spot rate 计算，就是通过用spot rate 折现的价格应该和用YTM折现的价格相等得到的，为什么还要先求spot rate，再求价格，而不是直接用YTM求价格呢？
问题如下图：
选项：
A. 
B. 
C. 
发亮_品职助教 · 2018年04月11日
同学你好。
先理顺一下这里的几个rate。
Spot rate是通过政府发行的零息债券（Government zero coupon bond）求出来的折现率。例如，1年的ZCB对应的是1年期的spot rate，2年期的ZCB折现率对应的是2年期的spot rate，3年对应的是3年的spot rate等等。由于不一定有每一个maturity到期的ZCB，我们就引入了Bootstrapping的方法求spot rate。本题中用par yield curve求spot rate就用到了bootstrapping的方法。
本题，我们知道benchmark的par yield curve。par yield是使得债券以面额发行的coupon rate。于是我们就知道：以par yield为coupon rate的1年期债券现值是100，那么我们自然可以求出来1年期的折现率，这里的折现率就是benchmark spot rate。利用本题Exhibit 2就可以求出来1年期、2年期、3年期的benchmark spot rate。spot rate和maturity一一对应的关系，我们就可以得到spot rate curve。
知道了spot rate我们就可以求得债券的【内在价值】，在本题中我们知道了benchmark spot rate。我们以他为折现率，就可以知道债券合理定价下的价格。
再来说说YTM。
YTM是债券持有至到期的投资收益率。既然是投资收益率，那么不同的购买价格会产生出不同的YTM。债券购买价格和YTM一一对应的关系，可以构成YTM curve。
单单从计算YTM来看，我们是用不上spot rate的。如果你还记得按计算器求YTM，我们需要的数据有：债券的购买价格（PV），债券的每期现金流(PMT)，债券的到期价格（FV），债券的maturity（N）。用以上4个输入，可以求得债券的YTM，缺一不可。
那么什么时候YTM可以和Spot rate联系起来呢？
那就是债券的购买价格（市场价格）等于债券的内在价值。
如果题目给定spot rate，再让你求债券的YTM。实际上是假设投资者能以内在价值在市场上购入债券。所以，我们可以先用spot rate求出来债券的内在价格，然后让债券的购买价格PV等于内在价值求YTM。
本题中，我们连Spot rate都不知道，他只给定了Par yield，所以我们先要通过benchmark par yield求出来benchmark spot rate。然后用spot rate求出来债券的内在价值。
在Exhibit 3中，我们可以直接读取债券的购买价格，上面我们求出来了债券的内在价值，所以misprice多少可以直接求出。
NO.PZ201602270200001902 问题如下 2. Baseon Exhibits 2 an3 anusing Metho1, the amount (in absolute terms) whithe Hutto-Barkley corporate bonis mispriceis closest to: A.0.3368 per 100 of pvalue. B.0.4682 per 100 of pvalue. C.0.5156 per 100 of pvalue. C is correct.The first step in the solution is to finthe correspot rate (zero-coupon rates) for eayear’s cash flow. The benchmark bon in Exhibit 2 are conveniently pricepso the yiel to maturity anthe coupon rates on the bon are the same. Because the one-yeissue honly one cash flow remaining, the YTM equals the spot rate of 3% (or z1z_1z1= 3%). The spot rates for Ye2 ( z2z_2z2) anYe3 (z3z_3z3 ) are calculatefollows: beginarrayl100=41.0300+104(1+z2)2;z2=4.02%100=51.0300+5(1.0402)2+105(1+z3)3;z3=5.07%begin{array}{l}100 = \frac{4}{{1.0300}} + \frac{{104}}{{{{(1 + {z_2})}^2}}};{z_2} = 4.02\% \\100 = \frac{5}{{1.0300}} + \frac{5}{{{{(1.0402)}^2}}} + \frac{{105}}{{{{(1 + {z_3})}^3}}};{z_3} = 5.07\% beginarrayl100=1.03004+(1+z2)2104;z2=4.02%100=1.03005+(1.0402)25+(1+z3)3105;z3=5.07%The correarbitrage-free prifor the Hutto-Barkley Inbonis:P0=3(1.0300)+3(1.0402)2+103(1.0507)3=94.4828P_0=\frac3{(1.0300)}+\frac3{{(1.0402)}^2}+\frac{103}{{(1.0507)}^3}=94.4828P0=(1.0300)3+(1.0402)23+(1.0507)3103=94.4828Therefore, the bonis misprice94.4828 – 94.9984 = –0.5156 per 100 of pvalue.A is incorrebecause the correspot rates are not calculateaninstethe Hutto-Barkley Inbonis scounteusing the respective YTM for eamaturity. Therefore, this lea to incorremispricing of 94.6616 – 94.9984 = –0.3368 per 100 of pvalue.B is incorrebecause the spot rates are riveusing the coupon rate for Ye3 (maturity) insteof using eayear’s respective coupon rate to employ the bootstrmethology. This lea to incorremispricing of 94.5302 – 94.9984 = –0.4682 per 100 of pvalue. 公式完全正确，但我每步计算都保留了小数点后四位，结果算出来的结果更接近B。后来我把中间步骤保留小数点后两位，再计算基本是C。所以到底怎么保留小数点?
