Modeling electricity spot and futures price dependence: A multifrequency approach

Physica A 388 (2009) 4763–4779

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Physica A journal homepage: www.elsevier.com/locate/physa

Modeling electricity spot and futures price dependence: A multifrequency approach Pekka Malo Department of Business Technology, Mathematics and statistics, Helsinki School of Economics, 00101, Helsinki, Uusimaa, Finland

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Article history: Received 7 May 2009 Received in revised form 1 July 2009 Available online 11 August 2009 Keywords: Electricity Markov-Switching Copula Risk Multifrequency Econophysics

abstract Electricity prices are known to exhibit multifractal properties. We accommodate this finding by investigating multifractal models for electricity prices. In this paper we propose a flexible Copula-MSM (Markov Switching Multifractal) approach for modeling spot and weekly futures price dynamics. By using a conditional copula function, the framework allows us to separately model the dependence structure, while enabling use of multifractal stochastic volatility models to characterize fluctuations in marginal returns. An empirical experiment is carried out using data from Nord Pool. A study of volatility forecasting performance for electricity spot prices reveals that multifractal techniques are a competitive alternative to GARCH models. We also demonstrate how the Copula-MSM model can be employed for finding optimal portfolios, which minimizes the Conditional Value-at-Risk. © 2009 Elsevier B.V. All rights reserved.

1. Introduction Electricity markets have attracted increasing attention within the last few years. The markets are not only of considerable economic importance but also highly volatile and difficult to model due to non-storability, strong seasonal fluctuations and price spikes [1–4]. In addition to these stylized facts, recent research has documented multifractal properties in electricity spot prices. Erzgräber et al. [5] demonstrate using detrended fluctuation analysis that the observed variation in Hurst exponents can be regarded as a signal of multifractality in electricity prices. Multifractal modeling techniques have been successfully applied in various areas of physics, computer networking and biology [6,7]. Recently, they have been recognized in financial econometrics as appropriate tools for modeling volatility and risk [8–10]. The models based on continuous cascades have been successfully applied to characterize random fluctuations of asset prices in stock and currency markets; see, for example, Bacry et al. [11] for an excellent review on multifractals, multiscaling and self-similarity. As suggested by the recent studies, multifractals present an interesting alternative for classical econometric tools due to their ability to replicate the stylized statistical properties of financial markets in a very parsimonious way. However, although the popularity of multifractal modeling techniques has recently increased due to their ability to replicate stylized statistical properties of financial markets in a very parsimonious way, the majority of models applied in empirical finance is still grounded on more classical techniques. An essential limitation of the currently available multifractal models is that they are mainly designed for univariate modeling. As discussed by Bacry et al. [11], only a few attempts have been proposed to extend multifractal models for multivariate time series. An early paper by Muzy et al. [12] was a pioneering proposal to extend their Multifractal Random Walk (MRW) model to the multidimensional case. More recent papers by Liu and Lux [13] and Calvet et al. [14] have considered the possibilities of extending and estimating the Markov-Switching Multifractal (MSM) model of Calvet and Fisher [15] for bivariate time series. It is by now well acknowledged that before the multifractal modeling techniques can gain broad acceptance within financial industry, there must be frameworks which enable their use in portfolio management applications. E-mail address: [email protected]. 0378-4371/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2009.07.048

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The need for improved portfolio management tools is best manifested in developing markets such as electricity exchanges, where the assets are characterized by extreme properties rarely encountered in other commodities and financial instruments. Especially, given the violent behavior of spot prices, the need to hedge spot risks is of critical importance. Currently, the main-stream of the electricity risk management literature on dynamic hedging is dominated by variations of jump diffusions and GARCH-based univariate and multivariate models; see for example Refs. [16,17,4,18,19]. However, the recent recognition of multifractal properties in electricity prices motivates one in examining the performance of alternative modeling approaches. One interesting and so far unexplored path would be to consider designing hybrid models, where some properties of classical and multifractal models are joined together. Clearly, both multifractal and classical models have their own strengths and weaknesses. Whereas, the multifractals have shown strong evidence on their excellent volatility forecasting performance for univariate processes, the classical models have a long history as successful tools for multivariate modeling. In particular, when modeling dependence in conditional expectations of asset prices, the classical frameworks are more established. It is also evident that in portfolio applications we often need to model assets with widely different characteristics. For example, in the case of electricity markets the futures prices exhibit considerably milder behavior than spot prices, yet they still share certain dependencies. The situation would be much simpler, if we could assume that all marginal distributions of the portfolio assets are similar. However, in practice it often appears that we need further flexibility to allow for different marginal distributions. Therefore, in order to benefit from multifractal models in portfolio applications, we consider it worthwhile to study the possibility of designing a combined model, where we employ a copula function to join together different marginal distributions with possibly multifractal properties. Recently, the copula-based models have been broadly accepted in various financial applications due to their flexibility in modeling dependence structure between heterogenous assets; see Patton [20], Chen and Fan [21], Granger et al. [22]. The purpose of this paper is to contribute to the existent literature in two ways. In the first part, we compare the ability of multifractal models to forecast spot return volatility against classical GARCH models. Motivation for this analysis stems from the special features of the electricity market, which set it apart from all other commodities and financial instruments. In the second part of our paper we consider the problem of modeling joint dynamics between spot and weekly futures prices. We propose a Copula-MSM model, where a conditional copula function is used to join together marginal distributions with possibly multifractal stochastic volatility processes. The model is constructed in three parts: conditional means, marginal distributions and a copula-function which defines the dependence structure. Following comparison of various alternatives we ended up using a combination, where dynamics of conditional expectations of spot and futures prices are modeled through a linear cointegration framework but the innovations, joined by a copula-function, are assumed to follow different model families. Whereas our analysis suggested that the spot innovations appeared well defined by MSM model, the futures innovations are better characterized by a GARCH(1,1) process with t-distribution. The two distinctly different marginals are coupled by static Student’s t-copula, which provided the best fit as compared to a set of static and dynamic alternatives. All applications are carried out using data from the Nordic electricity markets, Nord pool. Naturally, however, when proposing the hybrid model we have to give up some elegant properties of multifractal models at the prospect of gaining the added flexibility. As a result, the minimum time resolution is restricted to the frequency of the original time series, which is one day in our application. However, we do benefit from the ability of the multifractal MSM model to produce accurate volatility forecasts, which is perhaps the most valuable property of the MSM framework. Indeed, from the perspective of a risk manager, the model performance is best measured in its applicability for dynamic portfolio optimization. In order to illustrate the benefits of the Copula-MSM model, we consider an application of finding Conditional Value-at-Risk (CVaR) optimal portfolios for different holding periods and confidence levels. In this study we examine a simple electricity portfolio, which consists only of spot contracts and a futures strategy with three weeks left to maturity. To prevent the inclusion of thin market and expiration effect, we construct the futures strategy by rolling over the contract one week prior to its expiration. Our empirical results suggest that significant risk reduction relative to an unhedged portfolio can be achieved by using dynamic portfolio optimization. As such the result lends support for the efficiency of electricity futures markets. We also recognize that the functioning of our model in terms of CVaR forecasting accuracy is reasonably stable when holding period lengths of less than two weeks are considered. The disposition of the paper is following. Section 2 outlines the dominant properties of the Nordic electricity markets. Section 3 shows the construction of Copula-MSM model. Section 4 presents the dynamic CVaR optimization framework in the spirit of Rockafellar and Uryasev [23,24]. Section 5 defines the data set and discusses the numerical results of our experiments. We conclude in Section 6. 2. Dominant properties in electricity markets As discussed by Byström [19], among others, one of the main reasons for electricity producers and consumers to trade in electricity markets is to control their portfolio risks using spot and futures contracts. Therefore, when considering dynamic portfolio optimization, we find it necessary to understand not only the dominant properties of individual spot or futures price series but also their possibly time-varying interdependence structure. Below we will discuss the dominant properties in spot prices and the factors determining their relationship to futures prices. A recent paper by Granger et al. [22] introduces the idea of a dominant property (DP) in modeling conditional distributions of time series. In general terms, we understand the DP as a component of a process, which determines the relationship of the variable with others and how it fits into models and equations. The DPs become a relevant consideration especially when

