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
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
|
{
"cells": [
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"from ipywidgets import interact, interactive, fixed, FloatSlider\n",
"from IPython.display import display\n",
"import ipywidgets as widgets\n",
"import numpy as np\n",
"import math\n",
"import cvxpy\n",
"from inspect import signature\n",
"%matplotlib inline"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We model the assets in a 1-factor model setting. Namely conditionally on the factor $F$, the assets\n",
"are independent, and $r_i$, the return of asset $i$ is given by:\n",
"$$r_i = \\mu_i + \\delta_i(F-\\bar F) + \\sigma_i \\varepsilon_i$$\n",
"with $\\forall i,\\,\\textrm{Cov}(F, \\varepsilon_i)=0$, and $\\varepsilon\\sim\\mathcal{N}(0, \\mathbf{1})$ \n",
"\n",
"We can show that:\n",
"$$\\textrm{Corr}(r_i, F)=\\rho_i = \\frac{\\delta_i \\sqrt{\\textrm{Var}(F)}}{\\sqrt{\\textrm{Var}(r_i)}}$$\n",
"and with $\\textrm{Var(r_i)}=\\delta_i^2\\textrm{Var}(F) + \\sigma_i^2$, we get:\n",
"$$\\rho_i = \\frac{1}{\\sqrt{1+\\frac{\\sigma_i^2}{\\delta_i^2\\textrm{Var(F)}}}}$$\n",
"$$\\forall i\\neq j,\\;\\textrm{Cov}(r_i, r_j)=\\delta_i\\delta_j\\textrm{Var}(F)$$"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"def cor2cov(Rho, vol):\n",
" return np.diag(vol) @ Rho @ np.diag(vol)\n",
"\n",
"def rho(sigma, delta, volF):\n",
" \"\"\" computes the correlation between the asset and the factor \"\"\"\n",
" return 1/math.sqrt(1+sigma**2/(delta**2*volF**2))\n",
"\n",
"def resid_vol(rho, delta, volF):\n",
" \"\"\" computes the residual of the asset \"\"\"\n",
" return math.sqrt(delta**2*volF**2*(1/rho**2-1))\n",
"\n",
"def var(rho, delta, volF):\n",
" \"\"\" computes the variance of the asset \"\"\"\n",
" return delta**2*volF**2+resid_vol(rho, delta, volF)**2\n",
"\n",
"volHY = 0.07\n",
"## Edwin's parameters\n",
"rho = {'CLO': 0.6,\n",
" 'CSO': 0.5,\n",
" 'Subprime': 0.3}\n",
"delta = {'CLO': 0.4,\n",
" 'CSO': 0.2,\n",
" 'Subprime': 0.6}\n",
"mu = np.array([0.01, 0.075, 0.065, 0.25])"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"from matplotlib import pyplot as plt\n",
"plt.style.use('ggplot')"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"def compute_allocation(rho_clo = 0.9, rho_cso=0.6, rho_subprime=0.2,\n",
" delta_clo=1.2, delta_cso=0.4, delta_subprime=0.8,\n",
" mu_hy=0.02, mu_clo=0.07, mu_cso=0.08, mu_subprime=0.25):\n",
" rho = {'CLO': rho_clo,\n",
" 'CSO': rho_cso,\n",
" 'Subprime': rho_subprime}\n",
" delta = {'CLO': delta_clo,\n",
" 'CSO': delta_cso,\n",
" 'Subprime': delta_subprime}\n",
" assets = ['CLO', 'CSO', 'Subprime']\n",
" mu = np.array([mu_hy, mu_clo, mu_cso, mu_subprime])\n",
" u = volHY * np.array([delta[a] for a in assets])\n",
" Sigma = np.outer(u, u) + np.diag([resid_vol(rho[a], delta[a], volHY)**2\n",
" for a in assets])\n",
" v = volHY**2 * np.array([1] + [delta[a] for a in assets])\n",
" Sigma = np.vstack((v, np.c_[v[1:], Sigma]))\n",
"\n",
" sharpe = mu/np.sqrt(np.diag(Sigma))\n",
"\n",
" gamma = cvxpy.Parameter(sign='positive')\n",
" w = cvxpy.Variable(4)\n",
" ret = mu.T*w\n",
" risk = cvxpy.quad_form(w, Sigma)\n",
" prob = cvxpy.Problem(cvxpy.Maximize(ret-gamma*risk),\n",
" [cvxpy.sum_entries(w[1:]) - 0.1*w[0] == 1,\n",
" w[1:] >= 0,\n",
" w[0] <= 0])\n",
"\n",
" gamma_x = np.linspace(0, 20, 500)\n",
