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洁1017 · 2024年08月05日

老师，这道题A和B选项能否讲解一下？尤其是B选项

  1. CFA III
  2. Fixed Income

NO.PZ2023032703000027

问题如下：

Berendsen explains to Adams, “I plan to continue saving for retirement, regularly adding funds to the portfolio until I retire, and I would like a low-risk solution to provide additional retirement income.”

Adams replies to Berendsen, “We focus on the ability of the portfolio to meet future cash flow needs and seek to immunize the liabilities as an objective in the management of the portfolio. If the fixed-income portfolio achieves an average annual investment return of at least 4% for the next four years, the proceeds of its liquidation will be enough to purchase an annuity sufficient to provide the funds needed to supplement your Social Security benefits.

Adams has summarized Berendsen’s (the client) fixed-income portfolio consisting of three government bonds in Exhibit 1. The yield curve has steepened since the bonds were purchased, which can be seen by comparing their respective yield to maturities (YTMs) of the purchase price yield to today’s yield.


According to the information in Exhibit 1 and assuming Berendsen retires in four years, the fixed-income portfolio most likely: (2019 mock AM)

选项：

A.

should have a shorter duration.

B.

needs a higher cash flow yield.

C.

has currently achieved zero replication.

解释：

C is correct. The portfolio’s Macaulay duration of approximately 4.0 matches the time horizon of the liability and can be calculated as follows:

[(Portfolio weightBond 1×DurationBond 1) + (Portfolio weightBond 2×DurationBond 2) + (Portfolio weightBond 3 × DurationBond 3)] = 3.99.

When compared with the single liability due in four years, the portfolio has the same return and duration characteristics of a single zero-coupon bond maturing in four years. The interest rate risk has been immunized, which is known as zero replication.

A is incorrect because the portfolio’s current duration matches the duration of the liability, or retirement date.

B is incorrect because the cash flow yield matches the required investment return. Although not equivalent to investment return, it is likely the portfolio’s return will meet the required rate of return.

如题：

1.A的解题思路是？

2.B选项：当收益率曲线变陡峭时，Cash flow yield作为统一价格，包含了更高的折现率水平，确实应该是CFY＞YTM(加权)，这块错在哪里？

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已采纳答案

发亮_品职助教 · 2024年08月08日

B我是否可以理解为：当收益率曲线是向上倾斜的时候，确实CFY＞YTM加权，但这大于已经反映在预测里了（原文“The yield curve has steepened since the bonds were purchased...”）,所以题干中呈现出的CFY=4.15%，是足够的?


这个想复杂了哈。原文的the yield curve has steepened since the bonds were purchased，这句话对做题没有帮助。

因为表格其实给了2个YTM，一个是YTM at time of purchase，另外一个是YTM at current price。

因为利率已经发生过改变了，所以导致债券的YTM会变，于是，表格里面的YTM at time of purchase这个期初数据不能用了哈，这句话就是这个目的，告诉我们期初的YTM已经作废了。但其实整道题在计算的时候也没用上YTM。


然后题目的CFY=4.15%，这个折现率也是在利率steepened之后，根据最新的组合债券价格算的哈。这个收益率已经超过目标值4%了，所以组合是OK的。


那改为：当前预测未来4年收益率是陡峭的，就需要考虑到CFY needs to be higher？


因为现在已经构建好了duration-matching策略， 如果将来利率发生steepen，具体是啥情况得将来再看，需要用将来的组合价格数据算新的CFY。但一定要保证将来新的CFY也是满足4%的最低要求，才能cover住负债。

不过当前的4.15% CFY反映不出来预测的将来利率曲线变陡峭哈！

如果将来算出来新的CFY为3.9%，低于4%的最低要求，那就是新CFY needs to be higher

发亮_品职助教 · 2024年08月06日

cash flow yield就是债券组合的YTM，可以当成单个债券的YTM来分析。


题干下面这句说，只要未来4年，组合的平均年化收益率达到4%，那么就可以满足目标...

If the fixed-income portfolio achieves an average annual investment return of at least 4% for the next four years...

也就是说，只要债券组合的收益率达到4%，就可以实现偿还负债的目标。这个4%可以看成是实现目标的最低标准。


表格下面的小字说portfolio cash flow yield是4.15%，即，构建好的组合预期收益率是4.15%，这个收益率明显要大于最低标准4%，实际组合的收益率还要更高。所以这个4.15%的cash flow yield是足够的。选项B说need a higher cash flow yield就是错误的。


选项A就是在判断组合的duration是否满足要求。已知负债是4年期，所以债券组合的Macaulay duration应该等于4，这样才符合duration-matching的条件。

可以对组合内部3个债券的macaulay duration进行加权平均，计算组合的Macaulay duration：

28%×1.49 + 35%×3.48 + 37%×6.43 = 4.0143

组合的这个macaulay duration=4.0143就差不多等于负债的到期日、负债的macaulay duration 4年。所以满足duration-matching。选项A说的should have a shorter duration就是错误的。


当收益率曲线变陡峭时，Cash flow yield作为统一价格，包含了更高的折现率水平，确实应该是CFY＞YTM(加权)，这块错在哪里？


这句话没错。

一般收益率曲线向上倾向的话，组合的Cash flow yield算下来要比YTM的加权大。


比如，1-year, 5-year, 15-year组成一个10-year的组合。

如果是YTM加权的话，是短期利率1-year，5-year，和长期利率的15-Year的加权，一般短期的权重也不会太小，且这里面有短期利率1/5-year的出现，所以加权出来的YTM不会太大。


如果是算Cash flow yield的话，是把组合当成一个10年期的债券计算折现率，这是一个长期债券组合，算出来的折现率对应长期利率，收益曲线向上倾斜，cash flow yield就自然比较高。

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