这个作业是优化投资组合中资产的相关权重以达到最大化收益
Guidance of Portfolio Rebalancing (MANG 2004)
1个步骤投资组合再平衡
就像我在教程中所说的，与投资组合中的资产相关的权重与初始权重不同
由于资产价格的变化，随着时间的流逝而分配。重新平衡投资组合意味着我们需要调整
由于风险控制或其他动机，权重恢复到最佳权重。通常有三个步骤
在此过程中：
•确定最佳权重。 （许多方法可以做到这一点，其中之一是通过MV优化和
已在“教程－作业－2”中进行了说明。）
•确定每个组成资产的当前头寸。
•相应地调整投资组合。
本说明的重点是找出资产偏离最佳权重的程度（即
第二步）。如果我们知道此信息，则可以出售（购买）超重（称重不足）的商品，
投资组合余额再次出现（在本说明中，我假设我们不会在投资组合中向我们的投资组合添加/提取资金）
重新平衡一天）。
2问题设置
为了与我们的教程保持一致，我将假设我们的投资组合中有三个资产，资产1、2和3。
然后可以通过下表总结该问题：
表1：问题设置
资产价格分配的权重投资组合价值
Pt,1 Pt,2 Pt,3 Wt,1 Wt,2 Wt,3 Vt
Pt+1,1 Pt+1,2 Pt+1,3 Wt+1,1 Wt+1,2 Wt+1,3 Vt+1
Pt,1, Pt,2, and Pt,3 are the prices for Asset 1, 2, and 3 at time t; Pt+1,1, Pt+1,2, and Pt+1,3 are the prices
for Assets 1, 2, and 3 at time t + 1; Wt,1, Wt,2, and Wt,3 are the weights we assigned to the assets at time
t (optimal/target weights); Wt+1,1, Wt+1,2, and Wt+1,3 are the weights associated with the assets at time
t + 1. These weights at t + 1 are different from those at t due to the price variation of the assets. Finally,
Vt and Vt+1 denote the value of our portfolio at times t and t + 1.
So the problem now is to solve the values for Wt+1,1, Wt+1,2, Wt+1,3, and Vt+1 given other entries of the
table.
3解决方案
尽管资产价格会随时间变化，但我们在每种资产中投资的数量（例如，股份数量）
保持稳定，让我们用Q1表示资产1的数量：
Q1 =
Wt,1Vt
Pt,1
where Wt,1Vt is the amount of money we invested in Asset 1 at time t. If we divide it by the share price, we
have the number of shares.
Therefore, Asset 1 we hold in our portfolio is worth Pt+1,1Q1 = Pt+1,1
Wt,1Vt
Pt,1
at time t + 1. This is true
for all the assets in our portfolio. So the whole portfolio at t + 1 is worth:
Vt+1 = Pt+1,1
Wt,1Vt
Pt,1
+ Pt+1,2
Wt,2Vt
Pt,2
+ Pt+1,3
Wt,3Vt
Pt,3
Then we can easily solve the values for Wt+1,1, Wt+1,2, and Wt+1,3:
Wt+1,1 =
Q1Pt+1,1
Vt+1
=
Wt,1Vt
Pt,1
Pt+1,1
Pt+1,1
Wt,1Vt
Pt,1
+ Pt+1,2
Wt,2Vt
Pt,2
+ Pt+1,3
Wt,3Vt
Pt,3
Wt+1,2 =
Q2Pt+1,2
Vt+1
=
Wt,2Vt
Pt,2
Pt+1,2
Pt+1,1
Wt,1Vt
Pt,1
+ Pt+1,1
Wt,2Vt
Pt,2
+ Pt+1,2
Wt,3Vt
Pt,3
Wt+1,3 =
Q3Pt+1,3
Vt+1
=
Wt,3Vt
Pt,3
Pt+1,3
Pt+1,1
Wt,1Vt
Pt,1
+ Pt+1,2
Wt,2Vt
Pt,2
+ Pt+1,3
Wt,3Vt
Pt,3
Now we have got values for all entries in the table. To rebalance the portfolio we just need to sell the
overweighted and buy the underweighted by the difference between current position and optimal position.
4 Transaction costs
Another issue you may want to consider is the transaction costs incurred during the rebalancing. Since the
transaction costs are paid regardless of trading direction, we just need to compute the turnover in absolute
term and multiply that with overall value of our portfolio and the transaction cost ratio. That is:
T C = (|Wt,1 − Wt+1,1| + |Wt,2 − Wt+1,2| + |Wt,3 − Wt+1,3|) × Vt+1 × 0.5%
2