用python实现马科维茨有效边界图

要实现马科维茨有效边界图，需要以下步骤：

1. 安装必要的Python库，如numpy、pandas、matplotlib等。

2. 创建一个投资组合的收益率和波动率数据集。可以使用历史数据或模拟数据。

3. 计算投资组合的预期收益率、预期波动率和协方差矩阵。

4. 使用numpy中的linspace函数生成各个预期收益率水平下的最小方差投资组合的权重组合。

5. 计算每个最小方差投资组合的预期收益率和波动率。

6. 使用matplotlib中的scatter函数绘制最小方差投资组合的预期收益率和波动率散点图。

7. 使用scipy.optimize库中的minimize函数计算每个预期收益率水平下的最优投资组合。

8. 计算每个最优投资组合的预期收益率和波动率。

9. 使用matplotlib中的plot函数绘制最优投资组合的有效边界。

下面是示例代码：

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from scipy.optimize import minimize

# 创建数据集
returns = np.random.normal(0.05, 0.1, 100)
volatility = np.random.normal(0.1, 0.05, 100)
data = pd.DataFrame({'returns': returns, 'volatility': volatility})

# 计算预期收益率、预期波动率和协方差矩阵
mu = data['returns'].mean()
sigma = data['volatility'].std()
cov_matrix = np.cov(data['returns'], data['volatility'])

# 生成权重组合
num_portfolios = 1000
results = np.zeros((3 + len(data.columns), num_portfolios))
for i in range(num_portfolios):
    weights = np.random.random(len(data.columns))
    weights /= np.sum(weights)
    portfolio_return = np.dot(weights, data['returns'])
    portfolio_volatility = np.sqrt(np.dot(weights.T, np.dot(cov_matrix, weights)))
    results[0, i] = portfolio_return
    results[1, i] = portfolio_volatility
    results[2, i] = portfolio_return / portfolio_volatility
    for j in range(len(weights)):
        results[j+3, i] = weights[j]

# 绘制最小方差投资组合散点图
min_volatility_idx = np.argmin(results[1])
min_volatility_return = results[0, min_volatility_idx]
min_volatility_volatility = results[1, min_volatility_idx]
plt.scatter(results[1,:], results[0,:], c=results[2,:], cmap='YlOrRd', alpha=0.5)
plt.scatter(min_volatility_volatility, min_volatility_return, c='b', marker='*', s=500)
plt.xlabel('Volatility')
plt.ylabel('Returns')
plt.colorbar()

# 计算最优投资组合
def objective(weights):
    portfolio_return = np.dot(weights, data['returns'])
    portfolio_volatility = np.sqrt(np.dot(weights.T, np.dot(cov_matrix, weights)))
    return -portfolio_return / portfolio_volatility

constraints = ({'type': 'eq', 'fun': lambda x: np.sum(x) - 1})
bounds = [(0, 1)] * len(data.columns)
initial_weights = np.ones(len(data.columns)) / len(data.columns)
optimal_weights = minimize(objective, initial_weights, method='SLSQP', bounds=bounds, constraints=constraints)
optimal_weights = optimal_weights.x

# 计算最优投资组合的预期收益率和波动率
optimal_return = np.dot(optimal_weights, data['returns'])
optimal_volatility = np.sqrt(np.dot(optimal_weights.T, np.dot(cov_matrix, optimal_weights)))

# 绘制最优投资组合的有效边界
frontier_returns = []
frontier_volatilities = []
for r in np.linspace(min(results[0,:]), max(results[0,:]), num=100):
    constraints = ({'type': 'eq', 'fun': lambda x: np.sum(x) - 1},
                   {'type': 'eq', 'fun': lambda x, r=r: np.dot(x, data['returns']) - r})
    bounds = [(0, 1)] * len(data.columns)
    initial_weights = np.ones(len(data.columns)) / len(data.columns)
    optimal_weights = minimize(objective, initial_weights, method='SLSQP', bounds=bounds, constraints=constraints)
    optimal_weights = optimal_weights.x
    frontier_returns.append(r)
    frontier_volatilities.append(np.sqrt(np.dot(optimal_weights.T, np.dot(cov_matrix, optimal_weights))))
plt.plot(frontier_volatilities, frontier_returns, 'g--', linewidth=2)

plt.show()

此代码将生成一个包含最小方差投资组合散点图和有效边界图的matplotlib图形。