Valuation of wind energy projects: A real options approach

We address the valuation of an operating wind farm and the finite-lived option to invest in it under different reward/support schemes: a constant feed-in tariff, a premium on top of the electricity market price (either a fixed premium or a variable subsidy such as a renewable obligation certificate or ROC), and a transitory subsidy, among others. Futures contracts on electricity with ever longer maturities enable market-based valuations to be undertaken. The model considers up to three sources of uncertainty: the electricity price, the level of wind generation, and the certificate (ROC) price where appropriate. When analytical solutions are lacking, we resort to a trinomial lattice combined with Monte Carlo simulation; we also use a two-dimensional binomial lattice when uncertainty in the ROC price is considered. Our data set refers to the UK. The numerical results show the impact of several factors involved in the decision to invest: the subsidy per MWh generated, the initial lump-sum subsidy, the maturity of the investment option, and electricity price volatility. Different combinations of variables can help bring forward investments in wind generation. One-off policies, e.g., a transitory initial subsidy, seem to have a stronger effect than a fixed premium per MWh produced.


Introduction
Public support to renewable energies is usually justi…ed on three grounds: climate change, security of supply, and industrial policy. Some of the positive e¤ects from renewables' development are global, e.g. the abatement of greenhouse gas emissions, and the reduction of investment unit costs (because of the learning e¤ect). Impacts from enhanced energy security and industrial policy, instead, are derived at the national level.
Renewable sources are getting ever more relevant in the generation of electric energy. Major drivers are the decreasing costs of renewable technologies and strong support from government agencies. This trend is expected to continue in the years ahead (European Commission [4]). Pérez-Arriaga and Batlle [13] analyze the impact of a strong penetration of renewable, intermittent generation on the planning, operation, and control of power systems. See also EWEA [5] and NREL [10].
Within this set of technologies wind stands apart, with solar photovoltaic (PV) and concentrated solar power (CSP) somewhat behind. The increasing role of these intermittent generation technologies gives rise to important challenges in the operation of the electric system. Regarding solar energy, it is more predictable than wind over short periods of time. It also displays a diurnal seasonality which overlaps with the hours of strongest load thus coinciding with the times of highest prices. This suggests that the prices at the times of strongest operation of solar plants will approach peak prices.
One of the problems a-icting wind energy is certainly intermittence. However, this problem is less acute when dealing with a large balancing area since the behavior of wind correlates less than perfectly across all the sites in the area (provided there is enough transmission capacity). Further deployment of renewable energies (wind in particular) would also bene…t signi…cantly from greater storage capacities. A minor concern is that wind energy is not quite carbon free. 1 Large-scale deployment of turbines can also disrupt local wildlife and fauna, a¤ect local temperature and even global weather. These negative impacts are hard to quantify but this does not render them less real.
Despite these shortcomings the fact remains that in principle the potential of wind goes well beyond global needs. Marvel et al. [9] use a climate model to estimate the amount of power that can be extracted from both surface and highaltitude winds, considering only geophysical limits. According to their results, surface wind turbines alone could extract kinetic energy at a rate of at least 400 terawatts (TW, one trillion watts) while the level of present global primary power demand approaches 18 TW. On the other hand, Jacobson and Archer [6] de…ne the saturation wind power potential as the maximum wind power that can be extracted upon increasing the number of wind turbines over a large geographic region, independent of societal, environmental, climatic, or economic considerations. This saturation potential is over 250 TW at 100 meters up globally (100 m above ground is the hub height of most modern wind turbines), assuming conventional wind turbines distributed everywhere on Earth.
Empirical evidence shows, though, that actual deployment of this technology is well below that potential. Several barriers (whether economic, social, or other type) are probably playing a role in hampering adoption across the globe. Regarding economic barriers, casual observation allows to identify a number of support schemes which are presumably aimed at providing greater certainty to potential investors in this technology; see Klessmann et al. [7]. In other words, uncertain returns on these investments are considered a major cause for concern both to developers and investors alike (alongside others like electricity grid-and market-related barriers).
