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执瑞 Zhirui · 2024年07月22日

这一题为什么不能用1.7% three year来折现

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NO.PZ201602270200001802

问题如下:

2. Based on Exhibits 1 and 2, the exchange that reflects the arbitrage-free price of the bond is:

选项:

A.

Eurex.

B.

Frankfurt.

C.

NYSE Euronext.

解释:

C is correct.

The bond from Exhibit 1 is selling for its calculated value on the NYSE Euronext exchange. The arbitrage-free value of a bond is the present value of its cash flows discounted by the spot rate for zero coupon bonds maturing on the same date as each cash flow. The value of this bond, 103.7815, is calculated as follows:

Notes:

1. Spot rates calculated using bootstrapping; for example: Year 2 spot rate ( Z2 ): 100=1.5/1.0125+101.5/(1+Z2)2=0.015019 100=1.5/1.0125+101.5/{(1+Z_2)}^2=0.015019

2. Present value calculated using the formula PV=FV/(1+r)n PV=FV/{(1+r)}^n,where n= number of years until cash flow, FV= cash flow amount, and r= spot rate.

A is incorrect because the price on the Eurex exchange, €103.7956, was calculated using the yield to maturity rate to discount the cash flows when the spot rates should have been used. C is incorrect because the price on the Frankfurt exchange, €103.7565, uses the Year 3 spot rate to discount all the cash flows.

考点:Introduction of Arbitrage Free Valuation

债券的无套利价格是用spot rate对债券的现金流进行折现得到的。

Exhibit 2中给的是1,2,3年期的Par rates,因此通过Bootstrapping的方式,由前向后推导出各个spot rate。已知1-year par rate等于1.25%,则1-year spot rate也等于1.25%

第二年spot rate计算:

100= 1.5/1.0125 + 101.5/(1+ Z2 )^2,所以Z2 =1.5019%

同理,我们可以计算出第三年的Spot rate:

100= 1.7/1.0125 + 1.7/ (1.015019)^2 + 101.7/(1+ S3)^3,所以Z3 =1.7049%

算得债券的价值为103.7815,所以NYSE Euronext这个交易所定价是合理的。

PMT=3

FV=100

I/Y=1.7%

n=3

1 个答案

吴昊_品职助教 · 2024年07月23日

嗨,从没放弃的小努力你好:


现在题目要我们求的是the arbitrage-free price,债券的无套利价格是用spot rate对债券的现金流进行折现得到的,不能用YTM来折现。

所以我们要通过表格2给出的par rate,利用bootstrapping把各期的spot rate求出来,再进行折现。并且表格2的表头说明给出的利率是各期的par rate

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就算太阳没有迎着我们而来,我们正在朝着它而去,加油!

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