My bad, it wasn’t really an assumption so much as faith that the true motivator is cost savings, not “want.”
The only inaccuracy that I can point out in your proof is that the BMW won’t be worth zero at the end of the year he’ll keep it for. So say he sells it for $4000, then he’s only 400 in the hole. That $400 may be saved in slowing the depreciation of the truck having fewer miles. So he could break even.
Except that I’d bet that the insurance for a year of a 20 year old with a BMW would be more like $1000, which puts him at $600 in the hole.
But there are risks. But I agree with the gist of what you’re saying. It doesn’t make sense financially. If he just “wants” it, it might not be the worst thing ever financially. But I’ve learned over the last year or two that if you’re uncomfortable admitting to yourself the true reason you’re doing something then you probably shouldn’t do it.