WEBVTT 00:00:00.000 --> 00:00:01.980 align:middle line:90% [SQUEAKING] 00:00:01.980 --> 00:00:03.960 align:middle line:90% [RUSTLING] 00:00:03.960 --> 00:00:07.920 align:middle line:90% [CLICKING] 00:00:07.920 --> 00:00:16.088 align:middle line:90% 00:00:16.088 --> 00:00:17.630 align:middle line:84% RICARDO CABALLERO: Today, we're going 00:00:17.630 --> 00:00:22.080 align:middle line:84% to talk about a very important topic in economics, 00:00:22.080 --> 00:00:23.390 align:middle line:90% which is expectations. 00:00:23.390 --> 00:00:25.490 align:middle line:84% We have barely mentioned expectations 00:00:25.490 --> 00:00:27.200 align:middle line:84% when we talk about the Phillips curve, 00:00:27.200 --> 00:00:32.870 align:middle line:84% we talked about expectations when we discussed the UIP and so 00:00:32.870 --> 00:00:35.840 align:middle line:84% on, but expectation is a much bigger issue in economics. 00:00:35.840 --> 00:00:37.940 align:middle line:84% In fact, most decisions by firms, 00:00:37.940 --> 00:00:41.810 align:middle line:84% by consumers, governments involve 00:00:41.810 --> 00:00:43.370 align:middle line:90% considerations of the future. 00:00:43.370 --> 00:00:46.880 align:middle line:84% And it plays an even bigger role in finance in which essentially, 00:00:46.880 --> 00:00:48.440 align:middle line:90% everything is about the future. 00:00:48.440 --> 00:00:52.400 align:middle line:84% The price of an asset today is meaningless in itself. 00:00:52.400 --> 00:00:54.650 align:middle line:84% You have to compare it with what you expect to get out 00:00:54.650 --> 00:00:55.860 align:middle line:90% of that asset in the future. 00:00:55.860 --> 00:00:57.900 align:middle line:84% So it's all about expectations and so on. 00:00:57.900 --> 00:01:00.050 align:middle line:84% So that's what we're going to do today 00:01:00.050 --> 00:01:04.120 align:middle line:84% we're going to talk about expectations, how 00:01:04.120 --> 00:01:08.650 align:middle line:84% to value things that you expect to receive in the future, 00:01:08.650 --> 00:01:11.350 align:middle line:84% and how to compare those things with things that you 00:01:11.350 --> 00:01:14.570 align:middle line:90% have in the present. 00:01:14.570 --> 00:01:17.420 align:middle line:84% But before doing that, actually, let's talk a little bit 00:01:17.420 --> 00:01:18.360 align:middle line:90% about the news. 00:01:18.360 --> 00:01:21.950 align:middle line:84% Who knows who first Republic Bank is? 00:01:21.950 --> 00:01:24.230 align:middle line:84% Remember that a few weeks ago, I told you 00:01:24.230 --> 00:01:26.970 align:middle line:90% that the Silicon Valley Bank-- 00:01:26.970 --> 00:01:28.400 align:middle line:90% I mean, you read it. 00:01:28.400 --> 00:01:33.120 align:middle line:90% I just mentioned it, that the-- 00:01:33.120 --> 00:01:37.730 align:middle line:84% or discuss it that we had the second-largest bank by asset 00:01:37.730 --> 00:01:40.070 align:middle line:90% in US history-- 00:01:40.070 --> 00:01:44.660 align:middle line:84% it was-- Silicon Valley Bank was the second-largest asset-- 00:01:44.660 --> 00:01:47.840 align:middle line:84% bank in terms of assets to collapse in the US. 00:01:47.840 --> 00:01:50.360 align:middle line:84% The first one was many years ago, 00:01:50.360 --> 00:01:54.350 align:middle line:84% and then we had this bank that had more than $200 billion 00:01:54.350 --> 00:01:57.410 align:middle line:84% in assets that essentially collapsed in a few days. 00:01:57.410 --> 00:01:58.850 align:middle line:90% It was a run on deposits. 00:01:58.850 --> 00:02:02.270 align:middle line:84% They had problems before, but what really 00:02:02.270 --> 00:02:04.430 align:middle line:84% did it, as is always the case with banks, 00:02:04.430 --> 00:02:08.090 align:middle line:84% is they have a run on deposits-- funding. 00:02:08.090 --> 00:02:09.690 align:middle line:90% Well, it's no longer. 00:02:09.690 --> 00:02:13.160 align:middle line:84% The second-largest collapse in US bank history, 00:02:13.160 --> 00:02:17.780 align:middle line:84% now we have on the weekend the new second-largest bank 00:02:17.780 --> 00:02:20.930 align:middle line:84% to collapse, which is First Republic Bank that was 00:02:20.930 --> 00:02:25.790 align:middle line:84% essentially liquidated and sold to JPMorgan over-- 00:02:25.790 --> 00:02:28.760 align:middle line:84% today morning, very, very early in the morning. 00:02:28.760 --> 00:02:31.640 align:middle line:84% So you have an account in First Republic Bank, 00:02:31.640 --> 00:02:34.490 align:middle line:84% you soon are likely to have an account in JPMorgan. 00:02:34.490 --> 00:02:37.760 align:middle line:84% But again, what made it collapse was 00:02:37.760 --> 00:02:40.820 align:middle line:84% something very similar to what made Silicon Valley Bank 00:02:40.820 --> 00:02:44.790 align:middle line:84% collapse, which is that they had invested on a series of things 00:02:44.790 --> 00:02:50.400 align:middle line:84% that were very vulnerable to the fast pace of hikes in interest 00:02:50.400 --> 00:02:51.660 align:middle line:90% rates in the US. 00:02:51.660 --> 00:02:53.910 align:middle line:84% And when they had those losses, depositors 00:02:53.910 --> 00:02:56.610 align:middle line:84% became worried about it, and eventually they 00:02:56.610 --> 00:03:00.420 align:middle line:84% decided not to wait, just run and see what happened. 00:03:00.420 --> 00:03:03.030 align:middle line:84% First Republic Bank lost about $100 billion 00:03:03.030 --> 00:03:05.580 align:middle line:90% in deposits just last week. 00:03:05.580 --> 00:03:07.170 align:middle line:90% The last few days of last week. 00:03:07.170 --> 00:03:15.080 align:middle line:84% So it was obvious that it was not going to survive, 00:03:15.080 --> 00:03:18.490 align:middle line:84% and that's the reason something was arranged over the weekend 00:03:18.490 --> 00:03:21.650 align:middle line:84% to avoid the panics associated to collapse of the bank 00:03:21.650 --> 00:03:22.720 align:middle line:90% and so on. 00:03:22.720 --> 00:03:25.690 align:middle line:84% But anyways, by the way, this is all about expectations, 00:03:25.690 --> 00:03:30.608 align:middle line:84% is if people had expected the deposits to remain in the bank, 00:03:30.608 --> 00:03:32.650 align:middle line:84% then probably this bank would not have collapsed. 00:03:32.650 --> 00:03:35.380 align:middle line:84% It's all about people anticipating what other people 00:03:35.380 --> 00:03:38.660 align:middle line:90% will do and so on and so forth. 00:03:38.660 --> 00:03:43.170 align:middle line:84% OK, but now let me get into the specifics of this lecture. 00:03:43.170 --> 00:03:47.270 align:middle line:84% So there, you have-- this is the most important index of equity-- 00:03:47.270 --> 00:03:49.700 align:middle line:90% equity index in the US, S&P 500. 00:03:49.700 --> 00:03:53.390 align:middle line:84% It's a very inclusive index that captures all the large-- 00:03:53.390 --> 00:03:56.150 align:middle line:84% most of the large companies in the US-- 00:03:56.150 --> 00:03:57.770 align:middle line:90% all of the large, I think-- 00:03:57.770 --> 00:03:59.840 align:middle line:90% companies in the US. 00:03:59.840 --> 00:04:02.660 align:middle line:84% And that's an index, an average-- weighted average 00:04:02.660 --> 00:04:07.790 align:middle line:84% by this capitalization value of each of these shares, 00:04:07.790 --> 00:04:10.670 align:middle line:84% the weighted average of the major-- main shares in the US, 00:04:10.670 --> 00:04:12.230 align:middle line:90% equity shares in the US. 00:04:12.230 --> 00:04:15.470 align:middle line:84% And one thing you see is that it moves a lot around. 00:04:15.470 --> 00:04:19.250 align:middle line:84% Here, for example, when we became aware 00:04:19.250 --> 00:04:21.649 align:middle line:84% that COVID was going to be a serious issue, 00:04:21.649 --> 00:04:24.920 align:middle line:84% the US equity market collapsed by 35% or so. 00:04:24.920 --> 00:04:27.860 align:middle line:84% That's a very large collapse in a very short period of time. 00:04:27.860 --> 00:04:32.630 align:middle line:84% And then, as a result of lots of policy support, 00:04:32.630 --> 00:04:34.880 align:middle line:90% actually we had a massive rally. 00:04:34.880 --> 00:04:37.670 align:middle line:84% So up to the end of 2021, the equity market 00:04:37.670 --> 00:04:41.250 align:middle line:84% had rallied by 114%, so big rally. 00:04:41.250 --> 00:04:44.010 align:middle line:84% Then we got inflation and the Fed 00:04:44.010 --> 00:04:47.910 align:middle line:84% began to worry about inflation, so they began to hike interest 00:04:47.910 --> 00:04:49.890 align:middle line:84% rates, and when they hike interest rates, 00:04:49.890 --> 00:04:52.680 align:middle line:84% that eventually led to a very large decline 00:04:52.680 --> 00:04:56.280 align:middle line:84% in asset prices of the order of 30% or so-- 00:04:56.280 --> 00:05:00.300 align:middle line:84% 25% or so, actually, from the peak to the bottom. 