ECON 251: Financial Theory
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ECON 251 - Lecture 17 - Callable Bonds and the Mortgage Prepayment Option
Chapter 1. Introduction to Callable Bonds and Mortgage Options [00:00:00]
Professor John Geanakoplos: I think I’ll try to start. So last time we found out how to evaluate options, especially stopping options where you don’t have so many options. You either do something or you don’t do it. That’s the simplest kind of option.
And we discovered through two examples, at least, that the option is worth more than people realize. So this time we’re going to apply that–and that the way to figure out exactly what it’s worth and exactly what to do is to work by backward induction.
So this time we’re going to apply that reasoning to the two most important options in the economy, one is callable bonds and a much more important one is the mortgage option. And all of you at some time in your lives will probably own houses and have a mortgage option and have to think about what kind of mortgage to get and what the option is. So I want to teach you how to think about that problem.
So let’s start with the callable bond problem. Uh-oh, I forgot to turn this on. So let’s start here with the callable bond problem. So callable bonds are issued by corporations and they pay, usually, an interest rate, say 9 percent. So the bond pays 9 percent, 9 percent and then some years later it pays 109, but at any point in time you have the option, the company has the option of calling the bond, and calling it for–so it’s going to pay 9 + 100 at the end, so here it’s 9. So at any time the company has the option of calling the bond. So what does that mean?
It means after it’s paid the 9–the company’s issued a bond promising to pay 9 for say 10 years and then the principal in the 10th year, this is year 10, the total of 109. So that’s the simplest possible bond. And the company occasionally has the option, we’ll see in a minute why it would want this option, has the option of saying, “Okay, we don’t want to make the rest of those payments. We want to get out of our promise. We’ve just paid you 9. We’ll pay off the extra 100 that we’re eventually going have to owe and we’ll call it a day.” So this is the payment and this is the remaining balance. So for a callable bond the remaining balance is always 100.
So it’s pretty obvious that if you’ve made an arrangement, so I owe you–like for example the prototypical mortgage was exactly of this kind. Somebody borrows money from you and they say, “I promise to pay you 9 dollars a year until the last year when I’m going to pay you 109. This is called, for those old mortgages it was called the balloon payment, but in typical bonds it’s just the principal payment, the 100 face value of the bond.
So the person who borrowed the money and has agreed to pay off over the years, he might have a reason why the house he put up as collateral is no longer going to be his house. He might want to move in which case the lender doesn’t have a house anymore backing the loan and they have to have some way of resolving the loan and ending it. So the question is, after you’ve made a payment, how can you resolve the rest of the loan which is supposed to go on for 6 more years?
Well, you just agree to pay 100 and then you call it quits. So that’s a typical kind of bond. A non-callable bond is, you’ve committed to paying the 9 dollars and the 109 at the end and you have no option to get out of the thing. So we want to study what the difference is in value between the non-callable bond where you’re obliged to pay the whole thing until the end, and the callable bond where you can get out of it by paying 100. And it’s easy to see why a callable bond might have been invented, especially for a mortgage when you’ve got this balloon payment because you might want to dissolve the debt.
But by putting in this option to dissolve the debt you drastically change the value of the promise, and that’s what we want to calculate, how much that changes. So why might it change? Well, the interest rates might change. They might go up or down.
Soon we’re going to have them going up or down in a more complicated way, but suppose that the interest rate starts somewhere like at 8 percent, maybe, and it could go up to some number. Now, I’m always going to do a geometric random walk. This has become very fashionable in finance and you should be asking the question, is it special that we only have two possibilities? Life has many more than two possibilities. Suppose we had hundreds of possibilities, would that make a difference?
The answer is it’s not going to make a difference, but we’re going to have to see why. So in a geometric random walk literally the thing can go up or down each period. So what does it go to if this is r0? The next period we say it goes to r0times e to the volatility, which I usually call sigma, plus maybe a drift, plus d. And here it’s going to go to r0e–let’s write it with a little more room–r0 e to the minus sigma plus d, so what’s happened here?
The interest rate started at 8 percent. It gets multiplied by e to the d, which is just a number, so it’s maybe tending to go up over time if d is positive or if d is 0–think of d = 0–r0 on average is going to stay at r0, but there’s some uncertainty. Maybe the interest rate goes up. We multiply it by some number, e to the sigma, and then we multiply it by the reciprocal of the same number or we divide it by e to the sigma. So using the exponential notation just makes the computations in the computer easier because the computer’s adding numbers. This thing over here is going to be r0 e to the 2 sigma + 2d, right? And this will just be r0 e to the 2 d, and this will be r0 e to the minus 2 sigma + 2 d.
So the computer in calculating the interest rate at every step is adding exponents and it makes the calculation when you’re doing gazillions of nodes much faster. So that’s why it’s traditional to use this notation, but really all I’m saying is you multiply or divide by the same number to see what the next interest rate is, and on top of that you might be sort of increasing all the things over time just because you think the interest rates are going up over time.
So that’s how we’re modeling uncertainty, and everybody is supposed to understand what the probabilities are of these moves. So for now that’s going to be our model of uncertainty and in a few minutes I’m going to try and indicate why complicating it won’t have much effect. So are there any questions? This notation I hope isn’t too complicated. Are there any questions about the interest rate process I’m assuming, the uncertainty that people are facing? Yeah?
Student: Why is that middle probability on the far right, r0 is always equal to <>?
Professor John Geanakoplos: Because I’m assuming that if there weren’t uncertainty the interest rate was going to keep steadily rising by multiplying by the constant e to the d. So here I’ve multiplied it by e to the d. Here I’m multiplying it again by e to the d, and I’m going to multiply it again by e to the d. So if d is 1 tenth of a percent it means that the interest rate is rising at 1 tenth of a percent every period.
You might just think interest rates are going to get higher in the future. You might have a more complicated function than steadily rising or steadily falling. It’s just that this is the simplest to add a little complication without making it too complicated. Steadily rising or steadily falling and binomial uncertainty were the easiest simple things I could do. Of course a more realistic thing would do more complicated things.
Any other questions about what this means? I know this can be confusing, this notation, so let’s just figure out the notation. Make sure you understand it before I move on because it’s really nothing but notation. I’m just saying the interest rate is 8 percent today. It can go up a little or down a little and then after that it can go up or down and the percentage rise and fall is always going to be the same relative to a drift and the drift might be 0, the drift might be positive.
