NO.PZ2023100703000107
问题如下：
The manager of the fixed-income desk of an investment bank is examining the current term structure of swap rates and believes that the 5-year swap rate is too low relative to the 2-year and 10-year swap rates. The manager asks a risk analyst to design a hedged butterfly trade in which the bank is the payer in a 5-year swap contract and the receiver in 2-year and 10-year swap contracts. The analyst decides to perform a principal components analysis (PCA) of the term structure of swap rates and use the results of the PCA to construct the butterfly trade. The principal components (PCs) identified as having the greatest impact are the level, the slope, and the short rate. The results of the PCA, stated as the change in bps in the swap rates due to a 1 standard deviation increase in the PC, are given in the table below:
The analyst also notes that these three PCs explain over 99.5% of the variability in the swap rates, with the level PC having the greatest impact, the slope PC having a smaller impact, and the short rate PC only having an impact on very short-term swap rates.
To construct the hedged butterfly position, the analyst collects the current swap rates and DV01s of the 2-year, 5-year, and 10-year swaps, shown in the table below:
After receiving this information from the analyst, the manager instructs the analyst to construct a butterfly position with a notional amount of EUR 100 million in the 5-year swap in such a way that exposures to the level and slope PCs are neutralized. What notional amounts of the 2-year swap and the 10-year swap should be included in the butterfly and what are the risk weights of the two swaps relative to the DV01 of the 5-year swap?
选项：
A.Choice A B.Choice B C.Choice C D.Choice D解释：
Solving
for the face values of the 2-year and 10-year swaps requires using a system of
two equations and two unknown variables.
Notional
amount of 5-year swap is 100.
Equation1
that neutralizes exposure to level PC is:
F(2)
* (DV01(2)/100) * LevelPC(2) + F(10) * (DV01(10)/100) * LevelPC(10) + 100*(DV01(5)/100)
* LevelPC(5) = 0
Equation1
using the information in the tables is:
F(2)
* 0.0014421 + F(10) * 0.00396933 + 100 * 0.00296112 = 0
Solving
for F(2):
F(2) = ( -0.296112 - F(10) * 0.00396933) / 0.0014421
Equation2
that neutralizes exposure to slope PC:
F(2)
* (DV01(2)/100) * SlopePC(2) + F(10) * (DV01(10)/100) * SlopePC(10) + 100*(DV01(5)/100)
* SlopePC(5) = 0
Equation2
using the information in the table is:
F(2)
* -0.00083505 + F(10) * 0.00001462 + 100 * -0.00063488 = 0
Substitute
the previously solved for F(2) into Equation2 and solve for F(10)
-0.579051383 * (-0.296112 - F(10) * 0.00396933) + F(10) * 0.00001462 - 0.063488=
0
0.171464063
+ F(10) * 0.002298446 + F(10) * 0.00001462 = 0.063488
F(10)
* 0.002298446 + F(10) * 0.00001462 = -0.107976063
F(10)
* 0.002313066 = -0.107976063
F(10)
= -46.68092564 , or a face value of EUR 46.68 million
Substituting
this value of F(10) into the equation for F(2) and solving for F(2):
F(2)
= ( -0.296112 – -46.68093 *0.00396933) / 0.0014421
F(2)
= -76.84626685 , or a face value of EUR 76.85 million
The
face value of the 2-year swap receiving should be EUR 76.85 million
The
face value of the 10-year swap receiving should be EUR 46.68 million
The
risk weight of the 2-year swap relative to DV01 of the 5-year swap is equal to:
(F(2) * (DV01(2)/100))/DV01(5) =0.44155617
or 44.2%
And
the risk weight of the 10-year swap relative to DV01 of the 5-year swap is:
0.687978965
or 68.8%
A is
incorrect. This answer choice is incorrect in two ways. First, the notional
amount of the 2-year swap is incorrectly adjusted so that the notional amounts
of the 2-year swap and the 10-year swap add up to 100 to offset the notional
amount of the 5-year swap. Second, the risk weights are assumed to sum to 100%,
so the percentages are incorrectly adjusted to weight out of 100.
B is
incorrect. The notional amount of the 10-year swap is incorrectly adjusted so that
the notional amounts of the 2-year swap and the 10-year swap add up to 100 to offset
the notional amount of the 5-year swap.
D is
incorrect. The risk weights are assumed to sum to 100%, so the percentages are
incorrectly adjusted to weight out of 100.
看了经典题视频，还是不太懂