Actual support programs typically rely on a combination of di¤erent measures such as special tax regimes, cash grants, or …nancial incentives; an overview can be found in Daim et al. [3] and Snyder and Kaiser [15]. So-called Renewable Energy Feed-in Tari¤s (REFIT) are a guaranteed payment to generators of renewable electricity (say 90 e/MWh) over a certain period of time (e.g. 20 years). This instrument is typical in several EU countries, among them Germany. Spain allows similarly this remuneration option. Nonetheless, wind power generators seem to prefer the alternative option, namely a premium on top of the electricity market price. The UK instead incentivizes renewable electricity through the use of renewable energy credits (the Renewables Obligation Certi…cates, or ROCs) which are further traded in their speci…c market. EU nations also grant some tax exemptions (for instance, from carbon taxes) and subsidies (to capital expenditure). In the US there is a production tax credit at the federal level. The fact that it has expired three times over the last ten years is not reassuring, however. A number of States have set renewable portfolio standards whereby a certain fraction of the State's electricity must come from renewable sources. Some States also take part in a regional greenhouse gas initiative, a cap-and-trade market for carbon. Regarding subsidies, they are both lower and less certain than those in Europe.
A suitable valuation approach for wind projects must not only account for intermittence and uncertainty. It must also take account of their irreversible character and the ‡exibility enjoyed by project managers (e.g. the option to delay investment). Under these circumstances, traditional valuation techniques based on discounted cash ‡ows have been found inferior to contingent claims or real options analysis.
Following the latter approach, Boomsma et al. [1] assess both the time and the size of the investment in renewable energy projects under di¤erent support schemes. They consider up to three sources of uncertainty: steel price, electricity price, and subsidy payment, all of which are assumed to follow uncorrelated geometric Brownian motions (with the last one modulated by Markov switching). For illustration purposes, they focus on a Norwegian case study. According to their results, a …xed feed-in tari¤ encourages earlier investment than renewable energy certi…cates. The latter, though, create incentives for larger projects.
Reuter et al. [14] instead pick Germany as a case study. In their model the electric utility decides whether to add new generation capacity or not once a year over the planning horizon. The new capacity can be either a fossil fuel power plant (with a constant load factor) or a wind power plant (with a normally distributed load factor), both equally sized. The yearly electricity price is subject to (normally distributed) exogenous shocks (assumed independent from wind load factor). The third source of uncertainty concerns climate policy; it is represented by the feed-in tari¤ which is a Markov chain with two possible values and a given transmission matrix. This risk factor is also assumed independent from the other two. Their results stress the importance of explicitly modeling the variability of renewable loads owing to their impact on pro…t distributions and the value of the …rm. Besides, greater uncertainty about the future behavior of the feed-in tari¤ requires much higher trigger tari¤s for which renewable investments become attractive (i.e. equally pro…table as a coal-…red station of equal capacity).
Here we address the present value of an investment in a wind park and the optimal time to invest under di¤erent payment settings: (a) A …xed feed-in tari¤ for renewable electricity over 20 years of useful life. (b) Electricity price as determined by the market. (c) A combination of the market price and a constant premium. (d) A transitory subsidy available only at the initial time. We also develop sensitivity analyses with respect to changes in the investment option's maturity and electricity price volatility.
Our paper di¤ers from others in several respects. We consider two sources of uncertainty. We assume more general stochastic processes for the state variables; in particular, we account for mean reversion in commodity prices (this …ts better the sample data as shown in Appendix 1). We develop a trinomial lattice that supports this behavior. We also make room for seasonal behavior in the price of electricity and in wind load factor. Indeed, they turn out to be correlated to some degree, which has been typically overlooked despite its impact on project value. The underlying dynamics in the price of electricity is estimated from observed futures contracts with the longest maturities available (namely, up to …ve years into the future); this includes the market price of electricity price risk. The dynamics of wind load factor is also estimated from actual (monthly) time series alongside seasonality. The riskless interest rate is also taken from (…nancial) markets. Both the project's life and the option's maturity are …nite; in our simulations below the size of the time step is not t = 1 (or one step per year), but a much shorter t = 1/60 (…ve steps per month). In addition to a …xed feed-in tari¤ and a premium over electricity price, another support scheme that we consider is an investment subsidy that is only available at the initial time but is foregone otherwise. We further provide numerical estimates of the trigger investment cost below which it is optimal to invest immediately.
The paper is organized as follows. First we introduce the stochastic processes for electricity price and wind load factor. Next we estimate these processes with sample data from the UK. Valuation is then undertaken under two scenarios. The …rst one adopts a now-or-never perspective. This is the setting where the traditional Net Present Value rule applies. Numerical solutions are derived from exact formulas when possible but also from Monte Carlo simulation. The second scenario allows for optimally choosing the time to invest. In our case this is accomplished by means of a trinomial lattice which supports mean reversion. Several cases and sensitivity analyses are then addressed. A number of them involve running whole simulations of electricity price and load factor at each node in the lattice. We thus combine two numerical methods that are frequently used in isolation. A section with the main results concludes.

