Mr. Edward Witten has chosen to compare stock price variances versus related indexes before delving into the world of day trading. To this end, he has monitored weekly price changes of selected Stocks A and B for forty weeks, juxtaposing results with price movements of market, industry and economic indexes over the same observation period. Mr. Witten seeks to determine a correlation (or absence thereof) between his individual stocks and overall market behavior, results of which are presented in the attached spreadsheet. As statistical calculations are meaningless by themselves, Mr. Witten now faces the additional task of interpreting these results to determine if one, both, or neither of his stocks should be traded daily. Interpretation may likewise be the key to validating his decision to become a day trader as opposed to being an investor or longer-term trader.
Significance of Selected Indexes
Mr. Witten selected the S&P 500 as his Market Index. This index is a popular benchmark in stock market analysis because it is composed of the top 500 U.S. companies in terms of market capitalization and trading volume (Schick, 2016). About a dozen sectors comprise this index, with several industries falling industries under each sector. Industries within a sector will move together in the same direction, and the sectors will in turn move in the same direction as the S&P 500.

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The Industry Index to which Stocks A and B belong is not specified. However, an eyeballing of data under the columns for Market and Industry indices show that Mr. Witten has chosen an industry very much in tandem with the S&P 500 as far as index prices being up or down for the week, with only 4 out of the 40 weeks being an exception. Therefore, the chosen Industry Index is a valid statistical component in analyzing specific stock movement.

The Economic Index is likewise unspecified but behaves similarly. In the sample data, it is even slightly more in tandem with S&P 500 than the Industry index, with only 3 instances out of the 40 weeks wherein it moved opposite of the overall market. Therefore, it is an educated observation that the economic indicators reported by government agencies during the sampled weeks directly impacted the behavior of both the overall market and the chosen industry.

It would be remiss not to mention that the averages of all 3 indices were down overall for the observation period – minus 9%, minus 5% and minus 12% for the Market, Industry and Economic indexes, respectively. That Mr. Witten’s chosen industry holds up relatively better than the S&P and the economy during market downturns is significant and possibly attributable to his choice of stocks.

The Role of Statistical Tools in Analyzing Stock Market Data
The mutual impact of indexes on each other and on individual stocks can be verified through statistical tools which calculate relationships between gathered raw data. Specifically for this case study, calculations for Correlation and Beta prove useful in determining whether Mr. Witten has selected the right indexes and stocks to base future trades on.

Correlation Coefficient
Correlation “measures the degree to which two variables move in relation to each other” (Investopedia, 2015). Through its correlation function, Excel returns a coefficient falling in the range of plus 1 to negative 1 inclusive, with “+1” indicating not only a positive but also perfect correlation. A coefficient of “-1” would indicate an absolutely inverse or negative relationship between variables while a result of “0” would mean no relationship at all.

The Correlation column in Tab 1 of the attached spreadsheet indicates a relatively high, albeit not perfect, correlation among the S&P 500, Industry and Economic indexes. Range results from 0.90 to 0.94 confirm the earlier observation that the indices selected by Mr. Witten move in tandem, whether up or down, over 90% of the time.

Coefficient results are somewhat lower, in the 80%+ range, when each index is compared to Stock A or to Stock B. The lower range could be interpreted as both stocks being strong performers in the industry but not overwhelmingly strong enough to alter the movements of the broader indexes.

Of the three indices, price movements in the S&P 500 have the highest correlation to individual Stocks A and B and will best impact the projected rise and fall of these stocks.

The Correlation column in Tab 2 of the spreadsheet considers possible returns if Mr. Witten analyzes his stocks versus a combination of two indices, or even all three, for greater accuracy. It is interesting to note that regardless of what indices were combined, the relationship with Stock A resulted in exactly the same coefficient of 0.89 each time during the 40-week observation period. For Stock B, the coefficients were slightly lower, with the combination of S&P-and-Industry indexes providing the best correlation.

Calculation of Beta
Beta is a statistical tool used to calculate the risk and volatility of a stock relative to an index (Khan, 2011). It is the quotient obtained when dividing Covariance by Variance. Similar to Correlation, Covariance measures how two variables – such as stock and a benchmark index – are related, but Correlation goes one step further in measuring to what degree this relationship exists. Variance in Excel simply calculates a percentage increase or decrease in price.

The calculation of Beta returns one of the following results: “0”, “1”, “> 1” or “< 1”. Zero indicates no relationship between the stock and the index, while a Beta of one indicates that the stock has the same risk and volatility as its benchmark index. A Beta of less than one makes the stock less volatile, therefore entailing less risk for the trader or investor. Greater than one is significant because it means the stock is more volatile than the rest of the industry or even the overall market, but can be attractive to the trader because higher risk means higher returns. It is in the calculation of Beta that the distinction between the two stocks Mr. Witten has chosen becomes pronounced and worth noting. Stock A consistently renders Beta > 1, regardless of the individual index it is compared to and even higher when the stock analysis utilizes two or more indices. Stock A’s Beta ranges from 1.09 to 1.22, with the most profitable result occurring when the stock is compared to the S&P 500 Index, or a combination of S&P-Industry indices.

On the other hand, Stock B consistently results in Beta <1, ranging from 0.79 to 0.90, with a combination of the S&P-Industry indices as benchmark rendering the best possible results for the stock. Mr. Written will incur less risk with this stock, but will receive lesser returns as well. Edward Witten’s Decision to Trade Stocks A and B Going Forward Both the S&P 500 and Stock B were down by the same average of -9% during the observation period. While historical data is not necessarily indicative of future stock performance, it seems reasonable to predict that Stock B’s price movement is not only in tandem with the S&P’s but will likely render the same percentage results. By contrast, Stock A’s overall average was up 19%, a far cry from any of the indices despite the fact that it, too, moves in tandem with the overall market. Clearly, Stock A is the stronger performer of the two stocks. Having established that all three indices move in tandem with each other and that the selected stocks are positively correlated to the indices, a future change in one index will impact individual stocks proportionately. For instance, an upward future change of 5% in the S&P 500 will reasonably result in Stock A profiting by around 6%, i.e. highest Beta of 1.22 multiplied by .05. Similarly, Stock B with its highest Beta at 0.90 multiplied by .05 will render a return of 4.5%. The more significant factor is Mr. Witten’s decision on which stock he will trade and how often. Day trading, as the name implies, requires buying and selling frequently within the day quality stocks with high volatility. Stock A falls in this category and Mr. Witten will have to decide if he is willing to take the risk. Stock B, on the other hand, is not ideal for high-frequency trading but is comparable to holding any stock in the S&P 500 for a longer term.

  • Khan, S. (2011). Calculate stock beta with Excel. Retrieved from
  • Schick, K. (2016). An introduction to stock market indexes. Investopedia. Retrieved from