00:05:00.300 --> 00:05:03.550 align:middle line:84% And then since the bottom, which was more or less October 00:05:03.550 --> 00:05:07.500 align:middle line:84% of last year, we have seen a recovery of about 16% or so 00:05:07.500 --> 00:05:08.910 align:middle line:90% of the equity market. 00:05:08.910 --> 00:05:11.970 align:middle line:84% And if you look at the NASDAQ, which is another one index that 00:05:11.970 --> 00:05:15.820 align:middle line:84% is very loaded towards technology companies, 00:05:15.820 --> 00:05:19.300 align:middle line:84% and you can see swings are even larger than that. 00:05:19.300 --> 00:05:22.650 align:middle line:84% Now why do these prices move so much? 00:05:22.650 --> 00:05:27.140 align:middle line:84% Well, a lot of it has to do with expectations. 00:05:27.140 --> 00:05:30.120 align:middle line:84% Are things going to get worse in the future? 00:05:30.120 --> 00:05:33.200 align:middle line:90% Will the Fed cause a recession? 00:05:33.200 --> 00:05:35.570 align:middle line:84% How much higher will be the interest rates? 00:05:35.570 --> 00:05:38.750 align:middle line:84% And things like that matter a great deal. 00:05:38.750 --> 00:05:41.720 align:middle line:84% Another thing that matters a great deal 00:05:41.720 --> 00:05:45.543 align:middle line:84% is how much people want to take risk at any moment in time. 00:05:45.543 --> 00:05:47.210 align:middle line:84% If you're very scared about environment, 00:05:47.210 --> 00:05:49.520 align:middle line:84% you're unlikely to want to have something that-- 00:05:49.520 --> 00:05:52.640 align:middle line:84% to invest on something that can move so much. 00:05:52.640 --> 00:05:56.120 align:middle line:84% And so risk is well-known, so it's called risk-off, when 00:05:56.120 --> 00:05:58.640 align:middle line:84% people don't want to take risk, these asset prices 00:05:58.640 --> 00:06:00.200 align:middle line:90% tend to collapse. 00:06:00.200 --> 00:06:03.710 align:middle line:84% Of Risky assets-- equity is a very risky asset. 00:06:03.710 --> 00:06:06.320 align:middle line:84% But that's not the only thing that moves these assets around. 00:06:06.320 --> 00:06:08.990 align:middle line:84% It's not just the risk that the companies underlying companies 00:06:08.990 --> 00:06:11.270 align:middle line:84% may go bankrupt or anything like that. 00:06:11.270 --> 00:06:14.870 align:middle line:84% Here, you have, for example, the movement of a-- 00:06:14.870 --> 00:06:16.640 align:middle line:84% it's an ETF, but it doesn't matter. 00:06:16.640 --> 00:06:18.710 align:middle line:90% It's a portfolio of bonds-- 00:06:18.710 --> 00:06:20.840 align:middle line:84% of US Treasury bonds of very long duration, 00:06:20.840 --> 00:06:24.170 align:middle line:84% maturities beyond 20 years and so. 00:06:24.170 --> 00:06:27.890 align:middle line:84% So this is incredibly safe bonds because it's US treasuries. 00:06:27.890 --> 00:06:30.920 align:middle line:84% So there's no risk of default or anything like that. 00:06:30.920 --> 00:06:33.780 align:middle line:84% Still, the price swings can be pretty large. 00:06:33.780 --> 00:06:37.760 align:middle line:84% I mean, over this period, you have seen an increase in value 00:06:37.760 --> 00:06:44.240 align:middle line:84% of 45%, then a decline in value of about 20%, 00:06:44.240 --> 00:06:45.740 align:middle line:90% another increase in 15%. 00:06:45.740 --> 00:06:51.410 align:middle line:84% Here, there was a huge decline, 40% since essentially-- 00:06:51.410 --> 00:06:52.820 align:middle line:90% what do you think happened here? 00:06:52.820 --> 00:06:55.825 align:middle line:84% Why is this big decline in bonds? 00:06:55.825 --> 00:06:57.200 align:middle line:84% You're going to be able to answer 00:06:57.200 --> 00:06:59.990 align:middle line:84% that very precisely later on, but I can tell you 00:06:59.990 --> 00:07:03.170 align:middle line:84% in advance, that was essentially the result of monetary policy 00:07:03.170 --> 00:07:04.200 align:middle line:90% tightening. 00:07:04.200 --> 00:07:09.030 align:middle line:84% Increasing interest rates caused the bonds to decline. 00:07:09.030 --> 00:07:11.300 align:middle line:84% So even these instruments are very safe in the sense 00:07:11.300 --> 00:07:13.190 align:middle line:84% that if you hold them to maturity, 00:07:13.190 --> 00:07:15.650 align:middle line:84% you'll get your money back and all the promised coupons 00:07:15.650 --> 00:07:19.070 align:middle line:84% along the path, well still, the price can move a lot. 00:07:19.070 --> 00:07:21.800 align:middle line:84% And it's obvious that that movement in price 00:07:21.800 --> 00:07:23.570 align:middle line:84% is something that you need to explain 00:07:23.570 --> 00:07:27.560 align:middle line:84% in terms of expectations, what people expect things to happen. 00:07:27.560 --> 00:07:30.020 align:middle line:84% In this case, it's not whether people expect to get paid 00:07:30.020 --> 00:07:33.140 align:middle line:84% or not because you will get paid, but it's expected-- 00:07:33.140 --> 00:07:35.000 align:middle line:84% but in this particular case, it's 00:07:35.000 --> 00:07:37.293 align:middle line:84% about expectations about future interest rate. 00:07:37.293 --> 00:07:39.210 align:middle line:84% You think the interest rate will be very high, 00:07:39.210 --> 00:07:41.700 align:middle line:84% then the price of bonds will tend to be very low and so on, 00:07:41.700 --> 00:07:43.190 align:middle line:90% but it's all about the future. 00:07:43.190 --> 00:07:45.740 align:middle line:90% 00:07:45.740 --> 00:07:50.420 align:middle line:84% So a key concept that we're going to discuss today, 00:07:50.420 --> 00:07:55.010 align:middle line:84% and then you're going to use it to price specific assets, 00:07:55.010 --> 00:07:58.340 align:middle line:84% is a concept of expected present discounted value, 00:07:58.340 --> 00:07:59.930 align:middle line:90% and this is a loaded concept. 00:07:59.930 --> 00:08:02.060 align:middle line:84% There's lots of terms in there and we 00:08:02.060 --> 00:08:06.560 align:middle line:84% need to understand what each of these terms means. 00:08:06.560 --> 00:08:09.350 align:middle line:84% So the key issue that we're going to discuss 00:08:09.350 --> 00:08:10.805 align:middle line:90% is how do we decide-- 00:08:10.805 --> 00:08:12.680 align:middle line:84% for example, if you see the price of an asset 00:08:12.680 --> 00:08:15.560 align:middle line:84% out there that is 100, how do you 00:08:15.560 --> 00:08:19.640 align:middle line:84% decide whether that price is fair or not, it looks cheap 00:08:19.640 --> 00:08:21.740 align:middle line:90% or not? 00:08:21.740 --> 00:08:25.190 align:middle line:84% And that question means you have to decide 00:08:25.190 --> 00:08:27.440 align:middle line:84% whether that price that you're paying today 00:08:27.440 --> 00:08:29.930 align:middle line:84% is consistent with the future cash flows 00:08:29.930 --> 00:08:31.815 align:middle line:84% that you're going to get from these assets. 00:08:31.815 --> 00:08:33.440 align:middle line:84% And that's the reason you buy an asset, 00:08:33.440 --> 00:08:37.520 align:middle line:84% is because you'll get something in return in the future. 00:08:37.520 --> 00:08:38.780 align:middle line:90% But how do we compare that? 00:08:38.780 --> 00:08:40.608 align:middle line:84% How do we compare the price today 00:08:40.608 --> 00:08:42.650 align:middle line:84% with those things that will happen in the future? 00:08:42.650 --> 00:08:46.700 align:middle line:90% 00:08:46.700 --> 00:08:48.820 align:middle line:84% So answering that question, which 00:08:48.820 --> 00:08:50.590 align:middle line:84% is what we're going to do in this lecture, 00:08:50.590 --> 00:08:53.710 align:middle line:90% involves the following concepts. 00:08:53.710 --> 00:08:55.570 align:middle line:90% First, expectations. 00:08:55.570 --> 00:08:57.380 align:middle line:90% Big thing. 00:08:57.380 --> 00:08:58.280 align:middle line:90% That's it. 00:08:58.280 --> 00:09:00.680 align:middle line:84% This is expected present discounted value. 00:09:00.680 --> 00:09:02.690 align:middle line:90% The E part is for expectations. 00:09:02.690 --> 00:09:05.340 align:middle line:90% That comes there. 00:09:05.340 --> 00:09:07.850 align:middle line:84% The expectations are very crucial because these are things 00:09:07.850 --> 00:09:09.140 align:middle line:90% that happen in the future. 00:09:09.140 --> 00:09:10.280 align:middle line:90% You need to expect-- 00:09:10.280 --> 00:09:14.630 align:middle line:84% even if it's a bond that promises you to pay $0.50 per 00:09:14.630 --> 00:09:18.990 align:middle line:84% dollar every six months, you still may have an expectation 00:09:18.990 --> 00:09:21.600 align:middle line:84% that if it is a bond issue by First Republic Bank, 00:09:21.600 --> 00:09:22.620 align:middle line:90% it may not pay. 00:09:22.620 --> 00:09:27.750 align:middle line:84% So you need to have an expectation about that. 00:09:27.750 --> 00:09:30.240 align:middle line:84% So a crucial term is expectation. 00:09:30.240 --> 00:09:33.720 align:middle line:84% Then you need some method to compare 00:09:33.720 --> 00:09:36.630 align:middle line:84% payments received in the future with payments made today. 