Black-Litterman 模型：桥接市场均衡与投资者观点的资产配置框架

import numpy as np
import pandas as pd
from scipy.optimize import minimize

class BlackLitterman:
    """
    Black-Litterman 模型实现
    
    参数：
    - returns: 资产历史收益率 DataFrame (T x N)
    - market_caps: 资产市值权重 Series (N,)
    - risk_aversion: 风险厌恶系数 λ
    - tau: 缩放因子
    """
    
    def __init__(self, returns, market_caps, risk_aversion=2.5, tau=0.05):
        self.returns = returns
        self.market_caps = market_caps
        self.risk_aversion = risk_aversion
        self.tau = tau
        
        # 计算协方差矩阵
        self.Sigma = returns.cov() * 252  # 年化
        self.N = len(market_caps)
        
    def compute_implied_returns(self):
        """
        逆向优化：从市场权重计算均衡收益率
        Π = λ * Σ * w_mkt
        """
        Pi = self.risk_aversion * self.Sigma @ self.market_caps
        return Pi
    
    def bl_formula(self, P, Q, Omega):
        """
        Black-Litterman 公式：计算后验预期收益率
        
        E[R] = [(τΣ)^(-1) + P'Ω^(-1)P]^(-1) [(τΣ)^(-1)Π + P'Ω^(-1)Q]
        """
        Pi = self.compute_implied_returns()
        
        # 转换 to matrix
        tau_Sigma = self.tau * self.Sigma.values
        P_mat = P.values
        Q_vec = Q.values
        Omega_mat = Omega.values
        
        # 后验预期收益率
        M_inv = np.linalg.inv( np.linalg.inv(tau_Sigma) + P_mat.T @ np.linalg.inv(Omega_mat) @ P_mat )
        posterior_returns = M_inv @ ( np.linalg.inv(tau_Sigma) @ Pi.values + P_mat.T @ np.linalg.inv(Omega_mat) @ Q_vec )
        
        return pd.Series(posterior_returns, index=self.Sigma.columns)
    
    def optimize_portfolio(self, expected_returns, risk_aversion=None):
        """
        给定预期收益率，用均值方差优化计算权重
        """
        if risk_aversion is None:
            risk_aversion = self.risk_aversion
        
        N = len(expected_returns)
        Sigma = self.Sigma.values
        
        # 目标函数：最大化效用 = μ'w - (λ/2) * w'Σw
        def objective(w):
            port_return = expected_returns.values @ w
            port_risk = w @ Sigma @ w
            utility = port_return - (risk_aversion / 2) * port_risk
            return -utility  # 最小化负效用
        
        # 约束：权重和为1
        constraints = ({'type': 'eq', 'fun': lambda x: np.sum(x) - 1.0})
        bounds = tuple((0, 1) for _ in range(N))  # 只允许做多
        
        result = minimize(objective, np.ones(N)/N, method='SLSQP', 
                         bounds=bounds, constraints=constraints)
        
        return pd.Series(result.x, index=expected_returns.index)

# 使用示例
# 假设有5个资产：消费、科技、金融、医药、新能源
assets = ['Consumer', 'Tech', 'Finance', 'Healthcare', 'NewEnergy']

# 模拟历史收益率（均值和协方差）
np.random.seed(42)
returns = pd.DataFrame(
    np.random.randn(1000, 5) * 0.15 / np.sqrt(252) + 0.0005,
    columns=assets,
    index=pd.date_range('2020-01-01', periods=1000)
)

# 市场市值权重（假设）
market_caps = pd.Series([0.25, 0.30, 0.20, 0.15, 0.10], index=assets)

# 初始化 BL 模型
bl = BlackLitterman(returns, market_caps, risk_aversion=2.5, tau=0.05)

# 计算市场均衡收益率
Pi = bl.compute_implied_returns()
print("市场均衡收益率（年化）:")
print(Pi)

# 定义投资者观点
# 观点1：科技板块收益率 = 18%
# 观点2：新能源 - 金融 = 10%
P = pd.DataFrame([
    [0, 1, 0, 0, 0],   # 科技
    [0, 0, -1, 0, 1]   # 新能源 - 金融
], columns=assets, index=['View1', 'View2'])

Q = pd.Series([0.18, 0.10], index=['View1', 'View2'])

# 观点不确定性（对角线矩阵）
Omega = np.diag([0.05**2, 0.08**2])  # 观点1不确定性5%，观点2不确定性8%
Omega = pd.DataFrame(Omega, index=['View1', 'View2'], columns=['View1', 'View2'])

# BL 后验预期收益率
posterior_returns = bl.bl_formula(P, Q, Omega)
print("\nBL后验预期收益率（年化）:")
print(posterior_returns)

# 优化投资组合
optimal_weights = bl.optimize_portfolio(posterior_returns)
print("\nBL最优投资组合权重:")
print(optimal_weights)