# Journal of Risk

**ISSN:**

1465-1211 (print)

1755-2842 (online)

**Editor-in-chief:** Farid AitSahlia

# Empirical analysis of oil risk-minimizing portfolios: the DCC–GARCH–MODWT approach

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Abstract

This paper strives to analyze hedging strategies between Brent oil and six other heterogeneous assets – American ten-year bonds, US dollars, gold, natural gas futures, corn futures, and Europe, Australasia and Far East exchange-traded funds (EAFE- ETFs) – observing five wavelet time horizons and considering three different risk metrics: variance, value-at-risk (VaR) and conditional value-at-risk (CVaR). The authors constructs two-asset portfolios, whereby conditional variances and covariances are obtained via a bivariate rolling dynamic conditional correlation–generalized autore- gressive conditional heteroscedasticity (DCC–GARCH) model. Results indicate that gold is the best combination with Brent for minimum-variance investors, while the Brent–natural gas pair produces the worst minimum-variance results due to the very high unconditional variance of gas. As for VaR and CVaR results, the authors find that Brent with gold gives relatively good outcomes, but the portfolio with gas heavily outperforms the portfolio with gold when one views longer time horizons. This happens because the Brent–gas portfolio has very low skewness and kurtosis on longer time horizons compared with the unhedged portfolio, and these characteristics favor good VaR and CVaR results. These findings could help global portfolio managers and investors who seek various ways to diversify their Brent oil investments, who act on different time horizons, and who target different risk-minimizing goals.

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Introduction

## Abstract

This paper strives to analyze hedging strategies between Brent oil and six other heterogeneous assets – American ten-year bonds, US dollars, gold, natural gas futures, corn futures, and Europe, Australasia and Far East exchange-traded funds (EAFE-ETFs) – observing five wavelet time horizons and considering three different risk metrics: variance, value-at-risk (VaR) and conditional value-at-risk (CVaR). We construct two-asset portfolios, whereby conditional variances and covariances are obtained via a bivariate rolling dynamic conditional correlation–generalized autoregressive conditional heteroscedasticity (DCC–GARCH) model. Results indicate that gold is the best combination with Brent for minimum-variance investors, while the Brent–natural gas pair produces the worst minimum-variance results due to the very high unconditional variance of gas. As for VaR and CVaR results, we find that Brent with gold gives relatively good outcomes, but the portfolio with gas heavily outperforms the portfolio with gold when one views longer time horizons. This happens because the Brent–gas portfolio has very low skewness and kurtosis on longer time horizons compared with the unhedged portfolio, and these characteristics favor good VaR and CVaR results. These findings could help global portfolio managers and investors who seek various ways to diversify their Brent oil investments, who act on different time horizons, and who target different risk-minimizing goals.

## 1 Introduction

Since the 1970s, the oil market has significantly increased in importance to become one of the world’s largest commodity markets in terms of raw production as well as in dollar amount traded (see Chang et al 2011). Some authors, such as Alexis (2011), Broadstock et al (2014), Arouri and Rault (2012), Cuestas and Gil-Alana (2018) and Z̆ivkov et al (2019b), have contended that oil is an exceptionally important commodity for industry and for the transportation and agricultural sectors, and that its impact on the world economy is more influential than that of other commodities. However, it is well known that the oil market is susceptible to volatile and erratic behavior, as demonstrated by huge rises and falls in the market. Aloui and Mabrouk (2010) asserted that the average volatility of the crude oil prices is 2.5 times that of the US stock market and more than 3 times that of the world’s major foreign exchange rates, with average volatility sometimes exceeding 37% per year. Risk in the oil market is present for numerous reasons, eg, unexpected jumps in global oil demand, global oil reserve policies, decreases (increase) in the capacity of crude oil production, major global economic and political crises, and so on. According to Chang et al (2011), the increasing accessibility to financial and commodity markets for a broad range of participants allows global investors extended opportunities to hedge oil price risk. Some recent papers have investigated how oil price risk can be hedged in order to manage the inevitable and frequent periods of turbulence in the oil market more successfully (see, for example, Chang et al 2011; Cifarelli and Paladino 2012; Pan et al 2014; Khalfaoui et al 2015; Mirović et al 2017). However, the majority of the available studies have only considered a limited number of instruments for the purposes of diversification or hedging. In addition, it is commonly argued in the financial literature that the risk of a financial asset or portfolio is uniquely shaped by its observed time horizon. However, due to the fact that efficient portfolio construction depends on the sample periods, data frequency and technical specifications of each time series, very few empirical studies have considered the issue of different hedging horizons. Conlon and Cotter (2012) explained that the sample reduction problem arises when researchers try to match the frequency of data with the hedging horizon. Analysis of dynamic hedging for different time horizons has therefore not been sufficiently studied.

This paper endeavors to contribute to the literature by considering a dynamic hedging strategy with the two-asset portfolios, in which spot Brent oil is a primary investment. Brent oil is coupled with six heterogeneous types of auxiliary assets: American ten-year bonds, US dollars, gold, natural gas futures, corn futures, and Europe, Australasia and Far East exchange-traded funds (EAFE-ETFs). In order to deal with investors who have different term objectives, we use maximum overlap discrete wavelet transformation (MODWT), which is a handy mathematical tool for a time-frequency representation of a time series. Unlike traditional methodologies, this technique observes different time horizons without shrinking the sample size and without the wastage of valuable information. According to Dewandaru et al (2014), the wavelet function allows temporality, nonstationarity and volatility changes over time. The idea of using wavelets in this study came from recent papers such as Dajčman (2012, 2013), Altar et al (2017), Z̆ivkov et al (2018), Jiang et al (2018) and Si et al (2018), all of which used wavelet-transformed series in the context of different time-horizon observations.

Our intention is to combine the wavelet approach with the dynamic conditional correlation–generalized autoregressive conditional heteroscedasticity (DCC–GARCH) model. However, in order to avoid look-ahead bias in the process of portfolio construction, as well as possible structural changes, we use the rolling DCC–GARCH approach. In other words, we set a fixed-width window of two years, and this window is then rolled over in the DCC–GARCH estimation process throughout the whole sample. By doing this, we acquire rolling average conditional variances and conditional covariances, from which, in combination with a MODWT methodology, we can see how these conditional variances and covariances vary over time and over different investment horizons. The idea of using the rolling DCC–GARCH approach came from the paper of Ahmad et al (2018). Subsequently, using these estimates as inputs, we can calculate dynamic optimal in-sample portfolio weights via the Kroner and Ng (1998) equation, which computes the minimum-variance portfolio by default. Knowing that various market participants have different risk-minimizing goals (see, for example, Z̆iković and Filer 2013; Kabaila and Mainzer 2018; Bastin 2018), we also calculate two more portfolio risk metrics: value-at-risk (VaR) and conditional value-at-risk (CVaR). To the best of our knowledge, this paper is the first to comprehensively analyze oil portfolio diversification strategies using a broad range of auxiliary instruments, observing different term horizons and considering three different risk metrics.

