The optimal timing of investment in a Carbon Capture & Utilization plant
–
A real option approach with two uncertain variables
Abstract
The purpose of this paper is to find the optimal investment timing in a carbon, capture and utilization (CCU) plant, given that the CO2 price and variable costs are uncertain in a
time-dependent model. Three valuation models are evaluated, the NPV model, the analytical real option model with one stochastic variable, and the numerical real option model with two stochastic variables and finite time to maturity. The main findings are that the numerical real option model is preferred, as it is closest to reality. In this model, the planned CO2 price of
€125 per ton in 2030 is not sufficient to obtain a positive value, given the base case variable costs. Whereas, the other models provide different outcomes.
Keywords: numerical real options, finite time, two stochastic variables, CO2 price
University of Groningen Faculty of Economics and Business
Supervisor: Dr. G.T.J. Zwart Saar van Neerbos
1. Introduction
The Dutch government aims to reduce greenhouse gas emissions in 2030 by 49% compared to 1990. The measures to do this will enable the Netherlands to fulfil its commitments under the Paris Agreement and the National Climate Act. One of the primary greenhouse gasses in the earth's atmosphere is carbon dioxide (CO2), which poses a threat to the planet as it causes global
warming. To reduce CO2 and greenhouse gas emissions, the Dutch government imposes a
carbon tax on every ton emitted excess CO2. This tax starts at €30 per ton in 2021 and is
expected to rise to €125-150 per ton in 2030(Dutch Goverment, 2019).
The reduction of greenhouse gasses is a central theme in today's society and faces many challenges to overcome. At the same time, it creates a variety of opportunities too. The company +Earth stepped into the carbon dioxide challenge and turned it into an opportunity. Their business model seems very attractive; they turn captured pure CO2 into carbon black,
which is a commodity used in the tire and paint industry, for example. +Earth targets the tire industry as their main outlet with their marketing; imagine producing 'green' tires made from CO2 rather than fossil fuel oil. This is a form of Carbon Capture & Utilization (CCU), a process
that intends to use captured CO2 as recycled raw material with higher economic value. Thus,
because it creates new products with economic value, it is a circular technology. Besides that the CCU technology creates sustainable and CO2-negative carbon black, it also helps emitting
companies to mitigate taxes imposed on emitted CO2 and obtain a good reputation and reduce
their carbon footprint (+Earth , 2020). +Earth is keen to understand when the investment in a CCU plant would be optimal for them. For this reason, I was asked to provide them with in-depth analysis for my internship.
The simplest model to value an investment is the net present value (NPV) approach. However, this approach ignores irreversibility, the uncertainty of future rewards, and the choice of timing the investment, which are important characteristics affecting the decision to invest in reality and this could thus lead to undervalued investment opportunities (Dixit & Pindyck, 1994). Deploying a new CCU plant depends on uncertainty in CO2 prices, uncertainty in variable
costs, and relies on the right investment timing. To deal with these challenges, we examine the real option approach, which is considered more detailed than the NPV model and should give a more realistic optimal investment decision. We analyze two real option models; the analytical model and the numerical model.
a complex investment environment and deals with complex items that we cannot do analytical but need a numerical approach. First, because the option to invest does exist forever, but is limited to a given time to maturity, and we need to understand the impact. Second, because we deal with two stochastic variables that influence the investment decision. For the analysis of the optimal investment timing, we will evaluate three investment models and provide innovative companies like +Earth to make better decisions.
+Earth has the option to defer their investment, which refers to the possibility to decide whether to invest immediately or to postpone the investment. So, the real option approach distinguishes two regimes throughout the paper. The first regime is called the waiting regime, in which +Earth has an option to invest, the company has the right but not the obligation to invest in the technology at some future time of its choosing. In the second regime, the value of the firm equals the NPV of the investment. When the company invests, it gives up the option of waiting for new information to arrive that might affect the desirability or timing of the expenditure (Dixit & Pindyck, 1994). The research aims to detect the point where the two regimes meet each other. This point is called the threshold value; from this point it is optimal to invest in the CCU plant.
The analytical solution to the model is found by discussing two models depending on one stochastic variable. The most interesting about this paper are the numerical solutions and conclusions from the model, which could not be achieved analytically, but are determined by using a computer program. With this numerical solution, we could explore the impact of an option with a finite time horizon to invest in the CCU plant, instead of an infinite one obtained from the analytical model. With the finite time horizon, we can explore the impact on the investment if the option matures within five or ten years. Moreover, we could observe the effects of two stochastic variables instead of one. Here, the price of CO2 is uncertain, since it
depends on the European and Dutch government's decisions and on the reputation costs of emitting firms. Moreover, the variable operational costs are uncertain, mainly because this depends mostly on the electricity price. Therefore, this paper's main question states: What is
the optimal timing of investment in a CCU plant, given that the price of CO2 and the variable
costs are uncertain?
We find that the numerical real option approach is closes to reality for such investment decision as it considers finite time to maturity and deals with two uncertain parameters, which can be done by neither the analytical model nor the NPV model in such an integrated way. We observe that the NPV model provides too optimistic results, which could lead to poor investment decisions. Furthermore, we find that in the numerical real option model, the CO2 price of €125
The remainder of this paper is organized as follows. Chapter 2 describes a brief background on the business model of +Earth. In chapter 3 relevant literature studies based on the real option approach are summarized. Chapter 4 defines the basic model. Chapter 5's methodology for the real option approach is discussed, and the theoretical model will be elaborated, both for the one and the two stochastic variable cases. In chapter 6 the financial data will be interpreted. Chapter 7 applies this data in a computer model and explores the results for different scenarios. Moreover, a comparison with the NPV model is made. Chapter 8 concludes the overall results.
2. A short background on the CCU technology
In this chapter, the background of the investment in the CCU technology is discussed. The company +Earth, an innovative company founded in 2019 to address climate change, has adopted this technology. Their purpose is “to drive carbon responsibility and to collectively
remove excess carbon dioxide (CO2) from the atmosphere at speed and scale”. +Earth invests
in CO2 removal and upcycling technologies to generate both environmental and financial
returns. They aspire to accelerate the use of CO2 as industrial feed-through design and
innovation of CCU technologies.
In 2020 +Earth established a partnership with SkyNano LLC, a technology company in Knoxville, Tennessee, in the United States of America. SkyNano, together with the Vanderbilt University of Nashville, Tennessee, developed a patented technology that captures carbon dioxide from the air, turning it through electrochemistry, into useful nanomaterials. This is known as a negative emission CCU technique since it captures and reduces CO2 from the air,
utilizing it as a raw material for another product.
+Earth and SkyNano researched and developed adjacent technology that could transfer pure carbon dioxide into carbon black and oxygen (O2), which resulted in their proprietary
technology to turn CO2 into carbon materials, including carbon black. Carbon black is
commonly used as a reinforcing filler in rubber products, especially in tires. +Earth and SkyNano established a joint venture, Aeroborn, which exclusively owns this CCU-technology (patent pending). Their goal is to further engineer, scale and commercialize this sustainable carbon technology globally.
carbon per year into the North Sea (Porthos, 2020). For example, big multinationals, such as Shell, ExxonMobil, and Air Liquide, will capture CO2 from their industrial processes, transport
it through the Porthos pipeline, and eventually store it in empty gas fields beneath the North Sea. This project, also known as Carbon Capture & Storage (CCS), is an initiative to reduce CO2 emissions and therefore will reduce greenhouse gasses. However, at +Earth, they consider
this as a very expensive garbage bin for CO2, whereas this CO2 could and should be seen as a
valuable source of raw material for new products. So, this is how their business model has been developed.
