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smiledog · 2021年02月28日

对某一个回答解释中的疑问

NO.PZ2019103001000031

问题如下:

Doug Kepler, the newly hired chief financial officer for the City of Radford, asks the deputy financial manager, Hui Ng, to prepare an analysis of the current investment portfolio and the city’s current and future obligations. The city has multiple liabilities of different amounts and maturities relating to the pension fund, infrastructure repairs, and various other obligations.

Ng observes that the current fixed-income portfolio is structured to match the duration of each liability. Previously, this structure caused the city to access a line of credit for temporary mismatches resulting from changes in the term structure of interest rates

Kepler asks Ng for different strategies to manage the interest rate risk of the city’s fixed-income investment portfolio against one-time shifts in the yield curve. Ng considers two different strategies:

Strategy 1: Immunization of the single liabilities using zero-coupon bonds held to maturity

Strategy 2: Immunization of the single liabilities using coupon-bearing bonds while continuously matching duration.

Which duration measure should be matched when implementing Strategy 2?

选项:

A.

Key rate

B.

Modified

C.

Macaulay

解释:

C is correct.

An investor having an investment horizon equal to the bond’s Macaulay duration is effectively protected, or immunized, from the first change in interest rates, because price and coupon reinvestment effects offset for either higher or lower rates.

对选项没有疑问,请问老师在其中一个回答中提到,卖出债券的价格风险price risk和再投资风险Reinvestment risk可以完全抵消,这个抵消是定性意义上的抵消,还是可以通过数学公式做到定量的完全抵消?
1 个答案

发亮_品职助教 · 2021年02月28日

嗨,从没放弃的小努力你好:


“这个抵消是定性意义上的抵消,还是可以通过数学公式做到定量的完全抵消?”


在数学上可以证明相互抵消。Google上可以搜到相关的数学证明,证明比较复杂,超过了我们CFA的范围了;感兴趣可以搜关键词看看。


不过Immunization属于比较成熟的技术了,虽然现实中可能无法满足一些假设条件,但免疫策略在实务中也是经受住考验的,所以我们可以放心大胆的用结论即可。理论依据就是Price risk与Reinvestment risk相互抵消。


不用数学公式,反过来我们找个债券,自己带入数据也能检验出他是正确的。


由于债券的Macaulay duration一般都小于债券的Maturity;所以,假设我们现在就有一只10年期的债券,知道他的一切数据,假设他的Macaulay duration = 7;


此时,我们就让投资期Investment horizon=7,此时满足免疫条件。


首先,我们假设利率没变,由于知道债券的现金流,我们可以算债券这7年Coupon与Coupon的再投资收益;且可以算出来第7年年卖出债券时的价格。利率没变时的情景为Benchmark。


第二种情景,我们假设利率上升100bps,有了新的利率,我们可以算出债券Coupon的再投资收益;我们也可以算出来在第7年年末卖出债券时的价格。此时,与第一种情景Benchmark情况相比,我们可以算出来利率上升100bps时,再投资收益的变化金额,也可以算出来此时债券卖出价格的变动金额,最终可以发现再投资的变动金额与债券卖出价格的变动金额,这两个相加是基本等于0的。


第三种情景,我们假设利率下降100bps,有了新的利率,我们可以算出债券Coupon的再投资收益;我们也可以算出来在第7年年末卖出债券时的价格。此时,以Benchmark为标准,我们依然可以算出来再投资的变化金额与债券卖出价格的变动金额,最终发现,两个变动金额的加总为0;


因此,可以从实际数据上发现,免疫之后,利率平行移动时,Price risk(卖出价格的变动金额)与Coupon reinvestment risk(再投资的变动金额)可以相互抵消。


实际上在1级的固定收益里面,这个结论就是用以上这种带入实际数据的办法证明出来的,只不过在一级他是证明债券的投资收益率在三种情况下没有改变,即,利率的平行移动不影响免疫组合的收益。这和证明Price risk与Reinvestment risk相互抵消本质上是一样的。

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虽然现在很辛苦,但努力过的感觉真的很好,加油!

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2022-03-24 20:01 1 · 回答

NO.PZ2019103001000031 能否下KEY RATE RATION不选的原因,谢谢!

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NO.PZ2019103001000031 MofieMacaul C is correct. investor having investment horizon equto the bons Macaulration is effectively protecte or immunize from the first change in interest rates, because priancoupon reinvestment effects offset for either higher or lower rates. immunization的2个条件,PVa=PVL,=,是为了hee pririsk和ri risk带来的价格变动的影响,所以不是应该用的mify ration吗?mofy ration一样,这也利率变动,asset和liab的price变动一样,这也也能继续match了。为什么是mration,平均还款期呢?

2021-12-11 14:46 2 · 回答

NO.PZ2019103001000031 这道题我懂了,但是这两策略有什么区别呀

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NO.PZ2019103001000031 MofieMacaul C is correct. investor having investment horizon equto the bons Macaulration is effectively protecte or immunize from the first change in interest rates, because priancoupon reinvestment effects offset for either higher or lower rates. 如果前者匹配,后者不也一样匹配吗?毕竟就差个1/(1+y)

2021-04-20 18:51 18 · 回答