Relating Market Portfolio Variance to Constituent Betas

Summary

The document poses a portfolio variance identity for a market portfolio formed as a weighted sum of asset returns. It asks how to show that the market portfolio's variance equals the weighted sum of each constituent's covariance with the market return, then proposes substituting the beta relationship between each asset's covariance and market variance.

This is a question rather than a completed derivation: it supplies the portfolio return definition and the identities it wants to establish, but includes no answer, proof, assumptions, or numerical example. The underlying idea connects portfolio covariance algebra with beta-based risk attribution. Its use depends on consistent definitions of returns, weights, and beta, and the document alone does not establish the proposed final expression or discuss what happens under different portfolio or return conventions.

Key ideas

• The market return is defined as a weighted sum of constituent returns.
• The question seeks to express market variance as the weighted sum of constituent covariances with the market.
• It proposes using the beta relationship between constituent covariance and market variance.
• No proof, assumptions, or worked example are provided in the document.

Tags

# Show that the variance of the portfolio market portfolio is function of the betas of its consituents


# Show that the variance of the portfolio market portfolio is function of the betas of its consituents












Let us assume that the market portfolio consists of n assets. Given that the return of the market portfolio can be written as $r_m = \sum_{j=1}^{n} w_jr_j$, we have that $\sigma^2_m = E(\sum_{j=1}^{n} w_jr_j - E(\sum_{j=1}^{n} w_jr_j))^2$, but how do I show that $$E(\sum_{j=1}^{n} w_jr_j - E(\sum_{j=1}^{n} w_jr_j))^2 = \sum_{j=1}^{n} w_jCov(r_j,r_m)$$? If I show that the equation above is true, than I can claim that $$E(\sum_{j=1}^{n} w_jr_j - E(\sum_{j=1}^{n} w_jr_j))^2 = \sum_{j=1}^{n} w_jCov(r_j,r_m) = \sum_{j=1}^{n} w_j\beta\sigma^2_m$$

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