Value-at-Risk Based Portfolio Management in Electric Power Sector Ran SHI, Jin ZHONG Department of Electrical and Electronic Engineering University of Hong Kong, HKSAR, China ABSTRACT In the deregulated electricity market, highly volatile electricity price leads to the large skew and kurtosis in the price return distribution. Risk management is essential in the portfolio management for market participants. In this paper, we proposed a portfolio optimization model which is capable of managing non-normality in the electricity portfolio return distribution. The optimal portfolio is found by maximizing a performance index resembling to the Sharpe ratio, whereas the risk is defined using Valueat-Risk technique instead of standard deviation. In this model, we can automatically determine the risk-free asset allocation without using utility function, and, as a result, determine how much Contract for Difference(CfD) should be included in the portfolio. After some explanations of the market structure and portfolio elements in Nordic power market, we give an example of a hypothetic electricity company in Oslo, managing its portfolio following the proposed strategy. Keywords: Porfolio Management, Risk Measure, Nonnormality, Value-at-Risk, Nordic Power Market, Risk- Averseness 1. INTRODUCTION Deregulation in electric power sector makes electricity spot prices highly volatile hence induce substantial risk for utilities and power trading companies[1, 2]. In the Nordic electricity market, forward contract hedged by Contract for Difference(CfD) could be considered as riskfree investment in a short term, assuming fuel costs are known when the portfolio managers are making the decision. The risky transactions include forwards, futures, options, CfDs, and other derivatives in both exchange and over-the-counter(otc) market [3]. Further more, the company could also own or lease generation assets in order to get involved in spot trading which is also risky. Hence, how to optimize the portfolio in certain time horizon deserves more treatments. One initiative is applying traditional mean variance approach in finance to electricity market[4, 5]. However, the underlying assumption of mean variance framework is the normality of assets returns which is widely known far from satisfactory in electricity business[6]. Also the concept of utility function is not straightforward in the sense that we do not know how to determine risk averseness and the utility function itself does not have explicit economic meaning. Value-at-Risk(VaR) is a commonly used risk measurement, which is the maximum loss value of the investment over a specified period of time at certain confidence level [7]. To our knowledge, VaR based portfolio management [8, 9] application in electricity market is scarce in literatures. Though a VaR constrained approach has been applied to electric power sector in [10], the method uses the utility function [11], and basis risk is not accounted for. A VaR based optimization framework is proposed in this paper. The method does not required utility function, and it is capable of applying any underlying distribution assumptions. The framework is also applied to a hypothetic company at Oslo Norway in Nordic market. The impacts of degree of risk-averseness are illustrated in the paper. 2. RISK MEASURE Intuitively, risk has two essential components: exposure and uncertainty. Risk, then, is the exposure to a proposition of which one is uncertain. Although we can not control the uncertainty, we can usually control the risk according to the degree of risk-averseness. However, the definition of risk depends on the notion of exposure and uncertainty. Portfolio theory first developed by Markowitz use standard deviation of the expected return to measure the uncertainty, while use the utility function to capture the degree of risk-averseness. Figure 1 plots the standard deviation under normal return distribution. We can see that the risk measure here is just the uncertainty represented by standard deviation. However, when the return distribution is far from normality, which might have flat left tail and be asymmetrical, the standard deviation will not be able to capture the asymmetrical uncertainty which is critical to profitability. Meanwhile, it is not easy to determine the risk-averseness in the utility function, since it does not have explicit economic explanation. Value-at-Risk(VaR) has been adopted by banks and financial institutions as the measure for market risk. VaR is defined as the maximum expected loss on an investment over a specified horizon given some confidence level. The definition of VaR can be given mathematically as follows: Given some confidence level α ( 0,), 1 the VaR of the portfolio at the confidence level α is given by the smallest number l such that the probability that the loss L exceeds l is not larger than (1 α)
