EUROPEAN TRANSACTIONS ON ELECTRICAL POWER Euro. Trans. Electr. Power 2011;21:1825–1841 Published online 28 May 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/etep.429
A dynamic CVaR-portfolio approach using real options: an application to energy investments Jana Szolgayova´1,2, Sabine Fuss1*,y, Nikolay Khabarov1 and Michael Obersteiner1 2
1 International Institute for Applied Systems Analysis (IIASA), Laxenburg, Austria Department of Applied Mathematics and Statistics, Faculty of Mathematics, Physics and Informatics, Comenius University, Bratislava, Slovakia
SUMMARY Even though portfolio theory has increasingly been applied to analyze large-scale investments under uncertainty—and especially so in the electricity sector—most analysis so far has been based on the static mean-variance approach. Such an approach has two shortcomings: on the one hand, it fails to take into account irreversibility in the form of high sunk costs and the associated implications for optimal dynamic behavior. On the other hand, variance is not always the ideal risk measure, given that return or cost distributions are not necessarily normal. In fact, if large, potential losses are involved, it makes more sense to adopt a risk measure that can also take into account fat tails. In this paper, we generate these distributions arising from the investment behavior optimized in a real options model, thus accounting for uncertainty and irreversibility at the plant level, and use them in a dynamic portfolio model, where the conditional value-atrisk (CVaR) is the risk measure. More specifically, we look at the dynamics of the (CVaR-) optimal technology mix over a future time period conditional on the initial distribution of technologies, such that given energy demand is met. The application to investment in the electricity sector with uncertain climate change policy shows that this approach is not only useful from the aggregate investment point-of-view but also for the purpose of evaluating the effects of policy on investment patterns and the resulting energy mix. Copyright # 2010 John Wiley & Sons, Ltd. key words:
portfolio optimization; CVaR; climate change policy; uncertainty; real options; electricity investments
C61, D81, D92, G11, Q4, Q56, Q58
1. INTRODUCTION Climate change policy has mainly been centering around a more sustainable use of fossil fuels aiming at low-carbon technologies and a shift away from an energy mix, which increases cumulative emissions at the current pace. The most commonly advocated policy tool suggested is the so-called cap-and-trade system or hybrid forms of the same. The main idea behind cap-and-trade is to determine the ‘‘desirable’’ or sustainable amount of emissions based on information about climate sensitivity and other indicators and to allocate the corresponding amount of allowances accordingly through e.g., auctioning. The price at which these allowances will be traded is obviously a source of uncertainty and this will represent an incentive for investors to diversify their capacity. It is therefore evident that portfolio theory offers scope for additional insight into investment decisions in the energy sector, which is already facing CO2 price uncertainty through permit trading in some European countries. Moreover, it is not far-fetched to assume that energy sectors in other countries will soon be affected by similar arrangements with respect to the combustion of fossil fuels *Correspondence to: Sabine Fuss, International Institute for Applied Systems Analysis (IIASA), Laxenburg, Austria. y E-mail:
[email protected] Copyright # 2010 John Wiley & Sons, Ltd.
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for energy generation. In addition, volatile CO2 prices are not the only source of uncertainty: fossil fuel prices, for example, have been fluctuating widely over the past and the technological progress forecast for renewable energy technologies is not a steady, certain process either. Diversification thus seems to be a valuable strategy. As Reference [1] points out, diversity in the energy mix is often seen as a means to foster energy security [2], where countries are mainly concerned about their dependence on fossil fuel supplies from other countries. As a bonus, diversity can thus also be a means to realize sustainable energy supply [3], where the adoption of renewable energy carriers would also reduce the portion of fossil-fuel-based technologies in the overall energy portfolio. Having thus established the relevance of portfolio considerations in the energy context, let us now turn to a brief introduction of the underlying theory. Originally being designed for and applied to deal with financial assets [4,5], portfolio selection has also been adapted to investment problems involving real assets in the meantime. The earliest work goes back to Reference [6], who find that in the United States electric utilities are more or less efficiently diversified in each region; at the same time, however, utilities appear to have a tendency for combining high rates of expected return with high levels of risk. This willingness to accept more risk is attributed to the large number of regulations, which the authors claim to be a threat to investors, so that they feel forced to take on higher risks. Another early study by Reference [7] refines this approach by building a GARCH-type model, which allows the covariance matrix to be systematically updated over time as new events occur. They find that in times of e.g., oil price shocks the strategy to diversify away from oil-intensive generation equipment is efficient. In addition, although the electric utilities in the United States have been operating at a minimum variance position by the end of the 1990s according to the authors, overall energy consumption is evaluated to have been very inefficient during that time. In recent years, portfolio theory has also been used to look at issues of energy security, see e.g. References [8,9]. The first study [8] concludes that the existing portfolio of EU power generating technologies is suboptimal and therefore inefficient, i.e., that there are portfolios combining lower risks with higher returns. Another result is that fuel price risk always dominates the other types of risk (e.g., about O&M costs) analyzed, which has important implications for energy security. The other article [9] makes a case for a higher proportion of renewable energy to be chosen for the overall energy mix. Assessing EU countries as well as Mexico and other developing countries, Reference [9] concludes from his empirical results that the latter have or plan to have much less efficient generating portfolios in the future and makes some corresponding policy recommendations. A compilation of recent portfolio applications to energy investments and some frontier research can be found in References [10,11], which will be reviewed in detail below, provide an in-depth literature review focusing on the choice of risk measures as well. Much of this literature still relies on the original framework sketched out by Reference [4], even though this model has two major disadvantages when it comes to investment decisions in the energy sector: firstly, the analysis rests on the mean and the variance to assess profitability and variability of a project. Using the variance as a risk measure ignores the possibility that return or profit distributions might not be shaped normally. In fact, it has been found that the return distributions for some technologies might actually have fat tails, which implies that there can be potentially high losses, which will not be captured by considering only the variance. However, recent research has proposed alternative risk measures, such as the value-at-risk (VaR) or the conditional value-at-risk (CVaR) (see e.g., Reference [12]). Secondly, portfolio theory—as applied to energy investments so far—is largely static and thus myopic.1 Since we are concerned with investment into real assets, it is in fact not as straightforward to extend the static Markowitz portfolio to a dynamic setting as this might be the case for financial assets.2 The reason for this is the special feature of irreversibility: once resources have been committed to install a new power plant, for example, this asset can hardly be removed from the generating portfolio at zero transaction cost. In a recent article, Fortin et al. [11] have therefore integrated real options analysis— for investment decisions and operations at the plant level—with a portfolio framework—where a large investor or a region diversifies over different plant types. However, the analysis remained inherently 1 There are exceptions using elements from portfolio selection in the context of dynamic models such as Reference [13], who integrate portfolio features into a vintage model of the electricity sector with an application to UK energy policy. 2 Actually, in finance, dynamic portfolios have already been in use for decades, see Reference [14].
Copyright # 2010 John Wiley & Sons, Ltd.
