.

An in-depth analysis of ex-post risk and performance makes it possible to examine the validity and suitability of the monthly optimization problem and to identify which portfolio does best from a strictly financial view. Following the recent literature on the relevance of investment in safe havens during periods of high instability, our findings reveal the strong diversifying role of Gold over the course of the current pandemic and for different timescales or frequencies. On the one hand, we demonstrate that in terms of volatility, the higher the wavelet decomposition frequencies, the greater the risk mitigation effects provided by Gold. On the other hand, regarding the results in terms of risk-adjusted returns, we find that almost all of the active management strategies designed outperform passive management relative to buy and hold the single MSCI indices, and for the different timescales under study. This highlights the pertinency of the rebalance technique in designing optimal active investment strategies. Besides, and in line with prior portfolio rebalance findings, the realized performance experiment concludes that d3 frequencies report on the best results, revealing the overperformance of the mid-run. Thus, this study contributes to the existing literature in portfolio formation by reinforcing the need to decompose returns into different frequencies to find patterns -not available within the aggregate data- that allow to construct leading strategies in terms of risk-return. Furthermore, based on our robustness performance study, we suggest that gold always acts as a reliable diversifier for equity investments, irrespective of currency or assessment period, but it is especially valuable during recessions, when it acts as a safe haven.

The remainder of this paper proceeds as follows. Section 2 details the literature review. Section 3 describes the conditional univariate and multivariate methodology to be implemented in the calibration of the minimum VaR portfolios. Section 4 explains the main particularities of the database and the summary statistics. Empirical findings related to portfolio weighting, performance assessment and risk management are reported in Section 5. Robustness checks are provided in Section 6, while Section 7 concludes the research.

2. Literature review

According to Jammazi et al., (2017), most previous literature primarily focuses on the time domain aspect of the financial data, ignoring the evidence from the frequency domain. However, some recent studies explore potential interdependences between financial series in the context of the wavelet framework, such as Maghyereh et al., 2019, Ghosh et al., 2021, Raath and Ensor, 2020, among others. These authors confirm that the connectedness between financial markets are time-varying and horizon dependent, so methodologies that take this into account should be applied.

A wide strand of financial literature investigates the connectedness at different timescales between oil price and stock returns using wavelet methods. These interdependences would seem to be more pronounced during periods of economic turbulence, such as the 2008 global financial crisis. Some examples are Jammazi et al., 2017, Reboredo and Rivera-Castro, 2014, Martín-Barragán et al., 2015, Baruník et al., 2016, Khraief et al., 2021, among others, that apply the MODWT, widely employed in financial series, as well as other alternative transform methods. Some studies propose a hybrid methodology, such as Khalfaoui et al. (2015) that explore interdependences between oil prices and stock markets using a wavelet-based bivariate GARCH model, and Huang et al. (2015) that apply a wavelet-VAR model for Chinese sectors. To examine the connectedness between oil and stock market returns, Jammazi and Reboredo (2016) merge the HTW wavelet transform with time-varying copulas. Mishra et al. (2019) proposes a fresh combined methodology of wavelets and Quantile-on-Quantile regression to study potential interdependencies between changes in oil prices and the Islamic stock market index. The observed impact is opposite in the short term (positive) to that shown in the long term (negative). A more recent study is Karim et al. (2021) that explores the connectedness between oil and stock market returns for BRICS, using a combined methodology consist in wavelets and MGARCH-DCC methods. All of them are positively correlated and more volatile in the coronavirus pandemic crisis. Other challenging studies such as Nguyen et al. (2020) focus on the connectedness of renewable versus conventional energy stock markets, applying a rolling window wavelet methodology. They check a relevant impact of the global financial crisis on the interdependencies explored in this study, evidencing a diversification role of renewable energy.

In the previous literature we observe some studies that performs a time–frequency analysis of dynamic correlations between different traded assets, remarking relevant implications to manage investment strategies and, mainly, considering Gold’s traditional role as a safe-haven asset. Baruník et al. (2016) select assets such as Gold, oil and stocks and find heterogeneity in correlations by investment horizons and assets, mainly during periods of financial downturn, and homogeneity in calm periods. Thus, Gold, oil and stocks could be used in a well-diversified portfolio only during moderately short periods. Beckmann et al. (2019) apply a wavelet decomposition combined with a copula approach to explore the dynamic connectedness between Gold price and bonds, stock and exchange rates across different frequencies. Gold would serve as a safe haven before the 2008 global financial crisis, being unable to act as a hedge after the subprime mortgage collapse. Salisu et al. (2021) deepens the idea about Gold as a safe haven to changes in crude oil prices, using the asymmetric VARMA-GARCH model. In addition, this study explores the impact of the COVID-19 pandemic crisis, verifying Gold (and other precious metals) as a significant safe haven and its hedging effectiveness against oil price risks. Reboredo and Rivera-Castro (2014) apply wavelet multi-resolution analysis to examine hedging benefits of using Gold to changes in exchange rates for different timescales, that is, investment horizons. They corroborate the effectiveness of Gold for currency risk management at different timescales.

Recent studies such as Berger and Czudaj (2020) propose the fresh application of a wavelet-based portfolio strategy for commodity futures, highlighting the suitability of the wavelet methodology to manage investment portfolios. In the same vein, Dai et al. (2020) explore different time horizons between oil, Gold and stock markets by applying wavelet techniques that allow to distinguish frequency scales related to short-, medium- and long-run. Thus, this sort of studies could have relevant implications for short- and long-run investments. Paul et al. (2019) study interrelations over time and across time horizons between Gold mining stocks with Gold and equities in UK and US, applying the conventional MODWT wavelet approach. They find interesting results to manage multi-horizon investing portfolios. At the international investment level, an interesting study is the one by Rehman (2020), who decomposes international stock market returns using the MODWT wavelet technique and it deepens the study of contagion between world economic zones during episodes of economic and financial crisis. Al-Yahyaee et al. (2020) also explore the connectedness between Gold and silver (previous metals) and other non-ferrous metals, by applying a combined methodology of wavelets and the spillover methodology by Diebold and Yilmaz (2012). Relevant results confirm that interdependencies between them depend on frequencies, offering interesting implications for investors. Some recent studies, such as Jareño et al., 2020, Kumah and Mensah, 2020, Rehman and Kang, 2020, González et al., 2021 explore potential hedging properties of cryptocurrencies (like Gold) using varied methodologies (wavelets, quantile regression, NARDL approach, etc.). These studies offer useful information to portfolio managers over time and market states.

Other interconnections are examined in studies such as Raath and Ensor (2020) that preliminarily analyze the complexity of the water-energy nexus, by applying wavelet techniques and corroborating a time-varying relationship between them. One of the first studies to apply methodologies proposed in the present study (DCC-GARCH and wavelet multiscale analysis) is the one by Dajcman et al. (2012), but they compare them to explore interdependencies between European stock markets, concluding, as expected, that they are time-varying and scale dependent. Similar results are found by Sun et al. (2021) for the Chinese energy stock market. According to Raath and Ensor (2020), some previous applications of time-varying techniques to the portfolio management separating different investment horizons are, for instance, Ftiti et al., 2017, Wang et al., 2017, Kumar and Anandarao, 2019, among others. Recently, Maghyereh et al., 2019, Bhatia et al., 2020, Raath and Ensor, 2020, Ghosh et al., 2021 use similar techniques. Thus, Bhatia et al. (2020) propose a combined methodology of wavelets and DCC approach to explore potential dynamic interdependencies between precious metals and selected international stock markets. This connectedness is timescale dependent, showing very useful information for portfolio managers.

Regarding the main topic developed in this research, it is important to highlight other methodologies that, independently, have been implemented in recent works, such as the ADCC model, that is, modelling with asymmetric t-student distribution, minimum VaR portfolios, as well as the evaluation in out-of-sample periods to give robustness to the results. First, the ADCC model has been previously implemented for modelling volatilities and conditional correlations between financial markets (Basher & Sadorsky, 2016), for testing optimal hedge ratios for clean energy stocks (Ahmad et al., 2018), and for estimating the contagion effect during the COVID-19 pandemic (Banerjee, 2021), among others. Sahamkhadam et al. (2018) propose the use of the minimum Conditional VaR (Min-CVaR) for portfolio management, and Aziz et al., 2019, Fang et al., 2018, among others, compare the out-of-sample performance in the context of portfolio management. The great contribution of this paper consists of combining all these novel techniques in a single investigation, proposing an analysis in different stages, which we will be explained below.

3. Methodology

This study proposes to obtain minimum VaR portfolios assuming the asymmetric skewed student-t distribution to characterize the multivariate dependencies between country-Gold pairs. This time-varying optimization problem is conducted across time and frequencies on the basis of a hybrid MODWT-ADCC-GARCH model. Section 3.1. describes the particularities of multivariate distributions and portfolio formation. Section 3.2. reports on the MODWT wavelet procedures whereas the different univariate and multivariate GARCH specifications can be found in the Appendix. Section 3.3. defines a number of performance measures to assure portfolio assessment.

3.1. Portfolio construction

Given the bearish market context of this study, a minimization of downside risk measures is proposed as optimal problem to perform our portfolio rebalancing strategies. Additionally, it is assumed that returns follow an asymmetric t-student distribution, to get a better fit with the heavy tails and skewness patterns of the empirical series. Following Consigli (2002), we select the VaR as a measure to be minimized. The VaR of a portfolio is defined as the maximum expected loss due to an adverse market movement, within a given confidence level over a given time horizon (Boyle et al., 1997, Duffie and Pan, 1997, Danielsson and De Vries, 2000, Jorion, 2007). Analytically, the VaR at the confidence level of $\left(1-p\right)$ is the quantity such that the probability that the return will be lower or equal to this quantity is:

$$VaR\left(1-p\right)=-inf\left\{r|F\left(r\right)\ge \left(1-p\right)\right\}$$ (1)

Very different studies along the extensive financial literature consider the inclusion of time-varying distribution moments and assume a variety of distributions to fit the VaR risk measure (Engle, 1982, Bollerslev, 1986, Hull and White, 1998, Bali et al., 2008). Current research follows the study of Lambert and Laurent (2002) and assumes that the multivariate between pairs of returns is distributed as an asymmetric student-t. Then, VaR for long positions is given by:

$${\mathit{VaR}}_t\left(1-p\right)={\mathit{skst}\left(p\right)}_{\alpha ,\nu ,\xi }{\sigma }_{P,t}$$ (2)

with ${\mathit{skst}\left(p\right)}_{\alpha ,\nu ,\xi }$ being the left quantile at $\alpha \%$ of the skewed-Student distribution with $\nu$ degrees of freedom and asymmetry coefficient $\xi$. The quantile function ${\mathit{skst}}_{\alpha ,\nu ,\xi }$ of a non standardized skewed-Student density (Fernández & Steel, 1998) is given by:

$$\left\{\begin{array}{c}{\mathit{skst}}_{\alpha ,\nu ,\xi }=\frac{1}{\xi }{\mathit{st}}_{\alpha ,\nu }\left[\frac{\alpha }{2}(1+{\xi }^2)\right]if\alpha <\frac{1}{1+{\xi }^2}\\ {\mathit{skst}}_{\alpha ,\nu ,\xi }=-\xi {\mathit{st}}_{\alpha ,\nu }\left[\frac{1-\alpha }{2}\left(1+{\xi }^{-2}\right)\right]if\alpha \ge \frac{1}{1+{\xi }^2}\end{array}\right)$$ (3)

where $\xi$ is the asymmetry coefficient, $\nu$ the degree of freedom and $\alpha$ is the quantile probability. ${\mathit{st}}_{\alpha ,\nu }$ is the quantile function of the (unit variance) Student-t density.

Thus, the optimal portfolio rebalancing can be established as a dynamic optimization problem in which the VaR of the strategy is minimized:

$$\left\{\begin{array}{c}\underset{w_{p,t}}{\min }VaR\left(1-p\right)=\mathit{skst}{\left(p\right)}_{\alpha ,\nu ,\xi }{w}_{p,t}^{\text{'}}{\mathrm{\Sigma }}_t{w}_{p,t}\hfill \\ constrainedto:{\sum }_{i=1}^n{w}_{i,t}=1\hfill \end{array}\right)$$ (4)

where ${\mathrm{\Sigma }}_t$ is the $\mathit{nxn}$ dynamic covariance matrix calculated by modelling conditional correlations and volatilities over time, $w_{p,t}$ is the vector of dynamic weights invested in each of the included assets.

The conditional covariance matrix, truly essential in the minimum VaR problem, is obtained as follows for each country-Gold pair:

$${\mathrm{\Sigma }}_t=D_t{\mathrm{\Gamma }}_t{D}_t;{\mathrm{\Sigma }}_t=\left(\begin{array}{cc}{\sigma }_{i,t}& 0\\ 0& {\sigma }_{j,t}\end{array}\right)\left(\begin{array}{cc}1& {\rho }_{\mathit{ji},t}\\ {\rho }_{\mathit{ij},t}& 1\end{array}\right)\left(\begin{array}{cc}{\sigma }_{i,t}& 0\\ 0& {\sigma }_{j,t}\end{array}\right)=\left(\begin{array}{cc}{\sigma }_{i,t}^2& {\sigma }_{\mathit{ij},t}\\ {\sigma }_{\mathit{ji},t}& {\sigma }_{j,t}^2\end{array}\right)$$ (5)

where $D_t$ is the conditional deviation array, which is expressed as a diagonal matrix with the diagonal elements as conditional volatilities obtained from the univariate models. ${\mathrm{\Gamma }}_t$ is the dynamic correlation matrix obtained via different ADCC-GARCH specifications. This array has its main diagonal composed by ones, and $n\left(n-1\right)$ conditional correlations out of the diagonal.

This conditional covariance matrix, ${\mathrm{\Sigma }}_t$ is structured on the basis of two issues, a main diagonal composed of dynamic variances and the rest of the elements as time-varying covariances. On the one hand, the conditional variances of the main diagonal come from the fitting of different univariate autoregressive heteroscedasticity specifications as the standard GARCH (1,1), the E-GARCH (1,1) or the GJR-GARCH (1,1).3 Overall, we estimate the $n$ univariate GARCH models from the innovations or residuals of prior AR (1) specifications. On the other hand, the dynamic covariances are computed from the two-stage estimation of the ADCC model.4 We fit a total of $\frac{n\left(n+1\right)}{2}$ autoregressive models for each t.

3.2. Wavelet methods

According to Jammazi et al. (2017), the wavelet tool provides an intuitive way to examine dynamic interactions between non-stationary financial time series at different frequencies separately, that is, on a scale-by-scale basis. In addition, Maghyereh et al. (2019), among others, affirms that, unlike the discrete wavelet transform method (DWT), the MODWT can handle input data of any length and does not require the length of the series to be power of two (Risse, 2019, Jiang and Yoon, 2020, Kumah and Mensah, 2020, Karim et al., 2021). Furthermore, the MODWT does not introduce phase shifts in the wavelet coefficients (Gençay et al., 2002), therefore any peaks or troughs in the original time series will be correctly aligned with similar events in the multiresolution analysis (Masset, 2011, Mensi et al., 2018). Furthermore, the MODWT produces more asymptotically efficient wavelet variance than that based on the DWT (Shafaai & Masih, 2013).5

Similar to Živkov et al., 2020, Živkov et al., 2020, Lim, 2020, among others, we use wavelet technique, which is capable of decomposing time-series into their time–frequency components without wasting of valuable information, by getting an appropriate trade-off between resolution in the time and frequency domains. Wavelet theory usually uses two key wavelet functions: the father wavelet (ϕ) and the mother wavelet (ψ). Father wavelets augment the representation of the smooth or low frequency parts of a signal with an integral equal to 1, whereas the mother wavelets can describe the details of high frequency components with an integral equal to 0. In other words, father wavelet portrays the long-term trend over the scale of the time-series, whereas the mother wavelet delineates fluctuations in the trend. Father wavelet ϕ_j,k(t) and mother wavelet ψ_j,k(t) functions can be calculated as in Eq. (6):

$${\varphi }_{J,k}\left(t\right)=2^{-J/2}\varphi \left(\frac{t-2^{J}k}{2^J}\right){\psi }_{j,k}\left(t\right)=2^{-J/2}\psi \left(\frac{t-2^{j}k}{2^j}\right)$$ (6)

We make use of particular type of wavelet transformation – the maximum overlap discrete wavelet transformation (MODWT) algorithm, which is based on a highly redundant non-orthogonal transformation. In that sense, a signal-decomposing procedure in MODWT is given in the following way:

$$S_{J\left(t\right)}={\mathrm{\Sigma }}_k{S}_{j,k}{\varphi }_{j,k\left(t\right)}$$ (7)

$$D_{J\left(t\right)}={\mathrm{\Sigma }}_k{S}_{j,k}{\psi }_{j,k\left(t\right)}j=1,2,\dots ,J$$ (8)

where symbols S_j(t) and D_j(t) denote the fluctuation and scaling coefficients, respectively, at the j-th level wavelet that reconstructs the signal in terms of a specific frequency (trending and fluctuation components). Accordingly, an empirical time series r(t) can be expressed in terms of those signals as:

$$r\left(t\right)=S_J\left(t\right)+D_j\left(t\right)+D_{j-1}\left(t\right)+\dots +D_1\left(t\right)$$ (9)

According to Lim (2020), the Maximum Overlap Discrete Wavelet Transform (MODWT) is used with its advantage on the flexibility of the length of data which means not requiring the integral power of 2, as well as the time invariant property.

The transformed return series x(t) is represented as a linear combination of wavelet functions as follows:

$$x\left(t\right)=\sum _k{S}_{J,k}{\varphi }_{J,k}\left(t\right)+\sum _k{D}_{J,k}{\psi }_{J,k}\left(t\right)+\sum _k{D}_{J-1,k}{\psi }_{J-1,k}\left(t\right)+\cdots +\sum _k{D}_{1,k}{\psi }_{1,k}\left(t\right)$$ (10)

3.3. Performance assessment

Similarly to González-Pedraz et al. (2015), we conduct an ex-post performance analysis in which a variety of portfolios are built from historical data, but examined in terms of returns one day ahead. Then, we examine the overall performance along the entire out-of-sample period from previous continuous daily returns, so that we can test whether the technique used in portfolio construction has worked properly or not. The process of portfolio evaluation is described in two stages:

At the first stage, from the dynamic optimization process described in Section 3.1. (Eqs. (1), (2), (3), (4)), we calibrate the time-varying minimum VaR asset weights, $w_{P,t}$. Then, we define the portfolio return realized over the next trading day, $r_{P,t+1}$, as:

$$r_{P,t+1}=w_{P,t}^{\text{'}}{r}_{\mathit{assets},t+1}$$ (11)

where $w_{P,t}^{\text{'}}$ denotes the transpose vector of weights. This vector comes from the time-varying portfolio rebalance strategy at each given point in time. $r_{\mathit{assets},t+1}$ depicts the conditional column vector of asset returns realized one day later of portfolio construction.

At the second stage, the ex-post performance of both, single MSCI indices and Gold-indices combined strategies, is examined over the out-of-sample period. Two different measures are implemented: the Sharpe ratio (SR) and a number of Kappa indices (K). On the one hand, the SR ratio (Sharpe, 1966, Sharpe, 1975, Sharpe, 1994) assess the mean-excess return of the strategy per unit of total risk exposure:

$${\mathit{SR}}_{{(r}_p)}=\frac{E_{{(r}_p)}-rf}{{\sigma }_{{(r}_p)}}$$ (12)

where $E_{{(r}_p)}$ and ${\sigma }_{{(r}_p)}$ denote the mean and the standard deviation of the strategy under study. $\mathit{rf}$ depicts the risk-free rate.

On the other hand, another vein of the financial literature is especially concerned about the left tail of the distribution, i.e. results under a minimal acceptable return or threshold (Roy, 1952, Markowitz, 1959). The concept of Lower Partial Moments ($\mathit{LPM}$) is introduced by Fishburn (1977) to account the negative deviations from a certain threshold, $h$.6 Regarding the LPM, a number of K indices of different orders are defined (Kaplan & Knowles, 2004):

$$K(h,m)=\frac{E_{{(r}_p)}-h}{{\mathit{LPM}}_{m,h}{\left(r\right)}^{1/m}}$$ (13)

where ${\mathit{LPM}}_{m,h}\left(r\right)$ is the LPM of order, $m$ and threshold, $h$.