The goal and vision of +Earth are to transform (already) captured CO2 into carbon black and
oxygen. Both products are commodities which can be sold on the market with a green label. At this moment, +Earth could collect a gate fee for CO2 equal to the price the multinationals
would pay to store their CO2. In 2021 this price is set at €30 per ton, which is equal to the tax
on emitted CO2 in the Netherlands, plus the price multinationals pay for a ‘green’ reputation.
The tax rate is expected to rise rapidly and will be about €125-150 per ton CO2 or higher in
2030 (Dutch Goverment, 2019). Polluters in Europe are already paying the price for their CO2
emissions through a European trading system (ETS), of which the price per ton CO2 is about
€25 in 2020. The European tax will be deducted from the tax proposed by the Dutch government, to maintain the competitive position of Dutch companies (McDonald O., 2020).
3. Literature review
In this chapter, earlier research on real options will be discussed. First, the definition and assumptions behind real options will be considered. Then we elaborate on the real options with two uncertain variables. A general reference to this literature is Dixit & Pindyck (1994).
3.1 Real option valuation theory
mean that the NPV model assumes that the investment is either reversible, meaning that one could undo the investment, or that the investment cannot be delayed (Dixit & Pindyck, 1995). For these reasons, the NPV approach could lead to undervalued investment opportunities, sub-optimal decisions and underinvestment (Trigeorgis, 1993) (McDonald & Siegel, 1986). To deal with irreversibility, uncertainty, and the choice of timing the investments, we consider the real option approach. This approach is introduced by Myers (1977) and is originated from financial option theory to value an investment decision on a real asset such as land, buildings, plant and equipment. A financial option is the right, but not the obligation, to buy or sell a financial asset. A company like +Earth, with its patented technology holding an option, has the opportunity to invest. When making an irreversible investment, the firm exercises the option to invest and gives up the possibility of waiting for new information that might affect the value or timing of the expenditure. This lost option value is an opportunity cost that must be included as part of the investment cost, which is not considered in the NPV model.
they install a ‘heat exchanger’ after an initial ‘distribution generation’ unit is purchased that serves the base electricity load. They find that sequential investments increase the project's value, as it allows more precision over the timing of the upgrade. Furthermore, they argue that when a firm expands, it loses the option value of expanding, which means that the first phase is more valuable.
The option to choose is the option to choose between two projects. This is discussed by Murto et al. (2002), who consider an energy production investment when an investor faces the choice between fossil fuel and a biomass-fired plant. They assume that the fossil fuel price moves stochastically, while the biomass price is constant, and that investment could be delayed. So, they emphasize the effect of different input price uncertainties of the two projects, and the effect of the investment timing. They find that it might be optimal to postpone investment when the choice of the project is irreversible, and higher volatility causes the waiting regime to be wider. Siddiqui et al. (2010) analyze a firm that also has a choice between two technologies. The firm may choose to commercialize a new energy technology to meet given electricity demand or deploy more of an existing renewable energy technology. They assume that both the electricity price and the new technology's operating costs are uncertain and use a real option approach to examine the choice of technology and the timing decision. They provide numerical examples and show that the option to commercialize new technology increases the firm's value. The addition of the existing technology also increases the value, but it delays the potential initiation of the new technology.
Another option is to defer a project, which refers to the possibility of deciding whether to invest immediately or postpone the investment. This option is equivalent to an American call option on the value of the project; it has the right but not the obligation to invest at any time up to the date the option expires. Once the investment is made, it is irreversible, and the company gives up the option of waiting for new information. McDonald et al. (1986) find that the option of waiting to invest could add significant value. The postponement of the investment could be beneficial if the firm can get additional information about the project. The benefits must be weighed against the possible cost of deferring an investment, for example, the cost of the risk of entry by other firms or foregone cash flows. In this paper, we will discuss this option to defer for a CCU project.
impact on the return of the investment. This option concerning carbon prices and renewable energy projects will be discussed in further detail in the next section.
3.2 Real option analysis with two underlying stochastic variables
The real option analyses we discussed so far mostly assume one stochastic variable. In this section real option models with two stochastic variables will be discussed, in particular with respect to carbon prices and renewable energy projects.
A paper discussing the effects of the interaction between two uncertain variables on the optimal investment timing of an irreversible investment is presented by Murto et al. (2006). They find that when combining technological uncertainty with revenue uncertainty, increasing technological uncertainty makes investment less attractive relative to waiting. However, technological uncertainty alone has no impact. Zhang et al. (2014) use a trinomial tree model on real option theory to evaluate the investment decision of CCS for two types of power plants. They consider uncertainties in carbon prices, government incentives, annual running time, power plant lifetime, and technological improvements. They find that current carbon prices were too low for immediate investment in CCS technology. However, when introducing a CCU technology, they obtain that the critical carbon price falls, but is still not optimal for investment. This shows that CCU technologies could be advantageous in overcoming the barrier of the high investment costs of CCS, because it creates additional revenues. Another real option model of CCS investment with two stochastic variables, the carbon price and technological improvements, is discussed by Zhang et al. (2011). They use the dynamic program method to obtain the optimal investment timing. Furthermore, their results from numerical simulations illustrate that power producers delay investment in CCS technologies if the carbon price is more volatile and if the technological improvement is higher. Moreover, there will be no investment if the volatility of carbon price is high enough. Heydari et al. (2010) also argue that the investment option is highly sensitive to the volatility of carbon price. They emphasize that an increase in volatility cause a wider waiting area. In their paper, they present a real option model that values the choice between two CCS technologies, given uncertain electricity, CO2,
plant. Most papers conclude that with the current carbon price it is not optimal to invest in a CC(U)S plant.
In this paper we will analyze a real option model with two stochastic variables, the carbon price and the variable costs (greatly dependent on the electricity price). To find the optimal investment timing in a CCU technology with two stochastic variables, we will make use of a numerical method, as a real option with two stochastic variables is difficult to solve analytically. To value an option numerically, the binomial method (Cox, Ross, & Rubinstein, 1979), the trinomial method (Zhang, et al., 2014), implicit finite method (Clewlow & Strickland, 1998), and the Monte Carlo approach (Boyle, 1976) are the most applicable. In this paper, the implicit finite method will be used to value the option to invest with two underlying stochastic variables. The comparison between different valuation models, and in particular the numerical real option method with two stochastic variables and a finite time to maturity is an important addition to existing energy related literature.
4. Theoretical model
In this section, we will discuss the theoretical model. To find the optimal investment timing in the CCU plant, we should start by looking at a basic model and its parameters. First, the most important assumptions for the model should be made clear. The first assumption is that the CO2
price and the variable costs have uncertain values. The second assumption is that once the investment is made, it is assumed to be irreversible in the CCU plant case. This means that the investment costs of +Earth are sunk costs once the investment is made. The third assumption states that +Earth has the option to defer its investment in the CCU plant, meaning that it could postpone investment, which is discussed in more detail in section 3.1. It is assumed that the possible costs of postponement, such as the risk of entry by other firms are limited, as +Earth has a patent on the technology. The fourth assumption is that the theoretical model is a continuous-time model, meaning that the option to invest in the CCU plant has an infinite lifetime. It should be noted that the numerical model is a finite time model.
In our continuous-time model, we can at each moment (and price P and/or cost C) choose whether to invest right away, obtain the NPV of the project, or wait and gain the (discounted) expected continuation value. Two regions could be distinguished for the model with a stochastic variable price (for stochastic variable costs it is different):
Waiting regime Continuation value
Continuation value
In the second regime, it is optimal to invest, as the value of investing is higher than the value of waiting. Once the project is financed, the continuation value equals the NPV of the project. The continuation value of the CCU technology contains different cash flow parameters. One could distinguish variable parameters, which variate with the quantity of CO2 and fixed
parameters. The variable cash flow parameters consist of the purchase price of CO2, the
variable operational expenses (OPEX) also called variable costs and the sales prices of carbon black and oxygen. The fixed cash flow parameters include the yearly fixed OPEX, and the once-off investment costs made at the beginning of the period.