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modeling joint dynamics of series. For example trends, persistence, strong seasonal components, business cycle components and transitions in conditional mean can be viewed as dominant properties in component processes. As discussed by Granger et al. [22] a particularly interesting case involving DPs and common factors is in the form of co-trending and co-integration. When inspecting dominant properties, the electricity market makes a particularly interesting case. It is well acknowledged that the electricity price process has unique characteristics, which sets it apart from other energy commodities such as oil and gas [25]. Therefore, it is commonly found that models for financial assets such as stocks, bonds and currencies do not work so well in electricity markets [3,18,26–29,17,30]. Recognizing the fact that electricity prices show remarkably different behavior depending on the time-scale, we have restricted the scope of our study to only short-term contracts with less than 4 weeks to maturity. It is also evident that the shorter contracts are more liquid and more strongly connected to the spot price. Provided our interest in finding dynamic risk minimizing portfolios, we are concerned about the dependence between spots and futures. As often argued, it is possible that the link between spot and futures holds only on a time-scale of weeks due to the fact that electricity is difficult and costly to store. Therefore, we prefer to work with short futures contracts, which can be assumed to better reflect the events that affect today’s spot prices. 2.1. Dominant properties in spot prices In this section we review shortly the best known features of electricity prices [18,31,16,32,33]: Price spikes: Currently, there are many issues which are not clearly understood about energy markets. One of the most salient properties is the extreme volatility of electricity. Typically volatility is attributed to the expensive storage of electricity, which combined with balancing constraints, inelastic demand and supply creates conditions for violent price spikes [34]. The fact that supply or demand shocks cannot be immediately covered from inventories creates jumps or price spikes [27,28]. Typically the spikes are short-lived, but their significance is emphasized by the fact that they account for a considerable fraction of price variation. Seasonality: Another important feature of electricity spot prices is the seasonality [35,3]. Nord Pool is essentially a hydroelectric market, where both demand and capacity are driven by weather. The availability of water reservoirs sets supply restrictions, which can cause enormous price changes during high demand seasons such as a cold winter. Also, the reverse can be true when the reservoirs are flowing over. For studies on seasonal patterns and the behavior of electricity prices see e.g. Lucia and Schwartz [18] and Bhanot [31]. Mean-reversion and anti-correlation: Several studies have confirmed that electricity prices exhibit mean-reversion. Indeed, electricity is considered as one of the best-known examples of antipersistent data. Simonsen [36], among others, has studied anticorrelations in the Nordic electricity markets using wavelet techniques. Various explanations for mean-reversion have been proposed. One argument is by Knittel and Robberts [32], who interpret mean-reversion as a consequence of weather cycles and the tendency of weather to revert to its mean level. 2.2. Common factors in spot and short-term futures prices Until now, we have discussed DPs only as related to spot prices. When modeling joint dynamics of spots and a futures strategy, it becomes interesting to identify the existence of dominant common factors. As pointed out by Granger et al. [22] common factors may be either directly observed or derived from other series in the system. As an important example they consider a co-integrating relationship in modeling conditional distribution of a bivariate time series. Indeed, when considering electricity spot prices and short-term futures contracts, it is possible to establish a cointegrating relationship [4]. In the risk management literature, a combination of vector autoregressive models with error correction mechanism based approaches have been successfully used to account for serial correlations and cointegration between spot and futures prices; see, for example, Ghosh [37,38], Lien and Luo [39], Lien [40], among others. Having limited the consideration of our study to portfolios consisting of spot contracts and weekly futures strategy our goal is to establish a bivariate relationship between the spot price X and futures price Y . Using the theorem of Sklar [41], we can decompose the problem of finding a joint distribution into three parts: two marginal densities and a copula function. Although, the original version of the theorem was presented in a static context, several recent extensions of theorem for conditional information have been proposed; see Refs. [22,20]. In the bivariate case, if we let X ∼ FX , Y ∼ FY and (X , Y ) ∼ F , we can write the relationship between X and Y , conditional on W as f (x, y|W ) = fX (x|W )fY (y|W )c (FX (x|W ), FY (y|W )|W ) F (x, y|W ) = C (FX (x|W ), FY (y|W )|W ) where c is the conditional copula function, C is the copula distribution, FX is the conditional marginal distribution function of X and FY is the conditional marginal distribution function of Y . It can be shown that this decomposition is available for every joint distribution F . The copula links marginal distributions together to form a joint distribution, and defines the dependence between the spot and futures price. We can now formalize the notion of dominant common property, or a common factor W , in the sense of Granger et al. [22]: Definition 1. Let Xt and Yt be the spot and futures prices processes, respectively, and denote X¯ t = Xt −j , j ≥ 0 and similarly Y¯t is the present and the past of the Y ’s. A process Wt will be considered as one with a dominant common property, or a

