The optimal investment decision of a
subsidized PV energy project
-a re-al option -appro-ach of two uncert-ain v-ari-ables
Jelmer Mulder
s2582597
Abstract
1.
Introduction
The energy transition includes multiple aspects. This transition from a world based on fossil energy to a world based on renewable energy has many barriers to overcome. Institutions have an important role in this transition. In the Netherlands, the involvement on an institutional level aims to support the energy transition via co-evolution between public and private influences. The most salient influence of institutions on private participants is via finan-cial policies. In this context, an efficient policy to support this co-evolution is the feed-in tariff subsidy system (Couture and Gagnon, 2010). It encour-ages numerous participants to contribute to the energy transition. These participants consider these feed in tariff subsidies in their financial decisions regarding renewable energy projects. However, the political view on poli-cies regarding the energy transition can vary over time (Verbong and Geels, 2007). As such the financing considerations of renewable energy projects will vary over time. This variation of subsidy policies leads to uncertain financing considerations in the future.
investment decisions due to the uncertainty factors related to these invest-ments (McDonald and Siegel, 1986; Gazheli and van den Bergh, 2018; Putten and MacMillan, 2004).
Why are PV energy projects relevant to valuate with a real option ap-proach? In 2018, the contribution of PV energy projects to renewable en-ergy production in the Netherlands increased by 5 percent compared to 2017 (Centraal Bureau voor de Statistiek, 2019). This increase makes PV energy a salient factor in the energy transition for the coming years (Wustenhagen and Menichetti, 2012). This trend will increase investments in PV energy in the coming years. This increased interest and the ability to more accurately valuate renewable energy projects lead to the research topic discussed in this paper.
solar field investment?
The next section will provide a literature review of real option theory and PV energy developments. The third section explains the methodology used in this paper. The fourth and fifth section describe the model and the data that structures the analysis of the sixth section. The conclusion of this paper is outlined in the last section.
2.
Literature review
2.1.
Introduction into real option theory
To gain an understanding of real option theory, this paper first describes the relation between the traditional valuation methods and valuation with real option theory. In general one could argue that investments have three characteristics that are important to consider. Firstly, irreversibility, when an investment is made, there is no way back. Secondly, uncertainty, the outcome of the investment is not known. Thirdly, timing, the moment on which you invest is not set.
The conventional way to evaluate investments is by using the net present value (NPV) method. It is based on two concepts: calculate the value of the expected profits and the expenditures. Simplified, if the difference between both is positive, an investment should be made. However, this conventional evaluation of investments does not consider irreversibility and timing (Dixit and Pindyck, 1994). Nevertheless, the real option method is an addition to the NPV. The NPV method is a necessary input to determine the real option value (Trigeorgis, 1993).
could be explained because irreversibility, uncertainty and timing are not con-sidered. Real option theory could provide a better explanation why certain positive NPV projects are not exercised immediately in practice (Dixit and Pindyck, 1994). The variety in different decisions and options that a manager could potentially encounter is wide. The theory of real option methods es-sentially values these different decisions. It provides the manager with more information. Some examples of options discussed in literature are (Kozlova, 2017):
1. The option to start/stop operations, the option gives the opportunity to be flexible when demand or market conditions change.
2. The option to stage, this option helps to break down your investment decision in multiple stages.
3. The option to abandon, this option determines whether the investment currently running should be abandoned or not.
The type used in this paper is the option to wait with investing. An option to wait is an choice between either continuing the investment at the present moment or postponing the investment until a later date. Furthermore, an irreversible investment is in this case defined as effort or (financial) resources that no longer can be used for other purposes. Combining these concepts with the uncertainty of the investment outcome, the value of waiting becomes more evident. You have the option to choose between investing now or to wait and invest later which could provide you with more information and hence more value(McDonald and Siegel, 1986). The manager waits until the option value is smaller than the value to invest immediately.
salient works regarding analytic and dynamic programming of real options. They present analytic examples of multiple real option scenarios. McDon-ald and Siegel (1986) and Brennan, Schwartz, et al. (1985) describe analytic examples of the use of real options in timing decisions. Additionally, Murto and Nese (2002) give, analytic examples of real option valuation to support a choice between two different capital investments. Examples of different methods in the context of subsidy policy and renewable energy projects are presented by Zhang, Zhou, and Zhou (2016, 2014b); Torani, Rausser, and Zilberman (2016); Lin and Wesseh Jr (2013). They use a monte carlo sim-ulation, the lattice method, the analytic method and dynamic programming to determine numerical solutions respectively. To solve difficult real option decisions numerically the lattice and monte carlo simulation methods are convenient. This paper utilizes the trinomial tree (a lattice method) to nu-merically compute an option with two underlying variables. An advantage of the trinomial tree method is the easy approximation of the value of real op-tions that have the feature to be exercised continuously Boyle and P. (1988). This advantage is usefull in this research, because the manager should be able to invest continuously.
2.2.
Real option analysis of subsidy policies and two underlying
stochastic values
system by using a dynamic programming real option approach (Kozlova, Fleten, and Hagspiel, 2019). This comparison of subsidy schemes in relation to investment timing is also done by Boomsma, Meade, and Fleten (2012). They provide an analytic and a numerical solution for multiple sources of uncertainty. A recommendation is given for an extension of their framework to existing projects of wind power energy. Furthermore, according to the au-thors, their work improves decision considerations for investors, because the uncertainty in energy support schemes influences timing of investment signif-icantly. The combination of existing projects, real option theory and policies can be related to the topic of this research. Moreover, Martinez-Cesena, Azzopardi, and Mutale (2013) discuss a realistic PV energy investment in combination with an English feed-in tariff system. They also suggest further research in the real option assessment of realistic projects. Interestingly, they describe the problem that feed-in tariff systems are susceptible to changes.
2.3.