00:09:36.630 --> 00:09:38.760 align:middle line:84% I mean, if you buy an asset you pay today, 00:09:38.760 --> 00:09:41.010 align:middle line:84% but you're going to receive things returns 00:09:41.010 --> 00:09:42.460 align:middle line:90% for that asset in the future. 00:09:42.460 --> 00:09:44.970 align:middle line:90% So how do you compare that? 00:09:44.970 --> 00:09:49.500 align:middle line:84% Suppose I pay one today and I receive one one year from now. 00:09:49.500 --> 00:09:52.610 align:middle line:84% Does that seem like a good asset? 00:09:52.610 --> 00:09:53.450 align:middle line:90% Probably not. 00:09:53.450 --> 00:09:57.790 align:middle line:90% I mean, probably not. 00:09:57.790 --> 00:10:01.030 align:middle line:84% And that's what the word "discounted" really means. 00:10:01.030 --> 00:10:04.330 align:middle line:84% When you say expected present discounted value, 00:10:04.330 --> 00:10:08.170 align:middle line:84% it says, somehow that things I receive in the future 00:10:08.170 --> 00:10:10.782 align:middle line:84% are valued less than things I have today. 00:10:10.782 --> 00:10:12.490 align:middle line:84% So if you're going to tell me that you're 00:10:12.490 --> 00:10:16.060 align:middle line:84% going to pay me $1 in the future and I have to pay you $1 today, 00:10:16.060 --> 00:10:18.670 align:middle line:84% most likely I won't take that deal. 00:10:18.670 --> 00:10:21.930 align:middle line:84% So in other words, I'm discounting the future. 00:10:21.930 --> 00:10:23.370 align:middle line:90% How do we discount the future? 00:10:23.370 --> 00:10:27.110 align:middle line:84% Well, something that we now have to figure out. 00:10:27.110 --> 00:10:31.868 align:middle line:84% So let me first shut down this part, the expectations, and then 00:10:31.868 --> 00:10:32.660 align:middle line:90% we'll introduce it. 00:10:32.660 --> 00:10:35.200 align:middle line:84% So assume for now that the future. 00:10:35.200 --> 00:10:38.395 align:middle line:84% And I'm going to derive all the equations with assuming 00:10:38.395 --> 00:10:39.650 align:middle line:90% that you know the future. 00:10:39.650 --> 00:10:41.920 align:middle line:84% So there's no issue of trying to figure out what 00:10:41.920 --> 00:10:43.480 align:middle line:90% the future is, you know it. 00:10:43.480 --> 00:10:46.720 align:middle line:84% But still, you have to decide whether-- 00:10:46.720 --> 00:10:48.835 align:middle line:84% what is the right value for an asset. 00:10:48.835 --> 00:10:52.020 align:middle line:90% 00:10:52.020 --> 00:10:57.130 align:middle line:84% OK, so let's start with the case where you know the future. 00:10:57.130 --> 00:10:58.370 align:middle line:90% Sorry. 00:10:58.370 --> 00:11:02.500 align:middle line:90% And let's do the comparison-- 00:11:02.500 --> 00:11:04.780 align:middle line:84% let's try to understand how do we move flows-- 00:11:04.780 --> 00:11:07.480 align:middle line:84% how do we value flows at different points in time? 00:11:07.480 --> 00:11:09.550 align:middle line:84% And the easiest thing is think first 00:11:09.550 --> 00:11:11.740 align:middle line:84% about comparing an asset that gives you 00:11:11.740 --> 00:11:15.980 align:middle line:84% $1 in the future, how much do you think is worth today? 00:11:15.980 --> 00:11:18.850 align:middle line:84% Well, the easiest way to get to that value 00:11:18.850 --> 00:11:21.040 align:middle line:90% is to think on the alternatives. 00:11:21.040 --> 00:11:26.180 align:middle line:84% Suppose I have a dollar today, what can I do with it? 00:11:26.180 --> 00:11:28.570 align:middle line:90% Well-- in terms of investment? 00:11:28.570 --> 00:11:32.260 align:middle line:84% Well, suppose that you have available 1-year bonds, 00:11:32.260 --> 00:11:35.140 align:middle line:84% Treasury bonds, and that the interest rate is it. 00:11:35.140 --> 00:11:37.850 align:middle line:84% That's the interest rate on a 1-year bond. 00:11:37.850 --> 00:11:42.170 align:middle line:84% So if you want-- if you have a dollar, 00:11:42.170 --> 00:11:44.590 align:middle line:84% you have the option to invest it in that asset, 00:11:44.590 --> 00:11:51.530 align:middle line:84% in that bond, which will give you 1 plus i dollars next year. 00:11:51.530 --> 00:11:58.370 align:middle line:84% Well, that means that I can get $1 next year by investing 00:11:58.370 --> 00:12:02.260 align:middle line:90% 1 over 1 plus i dollars today. 00:12:02.260 --> 00:12:05.200 align:middle line:90% Because if I invest 1 plus 1-- 00:12:05.200 --> 00:12:08.830 align:middle line:84% rather than $1, I invest 1 over 1 plus i today, 00:12:08.830 --> 00:12:11.500 align:middle line:84% then I multiply this by 1 plus i, 00:12:11.500 --> 00:12:14.120 align:middle line:84% and I get my dollar in the future. 00:12:14.120 --> 00:12:16.210 align:middle line:84% So that tells me that-- say the interest 00:12:16.210 --> 00:12:22.430 align:middle line:84% rate is 10%, then with $1 today, I can get $1.1 in the future. 00:12:22.430 --> 00:12:27.100 align:middle line:84% That means that investing $0.90 today, more or less, 00:12:27.100 --> 00:12:29.320 align:middle line:84% I can get a bucket in the future. 00:12:29.320 --> 00:12:33.760 align:middle line:84% That tells me that $1 in the future is equivalent to $0.90 00:12:33.760 --> 00:12:36.790 align:middle line:90% today with that assumption. 00:12:36.790 --> 00:12:39.700 align:middle line:84% So that's the reason when I told you the deal of, look, 00:12:39.700 --> 00:12:42.220 align:middle line:84% I have an asset that costs you $1, 00:12:42.220 --> 00:12:44.110 align:middle line:84% but it gives you $1 in the future, 00:12:44.110 --> 00:12:47.480 align:middle line:84% and that's not a good deal if the interest rate is positive. 00:12:47.480 --> 00:12:51.530 align:middle line:84% If the interest rate is 10%, then a fair comparison is $0.90 00:12:51.530 --> 00:12:54.590 align:middle line:90% with $1, not $1 with $1. 00:12:54.590 --> 00:12:57.290 align:middle line:84% So that's a discounting of the future. 00:12:57.290 --> 00:12:59.810 align:middle line:84% The most obvious way of discounting the future 00:12:59.810 --> 00:13:04.150 align:middle line:84% is to discount it by the interest rate. 00:13:04.150 --> 00:13:05.830 align:middle line:90% Which interest rate to pick? 00:13:05.830 --> 00:13:06.820 align:middle line:90% That's more subtle. 00:13:06.820 --> 00:13:09.190 align:middle line:84% That depends on risk, it depends on many other things, 00:13:09.190 --> 00:13:12.320 align:middle line:84% which we're going to discuss to some extent here. 00:13:12.320 --> 00:13:14.837 align:middle line:84% But for now, let's make it very simple. 00:13:14.837 --> 00:13:17.170 align:middle line:84% And in a world which you really know the future, really, 00:13:17.170 --> 00:13:19.330 align:middle line:84% the right interest to use is the safe interest-- 00:13:19.330 --> 00:13:23.680 align:middle line:84% the interest rate of Treasury bonds and things like that. 00:13:23.680 --> 00:13:25.540 align:middle line:90% So that's that. 00:13:25.540 --> 00:13:27.640 align:middle line:84% What about the dollar that you receive-- 00:13:27.640 --> 00:13:29.500 align:middle line:84% what about if you are thinking about what 00:13:29.500 --> 00:13:31.660 align:middle line:84% is the value of $1 two years from now? 00:13:31.660 --> 00:13:34.210 align:middle line:90% 00:13:34.210 --> 00:13:36.460 align:middle line:84% Well, if I get a $1, can do the same logic. 00:13:36.460 --> 00:13:39.160 align:middle line:90% If I can use the same logic. 00:13:39.160 --> 00:13:43.390 align:middle line:84% If I get a $1 today, I can convert that into 1 00:13:43.390 --> 00:13:48.490 align:middle line:90% plus it times 1 plus it plus $1. 00:13:48.490 --> 00:13:52.360 align:middle line:84% So say 10% and 10%, I get 1.1 next year, 00:13:52.360 --> 00:13:56.740 align:middle line:84% and then I get 1.1 times 1.1, 1.21 or something like that. 00:13:56.740 --> 00:14:01.380 align:middle line:90% That's my final result. 00:14:01.380 --> 00:14:04.770 align:middle line:84% Well, then, how much is it worth to have $1-- 00:14:04.770 --> 00:14:09.110 align:middle line:84% $1 asset that gives you $1 two years from now? 00:14:09.110 --> 00:14:11.800 align:middle line:84% Well, it's going to be $1 divided 00:14:11.800 --> 00:14:15.060 align:middle line:84% by the product of this interest rate. 00:14:15.060 --> 00:14:15.930 align:middle line:90% Why is that? 00:14:15.930 --> 00:14:20.166 align:middle line:84% Well, because with this amount of dollars today-- 00:14:20.166 --> 00:14:22.950 align:middle line:84% at this point, $0.80, something like that, 00:14:22.950 --> 00:14:25.980 align:middle line:84% I can generate $1 two years from now. 00:14:25.980 --> 00:14:35.820 align:middle line:84% That means $1 two years from now is worth about $0.80 today. 00:14:35.820 --> 00:14:38.790 align:middle line:84% We're going to use a lot this type of logic. 00:14:38.790 --> 00:14:42.180 align:middle line:84% And I know that it may not be that intuitive the first time 00:14:42.180 --> 00:14:44.280 align:middle line:90% you see it, but ask questions. 