Derivation of optimal portfolio weights using Risk Budgeting approach

Question

In Thierry Roncalli's book Introduction to Risk Parity and Budgeting (2013), he gives an example of particular solutions to the Risk Budgeting portfolio such as for the $n=2$ asset case.

The risk contributions are:

$$ \frac{1}{\sigma(x)} \cdot \begin{bmatrix} w^2\sigma_1^2 + \rho w(1-w) \sigma_1 \sigma_2 \\ (1-w)^2\sigma_2^2 + \rho w(1-w) \sigma_1 \sigma_2 \\ \end{bmatrix} $$

The vector $[b,1-b]$ are the risk budgets.

He presents the optimal weight $w$ as:

$$ w^* = \frac {(b - \frac{1}{2}) \rho \sigma_1\sigma_2 - b\sigma_2^2 +\sigma_1\sigma_2\sqrt{(b - \frac{1}{2})^2\rho^2 + b(1-b)}} {(1-b)\sigma_1^2 - b\sigma_2^2 + 2(b - \frac{1}{2})\rho\sigma_1\sigma_2} $$

How are these weights derived ? I don't need a full derivation (it would be helpful though), I just don't know how it is derived.

Is it done by setting the risk contributions equal to the budgets?

$$ \begin{bmatrix} b \\ 1-b \\ \end{bmatrix} = \frac{1}{\sigma(x)} \cdot \begin{bmatrix} w^2\sigma_1^2 + \rho w(1-w) \sigma_1 \sigma_2 \\ (1-w)^2\sigma_2^2 + \rho w(1-w) \sigma_1 \sigma_2 \\ \end{bmatrix} $$

Answer 1

You are correct in your assumption, this is specified at the start of section 2.2.1 Definition of a risk budgeting portfolio.

We consider a set of given risk budgets $\{B_1,\dots,B_n\}$. Here $B_i$ is an amount of risk measured in dollars. We denote $\mathcal{RC}_i(x_1,\dots,x_n)$ the risk contribution of asset $i$ with respect to portfolio $x=(x_1,\dots,x_n)$. The risk budgeting portfolio is then defined by the following constraints:

$$ \mathcal{RC}_1(x_1,\dots,x_2)=B_1 \\ \vdots \\ \mathcal{RC}_i(x_1,\dots,x_2)=B_i \\ \vdots \\ \mathcal{RC}_n(x_1,\dots,x_2)=B_n \\ $$

The two asset case

We can rewrite the two equations into a singular formula by solving for $\sigma(x)$ for both of them and subsequently eliminating $\sigma(x)$ by setting the two formulations of $\sigma(x)$ equal to each other:

$$ \frac{w^2\sigma_1^2+w(1-w)\rho\sigma_1\sigma_2}{b}= \frac{(1-w)^2\sigma_2^2+w(1-w)\rho\sigma_1\sigma_2}{1-b} $$

After rearranging this becomes a (complicated) quadratic equation in $w$, which can be solved via the quadratic formula. By cancelling terms in the resulting fraction and observing that $0\leq w\leq 1$ (and thus eliminating one of the solutions of the quadratic formula) you should arrive at the optimal $w^*$.