上小学 · 2023年03月02日
NO.PZ2020021204000012
问题如下：
The six-month and one-year zero rates are 3% and 4% (both compounded semi-annually) and a 1.5-year bond paying a coupon of 4% per annum semi-annually has a yield of 5%. What is the 1.5-year zero-coupon interest rate?
解释：
The price of the 1.5-year bond with a face value of 100 is:
If the 1.5-year zero rate is R we must have:
The solution to this equation is R = 0.05027. The 1.5-year zero rate is therefore 5.027%.
麻烦介绍下该类题型怎么做？谢谢
李坏_品职助教 · 2023年03月02日
嗨，爱思考的PZer你好：
题目问的是1.5年的zero rate是多少，为了求这个结果，我们需要知道6个月的、1年的零息利率，还有1.5年债券的价格和收益率，恰好题目都有提供。
遇到这种题目，先找最长期限的债券相关的数据，题目里给了1.5年附息债券的票面利率是4%，半年付息一次，也就是每次利息是2元。yield是5%。
先把1.5年债券定价的公式写出来：price = 利息/ (1+yield / 2) + 利息/ (1+yield / 2)^2 + （本金+利息）/ (1+yield/2)^3，这里的利息是2元,yield=5%，带入之后可以得出price = 98.572.
题目最后问的是1.5年的zero rate，我们需要把上面债券定价公式分母里面的yield全都替换为zero rate：
这样求出里面的R就行了。
遇到这种题目，先找最长期限的债券相关的数据，先把债券定价结果求出来，再把分母的利率去和题目问的利率结合起来。
----------------------------------------------努力的时光都是限量版，加油！
NO.PZ2020021204000012问题如下The six-month anone-yezero rates are 3% an4% (both compounsemi-annually) ana 1.5-yebonpaying a coupon of 4% per annum semi-annually ha yielof 5%. Whis the 1.5-yezero-coupon interest rate? The priof the 1.5-yebonwith a favalue of 100 is:21+0.05/2+2(1+0.05/2)2+102（1+0.05/2）3=98.572\frac2{1+0.05/2}+\frac2{{(1+0.05/2)}^2}+\frac{102}{（1+0.05/2）^3}=98.5721+0.05/22+(1+0.05/2)22+（1+0.05/2）3102=98.572If the 1.5-yezero rate is R we must have:21+0.03/2+2(1+0.04/2)2+102（1+R/2）3=98.572\frac2{1+0.03/2}+\frac2{{(1+0.04/2)}^2}+\frac{102}{（1+R/2）^3}=98.5721+0.03/22+(1+0.04/2)22+（1+R/2）3102=98.572The solution to this equation is R = 0.05027. The 1.5-yezero rate is therefore 5.027%.同上，为什么这两个rate会是不同的
NO.PZ2020021204000012问题如下The six-month anone-yezero rates are 3% an4% (both compounsemi-annually) ana 1.5-yebonpaying a coupon of 4% per annum semi-annually ha yielof 5%. Whis the 1.5-yezero-coupon interest rate? The priof the 1.5-yebonwith a favalue of 100 is:21+0.05/2+2(1+0.05/2)2+102（1+0.05/2）3=98.572\frac2{1+0.05/2}+\frac2{{(1+0.05/2)}^2}+\frac{102}{（1+0.05/2）^3}=98.5721+0.05/22+(1+0.05/2)22+（1+0.05/2）3102=98.572If the 1.5-yezero rate is R we must have:21+0.03/2+2(1+0.04/2)2+102（1+R/2）3=98.572\frac2{1+0.03/2}+\frac2{{(1+0.04/2)}^2}+\frac{102}{（1+R/2）^3}=98.5721+0.03/22+(1+0.04/2)22+（1+R/2）3102=98.572The solution to this equation is R = 0.05027. The 1.5-yezero rate is therefore 5.027%.这题里面的Yiel有效年利率的意思吗？为什么债券的定价，每一次现金流都用这个yiel折现？谢谢。
NO.PZ2020021204000012问题如下The six-month anone-yezero rates are 3% an4% (both compounsemi-annually) ana 1.5-yebonpaying a coupon of 4% per annum semi-annually ha yielof 5%. Whis the 1.5-yezero-coupon interest rate? The priof the 1.5-yebonwith a favalue of 100 is:21+0.05/2+2(1+0.05/2)2+102（1+0.05/2）3=98.572\frac2{1+0.05/2}+\frac2{{(1+0.05/2)}^2}+\frac{102}{（1+0.05/2）^3}=98.5721+0.05/22+(1+0.05/2)22+（1+0.05/2）3102=98.572If the 1.5-yezero rate is R we must have:21+0.03/2+2(1+0.04/2)2+102（1+R/2）3=98.572\frac2{1+0.03/2}+\frac2{{(1+0.04/2)}^2}+\frac{102}{（1+R/2）^3}=98.5721+0.03/22+(1+0.04/2)22+（1+R/2）3102=98.572The solution to this equation is R = 0.05027. The 1.5-yezero rate is therefore 5.027%.否则就直接100/（1+R）^1.5=P，求出R
NO.PZ2020021204000012 问题如下 The six-month anone-yezero rates are 3% an4% (both compounsemi-annually) ana 1.5-yebonpaying a coupon of 4% per annum semi-annually ha yielof 5%. Whis the 1.5-yezero-coupon interest rate? The priof the 1.5-yebonwith a favalue of 100 is:21+0.05/2+2(1+0.05/2)2+102（1+0.05/2）3=98.572\frac2{1+0.05/2}+\frac2{{(1+0.05/2)}^2}+\frac{102}{（1+0.05/2）^3}=98.5721+0.05/22+(1+0.05/2)22+（1+0.05/2）3102=98.572If the 1.5-yezero rate is R we must have:21+0.03/2+2(1+0.04/2)2+102（1+R/2）3=98.572\frac2{1+0.03/2}+\frac2{{(1+0.04/2)}^2}+\frac{102}{（1+R/2）^3}=98.5721+0.03/22+(1+0.04/2)22+（1+R/2）3102=98.572The solution to this equation is R = 0.05027. The 1.5-yezero rate is therefore 5.027%.
