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Suppose that we are making up a portfolio by investing a fraction 1/N of the portfolio in each of N securities, with each individual security indexed by a different value of i, i=1...N.
Suppose further that the realized return on an individual security i is:
where ui denotes the particular "unique" risk associated with each particular security i alone (or, at least, not associated with too many of the other securities), r*i denotes the required rate of return on the i'th of the N securities, and rm and r*m denote the realized and required rates of return on the "market" portfolio.
Add up all the N positions in each of the N securities to get the total realized return on the porfolio:
Take expected values, note that the expected deviation of the realized market from the required market return is zero, note that the expected value of the "unique" risk of the i securities is zero, and find that the expected return on the portfolio is merely the average required return on each of the N securities:
and the variance of the portfolio is simply the expected value of the squared difference between the realized return and on the portfolio and the expected return on the portfolio:
Now note that (i) the individual ui's have no correlation with each other, and (ii) the excess return on the market has no correlation with any of the ui's, so the equation above reduces to:
As N grows large, the second term shrinks down toward zero, and so the variance and standard deviation are approximately:
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