Is there a danger of my pension provider not being able to pay my monthly pension Annuity due to the falling bond market. In particular the falling American bond market
Is it a sovereign annuity where payments are linked to the performance of certain government bonds, or a "regular" annuity with a guaranteed payment for the rest of your life?
No, you should be fine. Unless the life insurance company goes bust and defaults on its obligations (and no Irish life insurance company has ever doen this since before the foundation of the state) your annuity will be paid as promised.