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Victutu · 2020年07月26日

问一道题：NO.PZ2016031001000067

  1. CFA I
  2. Fixed Income

问题如下：

Based upon the given sequence of spot rates, the yield-to-maturity of Bond Z is closest to:

选项：

A.

9.00%.

B.

9.92%.

C.

11.93%

解释：

B is correct.

The yield-to-maturity is closest to 9.92%. The formula for calculating the price of Bond Z is:

PV=PMT(1+Z1)1+PMT(1+Z2)2+PMT+FV(1+Z3)3PV=\frac{PMT}{{(1+Z_1)}^1}+\frac{PMT}{{(1+Z_2)}^2}+\frac{PMT+FV}{{(1+Z_3)}^3}

where:

PV = present value, or the price of the bond

PMT = coupon payment per period

FV = future value paid at maturity, or the par value of the bond

Z1= spot rate, or the zero-coupon yield, or zero rate, for period 1

Z2= spot rate, or the zero-coupon yield, or zero rate, for period 2

Z3=spot rate, or the zero-coupon yield, or zero rate, for period 3

PV=6(1+0.08)1+6(1+0.09)2+6+100(1+0.10)3PV=\frac6{{(1+0.08)}^1}+\frac6{{(1+0.09)}^2}+\frac{6\text{+}100}{{(1+0.10)}^3}

PV = 5.56 + 5.05 + 79.64 = 90.25

Using this price, the bond’s yield-to-maturity can be calculated as:

PV=PMT(1+r)1+PMT(1+r)2+PMT+FV(1+r)3PV=\frac{PMT}{{(1+\text{r})}^1}+\frac{PMT}{{(1+\text{r})}^2}+\frac{PMT+FV}{{(1+\text{r})}^3}

90.25=6(1+r)1+6(1+r)2+6+100(1+r)390.25=\frac6{{(1+\text{r})}^1}+\frac6{{(1+\text{r})}^2}+\frac{6\text{+}100}{{(1+\text{r})}^3}

r = 9.92%

我没算出这个题目的三个答案，我的思路是，P=7/(1+9%)+8(1+8%)^2+106/(1+10)^3=92.9201，然后再用计算器PV=-92.9201，PMT=6，N=3，FV=100，算出I/Y=8.7863，请帮我剖析一下我的错误啊，谢谢！

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1 个答案

吴昊_品职助教 · 2020年07月26日

同学你好：

这道题的表格得分开来看。前三列合并在一起，代表的是债券X，coupon rate=8%；债券Y，coupon rate=7%；债券Z，coupon rate=6%。后两列放在一起看，代表的是一年期spot rate=S1=8%，S2=9%，S3=10%。

我们这里用到的原理就是用YTM给债券等价和用spot rate给债券定价，得到的债券价格一样。所以我们先利用spot rate折现得到债券的价格，然后再通过刚才求得的债券价格，反过来求出YTM。

第一步，用spot rate给债券定价时，分子上的coupon payment应该是恒定的6，而分母上分别是一年期、两年期、三年期的spot rate。列式可得：P=6/(1+8%)+6(1+9%)^2+106/(1+10%)^3=90.25。有了价格90.25，再利用计算器第三行求得YTM，PV= -90.25，PMT=6，N=3，FV=100，求得I/Y=9.92

虽然说这道题的表格排版看上去很奇怪，但这是原本书课后题，所以我们得习惯这样的表现形式。


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NO.PZ2016031001000067 问题如下 Baseupon the given sequenof spot rates, the yielto-maturity of BonZ is closest to: A.9.00%. B.9.92%. C.11.93% B is correct.The yielto-maturity is closest to 9.92%. The formula for calculating the priof BonZ is:PV=PMT(1+Z1)1+PMT(1+Z2)2+PMT+FV(1+Z3)3PV=\frac{PMT}{{(1+Z_1)}^1}+\frac{PMT}{{(1+Z_2)}^2}+\frac{PMT+FV}{{(1+Z_3)}^3}PV=(1+Z1​)1PMT​+(1+Z2​)2PMT​+(1+Z3​)3PMT+FV​PV=6(1+0.08)1+6(1+0.09)2+6+100(1+0.10)3PV=\frac6{{(1+0.08)}^1}+\frac6{{(1+0.09)}^2}+\frac{6\text{+}100}{{(1+0.10)}^3}PV=(1+0.08)16​+(1+0.09)26​+(1+0.10)36+100​PV = 5.56 + 5.05 + 79.64 = 90.25Using this price, the bons yielto-maturity ccalculateas:PV=PMT(1+r)1+PMT(1+r)2+PMT+FV(1+r)3PV=\frac{PMT}{{(1+\text{r})}^1}+\frac{PMT}{{(1+\text{r})}^2}+\frac{PMT+FV}{{(1+\text{r})}^3}PV=(1+r)1PMT​+(1+r)2PMT​+(1+r)3PMT+FV​90.25=6(1+r)1+6(1+r)2+6+100(1+r)390.25=\frac6{{(1+\text{r})}^1}+\frac6{{(1+\text{r})}^2}+\frac{6\text{+}100}{{(1+\text{r})}^3}90.25=(1+r)16​+(1+r)26​+(1+r)36+100​r = 9.92%考点Pricing Bon with Spot Rates解析表格前三列代表的是债券X按年支付的Coupon rate是8%，还有3年时间到期。债券Y按年支付的Coupon rate是7%，还有3年时间到期。债券Z按年支付的Coupon rate是6%，还有3年时间到期。三个债券都有三年的期限。表格后两列代表的是第一年的Spot rate是8%，第二年的Spot rate是9%，第三年的Spot rate是10%。所以相对应的，第一年现金流用8%折现，第二年用9%，第三年用10%。通过未来现金流折现求和，第一年的现金流(6)用S1(8%)折现，第二年的现金流(6)用S2(9%)折现，第三年的现金流(6+100)用S3(10%)折现，可得债券价格为90.25。利用计算器N=3，PMT=6，PV= -90.25，FV=100，求得I/Y=9.915，故B正确。 N=3 PV=-90.245004 PMT=6 FV=100 - I/Y=-2.937*

