تجزیه و تحلیل ریسک بازار انرژی فضایی با استفاده از اندازه گیری ریسک نیمه واریانس

The paper concentrates on the analysis of semivariance (SV) as a market risk measure for market risk analysis of mean–semivariance (MSV) portfolios. The advantage of MSV over variance as a risk measure is that MSV provides a more logical measure of risk than the MV method. In addition, the relationship of the SV with the lower partial movements is discussed. A spatial risk model is proposed as a basis of risk assessment for short-term energy markets. Transaction costs and other practical constraints are also included. A case study is provided to show the successful application of the model.

There have been many studies on risk in portfolio selection and practical applications based on the Markowitz mean–variance (MV) method [1] and [2]. The most important aspect of the MV method is that it introduces an important concept of portfolio efficient frontier on which each point is an efficient portfolio point with the variance minimized at a given level of return expectation. After the introduction of a risk-free asset, Sharpe proposed the capital asset pricing model (CAPM) [3] that has also attracted extensive attention in academics but with limited applications in practice [26]. The logical connection of CAPM and MV is explained in detail in [3]. The problem with the MV method is that variance may not be a proper risk measure because it contains the effect of return deviations above the mean. By common sense, portfolio returns above the mean should be regarded as beneficial not as risk. Hence, using variance as a risk measure does not make a good logical sense, as argued by many. A simple improvement over the MV method for measuring risk is to use deviations below the mean, and lower semivariance (SV) is hence a simple and good choice for the purpose. The SV method, however, has attracted less attention in practice due to more computation requirement. The other problem with the traditional MV method is that it cannot take into consideration fixed transaction cost, which usually over estimates the mean return of a portfolio. Since the fixed transaction cost is a lumpy sum no matter how big the transaction deal is, a binary integer variable is needed for properly modeling it. If a portfolio does not contain asset, the binary variable is zero, the fixed transaction cost is not counted for after optimization. Since the model contains both integer and continuous variables, it is called the mixed integer programming (MIP). The SV is also called the second lower partial movement (LPM) and is one of the several downside risk measures [10]. Other downside risk measures may include the absolute downside deviation (also called semideviation (SDV) or the first LPM), etc. The SDV as a risk measure has a computation advantage but cannot take into consideration correlations of returns on assets. LPM plays an important role in our discussion of downside risk measures. Therefore, it is appropriate that the definition of the k-degree LPM be provided: where R is a stochastic process such as the return of a portfolio, τ is a target value that an investor would use for measuring his/her preference of risk. F(R) is the cumulative probability distribution of R. τ is the expected return of a portfolio for both the SDV(k = 1) and the SV(k = 2). LPM is closely related to the value at risk (VaR) measure where τ is defined as a percentile on the lower tail, say, 1% of the probability distribution. When k ⩾ 1, LPM is used for measuring the risk preference of those risk averters. However, the k-degree LPM must be correctly related to the standard statistical movements of the distribution where investors have a preference for higher values of odd movements (skewness) and a dislike of higher values of even movements (variance, kurtosis and the like). In short, LPM is used for measuring an investor’s risk attitude towards the below-target returns. The motivation for the research is based on the reasoning that the variance of a portfolio return may not be an appropriate measure of risk. It is logical to think that variance is a measure of uncertainty rather than risk and only the part of the variance with returns less than the mean return is relevant to risk. A variance also includes the effect of the returns greater than the mean return, and a rational investor should love but not to avert higher returns. Therefore, the part of the variance reflecting greater returns than the mean should not be regarded as risk. It is generally agreed that this argument offers a logical thinking and it is consistent with the framework of VaR measure being widely used in the financial industry [4]. Note that VaR is also a downside risk measure. The paper is centered on the minimization of SV subject to practical constraints currently overlooked by many portfolio software packages. However, models based on the minimization of VaR and SDV are not excluded as alternatives. In the proposal, we do not assume symmetry in distribution and normality of returns. We also incorporate integers, fixed and proportional transaction costs, and other practical constraints. We do not propose CAPM as a risk management tool for electricity markets because CAPM is for efficient markets where the characteristics do not apply for electricity markets with imperfections such as games. For symmetrical distributions of asset returns, the MSV and MV only have a minor difference between their efficient frontiers for small values of the variance of portfolio returns [5]. However, for energy portfolios, especially with heavy option derivatives, asset returns are well known to be asymmetrical. The MSV method is hence a more general and better choice than the MV method for energy portfolio optimization. The following example shows how skewed the returns of the PJM electricity market can be for a gas turbine plant. Assume the plant has an average heat rate of 11 MMBTU/MWh, a variable O&M cost of $1/MWh and a capacity of 50 MW. Suppose that the peak plant ran from hour 15 to hour 18 at full capacity. A simple estimation of the discrete distribution of the expected returns for the plant is illustrated in Table 1 given a natural gas price of $2/MWh in August 1998. The market price for the plant to make nonnegative returns is about $23/MWh ignoring the start-up cost. The energy weighted average price is about $55.5/MWh and the root mean square error is about $29.8/MWh. It can be seen from Table 1 that the return distribution is seriously skewed and with a fat tail to the higher price range. Furthermore, the proposed model is extended to include the spatial nature of the portfolio optimization with downside risks. The introduction of integers will make portfolio selection more practical. Portfolio selection models with integers are often NP-complete and very difficult to solve. However, small portfolios can be solved within a short time using the nested LaGrange relaxation (LR) method and even the Brunch & Bound method. We will show that the objective function of the MSV portfolio model is in the form of multiplication of variables and a simple decomposition using the LR may not work well. In case there is a need to solve large energy portfolio problems, algorithms can be developed for computation time reduction. For example, several algorithms can be used to speed up computation, including the modified Benders decomposition (MBD) method, the genetic algorithms (GAs), etc. The remainder of the paper is arranged as follows. Section 2 introduces the Markowitz MV, discusses the Markowitz MSV method, and presents the MSV model with transaction costs. Section 3 presents the spatial risk model with detailed explanations for the major constraints, and it also discusses potential algorithms for solving the problem. Section 4 illustrates the use of the model via a case study. And Section 5 concludes the paper.

The paper presents a spatial energy market risk model based on the Markowitz MSV method. The model is for assessing the risk of profit making of power producers in a multi-pool market setting. The model also includes practical constraints such as transaction costs and wheeling contracts, leading to a mixed integer formulation. The case study shows the successful application of the model. An interesting observation is that the Markowitz MSV efficient frontier is neither smooth nor concave mainly due to the addition of those physical constraints of the plants and fixed costs.