WEBVTT 00:02.160 --> 00:03.760 Prof: So time to start. 00:03.760 --> 00:07.280 So let me begin by reviewing a little bit of the dynamic 00:07.280 --> 00:10.670 hedging which I could have described a little bit more 00:10.672 --> 00:12.082 clearly last time. 00:12.080 --> 00:17.150 It's a very important idea and I made a little bit of a mess of 00:17.151 --> 00:17.561 it. 00:17.560 --> 00:19.560 I didn't make that big a mess of it, but a little bit of a 00:19.555 --> 00:20.005 mess of it. 00:20.010 --> 00:24.300 So let's be a little bit careful now about who knows what 00:24.301 --> 00:26.141 and what you're doing. 00:26.140 --> 00:30.770 So we said imagine somebody who knows that the probability the 00:30.765 --> 00:35.235 Yankees are going to win the World Series is 60 percent, 00:35.240 --> 00:39.600 and therefore each game of the World Series is 60 percent, 00:39.600 --> 00:44.020 and knows that the probability the Phillies win the game is 40 00:44.024 --> 00:44.754 percent. 00:44.750 --> 00:49.400 So suppose that he finds somebody who is willing to bet 00:49.403 --> 00:51.733 with him on the Phillies. 00:51.730 --> 00:54.210 So he can win 100 dollars if the Yankees win. 00:54.210 --> 00:57.410 He has to pay 100 dollars if the Yankees lose. 00:57.410 --> 01:00.370 Now, his expected payoff is 20. 01:00.368 --> 01:04.178 He should take the bet because if the person's willing to bet 01:04.183 --> 01:06.623 at even odds, the Phillies fan's willing to 01:06.620 --> 01:09.120 bet at even odds and he knows the odds are 60/40, 01:09.120 --> 01:11.520 on expectation he's going to make 20. 01:11.519 --> 01:14.579 So it's clear what he should do, but the problem is he's 01:14.578 --> 01:15.578 subject to risk. 01:15.578 --> 01:19.848 Although he's right about the odds he could still, 01:19.849 --> 01:22.129 even though he's done the smart thing, 01:22.129 --> 01:25.559 he could still end up losing 100 dollars which could be a 01:25.563 --> 01:26.733 disaster for him. 01:26.730 --> 01:30.630 So what he would like to do is to hedge his bets. 01:30.629 --> 01:32.469 Now, what does it mean hedge his bet? 01:32.470 --> 01:36.900 Well, suppose there was another bookie who was willing to bet at 01:36.900 --> 01:40.840 60/40 odds in either direction then what should he do? 01:40.840 --> 01:46.140 Well, if he can find another bookie who was willing to bet at 01:46.144 --> 01:51.094 60/40 odds he should try and lock in 20 dollars no matter 01:51.093 --> 01:51.893 what. 01:51.890 --> 01:59.500 So he should do this bet. 01:59.500 --> 02:02.860 He should bet with the other bookie 80 dollars. 02:02.858 --> 02:06.698 He'd be willing to give up 80 dollars if the Yankees won in 02:06.700 --> 02:09.880 order to win 120 dollars if the Phillies win, 02:09.878 --> 02:13.398 and that's a fair bet according to the other bookie because it's 02:13.397 --> 02:15.487 60/40 odds, 60 percent of this and 40 02:15.492 --> 02:18.992 percent of that, this is 3 to 2 odds so it's a 02:18.990 --> 02:19.870 fair bet. 02:19.870 --> 02:22.750 So in other words, he makes money by taking 02:22.752 --> 02:26.122 advantage of the Philly fan to bet on the Yankees, 02:26.116 --> 02:28.446 but that subjects him to risk. 02:28.449 --> 02:32.149 So in order to minimize the risk he hedges his bet. 02:32.150 --> 02:33.830 That's where the expression came from. 02:33.830 --> 02:37.630 He hedges that bet by betting in the opposite direction, 02:37.628 --> 02:40.548 on--betting on the Phillies with his bookie, 02:40.550 --> 02:45.450 in the opposite direction, but standing to lose less than 02:45.454 --> 02:48.024 he, you know, he's betting in this 02:48.024 --> 02:50.524 proportion, in this amount so that he gets 02:50.520 --> 02:51.540 20 no matter what. 02:51.538 --> 02:53.988 So he's locked in his 20-dollar profit. 02:53.990 --> 02:55.950 So that was where we began. 02:55.949 --> 02:58.189 Now, let's just think about it a little bit more carefully than 02:58.187 --> 02:58.727 I did before. 02:58.729 --> 03:02.069 Somebody, in fact, basically asked this question. 03:02.068 --> 03:05.488 Is he really making money because he understands better 03:05.492 --> 03:08.222 the odds of the Yankees winning the game? 03:08.218 --> 03:11.798 Is he making money because he knows the odds are 60/40 and the 03:11.800 --> 03:14.090 poor Philly fan thinks they're 50/50? 03:14.090 --> 03:15.370 The answer is no. 03:15.370 --> 03:16.750 That's not why he's making money. 03:16.750 --> 03:20.170 It doesn't matter what the odds are of the Yankees winning. 03:20.169 --> 03:23.959 He's winning because he's managed to arbitrage two 03:23.955 --> 03:27.245 different betters, namely the Philly fan and the 03:27.247 --> 03:29.567 other bookie, and the fact that they differ 03:29.566 --> 03:32.526 in their beliefs, the fact that he can trade with 03:32.532 --> 03:35.782 the bookie at 3 to 2 odds whatever bet he wants, 03:35.780 --> 03:40.030 and the Philly fan is trading at different odds with him. 03:40.030 --> 03:42.440 That's what enables him to make his profit. 03:42.440 --> 03:46.300 So how did he make the arbitrage--so what is an 03:46.301 --> 03:47.311 arbitrage? 03:47.310 --> 03:50.470 An arbitrage is, you find two bets which are 03:50.466 --> 03:54.646 more or less the same thing, which in fact are exactly the 03:54.650 --> 03:58.100 same thing, but trade at a different price. 03:58.099 --> 04:03.079 So what are the two bets? 04:03.080 --> 04:12.690 Well, there's the Philly bet, the Philly fan bet, 04:12.693 --> 04:24.713 and then there's the bookie bet plus borrowing 20 dollars. 04:24.709 --> 04:27.539 So he's going to make 20 for sure. 04:27.540 --> 04:30.730 So he could make the 20 at the beginning just by borrowing 04:30.732 --> 04:32.022 against his winnings. 04:32.019 --> 04:37.829 Then he'd have 20 at the very beginning and his payoffs from 04:37.829 --> 04:43.639 the Philly fan bet would be exactly--he'd owe the 20 so he'd 04:43.639 --> 04:45.509 have minus 100. 04:45.509 --> 04:52.089 So this bet is he'd be able to get 20 at the beginning, 04:52.091 --> 04:57.091 but he'd have to pay minus 100 and 100. 04:57.089 --> 05:05.949 He'd pay 100 dollars if the Yankees won, and get 100 dollars 05:05.947 --> 05:09.247 if the Phillies won. 05:09.250 --> 05:14.140 And so that way he'd cancel his bet at the end and end up with 05:14.144 --> 05:15.914 20 dollars for sure. 05:15.910 --> 05:18.610 So just to say it again, he's made money apparently 05:18.608 --> 05:21.358 because he knows more than the Philly fan knows, 05:21.360 --> 05:24.950 but actually in order to guarantee that he doesn't run a 05:24.949 --> 05:27.429 risk he has to find another trader, 05:27.430 --> 05:30.990 and it's the presence of the other trader who's really giving 05:30.990 --> 05:34.630 him his profit opportunity, not his superior knowledge. 05:34.629 --> 05:37.089 And the fact that he has two different traders--and so why 05:37.093 --> 05:38.653 does that give him an opportunity? 05:38.649 --> 05:43.419 Because he can arrange a bet with the Philly fan and a bet 05:43.418 --> 05:48.588 with the other bookie, but combined with borrowing the 05:48.591 --> 05:54.461 money so that in the end he's going to just get 0 payoffs. 05:54.459 --> 05:56.789 Putting together the Philly fan bet, 05:56.790 --> 05:59.560 and the bookie fan bet, and borrowing 20 dollars he 05:59.564 --> 06:01.844 gets 20 dollars at the very beginning, 06:01.838 --> 06:05.098 and on net he has nothing at the end, 06:05.100 --> 06:08.430 so whatever losses he got from the Phillies he made up from the 06:08.430 --> 06:09.130 bookie bet. 06:09.129 --> 06:12.099 Whatever losses he had from the bookie bet he made up for it 06:12.100 --> 06:13.410 with the Philly fan bet. 06:13.410 --> 06:17.570 So arbitrage relies on replication. 06:17.569 --> 06:24.439 Somehow through some other bet--he was able to hedge his 06:24.439 --> 06:25.939 other bet. 06:25.939 --> 06:29.839 He was able to hedge his Phillies bet by replicating the 06:29.843 --> 06:33.823 bet with somebody else combined with borrowing to produce 06:33.819 --> 06:36.729 exactly the same payoffs in the end, 06:36.730 --> 06:39.080 or the negative of the payoffs in the end, 06:39.079 --> 06:41.859 and yet for a different price at the beginning. 06:41.860 --> 06:44.930 The Philly fan didn't demand any money upfront to make this 06:44.934 --> 06:46.634 (100, minus 100) bet with him. 06:46.629 --> 06:51.579 And with the bookie he's able to get 20 dollars upfront and 06:51.584 --> 06:56.204 make this the opposite bet, the (minus 100,100) bet. 06:56.199 --> 06:59.709 So by getting more money upfront and doing the opposite 06:59.711 --> 07:03.421 bet with the bookie he can end up completely canceling the 07:03.418 --> 07:05.268 Philly bet, Philly fan bet, 07:05.274 --> 07:07.234 and that's how he makes his profit. 07:07.230 --> 07:10.070 So that was the first idea. 07:10.069 --> 07:11.439 So what have I clarified? 07:11.439 --> 07:14.409 What I've clarified is that it wasn't his superior knowledge, 07:14.410 --> 07:17.110 it was the presence of two different traders that enabled 07:17.113 --> 07:20.123 him to make his profit, and I've clarified the method 07:20.115 --> 07:21.615 that he made the profit. 07:21.620 --> 07:25.490 The profit wasn't just kind of mixing in another bet, 07:25.488 --> 07:29.278 it was really cleverly arranging to undo totally the 07:29.283 --> 07:30.553 original bet. 07:30.550 --> 07:34.630 So he got actually 0 in the end and ended up with 20 dollars at 07:34.627 --> 07:35.677 the beginning. 07:35.680 --> 07:41.290 So let's just see now what you would do in a dynamic situation. 07:41.290 --> 07:41.990 Yes? 07:41.990 --> 07:42.600 Student: I apologize. 07:42.600 --> 07:45.410 I understand how this locks him into that $20 dollars, 07:45.410 --> 07:47.250 and how it makes sure he doesn't incur the losses that he 07:47.245 --> 07:49.175 might have occurred, but why would he do this if he 07:49.180 --> 07:51.580 has 20 dollars at the beginning and all this does is make sure 07:51.579 --> 07:53.939 no matter what he ends up with the 20 dollars at the end? 07:53.940 --> 07:55.320 Prof: Why would he do what at all? 07:55.319 --> 07:56.919 Student: This arbitrage at all, any of the bets. 07:56.920 --> 07:59.320 It seems like no matter what whether the Yankees or lose he 07:59.322 --> 08:00.402 ends up with 20 dollars. 08:00.399 --> 08:02.549 Why didn't he just keep his 20 dollars and not bet? 08:02.550 --> 08:04.230 Prof: He didn't have 20 dollars to begin with. 08:04.230 --> 08:05.560 He had no dollars. 08:05.560 --> 08:08.920 So he went and found the Philly fan and he made a bet against 08:08.915 --> 08:10.645 the Philly fan with no money. 08:10.649 --> 08:14.519 The Philly fan was willing to bet him 100 dollars on the 08:14.524 --> 08:15.304 Phillies. 08:15.300 --> 08:16.740 So he gets 100 if the Yankees win. 08:16.740 --> 08:19.040 He pays the fan 100 if the Phillies win. 08:19.040 --> 08:20.740 It didn't cost any money to do that. 08:20.740 --> 08:24.120 Then he goes out and makes another bet with the bookie. 08:24.120 --> 08:25.680 Again, it doesn't cost him any money. 08:25.680 --> 08:29.110 He gives up 80 dollars if the Yankees win, but he gets $120 if 08:29.110 --> 08:30.180 the Phillies win. 08:30.180 --> 08:33.400 So he's spent no money at all and his total profit is 20 08:33.398 --> 08:34.508 dollars for sure. 08:34.509 --> 08:36.819 So what could he do since he knows he's going to make 20 08:36.816 --> 08:38.196 dollars for sure in the future? 08:38.200 --> 08:40.600 He could just borrow it at the beginning if the interest rate's 08:40.596 --> 08:42.216 0, have 20 dollars at the 08:42.215 --> 08:45.905 beginning, end up with 0 no matter what the uncertainty is 08:45.908 --> 08:48.498 and put the 20 dollars in his pocket. 08:48.500 --> 08:52.980 So that's another way of saying the same thing which is that he 08:52.979 --> 08:54.719 could, by combining the bets, 08:54.721 --> 08:57.751 put money in his pocket in at the beginning and no matter what 08:57.746 --> 09:00.766 happens in the future he doesn't have to give anybody any more 09:00.774 --> 09:01.324 money. 