# Is The S&P 500 Statistically Different?

July 22nd, 2014

in contributors, syndication

*Written by Monty Agarwal, MA Capital Management*

The second quarter of 2014 finished on a high note with the S&P 500 up 6.1% for the year and continuing the 195%+ rally since the lows of March 2009. This 5 year bull run has invited plenty of speculation not only on its length but also on the lack of a meaningful correction. A recent article pointed to this fact saying that,

"

Wall Street's 10% correction-free run is now 1,000 days long - and counting."

Follow up:

This and other such statements made us wonder if something has changed about the distribution of the S&P 500 monthly returns when compared to historical patterns. Therefore we did a statistical analysis to look at the risk adjusted returns of the S&P 500 as well as the distribution of those returns. Often a comparative study of the risk adjusted returns will tell us if the market is complacent, risky or operating within historic parameters.

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.

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 that 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.

The following table compares the risk adjusted returns of the S&P 500 since 1950. The table looks at the average monthly return, percentage of positive months as well as volatility measures and risk ratios. We use two volatility measures, standard deviation of all the months and the standard deviation of just the down months. As most investors do not mind large positive months but care about the large down months, it made sense to make that distinction. The Sharpe ratio compares the average months returns to the standard deviation of all the months while the Sortino ratio compares the average monthly return to the standard deviation of just the negative months.

*Raw Data Source: Yahoo Finance*

## Conclusions

- The decade from 1990-2000 was by comparison a very sanguine decade with the lowest number of negative months (31%), low volatility and the highest average monthly return (+1.90%).
- The current bull run that started on March 2009 has produced good risk adjusted returns but this pace has increased significantly since 2013.
- The past 18 months have produced the highest risk adjusted returns as well as the lowest negative months with the lowest volatility. This pattern is generally seen towards the end of a bull run and something to be aware of.

Now let us look at the distribution of these returns to see if the general commentary about the lack of a big correction is justified under the present low volatility environment. The following table shows the skew of the distribution when compared to a standard Gaussian distribution. A negative skew means that there are more and larger negative months than expected and vice-versa for a positive skew. The other 4 columns (-1sigma, -2sigma, etc.) show the percentage of negative months by size. According to a Gaussian distribution as explained above, we should see a:

- standard deviation down month every 3 months (34% of the time)
- standard deviation down month every 7 months (13.6% of the time)
- standard deviation down month every 48 months (2.1% of the time)
- standard deviation down month every 1000 months (0.1% of the time)

*Raw Data Source: Yahoo Finance*

## Conclusions

- The decade from 1990-2000 was indeed very sanguine. We did not see any down month larger than 2σ.
- But that has also been true since the last bull run began in 3/2009. In the past 63 months we have not seen any negative month larger than 2σ.

With volatility running at 2.42% and downside volatility at only 0.88%, for the market to see a 10% correction would imply a few months of 3σ and 4σ drawdowns. This is very unlikely under the current market environment. First we would have to see a general pickup in volatility, not just downside but overall volatility. Therefore we would tell investors to keep an eye on daily S&P 500 volatility as a pickup in that will be the first indicator of a market correction. And as the following chart shows, that daily volatility is still very low.

*Raw Data Source: Yahoo Finance*

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