4. Data sample and summary statistics

This paper selects the MSCI indices for 8 countries, 4 among the BRICs and 4 within the G7, which were the most affected countries during the first wave of the COVID-19 pandemic crisis. In concrete, we have carefully chosen these countries according to an indicator that reflects the total number of deaths, from the Coronavirus Disease (COVID-19) Situation Reports, prepared by the World Health Organization (2020). Thus, we analyze the following MSCI indices for the BRICS (Brazil, India, Russia and China) and the G7 (USA, UK, Italy and France) countries.7

Table 1 collects some relevant information about the MSCI indices selected in this paper for the G7 (US, UK, France and Italy) and BRICS (Russia, India, Brazil and China) countries. Thus, regarding the average market capitalization (in USD millions), US shows the highest values while China reveals the lowest level. In addition, as expected, all the BRICS have lower average market capitalization than G7 countries. About the constituents of each MSCI index, the sector weights distribution is quite different among the countries included in this research. Thus, there are IT-intensive countries (such as US and India), as well as others where the financial sector is a major player (Italy, Russia and India, among others). Finally, the last column informs about the sad ranking based on the total deaths caused by the SARS-CoV-2 coronavirus. Thus, US and Brazil are the most affected countries during the COVID-19 pandemic for the G7 and BRICS groups, respectively.

Table 1.

Information about the MSCI indices for the selected G7 and BRICS countries explored in this study.

MSCI Indices	Av Mkt Cap (USD Millions) *	Constituents: Sector Weights (%) **	COVID-19 Ranking (Total deaths) ***
G7 countries
US	57,570.72	IT(27.51), HC(13.19), CD(12.11), CS(11.1), F(10.96), I(8.73), CST(5.94), M(2.65), U(2.63), RE(2.6), E(2.58)	1st
UK	26,461.83	CST(20.16), F(19.37), M(12.13), I(10.48), HC(10.41), E(9.91), CD(7.03), CS(4.74), U(3.4), RE(1.21), IT(1.16)	2nd
France	24,859.54	CD(21.55), I(21.26), CST(10.69), F(10.09), HC(8.32), IT(6.65), E(6.49), M(5.88), CS(4.56), U(3.06), RE(1.46)	4th
Italy	15,726.24	F(29.89), U(25.2), CD(18.06), E(8.92), I(8.07), HC(3.5), CS(2.88), IT(1.88), CST(1.59)	3rd
BRICS countries
Russia	10,443.06	F(27.26), M(25.06), E(11.97), CST(9.76), CD(8.2), I(6.94), U(4.94), HC(3.53), CS(1.64), IT(0.7)	3rd
India	8,099.95	F(26.15), IT(17.62), E(12.16), M(10.04), CST(9.87), CD(8.5), HC(5.37), I(3.55), CS(3.37), U(3.05), RE(0.31)	2nd
Brazil	6,780.23	E(45.72), F(21.32), M(17.9), CS(10.35), CST(3.71), U(0.99)	1st
China	4,306.13	CD(33.95), CS(20.45), F(14.45), HC(6.69), IT(5.97), I(4.68), CST(4.36), RE(4.03), M(2.24), U(1.97), E(1.21)	4th / 5th

This table reports on the considered components of the investment opportunity set. By columns, we report on the average market capitalization, the major holdings and the overall position in the COVID-19 ranking of each of the selected MSCI indices. *Average market capitalization (March 31, 2021). ** IT: Information Technology, HC: Health Care, CD: Consumer Discretionary, CS: Communication Services, F: Financials, I: Industrials, CST: Consumer Staples, M: Materials, U: Utilities, RE: Real Estate, E: Energy. *** Ranking during the out-of-sample period analyzed in this study (year 2020).

The full data sample spans from January 2018 to December 2020, including 784 observations of daily MSCI traded prices (see Fig. 1 ). All initial data processed have been downloaded from Bloomberg in US dollar currency.8 The sample interval is selected with the aim of studying the diversification properties of gold during the year 2020 or pandemic period in an out-of-sample analysis. Notwithstanding, to properly conduct it – as carefully explained in Section 5. –, we first need to perform an initial estimation (calibration) of the GARCH models from January 2018 to December 2019 (522 daily return observations which give name to the in-sample period). Then, the re-estimates that are made throughout the out-of-sample period have the same sample length (522 observations).9 The use of this window size is based on what is suggested by Hwang and Valls Pereira (2006), who state that the parameter estimation of GARCH specifications can be biased with small samples and therefore, a window as the one rolled in the present research of at least 500 workdays is highly recommended.

Fig. 1.

The figure plots the traded prices of the different MSCI indices, representing the major developed (G7) and emerging countries (BRICS), and Gold. All traded prices are expressed in terms of 100 base indices to ensure proper comparison. Left subplot reports on the trends described by the BRICS countries, while right subplot describes the time evolution of the G7 indices. Regarding the first subplot, yellow color is used to describe the Gold trend, cyan blue is chosen for Brazil, purple for Russia, black for India and orange for China. With respect to the second subplot, maroon color describes the trend of France, green is associated to Italy, grey to UK and dark blue to USA. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

We assess the trends and patterns described by the different equity indices and the commodity one (Gold) from Fig. 1 and under a pre-COVID and COVID viewpoint. First, from the beginning of 2018 to the early 2020 we observe that for the BRICS a clearly upward trend is depicted for the case of Brazil and Russia. Conversely, we find decreasing or stable patterns for the cases of India and China. Regarding the G7, it is reported that overall, relative indices show high growth, with special mention to the USA. In mid-March 2020, a common decline is observed for all indices and Gold, suggesting a contagion effect between very different markets in times of deep recession (Banerjee, 2021, Davidovic, 2021, Guo et al., 2021, So et al., 2021). However, Gold hardly drops at all, keeping or barely losing its intrinsic value, which highlights its role as a safe-haven asset in times of high market instability. China, the other exception that declined slightly during the pandemic, may be because it took the first measures to stop the coronavirus in the first days of January, when the outbreak had not yet been officially declared. From mid-April 2020, all the indices and Gold begin to grow and describe a bullish rally until the end of December 2020. However, a slight decline or flattening of values is observed during the months of September and October, coinciding with the second wave of the pandemic. In econometric terms, we report that all MSCI indices behave as random walks with stochastic trends or unit roots. To properly control these patterns and conduct our empirical study, a total of 783 log-returns are computed.

Table 2 provides the summary statistics of the log-returns conducted for the different MSCI indices. Regarding average annualized returns, we note that emerging Russia sufficiently dominates the rest of the equity indices and Gold. The reverse occurs in terms of volatility. It is reported that despite being a period of relative calm in the financial markets, Gold is already outperforming the equity indices (less risk exposure). This undoubtedly motivate its study as a hedging asset or safe haven during the pandemic out-of-sample period. Additionally, from higher order distribution moments (skewness and kurtosis) and the Jarque-Bera test conducted over the different financial series, we report clear evidence of non-normality. This leptokurtic and heavy tailed behavior will be considered in subsequent sections by assuming the skew student’s t distribution in the multivariate model fitting and for the different frequencies under analysis. Overall, we find low autoregressive presence or weak autocorrelation with regard to the Ljung-Box test for 1, 2 and 5 lags. This autoregressive presence in the first moments rises as we increase the lags, reaching its highest peaks for the fifth order and especially for the case of emerging countries and Gold. Lastly, the Lagrange Multiplier test reveals the presence of strongly significant autocorrelation in squared returns for almost all the indices and lags (with a few exceptions, such as Gold and Russia). A number of skewed GARCH models will be fitted in subsequent sections, to control these volatility and clustering patterns.

Table 2.

Descriptive statistics for the continuous returns over the period in-sample period (Jan 18-Dec 19).

	GOLD	BRAZIL	RUSSIA	INDIA	CHINA	FRANCE	ITALY	UK	USA
$E_{{(r}_j)}$	0.0661	0.1131	0.2065	−0.0007	0.0023	0.0540	0.0319	0.0191	0.1103
${\sigma }_{{(r}_j)}$	0.1080	0.2705	0.2110	0.1589	0.1926	0.1395	0.1730	0.1350	0.1473
${\tau }_{{(r}_j)}$	0.2052	−0.0179	−2.0196	0.0701	−0.1597	−0.4311	−0.2850	−0.1397	−0.6164
$k_{{(r}_j)}$	5.7930	4.1020	21.9863	5.0058	3.4968	3.9519	4.0210	5.2116	6.8871
$\mathit{JB}$	165.7573	24.6254	7949.90	83.4974	6.9288	34.1834	28.0159	102.8885	348.1419
	(0.0000)***	(0.0000)***	(0.0000)***	(0.0000)***	(0.0313)**	(0.0000)***	(0.0000)***	(0.0000)***	(0.0000)***
$Q\left(1\right)$	2.5364	0.3050	0.7269	0.1433	15.7753	1.8123	0.1891	0.0010	0.1314
	(0.1112)	(0.5808)	(0.3939)	(0.705)	(0.0001)***	(0.1782)	(0.6637)	(0.9746)	(0.717)
$Q\left(2\right)$	5.0043	6.6073	1.0837	4.1003	15.8939	2.5893	0.2228	0.5302	3.7469
	(0.0819)*	(0.0367)**	(0.5817)	(0.1287)	(0.0004)***	(0.274)	(0.8946)	(0.7671)	(0.1536)
$Q\left(5\right)$	10.4804	10.2647	10.0896	8.9838	17.1376	3.6133	3.0759	1.1819	9.4914
	(0.0627)*	(0.0681)*	(0.0727)*	(0.1097)	(0.0042)***	(0.6063)	(0.6883)	(0.9466)	(0.091)*
$\mathit{LM}\left(3\right)$	3.9192	10.8856	2.8766	5.4949	10.8482	13.9258	3.3296	20.7957	32.4959
	(0.2703)	(0.0124)**	(0.411)	(0.1389)	(0.0126)**	(0.003)***	(0.3435)	(0.0001)***	(0.0000)***
$\mathit{LM}\left(5\right)$	5.7502	18.3721	3.1454	11.5951	11.9568	29.7531	4.7741	20.9542	36.3807
	(0.3313)	(0.0025)***	(0.6776)	(0.0408)**	(0.0354)**	(0.0000)***	(0.4441)	(0.0008)***	(0.0000)***
$\mathit{LM}\left(7\right)$	11.5464	19.2635	5.2420	11.5044	26.5526	30.1553	16.8819	23.6023	49.4383
	(0.1165)	(0.0074)***	(0.6305)	(0.1181)	(0.0004)***	(0.0001)***	(0.0182)**	(0.0013)***	(0.0000)***

This table provides information of the continuous log-returns for the different MSCI indices and Gold. By columns we find the different indices, while each row reports on the different statistics. Additionally, non-normality is assessed by the Jarque-Bera test (JB), and autocorrelation in the first and second moments by the Ljung-Box ($Q$) and Lagrange-Multiplier ($\mathit{LM}$) tests, respectively. Note that different lag orders are evaluated, and P-values are presented in parentheses. Additionally, *, **, and *** reveal significance at the 10%, 5%, and 1% levels, respectively.

5. Empirical evidence

This section is understood in terms of two implemented techniques, the MODWT wavelet method and the ADCC-GARCH, and two different analysis periods, in-sample and out-of-sample. First, wavelet-based technique is employed to decompose the stock indices and Gold returns series over different time horizons from January 2018 to December 2020, thus including both, the in-sample and out-of-sample period (whole data sample). In the second step, ADCC–GARCH is implemented to investigate the dynamic dependences among decomposed series at respective frequencies and time horizons over the period that spans from January to December 2020 (out-of-sample period). Therefore, the years 2018 and 2019 are merely a period of parameter calibration (in-sample period).

5.1. In sample parameter calibration

The multivariate estimation process of the skewed dependence structure among assets and across frequencies consists of four steps. In a first step, and similar to Maghyereh et al. (2019), among others, our return series are decomposed into a number of scales (see Eqs. (6), (7)) by implementing the MODWT wavelet technique.10 Thus, the wavelet scales considered in this study are d1 = 2–4 days, d2 = 4–8 days, d3 = 8–16 days, d4 = 16–32 days, d5 = 32–64 days, d6 = 64–128 days, d7 = 128–256 days. The short-term horizon is defined as {D1 = (d1 + d2)} and it represents the Gold price and international stock market returns due to shocks happening from 2 to 8 days. The medium-term horizon is defined as {D2 = (d3 + d4)} which shows variations due to shocks occurring from 8 to 32 days. Finally, the long-term horizon is defined as {D3 = (d5 + d6 + d7)} which represents fluctuations occurring from 32 to 256 days.

Second, we implement a number of AR(1) models to correct for the time dependence in the mean equation (see Equation (14)) for the case of returns’ decompositions. Conversely, for the undecomposed returns we assume unconditional mean due to the low significance of the AR(1) parameters. Third, from the residuals of AR fitting (innovations), we estimate a number of univariate GARCH specifications to model the dynamic volatility of the returns and decompositions computed in a previous step. This is the first stage of the ADCC model, where the univariate parameters are estimated assuming that the standardized innovations are Gaussian distributed. As in recent dependence studies as Urquhart and Zhang (2019), the most accurate variance process for each asset has been selected from the different univariate GARCH models described in Appendix (Eqs. (14), (15), (16)) and based on the minimization of the Akaike Information Criterion (AIC) and the Bayes Information Criterion (BIC). Fourth, the second stage of the ADCC model is conducted from the standardized innovations of the prior univariate GARCH fitting, reporting as output the conditional correlation arrays. At this stage, the joint distribution of each pair of standardized innovations is assumed to be a skewed student’s t.

The analysis of the estimated GARCH parameters is divided into two tables (Table 3, Table 4 ), from which we report the parameter estimation for the log-returns with and without frequency decomposition during the in-sample period (January 2018-December 2019).11 This period serves as a model calibration in which we determine the optimal specifications and parameters to consider in order to adjust the mean, variance and correlation among processes. Subsequently, in Section 5.2. we begin with the specifications selected in this in-sample period to perform an out-of-sample forecasting experiment with rolling re-estimation of the initial parameters.

Table 3.

GARCH in-sample estimates for log-returns.

Panel A: Univariate constant mean-GARCH fitting
A.1. Country (C) distribution parameters & statistics							A.2. Gold (G) distribution parameters & statistics
	${\mu }_C$	${\omega }_C$	${\alpha }_C$	${\beta }_C$	${\gamma }_C$	${\mathit{LM}\left(5\right)}_C$	${\mu }_G$	${\omega }_G$	${\alpha }_G$	${\beta }_G$	${\gamma }_G$	${\mathit{LM}\left(5\right)}_G$
BRICS Countries & associated Gold parameters
Brazil	0.0008	−0.5591	−0.0709	0.9317	0.1674	4.5980	0.00033	−0.1995	0.0379	0.9806	−0.0772	3.3890
(EGARCH)	(0.8396)	(−1.9212)*	(−2.007)**	(26.2638)***	(2.4756)**	(0.1272)	(2.5786)***	(−15.39)***	(0.4043)	(9.3366)***	(−14.09)***	(0.2381)
Russia	0.0007	−0.2559	−0.0915	0.9705	0.0415	0.0370	0.00033	−0.1995	0.0379	0.9806	−0.0772	3.3890
(EGARCH)	(5.6957)***	(−28.23)***	(−2.208)**	(263090)***	(1.8467)*	(0.9968)	(2.5786)***	(−15.39)***	(0.4043)	(9.3366)***	(−14.09)***	(0.2381)
India	−0.0001	0.0000	0.0000	0.9080	0.1147	0.7330	0.00027	0.0000	0.0057	0.9986	−0.0147	2.3856
(GJRGARCH)	(−0.166)	(7.2717)***	(0.0000)	(69.711)***	(3.2258)***	(0.8134)	(11.989)***	(12.074)***	(3.3458)***	(1281600)***	(−3.394)***	(0.3921)
China	0.0002	−0.4701	−0.0930	0.9473	0.0854	3.3890	0.00033	−0.1995	0.0379	0.9806	−0.0772	3.3890
(EGARCH)	(0.465)	(−29.54)***	(−4.2995)***	(616.6024)***	(2.829)***	(0.2381)	(2.5786)***	(−15.39)***	(0.4043)	(9.3366)***	(−14.09)***	(0.2381)
G7 Countries & associated Gold parameters
France	0.0003	−1.1990	−0.2452	0.8752	0.0807	0.0407	0.00033	−0.1995	0.0379	0.9806	−0.0772	3.3890
(EGARCH)	(2.0573)**	(−53.70)***	(−7.6179)***	(534.4981)***	(1.4674)	(0.9963)	(2.5786)***	(−15.39)***	(0.4043)	(9.3366)***	(−14.09)***	(0.2381)
Italy	0.0001	−0.2242	−0.0867	0.9756	−0.0005	0.3756	0.00033	−0.1995	0.0379	0.9806	−0.0772	3.3890
(EGARCH)	(0.1908)	(−45.29)***	(−6.1997)***	(7302800)***	(−0.2672)	(0.9196)	(2.5786)***	(−15.39)***	(0.4043)	(9.3366)***	(−14.09)***	(0.2381)
UK	0.0002	−1.1565	−0.1541	0.8792	0.1701	1.4488	0.00033	−0.1995	0.0379	0.9806	−0.0772	3.3890
(EGARCH)	(0.5013)	(−5.076)***	(−4.33)***	(36.6497)***	(3.3269)***	(0.6059)	(2.5786)***	(−15.39)***	(0.4043)	(9.3366)***	(−14.09)***	(0.2381)
USA	0.0005	−0.5216	−0.2456	0.9463	0.1437	4.9047	0.00033	−0.1995	0.0379	0.9806	−0.0772	3.3890
(EGARCH)	(1.5807)	(−29.11)***	(−7.07)***	(4028.4632)***	(5.5608)***	(0.1081)	(2.5786)***	(−15.39)***	(0.4043)	(9.3366)***	(−14.09)***	(0.2381)
Panel B: Multivariate ADCC GARCH fitting
Joint (C-G)BRICS-Gold	Brazil	Russia	India	China			Joint (C-G)G7-Gold	France	Italy	UK	USA
${\alpha }_{C-G}$	0.0070	0.0666	0.0654	0.0390	Gold (G)		${\alpha }_{C-G}$	0.0568	0.0468	0.062751	0.049795	Gold (G)
	(0.2384)	(0.4917)	(1.3187)	(0.1468)				(0.8372)	(0.5913)	(1.4328)	(0.6728)
${\beta }_{C-G}$	0.9856	0.8293	0.5426	0.8872			${\beta }_{C-G}$	0.8999	0.9170	0.897951	0.908099
	(8.8665)***	(8.871)***	(2.1103)**	(3.4297)***				(2.1049)**	(3.2944)***	(2.4572)**	(2.5911)***
$\xi$	0.0000	0.0000	0.0000	0.0224			$\xi$	0.0000	0.0000	0.0000	0.0000
	(0.0000)	(0.0000)	(0.0000)	(0.0586)				(0.0000)	(0.0000)	(0.0000)	(0.0000)
$\nu$	7.2454	5.4238	7.2507	6.7468			$\nu$	6.0514	6.9733	5.178503	5.535973
	(0.4426)	(0.5899)	(5.925)***	(0.3579)				(0.3425)	(0.3313)	(0.3524)	(0.2471)

This table provides information about the estimation of the different bivariate ADCC specifications implemented to model the dependences between each MSCI index and Gold over the in-sample period (January-December 2020) for the aggregate data series (log-returns). The parameters relative to the univariate volatility processes are reported in Panel A, while those that describe the persistence and asymmetry of the different structures of dependence are reported in Panel B. Lagrange-Multiplier (LM) statistics computed under different lags are reported to assess persistence in second orders. Note that *, **, and *** indicate significance at the 10%, 5%, and 1% levels, respectively.

Table 4.

GARCH in-sample estimates for wavelet returns decompositions.

