In the following section the value of the wind farm is found including both events and market uncertainty, by use of the quadranomial approach. The quadranomial approach is an extension to the binomial approach, and requires relatively little effort once the binomial option parameters have been estimated.
4.5.1. Introduction
We have now calculated the value of a wind farm under development, including events in the ENPV model and market uncertainty in the binomial ROV. Both of the two models provide misguiding results, as it is more realistic to include both events and market uncertainty to value the wind farm under development.
What should be clear from the two previous valuations is the distinct nature of market uncertainty and events. Where market uncertainty is modeled as a continuous process, as it is resolved continuously over time, the events are model on a non-continuous basis. The difference between the two is demonstrated in Figure 4.17 below, where the market uncertainty develops smoothly through time, whereas the outcome of the event is resolved when the information becomes available.
Figure 4.17 Resolution of Uncertainty and Events over Time
Source: Own construction
Furthermore the figure shows how market uncertainty increases over time, whereas the uncertainty with regard to the events is resolved over time as the wind farm is either approved or denied. The distinct character of the market uncertainty and events is the reason that they have been treated separately until this point in the valuation. As mentioned in chapter 3, instead of combining the two into a single measure, we continue to keep them separated in our ROV – in line with Copeland and
Time Uncertainty
Event Market Uncertainty
92 Antikarov (2003), Kodukula and Papudesu (2006), Méndez et al. (2009), Shockley (2007) and Villiger and Bogdan (2005a). To do this we will use the quadranomial approach (QROV).
4.5.2. Quadranomial Approach
While the market uncertainty and the events are distinct in nature, they both influence the decisions taken by the management. As management cannot decide to exercise their option to continue if the project has failed, the option value should be adjusted for such events. The QROV handles this interaction by letting the market uncertainty and events interact in the tree, as illustrated below.
Figure 4.18 Illustration of Quadranomial Tree
𝑽𝟎: Value of underlying asset 𝑶𝒏: Outcome
Source: Own construction, inspired by Kodukula and Papudesu 2006: 72-73
In Figure 4.18, the asset value tree for the quadranomial approach shows how the quadranomial tree does not only let the value of the underlying asset move up and down, but for each movement it can also succeed or fail. The quadranomial tree shown above has 4 outcomes (O1 to 4) for the underlying asset value, compared to the two of a binomial tree. To obtain the value of the option, the four outcomes must be adjusted for the probabilities of up and down movements, success and failure, and be discounted to year zero as shown in the formula below.
Formula 4.8 Option Value Quadranomial Approach
𝐎𝐕=𝐌𝐀𝐗 �𝐬(𝐎𝟏𝐩+𝐎𝟐𝐪
𝐞𝐫𝐟∆𝐭 ) +𝐟(𝐎𝟑𝐩+𝐎𝟒𝐪
𝐞𝐫𝐟∆𝐭 )− 𝐄𝐗;𝟎�
Success Failure
𝑶𝑽: Option value 𝑶𝒏: Outcome number n 𝒑/𝒒: Risk-neutral probability 𝒔: Probability of success 𝒇: Probability of failure 𝑬𝑿: Exercise price
𝒆𝒓𝒇∆𝒕: One period discount rate Source: Own construction based on Copeland and Antikarov 2003: 332-333
V0
O1 Success
Failure
Up Down
Up Down
O2 O3 O4
93
Because the value of the underlying asset in the two failure outcomes O3 and O4 is equal to zero when a wind farm fails, the entire failure bracket can be removed, and the formula can be reduced to Formula 4.9 below.
Formula 4.9 Option Value in the Quadranomial Approach (f=0)
𝐎𝐕=𝐌𝐀𝐗 �𝐬(𝐎𝟏𝐩+𝐎𝟐𝐪
𝐞𝐫𝐟∆𝐭 )− 𝐄𝐗;𝟎�
Source: Own construction based on Villiger and Bogdan 2005a: 425.
The formula above differs from the one used in the binomial tree by including the probability of success. Hence the quadranomial tree can be reduced to a binomial tree where the individual stages are multiplied with the probability of success as suggested by Shockley (2007: 347) and Villiger and Bogdan (2005a: 425). This is demonstrated graphically in Figure 4.19, where the quadranomial tree of our wind farm under development has been illustrated. As can be seen from the blue
“success” section of the figure, the nodes with successful events form a regular recombining binomial tree. However, if development fails at the end of a stage, the tree ends up in the red
“failure” section, meaning that the project is worth nothing.
Figure 4.19 Illustration of Multi Step Quadranomial Tree
Source: Own construction, inspired from Copeland and Antikarov 2003: 334-335.