So to take an example, let’s take an example here. Did I plug everything in? Yes. This is a spreadsheet that you have called callable bond. I did a little work on it, but not much. It’s basically your spreadsheet. Everything is plugged in here. You’re seeing that, right? So what I’ve done is I’ve said let’s take over here the year, so here are the years, 0, 1, 2, 3, blah up to 31 or something.
Then the bond coupon rate is 9 percent, so the bond is going to pay 9 dollars, 9 percent. The face is always 100. It’ll pay 9 dollars until the end. So if I take the bond maturity of 30 down here I’ve got the 9 dollar payments and I’ve got the remaining principal, the remaining balance, what you could pay to dissolve the contract. You could after the first year, after paying 9, immediately afterwards you could pay 100 and say forget about the whole thing.
So that’s that, and so it goes all the way to year 30, and year 30 you see you’re paying 109, and then the bond is over, 109 and by then the bond is over, there’s nothing to talk about. So now I start off the interest rate somewhere at 8 percent, and I take the volatility, it’s always said in hundreds, I take the volatility of 16. That means sigma = 16 over 100, because that’s the way people talk about volatility of 16 percent.
So that is a sort of standard volatility, 12 or 16 percent is about the annual volatility of these things traditionally, so I’ve taken the volatility at 16 percent. So I take the drift of 0 to make that simple. The d’s disappearing, so what does that mean? It means that whatever the thing was last period when it moves to here I take the old number–at the top there if you read this, this is the up multiplier. I just take e to the sigma + d and that’s what this says over here, exponent of–it just says that. So that’s the up multiplier and that’s the down multiplier.
Let’s just see how that looks. So if I look at the interest rates I start at 8 percent and they can go up by that multiplier, 16 percent up or 16 percent down. And see if you take 8 percent you add 16 percent to it, slightly more than 16 percent, because e to the .16 is a little bit more than 16 percent. You take 8, 16 percent up is about 9.3 percent interest and 16 percent down you go from 8 percent to 6.8 percent. And things just go up or down so that’s the uncertainty. So that’s what interest rates are going to do. So everybody knows that you start at 8 percent and interest rates can go up or down by 16 percent for the next 30 years.
Chapter 2. Assessing Option Value via Backward Induction [00:12:14]
The question is, what’s the value of the bond? Well, if you couldn’t call it–please interrupt me if you’re not following. If you couldn’t call it, non-callable bond, why would you expect it to have such a high price? Well, because the interest rate is, sorry, the interest rate, remember, is only 8 percent and on average it’s going to stay at 8 percent.
It could go up or down, but it’s always coming back–the middle thing there is always 8 percent. So if you look in the middle it’s 8 percent, 8 percent, 8 percent, 8 percent, so sort of the geometric average is 8 percent. On average it’s going to be 8 percent. That’s less than the payment, 9, so the poor bondholder is overpaying, paying 9 every year when the interest rate is only 8 percent and on average is going to be 8 percent.
So obviously the bond issuer is giving a very good deal to the buyer by paying 9 when the interest rate’s only 8 percent. And you know how you calculate this. At every step you’d have to do it by backward induction. So you’d write down, and we’ve done this before, the value at every node VS, so if I take a node here what’s the value of it?
I don’t know if I need to do a concrete example. How do I do this by backward induction? So I keep going like that, right? And then life is going to end. This is the last payment somewhere. Let’s say the last payment is here so you’re going to get 109 no matter what because that’s the last period 109, 109, 109, 109. So we know what the value is here at the end because it’s 0 beyond this point, and we know what the value is here. You’re going to pay 109 no matter what so you just take the interest rate back here. So we know what the value is here.
So the question is if you figured out the value at the end of the tree, which is trivial because at the very end you’re just going to get 109 for sure, can you figure out what the value is at the beginning of the tree. And the way you do that is for every node you say to yourself, it doesn’t matter which node you are like this one or that one, let’s call is node S. So the value of node S, this is the value just after paying the coupon. So you’ve paid the coupon, you’ve paid the 9, now what’s the value.
Well, the non-call price, so let’s do non-call, non-call bond, what is the value going to be? Well, just after paying the 9 here what’s the value of what’s left? So you’re going to have to go up here. First of all you’re going to discount by the interest rate here, so it’s 1 over (1 + rS), so you’re discounting by the interest rate there.
Then half the time you’ll go up. So what will you have to do if you go up? You’ll have to pay 9 because that’s the required payment, and then you’ll be left in the same situation except that you’ll be at the up state, whatever the value is there–plus VSup. If this node is S this node we call Sup and this node we call Sdown. So I’m doing this by backward induction, remember.
I’m saying, suppose towards the end of the tree I figured out what is the value of the bond just after making the coupon payment. Since there are no coupon payments at the end I know the value is 0, so having done it at the end, and working my way backwards now by induction, I can figure out the value at any previous node.
So suppose I’m here and I’ve already figured out the value of all future nodes. The value here is simply going to be, value here is, just after the coupon payment, is going to be–I’m going to discount at the interest rate here, 1 + rS. Half the time I’m going to have to go up here, in which case I’m going to pay 9 and then what am I going to do? I’m going to be at a situation where I’ve just made the coupon payment and I see what the value left is. Well, that’s the number we already calculated. So VSup + 1 half–if I go down I’m down to here. It’s 9 again + VSdown.
So that’s pretty simple, and now I just have to solve it by backward induction. So that’s what I did in this tree. Here are the interest rates, and now back here here’s the present value of a non-callable bond. So at any node, let’s pick one like this one, what’s the value?
It says it’s 1 over (1 + the interest rate) times the probability .5, that’s 1 half, times the payment, that’s 9 dollars, plus what the value would be if I went up because J 149 is this node. That’s J 149 here, so it’s just exactly what I wrote, 50 percent probability J 4 is the 9 dollars at the top plus the value of what I would get here, plus 1 half times the same 9 dollars I’d have to pay if I went down, plus the value that I’d get down here.
So that’s an extremely simple calculation to do, and because Excel is a brilliant–as I said created by Kapor whose sister was my classmate and was two years ahead of me at Yale–Excel immediately understands–he called it Lotus– immediately understands that if you give all these directions you can’t figure out the value here unless you know what the value is forward. So Excel understands that you should keep going forward and starting at the end, and at the end it knows what the value is because it’s just 0. So having got the value 0 at the end it then works backwards all the way through the tree and gets the value at the beginning and gets 113.
