Price risk and hedging strategies in Nord Pool electricity market evidence with sector indexes

Energy Economics 80 (2019) 635–655

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Energy Economics journal homepage: www.elsevier.com/locate/eneeco

Price risk and hedging strategies in Nord Pool electricity market evidence with sector indexes Ben Amor Souhir a, Boubaker Heni a,b,⁎, Belkacem Lotfi a a b

Institute of High Commercial Studies of Sousse (IHEC), LaREMFiQ, 4054 Sousse, Tunisia Rabat Business School, BEAR LAB (UIR), Technopolis Rabat-Shore, 11100 Rabat-Salé, Morocco

a r t i c l e

i n f o

Article history: Received 25 June 2018 Received in revised form 29 January 2019 Accepted 6 February 2019 Available online 10 February 2019 Keywords: Electricity market Sector indexes Long run dependence Risk management Conditional VaR-optimal hedge ratio

a b s t r a c t Electricity markets become more competitive due to their liberalization; therefore, electricity prices are considerably more volatile compared to other commodity prices. As the electricity is an integral part of production and economic growth processes, the electricity price may influence the stock market through affecting the real output and consequently the sum of cash flows. Hence, investors are facing electricity price risks, and need to protect their benefits. This paper investigates the impacts of electricity market variations on the Nordic stock market returns using hourly observations of electricity spot prices pairwise in aggregate market index and some sector indexes. Our sample is divided into three sub-periods according to the electricity volatility structure. A generalized long memory model is adopted to estimate the conditional mean of the studied time series, and the FIGARCH process is used to model the conditional variance. Thereafter, a VaR, c-DCC-FIGARCH, CVaR and ΔCVaR models are applied to assess electricity market exposure. Moreover, in order to evaluate the optimal portfolio, we calculated the optimal portfolio weights, the optimal hedge ratios and the hedge effectiveness index of the electricity market commodity in several sectors stock portfolios. Our results show evidence of long run dependence between electricity market returns and sectoral stock market returns, and they indicate that the tail dependence is significant and varies across sectors and over periods. Finally, the optimal weights and hedge ratios for electricity/stock portfolio holdings are sensitive to the considered sectors. Therefore, electricity market commodities can be adopted to diversify and hedge against stock market risks. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Electricity is important and complementary to other production factors, and it is considered as an integral part of production as well as of economic growth process. Indeed, the economic growth requires an increase in investment to amplify the production capacity in order to satisfy the real time demand; as a result, the electricity consumption will rise. Therefore, economic growth is a key factor of electricity demand and any fluctuation in electricity supply may affect the macroeconomic factors, such as gross domestic product (GPD) growth. In this vein, as a crucial industry input nowadays, electricity is a fundamental part of the modern economy. Due to deregulation movement, that reforms the electricity industry worldwide, electricity markets have become more integrated and competitive. Indeed, the electricity industry is converging toward a competitive framework, since traditional monopolistic electricity market is replaced by deregulated market environment. This liberalization of

⁎ Corresponding author. E-mail address: [email protected] (B. Heni).

https://doi.org/10.1016/j.eneco.2019.02.001 0140-9883/© 2019 Elsevier B.V. All rights reserved.

electricity markets has led to the creation of an organised market where electricity is traded like other commodities. The countries that have liberalized their power market have similar goals. They seek to create competition in the electricity market to reach economic efficiency, lower consumer prices of electricity and higher quality services. In this deregulated market, electricity, like other commodities, is sold and bought by market participants. The price and quantity trades are fixed by the market operator referring to the supply and demand bids, submitted by the Load Serving Entities (LSE) and generators. However, in contrast to commodities markets, the impossibility of storing electricity to deliver in later periods in addition to the physical and technical restrictions imposed on the transportation of electrical energy, accompanying with the necessity of assuring a constant equilibrium between demand and production, making the market more complicated and hence it should be operated differently. Moreover, in electricity markets, retailers are uncertain about the quantity of electricity consumed by their customers at any hour of the day until they turn it on. While the uncertainty of demand is a mutual feature of all commodity markets, retailers usually rely on stored quantities to manage demand uncertainty. However, in electricity markets,

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retailers are exposed to both quantity and price risk on hourly basis, due to the physical singularity that characterise the electricity. On the supply side, the non-storability of electricity makes this market very specific. Accordingly, electricity must be produced the moment it is consumed. Therefore, intertemporal arbitrage is impossible, and the electricity price is fixed based on the supply and demand conditions at each given hour. These difficulties, added to the impact of meteorological conditions, significantly affect the electricity market organizations, and introduce special features to electricity prices, in particular to the spot price, such as high volatility, price spikes and mean-reversion.1 The main reason for these particular features is attributed to the non-storability of electricity, peak demand at certain periods, generator outages and fuel uncertainty for renewable energy generators. Consequently, electricity prices are considerably more volatile than other commodities prices.2 Confronting with this extreme price volatility, market participants are exposed to trading risks, and thus need to protect their benefits. The main objective of market participants is to incur the lowest expected costs for the expected electricity load, given a specific risk target. Therefore, in order to deal with price risk, market participants can apply risk management techniques to control risk while maximizing their profits. In this vein, the design of more accurate risk measurement models has become the central issue for more effective risk management in the electricity market (Deng and Oren, 2006). However, an efficient strategy of risk management must be supported with a robust methodology for forecasting the prices. More accurate forecasting reduces the risk related to the high price volatility, and consequently, provides better risk management (Bastian et al., 1999). On the other hand, forecast errors have significant implications for profits. As a result, forecasting prices in such markets become an essential issue for the risk management. Risk management is the process of accomplishing a desired profit/ return, considering the risks, by means of a strategy. In the financial field, there are two strategies to control risk. One is through risk financing by adopting hedging strategies to compensate losses that may occur and the other one is whereby risk reduction by applying diversification to decrease exposure to risks. Diversification is to engage in different markets in order to limit the exposure to the risk of any specific market. To the extent that electricity serves as an essential input for industries as well as the economic activity, the price of electricity may affect the aggregate stock prices through affecting the real output which, in turn, influence earnings, and therefore, the sum of discounted future cash flows. By the relations that exist between electricity and stock markets, investors may make appropriate portfolio based on electricity commodity and some sectoral indexes from the stock market in order to reduce the risk related to the electricity market. In this vein, we are addressing the problem of establishing a framework for risk management and hedging strategies in the Nordic market and how to make an optimal portfolio based on electricity and sector indexes from the point of view of risk control. In order to tackle these research questions, our study aims to investigate empirically the temporal dependence between the electricity market and the stock market. Two questions motivate this research: (i) how does electricity market valuation affect the sector indexes on the Nordic stock market? (ii) Which alternatives should be considered as safe havens by investors during the period of high fluctuations in the electricity market? To answer the research questions, the generalized long memory process (k-factor GARMA-FIGARCH) is adopted in this study in order to

1 We refer to Bunn and Karakatsani (2003) and Huisman et al. (2007), among others, for an overview on (hourly specific) day-ahead price characteristics. 2 Statistical data of US department of Energy indicates that in the US, the average annual volatility of electricity prices is 359.8%, while for Natural Gas and Petroleum, Financial, Metals, Agriculture, and Meat it is just 48.5%, 37.8%, 21.8%, 49.1% and 42.6% respectively (Energy Information Administration, 2002).

describe the data generating process of the electricity market return series. This model proves its effectiveness in fitting the different feature especially related to the seasonal long memory behavior in the data. In addition, in order to capture the dynamic of time varying conditional correlation between the electricity returns and each sector index returns, we apply a bivariate corrected dynamic conditional correlation FIGARCH model c-DCC-FIGARCH that allows examining the dynamic conditional correlations (short-run links) among the variables considering the effects of long-run interactions and volatility persistence. Moreover, this study is conducted by implementing a Conditional Value at Risk (CVaR) approach. It represents the Value at risk (VaR) of some stock market returns conditional on electricity market returns. The study also focuses on exposure CVaR or ΔCVaR measurement, which represents the variation of CVaR under distress state and the CVaR in its benchmark state. This measurement allows gauging the size of potential tail spillover effects from electricity market to each sectoral stock market. Finally, we evaluate the optimal portfolio, referring to three criteria: the optimal portfolio weight; the optimal hedge ratio, and the hedge effectiveness index. In sum, our study introduces the dual generalized k − factor GARMA − FIGARCH model and the (c − DCC − FIGARCH) model in the field of analysing the electricity time series and stock indexes and adoption of the VaR, the CVaR and the ΔCVaR in the field of analysing tail dependence between the electricity market returns and each sectoral stock returns. It is worth noting that our study is the first to describe the relationship between the electricity market returns and each sectoral stock returns. This investigation is crucial since electricity is a key factor of production and economic growth. In addition, we propose an empirical framework to build an optimal portfolio in order to hedge against the variation of electricity markets. However, as we know, there are no researches that propose a portfolio optimization based on intraday hedging for electricity market using stock indexes, despite that the frequency of spot hourly price spikes reinforces the necessity of intraday hedging strategies (Homayoun Boroumand et al., 2015). More precisely, since the electricity is a non-storable commodity, any variation of the demand in the matter of minutes or hours can generate an enormous effect on the prices, and consequently, this variation can affect immediately the sector that relies on electricity in their activities. Therefore, this variation can be observed in the very short term, and for this reason, our investigation is limited to a period of five months from which we adopt hourly observations (Fig. 1). The rest of this paper proceeds as follows: Section 2 provides a brief review of the literature. Section 3 presents the econometric methodology; which includes the generalized long memory (k − factor GARMA) model for conditional mean modelling, illustrates the multivariate conditional volatility and dynamic conditional correlations modelling (c − DCC − FIGARCH) and presents the VaR, the CVaR and the ΔCVaR measures. Section 4 deals with the empirical framework and describes the Portfolio Designs and Hedging Strategies. Section 5 provides policy recommendations before drawing conclusions of the general scope. 2. Literature review In a competitive electricity market, risk management is an essential part of a generation company and can intensely influence the profitability of companies. Several hedging instruments have been developed to allow economic agents to manage their risks (Hull, 2005; Geman, 2008; Xu et al., 2006; Oum et al., 2006). To exemplify, Oum and Oren (2010) attained the optimal hedging strategy with derivatives of electricity, through maximizing the expected utility of the hedged profit. Vehviläinen and Keppo (2003) studied the optimal hedging strategy related to the price risk by a mixture of electricity derivatives. For the investors who aim to hedge against the price volatility and at the same time to maximize their profits, they can adopt a hedging strategy based on financial instruments such as forward, options and futures

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Fig. 1. Graphical abstract of the adopted methodology.