NO.PZ201602270200001902 问题如下 2. Baseon Exhibits 2 an3 anusing Metho1, the amount (in absolute terms) whithe Hutto-Barkley corporate bonis mispriceis closest to: A.0.3368 per 100 of pvalue. B.0.4682 per 100 of pvalue. C.0.5156 per 100 of pvalue. C is correct.The first step in the solution is to finthe correspot rate (zero-coupon rates) for eayear’s cash flow. The benchmark bon in Exhibit 2 are conveniently pricepso the yiel to maturity anthe coupon rates on the bon are the same. Because the one-yeissue honly one cash flow remaining, the YTM equals the spot rate of 3% (or z1z_1z1= 3%). The spot rates for Ye2 ( z2z_2z2) anYe3 (z3z_3z3 ) are calculatefollows: beginarrayl100=41.0300+104(1+z2)2;z2=4.02%100=51.0300+5(1.0402)2+105(1+z3)3;z3=5.07%begin{array}{l}100 = \frac{4}{{1.0300}} + \frac{{104}}{{{{(1 + {z_2})}^2}}};{z_2} = 4.02\% \\100 = \frac{5}{{1.0300}} + \frac{5}{{{{(1.0402)}^2}}} + \frac{{105}}{{{{(1 + {z_3})}^3}}};{z_3} = 5.07\% beginarrayl100=1.03004+(1+z2)2104;z2=4.02%100=1.03005+(1.0402)25+(1+z3)3105;z3=5.07%The correarbitrage-free prifor the Hutto-Barkley Inbonis:P0=3(1.0300)+3(1.0402)2+103(1.0507)3=94.4828P_0=\frac3{(1.0300)}+\frac3{{(1.0402)}^2}+\frac{103}{{(1.0507)}^3}=94.4828P0=(1.0300)3+(1.0402)23+(1.0507)3103=94.4828Therefore, the bonis misprice94.4828 – 94.9984 = –0.5156 per 100 of pvalue.A is incorrebecause the correspot rates are not calculateaninstethe Hutto-Barkley Inbonis scounteusing the respective YTM for eamaturity. Therefore, this lea to incorremispricing of 94.6616 – 94.9984 = –0.3368 per 100 of pvalue.B is incorrebecause the spot rates are riveusing the coupon rate for Ye3 (maturity) insteof using eayear’s respective coupon rate to employ the bootstrmethology. This lea to incorremispricing of 94.5302 – 94.9984 = –0.4682 per 100 of pvalue. 老师这题我用的题目要求的hb公司处以benchmark'的yiel2年算出的spot rate和3年的spot rate，但是我看到答案写的是用2.3年的yiel作为2年，3年的cupon然后分别求出的spot rate，嗯我有些分不清什么时候用benchmark的coupon什么时候用给的公司的coupon
NO.PZ201602270200001902 问题如下 2. Baseon Exhibits 2 an3 anusing Metho1, the amount (in absolute terms) whithe Hutto-Barkley corporate bonis mispriceis closest to: A.0.3368 per 100 of pvalue. B.0.4682 per 100 of pvalue. C.0.5156 per 100 of pvalue. C is correct.The first step in the solution is to finthe correspot rate (zero-coupon rates) for eayear’s cash flow. The benchmark bon in Exhibit 2 are conveniently pricepso the yiel to maturity anthe coupon rates on the bon are the same. Because the one-yeissue honly one cash flow remaining, the YTM equals the spot rate of 3% (or