More General Formulas for Variances

More General Formulas for Variances - the same size, and...

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More General Formulas for Variances: Unpack this formula: what does it mean? An especially interesting special case: Suppose we have a number of securities, indexed by i, all of whose returns look something like this: and suppose that the u i 's are uncorrelated with each other and with the market--that their covariance is zero, or is at least not too large. Now suppose that we take our portfolio, and our portfolio shares x i , and set each x i =1/N: put 1/N of our portfolio's wealth into each of the first N of these securities. What does our variance look like? Well, we know that: And we can substitute in: Noticing that the first and third terms look awfully familiar: is the variance of the portfolio: Now the first term is simply the average beta of the stocks in the portfolio, squared, times the variance of the "market" portfolio. .. The second term is the sum of the idiosyncratic risks u associated with the securities i--all divided by N-squared.
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Now a funny thing happens as N gets large: the second term vanishes: the sigmas stay (roughly)
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Unformatted text preview: the same size, and you are adding up more (N) of them, but each one is divided by N-squared. So as N approaches infinity, the second term gets small--eventually small enough to ignore. .. An even funnier thing happens if we consider the variance of a portfolio made up of a large number of stocks (N large), for which the average beta happens to be one. Then the first term is simply sigma-squared-m, and the second term is negligible: the variance of the portfolio is the variance of the "market" Conclusion: If an investor is doing his or her job--trying to get to a portfolio that has the minimum risk for a given expected return--then "idiosyncratic" risk can be diversified away: by putting all your eggs into many baskets, you essentially eliminate any idiosyncratic risk factors from your portfolio. Hence a security's "riskiness"--from the point of view of its impact on the overall riskiness of your portfolio as a whole--is summarized by its beta. .....
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This note was uploaded on 11/10/2011 for the course GEB GEB1011 taught by Professor Henn during the Fall '10 term at Broward College.

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