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evelynql · 2023年09月12日

duration 和convexity的计算用的delta Y 应该是periodic delta Y么？

  1. FRM I
  2. Valuation And Risk Models

NO.PZ2020011303000223

问题如下：

What is the effective duration and convexity of a three-year Treasury bond with a face value of 1 million and a coupon of 4% when the term structure is flat at 5%? Express interest rates in decimals and consider five-basis-point changes.

选项：

解释：

The value of the bond is 97.245937. When there is five-basis-point increase in all rates so that the term structure is flat at 5.05%, the value falls by 0.135287 to 97.110650. When there is a five-basis-point decrease in all rates so that the term structure is flat at 4.95%, the value rises by 0.135514 to 97.381452. The duration is

0.5×(0.135287+0.135514)/(97.245937×0.0005)=2.784703

The convexity is

(97.110650+ 97.381452-2×97.245937)/(97.245937× 0.00052)=9.35

Note that even more decimal places than those indicated is necessary to provide this estimate of convexity.

题目问：3年期的treasury bond，面值=1m，coupon rate=4%，利率=5%，利率的期限结构是flat的。求这个债券的effective duration和convexity。利率的变动幅度是5bp。

treasury bond一般每半年付息一次。

首先利用计算器求债券的价格V0：PMT=100*4%/2=2，FV=100，I/Y=5%/2=2.5%，N=3*2=6，得PV=97.245937

利率上升5bp时，债券价格V-：PMT=100*4%/2=2，FV=100，I/Y=5.05%/2=2.525%，N=3*2=6，得PV=97.110650

价格下降0.135287

利率下降5bp时，债券价格V+：PMT=100*4%/2=2，FV=100，I/Y=4.95%/2=2.475%，N=3*2=6，得PV=97.381452

价格上升0.135514

duration=0.5*(价格上升幅度+价格下降幅度)/(V0*2)

0.5×(0.135287+0.135514)/(97.245937×0.0005)=2.784703

convexity=(V+ + V- -2*V0)/(V0*利率变动^2)

=(97.110650+ 97.381452-2×97.245937)/(97.245937× 0.00052)=9.35

What is the effective duration and convexity of a three-year Treasury bond with a face value of 1 million and a coupon of 4% when the term structure is flat at 5%? Express interest rates in decimals and consider five-basis-point changes.

解析

The value of the bond is 97.245937. When there is five-basis-point increase in all rates so that the term structure is flat at 5.05%, the value falls by 0.135287 to 97.110650. When there is a five-basis-point decrease in all rates so that the term structure is flat at 4.95%, the value rises by 0.135514 to 97.381452. The duration is

0.5×(0.135287+0.135514)/(97.245937×0.0005)=2.784703

The convexity is

(97.110650+ 97.381452-2×97.245937)/(97.245937× 0.00052)=9.35

Note that even more decimal places than those indicated is necessary to provide this estimate of convexity.

解析中duration和convexity计算都是直接用5BP，但是计算债券价格时又是假设semi-annual coupon payment. 所以delta Y应该用2.5bp 还是5bp呢？

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1 个答案

DD仔_品职助教 · 2023年09月12日

嗨，努力学习的PZer你好：


同学你好，

因为题目给出的条件是five-basis-point changes，所以在计算的时候用5bp。跟是不是semi没关系，因为这是利率变动，不是说semi的话，利率变动就会减半，无论针对什么债券利率变动都不会变。

计算D和CX的时候I/Y是要除2的，因为是semi的债券。


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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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NO.PZ2020011303000223问题如下 Whis the effective ration anconvexity of a three-yeTreasury bonwith a favalue of 1 million ana coupon of 4% when the term structure is fl5%? Express interest rates in cimals anconsir five-basis-point changes. The value of the bonis 97.245937. When there is five-basis-point increase in all rates so ththe term structure is fl5.05%, the value falls 0.135287 to 97.110650. When there is a five-basis-point crease in all rates so ththe term structure is fl4.95%, the value rises 0.135514 to 97.381452. The ration is0.5×(0.135287+0.135514)/(97.245937×0.0005)=2.784703The convexity is(97.110650+ 97.381452-2×97.245937)/(97.245937× 0.00052)=9.35Note theven more cimplaces ththose incateis necessary to provithis estimate of convexity. 题目问3年期的treasurybon面值=1m，coupon rate=4%，利率=5%，利率的期限结构是flat的。求这个债券的effective ration和convexity。利率的变动幅度是5bp。treasury bon般每半年付息一次。首先利用计算器求债券的价格V0PMT=100*4%/2=20，FV=100，I/Y=5%/2=2.5%，N=3*2=6，得PV=97.245937利率上升5bp时，债券价格V-PMT=100*4%/2=20，FV=100，I/Y=5.05%/2=2.525%，N=3*2=6，得PV=97.110650价格下降0.135287利率下降5bp时，债券价格V+PMT=100*4%/2=20，FV=100，I/Y=4.95%/2=2.475%，N=3*2=6，得PV=97.381452价格上升0.135514ration=0.5*(价格上升幅度+价格下降幅度)/(V0*2)0.5×(0.135287+0.135514)/(97.245937×0.0005)=2.784703convexity=(V+ + V--2*V0)/(V0*利率变动^2)=(97.110650+97.381452-2×97.245937)/(97.245937× 0.00052)=9.35 请问，用PMT20，FV100，用2，5利率，六期这几个数字反复按计算器都等于196。实在不知错在哪了？都是这么按的，烦请示范一下，感谢

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