Monte Carlo

How To Achieve Financial Independence and Live Your Passion Regardless of Age or Income
I just finished writing a new book called How to Achieve Financial Independence and Live Your Passion Regardless of Age or Income: 10 Paths to Financial Independence Analyzed (available on Kindle and a more expensive full color, printed book version from Amazon). In the book, I explained to my 22-year-old son 10 different paths to achieve financial independence and analyzed how quickly he could achieve financial independence with each one.

However, in the book, I used assumptions that remain static throughout each plan. As one example, I assumed the stock market returned 8.97% per year, every year without variation. My assumption was based on the performance of the entire stock market, VTSMX, from 1871 through 2017. It was based on historical reality, but in my opinion it has serious flaws.

Flaws (Already) in My New Book?!

What flaws? Imagine my son saving up enough money to achieve financial independence and being fully invested in an S&P 500 index fund. The day after he achieves financial independence and quits his job, the S&P 500 drops by the same percentage that it did on its worst day historically. That would be a 20.47% drop in value. If he had saved a million dollars, in one day he lost about $200,000. With the simple model of assuming the stock market averages 8.97% per year, every year, consistently I over-simplified the model for the book; that’s flawed.

One possible alternate way to model it would be to look at the historical returns from the stock market and try to make the return random to simulate what historical range of returns have been. Before I go too far down this path, I want to confess that this is certainly not the only way to model this. Some very smart folks would even argue it may not even the best way to model it. It is just one way and the way I will discuss here now. I am sure, time permitting, I will loop back around and discuss alternative ways to model this in the future. Back to our story now.

What if instead of saying the stock market returned 8.97% per year, every year, consistently, we pulled from the historical data from Simba’s backtesting spreadsheet and determined what the mean return was for VTSMX and what the standard deviation was for returns that it saw? If we did that, we might find (and you can check my math and let me know in a comment below if you think I am wrong) that the mean is about 10.57% and that it had a standard deviation of 18.30.

If we used the mean and standard deviation and used those parameters to come up with 5,000 random numbers based on a normal distribution with those assumptions, we might see the values be distributed like the following summary.

Frequency of Distribution of VTSMX Yearly Rates of Return
Frequency of Distribution of VTSMX Yearly Rates of Return

The chart above shows the most of the time the return is in the approximately 10% per year range, but there are definitely times when the return can be in the neighborhood of -40% or as high as positive 60%. It just tends not to be that high or low most of the time.

If we ran the same Scenario that we did for plan 1 in the book, but varied the stock market rate of return, we might get a yearly rate of return for the VTSMX account to look like this.

VTSMX Yearly Rate of Return
VTSMX Yearly Rate of Return with Variable Return

If we looked at the net worth with this variable return for the stock market, it might look like the following chart.

Net Worth
Net Worth

Compare this to the chart in the book where we ran it with a static 8.97% yearly rate of return and you can see the difference.

Net Worth Comparison
Net Worth Comparison

If we look at the amount time it takes for my son to achieve financial independence with a static 8.97% rate of return and the variable rate of return we can see that in the chart below. Which, purely by coincidence, happens to be the same exact 599 months.

Comparing Goals
Comparing Goals

Copy the scenario with a variable rate of return to your own Planner™ and see for yourself when you run it multiple times and compare the results.

Please Login to copy this Scenario to a Free Plan, Plus Plan, Copper Plan or better.

Copy Scenario into my Real Estate Financial Planner™ Software

A-01 100% Stocks (Variable), 10% Savings, with 2 Accounts, 0 Properties, 2 Rules, 1 Goal

Alternate Universes

Do you find yourself asking, “can I see what this looks like if I run it multiple times?” The answer is yes. First, you could make multiple copies of the Scenario above to your Planner™ and plot them all on the chart to see the results.

Or, we could use the Monte Carlo features of the Planner™ which allows us to create “alternate universes” to see what might have happened and summarize all the results together automatically.

For example, I took the same Scenario we have been playing with that has the variable rate of return and I set it up to run 100 times. Let’s look first at net worth for the first run.

Net Worth with 1 Monte Carlo Run
Net Worth with 1 Monte Carlo Run

In that “alternate universe” with the random stock market rate of return based on the characteristics of the actual historical returns it has had from 1871 to 2017, my son’s net worth in raw, inflated dollars, would be about $22.3 million dollars. Adjusted back to today’s dollars that’s like having just under $4 million dollars.

But there is an “alternate universe” somewhere where the stock market returns in the future are different. They are still based on the characteristics of the actual historical returns, but the future unfolds differently. In that “alternate universe” the net worth of my son might look like this.

Net Worth with 2 Monte Carlo Runs
Net Worth with 2 Monte Carlo Runs

In that “alternate universe” my son ends up with almost $30 million dollars (or about $5 million in today’s inflation adjusted dollars).

