Using Monte Carlo for portfolio optimisation but must satisfy constraints

Question

I have a problem trying to find portfolio weights and optimise them while satisfying constraints. I require some help on designing a code that would allow me to simulate multiple arrays of weights while satisfying a range of constraints, please see example below for an explanation:

Problem - simulate various weights while satisfying constraints:

Instruments = ['Fixed Income', 'Equity 1', 'Equity 2', 'Multi-Asset', 'Cash']

Constraints:

1. each weight between 0.1 and 0.4
2. cash = 0.05
3. Equity 1 less than 0.3

At the moment I have code:

import numpy as np:

instruments = ['Fixed Income', 'Equity 1', 'Equity 2', 'Multi-Asset', 'Cash']

weights = np.random.uniform(0.1, 0.4, len(instruments)) # random number generation
weights = weights / weights.sum() # normalise weights
# I have done test runs, and normalised weights always fit the constraints

if weights[-1] > 0.03:
   excess = weights[-1] - 0.03
   # distribute excess weights equally
   weights[:-1] = weights[:-1] + excess / (len(weights) - 1)

and I'm stuck, I also realised that when I distribute the excess weights I have effectively broken my constraint.

Is there anyway to do it? and I have to do it via monte-carlo

Thanks everyone for your help.

Answer 1

Here is one solution:

import numpy as np
N = 1000000
instruments = ['Fixed Income', 'Equity 1', 'Equity 2', 'Multi-Asset', 'Cash']

# each row is set of weights in order of instruments above
weights = np.zeros(shape=(N, len(instruments)))

weights[:, -1] = 0.05
weights[:, 1] = np.random.uniform(0, 0.3, N)

cols = (0, 2, 3)

# fill columns with random numbers
for col in cols[0:-1]:
    w_remaining = 1 - np.sum(weights, axis=1)
    weights[:, col] = np.random.uniform(0.1, 0.4, N)

# the last column is constrained for normalization
weights[:, cols[-1]] = 1 - np.sum(weights, axis=1)

# select only rows that meet condition:
cond1 = np.all(0.1 <= weights[:, cols], axis=1)
cond2 = np.all(weights[:, cols] <= 0.4, axis=1)
valid_rows = cond1*cond2

weights = weights[valid_rows, :]

# verify sum of weights == 1:
np.testing.assert_allclose(np.sum(weights, axis=1), 1)

This solution is performant, but discards generated examples that don't satisfy the constraints.

Answer 2

In my PoV using montecarlo simulation for portfolio optimization is very inefficient due to the problem is very hard when you increase the number of assets even if you use sobol sequence.

On the other, hand the majority of portfolio optimization problems are quadratic or linear programming problems, so you can use cvxpy to solve it but it takes time to model each problem. Finally, you can try Riskfolio-Lib a library based on cvxpy that simplify the implementation of portfolio optimization models, even have a format to implement asset classes constraints like you are trying to implement.

You can check the first example in this link: https://riskfolio-lib.readthedocs.io/en/latest/examples.html