When financial planners talk about how the profession has advanced over the years, one of the prime examples given is that of Monte Carlo simulation (MCS), which over the past 15 years has found its way into many financial planning software packages.

The software has its origins in the 1940s at the Los Alamos National laboratory in New Mexico, where scientists created a computer program to model the range of possible outcomes of a nuclear explosion. They named the program Monte Carlo after the quarter of Monte Carlo, in the Principality of Monaco, where many try their luck at the famous roulette wheels (which, of course, are based on chance).

These days, the majority of planning programs incorporate MCS to some degree, but this method of calculating probability may still be in the earlier stage of developmental use. Like the American pioneers of old, there are several new frontiers to explore and questions to address before planners understand and apply MCS appropriately. These would include determining how many trials should be run and which variables to include in simulations, and whether some planning software might be yielding an incomplete picture by applying MCS only to investment returns. It’s also important to explore the probability of the linear forecast and determine why such a forecast is important. This article will attempt to address these issues in a practical manner, using examples that we all might see with our clients.

**The benefits of Monte Carlo**

There are several different ways to “project” future outcomes. These methods include linear forecasting, time series forecasting, Latin hypercube, time path analysis (looping or rolling), and Monte Carlo simulation (MCS). In terms of MCS, three frequently used forecasting methods are parametric, non-parametric, and economic modeling. We will focus on the parametric method of forecasting.

Risk is a primary reason why we use MCS. Risk can either be subjective or objective, significant or insignificant. Some risks are objective, such as flipping a coin. It doesn’t matter if the first 10 flips were heads because the next flip of the coin still has a 50/50 chance of being either heads or tails. An example of subjective risk would be predicting the weather. Given the same data, two different weathermen may forecast different chances for rain. A significant risk would be a tightrope walker performing 500 feet above the ground without a net. An insignificant risk would be the same tightrope walker traveling only one foot above the ground.

Risk stems from our inability to see into the future. Newton’s third law of motion states that “for every action there is an equal and opposite reaction.” There may be a modified application here for Newton’s law outside the realm of physics. Here are some examples: If I increase my investment risk, I expect to increase my return by some factor; If I don’t buy that new car today, but instead invest the money, I should have more money tomorrow. These are just a few of the many decisions that we and our clients face. MCS can be of great help in making these decisions.

**Average Isn’t Good Enough**

Many of today’s software programs tend to use MCS primarily around investment returns. Additionally, they may categorize a particular holding as a large-cap stock, small-cap stock, intermediate term bond, etc., and impose the standard deviation (risk) for the entire category on that particular holding. While it’s prudent to develop sound assumptions around risk and return, this approach can be problematic. For instance, the stock of a company with a $50 billion market cap would be considered a large-cap stock by most observers. Let’s say the average standard deviation for large-cap stocks was 20%. What if that particular stock’s standard deviation was 40%, 50%, 60%, or higher? In such a case, we would be greatly understating the risk. Here’s the irony. The reason our industry has gravitated to MCS in the first place is because forecasting using linear assumptions can be misleading! Most planners would recognize the problem in relying on averages when forecasting, since the forecasted results rarely materialize as expected. Yet some continue to use averages by categorizing an asset and using the category average as a proxy for a particular holding.

To further illustrate this problem, consider the “Averages” table above, right. According to Morningstar, as of Dec. 31, 2005, there were 6,423 domestic stocks. This number included small, medium, and large companies. The average standard deviation for this universe was 27.40%, with a high of 964.9% and a low of 6.60%.

Of those 6,423 stocks, only 353 were large domestic companies. The average standard deviation of those large stocks was 23.80%, with a high of 69.60% and a low of 9.60%. With such a variance in values, the use of averages can be very misleading.

**First, Create a Linear Forecast**

Before you can employ MCS, you have to create a linear forecast. You have to make certain assumptions, such as “average return is x and inflation is y.” MCS enables us to venture beyond this basic analysis in an attempt to simulate reality.

One such approach is the parametric method, which essentially places a “parameter” or boundary around the variable. For instance, if the variable was the investment return, you might select a “normal” distribution curve (see “Before Using Monte Carlo” sidebar). With this, you would need to input the mean and the standard deviation. The standard deviation would set the parameter for that variable. Each time a simulation is performed, a return would be chosen, at random, from within this parameter.

But what type of parameter, i.e., distribution, should you use? The answer depends on the type of assumption you want to model. Some software allows the user to select from several different distributions. If you’re not sure which distribution is most suitable for a particular variable and you have historical data, some software will use the data to determine the most appropriate or “best fit” distribution. There are times, however, when the client will help determine this. For instance, a client may believe that his annual bonus in the coming year will be a minimum of $50,000 and a maximum of $100,000, with no certainty as to the most likely amount. In this case, a uniform distribution is best (see figure 2). If he believed that the most likely bonus is $75,000, then a triangular distribution may be more appropriate (see figure 3). In short, the information known about the particular variable will help direct you to the most appropriate distribution.

**Investment Return Modeling**

Now let’s explore further how this works. Assume we have an expected return of x and a standard deviation of y. The standard deviation determines the variance from the mean return, or in other words, how high and low we can expect the returns to vary based on historical data. One standard deviation (SD) encompasses approximately 68% of the returns. Two standard deviations takes in about 95% and three SDs about 99.7%. Then you have the outliers–the occasional returns that fall outside of the normal parameter. Using a normal distribution curve, MCS would choose numbers at random from within this parameter. Each time a new trial is run, the variable (return percentage) would change and a new ending portfolio value would be created. The program would then log, i.e., remember, the results of each trial.

Many of today’s financial planning software programs tend to employ MCS primarily around investment returns. Is this providing us with an incomplete picture? Let’s look at a very simplistic example to illustrate. Let’s assume we ran three trials on a portfolio. The first trial resulted in an ending value of $100, the second an ending value of $125, and the third an ending value of $75. In this example, each result (the ending value) occurred 33.3% of the time, and so MCS would declare that there was an equal probability of ending up with $75, $100, or $125. Let’s say a fourth trial was run and it also had an ending value of $100. Then there would be a 50% probability of having $100 (two trials out of four), a 25% probability of having $75 (one trial out of four), and a 25% probability of having $125 (one trial out of four). In reality you would run a much larger number of trials, perhaps as many as 500, 1,000, 3,000, or more. (See “How Many Trials” sidebar). After running those multiple trials, you can look at ranges of outcomes.