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Yoana · 2024年07月18日

priced at 85.784357 per 100 of par value

  1. CFA I
  2. Fixed Income

NO.PZ2016031001000124

问题如下：

A bond with exactly nine years remaining until maturity offers a 3% coupon rate with annual coupons. The bond, with a yield-to-maturity of 5%, is priced at 85.784357 per 100 of par value. The estimated price value of a basis point for the bond is closest to:

选项：

A.

0.0086.

B.

0.0648.

C.

0.1295.

解释：

B is correct.

The PVBP is closest to 0.0648. The formula for the price value of a basis pointis:

PVBP=(PV−)−(PV+)2PVBP=\frac{(PV-)-(PV+)}2

where:

PVBP = price value of a basis point

PV– = full price calculated by lowering the yield-to-maturity by one basis point

PV+ = full price calculated by raising the yield-to-maturity by one basis point

Lowering the yield-to-maturity by one basis point to 4.99% results in a bond price of 85.849134:

PV−=3(1+0.0499)1+⋯+3+100(1+0.0499)9=85.849134PV-=\frac3{{(1+0.0499)}^1}+\cdots+\frac{3+100}{{(1+0.0499)}^9}=85.849134

Increasing the yield-to-maturity by one basis point to 5.01% results in a bond price of 85.719638:

PV+=3(1+0.0501)1+⋯+3+100(1+0.0501)9=85.719638PV+=\frac3{{(1+0.0501)}^1}+\cdots+\frac{3+100}{{(1+0.0501)}^9}=85.719638

PVBP=85.849134−85.7196382=0.06475PVBP=\frac{85.849134-85.719638}2=0.06475

Alternatively, the PVBP can be derived using modified duration:

ApproxModDur=(PV−)−(PV+)2×(ΔYield)×(PV0)ApproxModDur=\frac{(PV-)-(PV+)}{2\times(\Delta Yield)\times(PV0)}

ApproxModDur=85.849134−85.7196382×0.0001×85.784357=7.548ApproxModDur=\frac{85.849134-85.719638}{2\times0.0001\times85.784357}=7.548

PVBP = 7.548 × 85.784357 × 0.0001 = 0.06475

考点：PVBP

解析：PVBP衡量的是利率变化1bp，带来的债券价格变化。

分别算出利率下降1个bp的债券价格PV-(85.849134)和利率上升一个bp的债券价格PV+(85.719638)，两者相减再除以2，即可得PVBP=0.0648，故选项B正确。

注意：如果直接算出PV+（或PV-），它和原本的PV之间本身就是由于利率变化1bp造成的，所以两者直接相减即可，不需要除以2。 如果分别算出PV-和PV+，那么它俩之间是由于利率相差2bps造成的，所以相减之后需要除以2。

老师想问一下这句话怎么理解 “priced at 85.784357 per 100 of par value” 在这道题中又怎么用呢。 如果我直接用 （PV_-PV+）/2 可以吗

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1 个答案

吴昊_品职助教 · 2024年07月18日

嗨，努力学习的PZer你好：


这句话的意思是面值为100的债券，现在价格是85.784357，也就是PV0。有两种思路：

1、既然有了PV0，那我们就只要算出PV+或者PV-其中的一个，然后直接和PV0相减，就可以得到PVBP。因为PVBP代表的是利率变动1bp带来的价格变化，PV+和PV0之间就差一个bp，PV-和PV0直接也相差一个bp。

2、我们也可以像解析中一样，分别求出PV-和PV+，然后相减再除以2.

