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.

Using MCS on investment returns can give us a more meaningful analysis than linear assumptions alone. But what if there are more variables in our forecast? What if investment returns are only one of the many uncertainties in our forecast? Which variables should you place parameters on?

All About Variables

First, let’s define a variable. A variable is any assumption in your analysis in which there is uncertainty (risk). Usually, uncertainty increases as the time horizon lengthens. The first question to ask, then, is which variables could have the greatest impact on the forecast. The answer will, of course, depend on the specific variables you are facing. Ideally, you should use MCS on as many variables as is possible.

To explore this further, let’s look at an example. Assume we have a 42-year-old client who is a business owner, married with children, and plans to send his children to college. Furthermore, he and his wife want to know if they’ll have enough money to live comfortably in retirement with a high degree of certainty that their money won’t expire before they do. What are the questions to ask and the assumptions to make? Here are a just a few: How long will they live? Which college and how much will that cost in the future? Do they need to reduce their current expenses to save more each year? How much investment risk are they comfortable with and how much do they need to assume to reach their goals? How much will Social Security provide or will the system be insolvent then? Should they invest in a Roth IRA or traditional IRA in addition to their company retirement plan? How would their portfolio be affected if the market declines at the same time they retire and begin withdrawing from their assets? Each decision they make today will have an impact on their future. Then, are the variables highly predictable or very uncertain? Significant or insignificant? If an assumption is highly predictable or varies little, then MCS probably won’t add much. On the other hand, if the assumption is very unpredictable, then MCS could provide a great deal of insight.

Next, let’s make a list of possible assumptions for this case. Investment returns are the obvious one. Next, we have to consider cash flows. Let’s start with cash outflows. Assume the couple would like to receive $100,000 annually after taxes, which includes a mortgage of $20,000. If their mortgage is a fixed rate, it should be separated and only $80,000 of their expenses would be subject to inflation. If the mortgage is a variable rate, MCS can help model this, and can include a minimum and maximum rate. The rest of the expenses are subject to inflation (which will vary). There could also be medical cost such as health insurance premiums or ongoing expenses associated with a chronic condition. Since medical costs have been rising faster than the general inflation rate, we could separate these expenses and include a separate medical inflation rate. Current education costs with a separate education inflation rate could also be included. Other costs may include new car purchases every few years, support payments, and future savings. Cash inflows could include earnings until retirement, annual bonuses, net rental income, alimony, an inheritance, Social Security, and pensions, to name just a few. Some of these inflows may increase with inflation and some may not. For those that will increase, you could attach a separate inflation rate. To be more conservative with the analysis, you could use a higher inflation rate on expenses and a lower one on inflows. What if the couple plan to sell their rental property in the future? Then consideration should be given to the rate of growth on the property, its cost basis, their tax bracket, and the capital gains tax rates.

If you’re trying to forecast future estate values, you would need to include the value of their tangible assets such as their primary residence, vacation home, personal property, automobiles, business interests, or anything else they own, such as life insurance. There could easily be in excess of 20 different assumptions to be modeled using MCS. Remember, some of these variables will have a greater impact on the forecast than others.

With the changing federal estate exemption in the next few years–$2 million in 2006-08, then $3.5 million in 2009–MCS can help assess the probability of someone’s estate exceeding these amounts.

The Importance of Linear Forecasts

We’ve all seen linear forecasts. Let’s assume we start with a portfolio of $100,000 and expect it to grow at 6% for the next five years. If these assumptions held true, our ending balance would be $133,822. What if we don’t earn 6% each and every year? In the past, all financial projections were based on this methodology. What is the probability of achieving a number equal to or greater than the linear forecast? Is there a normal range of probability and why is this important? When a linear forecast is calculated and a final value is determined, its probability is dependent on the amount of risk or uncertainty in the forecast. When the risk is greater, the probability is lower. This type of comparison can assist in quantifying the risk in the forecast. For instance, using a normal distribution curve, the majority of outcomes occurs around the mean, or 50th percentile. If a high degree of risk is present, as there may be with a stock concentration, the probability of the linear outcome would be much less than 50%. This number could be 20-30% or lower, depending on the amount of risk and the time horizon analyzed.

To summarize our exploration of Monte Carlo simulation, we can argue that MCS can enhance our understanding of and ability to manage risk in a number of ways. You could focus on the probability of running out of money at different ages, probabilities within a specified range, probabilities of achieving a return greater than x, and more. This technique can improve our decision making by allowing us to achieve a better understanding of the ranges of outcomes. When used properly, the client wins, the practitioner wins, and the financial services industry wins.

Mike Patton, a CFP and AEP (accredited estate planner), is a senior financial planner with JP Morgan Private Client Services in Baton Rouge, Louisiana. He specializes in retirement and investment planning, and wealth transfer strategies, with an emphasis on probability analysis using Monte Carlo simulation. He can be reached at michael.j.patton@jpmorgan.com.

How Many Trials?

The number of trials you should run in Monte Carlo simulations is related to the number of variables and the number of occurrences per trial.

Let’s look at an example. Assume you’re trying to determine the future value of a portfolio you will hold for five years. Assume also that there are no cash inflows or outflows to consider. Furthermore, assume the only variable in the analysis is the return on the portfolio. In this case the number of variables = 1 (the return), and the number of occurrences per trial = 5 (1 variable x 5 years). If you ran 100 trials you would have 500 total occurrences (1 variable x 5 years x 100 trials). Each time a trial is run the variable changes (randomly), not just in the first year but in each year it occurs. Since MCS selects numbers at random, it may select bad returns in the early years and good returns later, or the reverse may be true. In any event, each time a trial is run the ending portfolio value is changed. You’ll need to run a sufficient number of trials to assure you are modeling as many outcomes as possible. There is a point of diminishing returns, however. You’ll know you’ve reached this point whenever the additional numbers of trials yields little or no effect on the outcome (probabilities). In the above example, 250 to 500 trials may suffice.

In the retirement example used earlier, let’s assume we identified 20 variables. If the time horizon we’re modeling is 30 years and we ran 1,000 trials, then there would be 600 occurrences per trial (20 variables x 30 years) and 600,000 total occurrences (600 occurrences per trial x 1,000 trials).

So how many trials should we run? Let’s say we increased our trials from 1,000 to 3,000 and the probabilities changed significantly, but at 5,000 trials there was little or no change. In this case, then, the ideal number of trials is probably somewhere between 3,000 and 5,000. It’s difficult to say with any certainty how many trials should be run. You’ll have to experiment with this on a case-by-case basis. A simple rule of thumb is that the more variables you include, the more trials you should run.