	Panel A: Univariate distribution parameters						Panel B: Multivariate distribution parameters
d1 = 2–4 days	${\mu }_C$	${\varphi }_C$	${\omega }_C$	${\alpha }_C$	${\beta }_C$	${\gamma }_C$	${\alpha }_{C-G}$	${\beta }_{C-G}$	$\xi$	$\nu$
BRICS Countries
Brazil	−0.0903	−0.2497	0.0439	0.1064	0.2636	1.0000	0.0027	0.4367	0.1808	13.3405
(GJRGARCH)	(−2.6186)***	(−2.5355)**	(2.372)**	(0.6179)	(1.4717)	(1.7656)*	(0.1939)	(4.3937)***	(1.862)*	(3.7902)***
Russia	0.0000	−0.6268	0.0000	0.2690	0.4277		0.1926	0.4609	0.0059	18.6072
(GARCH)	(−0.3445)	(−18.6272)***	(46.9873)***	(6.7093)***	(7.7481)***		(3.7709)***	(5.0309)***	(0.0665)	(2.8234)***
India	−0.0001	−0.5473	0.0000	0.2065	0.6573		0.1233	0.4921	0.1834	17.2494
(GARCH)	(−0.5828)	(−15.0737)***	(8.6938)***	(6.5509)***	(14.1823)***		(1.6566)*	(5.7018)***	(1.344)	(2.7806)***
China	0.0000	−0.5412	0.0000	0.1985	0.6258		0.1772	0.5820	0.0933	29.9833
(GARCH)	(−0.1492)	(−14.8686)***	(7.7247)***	(3.7797)***	(11.2646)***		(2.9827)***	(6.8029)***	(1.1278)	(1.8398)*
G7 Countries
France	0.0000	−0.6249	0.0000	0.1691	0.7409		0.1233	0.5624	0.1481	18.8517
(GARCH)	(−0.2646)	(−18.6323)***	(2.7924)***	(6.0231)***	(18.1784)***		(2.5245)**	(7.0521)***	(1.7653)*	(2.6798)***
Italy	0.0000	−0.5995	0.0000	0.2939	0.5540		0.1444	0.4319	0.1865	10.5573
(GARCH)	(−0.1759)	(−16.8862)***	(12.5764)***	(5.4104)***	(9.8564)***		(2.8734)***	(5.28)***	(2.0255)**	(4.5697)***
UK	−0.0001	−0.6197	0.0000	0.2847	0.6155		0.1561	0.5775	0.1311	18.5713
(GARCH)	(−0.6576)	(−18.5673)***	(2.1274)**	(4.8217)***	(10.2707)***		(3.1576)***	(7.8316)***	(1.4051)	(2.524)**
USA	0.0000	−0.6210	0.0000	0.2303	0.5965		0.1760	0.5673	0.0864	11.4487
(GARCH)	(−0.3792)	(−19.6373)***	(8.2767)***	(6.1703)***	(10.6037)***		(3.2445)***	(7.1715)***	(1.0412)	(3.8749)***
d3 = 8–16 days	${\mu }_C$	${\varphi }_C$	${\omega }_C$	${\alpha }_C$	${\beta }_C$	${\gamma }_C$	${\alpha }_{C-G}$	${\beta }_{C-G}$	$\xi$	$\nu$
BRICS Countries
Brazil	−0.0008	−0.2977	0.0001	0.3935	0.1674	0.2379	0.0000	0.9830	0.0000	25.9061
(GJRGARCH)	(−0.5071)	(−5.48)***	(7.8817)***	(3.0504)***	(3.3712)***	(0.7036)	(0.0000)	(26.0892)***	(0.0000)	(1.7838)*
Russia	−0.0001	0.3235	0.0000	0.2336	0.6710		0.2162	0.5118	0.0737	49.9966
(GARCH)	(−0.4045)	(8.3868)***	(0.1413)	(1.6811)*	(2.5401)**		(4.6217)***	(10.1436)***	(0.76)	(2.1239)**
India	0.0000	0.3507	0.0000	0.2909	0.6043		0.2310	0.5050	0.2672	31.4340
(GARCH)	(0.1605)	(9.3257)***	(2.9325)***	(7.3048)***	(12.7107)***		(3.4866)***	(12.6904)***	(2.2106)**	(1.3518)
China	0.0001	0.2708	0.0000	0.3210	0.5972		0.2775	0.5670	0.1274	49.9996
(GARCH)	(0.1833)	(7.3925)***	(3.6954)***	(7.7889)***	(11.9193)***		(6.2213)***	(14.3672)***	(1.286)	(1.8508)*
G7 Countries
France	−0.0001	0.3119	0.0000	0.3066	0.6335		0.2386	0.4959	0.0750	31.0928
(GARCH)	(−0.3329)	(8.2905)***	(0.6804)	(4.9175)***	(10.3328)***		(5.5939)***	(12.2357)***	(0.8278)	(1.4635)
Italy	−0.0001	0.2833	0.0000	0.3346	0.6081		0.2560	0.5437	0.0000	28.4533
(GARCH)	(−0.5445)	(5.0297)***	(0.1711)	(2.2244)**	(2.1144)**		(5.1635)***	(12.8303)***	(0.0000)	(1.5531)
UK	−0.0002	0.3708	−1.9420	−0.0648	0.8261	0.5838	0.1988	0.6072	0.0000	24.0631
(EGARCH)	(−0.6522)	(10.5988)***	(−5.3521)***	(−1.0175)	(25.1751)***	(6.0297)***	(3.551)***	(10.717)***	(0.0000)	(1.8561)*
USA	−0.0001	0.3448	−1.7135	−0.0128	0.8473	0.5619	0.2083	0.5507	0.1579	15.0975
(EGARCH)	(−0.3715)	(9.7788)***	(−5.2242)***	(−0.1987)	(29.0059)***	(6.7435)***	(4.4273)***	(7.9928)***	(1.5439)	(2.5264)**
d6 = 64–128 days	${\mu }_C$	${\varphi }_C$	${\omega }_C$	${\alpha }_C$	${\beta }_C$	${\gamma }_C$	${\alpha }_{C-G}$	${\beta }_{C-G}$	$\xi$	$\nu$
BRICS Countries
Brazil	−0.0095	0.9731	−0.3854	0.0585	0.9762	2.2198	0.5623	0.3959	0.0324	50.0000
(EGARCH)	(−2.5732)**	(87.0524)***	(−4.2098)***	(0.6297)	(131.4491)***	(6.4101)***	(11.2794)***	(7.0058)***	(0.8425)	(6.8684)***
Russia	0.0020	0.9831	−7.8523	0.0622	0.5926	1.9665	0.7662	0.1480	0.0584	15.1074
(EGARCH)	(3.1621)***	(48.0129)***	(−1.014)	(0.8238)	(1.4394)	(2.9579)***	(18.2765)***	(3.779)***	(1.1124)	(1.5956)
India	−0.0034	1.0000	−7.5995	−0.0171	0.6040	1.9623	0.8354	0.0573	0.0000	50.0000
(EGARCH)	(−118.2775)***	(246.0305)***	(−7.8743)***	(−0.2561)	(11.343)***	(19.3033)***	(15.7494)***	(1.239)	(0.0000)	(5.0748)***
China	−0.0060	0.9966	−6.0660	0.1357	0.6699	1.8662	0.8157	0.0756	0.0950	17.8968
(EGARCH)	(−0.5055)	(332.2116)***	(−2.7832)***	(0.7109)	(5.5584)***	(7.8997)***	(24.3479)***	(1.9688)**	(2.2326)**	(0.8087)
G7 Countries
France	−0.0026	0.9990	−7.3367	0.1633	0.6175	1.8388	0.7462	0.1615	0.0378	14.8516
(EGARCH)	(−0.3329)	(160.7312)***	(−5.4545)***	(1.906)*	(8.5376)***	(11.4624)***	(16.2177)***	(3.0014)***	(0.8004)	(1.5661)
Italy	−0.0102	0.9995	−4.4547	0.0367	0.7665	1.8335	0.7171	0.2228	0.0000	50.00
(EGARCH)	(−190.6373)***	(341.1936)***	(−3.412)***	(0.796)	(10.8386)***	(9.0761)***	(7.6198)***	(0.6604)	(0.0000)	(2.1863)**
UK	−0.0067	0.9987	−6.2550	0.0811	0.6825	2.0262	0.6602	0.2738	0.0262	25.0863
(EGARCH)	(−3.6064)***	(120.2778)***	(−1.7648)*	(0.4826)	(3.604)***	(9.101)***	(19.4259)***	(3.0481)***	(0.6828)	(0.4477)
USA	−0.0080	0.9991	−4.1396	0.0369	0.7931	1.8346	0.6561	0.2700	0.0225	50.00
(EGARCH)	(−158.4261)***	(259.5462)***	(−1.2891)	(0.709)	(4.7657)***	(3.5681)***	(8.9373)***	(4.7098)***	(0.2379)	(9.0311)***

This table provides information about the estimation of the different bivariate ADCC specifications implemented to model the dependences between each MSCI index and Gold over the in-sample period (January-December 2020) for different wavelet decompositions (d1 or short-run, d3 or mid-run and d6 or long-run). The parameters relative to the univariate volatility processes are reported in Panel A, while those that describe the persistence and asymmetry of the different structures of dependence are reported in Panel B. Lagrange-Multiplier (LM) statistics computed under different lags are reported to assess persistence in second orders. Note that *, **, and *** indicate significance at the 10%, 5%, and 1% levels, respectively.

From Table 3 we draw the general conclusion that the univariate modeling of Gold as well as of the different MSCI indices using GARCH, E-GARCH and GJR-GARCH models is pertinent and necessary, since the Lagrange Multiplier test (LM) for the case of squared residuals and 5 lags confirms the null significance (we do not reject “H₀ = No time dependence” in any case). Thus, the calibration of the parameters and the choice of the suitable models has succeeded in diluting the time dependence in the univariate variance processes. Interestingly, and similar to Chen and Kuan (2002), negative or very low values of $\omega$ and $\alpha$ (in the range −1.2, −0.2) for the E-GARCH model estimates indicate that these parameters have a little influence on the univariate conditional heteroscedasticity processes. In contrast, the persistence driven by $\beta$ increases their influence exponentially. Notwithstanding, univariate $\alpha$ parameters are overall fairly significant, while the multivariate ones lack such significance, implying that variance processes adequately capture new information shocks, while in the multivariate case, this parameter has little influence in explaining the inter-asset dependence process. In contrast, for the $\beta$ parameters we report high significance at both the univariate and multivariate levels. The leverage effects (in the range −0.08, 0.17) of the E-GARCH and GJR-GARCH models are quite significant, which justifies the use of these models and the inclusion of these parameters in the variance equation.12

Regarding Table 3 and Table 4, we find that most of the parameters that characterize the univariate distributions of each of the country return decompositions are highly significant, which evidences the high importance of calibrating these parameters and their inclusion in the analyzed models. It is worth pointing out certain variations among parameters for the different domains in contrast to the log-returns format. On the one hand, regarding the mean equation of the different univariate processes, we find that the returns decompositions are highly and significantly dependent on their most recent past, ${\varphi }_C$, form most returns and decompositions. On the other hand, especially interesting is the case of the influence of new information on volatility processes, ${\alpha }_C$, a parameter that has increased its significance notably at high and mid frequencies, but not as much at low frequencies or long-term domains -this surge is understood regarding the continuous returns-. On the other hand, ${\beta }_C$ keeps similar levels of significance to those found in the format of log-returns without decomposition. This parameterization indicates the lower influence of new information shocks, but a higher persistence of their effects in the univariate processes of the G7 in contrast to the BRICS, for the short and mid-term decompositions. The opposite occurs at low frequency decompositions, the shocks persist more in the BRICS processes.

At the multivariate level and once again making a side-by-side comparison of Table 3, Table 4, we reveal a higher significance of the different parameters that characterize the ADCC dependence structure of the decompositions with respect to the logarithmic returns. The new information shocks, ${\alpha }_{C-G}$, as well as the shape parameter, $\nu$, exponentially increase their significance, while the skew parameter, $\xi$, only enhances its significance in a few specific cases. Besides, we find that in general the persistence in the correlation process of each country with Gold, ${\beta }_{C-G}$, remains constant or increases slightly at short-mid horizons, while at low frequencies the initial effect of new information on the dependence process becomes stronger and stronger, and at the same time dilutes more rapidly over time. Additionally, our results for the dependence structure highlight the sharp asymmetric and bumpy nature of the different log-return decompositions, as opposed to the smoother trends described by the correlation processes of the undecomposed returns. This is evidenced by the high values of the shape parameter, growing exponentially, and reaching extreme values as we lower the frequency of the decomposition. Overall, the correlation process of the BRICS with Gold exhibits more jumps in the short and mid-term decompositions, while the opposite occurs at longer horizons.

5.2. Out-of-Sample experiment

The out-of-sample experiment is conducted in four steps. First, we re-estimate the different wavelet-GARCH parameters over time and domains. In the second step, we investigate the time-varying correlations between decomposed series at the various frequencies under study. Third, a minimum VaR portfolio rebalancing experiment is conducted on the basis of the forecasted correlation dynamics. Lastly, we assess the risk-return relations reported by the different strategies, highlighting the contrast between single stock indices and composed strategies for the different frequencies and domains.

5.2.1. MODWT-ADCC dynamic re-estimation and forecasting

Fig. 2 shows the process of out-of-sample parameter re-estimation. The first set of parameters (${\mathrm{\Omega }}_{\mathit{ij},1}={\varphi }_{i,1},{\mu }_{i,1},{\omega }_{i,1},{\alpha }_{i,1},{\beta }_{i,1},{\gamma }_{i,1},{\varphi }_{j,1},{\mu }_{j,1},{\omega }_{j,1},{\alpha }_{j,1},{\beta }_{j,1},{\gamma }_{j,1},{\alpha }_{\mathit{ij},1},{\beta }_{\mathit{ij},1},{\xi }_{\mathit{ij},1},{\nu }_{\mathit{ij},1}$), i.e. the one that corresponds to January 2020, is estimated regarding the data that belongs to the in-sample timeframe (January 2018 to December 2019, 522 observations). Then, we recursively re-estimate the different univariate and multivariate GARCH models via a rolling window procedure with associated constant window size of 522 observations, such that we obtain a vector of parameter estimates throughout the out-of-sample period (11 re-estimates).13 Finally, we reach a time-varying vector for each of the univariate and multivariate GARCH estimated parameters (see Equations from 14 to 19) of $12x1$.

Fig. 2.

This figure explains the univariate and multivariate re-estimation processes. Each model is initially estimated for the in-sample period of 522 workdays (January 2018 – December 2019). ${\mathrm{\Omega }}_1$ is the 1st set of univariate GARCH (1,1) and multivariate ADCC-GARCH estimated parameters for the daily log-returns of each Country-Gold pair, C-G. Each model is re-estimated monthly using a rolling-window of 522 workdays (window size, ws, is always 522 observations) during the out-of-sample period (January 2020 – December 2020) to calibrate the stated parameters, being ${\mathrm{\Omega }}_m$ the set of parameters for the pair C-G fitted at month m. A time series of 12 observations (m = 1, 2…, 12) with respect to each parameter is calibrated. Regarding the conditional dependences, we conduct an additional one-day ahead forecast over the subsequent 21 days of the dynamic correlations from each re-estimated parameter, obtaining time-varying series of 262 observations for the correlations and hence, the covariances arrays.

From Fig. 3 we assess the evolution of the parameters characterizing the conditional dependence structure between the different MSCI indices and Gold.14 Overall, the time-varying patterns described by the different country-Gold pairs are more homogeneous for the G7 countries. Conversely, most of the parameters characterizing the multivariate conditional distribution of the BRICS-Gold pairs reveal very disparate trends, making it extremely complex to find a common pattern for these markets. The parameters defining the intensity of new information shocks on the correlation process between countries and Gold are more variable for BRICS pairs than for G7 pairs. Except for the case of Brazil, which stands apart from the rest, alphas are lower in the BRICS for high frequencies, reversing the above and exhibiting higher impacts in the BRICS for mid and low frequencies. The parameters measuring persistence in the dependency process are less stable in the BRICS than in the G7 countries. We find a higher beta in the BRICS-Gold dependence processes for high frequencies. The opposite occurs for lower frequencies, in G7-Gold pairs the effects of new information shocks take longer to disappear in conditional correlation processes. The parameters controlling the asymmetry of dependence processes are unstable for both BRICS and G7 countries for most of the frequencies analyzed. Interestingly, this parameter becomes more stable for lower frequencies, the so-called long-run, implying that the series present more symmetric patterns in their long-run decompositions. The shape is overall more stable for the G7-Gold pairs, implying that the largest number of jumps occurs for the BRICS-Gold pairs for the different frequencies under study. For high frequencies, a decreasing trend is found, whereas for the mid and long term these parameters jump to extreme values.

Fig. 3.

This figure shows the calibrated parameters of the multivariate ADCC model. We re-estimate the model over the out-of-sample period to calibrate the stated parameters from the last 522 observations of daily returns, obtaining a time series of 12 observations with regard to each parameter. Panel A depicts the trend of the rolling re-estimated parameters of the ADCC model with regard to the aggregate data (log-returns). Rest of the Panels (from B to D) reports on the different multivariate re-estimated parameters for the various wavelet scales under study. First row of each Panel shows those parameters related to the persistence in the dependence structure ($\alpha ,\beta$) for the BRICS and G7 countries, respectively. Second row of each Panel displays the parameters that capture the skewed behavior of the correlation process among MSCI-Gold pairs ($\xi ,\nu$) for the BRICS and G7 countries, respectively. Regarding the BRICS, cyan blue is chosen to describe the temporal evolution of the parameters for Brazil, purple for Russia, black for India and orange for China. With respect to the G7 countries, maroon color describes the trend of France parameters, green is associated to Italy, grey to UK and dark blue to USA. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

5.2.2. Conditional correlation analysis

From the parameters re-estimated in the previous section (see Fig. 2 and Fig. 3), we perform a one-day ahead forecast of correlations over the subsequent 22 days, just up to the following parameter recalibration. This process is repeated until the end of the out-of-sample period, obtaining a predicted sample of correlations or asymmetric dynamic dependencies of 262 working days.

From Fig. 4 and Table 5 , we report on the forecasted dynamic conditional correlations pairs among the various MSCI indices and Gold for different return cases and wavelet decompositions: log-returns, d1 (short-term horizon), d3 (middle-run) and d6 (long-run).15 ^, 16 On contrasting the different timescales, we find some degree of homogeneity between the patterns described by the G7-Gold correlation pairs, whereas BRICS fail to exhibit such attribute. Overall, we report that the ADCC structure of dependence among Gold and MSCI equity indices for log-differences and different timescales, dj, increases considerably as the timescale or frequency of shocks decreases. Additionally, the correlation trend moves in clusters for lower timescales. On assessing the correlation between log-returns, d1, d3 and d6, a reduction in the potential diversification properties of Gold seems to be suggested with the decrease in frequency (especially when we approach the long-run, d6). This declining pattern is understood in terms of increased portfolio volatility (risk exposure), as the risk-return ratios could report just the opposite. To truly assess the direction of these effects, we need to go further a portfolio rebalancing and performance experiment as the one proposed in Sections 5.2.3. and 5.2.4., respectively.

Fig. 4.

This figure represents the out-of-sample forecasted conditional correlations among MSCI-Gold pairs in terms of unified-dollar currency and over the period that spans from January 2020 to December 2020. Panel A shows the time-varying correlations corresponding to the set of assets in the log-returns form. Panel B displays the conditional correlations for the different pairs in the d1 decomposition form (short run). Panel C reports on the dynamic correlations for the case of the d3 decomposition. Panel D plots the time-varying structure of dependence for the long run (d6). Whole Panels are divided into two subsections: the left subsection depicts the dynamic relation between BRICS MSCI indices and Gold, while the right one shows the dependence among G7 MSCI indices and Gold. Regarding the BRICS, cyan blue is chosen to describe the temporal evolution of the correlations for Brazil, purple for Russia, black for India and orange for China. With respect to the G7 countries, maroon color describes the trend of France correlations, green is associated to Italy, grey to UK and dark blue to USA. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Table 5.

Conditional correlations. Out-of-sample descriptive statistics.

