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:

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 $$41$$?

• Some unemployed genius should use his spare time to re-type this wonderful article with all the typos corrected. Jan 14 at 14:20
• 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). Jan 15 at 15:12