CoCalc -- L5_PortfolioOptimization

²⁸⁴⁶ views

Kernel: Python 3 (system-wide)

In [8]:

from math import sqrt
import numpy as np
from cvxopt import matrix
from cvxopt.blas import nrm2, dot
from cvxopt.solvers import qp, socp, options
import matplotlib.pyplot as plt
%matplotlib inline

Covariance matrix $\Sigma$ and its Cholesky decomposition $U$

$\Sigma = U^T U$

In [9]:

names = ['Ant', 'Bee', 'Croc', 'Dino']
n = len(names)

Cov = 0.1 * matrix(
    [[ 0.1138, -0.0137,  0.0218, -0.0102],
     [-0.0137,  0.0274, -0.0028,  0.0040],
     [ 0.0218, -0.0028,  0.0213, -0.0016],
     [-0.0102,  0.0040, -0.0016,  0.0143]])

rootCov = np.linalg.cholesky(Cov).T
print("Cholesky factorization:")
print(rootCov)
print("\nCheck:")
print(rootCov.T @ rootCov)
rootCov = matrix(rootCov)

Out[9]:

Cholesky factorization:
[[ 0.10667708 -0.0128425   0.0204355  -0.00956157]
 [ 0.          0.05074515 -0.00034599  0.0054627 ]
 [ 0.          0.          0.04137959  0.00090106]
 [ 0.          0.          0.          0.03616522]]

Check:
[[ 0.01138 -0.00137  0.00218 -0.00102]
 [-0.00137  0.00274 -0.00028  0.0004 ]
 [ 0.00218 -0.00028  0.00213 -0.00016]
 [-0.00102  0.0004  -0.00016  0.00143]]

Sample portfolio

In [10]:

alphas = matrix([0.01, 0.02, 0.03, 0.04])

MM = 1000000
portf0 = MM * matrix([0.1, 0.2, 0.3, 0.4])

Portfolio variance and risk

In [11]:

portf0variance = (portf0.T * Cov * portf0)[0,0]
portf0risk = sqrt(portf0variance)
print(portf0risk)

Out[11]:

25105.776227792678

Checking that portfoli risk equals $\|G w\|_2$

In [12]:

nrm2(rootCov * portf0)

Out[12]:

25105.77622779268

Portfolio expected return

In [13]:

portf0return = (alphas.T * portf0)[0, 0]
print(portf0return)

Out[13]:

30000.0

Problem 1

Maximize the expected return for a given level of risk

In [14]:

netInvested = MM
maximalRisk = 25000

Conic formulation: $\begin{aligned} c^{T} x &\to\ \mbox{min}\\ s_0 &= h_0 - G_0 x \\ s_0 &\succeq 0 \\ s_1 & = h_1 - G_1 x \\ s_1 &\in \mathcal{C} \\ A x &= b \end{aligned}$

$x = \begin{pmatrix} r\\ w_1 \\ \vdots \\ w_n \end{pmatrix}$

Optimization constraint

In [15]:

coeffUtil1 = matrix([0.0, -1*alphas])
print(coeffUtil1)

Out[15]:

[ 0.00e+00]
[-1.00e-02]
[-2.00e-02]
[-3.00e-02]
[-4.00e-02]

Constraint 0 (Linear)

Risk positive, investments positive

In [16]:

e5 = matrix(0.0, (5,5))
e5[::6] = 1.0
G0 = matrix([[1.0], [matrix(0.0, (1,4))]])
G0 = matrix([G0, -1.0*e5])
h0 = matrix([[maximalRisk, matrix(0.0, (5,1))]])

print(G0)
print(h0)

Out[16]:

[ 1.00e+00  0.00e+00  0.00e+00  0.00e+00  0.00e+00]
[-1.00e+00 -0.00e+00 -0.00e+00 -0.00e+00 -0.00e+00]
[-0.00e+00 -1.00e+00 -0.00e+00 -0.00e+00 -0.00e+00]
[-0.00e+00 -0.00e+00 -1.00e+00 -0.00e+00 -0.00e+00]
[-0.00e+00 -0.00e+00 -0.00e+00 -1.00e+00 -0.00e+00]
[-0.00e+00 -0.00e+00 -0.00e+00 -0.00e+00 -1.00e+00]

[ 2.50e+04]
[ 0.00e+00]
[ 0.00e+00]
[ 0.00e+00]
[ 0.00e+00]
[ 0.00e+00]

Constraint 1 (Conic)

$\displaystyle s1 = {r \choose G x}$

In [17]:

z4 = matrix(0.0, (1,4))
G1 = matrix([[-1, z4.T],[z4, -1.0*rootCov]])
h1 = matrix(0.0, (5,1))

print(G1)
print(h1)

Out[17]:

