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he123456 · 2022年03月25日

关于0时刻value=0的计算思路不太理解

  1. FRM I
  2. Financial Markets And Products

NO.PZ2020021204000049

问题如下：

Suppose that the six-month Libor rate is 5%, the forward Libor rate for the period between 0.5 and 1.0 year is 5.6% and the forward Libor rate for the period between 1.0 and 1.5 years is 6.0. The two-year Libor swap rate is 5.7%. All risk-free rates are 4.5%. What is the forward Libor rate for the period between 1.5 and 2.0 years? All rates are expressed with semi-annual compounding.

选项：

解释：

A swap where 5.7% is paid and Libor is received is worth zero. Per 100 of principal, first FRA is worth:

0.5  X  (0.05  −  0.057)  X  1001  +  0.045/2\frac{0.5\;X\;(0.05\;-\;0.057)\;X\;100}{1\;+\;0.045/2}= -0.342

The second FRA is worth:

0.5  X  (0.056  −  0.057)  X  100(1  +  0.045/2)2\frac{0.5\;X\;(0.056\;-\;0.057)\;X\;100}{{(1\;+\;0.045/2)}^2}= -0.048

The third FRA is worth:

0.5  X  (0.060  −  0.057)  X  100(1  +  0.045/2)3\frac{0.5\;X\;(0.060\;-\;0.057)\;X\;100}{{(1\;+\;0.045/2)}^3}

= 0.14

If the required forward rate is R then:

0.5  X  (R  −  0.057)  X  100(1  +  0.045/2)4\frac{0.5\;X\;(R\;-\;0.057)\;X\;100}{{(1\;+\;0.045/2)}^4}- 0.342 - 0.048 + 0.140 = 0

This can be solved to give R = 0.0625. The forward rate for the period between 1.5 and 2.0 years is 6.25% (semiannually compounded).

看了助教在别的问题下边的解释说：无所谓哪边收哪边付的，这题算的浮动端和固定端的PV是一致的就可以了，所以才有了最后的那个等式。等式的成立条件就是两个固定和浮动两端现金流相反且PV之和为0。FRA的思路是在swap签订的t=0时刻，value=0这个思路来计算的，这个swap在t=0.5,t=1,t=1.5 t=2.0四个时刻都会产生现金流支付的，所以fra的解法是围绕这个思路来进行的。可是如果按照bond的思维在0时刻浮动利率债券就回归面值NP了呀

