Python: Optimize weights in portfolio

Question

I have the following dataframe with weights:

df = pd.DataFrame({'a': [0.1, 0.5, 0.1, 0.3], 'b': [0.2, 0.4, 0.2, 0.2], 'c': [0.3, 0.2, 0.4, 0.1],
           'd': [0.1, 0.1, 0.1, 0.7], 'e': [0.2, 0.1, 0.3, 0.4], 'f': [0.7, 0.1, 0.1, 0.1]})

and then I normalize each row using:

df = df.div(df.sum(axis=1), axis=0)

I want to optimize the normalized weights of each row such that no weight is less than 0 or greater than 0.4.

If the weight is greater than 0.4, it will be clipped to 0.4 and the additional weight will be distributed to the other entries in a pro-rata fashion (meaning the second largest weight will receive more weight so it gets close to 0.4, and if there is any remaining weight, it will be distributed to the third and so on).

Can this be done using the "optimize" function?

Thank you.

UPDATE: I would also like to set a minimum bound for the weights. In my original question, the minimum weight bound was automatically considered as zero, however, I would like to set a constraint such that the minimum weight is at at least equal to 0.05, for example.

user_2538939 · Answer

Unfortunately, I can only find a loop solution to this problem. When you trim off the excess weight and redistribute it proportionally, the underweight may go over the limit. Then they have to be trimmed off. And the cycle keep repeating until no value is overweight. The same goes for underweight rows.

# The original data frame. No normalization yet
df = pd.DataFrame(
    {
        "a": [0.1, 0.5, 0.1, 0.3],
        "b": [0.2, 0.4, 0.2, 0.2],
        "c": [0.3, 0.2, 0.4, 0.1],
        "d": [0.1, 0.1, 0.1, 0.7],
        "e": [0.2, 0.1, 0.3, 0.4],
        "f": [0.7, 0.1, 0.1, 0.1],
    }
)


def ensure_min_weight(row: np.array, min_weight: float):
    while True:
        underweight = row < min_weight
        if not underweight.any():
            break

        missing_weight = min_weight * underweight.sum() - row[underweight].sum()
        row[~underweight] -= missing_weight / row[~underweight].sum() * row[~underweight]
        row[underweight] = min_weight


def ensure_max_weight(row: np.array, max_weight: float):
    while True:
        overweight = row > max_weight
        if not overweight.any():
            break

        excess_weight = row[overweight].sum() - (max_weight * overweight.sum())
        row[~overweight] += excess_weight / row[~overweight].sum() * row[~overweight]
        row[overweight] = max_weight


values = df.to_numpy()
normalized = values / values.sum(axis=1)[:, None]
min_weight = 0.15 # just for fun
max_weight = 0.4

for i in range(len(values)):
    row = normalized[i]
    ensure_min_weight(row, min_weight)
    ensure_max_weight(row, max_weight)

# Normalized weight
assert np.isclose(normalized.sum(axis=1), 1).all(), "Normalized weight must sum up to 1"
assert ((min_weight <= normalized) & (normalized <= max_weight)).all(), f"Normalized weight must be between {min_weight} and {max_weight}"
print(pd.DataFrame(normalized, columns=df.columns))

# Raw values
# values = normalized * values.sum(axis=1)[:, None]
# print(pd.DataFrame(values, columns=df.columns))

Note that this algorithm will run into infinite loop if your min_weight and max_weight are illogical: try min_weight = 0.4 and max_weight = 0.5. You should handle these errors in the 2 ensure functions.

Python: Optimize weights in portfolio

1 Answers