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# ECON 251: Financial Theory

## Lecture 21

## - Dynamic Hedging and Average Life

### Overview

This lecture reviews the intuition from the previous class, where the idea of dynamic hedging was introduced. We learn why the crucial idea of dynamic hedging is marking to market: even when there are millions of possible scenarios that could come to pass over time, by hedging a little bit each step of the way, the number of possibilities becomes much more manageable. We conclude the discussion of hedging by introducing a measure for the average life of a bond, and show how traders use this to figure out the appropriate hedge against interest rate movements.

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html## Financial Theory## ECON 251 - Lecture 21 - Dynamic Hedging and Average Life## Chapter 1. Review of Dynamic Hedging [00:00:00]
I didn’t make that big a mess of it, but a little bit of a mess of it. So let’s be a little bit careful now about who knows what and what you’re doing. So we said imagine somebody who knows that the probability the Yankees are going to win the World Series is 60 percent, and therefore each game of the World Series is 60 percent, and knows that the probability the Phillies win the game is 40 percent. So suppose that he finds somebody who is willing to bet with him on the Phillies. So he can win 100 dollars if the Yankees win. He has to pay 100 dollars if the Yankees lose. Now, his expected payoff is 20. He should take the bet because if the person’s willing to bet at even odds, the Phillies fan’s willing to bet at even odds and he knows the odds are 60/40, on expectation he’s going to make 20. So it’s clear what he should do, but the problem is he’s subject to risk. Although he’s right about the odds he could still, even though he’s done the smart thing, he could still end up losing 100 dollars which could be a disaster for him. So what he would like to do is to hedge his bets. Now, what does it mean hedge his bet? Well, suppose there was another bookie who was willing to bet at 60/40 odds in either direction then what should he do? Well, if he can find another bookie who was willing to bet at 60/40 odds he should try and lock in 20 dollars no matter what. So he should do this bet. He should bet with the other bookie 80 dollars. He’d be willing to give up 80 dollars if the Yankees won in order to win 120 dollars if the Phillies win, and that’s a fair bet according to the other bookie because it’s 60/40 odds, 60 percent of this and 40 percent of that, this is 3 to 2 odds so it’s a fair bet. So in other words, he makes money by taking advantage of the Philly fan to bet on the Yankees, but that subjects him to risk. So in order to minimize the risk he hedges his bet. That’s where the expression came from. He hedges that bet by betting in the opposite direction, on–betting on the Phillies with his bookie, in the opposite direction, but standing to lose less than he, you know, he’s betting in this proportion, in this amount so that he gets 20 no matter what. So he’s locked in his 20-dollar profit. So that was where we began. Now, let’s just think about it a little bit more carefully than I did before. Somebody, in fact, basically asked this question. Is he really making money because he understands better the odds of the Yankees winning the game? Is he making money because he knows the odds are 60/40 and the poor Philly fan thinks they’re 50/50? The answer is no. That’s not why he’s making money. It doesn’t matter what the odds are of the Yankees winning. He’s winning because he’s managed to arbitrage two different betters, namely the Philly fan and the other bookie, and the fact that they differ in their beliefs, the fact that he can trade with the bookie at 3 to 2 odds whatever bet he wants, and the Philly fan is trading at different odds with him. That’s what enables him to make his profit. So how did he make the arbitrage–so what is an arbitrage? An arbitrage is, you find two bets which are more or less the same thing, which in fact are exactly the same thing, but trade at a different price. So what are the two bets? Well, there’s the Philly bet, the Philly fan bet, and then there’s the bookie bet plus borrowing 20 dollars. So he’s going to make 20 for sure. So he could make the 20 at the beginning just by borrowing against his winnings. Then he’d have 20 at the very beginning and his payoffs from the Philly fan bet would be exactly–he’d owe the 20 so he’d have minus 100. So this bet is he’d be able to get 20 at the beginning, but he’d have to pay minus 100 and 100. He’d pay 100 dollars if the Yankees won, and get 100 dollars if the Phillies won. And so that way he’d cancel his bet at the end and end up with 20 dollars for sure. So just to say it again, he’s made money apparently because he knows more than the Philly fan knows, but actually in order to guarantee that he doesn’t run a risk he has to find another trader, and it’s the presence of the other trader who’s really giving him his profit opportunity, not his superior knowledge. And the fact that he has two different traders–and so why does that give him an opportunity? Because he can arrange a bet with the Philly fan and a bet with the other bookie, but combined with borrowing the money so that in the end he’s going to just get 0 payoffs. Putting together the Philly fan bet, and the bookie fan bet, and borrowing 20 dollars he gets 20 dollars at the very beginning, and on net he has nothing at the end, so whatever losses he got from the Phillies he made up from the bookie bet. Whatever losses he had from the bookie bet he made up for it with the Philly fan bet. So arbitrage relies on replication. Somehow through some other bet–he was able to hedge his other bet. He was able to hedge his Phillies bet by replicating the bet with somebody else combined with borrowing to produce exactly the same payoffs in the end, or the negative of the payoffs in the end, and yet for a different price at the beginning. The Philly fan didn’t demand any money upfront to make this (100, minus 100) bet with him. And with the bookie he’s able to get 20 dollars upfront and make this the opposite bet, the (minus 100, 100) bet. So by getting more money upfront and doing the opposite bet with the bookie he can end up completely canceling the Philly bet, Philly fan bet, and that’s how he makes his profit. So that was the first idea. So what have I clarified? What I’ve clarified is that it wasn’t his superior knowledge, it was the presence of two different traders that enabled him to make his profit, and I’ve clarified the method that he made the profit. The profit wasn’t just kind of mixing in another bet, it was really cleverly arranging to undo totally the original bet. So he got actually 0 in the end and ended up with 20 dollars at the beginning. So let’s just see now what you would do in a dynamic situation. Yes?