Electricity price
We specify the long-term price of electricity in a risk-neutral world as a mean reverting stochastic process governed by the following di¤erential equation: or, rearranging, E t is the time-t price of electricity while E m is the level to which the deseasonalized price tends in the long run. f (t) is a deterministic function that captures the e¤ect of seasonality in electricity prices. This function is de…ned as f (t) = cos(2 (t + ')), where the time t is measured in years and the angle in radians; when t = ' we have f (t)= and the seasonal maximum value is reached. k E is the speed of reversion towards the "normal"level E m . It can be computed as k E = ln 2=t E 1=2 , where t E 1=2 is the expected half-life, i.e. the time required for the gap between E 0 f (0) and E m to halve. E is the instantaneous volatility of electricity price changes; it determines the variance of E t at t. And dW E t is the increment to a standard Wiener process; it is normally distributed with mean zero and variance dt. Last, E E t is the market price of electricity price risk.
The mathematical expectation (under the risk-neutral probability measure Q) at time t 0 , or equivalently the futures price with maturity t, is: For a time arbitrarily far into the future (t ! 1) Thus, (deseasonalized) electricity price in the long run is anticipated to reach the long-term equilibrium level.

Wind electricity
Reuter et al. [14] consider wind stations and address the impact of uncertainty in the load factor on their pro…ts. As expected, the distribution of yearly pro…ts is more variable than under a constant load factor (equal to the long-term average).
In addition, the expected pro…t is smaller under a changing load factor. They …nd similar evidence when analyzing the value of the …rm. Thus assuming a constant load factor leads to overestimating this technology's pro…tability.
Intermittence per se drives a sizeable wedge between installed capacity and metered electricity; it can be measured through the load factor, W . We explicitly recognize the uncertain character of wind energy. All the interruptions (whatever their reasons) are modeled through the stochastic behavior of the load factor. The theoretical model assumed is: Generation from wind stations shows a seasonal pattern. Our simulations below assume this behavior in wind electricity, g(t), so the seasonality in the load factor must be previously identi…ed (from historical time series). W t evolves around a long-run average value W m . And dW W t is the increment to a standard Wiener process; it is normally distributed with mean zero and variance dt.

ROC price
The UK government has committed to meeting a legally binding EU target of generating 15% of energy from renewable sources by the year 2020. The main …nancial mechanism for supporting large-scale renewable generation is the Renewables Obligation (RO); it is a market-based mechanism similar to a renewable portfolio standard. The RO places a mandatory requirement on electricity suppliers to source a proportion of electricity from renewable sources. The RO level increases every year (each beginning on April 1 and running to March 30); it started in 2002/03 at 3% and will reach 15.8% in 2012-13. Support is granted for 20 years. In April 2010, the RO end date was extended from 2027 to 2037 for new projects.
Renewables Obligation Certi…cates (ROCs) are issued by the regulator Ofgem to accredited renewable generators. They are issued into the ROC Register along with electronic certi…cates. ROCs store details of how electricity was generated, who generated it, and who eventually used it. Regarding the demand for ROCs, it is created by the above quota obligations for electricity suppliers (which are the same at any given time for all of them).
For each MWh of green electricity that a utility generates it receives one ROC. 2 If the utility generates more ROCs than needed to meet its obligation then it can sell the spare ROCs to other suppliers who are struggling to meet theirs via purchase agreements (it can also sell them at the quarterly auctions organized by the Non-Fossil Purchasing Agency, or through a broker for a fee). This allows them to receive a premium in addition to the wholesale electricity price (in addition to being exempted from the climate change levy on fossil fuels for industry; Toke [16]). During the period 2002-11 the proportion of the RO met by ROCs has ranged between 56% and 76%. Those suppliers who do not have enough ROCs to cover their obligation must incur a payment into the buyout fund. This penalty or 'buy-out price'is set every year in the same order that establishes the RO level; it is a …xed price per MWh shortfall at which Ofgem purchase ROCs (it is adjusted every year in line with the Retail Price Index). The buy-out price has passed from 30 £ /MWh in 2002-03 to 40.71 £ /MWh in 2012-113. The cash in ‡ows to the buy-out fund are later aggregated annually and recycled as an extra reward on a pro-rata basis to electricity suppliers who surrendered ROCs. The total buy-out fund thus redistributed has gone from 90 M£ in 2002-03 to 357 M£ in 2010-11. The nominal ROC price naturally tends to rise or fall depending on the size of the buy-out fund (or the percentage of the RO target that is actually met through ROCs). In sum, the ROC's value to the utility equals the buy-out price (the penalty that it avoids) plus the recycling payment (that it entitles to). Its price has moved between 42.5 £ /MWh and 54.5 £ /MWh.
We propose the following continuous-time stochastic model for the ROC price: We are implicitly assuming that the electricity price, the wind load factor and the ROC price are uncorrelated. Based on past data one can get a numerical estimate of the above parameters fR m ; R g. Later on they can be used to simulate random paths over a number of periods.

