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比如世界 · 2020年03月10日

问一道题:NO.PZ2020011303000225

问题如下:

Suppose that the five-, ten-, and 30-year rates are 4%, 5%, and 6% with semiannual compounding. Calculate the duration and convexity of zero-coupon bonds with five-, ten-, and 30-years to maturity. What position in five- and 30-year bonds would have a duration equal to that of the ten-year bond? Compare the convexities of (a) the positions in the ten-year bond and (b) the position in the five- and 30-year bonds? Which of these positions will give the better return if (a) rates remain the same and (b) there are parallel shifts in the term structure?

选项:

解释:

The duration and convexities calculated by making one-basis-point changes are

We can construct a bond with a duration of 9.756 by investing β in the five-year

bond and 1β in the 30-year bond where:

4.902β+29.126(1-β)=9.756

β is 0.7996, which we round to 0.8. We therefore invest 80% in the five-year bond and 20% in the 30-year bond. The ten-year bond investment (a bullet) has a convexity of 99.941 whereas the portfolio of five- and 30-year bonds (a barbell) has a convexity of about:

0.8×26.423+ 0.2×862.472 = 193.6

If rates remain the same the bullet will provide a yield of 5%, whereas the barbell will provide a weighted average yield of 0.8 × 4 + 0.2 × 6 or 4.4%. The bullet will perform better. When there are parallel shifts to the term structure, this effect is mitigated somewhat by the barbells higher convexity, which leads to an immediate improvement in the value of the barbell position. However, the bullet will perform better for some non-parallel shifts.

老师您好,图表中三个债券的d和c是用怎样可行的方法计算出来的?谢谢

3 个答案

小刘_品职助教 · 2020年08月07日

同学你好,

因为水平移动的时候,其实所有期限利率上升的幅度是一样的,所以久期相同的话在这一部分影响是一样的,需要转化为凸性的影响,现金流越分散,凸性越大,表现会更好,barbell的凸性大于bullet。

非水平移动的时候,比如曲线变得更陡峭了,那长端调整幅度变化会更大,所以bullet表现更好。

还是星宇好 · 2020年07月28日

老师这个答案算V+和V-的时候都是用了半个bps,怎么在计算Convexity的时候,\delta y^2这里用的是1bps啊,不应该是0.005^2?

小刘_品职助教 · 2020年03月12日

同学你好,这道题计算c的时候对于计算的精度要求很高,我选的是8位小数

以第一个5年期债券为例

在收益率没有变动之前,N=10,PMT=0,I/Y=2,FV=100,反求PV=-82.03482999

当收益率上涨1个BP,N=10,PMT=0,I/Y=2.005,FV=100,反求PV=-81.99462768

当收益率下跌1个BP,N=10,PMT=0,I/Y=1.995,FV=100,反求PV=-82.07505398;

duration=(82.07505398-81.99462768)/(2*82.03482999*0.0001)=4.902(基础班178页讲义)

c=(82.07505398+81.99462768-2*82.03482999)/((0.0001)^2*82.03482999)=26.428

如果保留的小数位更多,结果就跟答案一致,我用excel算一下,如下图

柚柚_柚 · 2020年08月06日

"When there are parallel shifts to the term structure, this effect is mitigated somewhat by the barbell’s higher ... However, the bullet will perform better for some non-parallel shifts."答案最后一句,请问为什么水平移动的时候barbell好,不水平的时候bullet好呢?

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2023-05-01 17:29 1 · 回答

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2022-07-02 11:26 1 · 回答

老师这个答案算V+和V-的时候都是用了半个bps,怎么在计算Convexity的时候,\lta y^2这里用的是1bps啊,不应该是0.005^2?

2020-07-28 16:24 1 · 回答

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