The Philly fan was willing to bet him 100 dollars on the Phillies. So he gets 100 if the Yankees win. He pays the fan 100 if the Phillies win. It didn’t cost any money to do that. Then he goes out and makes another bet with the bookie. Again, it doesn’t cost him any money. He gives up 80 dollars if the Yankees win, but he gets $120 if the Phillies win. So he’s spent no money at all and his total profit is 20 dollars for sure. So what could he do since he knows he’s going to make 20 dollars for sure in the future? He could just borrow it at the beginning if the interest rate’s 0, have 20 dollars at the beginning, end up with 0 no matter what the uncertainty is and put the 20 dollars in his pocket. So that’s another way of saying the same thing which is that he could, by combining the bets, put money in his pocket in at the beginning and no matter what happens in the future he doesn’t have to give anybody any more money. So that’s what he’s done. That’s what an arbitrage is. You put money in your pocket in the present and in the future you never have to face a problem. You’ve always got money that’s going out in one hand you’re getting from the other hand. ## Chapter 2. Dynamic Hedging as Marking-to-Market [00:09:15]Now, in a dynamic situation you can do the same thing. So I’m just reinterpreting what we did before a little bit. So in the dynamic situation what are you doing? The Philly fan now is betting on the whole World Series even though you know the odds are 60/40. So now that’s an even worse bet by the Philly fan if you’re right about the odds because the chances are even higher that the Yankees are going to win the series than one game. The odds, we know, are 71 percent. So by backward induction we did this calculation to say your expected payoff is 42 dollars. So let’s suppose now–but now the key is that you’re making money not just because you know better what the odds are, you’re making money because there’s another bookie standing there always ready to trade with you at 60/40 odds in any game. So the presence of the second bookie, even if you knew nothing, would enable you to make money in this case, lock in a profit for sure of 42 dollars. Now, how do you know that you can lock in a profit for sure of 42 dollars? Well, it’s now done by dynamic hedging. You have to lay off part of your bet, a little part each day, each game and the amount that you lay off is going to depend on what happens, so that’s why it’s dynamic. It’s a complicated calculation. So it seems almost hopeless to figure out what to do. I’m betting on the whole series. Many different things can happen. The Phillies could win the first game. The Yankees could win the first game. Lots of things could happen. By the end I want to make sure I’ve locked in my profit for sure. How can I do that? Well, the key is to think about marking to market. Just as we said, if you’re assessing your position correctly, even though the World Series takes 7 games, if you were reporting to your investors, your hedge fund, you’re reporting to your investors how you’re doing, you should be telling them after the first game, you know, at the beginning you’re telling them, “Look, I’m going to make 42 dollars. I expect to make 42 dollars.” After the Yankees win the first game it looks like you’re going to make 60, you know, now things are better, you expect to make 64 dollars on average, but if the Yankees lost the first game you’d only expect to make 9 dollars on average. So after the first game you go from thinking you’re going to make 42 to being even more optimistic now, you’re going to make 64 or you’re going to make 9. So 60 percent times 64 + 40 percent of 9 is 42. So if you were telling your investors properly how you stood after the very first game you might have to tell them you’ve lost 33 of the 42 dollars that you told them before you were going to make. So you’d already be running a risk. You really only care, maybe, about getting money for sure by the end, but if you’re doing proper accounting you should be worried all the way along whether you’re making or losing money. And that’s the key to how to have no risk by the end. You just run no risk at all the in between stages. So all you have to do is make a bet with the bookie who’s standing by to make 33 dollars in case the Phillies win the first game. You’re making a one game bet in exchange for losing 22 dollars if the Yankees win. This is still 3 to 2 odds, so the bookie’s willing to do it. So the night before the first game you’ve bet with the Philly fan on the entire series, which is giving you a huge advantage, but you’re willing to bet with the other bookie. The other bookie, we’re assuming, is only standing by game by game. You’re betting 33 to 22 on the Phillies. You’re betting on the team you expect to lose. You’re betting on that team with the other bookie, but in a smaller quantity. So this is going to mean that your profit after the first game, no matter what happens, is still going to be 42. So you can lock in 42 after every game. And so as I said last time the crucial thing is–so then one thing I didn’t say. So one thing which I forgot to observe last time, which I think is interesting, is so is that what you should do? Should you do that after every game, always bet 33 to 22 against the Yankees with the other bookie? Well, the answer is no. What should you do if the Yankees win the first game? By winning the first game your expectation goes up to 64. Now it could be either 82–and the night after the first game you’re much better off than you were before. Of course you laid off part of the bet with the other bookie, so you’ve had to pay off 22 so you’re still only 42 ahead, but the Yankees have now won the first game. So it’s now the night after the first game, before the second game. You’re subject to more risk. What should you do now? You should go back to the bookie and make another bet on the second game, but should the other bet be 22 to 33? Absolutely not, because 64 up here goes to whatever that number is, which I forgot already, 82 and 1 half to 36 and 1 half. So what should you do? What should you do now? So I’ve done some rounding here which is going to screw everything up a little bit, but now what should you do here? After the first game you’re here. Here you are after the first game. You’ve made your bet with the bookie. The first game you bet 33 against 22. Now what should you do the second game? What should your bet be with that bookie? He’s still willing to give you 3 to 2 odds on the Yankees either direction. What should you do? You’re not going to bet (33, 22) again. You’re going to make a different bet, and what should it be? Yes?
