# ## [Step 8](https://www.arpm.co/lab/redirect.php?permalink=s_bl_equilibrium_ret-implementation-step08): Compute Black-Litterman sure posterior parameters
mu_r_sure_bl = mu_r_equil + sig2_hat_r @ v.T @ \
np.linalg.solve(v @ sig2_hat_r @ v.T, i - v @ mu_r_equil)
sig2_r_sure_bl = (1 + 1 / tau) * sig2_hat_r - (1 / tau) * sig2_hat_r @ v.T\
@ np.linalg.solve(v @ sig2_hat_r @ v.T, v @ sig2_hat_r)
# ## [Step 9](https://www.arpm.co/lab/redirect.php?permalink=s_bl_equilibrium_ret-implementation-step09): Compare posterior parameters from point views
# +
k_ = len(v) # view variables dimension
v_point = v
z_point = i
mu_r_point, sig2_r_point = min_rel_entropy_normal(mu_r_equil, sig2_hat_r,
v_point, z_point, v_point,
np.zeros((k_)))
# -
# ## [Step 10](https://www.arpm.co/lab/redirect.php?permalink=s_bl_equilibrium_ret-implementation-step10): Compute posterior parameters from distributional views (Minimum Relative Entropy)
# +
v_mre = v
v_sig_mre = np.eye(n_)
imre = i
sig2_i_mumre = sig2_hat_r
mu_r_mre, sig2_r_mre = min_rel_entropy_normal(mu_r_equil, sig2_hat_r, v_mre,
imre, v_sig_mre, sig2_i_mumre)
# display_name: Python 3
# language: python
# name: python3
# ---
# # s_min_rel_ent_point_view [<img src="https://www.arpm.co/lab/icons/icon_permalink.png" width=30 height=30 style="display: inline;">](https://www.arpm.co/lab/redirect.php?code=s_min_rel_ent_point_view&codeLang=Python)
# For details, see [here](https://www.arpm.co/lab/redirect.php?permalink=ExViewTheoryPoint).
# +
import numpy as np
from arpym.views import min_rel_entropy_normal
# -
# ## [Input parameters](https://www.arpm.co/lab/redirect.php?permalink=s_min_rel_ent_point_view-parameters)
mu_base = np.array([0.26, 0.29, 0.33]) # base expectation
sig2_base = np.array([[0.18, 0.11, 0.13],
[0.11, 0.23, 0.16],
[0.13, 0.16, 0.23]]) # base covariance
v = np.array([[1, -1, 0], [0, 1, -1]]) # view matrix
z_view = np.array([1.02, -0.5]) # point view
# ## [Step 1](https://www.arpm.co/lab/redirect.php?permalink=s_min_rel_ent_point_view-implementation-step01): Compute point view updated parameters
# +
k_, n_ = v.shape # market and view dimension
mu_upd, sig2_upd = min_rel_entropy_normal(mu_base, sig2_base, v, z_view, v,
np.zeros((k_)))
# +
import numpy as np
from arpym.views import min_rel_entropy_normal
# -
# ## [Input parameters](https://www.arpm.co/lab/redirect.php?permalink=s_min_rel_ent_distr_view-parameters)
# +
mu_x_base = np.array([0.26, 0.29, 0.33]) # base case expectation
sig2_x_base = np.array([[0.18, 0.11, 0.13], [0.11, 0.23, 0.16],
[0.13, 0.16, 0.23]]) # base case covariance
v = np.array([[1, -1, 0], [0, 1, -1]]) # view matrix
mu_z_view = np.array([1.02, -0.50]) # expectation of the view variables
sig2_z_view = np.array([[0.35, -0.40],
[-0.40, 0.21]]) # covariance of the view variables
# -
# ## [Step 1](https://www.arpm.co/lab/redirect.php?permalink=s_min_rel_ent_distr_view-implementation-step01): Compute base parameters of the view variables
mu_z_base = v @ mu_x_base
sig2_z_base = v @ sig2_x_base @ v.T
# ## [Step 2](https://www.arpm.co/lab/redirect.php?permalink=s_min_rel_ent_distr_view-implementation-step02): Compute distributional view updated parameters
mu_x_upd, sig2_x_upd = min_rel_entropy_normal(mu_x_base, sig2_x_base, v,
mu_z_view, v, sig2_z_view)
# +
def eff_rank(s2):
lam2_n, _ = np.linalg.eig(s2)
w_n = lam2_n / np.sum(lam2_n)
return np.exp(-w_n @ np.log(w_n))
eff_rank_v_mu = eff_rank(cov_2_corr(v_mu @ sig2_x_base @ v_mu.T)[0])
eff_rank_v_sig = eff_rank(cov_2_corr(v_sig @ sig2_x_base @ v_sig.T)[0])
# -
# ## [Step 2](https://www.arpm.co/lab/redirect.php?permalink=s_min_rel_ent_partial_view-implementation-step02): Compute updated parameters
mu_x_upd, sig2_x_upd = min_rel_entropy_normal(mu_x_base, sig2_x_base, v_mu,
mu_view, v_sig, sig2_view)
# ## [Step 3](https://www.arpm.co/lab/redirect.php?permalink=s_min_rel_ent_partial_view-implementation-step03): Compute projectors
# +
k_ = len(mu_view) # view variables dimension
n_ = len(mu_x_base) # market dimension
v_mu_inv = sig2_x_base @ v_mu.T @ np.linalg.solve(v_mu @ sig2_x_base @ v_mu.T,
np.identity(k_))
v_sig_inv = sig2_x_base @ v_sig.T @\
(np.linalg.solve(v_sig @ sig2_x_base @ v_sig.T, np.identity(k_)))
p_mu = np.eye(n_) - v_mu_inv @ v_mu
p_mu_c = v_mu_inv @ v_mu
p_sig = np.eye(n_) - v_sig_inv @ v_sig
p_sig_c = v_sig_inv @ v_sig