	Mean	Std. dev.	Max.	Min.	Q1	Q2	Q3		Mean	Std. dev.	Max.	Min.	Q1	Q2	Q3
Panel A: log-returns
A.1. BRICS-Gold ADCC								A.2. G7-Gold ADCC
Brazil	0.0394	0.0801	0.2434	−0.1881	−0.0119	0.0433	0.1000	France	0.0166	0.2084	0.4916	−0.5277	−0.1051	0.0221	0.1910
China	0.0530	0.1239	0.4793	−0.2749	−0.0165	0.0498	0.1360	Italy	0.0240	0.1756	0.3997	−0.4642	−0.0837	0.0178	0.1622
India	0.0155	0.0809	0.6290	−0.3601	−0.0176	−0.0083	0.0530	UK	0.0134	0.2090	0.4883	−0.5769	−0.1092	0.0385	0.1728
Russia	0.0794	0.1382	0.4638	−0.2936	−0.0167	0.0929	0.1833	USA	0.0090	0.1557	0.3039	−0.4837	−0.1218	0.0326	0.1399
Panel B: d1 = 2–4 days
B.1. BRICS-Gold ADCC								B.2. G7-Gold ADCC
Brazil	−0.0033	0.0505	0.2394	−0.0819	−0.0445	−0.0097	0.0383	France	0.4352	0.1873	0.8948	−0.1216	0.3258	0.4511	0.5664
China	0.0059	0.0596	0.2434	−0.1608	−0.0355	−0.0037	0.0395	Italy	0.3686	0.1733	0.8383	−0.1993	0.2694	0.3693	0.4790
India	0.4588	0.1082	0.8174	0.1884	0.3900	0.4520	0.5163	UK	0.4658	0.1734	0.9014	−0.1054	0.3742	0.4848	0.5808
Russia	0.1465	0.1958	0.7597	−0.2979	0.0206	0.1339	0.2593	USA	0.4169	0.1835	0.8784	−0.1083	0.3084	0.4276	0.5359
Panel C: d3 = 8–16 days
C.1. BRICS-Gold ADCC								C.2. G7-Gold ADCC
Brazil	−0.0122	0.0217	0.0194	−0.0744	−0.0191	−0.0103	0.0013	France	0.6207	0.2092	0.9047	−0.1483	0.5072	0.6740	0.7865
China	0.4045	0.4015	0.9362	−0.6644	0.1544	0.4914	0.7304	Italy	0.4733	0.2782	0.9135	−0.4574	0.2901	0.5255	0.6959
India	0.5683	0.2710	0.9045	−0.2969	0.4266	0.6581	0.7681	UK	0.6539	0.2762	0.9599	−0.2745	0.5155	0.7685	0.8635
Russia	0.3537	0.3577	0.8408	−0.6508	0.1463	0.4166	0.6403	USA	0.6293	0.3045	0.9655	−0.5795	0.4805	0.7387	0.8671
Panel D: d6 = 64–128 days
D.1. BRICS-Gold ADCC								D.2. G7-Gold ADCC
Brazil	0.1404	0.3127	0.8992	−0.7439	−0.0103	0.1597	0.3781	France	0.3278	0.7645	0.9727	−0.9199	−0.6259	0.8692	0.9321
China	0.3243	0.7814	0.9729	−0.9351	−0.7239	0.8547	0.9253	Italy	0.4392	0.7126	0.9669	−0.8956	−0.2166	0.8986	0.9403
India	0.5909	0.6162	0.9836	−0.8421	0.6080	0.9396	0.9682	UK	0.7278	0.4880	0.9879	−0.7666	0.8520	0.9499	0.9677
Russia	0.3278	0.7645	0.9727	−0.9199	−0.6259	0.8692	0.9321	USA	0.5924	0.5980	0.9863	−0.8334	0.5124	0.9385	0.9621

This table shows a variety of descriptive statistics of the time evolution described by correlation pairs over the log-returns and the different wavelet frequencies under analysis and for the out-of-sample period. Panel A shows the statistics regarding the time-varying correlations corresponding to the log-returns form. Panel B displays the statistics for the different pairs in the d1 decomposition form (short run). Panel C reports on the correlation statistics for the case of the d3 decomposition. Panel D describes the structure of dependence for the long run (d6). Whole Panels are divided into two subsections: the left subsection depicts the dynamic relation between BRICS MSCI indices and Gold, while the right one shows the dependence among G7 MSCI indices and Gold. We report on the mean average correlation (Mean), the standard deviation of the correlation processes (Std. dev.), the maximum and minimum levels (Max. and Min.) and different quartiles, $Q_j$.

Overall, from Fig. 4 we find that Brazil's index performs very differently from the rest of the MSCI indices in terms of correlation. In fact, it displays a very stable and generally low or negative dependence structure with Gold throughout the different wavelet decompositions studied. This may be due to the low importance attached by Brazil to Gold investment and the scarce reserves it has of this precious metal (according to the World Gold Council, 2021, Brazil ranks 51st in world Gold reserves), which may be the reason why the country's economy and its stock markets are residually conditioned by it and therefore, reported very disparate trends with respect to the correlations of the rest of the countries with Gold.

Furthermore, from Fig. 4 and Table 5 we find mixed evidence regarding the variability of the dependence structure of the different countries and Gold depending on the wavelet scale. From Panel A (log-differences), we note that correlations between the different BRICS countries and Gold seem to be more stable than those of the G7-Gold pairs. Notwithstanding, the former dynamic correlation coefficients are higher on average, which may suggest lower diversification effects of Gold when included in portfolios comprised of BRICS equities. Contrary to Panel A, for the wavelet decomposition of returns driven by the 2–4-day shocks, d1 (Panel B), and except for Russia we show that correlations between BRICS and Gold are quite lower than those of the G7. In fact, these correlation pairs reach values very close to zero or even negative for long intervals over the time horizon, which may suggest diversifying properties of Gold in its short-run format, especially for the case of Brazil-Gold, China-Gold and Russia-Gold. Additionally, we find that the correlations of India-Gold and the global G7-Gold have grown more than twenty-fold with respect to those found in Panel A. In terms of stability, it appears that overall, the variability of BRICS-Gold correlations has increased slightly, while those of G7-Gold have decreased or remained invariant with respect to Panel A.

For the medium-term horizon (Panel C), we observe that the correlation between indices pairs and Gold continues to increase exponentially for most of the cases under study. The only correlation coefficient that seems to remain stable throughout the decompositions is the Brazil-Gold pair, reaching a negative average parameter. On the other hand, the variability of the correlations increases in almost all cases, doubling that of Panel B in some scenarios. As for the mean, in terms of standard deviation it is also the Brazil-Gold pair that keeps or even reduces its variability with respect to previous decompositions of the returns. Finally, it is reported that the wavelet decompositions relative to the long-run, d6 (Panel D), are the ones that displays the greatest distortion and jumps throughout the year 2020. As a proof of this, the highest peaks and troughs of all the cases under study are found in Panel D. This is reflected both in terms of mean and standard deviation. The mean appears very similar to that found for the middle-run (Panel C). In terms of stability, we find the largest oscillations and therefore the largest standard deviation of the correlation series in this case, d6. The biggest differences are undoubtedly found for the Brazil-Gold pair, which goes from being stable in the middle-run to showing a wide variability for the long-run, thus increasing its average correlation ostensibly.

Regarding the dynamic evolution of correlations, it is very complex to interpret a temporal analysis of correlations by frequencies, since the longer the horizon from the short to the long term (from higher to lower frequency shocks), the greater the distortion (see Panels B to D of Fig. 4). Either way, we find that the highest correlations are found around mid-March 2020, especially for developed countries and the frequencies d1 and d3 (short and medium term). Regarding the long term, interpretation is virtually unfeasible due to the continuous jumps. This high instability and distortion lead us to interpret Fig. 4 mostly in terms of log-returns. Thus, from Panel A of Fig. 4, we report on a widespread surge in the dependences among different assets and Gold by mid-March 2020, just around the major collapse of the markets following the first confinements caused by the COVID-19 outbreak. At this date -the first wave of the pandemic-, and similarly to recent studies as Banerjee, 2021, Davidovic, 2021, Guo et al., 2021, So et al., 2021, we also report on a clear contagion effect between geographically very distant markets in times of high instability or recession, as this dependence becomes very strong for both the BRICS and the G7. Subsequently, we find a period of relative calm (from mid-April to June), characterized by a smoothening of the infection curve and an overall decrease in the number of deaths caused by the COVID-19. In this period, a sharp drop in all correlation pairs is described, perhaps related to the recovery of certain economies and the consequent growth in financial markets.

From mid-June onwards, the dependencies become increasingly strong again, reaching correlation peaks similar to those of the first wave (range 0.2–0.5). These new peaks highlight the second wave of the pandemic (August-October), characterized mainly by new outbreaks due to the relaxation of sanitary measures during the summer and the increase in interterritorial mobility. From mid-October to the end of November 2020, we find another period of calm in the financial markets in which a new drop in correlations between pairs is shown. The last bars of the out-of-sample period suggest a new increase in dependencies, motivated by the surge of contagions following the Christmas holidays (beginning of the third wave of the pandemic). Notwithstanding, this new period of instability should be studied in greater depth in future research, considering not only the “Christmas effect” but also the effect of global vaccination.

5.2.3. Time-varying portfolio weights

In this section, different monthly portfolio rebalancing strategies are conducted based on the various wavelet timescales. Specifically, we rebalance our portfolios on the basis of the daily covariance in force at the end of each month. This covariance array varies regarding the considered return decomposition. Thus, four monthly rebalancing strategies are implemented: without decomposition, short-run, mid-run and long-run strategies. The construction of different time-varying portfolio choices is based on three stages. First, all the autoregressive specifications described in previous sections are calibrated over the in-sample period and then re-estimated monthly during the out-of-sample. These re-estimates allow to obtain one-day ahead forecasted dynamic volatilities and correlations ($D_t,{\mathrm{\Gamma }}_t$) over the entire year 2020. Second, the time-varying one-day ahead forecasted daily covariances (${\mathrm{\Sigma }}_t$) are computed as a cross product of the prior volatility and dependence arrays, as described in Eq. (5). Third, as stated in Eq. (4) we compute the weights such that the VaR of the portfolio, which is skewed student’s t distributed, is minimum. For the different monthly rebalancing strategies, the weights are calculated every 22 business days from January 2020 to December 2020, reaching a total of 12 rebalances for each portfolio. The first rebalance corresponds to the beginning of January, while the last occurs in the early December 2020.

From Fig. 5 we reveal that overall, going from the short- to the long-run or low-frequency shock strategies, increasingly higher weightings are found and, therefore, a greater trend to short-sell for certain countries or even Gold. From Panel A, we observe that for the log-returns there is hardly any swapping of positions between countries and Gold, except for China and the USA, suggesting that a monthly rebalancing such as the one carried out, or even others at lower frequencies such as the quarterly rebalancing of the large funds, might be more accurate in this case. Contrary to Panel A, in terms of return decompositions (Panels B to D), we find that as we shift from short-term shocks to long-term shocks, the weights become more unstable, and a greater diversification of portfolios is reached in the different rebalancing times. This greater instability of weights is driven by the large distortion in the forecasted correlations for the long-term wavelet timescales. The above seems to suggest that, despite the incursion of higher transaction costs, it would be interesting to propose rebalancing at a higher frequency for the mid- and long-run wavelet strategies in subsequent studies, in order to get a better fit to new market information and thus enhance the diversification benefits underlying in the autoregressive models.

Fig. 5.

This figure displays the different trends and patterns described by the minimum VaR weights of the various bivariate combinations among MSCI indices (BRICS and G7) and Gold over the year 2020 and for the different wavelet decompositions under study. The different combinations among pairs for the different wavelet scales are clearly divided into various panels: Panel A describes the combinations for the aggregate data (log-returns), Panel B represents the short-run decomposition weights, Panel C the mid-run and Panel D the long-run weights combinations over the time horizon. The left subplot of each Panel corresponds to the BRICS countries, while the right one to the G7. Regarding the BRICS, cyan blue is chosen to describe the temporal evolution of the correlations for Brazil, purple for Russia, black for India and orange for China. With respect to the G7 countries, maroon color describes the trend of France correlations, green is associated to Italy, grey to UK and dark blue to USA. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Interestingly, Table 6 reveals that it is in the intermediate or mid-run frequencies where we find the greatest diversification between the different pairs of countries and Gold. In this case we report on the highest weightings for the different countries (see Panel C of Table 6), constituting combinations very close to the equal-weighted portfolio, and in many cases exceeding 50% weighting. From Table 6 we also deduce that the lowest country weights are found in the short-run (Panel B) and long-run (Panel D), but with a clear difference. The long-run is where we find the biggest jumps, the most extreme weights and the highest weighting variability, whereas in the timescale corresponding to the shocks between 2 and 4 days, all the weights drop ostensibly compared to what is reported for the log-returns rebalancing. Regarding the shocks between 64 and 128 days, we evidence the propensity for extreme weightings and the abuse of short positions, since the global maximums and minimums are found in this type of strategies. Additionally, the lowest and even very negative weightings for countries -both BRICS and G7- are reached at these low frequencies of shocks. Similarly, Gold reaches the highest weightings, exceeding 100% in many cases. Table 6 also shows that for the case of the weights constructed on the basis of log-returns (Panel A), we find the lowest instability in weighting. In this case, we observe that the highest weights are allocated in the combinations among G7 countries and Gold (around 30–70%).

Table 6.

Average portfolio weights.

Panel A: log-returns
Stat.	A.1. BRICS-Gold ADCC								A.2. G7-Gold ADCC
	Brazil + Gold		Russia + Gold		India + Gold		China + Gold		France + Gold		Italy + Gold		UK + Gold		USA + Gold
	Brazil	Gold	Russia	Gold	India	Gold	China	Gold	France	Gold	Italy	Gold	UK	Gold	USA	Gold
Mean	0.161	0.839	0.232	0.768	0.360	0.640	0.398	0.602	0.366	0.634	0.321	0.679	0.328	0.672	0.455	0.545
Std. dev.	0.088	0.088	0.118	0.118	0.156	0.156	0.129	0.129	0.192	0.192	0.152	0.152	0.155	0.155	0.256	0.256
Max.	0.348	0.932	0.427	0.994	0.676	0.921	0.559	0.824	0.670	0.928	0.591	0.977	0.551	0.907	0.819	0.882
Min.	0.068	0.652	0.006	0.573	0.079	0.324	0.176	0.441	0.072	0.330	0.023	0.409	0.093	0.449	0.118	0.181
Q1	0.108	0.796	0.187	0.709	0.310	0.588	0.312	0.514	0.261	0.481	0.244	0.586	0.238	0.580	0.244	0.301
Q2	0.125	0.875	0.235	0.765	0.363	0.637	0.432	0.568	0.327	0.673	0.298	0.702	0.318	0.682	0.409	0.591
Q3	0.204	0.892	0.291	0.813	0.412	0.690	0.486	0.688	0.519	0.739	0.414	0.756	0.420	0.762	0.699	0.756
Panel B: d1 = 2–4 days
Stat.	B.1. BRICS-Gold ADCC								B.2. G7-Gold ADCC
	Brazil + Gold		Russia + Gold		India + Gold		China + Gold		France + Gold		Italy + Gold		UK + Gold		USA + Gold
Brazil	Gold	Russia	Gold	India	Gold	China	Gold	France	Gold	Italy	Gold	UK	Gold	USA	Gold
Mean	0.001	0.999	0.358	0.642	0.414	0.586	0.001	0.999	0.219	0.781	0.258	0.742	0.293	0.707	0.291	0.709
Std. dev.	0.002	0.002	0.186	0.186	0.389	0.389	0.003	0.003	0.269	0.269	0.247	0.247	0.238	0.238	0.279	0.279
Max.	0.008	1.001	0.779	0.871	0.951	1.026	0.009	1.002	0.772	1.279	0.725	1.007	0.714	1.037	1.032	1.019
Min.	−0.001	0.992	0.129	0.221	−0.026	0.049	−0.002	0.991	−0.279	0.228	−0.007	0.275	−0.037	0.286	−0.019	−0.032
Q1	0.000	0.999	0.216	0.511	0.035	0.208	0.000	0.999	0.024	0.651	0.065	0.555	0.172	0.559	0.159	0.659
Q2	0.000	1.000	0.291	0.709	0.303	0.697	0.000	1.000	0.266	0.734	0.179	0.821	0.253	0.747	0.226	0.774
Q3	0.001	1.000	0.489	0.784	0.792	0.965	0.001	1.000	0.349	0.976	0.445	0.935	0.441	0.828	0.341	0.841
Panel C: d3 = 8–16 days
Stat.	C.1. BRICS-Gold ADCC								C.2. G7-Gold ADCC
	Brazil + Gold		Russia + Gold		India + Gold		China + Gold		France + Gold		Italy + Gold		UK + Gold		USA + Gold
Brazil	Gold	Russia	Gold	India	Gold	China	Gold	France	Gold	Italy	Gold	UK	Gold	USA	Gold
Mean	0.283	0.717	0.531	0.469	0.514	0.486	0.069	0.931	0.387	0.613	0.337	0.663	0.581	0.419	0.601	0.399
Std. dev.	0.412	0.412	0.353	0.353	0.485	0.485	0.399	0.399	0.388	0.388	0.360	0.360	0.695	0.695	0.586	0.586
Max.	1.000	0.999	1.111	0.927	1.203	1.261	0.780	1.490	0.888	1.400	0.929	1.316	1.827	1.446	1.517	1.417
Min.	0.001	0.000	0.073	−0.111	−0.261	−0.203	−0.490	0.220	−0.400	0.112	−0.316	0.071	−0.446	−0.827	−0.417	−0.517
Q1	0.009	0.626	0.278	0.336	0.160	0.061	−0.223	0.709	0.133	0.336	0.063	0.431	0.128	0.081	0.204	0.006
Q2	0.058	0.942	0.486	0.514	0.435	0.565	0.125	0.875	0.507	0.493	0.430	0.570	0.375	0.625	0.589	0.411
Q3	0.374	0.991	0.664	0.722	0.939	0.840	0.291	1.223	0.664	0.867	0.569	0.937	0.919	0.872	0.994	0.796
Panel D: d6 = 64–128 days
Stat.	D.1. BRICS-Gold ADCC								D.2. G7-Gold ADCC
	Brazil + Gold		Russia + Gold		India + Gold		China + Gold		France + Gold		Italy + Gold		UK + Gold		USA + Gold
Brazil	Gold	Russia	Gold	India	Gold	China	Gold	France	Gold	Italy	Gold	UK	Gold	USA	Gold
Mean	−0.005	1.005	0.035	0.965	−0.192	1.192	−0.123	1.123	−0.140	1.140	0.220	0.780	0.340	0.660	0.253	0.747
Std. dev.	0.024	0.024	0.635	0.635	0.872	0.872	0.489	0.489	0.840	0.840	0.622	0.622	1.313	1.313	1.231	1.231
Max.	0.018	1.060	1.222	1.975	1.680	2.360	0.561	1.830	1.215	2.318	1.041	1.933	2.109	3.182	1.795	3.303
Min.	−0.060	0.982	−0.975	−0.222	−1.360	−0.680	−0.830	0.439	−1.318	−0.215	−0.933	−0.041	−2.182	−1.109	−2.303	−0.795
Q1	0.000	0.995	−0.609	0.589	−0.795	0.758	−0.590	0.708	−0.730	0.523	−0.290	0.427	−0.584	−0.180	−0.197	0.013
Q2	0.000	1.000	0.121	0.879	−0.323	1.323	−0.150	1.150	−0.167	1.167	0.433	0.567	0.803	0.197	0.570	0.430
Q3	0.005	1.000	0.411	1.609	0.242	1.795	0.292	1.590	0.477	1.730	0.573	1.290	1.180	1.584	0.987	1.197

This table shows a variety of descriptive statistics for the time-varying weights over the out-of-sample period. Panel A shows the statistics regarding the portfolio rebalancing corresponding to the information extracted from log-returns. Panel B displays the statistics for the different pairs in the d1 decomposition form (short run). Panel C reports on the weighting statistics for the case of the d3 decomposition. Panel D describes the portfolio rebalancing techniques for the long run (d6). Whole Panels are divided into two subsections: the left subsection depicts the dynamic relation between BRICS MSCI indices and Gold, while the right one shows the potential combinations among G7 MSCI indices and Gold. We report on the mean average weights (Mean), the standard deviation of the weights (Std. dev.), the maximum and minimum levels (Max. and Min.) and different quartiles, $Q_j$.

5.2.4. Performance and diversification assessment

From the optimal portfolio weights rebalanced over the out-of-sample period (see Section 5.2.3.), the procedure to assess the ex-post performance of the various wavelet decomposition strategies during 2020 is fairly simple. Similar to González-Pedraz et al. (2015), we calculate the portfolio return from holding that position until the immediately upcoming working day. The same applies along the entire out-of-sample timeframe but considering that the weights to be used change every 22 working days. Subsequently, once we have obtained all the series of portfolio returns calculated at one point in time after portfolio formation (see Eq. (11)), we compute the different performance ratios for the year 2020 (see Eqs. (9), (10)), and annualize them in a straightforward manner multiplying by the pertinent factors ($252$ for the mean returns, and $\sqrt{252}$ for the standard deviations). Besides, the non-seasonally adjusted 3-Month Treasury Bills are considered as a risk-free asset and the threshold of the Kappa indices is set to zero.

From Table 7 , we assess which portfolio rebalancing technique provides the best risk-return ratios over the out-of-sample timeframe by way of examining the ex-post realized returns of the daily rebalanced portfolios and MSCI indices. The performance of each combined strategy is compared with this obtained by the single MSCI index. The analysis is organized into four main panels regarding each of the different portfolio strategies: log-returns, d1, d3 and d6. Additionally, it should be mentioned that in terms of risk and performance for the different strategies under analysis, no clear pattern is found that evidences significant differences between the BRICS and the G7.

Table 7.

Performance measures for passive (single MSCI indices) and active management (combined strategies MSCI-Gold) in dollars for 2020.