Dutch PV energy investments
In this section developments of subsidy, technology and the electricity price within the area of PV energy projects are discussed. These develop-ments provide a framework for the scenarios discussed in the analysis. The importance of subsidy policies to diffuse renewable energy is significantly described in literature (W¨ustenhagen, Wolsink, and B¨urer, 2007; Verbong and Geels, 2007). In the Netherlands the policy framework for renewable energy changes frequently (van Rooijen and van Wees, 2006). These changes influence business models of renewable energy investors (Huijben and Ver-bong, 2013). In appendix A, the current policy is explained in more detail. An example of a change in the policy is the recent proposal of the SDE++ scheme in 2020, see Lensink and Welle (2019). This scheme determines the amount of subsidy to be received by an investor based on the CO2 reduction
of that specific technology. Under this scheme the amounts for PV energy investment projects are susceptible to change. Moreover, the development of policy support is salient for the continuation of Dutch PV energy investors. Jacobsson and Bergek (2004) describe the situation of Dutch wind farm com-panies in the 1980’s. They had an advancement in relation to German wind farm companies. Due to social and political resistance, this advantage was lost and currently the German wind energy market is ahead of the Dutch market.
Mulder and Scholtens (2013) describe the influence of renewable energies on the electricity price. Conventionally the electricity price is highly corre-lated with the price of gas. The increase in renewable energy production shifts the dependence of the electricity price towards a more volatile situation. For example, the wind speed in Germany has a statistical significant effect on the electricity price in the Netherlands. Although, the effect is small, Mulder and Scholtens (2013) states that the increase in renewable energies can increase the impact of this effect. Contrary, in a society with 100% renewable energy production in which enough energy storage capabilities are present, should not have significant volatility problems (Lund and Mathiesen, 2009). More-over, the development of the electricity price is difficult to predict due to the effect of renewable energies on the merit-order effect Sensfu, Ragwitz, and Genoese (2008). Concluding, future electricity prices are difficult to predict and probably will be more volatile in the future.
3.
Methodology
3.1.
Geometric Brownian Motion (GBM) and the Bellman
equa-tion
Understanding the concept behind a real option based on an underlying value starts with understanding the underlying stochastic process. Appendix section B provides a general introduction of the stochastic process used in this paper. A special case of the GBM includes changes in the mean and the variance over time. The mean change of the GBM is described as the drift rate, the change in variance over time is defined as the variance rate. The GBM can be depicted by the following equation:
In which a is the drift rate or growth rate, b is the variance rate, P the stochastic variable, dt unit of time and dz the change related to the wiener process. Now the general stochastic process of the underlying value of an asset is described. To relate this process to the current value of the underlying asset the Itˆo process is needed. In this particular stochastic process the drift rate and the variance rate are related to the current value of the underlying asset and the time. A general Itˆo process can be depicted as:
dp = a(p, t)dt + b(p, t)dz (2)
In which, a(p,t) is the drift rate and b(p,t) is the variance rate. In this paper the change in dp is related to the log normal return of the asset. This is done to determine µ numerical based on historical and future expected values, see section 5.1. The log normal return is used because it resembles an accurate approximation of µ when dp reaches zero (becomes very small). Furthermore, in a risk-free world the drift rate is equal to µ, however, in our two variable numerical example the drift rate is equal to ν = µ − 0.5σ2. This
factor is also called the expected return. dP
P = νdt + σdz (3)
The concept of a log normal return will return in section 3.2. The next step is to relate the underlying stochastic value to the value of an asset. This is done through what is known as Itˆo’s lemma. It states that if the underlying stochastic value of an asset follows the stochastic process of equation 2 than the asset can be described as:
∂F = ∂F ∂t + a(x, t)p ∂F ∂x + 1 2σp 2∂2F ∂p2 dt + b(x, t)∂F ∂xdz (4)
differential equation (PDE). It is used in the field of dynamic programming to determine the value of an asset. The bellman equation is useful because it cuts the value of an asset in different parts. The bellman equation consists of two parts: the immediate payoff of a project and the continuation value.
rF = E ∂F ∂t + αp ∂F ∂x + 1 2σp 2∂2F ∂p2 dt (5)
In which E∂F∂t are cash flows (not discounted), αp∂F∂x is the value due to the change in the growth of P and σp2 ∂∂p2F2 is the value correction due to
uncertainty. The last term dz is stochastic, therefore it is expected to be zero, therefore the whole final term of eq. (4) drops.
Finally, in this paper two assumptions are made to diminish computa-tional complications. Firstly, the assumption of an infinite time investment. This results in time independence of the continuation value and therefore this reduces complexity in both numerical and analytic solutions. Secondly. the assumption of continuous time. The time steps of the underlying values δt, as depicted by eq. (2) are considered to be very small. Analytically the values are perfectly continuous. However, numerically, these steps are taken as small as possible. Furthermore, one can relate this equation to the Black and Scholes formula in which time also is continuous.
3.2.
The finite difference method
the option value is smaller than the payoff acquired when it is exercised. A numerical value for this PDE is found by taking discrete steps in time. The smaller these discrete time steps, the more accurate the value of the option is predicted. The real option model of this paper is described in section 4. It resembels a traditional American style call option following the Black and Scholes equation (Hull, 2003):
∂f ∂t + µS ∂f ∂S + 1 2σ 2S2∂2f ∂S2 = rf (6)
In this formula f represents the value of the option, t time, r is discount factor, in this case the risk free rate, S is the underlying value, σ is the volatility of the underlying value. Furthermore, the growth factor is ν. To make this PDE discrete one has to assume that ∂t and ∂S can be described by discrete intervals ∆t and ∆S. A forward difference is used to predict the term ∂f∂t, central difference to use to predict the terms: ∂f∂S and ∂S∂2f2 (Clewlow
and Strickland, 1998). As discussed above, it is more convenient to use x = ln(S) in a finite difference method. The growth rate therefore is represented by ν = µ − 0.5σ2. The explicit finite difference method depicts the discrete formula of the PDE as:
rfi+1,j = fi+1,j− fi,j ∆t + ν fi+1,j+1− fi+1.j−1 2∆x + 1 2σ
2fi+1,j+1− 2fi+1,j+ fi+1,j−1
∆x2
(7)
the trinomial tree method, the nodes are calculated backwards in time. In other words, first the most future nodes are calculated, the nodes one step closer to the present time are calculated based on these nodes. This contin-ues until the present time is reached. The number of steps and thus the size (∆t) taken, determine the accuracy of the PDE approximation. The smaller ∆t, the better the approximation.
