WEBVTT 00:02.340 --> 00:05.800 So last time we found out how to evaluate options, 00:05.799 --> 00:09.679 especially stopping options where you don't have so many 00:09.682 --> 00:10.532 options. 00:10.530 --> 00:12.260 You either do something or you don't do it. 00:12.260 --> 00:14.140 That's the simplest kind of option. 00:14.140 --> 00:18.680 And we discovered through two examples, at least, 00:18.680 --> 00:23.600 that the option is worth more than people realize. 00:23.600 --> 00:25.450 So this time we're going to apply that-- 00:25.450 --> 00:28.270 and that the way to figure out exactly what it's worth and 00:28.274 --> 00:30.954 exactly what to do is to work by backward induction. 00:30.950 --> 00:34.650 So this time we're going to apply that reasoning to the two 00:34.646 --> 00:37.256 most important options in the economy, 00:37.260 --> 00:42.400 one is callable bonds and a much more important one is the 00:42.397 --> 00:44.017 mortgage option. 00:44.020 --> 00:46.820 And all of you at some time in your lives will probably own 00:46.824 --> 00:49.534 houses and have a mortgage option and have to think about 00:49.532 --> 00:52.242 what kind of mortgage to get and what the option is. 00:52.240 --> 00:55.170 So I want to teach you how to think about that problem. 00:55.170 --> 01:01.530 So let's start with the callable bond problem. 01:01.530 --> 01:08.440 Uh-oh, I forgot to turn this on. 01:08.438 --> 01:11.278 So let's start here with the callable bond problem. 01:11.280 --> 01:25.660 01:25.659 --> 01:30.449 So callable bonds are issued by corporations and they pay, 01:30.447 --> 01:34.057 usually, an interest rate, say 9 percent. 01:34.060 --> 01:38.070 So the bond pays 9 percent, 9 percent and then some years 01:38.066 --> 01:42.746 later it pays 109, but at any point in time you 01:42.753 --> 01:46.663 have the option, the company has the option of 01:46.660 --> 01:50.750 calling the bond, and calling it for--so it's 01:50.753 --> 01:55.663 going to pay 9 100 at the end, so here it's 9. 01:55.660 --> 01:59.610 So at any time the company has the option of calling the bond. 01:59.610 --> 02:01.470 So what does that mean? 02:01.468 --> 02:05.548 It means after it's paid the 9--the company's issued a bond 02:05.551 --> 02:09.771 promising to pay 9 for say 10 years and then the principal in 02:09.772 --> 02:12.922 the 10th year, this is year 10, 02:12.920 --> 02:14.960 the total of 109. 02:14.960 --> 02:17.360 So that's the simplest possible bond. 02:17.360 --> 02:20.310 And the company occasionally has the option, 02:20.310 --> 02:22.490 we'll see in a minute why it would want this option, 02:22.490 --> 02:24.060 has the option of saying, "Okay, 02:24.060 --> 02:26.250 we don't want to make the rest of those payments. 02:26.250 --> 02:28.310 We want to get out of our promise. 02:28.310 --> 02:30.080 We've just paid you 9. 02:30.080 --> 02:33.670 We'll pay off the extra 100 that we're eventually going have 02:33.674 --> 02:36.054 to owe and we'll call it a day." 02:36.050 --> 02:47.030 So this is the payment and this is the remaining balance. 02:47.030 --> 02:52.630 So for a callable bond the remaining balance is always 100. 02:52.628 --> 02:56.178 So it's pretty obvious that if you've made an arrangement, 02:56.179 --> 02:59.849 so I owe you--like for example the prototypical mortgage was 02:59.853 --> 03:01.413 exactly of this kind. 03:01.408 --> 03:04.008 Somebody borrows money from you and they say, 03:04.008 --> 03:07.378 "I promise to pay you 9 dollars a year until the last 03:07.376 --> 03:09.676 year when I'm going to pay you 109. 03:09.680 --> 03:12.470 This is called, for those old mortgages it was 03:12.468 --> 03:16.448 called the balloon payment, but in typical bonds it's just 03:16.445 --> 03:20.285 the principal payment, the 100 face value of the bond. 03:20.288 --> 03:23.558 So the person who borrowed the money and has agreed to pay off 03:23.562 --> 03:26.202 over the years, he might have a reason why the 03:26.200 --> 03:29.880 house he put up as collateral is no longer going to be his house. 03:29.878 --> 03:33.378 He might want to move in which case the lender doesn't have a 03:33.375 --> 03:36.865 house anymore backing the loan and they have to have some way 03:36.870 --> 03:39.200 of resolving the loan and ending it. 03:39.199 --> 03:41.169 So the question is, after you've made a payment, 03:41.167 --> 03:43.807 how can you resolve the rest of the loan which is supposed to go 03:43.805 --> 03:44.805 on for 6 more years? 03:44.810 --> 03:48.980 Well, you just agree to pay 100 and then you call it quits. 03:48.979 --> 03:52.549 So that's a typical kind of bond. 03:52.550 --> 03:55.920 A non-callable bond is, you've committed to paying the 03:55.922 --> 03:59.802 9 dollars and the 109 at the end and you have no option to get 03:59.803 --> 04:01.143 out of the thing. 04:01.139 --> 04:03.989 So we want to study what the difference is in value between 04:03.990 --> 04:06.840 the non-callable bond where you're obliged to pay the whole 04:06.841 --> 04:09.571 thing until the end, and the callable bond where you 04:09.566 --> 04:11.156 can get out of it by paying 100. 04:11.158 --> 04:14.968 And it's easy to see why a callable bond might have been 04:14.967 --> 04:17.927 invented, especially for a mortgage when 04:17.925 --> 04:21.885 you've got this balloon payment because you might want to 04:21.891 --> 04:23.381 dissolve the debt. 04:23.379 --> 04:26.599 But by putting in this option to dissolve the debt you 04:26.596 --> 04:29.446 drastically change the value of the promise, 04:29.449 --> 04:30.719 and that's what we want to calculate, 04:30.720 --> 04:32.170 how much that changes. 04:32.170 --> 04:34.630 So why might it change? 04:34.629 --> 04:36.189 Well, the interest rates might change. 04:36.190 --> 04:38.070 They might go up or down. 04:38.069 --> 04:42.309 Soon we're going to have them going up or down in a more 04:42.312 --> 04:46.192 complicated way, but suppose that the interest 04:46.192 --> 04:49.692 rate starts somewhere like at 8 percent, 04:49.690 --> 04:53.530 maybe, and it could go up to some number. 04:53.529 --> 05:01.119 Now, I'm always going to do a geometric random walk. 05:01.120 --> 05:03.880 This has become very fashionable in finance and you 05:03.879 --> 05:07.249 should be asking the question, is it special that we only have 05:07.245 --> 05:08.455 two possibilities? 05:08.459 --> 05:10.869 Life has many more than two possibilities. 05:10.870 --> 05:13.000 Suppose we had hundreds of possibilities, 05:12.999 --> 05:14.649 would that make a difference? 05:14.649 --> 05:16.529 The answer is it's not going to make a difference, 05:16.529 --> 05:17.949 but we're going to have to see why. 05:17.949 --> 05:22.599 So in a geometric random walk literally the thing can go up or 05:22.601 --> 05:24.051 down each period. 05:24.050 --> 05:27.040 So what does it go to if this is r_0? 05:27.040 --> 05:33.440 The next period we say it goes to r_0 times e to the 05:33.444 --> 05:39.544 volatility, which I usually call sigma, plus maybe a drift, 05:39.535 --> 05:40.685 plus d. 05:40.690 --> 05:45.560 And here it's going to go to r_0e-- 05:45.560 --> 05:49.430 let's write it with a little more room-- 05:49.430 --> 05:59.030 r_0 e to the minus sigma plus d, 05:59.029 --> 06:00.639 so what's happened here? 06:00.639 --> 06:02.779 The interest rate started at 8 percent. 06:02.778 --> 06:06.258 It gets multiplied by e to the d, which is just a number, 06:06.259 --> 06:10.839 so it's maybe tending to go up over time if d is positive or if 06:10.841 --> 06:12.841 d is 0-- think of d = 0--r_0 06:12.838 --> 06:14.878 on average is going to stay at r_0, 06:14.879 --> 06:16.329 but there's some uncertainty. 06:16.329 --> 06:17.809 Maybe the interest rate goes up. 06:17.810 --> 06:20.700 We multiply it by some number, e to the sigma, 06:20.696 --> 06:24.666 and then we multiply it by the reciprocal of the same number or 06:24.673 --> 06:26.923 we divide it by e to the sigma. 06:26.920 --> 06:30.010 So using the exponential notation just makes the 06:30.014 --> 06:33.774 computations in the computer easier because the computer's 06:33.767 --> 06:35.017 adding numbers. 06:35.019 --> 06:41.309 This thing over here is going to be r_0 e to the 2 06:41.310 --> 06:43.230 sigma 2d, right? 06:43.230 --> 06:48.280 And this will just be r_0 e to the 2 d, 06:48.283 --> 06:54.603 and this will be r_0 e to the minus 2 sigma 2 d. 06:54.600 --> 06:58.290 So the computer in calculating the interest rate at every step 06:58.293 --> 07:01.873 is adding exponents and it makes the calculation when you're 07:01.865 --> 07:04.405 doing gazillions of nodes much faster. 07:04.410 --> 07:09.420 So that's why it's traditional to use this notation, 07:09.420 --> 07:12.220 but really all I'm saying is you multiply or divide by the 07:12.218 --> 07:14.818 same number to see what the next interest rate is, 07:14.819 --> 07:17.159 and on top of that you might be sort of increasing all the 07:17.163 --> 07:19.673 things over time just because you think the interest rates are 07:19.673 --> 07:20.623 going up over time. 07:20.620 --> 07:24.440 So that's how we're modeling uncertainty, and everybody is 07:24.444 --> 07:28.274 supposed to understand what the probabilities are of these 07:28.269 --> 07:28.939 moves. 07:28.939 --> 07:32.419 So for now that's going to be our model of uncertainty and in 07:32.418 --> 07:36.008 a few minutes I'm going to try and indicate why complicating it 07:36.014 --> 07:37.584 won't have much effect. 07:37.579 --> 07:39.249 So are there any questions? 07:39.250 --> 07:42.020 This notation I hope isn't too complicated. 07:42.019 --> 07:44.279 Are there any questions about the interest rate process I'm 07:44.276 --> 07:46.256 assuming, the uncertainty that people are facing? 07:46.259 --> 07:46.909 Yeah? 07:46.910 --> 07:50.140 Student: Why is that middle probability on the far 07:50.142 --> 07:52.342 right, r_0 is always equal to 07:52.336 --> 07:54.066 >? 07:54.069 --> 07:56.969 Prof: Because I'm assuming that if there weren't 07:56.973 --> 07:59.933 uncertainty the interest rate was going to keep steadily 07:59.930 --> 08:02.780 rising by multiplying by the constant e to the d. 08:02.778 --> 08:04.988 So here I've multiplied it by e to the d. 08:04.990 --> 08:06.990 Here I'm multiplying it again by e to the d, 08:06.992 --> 08:09.372 and I'm going to multiply it again by e to the d. 08:09.370 --> 08:13.540 So if d is 1 tenth of a percent it means that the interest rate 08:13.541 --> 08:16.841 is rising at 1 tenth of a percent every period. 08:16.839 --> 08:19.279 You might just think interest rates are going to get higher in 08:19.278 --> 08:19.798 the future. 08:19.800 --> 08:22.270 You might have a more complicated function than 08:22.269 --> 08:24.309 steadily rising or steadily falling. 08:24.310 --> 08:26.720 It's just that this is the simplest to add a little 08:26.723 --> 08:29.093 complication without making it too complicated. 08:29.089 --> 08:33.989 Steadily rising or steadily falling and binomial uncertainty 08:33.986 --> 08:37.636 were the easiest simple things I could do. 08:37.639 --> 08:40.899 Of course a more realistic thing would do more complicated 08:40.904 --> 08:41.424 things. 08:41.418 --> 08:42.818 Any other questions about what this means? 08:42.820 --> 08:44.550 I know this can be confusing, this notation, 08:44.554 --> 08:46.174 so let's just figure out the notation. 08:46.168 --> 08:49.328 Make sure you understand it before I move on because it's 08:49.333 --> 08:51.033 really nothing but notation. 08:51.029 --> 08:53.809 I'm just saying the interest rate is 8 percent today. 08:53.808 --> 08:57.328 It can go up a little or down a little and then after that it 08:57.331 --> 09:00.381 can go up or down and the percentage rise and fall is 09:00.381 --> 09:03.551 always going to be the same relative to a drift and the 09:03.551 --> 09:06.931 drift might be 0, the drift might be positive. 