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# ECON 251: Financial Theory

## Lecture 18

## - Modeling Mortgage Prepayments and Valuing Mortgages

### Overview

A mortgage involves making a promise, backing it with collateral, and defining a way to dissolve the promise at prearranged terms in case you want to end it by prepaying. The option to prepay, the refinancing option, makes the mortgage much more complicated than a coupon bond, and therefore something that a hedge fund could make money trading. In this lecture we discuss how to build and calibrate a model to forecast prepayments in order to value mortgages. Old fashioned economists still make non-contingent forecasts, like the recent predictions that unemployment would peak at 8%. A model makes contingent forecasts. The old prepayment models fit a curve to historical data estimating how sensitive aggregate prepayments have been to changes in the interest rate. The modern agent based approach to modeling rationalizes behavior at the individual level and allows heterogeneity among individual types. From either kind of model we see that mortgages are very risky securities, even in the absence of default. This raises the question of how investors and banks should hedge them.

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html## Financial Theory## ECON 251 - Lecture 18 - Modeling Mortgage Prepayments and Valuing Mortgages## Chapter 1. Review of Mortgages [00:00:00]
This was 3,500-3,800 years old and we have on these cuneiform tablets these mortgages. And so the idea of a mortgage is you make a promise, you back your promise with collateral, so if you don’t keep the promise they can take your house, and there’s some way of getting out of the promise because everybody knows the collateral, you might want to leave the home, and then you have to have some way of dissolving the promise because the promise involves many payments over time. So it’s making a promise, backing it with collateral, and finding a way to dissolve the promise at prearranged terms in case you want to end it by prepaying. And that prepaying is called the refinancing option. And because there’s a refinancing option it makes the mortgage a much more complicated thing, and a much more interesting thing, and something that, for example, a hedge fund could imagine that it could make money trading. So I just want to give you a slight indication of how that could happen. So as we said if you have a typical mortgage, say the mortgage rate is 8 percent–maybe this is a different answer than I did–so here we have an 8 percent mortgage with a 6 percent interest rate to begin with. Now, if it’s an 8 percent mortgage the guy’s going to have to pay much more than 8 percent a year because a mortgage, remember, there are level payments. We’re talking about fixed rate mortgages. You pay the same amount every single year for 30 years, now you’re really paying monthly and I’ve ignored the monthly business because it’s just too many months and there are 360 of them. So I’m thinking of it as an annual payment. You have to pay, of course, more than 8 dollars a year because if the mortgage rate were 8 percent and you had a balloon payment on the end, you’d pay 8, 8, 8, 108. That’s the way they used to work, but they were changed. So you could imagine the old fashioned mortgage would pay 8, 8, 8, 8, 8, 108; if you didn’t pay your 8 somewhere along the line they’d confiscate your whole house and then take what was owed out of it and you could get out of it by paying 100. The new mortgages instead of paying 8 every year for 30 years you pay 8.88 every year for 30 years because if you discount payments of 8.8 for 30 years at 8 percent you get 100. So the present value is 100 at the agreed upon discounting rate or mortgage rate 8 percent. And so you see how important this discount rate is. And the remaining balance, however, goes down because every time you’re paying you’re paying more than the 8 percent interest. You’re paying in the first year 8.8 instead of 8 and so that gap of .88 is used to reduce the balance from 100 to 99.117. And so you see the balance is going down over time and making the lender safer and safer because the same house is backing it. So it’s called an amortizing mortgage. ## Chapter 2. Complications of Refinancing Mortgages [00:03:18]Now, why is it difficult to value? Because you have the option, any time you want, and there’s a good reason for that option, any time you want you have the option of getting out of the mortgage and just saying, “Okay, I’ve paid 3 payments of 8.88, I don’t want to do it anymore. I want to pay off 97.13 and then let’s call it quits.” And they say, “Okay,” and there’s nothing they can do about it. Now, when are you going to exercise that option? You’re going to exercise that option either because you have to move, that’s the intention of it, or you’ll exercise it when it’s most advantageous to you. Now, why could it become advantageous to exercise it? Well, you don’t really want to exercise the option and this is the way most people think of it backwards. They think, “Oh, the interest rates are going down. That means I’ll get a new mortgage with a lower interest rate.” They’re hoping for exactly the wrong thing. If the interest rates go up what they’ve got is a much better mortgage because they’re continuing to buy at the same 8 percent interest and maybe interest rates in the economy have become 12 percent and they’re actually making money. So people who borrow in times of high inflation do better. When there are times of deflation the borrowers get crushed. Irving Fisher said one of the main reasons for the Depression being so bad is all the entrepreneurial people in the country, as usual, were borrowing, and then there was a deflation and so they were getting crushed. And the very people who drive the economy were being hurt the most. And so that feedback, he said, was responsible for part of the severity of the Depression. So you see interest rates can go up or down and what happens? When they go up, if they go up high enough to 19 percent you think, “My, gosh, I’ve made a fortune holding this mortgage. I’m still borrowing at 8 percent and I can invest my money at 19 percent.” So you’ve made a fortune and the poor lender’s gotten crushed. On the other hand if the interest rates go way down here, so the present value of what you owe if you kept paying it becomes huge, you don’t have to face that big loss because you just prepay at