Since our model is a continuous-time model, the discount factor of the constant cash flow parameters is equal to 𝑒'(). These constant variables are the variable costs, the sales price of
carbon black and oxygen, and the fixed OPEX. To calculate the expected present value of these parameters, we have to compute the integral, which is equal to:
𝑉 = + 𝑒/ '()𝑃𝑑𝑡 0 = 1−𝑒'()𝑃 𝑟 4 = 𝑃 𝑟
So, the variables are considered to be perpetual and discounted by r. Where r is the risk-adjusted discount rate, which will be calculated with the capital asset pricing model in section 6.3. Once the investment is made, it is assumed that further uncertainties in the evolution of the price of CO2 and the variable OPEX are remaining. The price of CO2 is depended on decisions
of the Dutch government and the European Union on CO2 taxation, and the reputation costs of
the emitting firms. Therefore, this price could variate even after the investment is made. Furthermore, the variable costs largely depend on electricity costs, which also remain uncertain even after investment. Therefore, the purchase price of CO2 and the variable costs are both
divided by the discount rate minus its growth rate. The integral for the expected present value of the growing cash flow parameters is equal to:
𝑉 = + 𝑒/ '()𝑒5)𝑃𝑑𝑡 0 = + 𝑒/ (5'())𝑃𝑑𝑡 0 = 1−𝑒(5'())𝑃 𝜇 − 𝑟 4 = 𝑃 𝑟 − 𝜇
𝑁𝑃𝑉 = ; 𝑃<=> 𝑟 − 𝜇?− 𝐶 𝑟 − 𝜇A+ 𝑃<C+ 𝑃=> 𝑟 D × 𝑄 − 𝐹𝐶 𝑟 − 𝐼
Where 𝑃<=> is the purchase price of CO2, 𝐶 is the variable costs, 𝑃<C equals the sales price of
carbon black, 𝑃=> is the sales price of pure oxygen, 𝑄 is the quantity of CO2, 𝐹𝐶 is the yearly
fixed OPEX, 𝐼 is the investment cost, 𝑟 is the discount rate, and 𝜇<=> and 𝜇A are the growth rates of the price of CO2 and of the variable OPEX.
All the variables and their values are discussed in more detail in chapter 6.
Waiting regime
In the waiting regime the manager of the firm has an option to invest in the CCU plant. This option to defer investment has value, investing now means throwing away that value. As long as the value of waiting is higher than the value of investing right now, the investment should be postponed and the firm should stay in the waiting regime. This is probably because the CO2
price is too low or the variable costs are too high. During this waiting regime, the value of the investment opportunity satisfies the Bellman equation, which will be discussed in chapter 5. Furthermore, the value of the CCU plant is uncertain in this waiting regime. This is because the price of CO2 and the variable costs are stochastic variables, meaning that the values of these
variables change randomly through time (Clewlow & Strickland, 1998).
At threshold
To find the optimal investment timing, we need to find the threshold values, 𝑃<=∗ > and 𝐶∗. These
threshold values give the value of the project where the two regimes meet. When 𝑃 (𝐶) is above (below) some threshold value 𝑃<=∗ > (𝐶∗), it is optimal to invest. On the other hand, when
𝑃 (𝐶) is below (above) the threshold value 𝑃<=>∗ (𝐶∗), it is optimal to stay in the waiting regime.
To obtain the threshold value we will first look at the methodology of the analytical model with one stochastic variable in chapter 5, then we investigate the results of the analytical model and the numerical model, with two stochastic variables, in chapter 7.
5. Methodology and solution to analytical model
5.1 Stochastic process and the Bellman equation
As mentioned in chapter 4, the price of CO2 and the variable OPEX are stochastic variables in
our model. A variable is said to follow a stochastic process if the value changes randomly through time. The stochastic process in this analytical model is considered to develop in a continuous-time framework, meaning that the value of the variables can change at any time. A Brownian motion, or Wiener process, is a continuous-time stochastic process with three main characteristics. Firstly, it is a Markov process that states that the process's current value is the only thing needed to make future value forecasts. This is in line with the weak form of market efficiency. Secondly, the probability distribution for the change in the process over a time interval is independent of other time intervals; this is called independent increments. Thirdly, the change in the process is normally distributed (Dixit & Pindyck, 1994).
A generalized variant of the Wiener process is the Brownian motion with drift; this is also called a Geometric Brownian Motion (GBM) (Black & Scholes, 1973). For the CO2 price (𝑃)
and the variable OPEX (𝐶) following a GBM, this is determined by the following equations: 𝑑𝑃 = 𝜇?𝑃𝑑𝑡 + 𝜎?𝑃𝑑𝑧 𝑑𝐶 = 𝜇A𝐶𝑑𝑡 + 𝜎A𝐶𝑑𝑧A Or 𝑑𝑃 𝑃 = 𝜇?𝑑𝑡 + 𝜎?𝑑𝑧 𝑑𝐶 𝐶 = 𝜇A𝑑𝑡 + 𝜎A𝑑𝑧
Where the variable 𝜇 is known as the drift rate and 𝜎 is equal to the volatility, of the CO2 price
and variable OPEX. 𝑑𝑡 refers to the small interval of time. Moreover, the change 𝑑𝑧 is called a Brownian motion or Wiener process, which is usually distributed with a mean of zero and a variance 𝑑𝑡.
In the equations above, the drift rate and the volatility only depend on the current values of the underlying asset and time. Therefore, these equations are examples of an Itô process. The general equation for a variable, for example, 𝑃, following an Itô process equals:
𝑑𝑃 = 𝜇(𝑃, 𝑡)𝑑𝑡 + 𝜎(𝑃, 𝑡)𝑑𝑧
𝑑𝐹 = M𝜕𝐹𝜕𝑡 +𝜕𝐹𝜕𝑃𝜇(𝑡, 𝑃) +12𝜕
Q𝐹
𝜕𝑃Q𝜎Q(𝑡, 𝑃)R 𝑑𝑡 + M
𝜕𝐹
𝜕𝑃𝜎(𝑡, 𝑃)R 𝑑𝑧
Earlier we argued that 𝑃 follows a GBM, then it results from Itô’s lemma that the function 𝐹(𝑡, 𝑃) follows the process:
𝑑𝐹 = ;𝜕𝐹 𝜕𝑡 + 𝜕𝐹 𝜕𝑃𝜇 + 1 2 𝜕Q𝐹 𝜕𝑃Q𝜎Q𝑃QD 𝑑𝑡 + 𝜕𝐹 𝜕𝑃𝜎𝑑𝑧
Where the drift rate is equal to the part between the brackets before 𝑑𝑡, and the volatility is the part before 𝑑𝑧, and where 𝑑𝑧 is equal to the same Wiener process.