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1 个答案

DD仔_品职助教 · 2022年03月25日

嗨，努力学习的PZer你好：


同学你好，

FRA这个现金流在最后一笔到期时刻只有现金流的差值呀，并不是像债券一样还有面值的支付，所以这个和浮动利率债券价格回归面值是不一样的现金流形式。

同学你可以看下答案，蓝色框是四笔现金流，这四笔现金流全部都是利息的差值，没有本金交换。所以和浮动利率债券不一样，不是价格回归面值。

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努力的时光都是限量版，加油！

he123456 · 2022年03月26日

明白了，如果按照value的计算方法也根本没有办法算出f

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NO.PZ2020021204000049问题如下Suppose ththe six-month Libor rate is 5%, the forwarLibor rate for the periobetween 0.5 an1.0 yeis 5.6% anthe forwarLibor rate for the periobetween 1.0 an1.5 years is 6.0. The two-yeLibor swrate is 5.7%. All risk-free rates are 4.5%. Whis the forwarLibor rate for the periobetween 1.5 an2.0 years? All rates are expressewith semi-annucompounng.p.p1 {margin: 0.0px 0.0px 0.0px 0.0px; font: 8.5px Helveticcolor: #443e44}span.s1 {color: #4869}span.s2 {color: #65544b}span.s3 {color: #2f496b}A swwhere 5.7% is paianLibor is receiveis worth zero. Per 100 of principal, first FRA is worth:0.5  X  (0.05  −  0.057)  X  1001  +  0.045/2\frac{0.5\;X\;(0.05\;-\;0.057)\;X\;100}{1\;+\;0.045/2}1+0.045/20.5X(0.05−0.057)X100​= -0.342The seconFRA is worth:0.5  X  (0.056  −  0.057)  X  100(1  +  0.045/2)2\frac{0.5\;X\;(0.056\;-\;0.057)\;X\;100}{{(1\;+\;0.045/2)}^2}(1+0.045/2)20.5X(0.056−0.057)X100​= -0.048The thirFRA is worth:0.5  X  (0.060  −  0.057)  X  100(1  +  0.045/2)3\frac{0.5\;X\;(0.060\;-\;0.057)\;X\;100}{{(1\;+\;0.045/2)}^3}(1+0.045/2)30.5X(0.060−0.057)X100​= 0.14If the requireforwarrate is R then:0.5  X  (R  −  0.057)  X  100(1  +  0.045/2)4\frac{0.5\;X\;(R\;-\;0.057)\;X\;100}{{(1\;+\;0.045/2)}^4}(1+0.045/2)40.5X(R−0.057)X100​- 0.342 - 0.048 + 0.140 = 0This csolveto give R = 0.0625. The forwarrate for the periobetween 1.5 an2.0 years is 6.25% (semiannually compoun.p.p1 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #443f46}p.p2 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #453f47}p.p3 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #484047}p.p4 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #463f46}p.p5 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #453e45}p.p6 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #3f3945}p.p7 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #4435}p.p8 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #46434b}p.p9 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #473f45}p.p10 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #4b4247}p.p11 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #4c474e}p.p12 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #464247}span.s1 {color: #4b6f}span.s2 {color: #67564span.s3 {color: #635f67}span.s4 {color: #7f7b7b}span.s5 {color: #635251}span.s6 {color: #435263}span.s7 {color: #6f6b69}span.s8 {color: #615456}span.s9 {color: #7f7b7f}span.s10 {color: #6b6b6b}span.s11 {font: 6.0px Helvetica}span.s12 {color: #615454}span.s13 {color: #7f7f7f}span.s14 {color: #5c5b65}span.s15 {font: 9.0px Helvetica}span.s16 {color: #6f6b73}span.s17 {color: #7b7span.s18 {color: #4e5c70}那里有呢，沒有印象呢

2024-04-03 00:55 1 · 回答

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2023-03-07 14:36 1 · 回答

NO.PZ2020021204000049 问题如下 Suppose ththe six-month Libor rate is 5%, the forwarLibor rate for the periobetween 0.5 an1.0 yeis 5.6% anthe forwarLibor rate for the periobetween 1.0 an1.5 years is 6.0. The two-yeLibor swrate is 5.7%. All risk-free rates are 4.5%. Whis the forwarLibor rate for the periobetween 1.5 an2.0 years? All rates are expressewith semi-annucompounng.p.p1 {margin: 0.0px 0.0px 0.0px 0.0px; font: 8.5px Helveticcolor: #443e44}span.s1 {color: #4869}span.s2 {color: #65544b}span.s3 {color: #2f496 A swwhere 5.7% is paianLibor is receiveis worth zero. Per 100 of principal, first FRA is worth:0.5  X  (0.05  −  0.057)  X  1001  +  0.045/2\frac{0.5\;X\;(0.05\;-\;0.057)\;X\;100}{1\;+\;0.045/2}1+0.045/20.5X(0.05−0.057)X100​= -0.342The seconFRA is worth:0.5  X  (0.056  −  0.057)  X  100(1  +  0.045/2)2\frac{0.5\;X\;(0.056\;-\;0.057)\;X\;100}{{(1\;+\;0.045/2)}^2}(1+0.045/2)20.5X(0.056−0.057)X100​= -0.048The thirFRA is worth:0.5  X  (0.060  −  0.057)  X  100(1  +  0.045/2)3\frac{0.5\;X\;(0.060\;-\;0.057)\;X\;100}{{(1\;+\;0.045/2)}^3}(1+0.045/2)30.5X(0.060−0.057)X100​= 0.14If the requireforwarrate is R then:0.5  X  (R  −  0.057)  X  100(1  +  0.045/2)4\frac{0.5\;X\;(R\;-\;0.057)\;X\;100}{{(1\;+\;0.045/2)}^4}(1+0.045/2)40.5X(R−0.057)X100​- 0.342 - 0.048 + 0.140 = 0This csolveto give R = 0.0625. The forwarrate for the periobetween 1.5 an2.0 years is 6.25% (semiannually compoun.p.p1 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #443f46}p.p2 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #453f47}p.p3 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #484047}p.p4 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #463f46}p.p5 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #453e45}p.p6 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #3f3945}p.p7 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #4435}p.p8 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #46434b}p.p9 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #473f45}p.p10 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #4b4247}p.p11 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #4c474e}p.p12 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #464247}span.s1 {color: #4b6f}span.s2 {color: #67564span.s3 {color: #635f67}span.s4 {color: #7f7b7b}span.s5 {color: #635251}span.s6 {color: #435263}span.s7 {color: #6f6b69}span.s8 {color: #615456}span.s9 {color: #7f7b7f}span.s10 {color: #6b6b6b}span.s11 {font: 6.0px Helvetica}span.s12 {color: #615454}span.s13 {color: #7f7f7f}span.s14 {color: #5c5b65}span.s15 {font: 9.0px Helvetica}span.s16 {color: #6f6b73}span.s17 {color: #7b7span.s18 {color: #4e5c70} 分子就是浮动和固定的利率差 * 本金，那0.5是什么呢。