Now notice that’s still 3 to 2 odds. So the bookie’s still willing to do that for you and you’re locking in the 64 because it’s going to be 64 again, and adding this is 64 again, right? So you’ve locked in whatever this number is. So in the first step you made a bet to lock this number in. In the second step you made a bet to lock this number in. You’re always locking in where you were before. So that bet is changing each time, but you’re always locking in what you were before, because after the Yankees have won a game now the next game is not quite as important. Time is running out on the Phillies so it’s not as revealing the next game about what’s going to happen. So you make a smaller bet up here than you did before. So why is that locking everything in? Because this average is now 64, what it was here, but don’t forget you’d already made the initial bet to lose 22. So you’ve locked in 44 now here and also 44 here because you still owe the 22 from the first bet. So you just keep proceeding down the tree locking in where you were before and so by induction you’ve locked in 42. So that’s what you’re doing, and the bet is changing every time. So what do you think the bet is here, by the way? The bet here you could go from 9 to 36 or minus 32. So what should you do here? What should you do here? What’s the bet that you’re going to make with the bookie after the Phillies win the first game? So it’s going to be minus 27 and plus 41, and that’s 3 to 2 odds with some rounding, right, 13 and 1 half times 3 is 40 and 1 half. So it’s still fair odds, so the bookie’s willing to do that. You’ve locked in 9. So by induction you’ve always locked in where you were before and therefore you’ve locked in 42 all the way along. It sounds like we’re locking in 9, but we’re not locking in 9 because remember the first day you won 33 dollars from the bookie when the Phillies won the first game. So you’ve locked in the 9, but don’t forget the 33 from before, so it’s really you’ve locked in 42. So 42’s locked in the whole way. So that’s dynamic hedging. You have a complicated bet that depends on a lot of things along the way, but you bet each time along the way, maybe changing your bet each time depending on what’s happened so far, and that way you can lock in whatever profit you had. So that’s the thing that sports bookies have known for a long time. The expression “hedging your bets” comes from that and so that same logic can be applied to the market. But remember, the step that I didn’t emphasize very much is that although you seem to be locking in where you were before, there’s a slightly more abstract way of saying that, that at the end you’re going to get 42 dollars for sure. So of course you could borrow all the way back from the beginning and get 42 dollars immediately and then never owe anything in the future, because whatever bet you were winning or losing with the Philly fan you were making up for with your myriad bets with the bookie. So it’s a replication argument that by betting sequentially with the bookie different amounts, you’re undoing the single long bet you had with the Philly fan. ## Chapter 3. Dynamic Hedging and Prepayment Models in the Market [00:19:55]So now let’s move from sports, as we did last time, to the market. So here we had the prepayment spreadsheet. And so we had, remember, this was a bond, a mortgage, a 30-year mortgage, 8 percent mortgage, and the interest rate started off at 6 percent and it had a 20 percent volatility. So the interest rates could follow this process. There you see it. They start at 6 percent, but they can go up or down by 20 percent, basically every period. So you assume that everybody knows this and knows that the probabilities are 50/50. That’s like saying you know in advance there’s a bookie out there, that is the market, that’s willing to give you, no matter what day it is, 50/50 odds of the interest rate going up or down from then on. That’s just like the bookie standing by. You now are going to take advantage of the fact that you can anticipate that the market will stand there ready to bet with you. Now, if the whole market freezes or something you can be totally screwed. You’re doing your complicated dynamic betting with the market, one day you wake up and you can’t bet anymore because nobody’s willing to trade, something horrible has happened, 9/11 happened or something like that. You can be caught in a really bad situation and end up running a lot of risk. So you’re relying on the market, so it’s not surprising that when the market freezes up lots of people suddenly are exposed to more risk than they were before. So now in the market you’re not going to be able to find these pure arbitrages. In the sports example it was a pure arbitrage. There was a Philly fan who is betting at even odds and a bookie who was betting at 3 to 2 odds. Now, you shouldn’t expect that to happen. I mean, by what miracle did this fan come to you instead of to the other bookie? Why do you deserve to meet this fan and take advantage of him? What have you got that no other bookie has? It’s a miracle he came to you and not to one of the other hundred bookies. Well, it’s probably because you actually know something that the other bookies don’t that you can take advantage of. So in the background here there’s another thing that you knew. You knew the fan was going to pay off if he lost. Maybe the other bookies weren’t willing to bet with him because they didn’t think he was going to pay off if he lost. So that’s the extra thing that you knew, you knew that he was going to pay off. Now it makes sense that he came to you and not to the other bookies because you actually knew something that they don’t. And so that’s how it really works in the market. You’re relying on some piece of information you have that they don’t. Of course you could be wrong about that information and then you get into a lot of trouble. So here in this mortgage what’s the thing that you’re relying on? I’m assuming that the rest of the market is standing by ready, as I said, to trade, to make interest rate bets with you, with everybody, at even odds whether the interest rate’s going to go up or down. So no matter where you find yourself in the future, like here, someone is willing to bet at 50/50 odds that it will go up or down. Now, this is a year-long bet so you’re going to have to take into account the interest rate. It’s not just going to be 50/50. You have to discount all the cash flows by the interest rate, but it’s basically a 50/50 bet discounted by the interest rate. So now what’s new here? Well, you don’t know what the cash flows are. So here’s the extra piece of information you’re adding. You know that the mortgage homeowner is rational and is going to prepay rationally. So you know what these cash flows are. Some of the time the mortgage homeowner is going to pay the 8 dollars he owes, some of the time he’s just going to pay 8 dollars, in fact, here is what you know is going to happen. Down here where the 1s are he’s going to pay his coupon of 8, pay off the remaining principal and that’s the end of it. So in fact the very first day if interest rates went down the rational thing to do is to pay off the entire mortgage. So you know that’s going to happen. Now, that’s the extra information you have. You know what the cash flows are. If other people knew those cash flows they would figure out the price of the mortgage too. It’s just they don’t know those. You know the cash flows and you can now figure out that the price better be 98. Why–if there’s someone else willing to trade with you to buy that mortgage for more than 98, like why