Implementing the Sharpe's return-based style analysis on Python

Question

I am trying to implement the Sharpe's return-based style analysis on Python. The problem is formulated as follows:

min Var(M-(c1a1 + c2a2 + c3a3 + c4a4))
subject to c1 + c2 + c3 + c4 = 1
           c1 >=0, c2 >= 0, c3 >= 0, c4 >= 0
where M = monthly or daily return of an investor's portfolio
      a1, a2, a3, a4 = monthly or daily return of an index
      and c1, c2, c4, c4 are the optimization decision variables.

Of course, the objective function (Variance) makes the problem nonlinear. I am trying to use Scipy to implement this, but I cannot find a good example of quadratic/nonlinear optimization similar to this problem.

What Python library should I use to do this? The example above has only 4 indices, but I want to make it more general and flexible as to handle many indices.

I would also very appreciate it if someone could show a Python code for the optimization problem above.

Thank you!!!

Answer 1

You could restate your problem as:

$$ \min_x \quad Var \left (-\sum_{i=0}^n x_iR_i \right) $$ $$ \text{s.t.} \quad x_0 = -1, \quad \sum_{i=1}^n x_i = 1, \quad \text{non-negativity of }x_1:x_n $$ where $R_i$ are the expected returns of asset $i$ and $x_i$ are your solution variables.

The objective function can be expressed as: $$ \min_x \quad - \sum_{i,j} x_i x_j Cov(R_i, R_j)= - \mathbf{x^TQx} $$ where $\mathbf{x}=[x_0, ..., x_n]^T$ and $\mathbf{Q}$ is the covariance matrix of returns.