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小猫批脸 · 2023年01月17日

关于本题的公式问题

  1. FRM II
  2. Market Risk Measurement And Management

NO.PZ2018122701000073

问题如下：

For a 2-year zero-coupon bond, the 1-year rate is expected to remain at 5% for the first year. For the second year, it is foretasted the that 1-year spot rate will be either 7% or 3% at equal probability of 50%. If you are asked to reflect the convexity effect for this 2-year bond by Jensen’s inequality formula, which of the following inequalities is the best answer?

选项：

A.

$0.90736 > $0.90703.

B.

$0.90703 > $0.90000.

C.

$0.95238 > $0.90736.

D.

$0.95273 > $0.95238

解释：

A is correct.

考点：Jensen's inequality formula

解析：

不等式左边

E(11+r)=0.5×11.07+0.5×11.03=0.95273E(\frac1{1+r})=0.5\times\frac1{1.07}+0.5\times\frac1{1.03}=0.95273

0.95273/1.05 = 0.90736

不等式右边

$10.5×1.07+0.5×1.03=$11.05=0.95238\frac{\$1}{0.5\times1.07+0.5\times1.03}=\frac{\$1}{1.05}=0.95238

0.95238/1.05 = 0.90703

右边等式应该是 1/Er=1/E（1+r）但是 这1/E（1+r）应该怎么理解啊 李老师讲的时候我就没听懂 就是

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李坏_品职助教 · 2023年01月18日

嗨，爱思考的PZer你好：


这个是利率期限结构的凸性属性：

可以联想债券的凸性，债券的凸性指的是横坐标为收益率、纵坐标为价格的债券曲线的弯曲程度，曲线弯曲度越大，凸性越大。上图讲义中的convexity effect指的也是弯曲程度，只不过曲线不太一样，看下面的说明。


如何表示曲线的弯曲程度呢？参考原版书的说法：

这个图的横坐标是r，纵坐标是1/(1+r)。所以这个曲线表示的是函数1/(1+r)在第一象限的图象。


图中B的纵坐标是A和D的纵坐标的平均值。假定A和D两个点出现的概率相同，各占50%，那么B的纵坐标Yb = E(Ya + Yd)，而A和D都是1/(1+r)这个曲线上的点，所以可以认为Yb=E(1/(1+r))。也就是Yb是函数1/(1+r)上面不同的点的纵坐标的数学期望。


同理，点B的横坐标Xb可以看成是A和D的横坐标的数学期望，所以Xb = E(Xa + Xd)，所以Xb = E(r)。点C的横坐标=点B的横坐标，所以点C的纵坐标Yc = 1/(1+Xb) = 1 / (1+ E(r))。


从图上可以看出Yb > Yc, 所以E(1/(1+r)) > 1 / (1+E(r))。凸性越大，曲线越弯曲，差距越大。



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小猫批脸 · 2023年01月18日

懂了 老师 讲的很详细！！ 谢谢老师

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NO.PZ2018122701000073问题如下 For a 2-yezero-coupon bon the 1-yerate is expecteto remain 5% for the first year. For the seconyear, it is foretastethe th1-yespot rate will either 7% or 3% equprobability of 50%. If you are asketo reflethe convexity effefor this 2-yebonJensen’s inequality formulwhiof the following inequalities is the best answer? A.$0.90736 $0.90703.B.$0.90703 $0.90000.C.$0.95238 $0.90736.$0.95273 $0.95238 A is correct.考点Jensen's inequality formula解析不等式左边E(11+r)=0.5×11.07+0.5×11.03=0.95273E(\frac1{1+r})=0.5\times\frac1{1.07}+0.5\times\frac1{1.03}=0.95273E(1+r1​)=0.5×1.071​+0.5×1.031​=0.952730.95273/1.05 = 0.90736 不等式右边$10.5×1.07+0.5×1.03=$11.05=0.95238\frac{\$1}{0.5\times1.07+0.5\times1.03}=\frac{\$1}{1.05}=0.952380.5×1.07+0.5×1.03$1​=1.05$1​=0.952380.95238/1.05 = 0.90703 如题，怎么确定要不要再除以1.05%折现回0时刻呢？

2023-02-06 10:22 2 · 回答

NO.PZ2018122701000073 $0.90703 > $0.90000. $0.95238 > $0.90736. $0.95273 > $0.95238 A is correct. 考点Jensen's inequality formula 解析 不等式左边 E(11+r)=0.5×11.07+0.5×11.03=0.95273E(\frac1{1+r})=0.5\times\frac1{1.07}+0.5\times\frac1{1.03}=0.95273E(1+r1​)=0.5×1.071​+0.5×1.031​=0.95273 0.95273/1.05 = 0.90736 不等式右边 $10.5×1.07+0.5×1.03=$11.05=0.95238\frac{\$1}{0.5\times1.07+0.5\times1.03}=\frac{\$1}{1.05}=0.952380.5×1.07+0.5×1.03$1​=1.05$1​=0.95238 0.95238/1.05 = 0.90703 右侧 3%与7%平均是5% ，可否直接用5%作为第二年利率折现，这样更为简便 

2022-01-12 12:29 1 · 回答

$0.90703 > $0.90000. $0.95238 > $0.90736. $0.95273 > $0.95238 A is correct. 考点Jensen's inequality formula 解析 不等式左边 E(11+r)=0.5×11.07+0.5×11.03=0.95273E(\frac1{1+r})=0.5\times\frac1{1.07}+0.5\times\frac1{1.03}=0.95273E(1+r1​)=0.5×1.071​+0.5×1.031​=0.95273 0.95273/1.05 = 0.90736 不等式右边 $10.5×1.07+0.5×1.03=$11.05=0.95238\frac{\$1}{0.5\times1.07+0.5\times1.03}=\frac{\$1}{1.05}=0.952380.5×1.07+0.5×1.03$1​=1.05$1​=0.95238 0.95238/1.05 = 0.90703 为什么都需要再除以一个1.05呢？

2021-01-17 10:30 1 · 回答