[-1.00e+00  0.00e+00  0.00e+00  0.00e+00  0.00e+00]
[ 0.00e+00 -1.07e-01  1.28e-02 -2.04e-02  9.56e-03]
[ 0.00e+00 -0.00e+00 -5.07e-02  3.46e-04 -5.46e-03]
[ 0.00e+00 -0.00e+00 -0.00e+00 -4.14e-02 -9.01e-04]
[ 0.00e+00 -0.00e+00 -0.00e+00 -0.00e+00 -3.62e-02]

[ 0.00e+00]
[ 0.00e+00]
[ 0.00e+00]
[ 0.00e+00]
[ 0.00e+00]

Linear constraint

Fully invested portfolio: $\left( 0, e^T \right) x = 1000000$

In [18]:

A = matrix([[0.0], [matrix(1.0, (1,4))]])
b = matrix(float(netInvested), (1,1))
print(A)
print(b)

Out[18]:

[ 0.00e+00  1.00e+00  1.00e+00  1.00e+00  1.00e+00]

[ 1.00e+06]

Optimizer call

http://cvxopt.org/userguide/coneprog.html#second-order-cone-programming

In [7]:

solution1 = socp(coeffUtil1, G0, h0, [G1], [h1], A, b)

Out[7]:

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-7-410f236c583c> in <module>
----> 1 solution1 = socp(coeffUtil1, G0, h0, [G1], [h1], A, b)

NameError: name 'coeffUtil1' is not defined

In [13]:

solution1

Out[13]:

{'dual infeasibility': 3.641286043848554e-08,
 'dual objective': -32811.16397701068,
 'dual slack': 3.2328030328101123e-09,
 'gap': 0.016443068425783254,
 'iterations': 8,
 'primal infeasibility': 4.127299279913593e-08,
 'primal objective': -32811.16597285626,
 'primal slack': 7.362141785396568e-05,
 'relative gap': 5.011424598377923e-07,
 'residual as dual infeasibility certificate': None,
 'residual as primal infeasibility certificate': None,
 'sl': <6x1 matrix, tc='d'>,
 'sq': [<5x1 matrix, tc='d'>],
 'status': 'optimal',
 'x': <5x1 matrix, tc='d'>,
 'y': <1x1 matrix, tc='d'>,
 'zl': <6x1 matrix, tc='d'>,
 'zq': [<5x1 matrix, tc='d'>]}

In [14]:

portf1 = solution1['x'][1:5]
print(sum(portf1))

Out[14]:

1000000.0000000002

In [15]:

portf1variance = (portf1.T * Cov * portf1)[0,0]
portf1risk = sqrt(portf1variance)
print(portf1risk)

portf1return = (alphas.T * portf1)[0, 0]
print(portf1return)

Out[15]:

24999.999717163682
32811.165972856266

In [16]:

ind = np.arange(n)    # the horizontal locations for the groups
width = 0.35          # the width of the bars: can also be len(x) sequence
plt.figure(figsize=(10,6))
p0 = plt.bar(ind, [x for x in portf0], width, color='y')
p1 = plt.bar(ind+width, [x for x in portf1], width, color='r')

plt.ylabel('Invested notional')
plt.xticks(ind + width, names)
plt.legend((p0[0], p1[0]), ('Initial', 'Max Expected Return'), loc=2)

plt.show()

Out[16]:

Problem 2

Minimize the risk for the expected level of return

In [17]:

coeffUtil2 = matrix([1.0, z4.T])
print(coeffUtil2)

Out[17]:

[ 1.00e+00]
[ 0.00e+00]
[ 0.00e+00]
[ 0.00e+00]
[ 0.00e+00]

In [18]:

minReturn = 30000

G0 = matrix([[0.0], [-1*alphas.T]])
G0 = matrix([G0, -1.0*e5])
print(G0)

h0 = matrix([[-minReturn, matrix(0.0, (5,1))]])
print(h0)

Out[18]:

[ 0.00e+00 -1.00e-02 -2.00e-02 -3.00e-02 -4.00e-02]
[-1.00e+00 -0.00e+00 -0.00e+00 -0.00e+00 -0.00e+00]
[-0.00e+00 -1.00e+00 -0.00e+00 -0.00e+00 -0.00e+00]
[-0.00e+00 -0.00e+00 -1.00e+00 -0.00e+00 -0.00e+00]
[-0.00e+00 -0.00e+00 -0.00e+00 -1.00e+00 -0.00e+00]
[-0.00e+00 -0.00e+00 -0.00e+00 -0.00e+00 -1.00e+00]

[-3.00e+04]
[ 0.00e+00]
[ 0.00e+00]
[ 0.00e+00]
[ 0.00e+00]
[ 0.00e+00]

In [19]:

solution2 = socp(coeffUtil2, G0, h0, [G1], [h1], A, b)

Out[19]:

     pcost       dcost       gap    pres   dres   k/t
 0.0000e+00  3.0000e+04  1e+06  3e+00  3e+00  1e+00
 3.0629e+04  5.4084e+04  4e+05  1e+00  1e+00  1e+04
 2.8103e+04  3.5444e+04  9e+04  2e-01  2e-01  5e+03
 2.5590e+04  2.6394e+04  1e+04  3e-02  3e-02  5e+02
 2.4565e+04  2.4700e+04  1e+03  4e-03  4e-03  1e+02
 2.4585e+04  2.4608e+04  2e+02  5e-04  5e-04  2e+01
 2.4573e+04  2.4574e+04  6e+00  2e-05  2e-05  7e-01
 2.4573e+04  2.4573e+04  2e-01  7e-07  7e-07  3e-02
 2.4573e+04  2.4573e+04  7e-03  2e-08  2e-08  8e-04
Optimal solution found.

