Covariance Terms in Merton’s Security Market Line Derivation

Summary

The document raises questions about Merton’s derivation of the security market line from the tangency portfolio. Its focus is the covariance between an individual asset and the market portfolio: the reader asks why that covariance can be written as a weighted sum of covariances with portfolio constituents, and how to interpret the summation indices. It also flags a possible mismatch in an equation reference.

No answer or derivation is included, so the document does not resolve these questions or provide empirical evidence. The underlying portfolio identity is that covariance with a weighted portfolio equals the weighted sum of covariances with its component returns; a double sum appears when expanding portfolio variance. The entry is therefore most useful as a pointer to clarify notation in CAPM theory, rather than as a complete explanation. Readers should consult the cited paper’s equations to determine which equation the author intended to reference.

Key ideas

• Covariance between an asset and a portfolio can be expressed using the portfolio weights and constituent covariances.
• Expanding portfolio variance involves sums over pairs of asset indices.
• The document asks how to interpret the notation in Merton’s security market line derivation.
• It notes a potentially incorrect equation reference but does not establish the intended correction.

Tags

# Questions about Merton's derivation of the security market line


# Questions about Merton's derivation of the security market line












In Merton's "An Analytic Derivation of the Efficient Frontier" (PDF), he derives the security market line for the CAPM using the definition of the tangency portfolio. He writes:

Here, $m$ is the number of assets in the portfolio, $M$ denotes the market portfolio, and $x$ are the portfolio weights, so $x^M$ denote the market portfolio weights. I don't understand this for a couple reasons:

- I know that $\sigma_M = \sqrt{(w^M)^{\top} \Sigma w^M}$. But I don't understand why $\sigma_{kM}$, which I interpret to be the cross covariance between the $k$-th asset and the market portfolio ($\sigma_k \sigma_M$) is a sum of weighted cross-covariances.

- I don't understand what the sum is over. Is it over $i$ or $j$? I would guess $i$ because of $x_i^M$ but then what is $j$?

- Finally, he says "from $(44)$ but this is equation $42$. I don't think he's referencing a future equation. This must mean equation $41$?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.