fi,j = pufi+1,j+1+ pmfi+1,j+ pdfi+1,j−1 (8)
pu = ∆t σ2 2∆x2 + ν 2∆x pm = 1 − ∆t σ2 ∆x2 − r∆t pd= ∆t σ2 2∆x2 − ν 2∆x
Fig. 1. Every step in a trinomial tree is calculated based on three previous calcu-lated nodes. It works backwards in time, from the end values of the trinomial tree in the future to the present time. This is typical for the explicit finite difference method, it uses the previous calculated values to determine more recent values. The step towards fi+1,i+j can be seen as an upward step with probability pu; the step towards fi+1,j with probability pm; the step towards fi+1,j−1with probabil-ity pd (Hull, 2003).
3.3.
A numerical method for a trinomial tree with two variables
starting with a general PDE equation with two stochastic variables Clewlow and Strickland (1998): − ∂f ∂t = ν1 ∂f ∂x + ν2 ∂f ∂y + 1 2σ 2 1 ∂2f ∂x2 + 1 2σ 2 2 ∂2f ∂y2 − rf (9)
In which, again, ν1 = µ1− 0.5σ12 and ν2 = µ2− 0.5σ22. Applying the finite
different method result in:
f[t+1,i,j]− f[t,i,j]
∆t =
ν1
2∆x(f[t+1,i+1,j]− f[t+1,i−1,j]) + ν2
2∆y(f[t+1,i,j+1]− f[t+1,i,j−1]) + 1
2 σ2
1
∆x2(f[t+1,i+1,j]− 2f[t+1,i,j]+ f[t+1,i−1,j]) +
1 2
σ2 2
∆y2(f[t+1,i,j+1]− 2f[t+1,i,j]+ f[t+1,i,j−1]) −
rf[t+1,i,j]
(10)
In which t is a particular point in time and i and j are particular points in a grid structure. The grid is a cuboid with sizes t, i and j. As can be seen only 5 nodes are used. Intuitively it seems logically to use 9 nodes. However, the explicit finite different method and the forward different method do not require the usage of the corner nodes. Rewriting and simplifying eq. (10), results in the following payoff for a node at present time t and the corresponding probability equations:
f[t,i,j] = pum∗ f[t+1,i+1,j]+ pmu∗ f[t+1,i,j+1]+
pmm∗ f[t+1,i,j]+ pmd∗ f[t+1,i,j+1]+ pdm∗ f[t+1,i−1,j]
(11)
are calculated backwards through time. For example, for the value linked to probability pdm the values at node t+1, i-1, j is needed. The values of the
probabilities are calculated by:
pum= σ2 1 2∆x2 + ν1 2∆x Deltat pmu= σ22 2∆y2 + ν2 2∆y ∆t pmm = 1 − ∆t σ2 1 ∆x2 + σ2 2 ∆y2 − r∆t pmd= σ2 2 2∆y2 − ν2 2∆y ∆t pdm= σ12 2∆x2 − ν1 2∆x ∆t (12)
Concluding, at each time step t the node is calculated based on the option value at 5 different nodes at time t + 1. The trinomial tree steps backwards in time from t + n to t in which n is an integer number larger than 1.
4.
Model
4.1.
The basic model
In this section a model based on a realistic PV energy project is intro-duced. It explains the basis for the numerical calculations of the two variable model. Two one variable models are solved analytic to form an introduction for the numerical calculations of the two variable model. To model the op-tion value of a realistic project, first the NPV of the investment should be determined. The NPV is the continuation value, if the project is financed.
the investment. The option to wait has expired valueless. At this point in time the electricity price (subsidy level) has a certain value. Suppose that the investment continues when price threshold P* (or subsidy level S*) is reached. In other words, under price P* (S*) the project should not continue. Above price P* (S*) the PV energy field should be build. It is assumed that both the price and the subsidy follow a GBM, see section 3.1.
dP = µpP dt + σpP dz
dS = µsSdt + σsSdz
(13) In essence two regimes can be distinguished: a waiting regime and a regime in which the investment has been made. In this first regime the value of the option is determined by the underlying stochastic variables. It can be mimicked by the Bellman equation, which is discussed more elaborately below. The value of the project is uncertain, because it is determined by the two stochastic processes (or in the analytic cases one stochastic variable). Again, the value of the project in the second regime is determined by the continuation value.
The continuation value of a PV energy project consists of different cash flows. In general, the cash flows can be divided in three main categories: op-erating income, opop-erating costs and investment costs. The opop-erating income can be divided into two categories, namely income from energy production and income from subsidy. Lastly, the income, subsidy and costs are related to the quantity of production. Hence, the production quantity is multiplied by the price, subsidy and costs. In appendix A the income from subsidy broadly discussed.
Pindyck (1994). Therefore, the price, subsidy and costs are assumed to be perpetual cash flows discounted by a factor r. r is the risk adjusted discount factor and can be seen as the return of equity (or cost of equity) to be received by the investor.