whatever the remaining balance is there and then you’ve protected your downside. So by paying attention and deciding when the optimal time to prepay is, you can save yourself a lot of money and thereby cost the bank a lot of money. So when exactly should you prepay? When should you exercise your options? Well, in this example if you never exercised it you’d be handing the bank, effectively, 120 dollars even though they lent you 20 [correction: lent you 100]. So the bank would have made a 20 percent profit on you. But if you exercise your option optimally you’re going to make not 100–the bank is not going to get 100 dollars out of you, they’re going to even get less than 100 dollars. They’re going to get 98 dollars out of you. So when exactly should you be exercising your option? Well, we went over this last time. I’ll do it once again. So remember, the payment you owed was 8.88, 8.88, blah, blah, blah, 8.88. The remaining balance started, of course, at 100 and then it went down to 99.11 and then it kept going down from there. So since I can’t remember the numbers let’s just call this B Let’s call this B Now, you notice that if I had divided this by B There’s nothing fancy about 100, nothing important about 100. If the original loan was for 200 you just double all your payments and double all your remaining balances. What could be more obvious than that? So I want to think in those terms of a mortgage that always has 1 dollar left. So suppose at any stage you had 1 dollar left in your mortgage. Your remaining balance was 1. So let’s say at any node, let’s ask the question, what is the value of 1 dollar of remaining balance? So if you start at 100 and you haven’t prepaid, here you’ve got B And let’s say it’s in period 1, 2, 3, 4, 5, so I’m in period 5, B Well, what’s your payment? The payment is this 8.88 but divided by B Now, why is this right? I hope it is right by the way. I should have thought of this a little before. So this is the remainder of 1 dollar left. So if I divide by B And then with the other probability 1 half, plus I make the payment, but I go down instead of up and so I have B So I know now by working this backwards I can tell what 1 dollar at the beginning is worth. And so it’s exactly the same calculation I did before except I’m talking about 1 dollar. I’m always figuring out 1 dollar of remaining balance instead of the whole thing. Present value of callable, so here’s present value of 1 dollar of principal. And so remember the present value of a callable mortgage was 98.8. Here the present value of 1 dollar, figuring it out that way, is .98, obviously it’s divided by 100, but the key is that now you can see just by looking at it where the 1s are is where the guy decided to prepay. So it’s the same thing as before, but you see before you couldn’t tell very easily from the numbers when I did the 100. Sorry, that didn’t quite make it. Before when I did the present value with the 100 all these numbers were 98s and 97s. I mean, where has he prepaid? It’s hard to tell where the prepayment is. If I do it all in terms of 1 dollar of remaining balance then just by looking at the screen I can tell where the guy prepaid because there are 1s there. So I know where he’s prepaid. Wherever the 1s are that means he’s prepaid. So I can tell very easily what he did. All right, that’s the only purpose of doing the same calculation in a somewhat trickier way. So if you think about it a second you see I’ve just divided by–I’ve always reduced things to if you had 1 dollar left. All right, so this tells us what to do, when the guy should prepay and when he shouldn’t prepay. So if you’re now in the world looking at what’s happening you can find the historical record of how people have prepaid. So let’s just look at the historical record, for example. Here, if you can see this, this is blown up as big as it goes. So this is what you might see as the historical record of percentage prepayments annualized from ‘86 to ‘99, say. So you notice that they’re very low here, and then they get to be very high, and then they get low again, and then they get high again. So why do you think that happened? So what is this? This is prepayments for a particular mortgage, 8 percent. You take all the people in the country who started in 1986 with 8 percent mortgages. There’s a huge crowd of those because that was about what the mortgage rate was that year. So a huge collection of people got these mortgages in ‘86 and you keep track of what percentage of them prepaid, really every month, but you write the annualized rate, and then this is the record. So why do you think it changed so dramatically like that? What’s the explanation?
You got this shocking amount of prepayment. So this graph, which seems sort of surprising and looks like the stock market, turns out to have nothing to do with the stock market. It has to do with where the interest rates are. Well, do you think interest rates explain everything? No. What else could you notice about the–escape. What else have we learned here by doing these calculations? Well, what we’ve learned so far is that if the interest rates in the economy are at 6 percent, that’s where they started, remember we said they started at 6 percent and there was 16 percent volatility. Here I had 20 percent volatility. It doesn’t matter. I mean, that’s a plausible amount of volatility, a little high, but that volatility. The mortgage rate of 8 percent is not going to give a value of 100. It’s going to cheat the bank if the homeowners are acting rationally. The bank could get 120 if the people weren’t acting rationally. They were just never exercising their option. It they’re exercising their option optimally the thing was only worth 98. Now, I told you at that time the interest rates should have been around 7 and 1 half percent, not 8 percent given this 6 percent interest rate in the economy. The mortgage rate should be 7 and 1 half percent. So we deduced last time that obviously not everybody’s acting optimally. Well, you can tell that looking at this diagram. How do you know that not everybody’s acting optimally? Remember these are ‘86 mortgages, so everybody’s taking them out at the same time within a few months of each other, the same 8 percent mortgage. How can you tell from this graph that they’re not exercising their option optimally? It’s completely obvious. Just looking at it for one second you can say, “Oh, these people can’t be exercising their option optimally,” why is that? Yes?