00:14:44.280 --> 00:14:47.860 align:middle line:90% 00:14:47.860 --> 00:14:49.030 align:middle line:90% You want me to repeat it? 00:14:49.030 --> 00:14:54.460 align:middle line:90% 00:14:54.460 --> 00:14:55.390 align:middle line:90% OK. 00:14:55.390 --> 00:14:57.410 align:middle line:90% The final goal is the following. 00:14:57.410 --> 00:14:59.890 align:middle line:84% We're going to-- what comes next, we're going to see, 00:14:59.890 --> 00:15:02.630 align:middle line:84% which happens again with many decisions in life, 00:15:02.630 --> 00:15:05.050 align:middle line:84% but particularly for financial assets, 00:15:05.050 --> 00:15:08.700 align:middle line:84% we're going to try to value something that-- 00:15:08.700 --> 00:15:12.190 align:middle line:84% it was payoff happens at different times in the future. 00:15:12.190 --> 00:15:14.500 align:middle line:84% And the question is, how do I value 00:15:14.500 --> 00:15:18.730 align:middle line:84% an asset that pays me $5 one year from now, 00:15:18.730 --> 00:15:24.940 align:middle line:84% $25 three years from now, minus $10 10 years from now, 00:15:24.940 --> 00:15:27.280 align:middle line:90% plus $50 $100 from now? 00:15:27.280 --> 00:15:28.735 align:middle line:90% What is the value of that? 00:15:28.735 --> 00:15:31.100 align:middle line:90% Of having an asset like that? 00:15:31.100 --> 00:15:35.410 align:middle line:84% And so I need some method to bring it to today's value 00:15:35.410 --> 00:15:38.320 align:middle line:84% because today, I have a meaning of what $1 is, 00:15:38.320 --> 00:15:42.670 align:middle line:84% and therefore, I can compare it with whatever price. 00:15:42.670 --> 00:15:44.900 align:middle line:84% People are asking me for that asset. 00:15:44.900 --> 00:15:48.430 align:middle line:84% So what this is doing is it's doing that. 00:15:48.430 --> 00:15:51.760 align:middle line:84% It's telling you how to convert $1 at different parts 00:15:51.760 --> 00:15:54.550 align:middle line:90% in the future into $1 today. 00:15:54.550 --> 00:15:59.650 align:middle line:84% And by that logic, the recipe is, well, use the interest rate 00:15:59.650 --> 00:16:01.880 align:middle line:84% because you could always go the other way around. 00:16:01.880 --> 00:16:04.270 align:middle line:84% You could always-- you can ask a question, with $1 00:16:04.270 --> 00:16:08.010 align:middle line:84% today, how many dollars can I get two years from now, say, 00:16:08.010 --> 00:16:08.820 align:middle line:90% that? 00:16:08.820 --> 00:16:11.010 align:middle line:90% Well, say x? 00:16:11.010 --> 00:16:15.180 align:middle line:84% Well, then I need 1 over x, then $1 there is worth 1 00:16:15.180 --> 00:16:16.950 align:middle line:90% over x dollars today. 00:16:16.950 --> 00:16:18.030 align:middle line:90% That's the logic. 00:16:18.030 --> 00:16:21.120 align:middle line:90% Because 1 over x times x is 1. 00:16:21.120 --> 00:16:25.527 align:middle line:90% So let's do first problem here. 00:16:25.527 --> 00:16:30.300 align:middle line:90% 00:16:30.300 --> 00:16:32.215 align:middle line:90% With $1 today-- oops. 00:16:32.215 --> 00:16:35.550 align:middle line:90% 00:16:35.550 --> 00:16:46.740 align:middle line:84% I can generally say $1.1 at t equal 2. 00:16:46.740 --> 00:16:51.030 align:middle line:84% Then the question I want to know is, how much is $1 worth-- 00:16:51.030 --> 00:16:56.800 align:middle line:84% how much is $1 received at time t equal 2 worth today? 00:16:56.800 --> 00:16:58.660 align:middle line:84% That's the question I'm trying to answer. 00:16:58.660 --> 00:17:00.730 align:middle line:84% Because an asset will be something 00:17:00.730 --> 00:17:02.120 align:middle line:90% that will pay you in the future. 00:17:02.120 --> 00:17:06.430 align:middle line:84% So I want to know, how much is $1 received in the future worth 00:17:06.430 --> 00:17:08.200 align:middle line:90% today? 00:17:08.200 --> 00:17:12.460 align:middle line:84% And then the answer is, well, then it's-- 00:17:12.460 --> 00:17:14.470 align:middle line:84% I know the answer from this logic 00:17:14.470 --> 00:17:21.440 align:middle line:84% because I know that with 1, if I have 1 over $1.1 today, 00:17:21.440 --> 00:17:27.079 align:middle line:90% I can convert it into 1. 00:17:27.079 --> 00:17:28.369 align:middle line:90% How do I know that? 00:17:28.369 --> 00:17:44.130 align:middle line:84% Because 1 over 1.1 times 1.1 is equal to 1. 00:17:44.130 --> 00:17:47.840 align:middle line:84% If I invest this dollars today, I'm 00:17:47.840 --> 00:17:49.940 align:middle line:84% going to get this return on that. 00:17:49.940 --> 00:17:52.280 align:middle line:84% And the product of this thing gives me my dollar. 00:17:52.280 --> 00:17:54.830 align:middle line:90% 00:17:54.830 --> 00:17:58.040 align:middle line:84% So if I tell you, do you prefer to have $1 two days from-- two 00:17:58.040 --> 00:18:02.090 align:middle line:84% years from now or today, you say, I prefer it-- obviously 00:18:02.090 --> 00:18:07.630 align:middle line:84% I prefer it today because I can get $1.1 two years from now. 00:18:07.630 --> 00:18:09.940 align:middle line:84% But then the more relevant question is, no, no, 00:18:09.940 --> 00:18:13.660 align:middle line:84% but then do you prefer to have $0.90 today versus $1 00:18:13.660 --> 00:18:14.890 align:middle line:90% in the future? 00:18:14.890 --> 00:18:17.500 align:middle line:84% Then I need to do my multiplication because then I 00:18:17.500 --> 00:18:21.220 align:middle line:84% have to multiply the $0.90 by the 1.1 and see whether I get 00:18:21.220 --> 00:18:23.950 align:middle line:84% something comparable to $1 or not. 00:18:23.950 --> 00:18:27.970 align:middle line:84% But that's the logic behind that. 00:18:27.970 --> 00:18:32.980 align:middle line:84% So the interest rate is what we discount the future by. 00:18:32.980 --> 00:18:35.530 align:middle line:84% And it's natural because if the interest rate is very high-- 00:18:35.530 --> 00:18:38.300 align:middle line:84% if the interest rate is 0, say, then 00:18:38.300 --> 00:18:40.840 align:middle line:84% $1 received two years from now and $1 received today 00:18:40.840 --> 00:18:43.927 align:middle line:84% is the same because I can-- if I invest $1 today 00:18:43.927 --> 00:18:45.760 align:middle line:84% and the interest rate is 0, I'm going to get 00:18:45.760 --> 00:18:47.780 align:middle line:90% my $1 two years from now. 00:18:47.780 --> 00:18:50.843 align:middle line:84% If the interest rate is 50%, it makes a big difference 00:18:50.843 --> 00:18:52.760 align:middle line:84% receiving the dollar today versus receiving it 00:18:52.760 --> 00:18:54.110 align:middle line:90% two years from now. 00:18:54.110 --> 00:18:56.540 align:middle line:84% If you're in Argentina and the interest rate-- 00:18:56.540 --> 00:18:59.060 align:middle line:84% I don't know what it is, it's 700%. 00:18:59.060 --> 00:19:02.720 align:middle line:84% It makes a huge difference whether you receive it one year 00:19:02.720 --> 00:19:05.710 align:middle line:90% from now than today. 00:19:05.710 --> 00:19:09.920 align:middle line:84% And so that's the role of the interest rate. 00:19:09.920 --> 00:19:13.160 align:middle line:84% The higher the interest rate, the less 00:19:13.160 --> 00:19:15.620 align:middle line:84% is $1 received in the future worth 00:19:15.620 --> 00:19:18.110 align:middle line:84% relative to $1 received today because you 00:19:18.110 --> 00:19:21.080 align:middle line:84% can get a much higher return from the dollar you have today 00:19:21.080 --> 00:19:22.460 align:middle line:90% if the interest rate is high. 00:19:22.460 --> 00:19:26.143 align:middle line:84% The interest rate is low, you don't get that much. 00:19:26.143 --> 00:19:26.810 align:middle line:90% Much difference. 00:19:26.810 --> 00:19:28.160 align:middle line:90% OK, good. 00:19:28.160 --> 00:19:29.750 align:middle line:90% So this is a big principle. 00:19:29.750 --> 00:19:32.330 align:middle line:84% And, I mean, everything that I'll 00:19:32.330 --> 00:19:34.070 align:middle line:90% say next builds on this logic. 00:19:34.070 --> 00:19:38.590 align:middle line:90% 00:19:38.590 --> 00:19:41.110 align:middle line:84% So let me give you a general formula. 00:19:41.110 --> 00:19:45.060 align:middle line:84% So let's ask, what is the value of an asset that 00:19:45.060 --> 00:19:51.990 align:middle line:84% gives payouts of zt dollars this year, 00:19:51.990 --> 00:19:55.830 align:middle line:84% zt plus 1 one year from now, zt plus 2 two years 00:19:55.830 --> 00:20:01.890 align:middle line:84% from now, and so on and so forth for n periods more? 00:20:01.890 --> 00:20:05.790 align:middle line:84% Well, I just need to do several of these operations. 00:20:05.790 --> 00:20:09.390 align:middle line:84% I know that $1 received this year is worth $1. 00:20:09.390 --> 00:20:10.800 align:middle line:90% That's zt. 00:20:10.800 --> 00:20:16.500 align:middle line:84% $1 received one year from now is not the same as $1 00:20:16.500 --> 00:20:17.160 align:middle line:90% received today. 