09:01.320 --> 09:02.090 So that's what he's done. 09:02.090 --> 09:03.230 That's what an arbitrage is. 09:03.230 --> 09:06.960 You put money in your pocket in the present and in the future 09:06.961 --> 09:09.141 you never have to face a problem. 09:09.139 --> 09:13.679 You've always got money that's going out in one hand you're 09:13.681 --> 09:16.031 getting from the other hand. 09:16.028 --> 09:19.838 Now, in a dynamic situation you can do the same thing. 09:19.840 --> 09:22.260 So I'm just reinterpreting what we did before a little bit. 09:22.259 --> 09:25.019 So in the dynamic situation what are you doing? 09:25.019 --> 09:29.589 The Philly fan now is betting on the whole World Series even 09:29.586 --> 09:32.446 though you know the odds are 60/40. 09:32.450 --> 09:35.050 So now that's an even worse bet by the Philly fan if you're 09:35.046 --> 09:37.906 right about the odds because the chances are even higher that the 09:37.912 --> 09:40.332 Yankees are going to win the series than one game. 09:40.330 --> 09:42.830 The odds, we know, are 71 percent. 09:42.830 --> 09:46.800 So by backward induction we did this calculation to say your 09:46.803 --> 09:48.963 expected payoff is 42 dollars. 09:48.960 --> 09:52.260 So let's suppose now--but now the key is that you're making 09:52.263 --> 09:55.683 money not just because you know better what the odds are, 09:55.678 --> 09:59.558 you're making money because there's another bookie standing 09:59.559 --> 10:03.909 there always ready to trade with you at 60/40 odds in any game. 10:03.908 --> 10:07.758 So the presence of the second bookie, even if you knew 10:07.761 --> 10:11.541 nothing, would enable you to make money in this case, 10:11.541 --> 10:14.741 lock in a profit for sure of 42 dollars. 10:14.740 --> 10:17.370 Now, how do you know that you can lock in a profit for sure of 10:17.369 --> 10:17.929 42 dollars? 10:17.928 --> 10:22.448 Well, it's now done by dynamic hedging. 10:22.450 --> 10:26.990 You have to lay off part of your bet, 10:26.990 --> 10:29.920 a little part each day, each game and the amount that 10:29.916 --> 10:32.726 you lay off is going to depend on what happens, 10:32.730 --> 10:33.800 so that's why it's dynamic. 10:33.799 --> 10:35.559 It's a complicated calculation. 10:35.558 --> 10:38.678 So it seems almost hopeless to figure out what to do. 10:38.679 --> 10:40.699 I'm betting on the whole series. 10:40.700 --> 10:43.180 Many different things can happen. 10:43.178 --> 10:45.008 The Phillies could win the first game. 10:45.009 --> 10:46.199 The Yankees could win the first game. 10:46.200 --> 10:47.840 Lots of things could happen. 10:47.840 --> 10:51.560 By the end I want to make sure I've locked in my profit for 10:51.559 --> 10:52.009 sure. 10:52.009 --> 10:53.539 How can I do that? 10:53.538 --> 11:00.448 Well, the key is to think about marking to market. 11:00.450 --> 11:04.120 Just as we said, if you're assessing your 11:04.120 --> 11:07.690 position correctly, even though the World Series 11:07.686 --> 11:11.096 takes 7 games, if you were reporting to your 11:11.101 --> 11:13.131 investors, your hedge fund, 11:13.129 --> 11:16.439 you're reporting to your investors how you're doing, 11:16.440 --> 11:18.930 you should be telling them after the first game, 11:18.928 --> 11:20.528 you know, at the beginning you're telling them, 11:20.528 --> 11:22.028 "Look, I'm going to make 42 dollars. 11:22.028 --> 11:23.488 I expect to make 42 dollars." 11:23.490 --> 11:26.360 After the Yankees win the first game it looks like you're going 11:26.360 --> 11:28.120 to make 60, you know, now things are 11:28.116 --> 11:30.466 better, you expect to make 64 dollars on average, 11:30.470 --> 11:33.900 but if the Yankees lost the first game you'd only expect to 11:33.895 --> 11:35.545 make 9 dollars on average. 11:35.548 --> 11:39.168 So after the first game you go from thinking you're going to 11:39.168 --> 11:41.928 make 42 to being even more optimistic now, 11:41.928 --> 11:45.808 you're going to make 64 or you're going to make 9. 11:45.808 --> 11:51.228 So 60 percent times 64 40 percent of 9 is 42. 11:51.230 --> 11:55.160 So if you were telling your investors properly how you stood 11:55.155 --> 11:59.075 after the very first game you might have to tell them you've 11:59.081 --> 12:03.011 lost 33 of the 42 dollars that you told them before you were 12:03.009 --> 12:04.339 going to make. 12:04.340 --> 12:07.230 So you'd already be running a risk. 12:07.230 --> 12:11.080 You really only care, maybe, about getting money for 12:11.076 --> 12:13.766 sure by the end, but if you're doing proper 12:13.770 --> 12:16.880 accounting you should be worried all the way along whether you're 12:16.878 --> 12:18.138 making or losing money. 12:18.139 --> 12:23.029 And that's the key to how to have no risk by the end. 12:23.028 --> 12:25.658 You just run no risk at all the in between stages. 12:25.658 --> 12:29.588 So all you have to do is make a bet with the bookie who's 12:29.592 --> 12:33.672 standing by to make 33 dollars in case the Phillies win the 12:33.666 --> 12:34.716 first game. 12:34.720 --> 12:38.450 You're making a one game bet in exchange for losing 22 dollars 12:38.453 --> 12:39.743 if the Yankees win. 12:39.740 --> 12:42.720 This is still 3 to 2 odds, so the bookie's willing to do 12:42.719 --> 12:42.989 it. 12:42.990 --> 12:46.930 So the night before the first game you've bet with the Philly 12:46.929 --> 12:50.859 fan on the entire series, which is giving you a huge 12:50.855 --> 12:53.555 advantage, but you're willing to bet with 12:53.557 --> 12:54.597 the other bookie. 12:54.600 --> 12:56.350 The other bookie, we're assuming, 12:56.347 --> 12:58.257 is only standing by game by game. 12:58.259 --> 13:01.039 You're betting 33 to 22 on the Phillies. 13:01.038 --> 13:03.028 You're betting on the team you expect to lose. 13:03.028 --> 13:05.498 You're betting on that team with the other bookie, 13:05.504 --> 13:06.924 but in a smaller quantity. 13:06.918 --> 13:11.998 So this is going to mean that your profit after the first 13:11.995 --> 13:17.065 game, no matter what happens, is still going to be 42. 13:17.070 --> 13:20.520 So you can lock in 42 after every game. 13:20.519 --> 13:25.669 And so as I said last time the crucial thing is--so then one 13:25.674 --> 13:27.514 thing I didn't say. 13:27.509 --> 13:29.879 So one thing which I forgot to observe last time, 13:29.875 --> 13:32.725 which I think is interesting, is so is that what you should 13:32.734 --> 13:33.084 do? 13:33.080 --> 13:37.370 Should you do that after every game, always bet 33 to 22 13:37.368 --> 13:40.798 against the Yankees with the other bookie? 13:40.799 --> 13:42.909 Well, the answer is no. 13:42.908 --> 13:46.538 What should you do if the Yankees win the first game? 13:46.538 --> 13:50.978 By winning the first game your expectation goes up to 64. 13:50.980 --> 13:56.060 Now it could be either 82--and the night after the first game 13:56.057 --> 13:59.947 you're much better off than you were before. 13:59.950 --> 14:02.420 Of course you laid off part of the bet with the other bookie, 14:02.418 --> 14:06.238 so you've had to pay off 22 so you're still only 42 ahead, 14:06.240 --> 14:08.910 but the Yankees have now won the first game. 14:08.908 --> 14:12.128 So it's now the night after the first game, before the second 14:12.133 --> 14:12.513 game. 14:12.509 --> 14:14.609 You're subject to more risk. 14:14.610 --> 14:17.430 What should you do now? 14:17.428 --> 14:20.648 You should go back to the bookie and make another bet on 14:20.649 --> 14:23.339 the second game, but should the other bet be 22 14:23.344 --> 14:23.934 to 33? 14:23.928 --> 14:30.488 Absolutely not, because 64 up here goes to 14:30.493 --> 14:38.023 whatever that number is, which I forgot already, 14:38.017 --> 14:43.617 82 and 1 half to 36 and 1 half. 14:43.620 --> 14:44.620 So what should you do? 14:44.620 --> 14:55.590 14:55.590 --> 14:56.720 What should you do now? 14:56.720 --> 14:58.950 So I've done some rounding here which is going to screw 14:58.948 --> 15:01.138 everything up a little bit, but now what should you do 15:01.138 --> 15:01.508 here? 15:01.509 --> 15:03.389 After the first game you're here. 15:03.389 --> 15:08.409 Here you are after the first game. 15:08.408 --> 15:12.478 You've made your bet with the bookie. 15:12.480 --> 15:15.250 The first game you bet 33 against 22. 15:15.250 --> 15:18.830 Now what should you do the second game? 15:18.830 --> 15:24.260 What should your bet be with that bookie? 15:24.259 --> 15:27.629 He's still willing to give you 3 to 2 odds on the Yankees 15:27.634 --> 15:28.784 either direction. 15:28.779 --> 15:34.789 What should you do? 15:34.788 --> 15:37.028 You're not going to bet (33,22) again. 15:37.029 --> 15:38.999 You're going to make a different bet, 15:38.998 --> 15:40.308 and what should it be? 15:40.309 --> 15:41.189 Yes? 15:41.190 --> 15:43.530 Student: > 15:43.529 --> 15:44.699 Prof: Exactly. 15:44.700 --> 15:51.770 You should give up 18 if Yankees win and get 27 rounding 15:51.767 --> 15:52.407 up. 15:52.408 --> 15:55.628 She conveniently rounded a little bit because I had rounded 15:55.634 --> 15:56.474 a little bit. 15:56.470 --> 15:58.810 So you should do 18 and 27. 15:58.808 --> 16:01.288 Now notice that's still 3 to 2 odds. 16:01.288 --> 16:06.028 So the bookie's still willing to do that for you and you're 16:06.027 --> 16:10.187 locking in the 64 because it's going to be 64 again, 16:10.193 --> 16:13.383 and adding this is 64 again, right? 16:13.379 --> 16:15.589 So you've locked in whatever this number is. 16:15.590 --> 16:18.900 So in the first step you made a bet to lock this number in. 16:18.899 --> 16:22.179 In the second step you made a bet to lock this number in. 16:22.178 --> 16:26.218 You're always locking in where you were before. 16:26.220 --> 16:28.210 So that bet is changing each time, 16:28.210 --> 16:30.040 but you're always locking in what you were before, 16:30.038 --> 16:34.348 because after the Yankees have won a game now the next game is 16:34.351 --> 16:36.121 not quite as important. 16:36.120 --> 16:39.960 Time is running out on the Phillies so it's not as 16:39.958 --> 16:44.268 revealing the next game about what's going to happen. 16:44.269 --> 16:52.749 So you make a smaller bet up here than you did before. 16:52.750 --> 16:55.760 So why is that locking everything in? 16:55.759 --> 16:58.519 Because this average is now 64, what it was here, 16:58.515 --> 17:02.245 but don't forget you'd already made the initial bet to lose 22. 17:02.250 --> 17:06.140 So you've locked in 44 now here and also 44 here because you 17:06.135 --> 17:08.635 still owe the 22 from the first bet. 17:08.640 --> 17:12.350 So you just keep proceeding down the tree locking in where 17:12.352 --> 17:16.132 you were before and so by induction you've locked in 42. 17:16.130 --> 17:19.020 So that's what you're doing, and the bet is changing every 17:19.015 --> 17:19.365 time. 17:19.368 --> 17:21.728 So what do you think the bet is here, by the way? 17:21.730 --> 17:32.180 The bet here you could go from 9 to 36 or minus 32. 17:32.180 --> 17:41.720 So what should you do here? 17:41.720 --> 17:45.350 What should you do here? 17:45.348 --> 17:49.048 What's the bet that you're going to make with the bookie 17:49.048 --> 17:51.738 after the Phillies win the first game? 17:51.740 --> 17:57.790 So it's going to be minus 27 and plus 41, and that's 3 to 2 17:57.787 --> 18:03.517 odds with some rounding, right, 13 and 1 half times 3 is 18:03.522 --> 18:05.402 40 and 1 half. 18:05.400 --> 18:10.000 So it's still fair odds, so the bookie's willing to do 18:10.001 --> 18:10.611 that. 18:10.609 --> 18:11.659 You've locked in 9. 18:11.660 --> 18:15.100 So by induction you've always locked in where you were before 18:15.096 --> 18:18.186 and therefore you've locked in 42 all the way along. 18:18.190 --> 18:20.440 It sounds like we're locking in 9, 18:20.440 --> 18:23.100 but we're not locking in 9 because remember the first day 18:23.102 --> 18:25.912 you won 33 dollars from the bookie when the Phillies won the 18:25.905 --> 18:26.615 first game. 18:26.618 --> 18:29.458 So you've locked in the 9, but don't forget the 33 from 18:29.455 --> 18:31.815 before, so it's really you've locked in 42. 