Minimum variance portfolio in Python

Question

I have a portfolio or $N$ assets in $t=10$ days.

import numpy as np
import pandas as pd
n= 10
A = pd.DataFrame([np.random.randn(5) for i in range(n)],columns=['a', 'b', 'c', 'd', 'e'])
A  

T = A.shape[0]
k = A.shape[1]
print(T,k)

The covariance matrix

Σ = A.cov().to_numpy()
Sigma = Σ
print(Sigma)

I want to minimize the variance with convex optimization in Python.

Actually I want to solve the

$$\min \quad (1/2) w^T \Sigma w$$ s.t $$w_{i}\geq 0,\sum_{i=1}^{n}w_{I} =1$$

So I do :

import cvxpy as cp
w = cp.Variable(n)

# Defining risk objective
risk = cp.quad_form(w,Sigma)
objective = cp.Minimize((1/2)*risk)


# Budget and weights constraints
constraints = [cp.sum(w) == 1, 
                w >= 0]
# Solver 
prob = cp.Problem(objective, constraints)
prob.solve()

but I receive an error:

Exception: Invalid dimensions for arguments.

what is my mistake here ? Anybody ?

Answer 1

your Sigma matrix is 5x5 and not 10x10, try this

A = pd.DataFrame(
      [np.random.randn(n) for i in range(5*n)],
      columns=[chr(65+i) for i in range(n)]
   )

it will work.

[ADDITION following a remark] I assumed that you expected the portfolio to be of dimension 10 (because you write n=10;w = cp.Variable(n)), hence your covariance matrix should have the dimension of the portfolio, ie 10.