How to Use Monte Carlo Analysis to Assess the Probability of Meeting Your Lifetime Goals

Posted by Alex Frey (@alexhfrey )

In a previous article, we discussed how risk in the investing world can take on many different forms. One type of risk that often gets left out of the discussion is the ultimate risk that you really should care about: the possibility that you will run out of money before you die. One reason that this ultimate risk is so frequently ignored is that it can be quite difficult to wrap your head around. 

Confusingly, the ultimate "risk" of running out of money before you die can actually conflict with the intuitive "risk" of an investment losing a lot of money. This is because if you fail to take a sufficient amount of "investment risk," you might also fail to generate the degree of returns that you will need to generate a sufficient retirement wealth egg. Accepting a significant chance of losing money in the short-term might be necessary to avoid taking a huge risk of running out of money before you die. 

As an example, it might seem appealing to hold most of your money in a bank account in order to avoid the risks of investing in the stock market. But while this certainly seems has a very low volatility in the short-term, it is less clear how "risky" it is in the long term, because without the higher returns of the stock market our side, you might be taking a huge risk of being unable to fund future retirement expenses. 

Monte Carlo simulations are a great way to assess this tradeoff between short-term volatility and long-term solvency and quantify the probability that you will hit your investing goal - whether that goal is a prosperous retirement or a round-the-world vacation. This article will look at how they work, go over some ways you can use them to assess your retirement plan, and finally touch upon some dangers and limitations of Monte Carlo analysis.

How Monte Carlo Simulations Work

Monte Carlo analyses work by running thousands of different simulations of what the returns on your portfolio might look like over the simulation period. The machinery for all of this has two parts to it: a random number generator, and a probability distribution of the returns of the assets in your portfolio. 

Let's start with the probability distribution and focus on the stock market. Using historical data, we can classify the historical annual returns of the S&P 500 into buckets. So for instance, we might find that in the past 100 years the stock market lost more than 30% of its value on 3 different occasions, lost between 25 and 30% on 3 more occasions, etc. This implies that the chance of the stock market losing more than 30% in any given year is about 3% (3 divided by 100). 

Using this data, we can plot a graph in which we have 0 to 1 in increments of .01 (for 0% to 100% on the horizontal axis, and the annual return of a portfolio on the vertical axis (see below). For each increment on the horizontal axis, we plot the level of return that that percentage of portfolios performed worse than. For instance, if 1% of the time the stock market went down 50% or more and 5% of the time the stock market went down 20% or more than we would plot 50% for 1% and 

For each period of every simulation, the computer picks a random number between 0 and 1 and looks at the probability distribution to determine the annual return that this corresponds to. That return is then used in the analysis for that year. 

For instance if the number is .5, that might signal an "average" return, and if it is .9, that might signal a return better than 90% of other years. This is done for every year of the simulation. At the end of the simulation, the program figures out how much wealth is left in the portfolio (if any) and records this figure. 

Using Monte Carlo to Assess Your Retirement Plan

The output from a Monte Carlo analysis can lead to a number of insights.

Imagine a Monte Carlo analysis used in the context of assessing your retirement portfolio. Imagine that you are planning on saving an inflation-adjusted $10,000 every year until you retire at age 65, at which point you will withdrawal $40,000 a year until you die. You want to know what your odds of success are with a portfolio of 60% stocks and 40% bonds. Three things that you will want to look at in the results are:

  • Probability of Success. The percentage of all simulations in which the portfolio successfully met its goals, which in this context means the percentage of times that the portfolio lasted through your retirement years.
  • Average wealth at end of simulation. Higher wealth is obviously better, but if a portfolio has too high of an average ending wealth it might be a sign that either you were not withdrawing as much as you should have been able to and were therefore forgoing a higher lifestyle needlessly, or that you could have been using some of this "upside" to reduce your downside through - for instance - a strategy of partial annuitization. 
  • Shortfall risk. Measures the average amount the portfolio fell below its goal during the worst 5% of simulations. This is an important measure of "tail risk" as a large number will indicate a situation that could truly be disastrous. If your portfolio has a large percentage chance of meeting its goal, but also a large shortfall risk, you might want to look into whether some tail-loss reduction strategies could smooth things out.  

The true value of Monte Carlo lies in comparing these values across different runs. For instance, using the above example, you might find that by shifting to a portfolio of 40% stocks and 40% bonds and 20% real estate that you are able to meet your goal a greater percentage of the time while also reducing the shortfall risk if markets really tank. 

Dangers and Limitations of Monte Carlo Analysis

A danger of any kind of quantitative model is that the results might get taken a bit too literally. Using Monte Carlo on a modern computer, you can pretty easily run 100,000 simulations of your retirement and predict your portfolio's degree of success down to a tenth of a percentage point. This apparent precision belies the rather large potential sources of error in the forecast: 

  • It uses the wrong model for investment returns. Most Monte Carlo analyses assume that investment returns follow the same random process as a sequence of tosses of two dice (the Ivyvest model is a bit more sophisticated than this as we account for momentum and mean-reversion effects). It is probably not true that investment returns are totally independent and random like this, but the reality is that we do not have much of a better model to operate with, and most feel that something is better than nothing. 
  • It uses the wrong estimates of future returns or volatilities. Even if the model for investment returns used in a Monte Carlo analysis is approximately correct, the output can still be quite sensitivity to the parameters of that model. For instance, we must usually input expected average returns for every asset class along with the expected correlations between asset classes. The usual approach uses long-term historical averages to estimate these parameters, but the reality is that this approach is almost certainly wrong (though again, it is hard to come up with better alternatives). This is because the conditions that exist today are totally different than those in, say, 1961 when it was unclear whether the US and the Soviet Union were going to start a nuclear war over Cuba or not. 

The reality is that Monte Carlo should be thought of as a "back of the envelope estimate" that just happens to be done by a supercomputer.  We are all faced with a huge degree of making important decisions, both in life and in finance, and Monte Carlo provides one tool that can help confront it in a rational way. 

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By Alex Frey