18:31.819 --> 18:34.119 So 42's locked in the whole way. 18:34.119 --> 18:36.629 So that's dynamic hedging. 18:36.630 --> 18:41.820 You have a complicated bet that depends on a lot of things along 18:41.823 --> 18:45.053 the way, but you bet each time along the 18:45.049 --> 18:46.909 way, maybe changing your bet each 18:46.906 --> 18:49.066 time depending on what's happened so far, 18:49.068 --> 18:52.398 and that way you can lock in whatever profit you had. 18:52.400 --> 18:57.910 So that's the thing that sports bookies have known for a long 18:57.906 --> 18:58.546 time. 18:58.548 --> 19:01.828 The expression "hedging your bets" 19:01.826 --> 19:06.316 comes from that and so that same logic can be applied to the 19:06.320 --> 19:07.160 market. 19:07.160 --> 19:10.030 But remember, the step that I didn't 19:10.025 --> 19:14.765 emphasize very much is that although you seem to be locking 19:14.772 --> 19:18.552 in where you were before, there's a slightly more 19:18.548 --> 19:22.728 abstract way of saying that, that at the end you're going to 19:22.732 --> 19:24.722 get 42 dollars for sure. 19:24.720 --> 19:28.140 So of course you could borrow all the way back from the 19:28.144 --> 19:31.824 beginning and get 42 dollars immediately and then never owe 19:31.824 --> 19:35.404 anything in the future, because whatever bet you were 19:35.400 --> 19:38.820 winning or losing with the Philly fan you were making up 19:38.815 --> 19:41.605 for with your myriad bets with the bookie. 19:41.608 --> 19:45.788 So it's a replication argument that by betting sequentially 19:45.786 --> 19:48.446 with the bookie different amounts, 19:48.450 --> 19:55.690 you're undoing the single long bet you had with the Philly fan. 19:55.690 --> 20:01.860 So now let's move from sports, as we did last time, 20:01.864 --> 20:03.844 to the market. 20:03.838 --> 20:06.588 So here we had the prepayment spreadsheet. 20:06.588 --> 20:12.168 And so we had, remember, this was a bond, 20:12.170 --> 20:15.560 a mortgage, a 30-year mortgage, 8 percent mortgage, 20:15.558 --> 20:21.588 and the interest rate started off at 6 percent and it had a 20 20:21.585 --> 20:23.655 percent volatility. 20:23.660 --> 20:26.040 So the interest rates could follow this process. 20:26.039 --> 20:27.029 There you see it. 20:27.028 --> 20:30.058 They start at 6 percent, but they can go up or down by 20:30.061 --> 20:32.181 20 percent, basically every period. 20:32.180 --> 20:36.000 So you assume that everybody knows this and knows that the 20:35.998 --> 20:37.738 probabilities are 50/50. 20:37.740 --> 20:41.220 That's like saying you know in advance there's a bookie out 20:41.220 --> 20:42.820 there, that is the market, 20:42.819 --> 20:46.089 that's willing to give you, no matter what day it is, 20:46.089 --> 20:50.249 50/50 odds of the interest rate going up or down from then on. 20:50.250 --> 20:53.830 That's just like the bookie standing by. 20:53.828 --> 20:59.478 You now are going to take advantage of the fact that you 20:59.484 --> 21:05.554 can anticipate that the market will stand there ready to bet 21:05.553 --> 21:06.893 with you. 21:06.890 --> 21:09.840 Now, if the whole market freezes or something you can be 21:09.843 --> 21:10.813 totally screwed. 21:10.808 --> 21:15.768 You're doing your complicated dynamic betting with the market, 21:15.769 --> 21:18.179 one day you wake up and you can't bet anymore because 21:18.182 --> 21:20.502 nobody's willing to trade, something horrible has 21:20.500 --> 21:22.620 happened, 9/11 happened or something like that. 21:22.618 --> 21:25.908 You can be caught in a really bad situation and end up running 21:25.906 --> 21:26.766 a lot of risk. 21:26.769 --> 21:31.339 So you're relying on the market, so it's not surprising 21:31.340 --> 21:36.250 that when the market freezes up lots of people suddenly are 21:36.250 --> 21:40.230 exposed to more risk than they were before. 21:40.230 --> 21:43.090 So now in the market you're not going to be able to find these 21:43.086 --> 21:43.926 pure arbitrages. 21:43.930 --> 21:46.830 In the sports example it was a pure arbitrage. 21:46.828 --> 21:50.318 There was a Philly fan who is betting at even odds and a 21:50.315 --> 21:52.845 bookie who was betting at 3 to 2 odds. 21:52.848 --> 21:55.228 Now, you shouldn't expect that to happen. 21:55.230 --> 21:59.170 I mean, by what miracle did this fan come to you instead of 21:59.165 --> 22:00.655 to the other bookie? 22:00.660 --> 22:04.620 Why do you deserve to meet this fan and take advantage of him? 22:04.618 --> 22:07.238 What have you got that no other bookie has? 22:07.240 --> 22:09.600 It's a miracle he came to you and not to one of the other 22:09.601 --> 22:10.361 hundred bookies. 22:10.358 --> 22:13.838 Well, it's probably because you actually know something that the 22:13.835 --> 22:16.755 other bookies don't that you can take advantage of. 22:16.759 --> 22:21.429 So in the background here there's another thing that you 22:21.433 --> 22:22.033 knew. 22:22.028 --> 22:25.018 You knew the fan was going to pay off if he lost. 22:25.019 --> 22:28.089 Maybe the other bookies weren't willing to bet with him because 22:28.087 --> 22:30.807 they didn't think he was going to pay off if he lost. 22:30.808 --> 22:33.918 So that's the extra thing that you knew, you knew that he was 22:33.915 --> 22:34.895 going to pay off. 22:34.900 --> 22:37.770 Now it makes sense that he came to you and not to the other 22:37.769 --> 22:40.839 bookies because you actually knew something that they don't. 22:40.838 --> 22:44.758 And so that's how it really works in the market. 22:44.759 --> 22:47.769 You're relying on some piece of information you have that they 22:47.765 --> 22:48.155 don't. 22:48.160 --> 22:50.160 Of course you could be wrong about that information and then 22:50.163 --> 22:51.253 you get into a lot of trouble. 22:51.250 --> 22:53.940 So here in this mortgage what's the thing that you're relying 22:53.944 --> 22:54.174 on? 22:54.170 --> 22:59.500 I'm assuming that the rest of the market is standing by ready, 22:59.500 --> 23:03.410 as I said, to trade, to make interest rate bets with 23:03.412 --> 23:05.112 you, with everybody, 23:05.114 --> 23:09.314 at even odds whether the interest rate's going to go up 23:09.306 --> 23:10.156 or down. 23:10.160 --> 23:12.680 So no matter where you find yourself in the future, 23:12.682 --> 23:15.562 like here, someone is willing to bet at 50/50 odds that it 23:15.558 --> 23:16.718 will go up or down. 23:16.720 --> 23:18.980 Now, this is a year-long bet so you're going to have to take 23:18.982 --> 23:20.252 into account the interest rate. 23:20.250 --> 23:22.680 It's not just going to be 50/50. 23:22.680 --> 23:25.870 You have to discount all the cash flows by the interest rate, 23:25.866 --> 23:29.316 but it's basically a 50/50 bet discounted by the interest rate. 23:29.319 --> 23:33.639 So now what's new here? 23:33.640 --> 23:36.040 Well, you don't know what the cash flows are. 23:36.038 --> 23:39.308 So here's the extra piece of information you're adding. 23:39.308 --> 23:43.968 You know that the mortgage homeowner is rational and is 23:43.968 --> 23:46.468 going to prepay rationally. 23:46.470 --> 23:48.550 So you know what these cash flows are. 23:48.548 --> 23:51.128 Some of the time the mortgage homeowner is going to pay the 8 23:51.132 --> 23:53.112 dollars he owes, some of the time he's just 23:53.112 --> 23:55.832 going to pay 8 dollars, in fact, here is what you know 23:55.832 --> 23:56.902 is going to happen. 23:56.900 --> 24:00.300 Down here where the 1s are he's going to pay his coupon of 8, 24:00.298 --> 24:03.638 pay off the remaining principal and that's the end of it. 24:03.640 --> 24:07.420 So in fact the very first day if interest rates went down the 24:07.423 --> 24:11.023 rational thing to do is to pay off the entire mortgage. 24:11.019 --> 24:13.039 So you know that's going to happen. 24:13.038 --> 24:15.508 Now, that's the extra information you have. 24:15.509 --> 24:18.039 You know what the cash flows are. 24:18.038 --> 24:21.508 If other people knew those cash flows they would figure out the 24:21.505 --> 24:23.065 price of the mortgage too. 24:23.069 --> 24:24.299 It's just they don't know those. 24:24.298 --> 24:29.098 You know the cash flows and you can now figure out that the 24:29.101 --> 24:30.841 price better be 98. 24:30.838 --> 24:35.048 Why--if there's someone else willing to trade with you to buy 24:35.048 --> 24:39.328 that mortgage for more than 98, like why might that happen? 24:39.328 --> 24:42.758 Because someone else might have the erroneous opinion that the 24:42.759 --> 24:45.119 homeowner will never call the mortgage, 24:45.118 --> 24:47.808 will never prepay because he's just going to be asleep all the 24:47.810 --> 24:48.120 time. 24:48.118 --> 24:52.298 So let's say the homeowner's a she. 24:52.298 --> 24:54.678 Let's say the homeowners are all shes. 24:54.680 --> 24:58.080 This other trader thinks they're all asleep and will 24:58.078 --> 24:59.078 never prepay. 24:59.078 --> 25:02.668 You know that these homeowners are all incredibly alert and on 25:02.671 --> 25:05.321 top of things, and so they're going to prepay, 25:05.320 --> 25:06.970 and the value's only 98. 25:06.970 --> 25:10.840 So because of that extra information you know the actual 25:10.836 --> 25:11.746 cash flows. 25:11.750 --> 25:13.240 Sorry, we're here. 25:13.240 --> 25:14.420 You know the cash flows. 25:14.420 --> 25:18.390 You know how to value the bond, and you know it's worth 98. 25:18.390 --> 25:22.010 So, and let's say the other guy's willing to pay you 120 for 25:22.011 --> 25:22.321 it. 25:22.319 --> 25:23.919 So you can sell it short. 25:23.920 --> 25:27.540 You can sell it to the guy short for 120. 25:27.538 --> 25:30.578 That means you owe whatever cash flows the mortgage pays, 25:30.578 --> 25:33.238 and you know what it's going to pay down there. 25:33.240 --> 25:36.650 So how can you lock in your profit? 25:36.650 --> 25:40.770 Well, because with 98 dollars at the very beginning you can 25:40.769 --> 25:42.829 replicate those cash flows. 25:42.828 --> 25:46.118 You can go out into the market and buy those cash flows, 25:46.119 --> 25:49.769 the very cash flows you know the mortgage is going to make. 25:49.769 --> 25:50.879 How can you do that? 25:50.880 --> 25:54.630 Well, because at the first step--remember what this says. 25:54.630 --> 25:58.130 The first step the price is 98. 25:58.130 --> 25:59.920 The interest rate could go up or down. 25:59.920 --> 26:03.980 If the interest rate goes up the payment is 8. 26:03.980 --> 26:07.290 That's the coupon plus 92 something; 26:07.288 --> 26:14.208 92.6, and if it goes down the payment is 8. 26:14.210 --> 26:15.310 So what is this? 26:15.309 --> 26:16.849 The payment is actually 8. 26:16.848 --> 26:18.628 That's the cash flow that has to be made. 26:18.630 --> 26:21.950 The mortgage wasn't called here at the beginning so no matter 26:21.948 --> 26:23.498 what she does, the homeowner, 26:23.496 --> 26:24.986 she's got to pay the 8. 26:24.990 --> 26:27.680 So you know you're going to owe the guy 8 the first period. 26:27.680 --> 26:31.250 Now, down here she's actually going to prepay. 26:31.250 --> 26:34.800 So she's actually going to pay the guy the whole 99.1. 26:34.798 --> 26:37.258 Up here she's not going to prepay. 26:37.259 --> 26:38.919 She's going to hang onto the mortgage. 26:38.920 --> 26:41.170 So what is this 92.6? 26:41.170 --> 26:44.460 This is the present value, assuming those cash flows, 26:44.458 --> 26:46.038 of her future payments. 26:46.038 --> 26:51.188 So if you get 98 dollars you know that 1 half times this-- 26:51.190 --> 26:55.070 this by the way, is 102.6, so 1 half times this 26:55.074 --> 27:01.294 divided by 1.06 plus this thing, remember is 107.11 divided by 27:01.286 --> 27:03.416 1.06 times 1 half. 27:03.420 --> 27:07.440 That's how we got the number 98,98.8 whatever it was. 27:07.440 --> 27:11.550 That's how you calculated 98.8 by backward induction. 27:11.548 --> 27:15.728 You took this value times 1 half discounted plus this value 27:15.733 --> 27:17.613 times 1 half discounted. 