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luojy · 2023年07月07日

解析中看不懂的地方

  1. CFA II
  2. Derivative

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. 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：

A.

–$602,250.

B.

–$2,875,000.

C.

–$2,408,880.

解释：

The present value factors for Exhibit 2 are provided below:

For example, PV(180) is calculated as:

11+0.0262×(180/360)=0.987069\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∑i=1nPVt,ti(1)+PV0,tn=0.008396(0.994505+0.987069+0.972786)+0.972786=0.997591FB=C\sum_{i=1}^nPV_{t,ti}(1)+PV_{0,t_n}=0.008396(0.994505+0.987069+0.972786)+0.972786=0.997591

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 × (0.997591 – 1.000) = –$602,250.

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

rFIX=1.0−PV0,tn(1)∑i=1nPV0,tn(1)=1.0−0.972786/0.994505+0.987069+0.972786=0.009211r_{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=0.009211

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=(FS0−FSt)∑i=1nPVt,ti=(0.008396−0.009211)×(0.9945+0.9871+0.9728)=−0.002409V=(FS_0-FS_t)\sum_{i=1}^nPV_{t,t_i}=(0.008396-0.009211)\times(0.9945+0.9871+0.9728)=-0.002409

The swap value = $250,000,000 × –0.002409 = –$602,250

0.008396是什么，是怎样得到的这个数

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1 个答案

Lucky_品职助教 · 2023年07月07日

嗨，努力学习的PZer你好：


本题是大题的拆解，0.008396是前面有一问算出来的答案

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加油吧，让我们一起遇见更好的自己！

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NO.PZ2023020101000014 问题如下 Whitney’s first meeting is with Novatel, aUS basecompany thcurrently houtstanng loof $250,000,000 thatcarries a 5.15% fixeinterest rate. Novatel’s managers feel ththe current interestrate on the lois high anthey also believe thinterest rates are poiseo cline. Whitney aises Novatel to enter into a one-yepay-floating Liborreceive-fixeinterest rate swwith quarterly payments. The notionalprincipon the swwill $250,000,000，anthe annualizeswrate is 0.016792. Whitney’s first task is totermine the appropriate swrate.Ninety ys have passesinWhitney’sinitimeetings, anin the interim interest rates have increaseramatically. Whitney’s clients have asketo meet with her to review theirpositions.In orr to prepare for the meeting,Whitney hobtaineupteinterest rate ta this presentein Exhibit 2.Exhibit2 Term Structure of Rates 90 ys Later (%)Using ta in Exhibit 2 ana 30/360 ycount, the market value of Novatel’s swafter 90 ys is closest to: A.–$3,702,900. B.–$2,875,000. C.–$2,408,880. Thepresent value factors for Exhibit 2 are provibelow:For example, PV(180) is calculateas:11+0.0262×(180/360)=0.987069\frac1{1+0.0262\times(180/360)}=0.9870691+0.0262×(180/360)1​=0.987069 Other present value factors are calculaten a similmanner. Using the fixerate initially termineor the swanthe current PV factors, the current value of the fixebons:FB=C∑i=1nPVt,ti(1)+PV0,tn=0.004198(0.994505+0.987069+0.972786)+0.972786=0.9851884FB=C\sum_{i=1}^nPV_{t,ti}(1)+PV_{0,t_n}=0.004198(0.994505+0.987069+0.972786)+0.972786=0.9851884FB=C∑i=1n​PVt,ti​(1)+PV0,tn​​=0.004198(0.994505+0.987069+0.972786)+0.972786=0.9851884 The value of the floating rate bonreset is 1. The marketvalue of the pay-floating, receive fixerate swis the value of thefixerate bonless the value of the floating-rate bon or $250,000,000 ×(0.9851884 – 1.000) = –$3,702,900.Using alternative approach, the newfixeswrate woulberFIX=1.0−PV0,tn(1)∑i=1nPV0,tn(1)=1.0−0.972786/0.994505+0.987069+0.972786=0.00921147r_{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=0.00921147rFIX​=∑i=1n​PV0,tn​​(1)1.0−PV0,tn​​(1)​=1.0−0.972786/0.994505+0.987069+0.972786=0.00921147 Anthe value of the swis thefferenbetween the value the olrate anthe value the new rate, orV=(FS0−FSt)∑i=1nPVt,ti=(0.004198−0.00921147)×(0.9945+0.9871+0.9728)=−0.0148116V=(FS_0-FS_t)\sum_{i=1}^nPV_{t,t_i}=(0.004198-0.00921147)\times(0.9945+0.9871+0.9728)=-0.0148116V=(FS0​−FSt​)∑i=1n​PVt,ti​​=(0.004198−0.00921147)×(0.9945+0.9871+0.9728)=−0.0148116 The swvalue = $250,000,000 × –0.0148116 = –$3,702,900 用老师的画图方法怎么计算这道题目呢？为什么要重新算一遍swrate?