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static insofar as the large investor would allocate his/her funds once in the beginning of the planning period, using information from the real options model about the optimal, dynamic behavior of individual plants. The current paper builds on Reference [11] and seeks to remedy this deficiency by taking into account the possibility to diversify not only over assets but also over time. In other words, we should allow for the fact that the option of having a different portfolio at a future point in time might have a decisive impact on how we compose our portfolio today. Vice versa, the lock-in generated through today’s investments and its impact on the possibilities in the future are equally important considerations in the optimization process. We actually find that there is indeed an impact of the availability of future opportunities on today’s composition of the energy mix and demonstrate implications for the energy sector by focussing on two exemplary technologies: coal-fired power plants, which stand representative for fossil-fuel-based electricity generation and biomass-fired power plants as their renewable energy counterparts. The findings of the paper show, indeed, that the possibility to adapt the portfolio in the future already has an effect on today’s portfolio investment decisions. Optimizing dynamically taking future opportunities into account leads to diversification not only over technologies but also over time. It is therefore of paramount importance that policy-makers send clear signals about the future development of their policies to investors so that these can optimize under more complete information. The remainder of this introduction will present the technologies considered in our analysis, along with a motivation of why these are chosen, and then move on to define the parameters and further assumptions. The source of uncertainty considered is the carbon price, since the electricity sector is expected to undergo major changes, as governments resort to stricter climate change policies to curb emissions from fossil fuel combustion. Section 2 deals with the modeling framework, where we start out with the real options model, whose outcome will inform the portfolio investor. Particular emphasis will, of course, be on the dynamic aspects that constitute the novelty of this paper. The results will be analyzed in more detail in section 3, after which we conclude.
1.1. Conversion technologies: coal-fired versus biomass-fired power plants The current electricity system is mostly based on fossil-fuel-fired technologies, with a substantial part coming from coal- and gas-fired power plants at the EU-27 level. As these are ageing, new investments could be steered toward more climate-friendly technologies, which governments seek to entice by imposing a penalty on the emission of CO2 either through cap-and-trade or through direct taxes. We model this as a price on CO2 generated by operating the power plant in question. At the same time, the much hoped-for transition to renewable energy technologies is still being delayed by their high installation and operational costs (e.g., for offshore wind farms repairs are relatively expensive and for biomass-fired power plants fuel costs are very high). In spite of these factors, however, diversification considerations—especially in combination with climate change policy and the ensuing uncertainty (here modeled as a fluctuating CO2 price)—could lead to the inclusion of renewable energy in the current portfolio already today. Since the main purpose of the paper is to demonstrate the dynamic aspects of such portfolio investment in the electricity sector, we restrict our application to keep the analysis relatively straightforward. We do so by focussing on two technologies, which are thought to be representative of the established fossil-fuel-fired energy system and of the ‘‘desired’’ system based on a larger portion of renewable energy: the first technology is an average integrated gasification combined cycle (IGCC) coal plant, which has higher fuel efficiency than the traditionally installed, standard pulverized (PV) coal plant, and therefore emits less CO2 than the less modern plants. The other technology considered for new investment is a biomass-fired power plant, which has the advantage that as much CO2 is sequestered by the growing of biomass as is emitted by combusting it. Therefore, the CO2 emissions from the biomass plant are accounted for as being equal to zero. Both the coal-fired power plant and the biomass-fired power plant are in addition retrofittable with carbon capture modules (CCS). These modules capture part of the CO2 generated and can be added to the existing plant over the course of its lifetime. In the case of the biomass-fired power plant this Copyright # 2010 John Wiley & Sons, Ltd.
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implies that CO2 emissions will effectively be negative, since the sum of CO2 emissions sequestered and captured exceeds the amount released during power generation [15]. The choice for these technologies has been made for a number of reasons. Firstly, coal-fired capacity still features importantly in the energy mixes of many EU-27 countries, most notably Germany, Greece, Poland, the Czech Republic, etc. Even though much of the installed capacity is PV, the more modern IGCC technology, which is more efficient, will increasingly be considered for new investments as carbon policy becomes more stringent. Secondly, we want to examine the competitiveness of renewables. However, the potential is not very even across EU-countries and also the capacity factors are very different and typically low, which makes technologies such as wind farms and solar installations less comparable to their fossil-fuel-fired competitors. We have therefore decided for biomass-fired electricity, which is considered for as a zero-emission technology on account of the sequestration related to the growing of biomass as a fuel. In addition, biomass-fired electricity generation capacity retrofitted with carbon capture facilities is increasingly of interest in energy assessments looking at low stabilization targets implying sufficiently high carbon prices to make this technology commercially viable. The data used in this study are presented in Table I. They were gathered in a survey conducted by the International Energy Agency [16] and will be supplemented by data from IIASA’s GGI Scenario Database [17] for CO2 prices in the next section.
1.2. Carbon price uncertainty and other parameters Forecasting carbon prices is a difficult if no impossible task, since there are no suitable historical data, from which the CO2 price process could be calibrated. The combination of immature markets (and therefore insufficient data) and the uncertainty of the political process governing the magnitude of changes in the price level forces us to resort to scenario data to model carbon prices. IIASA’s GGI Scenario Database [17] provides scenarios based on different assumptions about population dynamics, diffusion of technology, economic development, technical progress, urbanization, etc (see Reference [18]). There are basically three scenarios and GHG shadow prices, which we can use as a proxy for CO2 permit prices. The ‘‘B1’’ scenario offers a very optimistic outlook, where the future is characterized by a population peak and subsequent decline mid-century. In addition, there is much technological progress leading to clean and resource-efficient technologies. Furthermore, economies transit to service and information economies, with policy-makers aiming at social, economic, and environmental sustainability. While this is a rather optimistic view of the future, scenario ‘‘A2r’’ is at the other side of the continuum, predicting that fertilities do not converge in the near term and population rises considerably. On top of this, economic development is very concentrated in specific regions and technical change is slow and fragmented. In the middle of these two extremes, ‘‘B2’’ is a scenario where technological change, population growth, economic development, etc all lie in between the assumptions of the optimistic B1 and the pessimistic A2r scenarios. For this study, we take the intermediate scenario B2, with a starting price of 7.91 s/ton CO2 in 2010 and an escalation rate just below 5%. The volatility of the CO2 price process can be estimated from the range of the scenarios as 4% and even though this is an imperfect measure, of course, we Table I. Power plant data for coal and biomass. Coal
Coal þ CCS
Bio
Bio þ CCS
7446 6047 39 510 43 710 1 1373
6475 576 39 510 60 110 1 1716
7446 0 135 655.4 43 269 1 1537
6475 6100 135 655.4 59 669 1 1880
Parameters Electricity output (MWh/year) CO2 emissions (t CO2/year) Fuel cost (s/year) O&M cost (s/year) Installed capacity (MW) Capital costs (1000 s) Source: IEA/OECD, 2005.
Copyright # 2010 John Wiley & Sons, Ltd.
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find that the results are not particularly different if the volatility parameter is not increased dramatically. The electricity price is left constant at its current level of 40 s/MWh, even though it can be argued that producers will pass their CO2 expenses on to consumers until the point where zero-emission technologies become sufficiently competitive to take over from fossil-fuel-based power plants. Moreover, the electricity price obviously also oscillates, even in the short run to a certain extent. Since we want to examine the impact of one source of uncertainty to make the impact of dynamic aspects more transparent, we disregard these factors and save them for future research. Note that both the CO2 and the electricity price in the model are explicitly exogenous and do not depend on the technological choice of the investor. This is because the investor is assumed to be small with respect to both the carbon and the electricity market.