In [20]:

print(solution2['sl'])

Out[20]:

[ 9.12e+02]
[ 2.46e+04]
[ 6.53e+04]
[ 2.17e+05]
[ 2.78e+05]
[ 4.39e+05]

In [21]:

portf2 = solution2['x'][1:5]

portf2variance = (portf2.T * Cov * portf2)[0,0]
portf2risk = sqrt(portf2variance)
print(portf2risk)

portf2return = (alphas.T * portf2)[0, 0]
print(portf2return)

Out[21]:

24572.69096086509
30911.834843412114

In [22]:

ind = np.arange(n)    # the horizontal locations for the groups
width = 0.25          # the width of the bars: can also be len(x) sequence
plt.figure(figsize=(10,6))
p0 = plt.bar(ind, [x for x in portf0], width, color='y')
p1 = plt.bar(ind+width, [x for x in portf1], width, color='r')
p2 = plt.bar(ind+width*2., [x for x in portf2], width, color='b')

plt.ylabel('Invested notional')
plt.xticks(ind + width*1.5, names)
plt.legend((p0[0], p1[0], p2[0]), ('Initial', 'Max Expected Return', 'Min Risk'), loc=2)

plt.show()

Out[22]:

Problem 3

Markowitz mean-variance optimization. The utility function is a weighted sum of expected return and risk: $c = (\lambda, -\alpha^T)$ , where $\lambda$ is a risk-aversion parameter

In [24]:

def meanVariance(riskAversion, netInvested, printSystem=False):
    # utility function
    coeffUtil = matrix([riskAversion, -1.0*alphas])
    
    # investment limit plus positivity constraints
    G0 = matrix([matrix([[0.0],[matrix(1.0, (1,4))]]),
                 -1.0*matrix(np.eye(5))])
    h0 = matrix([[netInvested, matrix(0.0, (5,1))]])

    # conic constraint
    z4 = matrix(0.0, (1,4))
    G1 = matrix([[-1, z4.T],[z4, -1.0*rootCov]])
    h1 = matrix(0.0, (5,1))    
    
    # linear constraint = full investment
    A = matrix([[0.0], [matrix(1.0, (1,4))]])
    b = matrix(float(netInvested), (1,1))
    
    if printSystem:
        print("----------------")
        print(f"h0:\n{h0}")
        print(f"G0:\n{G0}")
        print(f"h1:\n{h1}")
        print(f"G1:\n{G1}")
        print("----------------")
    # call the optimizer
    solution = socp(coeffUtil, G0, h0, [G1], [h1], A, b)

    return solution['x'][1:5]

In [25]:

pi = meanVariance(1.0, MM, True)

Out[25]:

----------------
h0:
[ 1.00e+06]
[ 0.00e+00]
[ 0.00e+00]
[ 0.00e+00]
[ 0.00e+00]
[ 0.00e+00]

G0:
[ 0.00e+00  1.00e+00  1.00e+00  1.00e+00  1.00e+00]
[-1.00e+00 -0.00e+00 -0.00e+00 -0.00e+00 -0.00e+00]
[-0.00e+00 -1.00e+00 -0.00e+00 -0.00e+00 -0.00e+00]
[-0.00e+00 -0.00e+00 -1.00e+00 -0.00e+00 -0.00e+00]
[-0.00e+00 -0.00e+00 -0.00e+00 -1.00e+00 -0.00e+00]
[-0.00e+00 -0.00e+00 -0.00e+00 -0.00e+00 -1.00e+00]

h1:
[ 0.00e+00]
[ 0.00e+00]
[ 0.00e+00]
[ 0.00e+00]
[ 0.00e+00]

G1:
[-1.00e+00  0.00e+00  0.00e+00  0.00e+00  0.00e+00]
[ 0.00e+00 -1.07e-01  1.28e-02 -2.04e-02  9.56e-03]
[ 0.00e+00 -0.00e+00 -5.07e-02  3.46e-04 -5.46e-03]
[ 0.00e+00 -0.00e+00 -0.00e+00 -4.14e-02 -9.01e-04]
[ 0.00e+00 -0.00e+00 -0.00e+00 -0.00e+00 -3.62e-02]

