A lower rate of interest per year is being paid. The bottom line is that you have made up your mind that annual percentage rate can only mean the interest rate assigned to the note, failing to understand that because this is how you view things does not turn it into fact.
No, a lower rate of interest is not being paid. The same interest rate of 4% per annum is being paid in both cases. Your calculation of what you are describing as the interest rate being paid ignores the fact that principal is being repaid prior to maturity in both scenarios (and at a faster rate in the second scenario). Earlier you got annoyed by the use of the word "absurd" to describe your argument, but there's no better word to describe what you are now arguing - that the "interest rate" (or "cost of credit" or whatever else you choose to call it) of a loan with a 4% per annum rate is actually 2.4%, or, if you repay really quickly, 2.19%. So if you repay the loan in full the instant you receive it, I guess you've now turned it into a 0% loan!
This isn't about semantics or hairsplitting. You are simply misunderstanding how the accrual of interest on a loan works.
Hello oneiota - fellow questioner here. Let me give you some backstory that's getting withheld from you. (it comes out when you start doing searches on people's posts and start to recognize their M.O.)
First: The notion of NOT prepaying a mortgage and INSTEAD investing is a kind of sacred cow here on the forums. That sacred cow has a number of Valiant Defenders (VD). You have asked questions causing the Valiant Defenders to rise up and attempt to strike you down with their Insulting Dynamic Insinuation Offensive Taunts (IDIOT).
You are now jousting with a number of these Valiant Defenders. They are throwing their Powerful Insult Exclamations (PIE) at you. They will keep throwing PIE until you finally stop and back up or quiet down. They will then believe they have won their VD acting out IDIOT by throwing PIE. Once again, the World will be Safe now that we all know we must Invest Invest Invest rather than paying down our mortgage. :-) :-) :-)
They'll hammer and hammer and hammer at you, "proving" you don't "really understand the math" of mortgages as a way to try to discredit your fundamental questions about the wisdom of investing vs. paying off a mortgage. They'll contrive things you never said, raise tangential, irrelevant arguments and insult you (then back up as if you were the offender) when you call them on it.
Now, the truth of their arguments lie in the fact that we are living through a period of time when mortgage interest rates are at historical, ungodly lows (and could someday STOP being at these historical, ungodly lows). That makes mortgage money cheaper than investing,
over reasonably long time periods.
There's a zillion companies out there who are climbing all over each other to throw mortgage money at us for these bargain rates. Surely, according to the VD arguments, these companies must be staffed by chumps who are giving away money at stupid low mortgage rates when THEY THEMSELVES should be investing in the market for higher returns. Right? And yet, they are not investing in the market. They are stupidly begging us to take on a mortgage at some insanely low interest rate.
And you know why it's happening? You quit paying your mortgage, the sheriff comes, kicks your ass out, throws your shit out on the street and padlocks your doors. That's a powerful incentive to make sure you pay that mortgage, no matter what.
It's such a serious difference that some frugalists can make an effective argument against buying a house at all and just renting. And a lot of mustachians do just that. They stay away from a mortgage even though it's some seriously cheap money right now.
Now, when Enron blew up, when GM declared bankruptcy, when Alcatel stock stayed in the crapper forever, when people lost their shirts investing in ATT or Thermo-whatever or any of a zillion other companies that simply disappeared overnight, did any investors get their money back? No. Did anyone at the company go to jail? No. Did the sheriff come padlock anything and throw any executives out on the street? No.
Anyone who raises a question about the insecurity of investing vs. the security of paying down a mortgage attracts the same set of players on the same topic using the same strategy to call you....stupid. When really, you're just some dude trying to figure out how to do this MMM thing.
Look at everyone who's FIRE and you will see nearly every one of them have no mortgage. What matters the most is what people in FIRE are actually doing. And what they are doing is paying off their mortgages.