The risk of a swap can be redistributed to more liquid swap maturities. The assignment of a bucket’s swap risk into two (or more) nodes is done on a linearly interpolated continuously compounded yield.
For example, if we consider the risk at a node of maturity t2 to be redistributed between the nodes t1 and t3, the associated DV01 in t2 (Δ2) can be expressed as a function of the maturities t1 and t3, the associated DV01s (Δ1 & Δ3) and the Par Swap rates.
The question is how to derive this appoximated formula.
Question) Show that Δ2 can be approximated using the following formula (in the example above r1 = 59% & r2 = 41%):
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Full outline of where question came from :
The example below shows the risk associated with trading GBP 100,000,000 notional of a 7 year swap. In this example, we are receiving the fixed rate for a 7 years swap.
Maturity Par Swap Rate GBP per 1bp Notional (m) DV01
1 Year 4.985% 0 0
2 Year 4.978% 0 0
3 Year 4.948% 0 0
4 Year 4.933% 0 0
5 Year 4.923% 0 0
6 Year 4.915% 0 0
7 Year 4.908% 0 58,000 100 580
8 Year 4.898% 0 0
9 Year 4.888% 0 0
10 Year 4.878% 0 0
11 Year 4.869% 0 0
12 Year 4.860% 0 0
15 Year 4.823% 0 0
20 Year 4.763% 0 0
25 Year 4.695% 0 0
30 Year 4.638% 0 0
35 Year 4.593% 0 0
40 Year 4.548% 0 0
50 Year 4.465% 0 0
60 Year 4.375% 0 0
Note that: t1, t2 and t3 represent the maturities in years; Y is the Par Swap rate and Δ is the DV01. Furthermore, we know that the following relationships are true for the discount factor, df, for a given maturity t & the continuously compounded zero yield y.
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Any help would be much appreciated!