간단한 최적화 예#1
In [1]:
from scipy.optimize import minimize
import numpy as np
In [3]:
def objective1(x):
return x + 1
def constraint(x):
return x - 3
x0= [ -1 ]
b = ( -1, 6 )
bnd = ( b, )
con = {'type':'ineq','fun':constraint}
sol= minimize(objective1, x0, method='SLSQP', bounds=bnd, constraints=con)
In [4]:
print(sol.x)
[3.]
간단한 최적화 예#2
In [6]:
def objective2(x):
x1=x[0]
x2=x[1]
x3=x[2]
x4=x[3]
return x1*x4*(x1+x2+x3)+x3
def constraint1(x):
return x[0]*x[1]*x[2]*x[3]-25
def constraint2(x):
sum_sq = np.sum(np.square(x))
return sum_sq-40
x0 = [ 1, 5, 5, 1 ]
b = ( 1, 5 )
bnds = ( b, b, b, b )
con1 = { 'type':'ineq','fun':constraint1 }
con2 = { 'type':'eq','fun':constraint2 }
cons = [ con1, con2 ]
sol = minimize(objective2, x0, method='SLSQP',bounds=bnds,constraints=cons)
In [7]:
print(sol)
fun: 17.01401724556073
jac: array([14.57227039, 1.37940764, 2.37940764, 9.56415081])
message: 'Optimization terminated successfully.'
nfev: 30
nit: 5
njev: 5
status: 0
success: True
x: array([1. , 4.74299607, 3.82115466, 1.37940764])
3장 평균-분산 포트폴리오 이론에서 이어지는 내용
In [8]:
import numpy as np
import pandas as pd
from pandas_datareader import data as web
import matplotlib.pyplot as plt
import matplotlib as mpl
tickers = ['AAPL', 'F', 'AMZN', 'GE', 'TSLA']
pxclose = pd.DataFrame()
for t in tickers:
pxclose[t] = web.DataReader(t, data_source='yahoo',start='01-01-2019', end='31-12-2019')['Adj Close']
In [9]:
ret_daily = pxclose.pct_change()
ret_annual = ret_daily.mean() * 250
cov_daily = ret_daily.cov()
cov_annual = cov_daily * 250
In [10]:
ret_daily.head()
Out[10]:
| AAPL | F | AMZN | GE | TSLA | |
|---|---|---|---|---|---|
| Date | |||||
| 2019-01-02 | NaN | NaN | NaN | NaN | NaN |
| 2019-01-03 | -0.099607 | -0.01519 | -0.025242 | 0.001242 | -0.031472 |
| 2019-01-04 | 0.042689 | 0.03856 | 0.050064 | 0.021092 | 0.057697 |
| 2019-01-07 | -0.002226 | 0.02599 | 0.034353 | 0.061968 | 0.054361 |
| 2019-01-08 | 0.019063 | 0.00965 | 0.016612 | -0.020595 | 0.001164 |
In [11]:
print(cov_annual.head())
AAPL F AMZN GE TSLA AAPL 0.068048 0.021624 0.035169 0.032125 0.042293 F 0.021624 0.074516 0.022533 0.039544 0.016574 AMZN 0.035169 0.022533 0.051708 0.030574 0.032658 GE 0.032125 0.039544 0.030574 0.161524 0.047065 TSLA 0.042293 0.016574 0.032658 0.047065 0.237425
In [12]:
p_returns = []
p_volatility = []
p_weights = []
n_assets = len(tickers)
n_ports = 30000
for s in range(n_ports):
wgt = np.random.random(n_assets)
wgt /= np.sum(wgt)
ret = np.dot(wgt, ret_annual)
vol = np.sqrt(np.dot(wgt.T, np.dot(cov_annual, wgt)))
p_returns.append(ret)
p_volatility.append(vol)
p_weights.append(wgt)
In [13]:
p_volatility = np.array(p_volatility)
p_returns = np.array(p_returns)
colors = p_returns/p_volatility
plt.style.use('seaborn')
plt.scatter(p_volatility, p_returns, c=colors, marker='o', cmap=mpl.cm.jet)
plt.xlabel('Volatility (Std. Deviation)')
plt.ylabel('Expected Returns')
plt.title('Efficient Frontier')
plt.show()
최소분산 포트폴리오(MVP, Minimum Volatility Portfolio)
In [14]:
from scipy.optimize import minimize
def obj_variance(weights, cov):
return np.sqrt(weights.T @ covmat @ weights)
In [15]:
covmat=cov_daily*250
weights =np.array([0.2, 0.2, 0.2, 0.2, 0.2])
bnds = ((0,1), (0,1), (0,1), (0,1), (0,1))
cons = ({'type': 'eq', 'fun': lambda x: np.sum(x) - 1})
res = minimize(obj_variance, weights,(covmat), method='SLSQP', bounds=bnds, constraints=cons)
In [16]:
print(res)
fun: 0.19535983060370768
jac: array([0.19517164, 0.19553577, 0.19532278, 0.19527321, 0.19548151])
message: 'Optimization terminated successfully.'