00:20:17.160 --> 00:20:21.910 align:middle line:84% It's the same as 1 over 1 plus it dollar received today. 00:20:21.910 --> 00:20:24.600 align:middle line:84% So that cash flow I'm going to receive from this asset 00:20:24.600 --> 00:20:26.250 align:middle line:90% is worth this amount. 00:20:26.250 --> 00:20:29.280 align:middle line:84% For something that I received two years from now, 00:20:29.280 --> 00:20:32.640 align:middle line:84% then it's not-- certainly it's much less 00:20:32.640 --> 00:20:34.050 align:middle line:90% than receiving $1 today. 00:20:34.050 --> 00:20:38.910 align:middle line:84% It's going to be 1 over 1 plus it 1 plus it plus 1. 00:20:38.910 --> 00:20:41.370 align:middle line:84% And that I have to multiply by the number of dollars 00:20:41.370 --> 00:20:43.830 align:middle line:84% I will receive two years from now. 00:20:43.830 --> 00:20:46.550 align:middle line:90% And I keep going. 00:20:46.550 --> 00:20:50.230 align:middle line:90% So that's the present value. 00:20:50.230 --> 00:20:52.450 align:middle line:84% Present discounted value-- present 00:20:52.450 --> 00:20:54.490 align:middle line:84% because I'm bringing all these future cash 00:20:54.490 --> 00:20:56.450 align:middle line:90% flows to the present. 00:20:56.450 --> 00:20:58.330 align:middle line:84% That's what each of these terms is doing. 00:20:58.330 --> 00:21:01.450 align:middle line:84% The 1 over that is bringing it to the present. 00:21:01.450 --> 00:21:04.760 align:middle line:84% Discounted because the interest rate is discounting things, 00:21:04.760 --> 00:21:06.190 align:middle line:90% making them smaller. 00:21:06.190 --> 00:21:08.890 align:middle line:84% And value because I'm trying to reduce them 00:21:08.890 --> 00:21:12.570 align:middle line:90% to the current value. 00:21:12.570 --> 00:21:14.550 align:middle line:84% That's a general formula, so it's a formula 00:21:14.550 --> 00:21:15.690 align:middle line:90% you need to understand. 00:21:15.690 --> 00:21:19.140 align:middle line:84% It's just-- so that was an asset that gives you 00:21:19.140 --> 00:21:23.800 align:middle line:84% z dollars today, zt plus 1 one year from now, 00:21:23.800 --> 00:21:25.050 align:middle line:90% so you use this formula. 00:21:25.050 --> 00:21:28.830 align:middle line:84% Zt plus 2 two years from now, so you use this formula, 00:21:28.830 --> 00:21:29.850 align:middle line:90% and then you keep going. 00:21:29.850 --> 00:21:33.190 align:middle line:90% 00:21:33.190 --> 00:21:35.950 align:middle line:84% What if we don't know the future? 00:21:35.950 --> 00:21:38.990 align:middle line:84% I had to remove the expected part. 00:21:38.990 --> 00:21:42.020 align:middle line:84% Well, if we don't know the future, then the best 00:21:42.020 --> 00:21:42.862 align:middle line:90% we can do-- 00:21:42.862 --> 00:21:44.570 align:middle line:84% in fact, we do fancier things, but that's 00:21:44.570 --> 00:21:47.970 align:middle line:84% what we want-- all that we'll do in this course. 00:21:47.970 --> 00:21:50.030 align:middle line:84% All that you can do is just replace 00:21:50.030 --> 00:21:54.980 align:middle line:84% the known quantities we have here for the expectations. 00:21:54.980 --> 00:21:56.090 align:middle line:90% So that's the closest. 00:21:56.090 --> 00:21:59.870 align:middle line:84% So I know zt, that's the cash flow I get now, 00:21:59.870 --> 00:22:04.040 align:middle line:84% but I don't know zt plus 1, so I can replace it by expectation. 00:22:04.040 --> 00:22:08.000 align:middle line:84% I do know the interest rate on a 1-year bond from today to one 00:22:08.000 --> 00:22:10.610 align:middle line:84% year, so that's the reason I don't need an expectation here. 00:22:10.610 --> 00:22:14.430 align:middle line:84% But I don't know what the 1-year rate will be one year from now, 00:22:14.430 --> 00:22:17.204 align:middle line:84% so that's the reason I need an expectation there. 00:22:17.204 --> 00:22:18.485 align:middle line:90% And so on. 00:22:18.485 --> 00:22:21.110 align:middle line:84% And I don't know what the cash flow will be two years from now. 00:22:21.110 --> 00:22:23.402 align:middle line:84% I have an expectation about what the cash flow will be, 00:22:23.402 --> 00:22:26.750 align:middle line:84% but I don't know it, so I have an expectation there. 00:22:26.750 --> 00:22:30.920 align:middle line:84% So all that I've done here is say, OK, 00:22:30.920 --> 00:22:33.493 align:middle line:84% acknowledge that this guy knew a little bit too much. 00:22:33.493 --> 00:22:35.660 align:middle line:84% He knew exactly what the cash flows were going to be 00:22:35.660 --> 00:22:38.660 align:middle line:84% in the future, and he knew what the 1-year rates were going 00:22:38.660 --> 00:22:40.480 align:middle line:90% to be in the future. 00:22:40.480 --> 00:22:42.010 align:middle line:90% This guy here knows less. 00:22:42.010 --> 00:22:45.070 align:middle line:84% He knows the cash flow today, he knows the interest rate today, 00:22:45.070 --> 00:22:48.100 align:middle line:84% but he doesn't know the cash flows-- really has a hunch, 00:22:48.100 --> 00:22:51.137 align:middle line:84% but he doesn't know the cash flows one year, two years, 00:22:51.137 --> 00:22:52.720 align:middle line:84% three years, and so on for the future, 00:22:52.720 --> 00:22:56.530 align:middle line:84% and it doesn't know the 1-year interest rate in the future. 00:22:56.530 --> 00:22:58.930 align:middle line:84% So all these expectations, here is important 00:22:58.930 --> 00:22:59.790 align:middle line:90% the concept of time. 00:22:59.790 --> 00:23:02.010 align:middle line:84% These are expectations as of time t. 00:23:02.010 --> 00:23:03.750 align:middle line:84% At time t, you have some information 00:23:03.750 --> 00:23:06.790 align:middle line:84% and you make forecasts about the future. 00:23:06.790 --> 00:23:07.980 align:middle line:90% Use whatever you want. 00:23:07.980 --> 00:23:09.190 align:middle line:90% Machine learning, whatever. 00:23:09.190 --> 00:23:11.190 align:middle line:84% But you have information at time t, 00:23:11.190 --> 00:23:14.640 align:middle line:84% and then you have a forecast for the future at t plus 1. 00:23:14.640 --> 00:23:17.050 align:middle line:84% You have more information, so you make another forecast, 00:23:17.050 --> 00:23:18.100 align:middle line:90% and so on and so forth. 00:23:18.100 --> 00:23:21.270 align:middle line:84% But in this-- we're valuing an asset at time t, 00:23:21.270 --> 00:23:25.440 align:middle line:84% then all these expectations are taken as of time t. 00:23:25.440 --> 00:23:29.803 align:middle line:84% That means given the information you have available at time t. 00:23:29.803 --> 00:23:31.220 align:middle line:84% That's the reason these guys don't 00:23:31.220 --> 00:23:32.690 align:middle line:84% have expectations in front of them 00:23:32.690 --> 00:23:36.630 align:middle line:90% because you know this at time t. 00:23:36.630 --> 00:23:38.730 align:middle line:84% How do we take in the value at t minus 1, 00:23:38.730 --> 00:23:41.610 align:middle line:84% we would have not known that and we had to-- an expectation 00:23:41.610 --> 00:23:46.740 align:middle line:84% because there would have been expectation of t minus 1. 00:23:46.740 --> 00:23:48.850 align:middle line:84% OK, so that's your big formula there. 00:23:48.850 --> 00:23:55.530 align:middle line:84% So there are some examples that are well-known, 00:23:55.530 --> 00:23:58.080 align:middle line:90% and so let me show you. 00:23:58.080 --> 00:24:00.330 align:middle line:90% They have nicer expressions. 00:24:00.330 --> 00:24:05.890 align:middle line:84% So that's an example of valuation of the same asset, 00:24:05.890 --> 00:24:08.990 align:middle line:84% but when the interest rate is constant. 00:24:08.990 --> 00:24:11.360 align:middle line:84% Then obviously I don't need all these products 00:24:11.360 --> 00:24:12.450 align:middle line:90% in the denominator. 00:24:12.450 --> 00:24:16.070 align:middle line:84% I have a constant interest rate, then 00:24:16.070 --> 00:24:18.540 align:middle line:84% I just get powers of that interest rate. 00:24:18.540 --> 00:24:21.000 align:middle line:84% That's one in which you have constant payments. 00:24:21.000 --> 00:24:23.740 align:middle line:84% So the interest rate may be different, 00:24:23.740 --> 00:24:27.390 align:middle line:84% but the payments are the same over time. 00:24:27.390 --> 00:24:29.180 align:middle line:90% So that's that. 00:24:29.180 --> 00:24:31.570 align:middle line:90% So those are two easy formulas. 00:24:31.570 --> 00:24:35.110 align:middle line:84% That's one in which you have both constants, the interest 00:24:35.110 --> 00:24:39.220 align:middle line:84% rate and the payment, then you get a nice expression 00:24:39.220 --> 00:24:42.070 align:middle line:90% that's just that. 00:24:42.070 --> 00:24:49.580 align:middle line:84% You recognize that-- if you have a constant interest rate here, 00:24:49.580 --> 00:24:53.640 align:middle line:84% you see that the value is declining-- 00:24:53.640 --> 00:24:55.670 align:middle line:90% is a geometric series. 