2023-03-11 05:43 1 · 回答

9.92%. 11.93% B is correct. The yielto-maturity is closest to 9.92%. The formula for calculating the priof BonZ is: PV=PMT(1+Z1)1+PMT(1+Z2)2+PMT+FV(1+Z3)3PV=\frac{PMT}{{(1+Z_1)}^1}+\frac{PMT}{{(1+Z_2)}^2}+\frac{PMT+FV}{{(1+Z_3)}^3}PV=(1+Z1​)1PMT​+(1+Z2​)2PMT​+(1+Z3​)3PMT+FV​ where: PV = present value, or the priof the bonPMT = coupon payment per perioFV = future value paimaturity, or the pvalue of the bonZ1= spot rate, or the zero-coupon yiel or zero rate, for perio1 Z2= spot rate, or the zero-coupon yiel or zero rate, for perio2 Z3=spot rate, or the zero-coupon yiel or zero rate, for perio3 PV=6(1+0.08)1+6(1+0.09)2+6+100(1+0.10)3PV=\frac6{{(1+0.08)}^1}+\frac6{{(1+0.09)}^2}+\frac{6\text{+}100}{{(1+0.10)}^3}PV=(1+0.08)16​+(1+0.09)26​+(1+0.10)36+100​ PV = 5.56 + 5.05 + 79.64 = 90.25 Using this price, the bons yielto-maturity ccalculateas: PV=PMT(1+r)1+PMT(1+r)2+PMT+FV(1+r)3PV=\frac{PMT}{{(1+\text{r})}^1}+\frac{PMT}{{(1+\text{r})}^2}+\frac{PMT+FV}{{(1+\text{r})}^3}PV=(1+r)1PMT​+(1+r)2PMT​+(1+r)3PMT+FV​ 90.25=6(1+r)1+6(1+r)2+6+100(1+r)390.25=\frac6{{(1+\text{r})}^1}+\frac6{{(1+\text{r})}^2}+\frac{6\text{+}100}{{(1+\text{r})}^3}90.25=(1+r)16​+(1+r)26​+(1+r)36+100​ r = 9.92% 老师，请问z1z2z3的数是哪里来的？没太理解

2020-03-18 08:48 1 · 回答

9.92%. 11.93% B is correct. The yielto-maturity is closest to 9.92%. The formula for calculating the priof BonZ is: PV=PMT(1+Z1)1+PMT(1+Z2)2+PMT+FV(1+Z3)3PV=\frac{PMT}{{(1+Z_1)}^1}+\frac{PMT}{{(1+Z_2)}^2}+\frac{PMT+FV}{{(1+Z_3)}^3}PV=(1+Z1​)1PMT​+(1+Z2​)2PMT​+(1+Z3​)3PMT+FV​ where: PV = present value, or the priof the bonPMT = coupon payment per perioFV = future value paimaturity, or the pvalue of the bonZ1= spot rate, or the zero-coupon yiel or zero rate, for perio1 Z2= spot rate, or the zero-coupon yiel or zero rate, for perio2 Z3=spot rate, or the zero-coupon yiel or zero rate, for perio3 PV=6(1+0.08)1+6(1+0.09)2+6+100(1+0.10)3PV=\frac6{{(1+0.08)}^1}+\frac6{{(1+0.09)}^2}+\frac{6\text{+}100}{{(1+0.10)}^3}PV=(1+0.08)16​+(1+0.09)26​+(1+0.10)36+100​ PV = 5.56 + 5.05 + 79.64 = 90.25 Using this price, the bons yielto-maturity ccalculateas: PV=PMT(1+r)1+PMT(1+r)2+PMT+FV(1+r)3PV=\frac{PMT}{{(1+\text{r})}^1}+\frac{PMT}{{(1+\text{r})}^2}+\frac{PMT+FV}{{(1+\text{r})}^3}PV=(1+r)1PMT​+(1+r)2PMT​+(1+r)3PMT+FV​ 90.25=6(1+r)1+6(1+r)2+6+100(1+r)390.25=\frac6{{(1+\text{r})}^1}+\frac6{{(1+\text{r})}^2}+\frac{6\text{+}100}{{(1+\text{r})}^3}90.25=(1+r)16​+(1+r)26​+(1+r)36+100​ r = 9.92%这道题可以想成与几个spot rate有关的YTM，可以看成是spot rate的打包价，最后一笔本金加利息的现金流最大，所以占比最大。所以最接近S3=10%，吗？

2020-03-15 15:30 1 · 回答

为什么这道题可以用XY在第一年和第二年的spot rate 来计算Z 的 YTM 呢？觉得XY coupon 和Z也不同诶

2020-03-12 18:05 1 · 回答