contracts. In this vein, several forward contracts that can offer hedging to the risk associated to spot prices for market participants are suggested (Zhang et al., 2000; and Yu, 2002). Willems and Morbee (2008) proposed an equilibrium model for the power market using some traded financial products, more precisely: forwards and options contracts. Carrion et al. (2007) developed a stochastic programming procedure for risk-constrained retailers in order to select the forward contracts that the retailer should sign and specify the selling price of electricity that maximizes its expected profit considering the risk exposure. Recently, Viswanath (2015) provided a set of risk management strategies adopted by investors in electricity markets by means of financial derivatives. In the literature, two other portfolio optimization techniques have received a lot of attention: mean-variance models, and CVaR models. These two techniques differ in their definition of risk. More precisely, the Mean-Variance models penalize risk in the objective function, whereas the CVaR model measures the risk based on the probability of realizing a minimum profit. The Mean-Variance optimization models have been applied successfully for electricity markets. The problem is to minimize the risk, which is defined as the variance of the portfolio returns, for a specified return (Bystrom, 2003; Yin and Zhou, 2004; and Yu, 2002). Roques et al. (2008) adopted Monte Carlo simulations of coal, nuclear plant and gas investment returns as inputs of a Mean–Variance Portfolio optimization in order to select optimal portfolios-based load generation for electricity generators in deregulated electricity markets. The Markowitz mean-variance technique has been frequently adopted in portfolio optimization (Markowitz, 1959; Bodie et al., 1999; and Skantze and Ilic, 2001). Moreover, the Mean Variance Criterion (MVC) models have been used to allocate electricity between

contract and spot markets in real time markets (Liu and Wu, 2007; and Mathuria and Bhakar, 2014; Gölgöz and Atmaca, 2012; Pindoriya et al., 2010). There are also other portfolio optimization techniques based on the Value at Risk (VaR). The VaR represent the monetary value that the portfolio will lose over a given period with a specified probability. This measure has been frequently applied in electricity markets (Dahlgren et al., 2003; and Denton et al., 2003). However, it presents the same limits compared to the CVaR measure since VaR is not subadditive and not convex. For this reason, the CVaR model seems more advantageous in portfolio optimization in a liberalized electricity market (Wang et al., 2005; Sun et al., 2005; and Lorca and Prina, 2014). To sum up, these existing pieces of literature aim at hedging against the risk of electricity price volatility in order to minimize the price risk and maximize profits. On the other hand, electricity can affect the sum of cash flows insofar as it serves as an essential input for industrial and economic activity. For this reason, stock market participants need to hedge against adverse price variations. In fact, this problem has been frequently treated in the oil market, where the researchers have studied the impact of variations (or shocks) of oil prices on stock markets indexes and/or sector indexes, in order to build a strategy for hedging against the oil market turmoil. Both oil and electricity commodities serve as important inputs in production, so they may have similar impacts on the stock markets, too. However, there are no researches that deal with this problem in the electricity market. Indeed, economic theory shows that any increase in oil price lead to increasing costs and restraining profits, which in turn cause a decrease in shareholders' value. Therefore, any oil price increase leads to a decrease in the stock prices. (Backus and Crucini, 2000; and Arouri and Nguyen, 2010).

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Previous empirical studies have analysed the relationship between crude oil price and financial market returns (Malik and Hammoudeh, 2007; Singh et al., 2010; Ono, 2011). More recently, Khalfaoui et al. (2015) showed the existence of significant relationship between oil price and stock markets in G-7 countries. Hamma et al. (2014) found unidirectional relationship between oil market and Tunisia stock market. These empirical results have motivated researchers to examine the relationship between oil price shocks and stock markets. To exemplify, Papapetrou (2001); Hong et al. (2002) and Park and Ratti (2008) found a negative relationship between the oil and stock market shocks. In contrast, Sadorsky (2001); Yurtsever and Zahor (2007); Gogineni (2008) and Lin et al. (2010) found that oil price has a positive impact on stock prices. Other researchers have studied the dynamic relationship between stock prices and oil prices through a sector-by-sector perspective (Faff and Brailsford (1999); El-Sharif et al. (2005); Jouini (2013); Sadorsky (2001); Boyer and Filion (2007); Nandha and Faff (2008); Nandha and Brooks (2009)) they showed that oil prices have positive influence on energy related industries. Several statistical techniques have been applied to analyse the relationship between oil price and stock price. Singh et al. (2010) applied the multivariate VAR − GARCH approach. Similarly, Sadorsky (2012) applied MGARCH. Bollerslev et al. (1998) proposed a Constant Conditional Correlation (CCC) model where the conditional correlations are assumed constant. This model reduces the number of parameters and thus considerably simplify the estimations. To exemplify, Cifarelli and Paladino (2010) adopted a multivariate CCC − GARCH. Engle (2002) proposes a generalisation of CCC model, to describe the time-varying correlations, identified as the Dynamic Conditional Correlation (DCC) model. Choi and Hammoudeh (2010) and Chang et al. (2010) applied a symmetric DCC − GARCH model. Filis et al. (2011) employed the DCC − GARCH − GJR to analyse the time-varying correlation between stock market and oil prices. Boubaker and Raza (2017) developed a bivariate ARMA − GARCH − DCC Student-t model. Trabelsi (2017) applied the DCC − MGARCH model and CVaR measure. The limits of the DCC model are constrained by the equal dynamics for the correlations of all the assets (Billio et al., 2006). Recently, Aielli (2013) suggests a more tractable dynamic conditional correlation model, known as a corrected DCC model or c − DCC model, which involves the three-step approach that is feasible with large systems and provides unbiased estimations (Kris Boudt and Laurent, 2013). Giving the existing relationship between oil price and aggregate and sectoral stock indexes, several authors suggest that oil commodity can be adopted to diversify stocks portfolios. In this vein, Arouri and Nguyen (2010); Arouri et al. (2011); Hamma et al. (2014); Khalfaoui et al. (2015) and Boubaker and Raza (2017) proposed a hedging strategy for oil stock portfolio against the risk of stock market prices variations. In sum, the results indicated the volatility transmission from oil market to stock sector market is a key factor for risk management and for portfolio designs. However, none of the former techniques approaches this problematic in liberalized electricity markets. Indeed, it is difficult to find in the literature scientific documents that deal with this problem, which is a very important subject in risk management in the electricity markets that is characterized by high price volatility. To our best knowledge, our research contains the first study that deals with this problematic. First, we investigated the volatility interaction between electricity returns and sector index returns, insofar as some sectors may be more exposed to electricity market volatility than other sectors, this depends on whether electricity is considered as an input or as an output of the industry. In addition, we studied the volatility transmission between electricity returns and the aggregate stock market index to check the robustness of our sector-level findings. Secondly, in contrast to the aforementioned researches that adopt the DCC − GARCH to study the

dynamic conditional correlation between oil market and stock market, we adopt a bivariate c − DCC − FIGARCH model that allows us to simultaneously examine short-run and long-run links between electricity and stock markets, and to analyse the volatility spillover effect between these two markets. Thirdly, none of the existing studies offers a conditional tail dependence analysis, for this purpose we adopt CVaR approach to examine the dependence structures between electricity returns changes and stock market indexes. We also adopt a delta CVaR measure that represent the change from the CVaR under distress state and the CVaR in a benchmark state. Our study is the first one that proposes a hedging strategy for stock market investors against the electricity market risk, by building an optimal portfolio using electricity commodity and stock indexes referring to the existing relationships between the electricity market and stock market. In this framework, electricity commodity can serve as diversifier and hedging asset to stock market risks. This investigation can be of interest to policymakers as unfavourable electricity market variations may have severe effects on the stock market performance through reducing corporate cash flows. On the other hand, while several researches adopt daily, weekly or monthly hedging strategies, we applied an intra-day hedging portfolio approach, using hourly data, insofar as electricity markets are hourly markets (Boroumand et al., 2015), and any price variations affect immediately the cash flows of related sectors, since the electricity is not a storable commodity. In conclusion, our study makes several contributions to the literature: first, this study examines the volatility transmission and tail dependence between electricity and aggregate market index as well as sector indexes over periods. Secondly, this paper provides an analysing of the optimal weights, hedge ratios and hedging effectiveness for an electricity-stock portfolio. 3. Methodology We assess the relationships between electricity market and stock market indexes using a bivariate analysis. The econometric specification adopted in our research has two components. For the univariate modelling, we adopt the generalized long memory models. Thereafter, this univariate analysis is used to estimate the unconditional VaR related to each sector indexes. Then, the bivariate c − DCC − FIGARCH model is adopted to study the volatility spillover between the electricity returns and each sector index returns. After this, the estimated residuals are used to compute the Conditional VaR(CVaR) and the ΔCVaR in order to gauge the size of potential tail spillover effects from electricity market to each sectoral stock index. More precisely, we follow three steps procedure to estimate the CVaR3: Step 1. The univariate generalized long memory models are fitted for the electricity spot price and for each sector index in order to estimate individual time series of VaRs. Step 2. In order to capture the dynamics of time-varying conditional correlation between the electricity spot price and each sector index, we estimate a bivariate FIGARCH model with corrected − DCC specification (termed the c − DCC − FIGARCH model) that enables to examine the dynamic conditional correlations (short-run links) among the considered variables under the effects of long-run interactions and volatility persistence. Hence, we estimate the bivariate c − DCC − FIGARCH for each pair. Step 3. Once we have estimated the bivariate density for each pair in step 2, in step 3 we proceed to obtain the CVaR measure for electricity returns and each sector index returns at time t. This methodology provides results related to the links between electricity returns and each sector index returns, notably as regards to their volatility spillover effects.

3

See Girardi and Ergün (2013) for more details.

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Because this study suggest using the electricity commodity as diversifier and hedge investment to sector index risks in Nordic markets, the implications for optimal design and hedging strategies are based on the optimal portfolio weights (ω); the optimal hedge ratio (β) and the hedge effectiveness index (HE). 3.1. The generalized long memory model The Generalized Autoregressive Moving Average (GARMA) model or k-frequency GARMA process, was developed by Woodward et al. (1998). It is considered as a generalized category of long memory models, which relaxes some characteristics of the ARFIMA model, by enabling for quasiperiodic or periodic movement in the time series. The multiple frequency GARMA model is defined as follows: ΦðLÞ

k  di Y I−2ν i L þ L2 ðyt −μ Þ ¼ ΘðLÞεt

ð1Þ

i¼1

Where Φ(L) and Θ(L) are the polynomials of delay operator L such that all the roots of Φ(z) and Θ(z) lie outside the unit circle. The parameters νi, |νi| b 1, i = 1, 2, … k, provides information about periodic movement in the conditional mean, εt is a white noise perturbation sequence with variance σε2, k is a finite integer, di are long memory parameters of the conditional mean showing how autocorrelations are slowly dampened, μ is the mean of the process, and λi = cos−1(νi), i = 1, 2, … k, represent the Gegenbauer frequencies (G-frequencies). The GARMA model with k-frequency is stationary when |νi| b 1, and di b 1/2 or when|νi| = 1, and di b 1/4, and the model exhibits a long memory when di N 0. The main characteristic of this process is given by introducing the Gegenbauer polynomial; P ðLÞ ¼

k  Y

I−2ν i L−L2

di

ð2Þ

i¼0

This polynomial is considered as a generalized long memory filter that estimates the cyclical long memory behavior at k + 1 different frequencies. Considering λi as the driving frequencies of a seasonal pattern of length S, λi = (2πi/S), and k + 1 = [S/2]+ 1, where [.] stands for the integer part. To highlight the contribution of P(L) at frequencies λ = 0 and λ = π, Eq. (2) can be written as: P ðLÞ ¼ ðI−LÞd0 ðI þ LÞdk IðEÞ

kþ1 Y

I−2ν i L−L2

di

ð3Þ

i¼1

where I(E) = 1 if S is even, and zero otherwise, and k + 1 = [S/2] + 1 − I(E). Cheung (1993) determines the spectral density function and proves that for d N 0 the spectral density function has a pole at λ = cos−1(ν), which varies in the interval [0, π]. It is interesting to note that when |ν| b 1, the spectral density function is bounded at the origin for GARMA processes, and thus does not suffer from many problems associated with ARFIMA models. Concerning the estimation of the k − factor GARMA model, we adopt an estimation procedure based on the wavelets following the methodology proposed by Whitcher (2004) and Boubaker and Sghaier (2015). Indeed, the complex behavior characterizes the spot price stimulated researchers to develop and test some methods for statistical analysis in order to realize a parsimonious model with greater accuracy. Recently, wavelet analysis has been proposed as a very practical tool adopted to deal with complex phenomena. The ability of wavelets to capture variability over both time and scale can offer more insight concerning the nature of data on electricity markets. These regimes can be effectively applied to decompose the time series into different