00:24:55.670 --> 00:24:59.420 align:middle line:84% The value of two years from now is a square of 1 over 1 00:24:59.420 --> 00:25:00.560 align:middle line:90% plus some-- 00:25:00.560 --> 00:25:04.100 align:middle line:84% is a square of a number less than 1. 00:25:04.100 --> 00:25:06.680 align:middle line:84% 1 over 1 plus i is some number less than 1. 00:25:06.680 --> 00:25:09.150 align:middle line:84% This is the square of that, then the cube, and so on. 00:25:09.150 --> 00:25:11.660 align:middle line:84% So it's a geometric series that is declining in the rate 1 00:25:11.660 --> 00:25:12.250 align:middle line:90% plus i-- 00:25:12.250 --> 00:25:13.760 align:middle line:90% 1 over 1 plus i. 00:25:13.760 --> 00:25:15.800 align:middle line:84% Or declining at the rate 1 plus i. 00:25:15.800 --> 00:25:19.520 align:middle line:90% So that's your geometric series. 00:25:19.520 --> 00:25:22.860 align:middle line:90% That's the value of that. 00:25:22.860 --> 00:25:25.950 align:middle line:84% Constant rate and payment forever. 00:25:25.950 --> 00:25:30.958 align:middle line:84% Suppose you have an asset that lives forever. 00:25:30.958 --> 00:25:32.250 align:middle line:90% There are some bonds like that. 00:25:32.250 --> 00:25:33.720 align:middle line:90% It's called perpetuities. 00:25:33.720 --> 00:25:36.580 align:middle line:90% 00:25:36.580 --> 00:25:38.530 align:middle line:90% The US has an issue-- 00:25:38.530 --> 00:25:40.730 align:middle line:90% the UK has and so on. 00:25:40.730 --> 00:25:42.850 align:middle line:84% So that's an asset, for example, that pays you 00:25:42.850 --> 00:25:45.790 align:middle line:90% a fixed amount forever. 00:25:45.790 --> 00:25:47.650 align:middle line:84% And it's the interest rate is constant, 00:25:47.650 --> 00:25:50.560 align:middle line:84% that's a trickier thing, then the value of that asset you 00:25:50.560 --> 00:25:55.260 align:middle line:84% can see, that this is going to 0, so the value of that asset 00:25:55.260 --> 00:25:56.880 align:middle line:90% is that. 00:25:56.880 --> 00:25:58.650 align:middle line:84% And actually, a formula that you may 00:25:58.650 --> 00:26:03.870 align:middle line:84% see that is very often used as a first approximation is this one. 00:26:03.870 --> 00:26:09.300 align:middle line:84% This is the same asset, but it's called x dividend or x coupon. 00:26:09.300 --> 00:26:13.920 align:middle line:84% It's after the coupon of this year has been paid. 00:26:13.920 --> 00:26:16.530 align:middle line:84% So it's an asset that starts paying at t plus 1, 00:26:16.530 --> 00:26:19.060 align:middle line:90% is zt plus 1, plus 2, and so on. 00:26:19.060 --> 00:26:22.770 align:middle line:84% Well, that is the same as this minus the first coupon, 00:26:22.770 --> 00:26:24.510 align:middle line:90% so it's equal to that. 00:26:24.510 --> 00:26:29.210 align:middle line:90% 00:26:29.210 --> 00:26:30.585 align:middle line:84% That's an interesting thing, huh? 00:26:30.585 --> 00:26:32.040 align:middle line:84% Look, what happened to this asset 00:26:32.040 --> 00:26:33.332 align:middle line:90% as the interest rate goes to 0? 00:26:33.332 --> 00:26:35.920 align:middle line:90% 00:26:35.920 --> 00:26:39.670 align:middle line:84% So this is an asset that lasts for a very long time. 00:26:39.670 --> 00:26:43.540 align:middle line:84% And look, we got to a valuation formula. 00:26:43.540 --> 00:26:45.370 align:middle line:84% What is happening as the interest 00:26:45.370 --> 00:26:47.745 align:middle line:90% rate goes to 0 to the line? 00:26:47.745 --> 00:26:49.370 align:middle line:84% AUDIENCE: The value becomes very large. 00:26:49.370 --> 00:26:49.780 align:middle line:90% RICARDO CABALLERO: Very large. 00:26:49.780 --> 00:26:51.130 align:middle line:90% It goes to infinity. 00:26:51.130 --> 00:26:55.060 align:middle line:84% And a lot of what has happened in global financial markets 00:26:55.060 --> 00:26:58.540 align:middle line:84% in the last few years has to do with that. 00:26:58.540 --> 00:27:01.190 align:middle line:84% Interest rates were very, very, very low. 00:27:01.190 --> 00:27:03.730 align:middle line:84% And so most assets that had long-duration 00:27:03.730 --> 00:27:04.715 align:middle line:90% had very high values. 00:27:04.715 --> 00:27:07.273 align:middle line:90% 00:27:07.273 --> 00:27:08.690 align:middle line:84% And it has a lot to do with that-- 00:27:08.690 --> 00:27:11.070 align:middle line:84% monetary policy has a lot to do-- 00:27:11.070 --> 00:27:13.550 align:middle line:84% whether it was a right monetary policy or not, 00:27:13.550 --> 00:27:15.110 align:middle line:84% that's something to be discussed. 00:27:15.110 --> 00:27:17.360 align:middle line:84% I think, on average, it was the right monetary policy, 00:27:17.360 --> 00:27:19.110 align:middle line:84% but one of the things it did, it increased 00:27:19.110 --> 00:27:20.810 align:middle line:90% the value of many assets. 00:27:20.810 --> 00:27:22.640 align:middle line:84% In fact, that's one of the mechanisms 00:27:22.640 --> 00:27:24.828 align:middle line:84% through which monetary policy works in practice. 00:27:24.828 --> 00:27:26.370 align:middle line:84% It's not something we have discussed, 00:27:26.370 --> 00:27:28.850 align:middle line:84% but you can begin to see here because if the value 00:27:28.850 --> 00:27:31.160 align:middle line:84% of all assets go up a lot, people feel wealthier 00:27:31.160 --> 00:27:33.398 align:middle line:84% and that will tend to consume more, and so on. 00:27:33.398 --> 00:27:35.690 align:middle line:84% Well, this is one of the channels monetary policy does. 00:27:35.690 --> 00:27:37.790 align:middle line:84% By the way this effect happens also 00:27:37.790 --> 00:27:39.540 align:middle line:84% to the assets that are finite end. 00:27:39.540 --> 00:27:41.030 align:middle line:90% It's just that this goes-- 00:27:41.030 --> 00:27:44.750 align:middle line:84% it's maximized when this asset lasts forever. 00:27:44.750 --> 00:27:47.760 align:middle line:84% This asset literally goes to infinity 00:27:47.760 --> 00:27:51.880 align:middle line:90% if the interest rate goes to 0. 00:27:51.880 --> 00:27:54.700 align:middle line:84% Well, if an acid lasts for 10 periods, 00:27:54.700 --> 00:27:58.980 align:middle line:84% it doesn't go to infinity, it goes to n times z. 00:27:58.980 --> 00:27:59.480 align:middle line:90% For some. 00:27:59.480 --> 00:28:02.775 align:middle line:84% If the interest rate is 0, just some things. 00:28:02.775 --> 00:28:04.610 align:middle line:90% You see that? 00:28:04.610 --> 00:28:08.540 align:middle line:84% If an asset lasts for n periods, and it gives me 00:28:08.540 --> 00:28:11.790 align:middle line:84% a payment of z in every single period, 00:28:11.790 --> 00:28:13.470 align:middle line:84% then when the interest is 0, that asset 00:28:13.470 --> 00:28:18.320 align:middle line:84% is worth n times z because I will receive z coupons. 00:28:18.320 --> 00:28:21.610 align:middle line:84% And don't discount the future because the interest rate is 0. 00:28:21.610 --> 00:28:23.890 align:middle line:84% What happens is, when the asset lasts forever, 00:28:23.890 --> 00:28:26.830 align:middle line:84% then n times z is a very large number. 00:28:26.830 --> 00:28:31.920 align:middle line:84% And that's what this expression captures here. 00:28:31.920 --> 00:28:34.210 align:middle line:90% OK. 00:28:34.210 --> 00:28:36.070 align:middle line:90% So let's talk about bonds now. 00:28:36.070 --> 00:28:39.510 align:middle line:84% We're going to start pricing bonds. 00:28:39.510 --> 00:28:44.150 align:middle line:84% So bonds differ along many dimensions, 00:28:44.150 --> 00:28:46.790 align:middle line:84% but one of them that's very important for bones 00:28:46.790 --> 00:28:49.640 align:middle line:84% is maturity the n that I had there 00:28:49.640 --> 00:28:53.120 align:middle line:90% in the previous expression. 00:28:53.120 --> 00:28:57.250 align:middle line:84% So maturity means, essentially, how long the bond lasts. 00:28:57.250 --> 00:28:59.510 align:middle line:84% When does it pay you back the principal? 00:28:59.510 --> 00:29:02.150 align:middle line:84% The bonds typically pay coupons, and then there 00:29:02.150 --> 00:29:04.910 align:middle line:84% is a final payment, which we call face value of the bond 00:29:04.910 --> 00:29:06.000 align:middle line:90% or something like that. 00:29:06.000 --> 00:29:09.590 align:middle line:84% And when that final payment takes place, 00:29:09.590 --> 00:29:12.530 align:middle line:90% that's a maturity of a bond. 00:29:12.530 --> 00:29:16.000 align:middle line:84% So a bond that promises to make a $1,000 final payment in six 00:29:16.000 --> 00:29:18.365 align:middle line:84% months has a maturity of six months. 00:29:18.365 --> 00:29:21.080 align:middle line:90% 00:29:21.080 --> 00:29:25.460 align:middle line:84% A bond that promises to pay $100 for 20 years and then $1,000-- 00:29:25.460 --> 00:29:28.930 align:middle line:84% a final payment in 20 years has a maturity of 20 years. 00:29:28.930 --> 00:29:30.430 align:middle line:84% Maturity is different from duration. 