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mushkc · 2023年07月10日

efficient frontier和Sharpe ratio

  1. CFA III
  2. Asset Allocation

* 问题详情，请 查看题干

NO.PZ202206210100000106

问题如下：

The most appropriate conclusion that can be drawn from Exhibit 3 is that:

选项：

A.management’s risk–return objectives may not have been achieved with the TAA portfolio. B.the current portfolio is a corner portfolio. C.the Sharpe ratios for the policy portfolio and the TAA portfolio are the same.

解释：

Solution

A is correct. The Sharpe ratio is the slope of the line drawn from the risk-free rate to a particular portfolio. The two portfolios of interest are the policy portfolio and the TAA portfolio because both are indicated as being efficient. The diagram to the right indicates that the policy portfolio/risk-free combination has a higher slope than the TAA/risk-free combination. Even though the TAA portfolio has a higher return than the policy portfolio, the additional return requires too much additional risk. In addition, the TAA portfolio may exceed management’s risk tolerance.

B is incorrect. Corner portfolios are efficient portfolios and represent a portfolio where an asset weight changes from zero to positive or positive to zero. No such behavior in weights is indicated for the current portfolio allocation in Exhibit 2. It is also an inefficient portfolio.

C is incorrect. The Sharpe ratio is the slope of the line drawn from the risk-free rate to a particular portfolio. The two portfolios of interest are the policy portfolio and the TAA portfolio because both are indicated as being efficient. The diagram to the right indicates that the policy portfolio/risk-free combination has a higher slope than the TAA/risk-free combination.

1）所以是不是只要在efficient frontier上面的任意一个组合，都应该有着相同的sharpe ratio？

2）但如果sharpe ratio代表的是Rf链接某一个有效前沿组合的斜率，那么应该会有千千万万个不同的sharpe ratio呀？有效前沿上越靠左边的组合理论上斜率就越高呀？


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lynn_品职助教 · 2023年07月10日

嗨，爱思考的PZer你好：


1）所以是不是只要在efficient frontier上面的任意一个组合，都应该有着相同的sharpe ratio？


是的，有效前沿这里的知识点其实和一级CAPM是一样的流程，情况如下。

 

1）在包含了所有风险资产后，我们会得到一条线，这条线在给定收益率情况下风险最小，在给定方差的情况下收益率最大，就是我们常说的有效前沿，如下图。

这条线上所有的点有着相同的sharpe ratio，且都是最大的。


2）但如果sharpe ratio代表的是Rf链接某一个有效前沿组合的斜率，那么应该会有千千万万个不同的sharpe ratio呀？有效前沿上越靠左边的组合理论上斜率就越高呀？


对，加上rf就变成了切线，切线当然斜率不同。


2）但是这条线中，缺少了一个东西，就是无风险资产，引入无风险资产后，我们可以拿无风险资产和有效前沿上的点做组合，得到无数多条CAL，其中有一条CAL，斜率最大，也就是CML，客观世界的可行集变大，如下图，原来客观世界的可行集就是EF，现在变成了CAL(P)，这条线就是CML。

到这里就可以引入无差异曲线来相切了。


1、如果没有rf资产，找无差异曲线与EF切点

 

因为无差异曲线是一组一组的，不是只有一根，就跟消费者行为理论中一样，预算线只有一根，无差异曲线有无数多条，其中有一条就会相切，这里也是一样，EF只有一根，IC有无数多条，有一根一定会切上，下图可以更直观地展现出来~

 

2、如果有rf资产，则是连线的切点即SR最大的点

 

引入无风险资产后由于客观世界的可行集变大了，所以我们的选择变多了，就能达到一个更高的效用了，这时候我们会选择的就是下图中的C点，这是无差异曲线和CML的切点，是那根客观世界和我们主观意愿结合效用最大的点。


 

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