With use of Itô's lemma we could derive the Bellman equation. The maximum value of the investment over time could be solved with dynamic programming by using the Bellman equation. Dynamic programming is especially useful when time and uncertainty appear together (Dixit, 1990). The Bellman equation is the sum of the immediate cash flow and the continuation value of the project. The expected value of the project is given by:
𝐹 = 𝐸 T𝐹 + ;𝜕𝐹 𝜕𝑡 − 𝑟𝐹 + 𝜕𝐹 𝜕𝑃𝜇 + 1 2 𝜕Q𝐹 𝜕𝑃Q𝜎Q𝑃QD 𝑑𝑡 + 𝜕𝐹 𝜕𝑃𝜎𝑑𝑧U
The stochastic term (𝑑𝑧) is in expectation zero. Therefore, rearranging this formula gives:
𝑟𝐹 =𝜕𝐹 𝜕𝑡 + 𝜇𝑃 𝜕𝐹 𝜕𝑃+ 1 2𝜎Q𝑃Q 𝜕Q𝐹 𝜕𝑃Q
Where 𝑟𝐹 is the growth in the value, VWV) represents the immediate cash flow, 𝜇𝑃VWVX is the growth in value due to expected growth in the CO2 price, and YQ𝜎Q𝑃Q V
>W
VX> is the correction for
uncertainty. This formula is similar to the general Black-Scholes partial differential equation in the real option theory. In this Black Scholes equation, the derivative's value must be independent of the investor's risk preferences; this means that all assets earn the riskless rate of interest. Therefore, the drift rate (𝜇) is replaced by the interest rate (𝑟). However, in the case of the CCU plant, we do not earn the risk-free rate and the growth rate of the price and costs are things that matter; therefore, we use 𝜇.
Eventually, we are interested in a model with two stochastic variables, the CO2 price Z𝑃<=Q[
and variable OPEX (𝐶). It is assumed that the CO2 price and variable costs are uncorrelated
𝑟𝐹(𝑃, 𝐶, 𝑡) =𝜕𝐹 𝜕𝑡 + 𝜇<=>𝑃 𝜕𝐹 𝜕𝑃+ 𝜇A𝐶 𝜕𝐹 𝜕𝐶+ 1 2𝜎YQ𝑃Q 𝜕Q𝐹 𝜕𝑃Q+ 1 2𝜎QQ𝐶Q 𝜕Q𝐹 𝜕𝐶Q
With this partial differential equation, the whole region of values of (𝑃, 𝐶, 𝑡) where investment occurs, and the whole region where investment will not occur, could be found. Furthermore, the threshold curve separating the two regions will be achieved. However, it is too complicated to obtain these values analytically. Therefore, a step-by-step approach is needed. We will first discuss two models with one stochastic variable. In the first model, the CO2 price is stochastic,
and the variable costs are given, and the second model is structured the other way around. Furthermore, to solve the analytical models with one stochastic variable, we should assume that the time to invest is infinite. This assumption ensures that the differential equation is independence from time 𝑡, which makes calculations easier (Dixit & Pindyck, 1994). Once we solved the analytical models, the numerical solution is examined using the finite difference method. With this numerical solution, we could explore the effect of a finite time horizon, and two stochastic variables instead of one.
5.2 Analytical solution with one stochastic variable
In this section, the two analytical models with one stochastic variable will be solved. In these models, we assume time independence. For both models, we determine the analytical threshold values. The threshold value is the project's value where the waiting regime and the continuation value meet, as discussed in chapter 4. In the waiting regime, the option value satisfies the Bellman equation. Moreover, once the investment is made, the continuation value is equal to the NPV of the option.
Stochastic CO2 price
First, we will discuss the model where only the price of CO2 is stochastic. In this model, it is
assumed that the variable cost is given and not stochastic. During the waiting regime, the value of the investment opportunity satisfies the Bellman equation:
𝑟𝑉Z𝑃<=>[ = 𝜇𝑃<=>𝑉^(𝑃 <=>) +
1
2𝜎Q𝑃<=>Q𝑉"(𝑃<=>)
Where 𝑉 is the value of the option, the cash flow part equals zero because there are no profit flows while waiting and the model is independent of time, 𝑃<=> is the CO2 price, 𝜇 the drift
rate, and 𝜎 is the volatility. To solve this equation, a general guess is that 𝑉Z𝑃<=>[ = 𝑃<=>` . This gives the general solution to the Bellman equation:
Where 𝐴 and 𝐵 are constants to be determined. And where it is important that the two roots satisfy the condition that 𝛽Y > 1 and 𝛽Q < 0. The general formulas for 𝛽 are:
𝛽Y = −𝜇 − 12𝜎 Q 𝜎Q + f(𝜇 − 12𝜎Q)Q+ 2𝑟𝜎Q 𝜎Q > 1 𝛽Q = −𝜇 − 12𝜎 Q 𝜎Q − f(𝜇 − 12𝜎Q)Q+ 2𝑟𝜎Q 𝜎Q < 0
The solution for the Bellman equation is valid over the range of CO2 prices for which it is
optimal to hold the option and stay in the waiting regime. Because higher prices make investment more attractive, the waiting regime starts at zero and ends at the investment threshold 𝑃<=∗ >: 𝑉Z𝑃<=>[ = 𝐴𝑃<=`b> + 𝐵𝑃<=`>> 𝑉Z𝑃<=>[ = g('5Xhi> hi>− < ('5j+ XhklXi> ( m × 𝑄 − W< ( − 𝐼 𝑃 = 0 𝑃<=∗ >
If 𝑃 is above some threshold value 𝑃<=∗ >, it is optimal to invest, and the continuation value of
the option is equal to the NPV of the project which we saw in chapter 4: 𝑉Z𝑃<=>[ = ; 𝑃<=> 𝑟 − 𝜇<=> − 𝐶 𝑟 − 𝜇A+ 𝑃<C + 𝑃=> 𝑟 D × 𝑄 − 𝐹𝐶 𝑟 − 𝐼
We have to figure out what the constants A and B are. And we have to find the investment threshold Z𝑃<=∗ >[, the point where the regimes meet. We will use three conditions to do this.
The first condition states that the value of the option should be zero if the price of CO2 equals
zero. If the price of CO2 equals zero, 𝑃`b also goes to zero because 𝛽Y > 1, so this satisfies the
condition. However, 𝑃`> goes to infinity because 𝛽
Q < 0, so this does not meet the condition.
Therefore, we need 𝐵 = 0. The solution of the Bellman equation in the waiting regime is now equal to 𝑉Z𝑃<=>[ = 𝐴𝑃<=>`b .
The second condition is called the value matching condition. Which states that, as we get near the point of investment, the value of the option must be near the net present value obtained by exercising it. The value of the option is continuous as we cross the point 𝑃<=∗ >. This gives:
The third condition is called the smooth pasting condition. Meaning that the derivatives must be equal at the threshold point, so the slopes must be equal at the threshold. Taking the derivatives with respect to the price of CO2 on both sides gives:
𝑉^Z𝑃 <=> ∗ [ = 𝐴𝛽 Y𝑃<=> `b'Y = 𝑄 𝑟 − 𝜇<=>
The value matching and smooth pasting conditions are used to determine the investment threshold price and the constant; these are calculated in appendix A:
𝑃<=∗ > 𝑟 − 𝜇<=> = g 𝐶 𝑟 − 𝜇A − 𝑃<C+ 𝑃=> 𝑟 m × 𝛽Y (𝛽Y− 1)+ 𝐹𝐶 𝑟 𝛽Y (𝛽Y− 1)𝑄+ 𝛽Y (𝛽Y− 1)𝑄 𝐼 𝐴 = ; 𝑃<=> ∗ 𝑟 − 𝜇<=> − 𝐶 𝑟 − 𝜇A+ 𝑃<C+ 𝑃=> 𝑟 D × 𝑄 𝑃<= > ∗`b − 𝐹𝐶 𝑟𝑃<= > ∗`b − 𝐼 𝑃<= > ∗`b
Stochastic variable OPEX
Now the model in which the variable OPEX is stochastic will be discussed. In this case, it is assumed that the CO2 price is given. The main difference with the stochastic price model is
that when the variable OPEX is too high, we should stay in the waiting regime. During the waiting regime, the value of the investment opportunity satisfies the Bellman equation:
𝑟𝑉(𝐶) = 𝜇𝐶𝑉^(𝐶) +1
2𝜎Q𝐶Q𝑉"(𝐶)
Where 𝐶 represents the variable OPEX, and the other variables have the same meaning as in the stochastic price model. Again, a general solution is found by plugging in 𝑉(𝐶) = 𝐶`:
𝑉(𝐶) = 𝐴𝐶`b + 𝐵𝐶`>
Where 𝐴 and 𝐵 are constants, and the conditions 𝛽Y > 1 and 𝛽Q < 0 should be satisfied. When the variable OPEX are above the threshold value, it is optimal to hold the option and stay in the waiting regime. Since lower costs make investment more attractive, the waiting regime starts at the investment threshold 𝐶∗ and goes to infinity. It is optimal to invest when 𝐶 is below
the threshold value 𝐶∗, the continuation value of the option is equal to the NPV of the project:
𝑉(𝐶) = g('5Xhi> hi>− < ('5j+ XhklXi> ( m × 𝑄 − W< ( − 𝐼 𝑉(𝐶) = 𝐴𝐶 `b+ 𝐵𝐶`> 𝐶 = 0 𝐶∗
To find the constants and the investment threshold (𝐶∗) we have to use the three conditions.