And over here when presumably you ought to prepay, in the entire year, right, they have 12 chances during the year. It takes them an entire year and only 60 percent of them have figured out that they should prepay. So you know they’re not acting optimally. So just from that graph that would tell you, and you have further evidence of that. That’s evidence that they aren’t acting optimally. Furthermore you have evidence that the banks don’t expect them to be acting optimally because the banks aren’t charging them 8 or 9 percent interest, which is what they would need to pay to get the thing worth 100, they’re charging them 7 and 1 half percent interest which for the optimal pre-payer is worth much less than 100 to the bank. So the banks wouldn’t do that. They would just go out of business if they did something stupid like that. They wouldn’t do that unless they thought that the homeowners weren’t acting, at least not all of them acting, optimally. ## Chapter 3. Non-contingent Forecasts of Mortgage Value [00:19:26]So suppose you had to predict how people are going to act in the future and you wanted to trade on that? What would you do? How would you think about predicting it? So this is the data that you have. What would you do? You have this data. These are 8 percent things. You also have 9 percent mortgages issued the year before, and then maybe a year before that there were 8 and 1 half percent interest and you have that history, and you’ve got all these different pools and all these different histories. How would you think about figuring out a prepayment–how would you predict prepayments? Well, the way economists, macro economists at least in the old days, used to make predictions, they would say, “Hum, the first quarter looks pretty good.” What are they predicting now? Now, they’re saying unemployment is probably going to keep rising for the next quarter or two well until the next year, but at that point things are going to turn around and we expect the economy to get stronger, come out of its recession and unemployment should gradually improve from its high which we expect will be 10 and 1 half percent to something back down to 6 percent by the end of 2011. That’s more or less the economists’ prediction. Now, can you make a prediction like that about prepayments? Would it make sense to make a prediction about that? Why is that an utterly stupid kind of prediction? What is the essence of good prediction? If you wanted to predict something and you were going to lose a lot of money if your prediction was wrong how would you refine your prediction compared to what I just gave as a sample prediction? Yep?
It’s a sure thing it’s going to be wrong. It’s just impossible that’s going to be right because the guy making the prediction has made no contingencies built in his prediction. You know that guy’s making a prediction for free. Someone may be paying him to hear him, but he’s not going to be penalized if his prediction is wrong. No one in their right mind would make such a prediction. So the first thing you should do in predicting prepayments is to realize that you’ve got a tree of possible futures, and given this tree of possible futures you’re going to predict different prepayments depending on where you go on the tree. So you see, prediction is not a simple one event–it’s not a one shot thing. Just as he so aptly put it, it’s a many scenario thing. You have to predict on many, many scenarios what you think will happen and that makes your prediction much better because, of course, if there is a war in Iraq, and if there is a catastrophe in Afghanistan, and if Iran does bomb Israel, and if the commercial real estate market collapses things are going to be a lot worse than this original guy’s prediction. So everybody knows that, so why not make the prediction more sensible? So, on Wall Street that’s what everybody’s done for 20 years. Now, they haven’t done it for 30 years. It’s just 20 years that they’ve been doing that. So when I got to Kidder Peabody in 1990 they were making these one scenario predictions. It’s a long story which I’ll tell maybe Sunday night. I ended up in charge of the Research Department and so we made, you know, other firms were doing this already, we made scenario predictions, okay? So now what kind of scenario predictions are you going to make? When you make contingent predictions there are an awful lot of them. You can’t even write them all down, so what you have to do is you have to have a model. So what kind of model should you have? I’ll tell you now what the standard guys were doing on Wall Street at the time. They were saying–here’s interest rate, sorry. Here’s the present value of a mortgage. Here’s the present value of a callable mortgage, present value of 1 dollar of principal, so realistic prepayments. So if we go over here we’ll see that people said, “Look, from this graph it’s clear,” they would say, “that when interest rates went down people prepay more so why don’t we have a function that looks like this?” So, prepay, that’s the percentage of remaining balance that is paid off. So what does that mean? Remember, after you’ve made your coupon payment you have a remaining balance, B Typically some people who are alert and think it’s a good time to prepay will pay all of B So we look at the aggregate prepayment. That’s the old fashioned way. And we say, “What percentage of the remaining balance is paid off?” So you’d make a function like this. You’d say, “Well, prepaid might equal 10 percent.” Why am I picking 10 percent? So if you go back to this picture you see that prepayments seem to be around 10 percent when nothing’s happening. So you say 10 percent plus maybe you’re going to get some more prepayments so you might write–well, I just wrote down a function plus the min. The min, say, of .60 because it never seems to get over 60 percent if you look at that you see it never gets over 60 percent really. So the min of 60 and 15 times the max of 0 and (M - r That would be a kind of prepayment function. So what does this say? What happens? You’re normally going to pay–so this is this whole function here, so I should write this as .1 plus, can you see that over there, maybe not, so this plus .1. So there’s a baseline of 10 percent and if the interest rate is high, so the interest rate is above the mortgage rate no one else is going to prepay because this is going to be a negative number and this will be 0. So you’re just going to do .1,10 percent. On the other hand, as the interest rate gets low and falls far enough below the mortgage rate people are going to say to themselves, “Ah-ha! I have a big incentive to prepay now. Maybe interest rates have gone down so far I can no longer hope they’re going to go back up above the mortgage rate. I should start prepaying more.” So more people are going to prepay and this thing is going to go