The rest of the paper is constructed as follows. Section 2 provides a review of the literature. Section 3 explains the wavelet approach, the rolling DCC–GARCH model and the methods for constructing the portfolio. Section 4 is reserved for discussing the data set. Section 5 presents the results of risk-minimizing performances across wavelet scales and using different risk metrics. Section 6 concludes.

## 2 A brief literature review

Many studies have analyzed the risk-reduction issue in the oil market, since increased global market integration has facilitated new ways of diversifying, hedging and managing risk. Pan et al (2014) explored the strategy of hedging crude oil using refined products and a regime-switching asymmetric DCC (RS–ADCC) model. They concluded that heating oil is better able to hedge crude oil than gasoline. Yun and Kim (2010) investigated the hedging effectiveness of different types of hedge by Korean oil traders. Both crude oil price and exchange rate risk were considered. They found that there exists an inverse relationship between hedging effectiveness and crude oil price sensitivity to exchange rate. In other words, hedging effectiveness tends to improve when crude oil prices become more volatile and/or exchange rates get less volatile. Chang et al (2010) scrutinized four major benchmarks in the international oil market: namely, West Texas Intermediate (WTI; the United States), Brent (the North Sea), Dubai/Oman (the Middle East) and Tapis (Asia-Pacific). They concluded that when trying to minimize risk by using a hedge, a long position of one dollar in the light sweet grade category (WTI) should be shorted by only a few cents in the heavier and less sweet grade category (Dubai/Oman and Tapis). Mnasri et al (2017) examined the motivations and the value effect of nonlinear hedges, using a new data set on the hedging activities of 150 US oil producers. They revealed that nonlinear hedging strategies are motivated by the sensitivities of firms’ investment expenditures and revenues to oil price fluctuations and by quantity–price correlation. The authors also claimed that investment opportunities, production uncertainty and changes in oil prices and volatilities play a significant role in hedging strategy choice.

Khalfaoui et al (2015) examined the linkages between the crude oil market (WTI) and the stock markets of the G7 countries using the wavelet-based multivariate GARCH approach. They found that hedging ratios and optimal weights vary across timescales, and they claimed that their results showed that investors and financial market participants should invest less in stocks than in crude oil. Reboredo (2013) assessed the role of gold as a hedge or safe haven against oil price movements using an approach based on copulas. Empirical evidence showed that gold can act as an effective safe haven against extreme oil price movements. Chang et al (2011) looked at the crude oil spot and futures returns of two major benchmark international crude oil markets – Brent and WTI – and calculated optimal portfolio weights and optimal hedge ratios using several multivariate GARCH models. Their empirical results showed that the optimal portfolio weights of all multivariate volatility models for Brent suggest holding futures in larger proportions than spot. On the other hand, for WTI, the DCC, BEKK (named after Baba, Engle, Kraft and Kroner) and diagonal BEKK models suggest holding crude oil futures rather than spot, but the constant conditional correlation (CCC) and vector autoregressive moving average–GARCH (VARMA–GARCH) models suggest holding crude oil spot rather than futures. Chkili et al (2014) investigated the dynamic relationships between the US stock market and two international crude oil markets (Brent and WTI), utilizing the DCC–fractionally integrated asymmetric power autoregressive conditional heteroscedasticity (FIAPARCH) model. They discussed some implications for portfolio management and risk hedging, contending that investors can improve the risk-adjusted performance of their portfolios by taking appropriate action in the crude oil futures market. They also claimed that American investors should allocate more weight to stocks than to crude oil.

## 3 Methodology

### 3.1 The MODWT approach

Wavelet methodology is a powerful mathematical instrument for time series analysis, and it has the capability to decompose a time series into its high- and low-frequency components, which are associated with different scales of resolution. In particular, it projects the original series onto a sequence of basic functions called wavelets. According to Gencay et al (2002), wavelets allow an appropriate trade-off between resolution in the time and frequency domains, while traditional Fourier analysis lacks this ability, in that it stresses the frequency domain at the expense of the time domain. Therefore, wavelet theory provides an efficient and convenient method to analyze complex signals. Recently, numerous financial articles have used discreet wavelet transformation for a range of empirical research: see, for example, the papers by Nikkinen et al (2011), Madaleno and Pinho (2012), Lee and Lee (2016), Njegić et al (2017) and Z̆ivkov et al (2019a).

From a theoretical perspective, there are two basic wavelet functions: the father wavelet ($\phi $) and the mother wavelet ($\psi $). Wavelets are nonlinear functions that can be rescaled and moved to form a basis in a Hilbert space of square integrable functions ($f\in {L}^{2}$). The father wavelets augment the representation of the smooth or low-frequency parts of a signal with an integral equal to $1$, and the mother wavelets are helpful in describing the details of high-frequency components with an integral equal to $0$. The long-term trend over the scale of the time series is portrayed by the father wavelet, while the mother wavelet delineates fluctuations in the trend. The most commonly used wavelets are the orthogonal ones, and the approximation to a continuous signal function $f(t)$ in ${L}^{2}(R)$ is as follows:

$$f(t)=\sum _{k}{s}_{J,k}{\varphi}_{J,k}(t)+\sum _{k}{d}_{J,k}{\psi}_{J,k}(t)+\sum _{k}{d}_{J-1,k}{\psi}_{J-1,k}(t)+\mathrm{\cdots}+\sum _{k}{d}_{1,k}{\psi}_{1,k}(t),$$ | (3.1) |

where $j$ denotes the number of multiresolution components or scales and where $k$ ranges from $1$ to the number of coefficients in the corresponding component. The coefficients ${s}_{J,k}$, ${d}_{J,k}$, $\mathrm{\dots}$, ${d}_{1,k}$ stand for the wavelet-transform coefficients that can be approximated by the following integrals:

${s}_{J,k}$ | $\approx {\displaystyle \int (t){\varphi}_{J,k}(t)dt},$ | (3.2) | ||

${d}_{j,k}$ | $\approx {\displaystyle \int (t){\psi}_{j,k}(t)dt},j=1,2,\mathrm{\dots},J.$ | (3.3) |

These coefficients represent a measure of the contribution of the corresponding wavelet function to the total signal, whereas the functions ${\varphi}_{J,k}$ and ${\psi}_{j,k}$ are the approximating wavelet functions. In other words, they are the scaled and translated versions of $\varphi $ and $\psi $. Generally, these functions are generated from $\varphi $ and $\psi $ in the following way:

${\varphi}_{J,k}(t)$ | $={2}^{-J/2}\varphi \left({\displaystyle \frac{t-{2}^{J}k}{{2}^{J}}}\right),$ | (3.4) | ||

${\psi}_{j,k}(t)$ | $={2}^{-j/2}\psi \left({\displaystyle \frac{t-{2}^{j}k}{{2}^{j}}}\right).$ |

According to (3.4), $j$ is typically referred to as the “level”, while the scale or dilation factor is ${2}^{j}$ and the translation or location parameter is ${2}^{j}k$. The scale factor ${2}^{j}$ grows as much as $j$ does; it is a measure of the width of the functions ${\varphi}_{J,k}(t)$ and ${\psi}_{j,k}(t)$ and it affects the underlying functions, making them shorter and more dilated. Also, when $j$ increases, the translation steps automatically get larger in order to accommodate the level of the scale parameter ${2}^{j}$.

The most commonly used types of wavelet transformations are the discrete wavelet transformation (DWT) and the MODWT.^{1}^{1} 1 Maximum overlap discrete wavelet transformation is done via R software, using the waveslim package. The former uses orthonormal transformations of the original series, while the latter is based on a highly redundant nonorthogonal transformation. For our empirical purposes we employ the MODWT, which is a linear filtering operation that transforms a series into coefficients related to variations over a set of scales.

As for our study, we perform multiresolution analysis with five timescales using MODWT with the Daubechies least asymmetric (LA) wavelet filter of length $L=8$, which is also known as the LA(8) wavelet filter. According to Khalfaoui et al (2015), the LA(8) wavelet filter has been widely used and applied in the financial literature because it has been shown to provide the best performance for the wavelet time series decomposition.

### 3.2 The rolling bivariate DCC–GARCH framework

In order to avoid look-ahead bias in the process of estimating minimum-variance portfolios, we use a bivariate rolling DCC–GARCH(1,1) model.^{2}^{2} 2 The DCC–GARCH model is calculated via R software, using the rmgarch package. The univariate GARCH specification is chosen because it shows the highest level of robustness – in comparison with asymmetric GARCH models (the threshold GARCH (TGARCH), exponential GARCH (EGARCH) and periodic GARCH (PGARCH) specifications are also considered) – when it comes to model convergence. As has been said earlier, the two-year window is rolled over (504 days) by adding one day at the beginning of the fixed window and dropping one day at the end of it. In this way, we lose two years of rolling estimates. More than 3100 rolling in-sample conditional variances and conditional covariances are estimated for every pair of assets, and they are subsequently used in the computation of minimum-variance portfolios. Generally, the DCC–GARCH methodology harnesses the flexibility of the univariate GARCH model but without the perplexity of conventional multivariate GARCH models (see, for example, Onay and Unal 2012; Z̆ivkov et al 2016; Antonakakis et al 2017). Applying the Student $t$ distribution, the mean equation specification for each wavelet scale (${2}^{j}$) is specified as an AR($p$) model, in order to take into account autocorrelation, whereby the lag order $p$ is determined by the Schwartz information criterion, using a maximum of five lags. The constant is not considered in the mean equation because the wavelet details have mean zero. The mean and GARCH specifications are presented in (3.5) and (3.6):

$${r}_{t}=\sum _{k=1}^{p}{\gamma}_{k}{r}_{t-k}+{\epsilon}_{t},{\epsilon}_{t}\sim \mathrm{St}(0,1,\nu ),$$ | (3.5) | ||

$${\sigma}_{t}^{2}=c+\alpha {\epsilon}_{t-1}^{2}+\beta {\sigma}_{t-1}^{2}.$$ | (3.6) |

Before undergoing wavelet transformation, all the selected empirical asset returns are calculated as $\mathrm{ln}({P}_{t}/{P}_{t-1})$, where ${P}_{t}$ is the closing price at time $t$. In the two-asset DCC–GARCH model, ${r}_{t}$ denotes the wavelet returns of both Brent oil and the selected assets at time $t$. The parameters $\alpha $ and $\beta $ capture the ARCH and GARCH effects, respectively. The symbol ${\epsilon}_{i,t}$ stands for the independently and identically distributed error terms of the selected time series at the particular scale ${2}^{j}$. We assume that wavelet series at all scales probably tend to report nonnormality features such as leptokurtosis, so we opt for the standard Student $t$ distribution. As for the joint multivariate distribution, we also opt for the Student $t$ distribution.

We standardize the asset-return residuals (at scale ${2}^{j}$) from (3.5) in order to estimate DCCs, ie, ${\nu}_{t}={\epsilon}_{t}/\sqrt{{h}_{t}}$, wherein ${\nu}_{t}$ is then used to estimate the parameters of the conditional correlation. The multivariate conditional variance is specified as ${H}_{t}={D}_{t}{C}_{t}{D}_{t}$, where ${D}_{t}=\mathrm{diag}(\sqrt{{h}_{11,t}}\mathrm{\cdots}\sqrt{{h}_{nn,t}})$ and ${h}_{nn,t}$ represents the conditional variance at some particular scale ${2}^{j}$, which is obtained from the univariate GARCH model in the first stage. The evolution of correlation in the DCC model is presented as

$${Q}_{t}=(1-a-b)\overline{Q}+a{\nu}_{t-1}{\nu}_{t-1}^{\prime}+b{Q}_{t-1},$$ | (3.7) |

where $a$ and $b$ are nonnegative scalar parameters of the DCC(1,1) model under the condition $$. These parameters capture the effects of previous shocks and previous DCCs on current DCCs, respectively. The symbol ${Q}_{t}=[{q}_{nm,t}]$ describes a $2\times 2$ time-varying symmetric positive definite covariance matrix of residuals at the scale ${2}^{j}$. The symbol $\overline{Q}=E[{\nu}_{t}{\nu}_{t}^{\prime}]$ stands for a $2\times 2$ time-invariant variance matrix of ${\nu}_{t}$. Since ${Q}_{t}$ does not have unit elements on its diagonal, it is scaled to obtain a proper correlation matrix (${C}_{t}$) according to the following equation:

$${C}_{t}={(\mathrm{diag}({Q}_{t}))}^{-1/2}{Q}_{t}{(\mathrm{diag}({Q}_{t}))}^{-1/2}.$$ | (3.8) |