2024-06-09 10:38 1 · 回答

NO.PZ2023020101000014 问题如下 Whitney’s first meeting is with Novatel, aUS basecompany thcurrently houtstanng loof $250,000,000 thatcarries a 5.15% fixeinterest rate. Novatel’s managers feel ththe current interestrate on the lois high anthey also believe thinterest rates are poiseo cline. Whitney aises Novatel to enter into a one-yepay-floating Liborreceive-fixeinterest rate swwith quarterly payments. The notionalprincipon the swwill $250,000,000，anthe annualizeswrate is 0.016792. Whitney’s first task is totermine the appropriate swrate.Ninety ys have passesinWhitney’sinitimeetings, anin the interim interest rates have increaseramatically. Whitney’s clients have asketo meet with her to review theirpositions.In orr to prepare for the meeting,Whitney hobtaineupteinterest rate ta this presentein Exhibit 2.Exhibit2 Term Structure of Rates 90 ys Later (%)Using ta in Exhibit 2 ana 30/360 ycount, the market value of Novatel’s swafter 90 ys is closest to: A.–$3,702,900. B.–$2,875,000. C.–$2,408,880. Thepresent value factors for Exhibit 2 are provibelow:For example, PV(180) is calculateas:11+0.0262×(180/360)=0.987069\frac1{1+0.0262\times(180/360)}=0.9870691+0.0262×(180/360)1​=0.987069 Other present value factors are calculaten a similmanner. Using the fixerate initially termineor the swanthe current PV factors, the current value of the fixebons:FB=C∑i=1nPVt,ti(1)+PV0,tn=0.004198(0.994505+0.987069+0.972786)+0.972786=0.9851884FB=C\sum_{i=1}^nPV_{t,ti}(1)+PV_{0,t_n}=0.004198(0.994505+0.987069+0.972786)+0.972786=0.9851884FB=C∑i=1n​PVt,ti​(1)+PV0,tn​​=0.004198(0.994505+0.987069+0.972786)+0.972786=0.9851884 The value of the floating rate bonreset is 1. The marketvalue of the pay-floating, receive fixerate swis the value of thefixerate bonless the value of the floating-rate bon or $250,000,000 ×(0.9851884 – 1.000) = –$3,702,900.Using alternative approach, the newfixeswrate woulberFIX=1.0−PV0,tn(1)∑i=1nPV0,tn(1)=1.0−0.972786/0.994505+0.987069+0.972786=0.00921147r_{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=0.00921147rFIX​=∑i=1n​PV0,tn​​(1)1.0−PV0,tn​​(1)​=1.0−0.972786/0.994505+0.987069+0.972786=0.00921147 Anthe value of the swis thefferenbetween the value the olrate anthe value the new rate, orV=(FS0−FSt)∑i=1nPVt,ti=(0.004198−0.00921147)×(0.9945+0.9871+0.9728)=−0.0148116V=(FS_0-FS_t)\sum_{i=1}^nPV_{t,t_i}=(0.004198-0.00921147)\times(0.9945+0.9871+0.9728)=-0.0148116V=(FS0​−FSt​)∑i=1n​PVt,ti​​=(0.004198−0.00921147)×(0.9945+0.9871+0.9728)=−0.0148116 The swvalue = $250,000,000 × –0.0148116 = –$3,702,900 每次因为小数点保留的问题都跟答案差一些怎么办呢？比如这道题我算的就是369925，和A查了一些

