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Most market analysis and projections of the market involve some degree of statistical analysis, and quite a few investors are wondering if the market rally has become over-extended - might it be time to get defensive?

Before I get into a statistical analysis of the S&P 500, I want to briefly go over some basic principles I will be discussing in this article.

The following is chart of a Gaussian distribution function, which basically shows the probability distribution of an event occurring in a data sample with a standard deviation of σ.

(click to enlarge)

This means that 34.1% of the data sample will be below the mean and 34.1% will be above the mean for all data within 1 σ of the mean and 47.70% for all data within 2 σ, etc.

This is important to understand, because investors use statistics to determine the probability of a big down move in the markets. Therefore, most down months in the S&P 500 should fall within a 3 σ band, which according to a Gaussian distribution implies a 99.6% probability. This means that any down move larger than a 3 σ should not happen more than 0.2% times (looking at just the probability of a negative month), which is basically once every 42 years.

Through our recent experiences in the market, we know that this is not the case. Down months that are larger than a 3σ happen with a lot higher frequency. This concept has often been termed and discussed under the Black Swan theory.

The following chart shows the distribution of the negative return months in the S&P 500 since 1950, broken out by decades.

(click to enlarge)

Raw Data Source: Yahoo Finance.

## Conclusions

1. The 90s were by comparison a very sanguine decade with the lowest number of negative months (31.1%) and the lowest number of down months greater than 2 σ (2.3%).

2. But, the number of months that the S&P 500 has been down 3 σ or greater has far exceeded the expectation from a Gaussian perspective. Expectation is once every 1,000 months, but as can be seen the frequency has been far greater than that.

3. The number of months that the S&P 500 has been down greater than 2 σ is happening with a higher frequency since 2000 than at any time since 1950.

Now let us look at the return side of the market, where we compare the average positive monthly return of the S&P 500 versus the downside deviations of the monthly returns.

Raw Data Source: Yahoo Finance.

## Conclusions

1. The average positive monthly return since 2000 has been the lowest off any previous period in our study at 3.00%/month.

2. The downside deviation as compared to the average return since 2000 has been very painful. Since 2000, investors have received:

• 1 -3.73% return month every 4 months
• 1 -7.46% return month every 12 months
• 1 -11.19% return month every 16 months
• 1 -14.92% return month every 4 years.

Such market moves that fall outside the standard statistical projections require pro-active portfolio management.

## Effects on Portfolio Management

1. Need for risk management to eliminate greater than a 2 σ negative months in a portfolio.

2. Recognition that 'Buy and Hold' needs to be replaced with 'Buy or Short' as the opportunities on the downside are more prevalent and instruments to capture it readily available through short selling, inverse ETFs, futures and options.

3. Need to look for assets that exhibit better risk/return parameters than the S&P 500, by considering commodities, frontier markets, real estate and alternatives.

And finally, let us see how the historical statistical breakdown of the S&P500 compares to what the market has delivered in the most recent bull run, i.e. since March 2009.

(click to enlarge)

Raw Data Source: Yahoo Finance.

## Conclusions

1. The number of down months at 33.3% is running in line with the sanguine 90s.

2. Negative months in the S&P 500 greater than 3 σ have not occurred in the past 4 years, which means that if history were to hold true, the probability of a -15% month occurring over the next 6 years has increased.

Markets are constantly evolving and the 2010-2020 decade might throw a few surprises of its own. The best way to traverse it, in my opinion, is to stick to a systematic approach which is not sullied by human emotions.