"""""" #
"""
Copyright (c) 2020-2024, Dany Cajas
All rights reserved.
This work is licensed under BSD 3-Clause "New" or "Revised" License.
License available at https://github.com/dcajasn/Riskfolio-Lib/blob/master/LICENSE.txt
"""
import numpy as np
import cvxpy as cp
import math
from scipy.special import binom
__all__ = [
"owa_l_moment",
"owa_gmd",
"owa_cvar",
"owa_wcvar",
"owa_tg",
"owa_wr",
"owa_rg",
"owa_cvrg",
"owa_wcvrg",
"owa_tgrg",
"owa_l_moment_crm",
]
[docs]
def owa_l_moment(T, k=2):
r"""
Calculate the OWA weights to calculate the kth linear moment (l-moment)
of a returns series as shown in :cite:`d-Cajas6`.
Parameters
----------
T : int
Number of observations of the returns series.
k : int
Order of the l-moment. Must be an integer higher or equal than 1.
Returns
-------
value : 1d-array
An OWA weights vector of size Tx1.
"""
w = []
for i in range(1, T + 1):
a = 0
for j in range(k):
a += (-1) ** j * binom(k - 1, j) * binom(i - 1, k - 1 - j) * binom(T - i, j)
a *= 1 / (k * binom(T, k))
w.append(a)
return np.array(w).reshape(-1, 1)
[docs]
def owa_gmd(T):
r"""
Calculate the OWA weights to calculate the Gini mean difference (GMD)
of a returns series as shown in :cite:`d-Cajas3`.
Parameters
----------
T : int
Number of observations of the returns series.
Returns
-------
value : 1d-array
An OWA weights vector of size Tx1.
"""
w_ = []
for i in range(1, T + 1):
w_.append(2 * i - 1 - T)
w_ = 2 * np.array(w_) / (T * (T - 1))
w_ = w_.reshape(-1, 1)
return w_
[docs]
def owa_cvar(T, alpha=0.05):
r"""
Calculate the OWA weights to calculate the Conditional Value at Risk (CVaR)
of a returns series as shown in :cite:`d-Cajas3`.
Parameters
----------
T : int
Number of observations of the returns series.
alpha : float, optional
Significance level of CVaR. The default is 0.05.
Returns
-------
value : 1d-array
An OWA weights vector of size Tx1.
"""
k = int(np.ceil(T * alpha)) - 1
w_ = np.zeros((T, 1))
w_[:k, :] = -1 / (T * alpha)
w_[k, :] = -1 - np.sum(w_[:k, :])
return w_
[docs]
def owa_wcvar(T, alphas, weights):
r"""
Calculate the OWA weights to calculate the Weighted Conditional Value at
Risk (WCVaR) of a returns series as shown in :cite:`d-Cajas3`.
Parameters
----------
T : int
Number of observations of the returns series.
alphas : list
List of significance levels of each CVaR model.
weights : list
List of weights of each CVaR model.
Returns
-------
value : 1d-array
An OWA weights vector of size Tx1.
"""
w_ = 0
for i, j in zip(alphas, weights):
w_ += owa_cvar(T, i) * j
return w_
[docs]
def owa_tg(T, alpha=0.05, a_sim=100):
r"""
Calculate the OWA weights to calculate the Tail Gini of a
returns series as shown in :cite:`d-Cajas3`.
Parameters
----------
T : int
Number of observations of the returns series.
alpha : float, optional
Significance level of TaiL Gini. The default is 0.05.
a_sim : float, optional
Number of CVaRs used to approximate the Tail Gini. The default is 100.
Returns
-------
value : 1d-array
A OWA weights vector of size Tx1.
"""
alphas = np.linspace(alpha, 0.0001, a_sim)[::-1]
w_ = [(alphas[1] - 0) * alphas[0] / alphas[-1] ** 2]
for i in range(1, len(alphas) - 1):
w_.append((alphas[i + 1] - alphas[i - 1]) * alphas[i] / alphas[-1] ** 2)
w_.append((alphas[-1] - alphas[-2]) / alphas[-1])
w_ = owa_wcvar(T, alphas, w_)
return w_
[docs]
def owa_wr(T):
r"""
Calculate the OWA weights to calculate the Worst realization (minimum)
of a returns series as shown in :cite:`d-Cajas3`.
Parameters
----------
T : int
Number of observations of the returns series.
Returns
-------
value : 1d-array
A OWA weights vector of size Tx1.
"""
w_ = np.zeros((T, 1))
w_[0, :] = -1
return w_
[docs]
def owa_rg(T):
r"""
Calculate the OWA weights to calculate the range of a returns series
as shown in :cite:`d-Cajas3`.
Parameters
----------
T : int
Number of observations of the returns series.
Returns
-------
value : 1d-array
A OWA weights vector of size Tx1.
"""
w_ = np.zeros((T, 1))
w_[0, :] = -1
w_[-1, :] = 1
return w_
[docs]
def owa_cvrg(T, alpha=0.05, beta=None):
r"""
Calculate the OWA weights to calculate the CVaR range of a returns series
as shown in :cite:`d-Cajas3`.
Parameters
----------
T : int
Number of observations of the returns series.
alpha : float, optional
Significance level of CVaR of losses. The default is 0.05.
beta : float, optional
Significance level of CVaR of gains. If None it duplicates alpha.
The default is None.
Returns
-------
value : 1d-array
A OWA weights vector of size Tx1.