2024-04-16 11:21 1 · 回答

NO.PZ2023020101000014问题如下 Whitney’s first meeting is with Novatel, aUS basecompany thcurrently houtstanng loof $250,000,000 thatcarries a 5.15% fixeinterest rate. Novatel’s managers feel ththe current interestrate on the lois high anthey also believe thinterest rates are poiseo cline. Whitney aises Novatel to enter into a one-yepay-floating Liborreceive-fixeinterest rate swwith quarterly payments. The notionalprincipon the swwill $250,000,000，anthe annualizeswrate is 0.016792. Whitney’s first task is totermine the appropriate swrate.Ninety ys have passesinWhitney’sinitimeetings, anin the interim interest rates have increaseramatically. Whitney’s clients have asketo meet with her to review theirpositions.In orr to prepare for the meeting,Whitney hobtaineupteinterest rate ta this presentein Exhibit 2.Exhibit2 Term Structure of Rates 90 ys Later (%)Using ta in Exhibit 2 ana 30/360 ycount, the market value of Novatel’s swafter 90 ys is closest to: A.–$3,702,900.B.–$2,875,000.C.–$2,408,880. Thepresent value factors for Exhibit 2 are provibelow:For example, PV(180) is calculateas:11+0.0262×(180/360)=0.987069\frac1{1+0.0262\times(180/360)}=0.9870691+0.0262×(180/360)1​=0.987069 Other present value factors are calculaten a similmanner. Using the fixerate initially termineor the swanthe current PV factors, the current value of the fixebons:FB=C∑i=1nPVt,ti(1)+PV0,tn=0.004198(0.994505+0.987069+0.972786)+0.972786=0.9851884FB=C\sum_{i=1}^nPV_{t,ti}(1)+PV_{0,t_n}=0.004198(0.994505+0.987069+0.972786)+0.972786=0.9851884FB=C∑i=1n​PVt,ti​(1)+PV0,tn​​=0.004198(0.994505+0.987069+0.972786)+0.972786=0.9851884 The value of the floating rate bonreset is 1. The marketvalue of the pay-floating, receive fixerate swis the value of thefixerate bonless the value of the floating-rate bon or $250,000,000 ×(0.9851884 – 1.000) = –$3,702,900.Using alternative approach, the newfixeswrate woulberFIX=1.0−PV0,tn(1)∑i=1nPV0,tn(1)=1.0−0.972786/0.994505+0.987069+0.972786=0.00921147r_{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=0.00921147rFIX​=∑i=1n​PV0,tn​​(1)1.0−PV0,tn​​(1)​=1.0−0.972786/0.994505+0.987069+0.972786=0.00921147 Anthe value of the swis thefferenbetween the value the olrate anthe value the new rate, orV=(FS0−FSt)∑i=1nPVt,ti=(0.004198−0.00921147)×(0.9945+0.9871+0.9728)=−0.0148116V=(FS_0-FS_t)\sum_{i=1}^nPV_{t,t_i}=(0.004198-0.00921147)\times(0.9945+0.9871+0.9728)=-0.0148116V=(FS0​−FSt​)∑i=1n​PVt,ti​​=(0.004198−0.00921147)×(0.9945+0.9871+0.9728)=−0.0148116 The swvalue = $250,000,000 × –0.0148116 = –$3,702,900 为什么要把给的FS0除以4？不要简单的说给的是年利率，按季度所以除4，按答案推这么简单的字面我能看不懂吗，为什么有的题FS0就不作处理直接用？为什么这道题就得除4？题目给的条件有什么差异？根源是在哪儿？我要听这个。而且为什么不把FSt年化，乘以4，直接用给的FS0去减乘4的FSt？

2023-10-21 21:41 2 · 回答

NO.PZ2023020101000014 问题如下 Whitney’s first meeting is with Novatel, aUS basecompany thcurrently houtstanng loof $250,000,000 thatcarries a 5.15% fixeinterest rate. Novatel’s managers feel ththe current interestrate on the lois high anthey also believe thinterest rates are poiseo cline. Whitney aises Novatel to enter into a one-yepay-floating Liborreceive-fixeinterest rate swwith quarterly payments. The notionalprincipon the swwill $250,000,000，anthe annualizeswrate is 0.016792. Whitney’s first task is totermine the appropriate swrate.Ninety ys have passesinWhitney’sinitimeetings, anin the interim interest rates have increaseramatically. Whitney’s clients have asketo meet with her to review theirpositions.In orr to prepare for the meeting,Whitney hobtaineupteinterest rate ta this presentein Exhibit 2.Exhibit2 Term Structure of Rates 90 ys Later (%)Using ta in Exhibit 2 ana 30/360 ycount, the market value of Novatel’s swafter 90 ys is closest to: A.–$3,702,900. B.–$2,875,000. C.–$2,408,880. Thepresent value factors for Exhibit 2 are provibelow:For example, PV(180) is calculateas:11+0.0262×(180/360)=0.987069\frac1{1+0.0262\times(180/360)}=0.9870691+0.0262×(180/360)1​=0.987069 Other present value factors are calculaten a similmanner. Using the fixerate initially termineor the swanthe current PV factors, the current value of the fixebons:FB=C∑i=1nPVt,ti(1)+PV0,tn=0.004198(0.994505+0.987069+0.972786)+0.972786=0.9851884FB=C\sum_{i=1}^nPV_{t,ti}(1)+PV_{0,t_n}=0.004198(0.994505+0.987069+0.972786)+0.972786=0.9851884FB=C∑i=1n​PVt,ti​(1)+PV0,tn​​=0.004198(0.994505+0.987069+0.972786)+0.972786=0.9851884 The value of the floating rate bonreset is 1. The marketvalue of the pay-floating, receive fixerate swis the value of thefixerate bonless the value of the floating-rate bon or $250,000,000 ×(0.9851884 – 1.000) = –$3,702,900.Using alternative approach, the newfixeswrate woulberFIX=1.0−PV0,tn(1)∑i=1nPV0,tn(1)=1.0−0.972786/0.994505+0.987069+0.972786=0.00921147r_{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=0.00921147rFIX​=∑i=1n​PV0,tn​​(1)1.0−PV0,tn​​(1)​=1.0−0.972786/0.994505+0.987069+0.972786=0.00921147 Anthe value of the swis thefferenbetween the value the olrate anthe value the new rate, orV=(FS0−FSt)∑i=1nPVt,ti=(0.004198−0.00921147)×(0.9945+0.9871+0.9728)=−0.0148116V=(FS_0-FS_t)\sum_{i=1}^nPV_{t,t_i}=(0.004198-0.00921147)\times(0.9945+0.9871+0.9728)=-0.0148116V=(FS0​−FSt​)∑i=1n​PVt,ti​​=(0.004198−0.00921147)×(0.9945+0.9871+0.9728)=−0.0148116 The swvalue = $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fixe所以最后计算value，方向应该是fixerate - florate吧2、题目中说float按年pay，annualizeswrate is 0.016792. fixerate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 florate-季度fixerate，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。

