求解释

问题如下：

A fixed-income portfolio manager currently holds a bullet 7-year US Treasury position with USD 60 million face value. The manager would like to create a cost matching barbell portfolio by purchasing a combination of a 2-year Treasury and a 15-year Treasury that would have the same duration as the 7-year US Treasury position. The data for the three US Treasuries are listed below:

Which of the following combinations correctly describes the weights of the two bonds that the manager will use to construct the barbell portfolio?

选项：

A.

14.22% 85.78%

B.

44.46% 55.54%

C.

55.54% 44.46%

D.

85.78% 14.22%

解释：

中文解析：

构建一个和bullet的cost与duration都一样的barbell组合：

bullet 的cost = (106.443/100)*USD 60,000,000 = USD 63,865,800

V2 + V15 = USD 63,865,800 等式1

6.272 = (V2/63,865,800)*1.938 + (V15/63,865,800)*11.687 等式2

通过等式1可得：V2 = 63,865,800 – V15

等式2即可变成：

6.272 = [(63,865,800 – V15)/63,865,800)]*1.938 + (V15/63,865,800)*11.687

400,566,297.6 = 123,771,920.4 – 1.938V15 + 11.687V15

276,794,377.2 = 9.749V15

V15 = USD 28,392,078.90

所以，V2 = 63,865,800 – V15 = 63,865,800 – 28,392,078.90 = USD 35,473,721.10

V2的权重 = 35,473,721.10/63,865,800 = 55.54%

V15的权重 = 28,392,078.90/63,865,800 = 44.46%

To construct a barbell portfolio with the same cost and same duration as the bullet:

Cost of bullet = (106.443/100)*USD 60,000,000 = USD 63,865,800

If V2 and V15 are values (costs) of the 2-Year and 15-Year Treasuries, respectively, then, V2 + V15 = USD 63,865,800 …………………………………………………………………….…… (1)

Therefore, to match duration:Duration of bullet = weighted-average duration of 2-year and 15-year Treasuries 6.272 = (V2/63,865,800)*1.938 + (V15/63,865,800)*11.687 ………………………… (2)

From Equation (1), V2 = 63,865,800 – V15.

Then, Equation (2) becomes: 6.272 = [(63,865,800 – V15)/63,865,800)]*1.938 + (V15/63,865,800)*11.687

400,566,297.6 = 123,771,920.4 – 1.938V15 + 11.687V15

276,794,377.2 = 9.749V15

And so, V15 = USD 28,392,078.90

And so, V2 = 63,865,800 – V15 = 63,865,800 – 28,392,078.90 = USD 35,473,721.10

Giving weight of 2-Year Treasury = 35,473,721.10/63,865,800 = 55.54%

And weight of 15-year Treasury = 28,392,078.90/63,865,800 = 44.46%