Deriving Asset–Portfolio Correlation from Pairwise Covariances

Summary

The document asks how a portfolio Sharpe-ratio rule can be understood through marginal risk contribution. The rule under discussion says to consider adding an asset when its Sharpe ratio exceeds the portfolio Sharpe ratio multiplied by the asset’s correlation with the portfolio. The question focuses on a derivation that rewrites that correlation using correlations between the candidate asset and the portfolio’s component assets.

The response explains that the covariance between the candidate asset and the portfolio can be expanded because portfolio returns are a weighted sum of asset returns. Covariance is linear in either argument, and the covariance of a variable with itself is its variance. Together with the usual covariance-over-volatility formula for correlation, these facts lead to a weighted expression using pairwise asset relationships. The excerpt gives the key identities but not the full algebra, assumptions, or a numerical example, so the derivation must be completed from the portfolio weights and return covariance definitions.

Key ideas

• Portfolio returns can be represented as a weighted sum of the returns of their component assets.
• Covariance with a weighted sum equals the weighted sum of the individual covariances.
• An asset’s correlation with a portfolio follows by dividing its covariance with that portfolio by both volatilities.
• Pairwise asset covariances can therefore be used to express asset–portfolio correlation.
• The excerpt provides core covariance identities but not the complete derivation or assumptions.

Tags

# Question about marginal risk contribution / portfolio volatility decomposition


# Question about marginal risk contribution / portfolio volatility decomposition












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?

## Answer by Richard Hardy (score 1)

https://quant.stackexchange.com/a/78700

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:

- $\text{Cov}(aX+bY,Z)=a \text{Cov}(X,Z) + b \text{Cov}(Y,Z)$

- $\text{Cov}(aZ,Z)=a \text{Var}(Z)$

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