Your (negative) duration is high, so you have high price risk.
Your payments are constant, so you have zero cash flow risk.
If you pay floating, then:
Your (negative) duration is low, so you have low price risk.
Your payments are variable, so you have high cash flow risk.
If you have issued a fixed-rate bond and you enter into a pay floating, receive fixed swap, you go from the first case above to the second: you decrease your price risk and increase your cash flow risk.
If you have issued a floating-rate bond and you enter into a pay fixed, receive floating swap, you go from the second case above to the first: you increase your price risk and decrease your cash flow risk.
The change of loan value on a floating from interest rate change is low compared to a fixed. Whether you’re a payer or receiver of a bond, given a change in interest rates, the bond value still changes the same for both parties. So you are adding duration.
I work for a bank. Should I think of it like this?:
We issue CDs to depositors, therefore we are short duration;
We use our liquidity portfolio to invest in government bonds (Treasuries), therefore we are long duration;
If we’re trying to manage to duration-neutral, then if I have a positive duration gap (e.g. I’m more long than short), then we would enter into a pay-fixed, receive floating interest rate swap to decrease duration;
If we’re trying to manage to duration-neutral, then if I have a negative duration gap (e.g. I’m more short than long), then we would enter into a pay-floating, receive fixed interest rate swap to increase duration.