Generalized Hyperbolic Distribution and Portfolio Efficiency in Energy and Stock Markets of BRIC Countries

Oil, also called black gold, is considered as the commodity which has the greatest impact on the world’s economy, and it has been studied in terms of its relationship and effects on macroeconomic variables such as Gross Domestic Product (GDP), inflation, trade balance, exchange rate and some others. Likewise, the relationship of oil with the financial market has been deepened and is very interesting in the case of emergent economies such as Brazil, Russia, India and China (BRIC) countries. There are many studies and approaches to this topic, but few of them focus on seeking investment opportunities through the diversification of these variables and therefore creating efficient portfolios using other distribution from the norm. This research proposes the construction of diversified portfolios with the returns of the indexes and oil mixes of the BRIC countries modeled under a Normal Inverse Gaussian (NIG) distribution, which is a notable member of the Generalized Hyperbolic (GH) family, and analyzing the effect on investment, by the inclusion of each variable into the portfolio. An important property of the GH family is that the correlations matrix of the returns is obtained from estimation of the parameters of empirical distribution through maximum likelihood. The results show in an optimal configuration, that each instrument of India, China and Brazil, contributes to the portfolio efficiency, in contrast to the index and oil mix of Russia, that do not contribute significantly.


Introduction
There are currently numerous investigations dealing with the relationship between the oil price and economic variables, including: Hamilton (1983Hamilton ( , 2003; Davis and Haltiwanger (2001); Lee et al. (2001); Lee and Ni (2002); Hooker (2002); Brown and Yücel (2002); Akram (2004); Jones et al. (2004); Hamilton and Herrera (2004); Cunado and Gracia (2005); Lardic and Mignon (2006); Chen and Chen (2007); Huang and Guo (2007) and Nandha and Hammoudeh (2007) among others. Hamilton's seminal work in 1983, where it is argued that most financial crises in the United States are preceded by a significant increase in the price of oil, establishes a line of research that continues to this day to be very fruitful.
Due to the appearance and evolution of oil financial instruments such as futures and derivatives in the last decades of this past century, the commodity adopted financial characteristics that led to the study of its links with the financial market (Alquist and Gervais 2013;Ma et al. 2019;Zhang and Wang 2015).
There are several works with various methodologies that have studied, and continue to study, the interaction of oil and financial markets. Lin and Su (2020) pointed to some subdivisions in this topic, including the spillover effect of crude oil price and the stock market (Du and He 2015;Ji et al. 2020;Wen et al. 2019a); the connections between oil price and stocks from the industry level or national level

Multivariate Normal Inverse Gaussian (MNIG) Distribution
The GH family distributions, specifically the NIG distribution has been a notable option to model the distribution of return of numerous financial instruments, such as commodities, equities, options and so on. Nevertheless, empirical studies frequently focus on univariate cases. In this paper we have extended previous works (Núñez et al. 2018;Sánchez-Ruenes et al. 2020) to model returns of BRIC Indexes and Crude Oil prices using an MNIG distribution.
In the paper of 1977, Barndorff-Nielsen proposed the family of the GH distributions, which is considered in the literature as a seminal paper (Barndorff-Nielsen 1977). The NIG distribution type resulted in a convenient alternative for modeling financial series due its properties of capturing some stylized features, like semi-heavy tails, asymmetry and high volatility, present in financial data. This kind of distribution is the only member of GH family that is closed under convolution, guaranteeing the conservation of the same NIG distribution even the logarithmic increments for the exponential process (Barndorff-Nielsen 1998).
On the other hand, the MNIG distribution is a mean-variance mixture combination of a Gaussian distribution with an inverse Gaussian distribution (Oigard and Hanssen 2002).
The Multivariate Normal (MNormal) distribution is defined as X = (X 1 , . . . , X d ) t if X = µ + AZ, where Z = (Z 1 , . . . , Z k ) t is an independent identically distributed vector of random variables with univariate Standard Normal distribution, A∈R d×k and µ∈R d . With this specification, we can express the multivariate distribution with the next density (Mardia et al. 1979): where Σ is a semidefinite positive and non-singular matrix representing the covariance. Once the parameters were determined, Press et al. (1992) proposed an algorithm to simulate random vectors that follows the same distribution by employing the Cholesky decomposition of Σ.
If we consider a random vector, expressed as X = µ + √ W AZ where W ≥ 0 is a random variable scalar. Assuming W has a Generalize Inverse Gaussian (GIG) distribution with density where K λ is the Bessel function of third kind defined as: then, the variance mixture of distributions has the joint density of: which is a special case of the GH case of: With this expression, particular cases may be extended, for the purpose of this paper, the MNIG distribution is obtained when λ = −0.5 which becomes close to the Hyperbolic distribution but with heavier tails (McNeil et al. 2005).
Although there are several parameterizations for the GH distributions, we selected the one corresponding to (λ, α, µ, Σ, γ) that allows a faster convergence when fitting the empirical data (Weibel et al. 2020). To realize this, we will use the following transformation: (λ, χ, ψ, µ, Σ, γ) → (λ, α, µ, Σ, γ) It should be mentioned that the applications and the potential of the family of the Generalized Hyperbolic are vast. Andresen et al. (2010) present a discrete random-field model for forward prices driven by the MNIG distribution. Likewise, O'Hagan et al. (2016) establish that a combination of MNIG distributions is a great alternative to a mixture of Gaussian distributions as a tool to develop a model of clustering.
On the other hand, Eberlein and Prause (2001) confirmed that the GH family allows a better description of the returns of financial assets. In particular, these authors derive a formula for valuing options for models that include the GH using the Esscher transformation as a possibility to determine prices in an incomplete market.
The remainder of this paper is structured as follows: in Sections 2 and 3 we explain the methodology applied to the data and results obtained. We present the conclusions in Section 4.