We assume the price of electricity is growing. Therefore the growth rate has to be subtracted from the discount factor. Furthermore, the costs and the subsidy are assumed to be stable if the investment is made, consequently they do not have a growth rate. The subsidy is stable because the amount of subsidy is determined at the time of investment. The costs are assumed to be stable because in practice they do not increase or diminish when the PV energy field has been build. The initial equity investment of the PV energy field is made when the project is built, consequently this investment does not have to be discounted. N P V = Q P r − µp +S − C r − I (14)
Equation 14 depicts the mathematical representation of the NPV with Q is the quantity of production, P is the electricity price, r is the discount factor, µp is the growth factor of the electricity price, S is the amount of
subsidy, C is the costs per quantity and I is the equity investment costs. In section 5.1, the background of these parameters are discussed in more detail and they are related to a realistic case.
4.2.
One uncertain valuable
Now the knowledge of section 3.1 is used to determine the Bellman equa-tion in continuous time that fits our case. Furthermore, the expected cash flow in the first regime is zero, because the investment is not yet made.
rF (P ) = 0 + µpP F0(P ) +
1 2σ
2P2F00
Now the Bellman equation of this case is known, the following steps will lead us towards the analytic solution. The first step that has to be taken is finding a general solution for equation 15. One can obtain the general solution by guessing the solution with the formula Pβ. It can be explained as follows: the relationship of eq. (15) is an Euler-Cauchy equation which can be depicted by another mathematical formula, namely one of the form Pβ. The second order homogeneous PDE should take the general solution:
F = APβ1 + BPβ2 (Dixit and Pindyck, 1994). The next step is to define
coefficients A and B. The powers β1 and β2 are of a more general form. β1
is always larger than 1 and β2 is always smaller than zero. They can be
described as: β1 = −(µ − 1 2σ 2) +q(µ − 1 2σ 2)2+ 2rσ2 σ2 > 1 β2 = −(µ − 1 2σ 2) −q(µ − 1 2σ2)2+ 2rσ2 σ2 < 0 (16)
To find the values of coefficients A and B we have to determine the bound-ary conditions of this specific case. Again, when price threshold P∗is reached, the project continues. The project will be worth the NPV, eq. (14). Below threshold P∗, which can be seen as the waiting period, the value follows the general solution of equation 15. When P reaches zero, the value of the plant should be zero, BPβ1 is very small. Consequently, B is assumed to be zero.
The general equation can be simplified to F = APβ1. To find the threshold
P∗ and the constant A for which the general solution below threshold P∗ is valid we generate two boundary conditions, also called smooth pasting and value matching conditions (Dixit and Pindyck, 1994):
Aβ1P∗ β1−1 =
Q r − µp
(18) These two boundary conditions ensure that the two parts fit together smoothly, because the derivative and the position of both parts are the same. Solving above equations gives P∗ and A:
P∗ = (r − µp)β1(S − C) (1 − β1)r − (r − µp)β1I (1 − β1)Q (19) A = QP ∗ (r − µp)P∗ β1 +Q(S − C) rP∗ β1 − I P∗ β1 (20)
The analytic solution of the option value for which only the subsidy is uncertain and the price is fixed follows a different Bellman equation. This bellman equation is based on the GBM of an uncertain subsidy value as described by eq. (13). Note that the growth rate µs and the volatility
pa-rameters σs and consequently β1, see eq. (16) differentiate from the case in
which the electricity price is uncertain. The following Bellman equation is the result: rF (S) = µsSF0(S) + 1 2σ 2 sS 2F00 (S) (21)
The result of this equation is similar to the solution of an uncertain elec-tricity price. The values of threshold S∗ and the value of constant A2, are
found by setting the value matching and smooth pasting conditions:
4.2.1. Two uncertain variables
In this section our main model is depicted analytic. The two uncertain values follow a GBM with drift, see eq. (13). The assumption is made that the two variables are independent. There is no correlation between both uncertain processes. Furthermore, as discussed, two regimes can be distin-guished. Below a certain option value F(S,P) it is more favorable to wait. Above a certain option value F(S,P) the value to invest exceeds the value to wait. The value can analytically be represented by the PDE:
−∂F (S, P ) ∂t =µsS ∂F (S, P ) ∂S + µpP ∂F (S, P ) ∂P + 1 2σ 2 sS 2∂2F (S, P ) ∂2S + 1 2σ 2 pP2 ∂2F (S, P ) ∂2P − rF (24)
5.
Data
5.1.
Base case
In this section the parameters used in the numerical analysis are discussed. The parameters in such a way that the NPV of the model represents the NPV of a realistic PV project. The parameters of the GBM are determined so they resemble the expected electricity and subsidy price.
Table 1: Base tariffs of the Dutch PV subsidy policy
Year 2015 2016 2017 2018 2019 2020 2021 2022
Base tariff (Eur/kWh) 0,141 0,128 0,125 0,107 0,099 0,08 0,07 0,069
Log return -5% -1% -7% -3% -9% -5% -1%
The base tariffs from 2016 to 2019 as described by Rijksdienst voor ondernemend Nederland (2015, 2016, 2017, 2018); Lensink, Pisca, and Strengers (2019). The base
tariffs of 2020 and 2021 are assumptions of the author fundamentally based on knowledge obtained from Lensink et al. (2019).
First the parameters of the GBM are discussed. Because the finite dif-ference method uses log normal returns, the volatility and the drift rate also have to be determined according to the log normal returns. Table 1 presents the historic and the predicted values for the base tariff. The assumption of the growth rate of the subsidy level is determined by looking at the parental change in base tariff of previous SDE+ rounds. The values of 2020, 2021 and 2020 are assumed based on data of Solarfields Netherlands c and Lensink et al. (2019). Based on these values, the growth rate and the volatility of the subsidy are calculated, see appendix C. The growth rate and volatility of the subsidy are µs= −0.05 and σs= 0.032. The growth rate and volatility of the
projection by Greetham and Tarasewicz (2019) 1. The values obtained for
the growth rate and volatility of the electricity price are mup = 0.03 and
σp = 0.03.