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time scales. Hence, wavelet analysis is likely to reveal seasonality, discontinuities, and volatility aggregation (Gençay et al. 2001aa, 2001b, 2001c). For these reasons, wavelet transforms are applied for energy market data in order to capture nonlinear patterns and hidden patterns that exist between variables. In this paper we focus on the discrete wavelet packet transform (DWPT), which deals with the existence of seasonalities and allows to decorrelate the spectrum of the process (see Boubaker and Sghaier (2015) and Guegan and Lu (2009), for more details). Note that for a GARMA model with a single frequency, when ν = 1 (i. e, λ = 0), the model is reduced to a ARFIMA model. Granger and Joyeux (1980) and Hosking (1981) introduce this model as a parametric tool to capture the dynamics of long rang dependence. It is a parsimonious method of modelling the long-term behavior of time series. Compared to existing models of long-term persistence, the family of ARFIMA models provide greater flexibility in estimating both short and long-term behavior of time series. In the spectral domain, ARFIMAprocesses have a peak for very low frequencies, near to zero frequency. ARFIMA(p, d, q) process is presented as follows: ΦðLÞð1−LÞd ðyt −μ Þ ¼ ΘðLÞεt ;

ð4Þ

where, εt is a white noise process, with zero mean and variance σ2, Φ(L) and Θ(L) are two polynomials of orders p and q, where ΘðLÞ ¼ p q X X φi Li and ΘðLÞ ¼ 1 þ θi Li, (L is the lag operator), (1 − L)d is 1þ i¼1

i¼1

a fractional differencing operator. On the other hand, for a GARMA model with a single frequency, when ν = 1 and d = 1/2, the process is an ARIMA model. Finally, when d = 0, we get a stationary ARMA model, given as follows: ΦðLÞðyt −μ Þ ¼ ΘðLÞεt :

ð5Þ

3.2. The multivariate conditional volatility and dynamic conditional correlations modelling The k − factor GARMA model supposes that the conditional variance is constant over time. Nevertheless, in the empirical studies, it is well recognized that many time series often exhibit volatility clustering, where time series exhibit both high and low periods of volatility. To reproduce these patterns, we extended the generalized long memory process, described above, by inserting a fractional filter in the equation of conditional variance. For this reason, we suggest the Fractional Integrated Generalized Autoregressive Conditional Heteroscedasticity (FIGARCH) model, which allows us to estimate the long memory behavior in the conditional variance. (Boubaker and Boutahar, 2011; and Boubaker and Sghaier, 2015). Baillie et al. (1996a, 1996b) and Bollerslev and Mikkelsen (1996) introduced the FIGARCH model, in order to estimate the finite persistence related to conditional variance. For this model, the autocorrelation function is characterized by a slow decay with hyperbolic rate. In the spectral domain, this model shows a peak correspondent to very low frequencies (so close to the zero frequency). Thus, we estimate k − factor GARMA process with FIGARCH type innovations in order to take into consideration the presence of long memory behavior in the conditional variance. This model is written as follow: yt = μt + εt = μt + σtzt; εt = σt zt   εt =It−1  N 0; σ 2t

ð6Þ

where μt is the conditional mean of yt, modelling using Eq. (1), σ2t is the conditional variance, It−1 being the information up to time t − 1, zt is an i. i. d random variable with zero mean and unitary variance. ht ¼ σ 2t ¼ ω þ ½1−ð1−ΒðLÞ−1 ΨðLÞð1−LÞδ ε2t :

ð7Þ

640

where ΨðLÞ ¼ 1−

B.A. Souhir et al. / Energy Economics 80 (2019) 635–655 r X

ψi Li and ΒðLÞ ¼ 1−

i¼1

s X

βi Li are suitable polyno-

i¼1

mials in the lag operator L with the roots assumed to lie outside the unit circle, 0 b δ b 1 is the fractional differencing (long memory) parameter. The FIGARCH(r, δ, s) model nests the GARCH(r, s)and the integrated GARCH (IGARCH) models in the sense that when δ = 0, FIGARCH model reduces to GARCH model while for δ = 1 it becomes a IGARCH model. In order to capture the dynamic of time-varying conditional correlation between electricity returns and each sectoral market index, we estimate a bivariate FIGARCHmodel with corrected-Dynamic Conditional Correlation specification termed the c − DCC − FIGARCH model that allows examining the dynamic conditional correlations (short-run links) among the considered variables under the effects of long-run interactions and volatility persistence. Aielli (2008) proposes a corrected Dynamic Conditional Correlation (c − DCC) modelling in order to correct both the lack of consistency and the potential bias in the estimated parameters of DCC − GARCH model of Engel (2002). In our study the conditional variancecovariance matrix, H t ¼ Eðεt ε t 0 j Ψt−1 Þ with εt ¼ ðε1t ; …; εkt Þ0 ¼ Ht 1=2 ηt where ηt~N(0, Ik), is modelled as. 8 Ht ¼ Dt Rt Dt > > > qffiffiffiffiffiffiffiffiffiffi > qffiffiffiffiffiffiffiffiffiffi > > > > D ¼ diag h ; …; hk; t > t 1; t > > <     −1=2 −1=2 −1=2 −1=2 > Rt ¼ diag q11; t ; …; qkk; t Q t diag q11; t ; …; qkk; t > > > > > Q ¼ ð1−θ −θ ÞQ þ θ η η0  þ θ Q > > 1 2 1 t−1 t−1 2 t−1 t > > > :  ηt ¼ diag fQ t g1=2 ηt

ð8Þ

where Xt is a (k × 1) vector of the system series; εt the vector of error terms estimated from the conditional mean equations (Eq. (1)); Ht the conditional variance-covariance matrix of system variables; Dt the (k × k) diagonal matrix of time-varying standard deviations computed from a univariate FIGARCH model; and Rt the (k × k) symmetric matrix of dynamic conditional correlations. Qt = (qijt) is a symmetric positive define matrix which is assumed to vary according to a FIGARCH process with Q being a (k × k) unconditional variance matrix of standardised residuals ηi, t. The parameters θ1 and θ2 are scalar parameters that capture the effects of shocks on dynamic correlations. θ1 and θ2 are positive and satisfy the following condition θ1 + θ2 b 1. The correlation estimator between variable i and variable j in the matrix Rt is defined as: qij; t ρij; t ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qii; t qjj; t

ð9Þ

Note that c − DCC model is similar to standard DCC model, except the correlation processQt, which was formulated, in the case of DCC model, as: Q t ¼ ð1−θ1 −θ2 ÞQ þ θ1 ηt−1 η0t−1 þ θ2 Q t−1

ð10Þ

It is worth noting that we use the univariate FIGARCHrepresentation (Eq. (7)) to model the conditional volatility of each variable and then compute their time-varying standard deviations. 3.3. Value at risk and conditional value at risk models The Value at Risk (VaR) is an aggregated measure of the total risk of a portfolio of contracts and assets. It summarizes the expected maximum loss (worst loss) of a portfolio over a target horizon within a given confidence interval (generally 95%). Thus, VaR is measured in monetary units. As the maximum loss of a portfolio, VaR(95%) is a negative number. Therefore, maximizing VaR is equivalent to minimizing the

portfolios loss. We rely on VaR because it is an efficient measure of downside portfolio risk. Given the return Rti of an individual market i at time t with a confidence level of q; VaRti, q is implicitly defined as qth quantile of the return distribution as follows:   P Rit ≤ VaRi;q ¼q t

ð11Þ

where VaRti, q is typically a negative number. However, the main drawback of this method is the failure to recognize potential losses that may exist above the value of VaR. To overcome this limitation, the Conditional Value-at-Risk CVaR method was also considered. This risk measure was introduced by Artzner et al. (1999) has been shown to share basic coherence properties (which is not the case of VaR(α)). Indeed, CVaR is based on the weighted average of losses with a higher probability compared to VaR. Hence, we strengthen the robustness of our results with the CVaR modelling. CVaR, is strongly related to the previous risk measure (VaR). More precisely, VaR at level α ∈ [0, 1] denoted VaR(α) of a given portfolio loss distribution represents the lowest value (or amount) that the loss does not exceed with a given probability α (generallyα ∈ [0.95,1]). On the other hand, the Conditional Value at Risk at level α denoted CVaR (α) represents the conditional expected portfolio losses beyond the VaR(α) level. CVaR shows better mathematical properties compared to VaR because it takes into consideration the thick tails in the portfolio loss distribution. Intuitively, VaR is the maximum possible loss at a certain level of confidence and the CVaR approach offers broad flexibility for describing risk contagion between markets or between stocks in the same market. According to Girardi and Ergün (2013), CVaRq, s/i t is computed as the q-quantile of the conditional distribution, which is defined as;   s=i s=i P Rst ≤ CVaRq;t =Rit ≤ VaRq;t ¼ q

ð12Þ

This change allows for more severe losses (farther in the tail). Along these lines, we follow Adrian and Brunnermeier (2016) and define ΔCoVaR as: s=i;q

ΔCVaRt

s=i;q

¼ CVaRt

i

s=b ;q

−CVaRt

ð13Þ

We can see that CVaR is an element of calculating ΔCVaR. For this equation, bi denotes the benchmark state, which represents the one standard deviation event around the mean: μti − σti ≤ Rti ≤ μti + σti, where μti and σti denote the conditional mean and standard deviation of the system, respectively. In our study, CVaR is determined using the following steps: Step 1. The univariate generalized long memory models (Eqs. (1) and (7)) are fitted for electricity return and for each sectoral index to approximate isolated time series of VaRs electricity returns and sectoral market indexes VaRs are given by: VaRsq;t ¼ Φ−1 ðqÞσ st

ð14Þ

Step 2. With the aim to estimate the dynamic conditional correlation existing between the electricity returns and each sectoral index, we adopt a bivariate FIGARCH process with corrected − DCC specification called c − DCC − FIGARCH model (Eq. (8)) that allows the identification of the dynamic conditional correlations (short-run links) between measured variables under the effects of volatility persistence and long-run interactions. Hence, we estimate the bivariate c − DCC − FIGARCH for each pair. Step 3. After estimating the bivariate density of each pair in step 2, in step 3 we proceed to find CVaRq, s/elec measure for electricity t

B.A. Souhir et al. / Energy Economics 80 (2019) 635–655

returns (elec) and each sectoral index "s" at time period t. CVaRqs/elec is defined by: s=elec

¼ Φ−1 ðqÞσ st

s=elec

¼ VaRsq;t

CVaRq;t CVaRq;t

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1−ρ2s=elec;t þ Φ−1 ðqÞρs=elec;t σ st

