# Higher coupon on interest rate swap has higher DV01

Why the higher the fixed rate of a swap is, the higher the DV01 will be?

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.

• 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! Jun 4, 2023 at 3:45
• 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. Jun 4, 2023 at 11:22
• 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.
– dm63
Jul 3, 2023 at 9:28
• 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.
– Attack68
Oct 31, 2023 at 13:52