Panel A: log-returns
	A.1. BRICS-Gold ADCC								A.2. G7-Gold ADCC
Stat.	Brazil	Brazil + Gold	Russia	Russia + Gold	India	India + Gold	China	China + Gold	France	France + Gold	Italy	Italy + Gold	UK	UK + Gold	USA	USA + Gold
$E_{{(r}_p)}$	−0.1983	0.0878	−0.11974	0.104035	0.1228	0.2398	0.2450	0.2216	0.0423	0.1395	0.0130	0.1131	−0.1124	0.0630	0.1831	0.1566
${\sigma }_{{(r}_p)}$.	0.5717	0.2068	0.401528	0.202545	0.3379	0.1786	0.2437	0.1810	0.3421	0.1968	0.3741	0.2008	0.3365	0.1922	0.3482	0.1942
${\tau }_{{(r}_p)}$	−1.2813	−0.2231	−0.9187	0.01889	−1.9281	−0.0991	−0.5252	−0.1212	−1.3729	−0.1084	−2.9599	−0.4212	−0.9759	0.0377	−0.9264	0.0449
$k_{{(r}_p)}$	11.2464	6.7645	8.303628	7.355225	17.4686	7.7925	4.7415	6.5466	13.4816	7.2176	27.5795	8.4193	13.7140	7.8006	11.7502	7.0127
$\mathit{SR}\left({\sigma }_{{(r}_p)}\right)$	−0.3719	0.3557	−0.33368	0.443312	0.3212	1.2624	0.9467	1.1457	0.0819	0.6364	−0.0033	0.4924	−0.3764	0.2539	0.4850	0.7334
$K\left(0,1\right)$	−0.0627	0.0884	−0.05053	0.10744	0.0776	0.2653	0.1879	0.2414	0.0273	0.1441	0.0086	0.1070	−0.0571	0.0697	0.1150	0.1611
$K\left(0,2\right)$	−0.4226	0.6390	−0.35967	0.779959	0.4397	1.8650	1.4259	1.7833	0.1790	1.0333	0.0504	0.7525	−0.3915	0.5086	0.7116	1.1266
$K\left(0,3\right)$	−0.6536	1.0877	−0.58211	1.325763	0.6380	3.0752	2.4818	3.0319	0.2686	1.7442	0.0685	1.2359	−0.5997	0.8583	1.0904	1.9354
$K\left(0,4\right)$	−0.7624	1.3516	−0.69544	1.638107	0.7125	3.7330	3.1097	3.7567	0.3002	2.1602	0.0732	1.4825	−0.6760	1.0600	1.2600	2.4381
${\mathrm{\Omega }}_{{(r}_p)}$	0.9373	1.0884	0.949468	1.10744	1.0776	1.2653	1.1879	1.2414	1.0273	1.1441	1.0086	1.1070	0.9429	1.0697	1.1150	1.1611
Panel B: d1 = 2–4 days
	B.1. BRICS-Gold ADCC								B.2. G7-Gold ADCC
Stat.	Brazil	Brazil + Gold	Russia	Russia + Gold	India	India + Gold	China	China + Gold	France	France + Gold	Italy	Italy + Gold	UK	UK + Gold	USA	USA + Gold
$E_{{(r}_p)}$	−0.1983	0.1830	−0.1197	0.1307	0.1228	0.2967	0.2450	0.1834	0.0423	0.3136	0.0130	0.1762	−0.1124	0.2286	0.1831	0.2482
${\sigma }_{{(r}_p)}$.	0.5717	0.2112	0.4015	0.2316	0.3379	0.2187	0.2437	0.2111	0.3421	0.2128	0.3741	0.1975	0.3365	0.1906	0.3482	0.1944
${\tau }_{{(r}_p)}$	−1.2813	−0.1986	−0.9187	−0.7078	−1.9281	−0.2761	−0.5252	−0.2019	−1.3729	0.1387	−2.9599	0.0540	−0.9759	0.1381	−0.9264	−0.1696
$k_{{(r}_p)}$	11.2464	6.7627	8.3036	11.0320	17.4686	7.9201	4.7415	6.7692	13.4816	10.0457	27.5795	6.7289	13.7140	7.5969	11.7502	9.0585
$\mathit{SR}\left({\sigma }_{{(r}_p)}\right)$	−0.3719	0.7992	−0.3337	0.5028	0.3212	1.2912	0.9467	0.8012	0.0819	1.4067	−0.0033	0.8203	−0.3764	1.1247	0.4850	1.2032
$K\left(0,1\right)$	−0.0627	0.1732	−0.0505	0.1255	0.0776	0.2856	0.1879	0.1737	0.0273	0.3244	0.0086	0.1726	−0.0571	0.2493	0.1150	0.2755
$K\left(0,2\right)$	−0.4226	1.2399	−0.3597	0.8115	0.4397	1.9280	1.4259	1.2428	0.1790	2.2363	0.0504	1.2892	−0.3915	1.8287	0.7116	1.8519
$K\left(0,3\right)$	−0.6536	2.0975	−0.5821	1.2519	0.6380	3.2116	2.4818	2.1016	0.2686	3.6156	0.0685	2.2232	−0.5997	3.0874	1.0904	3.0105
$K\left(0,4\right)$	−0.7624	2.5855	−0.6954	1.4571	0.7125	3.9497	3.1097	2.5897	0.3002	4.2975	0.0732	2.7743	−0.6760	3.7799	1.2600	3.6100
${\mathrm{\Omega }}_{{(r}_p)}$	0.9373	1.1732	0.9495	1.1255	1.0776	1.2856	1.1879	1.1737	1.0273	1.3244	1.0086	1.1726	0.9429	1.2493	1.1150	1.2755
Panel C: d3 = 8–16 days
	C.1. BRICS-Gold ADCC								C.2. G7-Gold ADCC
Stat.	Brazil	Brazil + Gold	Russia	Russia + Gold	India	India + Gold	China	China + Gold	France	France + Gold	Italy	Italy + Gold	UK	UK + Gold	USA	USA + Gold
$E_{{(r}_p)}$	−0.1983	0.4258	−0.1197	0.1875	0.1228	0.4158	0.2450	0.0874	0.0423	0.2881	0.0130	0.3688	−0.1124	−0.0465	0.1831	0.4477
${\sigma }_{{(r}_p)}$.	0.5717	0.2471	0.4015	0.2711	0.3379	0.2548	0.2437	0.2217	0.3421	0.2269	0.3741	0.2160	0.3365	0.3061	0.3482	0.2671
${\tau }_{{(r}_p)}$	−1.2813	0.1728	−0.9187	−0.2483	−1.9281	0.5986	−0.5252	−0.1157	−1.3729	−0.2773	−2.9599	−0.3171	−0.9759	−0.4668	−0.9264	−0.0959
$k_{{(r}_p)}$	11.2464	5.0097	8.3036	7.1705	17.4686	14.0032	4.7415	6.0306	13.4816	5.5156	27.5795	6.6785	13.7140	8.2322	11.7502	6.9668
$\mathit{SR}\left({\sigma }_{{(r}_p)}\right)$	−0.3719	1.6655	−0.3337	0.6389	0.3212	1.5760	0.9467	0.3297	0.0819	1.2069	−0.0033	1.6413	−0.3764	−0.1984	0.4850	1.6228
$K\left(0,1\right)$	−0.0627	0.3787	−0.0505	0.1560	0.0776	0.3669	0.1879	0.0757	0.0273	0.2563	0.0086	0.3659	−0.0571	−0.0324	0.1150	0.3729
$K\left(0,2\right)$	−0.4226	2.7189	−0.3597	1.1146	0.4397	2.4173	1.4259	0.5743	0.1790	1.8477	0.0504	2.5402	−0.3915	−0.2226	0.7116	2.4745
$K\left(0,3\right)$	−0.6536	4.7860	−0.5821	1.8302	0.6380	3.8832	2.4818	1.0058	0.2686	3.1662	0.0685	4.2335	−0.5997	−0.3652	1.0904	4.1641
$K\left(0,4\right)$	−0.7624	6.1224	−0.6954	2.2033	0.7125	4.6121	3.1097	1.2612	0.3002	3.9830	0.0732	5.1984	−0.6760	−0.4404	1.2600	5.1889
${\mathrm{\Omega }}_{{(r}_p)}$	0.9373	1.3787	0.9495	1.1560	1.0776	1.3669	1.1879	1.0757	1.0273	1.2563	1.0086	1.3659	0.9429	0.9676	1.1150	1.3729
Panel D: d6 = 64–128 days
	D.1. BRICS-Gold ADCC								D.2. G7-Gold ADCC
Stat.	Brazil	Brazil + Gold	Russia	Russia + Gold	India	India + Gold	China	China + Gold	France	France + Gold	Italy	Italy + Gold	UK	UK + Gold	USA	USA + Gold
$E_{{(r}_p)}$	−0.1983	0.1590	−0.1197	−0.2552	0.1228	0.3066	0.2450	0.0932	0.0423	−0.2068	0.0130	0.1076	−0.1124	0.1146	0.1831	0.0510
${\sigma }_{{(r}_p)}$.	0.5717	0.2126	0.4015	0.3262	0.3379	0.3640	0.2437	0.2504	0.3421	0.3603	0.3741	0.2729	0.3365	0.4509	0.3482	0.3915
${\tau }_{{(r}_p)}$	−1.2813	−0.1905	−0.9187	−1.2944	−1.9281	0.4986	−0.5252	0.0907	−1.3729	−1.0270	−2.9599	−1.4128	−0.9759	0.1894	−0.9264	−0.2259
$k_{{(r}_p)}$	11.2464	6.4958	8.3036	9.4591	17.4686	11.7442	4.7415	4.6771	13.4816	8.7002	27.5795	13.3405	13.7140	10.4465	11.7502	7.0588
$\mathit{SR}\left({\sigma }_{{(r}_p)}\right)$	−0.3719	0.6806	−0.3337	−0.8261	0.3212	0.8032	0.9467	0.3155	0.0819	−0.6137	−0.0033	0.3421	−0.3764	0.2226	0.4850	0.0938
$K\left(0,1\right)$	−0.0627	0.1451	−0.0505	−0.1520	0.0776	0.1562	0.1879	0.0722	0.0273	−0.1107	0.0086	0.0827	−0.0571	0.0397	0.1150	0.0311
$K\left(0,2\right)$	−0.4226	1.0538	−0.3597	−1.0667	0.4397	1.1048	1.4259	0.5797	0.1790	−0.7755	0.0504	0.5459	−0.3915	0.2879	0.7116	0.2253
$K\left(0,3\right)$	−0.6536	1.7972	−0.5821	−1.7129	0.6380	1.8331	2.4818	1.0501	0.2686	−1.2551	0.0685	0.8217	−0.5997	0.4739	1.0904	0.3762
$K\left(0,4\right)$	−0.7624	2.2273	−0.6954	−2.0239	0.7125	2.2316	3.1097	1.3596	0.3002	−1.4961	0.0732	0.9163	−0.6760	0.5625	1.2600	0.4538
${\mathrm{\Omega }}_{{(r}_p)}$	0.9373	1.1451	0.9495	0.8480	1.0776	1.1562	1.1879	1.0722	1.0273	0.8893	1.0086	1.0827	0.9429	1.0397	1.1150	1.0311

This table reports on the different performance measures and for the various single MSCI indices (passive management) and combined strategies MSCI-Gold (active management) under study in terms of dollar currencies for the year 2020. The information regarding the different rebalancing frequencies is clearly divided into three sections: Panel A describes the daily log-returns evaluation, while Panels B and C detail the short and mid-run assessments, respectively. Panel D reports on the long-run or low frequency analysis. By rows the information could be divided into three categories: statistics of the four order moments of the distribution, classical performance ratios (Sharpe) and downside risk measures (Kappa and Omega indices).

Overall, results from Table 7 shed some light on the safe haven role of Gold during the COVID-19 pandemic period. The evidence is very strong in favor of the outstanding diversifying role of Gold, which is found at practically all the wavelet frequencies analyzed and for the various risk-shaping parameters. In terms of second order risk, i.e., risk measure by volatility, we find that this goodness of Gold is only diluted in the long-run, d6, where risk exposure increases for strategies combined with countries such as Russia, China, France and the United States. In terms of higher risk, as measured by skewness and kurtosis, our results continue to reveal a clear reduction in risk when including Gold in the combined strategies, even though in this case the number of exceptions is higher than in volatility.

Additionally, some key and relevant findings are revealed in terms of performance. On the one hand, Panel A depicts that the rebalancing strategy based on log-returns outperforms the initial assets in risk-return and performance measures for all cases. Minor exceptions are the combined strategies for China, which worsens in terms of average observed return and kurtosis, and the USA, which is worse only in mean average return with respect to the initial MSCI index. On the other hand, considering rebalancing strategies based on return decompositions, we report that those based on short- and long-run exhibit poor results on average. Regarding the short-run, d1, we find that the combined strategies for Russia and China become significantly worse in kurtosis with respect to their underlying indices. In fact, the combined strategy for China performs poorer than those reported by the initial MSCI index in all risk-return parameters except for volatility and negative skewness, showing a residual risk reduction of order 2 and 3. Regarding the long-run, d6, we observe that the poor results are especially pronounced, as several countries as Russia, China, France and USA sufficiently outperform their respective combined strategies, thus showing the weakest performance among those presented in the study.

Our results demonstrate that the mid-run, d3, exhibits the best results on average and evidences the overperformance of the mid-term wavelet decomposition strategies over the rest of the portfolios based on log-returns, decompositions thereof and over the initial MSCI indices. This supremacy of the d3 strategies is understood both in terms of the first moments of the return distribution (mean average return, volatility, skewness and kurtosis), as well as traditional performance measures (Sharpe) and downside risk ratios (various Kappa indices). We note that in these mid-run timescales, the only strategy that performs worse than its underlying index is the China one (which only improves in volatility and skewness). Regarding the outperformance of d3, we observe that the differences in relative terms are generally larger based on the classical Sharpe measure than on measures that capture asymmetric patterns and jumps in the distribution such as higher order Kappa indices. Thus, this risk-return analysis allows us to clarify and rank the strategies from best to worst performers: mid-run, short-run, log-returns and long-run.

Lastly, the different patterns suggested by the dependence structure of Brazil MSCI index and Gold (see Section 5.2.2.), are what lead this country to be the one that enhances the most in terms of risk exposure -measured by volatility- when combined with Gold (around 62% drop in volatility) for the different timescales analyzed. Besides, considering the frequencies in which the best results are obtained (mid-run), it is the combined strategy of Brazil-Gold the one that outperforms the active management of buy and hold single market indices and the rest of the active management combined strategies, both in terms of classic performance and downside risk, and thus being the leading strategy.

6. Robustness analysis across currencies and time periods

In this section, we address two additional questions to provide larger robustness to our findings: Diversify our investments more by investing in dollars or different local currencies?17 Does gold diversify better before the COVID-19 outbreak (year 2019) or during the pandemic (year 2020)? This is examined in two ways: time-varying correlations and ex-post performance of the combined strategies.18

We first conduct an analysis of forecasted dynamic correlations in dollar currency very similar to the one developed for the pandemic period (see Section 5.2.2.), but now we include both, local currency and a pre-pandemic analyses. As explained in Section 5.2.2., the in-sample period to calibrate the models is from January 2018 to December 2019, while 2020 is left for an out-of-sample forecasting and assessment of dependencies. In this case we use returns data with an equivalent length (from January 2017 to December 2018) for calibrating the models in the in-sample period and 2019 is left for the out-of-sample assessment of gold-equities connectedness. In addition, we transform all data quoted in dollars from the Bloomberg database into local currency by applying the official exchange rates at any given time. Thus, in terms of the dynamic correlations based on local currency (see Fig. 8, Fig. 6 ) – both for the pre-pandemic and COVID-19 outbreak periods – we can infer that they follow a nearly similar trend in terms of returns for both the BRICS and G7 countries, but they are substantially more significant than those for the dollar (see Fig. 7 and Fig. 4 from Section 5.2.2.). For high and mid frequencies of wavelet decomposition, the correlation trends, levels, and degree of distortion differ greatly from those exhibited in dollars, whereas for low frequencies, d6, the continuous jumps describe patterns that are highly similar to those in dollars.

Fig. 8.

This figure represents the out-of-sample forecasted conditional correlations among MSCI-Gold pairs in terms of local currencies and over the period that spans from January 2019 to December 2019. Panel A shows the time-varying correlations corresponding to the set of assets in the log-returns form. Panel B displays the conditional correlations for the different pairs in the d1 decomposition form (short run). Panel C reports on the dynamic correlations for the case of the d3 decomposition. Panel D plots the time-varying structure of dependence for the long run (d6). Whole Panels are divided into two subsections: the left subsection depicts the dynamic relation between BRICS MSCI indices and Gold, while the right one shows the dependence among G7 MSCI indices and Gold. Regarding the BRICS, cyan blue is chosen to describe the temporal evolution of the correlations for Brazil, purple for Russia, black for India and orange for China. With respect to the G7 countries, maroon color describes the trend of France correlations, green is associated to Italy, grey to UK and dark blue to USA. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 6.

This figure represents the out-of-sample forecasted conditional correlations among MSCI-Gold pairs in terms of local currencies and over the period that spans from January 2020 to December 2020. Panel A shows the time-varying correlations corresponding to the set of assets in the log-returns form. Panel B displays the conditional correlations for the different pairs in the d1 decomposition form (short run). Panel C reports on the dynamic correlations for the case of the d3 decomposition. Panel D plots the time-varying structure of dependence for the long run (d6). Whole Panels are divided into two subsections: the left subsection depicts the dynamic relation between BRICS MSCI indices and Gold, while the right one shows the dependence among G7 MSCI indices and Gold. Regarding the BRICS, cyan blue is chosen to describe the temporal evolution of the correlations for Brazil, purple for Russia, black for India and orange for China. With respect to the G7 countries, maroon color describes the trend of France correlations, green is associated to Italy, grey to UK and dark blue to USA. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 7.

This figure represents the out-of-sample forecasted conditional correlations among MSCI-Gold pairs in terms of unified-dollar currency and over the period that spans from January 2019 to December 2019. Panel A shows the time-varying correlations corresponding to the set of assets in the log-returns form. Panel B displays the conditional correlations for the different pairs in the d1 decomposition form (short run). Panel C reports on the dynamic correlations for the case of the d3 decomposition. Panel D plots the time-varying structure of dependence for the long run (d6). Whole Panels are divided into two subsections: the left subsection depicts the dynamic relation between BRICS MSCI indices and Gold, while the right one shows the dependence among G7 MSCI indices and Gold. Regarding the BRICS, cyan blue is chosen to describe the temporal evolution of the correlations for Brazil, purple for Russia, black for India and orange for China. With respect to the G7 countries, maroon color describes the trend of France correlations, green is associated to Italy, grey to UK and dark blue to USA. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

From Fig. 4, Fig. 6, Fig. 7, Fig. 8, and Table 8 , we report on the forecasted dynamic conditional correlations pairs among the various MSCI indices and Gold for different return cases and wavelet decompositions, and for the two new analyses concern the cross-currency effect and the interaction between pre-pandemic and the COVID-19 outbreak periods. Overall, our contrasting conditional correlation analysis reveals that log-return dependencies among gold and equities are lower for dollar-unified investments (often close to zero or even negative) than for local currency investments, both pre-pandemic and post-pandemic, indicating that diversification is enhanced by the North American currency-denominated investments. In contrast, when we express returns as wavelet decompositions d1, d3, and d6, local currency investments show the lowest correlations, and thus signal greater diversification of risk. The dominance of diversification across periods is less evident. It is worthwhile to examine further in order to determine which periods and for which states of returns these correlations offer the greatest benefits.

Table 8.

Comparison between MSCI-Gold correlations across different currencies and periods.