Accordingly, the element of ${C}_{t}$ that is denoted by ${\rho}_{nm,t}$ can, for a bivariate case, be written as

$${\rho}_{nm,t}=\frac{{q}_{nm,t}}{\sqrt{{q}_{nn,t}{q}_{mm,t}}}.$$ |

All DCC–GARCH models were estimated using the quasi-maximum likelihood technique. Referring to Bollerslev and Wooldridge (1992), this procedure allows asymptotically consistent parameter estimates even if the underlying distribution is not normal.

### 3.3 Portfolio construction and hedging effectiveness

This section explains how the two-asset wavelet portfolios are constructed using the time-varying conditional correlations and conditional variances as inputs. Using this method, the time series are obtained from the rolling DCC–GARCH models. In the process of risk-reduction evaluation, we start with the minimum-variance hedge effectiveness ${\mathrm{HEI}}_{\mathrm{Var}}$. Although minimum-variance hedging undoubtedly reduces the standard deviation of portfolio returns, its effect on skewness and kurtosis is ambiguous, as Harris and Shen (2006) have contended. This poses a problem for investors who target different aspects of portfolio performance. Therefore, we consider two additional risk measures, ${\mathrm{HEI}}_{\mathrm{VaR}}$ and ${\mathrm{HEI}}_{\mathrm{CVaR}}$, that strive to stipulate whether the reduction in a portfolio’s standard deviation that can be gained from minimum-variance hedging offers a similar reduction in the VaR and CVaR of a portfolio. We construct a Kroner and Ng (1998) portfolio that minimizes variance without lowering expected returns, in which Brent oil is the basic investment and it is coupled with one of the other six auxiliary assets. Calculation of the dynamic portfolio weights at level $j$, with the following restrictions, gives us (3.9) and (3.10):

${W}_{t}^{\mathrm{Brent},\mathrm{asset}}(j)$ | $={\displaystyle \frac{{h}_{t}^{\mathrm{Brent}}(j)-{h}_{t}^{\mathrm{Brent},\mathrm{asset}}(j)}{{h}_{t}^{\mathrm{Brent}}(j)-2{h}_{t}^{\mathrm{Brent},\mathrm{asset}}(j)+{h}_{t}^{\mathrm{asset}}(j)}},$ | (3.9) | ||

${W}_{t}^{\mathrm{Brent},\mathrm{asset}}(j)$ | $$ | |||

$\mathrm{\hspace{1em}\hspace{1em}}\mathit{\hspace{1em}\hspace{1em}}0\le {W}_{t}^{\mathrm{Brent},\mathrm{asset}}(j)\le 1\vee 1\mid {W}_{t}^{\mathrm{Brent},\mathrm{asset}}(j)>1\},$ | (3.10) |

where ${W}_{t}^{\mathrm{Brent},\mathrm{asset}}$ represents the weight of the assets in a $\mathrm{\$}$1 portfolio of a two-asset holding (the portfolio combines Brent oil and the chosen asset) at time $t$ and level $j$. The symbols ${h}_{t}^{\mathrm{Brent}}$, ${h}_{t}^{\mathrm{asset}}$ and ${h}_{t}^{\mathrm{Brent},\mathrm{asset}}$ refer to the conditional variances of Brent oil and the selected assets, and to the conditional covariance, respectively. The weight of Brent oil in the considered $\mathrm{\$}$1 portfolio is ($1-{W}_{t}^{\mathrm{Brent},\mathrm{asset}}$).

The variance hedging effectiveness index (${\mathrm{HEI}}_{\mathrm{Var}}$) incorporates both upside and downside risk and assigns an equal weight to positive and negative returns, implying that the hedging effectiveness of a particular portfolio is greater when its ${\mathrm{HEI}}_{\mathrm{Var}}$ is higher (closer to 100), and vice versa. Following Arouri et al (2011), ${\mathrm{HEI}}_{\mathrm{Var}}$ is expressed as

$${\mathrm{HEI}}_{\mathrm{Var}}=\frac{{\mathrm{Var}}_{\mathrm{unhedged}}-{\mathrm{Var}}_{\mathrm{hedged}}}{{\mathrm{Var}}_{\mathrm{unhedged}}}\times 100,$$ | (3.11) |

where ${\mathrm{Var}}_{\mathrm{unhedged}}$ indicates the variance of the unhedged portfolio, which is composed of only Brent oil. ${\mathrm{Var}}_{\mathrm{hedged}}$ denotes the variance of the portfolio that is made up of Brent oil and some auxiliary asset according to the previously calculated optimal weights.

However, the variance gauges only the second moment of the returns distribution and cannot differentiate between positive and negative returns, while some investors prefer to know the tail risk of the hedged portfolio. In order to address higher moments of portfolio returns, we refer to Cao et al (2009), Cotter and Hanly (2006), Harris and Shen (2006) and Gregory and Reeves (2008) and calculate an additional indicator: the VaR hedge effectiveness index. Due to space restrictions, we only tested the hedging performance in the negative tail of the returns at the 95% confidence level. Thus, the VaR for the long (buy) trading positions at confidence level $\omega $ looks like

$${\mathrm{VaR}}_{\omega}=\widehat{\mu}+{Z}_{\omega}\widehat{\sigma},$$ | (3.12) |

where ${Z}_{\omega}$ designates the left quantile at $\omega \%$ of the Student $t$ distribution, while $\widehat{\mu}$ and $\widehat{\sigma}$ refer to the estimated mean and standard deviation of a particular portfolio. The hedge effectiveness from the point of VaR reduction can be measured as

$${\mathrm{HEI}}_{\mathrm{VaR}}=\frac{{\mathrm{VaR}}_{\mathrm{unhedged}}-{\mathrm{VaR}}_{\mathrm{hedged}}}{{\mathrm{VaR}}_{\mathrm{unhedged}}}\times 100,$$ | (3.13) |

where ${\mathrm{VaR}}_{\mathrm{unhedged}}$ indicates the VaR of Brent oil and ${\mathrm{VaR}}_{\mathrm{hedged}}$ indicates the VaR of the portfolio composed of Brent oil and an auxiliary asset.

