A Stock Solution; A Different Topping on Every Slice

When I was 14, my soccer coach would tell us that in order to be good defensive players we needed only to remember the three Ds: Delay, Direct, Destroy. Coincidentally, a similar axiom exists for investors hoping to maximize profits from a risk-averse portfolio: Diversity, Diversity, Diversity. Resource allocation, or where you choose to place your money, is one of the most fundamental cornerstones of investing, and rightfully so. Of course, certain types of investors or speculators have optimal portfolio compositions which are heavily skewed, and that works for their style. That being said, the average investor who is simply investing to grow their nest egg may be interested in the less risky investments.

At first blush it might seem like common sense. If your portfolio is composed solely of Nike, Under Armor, Lulu Lemon Athletica, and Adidas....

but of course, this is making a pretty big assumption that all of these companies essentially covary perfectly. We're assuming that when the whole athletic wear sector goes up, they all go up by the same number of points, and that when one has a successful period, they must naturally be taking an equal amount of value from their competitors. In addition to this concern, we should also consider more diversified companies, holding companies, and any other assets whose impact on a portfolio is not necessarily obvious. For example, how much do Amazon and Google co-vary? Facebook and Google? And now that Google is developing self-driving cars, are they in competition with the auto industry too?! When you take some of the projects at many companies into account, it can become overwhelming. Thus, in order to relax some assumptions and to account for complex contributions to a portfolio, we can use a number of statistical tools, the primary and most simple being a correlation analysis.

Correlation describes, on a scale of -1 to +1, the change in two stock's prices or one stock relative to an index (SP500, DJIA, etc.). In terms of readout, a +1 represents perfectly positively correlated stocks, a -1 would be a perfectly negatively correlation, and 0 indicates no correlation. In the case of constructing your portfolio, stocks with very little correlation to stocks you already own, as well as a stock with negative correlation to your other stocks are lend to the overall diversity of a portfolio. Perfectly positively correlated stocks do not complement each other and as a result provide zero diversity to a portfolio. Calculations such as these help investors to determine the diversity of a portfolio, but ultimately much more information and expertise go into determining the weighting of different sectors in a portfolio, as well as what price is a good price for a stock you are looking to add (asset pricing analysis is discussed in my previous post, if thats how you feel like spending the next 10 minutes of your life)

A Stock Solution; Asset Pricing Theory

Alright ya'll it's about to get stuffy in here. Something I have strong interest in is global markets and the stock exchange (well, strong for a biochemist with no economics training past high school). Naturally, this sector is perfect for exploring the widespread use of statistics and the unique and powerful ways they can be applied. While disciples of Benjamin Graham will warn that the valuation of a company's stock is not 100% tied to potential for an upward trend, this undoubtedly plays some role in the type of trading done on the market today. More than 50% of trading on the NYSE is done via something called High-Frequency Trading, which uses complex algorithms to buy and sell securities on the millisecond time scale for an overwhelming addition of small differences, resulting in a large profit for the companies employing these buying and trading algorithms. I'm not nearly savvy enough to describe these algorithms with sufficient detail, but I do want to discuss some aspects of these algorithms and other probability/statistics related applications in stock trading.

Investors deal with an immense amount of data, and more is generated every couple of milliseconds. As a result, a combination of instinct, experience, and sound statistical models are the professional trader's bread and butter. Various statistical tests are utilized to assess risk and confidence.

