02 现代投资组合理论

来源: 2022 CFA Program curriculum Reading 49

• 生成权向量

w1=[0.001*n for n in range(1001)]
w=[[w, .001*n, 1-w-.001*n] 
   for n in range(1001) 
   for w in w1 if w+.001*n<=1]

• 生成期望回报率向量与方差协方差矩阵

ret=stk.loc[:,['pct_chg_x','pct_chg_y',
               'pct_chg']].mean().to_numpy()
cov=stk.loc[:,['pct_chg_x','pct_chg_y',
               'pct_chg']].cov().to_numpy()

• 计算组合回报率与波动率

portfolio_return=[]
portfolio_risk=[]
for weight in w:
    #w=np.array([w1[n],w2[n],w3[n]])
    ww=np.array(weight)
    portfolio_return.append(ww.dot(ret.T))
    portfolio_risk.append(np.sqrt(ww.dot(cov).dot(ww.T)))

• 绘图

fig = plt.figure()
ax = plt.axes()
ax.scatter(portfolio_risk, portfolio_return)

• 导入必要的库

import random
import pandas as pd

• 准备数据

df=pd.read_csv('data/TRD_Dalyr.csv',sep='\t')
tic_vec=df.Stkcd.unique()
tic_vec=pd.DataFrame(tic_vec,columns=['tic'])

• 设定实验参数

n=50
sample_size=1000
x=np.linspace(1,n,n)

• 生成数据

y=[]
for i in x:
    z=[]
    #smpl=pd.DataFrame({'Trddt':[],'Dretwd':[]})
    j=0
    while j < sample_size:
        # j-th sampling
        smpl_tic=tic_vec.sample(int(i),replace=True)
        smpl=pd.DataFrame({'Trddt':[],'Dretwd':[]})
        k=0
        while k<i:
            smpl=smpl.merge(df[['Trddt','Dretwd']]
                            [df.Stkcd==smpl_tic.tic.iloc[k]],
                            how='outer',on='Trddt')
            k+=1
        smpl.fillna(0)
        z.append(np.std(smpl.mean(axis=1,numeric_only=True)))
        j+=1
        #smpl.pivot_table()
    y.append(np.mean(z))

• 绘图

%matplotlib inline
import matplotlib.pyplot as plt
plt.style.use('seaborn-whitegrid')
fig = plt.figure()
ax = plt.axes()
ax.plot(x, y);

来源: 2022 CFA Program curriculum Reading 49

假设投资组合由一个无风险资产和一个风险资产组成。组合的收益和方差为：

由此，我们可以得到资本配置线(CAL),

CAL类似消费者均衡分析中的预算线

来源: 2022 CFA Program curriculum Reading 49

• Generate data for long only portfolio optimization.

import numpy as np
import scipy.sparse as sp
np.random.seed(1)
n = 10
mu = np.abs(np.random.randn(n, 1))
Sigma = np.random.randn(n, n)
Sigma = Sigma.T.dot(Sigma)

• Long only portfolio optimization.

import cvxpy as cp
w = cp.Variable(n)
gamma = cp.Parameter(nonneg=True)
ret = mu.T@w 
risk = cp.quad_form(w, Sigma)
prob = cp.Problem(cp.Maximize(ret - gamma*risk), 
               [cp.sum(w) == 1, 
                w >= 0])

• Compute trade-off curve.

SAMPLES = 100
risk_data = np.zeros(SAMPLES)
ret_data = np.zeros(SAMPLES)
gamma_vals = np.logspace(-2, 3, num=SAMPLES)
for i in range(SAMPLES):
    gamma.value = gamma_vals[i]
    prob.solve()
    risk_data[i] = cp.sqrt(risk).value
    ret_data[i] = ret.value

• Plot long only trade-off curve.

import matplotlib.pyplot as plt
%matplotlib inline
%config InlineBackend.figure_format = 'svg'

markers_on = [29, 40]
fig = plt.figure()
ax = fig.add_subplot(111)
plt.plot(risk_data, ret_data, 'g-')
for marker in markers_on:
    plt.plot(risk_data[marker], ret_data[marker], 'bs')
    ax.annotate(r"$\gamma = %.2f$" % gamma_vals[marker], 
                xy=(risk_data[marker]+.08, ret_data[marker]-.03))
for i in range(n):
    plt.plot(cp.sqrt(Sigma[i,i]).value, mu[i], 'ro')
plt.xlabel('Standard deviation')
plt.ylabel('Return')
plt.show()

• plot return distributions for two points on the trade-off curve.

import scipy.stats as spstats

plt.figure()
for midx, idx in enumerate(markers_on):
    gamma.value = gamma_vals[idx]
    prob.solve()
    x = np.linspace(-2, 5, 1000)
    plt.plot(x, spstats.norm.pdf(x, ret.value, risk.value), 
             label=r"$\gamma = %.2f$" % gamma.value)

plt.xlabel('Return')
plt.ylabel('Density')
plt.legend(loc='upper right')
plt.show()

课后阅读与练习

• 课后阅读：教材第十三章相关内容（pp221-233）
• 练习
  • 教材pp232：1-8
  • 在线练习

拓展学习

• Roy安全第一准则及其发展
• Black-Litterman模型及其发展
• 风险平价(Risk Parity)策略