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¯ t −1 , X¯ t −1 , Y¯t −1 ) and FY (Yt |Wt , W ¯ t −1 , X¯ t −1 , Y¯t −1 ) do common factor in distribution, if the conditional marginals FX (Xt |Wt , W ¯ t −1 , but the conditional copula function c (ut , vt |Wt , W ¯ t −1 , X¯ t −1 , Y¯t −1 ) = depend on Wt and possibly the lagged terms W c (ut , vt |X¯ t −1 , Y¯t −1 ) does not depend on Wt either directly or through the lags. It is worthwhile to note that in this definition of the common factor we assume that the effect of Wt on (xt , yt ) is through the marginal distributions but not through their relationship. This means that we can allow both spot and futures prices to have seasonal patterns, but the copula is not allowed to vary seasonally. As shown by Granger et al. [22] the above formulation is well designed to allow for cointegrating relationships present in the electricity market. In cointegrating model we assume that spot and futures prices have a dominant I(1) property due to a common I(1) factor W . In this case the common factor is not directly observable as such, but we can take it into account using a standard linear cointegration framework. 3. Models In this section we propose a new approach for modeling conditional dependence between electricity spot and futures prices by using a Copula-MSM model. The framework is based on a flexible multivariate copula, which allows for marginal distributions modeled through a Markov-Switching Multifractal (MSM) framework developed by Calvet and Fisher [15,21]. An important advantage in this approach is that, due to the separation of marginal models from the dependence structure, we can specify the spot and futures price processes using different model families. It is well acknowledged that the behavior of spot prices is considerably more violent than what is observed in electricity futures prices. Although, we may find the MSM model suitable for capturing the properties of the spot return distribution, we want to allow for the possibility of adopting a different technique for characterizing the futures returns. This is also beneficial for further extensions of the model, because it enables inclusion of other assets which do not share similar characteristics with the electricity. In order to simplify the model construction, we consider deseasonalized logarithmic spot and futures price processes, denoted by Xt and Yt , respectively. In what follows, we have divided the model into three parts and discuss the steps separately: (1) in the first step, we construct a model for conditional expectations of deseasonalized prices (µXt , µYt ); (2) as a second step we propose models for conditional volatilities (hXt , hYt ) and marginal distributions of spot and futures prices (FX , FY ); and finally (3) we consider the specification of the conditional copula distribution model C . The overall model is conveniently summarized as

1Xt − µXt 1Yt − µYt p , p hXt

hYt

! ∼ FXY = C (FX , FY ).

Throughout this section, we restrict the time resolution to one day, i.e. 1Xt = Xt − Xt −1 and 1Yt = Yt − Yt −1 , which is the observation frequency of the underlying time series. Although, by adding this restriction we clearly sacrifice many elegant properties of multifractal frameworks, we can still benefit from the forecasting power of multifractal stochastic volatility models as applied to the given time resolution. 3.1. Conditional expectations Following our discussion in the previous section, we characterize the dynamics in conditional means by taking advantage of the cointegrating relationship between spot and futures price. For estimation purposes, we write the model as a simple error-correction framework:

1Vt = (µXt , µYt ) + εt

(1)

p−1

= αβ T Vt −1 +

X

Φi 1Vt −i + εt

(2)

i=1

where Vt = (Xt , Yt ), εt = (εXt , εYt ), and β is the cointegrating vector. Consequently, we have

µXt = α1 (β1 Xt −1 − β2 Yt −1 ) +

p X (φi(1,1) 1Xt −i + φi(1,2) 1Yt −i ) i =1

p X µYt = α2 (β1 Xt −1 − β2 Yt −1 ) + (φi(2,1) 1Xt −i + φi(2,2) 1Yt −i ). i=1

In order to specify p, the number of historical return terms, we use likelihood ratio tests with Sims-correction and check that most of the remaining autocorrelation is captured. 3.2. Stochastic volatility and marginal distributions The most challenging part in our model is to find an appropriate specification for the marginal distributions of spot and futures prices. Whereas the spot prices are found to exhibit enormous short-term fluctuations, the futures price

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process is considerably more tempered. Therefore, we consider the possibility of modeling the two processes using different approaches. After experimenting with various alternatives from multifractal and GARCH families, we have chosen to consider a hybrid model, where spot innovations εXt are characterized using a Markov-Switching Multifractal Model (MSM) developed by Calvet and Fisher [15,21,14] and futures innovations εYt are modelled through GARCH(1,1) with Student’s t-distribution. 3.2.1. MSM model In construction of the MSM model we begin with the assumption that spot innovations are given by

εXt = ut · (hXt )1/2 where the innovations ut are drawn from a standard Normal distribution N (0, 1) and the instantaneous volatility hXt is defined through a first-order Markov state process. Following the formulations by Calvet and Fisher [15,21] and Lux [42,43], we let Mt denote a state vector with k positive elements, (1)

(2)

(k)

Mt = (Mt , Mt , . . . , Mt ) ∈ Rk+ , the instantaneous volatility is specified as a product of k volatility components and a constant scale factor σ > 0 hXt = σ 2

k Y

(i)

Mt .

(3)

i =1

The MSM framework permits any distribution M with positive support and unit mean. The intuition behind this formulation is to assume that fluctuations are driven by news with very heterogeneous degrees of persistence, ranging from daily to several decades. The low-frequency components deliver the feedback required to capture the most extreme daily returns switches, while the high-frequency components represent the typical day-by-day fluctuations. In the above framework this can be achieved by allowing each component M (i) to have dynamics at a separate frequency. Following Lux [42], among others, we assume that the multipliers are determined by random draws from a Lognormal distribution with location and shape parameters λ and s (i)

∼ LN (−λ, s2 ) √ (i) where s = 2λ due to requirement E [Mt ] = 1. Mt

(4)

Each component is renewed at time t with probability γi depending on its rank within the hierarchy of multipliers and remains unchanged with probability 1 − γi . The transition probabilities are defined as:

γi = 1 − (1 − γk )(b

i−k )

where γk ∈ [0, 1] and b ∈ (1, ∞). A typical choice amounts to setting γk = 0.5 and b = 2 [42]. Experiments with different parameterizations suggest that the performance of MSM models is robust with respect to these parameters. In order to implement the model we have to estimate only two parameters: the scale factor σ > 0 and the location parameter λ. For this purpose, we apply the GMM estimator proposed by Lux [42]. When considering specifications with large number of volatility multipliers k ≥ 10, GMM estimator appears as a flexible and versatile approach. Furthermore, the maximum likelihood approach considered by Calvet and Fisher [15,21], is not applicable to the case of lognormal multipliers. 3.2.2. Marginal distributions In the resulting model the standardized spot innovations are conditionally normal and the standardized futures innovations follow a Student’s t distribution with v degrees of freedom:

εX p t |Ft −1 ∼ N (0, 1) hXt

εY t

r hYt

v |Ft −1 ∼ t (v) (v − 2)

(5)

(6)

where hXt is obtained from the MSM model described above and hYt is characterized through a standard GARCH(1,1) model with Student’s t-distribution. Before arriving at this current model specification for marginals, we had experimented with several alternative GARCH models for (hXt , hYt ), including Hansen’s Skewed t-distribution, GED-distribution and normal distribution. We also considered an approach of modeling both processes using MSM-models. However, in the case of futures innovations, the parameters of the MSM model did not appear to be statistically significant at the 5% confidence level. Experiments were conducted using MSM models with various numbers of volatility components. Instead, the standard GARCH(1,1) model for futures appeared to do a reasonably good job. The results were quite the reverse for the spot innovations, where MSM models were successfully identified with better forecasting performance than offered through classical approaches.

