1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
|
import cvxpy
import numpy as np
import math
from matplotlib import pyplot as plt
plt.style.use('ggplot')
def cor2cov(Rho, vol):
return np.diag(vol) @ Rho @ np.diag(vol)
def rho(sigma, delta, volF):
""" computes the correlation between the asset and the factor """
return 1/math.sqrt(1+sigma**2/(delta**2*volF**2))
def resid_vol(rho, delta, volF):
""" computes the residual of the asset """
return math.sqrt(delta**2*volF**2*(1/rho**2-1))
def var(rho, delta, volF):
""" computes the variance of the asset """
return delta**2*volF**2+resid_vol(rho, delta, volF)**2
def compute_allocation(rho_clo = 0.9, rho_cso=0.6, rho_subprime=0.2,
delta_clo=1.2, delta_cso=0.4, delta_subprime=0.8,
mu_HY=0.02, mu_clo=0.08, mu_cso=0.07, mu_subprime=0.25):
rho = {'CLO': rho_clo,
'CSO': rho_cso,
'Subprime': rho_subprime}
delta = {'CLO': delta_clo,
'CSO': delta_cso,
'Subprime': delta_subprime}
assets = ['CLO', 'CSO', 'Subprime']
mu = np.array([mu_HY, mu_clo, mu_cso, mu_subprime])
u = volHY * np.array([delta[a] for a in assets])
Sigma = np.outer(u, u) + np.diag([resid_vol(rho[a], delta[a], volHY)**2
for a in ['CLO', 'CSO', 'Subprime']])
v = volHY**2 * np.array([1] + [delta[a] for a in assets])
Sigma = np.vstack((v, np.c_[v[1:], Sigma]))
sharpe = mu/np.sqrt(np.diag(Sigma))
gamma = cvxpy.Parameter(sign='positive')
w = cvxpy.Variable(4)
ret = mu.T*w
risk = cvxpy.quad_form(w, Sigma)
prob = cvxpy.Problem(cvxpy.Maximize(ret-gamma*risk),
[cvxpy.sum_entries(w[1:]) - 0.1*w[0] == 1,
w[1:] >= 0,
w[0] <= 0])
gamma_x = np.linspace(0, 20, 500)
W = np.empty((4, gamma_x.size))
for i, val in enumerate(gamma_x):
gamma.value = val
prob.solve()
W[:,i] = np.asarray(w.value).squeeze()
fund_return = mu@W
fund_vol= np.array([math.sqrt(W[:,i]@Sigma@W[:,i]) for i in range(gamma_x.size)])
return (W, fund_return, fund_vol)
def plot_allocation(W, fund_return, fund_vol):
gamma_x = np.linspace(0, 20, fund_return.size)
fig, ax1 = plt.subplots()
ax1.stackplot(fund_vol, W[1:,], labels=['CLO', 'CSO', 'Subprime'])
ax1.set_xlabel('risk factor')
ax1.set_ylabel('portfolio weights')
ax1.legend()
# ax1.text(0.3, 0.82, 'RMBS')
# ax1.text(0.5, 0.45, 'CSO')
# ax1.text(0.5, 0.15, 'CLO')
ax1.set_ylim([0, 1])
ax2 = ax1.twinx()
ax2.plot(fund_vol, fund_return, lw=1, color="grey")
ax2.set_ylabel('fund volatility')
plt.show()
if __name__=="__main__":
volHY = 0.07
rho = {'CLO': 0.6,
'CSO': 0.5,
'Subprime': 0.3}
delta = {'CLO': 0.4,
'CSO': 0.2,
'Subprime': 0.6}
mu = np.array([0.01, 0.075, 0.065, 0.25])
W, fund_return, fund_vol = compute_allocation(rho['CLO'], rho['CSO'], rho['Subprime'],
delta['CLO'], delta['CSO'], delta['Subprime'],
mu[0], mu[1], mu[2], mu[3])
plot_allocation(W, fund_return, fund_vol)
|
