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I am trying to understand the rule where you add a new asset to a portfolio if its Sharpe ratio is greater than the product of the portfolio sharpe ratio and the correlation between the portfolio and the new asset. There have been a few other questions on here regarding this topic, one of which linked this paper (https://hal.science/hal-03189299v2/file/Computation_Marginal_Contribution_Sharpe_ratio.pdf), which provides a multi-step proof. On page 12, the proof starts with the volatility of the portfolio, then taking the partial derivative with respect to w_i to express the marginal change in volatility contributed by a small change in the weight of asset i.

Then, in equation 17, the author expresses the correlation between the asset i and the portfolio p in terms of the correlation between i and j. I am having trouble understanding how correlation can be expressed this way, I usually think it would be expressed as Covar(i,P)/(sigma_i)(sigma_p). Does anyone know the derivation of this?

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  • $\begingroup$ The numerator in that formula is covariance between asset i and portfolio p where the latter is written as a weighted sum of assets. $\endgroup$ Commented Mar 18 at 7:14
  • $\begingroup$ $\text{Cov}(aX+bY,Z)=a \text{Cov}(X,Z) + b \text{Cov}(Y,Z)$. This is what is used there, but there are more than just a couple of assets in the portfolio, so you get a longer sum. Also, $\text{Cov}(aZ,Z)=a \text{Var}(Z)$, that is used there, too. $\endgroup$ Commented Mar 18 at 17:07
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    $\begingroup$ Please stop vandalizing the question! Let it stay in its original form, to which an answer has been posted. $\endgroup$ Commented Mar 21 at 7:45

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The numerator in that formula is the covariance between asset $i$ and portfolio $p$ where the latter is written as a weighted sum of assets. The fomula also employs the following facts:

  1. $\text{Cov}(aX+bY,Z)=a \text{Cov}(X,Z) + b \text{Cov}(Y,Z)$
  2. $\text{Cov}(aZ,Z)=a \text{Var}(Z)$
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