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天sama · 2022年02月23日

公式是加法，为什么这题是连乘

  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?

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%.

我看公式是R1*T1+Rfoward*（T2-T1）=R2*T2，但答案是（1+R1）*T1*（1+Rforward）*（T2-T1）=（1+R2）^2*T2

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1 个答案

李坏_品职助教 · 2022年02月23日

嗨，努力学习的PZer你好：


先看6个月和12个月的利率，因为之间间隔期都是半年，所以应该先把各自利率都除以2。复利的时候半年复利一次（题目说的semi-annually compounded），所以从6个月到12个月的公式是：（1+5%/2）* (1+R/2) = (1+5.5%/2)^2, 这里的R就是要求的6个月之后期限为6个月的远期利率（six-month periods beginning in six）， R = 0.060012.


你说的公式算出来R_forward也是0.06，是一种近似估算值，选择题也可以按照你的方法来算。

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