A major task in assessing risks of investment projects is defining the approach to calculating the project’s volatility. Looking at assorted estimation techniques, this paper calculates their volatilities. The techniques originate from authors in the area and involve project-specific variables of uncertainty. These techniques are applied to a case of electricity distribution through real options. Results are then compared. The difference between the calculated volatilities was low, leaving, in the case of the project evaluated here, the decision unchanged. The paper’s contribution consists of providing a detailed presentation of calculating volatility by the methods cited and by comparing the results obtained by its application.
The aim of this paper is to address the issue of calculating volatility on a project-specific distribution of the electricity sector. A method to be applied to the electricity sector is then presented based on the theory of real options. To obtain uncertainty (volatility), the literature offers various methods; this paper implements these methods and comes up with results and then compares them. For this analysis, traditional investment is conducted and compared with results obtained by these different methods of calculating volatility.
Why we study investments in the Brazilian Electricity Sector? We consider such a study to be justified by the following factors observed in the Brazilian Electricity Sector: the capital expenditure of most projects, the irreversibility of investments, the high volatility of energy prices on the spot market, and the vulnerability of energy prices to several exogenous variables that contribute to an increased volatility. Just as in some cases, long-term contracts are linked to ensuring the predictability of future scenarios. Therefore a detailed analysis of the issue of volatility—the variables of uncertainty—is imperative for each sector project, because volatility can vary according to these characteristics.
One way of assessing the future as a dynamic event is through the use of real options. Real options, a tool that considers all events, are able to provide managerial flexibility in the face of changing scenarios, analyzing for every moment what options are the most advantageous. How do real options differ from financial options? The underlying asset of financial options are the papers and exchange-traded securities. The underlying asset of real options is the physical assets of companies like the machines, designs, patents, and so forth [
In this context, real options emerged as an innovative way of thinking about the evaluation of real assets. Indeed, it uses the model of discounted cash flow while complementing it with the concept of financial options [
In calculating the flexibility of real options, the only variable added to those used in traditional assessment methods is volatility. Volatility, however, is the most difficult variable to determine and even more complex when it comes to input parameters [
Copeland and Tufano [
Risk neutral probability is represented by
To obtain the value of one share of an option to purchase on Date 0, it is necessary only to know the risk-free rate, the increments of ascent and descent, and the possible values of the call option on a date. The theory of risk-neutral probabilities shows that the objective probabilities do not interfere with the value found for the option price. Hence, there is no need to adjust the discount rate to calculate the risk value of the option [
To price real options, Cox et al. [
where
Equations (
According to Mun [
This method is used primarily when calculating the volatility of assets with cash flow. Its main advantage is that it includes the ability to accommodate certain negative flows, and, in applying a more rigorous analysis, it provides a more accurate and conservative estimate of the volatility of assets analyzed. However, there is a need to use this method to obtain simulation volatility.
According to Copeland and Antikarov [
As in most cases, the uncertainties are multiple projects. These can be combined, through the distribution of returns on the project, into a single uncertainty. This done using the Monte Carlo analysis. Such an approach is called a consolidated approach of uncertainty.
Copeland and Antikarov [
This implies that, as stated earlier, the uncertainties related to the projects can be combined into a single uncertainty. Such an uncertainty—the consolidated uncertainty—will directly influence future cash flows. So, the volatility needed to develop the binomial tree is the volatility of the rate of return that, as raised by Samuelson’s theorem, comes from the random events that together influence the future cash flows. It can thus be obtained from the ratio of (
For calculation purposes, it is supposed that
where VP1 is the present value of future cash flows in Period 1, and VP0 is the present value of future cash flows at Date 0. With the completion of the Monte Carlo simulation, the standard deviation of the variable
The system used by Herath and Park [
The correct analysis of the value depends on the flexibility of care in handling the data as well as the development of a series of steps to obtain the input values needed
Copeland and Antikarov [
Mun [
Based on these two methods, the present study outlines a specific method (seen in Figure
Assessment for Real Options for a case of distribution.
Step 1 applies the traditional metrics for evaluating investments and then analyzes such criteria as NPV, IRR, payback, and cost benefits. As a result, the value of the project is obtained, calculated through discounted cash flow methods.
Step 2 defines the design variables of uncertainty, which can be defined either through the knowledge of experts or by analyzing how sensitive the economic outcome of the project is to variations of each variable. In this phase, the historical data is examined to produce a forecast of future scenarios. From these parameters, one can perform a Monte Carlo simulation to obtain the value of volatility, which in this example will consider a consolidated approach to uncertainty, and their two models, Copeland and Antikarov methods (CA) and Herath and Park (HP). Finally, with the value of the project’s volatility, one is able to build an event tree, where all possible values for the next period are elucidated. The event tree is the product of Step 2.
In Step 3, the event tree is used to incorporate flexibility. Aware of the possible values that a project might develop over a period, the decision maker may resort to strategies and carry out more efficiently the process of decision making. For critical analysis of possible future optimal decisions, values associated with these are incorporated into the project tree. Applying risk-neutral probabilities, these values are brought to Date 0, and the value of the options in the project are being calculated. The result of this step is the generation of a decision tree with initiatives that maximize the value of the project according to the achievement of future scenarios.