2. REAL OPTIONS ANALYSIS AND DYNAMIC ASPECTS OF PORTFOLIO INVESTMENT Real options theory has been a very popular modeling approach to look at investment decisions in the electricity sector, which is characterized by three special features, which make real options suitable for such analysis: (1) the irreversibility of investments due to the high sunk costs associated with the installation of power plants and related equipment, (2) the various uncertainties the investor faces (e.g., fuel price volatility, uncertain technical change, regulatory uncertainty), and (3) the flexibility on behalf of the investor to exercise the investment option at a later point in time, which is explicitly valued in the real options framework. A good introduction into the topic is provided by Reference [19], where there is also a section devoted to applications to energy investments. In the meantime many applications have emerged, which focus either on the short run (e.g., Reference [20]) or on the long run, where our work falls more into the latter category. Most closely related are Reference [21], who also look at carbon capture retrofits under regulatory uncertainty [22,23], who focus on operational decisions under permit price fluctuations [24], who allow for jumps in the price of carbon and [25,26], where the same solution methodology (dynamic programming and Monte Carlo simulation) is used as in this paper.3 The combined real options and portfolio approach mentioned above was first established in Reference [11] and later applied to explore the effects of climate policy on the energy technology mix by Reference [28]. There are two important theoretical insights derived from the analysis in Reference [11] that we want to emphasize at this point: (a) return distributions from investment in electricity generation equipment are not necessarily normal and therefore VaR and CVaR mostly outperform the variance as a risk measure. More precisely, the optimal portfolio’s return distribution can be shifted considerably to the right in particular situations, if the CVaR approach is used instead of the traditional Markowitz, mean-variance approach. That implies that the distributions are extended to a more profitable region. (b) The difference between the mean-variance and the CVaR approach appears to diminish as the bound on the expected return is lowered. With respect to policy conclusions, Fortin et al. [11] find that in scenarios with relatively low CO2 prices, the investor’s incentive to diversify already leads to the inclusion of renewables into the energy mix because with an upward trend in CO2 prices (as is the case in the European Trading Scheme), renewables become relatively more profitable than investors might initially have believed. Fossil fuel technologies initially exhibit less risk, so they still constitute a large part of the portfolio, obviously. Fuss et al. [28] further contribute to the policy perspective of the new approach by applying the framework to IIASA’s GGI scenarios [17] for different stabilization targets.4 They find that scenarios where CO2 emissions reductions can easily be achieved (as in e.g., B1) in combination with not so strict stabilization targets imply that fluctuating CO2 prices allow big investors to realize large returns at low risk (in terms of CVaR). More ambitious targets, however, entice retrofitting with CCS and thereby 3 Of course, there is a multitude of other energy applications focussing on different types of risks or decisions, the review of which is beyond the scope of this paper. We refer the reader to the literature review in Reference [27] for more details. 4 For this purpose, Fuss et al. [28] use the basic, static framework of Reference [11], which will be extended to look at dynamic decisions in this paper.
Copyright # 2010 John Wiley & Sons, Ltd.
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reduce profitability and increase risk. At the same time, this will make the policy more efficient in terms of achieving stabilization targets, since stricter targets always leads to more diversification, which also implies more renewable energy. This means that policy-makers should remain committed to targets resulting in CO2 prices, which are able to trigger the investment that is necessary for stabilization, even if industries prefer less ambitious targets. In the following, Section 2.1 will first describe the real options model for the optimization of investment and operations at the plant level. The information generated by the real options model will then serve larger investors in their decision-making process, which is based on the portfolio criterion (i.e., maximize expected return given some maximum risk level or minimizing risk given some minimal return level). The portfolio part of the framework will then be explained in more detail in Section 2.2.
2.1. Decision-making at the plant level: real options model The framework used in this study is intended to compute the optimal investment plan for a single profitmaximizing electricity producer, thereby generating the return distributions for the portfolio model. The producer has to deliver a certain amount of electricity over the course of the planning period and faces a stochastic price on CO2. The real options model is used to value the options to invest into each of these plants and the options to retrofit them. More precisely, each plant will be considered independently of the other one, so as to avoid any impact of the presence of one option on the value of the other one. As already indicated above, CO2 prices, Pct , are the only source of uncertainty and will thus be modeled stochastically. In particular, the price follows a geometric Brownian motion (GBM) in order to allow for an oscillating price path with an upward trend (see Section 1.2). dPct ¼ mc Pct dt þ s c Pct dWtc
(1)
where m c is the parameter for the drift, s c is the volatility parameter, and dWtc is the increment of a Wiener process. Let xt be the state variable, which describes whether the (coal-fired or biomass-fired) power plant without CCS, the CCS module, or both have been built and whether the CCS module is currently running. at denotes the action, which in our case is the i.e., the control variable. at can be any of the following: (I) building the power plant without the CCS module, (II) building the power plant with the CCS module, (III) adding the CCS module, (IV) switching the CCS module off, (V) switching the CCS module on,5 and (VI) doing nothing. Aðxt Þ is the set of feasible actions and only feasible actions at 2 Aðxt Þ can be undertaken. The investor’s optimization problem can then be formulated as follows:6 Z max
Tþt fat gt 0 0
t0
Tþt0
ert E½pðxt ; at ; Pct Þdt
(2)
where starting values are known and r ¼ 6% the discount rate. Profits are denoted by pðÞ and consist of income from electricity production less the cost of fuel, CO2 expenses, operational and maintenance (O&M) costs, and costs for carrying out specific actions such as installation of the CCS module, cðat Þ.7 The following equation shows the composition of profits in more detail: pðxt ; at ; Pct Þ
¼ qe ðxt ÞPe qf ðxt ÞPf qc ðxt ÞPct O&Mðxt Þ cðat Þ
(3)
5 Please note that, with the magnitudes of volatility considered in this study and the given trend, there is not much switching occurring, in fact. 6 The restrictions on actions are that the investor has to invest into the power plant in year t0, and do nothing before. Once the CCS module has been built, it can be only switched off and on. 7 As an assumption, a power plant will produce continuously throughout the year, i.e., we have a fixed coefficients production function in the style of Leontieff mimicking output contracts between distributors and generators.
Copyright # 2010 John Wiley & Sons, Ltd.