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1−ρ2s=elec;t þ VaRsq;t ρs=elec;t

ð15Þ ð16Þ

where ρs/elec, t is the correlation coefficient between electricity returns and sectoral index. Furthermore, in our study, we adopt ΔCVaRq,s/elec which we label t “exposure ΔCVaR”, to evaluate stock exposure to electricity market turmoil. By definition, ΔCVaR is the difference between its CVaR when electricity market is, or is not, in distress (median state); s=elec

ΔCVaRs=elec ¼ CVaRs=elec −CVaR50% q q

ð17Þ

Compared to other measures, this model has the advantage of evaluating exposure degree of each sector to electricity market shocks. This seems interesting for portfolio managers and policy makers. Since Φ−1 (50%) = 0, we can reduce ΔCVaR at each time as following: s=elec

¼ Φ−1 ðqÞρs=elec;t σ st

ð18Þ

s=elec

¼ VaRsq;t ρs=elec;t

ð19Þ

ΔCVaRq;t ΔCVaRq;t

A weaker or positive ΔCVaR argued that the equity market is less exposed to the electricity market collapse. 4. Empirical analysis 4.1. Nord pool power market The Nordic electricity market, also known as Nord Pool, is a power market dedicated to the electrical products. It was established in 1992 and includes some Nordic countries such as Norway, Sweden, Denmark, Finland, etc. In fact, this market started operating officially in 1993, where Norway was the only area. Later, Sweden joined in January 1996. Finally, Finland was fully integrated in March 1999. The Nord Pool market contains a spot market and a secondary market. Its spot market, known as Elspot, allows market participant to sell and buy electricity for delivery within 24 h. The operation market system is interesting, as it balances the demand and supply in volumes. In addition, this market grew speedily; recently, 40% of the whole electricity consumption in the Nordic area is registered on this market. It is noticeable that the spot market is not compulsory: Only 25% of all electricity traded in Norway and Sweden is managed by Nord Pool, while the rest is covered by bilateral contracts. The derivatives market allows the sale and purchase of raw materials, energy, equities, and bonds in predefined conditions (Hjalmarsson, 2000; and Weron, 2006; Junttila et al., 2018). Nord Pool distributes information, regarding the tender areas that will be applicable during the following week and on the operator's database system in Norway, to all market participants. These participants are mainly retailers, suppliers, large customers, traders, and financial institutions. Today, there are N300 participants on Nord Pool market. It is an active and liquid market, which is generally regarded as the most mature, and powerful market worldwide. It is considered as a good example for other energy exchanges, seeing that the network structure is quite simple, and structure of the industry is highly fragmented with N350 production companies. Also, one can note that the level of cooperation between network operators, governments, and regulators is very high. Furthermore, it is relevant to state that the system used by the Nord Pool shares several common characteristics with other electricity

641

markets. At Nord Pool market, spot price is the result of an auction at uniform prices on both sides for hourly intervals. It is determined based on several offers presented on the market (Weron et al., 2004; Junttila et al. (2018)). Briefly, Nord Pool electricity spot markets is an eminent European power exchange market. In such markets, participants trade power contracts for physical delivery for the next day. As for any power market, the prices are fixed each day, for each of the 24 h of the next day.

4.2. Data description and preliminary statistics With the aim of examining the relationship between electricity returns and some sectoral indexes in the Nordic market, we consider the hourly series of electricity spot price (Euros/MWh) extracted from official website of Nord Pool Power Market. Concerning the Nordic stock market, we consider the aggregate sector Index termed “Nordic Market Index” to illustrate the overall performance of the Nordic Market. Furthermore, we consider 11 other sector indexes in this market from various sectors of the economy (Banks, Basic Materials, Construction and Materials, Consumer Goods, Financials, Industrials, Insurance, Real Estate, Technology, Telecommunications, and Travel and Leisure) these series are extracted from DataStream database. Each time series ranges from the 3rd of July 2017 to the 8th of December 2017, which yields T = 1150 hourly observations. It seems that the study period is short. However, since the electricity is a non-storable commodity and its transmission is limited by physical and reliability constraints, any variation in the electricity demand in the matter of minutes or hours can generate an enormous effect on the electricity price. Furthermore, changes made to the electricity industry are the outcomes of the technological and economic alterations affecting the sector. In this vein, any variation in electricity market can be transmitted directly to other sectors that rely on electricity in their activity. On the other hand, as the electricity market is an hourly market the use of the hourly data allows determining an intraday hedging strategy for electricity market using stock indexes, which is the second goal of our study. In this paper, we consider the return series, which is computed by taking the differences in the Logarithm of two consecutive prices (Rt = ΔLogPt). In fact, the use of log difference sometimes makes the series stationary (see Figs. 2 and 4). Therefore, we analyse the Logreturn of each time series, in order to study their statistical and econometric features. The historical values of the Log- return electricity price is plotted in Fig. 2, it shows that the price return increases over time, and reaches maximum values during the months of November and December, i.e. during the winter, due to the increase in electricity consumption during this period. Fig. 3 shows the consumption of electricity in the Nordic area, it shows that the two curves (Log-return of electricity prices and electricity consumption) have a similar evolution. This can be explained by the fact that the temperature decreases gradually during these periods (November and December), which results in an increase in electricity consumption and since the electricity is not storable so any increase in consumption leads to the price increase (Junttila et al. (2018)). The time evolution of Log-return for all-time series (see Figs. 2 and 4) indicates that these series seem stationary. This hypothesis can also be supported by the unit root tests (ADF, PP and KPSS). In addition, the series present a clustering of volatility since periods of low volatility are followed by periods of high volatility. This is a sign of the presence of ARCH effects in the series. We tested for stationarity by performing unit root tests, namely, augmented Dickey-Fuller (ADF), Phillips- Perron (PP) and Kwiatkowski, Phillips, Schmidt, and Shin (KPSS) tests, for all Log-returns time series. These tests differ in the null hypothesis. The null hypothesis of ADF and PP tests is that a time series contains a unit root, while KPSS test

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B.A. Souhir et al. / Energy Economics 80 (2019) 635–655

0.4

Log Return Elspot Price

0.3

Log-Return Spot Price (Euro/MWh)

0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6

0

28/07/2017

25/08/2017

22/09/2017

20/10/2017

17/11/2017

15/12/2017

Time/ Hours

Fig. 2. Hourly Log-Return of Electricity Spot Price for Nord Pool Electricity Market.

has the null hypothesis of stationarity. The results of these tests are reported in Table 1. ADF and PP unit root tests indicate that all Log-returns time series are stationary. In addition, the statistics of KPSS test support the acceptance of the null hypothesis of stationary. Thus, these series are stationarity and suitable for subsequent tests in this study. The descriptive statistics of time series are reported in Table 2. Table 2 shows that the average of hourly return of Travel and Leisure sector is the highest (4.30 × 10−5) compared to the other sectors. Hence, this sector is considered as one of the most promising economic areas in the Nordic Market. However, Technology sector gives the lowest average return (−1.83 × 10−4). In addition, values of standard deviation are quite small, while the kurtosis for all of the studied series ranges from 6.91 to 57.21, indicating fat tails in the distributions, which mean that underlying data are leptokurtic. For the skewness, most of series are negatively skewed, indicating a distribution with an asymmetric tail extending toward more negative values. This significant

departure from normality is also confirmed by the large value of the Jarque-Bera (JB) test. Table 2 also presented the correlation between hourly return of electricity markets and each sector index return, indicating that the correlation degree is higher and positive for the Consumer Goods, the Basic Materials and Telecommunications indexes (0,084, 0,066 and 0,060, respectively), while electricity return is negatively correlated with Banks sector (−0,0107). Finally, the results of ARCH − LM test indicate the presence of the conditional heteroscedasticity for all Log-return time series. We test for the long-range dependence in the conditional mean of electricity returns, using GPH (Geweke and Porter-Hudak, 1983) and LW (Robinson, 1995) statistics, which tests the null hypothesis of the presence of short memory versus the alternative hypothesis of long memory. Corresponding results reported in Table 3, indicating the presence of long memory, are given for three bandwidth levels. As shown in Fig. 5, for Log-return of electricity price series, spectral density, traced by periodogram, shows peaks at equidistant frequencies,

Fig. 3. Electricity consumption in the Nordic Area.

B.A. Souhir et al. / Energy Economics 80 (2019) 635–655

Nordic Market Index Returns

Basic Materials Returns

Consumer Goods Returns

643

Bank Returns

Construction Materials Returns

Financials Returns

Fig. 4. Hourly Log-Return of Sector Indexes for Nordic Stock Market.

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B.A. Souhir et al. / Energy Economics 80 (2019) 635–655

Industrials Stock Returns

Insurance Returns

Real Estate Returns

Technology Stock Returns

Telecommunication Stock Returns

Travel Leisure Stock Returns Fig. 4 (continued).

B.A. Souhir et al. / Energy Economics 80 (2019) 635–655

645

Table 1 ADF, PP, KPSS unit root testing results for all Log-return series. ADF test

PP test

KPSS test

Log-return series

Model (3)

Model (2)

Model (1)

Model (3)

Model (2)

Model (1)

Model (3)

Model (2)

Electricity spot price

−22.7711⁎⁎⁎ (0.0000) −33.2299⁎⁎⁎ (0.0000) −21.6369⁎⁎⁎ (0.0000) −32.2693⁎⁎⁎ (0.0000) −31.9965⁎⁎⁎ (0.0000) −32.6517⁎⁎⁎ (0.0000) −21.6467⁎⁎⁎ (0.0000) −32.1002⁎⁎⁎ (0.0000) −33.4833⁎⁎⁎ (0.0000) −21.1696⁎⁎⁎ (0.0000) −19.8967⁎⁎⁎ (0.0000) −32.3685⁎⁎⁎ (0.0000)

−22.7787⁎⁎⁎ (0.0000)

−22.7884⁎⁎⁎ (0.0000) −33.2117⁎⁎⁎ (0.0000) −21.5252⁎⁎⁎ (0.0000) −33.2117⁎⁎⁎ (0.0000) −31.9211⁎⁎⁎ (0.0000) −32.6462⁎⁎⁎ (0.0000) −21.5443⁎⁎⁎ (0.0000) −32.0503⁎⁎⁎ (0.0000) −33.4873⁎⁎⁎ (0.0000) −21.1094⁎⁎⁎ (0.0000) −19.818⁎⁎⁎ (0.0000) −32.3370⁎⁎⁎ (0.0000)

−94.1254⁎⁎⁎ (0.0001) −33.2905⁎⁎⁎ (0.0000) −33.8445⁎⁎⁎ (0.0000) −32.3837⁎⁎⁎ (0.0000) −32.3381⁎⁎⁎ (0.0000) −32.9262⁎⁎⁎ (0.0000) −32.9169⁎⁎⁎ (0.0000) −32.2615⁎⁎⁎ (0.0000) −33.4833⁎⁎⁎ (0.0000) −32.8627⁎⁎⁎ (0.0000) −19.928⁎⁎⁎ (0.0000) −32.4429⁎⁎⁎ (0.0000)

