Why the higher the fixed rate of a swap is, the higher the DV01 will be?
1 Answer
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It won't.
The par fixed rate $c$ of a fixed/float $N$-year spot starting IRS is $$ c=\frac{1-df_N}{dv01} $$ with $dv01=\sum_i^N \delta_i \cdot df_i$ where $df_i$ are discount factors and $\delta_i$ are day count fractions. Hence $dv01$ is essentially just the sum of discount factors.
The higher rates are the lower the $dv01$ because discount factors are inversely proportional to rates.
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$\begingroup$ Thank you for answering! Just wanted to clarify if the dv01 you refer to here means pv01 (1bp change in fixed rate). I think what I mean by dv01 is how NPV will change if the curve is bumped by 1bp. Thank you! $\endgroup$– Mini PPJun 4 at 3:45
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$\begingroup$ It doesn't matter what measure of duration u use or what ur annuity is (swap, bond, simple cashflow etc): higher rates mean reduced duration. Forget swaps, just consider the PV of $1 to be recd in 1y with rates at 10% (this is = 1/(1+10%)). Shift rates to 11% and take the difference to obtain the duration (called it delta_1). Then compare the same case with rates at 20% (and shifted to 21% to obtain the duration, call it delta_2). You will find delta_1 > delta_2 always. Again, this is because discount factors are inversely proportional to rates. $\endgroup$ Jun 4 at 11:22
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$\begingroup$ The question might be asking how the dv01 of a swap changes as a function of its coupon, whilst leaving market rates constant. In that case indeed the dv01 grows with coupon, since there is just more cash flow. $\endgroup$– dm63Jul 3 at 9:28
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$\begingroup$ Agreed with @dm63, an IRS with an off market coupon can be considered as an IRS with an ATM coupon plus a fixed annuity. The fixed annuity will have additional delta risk. $\endgroup$– Attack68 ♦Oct 31 at 13:52