xi_tilde = np.zeros((t_, n_))
for n in range(n_):
u = t.cdf(xi[:, n],
df=10**6,
loc=mu_marg[n],
scale=np.sqrt(sigma2_marg[n]))
u[u <= 10**(-7)] = 10**(-7)
u[u >= 1 - 10**(-7)] = 1 - 10**(-7)
xi_tilde[:, n] = t.ppf(u, df=10**6)
# ## [Step 6](https://www.arpm.co/lab/redirect.php?permalink=s_dcc_fit-implementation-step06): Estimate the unconditional correlation matrix via MLFP
# +
_, sigma2_xi_tilde = fit_locdisp_mlfp(xi_tilde, p=p, nu=10**6)
rho2_xi_tilde, _ = cov_2_corr(sigma2_xi_tilde)
rho2 = rho2_xi_tilde
beta, delta2 = factor_analysis_paf(rho2_xi_tilde, k_)
rho2 = beta @ beta.T + np.diag(delta2)
# -
# ## [Step 7](https://www.arpm.co/lab/redirect.php?permalink=s_dcc_fit-implementation-step07): Compute the time series of true invariants via DCC fit
params, r2_t, epsi, q2_t_ = fit_dcc_t(xi_tilde, p, rho2=rho2)
c, a, b = params
q2_t_nextstep = c*rho2 +\
b*q2_t_ +\
a*(np.array([epsi[-1, :]])[email protected]([epsi[-1, :]]))
r2_t_nextstep, _ = cov_2_corr(q2_t_nextstep)
sigma2_z = sigma2_xz[n_:, n_:]
mu_z = mu_xz[n_:]
mu_x = mu_xz[:n_]
beta = sigma_xz @ np.linalg.inv(sigma2_z)
alpha = mu_x - beta @ mu_z
# -
# ## [Step 2](https://www.arpm.co/lab/redirect.php?permalink=s_regression_lfm-implementation-step02): Compute expectation and covariance of prediction
mu_xreg_bar = alpha + beta @ mu_z
sigma2_xreg_bar = beta @ sigma2_z @ beta.T
# ## [Step 3](https://www.arpm.co/lab/redirect.php?permalink=s_regression_lfm-implementation-step03): Compute the r-squared
# +
c2_xz, _ = cov_2_corr(sigma2_xz)
sigma2_x = sigma2_xz[:n_, :n_]
r2 = np.trace(
sigma_xz @ np.linalg.inv(sigma2_z) @ sigma_xz.T) / np.trace(sigma2_x)
# -
# ## [Step 4](https://www.arpm.co/lab/redirect.php?permalink=s_regression_lfm-implementation-step04): Compute joint distribution of residulas and factors
a = np.zeros(n_ + k_)
a[:n_] = -alpha
b = np.eye(n_ + k_)
b[:n_, n_:] = -beta
mu_uz = a + b @ mu_xz
sigma2_uz = b @ sigma2_xz @ b.T
beta_hat_pc = sigma @ e_hat[:, :k_] # loadings
gamma_hat_pc = e_hat[:, :k_].T @ sigma_inv # construction matrix
# ## [Step 5](https://www.arpm.co/lab/redirect.php?permalink=s_pca_truncated_lfm-implementation-step05): Compute the factor realizations and their expectation and covariance
z_hat_pc = (x - m_x_hat) @ gamma_hat_pc.T # factors
m_z_hat, s2_z_hat = meancov_sp(z_hat_pc)
# ## [Step 6](https://www.arpm.co/lab/redirect.php?permalink=s_pca_truncated_lfm-implementation-step06): Compute the residuals and the joint sample covariance of residuals and factors
u = x - (alpha_hat_pc + z_hat_pc @ beta_hat_pc.T) # residuals
_, s2_uz_hat = meancov_sp((np.c_[u, z_hat_pc]))
# ## [Step 7](https://www.arpm.co/lab/redirect.php?permalink=s_pca_truncated_lfm-implementation-step07): Compute correlations among residuals
c2_uz_hat, _ = cov_2_corr(s2_uz_hat)
c2_u_hat = c2_uz_hat[:n_, :n_] # correlation among residuals
# ## [Step 8](https://www.arpm.co/lab/redirect.php?permalink=s_pca_truncated_lfm-implementation-step08): Compute the truncated covariance of the returns
s2_u_hat = s2_uz_hat[:n_, :n_]
s_u_hat = np.sqrt(np.diag(s2_u_hat))
s2_x_trunc = beta_hat_pc@s2_z_hat@beta_hat_pc.T +\
np.diag(np.diag(s_u_hat)) # truncated covariance