So it’s exactly what we want to do, and he’s just written this brilliant program that understands if there’s a dependency you can’t calculate this without knowing this. He says, “Okay, I don’t know this. I’ll go forward into the program to here. I can’t do these either. I’ll go all the way to the end. Now everything is 0, so I know all these numbers. Now step-by-step I can go backwards and get all of them.”
That’s what Excel is doing and it does it instantly. So there’s no doubt about that, right? You all are way ahead of me on that. So the value’s 113, that’s a huge number, so obviously it’s a very generous company. The company may not want to be so generous, so the company says, look, we want to be able to call the bond at 100. So that simple option, what’s it going to be? What’s the value going to be?
Well, obviously the company could call it right away at 100. So it’s going to make the value go all the way down to 100 because the company now has the option, whenever it wants to, like at the very beginning before making the first payment, it could pay 100 here, or it could pay the 9 dollars and then pay 100 here. So the question is, should the company do it, and when do you think the company will exercise the option?
Why wouldn’t the company exercise the option right away? So here it is. The bond’s worth 113. Every period the company’s promising to pay 9 dollars. That’s like a 9 percent interest. The interest rate’s in the economy only 8 percent at the beginning here. So it’s losing money right from the start.
Why shouldn’t the company just say, “Okay, we gave ourselves the option, we’re taking it right away. We’re going to just pay you 100 and cancel the whole thing at the beginning.” Why wouldn’t they do that? Okay, hang on for one second. Someone else, why wouldn’t they do that?
You see, the puzzle is, the company has promised to pay 9 dollars forever. That’s like 9 percent on a 100-dollar coupon. The interest rate is only 8 percent now, so the company’s paying more money, 9 dollars. It could have borrowed and only had to pay 8 dollars. So the guy buying it is getting this huge bonus. He’s getting 9-dollar coupons instead of 8-dollar coupons.
He’s probably telling his friends who aren’t marking to market, look at my return. I’ve got 9 dollars the first year. Look at that great 9 percent return I got on my investment of 100. Why won’t the company call it right at the beginning? What’s the company waiting for? Yep?
Student: It’s hoping that the interest rate will go back up.
Professor John Geanakoplos: Back up, exactly, because if the interest rates go way up–after a very short amount of time if the company gets lucky the interest rates are going to go up. So here are the interest rates, right? So after a very short amount of time, 2 years, you get the up thing, the interest rates are 11 percent. So all of a sudden the company is looking great. It’s borrowed at 9 percent and the interest rates are actually 11 percent. Effectively it’s borrowing from the guy, 9 percent, and it’s able to invest the money at 11 percent. To put it another way, the present value of the payment’s left from then are much less than 100. So it’s glad it made the deal.
And on the other hand, if the interest rate goes down far enough then the company will be able to call the thing. Present value of the payments if it couldn’t call would be way higher. Because it can call it can never suffer worse than having to pay 100.
So you see that if the interest rates are going up or down and you couldn’t call it then when interest rates go up that’s really good news for the company because now the interest rates are over 9 percent on average going forward, so actually the present value of payments is 98. That’s less than 100. On the other hand if interest rates go down then the company’s really screwed, now the present value of what it owes is 127. At the beginning it’s 113, so as interest rates go further and further down the company’s made a worse and worse bond issue, but it doesn’t suffer very much because of that because it can always get out of its deal by calling it.
So in fact, astonishingly, it doesn’t even call the first time. So here’s the value. When the company behaves optimally it won’t call the first time. When interest rates go up it says, “Ah-ha! We’re going to make money, that’s great, but when interest rates go down now the interest rate’s 6 percent and it’s still paying 9. The company is still not going to call the bond. It won’t call the bond until the interest rates go down all the way to 5 percent.
So anyway, by backward induction you get the value of the callable bond is 95, which is 95.529, which is a lot–let’s write those down, in fact. So the callable bond, 95.529, and non-call was 113 point something, which I forgot already. So I hope you’re following because this is going to get a lot more surprising when we do mortgages, but it’s the same logic, 113.236. So the call option is worth 18 percent to the bond.
It’s a huge thing. It sounds so simple. Of course you should dissolve the thing for a value of 100. That’s the only natural thing to do. So it’s perfectly reasonable, you would think, for the company to give itself that option. “Look guys we’re borrowing money, but there may be some reason why things get too complicated. Our company might close down. We want to keep our promise, so we just want to pay off the loan at 100, so give us that option even if our company doesn’t close down.” Well, the company has just taken an incredibly valuable option. Instead of making a promise worth 113, it’s made a promise worth 95. So any questions about that? You could all get that and you all see why that’s happening, and it’s a bigger number than you’d expect. Yep?
Student: When should they call it?
Professor John Geanakoplos: When should they call it? You can tell when they should call it because–you’ll know when they call it because if they do call it what’s the value? So I didn’t work out the–what’s the backward induction? Sorry. So let’s write down the formula when you do call it. How could I have forgotten to do that?
What do you do when you do call it? So value, how do you figure this out? I just skipped this, sorry. Value of callable bond, so VS, what is that going to equal? Now you’ve got an option. So VS, remember this is value just after coupon payment. So what is the value just after the coupon payment? That’s when you have to call. So in a second you’ll see why that’s the way the rule works. So just after the coupon payment what’s the value of the bond?
Well now that you have the option you can pay the minimum of the remaining balance. It’s 100 all the time. This is just going to be 100. You could pay 100 or you could keep going. So if you keep going it’s going to be 1 + rS times the same thing, 1 half, and then you’re going to have to make the coupon payment the next up–so just a second, where should we go?
So here this is the interest rate, so at any node, you can pick any one like here in the tree, this one. That’s the interest rate, so at that node you could, if you wanted to after making the coupon payment there you could pay the 100 and say, “Okay, we’ve made the payment. I’m making 100 more. Let’s just call the whole thing quits,” but if you don’t you’re going to move on, and when you move on you’re going to go either up to here, so that’s with probability 1 half. You have to make the coupon payment and then again you’ll have the option of paying your way out of it, dissolving the contract or continuing.
