1.4 Monte Carlo Simulation
Monte Carlo simulation in traditional capital budgeting use repeated random sampling from probability distributions of crucial primary variables underlying cash flows to arrive at output distributions or risk profiles of probable cash flows in the project NPV for a given management strategy. Simulation attempts to imitate a real‑world decision setting by using a mathematical model (consisting of operating equations or identities) to capture the important functioning characteristics of the project as it evolves through time encountering random events, conditional on management's preset operating strategy. A Monte Carlo simulation usually follows these steps:
1. Modeling the project through a set of mathematical equations and identities for all the important primary variables, including a description of interdependencies among different variables and across different time periods.
2. Specifying probability distributions for each of the crucial variables, either subjectively or from past empirical data. Sensitivity analysis should precede simulation to determine which variables are important so that special care be taken to obtain their precise probability distributions; and which are not so that a single estimate of the variables may suffice. To deal with dependencies between two variables, in principle a single probability distribution can be determined for the independent variable while several distributions can be specified for the dependent one, each conditional on the independent variable falling within a given range. In practice this may sometimes place unreasonable demands on management's ability to furnish realistic estimates.
3. A random sample is then drawn (using a computer random number generator) from the probability distribution of each of the important primary variables enabling (with the help of the modeling equations and identities) the calculation of net cash flows for each period (from which the NPV for the sample can also be determined).
4. The process is repeated many times (until, say, 500 random samplings are obtained), each time storing the resulting cash flow or NPV sample observations so that finally a probability distribution for the project's cash flows or of NPV can be generated (along with its expected value, standard deviation and other statistics).
Although simulation can handle complex decision problems under uncertainty with a large number of (dynamically interacting) input variables, it is not without its own limitations.
1. 1. First, even if those responsible for estimating the probability distributions were unbiased, it would still be very difficult and complex to correctly capture all the inherent interdependencies. The existence of substantial complexity might then induce management to delegate model building to experts, with the resulting danger that management's understanding of and consequently faith in and commitment to the simulation results may be substantially reduced.
2. 2. Second, when the outcome of simulation is a risk profile of NPV rather than of the intermediate cash flows, the meaning of an outcome probability distribution of NPV, is questionable since it is not clear what is the correct discount rate to be used.
a. a. Quoting Myers (1976): "If NPV is calculated using an appropriate risk adjusted discount rate, any further adjustment for risk is double‑counting. If a risk-free rate of interest is used instead, then one obtains a distribution of what the project's value would be tomorrow if all uncertainty about the project's cash flows were resolved between today and tomorrow. But since uncertainty is not resolved in this way, the meaning of the distribution is unclear."
b. b. Also, if a project can have many possible "present values", one for each point on the distribution, then we can no longer interpret present value as the price the project would command in competitive capital markets.
3. 3. Third, even if management wants to base a decision on the probability distribution or risk profile of NPV, it still has no rule for translating that profile into a clear-cut decision for action; it can only stare at the expected NPV (discounted at the riskless interest rate) and its surrounding variance until it receives inspiration from above on how to trade them off.
4. 4. Fourth, simulation users may be tempted to use as a relevant measure of risk the variability of project outcomes (i.e., the project's own, or total risk) instead of its systematic risk, which we saw to be the relevant risk from the point of view of the firm's shareholders who have opportunities to diversify away part of that risk by buying other securities in the market; Lewellen and Long (1972) point exactly to this inability to reveal how the resulting distribution interacts with the distribution of returns faced by the firm in its other projects or by investors in their personal portfolios as the major shortcoming of simulation. Furthermore, using the total variability of the NPV distribution violates value additivity and enables managers to successfully promote unrelated projects as a group, when each project alone might be unacceptable.
5. 5. Myers (1976) points to other problems of interpretation of simulation output, such as the unreliability of extreme values of simulated probability distributions.
6. 6. Finally, Monte Carlo simulation is a forward‑looking technique based on a predetermined (built‑in) operating strategy offering roughly symmetric probability distributions ‑‑and as such it may be an appropriate model for path (or history) dependent problems‑‑ but it cannot handle well the asymmetries in the distributions introduced by management's flexibility to review its own preconceived operating strategy when it turns out that, as uncertainty gets resolved over time, the realization of cash flows differs significantly from initial expectations. Management in reality can adapt to surprises (e.g., abandon a project if it turns out to perform surprisingly poorly), but a computer simulation model cannot: it will faithfully and blindly continue obeying the business‑as‑usual operating strategy that was programmed to follow based on management's expectations and information at the outset. Thus, as will become clear later, simulation is limited in dealing with the options or free‑boundary problems that enter the valuation of real investment opportunities (such as determining the optimal timing policies for undertaking or abandoning a project), whose solution requires a backward induction or dynamic programming approach.
Despite the above shortcomings, in many real-life problems when making use of dynamic programming is difficult (e.g. when there are many state variables), simulation and numerical lattice approaches constitute the primary practical approaches to valuation. The more appropriate role for simulation in traditional capital budgeting would be to assess the probability distribution not of NPV but of cash flows, from which the expected value of cash flows and the appropriate risk‑adjusted discount rates can be determined and used to derive a single‑value expected NPV that can be used for clear-cut decision‑making. Thus, simulation should not be instead of but rather as an aid to implementing NPV. When real options are involved, as pointed out later in numerical methods, simulation may be a powerful tool for determining the relevant "certainty-equivalent" or risk-neutral probability distributions within a backward risk-neutral valuation process.