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In this equation, P e is the electricity price, P f is the fuel price, O&M is the O&M cost per year, and q e, q c, and q f are the annual quantities of power produced, CO2 emitted and fuel combusted, respectively. Furthermore, we abstract from construction times. This is not too strong an assumption, since the concerned technologies’ construction times do not differ substantially at the sizes considered and are not long in the first place compared to other plant types such as nuclear power plants. As formulated, the problem is a discrete optimal control problem on a finite horizon and can be solved by dynamic programming. In order to find the optimal actions for the optimal control problem described above, let us first formulate the Bellman equation as Vðxt ; Pct Þ ¼ max fpðxt ; at ; Pct Þ þ er EðVðxtþ1 ; Pctþ1 Þjxt ; Pct Þg at 2Aðxt Þ
(4)
V() is the value function; T ¼ 50 is the planning horizon and the value function is equal to zero at the end of the lifetime of the plant. The investor’s/producer’s problem can be solved recursively making use of dynamic programming. In this case, EðVðxtþ1 ; Pctþ1 Þjxt ; Pct Þ will be computed using Monte Carlo simulation. Alternative methods are the formulation of partial difference equations, which are then discretized, or the set-up of binomial lattices. We use Monte Carlo simulation, since it adapts rather easily when there are changes to the framework (e.g., when other price processes are tested). More importantly, however, it remains computationally efficient for a high degree of complexity and is rather precise when the discretization is sufficiently fine. The output of the recursive optimization part is a multidimensional table, which lists the optimal action for each time period, for each possible state and for each possible carbon price in that period.8 These optimal actions can be called ‘‘strategies’’ and the output table can be regarded as a kind of ‘‘recipe’’ for the producer, so that he knows in each period, for each possible state occurring and for each possible realized price, what he shall optimally do. For the analysis of the final outcome, we can then simulate (10 000) possible CO2 price paths and extract the corresponding decisions from the output matrix (or the ‘‘recipe’’). By plotting the distributions of the optimal time of exercising the investment option for all 10 000 price paths, we obtain the final results needed. In addition, the optimal decisions at are used to compute total discounted income and total discounted costs so that we can produce the return distributions necessary as an input for the portfolio optimization in the following section. 2.2. Static vs. dynamic portfolio investment using CVaR as risk measure The return distributions generated with the real options framework described in the previous section are obviously not shaped normally, as can be seen from the example of biomass in Figure 1. Using the standard mean-variance approach would imply that we assume a normal distribution of the shape depicted in Figure 1 and therefore make suboptimal use of the available information. In contrast, the VaR and the conditional value-at-risk (CVaR) are able to account for these information. Using Figure 2, we can explain the CVaR and VaR more precisely. Let us first define the b-VaR of a portfolio as the lowest amount a such that, with probability b, the portfolio loss will not exceed a. The b-CVaR, on the other hand, is the conditional expectation of losses above that amount a. In this case, b is the specified, corresponding probability level (see also Reference [12] for a more comprehensive treatment of these risk measures and their implementation). In Figure 2, the b-VaR corresponds to the bth-percentile of the distribution, while the b-CVaR is the mean of the random values exceeding this VaR. It is clear that if the distribution in Figure 2 was skewed, then the CVaR would capture the extent of the losses, while the variance would not, so we choose the CVaR as the risk measure for our portfolio application. 8 Note that the price will be discretized, so if we talk about possible instances of the price, we mean each point in a grid between a pre-defined maximum and minimum price, where the latter are set in such a way that they encompass 95% of all simulated price paths.
Copyright # 2010 John Wiley & Sons, Ltd.
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Figure 1. Return distribution for the biomass-fired power plant (living from year 0 to 50) compared to a normal distribution with the same expected return and variance.
Figure 2. b-VaR and b-CVaR of a normal distribution.
CVaR may be considered as an extension of VaR,9 as it provides a sort of approximation of VaR because it can be interpreted as an upper bound of VaR. Minimizing CVaR instead of VaR in portfolio optimization might thus be seen as a more ‘‘conservative’’ approach. Moreover, CVaR makes more information available to the decision maker than VaR, since VaR denotes the maximum losses, an investor faces subject to some pre-specified probability (which corresponds to the most probable situation), while CVaR also provides information about the size of the potential losses in the case of the less probable event. In addition, CVaR is relatively easy to compute because the calculation can be reduced to a linear programming problem, which can effectively be solved using standard applied software, which is commercially available. Having clarified our choice of the measure of risk, we now want to give a brief overview of the portfolio optimization problem. Even though we will give a full mathematical account below, we first want to describe the framework in more general terms, so that the reader can also skip safely to Section 3 without going through the mathematical details. The idea is to create a setup, where the investor is optimizing his portfolio taking into account the flexibility of investments over time. This is achieved by reformulating the basic, static framework, so that it considers not only current portfolio shares but also future sub-portfolios. Risk is represented by the CVaR, which needs to be minimized subject to a constraint on expected return. In other words, we determine the smallest possible dotted area in Figure 2 for a given required return. This return constraint is chosen sufficiently high, however, so that it will not drive out one technology. This has
9
See Reference [11] for a complete literature review of the differences between VaR and CVaR and their applications.
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been done in order to enable the observation of the full diversification effect—without artificially keeping one technology out by maintaining high return constraints. The next step in the analysis is then to determine the optimal portfolios using this model, where the inclusion of future sub-portfolios makes it possible to capture the effect from the flexibility to change the portfolio at a future point in time. The outcome of the new framework can then be compared to the portfolios, which are optimized for the same two points in time separately. In other words, there will be two, independent static portfolio optimizations (now and in the future). By comparing these outcomes to the one from the dynamic optimization, we can then establish whether the option to diversify in the future has an impact on the composition of today’s portfolio. 2.2.1. Mathematical model description. Defining CVaR according to Reference [12], let f(x, y) be the loss function depending on the investment strategy x 2 Rn and the random vector y 2 Rm, and let p(y) be the density of y. The probability of f(x, y) not exceeding some fixed threshold level a is Z Cðx; aÞ ¼ pðyÞdy f ðx;yÞa
The b-VaR is defined by ab ðxÞ ¼ minfajCðx; aÞ bg, and b-CVaR is finally defined by Z 1 CVaRb ðxÞ ¼ fb ðxÞ ¼ ð1 bÞ f ðx; yÞpðyÞdy f ðx;yÞab ðxÞ
which is the expected loss given that the loss exceeds the b-VaR level, hence conditional expected loss, or conditional VaR, where the risk measure depends explicitly on the confidence level b. Please note that both VaR and CVaR are applicable to returns as well as to losses, because one may consider returns as negative losses (and losses as negative returns). In the following, losses are defined as negative returns and thus we will report –VaR and –CVaR to indicate respectively the lower threshold for returns and expected returns in case they are exceeded by that threshold. We consider n different technology chains that can be invested into. (In our application, for example, the first ‘‘chain’’ is a coal-fired power plant plus CCS, then we have a biomass-fired plant plus CCS as the second one). Values y i, i ¼ 1; . . . ; n reflect the return on investment (ROI) for each technology chain. We assume the vector y ¼ ½y1 ; . . . ; yn T 2 Rn of ROIs to be a random vector having some distribution and describe the investment strategy using the vector x ¼ ½x1 ; . . . ; xn T 2 Rn where the scalar value x i, i ¼ 1; . . . ; n reflects the fraction of capital invested into technology i. The return function depends on the chosen investment strategy and the actual ROIs; computed as xT y. As the actual ROI is unknown, there is a specific degree of risk associated with investment strategy x. To measure this risk and find the corresponding optimal x, our optimization is based on minimizing CVaR with a loss function f ðx; yÞ ¼ xT y, i.e., negative returns.10 Following e.g. Reference [12], we approximate the problem of minimizing CVaR by solving a piecewise linear programming problem and reduce this to a linear programming (LP) problem with auxiliary variables. A sample fyk gqk¼1 , yk 2 Rn of the ROI distribution is used to construct the LP problem. Concerning the investment strategy in the sense that it should deliver a specified minimum expected return (or limited expected loss), the LP problem is equivalent to finding the investment strategy minimizing risk in terms of CVaR: Pq 9 1 minðx;a;uÞ a þ qð1bÞ = k¼1 uk T T (5) s:t: e x ¼ 1; m x R; x 0; u 0; ; T yk x þ a þ uk 0; k ¼ 1; . . . ; q where u ¼ ½u1 ; . . . ; uq T , uk 2 R, k ¼ 1; . . . ; q are auxiliary variables, e 2 Rn is a vector of ones, q is the sample size, m 2 Rn is the expectation of the ROI vector y, i.e., m ¼ EðyÞ, R is the minimum expected portfolio return, a is a threshold of the loss function (and therefore also the upper bound of 10 Note that minimizing the expected right tail loss (CVaR) is equivalent to maximizing the expected left tail return (–CVaR) as further elaborated in Reference [11].