to infinity, which implies that 𝐴 = 0. The solution of the Bellman equation in the waiting regime is now equal to 𝑉(𝐶) = 𝐵𝐶`>.
Again, the value matching and smooth pasting conditions are used to determine the investment threshold costs and the constant, the equations obtained from Appendix B equal:
𝐶∗ 𝑟 − 𝜇A = ; 𝑃<=> 𝑟 − 𝜇<=>+ 𝑃<C + 𝑃=> 𝑟 D × 𝛽Q (𝛽Q − 1)− 𝐹𝐶 𝑟 𝛽Q (𝛽Q− 1)𝑄− 𝛽Q (𝛽Q− 1)𝑄 𝐼 𝐵 = ; 𝑃<=> 𝑟 − 𝜇<=> − 𝐶∗ 𝑟 − 𝜇A+ 𝑃<C+ 𝑃=> 𝑟 D × 𝑄 𝐶∗`> − 𝐹𝐶 𝑟𝐶∗`> − 𝐼 𝐶∗`> 5.3 The explicit finite difference method
In the previous section, we assumed that the differential equation for the analytical model was time-independent. However, we aim to examine the effects of time dependence on the investment decision. Moreover, the purpose of this paper is to investigate the effects of two stochastic variables. Both cannot be solved by using the analytical model, but we could solve them numerically by using the finite difference method. The finite difference method is used to simplify the Black Scholes partial differential equation, by replacing the partial differentials with finite differences.
Time dependence
In this section, the finite difference method will be discussed, and how we could use it to examine the effects of time dependence on the investment decision in the CUU plant. We could expect that the difference, between an infinite option to invest and an option that expires within five years, matters for the investment decision. For time dependence, we still look at the model with one stochastic variable. Here we will discuss the explicit finite difference method on the basis of the model with the stochastic CO2 price.
In section 5.1, we determined the Bellman equation for a model with a stochastic CO2 price
following a GBM. We argued that this equation was similar to the general Black-Scholes partial differential equation, which we need for the explicit finite difference method. However, for the finite difference method, it is more efficient to use 𝑥 = ln(𝑃). We could use Itô’s lemma to derive the process followed by 𝑝 = ln(𝑃) since 𝑃 follows a GBM (Hull, 2015):
Where 𝑣 = 𝜇 −s>
Q, in which 𝜇 and 𝜎 are constant. Furthermore, the drift rate (𝑣) and variance
rate (𝜎Q) are constant. Therefore, the change in ln(𝑃) overtime is normally distributed, with
mean 𝑣𝑑𝑡 and variance rate 𝜎Q𝑑𝑡. The Black Scholes partial differential equation becomes:
𝑟𝑓 =𝜕𝑓 𝜕𝑡 + 𝑣𝑝 𝜕𝑓 𝜕𝑝+ 1 2𝜎Q𝑝Q 𝜕Q𝑓 𝜕𝑝Q
Where 𝑓 is the value of the option and where the coefficients are constant, so they do not depend on 𝑝 or 𝑡, which makes the application of the finite difference method easier.
Unlike the model we discussed in section 5.2, the finite difference method assumes discrete time, ∆𝑡, steps. Moreover, within these time steps, the CO2 price could change with ∆𝑝, which
is called a space step. These discrete time and space intervals give rise to the finite difference grid (also called lattice). Where the (𝑖, 𝑗) point on the grid is the point that refers to the time 𝑖∆𝑡 and price 𝑗∆𝑝. The variable 𝑓x,y indicates the value of the option to invest at the (𝑖, 𝑗) point (Hull, 2015).
To obtain the explicit finite method, we divide the Black-Scholes partial differential equation by using forward differences and central differences. For VzV) the forward difference approximation is used, such that the value at time for 𝑖∆𝑡 is related to the value at time (1 + 𝑖)∆𝑡. Central difference approximations are used for Vz
V? and V>z
V?>, for which the values at
point (𝑖, 𝑗) on the grid are assumed to be the same as at point (𝑖 + 1, 𝑗) (Hull, 2015). So, the explicit finite difference method is represented by:
𝑟𝑓xlY,y = 𝑓xlY,y− 𝑓x,y
∆𝑡 + 𝑣
𝑓xlY,ylY− 𝑓xlY,y'Y
2∆𝑝 +
1 2𝜎Q
𝑓xlY,ylY− 2𝑓xlY,y+ 𝑓xlY,y'Y ∆𝑝Q
Within the time interval ∆𝑡, the price could go up (𝑗 + 1), stay the same (𝑗), or could go down (𝑗 − 1) by ∆𝑝, with probabilities 𝑝{, 𝑝| and 𝑝}. Therefore, we could rewrite the equation to obtain the value of the option at the (𝑖, 𝑗) point as:
𝑝} = 1 2∆𝑡 ;g 𝜎 ∆𝑝m Q − 𝑣 ∆𝑝D
Where ∆𝑝 = 𝜎√3∆𝑡, in which ∆𝑡 =•€, where 𝑇 represents the life of the option and 𝑁 is the interval of length. And where:
𝑝{+ 𝑝|+ 𝑝} = 1
This explicit finite formula is also illustrated in figure 1, where one could see that the method provides the relationship between one value of the option to invest (𝑓x,y) at time 𝑖∆𝑡 and three possible values of the option Z𝑓xlY,y'Y, 𝑓xlY,y, 𝑓xlY,ylY[ at time (𝑖 + 1)∆𝑡 (Hull, 2015).
∆𝑡
Figure 1: The explicit finite difference method, where fƒ,„ indicates the option value (Hull, 2015). 𝑖 + 1 indicated the time interval ∆𝑡
The finite difference method works backwards in time through the grid. So, it first calculates the nodes at time 𝑇, which is the total life of the option also called the time to maturity, then it steps back N times until it reaches all the column of present values of the option. With this method, we could insert different time values 𝑇 and observe the effects of this. By working backwards in time, the finite difference method finds the present value of the threshold price of CO2 for which it is optimal to invest in the CCU plant. If we assumed an infinite lifetime of
the option, this threshold value would be the same as in the analytical case.
