4. Real Options Analysis - Valuing Capital Investment Projects:
Real options analysis is generally more difficult than valuing American call or put options on assets such as stocks for several reasons. The time to expiration during real options analysis is generally far longer than that on securities, often spanning into decades of forecasting, which can greatly increase the spread of simulated price paths. Real options analysis often features case specific or path dependent characteristics of investment, increasing the complexity of the analysis required. Finally, multiple stochastic underlying assets can impact the cash flows resulting from investment, increasing the dimensionality of the option valuation problem. The LSM simulation method is able to consider each of these aspects of the investment problem.
The LSM simulation method evaluates investment under stochastic price uncertainty, determining at each time point the optimal investment decision between immediate investment, exercising the right to all future cash flows at a price of the initial investment cost, or to delay investment.
4.1 Example 1 - Project Value Of One Underlying Asset:
The following example is a relatively simple, arbitrary scenario of consideration into a capital investment project, evaluating the value of the option to invest through real options analysis. Real options analysis is performed through the ‘LSM_real_options’ function, and treats investment as an optimal stopping problem.
The following example considers a capital investment project, where operating costs of the project are fixed, but revenues are dependent upon a stochastically evolving underlying asset that follows a GBM process. The project can be invested at the beginning of each month (i.e. dt = 1/12) of the year. Cash flows are delivered at the end of each month. There is a construction time of 6 months considered in this investment project, which means there is 6 months between the initial capital investment and when net cash flows of the project are accrued. The initial capital investment cost also decreases exponentially at a rate of 1% p.a. to allow research and development into the project to reduce the cost of investment.
The cash flows accrued at each discrete time point \(t\) are dependent upon the simulated stochastic price paths. We assume that there are variable and fixed cash flows in this investment project:
# Step 1 - Simulate the underlying asset price:
## Initial underlying price:
initial_price <- 36
## discrete time step:
dt <- 1/12
## Project lifetime (in years):
project_lifetime <- 10
forecasting_periods <- seq(0, project_lifetime, dt)
revenue_prices <- GBM_simulate(n = 2e4, t = project_lifetime, mu = 0.06,
sigma = 0.2, S0 = initial_price, dt = dt)
# Step 2 - Evaluate cash flows:
## Fixed cash flow:
# FCF <- 1e4 * initial_price
FCF <- 1e4 * 36
## Net cash flow is equal to variable cash flows subtract fixed cash flows:
NCF <- (1e4 * revenue_prices - FCF) * dt
## Financial Parameters:
construction <- 0.5 / dt
rf <- 0.05
## Initial capital investment:
learning_rate <- 0.01
CAPEX <- 1e5 * exp(- learning_rate * dt * (1:nrow(revenue_prices)-1))
# Step 3 - Evaluate Project Value through Real Options Analysis:
project_value <- LSM_real_option(state_variables = revenue_prices,
NCF = NCF,
CAPEX = CAPEX,
dt = dt,
rf = rf,
construction = construction,
verbose = TRUE)
print(format(unlist(project_value[1:6]), big.mark = ","))
#> ROV NPV WOV ROV SE NPV SE
#> "946,213.798" "848,194.087" " 98,019.712" " 9,508.289" " 10,425.300"
#> WOV SE
#> " 2,884.085"The ‘ROV’ is also often referred to as the real option net present value (RO NPV) and is the average discounted payoffs of each price path when investment is exercised at the optimal time calculated through the backward induction process. The NPV is the average of the discounted payoffs considering immediate investment and is a numeric approximation of the standard NPV approach. The ‘WOV’ is the waiting option value, which is the difference between the ‘ROV’ and the ‘NPV’. The ‘WOV’ is considered the value of the option to delay initial investment, allowing for future uncertainty in price paths to reveal themselves. When the ‘WOV’ is greater than zero, the optimal investment decision is to delay investment, as immediate investment would result in the loss of the ‘WOV’.
4.2 Example 2 - Project Value Of One Underlying Asset With operational Flexibility:
The following example expands the investment scenario of section 4.1 by further allowing for the investment project to consider operational flexibility after investment into the project has occurred.
