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水瓶公主 · 2023年03月07日

用这个公式计算可以吗

  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.

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).

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

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1 个答案

pzqa27 · 2023年03月07日

嗨，从没放弃的小努力你好：


不可以，这个计算方法，直接用四个半年期利率去推算远期利率，前提是这期间不发生现金流，利息最后一次结清。但对于swap来说，它是每隔一段时间都会发生现金流的，这部分现金流再往前折现和往后算终值的时候都是需要用到无风险利率的，就不符合远期利率的公式了。

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努力的时光都是限量版，加油！

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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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2023-03-06 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