up. I just multiply it by some constant, but it’ll never go up more than 60 percent. That’s what this function says. And sigma, this is the volatility–all right, so let’s just leave that aside. So there’s a prepayment function that seems to sort of capture what’s going on. It’s usually around 10 percent when there’s no incentive. It never gets above 60 percent, but as the incentive to prepay, as interest rates get lower and the incentive to prepay increases, more and more people prepay. That’s kind of the idea. All right, and then you would fit fancier curves than that. You would look at M - r ## Chapter 4. The Modern Behavior Rationalizing Model of Mortgage Value [00:28:40]That was the old fashioned way and that’s how people would predict prepayments. Now, that’s not going to turn out to be such a great way, but it certainly teaches you something. So let’s look at what happens if you now–with those realistic prepayments you compute the value of a mortgage. So this is the prepayment that you’d get for the different rates and so you can see that as the rates go down the total prepayment is going up. And by the way, it’s more than 60 percent because you’ve got this 10 percent added to the 60 percent, so the most it could be is 70 percent, which it hits over here. So you get 70 percent as the maximum prepayments, and as interest rates get higher no one prepays except the 10 percent of guys. Now, by the way, why are people prepaying over here even when the rates are so high? It’s because some people are moving or they’re getting divorced and they have to sell their house. So obviously you’re going to get some prepayments no matter what. People have to prepay, and why is it that people never prepay more than 60 percent historically or 70 percent, because not everybody pays attention. Now, I called them the dumb guys last time, but as I said, I probably fit into that category. It’s people who are distracted and doing other things. They’re just not paying attention and so they don’t realize. They don’t know what’s going on, so they don’t realize they should be prepaying. So as interest rates go down more people prepay. As interest rates go up less people prepay. And if you did some historical thing and figured out the right parameters you’d get a prepayment function. So how did I figure out this was 15? How did I figure out this was .6? Why should I divide this by 133? What’s sigma? Once you get those parameters historically you now have a well-determined behavior rule of what people are going to prepay, and from that you can figure out what the prices are of any mortgage by backward induction. So how would you do it again by backward induction? The same we always did it. Over here, what would you do over here? How would you change this rule? Well, you would just be feeding in the prepayment function. So what would the prepayment function be? Well, people wouldn’t be doing a minimum here, right? They’re not deciding whether or not to prepay, they’re just prepaying. So let’s get rid of that. They’re prepaying. So this is the value of 1 dollars left of principal. So some of them are prepaying and that’s the function, so prepay, and that depends on what node you’re at. And here it says what percentage of the remaining balance is being prepaid. So that tells you, that rule, who’s prepaying, and then with the rest of the money that’s going on until next time 1 minus that same thing, 1 minus prepay times exactly what we had before. So this part of 1 dollar got prepaid immediately so that’s the cash that went to the mortgage holder. The rest of the cash got saved until next time and here’s what happens to it. You have to make your coupon, then you have a remaining balance, and then whatever is going to happen is going to happen. So you’ll study this and you’ll figure out I’m sure. It takes a little bit of effort to see that through, but with half an hour staring at it you’ll understand how this works and you’ll read it in a spreadsheet so you can figure out the value of a mortgage. You get a value of a mortgage, and now we can start doing experiments by changing the parameters and see how the mortgage works. Now, before I do that I want to say that there’s a better way to do this. I mean, maybe these numbers are estimated–what’s a better way of doing it? How did I do it at Ellington, how did we–I mean at Kidder Peabody? How did we predict prepayments? What’s another way at looking at prepayments? Let me tell you something that’s missing. I used to ask people who wanted to work at Kidder Peabody or Ellington the following little simple puzzle, and most of the genius mathematicians always got this answer wrong. Of course we hired them anyway, but they’d always get this wrong. So the question is, suppose you’ve got a group of people like this and you figure out what the value of the mortgage is, and interest rates have been constant all this time. Let’s suppose for one month interest rates shoot down, interest rates collapse and half the pool, 60 percent of the pool disappears. So now you’ve only got 40 percent of the people left you had before, and then interest rates return to exactly where they were to begin with. Should the pool that’s left be worth 40 percent of the pool that you had just here, or more than 40 percent, or less than 40 percent? So remember, you had 100 people here. You’re the bank who’s lent them the money. You’re valuing the mortgage payments they’re going to make to you, you’re getting a certain amount of money from them, 60 percent of them suddenly disappeared in 1 month leaving 40 left, but now interest rates are back exactly where they were before. Is the value of the mortgage starting here with the 40 percent pool worth 40 percent of what it was originally, more than 40 percent or less than 40 percent? What do you think? Yes?
Now, this function doesn’t capture that at all, right? It doesn’t say anything. It just says your prepayment’s depending on where you are. So whether you were here or here you’re going to get the same prepayment, but we know that that’s not going to be the case. In fact, it’s clear that over here there must have been a much bigger incentive than there was over there. So the prepayments are the same, but actually interest rates here were vastly lower than interest rates there. So this is not such a good function. So how would you improve? What would you do to take into account this adverse selection, or actually pro-verse selection? What is the opposite of adverse? Well, it doesn’t matter. What would you think to do? Your whole livelihood depends on it, millions, trillions of dollars at stake here. You’ve got to model prepayments correctly, so how would you think of doing this? Just give me some sense of what a hedge fund does or what anyone in this market would have to do. Well, most of them did this. So what would you do? Yeah?
