[QuantEcon]MATLAB混编FORTRAN语言

which satisfies $\sum_{i=1}^{N}{\sf RC}_i=\sigma\left(\mathbf{w}\right)$.

The **relative risk contribution (RRC)** is a normalized version:
$${\sf RRC}_i = \frac{w_i\left(\boldsymbol{\Sigma}\mathbf{w}\right)_i}{\mathbf{w}^{T}\boldsymbol{\Sigma}\mathbf{w}}$$

so that $\sum_{i=1}^{N}{\sf RRC}_i=1$.

The **risk parity portfolio (RPP)** attemps to “equalize” the risk contributions:
$${\sf RC}_i = \frac{1}{N}\sigma(\mathbf{w})\quad\text{or}\quad{\sf RRC}_i = \frac{1}{N}.$$

More generally, the **risk budgeting portfolio (RBP)** attemps to allocate the risk according to the risk profile determined by the weights $\mathbf{b}$ (with $\mathbf{1}^T\mathbf{b}=1$ and $\mathbf{b}\ge \mathbf{0}$):
$${\sf RC}_i = b_i \sigma(\mathbf{w})\quad\text{or}\quad{\sf RRC}_i = b_i.$$

In practice, one can express the condition ${\sf RC}_i = \frac{1}{N}\sigma(\mathbf{w})$ in different equivalent ways such as
$$w_i(\Sigma \mathbf{w})_{i} = w_j(\Sigma \mathbf{w})_{j}, \quad\forall i, j.$$ The budget condition ${\sf RC}_i = b_i \sigma(\mathbf{w})$ can also be expressed as
$$w_i (\Sigma \mathbf{w})_i = b_i \mathbf{w}^{T}\Sigma\mathbf{w}, \quad\forall i.$$

Assuming that the assets are uncorrelated, i.e., that $\boldsymbol{\Sigma}$ is diagonal, and simply using the volatilities $\boldsymbol{\sigma} = \sqrt{{\sf diag(\boldsymbol{\Sigma})}}$, one obtains
$$\mathbf{w} = \frac{\boldsymbol{\sigma}^{-1}}{\mathbf{1}^T\boldsymbol{\sigma}^{-1}}$$

or, more generally,
$$\mathbf{w} = \frac{\sqrt{\mathbf{b}}\odot\boldsymbol{\sigma}^{-1}}{\mathbf{1}^T\left(\sqrt{\mathbf{b}}\odot\boldsymbol{\sigma}^{-1}\right)}.$$

## Vanilla convex formulation
Suppose we only have the constraints $\mathbf{1}^T\mathbf{w}=1$ and $\mathbf{w} \ge \mathbf{0}$. Then, after the change of variable $\mathbf{x}=\mathbf{w}/\sqrt{\mathbf{w}^{T}\boldsymbol{\Sigma}\mathbf{w}}$, the equations $w_i (\Sigma \mathbf{w})_i = b_i \mathbf{w}^{T}\Sigma\mathbf{w}$ become $x_i\left(\boldsymbol{\Sigma}\mathbf{x}\right)_i = b_i$ or, more compactly in vector form, as
$$\boldsymbol{\Sigma}\mathbf{x} = \mathbf{b}/\mathbf{x}$$

with $\mathbf{x} \ge \mathbf{0}$ and we can always recover the portfolio by normalizing: $\mathbf{w} = \mathbf{x}/(\mathbf{1}^T\mathbf{x})$.

At this point, one could use a nonlinear multivariate root finder for $\boldsymbol{\Sigma}\mathbf{x} = \mathbf{b}/\mathbf{x}$. For example, in R we can use the package [rootSolve](https://CRAN.R-project.org/package=rootSolve).


With the goal of designing risk budget portfolios, Spinu proposed in [@Spinu2013] to solve the
following convex optimization problem:
$$\underset{\mathbf{x}\ge\mathbf{0}}{\textsf{minimize}} \quad \frac{1}{2}\mathbf{x}^{T}\boldsymbol{\Sigma}\mathbf{x} - \sum_{i=1}^{N}b_i\log(x_i),$$

where the portfolio can be recovered as $\mathbf{w} = \mathbf{x}/(\mathbf{1}^T\mathbf{x})$.

