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mx · 2022年05月28日

This can also be calculated as 0.0255 × 95.199.

  1. FRM I
  2. Valuation And Risk Models

NO.PZ2020011303000208

问题如下：

The term structure is initially flat at 5%, and an investor buys a five-year bond with a face value of USD 100 and a coupon of 4% at a spread of ten basis points. At the end of six months the term structure is flat at 6% and the spread is zero. Carry out a P&L decomposition.

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解释：

First we calculate the carry roll-down. The cash-carry is 2%. In this case, the assumption underlying the carry roll-down is that the term structure remains flat at 5%. (This is true for all three definitions of carry roll-down.) The initial price paid for the bond is

The price of the bond, if six months passes without rates changing or the spread changing, is

The carryroll-down is therefore: 2+95.626-95.199=2.427

This can also be calculated as 0.0255 × 95.199.

The value of the bond at the end of six months, assuming no spread change, is

After the spread change is considered, the value of the bond is

This leads to the following table

The bond price in six months is 92.214 and the investor receives a coupon of 2.000 just before the end of the six months. The initial bond price is 95.199. The gain is therefore:
92.214 + 2.000－95.199 = ﹣0.985
The P&L decomposition splits this into:
(a) A carry roll-down of 2.427,
(b) The impact of a term structure change of -3.782, and
(c) A spread change of 0.370.
﹣0.985 = 2.427－3.782 + 0.370

为什么可以直接用term structure利率与PV直接相乘？

这个计算公式的含义是什么？为什么算出来正好是carry roll

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2 个答案

李坏_品职助教 · 2022年06月22日

嗨，爱思考的PZer你好：


carry- roll-down其实是两个部分：carry(利息) + roll-down。其中roll-down是假设利率和spread都不变， 纯粹因为时间的推移导致的债券价格的变动（如果是折价发行的债券，等于价格随着时间推移而上涨）。

具体计算可以参考原版书：

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李坏_品职助教 · 2022年05月28日

嗨，爱思考的PZer你好：


这里计算的是carry roll- down的收益。这题就是要把95.199到94.214的这 -0.985分解开，看分别由什么带来的：


  1. 如果利率和spread都不变，半年后债券的价值算出来是95.626，再加上2块钱的利息，获得的收益就是carry roll-down ：2.427。因为carry roll-down是假设利率和spread都不变，那么也就可以直接用上面公式的折现利率0.0255和PV来计算了。
  2. 然后在1的基础上：如果利率变成6%，spread仍然保留也就是折现率用6.1%，求出来4.5年期的债券价值是91.844，也就是term structure变化造成的收益就是91.844-95.626 = -3.782
  3. 然后再2的基础上：如果利率是6%，spread没了，也就是spread change带来的收益就是92.214-91.844=0.37


这就把 -0.985拆解成了三个部分： 2.427 - 3.782 +0.37

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就算太阳没有迎着我们而来，我们正在朝着它而去，加油！

he123456 · 2022年06月22日

求carry roll down为什么还要加上coupon2？

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