Someone’s asking you to run a research department, make a model of forecasting prepayments. All the data you have is aggregate data like that. You can’t observe individual homeowners in those days. They wouldn’t give you the information. I’ll explain all that Sunday night. So this is the kind of data you have, what the whole group of people is doing every year, but what would you do to build the model? Adverse selection is very important or pro-verse selection. It’s embarrassing I don’t remember the word, favorable selection, a very important thing. So how would you capture that in your model? Yep?
What you ought to do is you ought to say, well, 8 percent–remember we had two kinds of people already. We’ve already got two kinds of people, sorry. We’ve got these guys, the guys who never call, so they’re people. That’s a kind of person. And suppose you go down here and you have the people who are optimally prepaying? Suppose you imagine that half the people were optimally prepaying and half the people never prepaid? Well, would that explain this favorable selection? Absolutely it would explain it because when you went through your little tree and you went here, and here, and here, and here, by the time you got down here all those people, all the optimal pre-payers they’re all prepaying. So you start off with half-optimal guys and half-asleep guys. Once you get down here all the optimal guys have disappeared and the pool that’s left is all asleep, so of course the pool is worth much more here given the interest rate than it was over here. In fact, if it goes back then again to here where it was before–sorry that’s same line. If it goes back to here–have I done this right? No, I’ve got to go back twice here and then here. So once it goes back to here if it goes here, here, here and here then the pool is going to be much more valuable here than it started there. There are half as many people, but it’s worth much more than half of what it was there. So the way to do this is to break–so then you’re looking at the individuals. You’re saying one class of people is very smart, or one class of people is very alert, it’s a much better word, one class of people is very alert. One class of people is very un-alert and as you go through the tree the alert people are going to disappear faster than the non-alert people and that’s why you’re going to have a favorable selection of people who’s left in the pool. Well, of course, there are no extremes of perfectly rational or perfectly asleep in the economy so what you can do is you can make people in between. How do you make them in between? Well, suppose that, for example, I only did one thing. Suppose it’s costly to prepay? Some people just say to themselves, “I’m going to have to take a whole day off of work. I’m not going to write my paper. I might lose some business that I was going to do that day. A whole bunch of stuff I’m losing, so I’m going to subtract that. I’m not going to prepay. I’m not going to even think about doing it unless I can get at least a certain benefit from having done it.” So you can add a cost of prepaying and people aren’t going to prepay unless the gain that they have by prepaying exceeds the cost of doing the prepayments. So to take the simplest case let’s suppose the very act of–never mind the thinking and all that–the very act of prepaying, going to the bank literally costs you money. So if you have a value, if the thing is 100 and you can prepay, you know, if you do your calculations and don’t prepay today it’s worth 98 and if you prepay today the remaining balance is 94 you’re saving 4 dollars, but if the cost of prepayment is 5 you’re still not going to do it. So you get a guy with a high cost of prepaying, an infinite cost of prepaying, he’s going to look like he’s totally un-alert. A guy with zero cost of paying is going to look like he’s totally alert. So you can have gradations of rationality, and you can have different dimensions. So you can have cost of prepaying and you can have alertness. What’s the percentage of time you’re actually paying attention that month? What fraction of the months do you actually pay attention, and you can have a distribution of people, different costs and different alertnesses. So that’s the model that I built. It’s a simplified form of it. It gives you an idea. So here’s this burnout effect that I showed that if you take the same coupons, but an older one rather than a–an older one that’s burned out will always prepay slower, so the pink one is always less than the blue one because it went through an opportunity to prepay. So here you start with a pool of guys on the right, and then after a while, after time has gone down a lot of them have prepaid. So here’s alertness and cost. So you describe a person by what his cost of prepaying is and how alert he is. The more alert he is and the lower the cost of prepaying the closer to rational he is. The less alert he is, the higher the cost of prepaying the closer to the totally dumb guy he is. And so you could have a whole normally distributed distribution of people and over time those groups are going to be reduced because a lot of them are prepaying, but they won’t be reduce symmetrically. The low cost high alertness guys are going to disappear much faster and the pool’s going to get more and more favorable to you. And so anyway, all you have to do is parameterize the cost, what the distribution of people in the population, what the standard deviation and expectation of cost is and of alertness is, and that tells you what this distribution looks like. So you’re fitting four numbers and you’ve got thousands of pools and hundreds and hundreds of months, and fitting four parameters you can end up fitting all the data. So look at what happens here. So here’s the same data. So I just tell you I know that in a population, given what I’ve calculated in the ’90s there, I know what fraction of the people have this cost and that alertness, what fraction of the people are so close to dumb that their costs are astronomical and their alertness is tiny, what fraction of the people have almost no cost and a very high alertness, so I’m only estimating four parameters because I’m assuming it’s normally distributed. Given that fixed pool of people I apply that to the beginning of every single mortgage and I just crank out