might that happen? Because someone else might have the erroneous opinion that the homeowner will never call the mortgage, will never prepay because he’s just going to be asleep all the time. So let’s say the homeowner’s a she. Let’s say the homeowners are all shes. This other trader thinks they’re all asleep and will never prepay. You know that these homeowners are all incredibly alert and on top of things, and so they’re going to prepay, and the value’s only 98. So because of that extra information you know the actual cash flows. Sorry, we’re here. You know the cash flows. You know how to value the bond, and you know it’s worth 98. So, and let’s say the other guy’s willing to pay you 120 for it. So you can sell it short. You can sell it to the guy short for 120. That means you owe whatever cash flows the mortgage pays, and you know what it’s going to pay down there. So how can you lock in your profit? Well, because with 98 dollars at the very beginning you can replicate those cash flows. You can go out into the market and buy those cash flows, the very cash flows you know the mortgage is going to make. How can you do that? Well, because at the first step–remember what this says. The first step the price is 98. The interest rate could go up or down. If the interest rate goes up the payment is 8. That’s the coupon plus 92 something; 92.6, and if it goes down the payment is 8. So what is this? The payment is actually 8. That’s the cash flow that has to be made. The mortgage wasn’t called here at the beginning so no matter what she does, the homeowner, she’s got to pay the 8. So you know you’re going to owe the guy 8 the first period. Now, down here she’s actually going to prepay. So she’s actually going to pay the guy the whole 99.1. Up here she’s not going to prepay. She’s going to hang onto the mortgage. So what is this 92.6? This is the present value, assuming those cash flows, of her future payments. So if you get 98 dollars you know that 1 half times this–this by the way, is 102.6, so 1 half times this divided by 1.06 plus this thing, remember is 107.11 divided by 1.06 times 1 half. That’s how we got the number 98, 98.8 whatever it was. That’s how you calculated 98.8 by backward induction. You took this value times 1 half discounted plus this value times 1 half discounted. That’s how you got 98.8. So therefore, if people are willing to trade at these prices, in other words what’s the price here? It’s 1 half times 1 over 1.06 and this is 1 half times 1 over 1.06. You can buy 1 dollar here at a cost of 1 half times 1.06. You can buy 1 dollar here at a cost of 1 half divided by 1.06. Paying this amount at the beginning will buy you 1 dollar here. Paying this amount at the beginning will buy you 1 dollar here. So by paying this amount you can get this payoff, 102.6, 100.6, sorry, 100.6. You can get this 100.6 and you can get 107.11 because precisely this is the cost that you have to pay for each dollar up here, and this is the cost that you have to pay here for each dollar down here. Why is that? Because we said the market is standing by recognizing that these are 50/50 odds that it has to discount. So that means the whole market is willing to trade you, will be willing to promise you 1 dollar here in exchange for this amount of dollars here. It’ll be willing to promise you 1 dollar here in exchange for this amount of dollars here. So with this amount of dollars you can buy 8 dollars and have 92.6 leftover and you can also buy 8 dollars here with 99.11 leftover. So all this together is enough to make the prepayment because you know she’s going to prepay here. That’s what we found out. She’s going to prepay at the first step down. Oh, what the hell does this say? She’s prepaying just as we said. At the very first step she’s prepaying. The first interest rate down she’s going to prepay. So we know you’re going to have to come up with all this money, but you’ve bought it over here. It cost you this amount. You bought all that. If things went up you’d have to pay 8, which you’ve also bought, but you’ve got 92.6 left over and that’s going to be enough to buy everything that comes after that because the next day you’re going to have to buy more stuff, and so you just keep buying going forward. So with 98.8 dollars you can buy all the future cash flows and you’ve sold the thing so you can make all your payments that you’re obliged to by selling it short and you received 120 to begin with. So you locked in 21.2 dollars. So that’s the gist of the whole thing. So I said it last time not very well. I said it a little bit better this time, still not brilliantly, but do you now get it? Someone ask me a question perhaps? Is it okay? So I’ll say the punch line again. The punch line is that if you can find a favorable–so here what was different about the World Series and the mortgage example? In the World Series you took advantage of the fact that there was a bookie and a Philly fan willing to bet at different odds on the same thing, and so even though the Philly fan had a very complicated 7 day bet, and the other bookies were only willing to make day by day bets, you could take advantage of that disparity to lock in a profit for sure, and that’s what you should try to do. ## Chapter 4. Appropriate Hedges against Interest Rate Movements [00:30:50]In the market you’re probably relying on something else. You’re relying on the fact that you know more about the cash flows than the rest of the traders do, the rest of the market does, maybe because you understand prepayments better than they do. And so that’s why I said it was so important to model prepayments. So let’s say you understand prepayments better than they do so you know what the cash flows are going to be much better than the market does. You can take advantage of that. You need now this advantage. Just like in the World Series you might have needed the advantage of knowing the Philly fan would actually pay off if he lost. So there’s that advantage that you have, and now you have to rely on the fact that the market or the bookie from then on day by day, year by year, is willing to make bets at these odds that you can anticipate. And so using that you can trade over and over again with the rest of the market and dynamically replicate what the payments you have to make by selling the bond short. So therefore you’ve locked in your profit because you can anticipate–assuming the market stays there willing to trade at even odds you’ll always be able to produce the cash flows with the 98 dollars you started with to exactly meet all your payment obligations you’ve incurred by selling the mortgage short. And since you sold it at the beginning for $120 you’ve made 21 dollars at the beginning and you’ll never need those dollars. You just put them in your pocket. So that’s how to dynamically hedge a winning position to eliminate the chance of loss. Of course you still bear a chance of loss. If you’re wrong about the prepayments you’re going to lose money, and if the rest of the market suddenly stops being there to make these year by year bets on which way interest rates are going to go you’ll also be subject to a lot of risk. So you’re relying on those two things, the whole market not collapsing and your being right about prepayments, but conditional on those two facts you’ve locked in your profit. So you got rid of a bunch of risk, the blind luck of (100, minus 100) from the roll of the dice of the Phillies. You got rid of that luck, but you still had to count on the bookie being there day after day after day, and in the mortgage case on the fact that you knew how to predict the prepayments. Yes?