Consider the capital investment project of example 1. Assume that it now can be temporarily suspended at a cost of 30% of the initial capital investment cost. Whilst suspended, it accrues annual costs of 5% of the initial capital investment to keep the project “mothballed”. Restarting the operations takes 10% of the initial investment and abandoning the project costs 50% of the initial capital investment to decommission equipment, pay severance to existing workers, etc.
Operational flexibilities are path dependent problems and greatly increase the computational complexity of the LSM simulation function, resulting in slower processing times:
## Evaluate Project Value with OF through ROA:
project_value_OF <- LSM_real_option_OF(state_variables = revenue_prices,
NCF = NCF,
CAPEX = CAPEX,
dt = dt,
rf = rf,
construction = construction,
suspend_CAPEX = 0.1 * CAPEX[1],
suspend_OPEX = 0.05 * CAPEX[1] * dt,
resume_CAPEX = 0.1 * CAPEX[1],
abandon_CAPEX = 0.2 * CAPEX[1],
save_states = TRUE,
verbose = TRUE,
debugging = TRUE
)
print(format(unlist(project_value_OF[1:7]), big.mark = ","))
#> ROV NPV OF NPV WOV ROV SE
#> " 958,921.473" "1,002,987.162" " 859,265.486" " -44,065.689" " 9,495.155"
#> NPV SE WOV SE
#> " 9,423.684" " 1,428.443"Allowing for operational flexibility has increased both the ROV and the NPV of a project that can suspend, resume and shut down. The waiting option value is now negative, meaning that contrary to example 1, the optimal decision is to invest immediately. This has shown that allowing for additional assumptions and flexibility into the investment project can lead to different investment decisions.
4.3 Plot the cumulative probability of investment:
The cumulative proportion of invested paths is returned by the ‘LSM_real_option’ and ‘LSM_real_option_OF’ functions when ‘verbose = TRUE’.
matplot(forecasting_periods, cbind(project_value$`Cumulative Investment Prob`,
project_value_OF$`Cumulative Investment Prob`), type = 'l', ylim = c(0,1),
xlab = "Forecasting Horizon", ylab = "Cumulative Investment Proportion",
main = "Cumulative Investment Prop. over Forecasting Horizon")
legend("right", c("ROV", "ROV + OF"),cex=0.8, col = 1:2, fill = 1:2)
The proportion of invested paths is higher when considering operational flexibility due to the addition to the project value resulting from the options to suspend and abandon operations.
4.3 Plot the Operating States of the Investment Project:
The proportion of simulated price paths in each available operating state at each discrete time point is returned by the ‘LSM_real_option_OF’ function when ‘save_states = TRUE’. These states can be plotted using the package ‘ggplot2’ (Wickham, 2009):
states_list <- apply(matrix(colnames(project_value_OF$`Project States`)), 1,
FUN = function(x) cbind.data.frame(x, project_value_OF$`Project States`[,x],
forecasting_periods))
states_ggplot <- suppressWarnings(dplyr::bind_rows(states_list))
states_ggplot[,1] <- factor(states_ggplot[,1], levels = rev(colnames(project_value_OF$`Project States`)))
colnames(states_ggplot) <- c("state", "count", "time")
library(ggplot2)
ggplot(states_ggplot, aes(x = time, y = count, fill = state)) +
geom_bar(position = "fill", stat = "identity", width = 1) +
scale_y_continuous(labels = scales::percent, breaks = seq(0,1,0.1)) +
scale_x_continuous(breaks = seq(0, project_lifetime, 1)) +
ggtitle("Proportion of Project States over Project Planning Horizon") + xlab("Planning Horizon (Years)")
In general, the suspension operating state is less prolific as the residual lifetime of the investment project decreases, as the value of being suspended with the potential to resume operations becomes less valuable and suspended states may instead become abandoned. The abandonment and suspension operating states generally make up a low proportion of simulated operating states, with the cost of switching and maintaining the suspended state greatly influencing the propensity to suspend and abandon operations respectively. The additional value of the investment project when the cost of switching operating modes is equal to zero can be considered the upper limit to the added value of the investment project that considering operating flexibility can have.
The plot above could be an ideal result to present to the management of an investment project, as it can be interpreted as the likelihood of project success after initial investment has occurred. Project Managers are likely to be less inclined to consider investment projects that are shown to have a high possibility of being “mothballed” or abandoned at a net operating loss.
