### Why Monte Carlo Analysis is Optimistically Biased

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Monte Carlo (MC) analysis is the most common tool planners employ when projecting clients’ finances, yet it contains an inherent optimistic bias that has largely, if not completely, gone unnoticed. This article outlines why bias exists in MC, provides two cases that demonstrate the potential impact of MC bias and describes the factors that influence the degree of bias.

The effects can be sizeable. For a 20-year-old with moderate risk tolerance who invests a lump sum to draw upon for income at age 80, MC underestimates the recommended investment amount by almost five times. For a very conservative investor, MC can underestimate required savings by 24 times.

**The cause of optimistic bias**

The bias occurs because MC presumes the input assumptions for return mean and standard deviation have no error. In some applications, this is true. For instance, the probability of pulling a particular card from a standard deck of playing cards is precisely known. However, in finance, we will never know the true returns distribution exactly, so we often estimate the underlying probability distribution by using historical mean and standard deviation as inputs to MC. Doing so is analogous to being shown only five cards from a deck of unknown size and composition. With financial data, the historical parameters always contain sample error of unknown size and direction.

Before going further, two concepts are often confused that should be made clear: sample error and bias.

Sample error gauges the inaccuracy of only one sample, whereas bias is an average of the errors. For instance, when a fair coin is flipped 10 times, the resulting number of heads is likely to have sample error, such as six heads instead of five (an error of +1). If the experiment is performed correctly and repeated many times, the sample error will have a mean value of zero, and thus no bias. In other words, individual sample errors can vary -- +1, -2, -4, etc. -- but when observing an infinite stream of sample errors, the average should be zero. If bias is present, the average sample error will not be zero.

As mentioned earlier, the historical market returns fed into MC represent a sample with unknown error. Even if the input is unbiased, the sample error can produce an optimistic bias in MC predictions.

It is easy to imagine how this can happen with a biased input, but how is it possible when the sample is unbiased? The concept is demonstrated with another coin example. Assume a person has three coins in her pocket, and heads are the preferred outcomes. One coin has two heads, another coin has two tails, and a third coin is fair. The two unfair coins represent an extreme form of sample error, either plus or minus 50%. On average, the sample error of all three is zero, and there is no bias.