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lz523 · 2024年08月03日

根据这个题,想再明确一下利率变动和duration以及BPV等的关系

NO.PZ2023032703000038

问题如下:

Beatriz Maestre is a fixed-income consultant who has been retained by Filipe Ruelas, the CFO of Cávado Produtos Agricolas, SA (Cávado). Cávado is a manufacturer of prepared foods headquartered in Braga, Portugal.

During the meeting, Maestre presents some information about Cávado’s pension fund, which is primarily invested in corporate bonds with a mixture of investment-grade and speculative-grade issues. This information is presented in Exhibit 1.


Ruelas explains that he uses futures contracts on euro-denominated German government bonds to reduce the duration gap between assets and liabilities. However, because the pension fund has only a small surplus and he would like to increase this surplus through active management of the portfolio, he employs a contingent immunization strategy. The fund is currently short 254 contracts based on a 10-year bond with a par value of EUR 100,000 and a basis point value (BPV) of EUR 97.40 per contract.

Given the futures position entered into by the pension fund, Ruelas most likely believes interest rates will:

选项:

A.fall. B.

rise.

C.

remain the same.

解释:

A is correct. The number of futures contracts needed to fully remove the duration gap between the asset and liability portfolios is given by Nf = (BPVL – BPVA)/BPVf, where BPV is basis point value (of the liability portfolio, asset portfolio, and futures contract, respectively). In this case, Nf = (59,598 – 91,632)/97.4 = –328.891, where the minus sign indicates a short position or selling of 329 futures contracts (328,891/1,000). Ruelas has under-hedged, leaving a net position that will benefit from a reduction in interest rates, just as the unhedged position would benefit from a reduction in interest rates. Thus, he must believe interest rates will fall.

B is incorrect because if Ruelas believed rates would rise, he would under-hedge, leaving a net position that would benefit from rising rates.

C is incorrect because if Ruelas believed rates wouldn’t change, he would hedge fully, in case rates moved in an unexpected way.

单独这个题目是理解了,但是利率变动的内在逻辑有点混乱。

比如利率上升,会导致债券价格下降,那么也会导致duration的下降,所以BPV也会下降。

那么为什么利率上升要降低duration才能获利?这个推导的中间过程是什么?是只有在hedge的时候才用得到吗?还是针对单一债券都适用的结论?

1 个答案

发亮_品职助教 · 2024年08月05日

比如利率上升,会导致债券价格下降,那么也会导致duration的下降,所以BPV也会下降。


利率变化虽然会导致duration改变以及BPV改变。但是这个影响只有一处需要考虑,就是做duration-matching时,原本已经做好了资产duration=负债duration,但由于利率改变,使得这种duration相等被打破,使得组合在将来不再满足duration-matching,于是不得不进行rebalancing,使得资产、负债重新回归duration-matching,保证组合将来是免疫的。


其他地方不考虑利率改变对duration的影响。只分析利率对债券价格的影响。


说下这道题的解题思路:

这里做的是contingent immunization或有免疫策略。就是当资产的value大于负债的value时,资产有surplus时,可以先退出duration-matching,组合先直接做主动的投资策略,目的是尽可能扩大资产与负债之间的value差异(surplus)

当资产有亏损,导致value低于某个门槛值时(threshold),资产组合才会退回去做duration-matching


这道题的背景是,liability value = 47.3, asset value= 49.8,资产的value更大,存在surplus;

资产的BPV=91632,负债的BPV=59598


由于资产、负债的BPV不相等,所以这里就需要利用futures来调平,这里是需要short futures来降低资产端的BPV。但因为存在surplus,可以执行continge immunization,所以可以基于未来的利率预期,进行overhedge,或者underhedge,不一定非得实现fully hedge。


但是这种题型要先分析fully hedge,因为判断underhedge, overhedge的基础是fully-hedge的份数。

如果是fully hedge彻底close duration gap时,需要的futures份数是:

(91632-59598)/97.4 = 329份

329份是分析underhedge与overhedge的基础。


如果实际使用的份数比329份多,这是overhedge;如果实际使用的份数比329少,这是underhedge。


已知实际short了254份,使用的份数过少,这是underhedge。由于使用的份数过少,所以导致(资产BPV+ short futures BPV)仍然是大于负债BPV的。


注意这样的Duration gap是基于利率预期故意构造的,现在就想想在哪种利率预期下,这样的策略会盈利。显然是预测利率下降,因为利率下降时,资产BPV更大,他的value上升更多,这会扩大资产端的surplus。所以本题选A。


这道题有一个疑问就是,原本资产端的BPV就大于负债的BPV,如果是预测利率下降的话,那么不short futures,资产端的BPV还会更大一点,利率下降享受到的value上升还会更多。那为什么本题还要先short一部分futures,要降低一部分资产端的BPV呢?


主要原因是,这块的大框架还是在duration-matching(contingent immunization),资产与负债的BPV缺口可能存在一定范围限制,这个缺口不能太大。所以,先使用short futures降低资产端的BPV,先使得资产与负债的BPV缺口回到限制范围内,然后再让资产的BPV继续大于负债的BPV,享受利率下降带来的好处。


那么为什么利率上升要降低duration才能获利?这个推导的中间过程是什么?是只有在hedge的时候才用得到吗?还是针对单一债券都适用的结论?


利率上升,债券价格下降,duration大的债券下降幅度更大,亏损更多。

为了尽量减少亏损,所以需要降低duration。所有债券的分析都是适用。

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