Moving I agree that the 80k provided by the loan 13years ago had equal worth to the OP's deposit but the DP had funding costs (the interest) in order to be able to provide the 80k in the first place.
The DP wants the OP to share his funding costs. Some of what he has paid in 13 years will have (hopefully!) been to pay down capital and some will have been interest payments. Therefore he has a share of equity in the house, but it will be less than 80k.
For example, hypothetically, if the house were sold today for 200k
What happens here? Is it 100k each? Or is it 80 to OP, plus half of the remaining 120k, giving her a total of 140. I'm not sure which way to read the original post. And this is where I get confused because I can see some scenarios that look fair and quite a few that don't.
Scenario 1:House is sold today for 200k. At present, assuming that the mortgage loan is still being paid off, the DP does not yet "own" all of the 80k he put in to the house. He's still paying that back to the bank. So there are 3 owners in a way - OP, who stumped up funds, DP who will own some but not all of the 80k equity and the bank which owns the rest of the equity. And there is surplus equity to be distributed.
I think the arrangement means the 80k deposit goes back to the OP. The amount of equity that the DP has actually paid for is probably around 40 as it's roughly half way through a 25 year loan. That means that the bank would need to be paid back the remaining 40 and this is the responsibility of DP as he took the loan out. So...80 deposit to OP, 40 loan to the bank, 40 paid for equity to DP...That takes us to 160, and leaves us a further 40 equity increase that is shared equally between the two. That does not give a straight 50/50 split but I think it's a fair outcome if that is what happens.
Scenario 2: House is sold for 200k in the future when the mortgage is paid off. OP gets her 80k, DP should get his 80k of equity and the remaining 40 equity increase should be split between them. If this does not happen, then I completely agree it would be unfair.
Scenario 3: House is sold today for 100k. In this scenario I assume that the bank will be first in line to recoup the outstanding value of the mortgage - say 40 - leaving 60 to be divided between OP and DP. Does OP get all of this, to reflect her privileged deposit? Is there a pro-rata to reflect the amount of her deposit compared to the amount of equity that the DP owns by this stage? Or does the deposit actually come to OP before the bank is paid? In this case, the house will not cover the outstanding loan and the DP will be left with 20k debt to the bank and no property. That's tough on the DP, but arguably what has happened to many people who lived through the downturns in the market that had negative equity.
Scenario 4: House is sold in the future for 100k but the mortgage has been paid off. It is most equitable for the losses to be shared equally, but I'm not sure if that's how the arrangement works.
I've probably missed tons of posts clarifying all this....