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3.3. Conditional copula While searching for the most appropriate specification of the conditional copula for spot and futures innovations, we have followed the advice of Patton [20] and Granger et al. [22] and examined for a few different alternatives. Given the vast literature on copula models this is not an easy task. However, while having no economic theory to guide us in the choice of the dependence structure, it is generally argued that due to the assumptions associated with the different copulas, it is important to examine the performance of multiple structures covering both constant and dynamic copula specifications. In our study twelve alternative copula functional forms were considered: normal, Student’s t, Clayton, rotated Clayton, Gumbel, rotated Gumbel, Plackett, Frank, symmetrized Joe–Clayton, time-varying normal, time-varying rotated Gumbel, and time-varying symmetrized Joe–Clayton copula. The comparison of their fits was done in terms of log-likelihood values. As a result, we found that the best fit is attained by using constant Student’s t-copula. The functional form of the t-copula density is defined as

Γ ((ν + 2)/2)/Γ (ν/2) p cT (u, v; ρ, ν) = νπ t (˜x; ν)t (˜y; ν) 1 − ρ 2

 1+

x˜ 2 + y˜ 2 − 2ρ x˜ y˜

−(ν+2)/2 !

ν(1 − ρ 2 )

(7)

for ρ ∈ (−1, 1) and ν > 0, where t (·; ν) is the pdf of a Student’s t random variable with ν degrees of freedom, x˜ = T −1 (u; ν), y˜ = T −1 (v; ν) where T −1 (·; ν) is the inverse cdf of a Student’s t random variable with v degrees of freedom. For further discussion on the properties, consider Patton [20]. Using the Student’s t-copula we can outline the Copula-MSM model for spot and futures prices, where the conditional means are modeled through a linear cointegration framework, the conditional volatilities through MSM and GARCH models

1Xt − µXt 1Yt − µYt p , p hXt

hXt = σ 2

! ∼ FXY

hYt

k Y

= CT (Normal,Student’s t; ρ, ν)

(8)

(i)

Mt

i=1

hYt

= a0 + a1 εY2t + a2 hYt−1

where CT denotes the copula distribution function associated with Student’s t-copula. 4. Minimizing risk for electricity portfolios As an application of the Copula-MSM model for electricity risk management, we consider the problem of finding an optimal Conditional Value-at-Risk (CVaR) portfolio. For good treatment of the underlying theory on finding CVaR minimizing portfolios, we refer to Alexander et al. [44] and Rockafellar and Uryasev [23,24], and references therein. 4.1. CVaR minimization problem For a given time horizon t¯ and confidence level β , CVaR is the conditional expectation of the loss above Value-at-Risk (VaR) for the same time horizon and confidence level. In our application we assume that the only assets in the portfolio consist of spot contracts and a futures strategy. The strategy is constructed by using futures contracts with three weeks left to strategy. Once the expiration of the futures contract is only one week away, we roll over the contract replacing it with a new futures contract with three weeks left to maturity. Let f (w, S ) denote the loss of a portfolio with decision variable w ∈ R2 and S ∈ Rd denote the value of underlying risk factors at t¯. Following Alexander et al. [44], we may assume that the random variable S has a probability density p(S ). The instrument values consisting of the spot price and the futures price at time t¯ are Vt¯ = (Xt¯ , Yt¯ ), which are assumed to be functions of the risk factors S. Then, the loss associated with the portfolio x at time t¯ is f (w, St¯ ) = −w T (Vt¯ − V0 ), which is a linear function of w . For a given portfolio, the probability of loss not exceeding a threshold α is given by R Ψ (w, α) = f (w,S )≤α p(S )dS. Consequently, we have a definition for VaR through

αβ = inf{α ∈ R : Ψ (w, α) ≥ β}. We can now define the CVaR minimization problem by considering the augmented performance function Fβ (w, α) = α + (1 − β)

−1

Z S ∈Rd

[f (w, S ) − α]+ p(S )dS ,

where [f (w, S ) − α]+ = max{f (w, S ) − α, 0}. According to Rockafellar [23,24], minimizing CVaR over any set of admissible portfolios W is equivalent to minimizing Fβ (w, α) over (w, α) ∈ W × R, which becomes a convex minimization problem

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min

(w,α)∈W ×R

Fβ (w, α)

4769

(9)

when the set of admissible portfolios is a convex set. In our application, we consider only linear constraints, which makes the set of admissible portfolios polyhedral and thereby convex. 4.2. Simulated CVaR As suggested by Alexander et al. [44], we can approximate the minimization problem (9) using a Monte Carlo approach. Assuming that {(1V )i }m i=1 are independent samples of 1V , which denotes the change in the values of spot and futures prices over the given horizon, we obtain the following approximation to the CVaR optimization problem: F¯β (w, α) = α +

1

m X

!

[−(1V ) w − α] . (10) m(1 − β) i=1 where the set W is specified by budget and return constraints along with bound constraints on positions w . That is, we define min

(w,α)∈W ×R

T i

+

W = {w : l ≤ w ≤ u, (V0 )T w = 1, E [(1V )]T w ≥ r¯ }. In this study we assume that no short selling is allowed by setting l > 0. By the same token we also require that the weight of both spots and futures in the portfolio must be nonzero. The set of portfolio horizons ranges from 1 to 20 days. Throughout, we assume that 1V follow the Copula-MSM model defined in the previous section by (8). 5. Application Our data set contains Nord Pool closing prices for spots between Jan-1996 and Jan-2006. However, due to the limited availability of futures price data we have the closing prices for all futures contracts only for the period between Jan-1996 and Nov-2002. Therefore, we have divided our analysis into two parts. In the first part we inspect the empirical scaling structure of spot prices using wavelet techniques proposed by Abry et al. [45,46]. Next, we estimate the multifractal models based on MSM processes to spot price data. The estimation is followed by an evaluation of volatility forecasting performance. The purpose of the first part is to provide the information needed for specification of an appropriate model for spot price marginal distribution, which is used as part of the CopulaMSM model. In the second part we consider problem of finding optimal CVaR portfolios, when the spot and futures prices follow the Copula-MSM model. Only one futures strategy is included. Since the short-term futures contracts are more liquid as well s more correlated with the underlying spot, we use futures with three weeks left to maturity. In order to avoid inclusion of thin market and expiration effects, the futures contract is rolled over one week prior to its expiration. The data set of 1699 observations between Jan-1996 and Nov-2002 is divided into insample and outsample period. We use the first 1400 observations to estimate the parameters of the Copula-MSM model, while reserving the outsample for evaluating the dynamically updated portfolios. 5.1. Multiscaling in electricity spot prices In this section our goal is to study the empirical scaling properties Nord Pool spot prices between Mar-1998 and Jan2006; see Figs. 1 and 2. This study will establish the basis for considering multifractal models in the specification of the joint dependence model between spots and futures prices. A visual inspection confirms the extremal behavior of electricity prices with frequent jumps and strong seasonal fluctuations [31,1–3]. In order to take this into account, we have removed most of the deterministic trend using a linear combination of x1 (t ) = 1, x2 (t ) = t, x3 (t ) = sin(λt ), x4 (t ) = cos(λt ), x5 (t ) = sin(2λt ), and x6 (t ) = cos(2λt ); see e.g. Ooms et al. [16]. The remaining autoregressive structure in spot returns is filtered with ARMA(1,1) process. Although, careful specification of seasonal components would be necessary for forecasting purposes, our main interest is in the study of spiky behavior and extreme volatility exhibited by electricity prices. Furthermore, the wavelet techniques are reasonably robust against seasonalities. 5.1.1. Analysis of scaling exponents An important feature of electricity prices is mean-reversion. Electricity is often considered as one of the best known examples of antipersistent data [36]. To describe the nature of the data set, we have considered estimates for Hurst exponents using several alternative methods; see Table 1. Although, some variation is observed, the findings are reasonably similar with an average estimate at 0.34 for both series. In order to inspect variations in Hurst exponents we have estimated pointwise exponents and Legendre singularity spectrum; see Fig. 3. Both discrete and continuous wavelet transforms are considered. The obtained results suggest time variations in the exponents, which indicate large error bounds for analysis of Hurst exponents. The findings are consistent with the study by Erzgräber et al. [5].