# ## [Step 9](https://www.arpm.co/lab/redirect.php?permalink=s_pca_truncated_lfm-implementation-step09): Estimate the standard deviations of the portfolio returns using the sample covariance and the truncated covariance
# +
w1 = 1 / n_ * np.ones((n_, 1)) # equal-weights portfolio
w2 = np.zeros((n_, 1)) # long-short portfolio
epsi_tilde[:, i, l] = tstu.ppf(u[:, i], nu_vec_cop[l])
# estimate copula parameters with maximum likelihood
_, sig2 = \
fit_locdisp_mlfp_difflength(epsi_tilde[:, :, l],
p=p_copula,
nu=nu_vec_cop[l],
threshold=10 ** -3,
maxiter=1000)
# shrinkage: factor analysis
beta, delta2 = factor_analysis_paf(sig2, k_)
sig2_fa = beta @ beta.T + np.diag(delta2)
# compute correlation matrix
rho2_copula_vec[:, :, l], _ = cov_2_corr(sig2_fa)
# compute log-likelihood at times with no missing values
llike_nu[l] = np.sum(p_copula_bonds * np.log(
mvt_pdf(epsi_bonds, np.zeros(i_), rho2_copula_vec[:, :, l],
nu_vec_cop[l])))
# choose nu that gives the highest log-likelihood
l_nu = np.argsort(llike_nu)[-1]
db_estimation_copula = {
'nu': np.int(nu_vec_cop[l_nu]),
'rho2': rho2_copula_vec[:, :, l_nu]
}
# -
# ## [Step 6](https://www.arpm.co/lab/redirect.php?permalink=s_checklist_scenariobased_step03-implementation-step06): Estimate the distribution of the credit structural invariant
parse_dates=True)
db_epsi = db_epsi.iloc[:, :i_]
dates = db_epsi.index
t_ = len(dates)
stocks_names = db_epsi.columns
epsi = db_epsi.values
# Location-dispersion
db_locdisp = pd.read_csv(path_temp + 'db_fit_garch_stocks_locdisp.csv')
mu_hat = db_locdisp.loc[:, 'mu_hat'].values[:i_]
sig2_hat = db_locdisp.loc[:, 'sig2_hat'].values
i_tot = int(np.sqrt(len(sig2_hat)))
sig2_hat = sig2_hat.reshape(i_tot, i_tot)[:i_, :i_]
sig2_hat = cov_2_corr(sig2_hat)[0]
phi2_hat = np.linalg.solve(sig2_hat, np.eye(i_))
# -
# ## [Step 1](https://www.arpm.co/lab/redirect.php?permalink=s_shrink_cov_glasso-implementation-step01): Glasso shrinkage
k = int(i_ * (i_ - 1)) # shrink all covariances to 0
sig2_glasso, _, phi2_glasso, lam, conv, _ =\
markov_network(sig2_hat, k, lambda_vec)
# ## Plots
# +
plt.style.use('arpm')
# Graph
i = v @ mu_r_equil + eta * np.sqrt(np.diag(v @ sig2_hat_r @ v.T))
sig2_i_mu = ((1 - c) / (tau * c)) * (v @ sig2_hat_r @ v.T)
# ## [Step 5](https://www.arpm.co/lab/redirect.php?permalink=s_bl_equilibrium_ret-implementation-step05): Compute effective rank corresponding to the pick matrix
# +
def eff_rank(s2):
lam2_n, _ = np.linalg.eig(s2)
wn = lam2_n / np.sum(lam2_n)
return np.exp(-wn @ np.log(wn))
eff_rank = eff_rank(cov_2_corr(v @ sig2_hat_r @ v.T)[0])
# -
# ## [Step 6](https://www.arpm.co/lab/redirect.php?permalink=s_bl_equilibrium_ret-implementation-step06): Compute Black-Litterman posterior parameters
mu_m_pos, cv_pos_pred = black_litterman(mu_r_equil, sig2_hat_r, tau, v, i,
sig2_i_mu)
# ## [Step 7](https://www.arpm.co/lab/redirect.php?permalink=s_bl_equilibrium_ret-implementation-step07): Compute Black-Litterman posterior parameters in the case of uninformative views
# +
# compute vector quantifying the views in covariance
sig2_unifview = ((1 - c_uninf) / c_uninf) * v @ sig2_hat_r @ v.T
mu_m_pos, cv_pos_pred = black_litterman(mu_r_equil, sig2_hat_r, tau, v, i,
sig2_unifview)
