Examining Stock Returns for Normal Distributions

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Examining Stock Returns for Normal Distributions

July11, 2012

Part A.
A1 (CRSP 2000-2008)| VW Daily| EW Daily| VW Monthly| EW Monthly| Mean| 0.00%| 0.05%| -0.12%| 0.50%|
σ| 1.35%| 1.12%| 4.66%| 6.14%|
Table A1 shows return means and standard deviations for the CRSP market portfolio from 2000-2008.
In comparing daily vs monthly returns in both cases, equally weighted (EW) and value weighted (VW), Table A1 shows the mean and standard deviation are bigger for the monthly data than daily. This is due to the collection interval of the data. There are higher mean returns in the monthly than daily simply because the prices moved more in a month than in a day. There is also more deviation from the mean on a monthly basis than on a daily basis, because of the time period, more movement is possible.

In comparing EW to VW, in both the monthly and daily case. EW seems to have higher means signaling that smaller firms in general have higher returns which bring up the mean significantly more in the EW than in the VW because they have less influence over the VW. But there is not a large difference in daily standard deviations. Meaning, small and large stocks differ from their means (distribution) in a similar fashion on a daily basis. But on a monthly basis smaller stocks move more than larger ones, as indicated by higher st. dev on the monthly basis attributed to the EW portfolio.

Part B.

In this experiment I analyzed stock returns to determine if they follow a normal distribution. The result of this experiment could be extremely beneficial. If they do follow such a distribution one could develop a strategy based on historical performance and forecast future performance. To develop an accurate conclusion I used market indexes’, and individual stock’s, performance over different time periods. After this analysis I have concluded that stock returns are not normal and therefore cannot be forecasted on the basis of distribution probabilities.

The conclusion that stocks do not follow a normal distribution is evident in table B1.In the table I show the actual frequency of daily returns for three stocks (Boeing, Coca-Cola, and McDonalds) between 2002 and 2007. Based on the number of observations, I compare expected frequencies of a normal distribution of stock returns to the actual frequencies of returns. In the bottom row any number besides 0 shows that actual frequency does not equal what is expected in a normal distribution. The actual average row shows that there are extreme outliers and also that there is a concentration in the middle ranges, which are inconsistent with normal distributions.

Table B1. illustrates that these three stocks daily returns are not normally distributed based on deviations from their means.  
Even though table B1. Seems to provide evidence that stocks do not follow a normal distribution, there is the possibility that those three companies are exceptions. To further investigate I examine all 30 companies from the Dow Jones Industrial Average, on the basis of the Studentized Range (SR). In the past others have examined normal distributions with approximately 1,000 samples, which yield SR’s of less than 7.99. It is safe to conclude that in our case of 1,510 samples a SR greater than 7.99 is from a non-normal distribution. In table B2. the SR’s of the 30 companies are all greater than 7.99 and most are greater than 10. With this evidence I continue to reject the normality of stock returns.

B2. (Daily Returns)| MIN| MAX| St.Dev.| SR|
ALCOA INC| -0.10| 0.09| 0.02| 9.76|
AMERICAN EXPRESS CO| -0.08| 0.11| 0.02| 11.68|
BOEING CO| -0.08| 0.07| 0.02| 9.10|
BANK OF AMERICA CORP| -0.10| 0.08| 0.01| 14.69|
CATERPILLAR INC| -0.15| 0.09| 0.02| 13.60|
CISCO SYSTEMS INC| -0.11| 0.24| 0.02| 15.25|
CHEVRON CORP NEW| -0.07| 0.05| 0.01| 8.89|
DU PONT E I DE NEMOURS & CO| -0.07| 0.10| 0.01|...
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