27:17.609 --> 27:19.649 That's how you got 98.8. 27:19.650 --> 27:22.130 So therefore, if people are willing to trade 27:22.125 --> 27:24.825 at these prices, in other words what's the price 27:24.830 --> 27:25.350 here? 27:25.348 --> 27:31.168 It's 1 half times 1 over 1.06 and this is 1 half times 1 over 27:31.170 --> 27:31.850 1.06. 27:31.848 --> 27:36.418 You can buy 1 dollar here at a cost of 1 half times 1.06. 27:36.420 --> 27:41.420 You can buy 1 dollar here at a cost of 1 half divided by 1.06. 27:41.420 --> 27:44.790 Paying this amount at the beginning will buy you 1 dollar 27:44.788 --> 27:45.208 here. 27:45.210 --> 27:47.920 Paying this amount at the beginning will buy you 1 dollar 27:47.921 --> 27:48.261 here. 27:48.259 --> 27:55.299 So by paying this amount you can get this payoff, 27:55.301 --> 27:59.411 102.6,100.6, sorry, 100.6. 27:59.410 --> 28:04.460 You can get this 100.6 and you can get 107.11 because precisely 28:04.457 --> 28:09.747 this is the cost that you have to pay for each dollar up here, 28:09.750 --> 28:14.790 and this is the cost that you have to pay here for each dollar 28:14.788 --> 28:15.778 down here. 28:15.779 --> 28:16.519 Why is that? 28:16.519 --> 28:19.829 Because we said the market is standing by recognizing that 28:19.829 --> 28:22.559 these are 50/50 odds that it has to discount. 28:22.558 --> 28:26.078 So that means the whole market is willing to trade you, 28:26.078 --> 28:29.918 will be willing to promise you 1 dollar here in exchange for 28:29.923 --> 28:32.013 this amount of dollars here. 28:32.009 --> 28:35.059 It'll be willing to promise you 1 dollar here in exchange for 28:35.063 --> 28:36.593 this amount of dollars here. 28:36.588 --> 28:41.348 So with this amount of dollars you can buy 8 dollars and have 28:41.353 --> 28:46.043 92.6 leftover and you can also buy 8 dollars here with 99.11 28:46.037 --> 28:47.067 leftover. 28:47.068 --> 28:50.038 So all this together is enough to make the prepayment because 28:50.038 --> 28:51.918 you know she's going to prepay here. 28:51.920 --> 28:52.800 That's what we found out. 28:52.798 --> 29:01.258 She's going to prepay at the first step down. 29:01.259 --> 29:05.839 Oh, what the hell does this say? 29:05.839 --> 29:07.929 She's prepaying just as we said. 29:07.930 --> 29:10.200 At the very first step she's prepaying. 29:10.200 --> 29:12.420 The first interest rate down she's going to prepay. 29:12.420 --> 29:15.260 So we know you're going to have to come up with all this money, 29:15.258 --> 29:16.768 but you've bought it over here. 29:16.769 --> 29:17.789 It cost you this amount. 29:17.789 --> 29:19.059 You bought all that. 29:19.058 --> 29:22.048 If things went up you'd have to pay 8, 29:22.048 --> 29:24.728 which you've also bought, but you've got 92.6 left over 29:24.726 --> 29:27.846 and that's going to be enough to buy everything that comes after 29:27.848 --> 29:31.318 that because the next day you're going to have to buy more stuff, 29:31.318 --> 29:33.468 and so you just keep buying going forward. 29:33.470 --> 29:36.640 So with 98.8 dollars you can buy all the future cash flows 29:36.637 --> 29:40.137 and you've sold the thing so you can make all your payments that 29:40.140 --> 29:43.470 you're obliged to by selling it short and you received 120 to 29:43.474 --> 29:44.424 begin with. 29:44.420 --> 29:47.160 So you locked in 21.2 dollars. 29:47.160 --> 29:49.690 So that's the gist of the whole thing. 29:49.690 --> 29:52.380 So I said it last time not very well. 29:52.380 --> 29:58.110 I said it a little bit better this time, still not 29:58.105 --> 30:02.425 brilliantly, but do you now get it? 30:02.430 --> 30:07.410 Someone ask me a question perhaps? 30:07.410 --> 30:09.170 Is it okay? 30:09.170 --> 30:12.750 So I'll say the punch line again. 30:12.750 --> 30:17.010 The punch line is that if you can find a favorable--so here 30:17.009 --> 30:21.199 what was different about the World Series and the mortgage 30:21.198 --> 30:22.078 example? 30:22.078 --> 30:26.028 In the World Series you took advantage of the fact that there 30:26.034 --> 30:30.054 was a bookie and a Philly fan willing to bet at different odds 30:30.054 --> 30:34.184 on the same thing, and so even though the Philly 30:34.180 --> 30:37.690 fan had a very complicated 7 day bet, 30:37.690 --> 30:43.060 and the other bookies were only willing to make day by day bets, 30:43.058 --> 30:46.648 you could take advantage of that disparity to lock in a 30:46.646 --> 30:50.016 profit for sure, and that's what you should try 30:50.018 --> 30:50.508 to do. 30:50.509 --> 30:54.809 In the market you're probably relying on something else. 30:54.808 --> 30:59.148 You're relying on the fact that you know more about the cash 30:59.152 --> 31:02.172 flows than the rest of the traders do, 31:02.170 --> 31:04.770 the rest of the market does, maybe because you understand 31:04.769 --> 31:06.349 prepayments better than they do. 31:06.348 --> 31:10.448 And so that's why I said it was so important to model 31:10.445 --> 31:11.545 prepayments. 31:11.548 --> 31:14.188 So let's say you understand prepayments better than they do 31:14.190 --> 31:17.060 so you know what the cash flows are going to be much better than 31:17.058 --> 31:17.968 the market does. 31:17.970 --> 31:23.490 You can take advantage of that. 31:23.490 --> 31:25.540 You need now this advantage. 31:25.538 --> 31:27.848 Just like in the World Series you might have needed the 31:27.854 --> 31:30.554 advantage of knowing the Philly fan would actually pay off if he 31:30.554 --> 31:30.944 lost. 31:30.940 --> 31:33.120 So there's that advantage that you have, 31:33.118 --> 31:36.448 and now you have to rely on the fact that the market or the 31:36.453 --> 31:39.313 bookie from then on day by day, year by year, 31:39.313 --> 31:42.723 is willing to make bets at these odds that you can 31:42.717 --> 31:43.687 anticipate. 31:43.690 --> 31:47.590 And so using that you can trade over and over again with the 31:47.589 --> 31:51.029 rest of the market and dynamically replicate what the 31:51.027 --> 31:54.727 payments you have to make by selling the bond short. 31:54.730 --> 31:57.260 So therefore you've locked in your profit because you can 31:57.258 --> 31:59.628 anticipate-- assuming the market stays there 31:59.626 --> 32:03.046 willing to trade at even odds you'll always be able to produce 32:03.046 --> 32:06.016 the cash flows with the 98 dollars you started with to 32:06.017 --> 32:09.317 exactly meet all your payment obligations you've incurred by 32:09.324 --> 32:11.234 selling the mortgage short. 32:11.230 --> 32:13.910 And since you sold it at the beginning for $120 you've made 32:13.913 --> 32:16.973 21 dollars at the beginning and you'll never need those dollars. 32:16.970 --> 32:18.390 You just put them in your pocket. 32:18.390 --> 32:25.350 So that's how to dynamically hedge a winning position to 32:25.354 --> 32:29.284 eliminate the chance of loss. 32:29.278 --> 32:31.628 Of course you still bear a chance of loss. 32:31.630 --> 32:35.140 If you're wrong about the prepayments you're going to lose 32:35.144 --> 32:37.104 money, and if the rest of the market 32:37.098 --> 32:39.988 suddenly stops being there to make these year by year bets on 32:39.994 --> 32:42.604 which way interest rates are going to go you'll also be 32:42.602 --> 32:44.052 subject to a lot of risk. 32:44.048 --> 32:45.598 So you're relying on those two things, 32:45.598 --> 32:48.468 the whole market not collapsing and your being right about 32:48.472 --> 32:50.932 prepayments, but conditional on those two 32:50.931 --> 32:53.121 facts you've locked in your profit. 32:53.118 --> 32:56.688 So you got rid of a bunch of risk, the blind luck of (100, 32:56.686 --> 33:00.126 minus 100) from the roll of the dice of the Phillies. 33:00.130 --> 33:03.440 You got rid of that luck, but you still had to count on 33:03.442 --> 33:06.512 the bookie being there day after day after day, 33:06.509 --> 33:10.039 and in the mortgage case on the fact that you knew how to 33:10.039 --> 33:11.679 predict the prepayments. 33:11.680 --> 33:12.870 Yes? 33:12.868 --> 33:14.458 Student: So in the first example, 33:14.460 --> 33:17.930 what's the point of betting the second time since the maximum 33:17.930 --> 33:20.070 amount of money we can make is 42, 33:20.068 --> 33:22.178 and we already made it in first bet, 33:22.180 --> 33:24.780 so why should I come to you betting? 33:24.778 --> 33:27.718 Prof: I never made 42 on the first bet. 33:27.720 --> 33:30.380 Here at the beginning nothing's happened yet. 33:30.380 --> 33:32.260 The series hasn't been played yet. 33:32.259 --> 33:35.159 So I say to myself I've made this 100-dollar bet with the 33:35.163 --> 33:38.073 Philly fan for the whole series on who's going to win the 33:38.068 --> 33:38.638 series. 33:38.640 --> 33:41.580 That's going to happen 7 days later. 33:41.578 --> 33:45.078 So right now I can anticipate, on average I'll make 42 33:45.078 --> 33:45.738 dollars. 33:45.740 --> 33:47.210 Some of the time I'll lose 100. 33:47.210 --> 33:50.320 Some of the time I'll win 100, but on average I'll come out 42 33:50.323 --> 33:51.143 dollars ahead. 33:51.140 --> 33:54.380 So I've done a smart thing betting with the naive Philly 33:54.382 --> 33:56.802 fan, but I'm still potentially screwed. 33:56.799 --> 33:58.759 I could still lose 100 dollars. 33:58.759 --> 34:01.289 I'll still have to face my investors and tell them I've 34:01.288 --> 34:02.318 lost all your money. 34:02.318 --> 34:06.958 They're going to close me down, and so I don't want to run that 34:06.964 --> 34:07.494 risk. 34:07.490 --> 34:09.640 They don't want me to run that risk and I don't want to run the 34:09.637 --> 34:09.877 risk. 34:09.880 --> 34:12.190 I still want to take advantage of the Philly fans. 34:12.190 --> 34:16.840 So what I do is every game I make another bet against my 34:16.844 --> 34:18.034 inclination. 34:18.030 --> 34:20.540 I bet not on the Phillies, I bet, I mean, 34:20.539 --> 34:22.229 not on the Yankees who I think they're going to win, 34:22.230 --> 34:25.290 I'm betting on the Phillies because I'm undoing part of the 34:25.286 --> 34:27.076 gamble that I did with this guy. 34:27.079 --> 34:30.549 I'm hedging that gamble, but by doing that in a clever 34:30.550 --> 34:33.110 way, changing how much money I bet 34:33.110 --> 34:35.740 each game-- so the night before the first 34:35.742 --> 34:39.452 game I bet 33 dollars to 22 dollars on the Phillies winning. 34:39.449 --> 34:43.439 After the Yankees win the first game I bet 27 dollars to 18 34:43.443 --> 34:46.203 dollars again on the Phillies winning. 34:46.199 --> 34:50.479 So I decrease my bet going up, I increase it going down. 34:50.480 --> 34:55.490 So by doing these clever game by game bets I lock in 42 34:55.487 --> 34:57.247 dollars for sure. 34:57.250 --> 34:58.040 Yes? 34:58.039 --> 35:01.159 Student: Do you actually lock in for sure because say at 35:01.161 --> 35:03.581 the end of the last round you don't actually-- 35:03.579 --> 35:05.689 like that node is going to be the end, right? 35:05.690 --> 35:08.560 You don't actually have two branches coming out of it, 35:08.563 --> 35:09.923 > 35:09.918 --> 35:12.628 favorable outcome and > 35:12.630 --> 35:15.170 Prof: So you have locked it in. 35:15.170 --> 35:16.440 So how could that be? 35:16.440 --> 35:18.640 So it's a good question, but you've overlooked 35:18.641 --> 35:19.231 something. 35:19.230 --> 35:23.510 So her question is the game's going to end after 7 days. 35:23.510 --> 35:26.090 What if I ended with the minus 100? 35:26.090 --> 35:28.310 The Phillies won all four games. 35:28.309 --> 35:30.969 Doesn't it look like I've lost? 35:30.969 --> 35:36.389 So why didn't I lose? 35:36.389 --> 35:38.699 I bet 100 dollars with the original Phillies fan. 35:38.699 --> 35:40.109 I bet 100 dollars on the Yankees. 35:40.110 --> 35:43.940 If the Phillies win all first four games I've got to give the 35:43.940 --> 35:45.090 guy 100 dollars. 35:45.090 --> 35:47.960 Of course I made these side bets with all these bookies on 35:47.956 --> 35:48.456 the way. 35:48.460 --> 35:51.270 So what were the side bets? 35:51.268 --> 35:52.508 Student: Sorry. 