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0

200

400

600

800

1000

1200

1400

1600

1800

2000

Time [Δt=1 day]

Fig. 1. Daily Nord Pool closing prices for spot contracts between Mar-1998 and Jan-2006.

0.9999 0.9995 0.999 0.995 0.99

Probability

0.95 0.9 0.75 0.5 0.25 0.1 0.05 0.01 0.005 0.001 0.0005 0.0001 –0.8

–0.6

–0.4

–0.2

0

0.2

0.4

0.6

0.8

1

1.2

Data

Fig. 2. Cumulative distribution function for logarithmic spot returns between Mar-1998 and Jan-2006. Comparison is against the Gaussian cumulative distribution. Table 1 The table reports Hurst exponents for logarithmic spot and detrended spot prices between Mar-1998 and Jan-2006. Several alternative estimators are considered. For more details on the methods see e.g. Taqqu, Terovsky and Willinger [47]. Estimation method

Spot

Detrended

Absolute moments Aggregate variance Boxed periodogram Difference variance Higuchi Peng RS

0.36 0.33 0.30 0.13 0.39 0.33 0.52

0.35 0.31 0.26 0.24 0.38 0.32 0.52

We corroborate our analysis by applying multiscale diagrams proposed by Abry et al. [45,46], where an empirical scaling function ζˆ (q) = αˆ q − q/2 is plotted against moments; see Fig. 4. The estimation of multiscale diagrams is based on wavelets [48–50]. The connection between fractal processes and wavelets arises from the fact that the increments involved in the study of the local regularity of a sample path can be seen as simple examples of wavelet coefficients. We compute the multiscale diagram for detrended electricity prices with q ranging from −1 to 7. The number of vanishing moments was set to 4 and L2 normalization was used. The purpose in the diagrams of Fig. 4 is to look for significant deviations from a simple linear function over a range of moments. A visual inspection reveals that the electricity prices do not obey any linear scaling function. A lack of alignment in the multiscale diagram strongly suggests scaling behavior, which cannot be explained by a linear scaling function ζ (q) = qH. For comparison, we have also provided the linear multiscale diagram with hq = αˆ q /q − 1/2, where linear forms appear

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as horizontal alignments. Indeed, the observed absence of horizontal regions indicates that a multifractal model might be appropriate. For comparison with the earlier estimates for Hurst exponents reported in Table 1, we can read an alternative Hurst exponent estimate from the scaling function by considering the second order statistic α2 given in Fig. 4 by recognizing that H = (α2 − 1)/2 = 0.338. 5.1.2. Comparison of spot models Having verified the existence of multiscaling properties in spot prices, it is of interest to compare the performance of different multifractal models against well-established classical GARCH models. For comparison we estimate three MSMmodels, and two GARCH models with normal and Hansen’s Skew t-distributed innovations. The first model, denoted by MSM10, has 10 volatility components/ multipliers (i.e. k = 10). We set the highest transition probability γ10 = 0.5 and b = 2. In order to check for the effects of increasing the number of volatility components, we estimate the second model, MSM15, with k = 15 multipliers. Other parameters are held constant. The third MSM20 model has k = 20 multipliers and the highest transition probability at 0.9. The GMM estimates of location parameters are reported in Table 2. The table presents also the parameter estimates of the two GARCH models. The model performance comparisons are based on their ability to produce volatility forecasts. A useful property of the MSM framework is that it can be used for volatility forecasting. The simple structure of the MSM models permits recovery

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ζˆ (q) = αˆ q − q/2; see Refs. [45,46]. Table 2 Parameter estimates. The first panel reports the parameter estimates for three Markov-Switching models, where k denotes the number of volatility components, σ is a sample estimate of volatility, and λ is a location parameter for the component distribution. The second panel reports parameter estimates for GARCH(1,1) model with normal and Skew-t-distributions, respectively. Panel A: GMM estimates for MSM models

λ σ k

MSM10

MSM15

MSM20

0.0990 (0.03525) 0.0831 10

0.0500 (0.02197) 0.0831 15

0.0637 (0.02549) 0.0831 20

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ω α β df Shape

GARCH

ST-GARCH

0.0021 (0.0015) 0.3448 (0.1018) 0.3833 (0.2096)

0.0003 (0.0002) 0.4234 (0.1565) 0.5766 (0.5766) 4.1000 (1.0806) 0.0688 (0.0237)

of analytical auto-covariances. Therefore standard approaches to construct linear forecasts can be utilized; see Lux [42]. In Table 3 we have presented an evaluation of volatility forecasts for the GARCH models and the three MSM models. The forecasting horizon ranges from 1 to 100 steps ahead. The study is carried out using the last 500 observations of the spot returns series. The time unit in forecasting is 1 day. The reported mean-squared errors are standardized by dividing each result by the MSE of unconditional volatility forecasts. The second panel of the table reports the reduction in MSE achieved by the alternative models with respect to the GARCH model with Gaussian errors. Based on the results in Table 3 the MSM models appear to outperform both GARCH models in terms of their forecasting ability. The attained reduction in MSE ranges from −1.6% to −8.02%. Furthermore, we find that the difference becomes more visible at longer forecasting horizons. At short 1–5 day forecast horizons the differences between models are relatively small. However, when considering forecasts for horizons over 10 days ahead, the MSM models yield a several percentage reduction in MSE. When comparing the GARCH models, we find that the normal GARCH model is slightly better than the skew-t version at horizons 1–25. However, when longer periods are considered the use of skew-t distribution yields benefits over normal errors. In summary, we find that MSM models appear as a competitive alternative to classical approaches when volatility forecasting performance is considered. The models also seem to be reasonably robust with respect to their parameterizations. Although, some improvement in long-term forecasting performance is achieved by increasing the number of volatility components in MSM, the differences between the various versions of MSM models are small.

