PortfolioVarianceWithManyRiskyAssets

PortfolioVarianceWithManyRiskyAssets - Portfolio Variance...

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Portfolio Variance with Many Risky Assets 1 Case 1: Unsystematic risk only. Recall that when the correlation ρ between two securities equals zero, the portfolio vari- ance is given by: σ 2 p = X 2 1 σ 2 1 + X 2 2 σ 2 2 A simple generalization of this formula holds for many securities provided that ρ = 0 between all pairs of securities : σ 2 p = X 2 1 σ 2 1 + X 2 2 σ 2 2 + ··· + X 2 N σ 2 N . (1) We will prove the following result. As N → ∞ , the portfolio standard deviation σ p 0. To make the notation simpler, assume that σ 1 = σ 2 = ··· = σ N = σ . 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 X i = 1 N . These assumptions allow us to rewrite expression (1) as σ 2 p = ± 1 N 2 σ 2 + ± 1 N 2 σ 2 + ··· + ± 1 N 2 σ 2 (2) There are N identical terms in expression (2), which means: σ 2 p = N ± 1 N 2 σ 2 σ 2 p = 1 N σ 2 The expression in (3) shows that as
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PortfolioVarianceWithManyRiskyAssets - Portfolio Variance...

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