35:52.514 --> 35:55.124 My question is the only nodes that are secured, 35:55.119 --> 35:57.719 that are locked in, are nodes with two branches 35:57.724 --> 35:58.974 coming out of it. 35:58.969 --> 36:01.569 There's going to be an end where there's no more branches 36:01.568 --> 36:04.028 coming out of the node and so what's the situation? 36:04.030 --> 36:05.980 Prof: The situation is you're up 42 dollars, 36:05.976 --> 36:07.726 but we're trying to understand why that is. 36:07.730 --> 36:09.810 That's your question, right? 36:09.809 --> 36:13.929 So you're going to be able to answer the question although you 36:13.931 --> 36:15.351 don't think so yet. 36:15.349 --> 36:16.789 So what happened the first time? 36:16.789 --> 36:20.889 The first time the Phillies won, that bet with the original 36:20.889 --> 36:23.009 Philly fan is looking worse. 36:23.010 --> 36:26.640 I had 42 dollars I expected to win. 36:26.639 --> 36:31.259 Now I only expect to win 9 dollars because the series is 36:31.257 --> 36:34.947 turning against me, but the bookie paid me 33 36:34.952 --> 36:35.962 dollars. 36:35.960 --> 36:38.540 So I'm--33 dollars already in my pocket. 36:38.539 --> 36:39.509 This is 9. 36:39.510 --> 36:44.650 Now let's say I got to the end to the last game where this was 36:44.654 --> 36:45.164 a 0. 36:45.159 --> 36:48.219 Let's say this was the last game where I've lost. 36:48.219 --> 36:49.859 Well, this is minus 32. 36:49.860 --> 36:52.060 Let's say this is the last game. 36:52.059 --> 36:54.119 I lose 32 dollars here. 36:54.119 --> 36:58.569 Let's say it was only a two game series and I give up 32 36:58.567 --> 37:02.767 dollars in case the Phillies win the first two games, 37:02.771 --> 37:03.581 right? 37:03.579 --> 37:05.949 So you're saying, "I'm down 32 dollars. 37:05.949 --> 37:07.579 That's the last node. 37:07.579 --> 37:09.989 How can I possibly be up 42?" 37:09.989 --> 37:11.919 Well, so what's the answer? 37:11.920 --> 37:13.670 Student: Because I hedged it with the bookies. 37:13.670 --> 37:14.260 Prof: What? 37:14.260 --> 37:15.590 Student: Because I hedged it with the bookies. 37:15.590 --> 37:18.280 Prof: Right, because I bet on the first game 37:18.284 --> 37:20.554 and the first bookie gave me 33 dollars. 37:20.550 --> 37:24.950 Then the second game I'm going to bet with the bookies and I'm 37:24.945 --> 37:27.175 going to bet minus 27 and 41. 37:27.179 --> 37:31.489 So although I now owe the Philly fan 33 dollars here, 37:31.485 --> 37:35.955 I made 41 dollars with the second bookie and 33 dollars 37:35.956 --> 37:38.106 with the first bookie. 37:38.110 --> 37:43.740 That's 74 dollars and I pay 32 to the original Philly fan so 37:43.739 --> 37:46.219 I'm still up 42 dollars. 37:46.219 --> 37:49.109 So it sounds like you have to do a lot of arithmetic. 37:49.110 --> 37:50.540 You don't have to do any arithmetic. 37:50.539 --> 37:53.239 I knew I was going to end up 42 dollars up. 37:53.239 --> 37:53.979 How is that? 37:53.980 --> 37:57.990 Because every bet I said let's look at the number here wherever 37:57.985 --> 37:59.985 we are on the tree like here. 37:59.989 --> 38:03.869 At this node after all the stuff that's happened before the 38:03.865 --> 38:08.005 bet I'm going to make with the bookie that one game is going to 38:08.010 --> 38:11.420 be enough to turn this random thing into 64/64. 38:11.420 --> 38:14.690 So I always take what happens further down the tree and turn 38:14.688 --> 38:17.788 it into what's happened at the beginning of the tree, 38:17.789 --> 38:21.749 but then I work it all the way back and it's 42. 38:21.750 --> 38:25.190 So that means, see this thing was 64 and 9, 38:25.190 --> 38:30.350 but I made a bet 22/33 with the bookie to turn it into 42/42. 38:30.349 --> 38:32.119 This thing was 64. 38:32.119 --> 38:37.229 I made a bet to turn this into 64/64, but with the 22 here 38:37.226 --> 38:38.746 that's also 42. 38:38.750 --> 38:43.500 So no matter how far I go on the tree I'm still exactly 42 38:43.501 --> 38:44.171 ahead. 38:44.170 --> 38:45.240 Any other questions? 38:45.239 --> 38:46.259 Yes? 38:46.260 --> 38:50.340 Student: In the example that we just did for the 38:50.335 --> 38:54.785 mortgage, could you explain why we are gaining 8 dollars? 38:54.789 --> 38:58.109 I thought we'd be losing 8 dollars every branch. 38:58.108 --> 38:59.448 It's below the... 38:59.449 --> 39:02.419 Prof: Sorry, what am I doing? 39:02.420 --> 39:06.440 Student: In each branch we've said that we're gaining 8 39:06.440 --> 39:07.430 dollars 92.6. 39:07.429 --> 39:11.889 Aren't we doing negative 8 92.6? 39:11.889 --> 39:15.149 Prof: So her question is in this case the mortgage--so 39:15.152 --> 39:18.362 now we're moving from the World Series to more real life. 39:18.360 --> 39:22.460 In this case I know the cash flows from this mortgage are 39:22.456 --> 39:24.666 worth 98.8, but somebody else not 39:24.666 --> 39:27.716 understanding the homeowner and how rational she is, 39:27.719 --> 39:31.189 someone else is willing to pay me 120 for that mortgage. 39:31.190 --> 39:36.660 So I sell the mortgage for 120 and I take 21 dollars and put it 39:36.664 --> 39:37.994 in my pocket. 39:37.989 --> 39:41.109 Now I've got 98.8 dollars left. 39:41.110 --> 39:43.950 So that's money now I have to do something active with. 39:43.949 --> 39:45.509 What do I have to do? 39:45.510 --> 39:47.820 I owe this guy a lot of payments. 39:47.820 --> 39:50.100 Every time she, let's say it's you, 39:50.097 --> 39:52.167 makes your payment, her payment, 39:52.172 --> 39:55.122 in the market I've sold the thing short. 39:55.119 --> 39:57.939 That means I owe--I've promised to deliver whatever that 39:57.936 --> 39:58.956 mortgage delivers. 39:58.960 --> 40:00.470 That's what it means to sell it short. 40:00.469 --> 40:03.109 So I've got to deliver the 8 here, say. 40:03.110 --> 40:07.090 I've got to deliver the 8 here, and here she's prepaying. 40:07.090 --> 40:10.800 I have to deliver the whole 107, right? 40:10.800 --> 40:16.360 So with 98.8 dollars I can afford to buy 100.6 dollars in 40:16.358 --> 40:20.128 this state and 107.11 in this state. 40:20.130 --> 40:22.580 How do I know that, because that's where the 98.8 40:22.577 --> 40:23.187 came from. 40:23.190 --> 40:27.280 It said this total amount 100.6 times the price, 40:27.280 --> 40:29.930 which is the odds of it happening discounted, 40:29.929 --> 40:34.719 that's the price plus the 107.11 times its price, 40:34.719 --> 40:38.789 the odds it happens discounted, is exactly 98.8. 40:38.789 --> 40:40.599 That's where I got 98.8. 40:40.599 --> 40:43.459 So therefore with 98.8 dollars, remember the bookie's willing 40:43.456 --> 40:46.366 to go either way, with 98.8 dollars I can buy all 40:46.373 --> 40:49.103 these cash flows and all these cash flows. 40:49.099 --> 40:50.249 So what do I do? 40:50.250 --> 40:56.290 With the 107.1 down here I pay the 8, her coupon, 40:56.291 --> 40:59.691 and her 99.11 prepayment. 40:59.690 --> 41:02.830 I pay it all to that guy I sold the bond short to and I've kept 41:02.831 --> 41:03.491 my promise. 41:03.489 --> 41:04.969 That's the end of the mortgage. 41:04.969 --> 41:10.409 Up here the mortgage only delivered 8 so I only owe him 8, 41:10.409 --> 41:13.559 but I've got another 92.6 left. 41:13.559 --> 41:15.549 Why is that exactly what I need? 41:15.550 --> 41:18.970 Because from here on I have to use that money to buy the future 41:18.967 --> 41:22.157 cash flows because she's going to continue to pay her 8 and 41:22.164 --> 41:24.154 maybe eventually it will prepay. 41:24.150 --> 41:27.100 But by induction, by repeating this step over and 41:27.099 --> 41:30.789 over again I can always make the payment and have the present 41:30.788 --> 41:34.478 value left over to afford to buy the future payments plus the 41:34.476 --> 41:36.316 present value left over. 41:36.320 --> 41:39.170 So by induction I'm always making the coupon. 41:39.170 --> 41:42.820 Like in this first step I make the coupon payment plus I've got 41:42.818 --> 41:45.878 enough money left over to match the present value. 41:45.880 --> 41:47.970 So that means as I go forward, and forward, 41:47.969 --> 41:51.009 and forward I keep making the payment and matching the present 41:51.005 --> 41:52.345 value that's left over. 41:52.349 --> 41:55.779 By the time I get to the 30th year I've made the coupon 41:55.775 --> 41:56.405 payment. 41:56.409 --> 41:58.829 The present value left over is 0 because there isn't going to 41:58.829 --> 41:59.999 be any payments after that. 42:00.000 --> 42:02.880 So I've made all the payments. 42:02.880 --> 42:05.400 Student: Could you do this one year ahead as well? 42:05.400 --> 42:07.450 Prof: Yes, I could. 42:07.449 --> 42:12.659 So I was hoping you wouldn't ask me that because it's--so 42:12.664 --> 42:15.834 what happens in the next period. 42:15.829 --> 42:19.719 So here we know that she didn't pay. 42:19.719 --> 42:20.719 She didn't prepay. 42:20.719 --> 42:25.549 The present value is 92.6, so we're here, 42:25.554 --> 42:26.404 92.6. 42:26.400 --> 42:27.970 Now, she didn't pay. 42:27.969 --> 42:32.529 Let's look at what she's doing, whether she's prepaying or not. 42:32.530 --> 42:34.830 So this is why I did the second graph. 42:34.829 --> 42:39.429 So if the interest rates go up the first time and then back 42:39.434 --> 42:42.534 down she's still not going to prepay. 42:42.530 --> 42:52.390 So we can now go up and see--oops. 42:52.389 --> 42:53.489 Here we are. 42:53.489 --> 42:56.499 So we can go up now and see what's happening here. 42:56.500 --> 42:57.890 It's 92.65. 42:57.889 --> 43:14.099 You go up to 83.95 or down to 97.2. 43:14.099 --> 43:15.309 97.17. 43:15.309 --> 43:16.929 So what does that mean? 43:16.929 --> 43:22.429 So with 98.8 dollars I bought, I'm the guy selling the thing 43:22.431 --> 43:28.401 short, you're the smart rational pre-payer, so you paid 8 dollars 43:28.400 --> 43:29.240 here. 43:29.239 --> 43:32.079 That means because I've sold the mortgage short and you've 43:32.077 --> 43:35.067 made that 8-dollar payment I owe the guy the 8-dollar payment 43:35.065 --> 43:36.805 just as if he had the mortgage. 43:36.809 --> 43:38.809 So I've got to come up with the 8 dollars. 43:38.809 --> 43:41.199 Down here if things had moved here you would have paid 8 43:41.197 --> 43:43.367 dollars plus the whole prepayment so you would have 43:43.367 --> 43:44.277 paid 107 dollars. 43:44.280 --> 43:50.720 But with 98.8 I can buy 8 92.6 up here and 8 99.11 down here. 43:50.719 --> 43:53.969 So I've bought all the cash flows I need to down here. 43:53.969 --> 43:55.699 I hand them over to the guy and I say, 43:55.699 --> 43:57.769 "You see, I kept my promise to deliver 43:57.769 --> 44:00.879 whatever that mortgage did, and I knew that she was going 44:00.882 --> 44:03.192 to prepay, I knew how smart she was and so 44:03.186 --> 44:04.336 I've kept my promise. 44:04.340 --> 44:05.560 You didn't think she'd prepay. 44:05.559 --> 44:06.329 I knew it. 44:06.329 --> 44:08.049 I've kept my promise." 44:08.050 --> 44:12.120 And then up here I've got the 8 dollars so I can make the 44:12.123 --> 44:13.583 payment, match it. 44:13.579 --> 44:16.719 I've got 92.6 left, now I don't give it to the guy 44:16.715 --> 44:19.015 because the payment's only been 8. 44:19.018 --> 44:25.488 I take the 92.6 and with the 92.6 that happens to equal this 44:25.490 --> 44:29.770 plus 8 times 1 half divided by 1.06, 44:29.768 --> 44:33.738 and it equals, also, this plus 8 times 1 half 44:33.744 --> 44:35.374 divided by 1.06. 44:35.369 --> 44:38.939 Not by 1.06, by the interest rate here which 44:38.942 --> 44:43.512 happens to be--you have to figure out what that interest 44:43.512 --> 44:45.342 rate was up there. 44:45.340 --> 44:50.610 So the interest rate went up to--it started at 6 percent and 44:50.614 --> 44:52.944 went up to 0732 percent. 44:52.940 --> 44:57.870 So this is .0732. 44:57.869 --> 45:00.819 How did this number 92.6 get calculated? 