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chris2.0🔱 · 2020年10月02日

问一道题：NO.PZ201602270200001802 第2小题 [ CFA II ]

  1. CFA II
  2. Fixed Income

* 问题详情，请 查看题干

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.

我保留2位小数，算出来也是a接近
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1 个答案

WallE_品职答疑助手 · 2020年10月02日

 

同学您好，

我不清楚您是怎么计算的哈，您先看看下面的思路

先求Exhibit 1这个三年期、coupon rate是3.0%，按年付息，par value=100债券的合理价值。

然后在对比哪个交易所的价格是合理的。

求债券现值就需要Spot rate。

但是注意Exhibit 2给的是Par rates。

所以我们需要用Par rates通过bootstrapping的方法求出来Spot rate。然后再用spot rate去折现。

这个Par rates其实就是一个付息债券的Coupon-rate，只不过这个Coupon rate很特殊会使得债券的现值等于其面值。

例如一年期的Par rate是1.25%，则代表这个1年期债券的Coupon rate=1.25%，所以用一年期spot rate折现这个债券，得到的价值等于去面值，所以就是：

这样可以反求出来一年期的Spot rate。当然这个肯定是1.25%，则第一年的spot rate等于第一年的par rate。

然后本题中2年期的par rate等于1.5%，它代表这个2年期的债券其Coupon rate等于1.5%的债券，用spot rate折现值后得到的现值是其面值；第一年的spot rate上面已经求出来了，所以未知数是第二年的spot rate：

这样就知道了第二年的Spot rate等于1.5019%。

同理，知道第三年的Par rate=1.70%，他代表这个三年期债券的coupon rate是1.7%，用spot rate折现，折现的现值是面值100：

同理S1，S2，我们在上面已经求出来了，只有一个未知数S3，求出来就是第三年的Spot rate。

这样，我们就有了三个Spot rate，再用spot rate去折现债券。

上面由par rate求Spot rate的过程就叫bootstrapping。这也是我们通常求Spot rate curve的方法。如果忘记了回看视频。

 

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