00:29:30.430 --> 00:29:32.597 align:middle line:84% I don't think I'm going to talk about duration here, 00:29:32.597 --> 00:29:34.790 align:middle line:84% but that's maturity, just when it's 00:29:34.790 --> 00:29:39.560 align:middle line:90% the final payment of a loan-- 00:29:39.560 --> 00:29:42.460 align:middle line:90% of a bond? 00:29:42.460 --> 00:29:44.020 align:middle line:90% Bonds of different maturities. 00:29:44.020 --> 00:29:46.930 align:middle line:84% Each have a price and an associated interest rate. 00:29:46.930 --> 00:29:48.550 align:middle line:84% We're going to look at those things. 00:29:48.550 --> 00:29:52.750 align:middle line:84% And the associated interest rate is called the yield to maturity, 00:29:52.750 --> 00:29:55.515 align:middle line:90% or simply the yield of a bond. 00:29:55.515 --> 00:29:58.140 align:middle line:84% This is terminology, but we are going to calculate these things 00:29:58.140 --> 00:30:00.190 align:middle line:90% later on. 00:30:00.190 --> 00:30:03.610 align:middle line:84% The relationship between maturity and yield 00:30:03.610 --> 00:30:05.440 align:middle line:90% is called the yield curve. 00:30:05.440 --> 00:30:06.820 align:middle line:90% Very important concept. 00:30:06.820 --> 00:30:08.870 align:middle line:84% Big fuss about the yield curve these days. 00:30:08.870 --> 00:30:11.840 align:middle line:84% I'll talk a little bit more about that. 00:30:11.840 --> 00:30:15.080 align:middle line:84% Or sometimes it's called the term structure of interest rate. 00:30:15.080 --> 00:30:19.560 align:middle line:84% Term, in the language of bonds, is really maturity. 00:30:19.560 --> 00:30:22.760 align:middle line:84% So term structure of interest rate really tells you what is 00:30:22.760 --> 00:30:26.660 align:middle line:84% the yield in a 1-year bond, 2-year bond, 3-year bond, 4-, 00:30:26.660 --> 00:30:27.470 align:middle line:90% 5-, 6-, 7-. 00:30:27.470 --> 00:30:31.340 align:middle line:84% You plot them, and that gives you a curve. 00:30:31.340 --> 00:30:35.678 align:middle line:90% So for example, look at those-- 00:30:35.678 --> 00:30:37.220 align:middle line:84% these are two different yield curves. 00:30:37.220 --> 00:30:43.670 align:middle line:84% This is November 2000 and this is June 2001. 00:30:43.670 --> 00:30:48.750 align:middle line:84% So this tells you what the yield is on a 3-month bond. 00:30:48.750 --> 00:30:51.020 align:middle line:84% So a bond that matures in three months. 00:30:51.020 --> 00:30:55.780 align:middle line:84% On a 6-month bond, so on and so forth, up to 30-year bonds. 00:30:55.780 --> 00:30:57.530 align:middle line:84% What is the big difference between these-- 00:30:57.530 --> 00:30:59.360 align:middle line:84% what do you think happened here in between? 00:30:59.360 --> 00:31:01.550 align:middle line:84% Notice that these two curves are more or less 00:31:01.550 --> 00:31:04.190 align:middle line:84% the same long-term interest rate, 00:31:04.190 --> 00:31:06.290 align:middle line:84% but they have very different-- these curves-- this 00:31:06.290 --> 00:31:09.420 align:middle line:84% is a very steep curve and this is a very flat or even inverted 00:31:09.420 --> 00:31:09.920 align:middle line:90% curve. 00:31:09.920 --> 00:31:12.810 align:middle line:90% 00:31:12.810 --> 00:31:17.420 align:middle line:84% What do you think may have happened there between November 00:31:17.420 --> 00:31:19.848 align:middle line:90% 2000 and June 2001? 00:31:19.848 --> 00:31:21.640 align:middle line:84% AUDIENCE: People changed their expectations 00:31:21.640 --> 00:31:22.330 align:middle line:90% about interest rates. 00:31:22.330 --> 00:31:23.330 align:middle line:90% RICARDO CABALLERO: Yeah. 00:31:23.330 --> 00:31:24.190 align:middle line:90% That's true. 00:31:24.190 --> 00:31:26.230 align:middle line:90% That's for sure true about that. 00:31:26.230 --> 00:31:28.120 align:middle line:90% But look also that-- 00:31:28.120 --> 00:31:30.280 align:middle line:84% but that these three months, there 00:31:30.280 --> 00:31:32.240 align:middle line:84% is very little uncertainty about three months. 00:31:32.240 --> 00:31:34.190 align:middle line:90% It was a lot lower than that. 00:31:34.190 --> 00:31:36.243 align:middle line:84% So yes, people change their expectations, 00:31:36.243 --> 00:31:38.410 align:middle line:84% but why do you think they change their expectations? 00:31:38.410 --> 00:31:42.440 align:middle line:90% 00:31:42.440 --> 00:31:45.440 align:middle line:84% AUDIENCE: [INAUDIBLE] rising [INAUDIBLE] 00:31:45.440 --> 00:31:47.690 align:middle line:84% RICARDO CABALLERO: Rising inflation from here to here. 00:31:47.690 --> 00:31:50.553 align:middle line:84% These are nominal interest rates. 00:31:50.553 --> 00:31:52.970 align:middle line:84% Up to now, we've been talking about nominal interest rate. 00:31:52.970 --> 00:31:56.510 align:middle line:90% 00:31:56.510 --> 00:32:00.290 align:middle line:84% What happens here is there was a mini-recession, 00:32:00.290 --> 00:32:03.240 align:middle line:90% so the Fed cut interest rates. 00:32:03.240 --> 00:32:05.370 align:middle line:84% When you are in recessions, the curve 00:32:05.370 --> 00:32:10.240 align:middle line:84% tends to look like this because the central bank is cutting 00:32:10.240 --> 00:32:12.550 align:middle line:84% interest rates in the short-run to deal 00:32:12.550 --> 00:32:13.990 align:middle line:90% with the current recession. 00:32:13.990 --> 00:32:15.880 align:middle line:84% What happened 30 years from now has nothing 00:32:15.880 --> 00:32:17.380 align:middle line:84% to do with the business cycle today, 00:32:17.380 --> 00:32:19.540 align:middle line:84% so that interest rate doesn't need to move a lot. 00:32:19.540 --> 00:32:22.270 align:middle line:84% But the Fed is bringing interest rates down a lot 00:32:22.270 --> 00:32:23.740 align:middle line:90% in the front end. 00:32:23.740 --> 00:32:26.890 align:middle line:84% So that's a typical shape of a curve in a recession. 00:32:26.890 --> 00:32:29.770 align:middle line:84% That's a typical shape of a curve 00:32:29.770 --> 00:32:31.780 align:middle line:84% in the opposite situation where the inflation is 00:32:31.780 --> 00:32:32.650 align:middle line:90% too high and so on. 00:32:32.650 --> 00:32:33.550 align:middle line:90% Because what happens? 00:32:33.550 --> 00:32:35.290 align:middle line:84% The Fed is trying to-- the Fed really 00:32:35.290 --> 00:32:37.630 align:middle line:84% controls the very front end of the curve. 00:32:37.630 --> 00:32:39.880 align:middle line:84% That's what the Fed really controls-- the central bank 00:32:39.880 --> 00:32:41.950 align:middle line:84% in general, but the Fed, they control the very front end 00:32:41.950 --> 00:32:43.690 align:middle line:84% of the curve because they're setting the very short-term 00:32:43.690 --> 00:32:44.750 align:middle line:90% interest rate. 00:32:44.750 --> 00:32:46.300 align:middle line:84% So this is a situation where they're 00:32:46.300 --> 00:32:49.360 align:middle line:84% tighten-- the monetary policy is very tight because they 00:32:49.360 --> 00:32:52.635 align:middle line:84% have a situation of overheating in the economy. 00:32:52.635 --> 00:32:54.260 align:middle line:84% And in fact, they got too carried away. 00:32:54.260 --> 00:32:56.885 align:middle line:84% That's the reason that we ended up in a recession here. 00:32:56.885 --> 00:32:57.385 align:middle line:90% OK. 00:32:57.385 --> 00:33:00.022 align:middle line:90% 00:33:00.022 --> 00:33:01.355 align:middle line:90% How do you think it looks today? 00:33:01.355 --> 00:33:06.330 align:middle line:90% 00:33:06.330 --> 00:33:09.730 align:middle line:84% Do you think it looks more like this or more like that? 00:33:09.730 --> 00:33:11.305 align:middle line:90% Is inflation low or high today? 00:33:11.305 --> 00:33:14.190 align:middle line:90% 00:33:14.190 --> 00:33:14.880 align:middle line:90% High. 00:33:14.880 --> 00:33:15.930 align:middle line:90% I mean, that's a problem. 00:33:15.930 --> 00:33:17.670 align:middle line:84% The Fed is trying to hike interest rates. 00:33:17.670 --> 00:33:21.180 align:middle line:84% Now recently, because of the mess in the banking sector, 00:33:21.180 --> 00:33:22.800 align:middle line:84% then expectations of interest rates 00:33:22.800 --> 00:33:26.100 align:middle line:84% have begun to decline a little, but the situation 00:33:26.100 --> 00:33:27.120 align:middle line:90% was very inverted. 00:33:27.120 --> 00:33:28.710 align:middle line:90% Here you are. 00:33:28.710 --> 00:33:30.790 align:middle line:90% The green line is today. 00:33:30.790 --> 00:33:32.130 align:middle line:90% OK, so it's very inverted. 00:33:32.130 --> 00:33:35.600 align:middle line:90% 00:33:35.600 --> 00:33:38.130 align:middle line:90% A year ago, it looked like that. 00:33:38.130 --> 00:33:40.490 align:middle line:84% So you see, the longer hasn't changed much, 00:33:40.490 --> 00:33:42.440 align:middle line:84% but a year ago, there was no sense 00:33:42.440 --> 00:33:46.468 align:middle line:84% that inflation was getting so much out of line. 00:33:46.468 --> 00:33:48.010 align:middle line:84% It happened a little later than that. 00:33:48.010 --> 00:33:52.127 align:middle line:84% There was some concern that interest rates would rise, 00:33:52.127 --> 00:33:54.460 align:middle line:84% but now it's very clear that the economy is overheating. 