Electricity price process
We have 26,057 prices of monthly UK Base Electricity Futures from the Intercontinental Exchange (ICE, London). The sample period goes from 12/01/2009 to 03/30/2012 thus comprising 604 trading days; see Table 1. The number of traded contracts on the last day of the sample is 59, i.e. we use futures contracts with maturities up to …ve years from now (thus they are long-term futures prices instead of short-term forward prices or day-ahead prices). These prices for successive months are assumed to re ‡ect all the information available to the market about generation costs and pro…t margins of power plants. In particular, they take account of fuel prices, allowance prices, decommissioning of old plants, new starts, etc. reviewing.
The stochastic model in discrete time is: We estimate the parameters underlying this model using all the futures prices on each day by non-linear least-squares.  Figure 1 displays the futures prices actually observed on the last day of the sample (03/30/2012) along with those implied by our numerical estimates using all the contracts traded every day. We can estimate the spot electricity price for day t 0 from the futures contract with the nearest maturity using equation (2): The seasonally adjusted spot price is: E t0 f (t 0 ). Thus we compute a spot price for every day. They behave more smoothly (or are less bumpy) than actual futures prices. Using the di¤erential equation describing price behavior in the physical (as opposed to risk neutral) world we get: Discretizing this formula we derive a regression model whose residuals allow us to compute their volatility: E = 0:255045: On the other hand, the risk-free interest rate considered is r = 2.05 %, which corresponds to the 10-year UK government debt in January 2012.

Wind electricity: load factor, seasonality, and drift rate
In discrete time we have: We are implicitly assuming that beyond (deterministic) seasonality the electricity price and the wind load factor are uncorrelated; in other words, they can be correlated but only through their seasonal patterns. Based on past (say, monthly) data one can get a numerical estimate of the above parameters fg(t); W m ; W g. Later on they can be used to simulate random paths over a number of periods. The sample comprises the monthly ratios between output electricity and installed capacity for the whole UK from April 2006 to December 2010, i.e. 57 observations. 3 As a …rst step the seasonal component is taken out of the original series. Estimation then proceeds on the deseasonalized series. The estimate of the average value is c W m = 24.0899 %; and that of the standard deviation is The results for the (dummy) monthly variables appear in Table  3. 4 They are depicted in Figure 2.

The joint e¤ect of seasonalities in electricity price and wind generation
As Table 3 suggests, the periods with (statistically) highest wind generation fall in January and November. The highest prices of electricity are reached between October and March; thus there is some overlapping. This time coincidence allows UK farms' owners to get a greater pro…tability from wind generation. Other papers overlook this feature 5 yet our model takes it into account.