VaR measures the worst expected loss at a given confidence level. In addition, VaR is not a coherent measure of risk if portfolio returns are not drawn from a multivariate elliptical distribution. Therefore, in order to overcome this shortcoming of VaR, we use an alternative measure of risk, CVaR, that measures the mean loss conditional on the fact that the VaR has been exceeded. Aloui and Ben Hamida (2015) showed that, in some cases, CVaR is a more effective measure for controlling risk than VaR. According to Berens et al (2018), CVaR is given as

$${\mathrm{CVaR}}_{\omega}=-\frac{1}{\omega}{\int}_{0}^{\omega}\mathrm{VaR}(x)dx,$$ | (3.14) |

where $\mathrm{VaR}(x)$ is the VaR of a particular two-asset wavelet portfolio. $\omega $ denotes the left quantile of the Student $t$ distribution, and we have applied a confidence level of $\omega =95\%$. Therefore, the performance metric used to assess the hedging effectiveness in CVaR is given as

$${\mathrm{HEI}}_{\mathrm{CVaR}}=\frac{{\mathrm{CVaR}}_{\mathrm{unhedged}}-{\mathrm{CVaR}}_{\mathrm{hedged}}}{{\mathrm{CVaR}}_{\mathrm{unhedged}}}\times 100,$$ | (3.15) |

where ${\mathrm{CVaR}}_{\mathrm{unhedged}}$ indicates the CVaR of Brent oil and ${\mathrm{CVaR}}_{\mathrm{hedged}}$ indicates the CVaR of the portfolio with the selected asset. By using the Kroner and Ng (1998) equation, we are able to test whether the minimum-variance portfolio can offer similarly good hedging performances to investors who target scale-dependent minimum VaR and minimum CVaR outputs.

## 4 The data set and descriptive statistics of wavelet time series

The data set used for the empirical analysis and the calculations of the optimal in-sample portfolio weights comprises the daily spot Brent oil commodity as the primary investment and six diverse types of assets as auxiliary investments: American ten-year bonds, US dollars, natural gas futures, gold futures, corn futures and EAFE-ETFs. We choose ETFs instead of stock indexes because ETFs can be bought and sold throughout the trading day, which some international investors may find appealing (see Blitz and Huij 2012). Also, we opt for Brent oil because this type of oil is one of the most traded in the current global oil market, and because of its liquidity Brent follows the evolution of global oil prices perfectly (see Wlazlowski et al 2011). All empirical series are transformed in five wavelet scales via the MODWT algorithm. We choose only five wavelet scales because rolling DCC–GARCH models have convergence difficulties at higher wavelet scales. By using five wavelet scales, we are able to assess the various hedging effects on different time horizons: scale 1 (D1) has a trading period of 2–4 days, scale 2 (D2) 4–8 days, scale 3 (D3) 8–16 days, scale 4 (D4) 16–32 days and scale 5 (D5) 32–64 days. In this way, we can evaluate hedging performances for short-term investors (scales 1 and 2) and medium-term investors (scales 3–5). The data span ranges from January 2004 to May 2018 for all asset series. All series were obtained from Datastream. Taking into account the lack of availability of some data, due to nonworking days in various energy and commodity markets, all pairs of time series are synchronized according to the existing observations. Due to space limitations, we present graphical images of wavelets only for Brent oil in Figure 1; wavelet plots for the other assets can be obtained by request.