α-var = inf{ l R : P( L > l ) 1 α} Eq. 1 In probabilistic terms VaR is a quantile of the loss distribution. For example, a 95%-VaR of $4 million under normal trading conditions the trader can be 95% confident that a change in the value of its portfolio would not result in a decrease in portfolio value of more than $4 million during 1 day. This is equivalent to saying that there is a 5% confidence level that the value of its portfolio will decrease by $4 million or more during 1 day. A 95% confidence level does not imply a 95% chance of the event happening, the actual probability of the event cannot be determined. The key point to note is that the target confidence level (95% in the above example) is the given parameter here; the output from the calculation ($4 million in the above example) is the maximum loss (the value at risk) at that confidence level. Figure 1 Standard deviation as risk measure The advantages of VaR lie in the fact that it can capture risk-averseness through pre-defined confidence level, and account for the fat tail in underlying distribution, thus cover the uncertainty more accurately. Also, VaR is essentially a monetary amount, which is easily to be perceived by investors. However, VaR itself does not necessarily represent the risk. Investors usually consider risk in a relative sense by comparing the expected return with other reference measure such as risk free return or industry average return. In this paper, the measure of risk is defined in terms of the VaR of portfolio plus the expected return subject to the level of risk. Figure 2 explained this definition graphically. Please also be advised that the portfolio distribution here is set to be non-normal, which can show the strength of incorporating VaR. ϕ (, c p) = W(0) E( r ) + VaR(, c p) Eq. 2 p where p : portfolio allocation W (0) : initial amount for investment c : confidence level (CL) r p : portfolio return VaR(, c p) : estimated Value-at-Risk of total assets The rationale of defining risk as Eq.2 is to capture the downside risk, and in the mean time, when it is pure riskfree asset, ϕ should be equal to zero. It is obviously satisfied in the risk-free case, where the portfolio Valueat-Risk is simply the negative final wealth, making the ϕ zero. In fact, this risk measure collapses to the standard deviation when the portfolio return follows normal distribution, because in this case, it is just a fixed multiple of standard deviation. Figure 2 Definition of risk associated with VaR 3. PORTFOLIO OPTIMIZATION Portfolio optimization is the rational process of seeking optimal risk-return trade off. Basically the trade-off is quantified by sort of yield per unit of risk. Markowitz is the father of modern portfolio theory by proposing meanvariance framework. In this framework, covariance plays an important role in this theory, because unsystematic risk can be diversified away by selecting negatively correlated assets, thus we can construct portfolio with the same expected return but less risk by considering the interactions between assets. In the mean-variance framework, the trade-off is defined by the Sharpe ratio [12] equal to the expected portfolio return in excess of risk-free return, divided by standard deviation of the expected return. This framework can be presented by a two-step approach. This first step is to obtain optimal risky portfolio by maximizing Sharpe ratio, which is the slope of the capital market line (CAL). The second step is to obtain the optimal allocation between risk-free investment and optimal risky portfolio. The two step process is graphically presented as in Figure 3 for the purpose of comparison with VaR-based approach later..
variance framework has now become efficient mean ϕ frontier. Since risk-free investment does not affect the allocation of risky investment, we will first obtain the optimal risky portfolio by maximizing S(p). Then, we will show that the total allocation will be automatically determined by the desired VaR without resorting to utility function. Figure 3 Portfolio selection under mean-variance framework We should note that risk preference comes in the second step as a parameter pre-determined in utility function. However, how to define proper utility function becomes another problem, and the solution is usually not clear. Since mean-variance framework relies on the covariance structure, it is only possible to calculate portfolio variance under the assumption that all asset returns follow normal distribution. This brings the major problem of applying mean-variance framework to electricity market because electricity price is highly volatile, with characteristics such as seasonality, mean-reversion, spikes, hence the return in electricity portfolio is far from normality. VaRbased portfolio selection strategy can cope with this very