Monte Carlo simulations. Now, Monte Carlo simulations are a very popular statistical method of analyzing return data, and in fact it randomizes return data, which makes it extremely useful. You might hear other names for it, stochastic modeling or probability analysis, but there are they are one in the same. Now, Monte Carlo simulations have rapidly gained adoption and popularity and is named actually after the city of Monte Carlo in Monaco which is famous for its gambling venues. Well, in the simulations we use a roulette wheel like generator of random returns, hence the origin of the name. So to do this we use a computer and that computer program selects the annual returns based upon all of our assumptions. And those assumptions will include a distribution of the returns, the volatility of the returns and the correlation between assets. And then this process gets repeated thousands and thousands of times, allowing us to see not just an estimated value but a range of estimated values and all possible outcomes. Monte Carlo does not use straight line estimates. In straight line estimate analysis, we're using just one expected return as the full range of possible returns and we're ignoring the potential gyrations and returns from year to year, as long as they equal the average. This may not seem like a big deal, but in the end it is very much a big deal because of the sequence of the returns matter dramatically. Let's look at an example. A retired couple has $1 million portfolio and wants to know if they have enough to withdraw $125,000 a year for the next 10 years. A very typical retirement question or issue. In the first example, we're going to look at a straight line analysis, and here you can see that every year we're assuming the average return of 6.6% and by year 10 not only are we able to successfully withdraw $125,000 per year, as the question stated we also still have about $200,000 left in the portfolio. A huge success. However, let's look at an alternate scenario. Here the average return is still the same 6.6% across all the years, but it's not the same in every year. As you can see in the first year, we have a large drawdown of 20%. It's made up in future years, especially in year five with a large increase of 33.2%. But in the end, by year 10, the couple runs out of money. They're not able to make that last $125,000 withdraw despite having the same level of average return across both scenarios. So the, the difference is pretty clear here, right? The straight line estimates use the same return every year and assume no volatility or variance within that return. On the other hand the Monte Carlo simulates a volatile sequence of returns to get a better sense of all possible outcomes. Now here's your typical Monte Carlo output. You can see quartile dollar values. So it gives you a sense of, you know how much the portfolio would be worth in good times and in in bad times. The 90th percentile is essentially the best case scenario, and this couple could have close to $400,000 left in their account after 10 years and after all the withdrawals. However, at the 10th percentile and 25th percentile you see that the value of the portfolio is zero. So a large difference in terms of potential ranges of results. You'll also see a graph dictating the success probability. So after 10,000 simulated portfolios and different sequences of returns, we're able to see that 50% of the time the couple does not reach their goal while 50% of the time it does. And this allows investors to see the trade off that occurs when creating specific goals around a goal like retirement and when they're creating their portfolios. If an investor decides to get more aggressive he may increase the portfolio value in the best case, for example, however the probability of success may not increase. Now, of course, Monte Carlo simulations aren't perfect. They have limitations. And the first being that assumes return streams are normal and that's not always the case. And if they're not, this could indicate a larger amount of tail risk that the simulation is not accounting for which means a larger probability of extreme returns both on the positive and negative side. Another issue is that it assumes what has happened historically will happen in the future. And obviously that's not always the case and if it's not the case. We need to adjust our inputs to incorporate today's environment around return distributions, volatility correlation, or any any other assumptions that we're making. And lastly, returns may be overly random. How could that be? Well, some studies have shown that market returns tend to be mean reverting to some extent anyway, which means that low returns tend to be followed by higher returns and vice versa. And given the random nature of the Monte Carlo process this might not be incorporated into the analysis.