答案是不是写反了

问题如下：

Suppose that the five-, ten-, and 30-year rates are 4%, 5%, and 6% with semiannual compounding. Calculate the duration and convexity of zero-coupon bonds with five-, ten-, and 30-years to maturity. What position in five- and 30-year bonds would have a duration equal to that of the ten-year bond? Compare the convexities of (a) the positions in the ten-year bond and (b) the position in the five- and 30-year bonds? Which of these positions will give the better return if (a) rates remain the same and (b) there are parallel shifts in the term structure?

解释：

The duration and convexities calculated by making one-basis-point changes are

We can construct a bond with a duration of 9.756 by investing β in the five-year

bond and 1−β in the 30-year bond where:

4.902β+29.126(1－β)=9.756

β is 0.7996, which we round to 0.8. We therefore invest 80% in the five-year bond and 20% in the 30-year bond. The ten-year bond investment (a bullet) has a convexity of 99.941 whereas the portfolio of five- and 30-year bonds (a barbell) has a convexity of about:

0.8×26.423+ 0.2×862.472 = 193.6

If rates remain the same the bullet will provide a yield of 5%, whereas thebarbell will provide a weighted average yield of 0.8 × 4 + 0.2 × 6 or 4.4%. The bullet will performbetter. When there are parallel shifts to the term structure, this effect is mitigatedsomewhat by the barbell’s higher convexity, which leads to an immediateimprovement in the value of the barbell position. However, the bullet willperform better for some non-parallel shifts.

题目问：现在有5年、10年、30年的利率分别是4%、5%、6%，半年付息一次。

（1）计算5年、10年、30年期的零息债券的duration和convexity。

（2）5年期的债券和30年期的债券做组合，头寸分别为多少可以等于10年期的债券？

（3）比较10年期债券的convexity，和5年期与30年期债券做组合的convexity。

（4）在利率不变的情况下，和利率平行移动的情况下，哪一个头寸可以获得更好的return？

回答：

（1）

求利率上升1bp和下降1bp的债券价格，然后利用以下convexity的公式即可计算出convexity。

convexity=(V+ + V--2*V0)/(V0*1bp^2)

（2）

5年期和30年期的债券组合，设5年期占比为x，30年期的占比为(1-x)

4.902*x+29.126*(1－x)=9.756

x=0.8

5年期的占比80%，30年期的占比20%。(现金流是barbell的形式)

（3）这个组合的convexity=0.8×26.423+ 0.2×862.472 = 193.6

（4）10年期债券(现金流是bullet的形式)的convexity是99.941

利率平行移动时，用barbell

利率非平行移动时，用bullet