The ROC price
The behavior of the ROC price in discrete time is described by: R m = 46.58 £ /MWh, and the annualized volatility is R =0.2882. 3 The maximum possible output for each month is calculated from the installed capacity of the wind farm: Maximum output (MWh) = Installed capacity (MW) * number of days * 24. The actual output is then expressed as a percentage of the maximum possible output over the same time interval. Source: CLOWD [2]. 4 This value of c Wm is slightly higher than the average of 23 % cited for Germany by Reuter et al. [14]. The dummy variables here do not display a symmetrical behavior; this is in contradiction with their assumption of a normal distribution. 5 This is the case, for example, when a constant annual capacity factor is chosen, say 35%. 6 Source: http://www.e-roc.co.uk/trackrecord.htm 4 Valuation in a now-or-never setting: Monte Carlo simulation Uncorrelated random variables are generated according to the following discretetime schemes (see equations (2) and (3)): Note that in both cases we start from known values, e.g. E t f (t) or g(t), and then add a random component t . Let us consider a wind farm with installed capacity C = 50 MW (think of a set comprising 25 turbines each 2 MW). The average load factor is c W m = 24.0899 % (see Table 3). Seasonality comes on top of this. Thus the expected availability in January would be c W m + b d 1 = 24.0899 + 8.7442 = 32.8341 %; or, in absolute terms 50 24 31 0:328341 = 12,214.29 MWh. 7 In general, wind generation over a time period t amounts to: Now, if there is a …xed feed-in tari¤ p in place then the present value of production in that period is computed as: If, instead, the farm owner receives as a payment the market price E t then the present value of the revenues is given by: Note that our simulations below are based on a risk-neutral drift. Consequently future cash ‡ows can be discounted at the risk-free rate r. Each simulation run s (with s = 1; :::; m) comprises a number of time steps denoted by j (with j = 1; :::; n). We denote the value of the wind park at any step by V sj . These values are aggregated over the n steps to derive the value under simulation s, denoted V s . Then we compute the average value over all the m simulations: We undertake m = 1,000 simulation runs, and consider a useful time of 20 years. The step size is t = 1/60, thus each simulation comprises n = 1,200 steps.

A constant feed-in tari¤
The feed-in tari¤ is generally claimed to be the most e¤ective method for promoting renewable energy. Let p denote the tari¤ applied to the electricity generated. According to Klessmann et al.  Table 4 shows the present value for a range of potential tari¤s when the riskless interest rate is r = 0:0205. The second column is directly computed from the above exact formula. These sums of money are to be set against the investment cost and the present value of …xed costs. 9 For consistency with next sections, the numerical estimates of the parameters in wind load factor fg(t); W m ; W g are also used here to simulate random paths, month after month, over a number of years. The third column in Table 4 comes from this Monte Carlo approach. It results from running 1,000 simulations each comprising 1,200 time steps (i.e. …ve steps per month) and then taking the average value. The amounts resemble pretty much those in the second column.

The electricity market price
Assume that the unit payment to the owner of the wind park strictly amounts to the market price of electricity; this can be thought of as the case of a generator who is ineligible for renewable energy support (or the feed-in tari¤ suddenly ceases to apply). In this case we resort to simulation in order to take account of the situations in which high electricity prices (due to strong demand) coincide with high wind generation (owing to seasonal weather). Discretization of equation (1) and equation (3) yields: p tu E;t ; We use the parameter values in Table 2 for generating electricity price paths. The (average) present value turns out to be: With a total investment cost I = 66,000,000 £ (see LGA [8]) the net present value amounts to V I = 56,196,833 £ .
For a now-or-never investment this present value V is equivalent to a …xed feed-in tari¤ of 70.52 £ /MWh. Note in Table 4 that, for p = 70, the corresponding values are slightly lower than present value stated here V = 122,196,833 £ . So a small increase in the level of p su¢ ces to reach that …gure.

The market price plus a …xed premium
Here we assume that the farm owner gets a payment that is composed of the electricity price plus an extra premium for each megawatt-hour generated. For example, onshore wind projects in Spain received a premium of 29.29 e/MWh in 2007 over their …rst 20 years of operation; Klessmann et al. [7]. There were, nonetheless, upper and lower limits to the total level of the market price plus the premium: 84.94 e/MWh and 71.27 e/MWh, respectively. 10 Again we run 1,000 simulations with 1,200 steps. Table 5 displays the results.
Each amount in the second column consists of two parts. The …rst one comes from MC simulation, namely V = 122,196,833. The second is derived as in Table 4; thus, with p = 50 we get some 86.6 M£ , so with a fraction 0.1 of that p we would get 10 % of that amount, or 8.66 M£ . In sum, for a price premium of 5 £ /MWh we derive V = 130,860,523 £ . Similarly for other premium levels.