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VaR) and b is the confidence level. The solution ðx ; a ; u Þ of Equation (5) yields the optimal x, so that the corresponding b-CVaR reaches its minimum. The problem (5) reflects a static setup where an investment takes place only once. So far, this setup is therefore similar to its predecessor in Fortin et al. [11]. Now we will consider a dynamic setup, where the investor has the possibility to invest into specific portfolios of sizes bi at different time points i ¼ 1; . . . ; l.11 This setup extends the problem (5) to handle technologies and respective returns at those time points as follows: minðx;a;uÞ s:t:
Pq 9 1 a þ qð1bÞ > k¼1 uk > = T i e x ¼ bi ; i ¼ 1; . . . ; l; > mT x R; x 0; u 0; > ; T yk x þ a þ uk 0; k ¼ 1; . . . ; q
(6)
where yk 2 Rnl are NPV returns, uk 2 R are auxiliary variables, e 2 Rn is a vector of ones, xi ¼ ½xi1 ; . . . ; xil T are technology sub-portfolio shares corresponding to constraint bi at the time point i of investment, q is the sample size, m 2 Rnl is the expectation of the ROI vector y, i.e., m ¼ EðyÞ. The solution ðx ; a ; u Þ of Equation (6) yields the optimal x ¼ ½x11 ; . . . ; x1n ; . . . ; xl1 ; . . . ; xln T , so that the corresponding b-CVaR reaches its minimum.12 In this setup, the investor is optimizing his portfolio taking into account the flexibility of investments over time—future sub-portfolios described by constants fbi g—and is further looking at the dynamics of the entire portfolio, optimizing it in a timeintegrated way.
3. RESULTS: RISK ASSESSMENT AND DIVERSIFICATION The real options model produces the return distributions that are used as an input for the portfolio optimization. Therefore, the results of the former will first be described. In Table II, the descriptive statistics for the return distributions generated for the coal-fired and biomass-fired power plants are listed. Note that each plant has a technical lifetime of 50 years, so plants expiring in year 60 have effectively been built in year 10 from now, for example, and likewise for plants, which were built in year 5 from now.
Table II. Descriptive statistics of return distributions of coal-fired and biomass-fired power plants for 97% confidence. Installation time Coal 0 5 10 Biomass 0 5 10
Exp. return
SD
VaR
CVaR
1.4211 1.3127 1.2231
0.0430 0.0515 0.0474
1.3454 1.2248 1.1453
1.3326 1.2085 1.1292
1.4140 1.5834 1.8211
0.1105 0.1629 0.2406
1.2391 1.3213 1.4436
1.2157 1.2868 1.3941
11 We therefore implicitly assume that there are pre-specified capacities that have to be installed at pre-specified points in time, which is a realistic perspective when we think of the current situation of many OECD countries, which will have to replace part of their existing capacity over the next decade. A 10-year planning horizon, where investment can happen in 5-year steps therefore seems a reasonable case. 12 Problem (2.6) has efficiently been solved with GAMS (http://www.gams.com) using available linear programming solvers as in Fortin et al [11]. Note that the computational complexity of the entire optimization problem is defined by the complexity of the underlying mathematical model, which produces the sample to be fed into the LP problem solving module, i.e., the real options model.
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Table II shows that coal has the highest expected return when installed today (i.e., in year 0). In addition, it is less risky in terms of –CVaR compared to its biomass counterpart (1.3326 > 1.2157). However, the later the coal-fired power plant is installed, the more do the expected return and also the – CVaR and –VaR fall. This implies that less return can be secured at the confidence level of 97%. In contrast, biomass gains both on the investment safety side (in terms of –CVaR) and in terms of expected returns as we move further into the future. The driver of these results is, of course, the CO2 price. The emissions of the coal-fired power plant will still be considerably higher than those of the biomass-fired power plant, so their returns will fall, as the CO2 price increases and the investment risk will get more pronounced as well. For the portfolio optimization we therefore expect the future to play a vital role: while a static analysis would only consider a portfolio of coal and biomass that are installed now and used for the coming 50 years, the dynamic analysis will also allow for funding to be spread across time, i.e., portfolios can be composed of coal installed today and bio installed 5 years later. The option of having bio in the portfolio at a later point in time should then affect the decisions made today. The results of the portfolio optimization indeed show that dynamic portfolios always outperform the corresponding static portfolios in terms of expected return and risk. Dynamic portfolios take all subportfolios spread over time into account in one optimization process, while static portfolios result from optimization at each time point separately, i.e., several independent, static optimizations. For example, if the investor only has the option to invest in power plants, which are installed today, he will choose for over 89.5% of the capacity to be coal-fired and invests 10.5% into biomass. This gives an expected return of 1.42 and a –CVaR of 1.337.13 However, if the investor has to allocate 60% of his resources today and can invest the rest in 5 years from now, then the difference between optimizing dynamically and optimizing separately in year 0 and year 5 gives different results. The share of biomass relative to the sub-portfolio size in the dynamic setting exceeds the relative share of biomass in the static setting both in year 0 and in year 5. This shows that if there is an opportunity to invest at a later point in time, then investors will do so—at the same time, this opens up scope for investment in the current period as well. The expected return is larger by 2.2%, the –CVaR improves by 0.15%. The overall share of biomass is also higher in the dynamic optimization. The interpretation of these results is that the dynamic optimization takes into account the value of flexibility that the future opportunity offers, while the static optimization fails to do so. As a result, the return and the –CVaR are lower as well, making the dynamic portfolio superior in terms of both returns and risk. Further comparisons of results across different sub-portfolio sizes show that the more similar the current (installed in year 0) and the future (installed in year 5) sub-portfolio sizes are, the more does the solution tend to ‘‘bang-bang’’ optimal investment, where investment shifts completely to biomass in the future if most diversification can be taken care of in the current period. The conclusion from this is that for the dynamic version of the portfolio optimization, there is not only a diversification effect across technologies (coal versus biomass) but also a benefit to be reaped from diversification over time.14 Another way of interpreting these results is to see them from a real options perspective: taking the flexibility to invest in the future into account explicitly by dynamically optimizing the portfolio creates further value, which is essentially an option value. Being able to reap the benefits of diversification today opens up the possibility to focus on biomass in the future. The biomass option thus offers flexibility, while not having this option would force the investor to diversify at each point in time separately. In other words, we are looking at a portfolio of options, which are based on different subportfolios. 13 A complete overview of all sub-portfolios calculated and a more detailed comparison of dynamic and static portfolios is given in the Appendix. We restrict ourselves to present some examples here, so that the major mechanisms at work in the new framework become more transparent. 14 This effect is similar to the conclusions of [13], who find that there is a value of waiting in the presence of ongoing technical change: if the volatility of efficiency-improving technological change is reduced, investment in the concerned technology also decreases because investors are then more certain that efficiency will be improved in the future and therefore it pays off to reduce that technology’s portfolio share now in order to be able to increase it toward the end of the planning period when the full benefits of ongoing technical progress can be reaped.