09:06.928 --> 09:21.578 So to take an example, let's take an example here. 09:21.580 --> 09:22.740 Did I plug everything in? 09:22.740 --> 09:23.860 Yes. 09:23.860 --> 09:27.470 This is a spreadsheet that you have called callable bond. 09:27.470 --> 09:31.340 I did a little work on it, but not much. 09:31.340 --> 09:34.820 It's basically your spreadsheet. 09:34.820 --> 09:37.160 Everything is plugged in here. 09:37.159 --> 09:38.519 You're seeing that, right? 09:38.519 --> 09:43.449 So what I've done is I've said let's take over here the year, 09:43.452 --> 09:47.402 so here are the years, 0,1, 2,3, blah up to 31 or 09:47.399 --> 09:48.549 something. 09:48.548 --> 09:52.948 Then the bond coupon rate is 9 percent, so the bond is going to 09:52.945 --> 09:54.855 pay 9 dollars, 9 percent. 09:54.860 --> 09:55.970 The face is always 100. 09:55.970 --> 09:58.470 It'll pay 9 dollars until the end. 09:58.470 --> 10:02.500 So if I take the bond maturity of 30 down here I've got the 9 10:02.504 --> 10:06.274 dollar payments and I've got the remaining principal, 10:06.269 --> 10:09.349 the remaining balance, what you could pay to dissolve 10:09.351 --> 10:10.241 the contract. 10:10.240 --> 10:12.350 You could after the first year, after paying 9, 10:12.350 --> 10:14.830 immediately afterwards you could pay 100 and say forget 10:14.826 --> 10:16.016 about the whole thing. 10:16.019 --> 10:19.389 So that's that, and so it goes all the way to 10:19.388 --> 10:22.578 year 30, and year 30 you see you're 10:22.582 --> 10:26.012 paying 109, and then the bond is over, 10:26.014 --> 10:29.054 109 and by then the bond is over, 10:29.049 --> 10:31.499 there's nothing to talk about. 10:31.500 --> 10:35.600 So now I start off the interest rate somewhere at 8 percent, 10:35.596 --> 10:39.416 and I take the volatility, it's always said in hundreds, 10:39.417 --> 10:41.637 I take the volatility of 16. 10:41.639 --> 10:45.649 That means sigma = 16 over 100, because that's the way people 10:45.649 --> 10:48.189 talk about volatility of 16 percent. 10:48.190 --> 10:51.270 So that is a sort of standard volatility, 10:51.269 --> 10:54.959 12 or 16 percent is about the annual volatility of these 10:54.958 --> 10:58.258 things traditionally, so I've taken the volatility at 10:58.255 --> 10:58.905 16 percent. 10:58.908 --> 11:01.798 So I take the drift of 0 to make that simple. 11:01.798 --> 11:06.088 The d's disappearing, so what does that mean? 11:06.090 --> 11:09.850 It means that whatever the thing was last period when it 11:09.846 --> 11:12.506 moves to here I take the old number-- 11:12.509 --> 11:17.849 at the top there if you read this, this is the up multiplier. 11:17.850 --> 11:22.900 I just take e to the sigma d and that's what this says over 11:22.902 --> 11:26.302 here, exponent of--it just says that. 11:26.298 --> 11:31.908 So that's the up multiplier and that's the down multiplier. 11:31.909 --> 11:33.809 Let's just see how that looks. 11:33.808 --> 11:38.928 So if I look at the interest rates I start at 8 percent and 11:38.929 --> 11:43.609 they can go up by that multiplier, 16 percent up or 16 11:43.609 --> 11:45.109 percent down. 11:45.110 --> 11:48.130 And see if you take 8 percent you add 16 percent to it, 11:48.125 --> 11:51.195 slightly more than 16 percent, because e to the .16 is a 11:51.197 --> 11:53.207 little bit more than 16 percent. 11:53.210 --> 11:57.420 You take 8,16 percent up is about 9.3 percent interest and 11:57.418 --> 12:01.478 16 percent down you go from 8 percent to 6.8 percent. 12:01.480 --> 12:04.310 And things just go up or down so that's the uncertainty. 12:04.308 --> 12:08.048 So that's what interest rates are going to do. 12:08.048 --> 12:11.378 So everybody knows that you start at 8 percent and interest 12:11.375 --> 12:14.925 rates can go up or down by 16 percent for the next 30 years. 12:14.928 --> 12:17.858 The question is, what's the value of the bond? 12:17.860 --> 12:22.070 Well, if you couldn't call it--please interrupt me if 12:22.067 --> 12:23.927 you're not following. 12:23.928 --> 12:27.578 If you couldn't call it, non-callable bond, 12:27.580 --> 12:32.100 why would you expect it to have such a high price? 12:32.100 --> 12:36.070 Well, because the interest rate is, sorry, the interest rate, 12:36.067 --> 12:40.227 remember, is only 8 percent and on average it's going to stay at 12:40.234 --> 12:41.164 8 percent. 12:41.158 --> 12:43.848 It could go up or down, but it's always coming 12:43.850 --> 12:46.900 back--the middle thing there is always 8 percent. 12:46.899 --> 12:48.819 So if you look in the middle it's 8 percent, 12:48.817 --> 12:50.867 8 percent, 8 percent, 8 percent, so sort of the 12:50.869 --> 12:52.429 geometric average is 8 percent. 12:52.428 --> 12:53.998 On average it's going to be 8 percent. 12:54.000 --> 12:58.280 That's less than the payment, 9, so the poor bondholder is 12:58.279 --> 13:00.659 overpaying, paying 9 every year when the 13:00.659 --> 13:03.609 interest rate is only 8 percent and on average is going to be 8 13:03.606 --> 13:04.126 percent. 13:04.129 --> 13:08.169 So obviously the bond issuer is giving a very good deal to the 13:08.168 --> 13:12.138 buyer by paying 9 when the interest rate's only 8 percent. 13:12.139 --> 13:14.099 And you know how you calculate this. 13:14.100 --> 13:17.060 At every step you'd have to do it by backward induction. 13:17.058 --> 13:22.638 So you'd write down, and we've done this before, 13:22.636 --> 13:27.026 the value at every node V_S, 13:27.028 --> 13:33.198 so if I take a node here what's the value of it? 13:33.200 --> 13:35.340 I don't know if I need to do a concrete example. 13:35.340 --> 13:38.040 How do I do this by backward induction? 13:38.038 --> 13:42.558 So I keep going like that, right? 13:42.559 --> 13:45.179 And then life is going to end. 13:45.178 --> 13:47.668 This is the last payment somewhere. 13:47.668 --> 13:54.448 Let's say the last payment is here so you're going to get 109 13:54.447 --> 14:00.427 no matter what because that's the last period 109,109, 14:00.433 --> 14:01.793 109,109. 14:01.788 --> 14:07.048 So we know what the value is here at the end because it's 0 14:07.049 --> 14:11.309 beyond this point, and we know what the value is 14:11.312 --> 14:12.132 here. 14:12.129 --> 14:14.749 You're going to pay 109 no matter what so you just take the 14:14.753 --> 14:15.933 interest rate back here. 14:15.928 --> 14:17.448 So we know what the value is here. 14:17.450 --> 14:20.320 So the question is if you figured out the value at the end 14:20.315 --> 14:22.675 of the tree, which is trivial because at the 14:22.677 --> 14:25.217 very end you're just going to get 109 for sure, 14:25.220 --> 14:27.950 can you figure out what the value is at the beginning of the 14:27.946 --> 14:28.266 tree. 14:28.269 --> 14:31.749 And the way you do that is for every node you say to yourself, 14:31.748 --> 14:34.998 it doesn't matter which node you are like this one or that 14:34.999 --> 14:36.709 one, let's call is node S. 14:36.710 --> 14:46.870 So the value of node S, this is the value just after 14:46.874 --> 14:50.864 paying the coupon. 14:50.860 --> 14:53.770 So you've paid the coupon, you've paid the 9, 14:53.769 --> 14:55.289 now what's the value. 14:55.288 --> 14:59.448 Well, the non-call price, so let's do non-call, 14:59.453 --> 15:03.713 non-call bond, what is the value going to be? 15:03.710 --> 15:12.030 Well, just after paying the 9 here what's the value of what's 15:12.028 --> 15:12.998 left? 15:13.000 --> 15:15.580 So you're going to have to go up here. 15:15.580 --> 15:19.140 First of all you're going to discount by the interest rate 15:19.138 --> 15:21.508 here, so it's 1 over (1 r_S), 15:21.510 --> 15:24.820 so you're discounting by the interest rate there. 15:24.820 --> 15:31.670 Then half the time you'll go up. 15:31.668 --> 15:33.248 So what will you have to do if you go up? 15:33.250 --> 15:37.340 You'll have to pay 9 because that's the required payment, 15:37.340 --> 15:40.370 and then you'll be left in the same situation except that 15:40.370 --> 15:44.190 you'll be at the up state, whatever the value is 15:44.191 --> 15:47.301 there--plus V_Sup. 15:47.298 --> 15:50.548 If this node is S this node we call S_up and this 15:50.548 --> 15:52.338 node we call S_down. 15:52.340 --> 15:55.990 So I'm doing this by backward induction, remember. 15:55.990 --> 15:59.410 I'm saying, suppose towards the end of the tree I figured out 15:59.408 --> 16:02.658 what is the value of the bond just after making the coupon 16:02.655 --> 16:03.335 payment. 16:03.340 --> 16:05.950 Since there are no coupon payments at the end I know the 16:05.946 --> 16:08.416 value is 0, so having done it at the end, 16:08.421 --> 16:11.421 and working my way backwards now by induction, 16:11.418 --> 16:13.938 I can figure out the value at any previous node. 16:13.940 --> 16:17.350 So suppose I'm here and I've already figured out the value of 16:17.350 --> 16:18.430 all future nodes. 16:18.428 --> 16:20.518 The value here is simply going to be, 16:20.519 --> 16:23.869 value here is, just after the coupon payment, 16:23.870 --> 16:26.190 is going to be--I'm going to discount at the interest rate 16:26.192 --> 16:28.132 here, 1 r_S. 16:28.129 --> 16:30.499 Half the time I'm going to have to go up here, 16:30.496 --> 16:33.596 in which case I'm going to pay 9 and then what am I going to 16:33.601 --> 16:33.971 do? 16:33.970 --> 16:36.650 I'm going to be at a situation where I've just made the coupon 16:36.649 --> 16:38.539 payment and I see what the value left is. 16:38.538 --> 16:40.248 Well, that's the number we already calculated. 16:40.250 --> 16:46.630 So V_Sup 1 half--if I go down I'm down to here. 16:46.629 --> 16:49.689 It's 9 again V_Sdown. 16:49.690 --> 16:53.610 So that's pretty simple, and now I just have to solve it 16:53.609 --> 16:55.319 by backward induction. 16:55.320 --> 16:57.130 So that's what I did in this tree. 16:57.129 --> 17:00.399 Here are the interest rates, and now back here here's the 17:00.400 --> 17:02.680 present value of a non-callable bond. 17:02.678 --> 17:05.458 So at any node, let's pick one like this one, 17:05.459 --> 17:06.659 what's the value? 17:06.660 --> 17:11.140 It says it's 1 over (1 the interest rate) times the 17:11.144 --> 17:13.684 probability .5, that's 1 half, 17:13.678 --> 17:16.168 times the payment, that's 9 dollars, 17:16.170 --> 17:21.520 plus what the value would be if I went up because J 149 is this 17:21.516 --> 17:22.116 node. 17:22.118 --> 17:27.408 That's J 149 here, so it's just exactly what I 17:27.411 --> 17:31.111 wrote, 50 percent probability J 4 is 17:31.109 --> 17:36.389 the 9 dollars at the top plus the value of what I would get 17:36.390 --> 17:39.220 here, plus 1 half times the same 9 17:39.221 --> 17:42.171 dollars I'd have to pay if I went down, 17:42.170 --> 17:44.030 plus the value that I'd get down here. 17:44.029 --> 17:46.849 So that's an extremely simple calculation to do, 17:46.848 --> 17:51.028 and because Excel is a brilliant--as I said created by 17:51.028 --> 17:55.128 Kapor whose sister was my classmate and was two years 17:55.127 --> 17:58.137 ahead of me at Yale-- Excel immediately 17:58.140 --> 18:00.220 understands--he called it Lotus-- 18:00.220 --> 18:03.310 immediately understands that if you give all these directions 18:03.305 --> 18:06.335 you can't figure out the value here unless you know what the 18:06.338 --> 18:07.418 value is forward. 18:07.420 --> 18:10.570 So Excel understands that you should keep going forward and 18:10.567 --> 18:13.167 starting at the end, and at the end it knows what 18:13.172 --> 18:15.182 the value is because it's just 0. 18:15.180 --> 18:18.280 So having got the value 0 at the end it then works backwards 18:18.281 --> 18:21.601 all the way through the tree and gets the value at the beginning 18:21.595 --> 18:22.485 and gets 113. 