2023-03-06 12:29 1 · 回答

NO.PZ2020021204000049 问题如下 Suppose ththe six-month Libor rate is 5%, the forwarLibor rate for the periobetween 0.5 an1.0 yeis 5.6% anthe forwarLibor rate for the periobetween 1.0 an1.5 years is 6.0. The two-yeLibor swrate is 5.7%. All risk-free rates are 4.5%. Whis the forwarLibor rate for the periobetween 1.5 an2.0 years? All rates are expressewith semi-annucompounng.p.p1 {margin: 0.0px 0.0px 0.0px 0.0px; font: 8.5px Helveticcolor: #443e44}span.s1 {color: #4869}span.s2 {color: #65544b}span.s3 {color: #2f496 A swwhere 5.7% is paianLibor is receiveis worth zero. Per 100 of principal, first FRA is worth:0.5  X  (0.05  −  0.057)  X  1001  +  0.045/2\frac{0.5\;X\;(0.05\;-\;0.057)\;X\;100}{1\;+\;0.045/2}1+0.045/20.5X(0.05−0.057)X100​= -0.342The seconFRA is worth:0.5  X  (0.056  −  0.057)  X  100(1  +  0.045/2)2\frac{0.5\;X\;(0.056\;-\;0.057)\;X\;100}{{(1\;+\;0.045/2)}^2}(1+0.045/2)20.5X(0.056−0.057)X100​= -0.048The thirFRA is worth:0.5  X  (0.060  −  0.057)  X  100(1  +  0.045/2)3\frac{0.5\;X\;(0.060\;-\;0.057)\;X\;100}{{(1\;+\;0.045/2)}^3}(1+0.045/2)30.5X(0.060−0.057)X100​= 0.14If the requireforwarrate is R then:0.5  X  (R  −  0.057)  X  100(1  +  0.045/2)4\frac{0.5\;X\;(R\;-\;0.057)\;X\;100}{{(1\;+\;0.045/2)}^4}(1+0.045/2)40.5X(R−0.057)X100​- 0.342 - 0.048 + 0.140 = 0This csolveto give R = 0.0625. The forwarrate for the periobetween 1.5 an2.0 years is 6.25% (semiannually compoun.p.p1 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #443f46}p.p2 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #453f47}p.p3 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #484047}p.p4 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #463f46}p.p5 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #453e45}p.p6 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #3f3945}p.p7 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #4435}p.p8 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #46434b}p.p9 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #473f45}p.p10 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #4b4247}p.p11 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #4c474e}p.p12 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #464247}span.s1 {color: #4b6f}span.s2 {color: #67564span.s3 {color: #635f67}span.s4 {color: #7f7b7b}span.s5 {color: #635251}span.s6 {color: #435263}span.s7 {color: #6f6b69}span.s8 {color: #615456}span.s9 {color: #7f7b7f}span.s10 {color: #6b6b6b}span.s11 {font: 6.0px Helvetica}span.s12 {color: #615454}span.s13 {color: #7f7f7f}span.s14 {color: #5c5b65}span.s15 {font: 9.0px Helvetica}span.s16 {color: #6f6b73}span.s17 {color: #7b7span.s18 {color: #4e5c70} 假设本金100，每期固定coupon 2.85，那么2.85/(1+5%/2)+2.85/(1+5.6%/2) 2 +2.85/(1+6%/2) 3 +102.85/(1+X/2) 4 =100倒算X，为什么不可以？