−84.1465⁎⁎⁎ (0.0001) −33.2733⁎⁎⁎ (0.0000) −33.8310⁎⁎⁎ (0.0000) −32.3768⁎⁎⁎ (0.0000) −32.3422⁎⁎⁎ (0.0000) −32.9280⁎⁎⁎ (0.0000) −32.8984⁎⁎⁎ (0.0000) −32.2314⁎⁎⁎ (0.0000) −33.4730⁎⁎⁎ (0.0000) −32.8657⁎⁎⁎ (0.0000) −19.912⁎⁎⁎ (0.0000) −32.4354⁎⁎⁎ (0.0000)

−78.8691⁎⁎⁎ (0.0001) −33.2849⁎⁎⁎ (0.0000) −33.8308⁎⁎⁎ (0.0000) −33.3893⁎⁎⁎ (0.0000) −32.3361⁎⁎⁎ (0.0000) −32.9408⁎⁎⁎ (0.0000) −32.9001⁎⁎⁎ (0.0000) −32.2377⁎⁎⁎ (0.0000) −33.4873⁎⁎⁎ (0.0000) −32.8365⁎⁎⁎ (0.0000) −19.987⁎⁎⁎ (0.0000) −32.4448⁎⁎⁎ (0.0000)

0.2877⁎⁎⁎ [0.2160] 0.0274⁎⁎⁎ [0.2160] 0.0523⁎⁎⁎ [0.2160] 0.0414⁎⁎⁎ [0.2160] 0.03617⁎⁎⁎ [0.2160] 0.0450⁎⁎⁎ [0.2160] 0.0369⁎⁎⁎ [0.2160] 0.0295⁎⁎⁎ [0.2160] 0.0632⁎⁎⁎ [0.2160] 0.0176⁎⁎⁎ [0.2160] 0.03806⁎⁎⁎ [0.2160] 0.0572⁎⁎⁎ [0.2160]

0.3501⁎⁎⁎ [0.7390] 0.1623⁎⁎⁎ [0.7390] 0.3665⁎⁎⁎ [0.7390] 0.1248⁎⁎⁎ [0.7390] 0.1504⁎⁎⁎ [0.7390] 0.1286⁎⁎⁎ [0.7390] 0.3212⁎⁎⁎ [0.7390] 0.2237⁎⁎⁎ [0.7390] 0.1404⁎⁎⁎ [0.7390] 0.0700⁎⁎⁎ [0.7390] 0.2846⁎⁎⁎ [0.2160] 0.2442⁎⁎⁎ [0.7390]

Nordic market Index Banks Basic materials Construction and materials Consumer goods Financials Industrials Insurance Technology Telecommunication Travel and leisure

−33.2003⁎⁎⁎ (0.0000) −21.5510⁎⁎⁎ (0.0000) −32.2511⁎⁎⁎ (0.0000) −31.9768⁎⁎⁎ (0.0000) −32.6323⁎⁎⁎ (0.0000) −21.5653⁎⁎⁎ (0.0000) −32.0481⁎⁎⁎ (0.0000) −33.4729⁎⁎⁎ (0.0000) −21.1602⁎⁎⁎ (0.0000) −19.8124⁎⁎⁎ (0.0000) −32.3288⁎⁎⁎ (0.0000)

Notes: Model (3) With an intercept and a trend, Model (2) With an intercept, and Model (1) Without an intercept. Levels of significance of the ADF and the PP tests are indicated between brackets and represent the p-value. Levels of significance of the KPSS test are indicated between squared brackets, and represent the critical values at level 1%; ⁎⁎⁎ Denotes significance at 1% level.

which proves the presence of several seasonalities that requires the use of generalized long memory model to estimate such process. 4.3. Empirical findings and interpretations 4.3.1. Generalized long memory process for conditional mean modelling The conditional mean of Log-return of electricity price as well as Logreturn of each sector index are estimated using the generalized long

memory model (k − factor GARMA). Estimation results are displayed in Table 4. Mean estimation results indicate that only Log-return of electricity price is estimated using the k − factor GARMA model, which confirm that this series is characterized by periodic long memory behavior (see Fig. 5). However, for sector indexes, k − factor GARMA model is reduced to ARFIMA model for consumer goods, and to ARMA model for all other sectors.

Table 2 Data statistic's descriptions for all Log-return series. Std. Dev

Skewness

Kurtosis

Jarque-Bera

Correlation

Electricity spot price

Mean 1.19 × 10−4

0,0523

−1.2313

34.8777

1.0000

Nordic market index

−2.01 × 10−5

0,0019

0.0728

14.0647

−5

0,0028

0.7821

21.2503

48,940.2 (0.0000)⁎⁎⁎ 5862.3 (0.0000)⁎⁎⁎ 16,063.1 (0.0000)⁎⁎⁎

1.72 × 10−5

0,0031

−0.0085

13.4657

−1.31 × 10−4

0,0027

−1.0819

18.2113

−9.19 × 10

−6

0,0022

−0.0765

13.8334

Financials

−7.52 × 10

−5

0,0024

0.7559

15.7624

Industrials

−5.07 × 10−5

0,0026

0.1220

12.1330

Insurance

9.02 × 10−6

Bank Basic materials Construction materials Consumer goods

−9.40 × 10

0.0025

0.15467

6.9144

Technology

−1.83 × 10

−4

0.0038

−3.2909

57.2168

Telecommunications

−3.08 × 10−5

0.0026

0.2026

9.1190

4.30 × 10−5

0.0031

1.1046

20.2894

Travel and leisure

Levels of significance of Jarque-Bera and ARCH tests are indicated between squared brackets. ⁎⁎⁎ Denotes significance at 1% level. ⁎⁎ Denotes significance at 5% level. ⁎ Denotes significance at 10% level.

0.0370 −0.0107

5243.7677 (0.0000)⁎⁎⁎ 11,301.6 (0.0000)⁎⁎⁎ 5619.8515 (0.0000)⁎⁎⁎

0.0668

7907.1935 (0.0000)⁎⁎⁎ 3996.2 (0.0000)⁎⁎⁎ 2291.4 (0.0000)⁎⁎⁎ 142,800.8 (0.0000)⁎⁎⁎

0.0118

1800.4 (0.0000)⁎⁎⁎ 14,544.6 (0.0000)⁎⁎⁎

0.0602

0.0122 0.0848

0.0421 0.0311 0.0458

0.0582

ARCH (10) 17.7800 (0.0000)⁎⁎⁎ 22.0658 (0.0336)⁎⁎ 10.5720 (0.0000)⁎⁎⁎ 16.1523 (0.0707)⁎ 7.5775 (0.0000)⁎⁎⁎ 13.141 (0.0000)⁎⁎⁎ 24.058 (0.0000)⁎⁎⁎ 25.105 (0.0000)⁎⁎⁎ 24.487 (0.0000)⁎⁎⁎ 2.1873 (0.0165)⁎⁎ 7.5930 (0.0000)⁎⁎⁎ 4.9150 (0.0000)⁎⁎⁎

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B.A. Souhir et al. / Energy Economics 80 (2019) 635–655

Table 3 The GPH and LW long memory tests results for all Log-return series. T = 1150

Bandwidth

GPH Test ^m d

0.6

T = 69 T0.7 = 139 T0.8= 281

Electricity returns

−0.4845⁎⁎⁎ −0.1736⁎⁎⁎ −0.2955⁎⁎⁎

LW Test Standard error 0.0855 0.0580 0.0400

p-value

^m d

(0.0000) (0.0000) (0.0000)

−0.4042⁎⁎⁎ −0.1361⁎⁎⁎ −0.399⁎⁎⁎

Standard error

p-value

0.0601 0.0424⁎⁎ 0.0298⁎

(0.0000) (0.0013) (0.0000)

Levels of significance of GPH and LW tests are indicated between squared brackets. ⁎⁎⁎ Denotes significance at 1% level. ⁎⁎ Denotes significance at 5% level. ⁎ Denotes significance at 10% level.

Indeed, for the Log-return of electricity, according to the k − factor GARMA estimation results, the seasonality can be observed in the frequency domain λi=1/T; where λ is frequency of the seasonality and T is the period of seasonality. As shown, spectral densities, represented by periodogram (see Fig. 5), are unbounded at equidistant frequencies, which proves the presence of several seasonalities. They show special _

peaks at frequencies λ m;1 ¼ 0:1334 (T1 = 7.49 ≃ 8hours = 1/3 day), _

and λ m;2 ¼ 0:2467 (T2 = 4.05 ≃ 4hours = 1/6 day), corresponding to cycles with 1/3 day, 1/6 day periods, respectively. For sector indexes, we use some specifications of ARFIMA(p, d, q) model and we consider all the possible combinations with p = 0; 1and q = 0; 1 to select the appropriate model for each sector index. Estimation results are reported in Table 4, indicating that the appropriate model of mean equation differ from one sector index to another. 4.3.2. The long memory process for conditional variance modelling The residuals from the conditional mean estimation are used as a proxy for the conditional variance, which are modelled using fractional integration FIGARCH process. For the residuals of the k − factor GARMA model, the spectral density, traced by periodogram (Fig. 6) shows peak so close to the zero frequency. Moreover, the results of long memory GPH and LW tests (Table 5) indicate the presence of long memory in the conditional

variance, which require the use of some fractionally integrated FIGARCH method to estimate such processes. The conditional variance estimation results (Table 6) indicate that the variance equations are always estimated using FIGARCH model, which allows us to estimate the bivariate c − DCC − FIGARCH model, in the second step, between electricity returns and each sector index returns. The FIGARCH process is estimated under the assumption of skewed-student distribution (see Appendix), this last one proves its efficiency compared to two other distributions methods (Normal and Student), since it's able to assess both the excess of skewness and the fat tails in the data. Estimation results indicate that the parameters of _ _

_

FIGARCH model δ , ψ and β are significantly different from zero. More_

_

over, sum of the estimated coefficients of ARCH and GARCH, ψ and β , is close to one. These results indicate the presence of long-rang dependence in the conditional mean and the persistence of the volatility process. Remind that these univariate estimation results are adopted to estimate in-sample dataset5 % − VaR of each sector index returns and electricity returns. 4.3.3. Testing for breaks-points in the electricity returns In order to identify change-points in the dependence structure in electricity returns volatility, we test the stability using Bai and Perron

Fig. 5. Autocorrelation Function and Periodogram of the Log-Return Electricity price.

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647

Table 4 Mean equation estimation results. • The k-factor GARMA model:ΦðLÞ

k Y

di

ðI−2νi L þ L2 Þ ðyt −μÞ ¼ ΘðLÞε t

i¼1 L)d(yt

• The ARFIMA model: Φ(L)(1 − − μ) = Θ(L)εt, • The ARMA model: Φ(L)(yt − μ) = Θ(L)εt. Parameters

_

Electricity returns

0.5234⁎⁎⁎

Nordic market index Banks

(0.0000) _ 0.7669⁎⁎⁎

Basic materials Construction and materials Consumer goods Financials

_

_

_

d m;2

_ λ m;1

_

_

0.3982⁎⁎⁎

0.2023⁎⁎⁎

0.1334⁎⁎⁎

(0.0000) _ _

(0.0000) _ _

(0.0000) _ _

0.2467⁎⁎⁎ (0.0000) _ _

_ _

_ _

_ _

_ _

−0.9134⁎⁎⁎ (0.0000) −0.7162⁎⁎⁎

0.0400⁎ (0.0670) _

_

_

_

_

_

_

(0.0000) _

_

_

_

_

_ _

_ _

_ _

_ _

_ _

θ

ϕ

_ −0.7314⁎⁎⁎ (0.0000) _ _

(0.0000) _ 0.0686⁎⁎ (0.0025) 0.8973⁎⁎⁎ (0.0000) 0.7491⁎⁎⁎

d m;1

λ m;2

Technology

(0.0000) 0.0617⁎⁎ (0.0118) _ 0.0569⁎⁎ (0.0336) −0.7648⁎⁎⁎

_

_

_

Telecommunications

0.7787⁎⁎⁎ (0.0000) −0.4880⁎⁎⁎

_

(0.0000) 0.4842⁎⁎⁎

_

_

_

_

Travel and leisure

(0.0000) _

(0.0010) _

_

_

_

_

Industrials Insurance Real estate

Levels of significance (p-value) are indicated between squared brackets. ⁎⁎⁎ Denotes significance at 1% level. ⁎⁎ Denotes significance at 5% level. ⁎ Denotes significance at 10% level.