So you just have VSU because that’s exactly the value just after making the coupon payment at node SU. That’s VSU, and so it’s going to take into account that you’re minimizing something. So you see, VS, the value at any node, takes into account you could minimize by possibly paying the balance and dissolving it or you could continue.
And so if you continue and things go up you have to make the coupon and then you’re going to face a choice, but we assume by backward induction you’ve already figured out the value of that thing. Maybe it’s paying the 100. Maybe it’s not paying the hundred whatever that is, plus 1 half times 9 + VSD. So the only difference between this callable–non-call case and the call case is you notice that the second term is exactly the same in the two cases, but here we’ve added the minimum of 100 in that because you either could continue or you could pay 100.
Of course the values, VSU up here and VS, are not the same as the VS’s here because they’ve all been concatenated by taking the minimum with 100. So the computer now is going to solve this by backward induction, and now how can you tell what the computer has decided to do?
If the computer said it’s a good time to call that means the value at that point is going to be 100 because it’s called by paying the remaining balance which is 100. So if you go to the present value of non-call you get all these present values. As the interest rate goes down you get crushed, of course, because you have the same payments and lower interest rates, you’re discounting by less.
But the callable bond, see, you shouldn’t call here. It’s 99.9 so that means when the computer took the minimum 100 in this thing it took this which was 99. It didn’t use the option to call at 100, but if things go down again then it’s going to call at 100. So it’s got a threshold here where the interest rate has to go pretty far down before it calls at 100. It has to go down twice, but you notice this threshold.
As time goes on the threshold is going to get tighter. So even if you’re at the same interest rate four years, you know, this is 1, 2, 3, 4, this is 5 years later, and the interest rate’s gone down once below the minimum bar here–five years later you will call when the interest rate is, instead of 8 percent, is 6.8 percent, but 1 year later you won’t call if it’s 6.8 percent, and why is that? Well, why is that?
How come if in the first year the interest rate went from–remember it started at 8 percent and it goes down to 6.8 percent here you don’t call, on the other hand if things had stayed the same for a few years and then in year 5, or year whatever this–how many years have we got here, 1, 2, 3, no, year 0, 1, 2, 3, 4, so if in year 4 the interest rate goes down to–oh, year 5 the interest rate goes down to 6.8 percent. In year 1 if the interest rate went down from 8 percent to 6.8 percent you wouldn’t call, but if the interest rate sort of bounces around and then in year 5 it’s back down at 6.8 percent you will call.
Why would you call here even though in exactly the same situation you didn’t call here? Why call here and not here? You can tell where you call because those are 100s and the computer’s picked the minimum equal to 100. So why is it that it’s going to call here, but not here even though the interest rates are exactly the same, 6.8 percent in both cases? Yes?
Student: Because you can’t get to the kind of really high interest rates from that point five years later. There’s not enough time left.
Professor John Geanakoplos: Exactly. So the point is the only reason not to call here–you’re paying through the nose, it’s horrible. The interest rate is 6 percent, you’re paying 9. It’s terrible, but you’re hoping maybe things are going to turn around. The interest rate’s going to soar then it’s going to be a great deal for me. So I’m willing to have some short run losses in case I get a gigantic gain on the upside, and I know that I’m protected on the downside because if things go down again I can cut my losses and just call it 100.
But if you run out of time, here the interest rate’s 6.8, it’s still way below 9 percent which is the coupon you’re paying and you don’t have much time for the interest rates to go back up, you better call then and cut your losses, exactly. So that’s the whole logic of the thing. So any questions about that?
And as I said, the option is worth much more than it seems. Could you all do a problem like this? You’ll find out. So now let me just do one more example. I want to do an aside here.
Suppose that we had a trinomial tree instead of binomial tree. So I have the same picture, but instead of two things happening I say three things could happen or five things could happen. So I would take the same tree and put this here. I’d always allow for something happening in the middle.
And then, of course, it gets more complicated and I’m not going to be able to draw it because now this thing, three things could happen from here, and from here three things could happen. From here I have three things, so I have trinomial tree like that, right, where always three things could happen. So from here I could go here, stay the same or go up.
Now, if I had a trinomial tree, and of course you could think of an N-nomial tree, would I keep the same numbers here? Well, I can’t keep the same numbers because if I have 1 half and 1 half that doesn’t allow any probability for going in the middle. So I’ll put 1 quarter here, and 1 quarter. Now, what do I have to do to these numbers?
Well, the drift I’m going to leave the same because on average it’s still going to go up by d. So this was, remember, r0 e to the minus sigma + d. So this is the probability 1 half. I just made up three numbers, 1 quarter, 1 quarter and 1 half for the trinomial tree. Suppose those are my numbers? If I want to make this tree comparable to the binomial tree–what does it mean to be comparable?
It means that the standard deviation and the expectation of the interest rate ought to be the same. So I’m using the fact that if they’re normally distributed random variables all I need to know–so if there were millions of successor interest rates and I took them with this–if there are millions of these successor interest rates then as long as the standard deviation and the expectation were the same, no matter how I put the probabilities I’d get almost the same variable because it would be normally distributed with that standard deviation and that expectation, and normally distributed, as I said, means it’s determined by standard deviation and expectation.
So I’ve just picked 1 quarter, 1 half and 1 quarter. It’s as if I had two moves where you could get two ups to go here, an up or a down, or a down or an up to go here, and two downs to go here. That’s why a picked 1 quarter, 1 half and 1 quarter. It’s like I added 2 six-month moves here and called it a 1-year move. So if I keep dividing the process and having it happen quicker and quicker, by the central limit theorem I’m going to get something normally distributed with a corresponding standard deviation and mean.
So what is going to end up as the standard deviation here and here? Well, I know to get the same variance before, I had 1 half sigma squared + 1 half sigma squared that equaled sigma squared, but now my variance is going to be 1 quarter sigma–this is the new sigma hat squared, + 1 quarter sigma hat squared because I’ll go down whatever this sigma, so now I should put a sigma hat here. So 1 quarter sigma hat squared.
So if I look at the variance of the exponent it’s going to be a 1 quarter sigma hat squared + 1 quarter sigma hat squared + 1 half times 0 because half the time you just get the average. So therefore it means 1 half sigma hat squared has to equal sigma squared. So therefore sigma hat squared has to equal 2 sigma squared, and therefore sigma hat has to equal the square root of 2 times sigma.