Two stochastic variables
The purpose of this research is to evaluate a model depending on two stochastic variables, the CO2 price and variable OPEX, and on time. In this case, we have to find the whole region of
numerically by using the finite difference method. It is assumed that the price of CO2 and the
variable costs are uncorrelated and that they both follow a geometric Brownian motion. The partial differential equation in the two stochastic variable case is given by:
𝑟𝑉(𝑃, 𝐶, 𝑡) =𝜕𝑉 𝜕𝑡 + 𝜇<=>𝑃 𝜕𝑉 𝜕𝑃+ 𝜇A𝐶 𝜕𝑉 𝜕𝐶+ 1 2𝜎YQ𝑃Q 𝜕Q𝑉 𝜕𝑃Q + 1 2𝜎QQ𝐶Q 𝜕Q𝑉 𝜕𝐶Q Rewriting gives: 𝑟𝑓 =𝜕𝑓 𝜕𝑡 + 𝑣<=> 𝜕𝑓 𝜕𝑝+ 𝑣A 𝜕𝑓 𝜕𝑐 + 1 2𝜎YQ 𝜕Q𝑓 𝜕𝑝Q+ 1 2𝜎QQ 𝜕Q𝑓 𝜕𝑐Q
Where 𝑓 is the value of the option depending on the CO2 price, variable OPEX, and time,
moreover, 𝑣<=> = †𝜇<=> −YQ𝜎<=Q >‡ and 𝑣
< = †𝜇<−YQ𝜎<Q‡ and where the natural logarithms of
the assets are used, 𝑝 = ln 𝑃 and 𝑐 = ln 𝐶, which provide constant coefficients.
The finite difference method depending on two stochastic variables also assume discrete time steps (∆𝑡). Within these time steps the CO2 price could change with ∆𝑝 and the costs could
change by ∆𝑐. To obtain the explicit finite method we approximate the Black-Scholes partial differential equation using forward differences †VzV)‡ and central differences †VzVˆ‡ & †VVˆ>z>‡. So, the explicit finite difference method based on two stochastic variables is defined by:
𝑟𝑓xlY,y,‰ =𝑓xlY,y,‰− 𝑓x,y,‰
∆𝑡 + 𝑣Y 𝑓xlY,ylY,‰− 𝑓xlY,y'Y,‰ 2∆𝑥Y + 𝑣Q 𝑓xlY,y,‰lY− 𝑓xlY,y,‰'Y 2∆𝑥Q +1 2𝜎YQ
𝑓xlY,ylY,‰− 2𝑓xlY,y,‰+ 𝑓xlY,y'Y,‰
∆𝑥YQ
+1
2𝜎QQ
𝑓xlY,y,‰lY− 2𝑓xlY,y,‰+ 𝑓xlY,y,‰'Y ∆𝑥QQ
Where the (𝑖, 𝑗, 𝑘) point on the grid is the point that corresponds to the time 𝑖∆𝑡 and the joint evolution of CO2 price 𝑗∆𝑝 and variable OPEX 𝑘∆𝑐. The variable 𝑓x,y,‰ is used to denote the
value of the option to invest at the (𝑖, 𝑗, 𝑘) point.
In this case, the explicit finite method gives the relationship between one value of the option Z𝑓x,y,‰[ at time 𝑖∆𝑡 and nine different value of the option at time (𝑖 + 1)∆𝑡 instead of three. This means that within the time interval ∆𝑡, the price and costs could go up, stay the same, or could go down by ∆𝑝 and ∆𝑐, with nine different probabilities. Therefore, we could rewrite the equation to obtain the value of the option at the (𝑖, 𝑗, 𝑘) point as:
However, the explicit finite difference method is not able to consider the corner nodes. Therefore, only the probabilities for the five middle nodes are considered in the numerical model, these are equal to:
𝑝{| =1 2∆𝑡 ;g 𝜎<=> ∆𝑝 m Q +𝑣<=> ∆𝑝 D 𝑝|{ = 1 2∆𝑡 g† 𝜎< ∆𝑐‡ Q +𝑣< ∆𝑐m 𝑝|| = 1 − ∆𝑡 ;g𝜎<=> ∆𝑝 m Q + †𝜎< ∆𝑐‡ Q D − 𝑟∆𝑡 𝑝|} = 1 2∆𝑡 g† 𝜎< ∆𝑐‡ Q −𝑣< ∆𝑐m 𝑝}| =1 2∆𝑡 ;g 𝜎<=> ∆𝑝 m Q −𝑣<=> ∆𝑝 D Where: 𝑝{{ + 𝑝{|+ 𝑝{} + 𝑝|{+ 𝑝||+ 𝑝|}+ 𝑝}{ + 𝑝}|+ 𝑝}} = 1
By working backwards in time, the finite difference method finds the whole region of present values Z𝑃<=>, 𝐶, 𝑡[ where investment in the CCU plant occurs, and the whole region where it will not occur. In this paper, we perform the finite difference method by using a computer programming language, called Python. The results of this numerical model are analyzed in chapter 7. We could predict that the investment region is located at high CO2 prices and low
variable costs. And the region where investment will not occur is where costs are high, and CO2 prices are low.
6. Data and calibration
In this chapter, the numerical input for the model is illustrated. Because of confidential reasons, the data in this section is not exactly the CCU plant's real data from +Earth. We examine the data with respect to the Dutch market. Firstly, we discuss the parameters of the stochastic variables. Secondly, the main cash flow variables for the firm are elaborated. Thirdly, the discount rate will be derived.
6.1 Parameters of the Geometric Brownian Motion
The price of CO2
The first stochastic variable in this model is the price of carbon dioxide (CO2). With the price
of CO2, we refer to the price an emitting firm pays to get rid of its excess CO2. It is anticipated
that +Earth will get this amount as "gate fee" for the off-take. For this model, we assume that the emitting firm pays for capturing CO2. Furthermore, we assume that the transport costs of
CO2, from the emitting firm to the CCU firm, is zero. This is because we assume for simplicity
that the CCU firm will be located next to the emitting firm.
In September 2020, the Dutch government proposed that companies have to pay a tax of €30 per ton for emitted CO2 in 2021. This is expected to increase to €125-150 per ton in 2030. This
tax should encourage heavy polluters to make their process more sustainable, by emitting less CO2. Heavy industries and energy producers in Europe are already paying a price for their CO2
emissions through a European trading system (ETS), of which the price per ton CO2 was around
€25 in 2020. The European tax will be deducted from the tax proposed by the Dutch government (McDonald O. , 2020).
The proposition of the Dutch government is in line with the Climate Agreement. Environmental parties insisted on additional CO2 tax in the Netherlands. However, the industrial sector stated
that the European tax was enough and feared that the competitive position would deteriorate if other European countries did not introduce such a national tax. The latter is also a fear of the Dutch government, so they decided that Dutch industries only have to pay extra tax on that part of the CO2 emissions that could have been avoided. To determine this, they look per sector at
the 10% companies in Europe with the lowest emissions. If a Dutch company in that sector emits more than this reference point, it must pay the Dutch tax on that part (McDonald O. , 2020).
As one could expect, the carbon tax creates a corporate risk for large emitting companies. These large emitting industries are not only facing heavy carbon taxation, but also reputations issues. These reputation issues arise from the fact that a company's climate policy will be affected by stakeholder expectations and social responsibility standards. Furthermore, there is a growing public interest in climate-friendly companies and products. Poor marks on reporting and managing climate impact could put a company's reputation at risk (Porter & Reinhardt, 2007). One way to decrease CO2 emissions and obtain a good reputation is to capture excess carbon
dioxide. The technology of +Earth is a CCU power plant which creates carbon black from (captured) pure carbon dioxide. In order to mitigate carbon taxes, heavy polluters want to pay a company like +Earth to get rid of their excess CO2. The maximum price the polluter wants
If we look from the point of view of a company like +Earth, the maximum price they get for a ton of carbon dioxide is the price a polluter wants to pay in order to get rid of their excess CO2.