The market price plus the ROC price
Here we assume that the developer of the wind park receives a total payment comprising the electricity price plus the ROC price for each megawatt-hour generated. Using the above estimates ( b R m = 46.58 £ /MWh, R =0.2882), the present value of the revenues amounts to V = £ 202,962,706.
We could assume, instead, that the ROC price decreases exponentially with time: In this case, with = 0:10 we would get a present value V = £ 159,325,987.

Valuation and investment timing: Trinomial lattice with mean reversion
The investment time horizon T is subdivided in n steps, each of size t = T =n.
Starting from an initial electricity price E 0 , in a trinomial lattice one of three possibilities will take place: either the price jumps up (by a factor u to E + ), remains the same (E = ), or jumps down (by a factor d to E ). At time i, after j positive increments, the price is given by Consider an asset whose risk-neutral, seasonally-adjusted behavior follows the di¤erential equation: This can also be written as: Since it is usually easier to work with the processes for the natural logarithms of asset prices, we carry out the following transformation: X = ln E. Thus X E = 1=E, X EE = 1=E 2 , and X t = 0. By Ito's Lemma: where E k E (Em Et) Et E 1 2 2 E depends at each moment on the asset price E t (so strict notation would read E (t)). See Appendix 2 in the Electronic Supplemental Material for further details on this lattice.
In a trinomial lattice, there are three probabilities p u , p m , and p d associated with a rise, maintenance, and a fall in the (seasonally adjusted) price of electricity. In comparison to a binomial lattice, we can choose the size of the time step t so as to avoid negative probabilities. If, despite this choice, they do appear then we adopt the formulae in Table 6. Now, at the end of the investment horizon (time T ) the value of the investment option in each of the …nal nodes is given by the maximum of two quantities, namely the value of an immediate investment (which presumes that we have not invested yet) and zero. As before, the present value of investing immediately is determined through MC simulation. This means that we run 1,000 simulations of 1,200 steps at each …nal node. Since the option to invest is akin to a "call" option we denote its value by C: At earlier times, however, the option to invest is worth the maximum of two other values: that of investing immediately and that of waiting to invest for one more period (thus keeping the option alive): V (i; j) is derived by simulation at each node. Here the symbols +, =, and stand for a rise, no change, and a fall in the price of the asset.

Valuation of the option to invest: Case studies
All the cases that follow rest on the same starting values of the underlying variables; see Table 2. We assume that the investment option expires 10 years from now. When building the lattice we take a time step t = 1/4. From Section 3.1 the price change volatility is E = 0.255045.

The electricity market price
Let I denote the present value of all the costs (…xed and variable) incurred by the investment owner over the whole useful life of the wind farm. Table 7 shows the value of investing immediately (NPV) alongside that of investing at the optimal time. The former can take on negative values (it decreases linearly as I increases), while the latter is bounded from below at zero. As usual, the value of the option to invest is the maximum of both amounts (bottom row).
For low investment costs (I = 75 and I = 100) the net present value of the immediate investment is positive (N P V > 0). Indeed it remains positive as long as I 122.2 M£ . Therefore, if there is no option to wait the right decision is to rush for the investment provided I does not surpass that threshold. Yet, if the investment can be delayed, investing immediately is far from optimal. For all the investment costs considered in the table, waiting for the optimal time to invest increases the value of the project. In fact, as suggested by the last two columns (I = 125 and I = 150), the value of waiting can be so high as to turn an otherwise uninteresting project (N P V < 0) into an attractive one. Of course, I might rise so high that it renders the option to invest worthless. And conversely, it could be so low that the NPV is higher than the continuation value in which case delay makes no sense. We can resort to continuity arguments and claim that there is some threshold or "trigger" investment cost I below which immediate investment becomes optimal. Clearly this I is lower than any of the values considered in Table 7. Hence there seems to be little point in investing immediately.