well. The framework is explained as follows. Suppose W (0) is the initial amount of fund to be invested in the horizon T, which we want to invest such that the portfolio meets a chosen VaR limit. This amount could be invested along with amount B representing borrowing (B>0) or lending (B<0) at the risk free rate. Therefore the final wealth is The optimal portfolio is chosen independently from the level of initial amount of fund. It is also independent from the desired VaR, since the risk measure depends on the estimated portfolio VaR rather than the desired VaR. However, as the investor s degree of the risk aversion is already captured by the desired VaR and confidence level, the amount of borrowing or lending required to meet the VaR constraint is therefore determined as following: * W(0)( VaR VaR( c, p')) B = Eq. 5 ϕ '( c, p') We can find that if the portfolio manager set the desired VaR equal to the expected portfolio VaR, no risk-free investment will be involved. In the other way around, if there is no risk free investment exists, the desired VaR could only be the expected portfolio VaR. The whole process is graphically presented in Figure 4 * VaR VaR(, c p ') E0 W T p W B r p B r f ( (, )) = ( (0) + )(1 + ( )) (1 + ) Eq. 3 The Value-at-Risk based portfolio optimization mathematically formulated as follows: is ϕ '( c, p') Figure 4 VaR-based portfolio optimization scheme Where r( p) rf p' = max S( p) = p ϕ ( c, p) S( p) : performance index p ' :optimal portfolio r f : risk free rate of return r( p) : total return on initial wealth Eq. 4 The denominator in Eq.4 is as defined in Eq.2. Obviously the performance index S( p) resembles Sharp Ratio assuming the return distribution is multivariate normality. We will still maximize the trade-off between risk and return. The efficient frontier in the mean- 4. NORDIC POWER MARKET Imagine an integrated utility company which may own or lease generation assets, and also has a trading division that can trade for forward, futures, options, CfDs as well as other exotic options. We refer all potential assets for the portfolio including owned or leased generation assets as instruments. In fact, all these instruments can be conceptualized as option contracts on certain capacities (in MW) which could be called or sold in a specific period. Each instrument has a strike price at which the option is exercised, and an option premium at which the underlying capacity is reserved. In this framework, owned generation assets have strike prices equal to marginal costs, and zero option premium. Leased
generation has strike price equal to marginal cost, and leasing rate as option premium. The forward, which is prepaid, is obligated to deliver power, hence it has zero strike price. The settlements of forwards, options are based on Systems Prices. However, the actual power procurement cost is based on area price of whatever area the power is delivered to. System price is some sort of average of area prices. When transmission congestion occurs, area prices will be different because cheaper power cannot be delivered to demand in other areas freely, hence the whole market can not achieve equilibrium where all prices should be overall marginal cost of all generation assets. In fact transmission congestion is not uncommon, thus areas prices differ very often. In this case, forwards become risky in the presence of basis risk, the difference between system price and area price of which the power is going to. CfD is based on the difference between the area prices and system prices, and it is designed to hedge the basis risk. To create a perfect hedge that includes the basis risk when area prices are not equal to the System Prices, a three-step process using CfDs must be followed: Hedge the required volume using forward contract. Hedge any price difference for the same period and volume through CfDs Accomplish physical procurement by trade in the spot market area of the member s location.[3] return is the return of CfD-hedged forward return, which is 41%. Using the empirical data of 9 years, we equivalently obtain 279 scenarios for each random portfolio allocation. The 95%-VaR of each random portfolio allocation is obtained by searching for the lowest 5 percent quantile in the portfolio return distribution. 