Mean | Variance | Skewness | Kurtosis | JB | LB($Q$) | LB(${Q}^{\text{?}}$) | |

Brent oil | |||||||

D1 | 0.000 | 2.330 | 0.013 | 6.088 | 1 447 | 0.000 | 0.000 |

D2 | 0.000 | 0.988 | 0.018 | 4.952 | 578 | 0.000 | 0.000 |

D3 | 0.000 | 0.506 | 0.167 | 7.568 | 3 183 | 0.000 | 0.000 |

D4 | 0.000 | 0.223 | 0.061 | 4.909 | 555 | 0.000 | 0.000 |

D5 | 0.000 | 0.114 | 0.005 | 3.395 | 23 | 0.000 | 0.000 |

American ten-year bonds | |||||||

D1 | 0.000 | 2.381 | 0.039 | 5.834 | 1 219 | 0.000 | 0.000 |

D2 | 0.000 | 1.073 | 0.094 | 5.774 | 1 174 | 0.000 | 0.000 |

D3 | 0.000 | 0.518 | 0.052 | 5.382 | 863 | 0.000 | 0.000 |

D4 | 0.000 | 0.227 | 0.107 | 4.204 | 227 | 0.000 | 0.000 |

D5 | 0.000 | 0.104 | 0.006 | 3.470 | 33 | 0.000 | 0.000 |

US dollars | |||||||

D1 | 0.000 | 0.195 | 0.062 | 4.427 | 315 | 0.000 | 0.000 |

D2 | 0.000 | 0.097 | 0.012 | 5.218 | 755 | 0.000 | 0.000 |

D3 | 0.000 | 0.043 | $-$0.083 | 4.192 | 222 | 0.000 | 0.000 |

D4 | 0.000 | 0.023 | 0.208 | 4.784 | 515 | 0.000 | 0.000 |

D5 | 0.000 | 0.011 | $-$0.095 | 4.768 | 485 | 0.000 | 0.000 |

Gold | |||||||

D1 | 0.000 | 0.609 | $-$0.008 | 7.076 | 2 528 | 0.000 | 0.000 |

D2 | 0.000 | 0.308 | $-$0.114 | 6.952 | 2 385 | 0.000 | 0.000 |

D3 | 0.000 | 0.136 | $-$0.101 | 6.442 | 1 809 | 0.000 | 0.000 |

D4 | 0.000 | 0.065 | $-$0.148 | 4.372 | 299 | 0.000 | 0.000 |

D5 | 0.000 | 0.038 | $-$0.105 | 3.932 | 138 | 0.000 | 0.000 |

Natural gas | |||||||

D1 | 0.000 | 4.965 | 0.171 | 6.984 | 2 430 | 0.000 | 0.000 |

D2 | 0.000 | 2.182 | 0.120 | 5.920 | 1 304 | 0.000 | 0.000 |

D3 | 0.000 | 1.097 | $-$0.128 | 4.829 | 518 | 0.000 | 0.000 |

D4 | 0.000 | 0.510 | $-$0.003 | 3.603 | 55 | 0.000 | 0.000 |

D5 | 0.000 | 0.281 | 0.189 | 4.064 | 193 | 0.000 | 0.000 |

Corn | |||||||

D1 | 0.000 | 2.147 | 0.246 | 31.824 | 125 316 | 0.000 | 0.000 |

D2 | 0.000 | 1.018 | $-$0.071 | 8.733 | 4 959 | 0.000 | 0.000 |

D3 | 0.000 | 0.502 | $-$0.044 | 7.755 | 3 410 | 0.000 | 0.000 |

D4 | 0.000 | 0.222 | 0.039 | 4.380 | 288 | 0.000 | 0.000 |

D5 | 0.000 | 0.129 | 0.006 | 4.286 | 249 | 0.000 | 0.000 |

EAFE-ETFs | |||||||

D1 | 0.000 | 1.115 | 0.602 | 16.650 | 28 183 | 0.000 | 0.000 |

D2 | 0.000 | 0.462 | 0.314 | 14.684 | 20 546 | 0.000 | 0.000 |

D3 | 0.000 | 0.212 | 0.212 | 8.841 | 5 148 | 0.000 | 0.000 |

D4 | 0.000 | 0.091 | $-$0.156 | 6.327 | 1 676 | 0.000 | 0.000 |

D5 | 0.000 | 0.037 | $-$0.114 | 5.415 | 882 | 0.000 | 0.000 |

Table 1 contains concise descriptive statistics of wavelet details for the selected assets. Means of all wavelet series converge to zero. All wavelet unconditional variances decrease at higher scales, which is expected since higher wavelet scales are smoother. The skewness is found to be low, with both positive and negative signs. Kurtosis exceeds the reference value of the normal distribution (equal to $3$) for all wavelet time series, which justifies the usage of the Student $t$ distribution for the mean equation. The high JB values of all the wavelet scales suggests that all presented wavelet series display nonnormal properties. The presence of serial correlation and heteroscedasticity in all the wavelet time series is confirmed by the Ljung–Box $Q$-statistics for the level and the squared residuals, and this indicates that some form of AR–GARCH parameterization would be appropriate. Wavelet series are stationary by default, so unit root tests are not presented in Table 1.

## 5 Empirical results

### 5.1 Minimum-variance hedge effectiveness

This subsection reveals the results of minimum-variance portfolios, observed via five wavelet scales. According to Khalfaoui et al (2015), two characteristics determine the suitability of an auxiliary asset in a portfolio: its magnitude of risk relative to the risk of the primary asset and its level of correlation with the primary asset. If the auxiliary asset has a significantly higher level of risk than the primary asset, and if the level of positive correlation is high, then such an asset should be excluded from a portfolio. Table 1 contains the unconditional variances of Brent oil and all the selected assets observed via the five wavelet frequencies. It can be clearly seen that natural gas has a significantly higher variance than its Brent oil counterpart. American ten-year bonds and corn have similar risk profiles to Brent oil, while all the other assets are lower risk than Brent oil, at all wavelet scales. This is the first requirement to be met in terms of the suitability of an auxiliary asset in a portfolio. The second requirement is the strength of positive correlation between the primary and secondary assets in a portfolio. Table 2 presents the average dynamic rolling conditional correlations observed in five wavelet scales: they reveal the level of correlation between Brent oil and the selected assets. Graphical presentations of the dynamic rolling conditional correlations, calculated for five wavelet scales, are presented in Figures 2–4. Table 2 suggests that each of the assets has a relatively low average rolling correlation in combination with Brent oil, which is a good characteristic when these assets are coupled with Brent oil. Therefore, according to the findings in Tables 1 and 2, it can be assumed that combining any of the assets with Brent oil in a two-asset portfolio could yield some diversification benefits.

D1 | D2 | D3 | D4 | D5 | |
---|---|---|---|---|---|

Brent versus American | 0.154 | 0.159 | 0.145 | 0.132 | 0.111 |

ten-year bonds | |||||

Brent versus US dollars | 0.235 | 0.182 | 0.173 | 0.148 | 0.118 |

Brent versus gold | 0.127 | 0.160 | 0.178 | 0.122 | 0.057 |

Brent versus natural gas | 0.194 | 0.153 | 0.064 | 0.000 | 0.077 |

Brent versus corn | 0.231 | 0.205 | 0.175 | 0.161 | 0.243 |

Brent versus EAFE-ETFs | 0.316 | 0.307 | 0.297 | 0.278 | 0.219 |

D1 | D2 | D3 | D4 | D5 | |

Brent versus American | 56.6 | 55.0 | 52.9 | 50.9 | 50.8 |

ten-year bonds | |||||

Brent versus US dollars | 6.0 | 7.8 | 11.6 | 15.1 | 20.5 |

Brent versus gold | 23.0 | 24.4 | 27.0 | 30.9 | 36.4 |

Brent versus natural gas | 0.2 | 0.7 | 4.8 | 14.9 | 31.3 |

Brent versus corn | 47.0 | 51.3 | 50.0 | 49.8 | 51.3 |

Brent versus EAFE-ETFs | 25.4 | 25.8 | 29.8 | 32.0 | 32.7 |

Table 3 presents the average weights of auxiliary assets in a portfolio, constructed via the Kroner and Ng (1998) equation. Figures 5–7 show the dynamic rolling weight of the selected auxiliary assets. Optimal weights are given for various term horizons, and these are presented by different wavelet scales. It can be seen that the lowest weight in a two-asset portfolio is for natural gas, because gas is by far the riskiest asset according to Table 1. The asset with the highest weight is the American ten-year bond, with corn coming next. Although American ten-year bonds are characterized by their relatively high risk, the Kroner and Ng (1998) equation proposes that they have the highest weight, probably because bonds have one of the lowest correlations with Brent oil (see Table 2).

Table 4 discloses the variance-reduction performances of the wavelet portfolios presented via ${\mathrm{HEI}}_{\mathrm{Var}}$ indicators. It can be seen that each of the examined assets, in combination with Brent oil, reduces the portfolio variance to some extent, except for natural gas on the D4 and D5 wavelet scales. Natural gas has the lowest variance-reduction performance in general because gas is the riskiest auxiliary asset.