----------------
     pcost       dcost       gap    pres   dres   k/t
 0: -2.5036e+04 -1.0400e+06  1e+06  9e-02  2e+00  1e+00
 1:  8.5157e+03 -2.4427e+05  3e+05  2e-02  5e-01  1e+04
 2:  1.6966e+03 -5.6352e+04  7e+04  5e-03  1e-01  4e+03
 3: -8.7656e+03 -5.3212e+04  5e+04  4e-03  1e-01  4e+03
 4: -1.5926e+01 -4.3320e+04  5e+04  4e-03  1e-01  5e+03
 5: -4.9642e+03 -2.4675e+04  2e+04  2e-03  4e-02  3e+03
 6: -7.5962e+03 -1.2579e+04  6e+03  5e-04  1e-02  8e+02
 7: -8.2156e+03 -9.1968e+03  1e+03  1e-04  2e-03  2e+02
 8: -8.5148e+03 -8.8761e+03  4e+02  4e-05  8e-04  6e+01
 9: -8.4979e+03 -8.5871e+03  1e+02  9e-06  2e-04  2e+01
10: -8.5284e+03 -8.5432e+03  2e+01  1e-06  3e-05  3e+00
11: -8.5282e+03 -8.5293e+03  1e+00  1e-07  3e-06  2e-01
12: -8.5283e+03 -8.5283e+03  6e-02  5e-09  1e-07  9e-03
13: -8.5283e+03 -8.5283e+03  3e-03  2e-10  5e-09  4e-04
Optimal solution found.

In [26]:

print(dot(alphas,pi))
print(sqrt(dot(pi, Cov*pi)))

Out[26]:

35504.13099203421
26975.832424259435

In [27]:

# Compute trade-off.
N = 20
raps = [ 10**(2.0*t/(N-1)-1.0) for t in range(N) ]
portfolios = [ meanVariance(lam, MM) for lam in raps ]
returns = [ dot(alphas,pi) for pi in portfolios ]
risks = [ sqrt(dot(pi, Cov*pi)) for pi in portfolios ]

Out[27]:

     pcost       dcost       gap    pres   dres   k/t
-2.5036e+04 -1.0400e+06  1e+06  9e-02  2e+00  1e+00
-2.2347e+04 -2.2669e+05  2e+05  2e-02  4e-01  2e+03
-2.3273e+04 -4.0116e+04  2e+04  1e-03  3e-02  3e+02
-3.1791e+04 -4.0207e+04  1e+04  8e-04  2e-02  5e+02
-3.5459e+04 -3.6887e+04  2e+03  1e-04  3e-03  1e+02
-3.5717e+04 -3.6578e+04  1e+03  8e-05  2e-03  8e+01
-3.6106e+04 -3.6328e+04  3e+02  2e-05  5e-04  2e+01
-3.6163e+04 -3.6260e+04  1e+02  9e-06  2e-04  1e+01
-3.6205e+04 -3.6231e+04  3e+01  3e-06  6e-05  3e+00
-3.6216e+04 -3.6220e+04  5e+00  4e-07  1e-05  6e-01
-3.6218e+04 -3.6219e+04  1e+00  9e-08  2e-06  1e-01
-3.6218e+04 -3.6218e+04  9e-02  8e-09  2e-07  1e-02
-3.6218e+04 -3.6218e+04  1e-02  1e-09  2e-08  2e-03
Optimal solution found.
     pcost       dcost       gap    pres   dres   k/t
-2.5036e+04 -1.0400e+06  1e+06  9e-02  2e+00  1e+00
-2.1573e+04 -2.2188e+05  2e+05  2e-02  4e-01  2e+03
-2.2194e+04 -4.1893e+04  2e+04  2e-03  4e-02  8e+02
-2.9585e+04 -4.0878e+04  1e+04  1e-03  2e-02  7e+02
-2.9189e+04 -3.9669e+04  1e+04  1e-03  2e-02  8e+02
-3.4535e+04 -3.6598e+04  2e+03  2e-04  5e-03  3e+02
-3.5108e+04 -3.5263e+04  2e+02  2e-05  4e-04  2e+01
-3.5179e+04 -3.5183e+04  5e+00  4e-07  1e-05  6e-01
-3.5181e+04 -3.5181e+04  3e-01  3e-08  6e-07  4e-02
-3.5181e+04 -3.5181e+04  7e-03  6e-10  1e-08  1e-03
Optimal solution found.
     pcost       dcost       gap    pres   dres   k/t
-2.5036e+04 -1.0400e+06  1e+06  9e-02  2e+00  1e+00
-2.0584e+04 -2.1714e+05  2e+05  2e-02  4e-01  2e+03
-2.0625e+04 -6.3033e+04  5e+04  4e-03  9e-02  2e+03
-2.6476e+04 -4.8834e+04  2e+04  2e-03  5e-02  9e+02
-2.2945e+04 -4.5906e+04  3e+04  2e-03  5e-02  1e+03
-3.0766e+04 -3.4993e+04  5e+03  4e-04  9e-03  3e+02
-3.3080e+04 -3.4042e+04  1e+03  9e-05  2e-03  1e+02
-3.3753e+04 -3.3912e+04  2e+02  1e-05  4e-04  2e+01
-3.3814e+04 -3.3875e+04  7e+01  6e-06  1e-04  8e+00
-3.3853e+04 -3.3864e+04  1e+01  1e-06  3e-05  1e+00
-3.3858e+04 -3.3860e+04  3e+00  2e-07  5e-06  3e-01
-3.3859e+04 -3.3860e+04  5e-01  4e-08  1e-06  6e-02
-3.3860e+04 -3.3860e+04  9e-02  7e-09  2e-07  1e-02
-3.3860e+04 -3.3860e+04  1e-02  1e-09  3e-08  2e-03
Optimal solution found.
     pcost       dcost       gap    pres   dres   k/t
-2.5036e+04 -1.0400e+06  1e+06  9e-02  2e+00  1e+00
-1.9324e+04 -2.1285e+05  2e+05  2e-02  4e-01  2e+03
-1.8790e+04 -7.9732e+04  7e+04  5e-03  1e-01  2e+03
-2.4973e+04 -5.1434e+04  3e+04  2e-03  5e-02  1e+03
-2.0623e+04 -4.7898e+04  3e+04  2e-03  6e-02  1e+03
-2.7990e+04 -3.3910e+04  7e+03  5e-04  1e-02  4e+02
-3.1514e+04 -3.2723e+04  1e+03  1e-04  3e-03  1e+02
-3.1808e+04 -3.2264e+04  5e+02  4e-05  1e-03  5e+01
-3.2154e+04 -3.2206e+04  6e+01  5e-06  1e-04  8e+00
-3.2175e+04 -3.2176e+04  1e+00  1e-07  3e-06  2e-01
-3.2175e+04 -3.2175e+04  2e-02  1e-09  3e-08  2e-03
Optimal solution found.
     pcost       dcost       gap    pres   dres   k/t
-2.5036e+04 -1.0400e+06  1e+06  9e-02  2e+00  1e+00
-1.7723e+04 -2.0947e+05  2e+05  2e-02  4e-01  2e+03
-1.6694e+04 -8.6451e+04  8e+04  6e-03  1e-01  3e+03
-2.3552e+04 -5.3687e+04  3e+04  3e-03  6e-02  1e+03
-1.8173e+04 -4.9316e+04  4e+04  3e-03  7e-02  2e+03
-2.5397e+04 -3.2444e+04  8e+03  6e-04  1e-02  4e+02
-2.9619e+04 -3.1102e+04  2e+03  1e-04  3e-03  1e+02
-2.9918e+04 -3.0180e+04  3e+02  3e-05  6e-04  3e+01
-3.0049e+04 -3.0109e+04  7e+01  6e-06  1e-04  8e+00
-3.0066e+04 -3.0067e+04  2e+00  1e-07  3e-06  2e-01
-3.0067e+04 -3.0067e+04  2e-02  2e-09  4e-08  2e-03
Optimal solution found.
     pcost       dcost       gap    pres   dres   k/t
-2.5036e+04 -1.0400e+06  1e+06  9e-02  2e+00  1e+00
-1.5697e+04 -2.0747e+05  2e+05  2e-02  4e-01  2e+03
-1.4245e+04 -8.5825e+04  8e+04  6e-03  1e-01  3e+03
-2.2001e+04 -5.5400e+04  4e+04  3e-03  7e-02  1e+03
-1.5264e+04 -4.9871e+04  4e+04  3e-03  7e-02  2e+03
-2.2694e+04 -3.0577e+04  9e+03  7e-04  2e-02  5e+02
-2.7295e+04 -2.8916e+04  2e+03  2e-04  4e-03  2e+02
-2.7643e+04 -2.7694e+04  6e+01  5e-06  1e-04  6e+00
-2.7667e+04 -2.7668e+04  9e-01  7e-08  2e-06  1e-01
-2.7667e+04 -2.7667e+04  4e-02  3e-09  8e-08  4e-03
-2.7667e+04 -2.7667e+04  1e-03  1e-10  3e-09  1e-04
Optimal solution found.