Secondly, the residual parameters needed for the NPV calculation are explained. The data is based on a fictional PV energy field of 5 hectares, the lease costs of the land are 7000 Eur/hectare, the capacity of the PV panels are 1.2 MWp/hectare and the PV energy field produces 950 hours of electricity each year. The total capacity of the PV energy field is 10.8 MWp per hour (the land can be used for 90%). The CAPEX of the project is 4 mln of which 80% is contributed to material costs, 6% to grid connection, 3% to financing, 4% t0 development costs, 5% to land reform and 2% to unforseen. The equity investment is 15% of the CAPEX, resulting in annual debt obligations of 79 % of the annual income. The annual OPEX are approximately 19% of the annual income. The OPEX are based on: lease costs, real estate tax, maintenance, grid connection costs, insurance and debt repayment plus interest. The marginal tax rate on equity is 20%. The quantity is multiplying the total capacity of the PV field times the amount of production hours, 950. Based on these parameters the assumption for respectively the cost, investment and quantity for the NPV are C = 80 Eur/MWh, I = 6 · 105 Eur
and Q = 5.13 · 103 MWh/year.
The assumption is made that for this base case the risk adjusted discount rate for the NPV is r = 0.11. In an NPV calculation the discount rate represents the return that you are able to earn with other investments. In our case this is the equity return required for a PV project. It represents the
return an investor wants to receive in relation with the risk of the project. The weighted average cost of capital (WACC) of the project is 3.7%. The WACC is determined based on a debt to value ratio of 85%, an equity return of 11%, a debt return of 3% and a marginal tax rate of 20%.
Furthermore, a maximum base tariff subsidy of 0.080 Eur/kWh or 80 EUR/MWh is assumed, see appendix A. The electricity in the base case is 40 Eur/MWh (Greetham and Tarasewicz, 2019). The subsidy level is 80-40 = 40 Eur/kWh. Including all these parameters in equation 14 result in a positive NPV of 99,545 Eur.
Lastly, the investment horizon is chosen to be 50 years. Practically this can be seen as the time frame in which the decision to invest can be made. The option to start investing can be exercised in 50 years.
Table 2: Numerical parameters base case
Parameter Basic case Unit
Time investment horizon (T) 50 Years
Quantity of production (Q) 5.13 · 103 MWh per year
Price of electricity (P) 40 Eur per MWh
Subsidy (S) 40 Eur per MWh
Cost of production (C) 80 Eur per MWh
Investment (I) 6 · 105 Eur
Growth rate (µp) 0.03
Growth rate (µs) -0.05
Volatility (σp) 0.030
Volatility (σs) 0,032
Discount rate (r) 0.11
The numerical parameters of the base case. It resembles a realistic fictional investment scenario of a PV energy field of 5 hectares.
5.2.
Scenario analysis
sce-narios. Next to the base case, which resembles the current situation, three other scenarios are determined: price elusiveness, technological utopia, in-stitutional turmoil. The scenarios are based on possible trends within the industry. Three prominent trends could impact future finance of PV projects: developments in the electricity price, advancements in technology and devel-opments in the subsidy level.
Table 3: Numerical parameters scenario analysis
Parameters Q I µp µs σp σs r
Scenario 1 (Base case) 5.130 · 103 6 · 105 0.03 -0.05 0.03 0.032 0.11
Scenario 2 5.130 · 103 6 · 105 0.02 -0.05 0.1 0.032 0.11
Scenario 3 6.156 · 103 4.86 · 105 0.03 -0.05 0.03 0.032 0.07
Scenario 4 5.130 · 103 6 · 105 0.03 -0.1 0.03 0.15 0.11
Four different scenarios are determined, named: base case scenario (1), price elusiveness (2), technological utopia (3), institutional turmoil (4). The time investment horizon (T), the price of electricity (P), the subsidy level (S), The cost of production (C) and the discount rate (r) are the same in every scenario. The values similar to the base case, see
table 2.
Scenario 2, price elusiveness, is a scenario in which the volatility of the electricity price increases. However, the general growth resembles the lower case of the projection of Greetham and Tarasewicz (2019). The electricity price is correlated with the price of gas. In this scenario, an increased volatil-ity of the price of gas leads to an increase in the volatilvolatil-ity of the electricvolatil-ity price. A volatility increase of the price of gas could for example be caused by political decisions regarding oil and gas policies.
Scenario 3, technological utopia, is a scenario in which the technology of PV energy improves significantly. The instillation costs decrease by 20% and the capacity of the panels improve by 20%. Furthermore, it is assumed the risk adjusted discount rate reduces to 7%. The WACC decreases to 3% due to a positive view on PV energy projects and lack of better investment opportunities.
decreases. It is assumed that the subsidy level is susceptible to political changes. These political changes increase the uncertainty and therefore the volatility of the subsidy level. Furthermore, it is assumed that the subsidy decreases more rapidly than the base case. This decrease could be related to political decisions to reduce the subsidies in sustainable energy.
6.
Analysis and Discussion
6.1.
A one variable analysis
The two analytic discussed cases in section 4.2 are briefly discussed nu-merically. This is done to build a case towards the scenario in which the two variables are uncertain, an analogue approach is followed by Bakke et al. (2016) and Gazheli and van den Bergh (2018). fig. 2 depicts the option value in a case in which the electricity price is fixed at 40 Eur/kWh. For all three different sigma’s the decreasing growth rate of the subsidy is 5% per year. The base scenario has a sigma of 0.032. One can see that the option value calculated with the base case parameters almost resembles the NPV line. The value of this option is almost zero, because the volatility is relatively small and the subsidy growth rate decreases. The longer an investor waits, the lower the subsidy becomes. Hence, the option to invest is close to zero. If the volatility increases or in other words, the subsidy becomes more un-certain, the option value increases. Hence it becomes more valuable to wait with the investment. The vertical straight line depicts the investment thresh-old. At this subsidy level, the price to invest immediately is higher than the option to wait. This vertical line separates the two regimes as discussed in section 4.1.
secured by the investor via hedging or binding contracts. In this scenario the investor would have fixed their electricity income already, however he still has to apply for a subsidy and still has to decide whether to continue the investment.
