> You haven't really gained 100k. You have gained 100k multiplied by your percentage of equity, i.e. the amount you have paid so far into your mortgage principal. Assuming it's a 30-year mortgage, that will be around 30%, so around 30k.
No, you've gained 100k. Your asset is worth $300k, your mortgage liability is $160k minus however much you've paid off; your book value is $140k (plus a bit) where before it was $40k. The bank may "own" the other 70% of your house but they don't get to increase your loan amount because it's gone up; you pocket 100% of the rise in the house's value (unless you end up defaulting on the mortgage).
No, you've gained 100k. Your asset is worth $300k, your mortgage liability is $160k minus however much you've paid off; your book value is $140k (plus a bit) where before it was $40k. The bank may "own" the other 70% of your house but they don't get to increase your loan amount because it's gone up; you pocket 100% of the rise in the house's value (unless you end up defaulting on the mortgage).