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Table 3 Forecasting performance. The table presents the mean-squared errors for three MSM stochastic volatility models and benchmark GARCH and Skew-tGARCH models. The reported figures have been scaled by the mean-squared error of unconditional volatility forecast to enhance the comparison. We use the 500 last observations of the electricity spot price process to evaluate the forecasts. Panel A: MSE Horizon

GARCH

ST-GARCH

MSM10

MSM15

MSM20

1 5 10 15 20 25 30 50 75 100

0.887 0.894 0.900 0.839 0.846 0.854 0.862 0.889 0.936 0.987

0.889 0.896 0.903 0.841 0.848 0.855 0.861 0.880 0.911 0.941

0.847 0.846 0.827 0.813 0.816 0.828 0.831 0.863 0.898 0.945

0.873 0.867 0.830 0.820 0.817 0.828 0.827 0.849 0.874 0.919

0.866 0.860 0.828 0.817 0.815 0.826 0.826 0.850 0.877 0.922

Panel B: Reduction in MSE with respect to GARCH Horizon

GARCH

ST-GARCH

MSM10

MSM15

MSM20

1 5 10 15 20 25 30 50 75 100

– – – – – – – – – –

0.20% 0.23% 0.35% 0.31% 0.21% 0.06% −0.11% −1.05% −2.71% −4.68%

−4.55% −5.34% −8.13% −3.03% −3.53% −3.12% −3.65% −2.98% −4.09% −4.25%

−1.60% −2.96% −7.83% −2.16% −3.45% −3.12% −4.10% −4.51% −6.62% −6.96%

−2.45% −3.71% −8.02% −2.57% −3.67% −3.35% −4.24% −4.42% −6.34% −6.65%

5.2. Copula-MSM model and dynamic CVaR In this second part of our application we consider the problem of finding optimal CVaR portfolios, when spot and futures prices follow the Copula-MSM model described in Section 3. First, we present the results of estimating the Copula-MSM model. In particular, we discuss the selection of an appropriate functional form for the copula function by comparing fits of several static and dynamic copula specifications. Finally, we use the model to compute optimal CVaR portfolios for different holding periods and confidence levels and evaluate the model performance by using an out-of-sample dataset to inspect whether the CVaR forecasts produced by the model are consistent with realized portfolio losses. We also examine the achieved reduction in CVaR with respect to an unhedged electricity portfolio. The data set used for studying the relationship between spot and futures strategy covers the period Jan-1996 and Nov-2002; see Fig. 5. The first 1400 observations are used as an insample for estimation of the models. The rest of the observations are reserved as an outsample to evaluate forecasting performance. 5.2.1. Estimation of Copula-MSM The estimation procedure for Copula-MSM model is carried out in multiple stages. As the first step, we estimate the mean equations (µXt , µYt ) using a simple error-correction model. Next, we inspect the behavior of the both marginal residuals and estimate chosen stochastic volatility models (hXt , hYt ) discussed in Section 3.2. In the final stage, we use the filtered standardized residuals obtained from the preceding steps to select and estimate an appropriate copula function. The parameter estimates obtained from the different steps along with model equations are collected in Table 4. Prior to estimating the mean equations we used a sequence of likelihood ratio tests for different lag orders, which effectively whitened most of the serial correlations. We also checked the Johansen trace and eigenvalue tests for cointegration, which suggested estimating the model with one cointegrating vector as we had described in Section 3.1. Indeed, the coefficients α1 and α2 for the error correction term appeared significant in both spot and futures equations. By considering Portmanteau tests for squared residuals we also rejected clearly the null of no serial correlation, which is consistent with the earlier studies where GARCH models have been applied in electricity markets. The estimation results for the mean equations are furnished in Panel A of Table 4. The estimates for volatility equations are presented in Panel B of Table 4 by considering the obtained innovations individually. For spot price innovations we estimate MSM model with k = 10 volatility components and assuming that the components are log-normally distributed. The location parameter for the distribution is found by applying the GMM estimation technique proposed by Lux [42]. The procedure is carried out by matching the analytical moment structure of the MSM model against empirical moments. Altogether, we used 8 moment conditions to identify the location parameter λ. The scaling parameter σ was estimated using the sample variance. For futures price innovations the GARCH(1,1) model was estimated using Student’s t-distribution. Before selecting this particular specification, we had compared the results

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against two GARCH models with Normal and Hansen’s Skewed t-distributions. We also experimented with various MSM specifications with different numbers of volatility components. However, the parameter estimates were not statistically significant. Instead, the alternative GARCH models appeared to yield better results in the futures price case. The final step in our estimation process was to find a suitable specification for the dependence structure between standardized spot and futures innovations. In order to evaluate the copula specification, we estimated 12 alternative copula specifications. The results from these estimations are presented in Table 5. To help with comparisons, the results are ranked in terms of the log-likelihood values followed by Akaike information criterion and Bayes information criterion. In case there is asymmetry in the dependence structure, it is advisable to evaluate the fits of copulas implying different types of dependence between the variables. Whereas, for example, the Plackett and Frank copulas are symmetric, the Clayton copula would be more suitable if negative changes in spot and futures appear more strongly correlated than positive changes. In addition to allowing for various types of asymmetry, we wanted to check for the possibility of having time variations in the dependence structure. For this purpose we had included specifications such as the time-varying normal copula and time-varying symmetrized Joe–Clayton copula. When we examined the results from dynamic copulas (omitted for brevity), we found the copula parameters to be quite noisy but without any patterns or structural breaks which would necessitate the use of a time-varying specification. Instead, in our case we find the static Student’s t-copula to provide a better fit than any other static or dynamic copula included in comparison. The parameter estimates for Student’s t-copula are reported in Panel C of Table 4. 5.2.2. Minimizing CVaR Given the fact that trading in electricity markets is driven by the need to reduce portfolio risks, we evaluate the performance of our model by considering how it can be applied for dynamic construction of CVaR-optimal portfolios. For this purpose we use the out-of-sample data set, which covers the last 299 observations between Aug-2001 and Nov-2002. Before considering the CVaR optimization problem, we define the set of admissible portfolios. In order to keep the problem simple, we impose short-selling restrictions and also set upper and lower limits for portfolio positions. As there are

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Table 4 Copula-MSM parameter estimates. This table shows the multistage estimates for the parameters of Copula-MSM model. In the first stage, a combination of Johansen maximum-likelihood estimator and least-squares is used to estimate the cointegrating vector and the coefficients of the conditional means equations. In the second stage, conditional volatility models are estimated for the marginal innovations separately. For the MSM model the parameters are obtained using the GMM estimator proposed by Lux [42]. Finally, the conditional copula parameters are estimated using maximum likelihood for the standardized residuals obtained from the preceding steps. Panel A: Conditional means

