Bond Redemption When?

I encounter a statement in Schewser Notes for bond redemption as below:
"A firm may choose to redeem bonds before maturity because interest rates have fallen,because the firm has generated surplus cash through operations, or because funds from the issuance of equity make it possible (and desirable)"

As I understand, from the above staement, when interest goes down -> firm may choose to redeem early bond to REISSUE other bonds with lower interest cost

However, I also understand that when interest goes up -> bond price goes down -> firm also may choose to redeem early bond to RETIRE them.

Are my two understandings correct?

Thanks!

Yes. There are two ways to redeem a bond. Pay the par or buy at the market value. If rates go up, market value will go down so you might redeem it at a discount by outright purchases. If rates go down. The market value of the bond will rise but will prolly be bounded on the upside by par value because of the increased likelihood it will be called and you will receive the par value.

“If rates go down. The market value of the bond will rise but will prolly be bounded on the upside by par value because of the increased likelihood it will be called and you will receive the par value.”

Could you explain more the latter statement?

Nobody said anything about callable bonds.

For callable bond:
i/r goes down -> bond price up, if higher than the exercise price -> firm might redeem the bond

But what about straight bonds, i know it is hard to recall a bond if there is no option embedded. However, what about repurchase in market with higher offer price than the market price.

I mean if interest rate go up -> bond price goes down => firm, having excessive cash might go to the market and offer a higher price to redeem. (of course, they do not have to issue another bonds)