Finally, Step 4 produces the best time to invest in the project. It also evaluates the project with flexibility, where the main output is the value of design options. The objective of this step is optimal decision making and the use of the tool applied to justify the drafting of an action plan for the project by senior management.
We developed real options analysis for a project to expand capacity of a power distributor. We took into account the uncertainties present in the initial project implementation.
The company already has a concession in one area of energy supply, and forecasts suggest the growth of local consumers. In this scenario the basic network of subtransmission would not support such growth in demand. It then becomes necessary to analyze the feasibility of implementing a new substation, expanding preexisting substations, and including and increasing the capacity of subtransmission lines. The option must be considered, however, of putting off investment until the time to exercise the option is more advantageous.
The project has a lifespan of 30 years, during which time the company will pay back the investment as it depreciates. The company will also receive a portion of the gain loss arising from the implementation of new structures. Over the early years of other inputs, the investment is related to additional revenue from selling energy to the new aggregate demand.
The investment program was divided into short, medium, and long term. The plan was prepared in accordance with the regulation regarding the attendance of demand, providing for penalties if consumer demand is not met.
Analysis by DCF.
Total investment in trans. + subtransmission* | 17,590.92 |
Total investment feeders on primary network + underground* | 2,217.68 |
Total investment primary and secondary networks* | 16,049.31 |
Total investment** | 35,857.91 |
Net present value (NPV) (discounted to 14.00%)* |
|
Internal rate of return (IRR) |
|
Modified internal rate of return (MIRR) (reinv. rate 14.00%) | 17.72% |
Benefit/cost | 1.27 |
*Amounts in R$1,000.00.
**Simple sum.
The demand growth continues until the ninth period. The current maximum capacity is 177.30 MW. It is important to note that the figures are calculated according to the new assessment methodology adopted by the investment company in 2009.
To define the demand forecast, historical data of the company were used, applying forecasting studies for the region by planning experts from the company. Based on the information used, it was possible to outline future demand scenarios. It was found that the average growth rate accords with Table
Demand growth in the region.
Year 2 | Year 3 | Year 4 | Year 5 | Year 6 | Year 7 | Year 8 | Year 9 | |
---|---|---|---|---|---|---|---|---|
Average annual growth | 4.8% | 3.9% | 3.9% | 4.2% | 3.9% | 4.4% | 3.9% | 3.9% |
Growth rates of demand after the ninth period are not significant, because the capacity demanded would exceed the system load that would be generated by these new investments. So if there is a subsequent increase later in the year, it would require furhter investment to meet demand.
The rate of energy distribution is defined by the government through the allowed annual revenue (AR), according to the remuneration of company assets, operating costs, and maintenance, also taking into account the efficiency of companies.
Initially, in area studies, the energy rate was considered a fixed parameter over the periods. Similarly to the study of variable demand, historical data and expert planning studies were considered, where there was a stationary trend of the variable price. Therefore, we chose to model the variable through a normal distribution with
The weighted average cost of capital is the rate of return set by Brazilian Electricity Regulatory Agency (ANEEL). This is to remunerate the capital invested by distribution companies while taking into account the composition of capital structure as well as the costs associated with third party capital. The rate is established in each review cycle, with a tendency to stay unchanged for as long as possible so as to avoid questions from distributors who complete the review process at different times.
During the research, a forecasting study was conducted using the programs ForecastPRO and Crystal Ball. These considered historical data and the methodology used by ANEEL in its last review cycle. As a result, the forecast for the next WACC review is 9.15% p.a., with a relatively small standard deviation of 0.01%, can be seen in Table
Weighted average cost of capital over time.
WACC (pa) | |
---|---|
1st cycle 2003/2006 | 11.26% |
2nd cycle 2007/2010 | 9.98% |
3rd cycle 2011/2014 | 9.15%* |
*Forecast.
Graph of the probability distribution forecast WACC.
So by analyzing the main exogenous variables of the project, we selected two main sources of uncertainty: demand and price. The combination of these variables will generate the volatility of the project.
However, prior to inserting the uncertainty of these variables into the project, we performed a correlation test between them, an essential step for effective modeling of uncertainties in the project. It was found, as expected, that the variables were independent and inelastic, characteristics that can be attributed to their satisfying basic needs for all consumers, industrial, commercial, and residential. This result was also obtained by Schmidt and Lima [
In spreadsheets of the project’s discounted cash flow it was possible to perform Monte Carlo simulation using the Crystal Ball program, considering the uncertainty of the consolidated approach. The results for volatility using Copeland and Antikarov methods (CA) and Herath and Park (HP) during this stage can be seen in Table
Volatility of the project.
CA method | 6.90% |
HP method | 9.89% |
The next step to obtain the value of flexibility is the creation of an event tree for each alternative.