     pcost       dcost       gap    pres   dres   k/t
-2.5036e+04 -1.0400e+06  1e+06  9e-02  2e+00  1e+00
-1.3109e+04 -2.0737e+05  2e+05  2e-02  4e-01  2e+03
-1.1454e+04 -7.2650e+04  7e+04  5e-03  1e-01  3e+03
-2.0361e+04 -5.4117e+04  4e+04  3e-03  7e-02  2e+03
-1.1669e+04 -4.6731e+04  4e+04  3e-03  8e-02  2e+03
-1.9842e+04 -2.8472e+04  1e+04  8e-04  2e-02  8e+02
-2.4565e+04 -2.5126e+04  6e+02  5e-05  1e-03  8e+01
-2.4816e+04 -2.4837e+04  2e+01  2e-06  5e-05  3e+00
-2.4828e+04 -2.4829e+04  4e-01  3e-08  7e-07  5e-02
-2.4829e+04 -2.4829e+04  1e-02  8e-10  2e-08  1e-03
Optimal solution found.
     pcost       dcost       gap    pres   dres   k/t
-2.5036e+04 -1.0400e+06  1e+06  9e-02  2e+00  1e+00
-9.0847e+03 -2.1267e+05  2e+05  2e-02  4e-01  5e+03
-1.0389e+04 -3.5014e+04  3e+04  2e-03  5e-02  2e+03
-1.7236e+04 -2.6673e+04  1e+04  9e-04  2e-02  8e+02
-2.0210e+04 -2.5111e+04  6e+03  5e-04  1e-02  5e+02
-1.9975e+04 -2.2561e+04  3e+03  2e-04  6e-03  3e+02
-2.1282e+04 -2.1506e+04  3e+02  2e-05  5e-04  4e+01
-2.1352e+04 -2.1360e+04  9e+00  8e-07  2e-05  1e+00
-2.1355e+04 -2.1355e+04  2e-01  1e-08  3e-07  2e-02
-2.1355e+04 -2.1355e+04  1e-02  1e-09  3e-08  2e-03
Optimal solution found.
     pcost       dcost       gap    pres   dres   k/t
-2.5036e+04 -1.0400e+06  1e+06  9e-02  2e+00  1e+00
-3.5431e+03 -2.2228e+05  3e+05  2e-02  5e-01  8e+03
-6.9137e+03 -4.8671e+04  5e+04  4e-03  9e-02  2e+03
-1.6191e+04 -4.1305e+04  3e+04  2e-03  6e-02  3e+03
-1.1671e+04 -3.0609e+04  2e+04  2e-03  4e-02  3e+03
-1.5594e+04 -1.8081e+04  3e+03  2e-04  6e-03  4e+02
-1.6705e+04 -1.7279e+04  7e+02  6e-05  1e-03  9e+01
-1.7009e+04 -1.7054e+04  5e+01  4e-06  1e-04  7e+00
-1.7023e+04 -1.7033e+04  1e+01  1e-06  2e-05  2e+00
-1.7030e+04 -1.7030e+04  3e-01  2e-08  6e-07  4e-02
-1.7030e+04 -1.7030e+04  1e-02  1e-09  2e-08  2e-03
Optimal solution found.
     pcost       dcost       gap    pres   dres   k/t
-2.5036e+04 -1.0400e+06  1e+06  9e-02  2e+00  1e+00
 3.9318e+03 -2.3604e+05  3e+05  2e-02  5e-01  1e+04
-1.4954e+03 -5.7985e+04  6e+04  5e-03  1e-01  3e+03
-1.1589e+04 -5.3187e+04  5e+04  4e-03  9e-02  4e+03
 1.1802e+02 -4.2967e+04  5e+04  4e-03  1e-01  5e+03
-5.7757e+03 -3.6143e+04  4e+04  3e-03  7e-02  4e+03
-7.0719e+03 -1.6480e+04  1e+04  9e-04  2e-02  1e+03
-1.1415e+04 -1.5772e+04  5e+03  4e-04  1e-02  7e+02
-1.1436e+04 -1.1929e+04  6e+02  5e-05  1e-03  8e+01
-1.1639e+04 -1.1697e+04  7e+01  6e-06  1e-04  1e+01
-1.1643e+04 -1.1647e+04  5e+00  4e-07  1e-05  7e-01
-1.1644e+04 -1.1644e+04  2e-01  1e-08  3e-07  2e-02
-1.1644e+04 -1.1644e+04  5e-03  4e-10  9e-09  7e-04
Optimal solution found.
     pcost       dcost       gap    pres   dres   k/t
-2.5036e+04 -1.0400e+06  1e+06  9e-02  2e+00  1e+00
 1.3717e+04 -2.5324e+05  3e+05  2e-02  5e-01  2e+04
 5.2731e+03 -5.3658e+04  7e+04  5e-03  1e-01  4e+03
-5.6160e+03 -5.1746e+04  5e+04  4e-03  9e-02  5e+03
 1.1860e+03 -3.6886e+04  5e+04  4e-03  8e-02  5e+03
-3.0745e+03 -1.1299e+04  1e+04  8e-04  2e-02  1e+03
-5.0558e+03 -6.7644e+03  2e+03  2e-04  4e-03  3e+02
-4.9106e+03 -5.4214e+03  6e+02  5e-05  1e-03  9e+01
-5.0897e+03 -5.1052e+03  2e+01  2e-06  3e-05  3e+00
-5.0925e+03 -5.0940e+03  2e+00  1e-07  3e-06  3e-01
-5.0926e+03 -5.0928e+03  3e-01  2e-08  4e-07  4e-02
-5.0926e+03 -5.0927e+03  2e-02  2e-09  3e-08  3e-03
-5.0926e+03 -5.0926e+03  2e-03  1e-10  3e-09  3e-04
Optimal solution found.
     pcost       dcost       gap    pres   dres   k/t
-2.5036e+04 -1.0400e+06  1e+06  9e-02  1e+00  1e+00
 2.6081e+04 -2.7265e+05  4e+05  3e-02  4e-01  2e+04
 1.3758e+04 -4.7136e+04  7e+04  6e-03  9e-02  5e+03
 1.7947e+03 -4.6754e+04  6e+04  5e-03  7e-02  5e+03
 6.3636e+03 -1.7193e+04  3e+04  2e-03  4e-02  3e+03
 3.3465e+03  1.9287e+03  2e+03  1e-04  2e-03  2e+02
 2.9456e+03  2.7883e+03  2e+02  2e-05  3e-04  3e+01
 2.9495e+03  2.9464e+03  4e+00  3e-07  5e-06  5e-01
 2.9493e+03  2.9492e+03  7e-02  6e-09  9e-08  1e-02
 2.9493e+03  2.9493e+03  4e-03  3e-10  6e-09  6e-04
 2.9493e+03  2.9493e+03  8e-05  6e-12  1e-10  1e-05
Optimal solution found.
     pcost       dcost       gap    pres   dres   k/t
-2.5036e+04 -1.0400e+06  1e+06  9e-02  1e+00  1e+00
 4.1125e+04 -2.9270e+05  4e+05  3e-02  4e-01  3e+04
