shanken correction的python代码

以下是shanken correction的Python代码：

import numpy as np
from scipy.stats import norm


def shanken_correction(R, R_m, C, rf):
    """
    Calculates the Sharpe ratio adjusted for the bias in using sample covariance
    matrices, using the Shanken correction.
    Inputs:
    R: numpy array of shape (n_obs, n_assets) with the observed returns
    R_m: numpy array of shape (n_obs,) with the benchmark returns
    C: numpy array of shape (n_assets, n_assets) with the sample covariance matrix of the asset returns
    rf: float with the risk-free rate
    Outputs:
    shp: float with the Sharpe ratio adjusted for the bias in using sample covariance matrices, using the Shanken correction
    """
    n_obs, n_assets = R.shape
    R_ex = R - np.tile(rf, (n_obs, n_assets))
    R_m_ex = R_m - rf
    mu_ex = np.mean(R_ex, axis=0)
    mu_m_ex = np.mean(R_m_ex)
    Sigma_ex = C / (n_obs - 1)
    var_mu_ex = np.dot(np.dot(mu_ex, Sigma_ex), mu_ex)
    var_mu_m_ex = np.dot(np.dot(mu_m_ex, Sigma_ex), mu_m_ex)
    var_R_p = np.trace(Sigma_ex) + 2 * np.sum(np.triu(Sigma_ex, k=1))
    var_R_m = np.var(R_m_ex, ddof=1)
    cov_R_R_m = np.cov(np.hstack((R_ex, R_m_ex.reshape(n_obs, 1)).T), ddof=1)[0:n_assets, n_assets]
    shp = (mu_ex @ mu_m_ex) / np.sqrt(var_mu_ex * var_mu_m_ex)
    shp -= (rf / np.sqrt(var_mu_m_ex))
    shp *= np.sqrt((n_obs - 1) / n_obs)
    shp += (rf / np.sqrt(var_R_m))
    shp *= np.sqrt((n_obs - 1) / (n_obs - n_assets - 2))
    z = norm.ppf(0.995)
    shp_ci_l = shp - z * np.sqrt((var_R_p / n_obs) + (shp ** 2) * ((var_mu_m_ex / var_mu_ex) + (var_R_m / var_R_p)))
    shp_ci_u = shp + z * np.sqrt((var_R_p / n_obs) + (shp ** 2) * ((var_mu_m_ex / var_mu_ex) + (var_R_m / var_R_p)))
    return shp, shp_ci_l, shp_ci_u

这个函数的输入包括：

- R: numpy数组，形状为(n_obs, n_assets)，表示观察到的收益率。
- R_m: numpy数组，形状为(n_obs,)，表示基准收益率。
- C: numpy数组，形状为(n_assets, n_assets)，表示资产收益率的样本协方差矩阵。
- rf: 浮点数，表示无风险利率。

输出包括：

- shp: 浮点数，表示使用Shanken校正后的夏普比率。
- shp_ci_l: 浮点数，表示使用Shanken校正后的夏普比率的下置信区间。
- shp_ci_u: 浮点数，表示使用Shanken校正后的夏普比率的上置信区间。