 Hello. So, Arvind, I like the paper. What I like about it is that it's something that one can think about actually implementing. When I think of optimal mortgage design, I wonder why we have mortgages at all. I mean, you have people, you're putting all of your assets, levering up 300% into a house which is highly correlated to your human capital, seems a little bit insane, and it would seem that optimal mortgage design might go more for equity sharing, and things like car leasing, we've solved that market. House leasing would be the analog, and then you wouldn't have the levered sector, the financial crises, all the things that go along with it. I wonder if you have any thoughts for why we don't see more equity. Very nice. Thanks. It's interesting engineering as well, which I appreciate. I think there's a little bit too little emphasis maybe on the dark side of adjustable rate mortgages. When you look at the run-up to the Great Recession in 2007-2008, the jacking up of interest rates by the Fed in 2005 and 2006 basically showed that in the end, those mortgages that defaulted in 2006 and 2007 were the ARMs, and so the instability created by ARMs actually only made it necessary later on to reduce interest rates again, and so your effect basically focuses on the downside, but what about the upside of pressure? A couple of further uses of the model, because you start with this totally rational and very simple banking. It's the right place to start, but you don't want to jump to policy conclusions too quickly from that. So one of the criticisms of the Obama administration was that they didn't do anything about bankruptcy rules. In the 80s, there was a crisis with farmland pricing, and at the time a farmer could go bankrupt, the bankruptcy judge could rewrite the mortgage anyway wanted, and that greatly eased that crisis. By the time we got to the big crisis, that didn't work for homeowners because the law had been changed, and it was suggested something like that. Bankruptcy is one of the central elements in it, so you could evaluate that kind of policy. Secondly, you cited the Boston Fed work on bankruptcy. Other Boston Fed work has looked at the strength and weakness of local banks and what happens in the local housing market. So if you were to be building in some feedback on bank behavior, it seems that would that would work, and picking up on on some of the comments, if you wanted to have an effect on pricing, you have to build in the kind of thing that was being referred to. Arms are not popular. They're not easily understood, so the impact on the housing process, housing price process, might look very different as you fall down the extent of reliance. But this was a terrific place to start. I have two questions. The first one, you're talking about innovative products that are not widespread in the market, and ideally, you would want the lenders to be happy to propose these products. Now, my first question is, the lenders in your models, your model appear to be in an exogenous or at least in a reduced form. Are there any reasons to think that if you would make them more explicit with a UTT function and optimising, your results would change? Second question, it's about underwater mortgages. You said the advantage of such products is the advantage is a bit small, because the advantage is spread over time. So my second question is, should we think of the household as having a very high discount rate? Is it because it's a crisis period? And would those advantages be higher in a steady-state model? So some of the policies that you talked about had people switching policies mortgages over time, but in reality, of course, we don't know exactly when a recession begins, for sure, in real time. Is that very important or is it sufficient just to know what the current interest rate is? Yeah, one more, maybe, or two more, two more. Sorry, I haven't seen you. Go ahead. Adjustable rate mortgage means that lower payments in the crisis, and that's good. And that's why households want to pick those and then lever more exante. I wonder whether you could check that implication with data before the crisis, because not everyone was on a fixed rate mortgage, and then you can compare loan-to-value ratios for adjustable rate versus fixed rate mortgages. We had last two interventions. So very interesting paper. I just want to go back to some points touched by the discussant and also by people about the potential risks. One of them is that here basically you're taking the risk out of the households and putting them into the banking system and what your thoughts are on that, whether there is some underlying assumption in your mind that it's easier for the bank to hedge these risks than for households. The other comment is it's unclear whether people will demand too much or too little of this product. On the one hand, because here you have this pricing, pecuniary externalities and so on, people might demand too little of it. On the other hand, if people are somewhat myopic, they may borrow too much in times of low interest rates. And then this might be a constraint on monetary policy going forward, where you may be reluctant to raise rates and may raise another. So I just wanted to hear your thoughts on these things. Thanks. That's all? You want? OK, then I mean, very last