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eva · 2024年04月03日

这个是新公式吗

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.

解释：

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

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

The second FRA is worth:

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

The third FRA is worth:

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

= 0.14

If the required forward rate is R then:

$\frac{0.5\;X\;(R\;-\;0.057)\;X\;100}{{(1\;+\;0.045/2)}^4}$ - 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).

那里有呢，沒有印象呢

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1 个答案

品职答疑小助手雍 · 2024年04月03日

同学你好，这题没有用什么奇怪的公式，就是单纯的用了swap两端的PV在0时刻是相等的这个一点。

过程如下，把swap拆成4个FRA：

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：

以t=0.5时刻为例，因为pay swap，receive libor ，所以当时的现金流情况应该是（5%/2-5.7%/2）×100，因为这个现金流是发生在t=0.5时刻，所以需要对他进行折现到t=0时刻，因为有(（5%/2-5.7%/2）×100)/(1+4.5%/2)=-0.342

后面t=1,t=1.5 t=2.0也按照这个思路来计算，这几个值在t=0时刻加起来应该为0，所以可解方程，得到题目中的结果。

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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}(1+5%/2)(1+5.6%/2)(1+6%/2)(1+X/2)=(1+5.7%/2) ^4

2023-03-07 14:36 1 · 回答

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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