z1z_1z1= 3%). The spot rates for Ye2 ( z2z_2z2) anYe3 (z3z_3z3 ) are calculatefollows: beginarrayl100=41.0300+104(1+z2)2;z2=4.02%100=51.0300+5(1.0402)2+105(1+z3)3;z3=5.07%begin{array}{l}100 = \frac{4}{{1.0300}} + \frac{{104}}{{{{(1 + {z_2})}^2}}};{z_2} = 4.02\% \\100 = \frac{5}{{1.0300}} + \frac{5}{{{{(1.0402)}^2}}} + \frac{{105}}{{{{(1 + {z_3})}^3}}};{z_3} = 5.07\% beginarrayl100=1.03004+(1+z2)2104;z2=4.02%100=1.03005+(1.0402)25+(1+z3)3105;z3=5.07%The correarbitrage-free prifor the Hutto-Barkley Inbonis:P0=3(1.0300)+3(1.0402)2+103(1.0507)3=94.4828P_0=\frac3{(1.0300)}+\frac3{{(1.0402)}^2}+\frac{103}{{(1.0507)}^3}=94.4828P0=(1.0300)3+(1.0402)23+(1.0507)3103=94.4828Therefore, the bonis misprice94.4828 – 94.9984 = –0.5156 per 100 of pvalue.A is incorrebecause the correspot rates are not calculateaninstethe Hutto-Barkley Inbonis scounteusing the respective YTM for eamaturity. Therefore, this lea to incorremispricing of 94.6616 – 94.9984 = –0.3368 per 100 of pvalue.B is incorrebecause the spot rates are riveusing the coupon rate for Ye3 (maturity) insteof using eayear’s respective coupon rate to employ the bootstrmethology. This lea to incorremispricing of 94.5302 – 94.9984 = –0.4682 per 100 of pvalue. 老师，1、首先说了coupon rate，然而在下面计算的时候，二年期的CF用的4，三年期用的5，这数字都是怎么来的？2、没明白题目暗示怎么体现出来的？按照表格指向，指的是YTM的表格，难道不是用3%、4%、5%分别折算3,3,103三年现金流吗？请帮忙翻译并且查看下，谢谢
NO.PZ201602270200001902 问题如下 2. Baseon Exhibits 2 an3 anusing Metho1, the amount (in absolute terms) whithe Hutto-Barkley corporate bonis mispriceis closest to: A.0.3368 per 100 of pvalue. B.0.4682 per 100 of pvalue. C.0.5156 per 100 of pvalue. C is correct.The first step in the solution is to finthe correspot rate (zero-coupon rates) for eayear’s cash flow. The benchmark bon in Exhibit 2 are conveniently pricepso the yiel to maturity anthe coupon rates on the bon are the same. Because the one-yeissue honly one cash flow remaining, the YTM equals the spot rate of 3% (or z1z_1z1= 3%). The spot rates for Ye2 ( z2z_2z2) anYe3 (z3z_3z3 ) are calculatefollows: beginarrayl100=41.0300+104(1+z2)2;z2=4.02%100=51.0300+5(1.0402)2+105(1+z3)3;z3=5.07%begin{array}{l}100 = \frac{4}{{1.0300}} + \frac{{104}}{{{{(1 + {z_2})}^2}}};{z_2} = 4.02\% \\100 = \frac{5}{{1.0300}} + \frac{5}{{{{(1.0402)}^2}}} + \frac{{105}}{{{{(1 + {z_3})}^3}}};{z_3} = 5.07\% beginarrayl100=1.03004+(1+z2)2104;z2=4.02%100=1.03005+(1.0402)25+(1+z3)3105;z3=5.07%The correarbitrage-free prifor the Hutto-Barkley Inbonis:P0=3(1.0300)+3(1.0402)2+103(1.0507)3=94.4828P_0=\frac3{(1.0300)}+\frac3{{(1.0402)}^2}+\frac{103}{{(1.0507)}^3}=94.4828P0=(1.0300)3+(1.0402)23+(1.0507)3103=94.4828Therefore, the