As U.S. equity investors know, the U.S. stock market has had an impressive average annual real return of about 7% under dividend reinvestment conditions over the past 200 years.
On the other hand, the expected return and risk (standard deviation) is calculated as a statistic mathematically determined from the price movements of the stock market over the years.
In this article, we examine the validity of the normal distribution assumption of stock prices and the probabilistic prediction of future returns through Monte Carlo simulations, which estimate future returns probabilistically by running a large number of trials using random numbers based on these statistics (appropriately obtained like a roll of the dice) as a mathematical consideration.
If you want to set aside the mathematics and know the conclusions, I recommend skipping to “Results and Discussion of Monte Carlo Simulations in the U.S. Stock Market” from the table of contents below.
Assumption: Stock market price movements follow a normal distribution
Nowadays, with the development of information technology, big data is in the spotlight, and statistics is becoming more and more important in the business world, you may have heard of the expression normal distribution.
A normal distribution is one that exists in nature in large numbers and is most often employed in statistical processing, and is a characteristic distribution of “number” relative to the “measure” we are interested in.
Normal Distribution, and this Normal literally means “normal” or “very common”. A person’s height, weight, test score distribution, and even the size of raindrops during rainfall are said to fall under the normal distribution.
SOURCE:Wikipedia
The above figure shows a graph of a normal distribution. The horizontal axis is the value and the vertical axis is the frequency (or probability).
The σ in the figure is called the standard deviation and refers to the degree of variation in this distribution. In a normal distribution, there is a 68% probability of falling within +1σ to -1σ, 95% probability within ±2σ, and 99.7% probability at ±3σ.
The reason why I explained the normal distribution is because the theory is built on the assumption that stock price trends, which investors may be concerned about, also follow a normal distribution in financial engineering.
The point is that there is a highly general-purpose distribution called a normal distribution, and stock prices are generally thought to follow it.
In fact, if you look at the website of a securities company, you will see that the risk and expected return for each stock or index are described. The risk is the standard deviation of a normal distribution (σ in the above figure) and the expected return is the expected value assuming a normal distribution.
U.S. Stock Market Statistics
So what are the risks and returns of our US stock market (S&P 500)?
Dr. Siegel, who is familiar to investors, has calculated the real returns and risks of the U.S. stock market under dividend reinvestment conditions, based on nearly 200 years of stock price data, as shown in the chart below (a bit outdated from the second edition, but probably not much different today).
SOURCE:Stocks for the long run (2nd Edition), Jeremy J. Siegel (1998).
The first row of the column labeled Total Real Returns in the middle (for the period 1802-1997) is of interest here. From this table we can see the following:
Real total return: 7.0% (geometric average)
Risk: 18.1%
Results and Discussion of Monte Carlo Simulations in U.S. Stock Markets
In the previous section, we found that the real total return of the U.S. stock market is 7.0% per year and the risk (standard deviation) is 18.1%. With these two pieces of information, we can predict the probabilistic results by Monte Carlo simulations that involve a large number of trials, waving random numbers assuming a normal distribution.
So, let’s get on with the Monte Carlo simulation!
The conditions are as follows.
-Annual changes in the index are assumed to be normally distributed.
-Real return is 7.0%.
-Risk is 18.1%.
-10,000 trials.
-Investment periods are 1, 3, 5, 10, 15, 20, 25, 30, 40, 50 years.
-The language used is Python.
For example, in the case of 5 years, the probability distribution of the final return multiplier is calculated by multiplying the values multiplied by a random number 5 times (i.e., the return after 5 years) 10,000 times to find the probability distribution of the final return multiplier. The following figure shows the relationship between the return multiplier (horizontal axis) and its probability (vertical axis) for each investment period. The probability on the vertical axis is 100% when it is 1 and 20% when it is 0.2, for example.
[1-year return probability distribution]
[3-year return probability distribution]
[5-year return probability distribution]
[10-year return probability distribution]
[15-year return probability distribution]
[20-year return probability distribution]
[25-year return probability distribution]
[30-year return probability distribution]
[40-year return probability distribution]
[50-year return probability distribution]
Now, what do you think of the above diagram?
One of the features is that as the number of years invested increases, the distribution spreads to the right side (the high multiple side). This is something to be happy about as a long-term investor.
Another important feature is that even with a long term investment of 30 years, it can lose about 5% of its principal (since the probability of 0-1x is 0.052), and the most probable maximum probability is about 2-3x, which is not a great multiple.
That’s a bit odd.
The U.S. stock market, under dividend reinvestment conditions, should be expected to invest for 20 years or more over the past 200 years with no loss of principal, and a real total return of about 7% on average should be expected to be about 7.6 times over 30 years (see table below).
As shown in the figure above, the median value of the 30-year investment is 4.96 times, while the expected value is 7.62 times. The median is the value of the case where the results of the trials are ordered from the bottom (or the top) and come right in the middle. In this case, there are 10,000 trials, so the 5000th falls into this category.
Why is there such a large discrepancy between the expected value and the median value? The reason for this is that the right side of the graph, on the high-powered side of the graph, has a wider sketchy field, so the high-performance results attract a much higher expected value (because the expected value is the sum of value x probability). This is because the expected value is the sum of “value x probability”.
It’s the same reason why the average annual income is different from the median annual income. It’s the one where people who make an order of magnitude more money, like Masayoshi Son, pull the average income of the nation to the high income side.
So, based on these results, if you’re going to invest for 30 years, you should probably expect five times the median return, which in turn is the median return that comes to the center of the market.
By the way, why is it that the U.S. stock market has never lost principal under the dividend reinvestment condition if you invest for 20 years or more over the past 200 years, whereas the Monte Carlo simulation shows that the U.S. stock market can lose about 5% of its principal even if you invest for the long term over 30 years?
The author is neither a statistics expert nor an economic expert, but my personal opinion is that the reason for this is probably that the long term trend of stock prices is not normally distributed, and mathematically speaking, stock prices are not independent events on an annual basis.
And I think the reason for this is because of an invisible force called the “law of mean reversion” that makes the stock price rise exponentially in the long term.
Summary
A Monte Carlo simulation, assuming a normal distribution, was used to obtain a hypothetical future return probability distribution under dividend reinvestment conditions in the US stock market.
There are some differences between the Monte Carlo simulations and the actual stock price trends over the past 200 years, and my personal opinion is that the reason for this is that the law of mean reversion is at work and stock price trends are not independent events on an annual basis, so the normal distribution does not apply.
Well, either way, I think we have shown probabilistically that the future is bright for long-term stock investors.
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