Fig. 2. Three option price lines are depicted for different volatility levels. The NPV value is represented by the straight black line. The price threshold is represented by the verti-cal light blue line. Input is according to base case, see table 2 and with n=10000, nj=1000.
Fig. 3. The relation between different sub-sidy thresholds and different volatility levels is depicted. Input is according to base case, see table 2 and with n=10000, nj=1000.
If the uncertainty of the subsidy increases, the threshold increases simul-taneous. See fig. 3, the price threshold changes from 38.5 to 40 Eur/kWh when the volatility increases from 0.04 to 0.08. This relation is not linear, the change in volatility has an increasing impact on the price threshold. The more volatile the subsidy level is, the more valuable the option to wait be-comes.
Furthermore, fig. 5 shows the three option prices with three different levels of production. The quantity of productions are equal to the base case, 110% of the base case and 130% of the base case. An improvement of the quantity of the base case by 10% could be the result of technological advancements of the PV panels(in general, more production on the same land). The price threshold decreases from 54 to 52.5 Eur/kWh. The threshold reduction is not significant, however, the value of the option changes seriously.
Practically, a fixed subsidy implies a scenario in which the subsidy already is requested and granted. The investor now has the choice to built the solar field immediately or wait until the electricity price increases. The project will produce energy later, however, the value of the project increases.
Fig. 4. Three different scenarios with differ-ent discount factors are depicted. Input is according to base case, see table 2 and with n=10000, nj=1000.
6.2.
Two variable analysis
The results above show the impact of three different parameters, the volatility, the discount rate and the quantity of production. In this section the main analysis of this paper is done: the analysis of the optimal invest-ment threshold in the presence of two uncertain valuables. This threshold is depicted in fig. 6. The figure is divided in three different areas. The first area depicts the nodes for which the NPV value is negative, the second area depicts the nodes for which the option value is larger than the NPV value, the last area depicts the nodes for which the NPV value is larger than the option value. Note: at the nodes of the first area, the option values are not zero.
The two lines represent the base case scenario (a subsidy level of 40 and an electricity price of 40). If one moves along the horizontal line from an electricity price of 35 Eur/kWh to an electricity price of 50 Eur/kWh, one moves through the three areas. Again, two regimes are depicted, the first and second areas represent the waiting regime. The third area represents the investment regime. Following the horizontal line, the investment price threshold is approximately 43 Eur/kWh. As such, above a price threshold of 43 Eur/kWh the investment should be made immediately. Below 43 Eu-r/kWh the investor should wait. Furthermore, the two lines are crossing in the second area. Hence, the base case has a positive NPV. However, it is still present in the waiting regime. At a subsidy level near to zero, the investment threshold is approximately 92 Eur/kWh.
Fig. 6. This figure shows the optimal investment threshold of a subsidy level S in relation with an electricity price P. Each node (of price [P] and subsidy [S]) in the yellow area represents a node for which the NPV value of the project is positive. At each node in the green area the option value is larger than the NPV value. At each node in the purple area, the NPV is larger than the option value. The yellow and green area represents the waiting regime. The purple area represent the investment regime. Note: at the nodes of the first area, the option values are not zero. The parameters of the base case scenario are used, see table 2 with n=10,000 and nj=1,000.
in the future. On the contrary, below a subsidy of 20 Eur/kWh the option value is significant, because a higher electricity price in the future will lead to a higher income.
Eur/kWh at a subsidy level of 40 Eur/kWh. The threshold line shifts to the left due to a decreasing investment risk.
Figure 8 displays the optimal investment threshold for different subsidy growth rates. The difference between a growth rate of 0% and -5% is clearly visible. However the difference between -5% and -7% is minor. The impact of the growth rate diminishes when the growth rate is lower.
Fig. 7. Three different scenarios with differ-ent discount factors are depicted. Input is according to base case, see table 2 and with n=10000, nj=1000.
Fig. 8. Three different growth rates for the subsidy level are depicted. Input is accord-ing to the base case, see table 2 and with n=10000, nj=1000.
be related to the volatility. Due to the higher volatility the option value increases in value, therefore changing the threshold line.
When PV panels decrease in costs, the investment reduces, while the quantity of production remains the same. The effect of a 10% and a 20% decrease can be seen in 10. The threshold line slightly shifts to the left. The impact of a decrease of 20% is minor. Note: reducing an investment by 10% results in an increase of NPV and option value, however, the optimal investment threshold only slightly changes.
Fig. 9. Based on the three possible cases determined by Greetham and Tarasewicz (2019), three different scenarios are de-picted. The lower case has a volatility of 5.5% and a growth rate of 2%. The upper case has a volatility of 4% and a growth rate of 3.7%. The base case has the parameters as presented in table 2 and with n=10000, nj=1000.
Fig. 10. Three different investment amounts are depicted. In the base case scenario an in-vestment of 600000 is made. An inin-vestment of 540000 is a decrease of 10%, an invest-ment of 480000 is a decrease of 20%. The base case has the parameters as presented in table 2 and with n=10000, nj=1000.
which is not taken into account in the one-variable case, results in the lower option value. It shows that due to the decreasing effect of the subsidy level the option to wait is exercised earlier. Moreover, the lower the subsidy level, the lower this effect, because the option value is more dependent on the electricity price. As discussed, this effect could also be seen in fig. 9.