P µXt = c1 + α1 (β1 Xt −1 − β2 Yt −1 ) + 2i=1 (φi(1,1) 1Xt −i + φi(1,2) 1Yt −i ) P2 Y µt = c2 + α2 (β1 Xt −1 − β2 Yt −1 ) + i=1 (φi(2,1) 1Xt −i + φi(2,2) 1Yt −i ) 1X t − 1 1X t − 2 1Yt −1 1Yt −2

Parameter

Std. error

Parameter

Std. error

−0.052 −0.042

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−0.014 −0.003

0.013 0.013 0.028 0.029 0.001 0.000

0.372 0.019 −0.021 0.002

EC term Constant

0.029

−0.033 −0.002 0.000

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(i )

hXt = σ 2 i=1 Mt , where Mt hYt = a0 + a1 εY2t + a2 hYt−1

Q10

∼ LN (−λ, 2λ)

σ2 λ a0 a1 a2

v

Parameter

Std. error

0.0059 0.1188 0.0000 0.0522 0.9316 3.3020

0.0002 0.0360 0.0000 0.0127 0.0128 0.3005

Panel C: Conditional copula



1Xt −µXt

√

hX t

Y

1Y −µ , √t Y t ht



∼ CT (Normal, Student; ρ, ν)

ρ ν Copula likelihood

Parameter

Std. error

0.198 4.222 −67.72

0.027 0.557

Table 5 Comparison of copula functions. This table shows the log-likelihood fits of several copula specifications estimated for the standardized spot and futures innovations. Copula

LL

AIC

BIC

Student’s t Time-varying symmetrized Joe–Clayton Symmetrized Joe–Clayton Gumbel Time-varying rotated Gumbel Rotated Gumbel Rotated Clayton Plackett Time-varying normal Normal Frank Clayton

−67.72 −55.41 −47.82 −46.77 −44.22 −39.53 −38.75 −34.46 −33.03 −32.34 −30.42 −25.52

−135.43 −110.82 −95.65 −93.54 −88.44 −79.06 −77.49 −68.93 −66.07 −64.69 −60.84 −51.05

−135.43 −110.8 −95.64 −93.54 −88.43 −79.06 −77.49 −68.92 −66.06 −64.68 −60.83 −51.04

only two assets, we require that the lower bound for the position is 10%, which along with the budget constraints guarantees that the optimal asset weights for spot and futures contracts can only fluctuate within the bound [0.1, 0.9]. For simplicity, we also assume that the trader does not have a prior portfolio, which would need hedging, but instead refer to the CVaR minimizing optimal portfolio as a hedge portfolio. As a return requirement constraint, we expect the optimized portfolio to yield at least as good return as the expected spot return for each particular holding period for which the forecast is being computed. We did not want to add stricter return requirements in order to ensure the existence of feasible solutions for most data points. As discussed by Alexander et al. [44], we could naturally include various other properties in the CVaR optimization problem as constraints to reduce the ill-posedness of the problem and possibly obtain also more desirable optimal portfolios. Ill-posedness, in this case, refers to the possibility of having many portfolios that have similar CVaR or VaR values to those of the optimal portfolio. In practice this means that even a slight perturbation in data could lead to significantly different optimal solutions, which is likely to be reflected as large variations in optimal portfolio weights and CVaR forecasts. However,

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Fig. 6. Portfolio loss and CVaR for 1 day holding period. This figure draws the forecasted outsample CVaR and realized portfolio loss function computed using the optimal portfolio weights for one day holding period and confidence level of β = 0.95.

we believe that the constraints used in our example application satisfactory for illustration purpose and fairly consistent in the sense that feasible solutions can be found. Provided the portfolio restrictions we consider minimizing CVaR for different holding periods t¯ = 1, . . . , 10. At each time instant in the out-of-sample set, we solve the CVaR optimization problem for the given holding period t¯ and confidence level β ∈ {0.9, 0.95, 0.99}. The optimization problem is approximated using the simulated CVaR approach, where apply the Copula-MSM models conditional on the observed information up to each time instant to generate 10 000 scenarios for the holding period. As a result, we obtain the forecasted portfolio CVaR for the period along with optimal weights. Using the optimal portfolio weights wt we can then compute the realized portfolio losses l(t , t + t¯) = −wtT (Vt +t¯ − Vt ) for each holding period. For illustration, Fig. 6 shows the forecasted CVaR at 95% confidence level for an optimized portfolio with holding period of one day. The time-variation in the forecasted CVaR and loss functions can be interpreted as a timing risk for taking the portfolio. When the length of the holding period increases, the timing risk also increases. The large variations also partially reflect the degree of ill-posedness of the optimization problem. In Table 6 we have reported realized CVaR for optimized portfolios for different confidence levels and holding periods. The realized CVaR and VaR measures are computed using the actual observed returns assuming that at each step we would fix the

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Table 6 Realized CVaR. This table shows the realized CVaR of the optimized portfolio for different confidence levels and holding periods. Relative differences in CVaRs of three comparison portfolios with respect to the optimized portfolio are reported in the last three columns.

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t¯

Optimized portfolio

RelDifCVaR

CVaR

VaR

Spot (%)

Future (%)

Equal weights (%)

0.99

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0.107 0.138 0.174 0.203

0.079 0.108 0.167 0.196

98.70 109.59 45.41 41.54

6.05 17.07 6.02 2.26

6.33 7.33 −0.46 2.66

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0.035 0.067 0.121 0.142

102.85 104.58 36.68 31.53

4.69 19.44 4.72 9.09

12.23 25.36 0.07 3.79

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0.026 0.057 0.099 0.118

97.56 93.61 32.81 25.37

4.00 14.78 5.44 11.49

18.22 23.70 −0.62 3.72

portfolio with the given weights for the given holding period. For comparison, we consider simple benchmark portfolios: an unhedged portfolio consisting of 100% spot, a future portfolio with 100% position, and an equal-weights portfolio. Inclusion of equal-weights portfolio for comparison is well justified by our ex-post analysis of optimal portfolio weights, which on average were not too far from 50–50 position when looking over different horizons. In order to compare the portfolios, we report relative differences in CVaR for each benchmark portfolio, i.e., RelDifCVaRi =