	Brazil	Russia	India	China	France	Italy	UK	USA	Brazil	Russia	India	China	France	Italy	UK	USA
Panel A: Dollar currency analysis
	A.1. 2019 ADCC Countries-Gold Log-Returns								A.5. 2020 ADCC Countries-Gold Log-Returns
Mean	0.1179	0.0791	0.0293	−0.0089	−0.0184	0.0058	0.0057	−0.1098	0.0392	0.0794	0.0151	0.0526	0.0166	0.0241	0.0133	0.0087
Std. dev.	0.0568	0.1088	0.0444	0.0726	0.1354	0.1317	0.1697	0.1216	0.0802	0.1385	0.0808	0.1239	0.2088	0.1760	0.2094	0.1559
Max.	0.1723	0.5433	0.1234	0.2125	0.3297	0.3602	0.3320	0.2062	0.2434	0.4638	0.6290	0.4793	0.4916	0.3997	0.4883	0.3039
Min.	−0.0080	−0.1954	−0.0945	−0.2047	−0.3467	−0.3150	−0.4136	−0.4113	−0.1881	−0.2936	−0.3601	−0.2749	−0.5277	−0.4642	−0.5769	−0.4837
	A.2. 2019 ADCC Countries-Gold d1decompositions								A.6. 2020 ADCC Countries-Gold d1decompositions
Mean	0.0470	0.3879	0.6724	0.2640	0.4315	0.3278	0.3016	0.7781	−0.0036	0.1463	0.4593	0.0060	0.4352	0.3686	0.4659	0.4171
Std. dev.	0.0732	0.1946	0.1591	0.1550	0.1638	0.1531	0.1646	0.0940	0.0505	0.1962	0.1081	0.0597	0.1877	0.1737	0.1737	0.1839
Max.	0.4861	0.7279	0.9255	0.7501	0.7210	0.6547	0.8000	0.9505	0.2394	0.7597	0.8174	0.2434	0.8948	0.8383	0.9014	0.8784
Min.	−0.0664	−0.4450	0.0066	−0.0932	0.0148	−0.1456	−0.2737	0.3659	−0.0819	−0.2979	0.1884	−0.1608	−0.1216	−0.1993	−0.1054	−0.1083
	A.3. 2019 ADCC Countries-Gold d3 decompositions								A.7. ADCC Countries-Gold d3 decompositions
Mean	−0.0441	0.6408	0.6881	0.3114	0.3798	0.5092	0.2390	0.8013	−0.0122	0.3535	0.5682	0.4037	0.6203	0.4721	0.6545	0.6298
Std. dev.	0.0183	0.2602	0.2302	0.4346	0.4090	0.3239	0.4656	0.1924	0.0217	0.3584	0.2715	0.4021	0.2095	0.2780	0.2766	0.3050
Max.	0.0037	0.9429	0.9414	0.8779	0.9299	0.8803	0.8874	0.9691	0.0194	0.8408	0.9045	0.9362	0.9047	0.9135	0.9599	0.9655
Min.	−0.0636	−0.3923	−0.2785	−0.7146	−0.6627	−0.5316	−0.7442	0.0649	−0.0744	−0.6508	−0.2969	−0.6644	−0.1483	−0.4574	−0.2745	−0.5795
	A.4. 2019 ADCC Countries-Gold d6 decompositions								A.8. 2020 ADCC Countries-Gold d6 decompositions
Mean	0.1336	0.3148	0.5426	0.3958	0.1560	0.2062	0.0042	0.7900	0.1406	0.3273	0.5933	0.3239	0.3273	0.4402	0.7278	0.5918
Std. dev.	0.3442	0.7039	0.6602	0.7110	0.7721	0.7643	0.8534	0.4308	0.3133	0.7659	0.6161	0.7829	0.7659	0.7138	0.4889	0.5991
Max.	0.8188	0.9690	0.9892	0.9711	0.9680	0.9860	0.9676	0.9950	0.8992	0.9727	0.9836	0.9729	0.9727	0.9669	0.9879	0.9863
Min.	−0.7229	−0.8739	−0.8801	−0.8806	−0.8892	−0.9397	−0.9263	−0.6782	−0.7439	−0.9199	−0.8421	−0.9351	−0.9199	−0.8956	−0.7666	−0.8334
Panel B: Local currency analysis
	B.1. 2019 ADCC Countries-Gold Log-Returns								B.5. 2020 ADCC Countries-Gold Log-Returns
Mean	0.2816	0.2410	0.2016	0.0606	0.0707	0.0937	0.2611	−0.1076	0.2243	0.1903	0.1147	0.0777	0.1327	0.1122	0.2244	0.0101
Std. dev.	0.0549	0.0357	0.0402	0.0597	0.1272	0.0652	0.1603	0.1223	0.0726	0.0642	0.0339	0.0926	0.1834	0.1726	0.1588	0.1558
Max.	0.4524	0.3733	0.2679	0.2308	0.3539	0.2706	0.4722	0.2196	0.4300	0.3341	0.2529	0.4271	0.6388	0.5745	0.5155	0.3034
Min.	0.1209	0.1682	0.0849	−0.1843	−0.2034	−0.1317	−0.3624	−0.4077	0.0539	−0.0127	0.0479	−0.1836	−0.3064	−0.3463	−0.2558	−0.4902
	B.2. 2019 ADCC Countries-Gold d1decompositions								B.6. 2020 ADCC Countries-Gold d1decompositions
Mean	0.4209	−0.0204	0.0307	0.1337	0.1509	0.1623	0.3186	−0.1741	0.3788	0.0304	−0.0689	0.1468	0.1550	0.2015	0.2858	−0.0757
Std. dev.	0.1788	0.2042	0.2053	0.1805	0.2211	0.2628	0.2302	0.2385	0.2214	0.2425	0.2169	0.2832	0.2180	0.2486	0.1901	0.2501
Max.	0.8607	0.5877	0.6985	0.6298	0.7108	0.7365	0.7705	0.3060	0.8234	0.7006	0.7011	0.7682	0.7240	0.7629	0.6920	0.4661
Min.	−0.1484	−0.4867	−0.6697	−0.3854	−0.4333	−0.5186	−0.4714	−0.7312	−0.1889	−0.4818	−0.7354	−0.6056	−0.3355	−0.2936	−0.2581	−0.6574
	B.3. 2019 ADCC Countries-Gold d3 decompositions								B.7. ADCC Countries-Gold d3 decompositions
Mean	0.0776	−0.1246	−0.1257	−0.0957	−0.0923	−0.1991	0.1097	−0.2029	0.0819	0.0392	−0.0709	0.2319	0.0490	0.0511	0.1747	0.0322
Std. dev.	0.3350	0.3949	0.3689	0.3356	0.3888	0.3696	0.3810	0.3567	0.2711	0.3860	0.3879	0.3803	0.4617	0.4717	0.4420	0.4149
Max.	0.7523	0.7418	0.7849	0.7737	0.6905	0.6609	0.8759	0.6410	0.7863	0.7278	0.7765	0.9129	0.8492	0.8800	0.8744	0.7427
Min.	−0.6049	−0.8306	−0.8637	−0.6958	−0.7454	−0.7824	−0.6595	−0.8167	−0.6771	−0.7501	−0.7708	−0.6504	−0.8384	−0.8555	−0.8115	−0.8151
	B.4. 2019 ADCC Countries-Gold d6 decompositions								B.8. 2020 ADCC Countries-Gold d6 decompositions
Mean	−0.4621	−0.3361	0.5075	0.3707	−0.1693	0.0861	0.1649	−0.1196	−0.3479	−0.0598	−0.1740	0.1167	0.0363	0.0853	0.0326	0.0045
Std. dev.	0.6853	0.6472	0.6879	0.7418	0.7510	0.7821	0.7290	0.7536	0.6662	0.7643	0.7836	0.7798	0.7616	0.7457	0.7483	0.7697
Max.	0.9240	0.9387	0.9618	0.9579	0.9241	0.9387	0.9625	0.9233	0.8984	0.9329	0.9599	0.9602	0.9672	0.9683	0.9571	0.9524
Min.	−0.9514	−0.9525	−0.9441	−0.9174	−0.9235	−0.9472	−0.9106	−0.9412	−0.9597	−0.9389	−0.9506	−0.9120	−0.9331	−0.9421	−0.9231	−0.9355

This table shows a comparison between the different currency and period analysis of the time evolution described by correlation pairs over the log-returns and the different wavelet frequencies under analysis and for the out-of-sample period. Panel A shows the statistics regarding the time-varying correlations corresponding to undecomposed and decomposed returns y unified-dollar currency. Panel B displays the statistics for the different correlations pairs that corresponds to undecomposed and decomposed returns in local currencies. Both Panels are divided into eight subsections: the left subsections (from 0.1 to 0.4) depict the dynamic relation between MSCI indices and Gold for the pre-pandemic year, while the right ones (from 0.5 to 0.8) show the dependence among MSCI indices and Gold across the COVID-19 pandemic year. We report on the mean average correlation (Mean), the standard deviation of the correlation processes (Std. dev.), and the maximum and minimum levels (Max. and Min.).

Specifically, from our dollar currency correlation analysis for the year 2019, we find that the evidence described in Section 5.2.2. for the pandemic is also broadly applicable for the pre-pandemic period. Thus, our results reveal a degree of homogeneity among the patterns depicted by the G7-Gold correlation pairs, while these characteristics are remarkable absent from the BRICS. We also find that the ADCC connectedness between Gold and MSCI equity indices becomes increasingly pronounced as the frequency of shocks decreases. Furthermore, it appears that the correlation trend moves in clusters at lower timescales. Additionally, as suggested by the cross correlations of the precious metal and equity log-returns, d1, d3 and d6 decompositions, gold's diversification properties are progressively diminished with decreasing frequency. As one obvious difference between the years 2019 and 2020, correlations are often lower in the former period when examining undecomposed returns, whereas they are lower during the pandemic one when examining returns decomposed at a high frequency (d1). Concerning the other frequencies, there is no sign that one period is substantially superior to the other for different countries under study.

On the contrary, in the local currency analysis, correlations between countries and gold decrease – and even become negative, shockingly – when shifting from undecomposed returns or the highest return decomposition frequencies (d1) to the lowest frequency (d6), both in the pre-pandemic and the outbreak periods. Clearly, the local currency low-frequency wavelet analysis makes sense from a diversification perspective, as it could produce potentially profitable strategies for portfolio managers in the years 2019 and 2020. This need to be contrasted by an ex-post performance analysis. In contrast with currency analysis using the dollar and looking for differences between periods, we note that for the case of local undecomposed returns, correlations between countries and gold are also lower during 2019, but with the caveat that these patterns only apply to countries in the G7. In terms of correlations between decomposed returns, we find that the evidence is the same for the lowest frequencies or d6, while it is reversed for the high and mid wavelet frequencies (gold-equity connectedness is lower during the pre-pandemic for d1 and d3).

The disparate evidence across correlation analyses in different or unified dollar currencies and periods, undoubtedly condition both the formation of portfolios and their subsequent performance. Thus, in view of this lack of irrefutable evidence in the correlation analysis, which fails to provide homogeneous and conclusive results regarding diversification benefits of gold and the differences between periods and currencies, we conduct an ex-post performance assessment in which, in addition to the diversification inferred by the correlation of gold with equities, the diversification induced by the precious metal’s intrinsic volatility plays a key role. Using the same approach from Section 5.2.4., we calculate the returns resulting from holding our positions for 22 days (approximately one month) and then compute the overall performance for the year 2019 or pre-pandemic period, both in US dollars and local currencies. In this regard, Table 9 provides an equivalent pandemic performance analysis to the one developed in Section 5.2.4. for Table 7, but for local currency returns. Then, Table 10, Table 11 report on the pre-pandemic performance analysis, both from the point of view of dollars and local currencies worldwide.

Table 9.

Performance measures for passive (single MSCI indices) and active management (combined strategies MSCI-Gold) in local currencies for 2020.

Panel A: log-returns
	A.1. BRICS-Gold ADCC								A.2. G7-Gold ADCC
Stat.	Brazil	Brazil + Gold	Russia	Russia + Gold	India	India + Gold	China	China + Gold	France	France + Gold	Italy	Italy + Gold	UK	UK + Gold	USA	USA + Gold
$E_{{(r}_p)}$	0.0319	0.3430	0.0444	0.2892	0.1539	0.2806	0.1812	0.1489	−0.0465	0.0800	−0.0758	0.0620	−0.1354	−0.0248	0.1831	0.1595
${\sigma }_{{(r}_p)}$.	0.5911	0.3012	0.4176	0.2610	0.3394	0.1951	0.2486	0.1851	0.3518	0.2039	0.3848	0.2073	0.3545	0.2223	0.3482	0.1940
${\tau }_{{(r}_p)}$	−0.7846	0.0087	0.4433	0.6353	−1.8755	0.0701	−0.4399	−0.0287	−1.2041	0.0124	−2.7576	−0.2131	−0.7469	0.3442	−0.9264	0.0425
$k_{{(r}_p)}$	8.4103	5.4149	10.0164	6.6848	16.3287	9.0148	4.7562	6.0180	11.6856	6.2694	23.8027	6.4853	11.1261	6.0496	11.7502	7.0349
$\mathit{SR}\left({\sigma }_{{(r}_p)}\right)$	0.0299	1.0914	0.0721	1.0532	0.4115	1.3650	0.6717	0.7272	−0.1727	0.3226	−0.2339	0.2303	−0.4220	−0.1754	0.4850	0.7490
$K\left(0,1\right)$	0.0148	0.2022	0.0253	0.2150	0.0978	0.2987	0.1331	0.1495	−0.0201	0.0817	−0.0355	0.0585	−0.0623	−0.0086	0.1150	0.1648
$K\left(0,2\right)$	0.1044	1.6304	0.1986	1.7494	0.5577	2.0930	1.0335	1.1604	−0.1355	0.6319	−0.2153	0.4451	−0.4430	−0.0693	0.7116	1.1501
$K\left(0,3\right)$	0.1677	2.8590	0.3357	3.0929	0.8165	3.4384	1.8163	2.0141	−0.2091	1.0889	−0.3009	0.7600	−0.6992	−0.1254	1.0904	1.9738
$K\left(0,4\right)$	0.2004	3.5318	0.4066	3.8613	0.9219	4.1479	2.2781	2.4970	−0.2380	1.3465	−0.3262	0.9330	−0.8073	−0.1612	1.2600	2.4856
${\mathrm{\Omega }}_{{(r}_p)}$	1.0148	1.2022	1.0253	1.2150	1.0978	1.2987	1.1331	1.1495	0.9799	1.0817	0.9645	1.0585	0.9377	0.9914	1.1150	1.1648
Panel B: d1 = 2–4 days
	B.1. BRICS-Gold ADCC								B.2. G7-Gold ADCC
Stat.	Brazil	Brazil + Gold	Russia	Russia + Gold	India	India + Gold	China	China + Gold	France	France + Gold	Italy	Italy + Gold	UK	UK + Gold	USA	USA + Gold
$E_{{(r}_p)}$	0.0319	0.2314	0.0444	0.2644	0.1539	0.1667	0.1812	0.0783	−0.0465	−0.0308	−0.0758	−0.0650	−0.1354	0.0261	0.1831	0.0705
${\sigma }_{{(r}_p)}$.	0.5911	0.3203	0.4176	0.2852	0.3394	0.2047	0.2486	0.1950	0.3518	0.2149	0.3848	0.2275	0.3545	0.2395	0.3482	0.2270
${\tau }_{{(r}_p)}$	−0.7846	−0.1373	0.4433	0.7226	−1.8755	−0.7299	−0.4399	−0.2340	−1.2041	−0.4067	−2.7576	−1.3195	−0.7469	0.0684	−0.9264	−0.5273
$k_{{(r}_p)}$	8.4103	6.1793	10.0164	10.7105	16.3287	9.3869	4.7562	6.0777	11.6856	7.2350	23.8027	13.0840	11.1261	9.2562	11.7502	9.4226
$\mathit{SR}\left({\sigma }_{{(r}_p)}\right)$	0.0299	0.6780	0.0721	0.8770	0.4115	0.7446	0.6717	0.3287	−0.1727	−0.2096	−0.2339	−0.3485	−0.4220	0.0493	0.4850	0.2477
$K\left(0,1\right)$	0.0148	0.1243	0.0253	0.1848	0.0978	0.1580	0.1331	0.0700	−0.0201	−0.0206	−0.0355	−0.0517	−0.0623	0.0319	0.1150	0.0581
$K\left(0,2\right)$	0.1044	0.9596	0.1986	1.4092	0.5577	1.0721	1.0335	0.5330	−0.1355	−0.1549	−0.2153	−0.3592	−0.4430	0.2360	0.7116	0.3907
$K\left(0,3\right)$	0.1677	1.6406	0.3357	2.4111	0.8165	1.7045	1.8163	0.9137	−0.2091	−0.2610	−0.3009	−0.5471	−0.6992	0.3906	1.0904	0.6313
$K\left(0,4\right)$	0.2004	2.0070	0.4066	2.9934	0.9219	2.0129	2.2781	1.1264	−0.2380	−0.3177	−0.3262	−0.6174	−0.8073	0.4657	1.2600	0.7512
${\mathrm{\Omega }}_{{(r}_p)}$	1.0148	1.1243	1.0253	1.1848	1.0978	1.1580	1.1331	1.0700	0.9799	0.9794	0.9645	0.9483	0.9377	1.0319	1.1150	1.0581
Panel C: d3 = 8–16 days
	C.1. BRICS-Gold ADCC								C.2. G7-Gold ADCC
Stat.	Brazil	Brazil + Gold	Russia	Russia + Gold	India	India + Gold	China	China + Gold	France	France + Gold	Italy	Italy + Gold	UK	UK + Gold	USA	USA + Gold
$E_{{(r}_p)}$	0.0319	0.1793	0.0444	0.4869	0.1539	0.1722	0.1812	0.2124	−0.0465	0.0620	−0.0758	−0.1819	−0.1354	−0.0248	0.1831	0.0837
${\sigma }_{{(r}_p)}$.	0.5911	0.3278	0.4176	0.2993	0.3394	0.2119	0.2486	0.2083	0.3518	0.2187	0.3848	0.2545	0.3545	0.2223	0.3482	0.2230
${\tau }_{{(r}_p)}$	−0.7846	0.1263	0.4433	0.2136	−1.8755	−0.5717	−0.4399	−0.0636	−1.2041	−0.2749	−2.7576	−1.2492	−0.7469	0.3442	−0.9264	−0.4528
$k_{{(r}_p)}$	8.4103	5.0429	10.0164	6.0196	16.3287	8.3146	4.7562	5.9256	11.6856	5.3892	23.8027	9.6712	11.1261	6.0496	11.7502	8.9688
$\mathit{SR}\left({\sigma }_{{(r}_p)}\right)$	0.0299	0.5036	0.0721	1.5792	0.4115	0.7457	0.6717	0.9511	−0.1727	0.2183	−0.2339	−0.7706	−0.4220	−0.1754	0.4850	0.3113
$K\left(0,1\right)$	0.0148	0.0888	0.0253	0.3336	0.0978	0.1513	0.1331	0.2034	−0.0201	0.0567	−0.0355	−0.1234	−0.0623	−0.0086	0.1150	0.0750
$K\left(0,2\right)$	0.1044	0.7329	0.1986	2.5776	0.5577	1.0409	1.0335	1.5482	−0.1355	0.4335	−0.2153	−0.8763	−0.4430	−0.0693	0.7116	0.5070
$K\left(0,3\right)$	0.1677	1.3219	0.3357	4.3985	0.8165	1.6996	1.8163	2.6252	−0.2091	0.7554	−0.3009	−1.4115	−0.6992	−0.1254	1.0904	0.8233
$K\left(0,4\right)$	0.2004	1.6919	0.4066	5.3551	0.9219	2.0439	2.2781	3.2132	−0.2380	0.9518	−0.3262	−1.6553	−0.8073	−0.1612	1.2600	0.9927
${\mathrm{\Omega }}_{{(r}_p)}$	1.0148	1.0888	1.0253	1.3336	1.0978	1.1513	1.1331	1.2034	0.9799	1.0567	0.9645	0.8766	0.9377	0.9914	1.1150	1.0750
Panel D: d6 = 64–128 days
	D.1. BRICS-Gold ADCC								D.2. G7-Gold ADCC
Stat.	Brazil	Brazil + Gold	Russia	Russia + Gold	India	India + Gold	China	China + Gold	France	France + Gold	Italy	Italy + Gold	UK	UK + Gold	USA	USA + Gold
$E_{{(r}_p)}$	0.0319	0.1987	0.0444	0.0008	0.1539	−0.0354	0.1812	0.0350	−0.0465	−0.0079	−0.0758	−0.1361	−0.1354	0.0945	0.1831	0.0518
${\sigma }_{{(r}_p)}$.	0.5911	0.3357	0.4176	0.2913	0.3394	0.3551	0.2486	0.2343	0.3518	0.2433	0.3848	0.2707	0.3545	0.2646	0.3482	0.2566
${\tau }_{{(r}_p)}$	−0.7846	−0.1088	0.4433	0.4801	−1.8755	−2.6440	−0.4399	−0.1450	−1.2041	−0.3576	−2.7576	−0.7397	−0.7469	0.5034	−0.9264	−0.2812
$k_{{(r}_p)}$	8.4103	5.1185	10.0164	7.3386	16.3287	23.1045	4.7562	5.5791	11.6856	5.5770	23.8027	7.0072	11.1261	7.7438	11.7502	6.2743
$\mathit{SR}\left({\sigma }_{{(r}_p)}\right)$	0.0299	0.5495	0.0721	−0.0462	0.4115	−0.1399	0.6717	0.0886	−0.1727	−0.0910	−0.2339	−0.5555	−0.4220	0.3033	0.4850	0.1465
$K\left(0,1\right)$	0.0148	0.0939	0.0253	−0.0137	0.0978	−0.0229	0.1331	0.0279	−0.0201	−0.0026	−0.0355	−0.0826	−0.0623	0.0707	0.1150	0.0380
$K\left(0,2\right)$	0.1044	0.7564	0.1986	−0.1158	0.5577	−0.1264	1.0335	0.2205	−0.1355	−0.0206	−0.2153	−0.6328	−0.4430	0.5566	0.7116	0.2821
$K\left(0,3\right)$	0.1677	1.3300	0.3357	−0.2111	0.8165	−0.1762	1.8163	0.3865	−0.2091	−0.0355	−0.3009	−1.0560	−0.6992	0.9716	1.0904	0.4873
$K\left(0,4\right)$	0.2004	1.6641	0.4066	−0.2713	0.9219	−0.1934	2.2781	0.4801	−0.2380	−0.0439	−0.3262	−1.2646	−0.8073	1.2068	1.2600	0.6055
${\mathrm{\Omega }}_{{(r}_p)}$	1.0148	1.0939	1.0253	0.9863	1.0978	0.9771	1.1331	1.0279	0.9799	0.9974	0.9645	0.9174	0.9377	1.0707	1.1150	1.0380

This table reports on the different performance measures and for the various single MSCI indices (passive management) and combined strategies MSCI-Gold (active management) under study in terms of local currencies for the year 2020. The information regarding the different rebalancing frequencies is clearly divided into three sections: Panel A describes the daily log-returns evaluation, while Panels B and C detail the short and mid-run assessments, respectively. Panel D reports on the long-run or low frequency analysis. By rows the information could be divided into three categories: statistics of the four order moments of the distribution, classical performance ratios (Sharpe) and downside risk measures (Kappa and Omega indices).