(2003) test, using absolute return as a proxy of volatility (Fig. 5). Indeed, Bollerslev and Mikkelsen (1996), Andersen and Bollerslev (1997), and Boutahar et al. (2008), adopt absolute returns as a proxy of volatility instead of squared returns because this latter measure can be noisy. In addition, the absolute changes in series have a stronger autocorrelation than the square of changes (see Taylor (1986) for more details).

The identification of volatility structure in electricity returns allows us to investigate the impact of this structure changes on sector indexes. Specifically, we check whether this dependence is constant over period. Concretely, we apply the Bai- Perron test, to check existence, number and localisation of unknown change points. Test results (Table 7) indicate the presence of two significant change-points.

Fig. 6. Periodogram of the squared residuals Log-Return Electricity price.

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Table 5 GPH and LW long memory tests results for residuals of Log-return electricity price. T = 1150

Bandwidth

GPH Test

LW Test

^δ

p-value

^δ

0.0284 0.0582 0.0403

(0.0001) (0.0001) (0.0000)

0.2443⁎⁎⁎ 0.2698⁎⁎⁎ 0.2639⁎⁎⁎

0.1642⁎⁎⁎ 0.2320⁎⁎⁎ 0.2121⁎⁎⁎

T0.6=69 T0.7=139 T0.8=281

Residuals electricity spot price

Standard error

Standard error

p-value

0.0601 0.0424⁎⁎ 0.0298⁎

(0.0000) (0.0000) (0.0000)

Levels of significance of GPH and LW tests are indicated between squared brackets. ⁎⁎⁎ Denotes significance at 1% level. ⁎⁎ Denotes significance at 5% level. ⁎ Denotes significance at 10% level.

Fig. 7 represents electricity returns volatility considering the presence of two breaks (in 18/08/2017 and 10/11/2017, respectively). These change points divide the sample into three sub-samples. The first period (pre-Beak 1) is characterized by a stability in electricity (calm period). The second period (between the two breaks) represent more variability compared to the first period, and the third period (post-Break 2) indicate high volatility compared to the previous periods. These periods can be associated to the electricity demand or meteorological conditions variations. In fact, the studied period includes three seasons; the summer (July and August), the autumn (September and October) and winter (November and December). Therefore, we can refer the break points to the change of temperature between summer and autumn (first break point) and between autumn and winter (second break point). Indeed, any drop-in temperature leads to an increase in the electricity consumption (see Fig. 3) which leads to an increase in the electricity price (that causes the high volatility), since the electricity is a non-storable commodity (Junttila et al., 2018). This change of periods can affect simultaneously sectors that rely on electricity in their activities. Therefore, we need to compare the values of dynamic conditional correlation, tail dependence coefficients and

Table 6 The FIGARCH(1,δ,1) model for conditional variance. Estimation FIGARCH(1, δ,1) Parameters

_

ψ

_

_

β

Ln(L)

δ

0.1724⁎⁎ (0.0217) 0.6220⁎⁎⁎

0.6420⁎⁎⁎ (0.0000) 0.8638⁎⁎⁎

0.5267⁎⁎⁎ (0.0000) 0.3821⁎⁎⁎

(0.0000) 0.5610⁎⁎⁎ (0.0000) 0.4659⁎⁎⁎

(0.0000) 0.8169 (0.0000) 0.7748⁎⁎⁎

(0.0000) 0.3460⁎⁎⁎ (0.0000) 0.3627⁎⁎⁎

5208.6610

(0.0000) 0.5445⁎⁎⁎

(0.0000) 0.7787⁎⁎⁎

(0.0000) 0.3033⁎⁎⁎

5437.1251

Industrials

(0.0000) 0.5827⁎⁎⁎ (0.0000) 0.5639⁎⁎⁎ (0.0000) 0.5785⁎⁎⁎

Insurance

(0.0000) 0.5438⁎⁎⁎

(0.0000) 0.8195⁎⁎⁎ (0.0000) 0.8286⁎⁎⁎ (0.0000) 0.8369 (0.0000) 0.8506⁎⁎⁎

(0.0000) 0.3301⁎⁎⁎ (0.0000) 0.3521⁎⁎⁎ (0.0000) 0.3621 (0.0000) 0.4268⁎⁎⁎

(0.0000) 0.5698⁎⁎⁎ (0.0000) 0.3587⁎⁎⁎

(0.0000) 0.8088⁎⁎⁎ (0.0000) 0.5893⁎⁎⁎

(0.0000) 0.3256⁎⁎⁎ (0.0000) 0.2460⁎⁎⁎

(0.0000) 0.5603⁎⁎⁎ (0.0000) 0.6153⁎⁎⁎ (0.0000)

(0.0000) 0.8598⁎⁎⁎ (0.0000) 0.8071⁎⁎⁎ (0.0000)

(0.0000) 0.4209⁎⁎⁎ (0.0000) 0.2901⁎⁎⁎ (0.0000)

Electricity spot price Nordic market index Banks Basic materials Construction and materials Consumer goods Financials

Real Estate Technology Telecommunications Travel and leisure

Levels of significance (p-value) are indicated between squared brackets. ⁎⁎⁎ Denotes significance at 1% level. ⁎⁎ Denotes significance at 5% level. ⁎ Denotes significance at 10% level.

2625.2200 5793.8851 5360.3620

hedging parameters during the overall period as well as the threeabovementioned sub-periods. 4.3.4. Dynamic conditional correlation and conditional VaR estimation results To investigate the reaction of sectors to electricity market variability, we adopt DCC, VaR, CVaR and ΔCVaR models in order to examine the presence of spillover effects and to recognize the level of resistance of each sector to electricity turmoil. Estimation results are reported in Table 8. According to Fig. 8, it can be observed that 5 % − VaR electricity returns and 5 % − VaR of some sector indexes display a similar tendency for most of the period. To exemplify, a significant trough for Technology VaR prices follows the trough observed in electricity market at the end of July and October 2017. It is worth noting that VaR measure is unable to confirm that these electricity returns turmoil caused these stock market failures, since there have been several other events that can cause stock market declines. However, c − DCC and CVaR measures can make it possible. Following Fig. 9, we can observe that c − DCC between extreme negative electricity market returns and selected sector indexes are not stable and vary over periods. A high positive correlation is observed, not only during the turbulence of electricity market (the 3rd period) market, but also for the period of its stability (the 1st period). This positive dynamic correlation between electricity and stock market may support that electricity market stress can spread to stock market and caused bearish periods. In order to study the interdependence between electricity market and Nordic stock market, we have taken into consideration the Nordic market index, as the global market index, and 11 sector indexes. Table 8 summarizes the results of VaR sector indexes, the corrected dynamic conditional correlations c − DCC between electricity returns and each sector index, which is determined through bivariate c − DCC − FIGARCH estimation, and the averages of CVaR and ΔCVaR. For the whole period, the results indicate on average that there exists a strong positive dynamic correlation between Nordic market

5631.0140 5562.6370

Table 7 Bai-Perron test for electricity returns volatility.

5472.2831 Sequential F-statistic determined breaks: 2 5396.2354 5355.2012 5129.4880 5371.7670⁎

Break Test

F-statistic

Scaled F-statistic

Critical Value⁎⁎

0 vs. 1⁎ 1 vs. 2⁎ 2 vs. 3

125.0329 17.75848 1.810090

125.0329 17.75848 1.810090

8.58 10.13 11.14

Sequential F-statistic determined breaks: 2 5222.6500

Break dates

Repartition

Date

1 2

348 950

18/08/2017 10/11/2017

⁎ Significant at the 0.05 level. ⁎⁎ Bai-Perron critical values.

B.A. Souhir et al. / Energy Economics 80 (2019) 635–655

649

Fig. 7. Change periods of electricity returns volatility.

index and electricity market (0.81), and a positive dynamic correlation between electricity market and all other sector indexes, with various levels. The highest positive c − DCC average was accomplished by Financials sector (0.89) followed by Industrials sector (0.85). However, the lowest positive c − DCC average was observed for Telecommunications and Travel and Leisure sectors (0.68). Concerning the VaR average values, the highest level of risk is observed for Technology sector (−4.84 × 10−3) followed by Travel

and Leisure sector (−4.37 × 10 −3 ), and Basic Materials sector (−4.19 × 10 −3 ), hence these sectors are the most risky in this market. However the lowest risk is observed for Consumer goods sector with level of (−3.04 × 10−3) followed by the Financial sector, Real Estate and Telecommunications sectors with a close level of (−3.5 × 10−3). Hence, these sectors are the least risky in Nordic market and can be advantageous especially for the risk averse invertors.

Table 8 c − DCC, VaR, CVaR and ΔCVaR averages for Nordic market index and sector indexes. Overall period c − DCC Nordic index Banks Basic materials Constr & Mat Consumer goods Financials Industrials Insurance Real Estate Technology Telecom travel & leis

0,8105 0,7900 0,8186 0,7929 0,7814 0,8904 0,8571 0,7349 0,7259 0,7303 0,6880 0,6850

03/07/2017–18/08/2017 (Pre-Break 1) VaR

CVaR

ΔCVaR

−3.01 × 10−3 −3.81 × 10−3 −4.19 × 10−3 −3.81 × 10−3 −3.04 × 10−3 −3.50 × 10−3 −3.76 × 10−3 −3.84 × 10−3 −3.52 × 10−3 −4.84 × 10−3 −3.56 × 10−3 −4.37 × 10−3

−4.21 × 10−3 −5.35 × 10−3 −5.84 × 10−3 −5.35 × 10−3 −4.27 × 10−3 −4.70 × 10−3 −5.15 × 10−3 −5.42 × 10−3 −4.98 × 10−3 −6.82 × 10−3 −5.02 × 10−3 −6.17 × 10−3

−2.44 × 10−3 −3.01 × 10−3 −3.43 × 10−3 −3.02 × 10−3 −2.37 × 10−3 −3.12 × 10−3 −3.23 × 10−3 −2.83 × 10−3 −2.56 × 10−3 −3.49 × 10−3 −2.45 × 10−3 −3 × 10−3

c − DCC 0,7187 0,7703 0,7992 0,7257 0,7870 0,8443 0,9177 0,7017 0,6859 0,7066 0,6328 0,5117

ΔCVaR

c − DCC

18/08/2017–10/11/2017 (between Break 1 and Break 2) c − DCC Nordic index Banks Basic materials Constr & mat Consumer goods Financials Industrials Insurance Real estate Technology Telecom Travel & leisure

