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mx · 2022年05月28日

a coupon of 4% at a spread of ten basis points

  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.

选项：

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

这10bps怎么理解？是coupon rate实际为4.1%？还是说折现时候利率加上10bps？

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1 个答案

DD仔_品职助教 · 2022年05月29日

嗨，爱思考的PZer你好：


是折现的时候要加上。

这句话的中间的黄色部分是描述债券基本情况的，前一句是用逗号隔开，代表关于利率还没描述完，利率刚开始是5%，会有一个10bp的spread，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.

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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！

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NO.PZ2020011303000208问题如下 The term structure is initially fl5%, aninvestor buys a five-yebonwith a favalue of US100 ana coupon of 4% a spreof ten basis points. the enof six months the term structure is fl6% anthe spreis zero. Carry out a P L composition. First we calculate the carry roll-wn. The cash-carry is 2%. In this case, the assumption unrlying the carry roll-wn is ththe term structure remains fl5%. (This is true for all three finitions of carry roll-wn.) The initipripaifor the bonis The priof the bon if six months passes without rates changing or the sprechanging, is The carryroll-wn is therefore: 2+95.626-95.199=2.427This calsocalculate0.0255 × 95.199.The value ofthe bonthe enof six months, assuming no sprechange, isAfter thesprechange is consire the value of the bonisThis lea tothe following tableThe bonpriin six months is 92.214 anthe investor receives a coupon of 2.000 just before the enof the six months. The initibonpriis 95.199. The gain is therefore:92.214 + 2.000－95.199 = ﹣0.985The P L composition splits this into:(A carry roll-wn of 2.427,(The impaof a term structure change of -3.782, anA sprechange of 0.370.﹣0.985 = 2.427－3.782 + 0.370 题目问利率的期限结构最开始时flat的，利率是5%，投资者买了一个5年期的债券，面值是100UScoupon rate是4%，sprea10bp。在6个月结束的时候，利率的期限结构依旧是flat的，利率是6%，sprea0，请carry out P L的分解。题目默认半年付息一次。首先计算5年期，coupon rate是4%，半年付息一次，YTM是（5%+0.1%）=5.1%，的债券的价格PMT=4%*100/2=2，I/Y=5.1%/2=2.55，N=5*2=10，FV=100利用金融计算器求出PV=95.1996个月之后，债券变成期限为4.5年，YTM=6%，其他条件不变的债券，这个债券的价格为PMT=2，I/Y=6%/2=3，N=4.5*2=9，FV=100金融计算器求出PV=92.214需要将95.199到94.214的这-0.985进行分解，看是由什么带来的1.如果利率和sprea不变，半年后债券的价值算出来是95.626，再加上2的利息，获得的收益就是carry roll-wn 也就是2.427。2.然后在1的基础上如果利率变成6%，sprea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 =-3.7823.然后再2的基础上如果利率是6%，sprea了，也就是spreahange带来的收益就是92.214-91.844=0.37。-0.985被拆解成了三个产生原因2.427 - 3.782 +0.37 请问整道题怎么理解呢？。。。。

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