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4. CONCLUSION The analysis conducted and presented in this paper has clearly shown that also in portfolio applications dynamics matter. While this has not been widely appreciated in the existing literature, which largely rests on mean-variance Markowitz-style implementations of portfolio optimization, we believe that a richer framework taking into account the option of future portfolio investment can deliver important insights for large-scale electricity planning and therefore also for policy-makers, who are interested to learn about the impact of their actions on investment behavior. In the energy sector, where most equipment is long-lived, such information can be of great value, since large-scale investment into particular technologies or a particular family of technologies can lead to further lock-in for decades. In order to derive the results for the dynamic portfolio, the optimization of shares is done dynamically, that is in one optimization process. For comparison, the corresponding static portfolios are computed at each point in time individually and without taking future options of having other portfolios into account. The fact that we observe the static portfolios to be inferior in terms of expected return and risk underlines the usefulness of an integrated optimization process and, thus, a dynamic framework for the assessment of investment opportunities at different points in time. The situation where the value of future opportunities is taken into account when making decisions about investment in the face of uncertainty has previously often been tackled by models based on real options theory. However, real options models have the disadvantage of being rather limited in terms of computational requirements and, in addition, they cannot reap the benefits of diversification. In fact, real options models rely on risk-neutral valuation, while portfolio optimization assumes that the investor will want to minimize risk for a given return or maximize return for constrained risk. The new framework developed in this paper overcomes the deficiencies of both approaches by integrating the dynamic character of real options theory with the valuation of diversification in portfolio models: sub-portfolios that can be allocated optimally in the future can be regarded as options. Having such an option, which can be exercised in the future, clearly has an impact on today’s allocation decisions. We have seen that the fact that the investor has the opportunity to invest more into biomass in the future allows him to diversify over technologies today and, in addition, across time. If the future biomass option had not been taken into account, the investor would have had to diversify again at the future point in time, foregoing the benefits of diversification across time in terms of return and risk. The most important result of this study is therefore the apparent impact of the future on present portfolio investment decisions. Dynamically valuing present and future portfolio options in an integrated framework is thus paramount to capture not only risks, but also opportunities. From the point-of-view of the policy-maker implementing incentives to reduce CO2 emissions by imposing a price on CO2 (either in the form of cap-and-trade or in the form of an escalating tax system), the message is two-fold: larger uncertainty about the development of CO2 prices will lead to diversification, so some less emission-intensive capacity will be included at the expense of the fossilfuel-based technology share. This is a standard result from many static portfolio applications to energy investments (e.g., References [8,9,11,28]). Adding the dynamic dimension changes the picture, however: our results show that the future plays an important role for decisions today and thus for the current and future energy mix. Policy-makers should take into account that being unclear about their aims will therefore have profound effects on investors, who will adapt their diversification behavior not only over technologies but also over time. The role that policy uncertainty plays in this context thus needs further investigation in future research.
6. LIST OF ABBREVIATIONS CCS CC-TAME CVaR EU-27 GAMS GARCH
Carbon Capture and Storage Climate Change: Terrestrial Adaptation & Mitigation in Europe Conditional Value-at-risk European Union General Algebraic Modeling System Generalized autoregressive conditional heteroskedasticity
Copyright # 2010 John Wiley & Sons, Ltd.
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GBM GGI GHG IGCC IIASA LP OECD O&M PV ROI VaR
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Geometric Brownian motion Greenhouse Gas Initiative Greenhouse Gas Integrated gasification combined cycle International Institute for Applied Systems Analysis Linear programming Organization for Economic Co-operation and Development Operation and maintenance Pulverized Return on Investment Value-at-risk 7. LIST OF SYMBOLS
at A(.) bi c(.) dW tc e E f(x,y) m n O&M(.) r Ptc Pe Pf p(y) qc(.) qe(.) qf(.) R R t T u V(.) x xt y a b fb(x) C(x,a) p(.) mc sc
Action at time t Set of feasible actions, depending on state size of the subportfolio invested into in time i Costs of action, function of action Increment of the Wiener process at time t vector of ones Expected value Loss function E(y), expectation of the ROI random vector Number of assets Yearly operation and maintenance costs, function of state Discount rate Carbon price at time t Electricity price Fuel price Density of the random vector y Yearly CO2 emissions, function of state Yearly electricity output, function of state Yearly fuel demand, function of state minimum expected return Set of Real numbers Time Planning horizon auxiliary variables Value function, function of state and carbon price Investment strategy State at time t ROI of all assets, random vector Threshold of the loss function Confidence level b-CVaR of the investment strategy x Probability of f(x; y) not exceeding a Yearly profit, function of state, action and carbon price Trend of the carbon price Volatility of the carbon price ACKNOWLEDGEMENTS
This research was supported by the EC-funded project CC-TAME (http://www.cctame.eu/) at IIASA. Copyright # 2010 John Wiley & Sons, Ltd.