This price is also called a gate fee, which is the charge levied upon a given quantity of, in this case, CO2, waste received at a waste processing facility. The height of the gate fee thus depends
heavily on the decisions of the Dutch government and the Paris Agreements on the taxation of CO2 emissions, and on the expectation of shareholders and consumers of emitting companies.
The uncertain decisions of the government, uncertain shareholder/consumer expectations, and the uncertain behavior, in terms of taxes on CO2, of other countries, create an uncertain price
of CO2 for a company like +Earth. The CO2 price is expected to increase over time, but by how
much is uncertain in the long run. As things stand at present, companies that fail to achieve the agreed reduction in CO2 emissions, risk paying €125 in 2030 for each excess ton of
CO2 emitted. If new insights emerge, for example, new technologies, the rate will be reviewed,
after a recalculation by the Netherlands Environmental Assessment Agency (Dutch Government, 2020). In this model, we investigate different lifetimes of the option to invest in the CCU plant, and we do not know whether or when new technologies will emerge, whether the tax rate changes or how the future expectations of shareholders/consumers will emerge, these factors make the price of CO2 uncertain. The CO2 price is considered to be uncertain in
other studies (see, e.g. Yu, Li, Wei, and Liu, 2019; Zhang and Wei, 2011; Heydari, Ovenden, and Siddiqui, 2010)
This uncertainty of the price has an enormous impact on the decision of investing in the CCU plant. The price the company gets for the CO2 is a large partof the total revenue of +Earth. As
one could expect, this revenue variates already a lot with a difference in CO2 price between
€30 or €125 per ton CO2, but it could increase a lot more if the taxes are getting higher or if
shareholders impose stricter requirements. In this model, we set the base price at €125, because this will be the expected tax per ton CO2 from 2030.
The volatility of the CO2 prices is obtained from a report done by AFRY (2020), who modelled
sensitivities for all European ETS markets, including the Netherlands. Therefore, we assume the same volatility of carbon dioxide prices, which is 𝜎<=> = 20%. Moreover, the growth rate
of the price of CO2 is assumed to be 𝜇<=> = 0.02 for the base case. The operational costs
These annual fixed costs include the rental costs of the building, maintenance costs, and overhead costs. The variable OPEX are costs which variate with the quantity produced, which consists of the production costs, such as human resources and utilities, and transport costs. We will elaborate further on the three categories of the variable OPEX because these are stochastic in this model. Firstly, the transport costs for the produced carbon black and oxygen are different for national or international transport. Secondly, although the cost of human resources is not entirely variable per ton, for the purpose of this research, it is assumed to be variable per ton. Human resources contain operators, engineers and some overheads. Thirdly, the utilities are variable; these consist of the price of electricity, the price of steam and the price of water.
The uncertainty in the variable OPEX has an enormous impact on the total costs of investing in the CCU plant. The total OPEX is equal to approximately €5 million. The variable OPEX is significant and estimated at 85% of the total OPEX, assuming that the price of electricity equals €0.06. Thus, these unpredictable variable OPEX have a significant impact on the bottom line, the business case and decision to invest. In the model, we set the base variable OPEX at €450 per ton CO2.
The growth rate of the variable OPEX is for the base case equal to 𝜇A = 0.02. Furthermore, the volatility for the variable OPEX is expected to be 𝜎A = 0.2, based on the sensitivity assumption of high OPEX projects (AFRY, 2020).
6.2 Cash flow parameters
In this subsection, the residual cash flow parameters to calculate the NPV will be elaborated. The CCU firm creates carbon black, and pure oxygen as a byproduct, from the carbon dioxide.
Sales price of carbon black
The CCU firm creates carbon black from the carbon dioxide, which they can sell in the market as it is a commodity. The sales price for carbon black is set at the market value of ‘green’ carbon black, which is equal to €600 per ton CO2. Carbon black is used for manufacturing tires,
plastics, mechanical rubber goods, printing inks, and toners. The rise in the use of these products causes the carbon black market to grow. However, government regulations impact the carbon black market trends, as the manufacturing process of carbon black usually releases harmful air pollutants (Choudhary & Prasad, 2020). This increasing demand for carbon black and government regulations is convenient for +Earth who create ‘green’ carbon black. From a macroeconomic perspective, increasing demand could lead to an increase in the sales price of carbon black in the future. Furthermore, it makes green carbon black more attractive from a polluting point of view, and for the regulations that come with it.
Pure oxygen (O2)
Fixed OPEX
As mentioned before, the OPEX is divided into two parts, the fixed and variable part. The fixed part is 15% of the total OPEX. The fixed OPEX are fixed annual costs which do not variate with the quantity produced and consists of the rental cost of the building, the maintenance costs, and overhead costs. Total fixed OPEX equals €500,000.
Investment costs
The investment costs are a once off payment at the beginning of the period if the firm decides to invest. The investment costs, also called the CAPEX in this model, is equal to €3 million. These expenses consist of costs for the investment in the design, engineering and building of the plant like the equipment, installation, the reactor, electrolyzer, piping, tanks control measures, etc.
Quantity
The quantity that the CCU firm could produce is set at 10,000 ton of CO2 per year. In the
formula, we take into account the conversion factor of the quantity of carbon black and pure oxygen produced per ton CO2.
6.3 The weighted average cost of capital
To value the firm using the discounted cash flow approach, we discount the forecast of free cash flows at the weighted average cost of capital (WACC). The WACC represents the rate of return that investors of a company expect to earn for investing their funds in one particular business instead of others with similar risk, also referred to as the opportunity cost. To determine this WACC, we follow the approach of Koller et al. (2015).
The WACC equals the weighted average of the after-tax cost of debt and cost of equity:
𝑊𝐴𝐶𝐶 =𝐷
𝑉𝑘}(1 − 𝑇|) + 𝐸 𝑉𝑘Ž
In which •• is the target level of debt to enterprise value, ‘• is the target level of equity to enterprise value both using market-based values, 𝑘} is the cost of debt, 𝑘Ž is the cost of equity,
and 𝑇| is the company’s marginal income tax rate.
capital asset pricing model (CAPM). The CAPM adjusts for company specific risk through the use of beta, which determines how the firm responds to movements in the market. The formula for the CAPM is used to estimate the cost of equity for +Earth, which equals:
𝑘Ž= 𝑟z+ 𝛽Ž× 𝑀𝑅𝑃
Where 𝑘Ž is equal to the levered cost of equity, 𝑟z is the risk-free rate, 𝛽‘ is the levered equity beta, and 𝑀𝑅𝑃 is the market risk premium which is equal to the expected return of the market minus the risk-free rate.
To estimate the risk-free rate in typical times, the current yield on long-term government bonds is used. For valuing European companies, it is typical to use 10-year German government bonds. However, currently, the yield on the 10-year Government bonds is below zero (Bloomberg, 2020). In this case, with current low interest rates, equity values may be too high, and valuation models do not lead to sensible results. To overcome this inconsistency, Koller et al. (2015) recommend using a synthetic risk-free rate. To set up this synthetic rate, we have to add the expected inflation rate to the run average real interest rate. The expected long-term inflation rate is set equal to 1.7% (ECB, 2020). Moreover, the long-run average real interest rate is determined with the 30-year Dutch government bond rate and is set at 0.85% (Market Watch, 2020). So, the risk-free rate in this model is equal to 2.55%. If the economy returns to historical levels, this perspective should be reevaluated.