z = (v_sector[1:, :] - v_sector[:-1, :]) / v_sector[:-1, :] # ## [Step 2](https://www.arpm.co/lab/redirect.php?permalink=s_reg_truncated_lfm-implementation-step02): Compute OLSFP estimates and residuals alpha, beta, s2, u = fit_lfm_ols(x, z) # ## [Step 3](https://www.arpm.co/lab/redirect.php?permalink=s_reg_truncated_lfm-implementation-step03): Compute the joint covariance and correlation # + # compute covariance [mu_uz, sig2_uz] = meancov_sp(np.hstack((u, z))) sig2_u = sig2_uz[:n_, :n_] sig2_z = sig2_uz[n_:, n_:] # compute correlation c2_uz, _ = cov_2_corr(sig2_uz) c_uz = c2_uz[:n_, n_:] c2_u = np.tril(c2_uz[:n_, :n_], -1) # - # ## [Step 4](https://www.arpm.co/lab/redirect.php?permalink=s_reg_truncated_lfm-implementation-step04): Compute standard deviations of two portfolios # + w_1 = np.ones(n_) / n_ # equal weight portfolio w_2 = np.zeros(n_) # long/shoft portfolio w_2[n_long] = 0.69158715 w_2[n_short] = np.array([-0.67752045, -0.01406671]) _, sig2_x = meancov_sp(x) sig2_x_trunc = beta @ sig2_z @ beta.T + np.diag(np.diag(sig2_u))
alpha_mlfp_t, beta_mlfp_t, sig2_u_mlfp_t, _ = fit_lfm_mlfp(x, z, p, nu)
# compute r-squared
u_mlfp_t = x - alpha_mlfp_t - z @ beta_mlfp_t.T
r2_mlfp_t = multi_r2(sig2_u_mlfp_t, sig2_hat)
# -
# ## [Step 5](https://www.arpm.co/lab/redirect.php?permalink=s_market_prediction_regression-implementation-step05): Bayesian loadings
# +
# Prior
beta_pri = np.zeros((n_, k_))
sig2_z_pri = sig2_z_hat
t_pri = pri_param_load * t_
sig2_pri = np.diag(cov_2_corr(sig2_hat)[1]**2)
nu_pri = pri_param_load * t_
# Posterior
t_pos = t_pri + t_
nu_pos = nu_pri + t_
beta_pos = (t_pri*beta_pri@sig2_z_pri + t_*beta_ols@sig2_z_hat) @\
np.linalg.solve(t_pri*sig2_z_pri + t_*sig2_z_hat, np.eye(k_))
sig2_z_pos = 1 / t_pos * (t_pri * sig2_z_pri + t_ * sig2_z_hat)
sig2_pos_load = 1 / nu_pos * (t_ * sig2_hat + nu_pri * sig2_pri +
t_pri * beta_pri @ sig2_z_pri @ beta_pri.T +
t_ * beta_ols @ sig2_z_hat @ beta_ols.T -
t_pos * beta_pos @ sig2_z_pos @ beta_pos.T)
kend_x = 2 / np.pi * np.arcsin(rho_12)
kend_x1v_put_h = -1
kend_x1v_call_h = 1
kend_x2v_put_h = -2 / np.pi * np.arcsin(rho_12)
kend_x2v_call_h = 2 / np.pi * np.arcsin(rho_12)
# ## [Step 5](https://www.arpm.co/lab/redirect.php?permalink=s_dependence_structure_call_put-implementation-step05): Compute Spearman rho
# compute grades scenarios
u_x, _, _ = cop_marg_sep(x)
u_v_p, _, _ = cop_marg_sep(v_put_h)
u_v_c, _, _ = cop_marg_sep(v_call_h)
# Spearman rho
_, cov_u = meancov_sp(np.c_[u_x[:, 0], u_v_p, u_v_c, u_x[:, 1]])
spear, _ = cov_2_corr(cov_u)
# ## [Step 6](https://www.arpm.co/lab/redirect.php?permalink=s_dependence_structure_call_put-implementation-step06): Compute correlation
cov = np.cov(np.c_[x[:, 0], v_put_h, v_call_h, x[:, 1]].T)
corr, _ = cov_2_corr(cov)
# ## Plots
# +
plt.style.use('arpm')
violet = [170/255, 85/255, 187/255]
teal = [60/255, 149/255, 145/255]
f = plt.figure()
mydpi = 72.0
# ## [Input parameters](https://www.arpm.co/lab/redirect.php?permalink=s_shrinkage_factor-parameters)
tau_hl = 180 # half life
k_ = 25 # number of hidden factors
# ## [Step 0](https://www.arpm.co/lab/redirect.php?permalink=s_shrinkage_factor-implementation-step00): Load data
path = '../../../databases/temporary-databases/'
xi = np.array(pd.read_csv(path + 'db_GARCH_residuals.csv', index_col=0))
t_, n_ = xi.shape