45:00.820 --> 45:03.210 It was exactly take the payment of 8, 45:03.210 --> 45:05.250 take the present value of what's left, 45:05.250 --> 45:10.150 multiply by 1 half and discount it at the interest rate here 45:10.150 --> 45:15.050 plus 1 half times this payment of 8 plus the value of what's 45:15.050 --> 45:18.540 left discounted by that interest rate. 45:18.539 --> 45:22.879 So this number 92.6 allows me to buy this and to buy that. 45:22.880 --> 45:25.700 So therefore, if things went up again I could 45:25.697 --> 45:29.607 make your 8-dollar payment that you make to the bank I have to 45:29.606 --> 45:33.446 make to the other guy because I sold the mortgage short. 45:33.449 --> 45:34.879 I've got the money to pay him. 45:34.880 --> 45:37.730 I've also got the money to pay him if things had gone down 45:37.730 --> 45:38.630 again 8 dollars. 45:38.630 --> 45:42.240 And I've got more cash in my pocket, this and this in the two 45:42.237 --> 45:42.717 cases. 45:42.719 --> 45:46.419 With this money I can continue to buy the future payments that 45:46.422 --> 45:47.822 you're going to make. 45:47.820 --> 45:53.400 So I always have enough money to make whatever payments it is 45:53.398 --> 45:56.838 that I have to, to keep my promise. 45:56.840 --> 45:57.880 Yep? 45:57.880 --> 46:02.200 Student: Is it sort of preferable from your point of 46:02.202 --> 46:06.522 view that the interest rate go up because your theory about 46:06.523 --> 46:10.253 prepayments isn't going to be put to the test? 46:10.250 --> 46:11.930 Prof: Well, don't forget, 46:11.927 --> 46:14.467 I'm locking in my 21 dollar profit for sure, 46:14.469 --> 46:17.919 so no matter what happens I'm going to just be able to make 46:17.923 --> 46:19.953 all the payments no matter what. 46:19.949 --> 46:22.339 Student: But now it seems that your theory about 46:22.338 --> 46:23.178 prepayments is... 46:23.179 --> 46:24.699 Prof: Correct, yes. 46:24.699 --> 46:28.499 Student: So it's like ambiguity, right? 46:28.500 --> 46:33.050 Not just the rest is like ambiguity, so it's sort of 46:33.050 --> 46:34.390 preferable... 46:34.389 --> 46:35.209 Prof: So good. 46:35.210 --> 46:37.540 So let me repeat his question, if he'll forgive me, 46:37.541 --> 46:38.941 in slightly different words. 46:38.940 --> 46:42.950 He's saying that I was able to take--this is what my hedge fund 46:42.945 --> 46:44.105 does, literally. 46:44.110 --> 46:46.910 We think we know better what the prepayments are going to be 46:46.914 --> 46:50.044 than the rest of the market, so we're willing to take on a 46:50.039 --> 46:52.779 commitment-- someone will pay us 120 dollars. 46:52.780 --> 46:58.830 We think 98 dollars are enough to buy all the promises that 46:58.833 --> 47:03.533 we've made by selling the thing short, say. 47:03.530 --> 47:04.460 We could be wrong. 47:04.460 --> 47:06.210 We have been wrong. 47:06.210 --> 47:08.710 The last two years we made a lot of mistakes. 47:08.710 --> 47:11.560 So you're saying, the question is, 47:11.559 --> 47:15.879 the reason why the guy is willing to pay so much is 47:15.876 --> 47:21.486 because down here he thinks he's going to get a lot of money. 47:21.489 --> 47:24.449 The interest rates are going down so, discounting at a lower 47:24.454 --> 47:27.124 rate, the mortgage is going to be worth a lot more. 47:27.119 --> 47:29.669 We don't think it's going to be more because we're expecting 47:29.672 --> 47:30.282 prepayments. 47:30.280 --> 47:33.550 Up here it's sort of obvious that there aren't going to be 47:33.554 --> 47:34.594 any prepayments. 47:34.590 --> 47:37.840 So if the world really goes, interest rates going up, 47:37.838 --> 47:41.588 then our theory of prepayments never gets put to the test and 47:41.585 --> 47:43.205 so we just make money. 47:43.210 --> 47:46.340 So our theory never gets put to the test. 47:46.340 --> 47:50.050 If interest rates go down then our theory's been really put to 47:50.047 --> 47:53.267 the test and we have to see whether she's really smart 47:53.268 --> 47:54.968 enough to prepay or not. 47:54.969 --> 47:55.669 So you're right. 47:55.670 --> 47:56.770 That's absolutely true, yep. 47:56.768 --> 48:00.058 But in any case we're going to assume our theory is correct and 48:00.061 --> 48:03.091 hedge in such a way so that if our theory is correct about 48:03.090 --> 48:06.010 prepayments we will exactly be in a position to keep our 48:06.010 --> 48:08.720 promises without ever running a risk of loss. 48:08.719 --> 48:09.699 Yes? 48:09.699 --> 48:12.369 Student: So with the World Series example. 48:12.369 --> 48:16.319 So let's say there are two situations where the Phillies 48:16.322 --> 48:19.992 win in 4 games or they win the series in 5 games. 48:19.989 --> 48:26.759 Either way you lose 100 dollars and you win 4 single game bets, 48:26.760 --> 48:31.120 but in one of the situations you've lost one single game bet 48:31.117 --> 48:34.957 because the Phillies won in 5 games rather than 4. 48:34.960 --> 48:38.250 How do the values reconcile? 48:38.250 --> 48:41.440 Prof: Because it couldn't have happened that 48:41.438 --> 48:44.498 there were four straight wins and then a loss. 48:44.500 --> 48:47.240 The series would have been over, right? 48:47.239 --> 48:53.619 So his question is I've asserted that no matter what way 48:53.619 --> 49:01.159 things go I'm going to end up 42 dollars ahead no matter what, 49:01.159 --> 49:04.499 and he's saying that's great, but he's a little skeptical it 49:04.496 --> 49:07.776 could be true because I'm always betting against this other 49:07.777 --> 49:08.397 bookie. 49:08.400 --> 49:13.150 I'm always betting on the Phillies with the other bookie, 49:13.153 --> 49:17.233 and so in one case the Yankees win 4 in a row. 49:17.230 --> 49:21.360 I lose 4 bets with the other bookie, with the bookie I'm 49:21.358 --> 49:25.258 betting on the Phillies, so I've had to take 4 losing 49:25.262 --> 49:27.592 bets with the other bookie. 49:27.590 --> 49:30.940 In the case where the Yankees win three lose one and then win 49:30.942 --> 49:34.242 in the end I've again lost 4 bets with the other bookie, 49:34.239 --> 49:36.219 but I've actually won a bet with the other bookie. 49:36.219 --> 49:39.539 So it seems like somehow things are better for me, 49:39.539 --> 49:43.739 so it seems unbelievable that I'm still at 42 in both cases. 49:43.739 --> 49:44.899 That's your question, right? 49:44.900 --> 49:54.230 So the reason is that my bets are changing. 49:54.230 --> 49:57.930 I don't make the same last bet. 49:57.929 --> 50:00.969 So the first three bets there's a Yankee win, 50:00.969 --> 50:03.699 Yankee win, Yankee win, each of those nights I'm making 50:03.699 --> 50:05.569 the same bet in the same scenario, 50:05.570 --> 50:10.750 but when the Phillies lose and then win again I've changed my 50:10.746 --> 50:15.176 bet, so I'm still going to end 50:15.177 --> 50:22.697 up--so the Yankees lost the, your scenario was the Yankees 50:22.699 --> 50:26.659 lost the 4th game, so that outcome is different 50:26.655 --> 50:30.865 from what it would have been in the first scenario because the 50:30.871 --> 50:33.291 Yankees lost instead of winning. 50:33.289 --> 50:37.079 So the bookie bet went the other way. 50:37.079 --> 50:41.559 But now the last bet I made is not the same bet as I would have 50:41.563 --> 50:43.303 made the night before. 50:43.300 --> 50:46.690 It's a different bet so that it just ends up with a different 50:46.692 --> 50:50.142 direction than the 4th bet plus this different 5th bet it adds 50:50.141 --> 50:53.421 back up to what would have happened had the Yankees won the 50:53.422 --> 50:54.442 fourth time. 50:54.440 --> 50:56.330 I don't know if that helps, but that's what happens. 50:56.329 --> 51:00.719 And the proof that it happens is just by induction. 51:00.719 --> 51:06.319 I gave a proof and you're saying the proof is amazingly 51:06.322 --> 51:07.052 good. 51:07.050 --> 51:08.660 Remember, the proof is very simple. 51:08.659 --> 51:11.729 You always make a bet so that no matter what happens your 51:11.731 --> 51:14.751 expectation is back to where it was the night before. 51:14.750 --> 51:17.770 And if you do that you have to end up--that's the proof that 51:17.766 --> 51:20.216 you've always locked in 42 from the beginning. 51:20.219 --> 51:22.839 And so you can ask a thousand questions like that and it's 51:22.842 --> 51:24.822 always going to be a complicated answer, 51:24.820 --> 51:29.390 but there has to be an answer because we've proved that there 51:29.387 --> 51:30.147 was one. 51:30.150 --> 51:31.990 Any other questions about this? 51:31.989 --> 51:32.909 Yes? 51:32.909 --> 51:34.529 Student: So with the mortgage example 51:34.534 --> 51:36.504 you're hedging against the interest rate going up. 51:36.500 --> 51:40.920 Prof: So with the mortgage example you're hedging 51:40.916 --> 51:43.966 against the interest rate going--no. 51:43.969 --> 51:47.929 When the interest rate goes up you actually pay less. 51:47.929 --> 51:49.569 You've made a promise. 51:49.570 --> 51:55.970 Remember, the price as the interest rate goes up--where's 51:55.965 --> 51:58.475 the mortgage, sorry. 51:58.480 --> 52:01.580 The value of the non-call, the value of the call, 52:01.579 --> 52:05.259 so as the interest rate goes up you owe less money because 52:05.262 --> 52:08.042 you're promising to deliver this stuff. 52:08.039 --> 52:10.649 So the interest rate's up, the mortgage you've sold short 52:10.648 --> 52:14.018 is less and less valuable, so that's the good scenario for 52:14.018 --> 52:15.978 you, but if the interest rate goes 52:15.981 --> 52:19.181 down that's the scenario where you could get into trouble. 52:19.179 --> 52:23.449 So you're trying to protect against the interest rate going 52:23.449 --> 52:26.829 down, and in order to do that you 52:26.826 --> 52:32.706 have to give money up in case the interest rate goes up. 52:32.710 --> 52:36.810 So now I want to--any other questions about this? 52:36.809 --> 52:42.709 It's quite an ingenious thing, I think. 52:42.710 --> 52:44.160 So we're going to have one more step of this, 52:44.159 --> 52:47.289 but this is the high watermark of the standard, 52:47.289 --> 52:51.379 you know, the finance guys who made finance such an important 52:51.380 --> 52:54.590 subject, and the rational expectations 52:54.594 --> 52:56.214 school of finance. 52:56.210 --> 52:58.460 This is kind of the most clever thing that they did. 52:58.460 --> 53:02.880 So this kind of reasoning, bookies maybe knew it for a 53:02.876 --> 53:05.646 long time, but Black-Scholes in 1973 53:05.652 --> 53:09.512 started this kind of thinking and then lots of hedge funds 53:09.512 --> 53:12.292 started imitating it including my own. 53:12.289 --> 53:16.239 So I certainly didn't invent this idea of dynamic hedging. 53:16.239 --> 53:20.469 Now, let's take a step back. 53:20.469 --> 53:23.509 The crucial idea is marking to market. 53:23.510 --> 53:26.410 The crucial idea is that you want to hedge something through 53:26.407 --> 53:29.547 millions of scenarios and really the outcome you won't know until 53:29.550 --> 53:30.140 the end. 53:30.139 --> 53:35.389 What you should do is hedge at each step of the way your mark 53:35.393 --> 53:36.973 to market value. 53:36.969 --> 53:39.489 So you only have to do things a step at a time. 53:39.489 --> 53:42.319 There's an exponentially growing number of paths, 53:42.318 --> 53:45.378 but there are only two possibilities from today until 53:45.382 --> 53:46.152 tomorrow. 53:46.150 --> 53:48.270 So it's very simple to hedge two things. 53:48.268 --> 53:52.198 It seems incredibly complicated to hedge 2 to the 100 things, 53:52.195 --> 53:55.595 2 to the 10th is a very big number, 2 is a very small 53:55.599 --> 53:56.319 number. 53:56.320 --> 53:57.520 You can hedge two things. 53:57.518 --> 54:00.658 It may be very complicated to hedge 2 to the 10 things, 54:00.657 --> 54:03.617 but you only have to hedge 2 things on every day. 54:03.619 --> 54:05.959 So once you've realized that, hedging is actually quite a 54:05.963 --> 54:06.763 simple operation. 54:06.760 --> 54:10.200 You don't have to do it by trading in the interest rate 54:10.201 --> 54:11.541 derivatives market. 54:11.539 --> 54:13.619 You could trade in a simpler market. 54:13.619 --> 54:18.409 Suppose, for example, that you knew that there were 54:18.405 --> 54:22.805 bonds being traded, let's say 30-year bonds. 