0,7855 0,7488 0,7497 0,6623 0,7170 0,8999 0,8187 0,7574 0,7236 0,6084 0,6071 0,6242

VaR

CVaR −3

−2.67 × 10 −3.82 × 10−3 −3.30 × 10−3 −3.28 × 10−3 −1.33 × 10−2 −3.09 × 10−3 −3.28 × 10−3 −3.54 × 10−3 −3.27 × 10−3 −4.72 × 10−3 −3.19 × 10−3 −4.54 × 10−3

−3

−3.72 × 10 −5.36 × 10−3 −4.64 × 10−3 −4.62 × 10−3 −1.89 × 10−2 −4.13 × 10−3 −4.57 × 10−3 −4.99 × 10−3 −4.62 × 10−3 −6.60 × 10−3 −4.47 × 10−3 −6.38 × 10−3

VaR

CVaR

ΔCVaR

−3.09 × 10−3 −3.61 × 10−3 −4.46 × 10−3 −4.27 × 10−3 −3.36 × 10−3 −3.08 × 10−3 −3.69 × 10−3 −3.48 × 10−3 −3.42 × 10−3 −4.37 × 10−3 −3.66 × 10−3 −3.91 × 10−3

−4.38 × 10−3 −5.08 × 10−3 −6.25 × 10−3 −6.04 × 10−3 −4.72 × 10−3 −4.25 × 10−3 −4.85 × 10−3 −4.92 × 10−3 −4.84 × 10−3 −6.18 × 10−3 −5.15 × 10−3 −5.36 × 10−3

−2.23 × 10−3 −2.78 × 10−3 −3.56 × 10−3 −3.09 × 10−3 −2.64 × 10−3 −2.60 × 10−3 −3.38 × 10−3 −2.44 × 10−3 −2.35 × 10−3 −3.08 × 10−3 −2.32 × 10−3 −2.01 × 10−3

10/11/2017–08/12/2017 (Post-Break 2)

−3

2.13 × 10 2.87 − × 10−3 −2.49 × 10−3 −2.19 × 10−3 −9.59 × 10−3 −2.78 × 10−3 −2.69 × 10−3 −2.68 × 10−3 −2.37 × 10−3 −2.86 × 10−3 −1.94 × 10−3 −2.85 × 10−3

0,6492 0,8713 0,8634 0,8829 0,7869 0,9500 0,9035 0,7087 0,8206 0,6515 0,7088 0,6917

VaR

ΔCVaR

CVaR −3

−3.25 × 10 −4.50 × 10−3 −5.56 × 10−3 −4.33 × 10−3 −4.03 × 10−3 −4.24 × 10−3 −5.43 × 10−3 −3.71 × 10−3 3.75 × 10−3 −6.36 × 10−3 −4.14 × 10−3 −5.06 × 10−3

−3

−4.58 × 10 −6.14 × 10−3 −7.60 × 10−3 −5.86 × 10−3 −5.66 × 10−3 −5.36 × 10−3 −7.24 × 10−3 −5.25 × 10−3 −5.22 × 10−3 −8.97 × 10−3 −5.86 × 10−3 −7.16 × 10−3

−2.11 × 10−3 −3.92 × 10−3 −4.80 × 10−3 −3.83 × 10−3 −3.17 × 10−3 −4.03 × 10−3 −4.91 × 10−3 −2.63 × 10−3 −3.07 × 10−3 −4.14 × 10−3 −2.94 × 10−3 −3.50 × 10−3

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Fig. 8. VaR of electricity spot price against VaR of sector indexes.

Average risk of the aggregate Nordic market index is about (−3.01 × 10−3), this value is smaller than the value of the least risky sector, which due to the diversification of sectors that compose this aggregate index and that's prove the importance of diversification in decreasing the value of risk. CVaR represents the conditional extreme losses, and it measure the contribution of electricity returns risk to the overall extreme losses (measured by the VaR) for each sector index returns. The results of the average conditional extreme losses (CVaR) indicates that the risk increases for the aggregate index as well as for all sector indexes, since

the conditional extreme losses are higher than the overall extreme losses estimated registered by these sector indexes during the same period (VaR). Indeed, the highest average CVaR is observed for Technology sector and it's about (−6.82 × 10−3) which represent 140% of the estimated univariate VaR, followed by the Basic Materials and Travel and Leisure sectors. Therefore, these sectors seem more exposed to the risk of electricity market variabilities. These results are confirmed by ΔCVaR estimations. ΔCVaR measure the sector index exposure to electricity market turmoil. Results indicate that Technology, Basic Materials and Industrials

Fig. 9. c-DCC between electricity returns and sector Index returns.

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sectors remain more exposed to electricity market risk than other sectors, with a significant overreaction measured by ΔCVaR which are close to (−3 × 10−3). In contrast, Consumer Goods and Telecommunications sectors are less exposed to the electricity market variability, with ΔCVaR evaluated at (−2.4 × 10−3), therefore, these sectors show more resistance to the electricity market variability. The aggregate market index and the rest of sectors indexes demonstrate a moderate spillover effect; which proves the stability of these sectors facing the risk of electricity market deviations. For the sub-periods, we notice that c − DCC reaches its maximum values in the last period (post-Break 2) for all sector indexes, notably for Financial, Industrials and Basic Materials indexes (0.95, 0.90 and 0.88, respectively). Moreover, sector indexes remain more exposed to electricity market variations during the last period, since CVaR and ΔCVaR in this period are higher compared to the previous period. To exemplify, the highest CVaR average is achieved by Technology sector (pass from −6.18 × 10−3 to −8.97 × 10−3), followed by Basic Materials and Industrials sectors (−7.60 × 10−3 and −7.24 × 10−3, respectively). The ΔCVaR average of Basic Materials pass from (−3.56 × 10−3) during the first sub-period, to (−4.64 × 10−3) during the second sub-period and reaches its maximal value (−4.8 × 10−3) in the last sub-period. These findings can be explained by the fact that the third sub-period is characterized by a high volatility of the electricity market (see Fig. 7), which increases the risk in the stock market, this result confirm the existing of tail dependence between electricity market and stock market. To sum up, Technology, Industrials and Basic Materials sectors are the most exposed to electricity market turmoil, since these sectors rely on electricity in their activities. However, the Nordic market index as well as the other sector indexes, notably Telecommunication, Insurance and Real Estate sectors, show more resistance to electricity market risks. 4.3.5. Implications for portfolio designs and hedging strategies Our previous findings suggest that the volatility transmission across electricity market and stock markets is a crucial element for efficient diversified portfolios and risk management. Investors in some stock sectors may need to hedge electricity risk more effectively than they have to if they hold stocks of companies in other sectors. Practically, portfolio managers are required to quantify the optimal weights and hedge ratios in order to hedge effectively against electricity market risk. For minimizing the risk without reducing expected returns, we now consider a portfolio construction of electricity commodity and sector stock indexes. At first, consider the problem of the portfolio optimal weight of electricity commodity and stocks holding subject to a no-shorting constraint, as given by Kroner and Ng (1998): ωelec;t ¼

hs;t −hs=elec;t helec;t þ hs;t −2hs=elec;t

ð20Þ

Secondly, we consider the problem of estimating a dynamic riskminimizing hedge ratio (βs/elec, t) using the bivariate c − DCC − FIAPARCH results. For minimizing the risk of this portfolio (electricity and sector stock markets), we measure how much a long position (buy) of one-euro unit in the electricity market should be hedged by a short position (sell) of (βs/elec, t) euro in the sector stock index, that is: βs=elec;t ¼ ρ 

σs σ elec

ð21Þ

where ωelec, t refers to the weight of electricity commodity in one-euro portfolio of the two assets defined above at time t, hs, t and helec; t are the conditional variances of the sectoral stock returns and the electricity extreme returns, respectively, and hs/elec; t is the conditional covariance between electricity and the sectoral stock returns at time t. The optimal weight of the sectoral stock returns in the considered portfolio is obtained by computing (1 − ωelec, t).

ð22Þ

where σs and σelec are the standard deviations of the sectoral stock returns and the electricity extreme returns, respectively, and ρ represent the correlation between the electricity market return and the sectoral stock return. Finally, we examine the effectiveness of hedging (HE) of the portfolio diversification, this measure is judged through comparing return characteristics and the realized risk of the considered portfolios. The effectiveness of hedging (HE) across constructed portfolios (proposed by Ederington (1979)) can be evaluated by: HE ¼

Varunhedged −Var hedged Var unhedged

ð23Þ

where variances of the hedge portfolio (Varhedged) are obtained from the variance of return on electricity-stock portfolios, whereas the variance of unhedged portfolio (Varunhedged) is the variance of return on the portfolio of 100% stocks. Estimation results (Table 9) show that the optimal portfolio weight of electricity commodity varies substantially across sectors. For the overall period, we observed that the highest average value of optimal weights is for Basic Materials- electricity portfolio, indicating that the optimal weight of electricity commodity is 18.32% and the remaining proportion of 81.68% is invested in the Basic Materials stocks. This is Table 9 Average values of the optimal weight, hedge ratio and hedge effectiveness of the electricity market commodity in stock portfolios. Overall period

Nordic market index Banks Basic materials Construction & materials Consumer goods Financials Industrials Insurance Real estate Technology Telecommunications Travel & leisure

Under the condition that 8 if ωelec;t b 0 < 0; ωs=elec;t ωelec;t ; if 0 ≤ ωelec;t ≤ 1 : 1; if ωelec;t N1

651

Nordic market index Banks Basic materials Construction & materials Consumer goods Financials Industrials Insurance Real Estate Technology Telecommunications Travel & leisure

03/07/2017–18/08/2017 (Pre-Break 1)

We

Bs/e

HE

We

Bs/e

HE

0,1051 0,1544 0,1832 0,1502 0,1195 0,1363 0,1385 0,1180 0,0974 0,1626 0,1023 0,1213

0,1012 0,1500 0,1729 0,1455 0,1166 0,1319 0,1375 0,1150 0,1100 0,1734 0,1070 0,1295

0,3201 0,4099 0,3376 0,4303 0,3762 0,2046 0,1505 0,101 0,1674 0,9204 0,2777 0,4026

0,0828 0,1349 0,1823 0,1280 0,1105 0,1224 0,1974 0,0997 0,0928 0,1544 0,0905 0,0659

0,0844 0,1367 0,1748 0,1368 0,1077 0,1156 0,1712 0,1068 0,0985 0,16,858 0,1027 0,0992

0,0396 0,8791 0,4706 0,1824 0,0872 0,4635 0,5784 0,0073 0,1505 0,6713 0,1890 0,0520

18/08/2017–10/11/2017 (between Break 1 and Break 2)

10/11/2017–08/12/2017 (Post-Break 2)