18:22.490 --> 18:24.890 So it's exactly what we want to do, 18:24.890 --> 18:27.500 and he's just written this brilliant program that 18:27.497 --> 18:30.427 understands if there's a dependency you can't calculate 18:30.430 --> 18:32.060 this without knowing this. 18:32.058 --> 18:33.678 He says, "Okay, I don't know this. 18:33.680 --> 18:35.520 I'll go forward into the program to here. 18:35.519 --> 18:36.479 I can't do these either. 18:36.480 --> 18:37.820 I'll go all the way to the end. 18:37.818 --> 18:40.128 Now everything is 0, so I know all these numbers. 18:40.130 --> 18:42.490 Now step-by-step I can go backwards and get all of 18:42.492 --> 18:43.122 them." 18:43.118 --> 18:45.568 That's what Excel is doing and it does it instantly. 18:45.568 --> 18:47.538 So there's no doubt about that, right? 18:47.538 --> 18:50.758 You all are way ahead of me on that. 18:50.759 --> 18:53.709 So the value's 113, that's a huge number, 18:53.711 --> 18:56.961 so obviously it's a very generous company. 18:56.960 --> 19:03.300 The company may not want to be so generous, so the company 19:03.295 --> 19:09.625 says, look, we want to be able to call the bond at 100. 19:09.630 --> 19:12.620 So that simple option, what's it going to be? 19:12.619 --> 19:14.419 What's the value going to be? 19:14.420 --> 19:17.960 Well, obviously the company could call it right away at 100. 19:17.960 --> 19:21.060 So it's going to make the value go all the way down to 100 19:21.063 --> 19:23.353 because the company now has the option, 19:23.348 --> 19:26.018 whenever it wants to, like at the very beginning 19:26.019 --> 19:29.469 before making the first payment, it could pay 100 here, 19:29.467 --> 19:32.977 or it could pay the 9 dollars and then pay 100 here. 19:32.980 --> 19:36.490 So the question is, should the company do it, 19:36.487 --> 19:40.467 and when do you think the company will exercise the 19:40.472 --> 19:41.352 option? 19:41.348 --> 19:46.488 Why wouldn't the company exercise the option right away? 19:46.490 --> 19:47.110 So here it is. 19:47.109 --> 19:48.949 The bond's worth 113. 19:48.950 --> 19:52.360 Every period the company's promising to pay 9 dollars. 19:52.358 --> 19:53.888 That's like a 9 percent interest. 19:53.890 --> 19:56.460 The interest rate's in the economy only 8 percent at the 19:56.463 --> 19:57.263 beginning here. 19:57.259 --> 19:59.609 So it's losing money right from the start. 19:59.608 --> 20:01.728 Why shouldn't the company just say, "Okay, 20:01.730 --> 20:04.450 we gave ourselves the option, we're taking it right away. 20:04.450 --> 20:06.920 We're going to just pay you 100 and cancel the whole thing at 20:06.923 --> 20:07.833 the beginning." 20:07.829 --> 20:11.259 Why wouldn't they do that? 20:11.259 --> 20:12.319 Okay, hang on for one second. 20:12.318 --> 20:21.328 Someone else, why wouldn't they do that? 20:21.328 --> 20:24.828 You see, the puzzle is, the company has promised to pay 20:24.825 --> 20:26.115 9 dollars forever. 20:26.118 --> 20:29.028 That's like 9 percent on a 100-dollar coupon. 20:29.028 --> 20:32.498 The interest rate is only 8 percent now, so the company's 20:32.499 --> 20:34.419 paying more money, 9 dollars. 20:34.420 --> 20:37.350 It could have borrowed and only had to pay 8 dollars. 20:37.348 --> 20:39.628 So the guy buying it is getting this huge bonus. 20:39.630 --> 20:42.940 He's getting 9-dollar coupons instead of 8-dollar coupons. 20:42.940 --> 20:46.070 He's probably telling his friends who aren't marking to 20:46.067 --> 20:47.687 market, look at my return. 20:47.690 --> 20:49.830 I've got 9 dollars the first year. 20:49.828 --> 20:53.518 Look at that great 9 percent return I got on my investment of 20:53.519 --> 20:53.889 100. 20:53.890 --> 20:58.370 Why won't the company call it right at the beginning? 20:58.369 --> 21:03.979 What's the company waiting for? 21:03.980 --> 21:04.680 Yep? 21:04.680 --> 21:06.850 Student: It's hoping that the interest rate will go 21:06.848 --> 21:07.228 back up. 21:07.230 --> 21:10.130 Prof: Back up, exactly, because if the 21:10.132 --> 21:15.072 interest rates go way up-- after a very short amount of 21:15.065 --> 21:20.735 time if the company gets lucky the interest rates are going to 21:20.741 --> 21:21.581 go up. 21:21.578 --> 21:23.198 So here are the interest rates, right? 21:23.200 --> 21:25.830 So after a very short amount of time, 2 years, 21:25.825 --> 21:28.505 you get the up thing, the interest rates are 11 21:28.509 --> 21:29.209 percent. 21:29.210 --> 21:31.380 So all of a sudden the company is looking great. 21:31.380 --> 21:40.660 It's borrowed at 9 percent and the interest rates are actually 21:40.661 --> 21:42.641 11 percent. 21:42.640 --> 21:45.910 Effectively it's borrowing from the guy, 9 percent, 21:45.913 --> 21:49.193 and it's able to invest the money at 11 percent. 21:49.190 --> 21:51.420 To put it another way, the present value of the 21:51.422 --> 21:53.852 payment's left from then are much less than 100. 21:53.849 --> 21:56.279 So it's glad it made the deal. 21:56.279 --> 21:59.419 And on the other hand, if the interest rate goes down 21:59.423 --> 22:03.113 far enough then the company will be able to call the thing. 22:03.108 --> 22:06.268 Present value of the payments if it couldn't call would be way 22:06.272 --> 22:06.742 higher. 22:06.740 --> 22:10.860 Because it can call it can never suffer worse than having 22:10.862 --> 22:11.822 to pay 100. 22:11.818 --> 22:16.428 So you see that if the interest rates are going up or down and 22:16.428 --> 22:20.728 you couldn't call it then when interest rates go up that's 22:20.732 --> 22:25.422 really good news for the company because now the interest rates 22:25.416 --> 22:29.266 are over 9 percent on average going forward, 22:29.269 --> 22:32.319 so actually the present value of payments is 98. 22:32.319 --> 22:33.889 That's less than 100. 22:33.890 --> 22:37.750 On the other hand if interest rates go down then the company's 22:37.746 --> 22:40.526 really screwed, now the present value of what 22:40.528 --> 22:41.728 it owes is 127. 22:41.730 --> 22:44.820 At the beginning it's 113, so as interest rates go further 22:44.817 --> 22:47.897 and further down the company's made a worse and worse bond 22:47.904 --> 22:50.374 issue, but it doesn't suffer very much 22:50.374 --> 22:53.944 because of that because it can always get out of its deal by 22:53.942 --> 22:54.792 calling it. 22:54.788 --> 23:01.298 So in fact, astonishingly, it doesn't even call the first 23:01.296 --> 23:02.106 time. 23:02.109 --> 23:04.589 So here's the value. 23:04.588 --> 23:07.968 When the company behaves optimally it won't call the 23:07.968 --> 23:08.828 first time. 23:08.828 --> 23:11.018 When interest rates go up it says, "Ah-ha! 23:11.019 --> 23:13.169 We're going to make money, that's great, 23:13.165 --> 23:16.245 but when interest rates go down now the interest rate's 6 23:16.248 --> 23:18.228 percent and it's still paying 9. 23:18.230 --> 23:20.620 The company is still not going to call the bond. 23:20.618 --> 23:23.678 It won't call the bond until the interest rates go down all 23:23.676 --> 23:24.886 the way to 5 percent. 23:24.890 --> 23:30.560 So anyway, by backward induction you get the value of 23:30.555 --> 23:34.905 the callable bond is 95, which is 95.529, 23:34.913 --> 23:40.583 which is a lot--let's write those down, in fact. 23:40.578 --> 23:56.948 So the callable bond, 95.529, and non-call was 113 23:56.953 --> 24:10.993 point something, which I forgot already. 24:10.990 --> 24:15.560 So I hope you're following because this is going to get a 24:15.561 --> 24:20.301 lot more surprising when we do mortgages, but it's the same 24:20.298 --> 24:21.848 logic, 113.236. 24:21.848 --> 24:26.078 So the call option is worth 18 percent to the bond. 24:26.079 --> 24:27.069 It's a huge thing. 24:27.069 --> 24:28.149 It sounds so simple. 24:28.150 --> 24:33.170 Of course you should dissolve the thing for a value of 100. 24:33.170 --> 24:35.050 That's the only natural thing to do. 24:35.048 --> 24:37.428 So it's perfectly reasonable, you would think, 24:37.430 --> 24:39.810 for the company to give itself that option. 24:39.808 --> 24:42.658 "Look guys we're borrowing money, but there may be some 24:42.657 --> 24:44.587 reason why things get too complicated. 24:44.589 --> 24:46.709 Our company might close down. 24:46.710 --> 24:49.640 We want to keep our promise, so we just want to pay off the 24:49.637 --> 24:52.367 loan at 100, so give us that option even if our company 24:52.365 --> 24:53.825 doesn't close down." 24:53.828 --> 24:57.348 Well, the company has just taken an incredibly valuable 24:57.352 --> 24:57.942 option. 24:57.940 --> 25:04.800 Instead of making a promise worth 113, it's made a promise 25:04.796 --> 25:06.116 worth 95. 25:06.119 --> 25:07.019 So any questions about that? 25:07.019 --> 25:09.299 You could all get that and you all see why that's happening, 25:09.299 --> 25:11.039 and it's a bigger number than you'd expect. 25:11.039 --> 25:11.669 Yep? 25:11.670 --> 25:13.310 Student: When should they call it? 25:13.308 --> 25:14.518 Prof: When should they call it? 25:14.519 --> 25:20.679 You can tell when they should call it because--you'll know 25:20.681 --> 25:26.521 when they call it because if they do call it what's the 25:26.519 --> 25:27.599 value? 25:27.598 --> 25:30.818 So I didn't work out the--what's the backward 25:30.820 --> 25:31.700 induction? 25:31.700 --> 25:32.150 Sorry. 25:32.150 --> 25:35.130 So let's write down the formula when you do call it. 25:35.130 --> 25:36.380 How could I have forgotten to do that? 25:36.380 --> 25:38.410 What do you do when you do call it? 25:38.410 --> 25:42.930 So value, how do you figure this out? 25:42.930 --> 25:44.230 I just skipped this, sorry. 25:44.230 --> 25:49.360 Value of callable bond, so V_S, 25:49.361 --> 25:53.211 what is that going to equal? 25:53.210 --> 25:57.480 Now you've got an option. 25:57.480 --> 26:06.370 So V_S, remember this is value just 26:06.365 --> 26:11.005 after coupon payment. 26:11.009 --> 26:14.059 So what is the value just after the coupon payment? 26:14.059 --> 26:15.409 That's when you have to call. 26:15.410 --> 26:18.470 So in a second you'll see why that's the way the rule works. 26:18.470 --> 26:22.840 So just after the coupon payment what's the value of the 26:22.843 --> 26:23.403 bond? 26:23.400 --> 26:28.630 Well now that you have the option you can pay the minimum 26:28.627 --> 26:31.147 of the remaining balance. 26:31.150 --> 26:34.190 It's 100 all the time. 26:34.190 --> 26:35.330 This is just going to be 100. 26:35.328 --> 26:39.318 You could pay 100 or you could keep going. 26:39.318 --> 26:43.448 So if you keep going it's going to be 1 r_S times the 26:43.446 --> 26:47.276 same thing, 1 half, and then you're going 26:47.282 --> 26:52.192 to have to make the coupon payment the next up-- 26:52.190 --> 26:56.380 so just a second, where should we go? 26:56.380 --> 26:59.060 So here this is the interest rate, so at any node, 26:59.056 --> 27:02.056 you can pick any one like here in the tree, this one. 27:02.058 --> 27:05.688 That's the interest rate, so at that node you could, 27:05.690 --> 27:08.430 if you wanted to after making the coupon payment there you 27:08.432 --> 27:10.642 could pay the 100 and say, "Okay, we've made the 27:10.638 --> 27:10.918 payment. 27:10.920 --> 27:12.470 I'm making 100 more. 27:12.470 --> 27:16.190 Let's just call the whole thing quits," but if you don't 27:16.186 --> 27:19.866 you're going to move on, and when you move on you're 27:19.873 --> 27:23.903 going to go either up to here, so that's with probability 1 27:23.897 --> 27:24.247 half. 27:24.250 --> 27:28.510 You have to make the coupon payment and then again you'll 27:28.509 --> 27:31.859 have the option of paying your way out of it, 27:31.855 --> 27:35.045 dissolving the contract or continuing. 