54:22.809 --> 54:29.869 So let's just take a look at a 30-year bond here, 54:29.869 --> 54:35.899 callable bond, so this is exactly the same 54:35.898 --> 54:37.368 thing. 54:37.369 --> 54:39.969 This is the 30-year bond under the same circumstance. 54:39.969 --> 54:43.699 This is a 9 percent bond, coupon bond, 54:43.699 --> 54:49.349 and it's starting with the same 6 percent interest rates, 54:49.347 --> 54:53.277 exactly the same process as before. 54:53.280 --> 54:56.050 So the interest rates, those are the same interest 54:56.050 --> 54:57.070 rates as before. 54:57.070 --> 54:58.250 So here's the bond. 54:58.250 --> 55:04.480 Now, the bond starts at 140 dollars, can go to 121 dollars 55:04.481 --> 55:06.671 or to 159 dollars. 55:06.670 --> 55:16.210 Remember this thing that you're trying to hedge goes from 98.8 55:16.206 --> 55:19.956 to 100.6 or to 107.11. 55:19.960 --> 55:24.700 So what you need to do now is--you've promised to make 55:24.701 --> 55:28.461 deliveries in the future, very complicated, 55:28.458 --> 55:33.198 but you know you can summarize that whole future. 55:33.199 --> 55:36.879 As long as you've got 100.6 dollars in the up state and 55:36.876 --> 55:41.026 107.11 dollars in the down state after the very first step you 55:41.027 --> 55:44.157 will have hedged your mortgage obligation. 55:44.159 --> 55:47.629 But you could just buy the right percentage of bonds and be 55:47.632 --> 55:49.792 able to accomplish the same thing. 55:49.789 --> 55:51.029 So why is that? 55:51.030 --> 55:52.640 Well, there's a gap here and here. 55:52.639 --> 55:56.599 You owe less money here than you do down there. 55:56.599 --> 56:00.049 So the bond is worth less up here than down here. 56:00.050 --> 56:04.760 In fact this bond is worth 38 dollars more here than here. 56:04.760 --> 56:07.180 So the gap in the bond is 38. 56:07.179 --> 56:17.649 The gap in here is something like 6.5. 56:17.650 --> 56:21.040 So this is like 1 sixth. 56:21.039 --> 56:27.039 So as long as you held 1 sixth of the bond and something that 56:27.039 --> 56:33.039 paid the right constant amount in each case you would get the 56:33.039 --> 56:36.239 same payments here and here. 56:36.239 --> 56:37.879 So what is the right constant amount? 56:37.880 --> 56:42.780 1 sixth of the bond, by the way, would cost you 56:42.782 --> 56:48.222 22,24, 23, about 23 dollars, 1 sixth of the bond. 56:48.219 --> 56:51.519 So if you took the other 98.8 dollars and put it to the 1 year 56:51.519 --> 56:54.199 Treasury, I claim it's obvious--so take 1 56:54.197 --> 57:00.987 sixth of the-- so 1 sixth of 30 year 9 percent 57:00.994 --> 57:02.124 bond. 57:02.119 --> 57:08.929 That's 1 sixth of this thing, which costs you what? 57:08.929 --> 57:12.139 It cost 23 dollars. 57:12.139 --> 57:13.609 The price of that is 23 dollars. 57:13.610 --> 57:24.920 Now, combine that with 75.8 dollars of 1 year Treasury, 57:24.922 --> 57:30.792 1 year 6 percent Treasury. 57:30.789 --> 57:32.569 So hold those two things. 57:32.570 --> 57:37.390 I claim that'll pay you 100.6 here and 107.11 here. 57:37.389 --> 57:40.409 How do I know that without having to even calculate 57:40.405 --> 57:41.305 anything out? 57:41.309 --> 57:44.249 Well, the Treasury's going to pay the same amount here and 57:44.248 --> 57:44.608 here. 57:44.610 --> 57:45.960 It's going to be a constant. 57:45.960 --> 57:49.990 This thing, since its gap is 38 dollars and I only hold 1 sixth 57:49.985 --> 57:52.345 of it, it's going to pay me exactly 6 57:52.349 --> 57:55.789 and 1 half more down here than it pays here because the gap 57:55.789 --> 57:58.279 instead of 38 will be 1 sixth of that, 57:58.280 --> 57:59.730 which will be 6 and 1 half. 57:59.730 --> 58:01.830 So the gap will exactly match that. 58:01.829 --> 58:05.079 So added that to the Treasury's constant payments I'm going to 58:05.083 --> 58:08.133 get something over here and 6 and 1 half dollars more than 58:08.126 --> 58:09.136 that down here. 58:09.139 --> 58:11.189 But since I've paid the same amount, 58:11.190 --> 58:15.890 98.8, it's going to have to be that that constant was exactly 58:15.889 --> 58:19.179 right to make this be 100.6 and 107.11, 58:19.179 --> 58:22.199 because both the 1 year Treasury and the 30 year bond 58:22.201 --> 58:25.571 were calculated by discounting whatever their payments were 58:25.570 --> 58:27.780 here and here by the same prices, 58:27.780 --> 58:29.430 this and this. 58:29.429 --> 58:31.309 So there's a second way. 58:31.309 --> 58:36.639 Instead of trading in the derivatives market I could just 58:36.639 --> 58:42.729 hold a combination of two bonds, the 1 year Treasury and the 30 58:42.726 --> 58:46.316 year 9 percent bond, and I'd still be perfectly 58:46.317 --> 58:46.797 hedged. 58:46.800 --> 58:49.530 But then when I got to here I'd want to change my mix. 58:49.530 --> 58:52.360 I'd want to hold a different 1-year bond at a different 58:52.358 --> 58:55.288 interest rate, so I have to buy a new 1-year 58:55.286 --> 58:59.706 bond and a different amount of the 30-year non-callable bond, 58:59.710 --> 59:01.970 9 percent bond, but I could always reproduce 59:01.965 --> 59:03.325 these same two payments. 59:03.329 --> 59:06.659 So I don't have to hedge by holding derivatives, 59:06.664 --> 59:10.714 interest rate derivatives, I can hedge by holding standard 59:10.710 --> 59:11.420 bonds. 59:11.420 --> 59:15.940 And I can tell at a glance how much of the longer term bond to 59:15.940 --> 59:20.020 hold because I just have to match this gap in the prices 59:20.016 --> 59:24.536 since the 1 year bond is always going to pay the same thing up 59:24.538 --> 59:25.648 and down. 59:25.650 --> 59:32.580 Did that go too fast? 59:32.579 --> 59:33.559 Yeah? 59:33.559 --> 59:34.839 Student: I know it's just 59:34.844 --> 59:36.054 like> 59:36.045 --> 59:38.735 but how did you figure out how much of the 1-year bond to buy? 59:38.739 --> 59:44.359 Prof: So I want to hold 1 sixth of the 30 year bond, 59:44.360 --> 59:49.310 which cost me 140, so 1 sixth of that is going to 59:49.306 --> 59:51.466 cost me 23 dollars. 59:51.469 --> 59:57.039 All right, now I want to also hold a 1 year Treasury so that 59:57.036 --> 1:00:02.786 the payment 1 sixth of 121 at the top plus whatever the 1 year 1:00:02.793 --> 1:00:07.983 Treasury pays me is equal to 100.6 and such that 1 sixth 1:00:07.983 --> 1:00:12.643 times this value at the bottom, oh, sorry. 1:00:12.639 --> 1:00:15.249 It's 1 sixth this coupon is paying. 1:00:15.250 --> 1:00:17.270 Actually it wasn't 121.6. 1:00:17.269 --> 1:00:19.219 It's a 9 percent bond. 1:00:19.219 --> 1:00:21.589 I forgot its coupon, very bad of me. 1:00:21.590 --> 1:00:24.320 It won't change the numbers, but. 1:00:24.320 --> 1:00:26.650 So it's paying--I didn't write it down. 1:00:26.650 --> 1:00:28.570 So this thing is not paying 121. 1:00:28.570 --> 1:00:29.750 It's also paying the coupon. 1:00:29.750 --> 1:00:31.420 This is the present value of what's left. 1:00:31.420 --> 1:00:32.590 So it's paying 9 plus that. 1:00:32.590 --> 1:00:35.440 So it's 130 and here it's 168. 1:00:35.440 --> 1:00:37.290 The gap is still 38. 1:00:37.289 --> 1:00:43.779 So I want 1 sixth of 130. 1:00:43.780 --> 1:00:52.360 So 1 sixth of 130 costs--it's still the same number. 1:00:52.360 --> 1:00:53.470 It's 22. 1:00:53.469 --> 1:00:56.019 I did bad arithmetic to begin with so it's still 22. 1:00:56.018 --> 1:00:59.928 No, sorry, it's 1 sixth of this thing which is 140, 1:00:59.927 --> 1:01:01.487 so it's 23, right. 1:01:01.489 --> 1:01:01.979 Sorry. 1:01:01.980 --> 1:01:02.540 That didn't change. 1:01:02.539 --> 1:01:07.049 So the payment here is 130 and the payment here is 168. 1:01:07.050 --> 1:01:12.010 So 1 sixth of 130, that's the up payment, 1:01:12.012 --> 1:01:15.612 and this is 1 sixth of 168. 1:01:15.610 --> 1:01:21.090 Now, I want that plus whatever the 1 year Treasury gives me to 1:01:21.088 --> 1:01:24.948 equal 100.6 up here and 107.11 down here. 1:01:24.949 --> 1:01:27.709 That way I would have replicated things and hedged 1:01:27.706 --> 1:01:29.166 myself perfectly, right? 1:01:29.170 --> 1:01:31.640 But I don't have to figure out what that 1 year Treasury is. 1:01:31.639 --> 1:01:32.739 I think it's obvious. 1:01:32.739 --> 1:01:38.009 All I have to do is say subtract 23 from 98.8, 1:01:38.010 --> 1:01:42.050 so I just have to buy 75.8 dollars worth of the 1 year 1:01:42.050 --> 1:01:46.320 Treasury and that'll give me exactly the right payoff. 1:01:46.320 --> 1:01:47.720 Why is that? 1:01:47.719 --> 1:01:57.459 Because this number 140.93 divided by 6 is exactly this 1:01:57.460 --> 1:02:06.160 evaluated at this price, 1 half over 1.06 times this, 1:02:06.164 --> 1:02:12.194 this is the price 1 half times 1 over 1.06. 1:02:12.190 --> 1:02:17.450 All right, so that bond which cost 140.53 how did I get 140.53 1:02:17.449 --> 1:02:18.139 there? 1:02:18.139 --> 1:02:20.879 I took the price of 1 dollar in the up state, 1:02:20.880 --> 1:02:23.350 which is it's 50 percent likely discounted, 1:02:23.349 --> 1:02:27.199 multiplying by its payoff 130, plus 1 half discounted 1:02:27.195 --> 1:02:29.705 multiplied by the payoff of 168. 1:02:29.710 --> 1:02:32.550 This is 168 because it's a 9 percent bond. 1:02:32.550 --> 1:02:37.270 So it's 1 sixth times 1 sixth--so if I get a sixth of 1:02:37.268 --> 1:02:42.438 the bond it'll cost me 140.43 divided by 6 which is 23. 1:02:42.440 --> 1:02:45.850 So what about the 1 year Treasury? 1:02:45.849 --> 1:02:49.879 That's going to give me some payment, an additional payment 1:02:49.882 --> 1:02:52.802 here and here, namely 1.06 times whatever I 1:02:52.804 --> 1:02:53.574 put in. 1:02:53.570 --> 1:02:59.630 So you notice that the gap between here and here since I've 1:02:59.626 --> 1:03:06.306 multiplied by 1 sixth is exactly this 6 and 1 half dollar gap. 1:03:06.309 --> 1:03:14.139 So the total I have to spend is 98.8 over here, 1:03:14.139 --> 1:03:17.529 so if I spend 23 dollars on the Treasury and the remaining 1:03:17.525 --> 1:03:20.175 amount-- 23 dollars on the 30 year 9 1:03:20.177 --> 1:03:23.267 percent bond, the remaining amount of money 1:03:23.268 --> 1:03:27.028 which is 75.8 on the Treasury I will have spent this 98.8 1:03:27.032 --> 1:03:29.612 dollars, but then I have to get the same 1:03:29.606 --> 1:03:30.116 payoffs. 1:03:30.119 --> 1:03:30.949 Why is that? 1:03:30.949 --> 1:03:33.379 Because they're all being priced at the same prices, 1:03:33.382 --> 1:03:35.772 so it's just the distributive law of arithmetic. 1:03:35.768 --> 1:03:40.848 If these payoffs times 1 sixth plus the Treasury payoffs equal 1:03:40.849 --> 1:03:45.759 100.6 and 107.11 then it must be that the sum of the money I 1:03:45.764 --> 1:03:49.184 spent on the 30 year 9 percent bond, 1:03:49.179 --> 1:03:52.629 plus the sum of the money I spent on the Treasury has to 1:03:52.632 --> 1:03:53.452 equal 98.8. 1:03:53.449 --> 1:03:59.639 Therefore, I know that the amount of money of the 98.8 I 1:03:59.637 --> 1:04:06.837 spent on the 1 year Treasury was 98.8 minus the 23 I spent on the 1:04:06.836 --> 1:04:08.746 30-year bond. 1:04:08.750 --> 1:04:13.200 So that's how you can--so to say it again just in a big 1:04:13.202 --> 1:04:14.442 picture look. 1:04:14.440 --> 1:04:16.120 There are two things that can happen. 1:04:16.119 --> 1:04:19.049 If you hold two bonds, a 1-year Treasury and a 30-year 1:04:19.052 --> 1:04:21.012 thing, you're obviously going to be 1:04:21.007 --> 1:04:24.187 able to match these two things by holding the right proportion 1:04:24.186 --> 1:04:24.756 of them. 1:04:24.760 --> 1:04:26.750 That is just obvious. 1:04:26.750 --> 1:04:30.250 And then less obvious is how quickly I'm figuring out how 1:04:30.246 --> 1:04:32.366 much of those two bonds to hold. 1:04:32.369 --> 1:04:35.629 And the way I'm figuring it out so quickly is because the 1-year 1:04:35.632 --> 1:04:38.172 Treasury is paying the same thing up and down, 1:04:38.170 --> 1:04:40.680 so I have to get all the variation out of the 30-year 1:04:40.682 --> 1:04:43.172 bond, so I have to hold the right 1:04:43.166 --> 1:04:47.666 proportion of the 30-year bond to get the same variation. 