So I should put the square root of 2 here and the square root of 2 here. So now if I do a trinomial, for the trinomial to be similar to the binomial the binomial would be sort of here and here, and the trinomial there’s a lot probability, you’re stuck in the middle. So to get the same kind of average spread I’m going to have to have this thing sticking further out and this thing sticking further out, but if I choose my numbers by multiplying by the square root of 2 here, and the square root of 2 here, the standard deviation of this trinomial is the same as the standard deviation of the binomial and the expectation of the two are the same. And I could do the same thing with an N-nomial or an any-number thing I wanted to and I could always pick the standard deviation properly, pick these nodes properly so that I had the same standard deviation and expectation as I did with the original binomial tree.
So just because I’ve got three nodes or five nodes on average they can turn out to be the same, and the average spread, the average spread squared can also be the same as the binomial. And if I do that I’m going to get a shockingly similar answer. So let’s just see what we get. So what do I have to do now? I have to go to trinomial.
So what did we do here? We did–9 percent was the coupon, and the interest rates start at 8 percent, and the volatility was 16. So if I go to trinomial, I put 9 percent to start, 30 years, 8 percent the starting rate and volatility is 16, and now I have to do the up multiplier, and by doing that you see I’ve multiplied up there in that formula there’s a square root of 2 there, 2 square root of 2.
So I’ve got just a slightly more complicated thing. I’ve just multiplied by the square root of 2 and that’s how I’ve got all my nodes, so otherwise it’s the same thing and I do the same backward induction except instead of being 1 half, 1 half I put 1 quarter here, and 1 quarter here, so it’ll be 1 quarter VSU + 1 quarter VSD + 1 half, you stay the same, which will be 9 + VSsame, whatever that is–so I probably lost you.
So let me just say I want to convince you that this binomial assumption is not such a special case. So you should be thinking to yourself, two things happening next year, that’s ridiculous. I could imagine a million things happening next year. So I say, okay I believe it. A million things could happen next year, so I’m going to write down a million things, not any million things, I’m going to say let’s suppose that instead of two things happening in a year I say I divide what can happen over the year into a million different up and down moves.
Now, in order to replicate sort of the original binomial, those million up and down moves, let’s say 250 of them, 1 every day, those daily moves, of course, are going to be much smaller than the 1-year move. But if I make the drift be the old drift divided by 250, and the new standard deviation of the tiny things be the right ratio of the standard deviation, then when I compute the standard deviation of all the nodes here I’ll get the same standard deviation as the binomial and the same drift in the binomial.
So if I ended up with 3 nodes here I would just be multiplying by the square root of 2. And so by having a trinomial thing with a slightly adjusted move up and down, slightly spreading it and putting some probability in the middle, I can have the same variance and the same expectation. If I had thousands of nodes here I’d also have the same variance and the same expectation, and so I did it just for the trinomial. So let’s see what happens to the value.
If we look now at the interest rates, it was an uglier thing. I couldn’t do it quite as–I was too lazy to do it as neatly, so the interest rates started at 8 percent and now they can go up to 10 or down to 6.3, and then the next period they can go up to 12 and 1 half. So you see there are different moves here.
Remember the other one was 6.8 and 9, so if you don’t get the same interest rate right away you move more violently up or down. So these three things, these two are spread out further than this. So what happens to the price? The present value of the non-call bond is 113.230 and what was it before, 113.236. It’s not so different.
And if I look now at the value of the callable bond it’s 95.142 and what was it before, 95.529, a little bit different. So you can see that the trinomial thing gives you almost the same answer as the binomial. So I don’t want to say anything more than that. So no matter how many nodes you have in the tree the binomial’s going to give a good approximation provided that we shrink the periods. Instead of looking at a year, looking at daily moves or minute by minute moves, as long as those are binomial over a year a million things can happen.
And so you can restrict yourself to binomial without any loss of generality just taking the time period short enough, because if it’s a daily binomial over a year there’s 250 outcomes or something and so for a minute there are thousands of outcomes over a year even though each move is binomial and you’ll get the same answer. So that’s that.
Chapter 3. Fixed Rate Amortizing Mortgage [00:42:44]
All right, so we’ve done the callable bond idea, and the call option is worth a lot. Now, let’s take a more concrete example, the one that you’ve all heard your parents talking about, if you don’t own a house yourself, which is a mortgage. And I just want to do the same thing and help you start to think about how to value mortgages. So what’s a mortgage?
A mortgage is, you don’t have the balloon payment at the end. You pay the same amount each period. Let’s take a mortgage. So which example have I done here? So I’ve got a mortgage rate of 7 percent, a 7 percent coupon, you can see, and it’s a 30 year fixed rate mortgage. Now, how much does that mean you have to pay every year? So, every year, what are you going to pay?
You’re not going to pay 7. What this means is, whatever X you pay–every year that’s your annual payment–the 30th year you’re still going to pay X, but if I took this all over 1.07 + X over 1.07 squared + this over 1.07 cubed + this over 1.07 to the 30th I would get 100, equals 100. So that’s how X is chosen, so that the mortgage is an annuity of constant X whose present value at the coupon is equal to 100.
That’s how you define the payment on a mortgage, but a mortgage has to be defined more completely than that. You also have to say what do you get by dissolving the mortgage, how much do you have to pay? So right away you have to pay 100. What would you have to pay here if you dissolved the mortgage right after the–so remaining balance equals payment just after coupon necessary to dissolve the mortgage. So at the very beginning the guy gave you 100. If you want to dissolve it and undo it you should be able to do it for 100, but what about after the very first year you’ve paid X?
By the way, X in this case is going to work out to 8.05. If you solve for X you get 8.05. That 8.05 discounted at 7 percent gives you 100. So obviously X has to be bigger than 7 because if it were 7 everywhere you’d need 107 at the very end to make it 100. Of course at the very end that’s 30 years later, so that extra 100 can be compensated by an extra 1, 1.05 all the way along. 1.05 all the way along is just the same as an extra 100 at the end.
So the payment is 8.05. So if you’re paying a 7 percent mortgage your annual payment is going to be more than 7 percent. It’s going to be 8.05. So that’s the 7 percent coupon of the mortgage, coupon of mortgage.