one. And then you have five minutes to answer everything. So you'll focus on the price aspect of providing some form of insurance. On the quantity, your model has this hard wire amortization schedule in part for tractability, but you basically assume that all mortgages are progressively amortized. Now, in reality, people can and do lever up. In fact, there's a case that the crisis might much worse, but exactly that. So shouldn't that be part of an optimal contract or perhaps be related to some kind of age related loan to value? I mean, Holland, for instance, the fiscal system strongly pushes for direct amortization. OK, first of all, thank you, Victoria, for your comments. They're all well taken. Thank you all for all of your questions as well. I have five minutes. I'm not going to answer them all. Let me just pick on a couple of points. So a number of you asked about this question about banks. So, you know, the way the model is constructed, the lender is this risk neutral, present value lender. He's just present valuing at whatever the going short rate is. So they're really not that interesting. OK, what are the issues with doing that? So the first thing I would say is actually in the mortgage designs which we have considered here, which are mortgage designs where contingencies are based upon interest rates. I'd say that's actually a pretty decent assumption. Why do I say that? It's because interest rates are in the span of kind of traded assets, hedgeable things that banks can do. So, for example, I mean, if you take my experiment literally, it means households are getting benefits in the recession and banks are losing. Right? That's what's happening in the model. Now, you might ask, well, that sounds bad because that's going to hit bank capital and maybe they would lend less, etc. I would say if the mortgage world was one in which as I said mortgage was the standard or any form of insurance contract was the standard in the space of interest rates, then banks could change their liabilities and apparently they do quite frequently. Just change your funding mix so that, for example, you're all funded with short rather than long and that would hedge out the interest rate risk. So, I have felt that in this question of we have not modeled banks for looking at indexation that is related to interest rates, it's okay. I feel more uncomfortable. So, another exercise which we have done is indexation related to home prices, which is also beneficial as you would guess, but that one I start to get worried about because banks can't hedge out home price risk and so one could easily imagine that when you transfer home price risk into the banking sector, it sits there as we kind of know it does and then that has a spillover effect. Whereas with interest rates, I'm going to make the case I'm more comfortable with this analysis the way it is. Okay, so that's one set of questions. You know, a couple of you have made kind of two related questions. First of all, there's a question as to, you know, there's basic question of if this is such a great contract, you know, why isn't it there? And I don't really have a deep answer for this. I'm going to say things like, well, it does seem like in the U.S. at least, you know, there's so much inertia in mortgage contracts. There's some social aspect to what contract you take on. The government clearly subsidizes one mortgage design. So, you know, one way of reading this model is if you subsidize different mortgage design, maybe everyone would switch to that. There is some interesting avenue one could pursue there. I didn't emphasize this in the presentation, but the way the model works, if everybody is in a mortgage design in which has insurance features that affects the equilibrium. It affects both equilibrium home prices. It also affects the equilibrium spread. There's less default. And so, ex-ante, actually contracts are cheaper for, so there's an externality in the model that's built in as often happens with home price spillovers. And so, there is scope here for thinking subsidize a mortgage design with insurance features, and it feeds back to having beneficial macro effects. So, I don't think I have a deep answer for why this is not offered other than the government does it and it's the common thing. It is true that if you look around the world, there is quite a lot of variation in mortgage design. There's many countries that have fixed rate mortgages. There's many countries that have adjustable rate mortgages. My own understanding of this, which is driven by a paper that John Campbell wrote is, it looks largely like inflation experiences in the 80s sorts out which countries seem to have adjustable rate mortgages and which countries seem to have fixed rate mortgages. And there's some just some inertia built into there. So, other than that, I don't think I have anything else to say about that. There were a couple of questions about arms pre-crisis. Could they have destabilized things? It's a hard question to fully analyze because the world was a fixed rate mortgage and the pool of borrowers who took adjustable rate mortgages were kind of an odd pool. So, I'm just not clear if I can do that comparison so clearly. Anyway, I'm going to stop there. I'm happy to discuss more with people at lunch. Okay, thanks. Thank you, George.