6.3.
Four scenarios
Section 5.2 explains different scenarios. The optimal investment thresh-olds of these scenarios are displayed in fig. 11. The difference between sce-nario 1 and 2 is more significant at lower subsidy levels. Due to a higher price volatility and a higher growth rate, the threshold line of scenario 2 is situated left of the base case. In this scenario with a higher uncertain elec-tricity price than the current projections it becomes less likely an investment will be made. At a subsidy level of 40, the optimal electricity price shifts from 42 Eur/kWh to 52 Eur/kWh.
In scenario 3 the technological progress is increased. This scenario is the most optimal. It has the largest effect on the threshold. At a price of 80 Eur/kWh the solar field could be financed without subsidy. Moreover, the shape of the threshold line does not change, it only shifts to the left. At a subsidy level of 40, the threshold line shifts from 42 to 32 Eur/kWh.
Fig. 11. Four different scenarios are determined, named: base case scenario (1), price elusiveness (2), technological utopia (3), institutional turmoil (4). The parameters of different scenarios are used, see table 3 with n=10,000 and nj=1,000.
2 and 3.
The scenario analysis shows the impact of different parameter changes. It can be concluded that technology advancement changes the threshold sig-nificantly. Based on this analysis one could argue that an investor should focus on reducing the costs of investment rather than focusing on changes in subsidy. Moreover, reducing the uncertainty of the electricity price could also benefit the investor. The investor could use price agreements to reduce the uncertainty and reduce the optimal investment threshold.
7.
Conclusion
on the investment threshold are calculated numerically. First the impact of several parameters are discussed considering a model with one uncertain variable. Secondly, the threshold of a base case scenario considering two uncertain variables is presented. The impact of different parameters on this base case is discussed. Lastly, a scenario analysis is done to elaborate on possible outcomes.
In general, answering the main question, the impact of two variables is significant. Especially the uncertain electricity price results in an additional option value for waiting. Figure 6 shows that the optimal investment thresh-old is explicitly different from the NPV threshthresh-old at lower subsidy values. At higher subsidy values, this difference diminishes due to the decreasing growth rate of the subsidy level. Furthermore, the effect of the growth rate of the electricity price has a higher influence at lower subsidy values.
Next to the impact of uncertainty on the threshold line, the scenario analysis reveals that the impact of technological progress is also significant. This technology progress scenario entails a 20% increase of capacity, a 20% decrease of investment costs and a decrease of equity return by 4% in relation to the base case. Due to these changes the threshold shifts by 10 Eur/kWh. Furthermore, the analysis presents that a change in growth rate and volatility of the electricity price is significant. In contrast, an increase in volatility of the subsidy level has a minor influence. The recommendation of this paper is that investors focus on technology improvement and the influence of this improvement on optimal investment decisions.
scenario for PV production. A more conservative conclusion could be made based on fig. 6 in which the option price threshold is reached at an electricity price of 90 Eur/kWh without technological improvement. In a medium case, this would be reached in 22 years Greetham and Tarasewicz (2019). Fur-ther conclusions would be predictions without fundamental argumentation. Therefore the reader should interpret the results themselves.
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Appendix A.
Dutch PV subsidy feed-in tariff
between a correction tariff and a base tariff. The value of both are determined by the dutch government. The correction tariff is an additional amount on top of the APX electricity price, also called grey electricity price (this additional amount differs per technology). The base tariff is determined based on the costs of the technology.
SDE (+) subsidy = maximum base tarif f − correction tarif f (25) The changes in subsidy of table 1 can be seen as varying the maximum base tariff. After receiving the SDE+ approval the subsidy is a fixed value on top of the grey electricity price. Thus, to place it in context: consider an electricity price of 42 Euro per MWh, a correction tariff of 5 Euro per MWh and a base tariff of 85 Euro per MWh. This results in a subsidy of 38 Euro per MWh. Reducing the maximum base tariff to 81, results in a decrease of subsidy of 4 Euro per MWh to 34 Euro per MWh. In this paper we consider the subsidy independent of the electricity price. It is assumed that a change in the electricity price will not change the subsidy. It is a fixed value on top of the grey electricity price.
Appendix B.
An underlying stochastic
pro-cess
The stochastic process used in this paper is based on the geometric Brow-nian motion (GBM) or wiener process. The wiener process is a stochastic process in continuous time and is described as a Markov process. A Markov process entails that the probability distribution of a variable xt+1 only
de-pends on the current value of x, xt. Thus, the values of variable x of the past
have no influence on the values of the variable in the future. A practical ex-ample could be given as follows. The price of a certain stock A has currently a value 60. Yesterday’s news let to an extreme appreciation of 20 percent (yesterdays price was 50). If this stock would follow a markov process, this news and the price of 50 would not influence the price of tomorrow. Only the value of today, 60 would influence the price. Two other characteristics of the wiener process: the probability distribution of the variables are independent of each other through time and the change of the process has a linear relation with time and is normally distributed (Hull, 2003; Dixit and Pindyck, 1994).
Appendix C.
Return and volatility based on
data
This appendix explains the methods used to determine the growth rate and volatility of the electricity price and the subsidy level. The calcula-tions of the electricity price are based on projeccalcula-tions done by Greetham and Tarasewicz (2019). The calculations of the subsidy level are based on the subsidy levels as depicted by table 1.
Hull (2003): ui = ln xi xi−1 f or i = 1, 2, 3, ..., n (26)
In this case x1 is the subsidy in 2015. The volatility of an asset is the
parameter that is related to the uncertainty of the stock. This uncertainty can be found by subtracting the mean of a sample from the value of a par-ticular value ui. This difference has to be squared and the total deviation
of the sample has to be summed and divided by the amount of the sample minus one. The volatility is calculated by the equation:
σ = v u u t 1 n − 1 n X i=1 (ui− ¯u)2 (27)
Using table 1 the growth rate, µs, and volatility, σs, are respectively -0,046
and 0,032.