what would those guys do. In the tree if they knew what the volatilities were when would they decide to prepay, and then I have to follow a scenario out in the future and I say, “Well, along this path which guy would prepay and which guy wouldn’t prepay and what would the total prepayments look along that path?” And so this has generated the pink line from the model with no knowledge of the world except I fit those parameters and look how close it is to what actually happened. So it turns out that it was incredibly easy to predict, contingently predict what prepayments were going to be and therefore to be able to value mortgages. And this was a secret that not many people, you know, a bunch of people understood, but not that many understood, and so for years we were trading at our hedge fund, first at Kidder and then at Ellington with this ability to contingently forecast prepayments at a very high rate. And why was it so stable, the prediction, and so reliable? It’s because the class of people stayed pretty much the same and every year there’d be the same kinds of people with the same kinds of behavior. Some were very alert. Some were very not alert, but the distribution of types was more or less the same and you could predict with pretty good accuracy what was going to happen from year to year. Of course, then after 2003 or so the class of people started to radically change and many more people who never got mortgages before got them and it became much harder to predict what they were going to do. But so in the old days it was pretty easy to predict. And why was it so easy to predict? Because it was an agent based model, agent based. So, by the way, I added this volatility here, so these guys who just ran regressions they had to have a volatility or something parameter. So you see as volatility goes up the prepayments are slower. Well, they just had to notice that and build it right into their function. I didn’t even have to think of that or burnout. None of those things did I have to think about because if you’re a guy optimizing here and volatility goes up, so you reset the tree so that the interest rates can change faster. The option is worth more so you’re going to wait longer. You’re not going to just exercise it right away because you’ve got a chance that prices will really go up so you can wait a little longer, afford to wait longer. So prepayments will slow down. So all I’m saying, all of this is just to say that if you have the right–so it’s agent based, it’s contingent predictions, those two things together enable you to make quite reliable predictions about the future if you’re in a stable environment. And so what seems like a bewildering amount of stuff turns out to be pretty easy to explain. So now what happens? So do you have any questions here or should I–yes?
They are the people with the really high alertness up there and the really low cost, so they’re the guys back there. They’re the hyper–or maybe it was the guys, you know, one of these corners with very high alertness and very low cost. I forgot which way the scale works. It might be going down. So anyway, the guys with very high alertness and very low costs are the hyper rational people. At the other corner you’ve got the guys who have very low alertness and very high costs. They’re the people who you’re going to make a lot of money on if you’re the bank. So how do I know how many people are of each type? Well, I don’t. I have to fit this distribution. But you see I have so much data. I’ve got this kind of curve. This kind of curve I’ve got for every starting year for the whole history and there’s so many different interest rates and so many different–so I’m applying that same population at the beginning of every single curve and then seeing what happens to my prediction versus what really happened. So I’ve got thousands, and thousands, and thousands of data points and only four parameters to fit. So I pick the four parameters to fit the data as much as possible. If I assumed everybody was perfectly alert instead of that curve that I showed you, I put a huge crowd here of perfectly rational people then I would have found that I would have gotten prepayments at 100 percent up there and at 0 all the way over here and so it wouldn’t have fit that curve. So that’s how I knew that there couldn’t be that many perfectly rational people. Yes?
So now if you want to trade mortgages what are some of the interesting things that happen? The first interesting thing to notice is that what do you think happens as the interest rate goes down? So the first thing to notice is–so I’ll just ask you two questions. Let’s go on the other side. I’m running out of room. Suppose that you have the mortgage value, what you get in the tree? So in this tree that we’ve built, here’s the tree, it’s going like that, and at every node we’re predicting–for each class of people we’re predicting where his 1s are. So that class is prepaying. The other class is not as smart so they’re not prepaying here, but maybe when things get really low they’ll start prepaying here. So each class of people, each cost, alertness type has its own tree. They’re the same tree, but it’s own behavior on the tree, and then I add them all together. So what happens with the starting interest rate? So here we had .06 and this value was 98 or something, right? Now, suppose the interest rate went down to .05. I drew this picture of interest and mortgage value. What do you think happens? So the interest starts–this is ‘98, 6 percent is there. As the interest rate goes down what do you think happens to the value of the mortgage? If you’re a bank and you’ve fixed–the mortgage rate is 8 percent. That’s a fixed mortgage rate, but now you’ve moved in the tree from here to here. Do you think your mortgage is going to go up in value or down in value?
But will it keep going up like this and this? If it were a bond it would go up like that, right? A bond, a 1 year bond which owed 1 over 1 + r would keep going up and up the value before it got negative, say. It would go up. As r got negative it would go way up like that. So does the mortgage keep going up like that? As the interest rate goes down is the value of the mortgage going to get higher and higher and higher? Suppose the guy’s optimal, what’s going to happen? This is 100 here. What’ll happen? Yep?
