## Summary

- A statistical analysis of the distribution of returns can be used to determine many key characteristics of a stock's behavior.
- Volatility is not the only useful measure of risk that can be determined via quantitative analysis. Kurtosis and skewness can be equally helpful.
- Quantitative measure can be used as a tool for comparing different stocks.

Investors today have access to more data than any single person could ever make use of. There are thousands of investments to choose from and an endless number of ways to try to sort through them. Even a basic statistical analysis of a stock can provide some insight into the stock's behavior and risk profile. I will use the daily log returns for the S&P 500 (hereinafter SPX) to demonstrate the potential utility of this approach for investors.

I explained the reason for log returns in one of my previous articles, and to quote myself:

The reason for logarithmic returns is that, unlike percentage changes, they are symmetric. So if the log return is -0.05 on a given day, then a log return of +0.05 the next day will leave the index with the same value it started.

I usually use the closing value and it is good practice to be consistent in order to ensure that the statistics are comparable. I used freely available daily prices from Yahoo Finance for all of the analysis in this article. Now, I've talked about the distribution of log returns before in the context of market efficiency, but my primary focus at the time was on the deviation from a normal distribution. Here, the focus is on the summary statistics rather than the best fit for the distribution. Should you find yourself lacking fancy statistical software, Excel has functions for all of the sample statistics. Let's start with SPX.

*Source: Data by Yahoo Finance. The data is the daily closing value of the S&P* 500 *between 1/02/1950 and 6/25/2014.*

Unless there is something seriously wrong with you, it is unlikely that you have an intuitive sense for how large of a movement is implied by log returns. Let L denote the log return, P denote the corresponding percentage return, and e denote the base of the natural logarithm (approximate value: e=2.71828). The formula for converting is: P=eL-1. The average daily log return is 0.0002939, that corresponds to an average gain of 0.02939432%. If you annualize that you get 7.40736% as the average annual percentage gain; as expected.

The largest one day decline was a 20.4669% loss (Black Monday) and the largest one day gain was 11.58%. The significant difference in magnitude between the minimum and maximum values is a clue that there is likely an underlying asymmetry in the distribution even though it is not easy to see that by looking at the histogram. The skewness indicates the magnitude and direction of a distribution's asymmetry. If the skew is zero, as is the case for a normal distribution, then the left and right sides (relative to the mean) of the distribution carry equal weight (note: the distribution need not be symmetric about the mean; it's more accurate to say a skewness of zero implies that there is an equal probability of a randomly selected value from the distribution being above or below the mean). A general rule of thumb for skewness is that a value below -1.0 or above +1.0 is significant. In this case we have -1.03076 and that is enough to say that the distribution has a negative skew. To understand what the negative skew means in the context of the distribution of returns is a little bit tricky. Think of the negative skew as implying that the left tail of the distribution is heavier than the right tail.

Interpreting that is a little bit tricky. I see the negative skew as saying that the market doesn't crash upwards. That seems a bit unusual to say given that there has been a one day gain of 11.58%; however, it becomes more clear when the data is shown over time.

*Source: Data by Yahoo Finance. Close is plotted with a log scale. Log returns plotted with a linear scale. Dates: 1/2/1950-6/27/2014.*

The bottom chart clearly shows volatility clustering. Periods of low volatility are close together and periods of high volatility are close together. What's interesting is that even during a time of crisis (say, 2008) the log return chart seems very highly symmetric. It makes sense when you think about the panic and uncertainty, but it still seems unusual that some of the largest one day gains happened at a time when the market lost about half its value. When the volatility is so high, it doesn't take much of an asymmetry to cause a significant long term impact.

The last significant statistic on the list is kurtosis. Actually, there is an annoying ambiguity about kurtosis that you should be aware of: kurtosis or excess kurtosis. The difference is that the excess kurtosis is kurtosis-3. This is done to make the (excess) kurtosis of a normal distribution equal to zero. The measure in the distribution above is in fact excess kurtosis (I'm just going to write kurtosis from now on). As you can see, it's 27.715155 for SPX. Kurtosis is often used as a proxy for how heavy (or fat) the tails of a distribution are. A kurtosis of 27 means that the distribution has a *much* sharper peak and much heavier tails than a normal distribution. Wikipedia has a great picture for this:

*Source: Wikipedia.*

Now, why bother going through all of that if there are other ways to reach the conclusions stated above? In part, it depends upon your preferred approach to investing. This sort of analysis can be done on any stock, and can provide additional metrics for comparison. None of the statistics described here are hard to calculate, so there is no reason to only use the volatility when analyzing a stock (I'm assuming you are not doing this by hand). In particular, using both skewness and kurtosis along with the more widely cited measure of risk, volatility, can provide a better description of the risk profile than the volatility alone can.

**Disclosure: **The author has no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. The author wrote this article themselves, and it expresses their own opinions. The author is not receiving compensation for it (other than from Seeking Alpha). The author has no business relationship with any company whose stock is mentioned in this article.