27:35.048 --> 27:38.668 So you just have V_SU because that's exactly the value 27:38.666 --> 27:42.166 just after making the coupon payment at node S_U. 27:42.170 --> 27:45.170 That's V_SU, and so it's going to take into 27:45.174 --> 27:47.664 account that you're minimizing something. 27:47.660 --> 27:50.340 So you see, V_S, the value at any node, 27:50.338 --> 27:54.388 takes into account you could minimize by possibly paying the 27:54.394 --> 27:57.834 balance and dissolving it or you could continue. 27:57.828 --> 27:59.918 And so if you continue and things go up you have to make 27:59.916 --> 28:01.926 the coupon and then you're going to face a choice, 28:01.930 --> 28:04.180 but we assume by backward induction you've already figured 28:04.182 --> 28:05.372 out the value of that thing. 28:05.369 --> 28:06.509 Maybe it's paying the 100. 28:06.509 --> 28:12.039 Maybe it's not paying the hundred whatever that is, 28:12.044 --> 28:16.144 plus 1 half times 9 V_SD. 28:16.140 --> 28:19.140 So the only difference between this callable-- 28:19.140 --> 28:23.310 non-call case and the call case is you notice that the second 28:23.305 --> 28:26.425 term is exactly the same in the two cases, 28:26.430 --> 28:29.490 but here we've added the minimum of 100 in that because 28:29.486 --> 28:32.256 you either could continue or you could pay 100. 28:32.259 --> 28:34.979 Of course the values, V_SU up here and 28:34.977 --> 28:37.357 V_S, are not the same as the 28:37.362 --> 28:41.022 V_S's here because they've all been concatenated by 28:41.019 --> 28:42.909 taking the minimum with 100. 28:42.910 --> 28:47.030 So the computer now is going to solve this by backward 28:47.028 --> 28:51.378 induction, and now how can you tell what the computer has 28:51.381 --> 28:52.781 decided to do? 28:52.779 --> 28:56.379 If the computer said it's a good time to call that means the 28:56.380 --> 29:00.040 value at that point is going to be 100 because it's called by 29:00.041 --> 29:02.851 paying the remaining balance which is 100. 29:02.848 --> 29:08.018 So if you go to the present value of non-call you get all 29:08.023 --> 29:10.153 these present values. 29:10.150 --> 29:12.670 As the interest rate goes down you get crushed, 29:12.674 --> 29:16.134 of course, because you have the same payments and lower interest 29:16.133 --> 29:18.223 rates, you're discounting by less. 29:18.220 --> 29:21.350 But the callable bond, see, you shouldn't call here. 29:21.348 --> 29:25.358 It's 99.9 so that means when the computer took the minimum 29:25.362 --> 29:28.602 100 in this thing it took this which was 99. 29:28.598 --> 29:33.958 It didn't use the option to call at 100, but if things go 29:33.963 --> 29:38.183 down again then it's going to call at 100. 29:38.180 --> 29:42.040 So it's got a threshold here where the interest rate has to 29:42.035 --> 29:44.955 go pretty far down before it calls at 100. 29:44.960 --> 29:47.660 It has to go down twice, but you notice this threshold. 29:47.660 --> 29:52.410 As time goes on the threshold is going to get tighter. 29:52.410 --> 29:55.080 So even if you're at the same interest rate four years, 29:55.078 --> 29:58.708 you know, this is 1,2, 3,4, this is 5 years later, 29:58.710 --> 30:03.250 and the interest rate's gone down once below the minimum bar 30:03.252 --> 30:05.792 here-- five years later you will call 30:05.788 --> 30:09.648 when the interest rate is, instead of 8 percent, 30:09.645 --> 30:13.325 is 6.8 percent, but 1 year later you won't call 30:13.326 --> 30:17.256 if it's 6.8 percent, and why is that? 30:17.259 --> 30:17.909 Well, why is that? 30:17.910 --> 30:20.920 How come if in the first year the interest rate went from-- 30:20.920 --> 30:24.300 remember it started at 8 percent and it goes down to 6.8 30:24.303 --> 30:28.993 percent here you don't call, on the other hand if things had 30:28.988 --> 30:33.738 stayed the same for a few years and then in year 5, 30:33.740 --> 30:36.740 or year whatever this--how many years have we got here, 30:36.740 --> 30:42.490 1,2, 3, no, year 0,1, 2,3, 4, so if in year 4 the 30:42.492 --> 30:48.252 interest rate goes down to-- oh, year 5 the interest rate 30:48.247 --> 30:50.247 goes down to 6.8 percent. 30:50.250 --> 30:53.510 In year 1 if the interest rate went down from 8 percent to 6.8 30:53.509 --> 30:56.719 percent you wouldn't call, but if the interest rate sort 30:56.721 --> 30:59.991 of bounces around and then in year 5 it's back down at 6.8 30:59.987 --> 31:01.417 percent you will call. 31:01.420 --> 31:04.180 Why would you call here even though in exactly the same 31:04.182 --> 31:05.872 situation you didn't call here? 31:05.869 --> 31:08.009 Why call here and not here? 31:08.009 --> 31:11.109 You can tell where you call because those are 100s and the 31:11.105 --> 31:13.545 computer's picked the minimum equal to 100. 31:13.548 --> 31:17.928 So why is it that it's going to call here, but not here even 31:17.932 --> 31:21.352 though the interest rates are exactly the same, 31:21.351 --> 31:23.581 6.8 percent in both cases? 31:23.579 --> 31:24.689 Yes? 31:24.690 --> 31:26.850 Student: Because you can't get to the kind of really 31:26.849 --> 31:28.899 high interest rates from that point five years later. 31:28.900 --> 31:29.770 There's not enough time left. 31:29.769 --> 31:30.639 Prof: Exactly. 31:30.640 --> 31:34.060 So the point is the only reason not to call here--you're paying 31:34.056 --> 31:35.926 through the nose, it's horrible. 31:35.930 --> 31:38.470 The interest rate is 6 percent, you're paying 9. 31:38.470 --> 31:40.660 It's terrible, but you're hoping maybe things 31:40.663 --> 31:42.013 are going to turn around. 31:42.009 --> 31:44.519 The interest rate's going to soar then it's going to be a 31:44.521 --> 31:45.421 great deal for me. 31:45.420 --> 31:49.400 So I'm willing to have some short run losses in case I get a 31:49.398 --> 31:52.858 gigantic gain on the upside, and I know that I'm protected 31:52.862 --> 31:55.512 on the downside because if things go down again I can cut 31:55.510 --> 31:57.120 my losses and just call it 100. 31:57.118 --> 32:01.658 But if you run out of time, here the interest rate's 6.8, 32:01.660 --> 32:05.130 it's still way below 9 percent which is the coupon you're 32:05.134 --> 32:08.864 paying and you don't have much time for the interest rates to 32:08.856 --> 32:11.456 go back up, you better call then and cut 32:11.455 --> 32:12.805 your losses, exactly. 32:12.808 --> 32:15.118 So that's the whole logic of the thing. 32:15.119 --> 32:16.639 So any questions about that? 32:16.640 --> 32:21.810 And as I said, the option is worth much more 32:21.807 --> 32:23.727 than it seems. 32:23.730 --> 32:29.280 Could you all do a problem like this? 32:29.279 --> 32:34.929 You'll find out. 32:34.930 --> 32:40.790 So now let me just do one more example. 32:40.789 --> 32:42.359 I want to do an aside here. 32:42.358 --> 32:48.938 Suppose that we had a trinomial tree instead of binomial tree. 32:48.940 --> 32:53.350 So I have the same picture, but instead of two things 32:53.351 --> 32:58.021 happening I say three things could happen or five things 32:58.017 --> 32:59.457 could happen. 32:59.460 --> 33:02.850 So I would take the same tree and put this here. 33:02.848 --> 33:06.838 I'd always allow for something happening in the middle. 33:06.838 --> 33:08.718 And then, of course, it gets more complicated and 33:08.720 --> 33:11.150 I'm not going to be able to draw it because now this thing, 33:11.150 --> 33:13.490 three things could happen from here, 33:13.490 --> 33:15.880 and from here three things could happen. 33:15.880 --> 33:19.370 From here I have three things, so I have trinomial tree like 33:19.373 --> 33:22.573 that, right, where always three things could happen. 33:22.568 --> 33:27.758 So from here I could go here, stay the same or go up. 33:27.759 --> 33:31.399 Now, if I had a trinomial tree, and of course you could think 33:31.404 --> 33:34.384 of an N-nomial tree, would I keep the same numbers 33:34.383 --> 33:34.933 here? 33:34.930 --> 33:37.520 Well, I can't keep the same numbers because if I have 1 half 33:37.523 --> 33:40.213 and 1 half that doesn't allow any probability for going in the 33:40.205 --> 33:40.685 middle. 33:40.690 --> 33:46.610 So I'll put 1 quarter here, and 1 quarter. 33:46.608 --> 33:50.148 Now, what do I have to do to these numbers? 33:50.150 --> 33:53.820 Well, the drift I'm going to leave the same because on 33:53.819 --> 33:56.659 average it's still going to go up by d. 33:56.660 --> 34:01.850 So this was, remember, r_0 e to 34:01.853 --> 34:04.453 the minus sigma d. 34:04.450 --> 34:06.660 So this is the probability 1 half. 34:06.660 --> 34:09.240 I just made up three numbers, 1 quarter, 1 quarter and 1 half 34:09.242 --> 34:10.322 for the trinomial tree. 34:10.320 --> 34:13.080 Suppose those are my numbers? 34:13.079 --> 34:16.489 If I want to make this tree comparable to the binomial 34:16.485 --> 34:19.245 tree--what does it mean to be comparable? 34:19.250 --> 34:24.510 It means that the standard deviation and the expectation of 34:24.510 --> 34:28.230 the interest rate ought to be the same. 34:28.230 --> 34:32.370 So I'm using the fact that if they're normally distributed 34:32.367 --> 34:35.197 random variables all I need to know-- 34:35.199 --> 34:41.409 so if there were millions of successor interest rates and I 34:41.414 --> 34:45.404 took them with this-- if there are millions of these 34:45.398 --> 34:48.538 successor interest rates then as long as the standard deviation 34:48.536 --> 34:50.456 and the expectation were the same, 34:50.460 --> 34:52.940 no matter how I put the probabilities I'd get almost the 34:52.940 --> 34:55.550 same variable because it would be normally distributed with 34:55.554 --> 34:57.814 that standard deviation and that expectation, 34:57.809 --> 35:01.289 and normally distributed, as I said, 35:01.289 --> 35:04.819 means it's determined by standard deviation and 35:04.815 --> 35:05.885 expectation. 35:05.889 --> 35:15.709 So I've just picked 1 quarter, 1 half and 1 quarter. 35:15.710 --> 35:18.820 It's as if I had two moves where you could get two ups to 35:18.818 --> 35:21.868 go here, an up or a down, or a down or an up to go here, 35:21.869 --> 35:23.479 and two downs to go here. 35:23.480 --> 35:25.230 That's why a picked 1 quarter, 1 half and 1 quarter. 35:25.230 --> 35:30.930 It's like I added 2 six-month moves here and called it a 35:30.929 --> 35:32.379 1-year move. 35:32.380 --> 35:35.820 So if I keep dividing the process and having it happen 35:35.818 --> 35:38.898 quicker and quicker, by the central limit theorem 35:38.898 --> 35:41.908 I'm going to get something normally distributed with a 35:41.905 --> 35:44.455 corresponding standard deviation and mean. 35:44.460 --> 35:50.770 So what is going to end up as the standard deviation here and 35:50.773 --> 35:51.513 here? 35:51.510 --> 35:55.810 Well, I know to get the same variance before, 35:55.809 --> 36:00.849 I had 1 half sigma squared 1 half sigma squared that equaled 36:00.847 --> 36:05.987 sigma squared, but now my variance is going to 36:05.989 --> 36:09.979 be 1 quarter sigma-- this is the new sigma hat 36:09.980 --> 36:13.320 squared, 1 quarter sigma hat squared because I'll go down 36:13.318 --> 36:16.898 whatever this sigma, so now I should put a sigma hat 36:16.902 --> 36:17.322 here. 36:17.320 --> 36:19.110 So 1 quarter sigma hat squared. 36:19.110 --> 36:23.480 So if I look at the variance of the exponent it's going to be a 36:23.483 --> 36:27.793 1 quarter sigma hat squared 1 quarter sigma hat squared 1 half 36:27.786 --> 36:31.946 times 0 because half the time you just get the average. 36:31.949 --> 36:39.219 So therefore it means 1 half sigma hat squared has to equal 36:39.215 --> 36:41.215 sigma squared. 