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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!!!

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  • $\begingroup$ try google searching cvxopt $\endgroup$ – Attack68♦ Jun 20 '18 at 19:19
  • $\begingroup$ Attack68, for personal reason, I cannot install cvxopt. Is it possible to use Scipy to solve this? $\endgroup$ – Jun Jang Jun 20 '18 at 19:30
  • $\begingroup$ yes just can use scipy.optimize.minimize and pass your own functions and constraints. You will want to use the 'SLSQP' method. With 4 variables you wont need to bother about calculating jacobian functions but if you had a lot more varianbles it would work much faster if you provide a function for calculating the derivatives of your objective function and the constraints (which aren't actually that complicated) $\endgroup$ – Attack68♦ Jun 20 '18 at 19:38
  • $\begingroup$ Thank you very much. So I am actually having a hard time writing the objective function, because it is variance.. $\endgroup$ – Jun Jang Jun 20 '18 at 19:39
  • $\begingroup$ you might be interested in this: stackoverflow.com/questions/44515880/…. Note also that $Var(aX-bY) = a^2Var(X) + b^2Var(Y) - 2ab Cov(X,Y)$ so you need to know all of the variances and covariances of your portfolio and indexes $\endgroup$ – Attack68♦ Jun 20 '18 at 19:46
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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.

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  • $\begingroup$ Thank you very much for your reply. So the first objective function (Var(- sum of xR)) is the same as -X^TQX? The second objective function seems pretty easy to calculate (just matrix multiplications). $\endgroup$ – Jun Jang Jun 20 '18 at 20:07
  • $\begingroup$ yes it is. and is why i suggested cvxopt since that is a library optimised for quadratic programs, and this is a (matrix) quadratic equation. You look at covariance in wikipedia for the properties of expanding linear sums. $\endgroup$ – Attack68♦ Jun 20 '18 at 20:14
  • $\begingroup$ Oh i see. this is great. Thank you so much! By the way, do you have experience with scipy optimization? $\endgroup$ – Jun Jang Jun 20 '18 at 20:15
  • $\begingroup$ i do but give it a go first, and use the link I posted above which shows some python code with scipy. $\endgroup$ – Attack68♦ Jun 20 '18 at 20:16
  • $\begingroup$ My only problem is that I do not know how to declare the decision variables (X = [x_0, x_1, ..., x_n]^T) using Scipy.. Once I can do that, then I can just do the matrix multiplications, right? $\endgroup$ – Jun Jang Jun 20 '18 at 20:22

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