# ## [Step 1](https://www.arpm.co/lab/redirect.php?permalink=s_shrinkage_factor-implementation-step01): Compute the HFP correlation
p = exp_decay_fp(t_, tau_hl)
_, sigma2 = meancov_sp(xi, p)
c2, _ = cov_2_corr(sigma2)
# ## [Step 2](https://www.arpm.co/lab/redirect.php?permalink=s_shrinkage_factor-implementation-step02): Compute the loadings and residual variances via PAF factor analysis
beta, delta2 = factor_analysis_paf(c2, k_)
# ## [Step 3](https://www.arpm.co/lab/redirect.php?permalink=s_shrinkage_factor-implementation-step03): Compute the factor analysis correlation and the Frobenius norm
c2_paf = beta @ beta.T + np.diag(delta2)
d_fro = np.linalg.norm(c2 - c2_paf, ord='fro') / \
np.linalg.norm(c2, ord='fro') * 100.
# ## Plots
# +
plt.style.use('arpm')
parse_dates=True) db_epsi = db_epsi.iloc[:, :i_] dates = db_epsi.index t_ = len(dates) stocks_names = db_epsi.columns epsi = db_epsi.values # Location-dispersion db_locdisp = pd.read_csv(path_temp + 'db_fit_garch_stocks_locdisp.csv') mu_hat = db_locdisp.loc[:, 'mu_hat'].values[:i_] sig2_hat = db_locdisp.loc[:, 'sig2_hat'].values i_tot = int(np.sqrt(len(sig2_hat))) sig2_hat = sig2_hat.reshape(i_tot, i_tot)[:i_, :i_] sig2_hat = cov_2_corr(sig2_hat)[0] phi2_hat = np.linalg.solve(sig2_hat, np.eye(i_)) # - # ## [Step 1](https://www.arpm.co/lab/redirect.php?permalink=s_bayesian_estimation-implementation-step01): Bayesian estimation # + # Prior mu_pri = np.zeros(i_) sig2_pri = np.diag(cov_2_corr(sig2_hat)[1]**2) t_pri = int(pri_t_ * t_) nu_pri = int(pri_t_ * t_) # Posterior t_pos = t_pri + t_ nu_pos = nu_pri + t_
from arpym.tools import add_logo # - # ## [Input parameters](https://www.arpm.co/lab/redirect.php?permalink=s_scen_prob_pdf-parameters) h2 = 0.01 # bandwidth x = np.array([[0, 0, 1, 1], [0, 1, 0, 1]]).T # joint scenarios p = np.array([0.33, 0.10, 0.20, 0.37]) # probabilities # ## [Step 1](https://www.arpm.co/lab/redirect.php?permalink=s_scen_prob_pdf-implementation-step01): Compute expectation and covariance m, s2 = meancov_sp(x, p) # ## [Step 2](https://www.arpm.co/lab/redirect.php?permalink=s_scen_prob_pdf-implementation-step02): Compute correlation matrix c2, _ = cov_2_corr(s2) # ## [Step 3](https://www.arpm.co/lab/redirect.php?permalink=s_scen_prob_pdf-implementation-step03): Compute the scenario-probability pdf # + # grid points for pdf evaluation x1_grid = np.arange(np.min(x[:, 0]) - 0.5, np.max(x[:, 0]) + 0.5, 0.025) x2_grid = np.arange(np.min(x[:, 1]) - 0.5, np.max(x[:, 1]) + 0.5, 0.025) x_grid = np.array([[x1, x2] for x1 in x1_grid for x2 in x2_grid]) # scenario-probability pdf f = pdf_sp(h2, x_grid, x, p) # - # ## Plots
mu_x, sig2_x = meancov_sp(x) beta_ = beta.T / np.diag(sig2_x) gamma = np.linalg.solve(beta_ @ beta, beta_) proj = beta @ gamma alpha_cs = mu_x - proj @ mu_x # ## [Step 3](https://www.arpm.co/lab/redirect.php?permalink=s_cross_section_truncated_lfm-implementation-step03): Compute cross-sectional factors and residuals z_cs = x @ gamma.T u_cs = x - alpha_cs - z_cs @ beta.T # ## [Step 4](https://www.arpm.co/lab/redirect.php?permalink=s_cross_section_truncated_lfm-implementation-step04): Estimate correlations between exogenous and cross-sectional factors _, sig2_zz = meancov_sp(np.hstack((z, z_cs))) c2_zz, _ = cov_2_corr(sig2_zz) # joint