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臧小玉 · 2022年01月12日

折现之后取平均大于平均之后取折现 

  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 
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1 个答案

李坏_品职助教 · 2022年01月12日

嗨，努力学习的PZer你好：



第二年这里因为恰好是3%和7%，各占50%的概率，这道题可以直接用5%作为第二年的折现率。如果条件改一下就不行了。

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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

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2023-01-17 22:51 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