本文记录在可投资集已知预期收益率和预期协方差矩阵的条件下，使用程序工具求解最优投资组合的方法。

问题描述

构建最优投资组合是一个带约束的优化问题，优化目标是最大化投资组合的夏普比率，而约束是资产权重相加等于1。

Python实现

# construct optimized portfolioimport numpy as npimport pandas as pdimport scipy.optimize as sco

fcst_style_sharpe = pd.read_csv('fcst_style_sharpe.csv')style_cov = pd.read_csv('style_cov.csv')

selected_col = ['mkt','smb','hml','vmg','rmw','gms','pmo','mom']noa = len(selected_col)

style_ret = fcst_style_sharpe['fcst_ret'] #* 100

weights = np.random.random(noa)weights /= np.sum(weights)

def port_ret(weights):    return np.sum(style_ret * weights) * 252

def port_vol(weights):    return np.sqrt(np.dot(weights.T, np.dot(style_cov * 252, weights)))

def min_func_sharpe(weights):    return -port_ret(weights) / port_vol(weights)

# maximize sharpe ratioeweights = np.array(noa * [1. / noa,])#bnds = tuple((0, 1) for x in range(noa))bnds = tuple((-2, 2) for x in range(noa))cons = ({'type': 'eq', 'fun': lambda x:  np.sum(x) - 1})

opts = sco.minimize(min_func_sharpe, eweights,                    method='SLSQP', bounds=bnds,                    constraints=cons)

opt_weight = opts['x']opt_weight.round(4)

opts_sharpe = port_ret(opt_weight) / port_vol(opt_weight)opts_sharpe

# plot the optimized portfoliofrom lets_plot import *LetsPlot.setup_html()

fzoo = pd.read_csv('data/fzoo.csv')

opt_style = fzoo[['trading_date','mkt']].copy()opt_style.columns = ['trading_date','style']opt_style['style'] = np.dot(fzoo[selected_col],opt_weight)opt_style['style_cum'] = np.log1p(opt_style['style']/100).cumsum()

x = pd.to_datetime(opt_style['trading_date'])[-1000:]y = opt_style['style_cum'][-1000:]/opt_style['style_cum'].iloc[-1000]ggplot({'date': x, 'port': y}, aes(x='date', y='port')) + geom_line()

# minimize volatilityoptv = sco.minimize(port_vol, eweights,                    method='SLSQP', bounds=bnds,                    constraints=cons)

bnds = tuple((0, 1) for x in weights)

cons = ({'type': 'eq', 'fun': lambda x:  port_ret(x) - tret},        {'type': 'eq', 'fun': lambda x:  np.sum(x) - 1})

%timetrets = np.linspace(0.05, 0.5, 100)tvols = []for tret in trets:    res = sco.minimize(port_vol, eweights, method='SLSQP',                       bounds=bnds, constraints=cons)    tvols.append(res['fun'])tvols = np.array(tvols)

# plot the frontfrom pylab import mpl, pltplt.style.use('seaborn-v0_8-whitegrid')mpl.rcParams['font.family'] = 'serif'%matplotlib inline

prets = []pvols = []for p in range (5000):    weights = np.random.random(noa)    weights /= np.sum(weights)    prets.append(port_ret(weights))    pvols.append(port_vol(weights))prets = np.array(prets)pvols = np.array(pvols)

plt.figure(figsize=(10, 6))plt.scatter(pvols, prets, c=prets / pvols,            marker='o', cmap='coolwarm')plt.xlabel('expected volatility')plt.ylabel('expected return')plt.colorbar(label='Sharpe ratio');

tvols = np.array(tvols)

plt.figure(figsize=(10, 6))plt.scatter(pvols, prets, c=prets / pvols,            marker='.', alpha=0.8, cmap='coolwarm')plt.plot(tvols, trets, 'b', lw=4.0)plt.plot(port_vol(opts['x']), port_ret(opts['x']),         'y*', markersize=15.0)plt.plot(port_vol(optv['x']), port_ret(optv['x']),         'r*', markersize=15.0)plt.xlabel('expected volatility')plt.ylabel('expected return')plt.colorbar(label='Sharpe ratio')