A constant feed-in-tari¤
Sometimes policy makers grant di¤erent subsidies to developers of renewable energy (e.g. to help pay for the capital costs of o¤shore wind farms). They are meant to enhance the appeal of investments which in their absence would not seem to pay o¤. The impact of these measures depends on their speci…c terms and the institutional environment in place.
In the case of a feed-in-tari¤ that is kept constant over time, the argument is straightforward: all the relevant information is available at the very outset. The (gross) present values in Table 4 outweigh the continuation value. As a consequence, investment seems to be certainly brought forward by this support measure.

An initial, transitory subsidy
Now we check how the decision to invest reacts to a public subsidy S ranging from 5 M£ to 20 M£ which is only available at the initial time; in other words, if the decision maker opts for postponing the investment the subsidy is foregone. Speci…cally, we look for the threshold I that triggers immediate investment under di¤erent values of S. Table 8 displays the numerical results. A subsidy S = 10 M£ prompts the option holder to invest immediately whenever the investment cost falls below 61.9 M£ . Note, though, that this would not be the case if the subsidy were available at any time over the whole 10-year investment horizon; though not shown in Table 7, even for values of I as low as 25 M£ it is better to wait. Thus, a one-time initial subsidy seems to be more e¤ective at prompting investment than higher subsidies that are available for long periods.

The market price plus a …xed premium
Consider the case in which the owner of the wind farm receives the market price of electricity augmented by a …xed premium. Table 9 shows the value of the option to invest for two di¤erent levels, namely 10 £ /MWh and 20 £ /MWh.
For I = 75, the presence of a premium raises the value of investing immediately in 64.5 -47.2 = 17.3 M£ (p = 10 £ /MWh) and 81.8 -47.2 = 34.6 M£ (p = 20 £ /MWh), respectively. However, this does not lead to bringing forward the decision to invest in the wind farm since the continuation value is higher in both cases. In other words, a subsidy granted over the whole useful life with a present value of 17.3 M£ (i.e. almost 25 % of the total disbursement I) falls short of triggering the immediate investment when I = 75 M£ . Note, though, that with a subsidy S = 15 M£ which is available only initially the trigger investment cost I goes as high as 98.4 M£ (see Table 8). Thus we infer that a subsidy per generated MWh which is spread over the farm's life (20 years) and is available up to the investment option's maturity (10 years) is less e¤ective than a subsidy at t = 0 available only at the initial time.

The market price plus the ROC price
Last, consider that the developer receives the electricity price plus the ROC price .The trigger investment cost would then be I = £ 115,703,607. This threshold is to be compared with prior levels that triggered the option to invest. For example, Table 8 shows the value of I for di¤erent subsidies S. Since I = 115.7 M£ falls in the range [98. 1, 122.4] we infer that this scheme is equivalent to receiving the market price of electricity augmented by a subsidy somewhere between S = 15 M£ and S = 20 M£ . As seen in the table, the additional increase of 5 M£ (from S = 15 to S = 20) raises the critical investment cost by 122.4 -98.1 = 24.3 M£ ; so, when it comes to pushing up I , a multiplier of almost 5 is in operation (at these levels of subsidy). In this sense, the ROC price seems to be an e¤ective support measure in stimulating investment. Note that delaying the investment entails foregoing cash revenues from ROCs.

Sensitivity to changes in the option' s maturity
Intuitively, if the investment option is available over a shorter time frame there is less to be gained from waiting to invest. As a consequence the continuation value will fall and investment will take place earlier. Table 10 shows the impact of a shorter maturity T on the option value for di¤erent levels of investment cost I.
As the time that the option is available shortens, the di¤erence between the continuation value (always positive) and the investment value falls. Consider, for example, I = 100. With T = 5 the di¤erence amounts to 31.8 -22.2 = 9.6 M£ . Instead, with T = 1 it drops to 24.6 -22.2 = 2.4 M£ .
A combination of short option maturities and transitory public subsidies only available at t = 0 can bring forward investments in wind energy. See Table  11, where expiration of the option is assumed to take place at T = 1. Initial subsidies of certain amount are very e¤ective in that they raise the investment threshold below which it is optimal to invest. Note, however, that the marginal e¤ect of each additional monetary unit decreases signi…cantly.
On the other hand, potential improvements in wind technology (with their ensuing drops in facilities'costs) would lead to delaying investments. Yet this e¤ect could be o¤set by other factors such as rising …nancial or personnel costs, or prior occupation of the best sites for wind farms.