36:41.219 --> 36:45.379 So therefore sigma hat squared has to equal 2 sigma squared, 36:45.380 --> 36:49.750 and therefore sigma hat has to equal the square root of 2 times 36:49.753 --> 36:50.463 sigma. 36:50.460 --> 37:01.440 So I should put the square root of 2 here and the square root of 37:01.440 --> 37:03.010 2 here. 37:03.010 --> 37:08.310 So now if I do a trinomial, for the trinomial to be similar 37:08.306 --> 37:14.056 to the binomial the binomial would be sort of here and here, 37:14.059 --> 37:16.069 and the trinomial there's a lot probability, 37:16.070 --> 37:17.110 you're stuck in the middle. 37:17.110 --> 37:19.530 So to get the same kind of average spread I'm going to have 37:19.530 --> 37:21.740 to have this thing sticking further out and this thing 37:21.744 --> 37:24.314 sticking further out, but if I choose my numbers by 37:24.307 --> 37:26.497 multiplying by the square root of 2 here, 37:26.500 --> 37:29.960 and the square root of 2 here, the standard deviation of this 37:29.958 --> 37:33.008 trinomial is the same as the standard deviation of the 37:33.014 --> 37:36.304 binomial and the expectation of the two are the same. 37:36.300 --> 37:39.370 And I could do the same thing with an N-nomial or an 37:39.371 --> 37:43.221 any-number thing I wanted to and I could always pick the standard 37:43.224 --> 37:46.474 deviation properly, pick these nodes properly so 37:46.472 --> 37:50.212 that I had the same standard deviation and expectation as I 37:50.208 --> 37:52.718 did with the original binomial tree. 37:52.719 --> 37:55.789 So just because I've got three nodes or five nodes on average 37:55.788 --> 37:57.628 they can turn out to be the same, 37:57.630 --> 38:00.840 and the average spread, the average spread squared can 38:00.840 --> 38:02.960 also be the same as the binomial. 38:02.960 --> 38:10.020 And if I do that I'm going to get a shockingly similar answer. 38:10.019 --> 38:14.759 So let's just see what we get. 38:14.760 --> 38:16.560 So what do I have to do now? 38:16.559 --> 38:19.229 I have to go to trinomial. 38:19.230 --> 38:20.940 So what did we do here? 38:20.940 --> 38:25.630 We did--9 percent was the coupon, and the interest rates 38:25.628 --> 38:29.718 start at 8 percent, and the volatility was 16. 38:29.719 --> 38:37.889 So if I go to trinomial, I put 9 percent to start, 38:37.889 --> 38:42.079 30 years, 8 percent the starting rate and volatility is 38:42.083 --> 38:44.233 16, and now I have to do the up 38:44.230 --> 38:46.760 multiplier, and by doing that you see I've 38:46.762 --> 38:50.152 multiplied up there in that formula there's a square root of 38:50.150 --> 38:51.890 2 there, 2 square root of 2. 38:51.889 --> 38:55.309 So I've got just a slightly more complicated thing. 38:55.309 --> 38:58.409 I've just multiplied by the square root of 2 and that's how 38:58.405 --> 39:00.865 I've got all my nodes, so otherwise it's the same 39:00.871 --> 39:03.441 thing and I do the same backward induction except instead of 39:03.440 --> 39:06.740 being 1 half, 1 half I put 1 quarter here, 39:06.744 --> 39:10.294 and 1 quarter here, so it'll be 1 quarter 39:10.286 --> 39:14.056 V_SU 1 quarter V_SD 1 half, 39:14.059 --> 39:18.179 you stay the same, which will be 9 39:18.177 --> 39:24.027 V_Ssame, whatever that is--so I probably 39:24.027 --> 39:25.067 lost you. 39:25.070 --> 39:29.460 So let me just say I want to convince you that this binomial 39:29.461 --> 39:32.441 assumption is not such a special case. 39:32.440 --> 39:34.990 So you should be thinking to yourself, two things happening 39:34.994 --> 39:36.364 next year, that's ridiculous. 39:36.360 --> 39:38.530 I could imagine a million things happening next year. 39:38.530 --> 39:40.160 So I say, okay I believe it. 39:40.159 --> 39:41.949 A million things could happen next year, 39:41.949 --> 39:44.109 so I'm going to write down a million things, 39:44.110 --> 39:47.640 not any million things, I'm going to say let's suppose 39:47.643 --> 39:51.183 that instead of two things happening in a year I say I 39:51.177 --> 39:55.377 divide what can happen over the year into a million different up 39:55.378 --> 39:56.778 and down moves. 39:56.780 --> 39:59.490 Now, in order to replicate sort of the original binomial, 39:59.489 --> 40:02.609 those million up and down moves, let's say 250 of them, 40:02.610 --> 40:04.120 1 every day, those daily moves, 40:04.123 --> 40:05.963 of course, are going to be much smaller 40:05.960 --> 40:06.890 than the 1-year move. 40:06.889 --> 40:11.919 But if I make the drift be the old drift divided by 250, 40:11.920 --> 40:17.640 and the new standard deviation of the tiny things be the right 40:17.644 --> 40:22.544 ratio of the standard deviation, then when I compute the 40:22.541 --> 40:26.531 standard deviation of all the nodes here I'll get the same 40:26.534 --> 40:30.394 standard deviation as the binomial and the same drift in 40:30.387 --> 40:31.647 the binomial. 40:31.650 --> 40:35.210 So if I ended up with 3 nodes here I would just be multiplying 40:35.210 --> 40:36.730 by the square root of 2. 40:36.730 --> 40:41.070 And so by having a trinomial thing with a slightly adjusted 40:41.065 --> 40:43.535 move up and down, slightly spreading it and 40:43.541 --> 40:45.151 putting some probability in the middle, 40:45.150 --> 40:48.460 I can have the same variance and the same expectation. 40:48.460 --> 40:51.160 If I had thousands of nodes here I'd also have the same 40:51.155 --> 40:53.595 variance and the same expectation, and so I did it 40:53.601 --> 40:54.951 just for the trinomial. 40:54.949 --> 40:57.049 So let's see what happens to the value. 40:57.050 --> 41:01.680 If we look now at the interest rates, it was an uglier thing. 41:01.679 --> 41:05.699 I couldn't do it quite as--I was too lazy to do it as neatly, 41:05.699 --> 41:09.309 so the interest rates started at 8 percent and now they can go 41:09.311 --> 41:12.651 up to 10 or down to 6.3, and then the next period they 41:12.650 --> 41:14.330 can go up to 12 and 1 half. 41:14.329 --> 41:16.639 So you see there are different moves here. 41:16.639 --> 41:20.159 Remember the other one was 6.8 and 9, so if you don't get the 41:20.164 --> 41:23.574 same interest rate right away you move more violently up or 41:23.570 --> 41:24.100 down. 41:24.099 --> 41:26.379 So these three things, these two are spread out 41:26.378 --> 41:27.368 further than this. 41:27.369 --> 41:31.719 So what happens to the price? 41:31.719 --> 41:37.989 The present value of the non-call bond is 113.230 and 41:37.985 --> 41:41.595 what was it before, 113.236. 41:41.599 --> 41:43.249 It's not so different. 41:43.250 --> 41:48.700 And if I look now at the value of the callable bond it's 95.142 41:48.699 --> 41:53.709 and what was it before, 95.529, a little bit different. 41:53.710 --> 41:59.070 So you can see that the trinomial thing gives you almost 41:59.067 --> 42:02.377 the same answer as the binomial. 42:02.380 --> 42:05.880 So I don't want to say anything more than that. 42:05.880 --> 42:09.170 So no matter how many nodes you have in the tree the binomial's 42:09.172 --> 42:11.832 going to give a good approximation provided that we 42:11.827 --> 42:13.047 shrink the periods. 42:13.050 --> 42:15.840 Instead of looking at a year, looking at daily moves or 42:15.836 --> 42:18.776 minute by minute moves, as long as those are binomial 42:18.778 --> 42:21.038 over a year a million things can happen. 42:21.039 --> 42:24.089 And so you can restrict yourself to binomial without any 42:24.092 --> 42:27.592 loss of generality just taking the time period short enough, 42:27.590 --> 42:31.150 because if it's a daily binomial over a year there's 250 42:31.153 --> 42:35.243 outcomes or something and so for a minute there are thousands of 42:35.237 --> 42:39.387 outcomes over a year even though each move is binomial and you'll 42:39.385 --> 42:41.065 get the same answer. 42:41.070 --> 42:44.470 So that's that. 42:44.469 --> 42:47.769 All right, so we've done the callable bond idea, 42:47.771 --> 42:50.371 and the call option is worth a lot. 42:50.369 --> 42:53.329 Now, let's take a more concrete example, 42:53.329 --> 42:56.819 the one that you've all heard your parents talking about, 42:56.820 --> 43:00.880 if you don't own a house yourself, which is a mortgage. 43:00.880 --> 43:03.820 And I just want to do the same thing and help you start to 43:03.820 --> 43:05.730 think about how to value mortgages. 43:05.730 --> 43:07.480 So what's a mortgage? 43:07.480 --> 43:09.940 A mortgage is, you don't have the balloon 43:09.938 --> 43:11.228 payment at the end. 43:11.230 --> 43:28.590 You pay the same amount each period. 43:28.590 --> 43:29.790 Let's take a mortgage. 43:29.789 --> 43:32.529 So which example have I done here? 43:32.530 --> 43:40.040 So I've got a mortgage rate of 7 percent, a 7 percent coupon, 43:40.039 --> 43:45.169 you can see, and it's a 30 year fixed rate 43:45.172 --> 43:46.802 mortgage. 43:46.800 --> 43:54.610 Now, how much does that mean you have to pay every year? 43:54.610 --> 43:56.030 So, every year, what are you going to pay? 43:56.030 --> 43:57.330 You're not going to pay 7. 43:57.329 --> 43:59.069 What this means is, whatever X you pay-- 43:59.070 --> 44:02.990 every year that's your annual payment-- 44:02.989 --> 44:05.539 the 30th year you're still going to pay X, 44:05.539 --> 44:14.289 but if I took this all over 1.07 X over 1.07 squared this 44:14.286 --> 44:23.966 over 1.07 cubed this over 1.07 to the 30th I would get 100, 44:23.969 --> 44:25.909 equals 100. 44:25.909 --> 44:30.849 So that's how X is chosen, so that the mortgage is an 44:30.849 --> 44:36.259 annuity of constant X whose present value at the coupon is 44:36.264 --> 44:37.884 equal to 100. 44:37.880 --> 44:41.590 That's how you define the payment on a mortgage, 44:41.592 --> 44:45.702 but a mortgage has to be defined more completely than 44:45.699 --> 44:46.409 that. 44:46.409 --> 44:50.559 You also have to say what do you get by dissolving the 44:50.557 --> 44:53.687 mortgage, how much do you have to pay? 44:53.690 --> 44:56.650 So right away you have to pay 100. 44:56.650 --> 45:00.530 What would you have to pay here if you dissolved the mortgage 45:00.530 --> 45:08.060 right after the-- so remaining balance equals 45:08.056 --> 45:19.716 payment just after coupon necessary to dissolve the 45:19.719 --> 45:22.519 mortgage. 45:22.518 --> 45:26.588 So at the very beginning the guy gave you 100. 45:26.590 --> 45:29.720 If you want to dissolve it and undo it you should be able to do 45:29.722 --> 45:32.912 it for 100, but what about after the very first year you've paid 45:32.905 --> 45:33.205 X? 45:33.210 --> 45:40.440 By the way, X in this case is going to work out to 8.05. 45:40.440 --> 45:43.890 If you solve for X you get 8.05. 45:43.889 --> 45:48.259 That 8.05 discounted at 7 percent gives you 100. 45:48.260 --> 45:52.710 So obviously X has to be bigger than 7 because if it were 7 45:52.710 --> 45:57.240 everywhere you'd need 107 at the very end to make it 100. 45:57.239 --> 46:00.459 Of course at the very end that's 30 years later, 46:00.456 --> 46:04.146 so that extra 100 can be compensated by an extra 1,1.05 46:04.152 --> 46:05.662 all the way along. 46:05.659 --> 46:09.209 1.05 all the way along is just the same as an extra 100 at the 46:09.210 --> 46:09.560 end. 46:09.559 --> 46:11.449 So the payment is 8.05. 46:11.449 --> 46:14.659 So if you're paying a 7 percent mortgage your annual payment is 46:14.659 --> 46:16.419 going to be more than 7 percent. 46:16.420 --> 46:23.900 It's going to be 8.05. 46:23.900 --> 46:27.670 So that's the 7 percent coupon of the mortgage, 46:27.668 --> 46:29.388 coupon of mortgage. 46:29.389 --> 46:32.969 Now, if you want to get out of the mortgage it's called 46:32.965 --> 46:33.755 prepaying. 46:33.760 --> 46:36.290 What would you have to pay here to get out of the mortgage do 46:36.289 --> 46:36.879 you suppose? 