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I am trying to build an efficient frontier as in the Markowitz problem. I have written the code below, but I get the error "ValueError: Objective function must return a scalar". I have tested 'fun' with some values, for example, I input to the console:

W = np.ones([n])/n # start optimization with equal weights cov_matrix = returns.cov() fun = 0.5*np.dot(np.dot(W, cov_matrix), W) # variance of the portfolio fun

The output is 0.00015337622774133828, which is a scalar. I don't know what might be wrong. Any help is appreciated.

Code:

from scipy.optimize import minimize import pandas as pd import numpy as np from openpyxl import load_workbook wb = load_workbook('path/Assets_3.xlsx') # in this workbook there is data for returns. # The next lines clean unnecessary first column and first row. ws = wb.active df = pd.DataFrame(ws.values) df1 = df.drop(0,axis=1) df1 = df1.drop(0) df1 = df1.astype(float) rf = 0.05 r_bar = 0.05 returns = df1.copy() def efficient_frontier(rf, r_bar, returns): n = len(returns.transpose()) W = np.ones([n])/n # start optimization with equal weights exp_ret = returns.mean() cov_matrix = returns.cov() fun = 0.5*np.dot(np.dot(W, cov_matrix), W) # variance of the portfolio cons = ({'type': 'eq', 'fun': lambda W: sum(W) - 1. }, {'type': 'ineq', 'fun': lambda W: np.dot(exp_ret,W) - r_bar }) bnds = [(0.,1.) for i in range(n)] # weights between 0..1. res = minimize(fun, W, (returns, cov_matrix, rf), method='SLSQP', bounds = bnds, constraints = cons) return res x= efficient_frontier(rf,r_bar,returns) x

Some Data

 1 2 3 1 0.060206 0.005781 0.001117 2 0.006463 -0.007390 0.001133 3 -0.003211 -0.015730 0.001167 4 0.044227 -0.006250 0.001225 5 -0.040571 -0.006910 0.001292 6 -0.007900 -0.006160 0.001208 7 0.068702 0.013836 0.001300 8 0.039286 0.009854 0.001350 9 0.012457 -0.007950 0.001358 10 -0.013758 0.001021 0.001283 11 -0.002616 -0.013600 0.001300 12 0.059004 -0.006090 0.001442 13 0.015566 0.002818 0.001308 14 -0.036454 0.001395 0.001283 15 0.058899 0.011072 0.001325 16 -0.043086 0.017070 0.001308 17 0.023156 -0.003350 0.001392 18 0.063705 0.000301 0.001417 19 0.017628 -0.001960 0.001508 20 -0.014567 -0.006990 0.001525 21 -0.007191 -0.013000 0.001425 22 -0.000815 0.014773 0.001450 23 0.046493 -0.001540 0.001542 24 0.051832 -0.008580 0.001742 25 -0.007151 0.001177 0.001633 26 -0.018196 -0.008680 0.001642 27 -0.013513 -0.008810 0.001675 28 -0.026493 -0.010510 0.001825 29 -0.003249 -0.014750 0.001800 30 0.001222 0.022258 0.001758
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  • 1
    fun doesn't seem to be a function. Commented Oct 18, 2017 at 18:14
  • It's supposed to be a function of the weights W and the covariance matrix cov_matrix. Commented Oct 18, 2017 at 18:16
  • 1
    Are you sure you are not returning an np.array with shape (1), intead of a float? Commented Oct 18, 2017 at 18:17
  • @MatteoRagni If i put instead fun = float(0.5*np.dot(np.dot(W, cov_matrix), W)), I get the same error. Commented Oct 18, 2017 at 18:18
  • 1
    Uh ma fun is not a callable object! Probably that is the issue.... Commented Oct 18, 2017 at 18:20

1 Answer 1

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3

This code is a mess and while i can show you something which runs, that does not mean anything.

You will see convergence to your starting-point, whatever that means in your task! It's a strong indicator that something is still very wrong (might be the underlying theory)!

Some additional remarks:

  • scipy's optimizers are build to work with numpy-arrays, not pandas Dataframes or Series objects!
    • the only things in your original question which hinted pandas-usage was a var-name df and returns.cov() which does not exist for numpy-arrays!
  • rf is never used anywhere!
  • there are multiple things in optimize's args, which are not used!
  • it does not feel like a problem one should use scipy's optimizers for! (but it's possible; here we are paying for numerical-differentiation for example)
    • cvxpy would probably a much much better approach (more clean, faster, more accurate) if interpret the problem correctly (did not analyze much)
    • but the same rules apply: some python-knowledge is needed!

Code:

from scipy.optimize import minimize import numpy as np import pandas as pd rf = 0.05 r_bar = 0.05 returns = pd.DataFrame(np.random.randn(30, 3), columns=list('ABC')) # PANDAS DF cov_matrix = returns.cov().as_matrix() # use PANDAS one last time # but result = np.array! returns = returns.as_matrix() # From now on: np-only! def fun(x, returns, cov_matrix, rf): return 0.5*np.dot(np.dot(x, cov_matrix), x) def efficient_frontier(rf, r_bar, returns): n = len(returns.transpose()) W = np.ones([n])/n # start optimization with equal weights exp_ret = returns.mean() cons = ({'type': 'eq', 'fun': lambda x: np.sum(x) - 1. }, # let's use numpy here {'type': 'ineq', 'fun': lambda x: np.dot(exp_ret, x) - r_bar }) bnds = [(0.,1.) for i in range(n)] # weights between 0..1. res = minimize(fun, W, (returns, cov_matrix, rf), method='SLSQP', bounds = bnds, constraints = cons) return res x= efficient_frontier(rf,r_bar,returns) print(x)

Output:

A B C A 0.813375 -0.001370 0.173901 B -0.001370 1.482756 0.380514 C 0.173901 0.380514 1.285936 fun: 0.2604530793556774 jac: array([ 0.32863522, 0.62063321, 0.61345008]) message: 'Optimization terminated successfully.' nfev: 35 nit: 7 njev: 3 status: 0 success: True x: array([ 0.33333333, 0.33333333, 0.33333333])
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3 Comments

@MatteoRagni It's a question hard to work with. Especially until one recognizes that np_array.cov() does not exist.
Looking further into the results, I realized that in spite of outputting a result, the code does not optimize anything. Whatever W I choose to input as initial weights will be the final result of the weights and 'fun' is exactly the objective function when inputting these values.
That's what i told in the first place. The second sentence of my answer is important and without a clear problem-statement (theory of the problem) there is not much to help.

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