They’re going to close me down, and so I don’t want to run that risk. They don’t want me to run that risk and I don’t want to run the risk. I still want to take advantage of the Philly fans. So what I do is every game I make another bet against my inclination. I bet not on the Phillies, I bet, I mean, not on the Yankees who I think they’re going to win, I’m betting on the Phillies because I’m undoing part of the gamble that I did with this guy. I’m hedging that gamble, but by doing that in a clever way, changing how much money I bet each game–so the night before the first game I bet 33 dollars to 22 dollars on the Phillies winning. After the Yankees win the first game I bet 27 dollars to 18 dollars again on the Phillies winning. So I decrease my bet going up, I increase it going down. So by doing these clever game by game bets I lock in 42 dollars for sure. Yes?
I bet 100 dollars with the original Phillies fan. I bet 100 dollars on the Yankees. If the Phillies win all first four games I’ve got to give the guy 100 dollars. Of course I made these side bets with all these bookies on the way. So what were the side bets?
Now let’s say I got to the end to the last game where this was a 0. Let’s say this was the last game where I’ve lost. Well, this is minus 32. Let’s say this is the last game. I lose 32 dollars here. Let’s say it was only a two game series and I give up 32 dollars in case the Phillies win the first two games, right? So you’re saying, “I’m down 32 dollars. That’s the last node. How can I possibly be up 42?” Well, so what’s the answer?
That’s 74 dollars and I pay 32 to the original Philly fan so I’m still up 42 dollars. So it sounds like you have to do a lot of arithmetic. You don’t have to do any arithmetic. I knew I was going to end up 42 dollars up. How is that? Because every bet I said let’s look at the number here wherever we are on the tree like here. At this node after all the stuff that’s happened before the bet I’m going to make with the bookie that one game is going to be enough to turn this random thing into 64/64. So I always take what happens further down the tree and turn it into what’s happened at the beginning of the tree, but then I work it all the way back and it’s 42. So that means, see this thing was 64 and 9, but I made a bet 22/33 with the bookie to turn it into 42/42. This thing was 64. I made a bet to turn this into 64/64, but with the 22 here that’s also 42. So no matter how far I go on the tree I’m still exactly 42 ahead. Any other questions? Yes?
Now I’ve got 98.8 dollars left. So that’s money now I have to do something active with. What do I have to do? I owe this guy a lot of payments. Every time she, let’s say it’s you, makes your payment, her payment, in the market I’ve sold the thing short. That means I owe–I’ve promised to deliver whatever that mortgage delivers. That’s what it means to sell it short. So I’ve got to deliver the 8 here, say. I’ve got to deliver the 8 here, and here she’s prepaying. I have to deliver the whole 107, right? So with 98.8 dollars I can afford to buy 100.6 dollars in this state and 107.11 in this state. How do I know that, because that’s where the 98.8 came from. It said this total amount 100.6 times the price, which is the odds of it happening discounted, that’s the price plus the 107.11 times its price, the odds it happens discounted, is exactly 98.8. That’s where I got 98.8. So therefore with 98.8 dollars, remember the bookie’s willing to go either way, with 98.8 dollars I can buy all these cash flows and all these cash flows. So what do I do? With the 107.1 down here I pay the 8, her coupon, and her 99.11 prepayment. I pay it all to that guy I sold the bond short to and I’ve kept my promise. That’s the end of the mortgage. Up here the mortgage only delivered 8 so I only owe him 8, but I’ve got another 92.6 left. Why is that exactly what I need? Because from here on I have to use that money to buy the future cash flows because she’s going to continue to pay her 8 and maybe eventually it will prepay. But by induction, by repeating this step over and over again I can always make the payment and have the present value left over to afford to buy the future payments plus the present value left over. So by induction I’m always making the coupon. Like in this first step I make the coupon payment plus I’ve got enough money left over to match the present value. So that means as I go forward, and forward, and forward I keep making the payment and matching the present value that’s left over. By the time I get to the 30th year I’ve made the coupon payment. The present value left over is 0 because there isn’t going to be any payments after that. So I’ve made all the payments.
So if the interest rates go up the first time and then back down she’s still not going to prepay. So we can now go up and see–oops. Here we are. So we can go up now and see what’s happening here. It’s 92.65. You go up to 83.95 or down to 97.2. 97.17. So what does that mean? So with 98.8 dollars I bought, I’m the guy selling the thing short, you’re the smart rational pre-payer, so you paid 8 dollars here. That means because I’ve sold the mortgage short and you’ve made that 8-dollar payment I owe the guy the 8-dollar payment just as if he had the mortgage. So I’ve got to come up with the 8 dollars. Down here if things had moved here you would have paid 8 dollars plus the whole prepayment so you would have paid 107 dollars. But with 98.8 I can buy 8 + 92.6 up here and 8 + 99.11 down here. So I’ve bought all the cash flows I need to down here. I hand them over to the guy and I say, “You see, I kept my promise to deliver whatever that mortgage did, and I knew that she was going to prepay, I knew how smart she was and so I’ve kept my promise. You didn’t think she’d prepay. I knew it. I’ve kept my promise.” And then up here I’ve got the 8 dollars so I can make the payment, match it. I’ve got 92.6 left, now I don’t give it to the guy because the payment’s only been 8. I take the 92.6 and with the 92.6 that happens to equal this plus 8 times 1 half divided by 1.06, and it equals, also, this plus 8 times 1 half divided by 1.06. Not by 1.06, by the interest rate here which happens to be–you have to figure out what that interest rate was up there. So the interest rate went up to–it started at 6 percent and went up to 0732 percent. So this is .0732. How did this number 92.6 get calculated? It was exactly take the payment of 8, take the present value of what’s left, multiply by 1 half and discount it at the interest rate here plus 1 half times this payment of 8 plus the value of what’s left discounted by that interest rate. So this number 92.6 allows me to buy this and to buy that. So therefore, if things went up again I could make your 8-dollar payment that you make to the bank I have to make to the other guy because I sold the mortgage short. I’ve got the money to pay him. I’ve also got the money to pay him if things had gone down again 8 dollars. And I’ve got more cash in my pocket, this and this in the two cases. With this money I can continue to buy the future payments that you’re going to make. So I always have enough money to make whatever payments it is that I have to, to keep my promise. Yep?