46:36.880 --> 46:40.730 What's the only logical thing to have written in the contract? 46:40.730 --> 46:42.290 Student: Penalty. 46:42.289 --> 46:43.359 Prof: Well, you could pay a penalty, 46:43.356 --> 46:44.546 but what if you don't want to have a penalty? 46:44.550 --> 46:48.220 Then what should the guy pay to get out of the mortgage? 46:48.219 --> 46:48.789 Yeah? 46:48.789 --> 46:49.909 Student: The remaining balance. 46:49.909 --> 46:51.379 Prof: The remaining balance, yes, 46:51.380 --> 46:52.740 but what is the remaining balance? 46:52.739 --> 46:55.239 That's the question, what is the remaining balance. 46:55.239 --> 46:56.939 Well, maybe you know what the answer is. 46:56.940 --> 46:59.170 Student: It's the present value at the time. 46:59.170 --> 47:00.600 Prof: Of what's left. 47:00.599 --> 47:03.429 So the guy here, the remaining balance, 47:03.429 --> 47:06.779 B_1, ought to be this number without 47:06.778 --> 47:07.448 this. 47:07.449 --> 47:10.669 You've only got 29 payments, only you're going to put a 1 47:10.672 --> 47:11.712 here now, right? 47:11.710 --> 47:15.720 So it's B_1 squared and B to the 29th because you've 47:15.717 --> 47:17.817 made the first coupon payment. 47:17.820 --> 47:21.350 You've got 29 years left assuming the same terms as 47:21.351 --> 47:24.251 before it still is a 7 percent coupon, 47:24.250 --> 47:28.520 so you should discount the remaining 29 years at 7 percent 47:28.516 --> 47:30.236 to get B_1. 47:30.239 --> 47:32.679 So that's what B_1 is. 47:32.679 --> 47:39.609 So as you can see in the thing B_1 is 98.94. 47:39.610 --> 47:43.960 Now, 98.94 is an interesting number. 47:43.960 --> 47:57.630 You notice that the 8.05 minus 7 happens to equal--that plus 47:57.630 --> 48:10.840 B_1 has to equal 100, which is B_0. 48:10.840 --> 48:12.660 So why is that? 48:12.659 --> 48:15.589 So you started owing the guy 100. 48:15.590 --> 48:17.290 You've borrowed 100, you own him 100, 48:17.293 --> 48:19.713 and he's sort of charging you 7 percent interest. 48:19.710 --> 48:21.140 That's the way the deal works. 48:21.139 --> 48:24.249 So you would have expected to only pay 7 dollars next period. 48:24.250 --> 48:26.790 That's to keep up with the 7 percent interest. 48:26.789 --> 48:31.559 Remember, why did the old mortgages go from the balloon 48:31.559 --> 48:33.149 payment to this? 48:33.150 --> 48:37.230 Because in the Depression in 1933 every single farmer, 48:37.226 --> 48:40.296 practically, who owed the 109 because his 48:40.302 --> 48:43.152 mortgage was coming up defaulted. 48:43.150 --> 48:46.580 So the lenders decided they didn't want to be facing that 48:46.579 --> 48:48.599 situation where a guy owed 109. 48:48.599 --> 48:51.299 They'd rather have the guy pay X every period where there's 48:51.297 --> 48:53.947 never this gigantic payment that he's going to default. 48:53.949 --> 48:57.869 In fact, by paying X every period that lender's in a safer 48:57.867 --> 48:59.377 and safer situation. 48:59.380 --> 49:00.410 Why is that? 49:00.409 --> 49:04.569 Because he's asking for X which is 8 dollars even though the 49:04.570 --> 49:07.950 interest is 7 dollars, so the homeowner is paying 49:07.954 --> 49:08.594 8.05. 49:08.590 --> 49:09.800 The interest was only 7. 49:09.800 --> 49:13.690 So he's making an extra payment of 1.05 dollars. 49:13.690 --> 49:15.990 So what do you do with the extra 1.05 dollars? 49:15.989 --> 49:17.269 You write down the balance. 49:17.268 --> 49:19.538 You say you've overpaid me by 1.05. 49:19.539 --> 49:21.699 I'm no longer going to say you owe 100. 49:21.699 --> 49:27.959 You're going to owe, now, 100 - 1.05. 49:27.960 --> 49:33.630 1.05 had a few decimal places after it, and so it's going to 49:33.632 --> 49:34.982 be 98.94136. 49:34.980 --> 49:39.230 So that's this gap, so B_0 - 49:39.231 --> 49:41.071 B_1. 49:41.070 --> 49:46.610 The balance went down by exactly what the overpayment was 49:46.605 --> 49:48.775 beyond the interest. 49:48.780 --> 49:52.310 And so the next time the balance is going to go down even 49:52.309 --> 49:55.339 further because the next time the balance is only 49:55.335 --> 49:56.465 B_1. 49:56.469 --> 50:00.089 The next time what the guy owes--if he was just paying the 50:00.085 --> 50:03.695 interest he ought to owe only .07 times B_1, 50:03.699 --> 50:06.819 and yet he's still going to be paying 8.05, 50:06.820 --> 50:10.450 so B_1 - B_2. 50:10.449 --> 50:13.429 So the balance B_2 is going to go down even more, 50:13.434 --> 50:16.064 so the gap from 100 to B_1 is a very small 50:16.057 --> 50:16.467 one. 50:16.469 --> 50:19.459 It's going to be bigger from B_1 to B_2 50:19.458 --> 50:22.288 and keep getting bigger and bigger because each time the 50:22.293 --> 50:24.533 guy's paying 8.05, but what he owes, 50:24.525 --> 50:27.675 the balance that he owes on, that he's basically borrowed 50:27.684 --> 50:29.894 the money on is a smaller and smaller number, 50:29.889 --> 50:33.069 so he's overpaying the 7 percent interest by a bigger and 50:33.065 --> 50:36.465 bigger amount and the balance goes down by more and more each 50:36.469 --> 50:36.979 time. 50:36.980 --> 50:41.630 All right, to put it another way, if you take the first thing 50:41.626 --> 50:45.186 is, B_1 is X times 1.07,1.07 squared, 50:45.188 --> 50:46.658 1.07 to the 29. 50:46.659 --> 50:51.569 B_2 is X 1.07 blah, blah, blah, X times 1.07 to the 50:51.567 --> 50:52.137 28th. 50:52.139 --> 50:55.409 So the difference between this and this is the last payment 50:55.414 --> 50:57.394 which is 1 over 1.07 to the 29th. 50:57.389 --> 51:01.039 The difference between this and this is the last payment which 51:01.043 --> 51:02.783 is 1 over 1.07 to the 30th. 51:02.780 --> 51:06.460 And so as that final maturity gets smaller and smaller the gap 51:06.460 --> 51:09.480 gets bigger and bigger, so I'm just saying the same 51:09.478 --> 51:11.168 thing in different ways. 51:11.170 --> 51:13.740 So are there any questions about this? 51:13.739 --> 51:16.629 Did I go too fast or are you with me? 51:16.630 --> 51:18.230 Somebody ask a question if you're lost. 51:18.230 --> 51:19.090 Yes? 51:19.090 --> 51:21.740 Student: Why don't we discount B_1? 51:21.739 --> 51:22.729 Prof: Why do I what? 51:22.730 --> 51:25.140 Student: Discount B_1? 51:25.139 --> 51:26.759 Prof: I'm sorry, why do I--say it louder. 51:26.760 --> 51:27.680 Student: Discount. 51:27.679 --> 51:29.059 Prof: Why do I discount what? 51:29.059 --> 51:31.989 Student: Why don't we because you are subtracting the 51:31.987 --> 51:34.267 amount from the 0 >? 51:34.268 --> 51:36.428 Prof: Yes this, so let me say this again. 51:36.429 --> 51:40.399 You borrowed 100. 51:40.400 --> 51:45.950 The coupon is the agreed upon interest, so you should owe the 51:45.951 --> 51:50.111 guy 7 dollars, or 107 dollars the next period, 51:50.114 --> 51:51.044 right? 51:51.039 --> 51:51.989 Not 100 any more. 51:51.989 --> 51:52.949 There's 7 percent interest. 51:52.949 --> 51:56.579 Now you owe the guy 107, right? 51:56.579 --> 52:02.379 But at that moment when you owed him 107 you paid him 8.05. 52:02.380 --> 52:05.940 So the guy says, look, I lent you 100, 52:05.938 --> 52:10.648 I expect 107 back next period, you pay me 8.05. 52:10.650 --> 52:18.960 That's 8.05, so what's left that you owe me? 52:18.960 --> 52:23.670 It's 100 minus the 1.05. 52:23.670 --> 52:32.550 So 98.4 something, so let me just say it again. 52:32.550 --> 52:33.870 You borrowed 100. 52:33.869 --> 52:36.669 The next year you owe 107. 52:36.670 --> 52:39.200 We've already taken into account the interest rate. 52:39.199 --> 52:41.449 It's 107, so we're sitting in next year. 52:41.449 --> 52:42.749 We're no longer sitting back here. 52:42.750 --> 52:45.200 B_1 is as of next year. 52:45.199 --> 52:48.989 So you should really by rights be owed now, the lender should 52:48.987 --> 52:52.017 be owed 107 now because he lent 100 last year. 52:52.018 --> 52:55.678 The agreed upon interest is 7 percent so he should be getting 52:55.682 --> 52:56.662 107 this year. 52:56.659 --> 52:57.819 What is he getting? 52:57.820 --> 53:05.860 He's getting 8.05, so what's left is 107 - 8.05 53:05.858 --> 53:13.368 and that's what we have, what's left is 107, 53:13.373 --> 53:16.873 is exactly that. 53:16.869 --> 53:21.409 Right, so this is 7 he got paid, so if I write it this way 53:21.414 --> 53:25.934 107 he got paid 8.05, so 107 I'm just rewriting the 53:25.925 --> 53:30.925 same thing, 107 - 8.05 is going to be the 53:30.925 --> 53:36.315 new B_1, which just happens to equal 90 53:36.324 --> 53:45.704 whatever that is, 98.05, something like that, 53:45.702 --> 53:49.662 98.95, right? 53:49.659 --> 53:53.909 So this is 1.05, so 1.05 plus this is 100. 53:53.909 --> 53:57.399 So 107 - 8.05 is 98.95. 53:57.400 --> 53:58.600 You agree with that, right? 53:58.599 --> 53:59.259 Student: Yeah. 53:59.260 --> 54:02.140 Prof: So what I'm saying is a year later you were owed 54:02.141 --> 54:02.431 107. 54:02.429 --> 54:05.579 You were paid 8.05, so obviously what's left that 54:05.579 --> 54:09.449 you're owed is the difference, 98.95 and that's exactly what 54:09.449 --> 54:12.139 the equation was that I wrote before. 54:12.139 --> 54:17.709 Any other questions? 54:17.710 --> 54:21.100 So you can always divide a mortgage payment into the 54:21.099 --> 54:25.429 interest part-- so any mortgage payment X can 54:25.425 --> 54:31.925 be divided into the interest so now let's erase this and I'm 54:31.925 --> 54:35.005 saying it in other words. 54:35.010 --> 54:37.990 So your payments are X everywhere to the last period 54:37.994 --> 54:41.454 and your remaining balance is going to be B_1, 54:41.449 --> 54:43.969 B_2, B_3 and 54:43.972 --> 54:47.802 B_30 which is obviously equal to 0. 54:47.800 --> 54:51.240 So you can always get out of the mortgage by paying--after 54:51.240 --> 54:54.740 you make a coupon payment X you can get out of it by paying 54:54.742 --> 54:55.832 B_2. 54:55.829 --> 55:04.589 So your mortgage payment is going to be divided into the 55:04.590 --> 55:13.350 interest, and then the scheduled principal reduction. 55:13.349 --> 55:15.409 So here you pay 8.05. 55:15.409 --> 55:18.879 The interest was only 7 percent so you've overpaid. 55:18.880 --> 55:20.470 So that's why the balance goes down. 55:20.469 --> 55:23.569 That's the scheduled balance--It's called principal, 55:23.572 --> 55:25.642 but scheduled balance reduction. 55:25.639 --> 55:30.049 And if you want you can prepay and pay off part of the 55:30.047 --> 55:33.537 remaining balance, any part you want to. 55:33.539 --> 55:35.189 So let me say it again. 55:35.190 --> 55:38.200 In some period, let's say 3, 55:38.201 --> 55:41.771 what could you do in period 3? 55:41.768 --> 55:45.588 Well, by the time you've waited to period 3 you've got to pay 55:45.594 --> 55:46.554 the coupon X. 55:46.550 --> 55:51.560 Now, the coupon X is way in excess of 7 percent times 55:51.557 --> 55:53.097 B_2. 55:53.099 --> 55:56.639 It's 8 and 7 percent of 100 is less than X. 55:56.639 --> 55:58.569 We're already down to a balance of B_2. 55:58.570 --> 56:01.910 So the X is bigger than 7 percent of B_2. 56:01.909 --> 56:04.529 So 7 percent of B_2 is the interest, 56:04.525 --> 56:06.965 but on top of that you paid more than that, 56:06.965 --> 56:10.275 so that extra you paid reduced the remaining balance. 56:10.280 --> 56:13.480 That's the scheduled reduction in the remaining balance. 56:13.480 --> 56:14.190 You paid that off. 56:14.190 --> 56:16.610 You had no choice about that, but now in addition, 56:16.608 --> 56:18.628 if you want, you can pay the B_3 56:18.632 --> 56:20.512 and get rid of the whole mortgage. 56:20.510 --> 56:22.010 That's the rules of a mortgage. 56:22.010 --> 56:23.920 Is that clear to everybody? 56:23.920 --> 56:32.370 So any mortgage that you get, works by those rules. 