To construct an event tree, the following variables are needed: present value (PV) of the underlying asset and the increments of ascent and descent. The latter are obtained through the volatility found in (
The results of this step helped develop an event tree. The event tree shows what the possible values are that the project will take on over time. These values, regarding the market, are uncertainties, which compound will influence the value of the project in ascending and descending movements. According to Copeland and Antikarov [
Therefore considering the data of Step 1, a binomial grid of value was created of the underlying asset, as shown in Figures
Event tree volatility (CA) (R$*000).
Event tree volatility (HP) (R$*000).
The event tree provides the decision maker an idea of what the future possibilities are for the project under review. So at this stage, given the observed values in each branch of the tree of events, one can define what are the optimal decisions that might maximize the return on investment.
The investment will be carried out in three phases: short, medium, and long term. Since each phase is directly linked to the one prior, investing in the shortterm enables one to invest in the medium term. Investment in the medium term enables one to invest in the long term. It is on this basis that the long-term investment can only be executed if the two earlier investments are carried to the option’s limit of expiration.
The short-term investment can be realized from Year 1 to Year 2. The medium-term investment can be realized from Years 4 to 5. The long-term investment can be realized from Years 7 to 9, as can be seen in Table
Investments in expansion plan.
1° Short-term investment* | R$5,193.71 |
2° Medium-term investment* | R$13,776.65 |
3° Long-term investment* | R$16,887.55 |
*Values in R$1,000.00.
Equation (
By eliminating any possibility that the project value is negative, the decision maker has the power to make decisions that, empirically, he already had but which until then was not measured. The operation of maximizing the value branch-to-branch is performed in a backward manner, that is, from back to front. This is because the event tree makes plain all the values
The result of this algorithm is trees of option values as seen in Figures
Tree CA values (R*$000).
Tree of values (HP) (U.S. $000*).
Decision tree (CA).
Decision tree (HP).
After performing the calculations, it is possible to obtain a value for flexibility. According to Minardi [
Table
Ross et al. [
Therefore, there is a basic difference to consider in the volatility of real options and that of financial options. Therefore, the higher the volatility, the greater the value of the option or sale. But when it comes to a real asset, this is not true. A real asset must have a small margin for error in planning.
According to Table
Value of flexibility.
Flexibility value CA* | R$5,505.91 |
Flexibility value HP* | R$5,642.59 |
Difference (%) | 2.5% |
*Values in R$1,000.00.
Net present value with flexibility.
Inelastic NPV | R$14,064.79 |
NPV with flexibility CA | R$19,570.70 |
NPV with flexibility HP | R$19,707.38 |
Values in R$1.000,00.
The choice to use a consolidated approach to uncertainty is justified mainly by its use in the literature. In addition to other works in the area, Brandão et al. [
According to Godinho [
Godinho [
Similarly, Smith [
So what has been perceived through this earlier work is that there are still doubts about the best approach to estimating volatility. The new proposal by Godinho [
Because of this reasoning and proof, the consolidated approach of uncertainty was selected for use in calculating the volatility of this issue of real options.
Analyzing the different methods based on this methodology (CA and HP), it was found that, as expected, each method’s results were distinct but had a difference of only 2.99 percentage points, relatively small compared to the examples in the literature.
Hence, in the case of this distribution, the different methods showed little variation in the definition of scenarios, precisely because the volatility had little variation. Consequently, the difference in the value of flexibility between the models is R$136,678, representing only 2.5% of the total flexibility.
The main reason for this small difference in the models is the low variance of demand and price variables, which are reasonably foreseeable and are not related to this case.
It is known that traditional forms of assessment do not reflect the real value of projects. An administrator who relies solely on the mathematical results they are presented will see their project portfolio erroneously.
This paper carried out the application and verification of the approaches used in the calculation of volatility. Since this aspect is paramount to calculating the net present value with flexibility, very little has been done so far. On one point all authors agree: the key to achieving flexibility is the effective modeling of uncertainties in future scenarios. But how such modeling is done still generates a great deal of discussion, and most authors have not explored this field that is essential to the success of the calculation.
The main objective of this paper was to present an analysis of investments with volatility, from their modeling to calculation and the conclusion. Volatility was calculated by the consolidated approach of uncertainty, for the CA and HP models. There was a slight difference between the methods in this case, which can be attributed to the good predictability of the market demand study and to price, which is strongly regulated by the department of electricity.
In conclusion, it turns out that the decision in this case was not altered to be in accordance with the results obtained for each type of volatility. However, it is necessary to conduct further experiments with different cases, cases where the variables have greater variability.
Through this analysis of investments having volatility, it can be concluded in this case that even when the variables have low volatility, value is expected. And that the different models to obtain the volatility in the problem analyzed produce very similar results.
The authors would like to express their gratitude to the Brazilian agencies q National Counsel of Technological and Scientific Development (CNP), post-graduate federal agency (CAPES), and foundation for the promotion of science of the state of minas gerais (FAPEMIG), which have been supporting the efforts for the development of this work in different ways and periods.