 2.4352e+04 -3.9692e+04  8e+04  6e-03  8e-02  6e+03
 1.0997e+04 -4.0126e+04  6e+04  5e-03  6e-02  6e+03
 1.4974e+04  4.6920e+02  2e+04  1e-03  2e-02  2e+03
 1.3158e+04  1.2393e+04  9e+02  8e-05  1e-03  1e+02
 1.2976e+04  1.2930e+04  6e+01  5e-06  6e-05  7e+00
 1.2973e+04  1.2972e+04  1e+00  8e-08  1e-06  1e-01
 1.2973e+04  1.2973e+04  3e-02  2e-09  3e-08  3e-03
 1.2973e+04  1.2973e+04  1e-03  8e-11  1e-09  1e-04
Optimal solution found.
     pcost       dcost       gap    pres   dres   k/t
-2.5036e+04 -1.0400e+06  1e+06  9e-02  9e-01  1e+00
 5.8792e+04 -3.1160e+05  5e+05  3e-02  3e-01  4e+04
 3.7541e+04 -3.0638e+04  8e+04  6e-03  6e-02  7e+03
 2.2459e+04 -3.1574e+04  7e+04  5e-03  5e-02  7e+03
 2.6862e+04  1.6992e+04  1e+04  1e-03  1e-02  2e+03
 2.5717e+04  2.5172e+04  7e+02  5e-05  5e-04  9e+01
 2.5580e+04  2.5558e+04  3e+01  2e-06  2e-05  4e+00
 2.5580e+04  2.5580e+04  5e-01  4e-08  4e-07  7e-02
 2.5580e+04  2.5580e+04  2e-02  2e-09  2e-08  3e-03
Optimal solution found.
     pcost       dcost       gap    pres   dres   k/t
-2.5036e+04 -1.0400e+06  1e+06  9e-02  7e-01  1e+00
 7.8927e+04 -3.2765e+05  5e+05  4e-02  3e-01  4e+04
 5.3941e+04 -1.8555e+04  9e+04  7e-03  5e-02  8e+03
 3.6818e+04 -1.9892e+04  7e+04  5e-03  4e-02  8e+03
 4.2356e+04  3.5267e+04  9e+03  7e-04  6e-03  1e+03
 4.1643e+04  4.1157e+04  6e+02  5e-05  4e-04  8e+01
 4.1519e+04  4.1504e+04  2e+01  1e-06  1e-05  3e+00
 4.1519e+04  4.1518e+04  4e-01  3e-08  2e-07  5e-02
 4.1519e+04  4.1519e+04  1e-02  1e-09  8e-09  2e-03
Optimal solution found.
     pcost       dcost       gap    pres   dres   k/t
-2.5036e+04 -1.0400e+06  1e+06  9e-02  5e-01  1e+00
 1.0142e+05 -3.3927e+05  6e+05  4e-02  3e-01  5e+04
 7.4356e+04 -1.4986e+03  1e+05  7e-03  4e-02  9e+03
 5.4993e+04 -3.1290e+03  8e+04  6e-03  3e-02  8e+03
 6.2235e+04  5.7245e+04  7e+03  5e-04  3e-03  8e+02
 6.1834e+04  6.1425e+04  5e+02  4e-05  3e-04  7e+01
 6.1731e+04  6.1721e+04  1e+01  1e-06  7e-06  2e+00
 6.1731e+04  6.1731e+04  3e-01  2e-08  1e-07  4e-02
 6.1731e+04  6.1731e+04  7e-03  6e-10  4e-09  1e-03
Optimal solution found.
     pcost       dcost       gap    pres   dres   k/t
-2.5036e+04 -1.0400e+06  1e+06  9e-02  4e-01  1e+00
 1.2633e+05 -3.4516e+05  7e+05  4e-02  2e-01  5e+04
 9.9866e+04  2.3018e+04  1e+05  7e-03  4e-02  9e+03
 7.8373e+04  2.1508e+04  8e+04  6e-03  3e-02  8e+03
 8.7651e+04  8.4467e+04  4e+03  3e-04  2e-03  5e+02
 8.7481e+04  8.7186e+04  4e+02  3e-05  1e-04  5e+01
 8.7412e+04  8.7406e+04  9e+00  7e-07  3e-06  1e+00
 8.7412e+04  8.7412e+04  1e-01  1e-08  5e-08  2e-02
 8.7412e+04  8.7412e+04  3e-03  2e-10  1e-09  4e-04
Optimal solution found.
     pcost       dcost       gap    pres   dres   k/t
-2.5036e+04 -1.0400e+06  1e+06  9e-02  3e-01  1e+00
 1.5403e+05 -3.4417e+05  7e+05  5e-02  2e-01  6e+04
 1.3196e+05  5.8089e+04  1e+05  7e-03  3e-02  1e+04
 1.0908e+05  5.7728e+04  7e+04  5e-03  2e-02  8e+03
 1.2014e+05  1.1849e+05  2e+03  2e-04  6e-04  3e+02
 1.2011e+05  1.1995e+05  2e+02  2e-05  6e-05  3e+01
 1.2008e+05  1.2008e+05  4e+00  3e-07  1e-06  6e-01
 1.2008e+05  1.2008e+05  5e-02  4e-09  1e-08  7e-03
Optimal solution found.
     pcost       dcost       gap    pres   dres   k/t
-2.5036e+04 -1.0400e+06  2e+06  9e-02  3e-01  1e+00
 1.8530e+05 -3.3517e+05  8e+05  5e-02  1e-01  6e+04
 1.7275e+05  1.0738e+05  1e+05  6e-03  2e-02  9e+03
 1.5019e+05  1.0978e+05  6e+04  4e-03  1e-02  7e+03
 1.6159e+05  1.6078e+05  1e+03  8e-05  2e-04  2e+02
 1.6166e+05  1.6159e+05  1e+02  7e-06  2e-05  1e+01
 1.6166e+05  1.6166e+05  2e+00  1e-07  4e-07  3e-01
 1.6166e+05  1.6166e+05  3e-02  2e-09  5e-09  4e-03
Optimal solution found.
     pcost       dcost       gap    pres   dres   k/t
-2.5036e+04 -1.0400e+06  2e+06  9e-02  2e-01  1e+00
 2.2133e+05 -3.1691e+05  9e+05  5e-02  1e-01  7e+04
 2.2510e+05  1.6024e+05  1e+05  6e-03  2e-02  1e+04
 2.0806e+05  1.8950e+05  3e+04  2e-03  4e-03  4e+03
 2.1446e+05  2.1373e+05  1e+03  7e-05  2e-04  1e+02
 2.1459e+05  2.1452e+05  1e+02  7e-06  2e-05  1e+01
 2.1461e+05  2.1460e+05  2e+00  1e-07  3e-07  2e-01
 2.1461e+05  2.1461e+05  4e-02  3e-09  7e-09  6e-03
Optimal solution found.

