A Summary on the Domestic Energy Resources of Turkey 4.3
5 THE QUANTIFICATION OF THE TRIGGER PRICES FOR INVESTMENTS
In this chapter, information is given about the dynamic programming approach for quantifying the investment trigger prices. The given information is based on the text book
“Investment under Uncertainty” by Dixit and Pindyck (1994) and the article “Investment and Hysteresis” by Dixit (1992) as the main references. Detailed information can be found in the Chapters 4 and 5 of the corresponding book and in the corresponding journal.
The trigger price for an investor’s flexible investment decision is defined to be indicating the price which is sufficient to make an investment decision considering the value of the project to be uncertain; whereas the investment costs, operational costs and the expected amount of annual electricity generation by RETs are assumed to be deterministic. The value of the RET projects are uncertain due to the uncertainty in the annual average wholesale market price of electricity which is assumed to follow a geometric Brownian motion (GBM):
𝑑𝑆𝑡= 𝛼𝑠 𝑆𝑡 𝑑𝑡 + 𝜎𝑠 𝑆𝑡 𝑑𝑧 (4.3.1)
The symbol “𝑆𝑡” indicates the annual average market price of electricity in year 𝑡. The symbols “𝛼𝑠” and “𝜎𝑠” are constants and represent the drift rate and the volatility of the corresponding electricity price respectively. In Eq. (4.3.1), the last term indicates the standard Brownian motion and is explicitly expressed in Eq. (4.3.2). The term “𝜖𝑡” is considered to be a normally distributed random variable with a mean of zero and a standard deviation of one.
The term “𝑑𝑡” denotes the time increment.
𝑑𝑧 = 𝜖𝑡√𝑑𝑡 (4.3.2)
The expected present value of the project ( 𝑉) is a linear function of the expected market price of electricity following GBM and is discounted by utilizing continuously compounding rate factor (𝜌) as represented below:
𝑉(𝑆) = 𝐺𝑒𝑙∫ (𝑆0
𝑇 0
𝑒𝛼𝑠 𝑡)𝑒−𝜌𝑡𝑑𝑡 (4.3.3)
42 After taking the integral, the Eq. (4.3.3) can be expressed as follows:
𝑉(𝑆) = 𝐺𝑒𝑙∙ 𝑆0∙𝑒(𝛼𝑠−𝜌)𝑇− 1 𝛼𝑠 − 𝜌
(4.3.4)
In Eq. (4.3.4), the symbols, “𝑆0”, “𝐺𝑒𝑙” and “𝑇” denote the annual average market price of electricity in the first year of operation, the expected amount of annual electricity generation42 and the economic lifetime of the RET respectively. In Eq. (4.3.4), the future cash inflows are discounted with the required rate of return 𝜌 of the project. The revenue streams continue for 𝑇 years, after commissioning of the power plant. The expected NPV of the project (Ω(𝑆0)) can then be calculated by taking the difference between the expected revenues and the total costs as expressed below:
Ω(𝑆) = 𝑉(𝑆0) − 𝑇𝐶 (4.3.5)
The total cost of electricity generation by each type of power plant is composed of investment cost (𝐼), discounted total fixed cost of operation and maintenance (𝑇𝐹𝐶𝑂&𝑀) and discounted total variable cost of O&M (𝑇𝑉𝐶𝑂&𝑀) as indicated below:
𝑇𝐶 = 𝐼+𝑇𝐹𝐶𝑂&𝑀+ 𝑇𝑉𝐶𝑂&𝑀 (4.3.6)
It is assumed that the fixed cost of operation and maintenance (𝑓𝑐𝑂&𝑀) and the variable cost of O&M (𝑣𝑐𝑂&𝑀) develop w.r.t. the given growth/decline rate 𝑖43 and are discounted back to time zero w.r.t. the risk free rate of interest 𝑟 as indicated below for the calculation of 𝑇𝐹𝐶𝑂&𝑀:
𝑇𝐹𝐶𝑂&𝑀 = ∫ (𝑓𝑐𝑂&𝑀
𝑇 0
𝑒𝑖 𝑡)𝑒−𝑟𝑡𝑑𝑡 (4.3.7)
After taking the integral, the Eq. (4.3.7) can be expressed as follows:
𝑇𝐹𝐶𝑂&𝑀 = 𝑓𝑐𝑂&𝑀∙𝑒(𝑖−𝑟)𝑇− 1 𝑖 − 𝑟
(4.3.8)
42 Note that the investment analyses are carried out for 1 MWel installed capacity. Therefore, the expected amount of annual electricity generation equals to the annual FLHs of operation.
