Build an Optimal Stock Portfolio with Markowitz Mean‑Variance Model in Python

Investment returns are evaluated using net present value (NPV) and diversification reduces risk. The Markowitz mean‑variance model (1952) treats asset returns as random variables characterized by mean, variance and covariance.

For three stocks A, B, C with historical prices (1943‑1954) and the S&P 500 index, the article shows how to compute each stock’s expected return, variance and the covariance matrix using Python.

<code>import pandas as pd

data = pd.read_excel('data/马科维兹.xlsx')
stock = data.iloc[:,1:-1]
ER = stock.mean()
COV = stock.cov()
</code>

The resulting covariance matrix is presented, and the diagonal entries give the variances of the three stocks while off‑diagonal entries indicate their pairwise covariances.

Quadratic Programming Model

Decision variables represent the proportion of capital allocated to each stock, with the constraint that the sum equals one. The objective is to minimize portfolio variance subject to a target expected return (e.g., at least 15%).

Using the cvxopt.solvers.qp function, the quadratic program is formulated and solved in Python:

<code>from cvxopt import matrix, solvers

P = matrix(2*COV.values)
q = matrix([0.0,0.0,0.0])
G = matrix([ER.values.tolist(),[1.0,1.0,1.0]],(2,3))
h = matrix([-0.15,0.0])
A = matrix([1.0,1.0,1.0],(1,3))
b = matrix([1.0])

result = solvers.qp(P,q,G,h,A,b)
print('x',result['x'])
</code>

The optimal solution yields the weight of each stock in the portfolio and the corresponding minimal risk.

Reference: Zhang Jingxin et al., *Mathematical Modeling Algorithms and Programming Implementation*.