D1 | D2 | D3 | D4 | D5 | |
---|---|---|---|---|---|

Brent–American | 6.080 | 9.033 | 6.202 | 2.410 | 12.621 |

ten-year bonds | |||||

Brent–US dollars | 10.376 | 11.722 | 17.428 | 21.320 | 26.291 |

Brent–gold | 24.737 | 28.187 | 30.438 | 28.631 | 36.249 |

Brent–natural gas | 1.202 | 1.696 | 2.148 | $-$20.911 | $-$72.053 |

Brent–corn | 10.625 | 10.014 | 7.229 | 11.164 | 11.831 |

Brent–EAFE-ETFs | 20.441 | 21.489 | 22.912 | 22.436 | 23.588 |

As for the other assets, Table 1 suggests that they all have lower or similar risk relative to Brent oil. Therefore, these preliminary findings indicate that all selected assets, other than natural gas, can be considered as potentially good hedging (diversification) instruments. Only portfolios with bonds have relatively poor diversification benefits; the reason for this lies in the relatively high risk of bonds, which makes the asset unsuitable in combination with Brent oil. The results in Table 4 are in line with these contentions, since all assets, except natural gas and bonds, contribute significantly to the reduction of portfolio variance, to a lesser or greater extent. Table 4 indicates that the combination of gold and Brent oil gives the best minimum-variance results for all wavelet scales. When Brent is combined with gold it reduces average portfolio variance between 24% and 36%, depending on the observed time horizon. These results are consistent with the claims of Mirović et al (2017) and Chkili (2016), who found that adding gold to a portfolio could increase the variance hedge effectiveness index markedly. The second-best asset in combination with Brent oil is EAFE-ETFs; this instrument reduces the variance of two-asset portfolios by between 20% and 23%. Corn and US dollars have similar ${\mathrm{HEI}}_{\mathrm{Var}}$ measures for the first two wavelet scales, while US dollars turn out to be a very good diversification instrument from the D3 scale onward.

### 5.2 Minimum VaR and minimum CVaR hedge effectiveness

Minimum-variance hedging is optimal from a risk-reduction perspective only when investors have quadratic utility or when returns are drawn from a multivariate elliptical distribution (see Cao et al 2009). However, in practice, these assumptions probably remain unfulfilled. In such circumstances, variance is no longer an appropriate measure of risk, since it ignores the higher moments of the return distribution. This has led to alternative measures of risk, such as the VaR and CVaR metrics, being used. This subsection therefore analyzes the effect of minimum-variance hedging on the VaR and CVaR performances of the scale-adjusted, two-asset portfolios. Table 5 gives the ${\mathrm{HEI}}_{\mathrm{VaR}}$ and ${\mathrm{HEI}}_{\mathrm{CVaR}}$ results. Comparing Table 4 with Table 5, it can be seen that in all cases except gas, the majority of the ${\mathrm{HEI}}_{\mathrm{VaR}}$ metrics are more than 40% lower than their ${\mathrm{HEI}}_{\mathrm{Var}}$ counterparts. As in the minimum-variance cases, gold in combination with Brent oil has the best minimum VaR performance on the first three time horizons, while natural gas outperforms gold at higher wavelet scales. We also find that the portfolio with EAFE-ETFs has good diversification characteristics for all wavelet scales, while for US dollars it applies on the longer time horizons. American ten-years bonds and corn have the worst ${\mathrm{HEI}}_{\mathrm{VaR}}$ performances at the lower frequency scales.

D1 | D2 | D3 | D4 | D5 | |

${\text{HEI}}_{\text{VaR}}$ results | |||||

Brent–American | 3.461 | 4.784 | 1.096 | $-$0.854 | 6.221 |

ten-year bonds | |||||

Brent–US dollars | 4.396 | 6.713 | 9.736 | 9.097 | 15.714 |

Brent–gold | 15.777 | 16.687 | 15.420 | 12.929 | 20.447 |

Brent–natural gas | 0.430 | 1.584 | 6.961 | 18.175 | 24.156 |

Brent–corn | 9.955 | 6.540 | 9.736 | 6.446 | 4.163 |

Brent–EAFE-ETFs | 12.836 | 13.429 | 11.983 | 12.225 | 12.453 |

${\text{HEI}}_{\text{CVaR}}$ results | |||||

Brent–American | 5.904 | 5.990 | $-$0.069 | $-$8.379 | 10.160 |

ten-year bonds | |||||

Brent–US dollars | 0.688 | 3.502 | 6.532 | $-$0.788 | 9.445 |

Brent–gold | 12.431 | 15.027 | 12.670 | 1.338 | 6.732 |

Brent–natural gas | $-$1.805 | $-$0.224 | 2.011 | 23.122 | 31.290 |

Brent–corn | $-$20.896 | $-$30.294 | $-$47.549 | $-$38.007 | $-$25.563 |

Brent–EAFE-ETFs | 11.689 | 12.734 | 9.092 | 4.747 | 1.909 |

As for the ${\mathrm{HEI}}_{\mathrm{CVaR}}$ findings, gold is the best performing asset at the first three wavelet scales. Table 5 suggests that the Brent–gas and Brent–EAFE-ETFs combinations outperform the Brent–gold pair at the longer time horizon (D4), while for the D5 scale, American ten-year bonds, US dollars and gas give better ${\mathrm{HEI}}_{\mathrm{CVaR}}$ results. In spite of the good ${\mathrm{HEI}}_{\mathrm{Var}}$ and ${\mathrm{HEI}}_{\mathrm{VaR}}$ results for corn, this commodity has pretty bad ${\mathrm{HEI}}_{\mathrm{CVaR}}$ outcomes, which means that investors who pursue CVaR will get better results when they invest solely in Brent oil. This is because ${\mathrm{HEI}}_{\mathrm{CVaR}}$ reported negative values when Brent is combined with corn at all wavelet scales. It means that for these time horizons, the size of a loss exceeds VaR by more in constructed portfolios than it does in the unhedged investment.

