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比如世界 · 2020年03月11日

问一道题：NO.PZ2020011303000223

  1. FRM I
  2. Valuation And Risk Models

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.

The value of the bond is 97.245937这个是怎么计算出来的？对face value，par value，market value ，bond price，pv，fv这几个名词，在做题的时候很晕，不知道什么时候要计算什么，麻烦老师再帮忙讲解一下，谢谢。

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1 个答案

小刘_品职助教 · 2020年03月12日

 同学你好，

这道题的债券价值是这样算的，因为对于美国债券来说，通常是半年付息一次，所以计算债券的时候可以如下按计算器：

N=3*2=6 I/Y=5/2=2.5 PMT=4/2=2 FV=100,按计算器PV=-97.2459

face value和 par value是指面值，一张债券面值多少钱的意思，一般都是100，就是一张债券值100块，这么理解就可以了。

market value是指债券的市价，因为随着市场收益率的变化，原来面值100的债券，有可能在市场上你只要花95或者要花105才能买到。

bond price 一般指债券发行的价格，在中国一般的债券发行价格都是100，所谓的平价发行，也有可能出现溢价或者折价的情况，因为票息和YTM之间的关系不同导致的。

PV基本可以和market value等价，在计算债券的时候，已知未来的现金流，倒算出现在的价值。

FV计算在债券计算中比较少见，因为债券在到期的时候会回归到面值100，所以不太需要用到这个。

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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=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0.135514ration=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 老师好，1、题目的FV明明是1million，也就是100万，为啥计算不用1000000，而用100？我计算的三个PV都是您的结果乘以10000。2、the ration is0.5×(0.135287+0.135514)/(97.245937×0.0005)=2.784703，为什么用BV的增加值和减少值，即0.135287，0.135514，课程里给出的公式明明是债券价格，也就是BV-和BV+？

2024-07-17 22:11 3 · 回答

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=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0.135514ration=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 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.解析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 provi this estimate of convexity.解析中ration和convexity计算都是直接用5BP，但是计算债券价格时又是假设semi-annucoupon payment. 所以lta Y应该用2.5还是5bp呢？

2023-09-12 11:14 1 · 回答

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

2023-03-29 17:24 2 · 回答

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 答案给的公式是相加

2023-03-16 13:10 1 · 回答

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. 这个对应的公式是什么？

2022-06-02 13:08 1 · 回答