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老师，请教这道题

NO.PZ2023020101000014

问题如下：

Whitney’s first meeting is with Novatel, a US based company that currently has an outstanding loan of $250,000,000 that carries a 5.15% fixed interest rate. Novatel’s managers feel that the current interest rate on the loan is high and they also believe that interest rates are poised to decline. Whitney advises Novatel to enter into a one-year pay-floating Libor receive-fixed interest rate swap with quarterly payments. The notional principal on the swap will be $250,000,000，and the annualized swap rate is 0.016792. Whitney’s first task is to determine the appropriate swap rate.Ninety days have passed since Whitney’s initial meetings, and in the interim interest rates have increased dramatically. Whitney’s clients have asked to meet with her to review their positions.In order to prepare for the meeting, Whitney has obtained updated interest rate data that is presented in Exhibit 2.Exhibit 2 Term Structure of Rates 90 Days Later (%)

Using data in Exhibit 2 and a 30/360 day count, the market value of Novatel’s swap after 90 days is closest to:

选项：

–$3,702,900.

–$2,875,000.

–$2,408,880.

解释：

The present value factors for Exhibit 2 are provided below:

For example, PV(180) is calculated as:

$\frac1{1+0.0262\times(180/360)}=0.987069$

Other present value factors are calculated in a similar manner.

Using the fixed rate initially determined for the swap and the current PV factors, the current value of the fixed bond is:

$FB=C\sum_{i=1}^nPV_{t,ti}(1)+PV_{0,t_n}=0.004198(0.994505+0.987069+0.972786)+0.972786=0.9851884$

The value of the floating rate bond at reset is 1. The market value of the pay-floating, receive fixed rate swap is the value of the fixed-rate bond less the value of the floating-rate bond, or $250,000,000 × (0.9851884 – 1.000) = –$3,702,900.

Using an alternative approach, the new fixed swap rate would be

$r_{FIX}=\frac{1.0-PV_{0,t_n}(1)}{\sum_{i=1}^nPV_{0,t_n}(1)}=1.0-0.972786/0.994505+0.987069+0.972786=0.00921147$

And the value of the swap is the difference between the value at the old rate and the value at the new rate, or

$V=(FS_0-FS_t)\sum_{i=1}^nPV_{t,t_i}=(0.004198-0.00921147)\times(0.9945+0.9871+0.9728)=-0.0148116$

The swap value = $250,000,000 × –0.0148116 = –$3,702,900

有几个点特别困惑，请老师帮助解答：

1、这个人付float，收fixed，所以最后计算value，方向应该是fixed rate - float rate吧

2、题目中说float按年pay，annualized swap rate is 0.016792. fixed rate计算出来按季度=0.0092.如果两者相减的话，为什么把float转化成0.008396呢？这明显是半年度的，也不是按季度呀。半年度的 float rate-季度fixed rate，这点我很困惑，一个是不匹配，一个是方向和我问题1中理解的也不一样

3、答案解析中的倒数第二步，(0.008396−0.009211)×(0.9945+0.9871+0.9728)=−0.002409 ，不太理解×(0.9945+0.9871+0.9728)这部分内容，乘以折现因子的和是什么意思呀老师，代表什么含义呢。按我的想法，应该是利率直接轧差乘以NP就可以了呀，为什么乘以这么多折现因子的和呢。

请老师解惑，谢谢。

嗨，努力学习的PZer你好：

1.这个人是收固定支付浮动，所以swap的value是用固定减去浮动是对的，在第一种方法中，我们先算出来新的fix rate，然后减去了浮动的value1，最后乘以NP，即这一步的公式 $250,000,000 × (0.9851884 – 1.000) = –$3,702,900.

2.annualized swap rate is 0.016792，按照季度来付息即为0.004198，题目解析已经做出修改。

3.答案解析中倒数第二步用的是重新定价法，所以这个计算(0.004198−0.00921147)×(0.9945+0.9871+0.9728)=−0.0148116是将所有的利差都折现到了90天这个时间点。

这道题使用了两种方法进行解析，Using an alternative approach这句话上面的部分采用的是画图法，即将未来的现金流折现，再用固定的value-浮动的value。

这句话的后面采用的是重新定价法，需要先算出当前时间点开始，270天后到期的swap的定价，再计算两个swap的利差最后折现到90时间点的价值。

同学在做题中可以选择一个自己比较容易理解的方法进行求解即可。

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虽然现在很辛苦，但努力过的感觉真的很好，加油！

老师，请教这道题

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