Q:

Marginal Risk Contribution Formula

I am trying to understand and implement the standard 'marginal risk contribution' approach to portfolio risk and hoping to reconcile the formulae provided for its calculation in different sources. Specifically I am trying to understand the difference of these two papers:

Page 4 of this paper by Roncalli (http://thierry-roncalli.com/download/erc.pdf)
Pages 2-3 of this paper by Kazemi (http://people.umass.edu/kazemi/An%20Introduction%20to%20Risk%20Parity.pdf)

What I'd really like more help understanding is:
1) on page 2 of the Kazemi PDF, where MC1 is defined, how is this partial derivative of the portfolio vol with respect to w1 taken? (how does the square root of the portfolio vol disappear and how does the portfolio vol appear in the denominator)
2) how does the alternative formulation of MC in terms of the covariance between the asset and the portfolio (page 3 of Kazemi - "beta") get derived? ideally, looking for a step-by-step guide which illuminates how MC can be thought of in this way

A:

concerning your first question: the derivative does not disappear: $\sigma(R_p)$ contains the square root.
To be more precise, set
$$
\sigma(R_p) = \sqrt{w_1^2\cdot\sigma(R_1)^2 + w_2^2\cdot\sigma(R_2)^2 + 2w_1w_2\text{Cov}(R_1, R_2)}.
$$
Then we get using the chain rule:
\begin{align}
\frac{\partial\sigma(R_p)}{\partial w_1} &= \frac 12 \cdot \biggl(\sqrt{w_1^2\cdot\sigma(R_1)^2 + w_2^2\cdot\sigma(R_2)^2 + 2w_1w_2\text{Cov}(R_1, R_2)}\biggr)^{-1} \cdot\Bigl(2w_1\cdot\sigma(R_1)^2 + 2w_2\text{Cov}(R_1, R_2)\Bigr) = \\
&= \frac{1}{\sigma(R_p)} \cdot\Bigl(w_1\cdot\sigma(R_1)^2 + w_2\text{Cov}(R_1, R_2)\Bigr).
\end{align}
So you see, the square root is still there, it is just hidden in $\sigma(R_p)$.
$MC_1$ can be obtained from this by just multiplying the derivative with $w_1$.
Concerning you second question: notice two things.
First, 
$$
R_p = \sum_{j =1}^N w_j R_j
$$
and second the covariance function is bilinear. This implies that
$$
\text{Cov}(R_i, R_p) = \text{Cov}\Bigl(R_i, \sum_{j =1}^N w_j R_j\Bigr) = \sum_{j = 1}^N w_j \text{Cov}(R_i, R_j).
$$
From this you can easily derive the alternative representation:
\begin{align}
MC_1 &= w_1  \cdot \frac{\sum_{j = 1}^N w_j \text{Cov}(R_1, R_j)}{\sigma(R_p)} = w_1\sigma(R_p) \cdot \frac{\sum_{j = 1}^N w_j \text{Cov}(R_1, R_j)}{\sigma(R_p)^2}  \\
&= w_1\sigma(R_p) \cdot \frac{\text{Cov}(R_1, R_p)}{\sigma(R_p)^2} = w_1\sigma(R_p) \cdot \beta_1.
\end{align}
I hope this helps a little.

