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粗眉毛辣椒油 · 2024年03月15日

请问这道题的考点是什么？

  1. FRM I
  2. Financial Markets And Products

NO.PZ2020021204000017

问题如下：

The six-month, 12-month, 18-month, and 24-month zero rates are 5%, 5.5%, 6%, and 6.5%, what are the (semi-annually compounded) forward rates for a six-month periods beginning in six, 12, and 18 months?

选项：

解释：

The forward rates are

2 X ( 1.02752 /1.025-1) = 0.060012

2 X ( 1.033 /1.02752- 1) = 0.070037

2 X ( 1.03254 /1.033 - 1)= 0.080073

If all rates were continuously compounded, the forward rates would be 6%, 7%, and 8%. Because we are dealing with a semi-annually compounded rate, they are slightly different: 6.0012%, 7 .0037%, and 8.0073%.

另外，zero rate不是一般是期限为1年以内债券的到期收益率吗？

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1 个答案

李坏_品职助教 · 2024年03月16日

嗨，努力学习的PZer你好：


考点是利用即期利率推算远期利率。可以参考讲义P260。讲义里写的是简便算法，本题答案给的是复利计算的精确算法。

题目给的是0-6，0-12，0-18，0-24这个四个时间段的零息债券的利率，所谓的zero rates指的是零息债券的到期收益率。也可以看做是spot rate（即期利率）。

现在让你求出6-12的远期利率、12-18的远期利率、18-24的远期利率.


先看第一个远期利率。既然0-12的spot rate是5.5%，而0-6的spot rate是5%。现在假设6-12的远期利率是r。那么(1+5%/2) * (1+r/2) = (1+5.5%/2)^2，所以r = 0.060012.

第二个远期利率，首先找到0-18的spot rate是6%，而0-12的spot rate是5.5%, 现在假设12-18的远期利率是r，那么(1+5.5% / 2)^2 * (1+r/2) = (1+6%/2)^3， r = 0.070037.


计算远期利率的时候，要按照半年进行复利（semi-annually compounded）。

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努力的时光都是限量版，加油！

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