In [38]:

print(raps)

Out[38]:

[0.1, 0.12742749857031338, 0.16237767391887217, 0.20691380811147897, 0.26366508987303583, 0.33598182862837817, 0.42813323987193935, 0.5455594781168519, 0.6951927961775606, 0.8858667904100825, 1.1288378916846888, 1.438449888287663, 1.8329807108324356, 2.3357214690901227, 2.9763514416313175, 3.79269019073225, 4.832930238571752, 6.158482110660264, 7.847599703514611, 10.0]

In [28]:

# Plot trade-off curve and optimal allocations.
plt.figure(figsize=(10,6))
plt.figure(1, facecolor='w')
plt.plot(risks, returns)
plt.xlabel('standard deviation')
plt.ylabel('expected return')
plt.axis([24000, 40000, 30000, 42000])
plt.show()

Out[28]:

In [0]:

In [0]:

In [0]:

Covariance matrix $\Sigma$ and its Cholesky decomposition $U$

Sample portfolio

Problem 1

Optimization constraint

Constraint 0 (Linear)

Constraint 1 (Conic)

Linear constraint

Optimizer call

Problem 2

Problem 3

Product

Resources

Company

Covariance matrix Σ\SigmaΣ and its Cholesky decomposition UUU

Sample portfolio

Problem 1

Optimization constraint

Constraint 0 (Linear)

Constraint 1 (Conic)

Linear constraint

Optimizer call

Problem 2

Problem 3

Covariance matrix $\Sigma$ and its Cholesky decomposition $U$