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Luhunlu · 2022年03月14日

老师，我用的是课上老师讲的等效债券的做法，结果算出来是6.2115%

  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).

2-year libor spot rate是两年期浮动债券的平均值，所以可以看做是一个固定利率债券，

本题的计算原理就是让一个浮动利率债券的现值与固定利率债券相等来求出最后一期的libor：


老师看看有错吗？


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1 个答案

品职答疑小助手雍 · 2022年03月14日

同学你好，这里没有+100的FV，两种方法我都用excel算了，答案是一样的。

Luhunlu · 2022年03月14日

好的谢谢老师，本金我忘记写上了

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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

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

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