The first technique (and arguably the most central) application of statistics in stock trading is in Asset Pricing Theory. This branch of investment theory uses the calculated effects (effect size!) of various macro-economic factors, or the behavior of theoretical indices (indices track many different stocks, or sectors and are like a mean value representing how a sector is doing. Common examples are the S&P 500 or the Dow Jones Industrial Average) to forecast the expected return of an asset. I realize that all sounds a bit vague, so let's focus on a specific example:

We are all familiar with the T-statistic (departure of a parameter from its notional value and its standard error) which we use in Student's T-Tests. In the case of investing, the utility of this statistic is almost exactly the same as when we would use it to compare means. In fact, one of the foundations of investment statistics is formation and testing of a null hypothesis. In Asset Pricing Theory, the null hypothesis would propose that "the expected return of the asset is not different from the risk-free rate of return". In other words, they compare the asset in question to the performance of risk-free investments like some bonds or savings accounts. Given the historical returns of an asset and the risk-free investment of choice, an investor may find a T-statistic describing the difference between their asset (our sample of interest) and the risk-free investment (background, WT, negative control, etc.). Investors will then use the T-statistic as an indicator of the probability of observing the asset's returns under the assumption of the null hypothesis. Another similarity is that, in this branch of economics, they set their statistically significant p-value as 0.05, and utilize confidence intervals to have a better understanding of how this asset is likely to behave. Similar to our experiments, these calculations also rely on a certain "N", where a single N could be individual transactions involving this asset (in this case, higher volume stocks would be advantaged, due to a greater number of values), but this is not always the case.

This type of statistical testing is vital to many stock analysts, who will use the likelihood that a stock will perform better than a risk-free investment as part of their assessment of how to score the stock (what they should advise their advisees or their firm to do with regard to the stock). It can also indicate when a stock may be "overpriced" or "on sale". In many trading circles, the direction that a stock is likely to go short-term is less important than the absolute value of the company and what a stock of that company should be "worth", hence "Asset Pricing Theory".

Challenges: Paper Compilation and Mad Dollar $igns

When I first looked at my assignment for this exam and found "Challenges in Statistics" next to my name I was a little dismayed. Isn't everything in statistics a challenge in statistics? I asked myself, but I realized this was the section of the text which dealt with gaussian distributions and outliers, probably the two statistical challenges I've actually faced since beginning this course. I have encountered both of recent in looking through a large old data set which I am trying to re-interpret for a paper I am writing. The data was generated by an old tech in the lab who isn't in the state anymore, and one challenge I had in approaching the data statistically and anticipating how I could display my data was that I didn't know about the distribution. I hadn't worked with these cells before, and I had only done a similar, but not exactly the same protocol. Reading about normality tests reminded me that I had that option, and now I remember what the "skewness" and "kurtosis" values are useful for in prism. Similarly, I struggled with outliers and how to treat potential outliers, because normally I would only reject an outlier if I had reason to believe that my own experimental error or some other thing outside my control had biased this outlier. However, since this wasn't my data, I also didn't perform the experiments! I had to make some educated choices with my PI, trying to remain informed with the Grubb's test.

 

In addition to the lab, the issue of distribution also affects my day-to-day money management. I'm a grad student who likes to invest my modest stipend in stocks. As someone who is fairly new to trading securities, one comfortable way into the world of Wall Street was via statistics. The same challenges in statistics which face scientists also face economists and stock traders. When there is real money on the line, it helps to have some of these statistical tools on my side, to help temper any bias. Given that I am trading in real companies with significant reputations, it's hard not to get emotionally involved in any of my holdings. However, if I can set some rules and use some of the tools in this section of the text, I can increase my overall profit through bullish and bearish. For example, check out this Forbes article which underscores how a normal distribution of the DJIA 30 indicates a healthy market (also see figure).

Displaying an individual stock's performance in what we hope is a normal distribution or finding standard deviation can indicate the volatility of a stock, correlated to risk. This value can also be similar to beta, the value many investors use to indicate volatility/risk, although it should be mentioned that this is not how beta is calculated. Lastly, many trading algorithms on Wall Street and financial theories anticipate a gaussian distribution of stock prices in order to maximize alpha or gains adjusted to the overall performance of the market. Thus, skew and kurtosis are important for investors to keep track of! The role of normal distribution, skew, and kurtosis in investing is simply summed up in this post, for those interested.