" W = np.empty((4, gamma_x.size))\n",
" for i, val in enumerate(gamma_x):\n",
" gamma.value = val\n",
" prob.solve()\n",
" W[:,i] = np.asarray(w.value).squeeze()\n",
" fund_return = mu@W\n",
" fund_vol = np.array([math.sqrt(W[:,i]@Sigma@W[:,i]) for i in range(gamma_x.size)])\n",
" return (W, fund_return, fund_vol)\n",
"\n",
"def plot_allocation(plot_vol=True, **kwargs):\n",
" W, fund_return, fund_vol = compute_allocation(**kwargs)\n",
" gamma_x = np.linspace(0, 20, 500)\n",
" fig, ax1 = plt.subplots()\n",
" ax1.stackplot(gamma_x, W[1:,], labels=['CLO', 'CSO', 'Subprime'])\n",
" ax1.set_xlabel('risk factor')\n",
" ax1.set_ylabel('portfolio weights')\n",
" ax1.legend()\n",
" # ax1.text(0.3, 0.82, 'RMBS')\n",
" # ax1.text(0.5, 0.45, 'CSO')\n",
" # ax1.text(0.5, 0.15, 'CLO')\n",
" ax1.set_ylim([0, 1])\n",
" ax2 = ax1.twinx()\n",
" if plot_vol:\n",
" ax2.plot(gamma_x, fund_vol, lw=1, color=\"grey\")\n",
" ax2.set_ylabel('fund volatility')\n",
" else:\n",
" ax2.plot(gamma_x, fund_return, lw=1, color=\"grey\")\n",
" ax2.set_ylabel('fund return')\n",
" plt.show()\n"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"def desc(greek, asset):\n",
" \"\"\"returns the latex label\"\"\"\n",
" return r'${0}_{{\\textrm{{{1}}}}}$'.format('\\\\'+greek, asset)\n",
"\n",
"desc_mapping = {}\n",
"assets = ['CLO', 'CSO', 'Subprime']\n",
"for greek in ['rho', 'delta', 'mu']:\n",
" if greek == 'mu':\n",
" assets.append('HY')\n",
" desc_mapping.update({desc(greek, a): '{0}_{1}'.format(greek, a.lower()) for a in assets})\n",
" \n",
"w = interactive(plot_allocation, rho_clo=FloatSlider(description=desc('rho', 'CLO'), min=0., max=1., step=0.1, value=0.9),\n",
" rho_cso=FloatSlider(description=desc('rho', 'CSO'), min=0., max=1., step=0.1, value=0.6),\n",
" rho_subprime=FloatSlider(description=desc('rho', 'Subprime'), min=0., max=1., step=0.1, value=0.2),\n",
" delta_clo=FloatSlider(description=desc('delta', 'CLO'), min=0., max=2., step=0.1, value=1.2),\n",
" delta_cso=FloatSlider(description=desc('delta', 'CSO'), min=0., max=2., step=0.1, value=0.4),\n",
" delta_subprime=FloatSlider(description=desc('delta', 'Subprime'), min=0., max=2., step=0.1, value=0.8),\n",
" mu_hy = FloatSlider(description=desc('mu', 'HY'), min=0., max=.1, step=0.01, value=0.02),\n",
" mu_clo = FloatSlider(description=desc('mu', 'CLO'), min=0., max=.12, step=0.01, value=0.07),\n",
" mu_cso = FloatSlider(description=desc('mu', 'CSO'), min=0., max=.12, step=0.01, value=0.08),\n",
" mu_subprime = FloatSlider(description=desc('mu', 'Subprime'), min=0.1, max=.35, step=0.01, value=0.25))\n",
"\n",
"## black magic to get back to the function order\n",
"s = signature(compute_allocation)\n",
"param_ordering = {p:i for i, p in enumerate(s.parameters)}\n",
"param_ordering[None] = -1\n",
"w_ordered = sorted([c for c in w.children], key=lambda x: param_ordering[desc_mapping.get(x.description)])\n",
"display(*w_ordered)"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"args = w.kwargs.copy()\n",
"args.pop('plot_vol')\n",
"W, fund_return, fund_vol = compute_allocation(**args)\n",
"import pandas as pd\n",
"df = pd.DataFrame(np.c_[W.T, fund_return, fund_vol], columns=['HY', 'CLO', 'CSO', 'Subprime', 'return', 'vol'])\n",
"df.to_csv('weights.csv', index=False)"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.8.0"
}
},
"nbformat": 4,
"nbformat_minor": 4
}
|