Now, if you want to get out of the mortgage it’s called prepaying. What would you have to pay here to get out of the mortgage do you suppose? What’s the only logical thing to have written in the contract?
Professor John Geanakoplos: Well, you could pay a penalty, but what if you don’t want to have a penalty? Then what should the guy pay to get out of the mortgage? Yeah?
Student: The remaining balance.
Professor John Geanakoplos: The remaining balance, yes, but what is the remaining balance? That’s the question, what is the remaining balance. Well, maybe you know what the answer is.
Student: It’s the present value at the time.
Professor John Geanakoplos: Of what’s left. So the guy here, the remaining balance, B1, ought to be this number without this. You’ve only got 29 payments, only you’re going to put a 1 here now, right? So it’s B1 squared and B to the 29th because you’ve made the first coupon payment. You’ve got 29 years left assuming the same terms as before it still is a 7 percent coupon, so you should discount the remaining 29 years at 7 percent to get B1. So that’s what B1 is. So as you can see in the thing B1 is 98.94.
Now, 98.94 is an interesting number. You notice that the 8.05 minus 7 happens to equal–that plus B1 has to equal 100, which is B0. So why is that? So you started owing the guy 100. You’ve borrowed 100, you own him 100, and he’s sort of charging you 7 percent interest. That’s the way the deal works. So you would have expected to only pay 7 dollars next period. That’s to keep up with the 7 percent interest.
Remember, why did the old mortgages go from the balloon payment to this? Because in the Depression in 1933 every single farmer, practically, who owed the 109 because his mortgage was coming up defaulted. So the lenders decided they didn’t want to be facing that situation where a guy owed 109. They’d rather have the guy pay X every period where there’s never this gigantic payment that he’s going to default.
In fact, by paying X every period that lender’s in a safer and safer situation. Why is that? Because he’s asking for X which is 8 dollars even though the interest is 7 dollars, so the homeowner is paying 8.05. The interest was only 7. So he’s making an extra payment of 1.05 dollars. So what do you do with the extra 1.05 dollars? You write down the balance. You say you’ve overpaid me by 1.05. I’m no longer going to say you owe 100. You’re going to owe, now, 100 - 1.05. 1.05 had a few decimal places after it, and so it’s going to be 98.94136. So that’s this gap, so B0 - B1. The balance went down by exactly what the overpayment was beyond the interest.
And so the next time the balance is going to go down even further because the next time the balance is only B1. The next time what the guy owes–if he was just paying the interest he ought to owe only .07 times B1, and yet he’s still going to be paying 8.05, so B1 - B2. So the balance B2 is going to go down even more, so the gap from 100 to B1 is a very small one. It’s going to be bigger from B1 to B2 and keep getting bigger and bigger because each time the guy’s paying 8.05, but what he owes, the balance that he owes on, that he’s basically borrowed the money on is a smaller and smaller number, so he’s overpaying the 7 percent interest by a bigger and bigger amount and the balance goes down by more and more each time.
All right, to put it another way, if you take the first thing is, B1 is X times 1.07, 1.07 squared, 1.07 to the 29. B2 is X 1.07 blah, blah, blah, X times 1.07 to the 28th. So the difference between this and this is the last payment which is 1 over 1.07 to the 29th. The difference between this and this is the last payment which is 1 over 1.07 to the 30th.
And so as that final maturity gets smaller and smaller the gap gets bigger and bigger, so I’m just saying the same thing in different ways. So are there any questions about this? Did I go too fast or are you with me? Somebody ask a question if you’re lost. Yes?
Student: Why don’t we discount B1?
Professor John Geanakoplos: Why do I what?
Student: Discount B1?
Professor John Geanakoplos: I’m sorry, why do I–say it louder.
Professor John Geanakoplos: Why do I discount what?
Student: Why don’t we because you are subtracting the amount from the 0 <>?
Professor John Geanakoplos: Yes this, so let me say this again. You borrowed 100. The coupon is the agreed upon interest, so you should owe the guy 7 dollars, or 107 dollars the next period, right? Not 100 any more. There’s 7 percent interest. Now you owe the guy 107, right?
But at that moment when you owed him 107 you paid him 8.05. So the guy says, look, I lent you 100, I expect 107 back next period, you pay me 8.05. That’s 8.05, so what’s left that you owe me? It’s 100 minus the 1.05. So 98.4 something, so let me just say it again.
You borrowed 100. The next year you owe 107. We’ve already taken into account the interest rate. It’s 107, so we’re sitting in next year. We’re no longer sitting back here. B1 is as of next year. So you should really by rights be owed now, the lender should be owed 107 now because he lent 100 last year. The agreed upon interest is 7 percent so he should be getting 107 this year. What is he getting? He’s getting 8.05, so what’s left is 107 - 8.05 and that’s what we have, what’s left is 107, is exactly that.
Right, so this is 7 he got paid, so if I write it this way 107 he got paid 8.05, so 107 I’m just rewriting the same thing, 107 - 8.05 is going to be the new B1, which just happens to equal 90 whatever that is, 98.05, something like that, 98.95, right? So this is 1.05, so 1.05 plus this is 100. So 107 - 8.05 is 98.95. You agree with that, right?
Professor John Geanakoplos: So what I’m saying is a year later you were owed 107. You were paid 8.05, so obviously what’s left that you’re owed is the difference, 98.95 and that’s exactly what the equation was that I wrote before.
Any other questions? So you can always divide a mortgage payment into the interest part–so any mortgage payment X can be divided into the interest so now let’s erase this and I’m saying it in other words. So your payments are X everywhere to the last period and your remaining balance is going to be B1, B2, B3 and B30 which is obviously equal to 0. So you can always get out of the mortgage by paying–after you make a coupon payment X you can get out of it by paying B2.
So your mortgage payment is going to be divided into the interest, and then the scheduled principal reduction. So here you pay 8.05. The interest was only 7 percent so you’ve overpaid. So that’s why the balance goes down. That’s the scheduled balance–It’s called principal, but scheduled balance reduction. And if you want you can prepay and pay off part of the remaining balance, any part you want to. So let me say it again.
