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凉茶325 · 2022年10月30日

modified duration

  1. CFA III
  2. Fixed Income

NO.PZ2018120301000031

问题如下：

Doug, the newly hired chief ­financial officer for the City of Radford, asks the deputy ­financial manager, Hui, 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.

Hui 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.

Doug asks Hui 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. Hui 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：

Correct Answer: C

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.

为什么single liability用mac duration match，multiple liability用modified duration match

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3 个答案

pzqa015 · 2024年02月03日

嗨，爱思考的PZer你好：


可以的

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加油吧，让我们一起遇见更好的自己！

Shafengler · 2024年02月04日

辛苦了，非工作日加班希望老板和老板娘给你加鸡腿

pzqa015 · 2023年11月18日

嗨，努力学习的PZer你好：


mac duration可以用来衡量利率变动对债券价格的影响，Mac d越大，则利率变动，债券价格变动越大，也就是price risk越大。

investment horizon代表再投资期限，再投资期限越长，面临的不确定性越大，则reinvestment risk越大。

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加油吧，让我们一起遇见更好的自己！

pzqa015 · 2022年10月31日

嗨，爱思考的PZer你好：


关于用什么duration match这里，结论是single liability用mac duration match，multiple liability用modified duration match。原理如下：

Single liability免疫：

负债端只有一笔现金流，不受收益率曲线变化的影响。那么我们构建的Portfolio也要尽量不受收益率曲线变化的影响，这是immunization这个词的含义，也是构建Portfolio的基本原则。

首先，回顾一下，持有一只债券的收益率是如何计算。

根据P0(1+r)=P1+coupon①这个公式，P0是期初买入债券的价格，P1是现在债券市场价格或者未来可以卖出的价格，r就是持有债券一段时间的投资收益率。如果投资期是一期，没有coupon再投资问题；如果投资期是多期，要考虑coupon的reinvestment问题，比如，投资3期，每期coupon为3，假定期间现金流以ytm再投资，那么根据PV=0，PMT=3，N=3，I/Y=ytm，求解出FV就是上面公式的coupon了，它包含的是coupon和coupon的在投资收益。（这是一级固收的知识点）。

其次，我们分析一下公式①，影响债券投资收益r的因素有两方面：

一是P1，它是债券的卖出价格，收益率曲线一旦变化，债券卖出价格不再确定，这是price risk，用duration衡量，duration越大，price risk越大；

二是coupon，它是持有债券期间的coupon以及coupon再投资收益，一旦收益率曲线变化，coupon确定，但coupon的再投资收益就不确定了（也就是不能假设以ytm再投资了），这是reinvestment risk，用investment horizon衡量，投资期越长，reinvestment risk越大。

收益率发生变动，price risk和reinvestment risk对投资者的影响是相反的，比如考虑收益率曲线上升的情形：

若投资期长，price risk对投资者不利，但reinvestment risk对投资者有利；若投资期短，price risk对投资者有利，但reinvestment risk对投资者不利。所以，一定存在一个条件，这个条件使得price risk与reinvestment risk相互抵消，收益率曲线变动对portfolio value的影响就被控制了，portfolio可以获得确定的return，这样就实现了免疫的初衷。

这个条件就是mac D=investment horizon。由于要与投资期比，只有mac D代表现金流的加权平均到期期限，是一个时间的概念，所以，要用mac D与investment horzion比较。

有下面结论：

Mac D＞investment horizon，则price risk＞reinvestment risk

Mac D=investment horizon，则price risk=reinvestment risk

Mac D＜investment horizon，则price risk＜reinvestment risk

所以，单笔现金流负债免疫，我们要让mac D=investment horizon。

Multiple liability 免疫：

与单笔现金流负债不受收益率曲线变动影响不同，多笔现金流负债受收益率曲线变动影响，所以，我们构建portfolio的目的也不是获得确定收益了，而是要让收益率曲线变动时，资产与负债的价格变动尽量相等，也就是BPVA=BPVL。BPV=-MD*1bp*P，所以用到的是modified duration，而不再是mac duration了。

 

总之，虽然single liability与multiple liabiliies都叫免疫，但两类免疫的目标是不一样的，前者的目标是追求realized return，后者的目标是追求资产与负债的value变化相同，因此，前者要用mac duration，后者要用modified duration。

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虽然现在很辛苦，但努力过的感觉真的很好，加油！

考拉 · 2023年11月18日

Mac D＞investment horizon，则price risk＞reinvestment risk, 这是为什么

Shafengler · 2024年02月03日

看过老师的回复之后我赞同。另一个角度，我是这样理解的，Mac.Duration代表平均还款期（以这个为标准，当投资期=平均还款期时，price risk和reinvestment risk均衡），当投资期<平均还款期，price risk，当投资期>平均还款期，reinvestment risk。

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