We

Bs/e

HE

We

Bs/e

HE

0,1090 0,1446 0.1845 0,1625

0,1047 0,1485 0,1600 0,1199

0,1863 8.93 × 10−2 0.1450 0.1590

0,0648 0,1852 0,1865 0,1521

0,0691 0,1702 0,1687 0,1381

0,0020 0,6316 0,1293 0,2114

0,1366 0,1780 0,1851 0,1690 0,1589 0,1419 0,1041 0,1393

0,1232 0,1535 0,1357 0,1552 0,1513 0,2257 0,1100 0,1761

0,5345 0,5797 0,1445 0,0985 0,7346 0,1123 −2.69 × 10−2 0,6032

0,1006 0,2166 0,1621 0,1937 0,1318 0,0866 0,1122 0,1054

0,1000 0,1530 0,1448 0,1065 0,1255 0,1217 0,1192 0,1164

0,0795 0,2940 0,2731 0,2360 0,2606 0,01429 0,0313 0,1120

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followed by Technology- electricity portfolio and Construction Materials- electricity portfolio, with optimal weights of electricity commodity close to 15% and 85% attributed to these stock sectors. Hence, for the Nordic stock market optimal portfolio is characterized by high weights of sector stocks instead of electricity commodity. For the sub-period's windows, we observed that the optimal weight of electricity commodity for Industrials - electricity portfolio decreases over periods (pass from 19.75% during the first period, to 18.50% in the second period and reach 16.20% in the last period). The same change is also observed for Technology- electricity portfolio and Construction Materials- electricity portfolio. However, the optimal weight of electricity for Financials- electricity portfolio, Bank-electricity portfolio and Insurance-electricity portfolio increases over periods. To exemplify, for Financials- electricity portfolio, the optimal weight of electricity rises from 12.24% during the first period to 21.66% in the third period, which characterized by high volatility compared to the previous periods. Hence, the weight of electricity in the portfolio increases in favour of sector stocks. In sum, the optimal weight of electricity commodity decreases for sectors that are more exposed to electricity turmoil. In contrast, this weight increases for sectors that are more stable to electricity market variability. Average values of the hedge ratios (Table 9 and Fig. 10) are low in general for all portfolios indicating that hedging effectiveness involving electricity and stock markets is quite good. We observe that, for the whole period, the lowest value of average hedge ratio is for Telecommunication - electricity Portfolio (0.10) followed by Insurance - electricity Portfolio and Real Estate- electricity Portfolio (0.11). This indicates that the investor can minimizes the risk, where a long position of oneeuro in the electricity market can be hedged by a short position of 0.10 euro in Telecommunication stocks. For instance, the hedging strategy for Telecommunication is buying electricity commodity and selling Telecommunication stocks. This inversed hedging position can be used also for investors in Insurance and Real Estate sectors. However, the highest value of average hedge ratio is observed for Technology electricity Portfolio (0.173) followed by Basic Materials - electricity Portfolio (0.172) and Construction Materials- electricity Portfolio (0.145). This indicates that the investors can minimizes the risk, where a long position of one-euro in the electricity market can be hedged by a short position of 0.17 euro in Technology stocks.

The values of the hedge effectiveness index HE are positive for all sector stocks and global index, due to the decreasing values of the hedged portfolio variance compared to unhedged portfolio variance. We observe that the reduction of risk is close to 92% for portfolios including Technology stocks and electricity commodity. These reductions can also apply to portfolios of Construction and Materials and Banks with hedge effectiveness value of 40%. Moreover, a close look at the hedge effectiveness index indicates also significant gains from adopting a dynamic hedging strategy, since the introduction of electricity commodity decreases the risk for all stock sectors and for the global market index, due to the decreasing of the hedged portfolio variance compared to the unhedged portfolio variance (100% stocks). These findings are important for the Nordic market investors seeking to reduce the effects on portfolio risk stemming from the volatility related to the electricity market returns. 5. Policy recommendations Our findings have several interesting policy implications for the heterogeneity of various types of investors, market participants and policymakers: Results on the unconditional correlation coefficients (Table 1) indicate that the stock market sectoral returns can viewed as having low to moderate positive correlation with electricity market returns. The only exception is the negative unconditional correlation observed for the Bank sector. However, a weaker positive correlation was noticed for Basic Materials, Telecommunications and consumer Goods. Based on these findings, it is thus apparent that investors in Nordic Market can achieve a diversified portfolio with Bank, Basic Materials, Telecommunications and consumer Goods. Continuing the analysis, our c − DCC results indicate that the lowest c − DCC average is observed for the Telecommunications and Travel and Leisure sectors. This relationship shows that electricity market assets can be considered as a hedger for these stock sectors. Hence, Telecommunications and Travel & Leisure stock sectors can boost safety for Nordic market investors during the period of electricity market turmoil. Concerning the optimal portfolio weights, our results indicate that in Nordic Market the investor should hold more sector stocks than electricity commodity to minimize the overall portfolio risk. This may be

Fig. 10. Optimal hedge ratios.

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explained by the fact that electricity market is more volatile than the stock market. For the optimal hedge ratios, we observe that the Financials -Electricity portfolio, Insurance-Electricity portfolio and Banks -Electricity portfolio have the lowest values of average hedge ratio. However, Technology- Electricity portfolio, Industrials -Electricity portfolio and Basic Materials -Electricity portfolio have the highest value of average hedge ratio. According to these findings, we suggest that risk lover traders can invest in the portfolio, which depicts minimum hedge ratio value. Conversely, risk aversion traders can invest in the portfolio, which depicts the maximum hedge ratio value. It is worth noting that optimal weights and hedge ratios change over the period according to the volatility structure of the electricity returns. In such cases, investors and market participants such as traders are highly recommended to be cautious and better understand the risk factor originating from the volatility of electricity returns. For that purpose, it is crucial for them to re-weight their electricity- stock portfolios according to the electricity market turmoil. The obtained results should be of interest to policymakers as unfavourable electricity market fluctuations may have severe impacts on the stock market performance through reducing corporate cash flows. Furthermore, investors and fund managers can use our results to build appropriate diversification and hedging strategies. If, for example, the electricity market exhibits high volatility (3rd sub-period), profitable investment strategies can be constructed based on sectors, which are less dependent on electricity such as Financials, Insurance and Bank sectors. Hence, electricity commodity can be used as a hedger and diversifier asset to these sectors. In summary, our findings provide an important guideline on building optimal portfolios using electricity commodity and sector indexes, which requires some information on the optimal diversifiable portfolio to minimize the electricity market risk without any impairment of expected returns of the portfolio. 6. Conclusion With the deregulation movement reforming the electricity industry worldwide, electricity prices are substantially more volatile than any other commodity prices. On the other hand, electricity is an integral part of production and economic growth process. Therefore, electricity market may affect the stock market through influencing the real output, and as consequence, the sum of cash flows. Accordingly, investors are facing electricity market risks, and thus need to find ways to protect their returns. In this study, we investigate the problem of establishing a framework for risk management and hedging strategies in the Nordic market. For that purpose, we considered the electricity return pairwise in aggregate market index and some sectoral stock returns in the Nordic market, for the period ranging from the 3rd of July to the 8th of December 2017. Specifically, we examined, firstly, the adequate model for each time series; and our results indicated that the k − factor GARMA − FIGARCH model seemed appropriate for the Log-return electricity price series. However, the ARFIMA(p, d, q) − FIGARCH(1, δ, 1) model seemed appropriate for the aggregate market index and the sectoral stock returns. Then, the univariate VaR estimation results indicated that Technology, Basic Materials sectors are the riskiest sectors in the Nordic market. Thereafter, the c − DCC − FIGARCH approach was adopted to assess the spillover effects of the electricity market variations among aggregate and sectoral stock returns. Then, to analyse the tail dependence between the electricity market and each sectoral stock markets, we applied the CVaR approach. Our empirical findings point to the existence of significant volatility spillover across electricity market and sectoral stock markets, but the intensity of volatility interaction differs from one sector to another and from one period to another. Particularly, the risk increases in cases of the Technology and Basic Materials sectors

653

since the conditional extreme losses are higher compared to the overall extreme losses estimated on these sector indexes during the same period (VaR). In addition, this study focused onΔCVaR, and the results vary over sectors and over time depending on the volatility of the electricity market returns for each period. In particular, the Technology, the Basic Materials and the Industrials sectors remain more exposed to the electricity market risk than other sectors. Finally, we extended our methodology to portfolio diversification strategies, which is the most important application to market participants and market makers. We simulated optimal intra-day portfolios, given that electricity markets are hourly markets, in terms of the optimal weight, the optimal hedge ratio and hedging effectiveness across different periods. Our findings showed that optimal weights and hedging ratios vary over periods and across sectors. More precisely, based on the optimal weights we remarked that investors should hold more stocks than electricity commodity. This may be referred to the fact that electricity markets are more volatile than the other stock markets. Likewise, the optimal hedge ratios between electricity and sector stock markets allows investors to effectively hedge the electricity risk by using the short position of sector stock indexes. Moreover, the hedge effectiveness index indicated significant gains, since the introducing of electricity commodity decreases the risk for all sector indexes as well as the aggregate market index, due to the decreasing of hedged portfolio variance compared to unhedged portfolio variance (100% stocks). Two important results that emerge from this research are as follows. First, our study shows evidence of long run dependence between electricity returns and sectoral stock returns, and the results indicate that the tail dependence is significant and varies across sectors and over periods. Secondly, the optimal weights and hedge ratios for electricity/ stock portfolio holdings are sensitive to the sectors considered. Therefore, electricity commodity can diversify portfolio of sector stocks and serves to hedge electricity risk more effectively. These findings are important for Nordic market investors seeking to reduce the risk related to the electricity market. Hence, our research offers useful and significant information for risk management and for optimal portfolio allocation decisions, which are the important objectives of financial market participants. Acknowledgments We wish to thank the Editor and anonymous referee for their helpful comments and suggestion which have led to substantial improvement in the presentation of this paper. Appendix A. Appendix The standardised t-distribution with a given degrees of freedom υ (N2), designated εt ∼ St(0, 1,υ) is given by the following equation:   υþ1 Þ

−ðυþ1 Γ 2 ε2 2 g ðεt =υÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi υ 1 þ t υ−2 πðυ−2ÞΓ 2

ð24Þ

where Γ(υ) denote the Euler's gamma function. Subsequently, Hansen (1994) was the first to develop a skew-student distribution for financial time series modelling. Later, Lambert and Laurent (2000, 2001) extended this distribution by proposing a standardised version and indicated that it may be assumed for the ARCH model innovations. Refering to Lambert and Laurent (2000, 2001), the random variable εt is assumed to be SKST(0, 1, ζ, υ), i.e., distributed by means of standardised skew-student with parameters υ N 2 (that represent the degrees of freedom) and ζ N 0 (a parameter of the skewness), if its

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probability density function is defined as follows: 8 2 m > > sg½ζ ðsεt þ mÞ=υ if εt b− > > 1 s > > <ζ þ ζ

f ðεt =ζ ; υÞ ¼ 2 ðsε t þ mÞ m > > > =υ if εt ≥− sg > > 1 ζ s > :ζ þ ζ

ð25Þ

where g(z/υ) is a symmetric student density with zero mean and unit variance for a given degrees of freedom denote υ. In Eq. (25), the conqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ç stants m = m(ζ,υ) and s ¼ s ðζ ; υÞ denote the mean and the standard deviation, respectively, associated to the non-standardised skewstudent density SKST(m, s², ζ, υ) of Fernández and Steel (1998), are given by the follows function:   υ−1 pffiffiffiffiffiffiffiffiffiffiffi  υ−2  Γ 1 2 ζ − mðζ ; υÞ ¼   pffiffiffi υ ζ πΓ 2

ð26Þ

and Ç

s ðζ ; υÞ ¼

Ç

ζþ

1 Ç

ζ

! Ç

−1 −m :

ð27Þ

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