The lower ${\mathrm{HEI}}_{\mathrm{VaR}}$ and ${\mathrm{HEI}}_{\mathrm{CVaR}}$ in comparison with the results for ${\mathrm{HEI}}_{\mathrm{Var}}$ are consistent with the findings of Cao et al (2009) and Harris and Shen (2006), who claimed that while minimum-variance hedging undoubtedly reduces the standard deviation of portfolio returns, the effect on skewness and kurtosis could be ambiguous. They asserted that the reduction in standard deviation that arises from portfolio creation can result in an increase in the negative skewness and/or kurtosis of a portfolio, which in turn creates a lower reduction in VaR and CVaR in comparison with the reduction in standard deviation. In order to test this contention, we calculate the third and fourth moments of the portfolios assembled via the Kroner and Ng (1998) equation and present these results in Table 6. It is obvious that in most cases across the wavelet scales, the skewness of all the hedged portfolios is negative and much higher than for the unhedged investment, ie, the investment in Brent oil only. In addition, at some wavelet scales and for some pairs, the kurtosis of the hedged portfolio is also higher than for the Brent-only investment. We find that the portfolio with gas has low kurtosis relative to the unhedged portfolio at the fourth and fifth wavelet scales, and Table 5 shows that the Brent–gas portfolio at these scales gives very good CVaR results. In addition, the Brent–corn portfolio has significantly higher skewness and kurtosis values than the unhedged portfolio, and this in turn gives very bad CVaR results at all wavelet scales, as can be seen in Table 5. All these findings confirm the assertion of Cao et al (2009) and Harris and Shen (2006) regarding why the ${\mathrm{HEI}}_{\mathrm{VaR}}$ and ${\mathrm{HEI}}_{\mathrm{CVaR}}$ values of the same portfolio are lower than the ${\mathrm{HEI}}_{\mathrm{Var}}$ value.

D1 | D2 | D3 | D4 | D5 | ||

Brent oil | Skewness | 0.013 | 0.018 | 0.167 | 0.061 | 0.005 |

Kurtosis | 6.088 | 4.952 | 7.568 | 4.909 | 3.395 | |

Brent–American | Skewness | $-$0.039 | $-$0.065 | $-$0.149 | $-$0.249 | $-$0.039 |

ten-year bonds | Kurtosis | 5.719 | 5.846 | 5.933 | 6.620 | 5.333 |

Brent–US dollars | Skewness | $-$0.165 | $-$0.103 | $-$0.105 | $-$0.305 | $-$0.130 |

Kurtosis | 6.375 | 6.820 | 6.832 | 7.005 | 8.014 | |

Brent–gold | Skewness | $-$0.066 | $-$0.066 | $-$0.176 | $-$0.349 | $-$0.292 |

Kurtosis | 7.712 | 6.789 | 6.175 | 7.291 | 9.226 | |

Brent–natural gas | Skewness | $-$0.117 | $-$0.106 | $-$0.169 | 0.224 | 0.369 |

Kurtosis | 6.310 | 6.217 | 5.993 | 5.010 | 3.680 | |

Brent–corn | Skewness | $-$0.758 | $-$0.670 | $-$0.312 | $-$0.815 | $-$0.517 |

Kurtosis | 18.208 | 15.170 | 22.878 | 16.572 | 10.832 | |

Brent–EAFE-ETFs | Skewness | $-$0.040 | $-$0.035 | $-$0.134 | $-$0.175 | $-$0.227 |

Kurtosis | 6.855 | 6.712 | 6.441 | 7.597 | 8.190 |

## 6 Summary and conclusion

This paper presents an analysis of portfolio construction, combining Brent oil with six heterogeneous assets (American ten-year bonds, US dollars, gold, natural gas, corn and EAFE-ETFs) in order to analyze the two-asset portfolio hedging performances via three different risk metrics (variance, VaR and CVaR) and across five wavelet scales. Portfolios are constructed via the Kroner and Ng (1998) equation, whereas conditional variances and covariances are obtained via the bivariate rolling DCC–GARCH model.

The results indicate that for investors who target minimum variance, the best combination at all time horizons is the portfolio that contains Brent oil and gold: investors could achieve variance reduction between 24% and 36% with this combination. The second-best minimum-variance portfolio is the combination of Brent oil with EAEF-ETFs, which yields variance reduction between 20% and 23%. Investors who target the minimum-variance portfolio should avoid the combination of natural gas with Brent oil, as it has high unconditional variance.

The VaR results indicate that portfolios with gold give good VaR performances at all time horizons. However, portfolios with gold are no longer dominant across all wavelet scales, since portfolios with natural gas give better VaR results at the fourth and fifth wavelet scales. Investors who seek minimum VaR targets can achieve relatively good results when they combine Brent oil with EAFE-ETFs over all time horizons, while Brent oil combined with US dollars produces relatively good VaR results for the longer time horizons. All other auxiliary assets give poor VaR performance when combined with Brent oil.

As for the ${\mathrm{HEI}}_{\mathrm{CVaR}}$ findings, these results are very similar to those for ${\mathrm{HEI}}_{\mathrm{VaR}}$. In other words, portfolios with gold give good ${\mathrm{HEI}}_{\mathrm{CVaR}}$ results for shorter time horizons, but the Brent–gas pair heavily outperforms the results for gold for longer time horizons. Brent with EAFE-ETFs gives relatively good CVaR results for the shorter time horizons, whereas investors who pursue CVaR as their goal should not combine Brent with corn, because this combination produces very bad CVaR results across all wavelet scales. In other words, investors will achieve much better CVaR results if they invest solely in Brent oil than if they combine Brent with corn. Referring to Cao et al (2009) and Harris and Shen (2006), we confirm that minimum-variance portfolios could have increased negative skewness and high kurtosis, and on that occasion ${\mathrm{HEI}}_{\mathrm{VaR}}$ and ${\mathrm{HEI}}_{\mathrm{CVaR}}$ values are lower in comparison to their ${\mathrm{HEI}}_{\mathrm{Var}}$ counterparts. Our findings regarding the skewness and kurtosis of the constructed portfolios coincide very well with our ${\mathrm{HEI}}_{\mathrm{VaR}}$ and ${\mathrm{HEI}}_{\mathrm{CVaR}}$ results, thus confirming the assertion of Cao et al (2009) and Harris and Shen (2006).

We believe that our findings could help portfolio managers and global investors who are looking for ways of diversifying their Brent oil investments, those who act across different time horizons and those who target different risk-minimizing goals. The presented results could help those managers to decide how they can reduce their Brent oil risk in the most effective way.

## Declaration of interest

The authors report no conflicts of interest. The authors alone are responsible for the content and writing of the paper.

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