[mckn1ght]

This feels like one of those PEMDAS posts on facebook where everyone comes to different results and argues endlessly about it. So let me add my 2¢:

if you have 100K in equity and owe 200K on your mortgage, then you’re net -100K on your house

that combines with the 100K in the retirement account to produce a net worth of $0

Where is my mistake?

[Tcepsa]

Net worth isn't equity - mortgage + 401k. Net worth is assets - liabilities. Equity is not an asset; the house is. So is the 401k. The mortgage is a liability. So Joe's net worth is 100k (IRA) + 300k (value of house) - 200k (mortgage) = 200k positive net worth.

(Another way of thinking about equity, specifically, is it is the real estate contribution to net worth, because it is what is left when you subtract the real estate liability (mortgage) from the real estate asset (value of house). That's why you shouldn't subtract the mortgage from the equity: equity is what's left after you've already subtracted the mortgage.)

(Edit: Adjusted sign in first equation to subtract mortgage. It's probably more technically accurate to keep it as addition and consider the mortgage to be a negative value, but I believe it's more straightforward and intuitive for most people as it is now represented.)