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REFERENCES 1. Stirling A. Diversity and sustainable energy transitions: multicriteria diversity analysis of elecricity portfolios. In Analytical Methods for Energy Diversity and Security, 1st edn. Bazilian M, Roques F (eds). Elsevier: Oxford, 2008 (A Tribute to the work of Dr Shimon Awerbuch). 2. Department of Trade and Industry. Energy White Paper 2003: Our Energy Future-Creating a Low-carbon Economy. Available online at: http://www.berr.gov.uk/files/file10723.pdf 3. Grubb M, Butler L, Twomey P. Diversity and security in UK electricity generation: the influence of low-carbon objectives. Energy Policy 2006; 34(18):4050–4062. 4. Markowitz H. Portfolio selection. Journal of Finance 1952; 7(1):77–91. 5. Markowitz H. Portfolio Selection: Efficient Diversification of Investments. Wiley: New York, 1959. 6. Bar-Lev D, Katz S. A portfolio approach to fossil fuel procurement in the electric utility industry. Journal of Finance 1976; 31(3):933–942. 7. Humphreys H, McClain K. Reducing the impacts of energy price volatility through dynamic portfolio selection. Energy Journal 1998; 19(3):107–131. 8. Awerbuch S, Berger M. Applying portfolio theory to eu electricity planning and policy-making. Working Paper EET/ 2003/03, International Energy Agency, 2003. 9. Awerbuch S. Portfolio-based electricity generation planning: policy implications for renewables and energy security. Mitigation and Adaptation Strategies for Global Change 2006; 11(3):693–710. 10. Bazilian M, Roques F. Analytical Methods for Energy Diversity and Security, 1st edn. Elsevier: Oxford, 2008 (A Tribute the work of Dr Shimon Awerbuch). 11. Fortin I, Fuss S, Hlouskova J, Khabarov N, Obersteiner M, Szolgayova J. An integrated cvar and real options approach to investments in the energy sector. Journal of Electricity Markets 2008; 1(2):61–85. 12. Rockafellar R, Uryasev S. Optimization of conditional value at risk. Journal of Risk 2000; 2:21–42. 13. van Zon A, Fuss S. Risk, embodied technical change and irreversible investment decisions in uk electricity production: an optimum technology portfolio approach. In Analytical Methods for Energy Diversity and Security, 1st edn. Bazilian M, Roques F (eds). Elsevier: Oxford, 2008 (A Tribute the work of Dr Shimon Awerbuch). 14. Merton R. Lifetime portfolio selection under uncertainty: the continuous-time case. Review of Economics and Statistics 1969; 51(3):247–257. 15. Uddin S, Barreto L. Biomass-fired cogeneration systems with co2 capture and storage. Renewable Energy 2007; 32:1006–1019. 16. International Energy Agency. Projected Costs of Generating Electricity 2005 Update. OECD Nuclear Energy Agency: Paris, 2005. 17. International Institute of Applied Systems Analysis. GGI scenario database 2009. Available at: //www.iiasa.ac.at/ Research/GGI/DB/ 18. Riahi K, Gru¨bler A, Nakicenovic N. Scenarios of long-term socio-economic and environmental development under climate stabilization. Technological Forecasting and Social Change 2007; 47:887–935. 19. Dixit A, Pindyck R. Investment Under Uncertainty. Princeton University Press: Princeton, 1994. 20. Tseng C, Barz G. Short-term generation asset valuation: a real options approach. Operations Research 2002; 50(2):297–310. 21. Reinelt P, Keith D. Carbon capture retrofits and the cost of regulatory uncertainty. Energy Journal 2007; 28(4):101– 127. 22. Laurikka H. The impact of climate policy on heat and power capacity investment decisions. In Proceedings of the Workshop ‘‘Business and Emissions Trading,’’ Hansju¨rgens B (ed.). Wittenberg: Germany, 2004. 23. Laurikka H, Koljonen T. Emissions trading and investment decisions in the power sector—a case stdy of finland. Energy Policy 2006; 34:1063–1074. 24. Blyth W, Yang M. Modeling investment risks and uncertainties with real options approach. Working Paper LTO/ 2007/WP 01, IEA, 2007. 25. Fuss S, Szolgayova J, Gusti M, Obersteiner M. Investment under market and climate policy uncertainty. Applied Energy 2008; 85:708–721. 26. Szolgayova J, Fuss S, Obersteiner M. Assessing the effects of co2 price caps on electricity investments—a real options analysis. Energy Policy 2008; 26(10):3974–3981. 27. Fuss S. Sustainable energy development under uncertainty. PhD Thesis, Universitaire Pers Maastricht, 2008. 28. Fuss S, Khabarov N, Szolgayova J, Obersteiner M. The effects of climate policy on the energy-technology mix: an integrated cvar and real options approach. In Modeling Environment-Improving Technological Innovations under Uncertainty, 1st edn. Golub A, Markandya A (eds). Routledge Explorations in Environmental Economics, Routledge: Oxon/New York, 2008.
APPENDIX: COMPLETE OVERVIEW OF RESULTS FOR ALL SUB-PORTFOLIOS In this appendix, we present the full portfolio results for all sub-portfolios. Table AI starts with subportfolio SP0 (i.e., all investment goes into plants installed in year 0), then reduces the share of SP0, as Copyright # 2010 John Wiley & Sons, Ltd.
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we go down in the table. Next to the sub-portfolios, the expected return, R, the –VaR, the –CVaR and the shares of coal and biomass at the different time points are displayed. Table A.1 shows the static case. It can be confirmed that the dynamic portfolios generally outperform the static portfolios. When the difference is computed, gains in excess of 3% can be observed for returns and also –CVaR and – VaR often improve by around one percentage point. In fact, the less emphasis is put on the presence (i.e., the lower SP0), the higher the gains are. This is intuitively plausible, since higher returns can be realized if diversification flexibility over time opens up new opportunities.
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1.42 1.437 1.46 1.453 1.477 1.5 1.469 1.493 1.517 1.541 1.486 1.509 1.533 1.557 1.581 1.486 1.526 1.549 1.573 1.597 1.621 1.486 1.538 1.566 1.59 1.613 1.637 1.661 1.485 1.54 1.582 1.606 1.63 1.653
1.349 1.367 1.383 1.374 1.4 1.402 1.375 1.407 1.419 1.415 1.372 1.408 1.426 1.431 1.423 1.365 1.406 1.428 1.439 1.437 1.429 1.357 1.401 1.426 1.442 1.448 1.442 1.431 1.348 1.393 1.423 1.442 1.45 1.454
1.337 1.354 1.37 1.359 1.386 1.386 1.358 1.393 1.402 1.395 1.353 1.392 1.409 1.41 1.399 1.346 1.387 1.409 1.417 1.414 1.401 1.337 1.379 1.406 1.419 1.422 1.416 1.401 1.327 1.37 1.399 1.417 1.425 1.424
0.895 0.81 0.797 0.714 0.711 0.71 0.63 0.633 0.625 0.623 0.532 0.542 0.542 0.535 0.544 0.433 0.437 0.447 0.456 0.451 0.457 0.347 0.337 0.343 0.37 0.375 0.372 0.372 0.25 0.248 0.255 0.276 0.287 0.288
0 0 0.1 0 0.1 0.2 0 0.1 0.2 0.3 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5
1 0.9 0.9 0.8 0.8 0.8 0.7 0.7 0.7 0.7 0.6 0.6 0.6 0.6 0.6 0.5 0.5 0.5 0.5 0.5 0.5 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.3 0.3 0.3 0.3 0.3 0.3
0 0.1 0 0.2 0.1 0 0.3 0.2 0.1 0 0.4 0.3 0.2 0.1 0 0.5 0.4 0.3 0.2 0.1 0 0.6 0.5 0.4 0.3 0.2 0.1 0 0.7 0.6 0.5 0.4 0.3 0.2
Coal at 0
SP0 SP5 SP10 Return VaR CVaR 0.105 0.09 0.103 0.086 0.089 0.09 0.07 0.067 0.075 0.077 0.068 0.058 0.058 0.065 0.056 0.067 0.063 0.053 0.044 0.049 0.043 0.053 0.063 0.057 0.03 0.025 0.028 0.028 0.05 0.052 0.045 0.024 0.013 0.012
Bio at 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.059 0 0 0 0 0 0.118 0.014 0 0 0 0 0 0.181 0.065 0 0 0 0
Coal at 5 0 0.1 0 0.2 0.1 0 0.3 0.2 0.1 0 0.4 0.3 0.2 0.1 0 0.441 0.4 0.3 0.2 0.1 0 0.482 0.486 0.4 0.3 0.2 0.1 0 0.519 0.535 0.5 0.4 0.3 0.2
Bio at 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0.1 0 0.1 0.2 0 0.1 0.2 0.3 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5
1.58 1.579 1.627 1.577 1.626 1.674 1.575 1.624 1.673 1.721 1.573 1.622 1.671 1.72 1.768 1.571 1.62 1.669 1.718 1.767 1.815 1.569 1.618 1.667 1.716 1.765 1.814 1.862 1.567 1.616 1.665 1.714 1.763 1.812
1.5 1.505 1.536 1.506 1.54 1.551 1.503 1.539 1.553 1.557 1.497 1.536 1.551 1.559 1.559 1.488 1.529 1.548 1.557 1.56 1.561 1.477 1.521 1.543 1.554 1.558 1.561 1.562 1.465 1.51 1.534 1.549 1.555 1.558
1.486 1.492 1.521 1.493 1.526 1.532 1.49 1.525 1.535 1.534 1.484 1.521 1.534 1.535 1.531 1.474 1.514 1.529 1.534 1.532 1.526 1.463 1.505 1.523 1.53 1.53 1.526 1.52 1.449 1.493 1.514 1.523 1.526 1.524
Coal Bio Return –VaR –CVaR at 10 at 10
Table AI. Dynamic vs. static portfolio results.