nfev: 63
nit: 9
njev: 9
status: 0
success: True
x: array([0.20785615, 0.29963679, 0.42386232, 0.02998727, 0.03865747])
In [17]:
p_returns = []
p_volatility = []
p_weights = []
n_assets = len(tickers)
n_ports = 30000
for s in range(n_ports):
wgt = np.random.random(n_assets)
wgt /= np.sum(wgt)
ret = np.dot(wgt, ret_annual)
vol = np.sqrt(np.dot(wgt.T, np.dot(cov_annual, wgt)))
p_returns.append(ret)
p_volatility.append(vol)
p_weights.append(wgt)
In [18]:
rets = np.sum(ret_daily.mean() * res['x']) * 250
vol = np.sqrt(res['x'].T @ covmat @ res['x'])
p_volatility = np.array(p_volatility)
p_returns = np.array(p_returns)
colors = p_returns/p_volatility
plt.style.use('seaborn')
plt.scatter(p_volatility, p_returns, c=colors, marker='o', cmap=mpl.cm.jet)
plt.scatter(vol, rets, marker="*", s=500, alpha=1.0)
plt.xlabel('Volatility (Std. Deviation)')
plt.ylabel('Expected Returns')
plt.title('Efficient Frontier')
plt.show()
Sharpe ratio 최적화
In [19]:
from scipy.optimize import minimize
def obj_sharpe(weights, returns, covmat, rf):
ret = np.dot(weights, returns)
vol = np.sqrt(np.dot(weights.T, np.dot(covmat, weights)))
return 1/((ret-rf)/np.sqrt(vol))
In [20]:
n_assets = len(tickers)
covmat=cov_daily*250
rf = 0.01
weights = np.ones([n_assets])/n_assets
bnds = tuple((0., 1.) for i in range(n_assets))
cons = ({'type': 'eq', 'fun': lambda w: np.sum(w) - 1})
res = minimize(obj_sharpe, weights,(ret_annual, covmat, rf), method='SLSQP', bounds=bnds, constraints=cons)
In [21]:
print(res)
fun: 0.7766010981552891
jac: array([-0.40010898, -0.20859025, -0.04480552, -0.34614637, -0.2510052 ])
message: 'Optimization terminated successfully.'
nfev: 21
nit: 3
njev: 3
status: 0
success: True
x: array([1.00000000e+00, 1.30104261e-16, 0.00000000e+00, 4.85722573e-17,
1.04083409e-17])
In [22]:
p_returns = []
p_volatility = []
p_weights = []
n_assets = len(tickers)
n_ports = 30000
for s in range(n_ports):
wgt = np.random.random(n_assets)
wgt /= np.sum(wgt)
ret = np.dot(wgt, ret_annual)
vol = np.sqrt(np.dot(wgt.T, np.dot(cov_annual, wgt)))
p_returns.append(ret)
p_volatility.append(vol)
p_weights.append(wgt)
In [23]:
rets = np.sum(ret_daily.mean() * res['x']) * 250
vol = np.sqrt(res['x'].T @ covmat @ res['x'])
p_volatility = np.array(p_volatility)
p_returns = np.array(p_returns)
colors = p_returns/p_volatility
plt.style.use('seaborn')
plt.scatter(p_volatility, p_returns, c=colors, marker='o', cmap=mpl.cm.jet)
plt.scatter(vol, rets, marker="*", s=500, alpha=1.0)
plt.xlabel('Volatility (Std. Deviation)')
plt.ylabel('Expected Returns')
plt.title('Efficient Frontier')
plt.show()