2022-05-26 01:51 1 · 回答

NO.PZ2020021204000049 问题如下 Suppose ththe six-month Libor rate is 5%, the forwarLibor rate for the periobetween 0.5 an1.0 yeis 5.6% anthe forwarLibor rate for the periobetween 1.0 an1.5 years is 6.0. The two-yeLibor swrate is 5.7%. All risk-free rates are 4.5%. Whis the forwarLibor rate for the periobetween 1.5 an2.0 years? All rates are expressewith semi-annucompounng.p.p1 {margin: 0.0px 0.0px 0.0px 0.0px; font: 8.5px Helveticcolor: #443e44}span.s1 {color: #4869}span.s2 {color: #65544b}span.s3 {color: #2f496 A swwhere 5.7% is paianLibor is receiveis worth zero. Per 100 of principal, first FRA is worth:0.5  X  (0.05  −  0.057)  X  1001  +  0.045/2\frac{0.5\;X\;(0.05\;-\;0.057)\;X\;100}{1\;+\;0.045/2}1+0.045/20.5X(0.05−0.057)X100​= -0.342The seconFRA is worth:0.5  X  (0.056  −  0.057)  X  100(1  +  0.045/2)2\frac{0.5\;X\;(0.056\;-\;0.057)\;X\;100}{{(1\;+\;0.045/2)}^2}(1+0.045/2)20.5X(0.056−0.057)X100​= -0.048The thirFRA is worth:0.5  X  (0.060  −  0.057)  X  100(1  +  0.045/2)3\frac{0.5\;X\;(0.060\;-\;0.057)\;X\;100}{{(1\;+\;0.045/2)}^3}(1+0.045/2)30.5X(0.060−0.057)X100​= 0.14If the requireforwarrate is R then:0.5  X  (R  −  0.057)  X  100(1  +  0.045/2)4\frac{0.5\;X\;(R\;-\;0.057)\;X\;100}{{(1\;+\;0.045/2)}^4}(1+0.045/2)40.5X(R−0.057)X100​- 0.342 - 0.048 + 0.140 = 0This csolveto give R = 0.0625. The forwarrate for the periobetween 1.5 an2.0 years is 6.25% (semiannually compoun.p.p1 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #443f46}p.p2 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #453f47}p.p3 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #484047}p.p4 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #463f46}p.p5 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #453e45}p.p6 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #3f3945}p.p7 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #4435}p.p8 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #46434b}p.p9 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #473f45}p.p10 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #4b4247}p.p11 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #4c474e}p.p12 {margin: 0.0px 0.0px 0.0px 0.0px; font: 9.5px Helveticcolor: #464247}span.s1 {color: #4b6f}span.s2 {color: #67564span.s3 {color: #635f67}span.s4 {color: #7f7b7b}span.s5 {color: #635251}span.s6 {color: #435263}span.s7 {color: #6f6b69}span.s8 {color: #615456}span.s9 {color: #7f7b7f}span.s10 {color: #6b6b6b}span.s11 {font: 6.0px Helvetica}span.s12 {color: #615454}span.s13 {color: #7f7f7f}span.s14 {color: #5c5b65}span.s15 {font: 9.0px Helvetica}span.s16 {color: #6f6b73}span.s17 {color: #7b7span.s18 {color: #4e5c70} 1、为什么用无风险利率折现？，固定利率产生的现金流折现不是应该用浮动利率么？2、为什么不能直接这么算:(1+5%/2)(1+5.6%/2)(1+6%/2)(1+X/2)=(1+5.7%/2) 4 这样算出来X=6.2%现金流不就是应该为了各期利率相乘和YTM一致一样的道理么？

2022-05-26 01:31 2 · 回答