In some period, let’s say 3, what could you do in period 3? Well, by the time you’ve waited to period 3 you’ve got to pay the coupon X. Now, the coupon X is way in excess of 7 percent times B2. It’s 8 and 7 percent of 100 is less than X. We’re already down to a balance of B2. So the X is bigger than 7 percent of B2. So 7 percent of B2 is the interest, but on top of that you paid more than that, so that extra you paid reduced the remaining balance. That’s the scheduled reduction in the remaining balance. You paid that off.
You had no choice about that, but now in addition, if you want, you can pay the B3 and get rid of the whole mortgage. That’s the rules of a mortgage. Is that clear to everybody? So any mortgage that you get, works by those rules. So we have to figure out the value of the mortgage. Somehow I feel I’m going too fast. Could someone ask a question if you’re lost? I need to pick out–yes?
Student: So what you just said applies that as time goes on you pay a bigger and bigger chunk of the principal with each successive payment?
Professor John Geanakoplos: Yes. So the mortgage is called an amortizing mortgage. This was a great invention at the time of the Depression, after the Depression.
It probably had been invented before, but was used in a big way after that. I don’t know the person’s name who invented it. First it has the property that you’re always paying more. Because you’re paying level payments you must always be paying your principal down.
That makes the lender safer and safer because it’s the same house protecting his loan as collateral. We haven’t come to collateral and how the whole world depends on collateral yet, but believe me we’re going to get there. And so we haven’t gotten to collateral yet, but when we get to it, the lender’s protected by the house. The amount that’s owed to him started at 100. It’s getting lower every period so he’s feeling safer and safer.
That contrasts to the old lenders during the Depression who were at the most vulnerable right at the end–oh, I’ve lost it–most vulnerable right at the end. Here these lenders are getting less and less vulnerable as time goes on. So that’s the advantage of an amortizing mortgage.
Chapter 4. How Banks Set Mortgage Rates for Prepayers [00:57:51]
Now, it so happens that the rate of amortization gets faster and faster. So these numbers start at 100 and stay pretty close to 100, and then they go down pretty fast, which is what his point is, and that’s absolutely right. And so if you look at the remaining balances here in this chart it started at 100, went to 98, went to 97, 96, 95, now 93, 92, 90. They’re going down faster and faster.
If you keep going you’re going to see that they start leaping down from 76 to 73, 73 to 70. Now they’re starting to go from 43 to 38. They’re really, by percentages, going way down really fast, 21 to 14 to 7 to 0. So they go down really fast at the end, the remaining balance. But that’s just the way the amortizing fixed interest mortgage works.
And now we want to think about how to value it, all right? So the payment is 8, the coupon that they agreed on in the mortgage was 7 percent, but that’s different from what the interest rate is in the whole economy which is 6 percent, say. So suppose the interest rate is 6 percent.
Why would it be that with a 6 percent interest rate today, one year interest rate today, why would it be that if the interest rate was 6 percent today–too complicated, this tree–if the interest rate were 6 percent today why would it be that the mortgage rate that people would agree to–so it could be 6 percent–why is it that the mortgage rate people could agree to–so this is 7.3 percent and this went down to 4.9 percent. That’s in that tree. So you tell me, why is it that when the interest rate is 6 percent today–so everybody knows on average it’s going to stay around 6 percent, and here I want to get a loan from a bank to buy a house and I put up my spectacular house as collateral so the bank should feel safe.
It’s amortizing and all that. They should feel safe as long as they haven’t lent me too much money, right? By the way, if my house is only worth 80 dollars and they lend me 100 then the bank is in trouble. If the house is worth 100 and they lend me 100 the bank might be in some vulnerable situation if the house loses value. So banks would have to be stupid, as they were, to make loans of the home value almost equal to the house value, but we’re going to come back to that.
So suppose the house is worth way more than the 100 they lend me. The amount I’m owing the bank is only going down, so the house is protecting the bank completely, has no worries of me defaulting because it’ll just take my house. Let’s take that case. The interest rate in the economy is 6 percent and here the bank is charging me 7 percent mortgage rate. That’s what we agree on, and of course when it charges me 7 percent mortgage rate I’m going to be paying 8 dollars every month.
How could that possibly be fair? Or to put it another way, why does the bank have to charge a mortgage rate that’s way above the starting interest rate? Why would it have to do that? Why does it do it and why is everyone willing to pay it? You see what the question is? Back there, yes?
Student: You have the option of prepaying it.
Professor John Geanakoplos: Right. I’ve got this really valuable option, which of course the bank realizes. I can always get out of it. I can pay the remaining balance and get out of it. Now, what am I hoping for when I take my 7 percent mortgage out from the bank? What am I hoping for? How am I going to make money, the homeowner? Yes?
Student: If the interest rate rises.
Professor John Geanakoplos: Right, so if there’s some tremendous inflation like there might well be now, some tremendous inflation in the next 5 years–this is a 30-year loan. If in the next 5 years the economy’s in so much trouble and we’ve got all these homeowners under water and we owe so much money to the Chinese we could well see the Fed engineering a gigantic inflation, so the interest rate might go up to 10 percent. I’m continuing to borrow effectively at 7 percent from the bank.
How can you see that? Because this remaining balance, remember, the whole way it’s structured is, the payments, after a year, the 100 I borrowed goes down to B1, but it’s like a 20 [correction: 29] year mortgage where I’ve borrowed B1, again, at 7 percent interest. After the third year, right, just after making my payment the 27 years left of the mortgage are treated explicitly by the law of the contract as if I borrowed B3 dollars at 7 percent interest.
So in other words, a mortgage is like a contract where every year I get to re-borrow at 7 percent, but I’m borrowing a smaller and smaller amount. So I perpetually get to borrow at 7 percent, but I have the option of canceling the whole thing by paying off the remaining balance. So I’m hoping that the interest rate’s going to go to 12 percent and I’m going to be borrowing still at 7 percent, borrowing at 7 percent and reinvesting it at 12 percent.
I’ll make a killing. I’m praying for interest rates to go up. Now, most homeowners think that they’re praying for interest rates to go down, but they just have it all backwards. They think if interest rates go down they’ll get a lower mortgage with a lower interest. They don’t realize that the present value of their future payments, are going to go up. So they should be hoping for the interest rates to go up. They should be hoping for a big inflation and that’s when they’re going to make money.
And the option that the homeowner gains if interest rates go up and the homeowner can always get out of the mortgage and pay it off if the interest rates go down by paying off the remaining balance, that’s tremendously valuable. So how do we compute that value?