2023-10-07 19:09 1 · 回答

NO.PZ2023020101000014问题如下 Whitney’s first meeting is with Novatel, aUS basecompany thcurrently houtstanng loof $250,000,000 thatcarries a 5.15% fixeinterest rate. Novatel’s managers feel ththe current interestrate on the lois high anthey also believe thinterest rates are poiseo cline. Whitney aises Novatel to enter into a one-yepay-floating Liborreceive-fixeinterest rate swwith quarterly payments. The notionalprincipon the swwill $250,000,000，anthe annualizeswrate is 0.016792. Whitney’s first task is totermine the appropriate swrate.Ninety ys have passesinWhitney’sinitimeetings, anin the interim interest rates have increaseramatically. Whitney’s clients have asketo meet with her to review theirpositions.In orr to prepare for the meeting,Whitney hobtaineupteinterest rate ta this presentein Exhibit 2.Exhibit2 Term Structure of Rates 90 ys Later (%)Using ta in Exhibit 2 ana 30/360 ycount, the market value of Novatel’s swafter 90 ys is closest to: A.–$3,702,900.B.–$2,875,000.C.–$2,408,880. Thepresent value factors for Exhibit 2 are provibelow:For example, PV(180) is calculateas:11+0.0262×(180/360)=0.987069\frac1{1+0.0262\times(180/360)}=0.9870691+0.0262×(180/360)1​=0.987069 Other present value factors are calculaten a similmanner. Using the fixerate initially termineor the swanthe current PV factors, the current value of the fixebons:FB=C∑i=1nPVt,ti(1)+PV0,tn=0.004198(0.994505+0.987069+0.972786)+0.972786=0.9851884FB=C\sum_{i=1}^nPV_{t,ti}(1)+PV_{0,t_n}=0.004198(0.994505+0.987069+0.972786)+0.972786=0.9851884FB=C∑i=1n​PVt,ti​(1)+PV0,tn​​=0.004198(0.994505+0.987069+0.972786)+0.972786=0.9851884 The value of the floating rate bonreset is 1. The marketvalue of the pay-floating, receive fixerate swis the value of thefixerate bonless the value of the floating-rate bon or $250,000,000 ×(0.9851884 – 1.000) = –$3,702,900.Using alternative approach, the newfixeswrate woulberFIX=1.0−PV0,tn(1)∑i=1nPV0,tn(1)=1.0−0.972786/0.994505+0.987069+0.972786=0.00921147r_{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=0.00921147rFIX​=∑i=1n​PV0,tn​​(1)1.0−PV0,tn​​(1)​=1.0−0.972786/0.994505+0.987069+0.972786=0.00921147 Anthe value of the swis thefferenbetween the value the olrate anthe value the new rate, orV=(FS0−FSt)∑i=1nPVt,ti=(0.004198−0.00921147)×(0.9945+0.9871+0.9728)=−0.0148116V=(FS_0-FS_t)\sum_{i=1}^nPV_{t,t_i}=(0.004198-0.00921147)\times(0.9945+0.9871+0.9728)=-0.0148116V=(FS0​−FSt​)∑i=1n​PVt,ti​​=(0.004198−0.00921147)×(0.9945+0.9871+0.9728)=−0.0148116 The swvalue = $250,000,000 × –0.0148116 = –$3,702,900 为什么不是折到第90天 而是折到0时点？

2023-10-05 15:13 1 · 回答