"""
if beta is None:
beta = alpha
w_ = owa_cvar(T, alpha) - owa_cvar(T, beta)[::-1]
return w_
[docs]
def owa_wcvrg(T, alphas, weights_a, betas=None, weights_b=None):
r"""
Calculate the OWA weights to calculate the WCVaR range of a returns series
as shown in :cite:`d-Cajas3`.
Parameters
----------
T : int
Number of observations of the returns series.
alphas : list
List of significance levels of each CVaR of losses model.
weights_a : list
List of weights of each CVaR of losses model.
betas : list, optional
List of significance levels of each CVaR of gains model. If None it duplicates alpha.
The default is None.
weights_b : list, optional
List of weights of each CVaR of gains model. If None it duplicates weights_a.
The default is None.
Returns
-------
value : 1d-array
A OWA weights vector of size Tx1.
"""
if betas is None or weights_b is None:
betas = alphas
weights_b = weights_a
w_ = owa_wcvar(T, alphas, weights_a) - owa_wcvar(T, betas, weights_b)[::-1]
return w_
[docs]
def owa_tgrg(T, alpha=0.05, a_sim=100, beta=None, b_sim=None):
r"""
Calculate the OWA weights to calculate the Tail Gini range of a returns
series as shown in :cite:`d-Cajas3`.
Parameters
----------
T : int
Number of observations of the returns series.
alpha : float, optional
Significance level of Tail Gini of losses. The default is 0.05.
a_sim : float, optional
Number of CVaRs used to approximate Tail Gini of losses. The default is 100.
beta : float, optional
Significance level of Tail Gini of gains. If None it duplicates alpha value.
The default is None.
b_sim : float, optional
Number of CVaRs used to approximate Tail Gini of gains. If None it duplicates a_sim value.
The default is None.
Returns
-------
value : 1d-array
A OWA weights vector of size Tx1.
"""
if beta is None:
beta = alpha
if b_sim is None:
b_sim = a_sim
w_ = owa_tg(T, alpha, a_sim) - owa_tg(T, beta, b_sim)[::-1]
return w_
[docs]
def owa_l_moment_crm(T, k=4, method="MSD", g=0.5, max_phi=0.5, solver="CLARABEL"):
r"""
Calculate the OWA weights to calculate a convex risk measure that considers
higher linear moments or L-moments as shown in :cite:`d-Cajas6`.
Parameters
----------
T : int
Number of observations of the returns series.
k : int
Order of the l-moment. Must be an integer higher or equal than 2.
method : str, optional
Method to calculate the weights used to combine the l-moments with order higher than 2.
The default value is 'MSD'. Possible values are:
- 'CRRA': Normalized Constant Relative Risk Aversion coefficients.
- 'ME': Maximum Entropy.
- 'MSS': Minimum Sum Squares.
- 'MSD': Minimum Square Distance.
g : float, optional
Risk aversion coefficient of CRRA utility function. The default is 0.5.
max_phi : float, optional
Maximum weight constraint of L-moments.
The default is 0.5.
solver: str, optional
Solver available for CVXPY. Used to calculate 'ME', 'MSS' and 'MSD' weights.
The default value is 'CLARABEL'.
Returns
-------
value : 1d-array
A OWA weights vector of size Tx1.
"""
if k < 2 or (not isinstance(k, int)):
raise ValueError("k must be an integer higher equal than 2")
if method not in ["CRRA", "ME", "MSS", "MSD"]:
raise ValueError("Available methods are 'CRRA', 'ME', 'MSS' and 'MSD'")
if g >= 1 or g <= 0:
raise ValueError("The risk aversion coefficient mus be between 0 and 1")
if max_phi >= 1 or max_phi <= 0:
raise ValueError(
"The constraint on maximum weight of L-moments must be between 0 and 1"
)
ws = np.empty((T, 0))
for i in range(2, k + 1):
w_i = (-1) ** i * owa_l_moment(T, i)
ws = np.concatenate([ws, w_i], axis=1)
if method == "CRRA":
phis = []
e = 1
for i in range(1, k):
e *= g + i - 1
phis.append(e / math.factorial(i + 1))
phis = np.array(phis)
phis = phis / np.sum(phis)
phis = phis.reshape(-1, 1)
a = ws @ phis
w = np.zeros_like(a)
w[0] = a[0]
for i in range(1, len(a)):
w[i, 0] = np.max(a[: i + 1, 0])
else:
theta = cp.Variable((T, 1))
n = ws.shape[1]
phi = cp.Variable((n, 1))
constraints = [
cp.sum(phi) == 1,
theta == ws @ phi,
phi <= max_phi,
phi >= 0,
phi[1:] <= phi[:-1],
theta[1:] >= theta[:-1],
]
if method == "ME":
theta_ = cp.Variable((T, 1))
obj = cp.sum(cp.entr(theta_)) * 1000
constraints += [
theta_ >= theta,
theta_ >= -theta,
]
objective = cp.Maximize(obj)
elif method == "MSS":
obj = cp.pnorm(theta, p=2) * 1000
objective = cp.Minimize(obj)
elif method == "MSD":
obj = cp.pnorm(theta[1:] - theta[:-1], p=2) * 1000
objective = cp.Minimize(obj)
problem = cp.Problem(objective, constraints)
if solver is not None:
problem.solve(solver=solver)
else:
problem.solve()
phis = phi.value
phis = phis / np.sum(phis)
phis = phis.reshape(-1, 1)
w = ws @ phis
return w