43 This rate is considered; in order to take into account the developments in the corresponding costs depending on the technological prospects.
43 The 𝑇𝑉𝐶𝑂&𝑀 of a power plant can be calculated as follows:
𝑇𝑉𝐶𝑂&𝑀= 𝐺𝑒𝑙∫ (𝑣𝑐𝑂&𝑀
𝑇 0
𝑒𝑖 𝑡)𝑒−𝑟𝑡𝑑𝑡 (4.3.9)
After taking the integral, the Eq. (4.3.9) can be expressed as follows:
𝑇𝑉𝐶𝑂&𝑀 = 𝐺𝑒𝑙∙ 𝑣𝑐𝑂&𝑀∙𝑒(𝑖−𝑟)𝑇− 1 𝑖 − 𝑟
(4.3.10)
If an investment opportunity on a RET is a now or never decision, an investor must give an immediate decision whether to invest now or not at all. According to the Marshall’ s analysis of long run equilibrium under competitive condition, if the market price exceeds long run average cost, investments are triggered not only for new entry but also for capacity expansion.
The trigger price (or break-even price) for an investor’s immediate investment decision (𝑆𝑖𝑚𝑚𝑒𝑑𝑖𝑎𝑡𝑒) can be calculated by equating the expected present value of the revenues to the total cost and then by solving for 𝑆𝑖𝑚𝑚𝑒𝑑𝑖𝑎𝑡𝑒 as expressed below:
𝐺𝑒𝑙∫ (𝑆𝑖𝑚𝑚𝑒𝑑𝑖𝑎𝑡𝑒
𝑇 0
𝑒𝛼𝑠 𝑡)𝑒−𝜌𝑡𝑑𝑡 = 𝑇𝐶 (4.3.11)
After taking the integral and rearranging, the Eq. (4.3.11) can be expressed as follows:
𝑆𝑖𝑚𝑚𝑒𝑑𝑖𝑎𝑡𝑒 = 𝑇𝐶 (𝛼𝑠− 𝜌)
𝐺𝑒𝑙 (𝑒(𝛼𝑠−𝜌)𝑇− 1) (4.3.12)
Accordingly, the investment rule is to invest now, if 𝑆0 > 𝑆𝑖𝑚𝑚𝑒𝑑𝑖𝑎𝑡𝑒 and otherwise reject the investment proposition. Note that in this context, 𝑆0 equals to the FiT rate of the considered RET. In analogy to the general approach, the threshold values for FLHs of operation, which are necessary to trigger investments, can be calculated due to an immediate investment decision. In this respect, the threshold FLHs of operation of a considered technology corresponds to the break-even price which is set as high as the corresponding FiT rate. The corresponding rearrangement of the Eq. (4.3.12) is expressed below:
𝐺𝑒𝑙 = 𝑇𝐶 (𝛼𝑠− 𝜌)
𝑆𝑖𝑚𝑚𝑒𝑑𝑖𝑎𝑡𝑒 (𝑒(𝛼𝑠−𝜌)𝑇− 1) (4.3.13)
44 After rearrangement, the term “𝐺𝑒𝑙” in Eq. (4.3.13) is redefined as the threshold FLHs of operation above/below which investments should be made/not be made given the FiT rates for RETs substituting for 𝑆𝑖𝑚𝑚𝑒𝑑𝑖𝑎𝑡𝑒.