We think we know better what the prepayments are going to be than the rest of the market, so we’re willing to take on a commitment–someone will pay us 120 dollars. We think 98 dollars are enough to buy all the promises that we’ve made by selling the thing short, say. We could be wrong. We have been wrong. The last two years we made a lot of mistakes. So you’re saying, the question is, the reason why the guy is willing to pay so much is because down here he thinks he’s going to get a lot of money. The interest rates are going down so, discounting at a lower rate, the mortgage is going to be worth a lot more. We don’t think it’s going to be more because we’re expecting prepayments. Up here it’s sort of obvious that there aren’t going to be any prepayments. So if the world really goes, interest rates going up, then our theory of prepayments never gets put to the test and so we just make money. So our theory never gets put to the test. If interest rates go down then our theory’s been really put to the test and we have to see whether she’s really smart enough to prepay or not. So you’re right. That’s absolutely true, yep. But in any case we’re going to assume our theory is correct and hedge in such a way so that if our theory is correct about prepayments we will exactly be in a position to keep our promises without ever running a risk of loss. Yes?
So his question is I’ve asserted that no matter what way things go I’m going to end up 42 dollars ahead no matter what, and he’s saying that’s great, but he’s a little skeptical it could be true because I’m always betting against this other bookie. I’m always betting on the Phillies with the other bookie, and so in one case the Yankees win 4 in a row. I lose 4 bets with the other bookie, with the bookie I’m betting on the Phillies, so I’ve had to take 4 losing bets with the other bookie. In the case where the Yankees win three lose one and then win in the end I’ve again lost 4 bets with the other bookie, but I’ve actually won a bet with the other bookie. So it seems like somehow things are better for me, so it seems unbelievable that I’m still at 42 in both cases. That’s your question, right? So the reason is that my bets are changing. I don’t make the same last bet. So the first three bets there’s a Yankee win, Yankee win, Yankee win, each of those nights I’m making the same bet in the same scenario, but when the Phillies lose and then win again I’ve changed my bet, so I’m still going to end up–so the Yankees lost the, your scenario was the Yankees lost the 4th game, so that outcome is different from what it would have been in the first scenario because the Yankees lost instead of winning. So the bookie bet went the other way. But now the last bet I made is not the same bet as I would have made the night before. It’s a different bet so that it just ends up with a different direction than the 4th bet plus this different 5th bet it adds back up to what would have happened had the Yankees won the fourth time. I don’t know if that helps, but that’s what happens. And the proof that it happens is just by induction. I gave a proof and you’re saying the proof is amazingly good. Remember, the proof is very simple. You always make a bet so that no matter what happens your expectation is back to where it was the night before. And if you do that you have to end up–that’s the proof that you’ve always locked in 42 from the beginning. And so you can ask a thousand questions like that and it’s always going to be a complicated answer, but there has to be an answer because we’ve proved that there was one. Any other questions about this? Yes?
Remember, the price as the interest rate goes up–where’s the mortgage, sorry. The value of the non-call, the value of the call, so as the interest rate goes up you owe less money because you’re promising to deliver this stuff. So the interest rate’s up, the mortgage you’ve sold short is less and less valuable, so that’s the good scenario for you, but if the interest rate goes down that’s the scenario where you could get into trouble. So you’re trying to protect against the interest rate going down, and in order to do that you have to give money up in case the interest rate goes up. So now I want to–any other questions about this? It’s quite an ingenious thing, I think. So we’re going to have one more step of this, but this is the high watermark of the standard, you know, the finance guys who made finance such an important subject, and the rational expectations school of finance. This is kind of the most clever thing that they did. So this kind of reasoning, bookies maybe knew it for a long time, but Black-Scholes in 1973 started this kind of thinking and then lots of hedge funds started imitating it including my own. So I certainly didn’t invent this idea of dynamic hedging. Now, let’s take a step back. The crucial idea is marking to market. The crucial idea is that you want to hedge something through millions of scenarios and really the outcome you won’t know until the end. What you should do is hedge at each step of the way your mark to market value. So you only have to do things a step at a time. There’s an exponentially growing number of paths, but there are only two possibilities from today until tomorrow. So it’s very simple to hedge two things. It seems incredibly complicated to hedge 2 to the 100 things, 2 to the 10th is a very big number, 2 is a very small number. You can hedge two things. It may be very complicated to hedge 2 to the 10 things, but you only have to hedge 2 things on every day. So once you’ve realized that, hedging is actually quite a simple operation. You don’t have to do it by trading in the interest rate derivatives market. You could trade in a simpler market. Suppose, for example, that you knew that there were bonds being traded, let’s say 30-year bonds. So let’s just take a look at a 30-year bond here, callable bond, so this is exactly the same thing. This is the 30-year bond under the same circumstance. This is a 9 percent bond, coupon bond, and it’s starting with the same 6 percent interest rates, exactly the same process as before. So the interest rates, those are the same interest rates as before. So here’s the bond. Now, the bond starts at 140 dollars, can go to 121 dollars or to 159 dollars. Remember this thing that you’re trying to hedge goes from 98.8 to 100.6 or to 107.11. So what you need to do now is–you’ve promised to make deliveries in the future, very complicated, but you know you can summarize that whole future. As long as you’ve got 100.6 dollars in the up state and 107.11 dollars in the down state after the very first step you will have hedged your mortgage obligation. But you could just buy the right percentage of bonds and be able to accomplish the same thing. So why is that? Well, there’s a gap here and here. You owe less money here than you do down there. So the bond is worth less up here than down here. In fact this bond is worth 38 dollars more here than here. So the gap in the bond is 38. The gap in here is something like 6.5. So this is like 1 sixth. So as long as you held 1 sixth of the bond and something that paid the right constant amount in each case you would get the same payments here and here. So what is the right constant amount? 