Sensitivity to changes in electricity price volatility
Again we consider T = 10 and t = 1/4; price volatility in the base case is E = 0.255045. To the extent that investments in wind energy are highly irreversible the volatility of electricity prices can be anticipated to play a major role (when wind park developers receive the electricity price alone). Thus, a low volatility pushes in favor of deploying wind turbines while the opposite is true for high volatilities. Unless there are good reasons for assuming that future volatility will deviate signi…cantly from past volatility (e.g. owing to regulatory or structural changes), an initial assessment based on historical volatility seems reasonable. 11 1 1 Pérez-Arriaga [12] anticipates the volatility of marginal prices to increase in deregulated electricity markets with substantial penetration of renewables. Of course, volatility can also be caused by a number of reasons, many of them falling beyond the realm of policy makers. One such example is the price of natural gas in the international markets, as long as it serves sometimes as a reference for establishing the price of electricity (in conjunction with other factors like the emission allowance price in those countries where electric utilities are subject to carbon restrictions).
Regarding the numerical results in Table 12, whatever the value of I assumed, the NPV rises when volatility falls. Thus, for I = 100 the NPV goes from 22.2 ( E = 0:20) to 22.4 ( E = 0:10). This e¤ect, however, is not strong enough to o¤set the incentive to wait: the value of investing immediately falls short of the continuation value in all the cases considered. The reason is that the value of waiting is quite signi…cant. This being clear for E = 0:10 and E = 0:20, it is easy to anticipate the results with E = 0.255045 or even higher volatilities.
Unlike the NPV, the continuation value falls when volatility falls. Take, for example, I = 100; it goes from 36.3 ( E = 0:20) to 34.4 ( E = 0:10). This e¤ect becomes stronger as the investment cost increases. With I = 150, it drops from 5.1 ( E = 0:20) to 1.2 ( E = 0:10). By itself, this e¤ect would push for investing early; the problem is, when investment costs are that high the NPV is in the red.

Conclusions
We have developed a valuation model for investments in wind energy in deregulated electricity markets when there are futures markets with long maturities. The results are thus focused on developed electricity markets where short-and long-term transactions take place regularly and it is possible to reward wind generation through a 'pure' scheme (i.e. at market rates alone) or a 'mixed' scheme (with one or more subsidies).
Looking at the UK futures market we …nd that contracts on electricity display mean reversion; this in turn has some implications for the valuation model. We estimate the parameters underlying the stochastic behavior of prices (including the seasonal e¤ect) from actual market data. We have also estimated another stochastic model (with seasonality) for electricity wind generation at any time as a function of the availability of wind.
The option to invest in a wind farm can be exercised up to some point into the future; thus it is an American-type option. Maximizing its value calls for exercising it at the optimal time. To assess this option we have built a trinomial lattice which supports mean reversion in prices. A new feature (to our knowledge) here is that the values involved in the decision to invest at each node are derived from Monte Carlo simulations where stochastic realizations of electricity price and ROC price are combined with those of wind availability (and thus generation level) at any time. We derive optimal exercise (investment) rules in terms of threshold investment costs below which it is optimal to invest immediately.
Our numerical results show the impact of a number of factors involved in the decision to invest in a wind farm. Thus, when the only source of revenue to developers is the electricity market price, delaying investment seems to be in their best interest. A …xed feed-in tari¤, however, certainly brings investment forward. When it comes to a lump-sum subsidy, a one-time or transitory initial subsidy seems to perform better at prompting investment than a higher sub-sidy available for long. The one-time subsidy can also outperform a constant premium per MWh received over the project's life. Augmenting the electricity price with the ROC price is proven to be somehow equivalent to a lump-sum subsidy high enough to raise signi…cantly the trigger investment cost; thus the ROC price contributes e¤ectively to early investment.
The sensitivity analyses show that di¤erent combinations of variables can have an in ‡uence in bringing forward the investments in wind generation. One such example is a short maturity of the option to invest and an initial subsidy available only for limited time. Regarding electricity price volatility, this is clearly a major driver when developers only receive the electricity price. As volatility falls, the NPV rises while the continuation value falls. These two e¤ects push for early deployment; the problem is, under this setting the former always falls short of the latter so it is optimal to wait.      Table 11. Trigger investment cost I for di¤erent subsidies S (M£ ) with T = 1.