如何使用cvxopt.qp爲二次編程設置因子暴露約束？

Question

嘗試優化投資組合權重分配，以最大限度降低經典Markowitz投資組合的風險。 比方說，如果我有一個因素暴露約束數據幀，其表示像

In [138]: exp_sub = pd.DataFrame(data=[[-10, 20],[-10, 20],[-10, 20],[-10, 20],[-10, 20]], columns=['lower','upper']) 

In [131]: exp_sub 
In [132]: lower upper 
0 -10  20 
1 -10  20 
2 -10  20 
3 -10  20 
4 -10  20 

我想在我的代碼添加此約束，但解決的辦法是不正確的，即使溶膠狀態是最佳的。任何人都可以幫忙嗎？謝謝。

我的代碼如下：

# -*- coding: utf-8 -*- 
### Portfolio Optiimization 
# Finds an optimal allocation of stocks in a portfolio, 
# satisfying a minimum expected return. 
# The problem is posed as a Quadratic Program, and solved 
# using the cvxopt library. 
# Uses actual past stock data, obtained using the stocks module. 

import sys 
import itertools 
from cvxopt import matrix, solvers, spmatrix, sparse 
from cvxopt.blas import dot 
import pandas as pd 
import numpy as np 
from datetime import datetime 

solvers.options['show_progress'] = False 

import logging 
logger = logging.getLogger() 
handler = logging.StreamHandler() 
formatter = logging.Formatter('%(asctime)s %(name)-12s %(levelname)-8s %(message)s') 
handler.setFormatter(formatter) 
logger.addHandler(handler) 
logger.setLevel(logging.DEBUG) 


# solves the QP, where x is the allocation of the portfolio: 
# minimize x'Px + q'x 
# subject to Gx <= h 
#   Ax == b 
# 
# Input: n  - # of assets 
#   avg_ret - nx1 matrix of average returns 
#   covs - nxn matrix of return covariance 
#   r_min - the minimum expected return that you'd 
#     like to achieve 
# Output: sol - cvxopt solution object 

dates = pd.date_range('2000-01-01', periods=6) 
industry = ['industry', 'industry', 'utility', 'utility', 'consumer'] 
symbols = ['A', 'B', 'C', 'D', 'E'] 
zipped = list(zip(industry, symbols)) 
index = pd.MultiIndex.from_tuples(zipped) 

noa = len(symbols) 

data = np.array([[10, 11, 12, 13, 14, 10], 
       [10, 11, 10, 13, 14, 9], 
       [10, 10, 12, 13, 9, 11], 
       [10, 11, 12, 13, 14, 8], 
       [10, 9, 12, 13, 14, 9]]) 

market_to_market_price = pd.DataFrame(data.T, index=dates, columns=index) 
rets = market_to_market_price/market_to_market_price.shift(1) - 1.0 
rets = rets.dropna(axis=0, how='all') 

# covariance of asset returns 
covs = matrix(rets.cov().values) 

# average yearly return for each stock 
rets_mean = rets.mean() 
avg_ret = matrix(rets_mean.values) 
n = len(symbols) 

factor_exposure = pd.DataFrame(np.ones((5,5)), 
           columns=list('ABCDE')) 


P = covs 
q = matrix(np.zeros((n, 1)), tc='d') 
asset_sub = matrix(np.eye(n), tc='d') 
asset_sub = matrix(sparse([asset_sub, -asset_sub])) 
exp_sub = matrix(factor_exposure.values) 
exp_sub = matrix(sparse([exp_sub, -exp_sub])) 
# set boundary vector for h 
df_asset_weight = pd.DataFrame({'lower': [0.0], 'upper': [1.0]}, 
           index=list("ABCDE")) 
df_asset_bnd_matrix = matrix(np.concatenate(((df_asset_weight.upper, 
               df_asset_weight.lower)), 0)) 


df_factor_exposure_bound = pd.DataFrame(data=[[-10, 20],[-10, 20],[-10, 20],[-10, 20],[-10, 20]], columns=['lower','upper']) 


df_factor_exposure_bnd_matrix = matrix(np.concatenate(((df_factor_exposure_bound.upper, 
                 df_factor_exposure_bound.lower)), 0)) 


G = matrix(sparse([asset_sub, exp_sub])) 
h = matrix(sparse([df_asset_bnd_matrix, df_factor_exposure_bnd_matrix])) 

# equality constraint Ax = b; captures the constraint sum(x) == 1 
A = matrix(1.0, (1, n)) 
b = matrix(1.0) 
sol = solvers.qp(P, q, G, h, A, b) 

Answer 1

這個庫cvxportfolio是很容易解決的因素暴露約束問題。 cvxpy