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Business Administration Derivation of the Variance of a Portfolio According to the Capital Asset Pricing Model 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 u i 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 r m 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

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Unformatted text preview: 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 u i 's have no correlation with each other, and (ii) the excess return on the market has no correlation with any of the u i '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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