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I'm trying to do portfolio optimization with cvxopt (Python), I'm able to get the efficient frontier with the following code, however, I'm not able to specify a Y value (mean or return) and get a corresponding X value (std or risk), if anyone has knowledge about this, I would be more than grateful if you can share it:

def optimal_portfolio(returns_vec, cov_matrix):
    n = len(returns_vec)

    N = 1000
    mus = [10 ** (5 * t / N - 1.0) for t in range(N)]

    # Convert to cvxopt matrices
    S = opt.matrix(cov_matrix)

    pbar = opt.matrix(returns_vec)

    # Create constraint matrices
    G = -opt.matrix(np.eye(n))  # negative n x n identity matrix
    h = opt.matrix(0.0, (n, 1))
    A = opt.matrix(1.0, (1, n))
    b = opt.matrix(1.0)

    #solvers.options["feastol"] =1e-9
    # Calculate efficient frontier weights using quadratic programming
    portfolios = [solvers.qp(mu * S, -pbar, G, h, A, b)['x']
                  for mu in mus]
    ## CALCULATE RISKS AND RETURNS FOR FRONTIER
    returns = [blas.dot(pbar, x) for x in portfolios]
    risks = [np.sqrt(blas.dot(x, S * x)) for x in portfolios]
    ## CALCULATE THE 2ND DEGREE POLYNOMIAL OF THE FRONTIER CURVE
    m1 = np.polyfit(returns, risks, 2)
    x1 = np.sqrt(m1[2] / m1[0])
    # CALCULATE THE OPTIMAL PORTFOLIO
    wt = solvers.qp(opt.matrix(x1 * S), -pbar, G, h, A, b)['x']
    return np.asarray(wt), returns, risks
   
return_vec = [0.055355,
    0.010748,
    0.041505,
    0.074884,
    0.039795,
    0.065079
]

cov_matrix =[[ 0.005329, -0.000572, 0.003320, 0.006792, 0.001580, 0.005316],
[-0.000572, 0.000625, 0.000606, -0.000266, -0.000107, 0.000531],
[0.003320, 0.000606, 0.006610, 0.005421, 0.000990, 0.006852],
[0.006792, -0.000266, 0.005421, 0.011385, 0.002617, 0.009786],
[0.001580, -0.000107, 0.000990, 0.002617, 0.002226, 0.002360],
[0.005316, 0.000531, 0.006852, 0.009786, 0.002360, 0.011215]]

weights, returns, risks = optimal_portfolio(return_vec, cov_matrix)


fig = plt.figure()
plt.ylabel('Return')
plt.xlabel('Risk')
plt.plot(risks, returns, 'y-o')
print(weights)
plt.show()

I would like to find the corresponding risk value for a given return value.

Thank you very much!

Itachi Kurosaki
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1 Answers1

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Not sure I properly understood your question but if you are looking to find a solution for a given return then you should use the min return required as a constraint

def optimal_portfolio(returns_vec, cov_matrix, Y):
h1 = opt.matrix(0.0, (n, 1))
h2 = np.matrix(-np.ones((1,1))*Y)
h = concatenate (h1,h2)

This will eventually return the optimal portfolio based on a lowerbound Y. You can then add an upper bound in the same way so that you can fix the value of Y

Marco_sbt
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