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View Full Version : Variance of a subset of stocks vs. variance of market


DcifrThs
03-05-2004, 12:48 PM
The stock market is comprised of n different investments (stocks or whatever but i'll stick with stocks). Assuming it has an overall variance (i.e. there IS some central tendency from which to calculate a variance. it could be the average daily movement from time t0-the first day of trading of that market- to time tp-or the present day.), how does this variance relate to a random subset of investments?

specifically, let the subset Sk be comprised of [s1,s2,s3....sk] stocks. further, let this subset of stocks be randomly selected from all n stocks.

now, what is the probability that the VARIANCE of that random subset of stocks is LESS THAN the total market variance of all n stocks??

that is to say each subset has its own variance, i.e. average daily movement from initial public offering to the present day of each individual stock would comprise the mean and then the given day's difference in terms of % change would be the individual observation and this would obviously be summed over the lifetime of the shortest investment in the subset (for comparison the 'market variance' would be similarly computed).

given all that, what is the probability of selecting a subset whose variance as defined above is LESS than the total market variance over the lifetime of the shortest investment in the subset (clearly, if a subset was chosen where the shortest investment lifetime was too short to be significant then either a new subset would be drawn or that investment would be thrown out)

OR, more generally, what is the relationship between the variance of the subset and the variance of the whole over the time period dictated by the lifetime of the shortest lived stock in the subset??

Further, is there a better way to identify this problem? (i may be off base here with the comparisons and what not so somebody please correct me).

Questions? Comments?
-Barron

DOTTT
03-05-2004, 02:29 PM
I may not understand your question, but I believe what your refering to is called beta. Beta is used to measure mutual funds, hedge funds, as well as stocks correlation and risk to that of the market. Beta is referred to as an index of the systematic risk due to general market conditions that cannot be diversified away. It is the measure of a stock's riskiness as compared to the market. A stock with a beta of 2 means that the stock's excess return is expected to move 2 times that of the market. Thus, for this particicular stock, if the market's excess returns are 5%, the expected stock's excess returns are 10%. The beta equation used for stocks is below.
{[(n)*(sum of (xy))]-[(sum of x)*(sum of y)]} / {[(n)*(sum of (xx))]-[(sum of x)*(sum of x)]}
In this equation:
n = # of observations (24-60 months)
x = rate of return for the S&P 500 Index
y = rate of return for the stock
Beta is the slope coefficient when you regress excess returns for stock y onto the excess returns for the market x.
Giving this information we can choose stocks or mutual funds based on a beta that is proper for the current market condtions. If you believe that we are in a bull market we would want to own a high beta stock, on the other hand if you feel we are in a bear market you would want to own low beta stocks. I hope this helps.