5. Real Options Analysis - Calculating Investment Trigger Values:
A key result of real options analysis is the ability to consider the critical “trigger” values of underlying, stochastically evolving assets at which immediate investment into the project is the optimal decision. Investment trigger values can be calculated by finding the root of the WOV of an investment project, i.e. the value of the state variable that results in the WOV of an investment project being approximately equal to zero, indicating that immediate investment into the project is the optimal investment decision at time \(t=0\). This can be effectively calculated by recursively running the ‘LSM.OptionValue’ function for different initial values of the state variable until the root of the WOV is found. The secant method is a root-finding algorithm that is functionally a finite-difference approximation of the well known Newton’s method that can efficiently converge to the root of the WOV. The secant recurrence relation is:
\[x_n = x_{n-1} - f(x_{n-1})\frac{x_{n-1}. - x_{n-2}}{f(x_{n-1}) - f(x_{n-2})} \] The critical value of the project according to the NPV criterion (i.e. the critical value where \(NPV=0\)) of the project can generally be calculated after 2 iterations as it is a linear function of the starting price.
The following section provides a worked solution of obtaining the critical trigger values of investment for the example presented in section 4.1:
# Instantiate iterations:
it <- 0
current_price <- initial_price
# Begin Investment Trigger Value Calculate:
repeat{
# Step 1: Calculate the ROV using real options analysis
LSM_results <- LSM_real_option(state_variables = revenue_prices,
NCF = NCF,
CAPEX = CAPEX,
dt = dt,
rf = rf,
construction = construction)
NPV <- LSM_results$NPV
WOV <- LSM_results$WOV
# Step 2: Evaluate the next initial asset price through the 'secant' method:
## For the first iteration, use an arbitrary initial price multiplier of 2
if(it == 0){
multiplier = 2
new_price = current_price * multiplier
}
if(it > 0){
## NPV - a linear function of initial prices, so we can find it exactly after two iterations:
NPV_gradient = (NPV - NPV_old) / (current_price - old_price)
NPV_new_price = current_price + (0 - NPV)/NPV_gradient
if(it == 2) NPV_crit_value = NPV_new_price
## ROV - Secant Method:
new_price = current_price - WOV * ((current_price - old_price) / (WOV - WOV_old))
## Which is a multiple of:
multiplier = new_price / current_price
## The WOV does not have to be exactly zero. Having it within a tolerance value
## can be adequate and decrease processing time:
WOV_tolerance <- abs(WOV) < 100
## If the price is identical within one cent, this can be considered the critical value:
price_tolerance <- round(new_price,2)==round(current_price, 2)
## If the underlying asset impacts costs, and the iteration has pushed the price of the asset
## below zero, it's never optimal to invest immediately:
negative_price <- new_price < 0
## Recursion break to ensure infinite loop does not occur:
loop_break <- it > 20
##Approximate the root of WOV to 2 significant figures:
if(price_tolerance || WOV_tolerance || negative_price || loop_break){
ROV_crit_value = new_price
break
}
}
# Step 3: Update values:
## Updating simulated prices:
revenue_prices <- revenue_prices * multiplier
## Updating the NCF of each period:
NCF <- (1e4 * revenue_prices - FCF) * dt
## Updating values
old_price <- current_price
current_price <- new_price
WOV_old <- WOV
NPV_old <- NPV
# Step 4: Re-iterate:
it <- it + 1
}
print(round(c(NPV = NPV_crit_value, ROV = ROV_crit_value),2))
#> NPV ROV
#> 27.54 39.76The ‘ROV’ value considers the value of the option to delay investment over the ‘NPV’ value. The ‘ROV’ is always consequently greater than, or equal to, NPV. This also means that the critical value of investment under the ROV is always greater than that under the NPV criterion. Calculating critical values can be appealing because it is easy to interpret and communicate these results, it provides a clear signal for when investment should be triggered and finally the waiting option value (WOV) is generally calculated with a greater level of certainty (i.e. lower standard error) than the real option value or net present value, providing greater certainty in results when interpreting them through critical values (Aspinall et al., 2020).