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Eric · 2023年07月26日

零利率的意义

  1. CFA I
  2. Derivative

NO.PZ2021061002000047

问题如下：

A portfolio manager observes the price and YTM of one-year (r1) and two-year ( r2) annual coupon government benchmark bonds currently available in the market. How the manager can determine a breakeven reinvestment rate in one year’s time to help decide whether to invest now for one or two years?

选项：

A.

Divide the square root of (1 + r2 ) by (1 + r1 ) and subtract 1 to arrive at a breakeven reinvestment rate for one year in one year’s time.

B.

Set (1 +r1) multiplied by 1 plus the breakeven reinvestment rate equal to (1 + r2) 2 and solve for the breakeven reinvestment rate.

C.

First substitute the one-year rate (r1) into the two-year bond price equation to solve for the two-year spot or zero rate (z2 ), then set (1 + r1 ) × (1 + breakeven reinvestment rate) = (1 + z2 ) 2 and solve for the breakeven reinvestment rate.

解释：

中文解析：

本题考察的是计算Implied forward rates(IFR)。

即题干说的：观察到市场上现有的一年期(r1)和两年(r2)年期国债的收益率，想确定一年内的盈亏平衡再投资率。

考察的公式为：


基于此公式，只有C选项描述正确。

看了其他回答还是不太明白，请再详细一点，为什么要求z2以及怎么求的？还有零利率这个概念是什么意思

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1 个答案

Lucky_品职助教 · 2023年07月26日

嗨，从没放弃的小努力你好：


在问题中，我们需要确定一年后的再投资率，以帮助决定是选择一年还是两年的投资期限。为了确定这个再投资率，我们需要使用隐含的远期利率（implied forward rates）概念。

选项C中提到了计算两年零息（spot）或者即期（zero coupon）利率（z2）的步骤，这是为了获得该公式的输入参数。实际上，我们可以通过市场上可观察到的利率数据推断出两年零息利率。

假设市场上存在一个两年零息债券，我们可以使用该债券的价格和面值之间的关系来计算两年零息利率。具体而言，我们可以使用以下公式：

价格 = 面值 / (1 + z2)^2

其中，价格是我们从市场上观察到的两年零息债券价格，面值是该债券的面值，z2是我们要计算的两年零息利率。

通过解这个方程，我们可以得到z2的值。

一旦我们得到了z2（两年零息利率），我们可以使用该值和一年利率（r1）来计算一年后的盈亏平衡再投资率。

选项C中的公式 (1 + r1 ) × (1 + breakeven reinvestment rate) = (1 + z2 )^2 就是用来计算这个再投资率的。我们可以将这个方程变形并解出 breakeven reinvestment rate 的值。

所以，选项C是正确的。

至于“零利率”（zero rate）的概念，它指的是没有固定或周期性付息的债券。在计算隐含远期利率时，我们可以使用零息债券的价格和面值之间的关系来推断出未来某个时间点的利率。

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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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