0.979 0.881 0.881 0.783 0.783 0.783 0.685 0.685 0.685 0.685 0.587 0.587 0.587 0.587 0.587 0.49 0.49 0.49 0.49 0.49 0.4900 0.392 0.392 0.392 0.392 0.392 0.392 0.392 0.294 0.294 0.294 0.294 0.294 0.294
Coal at 0 0.021 0.019 0.019 0.017 0.017 0.017 0.015 0.015 0.015 0.015 0.013 0.013 0.013 0.013 0.013 0.01 0.01 0.01 0.01 0.01 0.01 0.008 0.008 0.008 0.008 0.008 0.008 0.008 0.006 0.006 0.006 0.006 0.006 0.006
Bio at 0 0 0.065 0 0.13 0.065 0 0.196 0.13 0.065 0 0.261 0.196 0.13 0.065 0 0.326 0.261 0.196 0.13 0.065 0 0.391 0. 326 0.261 0.196 0.13 0.065 0 0.456 0. 351 0. 326 0.261 0.196 0.13
Coal at 5 0 0.035 0 0.07 0.035 0 0.104 0.07 0.035 0 0.139 0.104 0.07 0.035 0 0.174 0.139 0.104 0.07 0.035 0 0.209 0.174 0.139 0.104 0.07 0.035 0 0.244 0.209 0.174 0.139 0.104 0.07
Bio at 5
Copyright # 2010 John Wiley & Sons, Ltd.
0 0 0.1 0 0.1 0.2 0 0.1 0.2 0.3 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0.500 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5
Bio at 10
(Continues)
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Coal at 10
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Euro. Trans. Electr. Power 2011;21:1825–1841 DOI: 10.1002/etep
Copyright # 2010 John Wiley & Sons, Ltd.
1.447 1.434 1.338 1.383 1.418 1.44 1.452 1.458 1.458 1.451 1.437 1.328 1.373 1.41 1.436 1.452 1.461 1.466 1.462 1.454 1.441 1.316 1.364 1.4 1.428 1.45 1.462 1.469 1.471 1.466 1.457 1.444
1.416 1.4 1.315 1.359 1.39 1.412 1.424 1.428 1.425 1.415 1.399 1.303 1.348 1.38 1.405 1.42 1.428 1.43 1.425 1.414 1.397 1.29 1.336 1.369 1.396 1.415 1.425 1.43 1.43 1.424 1.412 1.394
0.01 0.01 0.036 0.031 0.027 0.035 0 0.002 0 0 0 0.029 0.016 0.022 0.018 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Bio at 0 0 0 0.214 0.107 0 0 0 0 0 0 0 0.255 0.147 0.019 0 0 0 0 0 0 0 0.288 0.177 0.057 0 0 0 0 0 0 0 0
Coal at 5 0.1 0 0.586 0.593 0.6 0.5 0.4 0.3 0.2 0.1 0 0.645 0.653 0.681 0.6 0.5 0.4 0.3 0.2 0.1 0 0.712 0.723 0.743 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
Bio at 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0.6 0.7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1.861 1.909 1.566 1.614 1.663 1.712 1.761 1.81 1.859 1.908 1.956 1.564 1.613 1.661 1.71 1.759 1.808 1.857 1.906 1.955 2.003 1.562 1.611 1.66 1.708 1.757 1.806 1.855 1.904 1.953 2.002 2.051
1.56 1.559 1.451 1.498 1.524 1.54 1.551 1.555 1.558 1.558 1.558 1.437 1.486 1.513 1.532 1.545 1.551 1.554 1.556 1.556 1.556 1.421 1.472 1.502 1.521 1.535 1.544 1.551 1.552 1.554 1.555 1.552
1.519 1.512 1.434 1.479 1.503 1.515 1.52 1.52 1.516 1.511 1.504 1.417 1.464 1.49 1.505 1.512 1.514 1.512 1.508 1.502 1.495 1.4 1.448 1.476 1.493 1.502 1.507 1.507 1.504 1.499 1.493 1.486
Coal Bio Return –VaR –CVaR at 10 at 10
Note: Columns 4–12 correspond to the dynamic framework and columns 13–21 to the static one.
1.677 1.701 1.493 1.545 1.598 1.622 1.646 1.67 1.694 1.717 1.741 1.498 1.551 1.609 1.638 1.662 1.686 1.71 1.734 1.757 1.781 1.506 1.559 1.616 1.655 1.678 1.702 1.726 1.75 1.774 1.797 1.821
0.29 0.29 0.164 0.169 0.173 0.165 0.2 0.198 0.2 0.2 0.2 0.071 0.084 0.078 0.082 0.1 0.1 0.1 0.1 0.1 0.1 0 0 0 0 0 0 0 0 0 0 0
0.6 0.7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.3 0.3 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0 0 0 0 0 0 0 0 0 0 0
0.1 0 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
Coal at 0
SP0 SP5 SP10 Return VaR CVaR
Table AI. (Continued)
0.294 0.294 0.196 0.196 0.196 0.196 0.196 0.196 0.196 0.196 0.196 0.098 0.098 0.098 0.098 0.098 0.098 0.098 0.098 0.098 0.098 0 0 0 0 0 0 0 0 0 0 0
Coal at 0 0.006 0.006 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0 0 0 0 0 0 0 0 0 0 0
Bio at 0 0.065 0 0.521 0.456 0. 391 0.326 0.261 0.196 0.13 0.065 0 0.587 0.521 0.456 0. 391 0. 326 0.261 0.196 0.13 0.065 0 0.652 0.587 0.521 0.456 0.391 0.326 0.261 0.196 0.13 0.065 0
Coal at 5 0.035 0 0.279 0.244 0.209 0.174 0.139 0.104 0.07 0.035 0 0.313 0.279 0.244 0.209 0.174 0.139 0.104 0.07 0.035 0 0.348 0.313 0.279 0.244 0.209 0.174 0.139 0.104 0.07 0.035 0
Bio at 5
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Coal at 10
0.6 0.7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Bio at 10
A DYNAMIC CVaR-PORTFOLIO APPROACH
1841
Euro. Trans. Electr. Power 2011;21:1825–1841 DOI: 10.1002/etep
