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A Monte Carlo simulation is a mathematical model for calculating the probability of a specific outcome by randomly testing or sampling a wide variety of scenarios and variables. First utilized by Stanilaw Ulam, a mathematician who worked on the Manhattan Project during World War II, the simulations provide analysts an avenue for making difficult decisions and solving complex problems that have multiple areas of uncertainty. Named after the casino-populated resort in Monaco, the Monte Carlo simulation uses historical statistical data to generate millions of different financial outcomes by randomly inserting components in each run that can influence the end result, such as account returns, volatility, or correlations. Once the scenarios are formulated, the method calculates the odds of reaching a particular outcome. Unlike standard financial planning analyses that use long-term averages and estimates of future growth or savings, the Monte Carlo simulation, available in software and web applications, can provide a more realistic means of handling variables and measuring the probabilities of financial risk or reward.
Monte Carlo methods are often used for personal financial planning, portfolio evaluation, valuation of bonds and bond options, and in corporate or project finance. Although probability computations are not new, David B. Hertz first pioneered them in finance in 1964 with his article, “Risk Analysis in Capital Investment,” published in the Harvard Business Review. Phelim Boyle applied the method to derivative valuation in 1977, publishing his paper, “Options: A Monte Carlo Approach,” in the Journal of Financial Economics. The technique is harder to use with American options, and with the results being dependent on the underlying assumptions, there are some events that the Monte Carlo simulation cannot predict.
Simulation offers several distinct advantages over other forms of financial analysis. In addition to generating the probabilities of the possible endpoints of a given strategy, the method of data formulation facilitates the creation of graphs and charts, fostering better communication of the findings to investors and shareholders. Monte Carlo simulation highlights the relative impact of each variable to the bottom line. Using this simulation, analysts can also see exactly how certain combinations of inputs affect and interplay with each other. Understanding of the positive and negative interdependent relationships between variables affords a more accurate risk analysis of any instrument.
Risk analysis by this method involves the use of probability distributions to describe the variables. A well-known probability distribution is the normal or bell curve, with users specifying the expected value and a standard deviation curve defining the variation. Energy prices and inflation rates may be depicted by bell curves. Lognormal distributions portray positive variables with unlimited potential to increase, such as oil reserves or stock prices. Uniform, triangular, and discrete are examples of other possible probability distributions. Values, which are randomly sampled from the probability curves, are submitted in sets called iterations.
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