1000 random allocations are plot in Figure 5, where the optimal risky allocation is obtained through Eq.4, which is at the tangential point in Figure 5. Table 2 below shows the optimal weights of elements 1-4 in the portfolio setting. The portfolio VaR is -0.26 million NoK, which means, without involving risk-free assets, the decision maker should have a desired VaR equal to -0.26 million NoK after the optimal allocation is made. Label 1 2 3 4 Weights 0.4882 0.0092 0.0119 0.4907 Table 2 Optimal weights Now, we will show how much CfD is needed to cater for the decision maker s desired VaR. Suppose the decision maker has a desired 95%-VaR different from -0.26 million NoK, the risk-free asset could be obtained directly using Eq.5, without resorting to utility function. Thereafter, the CfD portion and additional forwards are obtained since the risk-free asset is constructed as forward contract fully hedged by CfDs. 5. EXAMPLES We construct a company operating at Oslo Norway, using area prices of Oslo and Nordpool system prices from 1998 to 2006. All prices used are daily average prices obtained from nordpool website [13]. The company uses spot trading, forwards, CfDs, and options to make profit while meet the load demand. The trading period is August. In this month, the Oslo prices are usually lower than system prices due to the abundant water supply to the hydropower generators in Norway. Assuming the initial money W (0) is one million Norwegian Krone (NoK). In this example, two cases are considered to illustrate the procedure of the proposed portfolio management framework, and demonstrate the advantages of capturing risk-averseness of decision maker respectively. Case 1 Asset Label Strike Price Premium Generation 1 150 0 Aug Forward 2 0 210 Daily Call 3 270 30 Daily Put 4 180 15 Aug CfD 5 0 1 Table 1 Portfolio elements Here, the CfD is assumed to cover the same capacity of one monthly forward contract. Therefore, the risk-free Figure 5 Efficient frontier and optimal risky allocation Case 2 Varying the desired VaR can help us find the risk-free allocation, which captures a part of risk-averseness of decision maker. Also, varying the confidence level in the VaR setting provides us another way to incorporate riskaverseness. When the decision maker is more conservative, and has a larger confidence level, the portfolio VaR in the risk measure ϕ (, cp) will become larger, thus make the risk larger. Intuitively, if the return distribution is strictly normal, this will not change our optimal allocation, because our defined risk measure at any confidence level is just a multiple of standard
deviation. However, it s not the case when we have other leptokurtic distribution assumptions, or simply using the empirical distribution. Figure 6 shows the shift of efficient frontier towards RHS as the confidence level changed from 90% to 99%. Expected Return Expected Return 1.5 1.2 0.9 0.6 0.3 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Risk/ Million NoK( 99% CL) 1.5 1.2 0.9 0.6 0.3 [6] R. Weron and I. Simonsen, "Blackouts, risk, and fattailed distributions," Springer Verlag, 2006. [7] P. Jorion, Value at risk: the new benchmark for controlling market risk: McGraw-Hill, 1997. [8] A. Puelz, "Value-at-Risk Based Portfolio Optimization," 2001. [9] R. Campbell, R. Huisman, and K. Koedijk, "Optimal portfolio selection in a Value-at-Risk framework." vol. 25: Elsevier Science, 2001, pp. 1789-1804. [10] P. R. Kleindorfer and L. Li, "Multi-period VaRconstrained portfolio optimization with applications to the electric power sector." vol. 26, 2005, pp. 1 26. [11] H. M. Markowitz, "Foundations of portfolio theory," World Scientific, 1992. [12] W. F. Sharpe, "The Sharpe Ratio," Princeton University Press, 1998. [13] www.nordpool.com 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Risk/Million NoK ( 90% CL) Figure 6 Impact of varying confidence level 6. CONCLUSIONS The need for capturing downside risk motivates us to incorporate the Value-at-Risk in the risk measure. Since the risk-averseness of decision maker could be well accounted for in both the confidence level and desired VaR level, there is no need to use utility function to find the optimal allocation between risk-free and risky assets. The portfolio optimization criterion is designed as to maximizing the downside risk-return trade-off which is defined as. One special result in electric power sector is that the use of CfD can be determined automatically as the risk-free proportion is found, since the risk-free asset is considered as the forward contract fully hedged by CfDs. Since our framework is capable of incorporating any return distribution assumption, one natural extension of this paper could be the comparison among more leptokurtic distributions assumptions, in additional to empirical distribution and normal distribution. 7. REFERENCES [1] S. J. Deng and S. S. Oren, "Electricity derivatives and risk management." vol. 31: Elsevier, 2006, pp. 940-953. [2] J. J. Lucia and E. S. Schwartz, "Electricity Prices and Power Derivatives: Evidence from the Nordic Power Exchange." vol. 5: Springer, 2002, pp. 5-50. [3] A. S. A. Nord Pool, "Trade at Nord Pool s Financial Market," April, 2004. [4] M. Liu and F. F. Wu, "Portfolio optimization in electricity markets." vol. 77: Elsevier, 2007, pp. 1000-1009. [5] H. M. Markowitz, Mean-variance analysis in portfolio choice and capital markets: Basil Blackwell, 1987.