For the cost of equity, we also need the levered equity beta, which equals:
𝛽Ž= 𝛽{+
𝐷
𝐸× (𝛽{− 𝛽})
Where 𝛽{ is the unlevered equity beta, •‘ is the debt to equity ratio, and 𝛽• is the beta of debt. In practice, measurements of beta are highly imprecise. Therefore, use a set of peer company betas adjusted for financial leverage are used to estimate +Earth’s betas. To find the levered and unlevered betas, data from Damadaran (2020) is obtained. The CCU plant is assumed to be considered in the 'environmental and waste services' industry because the firm turns CO2
waste into environmental green tires. With this information, we find that 𝛽{ = 0.76.
To calculate the beta of debt, the cost of debt is used, which is given by: 𝑘• = 𝑟W+ 𝛽•× 𝑀𝑅𝑃 = 𝑟W+ 𝑐𝑠 Rewriting gives:
𝛽• × 𝑀𝑅𝑃 = 𝑐𝑠
𝛽• = 𝑐𝑠
Where 𝑟z is the risk-free rate, 𝛽• is the beta of debt, 𝑀𝑅𝑃 is the market risk premium, and 𝑐𝑠
represents the credit spread. The credit spread describes the difference between the yield a company's debt and on the risk-free debt. Companies with a higher probability of default have a higher credit spread than companies with a low probability of default. For +Earth, the credit spread is equal to 2.95% (Damodaran, 2020). The market risk premium is defined by the difference between the expected return on the market portfolio and the risk-free rate. The market risk premium used here equals to 6.1% (Damodaran, 2020).
With these inputs, we could calculate the answers to the formulas above, where 𝛽• = 0.49, 𝛽Ž = 0.88, 𝑘• = 5.50%, and 𝑘Ž = 7.84%.
To calculate the WACC, we now only need the marginal tax rate, which is equal to 25% in the Netherlands. The government announced earlier this year that high corporate tax rate will be reduced. However, this announcement will be scrapped because the funds will enable the government to strengthen the economy during the critical times of the coronavirus (Dutch Government, 2020). This gives the WACC, which is equal to 𝑊𝐴𝐶𝐶 = 6.50%.
Table 1 provides a summary of the parameters for our model. It should be noted that an infinite time is assumed to be equal to 50 years to maturity in the numerical model.
Table 1: Values of parameters in base case
Parameter Base case value Unit
Price of CO2 125 € per ton CO2
Variable costs (OPEX) 450 € per ton CO2
Sales price of carbon black 600 € per ton CO2
Sales price of oxygen (O2) 70 € per ton CO2
Fixed OPEX 500,000 €
Investment cost 3,000,000 €
Quantity 10,000 per year
Volatility CO2 20%
Volatility variable costs 20%
Drift rate CO2 2%
Drift rate variable costs 2%
Discount rate 6.5%
7. Results
In this chapter, the results of the models will be discussed. First, the traditional NPV method will be discussed. Second, we will determine the analytical results for the one stochastic variable cases and analyze different rates of volatility. Third, the numerical results will be explored for the case with two stochastic variables for indefinite period of time as well as for specific time values. Lastly, we will evaluate these three different analyses to value the investment in the CCU plant and conclude if it is necessary to compute a real option analysis, with one or two stochastic variables, for investment opportunities in a new plant.
7.1 Results for the NPV approach
The traditional approach to value an investment is the NPV approach, which states that if the NPV is positive, the investment should be made. As mentioned in section 3.1, critics of the NPV approach state that it could lead to undervalued investment opportunities as it ignores irreversibility, uncertainty of future rewards, and the choice of timing the investment. So, critics argue that one should use a real option approach to value an investment that deals with these elements. For the results, we use the NPV as benchmark in order to examine if the addition of these real option factors change the investment decision in the CCU plant.
Figure 2 presents the possible combinations of CO2 prices and variable costs which set the
NPV formula equal to zero, represented by the blue line. A higher CO2 price means that the
emitting firms pay more to +Earth in order to get rid of their excess CO2, this could occur
because of a higher tax rate for emissions or higher reputation costs for emitting firms. As one could observe from the figure, the higher the variable costs, the higher the CO2 price paid to
+Earth should be. Furthermore, it could be concluded that the part on the right-hand side of the line gives a positive NPV, as the price is the same, but the variable costs are lower for these points in the figure.
If the price of CO2 is equal to zero, which means that an emitting firm pays nothing to +Earth
to get rid of their excess CO2, the variable cost could be maximal €415 per ton CO2 in order to
obtain an NPV which is equal to or higher than zero. This is explained because +Earth is a CCU plant which transforms CO2 into carbon black and pure oxygen, which are sold in the
market, so the price of CO2 is not their only income stream. In the base case scenario, the
variable costs are equal to €450 per ton CO2, which means that the CO2 price should be at least
€34 to realize a positive NPV. If we assume that the CO2 price raises to the expected market
price of €125 per ton (base case), the variable costs could increase to €540 per ton to maintain a positive NPV.
It should be noted that this NPV model does not consider any volatility in the parameters, such as CO2 price and variable costs. This means that there is no uncertainty over time, which does
not reflect reality. Furthermore, it does not address irreversibility and the choice of timing of the investment.
7.2 Analytical results
The analytical model considers the irreversibility and the choice of timing of the investment. Furthermore, it considers one stochastic variable. The two different models with one stochastic variable will be discussed.
One stochastic variable – CO2 price
In this first analytical model, we assume that the CO2 price is uncertain and that the variable
costs are given and equal to €450, which represent the base case variable costs. Figure 3 demonstrates the effects of different levels of volatility on the value of the CCU plant. In this scenario, the values of the parameters are 𝑟 = 0.065, 𝜇<=> = 0.02, and 𝜇A = 0.02. Besides the base case value of volatility, 𝜎<=> = 0.2, the scenarios with a volatility equal to 0.1, 0.3, and 0.4 are included. The vertical lines represent the CO2 price threshold values, which are the
tangency points of the line of the option value and the NPV (dark blue line), or continuation value, line. The threshold price for the base case scenario (blue lines, 𝜎<=> = 0.2) is equal to a
CO2 price of €77 per ton CO2. If the price of CO2 is below this price, the value of the option to
wait is higher than the NPV, so there will be no investment in the CCU plant. The option value is not close to the NPV line, meaning that the option has value. Another observation is that the value of the investment option increases with volatility, as does the threshold price of CO2. The
reason is that higher uncertainty in the CO2 price increases the value of waiting with the
potential from volatility while its downside is limited to zero as the firm has the option and not the obligation to invest.
The relationship between the threshold CO2 price and the volatility is also illustrated in figure
4. One could observe that the threshold prices increases with volatility. This indicates that the higher the volatility of the CO2 price, the more valuable the option to postpone investment. So,
the higher the volatility, the longer +Earth waits before they invest. It is noted that this is not a linear relationship, meaning that the volatility has a progressing impact on the threshold.
One stochastic variable – variable costs
In this second analytical model, we assume stochastic variable costs and that the CO2 price is
The advantage of this analytical model, over the NPV, is that it addresses the irreversibility, uncertainty and choice of timing. However, the limitations of this model are that it deals with only one stochastic variable and that this assumes an infinite period of time to maturity for the option, which are both not in line with reality.
7.3 Numerical results
In this section, we focus on the numerical model for which the value depends on two stochastic variables and on specific time periods for the option. First, we will shortly discuss the time-dependent models in the case of one stochastic variable. Second, the grid for the model with two uncertain variables will be explained. Third, we investigate different scenarios for the model with two stochastic variables.
Time dependence on one stochastic variable models