An investment opportunity on a RET is considered to be a perpetual call option and an investor can decide whether to invest or to keep the option alive. Therefore, the dynamic programming approach is utilized for the comparison of the present value that results from the immediate investment decision and from waiting. The corresponding Bellman equation is expressed below:
𝐹(𝑆) = 𝑚𝑎𝑥 {𝑉(𝑆) − 𝑇𝐶, 1
1 + 𝜌𝑑𝑡𝐸[𝐹(𝑆 + 𝑑𝑆)|𝑆]} (4.3.14) The value of the investment opportunity (𝐹) is conditional on 𝑆 and equals to the maximum value among the termination value (i.e. the left term in the curly braces) and the continuation value (i.e. the right term in the curly braces). By using the dynamic programming approach, a whole sequence of decisions is divided into two components such as the immediate decision and the valuation function. The valuation function is utilized for evaluating the consequences of all subsequent decisions, starting with the state that arises from the immediate decision.
The optimal decision can be found by carrying out sequence of computations by initiating from the last decision.
If the payoff from the immediate investment is higher than the one from waiting, then the option is exercised and the investment opportunity equals the termination value. If the value of waiting is higher than the immediate payoff, then the option is held and the value of the investment opportunity equals to the continuation value as indicated below.
𝐹(𝑆) = 1
1 + 𝜌𝑑𝑡𝐸[𝐹(𝑆 + 𝑑𝑆)] (4.3.15)
In particular, the continuation value equals to the discounted expected value of all future optimal decisions according to considered stochastic process. The Eq. (4.3.15) can be expressed in rearranged form as follows:
𝜌𝐹(𝑆)𝑑𝑡 = 𝐸[𝑑𝐹] (4.3.16)
45 Accordingly, over a short period of time 𝑑𝑡, the total expected return on the investment opportunity, 𝜌𝐹(𝑆)𝑑𝑡, is equal to its expected rate of capital appreciation (i.e. its expected change in value) (Dixit & Pindyck, 1994, p. 140). It can be inferred from Eq. (4.3.16) that the investment opportunity (i.e. holding an option to invest) is supposed to yield no profits until the investment is under taken. The only return from holding it is its capital appreciation. The term “𝑑𝐹” can be expanded by using Ito’s Lemma as follows:
𝑑𝐹 =𝜕𝐹
𝜕𝑆𝑑𝑆 +1 2
𝜕2𝐹
𝜕𝑆2𝑑𝑆2 (4.3.17)
After substituting Eq. (4.3.1) for 𝑑𝑆, the Eq. (4.3.17) takes the rearranged form as indicated below (note that 𝐸(𝑑𝑧) = 0):
After dividing the Eq. (4.3.18) by 𝑑𝑡 and then rearranging it (considering Eq. (4.3.16)) yields a differential equation which is independent of time but depends on the current start price in the spot market (i.e. so called perpetual call option).
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The first boundary condition requires that if 𝑆 goes to zero, the option to invest will have a zero value as implied by the stochastic process for 𝑆 (see Eq. (4.3.1)). The second boundary condition is the value-matching condition. The value matching condition requires that upon investing, the firm receives a net pay-off which equals to expected net present value depending on the critical (trigger) price “𝑆𝑝𝑒𝑟𝑝𝑒𝑡𝑢𝑎𝑙”. The price 𝑆𝑝𝑒𝑟𝑝𝑒𝑡𝑢𝑎𝑙 is the price above
46 which it is optimal to invest according to the expected market price of electricity. The expected market price of electricity is simulated by utilizing the Monte Carlo simulation approach as explained in the next chapter. The third boundary condition is called the smooth pasting condition. The smooth pasting condition states that if 𝐹(𝑆𝑝𝑒𝑟𝑝𝑒𝑡𝑢𝑎𝑙) were not continuous and smooth at the price 𝑆𝑝𝑒𝑟𝑝𝑒𝑡𝑢𝑎𝑙, one could do better by exercising at a different point. The third boundary condition helps in finding the position of the second boundary by equating the slopes of the value of waiting and the value of investing at the price 𝑆𝑝𝑒𝑟𝑝𝑒𝑡𝑢𝑎𝑙 (see Figure 16).
