I tend to think that the psychological and physical theories of personal identity are both insufficient. There are cases where psychological continuity wouldn’t count as identity (say, we find a way to duplicate brains), and cases where physical continuity clearly fails (for example, most people would equate a complete brain-wipe with death.)
But if we assume that our theories are determinative for our own identity, we could do the following: we find a young, healthy physical continuity theorist and an elderly, unwell psychological continuity theorist, and offer them the following deal: the psychological continuity theorist gives all of his or her money to the physical theorist, and we then reprogram the physical theorist’s brain to be an exact match of the psychological theorist. The psych theorist then gets to live in the physical theorist’s body, at least on the psych theory, and the physical theorist gets a lot of money. Both would think of themselves as continuing to exist, so it’s a win-win.