56:32.369 --> 56:36.189 So we have to figure out the value of the mortgage. 56:36.190 --> 56:37.710 Somehow I feel I'm going too fast. 56:37.710 --> 56:40.380 Could someone ask a question if you're lost? 56:40.380 --> 56:41.800 I need to pick out--yes? 56:41.800 --> 56:44.570 Student: So what you just said applies that as time 56:44.572 --> 56:47.342 goes on you pay a bigger and bigger chunk of the principal 56:47.344 --> 56:48.954 with each successive payment? 56:48.949 --> 56:49.599 Prof: Yes. 56:49.599 --> 56:52.609 So the mortgage is called an amortizing mortgage. 56:52.610 --> 56:55.450 This was a great invention at the time of the Depression, 56:55.445 --> 56:56.605 after the Depression. 56:56.610 --> 56:59.410 It probably had been invented before, but was used in a big 56:59.409 --> 57:00.229 way after that. 57:00.230 --> 57:02.640 I don't know the person's name who invented it. 57:02.639 --> 57:08.089 First it has the property that you're always paying more. 57:08.090 --> 57:10.850 Because you're paying level payments you must always be 57:10.847 --> 57:12.327 paying your principal down. 57:12.329 --> 57:15.379 That makes the lender safer and safer because it's the same 57:15.380 --> 57:17.590 house protecting his loan as collateral. 57:17.590 --> 57:20.950 We haven't come to collateral and how the whole world depends 57:20.947 --> 57:23.577 on collateral yet, but believe me we're going to 57:23.576 --> 57:24.356 get there. 57:24.360 --> 57:26.530 And so we haven't gotten to collateral yet, 57:26.528 --> 57:29.158 but when we get to it, the lender's protected by the 57:29.163 --> 57:29.683 house. 57:29.679 --> 57:32.409 The amount that's owed to him started at 100. 57:32.409 --> 57:35.889 It's getting lower every period so he's feeling safer and safer. 57:35.889 --> 57:38.779 That contrasts to the old lenders during the Depression 57:38.782 --> 57:41.572 who were at the most vulnerable right at the end-- 57:41.570 --> 57:44.670 oh, I've lost it--most vulnerable right at the end. 57:44.670 --> 57:49.240 Here these lenders are getting less and less vulnerable as time 57:49.242 --> 57:49.982 goes on. 57:49.980 --> 57:52.080 So that's the advantage of an amortizing mortgage. 57:52.079 --> 57:56.309 Now, it so happens that the rate of amortization gets faster 57:56.307 --> 57:57.237 and faster. 57:57.239 --> 58:00.179 So these numbers start at 100 and stay pretty close to 100, 58:00.179 --> 58:03.169 and then they go down pretty fast, which is what his point 58:03.173 --> 58:04.773 is, and that's absolutely right. 58:04.768 --> 58:09.298 And so if you look at the remaining balances here in this 58:09.297 --> 58:13.337 chart it started at 100, went to 98, went to 97,96, 58:13.340 --> 58:15.120 95, now 93,92, 90. 58:15.119 --> 58:17.639 They're going down faster and faster. 58:17.639 --> 58:22.659 If you keep going you're going to see that they start leaping 58:22.659 --> 58:25.169 down from 76 to 73,73 to 70. 58:25.170 --> 58:28.710 Now they're starting to go from 43 to 38. 58:28.710 --> 58:30.860 They're really, by percentages, 58:30.862 --> 58:34.382 going way down really fast, 21 to 14 to 7 to 0. 58:34.380 --> 58:37.850 So they go down really fast at the end, the remaining balance. 58:37.849 --> 58:42.189 But that's just the way the amortizing fixed interest 58:42.190 --> 58:43.610 mortgage works. 58:43.610 --> 58:47.610 And now we want to think about how to value it, 58:47.614 --> 58:48.664 all right? 58:48.659 --> 58:53.649 So the payment is 8, the coupon that they agreed on 58:53.652 --> 58:58.982 in the mortgage was 7 percent, but that's different from what 58:58.981 --> 59:02.441 the interest rate is in the whole economy which is 6 59:02.436 --> 59:03.516 percent, say. 59:03.518 --> 59:05.698 So suppose the interest rate is 6 percent. 59:05.699 --> 59:09.949 Why would it be that with a 6 percent interest rate today, 59:09.949 --> 59:13.309 one year interest rate today, why would it be that if the 59:13.309 --> 59:15.529 interest rate was 6 percent today-- 59:15.530 --> 59:20.850 too complicated, this tree--if the interest rate 59:20.853 --> 59:27.423 were 6 percent today why would it be that the mortgage rate 59:27.423 --> 59:33.043 that people would agree to-- so it could be 6 percent--why 59:33.041 --> 59:36.681 is it that the mortgage rate people could agree to-- 59:36.679 --> 59:42.719 so this is 7.3 percent and this went down to 4.9 percent. 59:42.719 --> 59:46.069 That's in that tree. 59:46.070 --> 59:48.400 So you tell me, why is it that when the 59:48.396 --> 59:50.596 interest rate is 6 percent today-- 59:50.599 --> 59:53.649 so everybody knows on average it's going to stay around 6 59:53.650 --> 59:55.710 percent, and here I want to get a loan 59:55.710 --> 59:58.830 from a bank to buy a house and I put up my spectacular house as 59:58.833 --> 1:00:01.003 collateral so the bank should feel safe. 1:00:01.000 --> 1:00:02.620 It's amortizing and all that. 1:00:02.619 --> 1:00:05.789 They should feel safe as long as they haven't lent me too much 1:00:05.789 --> 1:00:06.569 money, right? 1:00:06.570 --> 1:00:09.240 By the way, if my house is only worth 80 dollars and they lend 1:00:09.239 --> 1:00:10.859 me 100 then the bank is in trouble. 1:00:10.860 --> 1:00:14.230 If the house is worth 100 and they lend me 100 the bank might 1:00:14.228 --> 1:00:17.538 be in some vulnerable situation if the house loses value. 1:00:17.539 --> 1:00:19.859 So banks would have to be stupid, as they were, 1:00:19.860 --> 1:00:23.040 to make loans of the home value almost equal to the house value, 1:00:23.039 --> 1:00:25.109 but we're going to come back to that. 1:00:25.110 --> 1:00:28.560 So suppose the house is worth way more than the 100 they lend 1:00:28.561 --> 1:00:28.851 me. 1:00:28.849 --> 1:00:31.549 The amount I'm owing the bank is only going down, 1:00:31.550 --> 1:00:33.930 so the house is protecting the bank completely, 1:00:33.929 --> 1:00:37.459 has no worries of me defaulting because it'll just take my 1:00:37.463 --> 1:00:37.963 house. 1:00:37.960 --> 1:00:40.290 Let's take that case. 1:00:40.289 --> 1:00:44.269 The interest rate in the economy is 6 percent and here 1:00:44.266 --> 1:00:48.016 the bank is charging me 7 percent mortgage rate. 1:00:48.018 --> 1:00:50.218 That's what we agree on, and of course when it charges 1:00:50.219 --> 1:00:52.629 me 7 percent mortgage rate I'm going to be paying 8 dollars 1:00:52.626 --> 1:00:53.286 every month. 1:00:53.289 --> 1:00:55.359 How could that possibly be fair? 1:00:55.360 --> 1:00:58.850 Or to put it another way, why does the bank have to 1:00:58.847 --> 1:01:03.027 charge a mortgage rate that's way above the starting interest 1:01:03.032 --> 1:01:03.662 rate? 1:01:03.659 --> 1:01:05.509 Why would it have to do that? 1:01:05.510 --> 1:01:10.330 Why does it do it and why is everyone willing to pay it? 1:01:10.329 --> 1:01:11.899 You see what the question is? 1:01:11.900 --> 1:01:13.120 Back there, yes? 1:01:13.119 --> 1:01:15.319 Student: You have the option of prepaying it. 1:01:15.320 --> 1:01:15.890 Prof: Right. 1:01:15.889 --> 1:01:19.469 I've got this really valuable option, which of course the bank 1:01:19.465 --> 1:01:20.105 realizes. 1:01:20.110 --> 1:01:21.530 I can always get out of it. 1:01:21.530 --> 1:01:24.150 I can pay the remaining balance and get out of it. 1:01:24.150 --> 1:01:28.130 Now, what am I hoping for when I take my 7 percent mortgage out 1:01:28.132 --> 1:01:29.162 from the bank? 1:01:29.159 --> 1:01:30.799 What am I hoping for? 1:01:30.800 --> 1:01:33.760 How am I going to make money, the homeowner? 1:01:33.760 --> 1:01:35.230 Yes? 1:01:35.230 --> 1:01:37.030 Student: If the interest rate rises. 1:01:37.030 --> 1:01:39.180 Prof: Right, so if there's some tremendous 1:01:39.182 --> 1:01:41.382 inflation like there might well be now, 1:01:41.380 --> 1:01:43.730 some tremendous inflation in the next 5 years-- 1:01:43.730 --> 1:01:45.160 this is a 30-year loan. 1:01:45.159 --> 1:01:48.329 If in the next 5 years the economy's in so much trouble and 1:01:48.331 --> 1:01:51.611 we've got all these homeowners under water and we owe so much 1:01:51.612 --> 1:01:54.842 money to the Chinese we could well see the Fed engineering a 1:01:54.838 --> 1:01:57.518 gigantic inflation, so the interest rate might go 1:01:57.518 --> 1:01:58.298 up to 10 percent. 1:01:58.300 --> 1:02:03.070 I'm continuing to borrow effectively at 7 percent from 1:02:03.070 --> 1:02:04.060 the bank. 1:02:04.059 --> 1:02:05.179 How can you see that? 1:02:05.179 --> 1:02:06.789 Because this remaining balance, remember, 1:02:06.789 --> 1:02:09.499 the whole way it's structured is, the payments, 1:02:09.500 --> 1:02:12.330 after a year, the 100 I borrowed goes down to 1:02:12.333 --> 1:02:14.403 B_1, but it's like a 20 [correction: 1:02:14.400 --> 1:02:15.240 29] year mortgage where I've 1:02:15.244 --> 1:02:17.724 borrowed B_1, again, at 7 percent interest. 1:02:17.719 --> 1:02:20.899 After the third year, right, just after making my 1:02:20.896 --> 1:02:25.066 payment the 27 years left of the mortgage are treated explicitly 1:02:25.068 --> 1:02:29.308 by the law of the contract as if I borrowed B_3 dollars 1:02:29.306 --> 1:02:31.156 at 7 percent interest. 1:02:31.159 --> 1:02:33.859 So in other words, a mortgage is like a contract 1:02:33.864 --> 1:02:36.634 where every year I get to re-borrow at 7 percent, 1:02:36.625 --> 1:02:39.555 but I'm borrowing a smaller and smaller amount. 1:02:39.559 --> 1:02:41.719 So I perpetually get to borrow at 7 percent, 1:02:41.719 --> 1:02:44.329 but I have the option of canceling the whole thing by 1:02:44.331 --> 1:02:46.191 paying off the remaining balance. 1:02:46.190 --> 1:02:48.980 So I'm hoping that the interest rate's going to go to 12 percent 1:02:48.978 --> 1:02:51.278 and I'm going to be borrowing still at 7 percent, 1:02:51.280 --> 1:02:53.940 borrowing at 7 percent and reinvesting it at 12 percent. 1:02:53.940 --> 1:02:54.890 I'll make a killing. 1:02:54.889 --> 1:02:57.309 I'm praying for interest rates to go up. 1:02:57.309 --> 1:02:59.539 Now, most homeowners think that they're praying for interest 1:02:59.541 --> 1:03:01.131 rates to go down, but they just have it all 1:03:01.130 --> 1:03:01.660 backwards. 1:03:01.659 --> 1:03:04.269 They think if interest rates go down they'll get a lower 1:03:04.271 --> 1:03:05.841 mortgage with a lower interest. 1:03:05.840 --> 1:03:08.370 They don't realize that the present value of their future 1:03:08.369 --> 1:03:09.769 payments, are going to go up. 1:03:09.768 --> 1:03:12.388 So they should be hoping for the interest rates to go up. 1:03:12.389 --> 1:03:15.199 They should be hoping for a big inflation and that's when 1:03:15.202 --> 1:03:16.712 they're going to make money. 1:03:16.710 --> 1:03:18.950 And the option that the homeowner gains if interest 1:03:18.945 --> 1:03:21.355 rates go up and the homeowner can always get out of the 1:03:21.360 --> 1:03:24.130 mortgage and pay it off if the interest rates go down by paying 1:03:24.134 --> 1:03:28.094 off the remaining balance, that's tremendously valuable. 1:03:28.090 --> 1:03:31.710 So how do we compute that value? 1:03:31.710 --> 1:03:35.180 We have to do it by backward induction. 1:03:35.179 --> 1:03:38.519 You can intuitively see there's some value, but you don't know 1:03:38.518 --> 1:03:41.748 exactly what it is until you do it by backward induction. 1:03:41.750 --> 1:03:45.530 So here the payments are always 8. 1:03:45.530 --> 1:04:01.500 The payments we said were 8.05 and 8.05 and then at the end 1:04:01.501 --> 1:04:03.431 8.05. 1:04:03.429 --> 1:04:09.419 So we know that at the end the present value of what the bank 1:04:09.422 --> 1:04:12.242 is going to owe, the present value of how much 1:04:12.235 --> 1:04:13.495 the homeowner is going to have to pay, 1:04:13.500 --> 1:04:16.280 let's do it that way, it going to be 0. 