Figure 16- The investment trigger prices w.r.t. the immediate decision and the decision at any time (own illustration according to Dixit (1992, p. 114)) discounted present value of the revenues. Accordingly, one root of it exceeds 1 (𝛽); whereas the other one is negative (i.e. 𝛾). The general solution of the PDE can be expressed as follows:
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𝐹(𝑆) = 𝐴𝑆𝛽+ 𝐵𝑆𝛾 (4.3.24)
In Eq. (4.3.24), the terms 𝐴 and 𝐵 are constants to be computed. The value of waiting should go to zero as 𝑆 goes to zero; since 𝛾 has a negative non-zero value, the constant 𝐵 must consequently be set to zero. Correspondingly, the solution to the Bellman equation takes the form as represented below:
𝐹(𝑆) = 𝐴𝑆𝛽 (4.3.25)
The root 𝛽 of the quadratic equation, Eq. (4.3.23), is represented below:
𝛽 =1 2− 𝛼
𝜎2+ √(𝛼 𝜎2−1
2)
2
+2𝜌
𝜎2 > 1 (4.3.26)
As a result, 𝐴 is determined by substituting Eq. (4.3.25) into Eq. (4.3.21) and 𝑆𝑝𝑒𝑟𝑝𝑒𝑡𝑢𝑎𝑙 is consequently determined by taking the derivative according to Eq. (4.3.22) and then rearranging as represented below respectively:
𝐴 =(𝑉(𝑆∗) − (𝐼 + 𝑇𝐶))
(𝑆𝑝𝑒𝑟𝑝𝑒𝑡𝑢𝑎𝑙)𝛽 (4.3.27)
𝑆𝑝𝑒𝑟𝑝𝑒𝑡𝑢𝑎𝑙 = 𝛽
𝛽 − 1∙ (𝐼 + 𝑇𝐶)
𝐺𝑒𝑙∫ 𝑒0𝑇 (𝛼𝑠−𝜌)𝑡𝑑𝑡 (4.3.28)
Accordingly, the derived Eqs. (4.3.27) and (4.3.28) are utilized for calculating the value of 𝐹(𝑆) and 𝑆𝑝𝑒𝑟𝑝𝑒𝑡𝑢𝑎𝑙. In this respect, the investment rule is to invest now, if 𝑆0 > 𝑆𝑝𝑒𝑟𝑝𝑒𝑡𝑢𝑎𝑙 and otherwise wait. Note that in this context, 𝑆0 equals to the expected market price of electricity. In Eq. (4.3.28), the term 𝛽/(𝛽 − 1) acts as a factor for the adjustment of the uncertainty (i.e. risk premium) which is not considered for the calculation of 𝑆𝑖𝑚𝑚𝑒𝑑𝑖𝑎𝑡𝑒. It can be inferred that as 𝜎 increases, 𝛽 decreases and as a consequence 𝛽/(𝛽 − 1) increases.
Thus, the greater is the amount of uncertainty over future values of 𝑉(𝑆), the larger is the wedge between 𝑉(𝑆∗) and 𝐼 + 𝑇𝐶, that is, the higher the excess rate of return which is demanded by firms. In addition, as 𝜌 increases, 𝛽 decreases, so a higher 𝜌 implies a larger wedge between 𝑉(𝑆∗) and 𝐼 + 𝑇𝐶 which increases value of waiting.
48 In analogy to the general approach, the threshold values for FLHs of operation, which are necessary to trigger investments, can be calculated for a flexible investment decision. In this respect, the threshold FLHs of operation of a considered technology corresponds to the break-even price which is set as high as the expected market price of electricity. The corresponding rearrangement of the Eq. (4.3.28) is expressed below:
𝐺𝑒𝑙 = 𝛽
𝛽 − 1∙ (𝐼 + 𝑇𝐶)
𝑆𝑝𝑒𝑟𝑝𝑒𝑡𝑢𝑎𝑙∫ 𝑒0𝑇 (𝛼𝑠−𝜌)𝑡𝑑𝑡 (4.3.29)
After rearrangement, the term “𝐺𝑒𝑙” in Eq. (4.3.29) is redefined as the threshold FLHs of operation above/below which investments should be made/not be made given the simulated market price of electricity substituting for 𝑆𝑝𝑒𝑟𝑝𝑒𝑡𝑢𝑎𝑙.
49