Indeed, Spinu realized in [@Spinu2013] that precisely the risk budgeting equation $\boldsymbol{\Sigma}\mathbf{x} = \mathbf{b}/\mathbf{x}$ corresponds to the gradient of the convex function $f(\mathbf{x}) = \frac{1}{2}\mathbf{x}^{T}\boldsymbol{\Sigma}\mathbf{x} - \mathbf{b}^T\log(\mathbf{x})$ set to zero:
$$\nabla f(\mathbf{x}) = \boldsymbol{\Sigma}\mathbf{x} - \mathbf{b}/\mathbf{x} = \mathbf{0}.$$

Thus, a convenient way to solve the problem is by solving the following convex optimization problem:
$$\underset{\mathbf{x}\ge\mathbf{0}}{\textsf{minimize}} \quad \frac{1}{2}\mathbf{x}^{T}\boldsymbol{\Sigma}\mathbf{x} - \mathbf{b}^T\log(\mathbf{x})$$

which has optimality condition $\boldsymbol{\Sigma}\mathbf{x} = \mathbf{b}/\mathbf{x}$.

## General nonconvex formulation
The previous methods are based on a convex reformulation of the problem so they are guaranteed to converge to the optimal risk budgeting solution. However, they can only be employed for the simplest risk budgeting formulation with a simplex constraint set (i.e., $\mathbf{1}^T\mathbf{w}=1$ and $\mathbf{w} \ge \mathbf{0}$). They cannot be used if

- we have other constraints like allowing shortselling or box constraints: $l_i \le w_i \le u_i$
- on top of the risk budgeting constraints $w_i\left(\boldsymbol{\Sigma}\mathbf{w}\right)_i = b_i \;\mathbf{w}^{T}\boldsymbol{\Sigma}\mathbf{w}$ we have other objectives like maximizing the expected return $\mathbf{w}^T\boldsymbol{\mu}$ or minimizing the overall variance $\mathbf{w}^T\boldsymbol{\Sigma}\mathbf{w}$.

For those more general cases, we need more sophisticated formulations, which unfortunately are not convex. The idea is to try to achieve equal risk contributions
${\sf RC}_i = \frac{w_i\left(\boldsymbol{\Sigma}\mathbf{w}\right)_i}{\sqrt{\mathbf{w}^{T}\boldsymbol{\Sigma}\mathbf{w}}}$
by penalizing the differences between the terms $w_{i}\left(\boldsymbol{\Sigma}\mathbf{w}\right)_{i}$.

There are many reformulations possible. For illustrative purposes, one such formulation is
$$\begin{array}{ll}
\underset{\mathbf{w}}{\textsf{minimize}} & \sum_{i,j=1}^{N}\left(w_{i}\left(\boldsymbol{\Sigma}\mathbf{w}\right)_{i}-w_{j}\left(\boldsymbol{\Sigma}\mathbf{w}\right)_{j}\right)^{2} \color{blue}{- \;F(\mathbf{w})}\\
\textsf{subject to} & \mathbf{w} \ge \mathbf{0}, \quad\mathbf{1}^T\mathbf{w}=1, \quad\color{blue}{\mathbf{w}\in\cal{W}}
\end{array}$$

where $F(\mathbf{w})$ denotes some additional objective function and $\cal{W}$ denotes an arbitrary convex set of constraints. More expressions for the risk concentration terms are listed in

## RPP with additional variance term
Similarly, the `riskParityPortfolio` package allows us to include the variance $\mathbf{w}^{T}\boldsymbol{\Sigma}\mathbf{w}$
as an objective term:
$$\begin{array}{ll}
\underset{\mathbf{w}}{\textsf{minimize}} &
R(\mathbf{w}) + \lambda_{\sf var} \mathbf{w}^{T}\boldsymbol{\Sigma}\mathbf{w}\\
\textsf{subject to} & \mathbf{1}^T\mathbf{w}=1, \mathbf{w} \ge \mathbf{0},
\end{array}$$

## RPP with general linear constraints
In version 2.0, we added support for general linear constraints, i.e., `riskParityPortfolio` is now able
to solve the following problem:
$$\begin{array}{ll}
\underset{\mathbf{w}}{\textsf{minimize}} &
R(\mathbf{w}) + \lambda F(\mathbf{w})\\
\textsf{subject to} & \mathbf{C}\mathbf{w} = \mathbf{c},~~\mathbf{D}\mathbf{w} \leq \mathbf{d}.
\end{array}$$