1 sixth of the bond, by the way, would cost you 22, 24, 23, about 23 dollars, 1 sixth of the bond. So if you took the other 98.8 dollars and put it to the 1 year Treasury, I claim it’s obvious–so take 1 sixth of the–so 1 sixth of 30 year 9 percent bond. That’s 1 sixth of this thing, which costs you what? It cost 23 dollars. The price of that is 23 dollars. Now, combine that with 75.8 dollars of 1 year Treasury, 1 year 6 percent Treasury. So hold those two things. I claim that’ll pay you 100.6 here and 107.11 here. How do I know that without having to even calculate anything out? Well, the Treasury’s going to pay the same amount here and here. It’s going to be a constant. This thing, since its gap is 38 dollars and I only hold 1 sixth of it, it’s going to pay me exactly 6 and 1 half more down here than it pays here because the gap instead of 38 will be 1 sixth of that, which will be 6 and 1 half. So the gap will exactly match that. So added that to the Treasury’s constant payments I’m going to get something over here and 6 and 1 half dollars more than that down here. But since I’ve paid the same amount, 98.8, it’s going to have to be that that constant was exactly right to make this be 100.6 and 107.11, because both the 1 year Treasury and the 30 year bond were calculated by discounting whatever their payments were here and here by the same prices, this and this. So there’s a second way. Instead of trading in the derivatives market I could just hold a combination of two bonds, the 1 year Treasury and the 30 year 9 percent bond, and I’d still be perfectly hedged. But then when I got to here I’d want to change my mix. I’d want to hold a different 1-year bond at a different interest rate, so I have to buy a new 1-year bond and a different amount of the 30-year non-callable bond, 9 percent bond, but I could always reproduce these same two payments. So I don’t have to hedge by holding derivatives, interest rate derivatives, I can hedge by holding standard bonds. And I can tell at a glance how much of the longer term bond to hold because I just have to match this gap in the prices since the 1 year bond is always going to pay the same thing up and down. Did that go too fast? Yeah?
It won’t change the numbers, but. So it’s paying–I didn’t write it down. So this thing is not paying 121. It’s also paying the coupon. This is the present value of what’s left. So it’s paying 9 plus that. So it’s 130 and here it’s 168. The gap is still 38. So I want 1 sixth of 130. So 1 sixth of 130 costs–it’s still the same number. It’s 22. I did bad arithmetic to begin with so it’s still 22. No, sorry, it’s 1 sixth of this thing which is 140, so it’s 23, right. Sorry. That didn’t change. So the payment here is 130 and the payment here is 168. So 1 sixth of 130, that’s the up payment, and this is 1 sixth of 168. Now, I want that plus whatever the 1 year Treasury gives me to equal 100.6 up here and 107.11 down here. That way I would have replicated things and hedged myself perfectly, right? But I don’t have to figure out what that 1 year Treasury is. I think it’s obvious. All I have to do is say subtract 23 from 98.8, so I just have to buy 75.8 dollars worth of the 1 year Treasury and that’ll give me exactly the right payoff. Why is that? Because this number 140.93 divided by 6 is exactly this evaluated at this price, 1 half over 1.06 times this, this is the price 1 half times 1 over 1.06. All right, so that bond which cost 140.53 how did I get 140.53 there? I took the price of 1 dollar in the up state, which is it’s 50 percent likely discounted, multiplying by its payoff 130, plus 1 half discounted multiplied by the payoff of 168. This is 168 because it’s a 9 percent bond. So it’s 1 sixth times 1 sixth–so if I get a sixth of the bond it’ll cost me 140.43 divided by 6 which is 23. So what about the 1 year Treasury? That’s going to give me some payment, an additional payment here and here, namely 1.06 times whatever I put in. So you notice that the gap between here and here since I’ve multiplied by 1 sixth is exactly this 6 and 1 half dollar gap. So the total I have to spend is 98.8 over here, so if I spend 23 dollars on the Treasury and the remaining amount–23 dollars on the 30 year 9 percent bond, the remaining amount of money which is 75.8 on the Treasury I will have spent this 98.8 dollars, but then I have to get the same payoffs. Why is that? Because they’re all being priced at the same prices, so it’s just the distributive law of arithmetic. If these payoffs times 1 sixth plus the Treasury payoffs equal 100.6 and 107.11 then it must be that the sum of the money I spent on the 30 year 9 percent bond, plus the sum of the money I spent on the Treasury has to equal 98.8. Therefore, I know that the amount of money of the 98.8 I spent on the 1 year Treasury was 98.8 minus the 23 I spent on the 30-year bond. So that’s how you can–so to say it again just in a big picture look. There are two things that can happen. If you hold two bonds, a 1-year Treasury and a 30-year thing, you’re obviously going to be able to match these two things by holding the right proportion of them. That is just obvious. And then less obvious is how quickly I’m figuring out how much of those two bonds to hold. And the way I’m figuring it out so quickly is because the 1-year Treasury is paying the same thing up and down, so I have to get all the variation out of the 30-year bond, so I have to hold the right proportion of the 30-year bond to get the same variation. So in the very big picture if you’re trying to hedge some instrument with another instrument and a portfolio of stuff, much of which is certain, and only one thing that’s risky, the amount of the risky thing you have to hold has to produce as much variation as the thing you’re trying to hedge. So the key is being able to tell at a glance how much variation each thing has then you can tell in what proportion to hold them. ## Chapter 5. Measuring the Average Life of a Bond [01:05:15]And so the very last thing I want to tell you today, which is unfortunately going to take me a few minutes, is how traders look at variation and can guess it incredibly quickly. So a trader could look at this and guess without doing almost any calculations what the right hedge is. So just like we did the doubling rule I want to