Methodology
In this research we worked with daily data obtained from Bloomberg of market indexes of BRIC economies: IBOV from Brazil, RTSI from Russia, NIFTY from India and SHCOMP from China. We also studied the main crude oil mixes in those countries, except in India, owing to the unavailability of data. The oil mixes included in this study were: Marlim from Brazil, Urals from Russia and Daqing from China, and the data were obtained from Platts.
The period of time selected for the stock and oil prices series was from 2004 to 2019, owing to include structural oil price breakpoints and important events for the oil market in the current millennium such as the beginning of the oil price surge until the financial crisis in 2005 and the oil price rebound in 2012 until 2014. The bubble periods identified by Perifanis (2019) are included in the studying intervals as well. Finally, the decline in oil prices since 2014 was considered due to oversupply and demand decrease in oil markets, besides Saudi Arabia's decision to maintain its oil production levels.
We applied the Augmented Dickey-Fuller test with Schwarz information criterion and the Phillips-Perron test with the Bartlett kernel spectral estimation method to find out the unit root of the variables. Then, we re-evaluated the stationarity using the logarithmic returns of the series and we obtained for every series a p-value less than 0.0001, that means the null hypothesis is rejected, that is, the series are stationary. Table 1 presents the descriptive statistics for those series. Taking into account the stylized facts, mainly the presence of high levels of kurtosis, this could lead intuitively to the empirical adjustment of a distribution with semi-heavy tails. In this case, the proposal is the NIG distribution, as it has praiseworthy properties for financial analysis (Barndorff-Nielsen 1997).
In previous research (Núñez et al. 2018), the returns of BRIC indexes: IBOV, RTSI, NIFTY and SHCOMP, adjusted significantly in a univariate approach to NIG distribution, even during periods of economic crisis. In this study, we first verified if NIG distribution also fits the series of crude oil returns: MARLIM, URALS and DAQING. The results confirm that NIG distribution reasonably adjusted to the empirical data in a univariate case for those crude oil mixes. However, the world is connected through different variables, including the financial variables, so it makes sense to look for a multivariate distribution.
For this research, we created some vectors of dimension four, five and six with returns of BRIC indexes and crude oil mixes, and then we confirmed their adjustment to a MNIG distribution using a goodness-of-fit test through Cramer criteria with significance levels of 0.95 and 0.99 (Baringhaus and Franz 2004). See Table 2. The parameters of the MNIG distributions that adjust the suggested vectors are presented in Appendix A. For every proposed vector, we did not reject the null hypothesis for NIG distribution-H 0 : the empirical data follow a MNIG distribution. We used R software to obtain these results, as well as the later ones.