1:04:16.280 --> 1:04:19.520 So we know that at this point the value that's left is 0 1:04:19.519 --> 1:04:22.939 because there are no payments after this final--that's year 1:04:22.936 --> 1:04:23.346 30. 1:04:23.349 --> 1:04:24.789 There are no payments after that. 1:04:24.789 --> 1:04:28.699 So assume that we've figured out the value and we're working 1:04:28.702 --> 1:04:29.502 backwards. 1:04:29.500 --> 1:04:32.640 So we are at some node like this one where we know the 1:04:32.639 --> 1:04:35.959 interest rate and we want to figure out what the value is 1:04:35.956 --> 1:04:38.736 here just after the coupon payment is made. 1:04:38.739 --> 1:04:40.119 So what's the value? 1:04:40.119 --> 1:04:44.219 Well, the homeowner has the option of making--remaining 1:04:44.219 --> 1:04:46.949 balance, remember, was B_1, 1:04:46.952 --> 1:04:50.372 B_2 and this is B_30. 1:04:50.369 --> 1:04:55.269 So at this node here this is 0, this is times 0. 1:04:55.269 --> 1:04:58.599 So we have B_0 = 100. 1:04:58.599 --> 1:05:02.499 At time 1 the homeowner, he's made the payment here, 1:05:02.500 --> 1:05:07.840 so at time 1 he could just pay B_1 and be out of the 1:05:07.836 --> 1:05:12.926 whole thing, or he could make his payment. 1:05:12.929 --> 1:05:16.619 So if he makes his payment then he owes the 8.05-- 1:05:16.619 --> 1:05:19.569 sorry, he's just made his payment, so he could either get 1:05:19.567 --> 1:05:22.197 out of the whole thing by paying B_1, 1:05:22.199 --> 1:05:25.019 or he could wait to see what happens next period. 1:05:25.018 --> 1:05:27.278 So half the time he's going to go up, 1:05:27.280 --> 1:05:31.090 so he's going to have to pay 8.05 and then he's going to have 1:05:31.092 --> 1:05:33.982 a decision then to make, but we're working by backward 1:05:33.976 --> 1:05:34.416 induction. 1:05:34.420 --> 1:05:36.990 We've already solved out for what the value of his decision 1:05:36.992 --> 1:05:37.482 is there. 1:05:37.480 --> 1:05:44.740 So it's V_Sup half the time he's going to have to 1:05:44.742 --> 1:05:46.122 pay 8.05. 1:05:46.119 --> 1:05:49.329 He's going to be down here, and then he's going to have the 1:05:49.329 --> 1:05:51.319 remaining-- he's going to have to decide 1:05:51.317 --> 1:05:53.577 whether to pay off the remaining balance or not, 1:05:53.579 --> 1:05:56.329 and V_Sdown is that decision, 1:05:56.329 --> 1:05:59.079 because we've already figured out whether he should prepay or 1:05:59.076 --> 1:05:59.576 not here. 1:05:59.579 --> 1:06:01.079 So that's it. 1:06:01.079 --> 1:06:02.509 It couldn't be simpler. 1:06:02.510 --> 1:06:07.600 At every node after the coupon payment is made the guy has the 1:06:07.599 --> 1:06:10.549 choice of paying or, I mean, of prepaying, 1:06:10.545 --> 1:06:13.085 paying off the remaining balance or waiting until the 1:06:13.086 --> 1:06:15.256 next period, and then he has to pay the 1:06:15.257 --> 1:06:17.337 coupon if he waits until next period, 1:06:17.340 --> 1:06:23.200 and if he doesn't pay the coupon then... 1:06:23.199 --> 1:06:24.309 Student: What happened to the interest rate? 1:06:24.309 --> 1:06:25.379 Prof: Oh, what happened to the interest 1:06:25.382 --> 1:06:25.552 rate? 1:06:25.550 --> 1:06:26.770 Well, I forgot it. 1:06:26.769 --> 1:06:28.229 So thank you. 1:06:28.230 --> 1:06:36.930 So I have to put this in brackets and multiply by 1 over 1:06:36.925 --> 1:06:38.185 1.049. 1:06:38.190 --> 1:06:39.410 That's what happened to the interest rate. 1:06:39.409 --> 1:06:40.649 I just forgot it. 1:06:40.650 --> 1:06:43.660 So either he pays the remaining balance or he waits, 1:06:43.659 --> 1:06:46.929 and of course he has to discount it by the interest rate 1:06:46.925 --> 1:06:50.365 there which is 1 over 1.049 times 1 half times what happens 1:06:50.369 --> 1:06:53.339 over here 1 half times what happens over here. 1:06:53.340 --> 1:06:55.910 And that's it, and that's what I've written 1:06:55.914 --> 1:06:57.144 down in the nodes. 1:06:57.139 --> 1:07:01.159 So here are the interest rate nodes and here are the present 1:07:01.161 --> 1:07:04.231 value--so here's the non-callable mortgage. 1:07:04.230 --> 1:07:08.220 So the non-callable mortgage, if the homeowner was so stupid 1:07:08.224 --> 1:07:12.024 that he never prepaid and the mortgage interest rate is 7 1:07:12.016 --> 1:07:15.736 percent and the interest rate starts at 6 percent that's 1:07:15.739 --> 1:07:18.109 obviously great for the bank. 1:07:18.110 --> 1:07:25.350 The guy's stupidly always going to pay, never prepays, 1:07:25.347 --> 1:07:33.127 the bank makes 109 dollars, but if the guy is much smarter 1:07:33.132 --> 1:07:37.642 than that he's going to call. 1:07:37.639 --> 1:07:42.289 And now, lo and behold, the value's only 95.55. 1:07:42.289 --> 1:07:47.899 The option is so valuable that even though the interest rate is 1:07:47.902 --> 1:07:50.912 6 percent, the mortgage rate is 7 percent, 1:07:50.909 --> 1:07:53.449 the banker's incredibly overcharging him, 1:07:53.449 --> 1:08:00.059 and the guy will never default, still the bank is getting a 1:08:00.056 --> 1:08:06.886 terrible deal because the option's so valuable to the guy. 1:08:06.889 --> 1:08:09.079 So what is a bank going to do in that case? 1:08:09.079 --> 1:08:14.759 What would a bank do? 1:08:14.760 --> 1:08:17.640 I mean, if you were the banker what would you do? 1:08:17.640 --> 1:08:18.850 The interest rate is 6 percent. 1:08:18.850 --> 1:08:20.170 You can't control the interest rates. 1:08:20.170 --> 1:08:21.320 That's the whole economy. 1:08:21.319 --> 1:08:23.469 Everybody's patience, and impatience and all the 1:08:23.469 --> 1:08:25.159 Fisher stuff, is determining all these 1:08:25.161 --> 1:08:26.031 interest rates. 1:08:26.029 --> 1:08:30.489 What would you do? 1:08:30.488 --> 1:08:33.108 You wouldn't charge 7 percent as your mortgage rate. 1:08:33.109 --> 1:08:35.259 You'd have to charge a higher mortgage rate. 1:08:35.260 --> 1:08:38.340 Maybe you'd charge 8 percent, and so we could just change the 1:08:38.342 --> 1:08:41.432 whole thing to 8 percent and redo it, and you'll see that the 1:08:41.426 --> 1:08:42.606 guy will do better. 1:08:42.609 --> 1:08:49.579 So the mortgage rate, let's put it as .075 instead of 1:08:49.576 --> 1:08:50.376 .07. 1:08:50.380 --> 1:08:52.780 So now the annual payment's gone up and everything is going 1:08:52.782 --> 1:08:53.282 to change. 1:08:53.279 --> 1:08:56.549 And so the interest rate process is the same. 1:08:56.550 --> 1:08:59.010 If the guy never prepays it's now worth 114, 1:08:59.010 --> 1:09:01.410 and if the guy does prepay optimally, well, 1:09:01.412 --> 1:09:03.932 you still haven't gotten it high enough. 1:09:03.930 --> 1:09:05.910 You have to make it 8 percent, maybe. 1:09:05.908 --> 1:09:12.128 So the interest rate looks like it'll have to get to a lot above 1:09:12.126 --> 1:09:13.306 6 percent. 1:09:13.310 --> 1:09:14.810 Maybe it'll have to go to 8 percent. 1:09:14.810 --> 1:09:16.830 Maybe 8 percent isn't enough. 1:09:16.828 --> 1:09:22.288 Well, on average--let's just try .08 and now we can see how 1:09:22.286 --> 1:09:23.976 that worked out. 1:09:23.979 --> 1:09:25.669 Interest rate process is the same. 1:09:25.670 --> 1:09:26.170 Look at this. 1:09:26.170 --> 1:09:28.250 If the guy never prepays it's worth 120. 1:09:28.250 --> 1:09:30.360 If he does prepay it's still not enough. 1:09:30.359 --> 1:09:32.309 So it's going to have to be 9 percent or something. 1:09:32.310 --> 1:09:36.150 Now, typically if the interest rate is 6 percent the mortgage 1:09:36.150 --> 1:09:39.480 rate will be something like 7 and 1 half percent, 1:09:39.479 --> 1:09:42.839 not the 9 percent or 10 percent I'd have to get up to. 1:09:42.840 --> 1:09:46.290 So why do you think that is? 1:09:46.289 --> 1:09:47.119 Yes? 1:09:47.118 --> 1:09:49.298 Student: Maybe there are enough dumb people who don't 1:09:49.302 --> 1:09:51.082 prepay to make up for the smart people who do? 1:09:51.079 --> 1:09:51.939 Prof: Exactly. 1:09:51.939 --> 1:09:54.849 You have to count on the dumb people. 1:09:54.850 --> 1:09:58.050 That's an important fact of life. 1:09:58.050 --> 1:10:00.480 You not only have to count on them, you have to count them. 1:10:00.479 --> 1:10:04.749 So you have to figure out what fraction of the population is 1:10:04.753 --> 1:10:08.453 this that's only going to pay you 98 in the end, 1:10:08.448 --> 1:10:10.868 and what fraction of the population is this, 1:10:10.868 --> 1:10:13.178 they're going to pay you 120 in the end. 1:10:13.180 --> 1:10:16.310 And if you knew which of the guys were the 120 guys, 1:10:16.310 --> 1:10:23.460 what fraction were the 120 guys and what fraction were the 98 1:10:23.462 --> 1:10:28.592 guys you'd know what the thing was worth. 1:10:28.590 --> 1:10:30.000 So how far am I going? 1:10:30.000 --> 1:10:32.030 All right, so if you're a mortgage hedge fund, 1:10:32.033 --> 1:10:33.753 my company is a mortgage hedge fund. 1:10:33.750 --> 1:10:35.590 That's what we started out with. 1:10:35.590 --> 1:10:39.510 We quickly did these calculations and now-- 1:10:39.510 --> 1:10:43.170 all right, so if you look at the data you're going to find-- 1:10:43.170 --> 1:10:44.590 oh, by the way, Sunday I'm having, 1:10:44.591 --> 1:10:46.471 remember, this extra class Sunday night 1:10:46.472 --> 1:10:48.732 to tell you a little bit about the real world, 1:10:48.729 --> 1:10:51.189 so I'm going to save the stories of all the data and 1:10:51.185 --> 1:10:53.915 stuff until then, but basically we look at how 1:10:53.921 --> 1:10:56.011 people have behaved in the past. 1:10:56.010 --> 1:10:58.600 So were doing this tree and we're figuring out, 1:10:58.604 --> 1:11:01.994 from this tree you can figure out when the people should have 1:11:01.988 --> 1:11:03.058 prepaid or not. 1:11:03.060 --> 1:11:06.790 Actually the next tree, if you have 1 dollar's worth of 1:11:06.791 --> 1:11:09.431 principal, if you assume that the original 1:11:09.427 --> 1:11:12.907 principal was 1 dollar instead of 100 and you always figure out 1:11:12.905 --> 1:11:16.045 at every node what would 1 dollar's worth of principal be 1:11:16.046 --> 1:11:19.506 there, you can find out in an easy way 1:11:19.510 --> 1:11:21.940 when people should prepay. 1:11:21.939 --> 1:11:25.379 So here you can see where all the 1s are is when they should 1:11:25.376 --> 1:11:26.246 have prepaid. 1:11:26.250 --> 1:11:27.670 I don't have time now to explain that, 1:11:27.666 --> 1:11:29.726 but anyway just a slight modification of the tree shows 1:11:29.734 --> 1:11:30.964 you when they should prepay. 1:11:30.960 --> 1:11:33.700 So you can look at when people should prepay and you can look 1:11:33.698 --> 1:11:35.978 in the data at how many of the people do prepay. 1:11:35.979 --> 1:11:37.809 So you can go house by house. 1:11:37.810 --> 1:11:39.610 It's public information. 1:11:39.609 --> 1:11:42.159 Are these people prepaying or aren't they prepaying? 1:11:42.159 --> 1:11:43.599 How smart are they? 1:11:43.600 --> 1:11:46.980 And you can deduce from having watched them in the past miss 1:11:46.981 --> 1:11:50.311 opportunity after opportunity to prepay you know that these 1:11:50.305 --> 1:11:51.945 guys-- now, it's not a matter of being 1:11:51.948 --> 1:11:53.278 stupid, in fact the smartest people 1:11:53.278 --> 1:11:56.258 might be the ones not prepaying, they're not paying attention or 1:11:56.260 --> 1:11:58.730 maybe it's a real hassle for them to prepay. 1:11:58.729 --> 1:12:00.749 So we're going to come to the reasons why they won't prepay. 1:12:00.750 --> 1:12:03.590 So you have to calibrate how many people are behaving 1:12:03.587 --> 1:12:05.767 optimally and how many people aren't, 1:12:05.770 --> 1:12:08.410 and then you can judge how high you have to set the interest 1:12:08.412 --> 1:12:09.982 rate to make a reasonable profit. 1:12:09.979 --> 1:12:13.099 And I'm going to talk more about that next time. 1:12:13.100 --> 1:12:18.000