Table 10.

Performance measures for passive (single MSCI indices) and active management (combined strategies MSCI-Gold) in dollars for 2019.

Panel A: log-returns
	A.1. BRICS-Gold ADCC								A.2. G7-Gold ADCC
Stat.	Brazil	Brazil + Gold	Russia	Russia + Gold	India	India + Gold	China	China + Gold	France	France + Gold	Italy	Italy + Gold	UK	UK + Gold	USA	USA + Gold
$E_{{(r}_p)}$	0.2198	0.1652	0.4050	0.1810	0.0803	0.0896	0.1988	0.1691	0.2282	0.1715	0.2514	0.1740	0.1789	0.1484	0.2665	0.1784
${\sigma }_{{(r}_p)}$.	0.2438	0.1035	0.1601	0.0953	0.1570	0.0910	0.1659	0.0882	0.1356	0.0826	0.1482	0.0872	0.1290	0.0808	0.1251	0.0762
${\tau }_{{(r}_p)}$	−0.4360	0.1516	−0.4235	0.3613	0.4634	0.4269	−0.4117	0.6372	−0.4953	0.4953	−0.2653	0.2918	0.3763	0.5144	−0.6117	0.5941
$k_{{(r}_p)}$	4.0616	5.8828	4.9185	6.9234	6.6459	6.0893	3.8370	7.3346	4.4609	6.6875	4.0650	6.7705	6.5379	6.3701	6.3380	8.0025
$\mathit{SR}\left({\sigma }_{{(r}_p)}\right)$	0.8934	1.5757	2.5172	1.8774	0.4986	0.9618	1.1864	1.8940	1.6680	2.0529	1.6825	1.9730	1.3710	1.8119	2.1146	2.3134
$K\left(0,1\right)$	0.1560	0.2672	0.5152	0.3327	0.0883	0.1499	0.2241	0.3373	0.3265	0.3772	0.3216	0.3677	0.2775	0.3201	0.4528	0.4372
$K\left(0,2\right)$	1.2420	2.0364	3.8823	2.5758	0.7274	1.2216	1.7475	2.6869	2.4065	2.9429	2.4990	2.7289	2.1397	2.5179	3.0351	3.3723
$K\left(0,3\right)$	2.2233	3.6242	6.4927	4.5360	1.3336	2.1993	3.1091	4.8512	4.1896	5.1954	4.4434	4.7910	3.7674	4.5176	4.9912	6.0060
$K\left(0,4\right)$	2.8253	4.6580	7.7351	5.7519	1.7097	2.8047	3.9554	6.2708	5.2751	6.6036	5.6717	6.1337	4.7397	5.8214	6.1011	7.6083
${\mathrm{\Omega }}_{{(r}_p)}$	1.1560	1.2672	1.5152	1.3327	1.0883	1.1499	1.2241	1.3373	1.3265	1.3772	1.3216	1.3677	1.2775	1.3201	1.4528	1.4372
Panel B: d1 = 2–4 days
	B.1. BRICS-Gold ADCC								B.2. G7-Gold ADCC
Stat.	Brazil	Brazil + Gold	Russia	Russia + Gold	India	India + Gold	China	China + Gold	France	France + Gold	Italy	Italy + Gold	UK	UK + Gold	USA	USA + Gold
$E_{{(r}_p)}$	0.2198	0.1608	0.4050	0.2488	0.0803	0.1548	0.1988	0.1697	0.2282	0.1808	0.2514	0.1941	0.1789	0.1723	0.2665	0.2066
${\sigma }_{{(r}_p)}$.	0.2438	0.1136	0.1601	0.1052	0.1570	0.1169	0.1659	0.1036	0.1356	0.0945	0.1482	0.0882	0.1290	0.0830	0.1251	0.1106
${\tau }_{{(r}_p)}$	−0.4360	0.1876	−0.4235	0.0355	0.4634	0.3023	−0.4117	0.3399	−0.4953	0.4606	−0.2653	0.2707	0.3763	0.0596	−0.6117	−0.0159
$k_{{(r}_p)}$	4.0616	5.8873	4.9185	5.3314	6.6459	7.8627	3.8370	6.2875	4.4609	8.0417	4.0650	6.3075	6.5379	5.6000	6.3380	9.5333
$\mathit{SR}\left({\sigma }_{{(r}_p)}\right)$	0.8934	1.3985	2.5172	2.3450	0.4986	1.3076	1.1864	1.6188	1.6680	1.8929	1.6825	2.1781	1.3710	2.0515	2.1146	1.8497
$K\left(0,1\right)$	0.1560	0.2359	0.5152	0.4504	0.0883	0.2568	0.2241	0.2832	0.3265	0.3685	0.3216	0.4342	0.2775	0.3911	0.4528	0.4008
$K\left(0,2\right)$	1.2420	1.7919	3.8823	3.4796	0.7274	1.9300	1.7475	2.1672	2.4065	2.6933	2.4990	3.2125	2.1397	2.8621	3.0351	2.6426
$K\left(0,3\right)$	2.2233	3.1514	6.4927	6.0959	1.3336	3.2367	3.1091	3.8155	4.1896	4.6593	4.4434	5.5910	3.7674	4.9171	4.9912	4.2903
$K\left(0,4\right)$	2.8253	4.0042	7.7351	7.6348	1.7097	3.9371	3.9554	4.8601	5.2751	5.9161	5.6717	7.1243	4.7397	6.1818	6.1011	5.1871
${\mathrm{\Omega }}_{{(r}_p)}$	1.1560	1.2359	1.5152	1.4504	1.0883	1.2568	1.2241	1.2832	1.3265	1.3685	1.3216	1.4342	1.2775	1.3911	1.4528	1.4008
Panel C: d3 = 8–16 days
	C.1. BRICS-Gold ADCC								C.2. G7-Gold ADCC
Stat.	Brazil	Brazil + Gold	Russia	Russia + Gold	India	India + Gold	China	China + Gold	France	France + Gold	Italy	Italy + Gold	UK	UK + Gold	USA	USA + Gold
$E_{{(r}_p)}$	0.2198	0.1407	0.4050	0.2094	0.0803	0.3922	0.1988	0.1724	0.2282	0.1062	0.2514	0.1351	0.1789	0.1835	0.2665	0.3209
${\sigma }_{{(r}_p)}$.	0.2438	0.1453	0.1601	0.1290	0.1570	0.1519	0.1659	0.1143	0.1356	0.1403	0.1482	0.1372	0.1290	0.1101	0.1251	0.1602
${\tau }_{{(r}_p)}$	−0.4360	−1.0407	−0.4235	0.6792	0.4634	0.8842	−0.4117	0.2422	−0.4953	−0.3175	−0.2653	0.4129	0.3763	0.7838	−0.6117	0.4918
$k_{{(r}_p)}$	4.0616	11.3123	4.9185	5.7212	6.6459	7.6560	3.8370	5.1726	4.4609	10.0017	4.0650	6.6520	6.5379	6.8154	6.3380	9.3913
$\mathit{SR}\left({\sigma }_{{(r}_p)}\right)$	0.8934	0.9541	2.5172	1.6084	0.4986	2.5684	1.1864	1.4908	1.6680	0.7426	1.6825	0.9699	1.3710	1.6488	2.1146	1.9910
$K\left(0,1\right)$	0.1560	0.1653	0.5152	0.2957	0.0883	0.5776	0.2241	0.2457	0.3265	0.1128	0.3216	0.1621	0.2775	0.3104	0.4528	0.4828
$K\left(0,2\right)$	1.2420	1.0686	3.8823	2.4282	0.7274	4.2417	1.7475	1.9241	2.4065	0.7715	2.4990	1.2580	2.1397	2.4152	3.0351	3.0931
$K\left(0,3\right)$	2.2233	1.6156	6.4927	4.4805	1.3336	7.3613	3.1091	3.4359	4.1896	1.2173	4.4434	2.1809	3.7674	4.3599	4.9912	4.9070
$K\left(0,4\right)$	2.8253	1.8384	7.7351	5.8717	1.7097	9.2470	3.9554	4.3943	5.2751	1.4142	5.6717	2.6964	4.7397	5.6479	6.1011	5.8658
${\mathrm{\Omega }}_{{(r}_p)}$	1.1560	1.1653	1.5152	1.2957	1.0883	1.5776	1.2241	1.2457	1.3265	1.1128	1.3216	1.1621	1.2775	1.3104	1.4528	1.4828
Panel D: d6 = 64–128 days
	D.1. BRICS-Gold ADCC								D.2. G7-Gold ADCC
Stat.	Brazil	Brazil + Gold	Russia	Russia + Gold	India	India + Gold	China	China + Gold	France	France + Gold	Italy	Italy + Gold	UK	UK + Gold	USA	USA + Gold
$E_{{(r}_p)}$	0.2198	0.1613	0.4050	0.3751	0.0803	−0.0156	0.1988	0.3038	0.2282	0.2294	0.2514	0.3529	0.1789	0.4593	0.2665	0.7795
${\sigma }_{{(r}_p)}$.	0.2438	0.1140	0.1601	0.1602	0.1570	0.2510	0.1659	0.1515	0.1356	0.1466	0.1482	0.1539	0.1290	0.1738	0.1251	0.3895
${\tau }_{{(r}_p)}$	−0.4360	0.1558	−0.4235	0.1354	0.4634	0.1831	−0.4117	−0.1696	−0.4953	0.4998	−0.2653	0.6613	0.3763	0.7239	−0.6117	0.4812
$k_{{(r}_p)}$	4.0616	5.8444	4.9185	6.0616	6.6459	5.1924	3.8370	5.6145	4.4609	8.0030	4.0650	6.1414	6.5379	7.3612	6.3380	6.2417
$\mathit{SR}\left({\sigma }_{{(r}_p)}\right)$	0.8934	1.3968	2.5172	2.3286	0.4986	−0.0701	1.1864	1.9912	1.6680	1.5507	1.6825	2.2799	1.3710	2.6312	2.1146	1.9961
$K\left(0,1\right)$	0.1560	0.2361	0.5152	0.5028	0.0883	0.0345	0.2241	0.3764	0.3265	0.2974	0.3216	0.5091	0.2775	0.6071	0.4528	0.4904
$K\left(0,2\right)$	1.2420	1.7838	3.8823	3.6734	0.7274	0.2737	1.7475	2.7358	2.4065	2.0709	2.4990	3.8455	2.1397	4.3764	3.0351	3.4857
$K\left(0,3\right)$	2.2233	3.1361	6.4927	6.1869	1.3336	0.4980	3.1091	4.6228	4.1896	3.4979	4.4434	6.7530	3.7674	7.5609	4.9912	5.8229
$K\left(0,4\right)$	2.8253	3.9871	7.7351	7.5482	1.7097	0.6462	3.9554	5.6311	5.2751	4.3614	5.6717	8.5578	4.7397	9.4222	6.1011	6.9662
${\mathrm{\Omega }}_{{(r}_p)}$	1.1560	1.2361	1.5152	1.5028	1.0883	1.0345	1.2241	1.3764	1.3265	1.2974	1.3216	1.5091	1.2775	1.6071	1.4528	1.4904

This table reports on the different performance measures and for the various single MSCI indices (passive management) and combined strategies MSCI-Gold (active management) under study in terms of dollar currencies for the year 2019. The information regarding the different rebalancing frequencies is clearly divided into three sections: Panel A describes the daily log-returns evaluation, while Panels B and C detail the short and mid-run assessments, respectively. Panel D reports on the long-run or low frequency analysis. By rows the information could be divided into three categories: statistics of the four order moments of the distribution, classical performance ratios (Sharpe) and downside risk measures (Kappa and Omega indices).

Table 11.

Performance measures for passive (single MSCI indices) and active management (combined strategies MSCI-Gold) in local currencies for 2019.

Panel A: log-returns
	A.1. BRICS-Gold ADCC								A.2. G7-Gold ADCC
Stat.	Brazil	Brazil + Gold	Russia	Russia + Gold	India	India + Gold	China	China + Gold	France	France + Gold	Italy	Italy + Gold	UK	UK + Gold	USA	USA + Gold
$E_{{(r}_p)}$	0.2651	0.2473	0.3054	0.1454	0.0985	0.1222	0.2171	0.1720	0.2615	0.2089	0.2847	0.1922	0.1518	0.0653	0.2665	0.1784
${\sigma }_{{(r}_p)}$.	0.2636	0.1633	0.1702	0.1191	0.1660	0.1157	0.1647	0.0932	0.1362	0.0941	0.1483	0.0992	0.1403	0.1109	0.1251	0.0762
${\tau }_{{(r}_p)}$	−0.0187	0.3658	−0.0906	0.1377	0.5841	0.5302	−0.2390	0.6296	−0.5060	0.2818	−0.3292	0.2199	−0.1283	0.1014	−0.6117	0.5941
$k_{{(r}_p)}$	3.6447	3.9953	3.7224	3.6726	6.2247	5.4388	3.4547	5.6923	4.1934	4.2167	3.6581	4.5894	3.9666	3.6427	6.3380	8.0025
$\mathit{SR}\left({\sigma }_{{(r}_p)}\right)$	0.9980	1.5017	1.7828	1.2037	0.5813	1.0391	1.3056	1.8233	1.9057	2.1986	1.9059	1.9161	1.0678	0.5701	2.1146	2.3134
$K\left(0,1\right)$	0.1800	0.2740	0.3313	0.1940	0.1025	0.1747	0.2422	0.3181	0.3741	0.4048	0.3651	0.3633	0.1886	0.0619	0.4528	0.4372
$K\left(0,2\right)$	1.4928	2.3359	2.7472	1.6828	0.9010	1.5136	1.9599	2.6809	2.7517	3.3409	2.8234	2.8828	1.5450	0.5433	3.0351	3.3723
$K\left(0,3\right)$	2.7408	4.4010	4.8718	3.0792	1.6895	2.7800	3.5609	5.0262	4.7950	6.1823	5.0140	5.2264	2.8057	1.0243	4.9912	6.0060
$K\left(0,4\right)$	3.5433	5.8316	6.0639	3.9284	2.1965	3.5335	4.5849	6.6263	6.0058	8.0913	6.4153	6.7504	3.5932	1.3496	6.1011	7.6083
${\mathrm{\Omega }}_{{(r}_p)}$	1.1800	1.2740	1.3313	1.1940	1.1025	1.1747	1.2422	1.3181	1.3741	1.4048	1.3651	1.3633	1.1886	1.0619	1.4528	1.4372
Panel B: d1 = 2–4 days
	B.1. BRICS-Gold ADCC								B.2. G7-Gold ADCC
Stat.	Brazil	Brazil + Gold	Russia	Russia + Gold	India	India + Gold	China	China + Gold	France	France + Gold	Italy	Italy + Gold	UK	UK + Gold	USA	USA + Gold
$E_{{(r}_p)}$	0.2651	0.2639	0.3054	0.1908	0.0985	0.1522	0.2171	0.2401	0.2615	0.2297	0.2847	0.2711	0.1518	0.2221	0.2665	0.1935
${\sigma }_{{(r}_p)}$.	0.2636	0.1777	0.1702	0.1209	0.1660	0.1172	0.1647	0.0968	0.1362	0.0973	0.1483	0.1051	0.1403	0.1188	0.1251	0.0770
${\tau }_{{(r}_p)}$	−0.0187	0.2040	−0.0906	0.3149	0.5841	0.3857	−0.2390	0.5134	−0.5060	0.3290	−0.3292	0.0692	−0.1283	−0.1888	−0.6117	0.3800
$k_{{(r}_p)}$	3.6447	3.9991	3.7224	3.7878	6.2247	5.6533	3.4547	5.3879	4.1934	5.0382	3.6581	3.4451	3.9666	4.1199	6.3380	6.8632
$\mathit{SR}\left({\sigma }_{{(r}_p)}\right)$	0.9980	1.4742	1.7828	1.5620	0.5813	1.2815	1.3056	2.4608	1.9057	2.3384	1.9059	2.5611	1.0678	1.8525	2.1146	2.4856
$K\left(0,1\right)$	0.1800	0.2726	0.3313	0.2678	0.1025	0.2279	0.2422	0.4747	0.3741	0.4511	0.3651	0.5132	0.1886	0.3286	0.4528	0.4962
$K\left(0,2\right)$	1.4928	2.2350	2.7472	2.3483	0.9010	1.8908	1.9599	3.9239	2.7517	3.6301	2.8234	4.1223	1.5450	2.5939	3.0351	3.6897
$K\left(0,3\right)$	2.7408	4.0923	4.8718	4.3483	1.6895	3.3632	3.5609	7.2675	4.7950	6.5153	5.0140	7.6039	2.8057	4.5873	4.9912	6.6057
$K\left(0,4\right)$	3.5433	5.3049	6.0639	5.5731	2.1965	4.1790	4.5849	9.4947	6.0058	8.2705	6.4153	9.9908	3.5932	5.7140	6.1011	8.5321
${\mathrm{\Omega }}_{{(r}_p)}$	1.1800	1.2726	1.3313	1.2678	1.1025	1.2279	1.2422	1.4747	1.3741	1.4511	1.3651	1.5132	1.1886	1.3286	1.4528	1.4962
Panel C: d3 = 8–16 days
	C.1. BRICS-Gold ADCC								C.2. G7-Gold ADCC
Stat.	Brazil	Brazil + Gold	Russia	Russia + Gold	India	India + Gold	China	China + Gold	France	France + Gold	Italy	Italy + Gold	UK	UK + Gold	USA	USA + Gold
$E_{{(r}_p)}$	0.2651	0.2005	0.3054	0.0721	0.0985	0.0198	0.2171	0.2332	0.2615	0.1819	0.2847	0.1884	0.1518	0.0474	0.2665	0.1687
${\sigma }_{{(r}_p)}$.	0.2636	0.1736	0.1702	0.1278	0.1660	0.1263	0.1647	0.1094	0.1362	0.1005	0.1483	0.1006	0.1403	0.1150	0.1251	0.0795
${\tau }_{{(r}_p)}$	−0.0187	0.3412	−0.0906	0.0166	0.5841	0.1332	−0.2390	0.3863	−0.5060	0.2612	−0.3292	−0.0114	−0.1283	−0.2280	−0.6117	−0.4863
$k_{{(r}_p)}$	3.6447	3.7795	3.7224	3.8500	6.2247	4.6699	3.4547	4.9535	4.1934	4.8410	3.6581	4.2128	3.9666	4.3978	6.3380	5.2286
$\mathit{SR}\left({\sigma }_{{(r}_p)}\right)$	0.9980	1.1432	1.7828	0.5481	0.5813	0.1406	1.3056	2.1123	1.9057	1.7890	1.9059	1.8530	1.0678	0.3942	2.1146	2.0975
$K\left(0,1\right)$	0.1800	0.2026	0.3313	0.0715	0.1025	0.0016	0.2422	0.3899	0.3741	0.3219	0.3651	0.3437	0.1886	0.0294	0.4528	0.3983
$K\left(0,2\right)$	1.4928	1.7351	2.7472	0.6208	0.9010	0.0139	1.9599	3.2378	2.7517	2.6255	2.8234	2.6860	1.5450	0.2451	3.0351	2.9074
$K\left(0,3\right)$	2.7408	3.2948	4.8718	1.1422	1.6895	0.0257	3.5609	5.9176	4.7950	4.6931	5.0140	4.7713	2.8057	0.4389	4.9912	4.9104
$K\left(0,4\right)$	3.5433	4.3922	6.0639	1.4498	2.1965	0.0333	4.5849	7.6274	6.0058	5.8913	6.4153	6.0400	3.5932	0.5493	6.1011	6.0050
${\mathrm{\Omega }}_{{(r}_p)}$	1.1800	1.2026	1.3313	1.0715	1.1025	1.0016	1.2422	1.3899	1.3741	1.3219	1.3651	1.3437	1.1886	1.0294	1.4528	1.3983
Panel D: d6 = 64–128 days
	D.1. BRICS-Gold ADCC								D.2. G7-Gold ADCC
Stat.	Brazil	Brazil + Gold	Russia	Russia + Gold	India	India + Gold	China	China + Gold	France	France + Gold	Italy	Italy + Gold	UK	UK + Gold	USA	USA + Gold
$E_{{(r}_p)}$	0.2651	0.1856	0.3054	0.2097	0.0985	0.1740	0.2171	0.2392	0.2615	0.1471	0.2847	0.2370	0.1518	0.2292	0.2665	0.2950
${\sigma }_{{(r}_p)}$.	0.2636	0.1734	0.1702	0.1601	0.1660	0.2229	0.1647	0.1818	0.1362	0.1216	0.1483	0.1217	0.1403	0.1507	0.1251	0.1372
${\tau }_{{(r}_p)}$	−0.0187	0.1922	−0.0906	0.4465	0.5841	0.0480	−0.2390	0.1866	−0.5060	−0.0663	−0.3292	−0.1141	−0.1283	0.5862	−0.6117	0.2631
$k_{{(r}_p)}$	3.6447	4.3960	3.7224	4.5878	6.2247	5.1187	3.4547	4.9111	4.1934	4.9434	3.6581	4.1253	3.9666	5.6611	6.3380	10.4202
$\mathit{SR}\left({\sigma }_{{(r}_p)}\right)$	0.9980	1.0587	1.7828	1.2970	0.5813	0.7717	1.3056	1.3048	1.9057	1.1928	1.9059	1.9308	1.0678	1.5077	2.1146	2.1344
$K\left(0,1\right)$	0.1800	0.1826	0.3313	0.2426	0.1025	0.1281	0.2422	0.2162	0.3741	0.2342	0.3651	0.4257	0.1886	0.2635	0.4528	0.5135
$K\left(0,2\right)$	1.4928	1.5538	2.7472	2.0260	0.9010	1.0430	1.9599	1.6934	2.7517	1.8362	2.8234	3.2883	1.5450	2.1894	3.0351	3.2972
$K\left(0,3\right)$	2.7408	2.8299	4.8718	3.6671	1.6895	1.8382	3.5609	3.0120	4.7950	3.2081	5.0140	5.8062	2.8057	3.9987	4.9912	5.2487
$K\left(0,4\right)$	3.5433	3.5575	6.0639	4.6817	2.1965	2.2834	4.5849	3.8350	6.0058	4.0112	6.4153	7.2188	3.5932	5.1393	6.1011	6.2899
${\mathrm{\Omega }}_{{(r}_p)}$	1.1800	1.1826	1.3313	1.2426	1.1025	1.1281	1.2422	1.2162	1.3741	1.2342	1.3651	1.4257	1.1886	1.2635	1.4528	1.5135

