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好好学习向前进 · 2024年06月11日

关于MVO的E(R)和reverse的E(R)

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NO.PZ201803130100000402

问题如下:

Contrast, using the information provided above, the results of a reverse optimization approach with that of the MVO approach for each of the following:

ii. The values of the expected returns for US equities and global bonds. Justify your response.

选项:

解释:

The values of the expected returns for US equities and global bonds
For the reverse optimization approach, the expected returns of asset classes are the outputs of optimization with the market capitalization weights, covariances, and the risk aversion coefficient used as inputs.
In contrast, for the MVO approach, the expected returns of asset classes are inputs to the optimization, with the expected returns generally estimated using historical data.
The computed values for the expected returns for global bonds and US equities using the reverse optimization method are 5.3% and 9.7%, respectively.
In contrast, the expected return estimates used in the MVO approach from Exhibit 1 for global bonds and US equities are 4.7% and 8.6%, respectively.


The output of the reverse optimization method are optimized returns which are viewed as unobserved equilibrium or imputed returns. The equilibrium returns are essentially long-run capital market returns provided by each asset class and are strongly linked to CAPM. In contrast, the expected returns in the MVO approach are generally forecasted based on historical data and are used as inputs along with covariances and the risk aversion
  coefficient in the optimization. The reverse-optimized returns are calculated using a CAPM approach. The return on an asset class using the CAPM approach is calculated as follows:
Return on Asset Class = Risk-Free Rate + (Beta) (Market Risk Premium)


Therefore, the implied returns for global bonds and US equities are calculated as follows:
Return on Global Bonds = 2.0% + (0.6) (5.5%) = 5.3%
Return on US Equities = 2.0% + (1.4) (5.5%) = 9.7%


The implied equilibrium returns for global bonds and US equities are 5.3% and 9.7%, respectively. These implied returns are above the forecasted returns based on historical data (from Exhibit 1) used as inputs in the MVO approach for global bonds and US equities of 4.7% and 8.6%, respectively.

老师,我理解的是,mvo方法下,根据历史数据得到E(R),方差等数据,根据E(R)来算出各类资产的weight;reverse下,是根据各类资产的市值先算出weight,再根据weight算出E(R),后面如果是BL的话,再在这个E(R)中加入主观判断,调整E(R),在用调整的E(R)计算weight。

但是本题中用了capm模型,是说reverse方法下,根据weight算出E(R)的这个过程其实是用CAPM算出来的吗?不太清楚是怎么根据weight算出E(R)的,为什么又用CAPM了?

2 个答案

lynn_品职助教 · 2024年06月12日

嗨,从没放弃的小努力你好:


加油!

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努力的时光都是限量版,加油!

lynn_品职助教 · 2024年06月11日

嗨,努力学习的PZer你好:


这个其实是一个“巧合”的结论,但是可以记下来,reverse MVO优点之一:relate assets return to systematic risk


这个优点是怎么和CAPM扯上关系的呢?通过计算


资产的收益率体现了系统性风险,根本原因是 reverse optimization 得到的implied return与各个资产通过CAPM计算出的return一致。


而CAPM假设了市场上所有投资者有相同的预期,market portfolio是以市值为权重的资产构成的组合。


所以单个资产收益率的计算公式是E(Ri)=Rf+βi(Rm-Rf),这里把每个资产的收益率E(Ri)通过CAPM模型 与投资者所承担的系统性风险βi联系起来了。


所以说 reverse optimization 中每个资产的implied return与承担的系统性风险是一致的。


教材原文有如下这样的描述,同学在这里先记下来,随着学习深入会慢慢把这两点结合起来,就像何老师一直说的要形成框架,把旧的知识和新的内容链接起来。


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虽然现在很辛苦,但努力过的感觉真的很好,加油!

好好学习向前进 · 2024年06月12日

谢谢老师,很清楚!

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