Appendix D.
Python code numerical values
This appendix presents the code used for the analysis.
D.1.
Code one stochastic variable, uncertain electricity price
t = 50 c o s t s = 80 i n v = 6 0 0 0 0 0 s i g = 0 . 0 3 r = 0 . 1 1 nu = 0 . 0 3 n = 1 00 00 n j = 1000 b a s e p r i c e = 40 s u b s i d y = 50 q = 5130 def t r i a m ( i n v , t , b a s e p r i c e , s u b s i d y , s i g , r , q , nu , n , n j ) : k = i n v + q ∗ ( c o s t s / r − s u b s i d y / r ) # C a l c u l a t i o n i n i t i a l p a r a m e t e r s d t = t /n dx = s i g ∗ np . s q r t ( 3 ∗ ( t / n ) ) edx = np . exp ( dx ) pu = 0 . 5 ∗ d t ∗ ( ( s i g / dx ) ∗∗2 + ( nu −0.5∗ s i g ∗ ∗ 2 ) / dx ) pm = 1 − d t ∗ ( s i g / dx ) ∗∗2 − d t ∗ r pd = 0 . 5 ∗ d t ∗ ( ( s i g / dx ) ∗∗2 − ( nu −0.5∗ s i g ∗ ∗ 2 ) / dx ) c o n s t a = q / ( r−nu ) # D e t e r m i n a t i o n o f a s s e t p r i c e a t m a t u r i t y
e d x 2 = np . exp ( d x 2 ) # Nodes p r o b a b i l i t i e s c a l c u l a t e d # puu = pu ∗ pu = 0 # pum = pu ∗ pm # pud = pu ∗ pd = 0 # pmu = pm ∗ pu # pmm = pm ∗ pm # pmd = pm ∗ pd # pdu = pd ∗ pu = 0 # pdm = pd ∗ pm # pdd = pd ∗ pd = 0 pum = ( ( s i g 1 / d x 1 ) ∗∗ 2 + ( mu 1 − 0 . 5 ∗ s i g 1 ∗∗ 2 ) / d x 1 ) ∗ 0 . 5 ∗ d t pmu = ( ( s i g 2 / d x 2 ) ∗∗ 2 + ( mu 2 − 0 . 5 ∗ s i g 2 ∗∗ 2 ) / d x 2 ) ∗ 0 . 5 ∗ d t pmm = 1 − ( ( s i g 1 / ( d x 1 ) ) ∗∗ 2 + ( s i g 2 / ( d x 2 ) ) ∗∗ 2 ) ∗ d t − r ∗ d t pmd = ( ( s i g 2 / d x 2 ) ∗∗ 2 − ( mu 2 − 0 . 5 ∗ s i g 2 ∗∗ 2 ) / d x 2 ) ∗ 0 . 5 ∗ d t pdm = ( ( s i g 1 / d x 1 ) ∗∗ 2 − ( mu 1 − 0 . 5 ∗ s i g 1 ∗∗ 2 ) / d x 1 ) ∗ 0 . 5 ∗ d t
v a l u e o p t i o n [ 0 , 0 , : ] = b o u n d a r y 2
v a l u e o p t i o n [ 0 , : , 2∗ n j ] = p a y o f f [ : , 2∗ n j ] v a l u e o p t i o n [ 0 , 2∗ n j , : ] = p a y o f f [ 2 ∗ n j , : ]
v a l u e o p t i o n [ 0 , : , : ] = np . maximum ( v a l u e o p t i o n [ 1 , : , : ] , p a y o f f [ : , : ] )
c o n t i n u e n o d e s 2 = 10 ∗ ( p a y o f f [ 1 : 2 ∗ n j +1 , 1 : 2 ∗ n j +1] == 0 ) c o n t i n u e n o d e s = 50 ∗ ( r e a l o p t i o n [ 0 , 1 : 2 ∗ n j +1 , 1 : 2 ∗ n j +1] > p a y o f f [ 1 : 2 ∗ n j +1 , 1 : 2 ∗ n j + 1 ] ) c o n t i n u e n o d e s = c o n t i n u e n o d e s 2 + c o n t i n u e n o d e s x g r i d = np . a r a n g e ( 2 ∗ n j −1) y g r i d = np . a r a n g e ( 2 ∗ n j −1) x g r i d , y g r i d = np . m e s h g r i d ( x g r i d , y g r i d ) p r i c e l e v e l g r i d , s u b s i d y l e v e l g r i d = np . m e s h g r i d ( p r i c e l e v e l [ 1 : 2 ∗ n j ] , s u b s i d y l e v e l [ 1 : 2 ∗ n j ] ) # B u i l d f i g u r e s : d i f f e r e n t g r i d and t h r e s h o l d l i n e s . # I t d e p i c t s t h e r e l a t i o n s h i p b e t w e e n s u b s i d y and p r i c e a t t = 0 . # I t d e p i c t s t h e w a i t i n g and c o n t i n u a t i o n r e g i m e r e g a r d i n g two v a r i a b l e s . p2 = s c i p . p o l y f i t (X, Y, 4 ) p 2 2 = s c i p . p o l y f i t ( X 2 , Y 2 , 4 ) p 2 3 = s c i p . p o l y f i t ( X 3 , Y 3 , 4 ) p 2 4 = s c i p . p o l y f i t ( X 4 , Y 4 , 4 ) f i g = p l t . f i g u r e ( 1 ) ax = f i g . a d d a x e s ( [ 0 . 1 1 , 0 . 1 , 0 . 8 5 , 0 . 8 2 ] ) # P l o t s n o d e s t h a t d e t e r m i n e p o l y n o m i a l l i n e :