We have to do it by backward induction. You can intuitively see there’s some value, but you don’t know exactly what it is until you do it by backward induction. So here the payments are always 8. The payments we said were 8.05 and 8.05 and then at the end 8.05. So we know that at the end the present value of what the bank is going to owe, the present value of how much the homeowner is going to have to pay, let’s do it that way, it going to be 0. So we know that at this point the value that’s left is 0 because there are no payments after this final–that’s year 30. There are no payments after that.
So assume that we’ve figured out the value and we’re working backwards. So we are at some node like this one where we know the interest rate and we want to figure out what the value is here just after the coupon payment is made. So what’s the value?
Well, the homeowner has the option of making–remaining balance, remember, was B1, B2 and this is B30. So at this node here this is 0, this is times 0. So we have B0 = 100. At time 1 the homeowner, he’s made the payment here, so at time 1 he could just pay B1 and be out of the whole thing, or he could make his payment.
So if he makes his payment then he owes the 8.05–sorry, he’s just made his payment, so he could either get out of the whole thing by paying B1, or he could wait to see what happens next period. So half the time he’s going to go up, so he’s going to have to pay 8.05 and then he’s going to have a decision then to make, but we’re working by backward induction. We’ve already solved out for what the value of his decision is there. So it’s VSup + half the time he’s going to have to pay 8.05. He’s going to be down here, and then he’s going to have the remaining–he’s going to have to decide whether to pay off the remaining balance or not, and VSdown is that decision, because we’ve already figured out whether he should prepay or not here. So that’s it. It couldn’t be simpler.
At every node after the coupon payment is made the guy has the choice of paying or, I mean, of prepaying, paying off the remaining balance or waiting until the next period, and then he has to pay the coupon if he waits until next period, and if he doesn’t pay the coupon then…
Student: What happened to the interest rate?
Professor John Geanakoplos: Oh, what happened to the interest rate? Well, I forgot it. So thank you. So I have to put this in brackets and multiply by 1 over 1.049. That’s what happened to the interest rate. I just forgot it. So either he pays the remaining balance or he waits, and of course he has to discount it by the interest rate there which is 1 over 1.049 times 1 half times what happens over here + 1 half times what happens over here.
And that’s it, and that’s what I’ve written down in the nodes. So here are the interest rate nodes and here are the present value–so here’s the non-callable mortgage. So the non-callable mortgage, if the homeowner was so stupid that he never prepaid and the mortgage interest rate is 7 percent and the interest rate starts at 6 percent that’s obviously great for the bank. The guy’s stupidly always going to pay, never prepays, the bank makes 109 dollars, but if the guy is much smarter than that he’s going to call. And now, lo and behold, the value’s only 95.55.
The option is so valuable that even though the interest rate is 6 percent, the mortgage rate is 7 percent, the banker’s incredibly overcharging him, and the guy will never default, still the bank is getting a terrible deal because the option’s so valuable to the guy. So what is a bank going to do in that case? What would a bank do?
I mean, if you were the banker what would you do? The interest rate is 6 percent. You can’t control the interest rates. That’s the whole economy. Everybody’s patience, and impatience and all the Fisher stuff, is determining all these interest rates. What would you do?
You wouldn’t charge 7 percent as your mortgage rate. You’d have to charge a higher mortgage rate. Maybe you’d charge 8 percent, and so we could just change the whole thing to 8 percent and redo it, and you’ll see that the guy will do better. So the mortgage rate, let’s put it as .075 instead of .07. So now the annual payment’s gone up and everything is going to change.
And so the interest rate process is the same. If the guy never prepays it’s now worth 114, and if the guy does prepay optimally, well, you still haven’t gotten it high enough. You have to make it 8 percent, maybe. So the interest rate looks like it’ll have to get to a lot above 6 percent. Maybe it’ll have to go to 8 percent. Maybe 8 percent isn’t enough. Well, on average–let’s just try .08 and now we can see how that worked out. Interest rate process is the same. Look at this. If the guy never prepays it’s worth 120. If he does prepay it’s still not enough. So it’s going to have to be 9 percent or something.
Now, typically if the interest rate is 6 percent the mortgage rate will be something like 7 and 1 half percent, not the 9 percent or 10 percent I’d have to get up to. So why do you think that is? Yes?
Student: Maybe there are enough dumb people who don’t prepay to make up for the smart people who do?
Professor John Geanakoplos: Exactly. You have to count on the dumb people. That’s an important fact of life. You not only have to count on them, you have to count them. So you have to figure out what fraction of the population is this that’s only going to pay you 98 in the end, and what fraction of the population is this, they’re going to pay you 120 in the end.
And if you knew which of the guys were the 120 guys, what fraction were the 120 guys and what fraction were the 98 guys you’d know what the thing was worth. So how far am I going?
All right, so if you’re a mortgage hedge fund, my company is a mortgage hedge fund. That’s what we started out with. We quickly did these calculations and now–all right, so if you look at the data you’re going to find–oh, by the way, Sunday I’m having, remember, this extra class Sunday night to tell you a little bit about the real world, so I’m going to save the stories of all the data and stuff until then, but basically we look at how people have behaved in the past. So were doing this tree and we’re figuring out, from this tree you can figure out when the people should have prepaid or not.
Actually the next tree, if you have 1 dollar’s worth of principal, if you assume that the original principal was 1 dollar instead of 100 and you always figure out at every node what would 1 dollar’s worth of principal be there, you can find out in an easy way when people should prepay. So here you can see where all the 1s are is when they should have prepaid. I don’t have time now to explain that, but anyway just a slight modification of the tree shows you when they should prepay. So you can look at when people should prepay and you can look in the data at how many of the people do prepay. So you can go house by house. It’s public information.
Are these people prepaying or aren’t they prepaying? How smart are they? And you can deduce from having watched them in the past miss opportunity after opportunity to prepay you know that these guys–now, it’s not a matter of being stupid, in fact the smartest people might be the ones not prepaying, they’re not paying attention or maybe it’s a real hassle for them to prepay. So we’re going to come to the reasons why they won’t prepay. So you have to calibrate how many people are behaving optimally and how many people aren’t, and then you can judge how high you have to set the interest rate to make a reasonable profit. And I’m going to talk more about that next time.
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