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**Unformatted text preview: **Gains from Diversification for 2 Risky Assets 1 Suppose you are deciding how to allocate your wealth between two risky assets. Recall that the expected return of a two-asset portfolio is: ¯ R p = X 1 ¯ R 1 + X 2 ¯ R 2 where ¯ R 1 and ¯ R 2 are the expected returns on Asset 1 and 2, and X 1 and X 2 are the weights invested in each. The standard deviation ( σ ) on the portfolio is: σ p = £ X 2 1 σ 2 1 + X 2 2 σ 2 2 + 2 X 1 X 2 σ 1 σ 2 ρ / 1 / 2 where ρ is the correlation between the two assets. Suppose we have the following information: ¯ R 1 = . 17 σ 1 = . 25 ¯ R 2 = . 10 σ 2 = . 12 We also know that the correlation between these assets is ρ = . 2. Notice that Asset 2 has a lower mean and standard deviation than Asset 1. Using the two formulas, we can compute the mean and standard deviation for a range of weights in Assets 1 and 2: X 1 X 2 ¯ R p σ p 1 .100 .120 .2 .8 .114 .117 .4 .6 .128 .134 .6 .4 .142 .166 .8 .2 .156 .206 1 .170 .250 For example, consider the portfolio with 20% in Asset 1 and 80% in Asset 2. From the formula for the mean: ¯ R p = . 2( . 17) + . 8( . 10) = . 114 . 1 Notes for Finance 100 (sections 301 and 302) prepared by Jessica A. Wachter. From the formula for the standard deviation: σ p = £ ( . 2) 2 ( . 25) 2 + ( . 8) 2 ( . 12) 2 + 2( . 2)( . 8)( . 25)( . 12)( . 2) / 1 / 2 = . 117 The table shows that as we go from 100% in the second asset ( X 2 = 1) to 100% in the first the mean increases. Not so for the standard deviation. Adding a little bit of Asset 1first the mean increases....

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