如何解决投资组合优化的方程组和约束条件？

Question

我有一个 DataFrame 如下：

Name  Volatility    Return
a      0.0243        0.212
b      0.0321        0.431
c      0.0323        0.443
d      0.0391        0.2123
e      0.0433        0.3123

我想要 0.035 的 Volatility 和该波动率的最大化 Return。

也就是说，我想在新的 Df 中 Name 以及将在我的 portfolio 中提供最大 [=14] 的资产百分比=] 对于 Volatility 等于 0.035.

因此，我需要求解具有多个条件的方程组，以获得固定结果（Volatility == 0.035）的最佳解（最高Return）。

条件是：

每项资产的权重都在 0 到 1 之间。
权重之和为1
每种资产的权重乘以波动率的总和是"Desired Volatility"。
每个资产的权重乘以 return 的总和就是 "Total Return"。这应该最大化。

Answer 1

这是一种使用 Z3Py, an open source SAT/SMT 求解器的方法。在 SAT/SMT 求解器中，您可以将代码编写为条件列表，程序会找到一个最佳解决方案（或者当 Z3 用作求解器时，只是满足所有条件的解决方案）。

最初 SAT 求解器仅使用纯布尔表达式，但现代 SAT/SMT 求解器还允许固定位和无限整数、分数、实数甚至函数作为中心变量。

为了将给定的方程式写入 Z3，它们会按字面意思转换为 Z3 表达式。下面的代码注释了每个步骤。

import pandas as pd
from z3 import *

DesiredVolatility = 0.035
df = pd.DataFrame(columns=['Name', 'Volatility', 'Return'],
                  data=[['a', 0.0243, 0.212],
                        ['b', 0.0321, 0.431],
                        ['c', 0.0323, 0.443],
                        ['d', 0.0391, 0.2123],
                        ['e', 0.0433, 0.3123]])

# create a Z3 instance to optimize something
s = Optimize()
# the weight of each asset, as a Z3 variable
W = [Real(row.Name) for row in df.itertuples()]
# the total volatility
TotVol = Real('TotVol')
# the total return, to be maximized
TotReturn = Real('TotReturn')

# weights between 0 and 1, and sum to 1
s.add(And([And(w >= 0, w <= 1) for w in W]))
s.add(Sum([w for w in W]) == 1)
# the total return is calculated as the weighted sum of the asset returns
s.add(TotReturn == Sum([w * row.Return for w, row in zip(W, df.itertuples())]))
# the volatility is calculated as the weighted sum of the asset volatility
s.add(TotVol == Sum([w * row.Volatility for w, row in zip(W, df.itertuples())]))
# the volatility should be equal to the desired volatility
s.add(TotVol == DesiredVolatility)
# we're maximizing the total return
h1 = s.maximize(TotReturn)
# we ask Z3 to do its magick
res = s.check()
# we check the result, hoping for 'sat': all conditions satisfied, a maximum is found
if res == sat:
    s.upper(h1)
    m = s.model()
    #for w in W:
    #    print(f'asset {w}): {m[w]} = {m[w].numerator_as_long() / m[w] .denominator_as_long() : .6f}')
    # output the total return
    print(f'Total Return: {m[TotReturn]} = {m[TotReturn].numerator_as_long() / m[TotReturn] .denominator_as_long() :.6f}')
    # get the proportions out of the Z3 model
    proportions = [m[w].numerator_as_long() / m[w] .denominator_as_long() for w in W]
    # create a dataframe with the result
    df_result = pd.DataFrame({'Name': df.Name, 'Proportion': proportions})
    print(df_result)
else:
    print("No satisfiable solution found")

结果：

Total Return: 452011/1100000 = 0.410919
  Name  Proportion
0    a    0.000000
1    b    0.000000
2    c    0.754545
3    d    0.000000
4    e    0.245455

您可以轻松添加其他约束，例如 "no asset can have more than 30% of the total":

# change 
s.add(And([And(w >= 0, w <= 1) for w in W]))`
# to
s.add(And([And(w >= 0, w <= 0.3) for w in W]))`

这将导致：

Total Return: 558101/1480000 = 0.377095
  Name  Proportion
0    a    0.082432
1    b    0.300000
2    c    0.300000
3    d    0.017568
4    e    0.300000

如何解决投资组合优化的方程组和约束条件？

How to solve a system of equations and constraints for portfolio optimization?

python

solver

z3py