CVaRi − CVaR0 CVaR0

where CVaR0 denotes the CVaR of the optimized portfolio and CVaRi is the CVaR of the comparison portfolio. The results in Table 6 suggest clear benefits from using futures contracts to reduce portfolio risk. This is quite expected recalling the fact that futures prices follow the spot price trend rather closely but exhibit considerably less variation. Indeed, as compared to an unhedged spot portfolio, we find over 100% reduction in CVaR due to optimization. However, the optimization benefits decrease rapidly with the holding period. The optimized portfolio has generally a clear advantage also with respect to the future and equal-weights portfolio. Again the differences decrease as longer holding periods are considered. This lends support for the view according to which strategies to reduce spot portfolio risk perform best with short holding periods only, perhaps less than one week. One interesting remark concerns the behavior of the optimal weights as a function of the holding period and confidence level. We observed that the average weight of futures contract increases with holding period and conversely the weight of spot contract decreases. Furthermore, we remarked that the higher the confidence level, the larger the weight of the spot contract in the portfolios. These findings appeared consistent across different confidence levels and holding periods. In order to evaluate the forecasting performance of our model, we have presented the mean-squared-error (MSE) and mean-absolute-error (MAE) for each holding period t¯ and confidence level β MSE(t¯, β) = MAE(t¯, β) =

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X 

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l(t , t + t¯) − CVaR(t , t + t¯)

2

X l(t , t + t¯) − CVaR(t , t + t¯) ,

|N (t¯, β)| t ∈N (t¯,β)

where N (t¯, β) denotes the set of failure points, when the realized portfolio loss has exceeded the forecasted VaR at the given confidence level, i.e. N (t¯, β) = {t : l(t , t + t¯) > α(t , t + t¯; β)}, and α(t , t + t¯; β) is the VaR forecast for the holding period at time t with confidence β . The CVaR forecasting performance measures, MSE and MAE, are reported in Table 7. The overall accuracy of the model appears to be reasonably uniform when holding period lengths of less than two weeks are considered. When looking at the size of the forecasting errors relative to the level of realized CVaR, the amount by which the model over- or underestimates the actual CVaR does not seem to increase with the holding period. This suggests that the observed reduction in the benefits of portfolio optimization cannot be entirely attributed to potential degradation in model performance at longer holding periods, but instead could be better explained by behavior of the spot and futures markets or issues concerned with the construction of the futures strategy. The analysis is, of course, deliberately restricted to holding periods of less than two weeks in order to avoid rollover effects in the futures strategy. In summary, the findings suggest that our Copula-MSM model can be a useful tool for modeling the dependence between spot and futures prices. Furthermore, we have provided support for the fact that considerable risk management benefits can be achieved by using electricity futures strategy as a part of portfolio optimization.

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Table 7 CVaR MSE and MAE. This table displays the mean-squared error and mean-absolute error of CVaR forecasts computed for different holding periods and confidence intervals β . Panel A: MSE Holding period

β = 0.99

β = 0.95

β = 0.9

1 3 6 9

0.0043 0.0005 0.0172 0.0010

0.0008 0.0011 0.0017 0.0016

0.0007 0.0009 0.0018 0.0020

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β = 0.99

β = 0.95

β = 0.9

1 3 6 9

0.0299 0.0172 0.0549 0.0255

0.0163 0.0203 0.0316 0.0313

0.0161 0.0193 0.0307 0.0343

6. Conclusions The paper has two main contributions. In the first part of our application, we wanted to examine the multifractal scaling properties of electricity spot prices and compare the performance of multifractal stochastic volatility models against a classical GARCH framework. The second contribution of our paper is to propose a Copula-MSM model for characterizing the dynamics between spot and weekly futures prices. The framework enables a flexible approach for combining classical volatility models such as GARCH with more recently developed multifractal stochastic volatility models. The approach is, however, limited to only such fractal models which have volatility covariance structure expressible in analytical form. In our application for electricity markets we demonstrated how a Markov-Switching Multifractal model can be joined together with t-GARCH(1,1) model using a T-copula to characterize the dependence structure. The empirical results suggest that electricity spot price is characterized by relatively strong multiscaling properties. Therefore, it appears that a multifractal modeling approach can be well justified. Indeed, a comparison of volatility forecasting performance across GARCH models and MSM models, shows that multifractal MSM models make a highly competitive alternative especially when longer forecasting horizons are considered. However, the futures price process, on the other hand, appears to have rather different characteristics from spot prices. The behavior of futures prices is considerably less violent than the erratic variations observed in spot prices. Therefore, instead of multifractal modeling techniques, we opted for a classical GARCH model to characterize futures volatility dynamics. In the second part of our application, we studied how the Copula-MSM model can be used for finding CVaR optimal electricity portfolios. The findings suggest that considerable risk reductions can be achieved by employing a futures strategy as part of an electricity portfolio. The efficiency of the electricity futures market for risk management purposes has been historically a debated subject. In particular, as suggested by studies of Yang and Awokuse [51], among others, it is widely held that hedging is weak for nonstorable commodities. However, although our study does not make any statements regarding market efficiency, it appears that in the case of electricity markets hedging benefits can be gained at least for short investment horizons (time scales of order weeks). Whether this is possible for longer horizons using forward contracts, we leave as an issue for further inspection. It would be also interesting to examine, how much better risk minimizing portfolios could be established by using several different futures and forward strategies as part of a portfolio. Acknowledgements The author would like to thank Dick Van Dijk, János Kertész, and anonymous referees for helpful comments. The financial support from Fortum foundation is acknowledged. References [1] R. Weron, B. Kozlowska, J. Nowicka-Zagrajek, Hurst analysis of electricity price dynamics, Physica A 283 (2000) 462–468. [2] R. Weron, B. Kozlowska, J. Nowicka-Zagrajek, Modelling electricity loads in California: A continuous time approach, Physica A 299 (2000) 344–350. [3] R. Weron, I. Simonsen, P. Wilman, Modeling highly volatile and seasonal markets: Evidence from the nord pool electricity market, in: The Application of Econophysics, Springer, Tokyo, 2004, pp. 182–191. [4] P. Malo, A. Kanto, Evaluating multivariate GARCH models in the nordic electricity markets, Communications in Statistics: Simulation and Computation 35 (2006) 117–148. [5] H. Erzgräber, F. Strozzi, J. Zaldivar, H. Touchette, E. Gutiérrez, D. Arrowsmith, Time series analysis and long range correlations of nordic spot electricity market data, Physica A 387 (2008) 6567–6574. [6] P. Chainais, R. Riedi, P. Abry, On non scale invariant infinitely divisible cascades, IEEE Transactions on Information Theory (2005) 51. [7] P. Chainais, R. Riedi, P. Abry, Warped infinitely divisible cascades: Beyond power laws, Traitement du Signal (2005) 22. [8] R. Mantegna, H.E. Stanley, An Introduction to Econophysics: Correlations and Complexity in Finance, Cambridge University Press, Cambridge, 2000. [9] F. Wang, K. Yamasaki, S. Havlin, H.E. Stanley, Multifactor analysis of multiscaling in volatility return intervals, Physical Review E 79 (2009). [10] J. Bouchoud, M. Potters, M. Meyer, Apparent multifractality in financial time series, The European Physical Journal B 13 (2000) 595–599.

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