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At 3:32 PM -0400 9/26/00, Chris Emery wrote:
>If one were to try to take a group of stocks and correlate that group
>of stocks to an index, mathematically how would you do it? I know how
>to correlate one stock to an index but how do you correlate a bunch
>of stocks to an index? I am assuming that you would have to create an
>pseudo-index out of the stocks then correlate that result to the
>existing index to see if you have a match. Is there any mathematical
>way to accomplish the same task so as to avoid minimize the trial and
>error process?
>
>I know the simple theory would be to just look at the stocks in the
>existing index but my goal is to look for a combination of more
>volatile stocks which correlate to a given index.
It is not clear what you are trying to accomplish. Making a
pseudo-index is full of problems about how you weigh each stock in
the index.
The usual process is to relate the price of each stock to an index,
such as the SPX, by the equation:
Stock = alpha + beta * Index + Random
where:
"Stock" is the annual % change in stock price, and
"Index" is the annual % change in the value of the Index, and
"alpha" is a constant, and
"beta" is a constant, and
"Random" is a random variable with zero mean that represents the
error in the estimate.
"alpha" and "beta" are derived by fitting a least-squares linear
regression line through the data points of Stock and Index. You have
probably heard of the "alpha" and "beta" of a stock. A stock with an
alpha of 5% and a beta of 1.2 will change 12% when the index changes
10% but will increase 5% per year even if the index does not change
in a year.
Once you combine the stocks into a portfolio, you get the effects of
diversification. The returns of each stock combine as you would
expect but the standard deviation of returns of the portfolio
increase far slower depending upon how well correlated the stock
prices are.
Harry Markowitz shared a Nobel Prize in economics in 1990 for his
work in this area so you can imagine that it is not real obvious what
happens. Almost any modern economics text will explain the principles
involved, however. This field of economics is now called "Modern
Portfolio Theory".
Bob Fulks
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