As the interest rate gets higher you get crushed, and as the interest rate gets lower you don’t get the full upside because he’s prepaying at 100. He’s never letting it go above 100, right? So if he’s not so optimal maybe your value will go up, but not so astronomically high. So this idea that the mortgage curve, instead of being like this goes like that, this is what was called negative convexity. ## Chapter 5. Risk in Mortgages and Hedging [00:54:07]Now, the next thing to know is suppose that the guys are partly irrational so it’s going above 100. So it’s starting to go like this. Then what do you think? As the interest gets really low what’s going to happen? All right, you just said it, so. If the guy was rational, perfectly rational it would go like that. He’d never let it go above 100, but now suppose guys are not totally rational? What’s going to happen is they’re going to, sort of–as rates get a little bit low they’re going to overlook the fact that they should prepay. So now it’s advantageous to you. Things are worth more than 100, but if rates get incredibly low even the dumbest guy, the highest cost guy is going to realize he has an advantage to prepay and so things are going to go back down like that. So the value’s going to be quite complicated. So this is the mortgage value as a function of interest rates. Just common sense will tell you this. In a typical bond as the interest rate gets lower the present value gets higher. You should expect a curve like that, but because of the option if it were rationally exercised the curve would never get above 100. It would have to go like that. But now if people are irrational you can take advantage of them and get more than 100 out of them. But if the situation gets so favorable to you it becomes blindingly obvious, eventually to them, that they’re getting screwed, and eventually they act and bring it all the way back to 100 again. So this value of the mortgage looks like that. So that’s a very tricky thing. I’ll even write, very tricky. So if you don’t know what you’re doing you could easily get yourself hurt holding mortgages. You could suddenly find yourself losing money holding mortgages. So that’s my next subject here. I want to talk about hedging. So we know something now about valuing mortgages. Now I want to talk about hedging, and what hedge funds do, and what everyone on Wall Street should be doing which is hedging. So if you hold a mortgage you’re going to hold it because maybe you can lend 100 to a bunch of people but actually get a value that’s more than 100. So it looks like you’re here, but if interest rates change a little bit suddenly this huge value you thought you had might collapse back down to 100, or the interest rates might go up and it might collapse to way below 100. So you look like you’re well off, but there are scenarios where you could lose money and you want to protect yourself against that. So how do you go about doing it? What does hedging mean? And I want to put it in the context, the old context of the World Series which we started with before. So it’s easier to understand there, and so many of you will have thought about this before so you’ll be able to answer it, but if I put it in the mortgage context it would seem just too difficult. I don’t know why I did that. So the World Series–I’m going to lower it in a second. So suppose that the Yankees have a 60 percent chance, I said beating the Dodgers, I thought the Dodgers would be in the World Series, a 60 percent chance of winning any game against the Phillies in the World Series. And you are a bookie and your fellow bookies all understand that it’s 60 percent. So some naïve Philly fan comes to you and says I want to bet 100 dollars that the Phillies win the World Series. Should you take the bet or not? Yes you should take the bet because 60 percent of the time you’re going to win 100 dollars–no. Yes you should take the bet. If he bet on one game you would make, with 60 percent probability you’d win 100 and with 40 percent probability you’d lose 100. So that means on average your expectation is equal to 20. So if he’s willing to bet 100 dollars on the Phillies winning the first game of the series with you, you know that your expected chance of winning is 20 dollars. You’re expecting to win 20 dollars from the guy. Now, suppose he’s willing to make the same bet, 100 dollars for the entire series? What’s your chance of winning and what’s your expected profit from him? Is it less than 20, 20, or more than 20?
In either case you might lose 100 dollars. The Phillies, they’re probably going to lose, but there’s a chance something goes crazy and some unknown guy hits five home runs in the first four games or something, and some other unknown guy hits another four home runs and you lose the World Series. You could lose 100 dollars, and maybe the guy’s not betting 100 dollars but 100 thousand dollars or a hundred million dollars. You know you’ve got a favorable bet, but you don’t want to run the risk of losing even though there’s not that high a chance you’re going to lose. What can you do about it? Well, you know that there are these other bookies out there who every game are willing to bet at odds 60/40 either direction on the Phillies or the Yankees because they just all know–they’re just like you. You all know that the odds are 60 percent for the Yankees winning every game. So suppose this naïve guy, the Phillies fan, comes up to you and bets 100 dollars on the World Series that the Phillies will win. You don’t want to run the risk of losing 100 dollars. You know there are these other bookies who are willing to take bets a game at a time 60/40 odds. What should you be doing? What would you do? Yes?
All right, so what happens is you’ll have a tree which looks like this and like this, like this and like this, like this and like this and let’s say we go out a few games like this. Now, this is a 1, 2, 3 game series. All right, so I’ve done it. Here’s the start of World Series. This is the World Series spreadsheet you had before. Now, here’s the start. Here’s game 1, 2, 3, 4, 5, 6, 7. So if the Yankees win the series they get 100 dollars. You get 100 dollars, sorry. Oh, what an idiot. So every time you end up above the start, win more than you lose, you get 100 dollars. On the other hand, if you lose more than you win you lose 100 dollars, and so ctrl, copy. Here is losing 100 dollars. So now this tree, remember from doing it before, is just by backward induction. If you look at this thing up there it says you get, 60 percent I think was the number we figured out over here, so right? So 60 percent is the probability of the Yankees winning a game. So you take any node like this one you’re always taking 60 percent of the value up here plus 40 percent of the value here. So if you do that you find out that the value to you is 42 dollars, just what we said. So let’s put that in the middle of the screen. So the value is 42 dollars. Now, if the Yankees win the first game you’re in much better shape. So winning the first game means you moved up to this node here. All of a sudden you went from 42 dollars to 64 dollars. And if the Yankees lost the first game you would have gone down to that value which is like 9 dollars. Your expected winnings when the Yankees are down a game, you know, they’re still a better team so actually it’s more likely even after losing the first game that the Yankees would still win the series. So you see the risk that you’re running and you can calculate this. So what should you do in the very first game? This tells you that your expected winnings is 42. Of course .6 times 64 that’s 38.4 + .4 times 9 is 3.6. That is 42 dollars. So that’s 42 because it’s the average of this and this, and 64 is the average of .6 of this and .4 of that. So what should you do? Well, on average you’re going to make 42 dollars. What’s the essence of hedging? You want to guarantee that you make 42 dollars no matter what happens. No matter who wins the series you want to end up with 42 extra dollars assuming the interest rate is 0 from the beginning to the end of the series. So how can you arrange that? What can you do? Well, so that’s the mystery. I’ll give you one second to try and think it through. You should get this. What would you do here? Are there no baseball bookies in the–yep?
