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Unformatted text preview: Portfolio Variance with Many Risky Securities William L Silber and Jessica A. Wachter Case 1: Unsystematic risk only. Recall that when the correlation between two securities equals zero, the portfolio vari ance is given by: 2 p = w 2 1 2 1 + w 2 2 2 2 A simple generalization of this formula holds for many securities provided that = 0 between all pairs of securities : 2 p = w 2 1 2 1 + w 2 2 2 2 + + w 2 N 2 N . (1) We will prove the following result. As N , the portfolio standard deviation p 0. To make the notation simpler, assume that 1 = 2 = = N = . This means that each asset is equally risky. Under those circumstances, we try the simple diversification strategy of dividing our wealth equally among each asset such that w i = 1 N . These assumptions allow us to rewrite expression (1) as 2 p = 1 N 2 2 + 1 N 2 2 + + 1 N 2 2 (2) There are N identical terms in expression (2), which means: 2 p = N 1 N...
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 Spring '08
 Miyakawa

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