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syfzln · 2025年07月03日

return decomposition

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:

A.

Carry roll-down: +2.427, Term structure change: -3.782, Spread change: +0.37

B.

Carry roll-down: +2.50, Term structure change: -3.80, Spread change: +0.40

C.

Carry roll-down: +2.30, Term structure change: -3.70, Spread change: +0.30

D.

Carry roll-down: +2.60, Term structure change: -3.90, Spread change: +0.50

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

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100USDcoupon rate4%spread10bp。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%spread0,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 rate4%,半年付息一次,YTM是(5%+0.1%=5.1%,的债券的价格:

PMT=4%*100/2=2I/Y=5.1%/2=2.55N=5*2=10FV=100

利用金融计算器求出PV=95.199

6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=2I/Y=6%/2=3N=4.5*2=9FV=100

金融计算器求出PV=92.214

需要将95.19994.214的这 -0.985进行分解,看是由什么带来的:

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的利息,获得的收益就是carry roll-down 也就是2.427

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

-0.985被拆解成了三个产生原因 2.427 - 3.782 +0.37

算carryrolldown 为什么要加上2?

1 个答案

李坏_品职助教 · 2025年07月04日

嗨,努力学习的PZer你好:


收益率分解,一共就是三个组成部分。 其中第二个rate changes是利率期限结构变化带来的影响,而第三个指的是spread变化的影响。


而只有第一项是不涉及外部变量变化的,2块钱的利息 无论外部的利率、spread怎么变化,始终都有。所以这2块钱只能放到第一个部分,carry roll down里面。



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