If there’s 2 “alternate universes” there probably are 5. Here’s what 5 might look like.

Net Worth with 5 Monte Carlo Runs
Net Worth with 5 Monte Carlo Runs

And if there are 5, there are surely 10 “alternate universes”.

Net Worth with 10 Monte Carlo Runs
Net Worth with 10 Monte Carlo Runs

Remember, these are all the same plan for my son. He’s saving 10% of his income each month and investing it in the stock market. The only thing that is varying is the rate of return he’s getting in the stock market in each of these “alternate universes” and it is making a huge difference. In the worst case of the 10 “alternate universes” we’ve looked at he ends up with just over $12 million in inflated dollars (just over $2 million in today’s dollars) and in another “alternate universe” he ends up with almost $38 million dollars (almost $6.5 million in today’s dollars). That’s a pretty big difference for taking the same action: investing the same 10% of his income each month and investing in the entire stock market index fund, VTSMX.

Each “alternate universe” is what we call a Monte Carlo run. The more Monte Carlo runs we do, we see more variation. Sometimes that means we see a wider range of values with new highs and/or new lows. Sometimes it reinforces a range of values we’ve already seen.

If we look at 20 “alternate universes”, you can see the results on net worth in the chart below.

Net Worth with 20 Monte Carlo Runs
Net Worth with 20 Monte Carlo Runs

And, 30 looks like this.

Net Worth with 30 Monte Carlo Runs
Net Worth with 30 Monte Carlo Runs

40 like this.

Net Worth with 40 Monte Carlo Runs
Net Worth with 40 Monte Carlo Runs

50 “alternate universes” might look like this.

Net Worth with 50 Monte Carlo Runs
Net Worth with 50 Monte Carlo Runs

I love my charts, but even I have limits. I think they’re beautiful, but you do get to the point where it is really hard to see what is going on in as you continue plotting more and more “alternate universes”.

Summarizing Monte Carlo Results

So, what should we do about that? Could we summarize all the “alternate universes” instead of plotting each one individually. Maybe… just maybe. What if we showed you what the median (or the middle most value of all 100 “alternate universes” for this Scenario from the book, but with variable stock market rates of return. The following chart shows up the median (or the middle most value).

Median Net Worth with 100 Monte Carlo Runs
Median Net Worth with 100 Monte Carlo Runs

But how does the median of 100 Monte Carlo runs with variable stock market rates of return compare to the original book Scenario where we had a fixed 8.97% yearly rate of return for the stock market. Glad you asked. Here’s a chart showing how the median compares to the median of the static one.

Median Net Worth with 100 Monte Carlo Runs Compared To Static
Median Net Worth with 100 Monte Carlo Runs Compared To Static

Average or Expected Value

In the last couple charts we have been talking about the median or the middle most value from all the “alternate universes”. That means for any given month, we sort all the values for that month from lowest to highest and find the middle most number and plot that.

However, what if you wanted to look at the average result. You can look at any given month, add up all the values and divide through by the total number of values. That would tell you what my son might get, on average, from any “alternate universe”. Turns out that is different from median because sometimes the values are much higher or much lower and those pull the average either up or or down. In this particular set of “alternate universes”, there are some values that are much higher and those pull the values up.

Average Net Worth with 100 Monte Carlo Runs Compared To Static
Average Net Worth with 100 Monte Carlo Runs Compared To Static

SIDE NOTE: We call the average line the Expected Value or EV for short. Because if we take each of the possible “alternate universes” and multiply the chance of having that “alternate universe” (in this case with 100 Monte Carlo runs the chance is 1 out of 100) by the value (in this case the net worth value), we can see what are average Expected Value of doing this entire Scenario is, on average.

Back to our story, you can see that the Expected Value of a random stock market rate of return (based on actual historical data) is higher than the Expected Value of having a static rate of return of 8.97% for the stock market.

How big of a difference is it? Glad you asked because I like to plot the EV difference on charts as well. Here is a chart showing the same chart as above except I also an additional dashed line with an axis on the right showing the dollar EV difference as well.

EV Difference in Net Worth
EV Difference in Net Worth

In the chart above you can see that the biggest difference between the two average lines is about $1.3 million dollars in inflated future dollars.

Let’s look back to summarizing the results of the 100 Monte Carlo runs.

90% of The Results

What if you wanted to quickly see where 90% of the results for all 100 of the Monte Carlo runs are? Couldn’t we take any month and list all the results for the 100 runs in order of highest to lowest then ignore the top 5 runs and the bottom 5 runs and look at what the 6th best run is and the 94th best run are and plot those making a shaded band of values on the chart to give you an idea of where you are 90% likely to be between? I think we could. Here’s a chart showing that.

Net Worth with 90% of Results Shaded
Net Worth with 90% of Results Shaded
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