correlation c_zz = np.diag(c2_zz[:k_, k_:]) # ## [Step 5](https://www.arpm.co/lab/redirect.php?permalink=s_cross_section_truncated_lfm-implementation-step05): Compute the joint covariance and correlation # + mu_uz, sig2_uz = meancov_sp(np.hstack((u_cs, z_cs))) sig2_u = sig2_uz[:n_, :n_] sig2_z = sig2_uz[n_:, n_:] c2_uz, _ = cov_2_corr(sig2_uz) c_uz = c2_uz[:n_, n_:] c2_u = np.tril(c2_uz[:n_, :n_], -1) # - # ## [Step 6](https://www.arpm.co/lab/redirect.php?permalink=s_cross_section_truncated_lfm-implementation-step06): Compute the risk premia
df = pd.read_csv(path + 'data.csv', index_col=0)
y = np.array(df.loc[:, tau.astype('str')])
y = y[1800:, ] # remove missing data
fx_df = pd.read_csv(path + 'data.csv', usecols=['dates'],
parse_dates=['dates'])
fx_df = fx_df[1801:]
# ## [Step 1](https://www.arpm.co/lab/redirect.php?permalink=s_yield_change_correlation-implementation-step01): Compute invariants
x = np.diff(y, n=1, axis=0)
t_, n_ = x.shape
# ## [Step 2](https://www.arpm.co/lab/redirect.php?permalink=s_yield_change_correlation-implementation-step02): Compute HFP mean, covariance, correlation and vector of standard deviations
m_hat_HFP_x, s2_hat_HFP_x = meancov_sp(x)
c2_HFP_x, s_vec = cov_2_corr(s2_hat_HFP_x)
# ## [Step 3](https://www.arpm.co/lab/redirect.php?permalink=s_yield_change_correlation-implementation-step03): Fit and compute the Toeplitz cross-diagonal form
c2_star, gamma_star = min_corr_toeplitz(c2_HFP_x, tau)
# ## [Step 4](https://www.arpm.co/lab/redirect.php?permalink=s_yield_change_correlation-implementation-step04): Save the data
# +
output = {
'tau': pd.Series(tau),
'n_': pd.Series(x.shape[1]),
'gamma_star': pd.Series(gamma_star),
'm_hat_HFP_x': pd.Series(m_hat_HFP_x),
's2_hat_HFP_x': pd.Series((s2_hat_HFP_x.reshape(-1))),
's_vec': pd.Series(s_vec),
# view quantification parameters
mu_view = np.array([1.02, -0.50])
sig2_view = np.array([[0.19, 0.09], [0.09, 0.44]])
# ## [Step 1](https://www.arpm.co/lab/redirect.php?permalink=s_min_rel_ent_partial_view-implementation-step01): Compute effective ranks corresponding to the pick matrices
# +
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,
# ## [Step 2](https://www.arpm.co/lab/redirect.php?permalink=s_capm_like_identity-implementation-step02): Generate the MC scenarios of the compounded returns # + c_tnow_thor = simulate_normal(mu, sigma2, j_) # compounded returns scenarios # - # ## [Step 3](https://www.arpm.co/lab/redirect.php?permalink=s_capm_like_identity-implementation-step03): Compute the scenarios of the linear returns # + # linear returns scenarios r_tnow_thor_j = np.exp(c_tnow_thor) - 1 # linear returns expectation and covariance mu_r, sigma2_r = meancov_sp(r_tnow_thor_j) # correlation and volatility vector c2_r, sigmavol_r = cov_2_corr(sigma2_r) # - # ## [Step 4](https://www.arpm.co/lab/redirect.php?permalink=s_capm_like_identity-implementation-step04): Compute the MC scenarios of P&L's # + # P&L scenarios pi_tnow_thor = r_tnow_thor_j * v_tnow # P&L expectation and covariance mu_pi, sigma2_pi = meancov_sp(pi_tnow_thor) # - # ## [Step 5](https://www.arpm.co/lab/redirect.php?permalink=s_capm_like_identity-implementation-step05): Compute the maximum Sharpe ratio portfolio # + # maximum Sharpe ratio portfolio
