12. Question

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:

Term (years)	Level PC	Slope PC	Short Rate PC
1	3.25	-2.51	1.27
2	5.06	-2.93	0.44
5	5.97	-1.28	-0.36
10	5.43	0.02	-0.18
20	4.84	0.64	0.25

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:

Term (years)	Swap rate	DV01
2	2.992%	0.0285
5	2.551%	0.0496
10	2.454%	0.0731

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?