bonis misprice94.4828 – 94.9984 = –0.5156 per 100 of pvalue.A is incorrebecause the correspot rates are not calculateaninstethe Hutto-Barkley Inbonis scounteusing the respective YTM for eamaturity. Therefore, this lea to incorremispricing of 94.6616 – 94.9984 = –0.3368 per 100 of pvalue.B is incorrebecause the spot rates are riveusing the coupon rate for Ye3 (maturity) insteof using eayear’s respective coupon rate to employ the bootstrmethology. This lea to incorremispricing of 94.5302 – 94.9984 = –0.4682 per 100 of pvalue. 1、他不是说了COUPON是3%吗？2、YTM和这个COUPON RATE为什么又等同了？为什么可以认为第二年、第三年的COUPON RATE是YTM？3、benchmark pcurve，为什么表格里面又说是YTM,？YTM和这个COUPON RATE为什么又等同了？4、为什么不能用YTM直接算，之前的答案并没有很清晰回答，感觉像是诡辩？他不是说了COUPON是3%吗？，那不是三年的话，3年的COUPO都是3%？，对应的YTM就是5%？
NO.PZ201602270200001902 0.4682 per 100 of pvalue. 0.5156 per 100 of pvalue. C is correct. The first step in the solution is to finthe correspot rate (zero-coupon rates) for eayear’s cash flow. The benchmark bon in Exhibit 2 are conveniently pricepso the yiel to maturity anthe coupon rates on the bon are the same. Because the one-yeissue honly one cash flow remaining, the YTM equals the spot rate of 3% (or z1z_1z1= 3%). The spot rates for Ye2 ( z2z_2z2) anYe3 ( z3z_3z3 ) are calculatefollows: beginarrayl100=41.0300+104(1+z2)2;z2=4.02%100=51.0300+5(1.0402)2+105(1+z3)3;z3=5.07%begin{array}{l}100 = \frac{4}{{1.0300}} + \frac{{104}}{{{{(1 + {z_2})}^2}}};{z_2} = 4.02\% \\100 = \frac{5}{{1.0300}} + \frac{5}{{{{(1.0402)}^2}}} + \frac{{105}}{{{{(1 + {z_3})}^3}}};{z_3} = 5.07\% beginarrayl100=1.03004+(1+z2)2104;z2=4.02%100=1.03005+(1.0402)25+(1+z3)3105;z3=5.07% The correarbitrage-free prifor the Hutto-Barkley Inbonis: P0=3(1.0300)+3(1.0402)2+103(1.0507)3=94.4828P_0=\frac3{(1.0300)}+\frac3{{(1.0402)}^2}+\frac{103}{{(1.0507)}^3}=94.4828P0=(1.0300)3+(1.0402)23+(1.0507)3103=94.4828 Therefore, the bonis misprice94.4828 – 94.9984 = –0.5156 per 100 of pvalue. A is incorrebecause the correspot rates are not calculateaninstethe Hutto-Barkley Inbonis scounteusing the respective YTM for eamaturity. Therefore, this lea to incorremispricing of 94.6616 – 94.9984 = –0.3368 per 100 of pvalue. B is incorrebecause the spot rates are riveusing the coupon rate for Ye3 (maturity) insteof using eayear’s respective coupon rate to employ the bootstrmethology. This lea to incorremispricing of 94.5302 – 94.9984 = –0.4682 per 100 of pvalue.何老师上课不是说YTM是sport rate的打包价么，为什么不可以直接用第三年的YTM 5%来进行折现呢？