Joint mortgage, split up, negative equity
If John agrees to buy Mary out, the following has to happen...
John buys Mary's share of the property for its value - €60k
John takes over €60k of Mary's mortgage
Mary would have to pay the lender €40k or take out a loan of €40k.
So the revised position would be
|total|John|Mary
Mortgage|€160k|€160k|0
Loan|€40k|0|€40k
Value|€120k|€120k|0
Negative equity|€80k|€40k|€40k
John would have to be able to service a mortgage of €160k on his own.
Before the split up, the bank has a joint loan. If Mary does not pay her share, John is responsible for it. So the bank is giving up a fair bit of security and will only agree to this if there is some compensation.
What might encourage the bank to agree to this?
If Mary can actually pay €40k cash off the loan, the bank may well consider it, as it is reducing their total loan and their negative equity.
The bank might agree to convert the €40k into a personal loan at personal loan interest rates. If Mary has a guarantor, they might be more open.
If the original mortgage is a cheap tracker, the bank might be open to replacing it with a mortgage at a higher interest rate.
If John can't afford a mortgage of €160k on his own...
John's could ask someone else (Anne) to buy Mary's share of the house for €60k. Mary would still have to find the €40k shortfall.
Anne should not agree to a simple replacement of Mary on the house and the mortgage. By doing so, she would be paying €100k for a share of a house worth €60k.
If John can afford a mortgage of €200k on his own...
The bank might allow John to take over the full mortgage on his own. If he does this, he would be paying €100k for €60k worth of house. He could have a side agreement with Mary that she repays the loan to John over time.