do the variation rule of thumb. And this is called duration and convexity, or average life. So I asked myself, suppose I have this 30-year bond, 30-year non-callable bond, that’s this one, 9 percent bond. How could I guess, kind of, what the variation in price would be without having to do, you know, I’ve had to do this very complicated backward induction calculation. Now, remember once the interest rates go up here it’s a random walk so on average they’re pretty much, not pretty much, they’re exactly what they were here. They start here. They go up and down, but on average they’re still here. And over here they’ve gone down so the average is lower going forward. In fact, it’s exactly equal to that. So if you’re a trader and there was no uncertainty you would know that the present value with no uncertainty is going to be the coupon divided by (1 + the interest rate) + the coupon over (1 + the interest rate) squared + (in the last year) the coupon over (1 + the interest rate) to the T + 100 times [correction: over] (1 + the interest rate) to the T. That’s the value of the bond. Now, you sort of want to know how much is that price going to change when the interest rate changes. Well, at a glance, I mean, you just take the derivative. What is dPV/dr? It’s going to be minus C over (1 + r) squared - 2 C over (1 + r) cubed - T times C over (1 + r) to the (T + 1) - T times 100 divided by (1 + r) to the (T + 1). That’s the derivative. It looks like I’m making things even more complicated, you would think, but this turns out to be not so complicated, because this is a number–so let’s put the minus on the outside, so I’ll get rid of this. This is a number which you can sort of get in your head, so let’s try writing it. Let’s pull out one of the (1 + r)s. So it’s minus 1 over (1 + r) times (C over (1 + r) + 2 C over (1 + r) squared + … + T times (C + 100) (that’s the last payment) over (1 + r) to the T). Now, what is that? Can you interpret that? And the answer is yes you can interpret that because what is that? That’s saying these are all the payments you’re getting over the life of the bond. It’s C, C and the last one’s C + 100. This number that keeps coming in front of them, though, is the year in which you’re getting the payment. This is in the first year. I could put a 1 there. This is the payment in the second year. There’s a 2 here. This is the payment in the Tth year. There’s a T here. You got that T from taking the derivative of the denominator here. So it’s like weighting the year in which the payments come by the value of those payments. So in the first year the value is C. The present value is C over (1 + r). In the second year the payment is C, but discounted, C over (1 + r) squared, and the last year the total payment is C + 100, but you had to discount it by (1 + r) to the Tth. So this is just like taking the average year in which the present value flows actually come. If I divided this now by the present value of–the present value, these numbers down here, C over this + C over this squared + C + 100 over this to the Tth, those are all weights multiplying the year in which the payments come. They add up to the present value. So basically I’ve just got a weighted average of what year the payments come in. This is called the average life. This thing in here is the average life because I’ve multiplied the year in which the payments come by the present value of the cash flows there and then I’ve made those multipliers add up to 1 by dividing by the PV. See, this in parenthesis, all these terms are the things multiplying the year in which the thing came. So if I add up all these coefficients here I’m just getting the present value. So now the coefficients add up to 1. So I’m just taking an average of the year in which the thing came and that’s what the average life is, but that’s equal to the derivative of the present value divided by the present value. So it’s–the percentage change in the present value is the average year in which the payments come. So let’s just think about it. Suppose you have a 10-year bond? It pays C, C, C, C, C, and then 100 at the very end. What do you think the average life is? Well, if there were no discounting the average life wouldn’t be 5 because there’s such a huge payment in the end. The average life would be 8 or 9, but there is discounting so this thing at the end won’t count as much. So the average life is probably 7. If it’s a 30 year bond, the average life, you’re going to get C, C, C, C, and then all the way at the end you’re going to get C + 100. Well, if the payments were all equal and no discounting it would be 15, but there’s a huge amount of discounting so that last payment is really pretty irrelevant, so probably–so it’s not clear which way it goes, probably the average life will be less than 15. So you can see just by looking at the bond, and I’m ending with this thought, by looking at the bond itself you can figure out what the average life is. So a trader, just by a little experience and common sense, can figure out the average life. I mean, here are all the payments the way they’re coming. They’re coming evenly across the whole history of the bond with something at the end, so the average life is probably a little bit over the average, but if you go way off, you push the final payment too far to the end then it’s becoming negligible, it’s almost irrelevant, and so the average life by discounting will probably be before the average. So a 10-year bond probably has an average life of 7, a 30-year bond maybe has an average life of 12 or 13. Once you know the average life, you just have to hold bonds in proportion to their average lives. The dollar amount you put in each bond to hedge it is just proportional to its average life. So that’s how a trader, without having to do any arithmetic, can kind of guess how much of the 30 year bond will it take to hedge a 10 year bond, and how much of a–so which has a higher average life? The longer the bond the higher the average life, the more sensitive it is to interest rate changes so the less of it you need to hedge your position, so if you hold the 30 year bond you don’t have to hold that much of it, 1 sixth of it, in order to hedge this mortgage. The mortgage has a very short average life because it’s prepaying so quickly. So it’s not going to last very long. So this is an amazing connection between average life and sensitivity to interest rate. And average life is something you can guess. Sensitivity to interest rate is something you have to do huge calculations for, but the two are almost the same. So in the problem set we’ll see if you can do it. I asked you a couple problems to do with average life, so. [end of transcript] Back to Top |
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