This table reports on the different performance measures and for the various single MSCI indices (passive management) and combined strategies MSCI-Gold (active management) under study in terms of local currencies for the year 2019. The information regarding the different rebalancing frequencies is clearly divided into three sections: Panel A describes the daily log-returns evaluation, while Panels B and C detail the short and mid-run assessments, respectively. Panel D reports on the long-run or low frequency analysis. By rows the information could be divided into three categories: statistics of the four order moments of the distribution, classical performance ratios (Sharpe) and downside risk measures (Kappa and Omega indices).

As we compare cross-currency investments, we find that diversification benefits appear to be more pronounced on US dollar-denominated investments when returns are undecomposed. Conversely, we report that the largest reductions in volatility are suggested by strategies denominated in local currencies worldwide when the different wavelet decompositions are considered. It is also noteworthy that the benefits of diversification become greater as the frequency of decomposition approaches to d6 (lower frequencies). Besides, though the overall volatility and performance results are better during the 2019 year or pre-pandemic period, it is more informative to study when the countries improve more by combining gold into their strategy: before the pandemic or during the COVID-19 outbreak. In other words, to study the relative differences (ups or downs) in terms of both risk and performance. Interestingly, overall, the results of the study indicate that gold – regardless of the difference between dollars and national currencies – is able to reduce equity volatility in a greater degree and also contributes to improve more the performance for all the countries, and return-decompositions during the 2020 pandemic period.

In terms of dollar returns, we observe that gold tends to contribute more to a relative risk reduction (compared to investments in country equities individually) in terms of volatility – except for China – and kurtosis. It does not appear to be the case for the portion of positive risk measured by skewness, which is incremented more by including gold in the pre-pandemic period.19 In addition, China’s exception for the volatility case appears to lead to its poorer performance in 2020 for all ratios analyzed. It must be noted that again this is the only exception, because in the rest of the countries the precious metal offers much more diversification benefit in 2020, thereby reducing volatility of the combined strategies by 10–20% in comparison with the pre-pandemic period, while the relative performance is multiplied by 1, 2, 10 or even 900 in the particular case of Italy. The findings exhibited by the wavelet decompositions of the dollar returns follow a very similar pattern to those observed for the undecomposed series, in that they show an enhance in relative risk reduction and performance during the pandemic period compared to the pre-pandemic one, with the caveat that in addition to the case of China (already observed for the undecomposed returns), the number of exceptions increases as the frequency of the decomposition is reduced up to d6. We also draw a very interesting conclusion regarding dollar investments: gold contributes significantly more to the reduction of the relative risk of almost all equity strategies when we work with wavelet decompositions rather than undecomposed returns. This is because the relative reduction in volatility of the combined strategies is much more pronounced in the different decompositions (d1, d3 and d6) than in the undecomposed returns. There is less evidence of a relative performance increase between returns and wavelet decompositions, so we cannot assert a preference for one over the other in regard to these parameters.

The results in local currency are consistent with those found in dollars, in the sense that they reflect the dominance of 2020 over 2019 in terms of relative risk reduction. The same cannot be said in terms of performance; as we move from undecomposed returns to wavelet decompositions, we report an increase in the number of exceptions. Indeed, the evidence reverses for the lowest frequencies (d6), and gold contributes more to the outperformance in five of the eight countries studied during the pre-pandemic year 2019. Additionally, the analysis of local currencies reports similar findings to that of dollars, as gold significantly reduces more the relative risk of almost all equity strategies when decomposing these returns with wavelets, but in this case this effect is more moderate and applies mainly to the d6 decompositions.

7. Concluding remarks

The feasibility of Gold as a safe-haven and diversifying asset in traditional portfolios is revisited during the current COVID-19 pandemic period. Specifically, in this paper we go beyond the investment opportunities suggested by Gold in its daily frequency regime, decomposing this time dimension into lower timescales. Thus, we propose a portfolio rebalance and ex-post performance assessment, in which BRICS and G7 MSCI indices and Gold-MSCI combined strategies are confronted by modelling the time-varying portfolio’s density over time. We conduct the empirical experiment in four different steps. First, the MODWT wavelet transform is conducted to decompose the initial daily return series into lower frequencies: d1 = 2–4 days, d2 = 4–8 days, d3 = 8–16 days, d4 = 16–32 days, d5 = 32–64 days, d6 = 64–128 days, d7 = 128–256 days. Second, on the basis of the minimum AIC and BIC criteria different univariate heteroscedasticity specifications are implemented to model the marginal distributions and the ADCC model is estimated to fit the dependence structure of the various MSCI indices and Gold over the period that spans from January 2018 to December 2019, the in-sample period (training timeframe to select the most accurate models and calibrate parameters). Third, to avoid spurious findings, a re-estimation and one-day ahead forecast procedure is conducted over the year 2020, the out-of-sample period. Fourth, we construct and rebalance portfolios on the basis of a minimum skewed student’s t VaR strategy and assess them from the view of risk and performance in the range January-December 2020. Complementarily, we conduct a robustness analysis in which we separate the study of pre- and during COVID-19 pandemic. We select returns data with an equivalent length to the one implemented in the main analysis (January 2017 – December 2018) for the in-sample calibration of the models and 2019 is left for the out-of-sample experiment.

From the preliminary analysis of the time-varying ADCC correlations we report on some of homogeneity between the patterns described by the various ADCC G7-Gold dependence structures, while BRICS fail to report such attribute (Brazil-Gold pair exhibit a vert disparate trend regarding the rest of the BRICS set). Our research reports interesting evidence regarding these correlation trends. On the one hand, we report on strong dependencies during the first wave of the COVID-19, suggesting a contagion effect between very distant geographical markets around the early March 2020. These effects are more pronounced for the log-returns and the d1 scale, while less so for lower frequencies. Notwithstanding, we find that these high correlation structures drop after the first wave and show that in average terms the diversifying patterns are very favorable to Gold during the whole pandemic period and for the different timescales under study. Besides, our results reveal that the higher the frequency of the wavelet decomposition, the weaker the dynamic correlation among the different MSCI indices and Gold. Thus, an increase in frequency could lead to surge in the potential diversification benefits of Gold, whereas a shrinkage of the timescale could mean just the opposite. On the other hand, going from the short- to the long-term decompositions, the variability of the portfolio weights becomes higher and a constant swapping of positions between Gold and MSCI indices is observed. Obviously, these bumpy patterns described by portfolio weights at low decomposition frequencies come from the dynamics of the time-varying correlations between wavelet decompositions of returns at very low frequencies (i.e., d6). In this regard, it may be of interest to develop a line of future research to show whether it is possible to take advantage of the potential diversification benefits for market participants with long investment horizons (represented by the low-frequency wavelet decompositions). The Markov Switching DCC GARCH model will be suitable to fit the regime switching behavior inherent to these frequencies.

Additionally, the comparison of time-varying correlations across different currencies reveals that log-return dependence among gold and equity is weaker for dollar-unified investments (often near zero or even negative) than for local currency investments, both pre-pandemic and post-pandemic, suggesting that diversification is improved by North American currency-denominated investments in terms of undecomposed returns. By contrast, when returns are expressed as wavelet decompositions d1, d3, and d6, local currency investments show the weakest correlations, and therefore indicate greater risk diversification. The dominance of diversification across periods is less evident. On the one hand, we find that correlations between undecomposed returns are often weaker in the pre-pandemic period both from the view of the unified dollar and the national currency returns, with the caveat that these patterns only apply to countries in the G7 for the latter case. On the other hand, the evidence in terms of wavelet decompositions is contrary. While the dynamic dependences are weaker during the pandemic period when examining dollar returns decomposed at high frequencies, the connectedness is lower during the pre-pandemic for d1 and d3 local decomposed returns.

Unlike the analysis of dynamic correlations, we report that in terms of ex-post risk and performance, there is no pattern that reveals significant differences between the combined strategies that include indices related to BRICS and G7 countries. Regarding risk as measured by volatility, our study follows the recent literature and highlights the strong diversifying role of Gold during a time of instability such as the current pandemic. Our study goes beyond the traditional financial literature, demonstrating that the strongly uncorrelated and diversifying nature of Gold is not only reflected in the continuous returns of this commodity, but also in the potential shocks that makes an influence for the short or long term. In short, we demonstrate the benefits of Gold as a safe-haven asset for different timescales. In line with previous dependence findings, we note that the risk reduction provided by Gold decreases with lower frequencies of the shocks. Our results go further and show that when we just consider the downside risk, or the risk-return relations (performance measures), mid-frequency wavelet decompositions, d3, exhibit the best results on average. Therefore, we conclude that a Gold dependence analysis across time and frequencies is pertinent and necessary to allow policy and portfolio makers to manage these assets time-efficiently during instable market periods.

Likewise, the cross-currency ex-post analysis shed some light on the greater diversification advantages when US returns are undecomposed, while strategies with local currency exposure seem to reduce volatility the most when different wavelet decompositions are considered. Furthermore, the comparative period study suggests that gold – regardless of the difference between unified-dollar and local currency investments – reduces equity volatility at a greater degree and also contributes to a higher outperformance for all countries and return-decompositions during the pandemic period. In this vein, our study fundamentally conclude that gold is always a reliable diversifier for equity portfolios – no matter the currency nor the assessment period – , but it is especially useful during recessions, when it acts as a safe haven and performs remarkably well.20

Funding

This work was supported by Ministerio de Economía, Industria y Competitividad (ECO 2017-89715-P), Universidad de Castilla-La Mancha (2020-GRIN-28832, co-financed with FEDER funds) and Excma. Diputación Provincial de Albacete – UCLM (DIPUAB-2021-4, Proyectos de I + D + i para jóvenes investigadores 2021).

CRediT authorship contribution statement

Carlos Esparcia: Data curation, Formal analysis, Funding acquisition, Investigation, Methodology, Software, Supervision, Validation, Writing – original draft, Writing – review & editing. Francisco Jareño: Formal analysis, Funding acquisition, Investigation, Project administration, Supervision, Validation, Writing – original draft, Writing – review & editing. Zaghum Umar: Data curation, Funding acquisition, Investigation, Methodology, Software, Validation.

Appendix Conditional heteroscedasticity and correlations: GARCH models

Regarding the time-varying modelling of the different parameters involved in the portfolio choice, we divide the explanation of the different methods into five subsections. The first subsection A.1. is focused on analyzing the AR models implemented to characterize the mean equation and to capture the possible autoregressive presence in the first moment of the time series distribution. These AR models are used to filter all the series, except for the log-returns series, where we do not find sufficient autoregressive presence, as it is shown in section 4. Moreover, from subsection A.2. to A..4., we describe the different specifications used to model the conditional heteroskedasticity of the series. The choice of the correct model as well as the selection criteria are described in the main text (section 5). Subsection A.5. reports the modeling of dependencies between asset pairs based on the ADCC GARCH model.

A.1. Modelling the mean equation: AR (1) specifications

For regression purposes, to model the various subsequent univariate conditional heteroscedasticity types (GARCH, E-GARCH, GJR-GARCH), we fit the residuals from a prior AR (1) specification:21

$$\begin{array}{c}{r}_{j,t}={\varphi }_j{r}_{j,t-1}+{\epsilon }_{j,t}\\ {\epsilon }_{j,t}={\sigma }_{j,t}{\eta }_{j,t}{\eta }_{j,t}\sim N\left(0,1\right)\end{array}$$ (14)

where $r_{j,t}$ denotes the assets’ return, ${\epsilon }_{j,t}$ are the residuals of current period, ${\sigma }_{j,t}$ is the returns’ volatility and ${\eta }_{j,t}$is the standardized innovation of the process.

A.2. Fitting the marginals: The standard GARCH model

The standard GARCH model of Bollerslev (1986) states that the current variance depends not only on squared innovations of the previous period, but also on its prior variance:

$${\sigma }_{j,t}^2={\omega }_j+{\alpha }_j{\epsilon }_{j,t-1}^2+{\beta }_j{\sigma }_{j,t-1}^2$$ (15)

where${\omega }_j$ is the constant of the model, $\alpha$ is the parameter of the ARCH component model, ${\beta }_j$ the parameter of the GARCH component model, ${\epsilon }_{j,t-1}$ is the model's residual at time $t-1$ and ${\sigma }_{j,t-1}^2$is the variance of the previous period.

A.3. Fitting the marginals: The Exponential GARCH (E-GARCH) model

The Exponential GARCH model of Nelson (1991) is an autoregressive specification that exhibits leverage effect and by means of its logarithmic specification, ensures a positive volatility process:

$$\ln \left({\sigma }_{j,t}^2\right)={\omega }_j+{\alpha }_j{\epsilon }_{j,t-1}+{\gamma }_j\left(\left|{\epsilon }_{j,t-1}\right|-\sqrt{2/\pi }\right)+{\beta }_j\mathrm{l}\mathrm{n}\left({\sigma }_{t-1}^2\right)$$ (16)

where ${\omega }_j$, the constant of the model, has less impact in the current volatility process as it turns more negative. ${\alpha }_j$, the parameter of the ARCH component model, captures the sign effect, while ${\gamma }_j$ measures the size effect of the process. ${\beta }_j$, the parameter of the GARCH component model, reports on the persistence in volatility. ${\epsilon }_{j,t-1}$ are the residuals of the prior period for each of the considered assets. $\sqrt{2/\pi }$ depicts the expected absolute value of residuals.

A.4. Fitting the marginals: The GJR-GARCH model

A negative return on a stock implies a drop in the market value of the company, which increases its financial leverage, increasing its level of risk (at the same level of debt). As proposed by Glosten et al. (1993), we can modify the GARCH(1,1) model to capture this leverage effect as follows:

$${\sigma }_{j,t}^2={\omega }_j+{\alpha }_j{\epsilon }_{j,t-1}^2+{\gamma }_j{I}_{t-1}{\epsilon }_{j,t-1}^2+{\beta }_j{\sigma }_{j,t-1}^2$$ (17)

where ${\gamma }_j$ measures the leverage effect and $I_t$is a function that returns 1 for $\mathrm{\epsilon }<0$ and 0 otherwise. Given the $I_t$ indicator function, the persistence of the process ($\widehat{P}={\alpha }_j+{\gamma }_{j}k+{\beta }_j$) critically depends on the skewed and heavy tailed patterns of the implemented distribution. $k$ reports on the expected value of the standardized residuals, $k=E\left[I_{t-1}{\eta }_{j,t-1}^2\right]={\int }_{-\infty }^{0}f\left(\eta ,0,1,\cdots \right)d\eta$.

A.5. Multivariate methodology

The seminal DCC GARCH model was first introduced by Engle (2002) to model conditional correlations over time. In this study, we implement the asymmetric variant proposed by Cappiello et al. (2006), the ADCC GARCH specification. This model depends on instrumental variables playing the role of covariances, $q_{\mathit{ij},t}$. The relative dependence process between each country-Gold pair can be expressed as follows:

$$q_{\mathit{ij},t}=\left({\overline{q}}_{\mathit{ij}}-{\alpha }_{\mathit{ij}}{\overline{q}}_{\mathit{ij}}-{\beta }_{\mathit{ij}}{\overline{q}}_{\mathit{ij}}-{\xi }_{\mathit{ij}}{\overline{q}}_{\mathit{ij}}\right)+{\alpha }_{\mathit{ij}}\left({\eta }_{i,t-1}{\eta }_{j,t-1}\right)+{\beta }_{\mathit{ij}}{q}_{\mathit{ij},t-1}+{\xi }_{\mathit{ij}}\left({\eta }_{i,t}^{-}{\eta }_{j,t}^{-}\right)$$ (18)

where $q_{\mathit{ij},t}$ are instrumental variables that play the role of covariances at each moment in time, $t$, while ${\overline{q}}_{\mathit{ij}}$ depict the same but under their unconditional form. ${\alpha }_{\mathit{ij}}$ is the parameter that measures the degree to which new information affects the dependency process among pairs. ${\beta }_{\mathit{ij}}$ is the parameter that reports on persistence in correlation. ${\xi }_{\mathit{ij}}$ is the parameter that controls the skewed behavior of the dependence structure. ${\eta }_{i,t}$ and ${\eta }_{j,t}$ are the standardized innovations of the assets, obtained from the univariate GARCH models. ${\eta }_{i,t}^{-}$ and ${\eta }_{j,t}^{-}$ are the zero-threshold standardized errors which are equal to ${\eta }_{i,t}$ and ${\eta }_{j,t}$ when less than zero else zero.

Then, to extract the conditional correlations, ${\rho }_{\mathit{ij},t}$, we standardize the prior instrumental variables, $q_{\mathit{ij},t}$:

$${\rho }_{\mathit{ij},t}=\frac{q_{\mathit{ij},t}}{\sqrt{q_{\mathit{ij},t}}\sqrt{q_{\mathit{ij},t}}}$$ (19)