Results
The results from the previous tests indicate that the logarithmic returns from the oil prices and indexes altogether adjust statistically, to an MNIG distribution.
We observed the strongest correlation between the oil mix and stock index of Russia. We could intuit, excluding Russia, that it is possible to create a diversified portfolio, and we proved it through an empirical study.
In this regard, we created a portfolio for every vector, according to the theory of Markowitz (1952), finding the optimal weights (ω's), as seen in Table 3, for each investment instrument, in our case market indexes and commodities, in order to obtain a higher return with a lower risk.
Therefore, we employed the relationship between the return-standard deviation to show the efficiency of every portfolio, obtaining the following ratios (Table 4): Table 3. Asset weights obtained for suggested portfolios 1 to 6 considering the 1st and 2nd moments of the adjusted MNIG distribution. From these weights, the expected returns and the standard deviation of each portfolio were calculated. Elaborated by authors with data from Bloomberg. It should be noted that the variance-covariance matrix plays a fundamental role in portfolio optimization theory (Markowitz 1952) and in the case of the family of GH distributions, it is calculated with the skewness (γ) and dispersion (Σ) parameters, through the variance equation according to McNeil et al. (2005). In Appendix B, we present the variance-covariance matrices obtained through the maximum likelihood criterion with a MNIG distribution for each portfolio.
Using those matrices, we obtained the following curves as a result of the optimization of the return-standard deviation relationship for those portfolios and we compared them with those obtained using the empirical sample variance-covariance matrix (see Figure 1).
In all cases, we observed that the curves of the MNIG portfolios exceed those in which the MNormal variance-covariance matrices were considered. The difference between both groups of portfolios is attributed to the better fit of MNIG distribution on the logarithmic returns of the asset series. The variance-covariance matrices estimated through the multivariate distribution better represent the return behavior than the traditional covariance matrix.
Regarding the efficient frontier curves for portfolios obtained using the MNIG distribution, the curves of portfolio 1 (oil mixes from China, Russia and Brazil; and indexes from China, Brazil and India) and portfolio 3 (oil mixes from China and Brazil and indexes of China, Brazil and India); as well as portfolio 2 (oil mixes and indexes from China, Russia and Brazil) and portfolio 4 (oil mixes and indexes from China and Brazil) would be almost overlapping, with a difference of less than 0.14% and 0.8%, respectively. Comparing the curves of portfolios 1 and 3, we observe that the inclusion of the Russian mix does not represent a significant increase in the efficiency of the portfolio. Similarly, if we compare curves 2 and 4, we also find that, incorporating the Russian mix and index, it does not have a significant impact on portfolio efficiency.
On the other hand, the curves of portfolios 1 and 3 are above the pair of curves 2 and 4, revealing that the incorporation of the Hindu index contributes more than 32.6% to the efficiency of the portfolio. However, the increase greater than 6.7% in the curves of portfolios 2 and 4 with respect to curve 5, responds to the inclusion of the Brazilian mix and index. Finally, the growth of more than 23.8% in the efficiency of portfolio 5 compared to 6, is due to the exchange of instruments from Brazil for those from China.

5
Daqing, Urals, SHCOMP, RTSI 0.02056 0.02026 6 Urals, Marlim, RTSI, IBOV 0.01660 0.01581 It should be noted that the variance-covariance matrix plays a fundamental role in portfolio optimization theory (Markowitz 1952) and in the case of the family of GH distributions, it is calculated with the skewness (γ) and dispersion (Σ) parameters, through the variance equation according to McNeil et al. (2005). In Appendix B, we present the variance-covariance matrices obtained through the maximum likelihood criterion with a MNIG distribution for each portfolio.
Using those matrices, we obtained the following curves as a result of the optimization of the return-standard deviation relationship for those portfolios and we compared them with those obtained using the empirical sample variance-covariance matrix (see Figure 1). In this sense, the instruments that had the greatest impact on portfolio efficiency (highest return with the lowest risk) are, in this order: NIFTY index (India); Daqing mix and the SHCOMP index (China); and Marlim mix and IBOV index (Brazil). On the opposite side, the contribution of the Urals mix and RTSI index (Russia) represents less than 1% in the return-standard deviation ratio of the portfolio.

Conclusions
In the present work, a multivariate approach for portfolio construction is employed using the market indexes IBOV, RTSI, NIFTY and SHCOMP, and crude oil mixes of Marlim, Urals and Daqing. Employing a Multivariate Normal Inverse Gaussian test for the logarithmic returns it was possible to determine that Normal Inverse Gaussian assumptions hold even in the multivariate case fitting the empirical data appropriately. In order to make efficient portfolios using MNIG distributions, the Markowitz theory was employed for some combinations of the oil mixes and market indexes of BRIC. The results show an insignificant diversification effect of including Russia's index and/or oil mix in the Portfolio. From an investment optimization perspective, the results of this work show the possibility to diversify with market indexes and oil mixes of India, China and Brazil, even in periods including extreme and crisis events, showing that the greatest impact on portfolio efficiency is due to the Hindu index and the Chinese mix and index.