The formulas work only if market uncertainty and events are assumed to be uncorrelated, which means that the market uncertainty does not affect the probability of success or failure in the event and vice versa. That uncertainties are uncorrelated implies that their unconditional probabilities are equal to the conditional probabilities, i.e. that that the probability of the combination can be found
Stage 1 Analysis and Preapproval
Stage 2 VVM and Final Approval
Stage 3 Complaints and
Compensation
Stage 4 Construction
Success Success Success Success
Failure Failure Failure Failure
Stage 1 Feasibility studies and
Pre-approval
Stage 2 VVM and Final Approval
Stage 3 Complaints and
Compensation
Stage 4 Construction
94 by multiplying them with each other (Copeland and Antikarov 2003: 283).68 That market uncertainty and events are uncorrelated means that an increase in value should not increase the possibility that the municipality approves the wind farm.
4.5.3. Calculation of Option Value
With the QROV explained, we are now ready to model the value of the wind farm under development including both market uncertainty and events. This approach, just like the regular binomial tree, requires the construction of an underlying asset value tree and an option value tree.
But to implement the model, we will need to modify our earlier presented six step model, as seen in Figure 4.20.
Figure 4.20 Modified Six Step Model for QROV
Source: Own construction
As shown with red above, we need to modify step 3 to include the events and probabilities estimations. Furthermore, step 5 now includes the construction of a quadranomial option value tree.
The rest of the stages all remain the same.
As we have already estimated the events and probabilities, we will now turn to the construction of the quadranomial option value tree.
The quadranomial option value tree is constructed from the asset value tree (in section 4.4.7.1), as this has not changed. The tree is solved recursively, just like the binomial option value tree in Figure 4.15. The major difference is that the events are included.
The logic behind the tree is as follows. EE enters stage 1 by purchasing an option primo 2010 for DKK 100,000. Ultimo Q2 2010 the pre-municipality approval will be given with a 50% chance, which means that this is the probability for EE to pass the stage. In the meantime, the market can either have moved up or down. If the value of the option (given the market condition) is worth more than the exercise price to enter stage 2, which is 500,000, then EE will exercise the option (invest
68 Copeland and Antikarov 2003: 279-86 also discuss a quadranomial approach with two correlated uncertainties and one with two uncorrelated uncertainties following a geometric Brownian motion.
Uncertainty Estimation
Step 3
Uncertainty Estimation Identify and estimate market
uncertainty Identify events and
determine probabilities
Step 1
Framing ROV
Justify ROV Identify relevant
options
Step 2
Underlying Asset
Calculate DCF value of operational
phase
Step 6
Sensitivity Analysis Identify and test main value drivers
Discuss results
Step 5
Calculate Option Value Build asset value tree Build quadranomial
option value tree Build asset value
tree Build quadranomial
option value tree Determine Op
Parameters
Step 4
Time to maturity Length of time
steps Risk neutral prob.
Determine Option Parameters
95
DKK 500,000) and move on to the next stage. From the option value tree in Figure 4.21 below, it is shown that in the end of stage 1, two out of the three nodes have a positive option value after subtracting the exercise price to enter stage 2, and development is therefore continued, whereas the last node is zero and development is therefore discontinued.
Figure 4.21 Option Value Tree
Source: Own construction. Figures in thousands DKK.
The value of a wind farm under development using QROV is DKK 431,289, which is approximately DKK 500,000 more than the value found using the ENPV previously. This clearly shows that a significant value can be added by perceiving a wind farm as a real option and this difference can be perceived as the value of decision making. In our particular case the difference between the results leads to two different strategies. Where the ENPV suggests not to invest – the ROV suggests to invest. The increase in value demonstrates that it makes sense to think of wind farms under development as real options to conduct a proper valuation.
4.5.4. Sensitivity Analysis
In the above valuation, we combined the events and the market uncertainty in the development phase. We will now conduct a sensitivity analysis of how the events and market uncertainty influence the value of the result of the QROV approach, as shown in Figure 4.22 below.
Primo 2010 Q1 2010 Q2 2010 Q3 2010 Q4 2010 Q1 2011 Q2 2011 Q3 2011 Q4 2011 Q1 2012 Q2 2012 431
770 1,082 3,419 4,294 5,276 6,346 15,478 17,882 20,410 23,067 196
334 1,629 2,224 2,965 3,839 10,167 12,332 14,610 17,006
527 801 1,199 1,761 5,472 7,251 9,300 11,457 83
141 240 409 2,007 3,012 4,437 6,375
- - 348 593 1,011 1,722
- - - -
- - -
- -
-
Prob. 100% 100% 50% 100% 100% 100% 50% 100% 100% 100% 80%
EX 100 500 500 61,000
Value t=0 Feasibility Studies and
Pre-approval VVM and Final Approval Complaints and Compensation
MAX[((0.5921*3,419+1,629*0.4079)/1.0091-500);0]*0.5 = 1,082
96
Figure 4.22 Sensitivity Analysis QROV
Source: Own construction
From the sensitivity analysis we can see that the changes in the probabilities have a large impact on the value. The volatility is also a very important value driver, and can potentially change the value significantly. Thus both market uncertainty and events should be included in a valuation, in order to obtain a superior valuation.