1:04:47.670 --> 1:04:51.940 So in the very big picture if you're trying to hedge some 1:04:51.938 --> 1:04:55.898 instrument with another instrument and a portfolio of 1:04:55.900 --> 1:04:58.430 stuff, much of which is certain, 1:04:58.429 --> 1:05:00.729 and only one thing that's risky, 1:05:00.730 --> 1:05:03.660 the amount of the risky thing you have to hold has to produce 1:05:03.657 --> 1:05:06.387 as much variation as the thing you're trying to hedge. 1:05:06.389 --> 1:05:09.639 So the key is being able to tell at a glance how much 1:05:09.643 --> 1:05:13.403 variation each thing has then you can tell in what proportion 1:05:13.396 --> 1:05:14.456 to hold them. 1:05:14.460 --> 1:05:18.040 And so the very last thing I want to tell you today, 1:05:18.039 --> 1:05:20.069 which is unfortunately going to take me a few minutes, 1:05:20.070 --> 1:05:24.540 is how traders look at variation and can guess it 1:05:24.543 --> 1:05:26.503 incredibly quickly. 1:05:26.500 --> 1:05:29.780 So a trader could look at this and guess without doing almost 1:05:29.784 --> 1:05:32.144 any calculations what the right hedge is. 1:05:32.139 --> 1:05:40.769 So just like we did the doubling rule I want to do the 1:05:40.766 --> 1:05:44.996 variation rule of thumb. 1:05:45.000 --> 1:05:54.060 And this is called duration and convexity, or average life. 1:05:54.059 --> 1:05:58.679 So I asked myself, suppose I have this 30-year 1:05:58.684 --> 1:06:03.624 bond, 30-year non-callable bond, that's this one, 1:06:03.617 --> 1:06:05.567 9 percent bond. 1:06:05.570 --> 1:06:08.650 How could I guess, kind of, what the variation in 1:06:08.648 --> 1:06:11.148 price would be without having to do, 1:06:11.150 --> 1:06:14.270 you know, I've had to do this very complicated backward 1:06:14.271 --> 1:06:15.661 induction calculation. 1:06:15.659 --> 1:06:19.639 Now, remember once the interest rates go up here it's a random 1:06:19.644 --> 1:06:22.394 walk so on average they're pretty much, 1:06:22.389 --> 1:06:24.989 not pretty much, they're exactly what they were 1:06:24.985 --> 1:06:25.375 here. 1:06:25.380 --> 1:06:26.480 They start here. 1:06:26.480 --> 1:06:28.940 They go up and down, but on average they're still 1:06:28.940 --> 1:06:29.300 here. 1:06:29.300 --> 1:06:32.390 And over here they've gone down so the average is lower going 1:06:32.385 --> 1:06:32.895 forward. 1:06:32.900 --> 1:06:35.270 In fact, it's exactly equal to that. 1:06:35.268 --> 1:06:39.668 So if you're a trader and there was no uncertainty you would 1:06:39.672 --> 1:06:44.222 know that the present value with no uncertainty is going to be 1:06:44.224 --> 1:06:48.554 the coupon divided by (1 the interest rate) the coupon over 1:06:48.550 --> 1:06:52.880 (1 the interest rate) squared (in the last year) the coupon 1:06:52.878 --> 1:06:57.128 over (1 the interest rate) to the T 100 times [correction: 1:06:57.132 --> 1:07:00.642 over] (1 the interest rate) to the T. 1:07:00.639 --> 1:07:02.829 That's the value of the bond. 1:07:02.829 --> 1:07:07.209 Now, you sort of want to know how much is that price going to 1:07:07.210 --> 1:07:10.130 change when the interest rate changes. 1:07:10.130 --> 1:07:13.440 Well, at a glance, I mean, you just take the 1:07:13.438 --> 1:07:14.438 derivative. 1:07:14.440 --> 1:07:19.560 What is dPV/dr? 1:07:19.559 --> 1:07:30.109 It's going to be minus C over (1 r) squared - 2 C over (1 r) 1:07:30.110 --> 1:07:41.200 cubed - T times C over (1 r) to the (T 1) - T times 100 divided 1:07:41.199 --> 1:07:45.849 by (1 r) to the (T 1). 1:07:45.849 --> 1:07:48.489 That's the derivative. 1:07:48.489 --> 1:07:51.229 It looks like I'm making things even more complicated, 1:07:51.230 --> 1:07:57.760 you would think, but this turns out to be not so 1:07:57.755 --> 1:08:02.025 complicated, because this is a number--so 1:08:02.027 --> 1:08:04.917 let's put the minus on the outside, 1:08:04.920 --> 1:08:10.040 so I'll get rid of this. 1:08:10.039 --> 1:08:13.629 This is a number which you can sort of get in your head, 1:08:13.630 --> 1:08:15.330 so let's try writing it. 1:08:15.329 --> 1:08:17.739 Let's pull out one of the (1 r)s. 1:08:17.738 --> 1:08:27.548 So it's minus 1 over (1 r) times (C over (1 r) 2 C over (1 1:08:27.546 --> 1:08:30.296 r) squared ... 1:08:30.300 --> 1:08:38.530 T times (C 100) (that's the last payment) over (1 r) to the 1:08:38.529 --> 1:08:39.239 T). 1:08:39.239 --> 1:08:42.669 Now, what is that? 1:08:42.670 --> 1:08:44.300 Can you interpret that? 1:08:44.300 --> 1:08:47.450 And the answer is yes you can interpret that because what is 1:08:47.454 --> 1:08:47.834 that? 1:08:47.828 --> 1:08:51.588 That's saying these are all the payments you're getting over the 1:08:51.586 --> 1:08:52.716 life of the bond. 1:08:52.720 --> 1:08:55.230 It's C, C and the last one's C 100. 1:08:55.229 --> 1:08:57.829 This number that keeps coming in front of them, 1:08:57.829 --> 1:09:01.109 though, is the year in which you're getting the payment. 1:09:01.109 --> 1:09:02.319 This is in the first year. 1:09:02.319 --> 1:09:03.249 I could put a 1 there. 1:09:03.250 --> 1:09:04.840 This is the payment in the second year. 1:09:04.840 --> 1:09:05.730 There's a 2 here. 1:09:05.729 --> 1:09:07.519 This is the payment in the Tth year. 1:09:07.520 --> 1:09:09.030 There's a T here. 1:09:09.029 --> 1:09:12.479 You got that T from taking the derivative of the denominator 1:09:12.479 --> 1:09:12.889 here. 1:09:12.890 --> 1:09:19.280 So it's like weighting the year in which the payments come by 1:09:19.283 --> 1:09:22.483 the value of those payments. 1:09:22.479 --> 1:09:24.369 So in the first year the value is C. 1:09:24.368 --> 1:09:26.198 The present value is C over (1 r). 1:09:26.198 --> 1:09:28.638 In the second year the payment is C, 1:09:28.640 --> 1:09:31.660 but discounted, C over (1 r) squared, 1:09:31.658 --> 1:09:34.748 and the last year the total payment is C 100, 1:09:34.750 --> 1:09:37.280 but you had to discount it by (1 r) to the Tth. 1:09:37.279 --> 1:09:42.019 So this is just like taking the average year in which the 1:09:42.020 --> 1:09:45.070 present value flows actually come. 1:09:45.069 --> 1:09:50.339 If I divided this now by the present value of-- 1:09:50.340 --> 1:09:55.270 the present value, these numbers down here, 1:09:55.270 --> 1:09:59.970 C over this C over this squared C 100 over this to the Tth, 1:09:59.970 --> 1:10:02.640 those are all weights multiplying the year in which 1:10:02.640 --> 1:10:03.710 the payments come. 1:10:03.710 --> 1:10:05.340 They add up to the present value. 1:10:05.340 --> 1:10:09.120 So basically I've just got a weighted average of what year 1:10:09.122 --> 1:10:10.652 the payments come in. 1:10:10.649 --> 1:10:18.679 This is called the average life. 1:10:18.680 --> 1:10:21.730 This thing in here is the average life because I've 1:10:21.730 --> 1:10:25.390 multiplied the year in which the payments come by the present 1:10:25.391 --> 1:10:28.631 value of the cash flows there and then I've made those 1:10:28.626 --> 1:10:31.796 multipliers add up to 1 by dividing by the PV. 1:10:31.800 --> 1:10:36.190 See, this in parenthesis, all these terms are the things 1:10:36.189 --> 1:10:39.939 multiplying the year in which the thing came. 1:10:39.939 --> 1:10:42.709 So if I add up all these coefficients here I'm just 1:10:42.708 --> 1:10:44.258 getting the present value. 1:10:44.260 --> 1:10:45.830 So now the coefficients add up to 1. 1:10:45.828 --> 1:10:48.768 So I'm just taking an average of the year in which the thing 1:10:48.774 --> 1:10:50.974 came and that's what the average life is, 1:10:50.970 --> 1:10:55.560 but that's equal to the derivative of the present value 1:10:55.563 --> 1:10:58.203 divided by the present value. 1:10:58.198 --> 1:11:00.758 So it's--the percentage change in the present value is the 1:11:00.756 --> 1:11:02.636 average year in which the payments come. 1:11:02.640 --> 1:11:03.970 So let's just think about it. 1:11:03.970 --> 1:11:05.910 Suppose you have a 10-year bond? 1:11:05.908 --> 1:11:09.158 It pays C, C, C, C, C, and then 100 at the 1:11:09.158 --> 1:11:10.028 very end. 1:11:10.029 --> 1:11:12.479 What do you think the average life is? 1:11:12.479 --> 1:11:15.189 Well, if there were no discounting the average life 1:11:15.186 --> 1:11:18.596 wouldn't be 5 because there's such a huge payment in the end. 1:11:18.600 --> 1:11:23.170 The average life would be 8 or 9, but there is discounting so 1:11:23.168 --> 1:11:26.518 this thing at the end won't count as much. 1:11:26.520 --> 1:11:28.410 So the average life is probably 7. 1:11:28.408 --> 1:11:30.768 If it's a 30 year bond, the average life, 1:11:30.773 --> 1:11:33.793 you're going to get C, C, C, C, and then all the way 1:11:33.787 --> 1:11:36.207 at the end you're going to get C 100. 1:11:36.210 --> 1:11:39.350 Well, if the payments were all equal and no discounting it 1:11:39.345 --> 1:11:41.735 would be 15, but there's a huge amount of 1:11:41.744 --> 1:11:45.294 discounting so that last payment is really pretty irrelevant, 1:11:45.288 --> 1:11:47.838 so probably--so it's not clear which way it goes, 1:11:47.840 --> 1:11:52.270 probably the average life will be less than 15. 1:11:52.270 --> 1:11:56.550 So you can see just by looking at the bond, 1:11:56.550 --> 1:12:00.270 and I'm ending with this thought, by looking at the bond 1:12:00.266 --> 1:12:03.846 itself you can figure out what the average life is. 1:12:03.850 --> 1:12:06.030 So a trader, just by a little experience and 1:12:06.028 --> 1:12:07.998 common sense, can figure out the average 1:12:08.003 --> 1:12:08.463 life. 1:12:08.460 --> 1:12:10.710 I mean, here are all the payments the way they're coming. 1:12:10.710 --> 1:12:13.550 They're coming evenly across the whole history of the bond 1:12:13.546 --> 1:12:16.356 with something at the end, so the average life is probably 1:12:16.356 --> 1:12:19.016 a little bit over the average, but if you go way off, 1:12:19.015 --> 1:12:22.065 you push the final payment too far to the end then it's 1:12:22.070 --> 1:12:24.520 becoming negligible, it's almost irrelevant, 1:12:24.519 --> 1:12:27.079 and so the average life by discounting will probably be 1:12:27.077 --> 1:12:28.117 before the average. 1:12:28.118 --> 1:12:31.228 So a 10-year bond probably has an average life of 7, 1:12:31.228 --> 1:12:34.578 a 30-year bond maybe has an average life of 12 or 13. 1:12:34.578 --> 1:12:38.638 Once you know the average life, you just have to hold bonds in 1:12:38.641 --> 1:12:41.041 proportion to their average lives. 1:12:41.038 --> 1:12:44.218 The dollar amount you put in each bond to hedge it is just 1:12:44.217 --> 1:12:46.167 proportional to its average life. 1:12:46.170 --> 1:12:48.770 So that's how a trader, without having to do any 1:12:48.765 --> 1:12:51.505 arithmetic, can kind of guess how much of 1:12:51.505 --> 1:12:55.295 the 30 year bond will it take to hedge a 10 year bond, 1:12:55.300 --> 1:12:59.940 and how much of a--so which has a higher average life? 1:12:59.939 --> 1:13:02.739 The longer the bond the higher the average life, 1:13:02.738 --> 1:13:06.558 the more sensitive it is to interest rate changes so the 1:13:06.556 --> 1:13:09.746 less of it you need to hedge your position, 1:13:09.750 --> 1:13:12.390 so if you hold the 30 year bond you don't have to hold that much 1:13:12.391 --> 1:13:13.471 of it, 1 sixth of it, 1:13:13.471 --> 1:13:15.131 in order to hedge this mortgage. 1:13:15.130 --> 1:13:17.940 The mortgage has a very short average life because it's 1:13:17.942 --> 1:13:19.142 prepaying so quickly. 1:13:19.140 --> 1:13:20.770 So it's not going to last very long. 1:13:20.770 --> 1:13:23.670 So this is an amazing connection between average life 1:13:23.667 --> 1:13:25.617 and sensitivity to interest rate. 1:13:25.618 --> 1:13:27.548 And average life is something you can guess. 1:13:27.550 --> 1:13:30.210 Sensitivity to interest rate is something you have to do huge 1:13:30.212 --> 1:13:32.122 calculations for, but the two are almost the 1:13:32.119 --> 1:13:32.519 same. 1:13:32.520 --> 1:13:35.520 So in the problem set we'll see if you can do it. 1:13:35.520 --> 1:13:40.800 I asked you a couple problems to do with average life, 1:13:40.801 --> 1:13:41.301 so. 1:13:41.300 --> 1:13:46.000