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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:

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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:

  1. 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.

  2. 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 $j$?

  3. 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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    $\begingroup$ Some unemployed genius should use his spare time to re-type this wonderful article with all the typos corrected. $\endgroup$
    – nbbo2
    Jan 14, 2022 at 14:20
  • $\begingroup$ In the first line it is probably $\sigma_{ik}$ and not $\sigma_{ij}$ and the summation index is i (i is the only free index; k is bound to the k on the LHS). This line is computing $\Sigma x$, which is an intermediate step in the calculation of $x^T \Sigma x$ and is also the derivative of $\frac{1}{2}x^T \Sigma x$ with respect to $x$. Yes, he is probably referring to equation (41). $\endgroup$
    – nbbo2
    Jan 15, 2022 at 15:12

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