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努力的时光都是限量版，加油！

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NO.PZ2016031001000124 问题如下 A bonwith exactly nine years remaining until maturity offers a 3% coupon rate with annucoupons. The bon with a yielto-maturity of 5%, is price85.784357 per 100 of pvalue. The estimateprivalue of a basis point for the bonis closest to: A.0.0086. B.0.0648. C.0.1295. B is correct.The PVis closest to 0.0648. The formula for the privalue of a basis pointis: PVBP=(PV−)−(PV+)2PVBP=\frac{(PV-)-(PV+)}2PVBP=2(PV−)−(PV+)​where:PV= privalue of a basis pointPV– = full pricalculatelowering the yielto-maturity one basis pointPV+ = full pricalculateraising the yielto-maturity one basis pointLowering the yielto-maturity one basis point to 4.99% results in a bonpriof 85.849134:PV−=3(1+0.0499)1+⋯+3+100(1+0.0499)9=85.849134PV-=\frac3{{(1+0.0499)}^1}+\cts+\frac{3+100}{{(1+0.0499)}^9}=85.849134PV−=(1+0.0499)13​+⋯+(1+0.0499)93+100​=85.849134Increasing the yielto-maturity one basis point to 5.01% results in a bonpriof 85.719638:PV+=3(1+0.0501)1+⋯+3+100(1+0.0501)9=85.719638PV+=\frac3{{(1+0.0501)}^1}+\cts+\frac{3+100}{{(1+0.0501)}^9}=85.719638PV+=(1+0.0501)13​+⋯+(1+0.0501)93+100​=85.719638PVBP=85.849134−85.7196382=0.06475PVBP=\frac{85.849134-85.719638}2=0.06475PVBP=285.849134−85.719638​=0.06475Alternatively, the PVcriveusing mofieration:ApproxMour=(PV−)−(PV+)2×(ΔYiel×(PV0)ApproxMour=\frac{(PV-)-(PV+)}{2\times(\lta Yiel\times(PV0)}ApproxMour=2×(ΔYiel×(PV0)(PV−)−(PV+)​ApproxMour=85.849134−85.7196382×0.0001×85.784357=7.548ApproxMour=\frac{85.849134-85.719638}{2\times0.0001\times85.784357}=7.548ApproxMour=2×0.0001×85.78435785.849134−85.719638​=7.548PV= 7.548 × 85.784357 × 0.0001 = 0.06475考点PVBP解析PVBP衡量的是利率变化1bp，带来的债券价格变化。分别算出利率下降1个bp的债券价格PV-(85.849134)和利率上升一个bp的债券价格PV+(85.719638)，两者相减再除以2，即可得PVBP=0.0648，故B正确。注意如果直接算出PV+（或PV-），它和原本的PV之间本身就是由于利率变化1bp造成的，所以两者直接相减即可，不需要除以2。 如果分别算出PV-和PV+，那么它俩之间是由于利率相差2bps造成的，所以相减之后需要除以2。 The estimateprivalue of a basis point for the bonis closest to，题目的问题是如何对应在PVBP计算知识点上的

2024-05-22 21:46 1 · 回答

NO.PZ2016031001000124问题如下A bonwith exactly nine years remaining until maturity offers a 3% coupon rate with annucoupons. The bon with a yielto-maturity of 5%, is price85.784357 per 100 of pvalue. The estimateprivalue of a basis point for the bonis closest to:A.0.0086.B.0.0648.C.0.1295. B is correct.The PVis closest to 0.0648. The formula for the privalue of a basis pointis: PVBP=(PV−)−(PV+)2PVBP=\frac{(PV-)-(PV+)}2PVBP=2(PV−)−(PV+)​where:PV= privalue of a basis pointPV– = full pricalculatelowering the yielto-maturity one basis pointPV+ = full pricalculateraising the yielto-maturity one basis pointLowering the yielto-maturity one basis point to 4.99% results in a bonpriof 85.849134:PV−=3(1+0.0499)1+⋯+3+100(1+0.0499)9=85.849134PV-=\frac3{{(1+0.0499)}^1}+\cts+\frac{3+100}{{(1+0.0499)}^9}=85.849134PV−=(1+0.0499)13​+⋯+(1+0.0499)93+100​=85.849134Increasing the yielto-maturity one basis point to 5.01% results in a bonpriof 85.719638:PV+=3(1+0.0501)1+⋯+3+100(1+0.0501)9=85.719638PV+=\frac3{{(1+0.0501)}^1}+\cts+\frac{3+100}{{(1+0.0501)}^9}=85.719638PV+=(1+0.0501)13​+⋯+(1+0.0501)93+100​=85.719638PVBP=85.849134−85.7196382=0.06475PVBP=\frac{85.849134-85.719638}2=0.06475PVBP=285.849134−85.719638​=0.06475Alternatively, the PVcriveusing mofieration:ApproxMour=(PV−)−(PV+)2×(ΔYiel×(PV0)ApproxMour=\frac{(PV-)-(PV+)}{2\times(\lta Yiel\times(PV0)}ApproxMour=2×(ΔYiel×(PV0)(PV−)−(PV+)​ApproxMour=85.849134−85.7196382×0.0001×85.784357=7.548ApproxMour=\frac{85.849134-85.719638}{2\times0.0001\times85.784357}=7.548ApproxMour=2×0.0001×85.78435785.849134−85.719638​=7.548PV= 7.548 × 85.784357 × 0.0001 = 0.06475考点PVBP解析PVBP衡量的是利率变化1bp，带来的债券价格变化。分别算出利率下降1个bp的债券价格PV-(85.849134)和利率上升一个bp的债券价格PV+(85.719638)，两者相减再除以2，即可得PVBP=0.0648，故B正确。注意如果直接算出PV+（或PV-），它和原本的PV之间本身就是由于利率变化1bp造成的，所以两者直接相减即可，不需要除以2。 如果分别算出PV-和PV+，那么它俩之间是由于利率相差2bps造成的，所以相减之后需要除以2。 请问算pv+和pv-能用计算器吗?