You’re already, in effect, suffering some risk at the very beginning. So this is one of the great ideas of finance. You shouldn’t hedge the final outcome. You should hedge next day’s outcome. If you’re marking to market that’s what you’d have to do. Marking to market you’d have to say my position now–my bet is worth 64 dollars. The Yankees lost the first game, the bet would be worth 9 dollars. So what does it mean to protect yourself? Not just protect yourself against what’s happening at the end, that’s really what you want to do, but in order to do that you should protect yourself every day against what could happen. So every day you should end up with 42 here and 42 there because, after all, that’s what you’re trying to lock in. No matter who wins the first game you should still say I’m 42 dollars ahead because I got myself in this position. So how could you do that? Well, let’s bet at 3 to 2 odds, right, 60/40 is 3 to 2 odds. Let’s make a bet with another bookie at 22 and 33 here. So 22–I put it in the wrong place. This is the 33 and this is 22, but plus 33 and minus 22. So what are you doing here? Notice that this is 2 times 11, this is 3 times 11. This is 60/40 odds. I’m betting on the Phillies. If the Phillies win one game I collect 33 dollars. That’s what I should do that he said. He said, “Go to the bookie across the street and bet on one game, not the whole series. Bet on one game with that bookie across the street, 33 dollars versus 22 dollars.” Let’s say you can only bet 1 game at a time with the other bookies, actually, maybe you were saying all along bet on the whole series, but let’s say you can only bet one game at a time with the other bookies. You’d bet 33 dollars on the Phillies in the first game. That naïve Philly fan has put up 100 dollars on the series. You’re, in the first game, going to put 33 dollars. You’ve taken his bet so you’re hoping the Yankees win, but that’s bad to be in a position where you have to hope. You don’t want to do that. So you take his bet on the Phillies because he’s given you 100 to 100 odds. That’s even odds even though you know the Yankees have a 60 percent change of winning. You go to the bookie across the street and you bet at 60/40 odds on the Phillies, but you don’t bet the whole 100. You only bet 33 dollars of it. So if you win you get 33 dollars. If you lose you only have to pay the guy 22. So what’s going to happen? After the first day this position is going to be worth 42 and this position is also going to be worth 42, exactly where you started. So because a win in the first game is going to put you so far ahead in your bet with the first naïve Philly better, and a loss in the first game is going to put you so far behind, you hedge that possibility by going 33/22 in favor of the Phillies. You take a big bet on the Yankees and then you make a smaller bet on the Phillies that cancels out part of the big bet on the Yankees, but you’ve made the two at different odds and so on net you’re still going to be 42 dollars ahead. Let’s just pause for a second and see if you got that. So by doing this you can’t possibly lose any money. And now you’re going to repeat this bet down here and here. So in the next–you see where do things go next? Here you’re down 8 dollars. If you lost again you’d be down 32 dollars. Now things would really be bad. After the Yankees lost two games in a row your original bet would look terrible, but things aren’t so bad because you bet on the Phillies here. You already made 33 dollars. So how much money do you think you should be betting on the Phillies down here? Well, you want to lock in 42 dollars at every node no matter what happens. This 42 dollars, by making the right offsetting bet you can keep 42 everywhere, here until the very end, and so no matter what happened you can always end up with 42 dollars. That’s the essence of hedging. So let’s just say it again what the idea is. It’s a great idea and we don’t have time to go through all the details, but the great idea is this. You’ve made some gigantic bet with somebody. Why do you bet with anybody? Because you think you know more than they do. The whole essence of trading and finance is you think you understand the world better than somebody else. So understand it means you think something’s going to turn out one way that the other guy doesn’t really know is going to happen. So you’re making a bet on whether you’re right or wrong. So when you say you know you don’t know for sure. You just have a better idea than he does, so you want to use your idea without running the risk. So how can you do it? If your idea is really correct there may be a way so that you can eliminate the luck. So here if you really know the odds are 60/40, your class of bookies knows the odds are 60/40, and some other guys who doesn’t know thinks the odds are 50/50 and is willing to bet against you, you can lock in your 42 dollars for sure. You don’t just take a bet and hope you win. You can take a bet and then hedge it to lock in your profit for sure, step by step, and that’s what we have to explain how that dynamic hedging works. So I have to stop. [end of transcript] Back to Top |
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