2022-08-04 03:47 1 · 回答

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2022-08-01 07:10 1 · 回答

NO.PZ2016031001000124 0.0648. 0.1295. B is correct. The PVis closest to 0.0648. The formula for the privalue of a basis pointis: PVBP=(PV−)−(PV+)2PVBP=\frac{(PV-)-(PV+)}2PVBP=2(PV−)−(PV+)​ where: PV= privalue of a basis point PV– = full pricalculatelowering the yielto-maturity one basis point PV+ = full pricalculateraising the yielto-maturity one basis point Lowering the yielto-maturity one basis point to 4.99% results in a bonpriof 85.849134: PV−=3(1+0.0499)1+⋯+3+100(1+0.0499)9=85.849134PV-=\frac3{{(1+0.0499)}^1}+\cts+\frac{3+100}{{(1+0.0499)}^9}=85.849134PV−=(1+0.0499)13​+⋯+(1+0.0499)93+100​=85.849134 Increasing the yielto-maturity one basis point to 5.01% results in a bonpriof 85.719638: PV+=3(1+0.0501)1+⋯+3+100(1+0.0501)9=85.719638PV+=\frac3{{(1+0.0501)}^1}+\cts+\frac{3+100}{{(1+0.0501)}^9}=85.719638PV+=(1+0.0501)13​+⋯+(1+0.0501)93+100​=85.719638 PVBP=85.849134−85.7196382=0.06475PVBP=\frac{85.849134-85.719638}2=0.06475PVBP=285.849134−85.719638​=0.06475 Alternatively, the PVcriveusing mofieration: ApproxMour=(PV−)−(PV+)2×(ΔYiel×(PV0)ApproxMour=\frac{(PV-)-(PV+)}{2\times(\lta Yiel\times(PV0)}ApproxMour=2×(ΔYiel×(PV0)(PV−)−(PV+)​ ApproxMour=85.849134−85.7196382×0.0001×85.784357=7.548ApproxMour=\frac{85.849134-85.719638}{2\times0.0001\times85.784357}=7.548ApproxMour=2×0.0001×85.78435785.849134−85.719638​=7.548 PV= 7.548 × 85.784357 × 0.0001 = 0.06475 考点PV解析PVBP衡量的是利率变化1bp，带来的债券价格变化。 分别算出利率下降1个bp的债券价格PV-(85.849134)和利率上升一个bp的债券价格PV+(85.719638)，两者相减再除以2，即可得PVBP=0.0648，故B正确。 注意如果直接算出PV+（或PV-），它和原本的PV之间本身就是由于利率变化1bp造成的，所以两者直接相减即可，不需要除以2。 如果分别算出PV-和PV+，那么它俩之间是由于利率相差2bps造成的，所以相减之后需要除以2。 如题。。。。。。。谢谢。

2022-01-14 17:52 1 · 回答