00:33:54.460 --> 00:33:59.540 align:middle line:84% And this-- I should have plot you something for a month ago, 00:33:59.540 --> 00:34:01.030 align:middle line:90% it would have been even steeper. 00:34:01.030 --> 00:34:03.340 align:middle line:90% OK. 00:34:03.340 --> 00:34:03.880 align:middle line:90% Anyways. 00:34:03.880 --> 00:34:06.710 align:middle line:84% But that's because the Fed is trying to slow down the economy. 00:34:06.710 --> 00:34:07.840 align:middle line:90% It's hiking interest rates. 00:34:07.840 --> 00:34:10.510 align:middle line:84% That's the reason the curve is very, very inverted today. 00:34:10.510 --> 00:34:13.300 align:middle line:90% 00:34:13.300 --> 00:34:14.800 align:middle line:90% So let me calculate these rates. 00:34:14.800 --> 00:34:15.760 align:middle line:90% How do we go about it? 00:34:15.760 --> 00:34:17.260 align:middle line:84% So the first thing we're going to do 00:34:17.260 --> 00:34:19.510 align:middle line:84% is we're going to use the expected present discounted 00:34:19.510 --> 00:34:24.070 align:middle line:84% value formula to calculate the price of a bond. 00:34:24.070 --> 00:34:27.460 align:middle line:84% And then we want to start doing it for different bonds, 00:34:27.460 --> 00:34:30.750 align:middle line:84% and we're going to construct the yield curve. 00:34:30.750 --> 00:34:36.620 align:middle line:84% So suppose you have a bond that pays $100, nothing in between, 00:34:36.620 --> 00:34:38.400 align:middle line:90% $100 one year from now. 00:34:38.400 --> 00:34:41.770 align:middle line:84% So this is a bond with maturity, 1-year maturity. 00:34:41.770 --> 00:34:45.790 align:middle line:84% I'm going to call that bond with 1-year maturity-- 00:34:45.790 --> 00:34:50.020 align:middle line:84% the price of a bond with a 1-year maturity at time t P1t. 00:34:50.020 --> 00:34:51.400 align:middle line:90% Well, that's easy to calculate. 00:34:51.400 --> 00:34:53.590 align:middle line:84% It's expected present discounted value for you. 00:34:53.590 --> 00:34:55.449 align:middle line:84% You have the interest rate, whatever it-- 00:34:55.449 --> 00:34:58.660 align:middle line:84% say 1-year interest rate, then I know that the price of the bond 00:34:58.660 --> 00:35:01.370 align:middle line:84% is 100 divided by 1 plus the interest rate, 00:35:01.370 --> 00:35:04.820 align:middle line:90% the 1-year interest rate today. 00:35:04.820 --> 00:35:05.540 align:middle line:90% That's the price. 00:35:05.540 --> 00:35:06.840 align:middle line:84% That's the expected discounted value. 00:35:06.840 --> 00:35:08.340 align:middle line:84% So I tell you-- what I'm showing you 00:35:08.340 --> 00:35:12.260 align:middle line:84% is the relationship between interest rates and prices. 00:35:12.260 --> 00:35:13.340 align:middle line:90% Our price of a bond. 00:35:13.340 --> 00:35:18.290 align:middle line:84% The price of that bond is just 100 divided by 1 plus the-- 00:35:18.290 --> 00:35:20.510 align:middle line:84% 1 plus-- the 1-year interest rate today. 00:35:20.510 --> 00:35:23.610 align:middle line:90% 00:35:23.610 --> 00:35:27.120 align:middle line:84% So important observation is that the price of a 1-year bond 00:35:27.120 --> 00:35:31.170 align:middle line:84% varies inversely with the current 1-year nominal interest 00:35:31.170 --> 00:35:31.720 align:middle line:90% rate. 00:35:31.720 --> 00:35:34.570 align:middle line:90% This is all nominal. 00:35:34.570 --> 00:35:36.490 align:middle line:84% Why is that inverse relationship? 00:35:36.490 --> 00:35:39.490 align:middle line:84% Why is it the price of a 1-year bond is inversely related 00:35:39.490 --> 00:35:40.900 align:middle line:90% to the 1-year interest rate? 00:35:40.900 --> 00:35:44.750 align:middle line:90% 00:35:44.750 --> 00:35:47.050 align:middle line:84% In other words, I'm asking, what do 00:35:47.050 --> 00:35:49.450 align:middle line:84% you think happens to the price as the nominal interest 00:35:49.450 --> 00:35:50.620 align:middle line:90% rate rises? 00:35:50.620 --> 00:35:55.010 align:middle line:84% And why do you think that's what happens to the price? 00:35:55.010 --> 00:35:56.998 align:middle line:84% Well, the first question doesn't have a-- 00:35:56.998 --> 00:35:58.790 align:middle line:84% I mean, it's very easy, you know the answer 00:35:58.790 --> 00:35:59.707 align:middle line:90% to the first question. 00:35:59.707 --> 00:36:01.110 align:middle line:90% What happens if i goes up? 00:36:01.110 --> 00:36:04.100 align:middle line:84% Well, it's obvious that this price comes down, but why? 00:36:04.100 --> 00:36:08.500 align:middle line:90% 00:36:08.500 --> 00:36:11.950 align:middle line:84% And use the concepts we have developed here. 00:36:11.950 --> 00:36:15.760 align:middle line:84% Remember, we spent like 20 minutes in one slide. 00:36:15.760 --> 00:36:17.530 align:middle line:90% Use that slide for that answer. 00:36:17.530 --> 00:36:22.800 align:middle line:90% 00:36:22.800 --> 00:36:26.220 align:middle line:84% Hint-- this $100 you are not receiving today, 00:36:26.220 --> 00:36:28.710 align:middle line:84% you are receiving a year from now. 00:36:28.710 --> 00:36:31.530 align:middle line:84% What happens with $1 received a year from now? 00:36:31.530 --> 00:36:35.250 align:middle line:84% What is the value of a dollar received one year from now 00:36:35.250 --> 00:36:36.375 align:middle line:90% when the interest is high? 00:36:36.375 --> 00:36:39.150 align:middle line:90% 00:36:39.150 --> 00:36:42.660 align:middle line:84% It's low because you'd much rather have the dollar today, 00:36:42.660 --> 00:36:46.180 align:middle line:84% invest it, and get this big return on the dollar. 00:36:46.180 --> 00:36:49.770 align:middle line:84% That means, naturally, a bond that is paying you $100 tomorrow 00:36:49.770 --> 00:36:52.985 align:middle line:84% is going to be worth less when the interest rate is very high. 00:36:52.985 --> 00:36:54.360 align:middle line:84% It's going to be worth less today 00:36:54.360 --> 00:36:55.350 align:middle line:90% when the interest is very high. 00:36:55.350 --> 00:36:58.080 align:middle line:84% You'd rather have the money today, invest it in the interest 00:36:58.080 --> 00:37:00.270 align:middle line:90% rate and get the interest rate. 00:37:00.270 --> 00:37:06.780 align:middle line:84% And I need to invest 1 over 1 plus i1t dollars to get $100. 00:37:06.780 --> 00:37:10.380 align:middle line:90% That's another way of saying it. 00:37:10.380 --> 00:37:15.110 align:middle line:84% What about the bond that pays $100 in two years? 00:37:15.110 --> 00:37:17.910 align:middle line:84% Well, I need to discount that by this, 00:37:17.910 --> 00:37:20.640 align:middle line:84% which is-- it's a product of the two interest rate. 00:37:20.640 --> 00:37:24.170 align:middle line:84% And since I don't know what the 1-year rate will be one year 00:37:24.170 --> 00:37:26.413 align:middle line:84% from now, I have to use expectation here rather than 00:37:26.413 --> 00:37:27.080 align:middle line:90% the actual rate. 00:37:27.080 --> 00:37:28.340 align:middle line:90% But look at the notation. 00:37:28.340 --> 00:37:34.910 align:middle line:84% I'm calling P2t-- $P2t, the price of a 2-year bond, 00:37:34.910 --> 00:37:39.090 align:middle line:84% a bond with maturity of two years, as of time t. 00:37:39.090 --> 00:37:42.100 align:middle line:84% And this is a bond that has no coupons. 00:37:42.100 --> 00:37:47.850 align:middle line:84% Yes, pays you $100 at the end of the two years. 00:37:47.850 --> 00:37:53.640 align:middle line:84% Now note that this price is inversely related to both 00:37:53.640 --> 00:37:58.890 align:middle line:84% the 1-year rate today and the expectation of the 1-year rate 00:37:58.890 --> 00:38:01.520 align:middle line:90% one year from now. 00:38:01.520 --> 00:38:05.650 align:middle line:84% If either one of these goes up, the bond is worthless today. 00:38:05.650 --> 00:38:11.780 align:middle line:84% You discount more $1 received two years from now. 00:38:11.780 --> 00:38:13.100 align:middle line:90% I don't care which one-- 00:38:13.100 --> 00:38:16.550 align:middle line:84% either of them that goes up is bad news 00:38:16.550 --> 00:38:18.020 align:middle line:90% for the price of a bond. 00:38:18.020 --> 00:38:24.440 align:middle line:90% 00:38:24.440 --> 00:38:25.050 align:middle line:90% It's clear? 00:38:25.050 --> 00:38:28.340 align:middle line:90% 00:38:28.340 --> 00:38:30.440 align:middle line:90% So there is an alternative. 00:38:30.440 --> 00:38:33.640 align:middle line:84% So this is the way you price bonds using just 00:38:33.640 --> 00:38:37.580 align:middle line:84% expected discounted value approach. 00:38:37.580 --> 00:38:39.790 align:middle line:84% Now it turns out that in practice, a lot of the asset 00:38:39.790 --> 00:38:42.130 align:middle line:84% pricing is done by arbitrage, meaning 00:38:42.130 --> 00:38:46.480 align:middle line:84% you compare different assets, and that have similar risks, 00:38:46.480 --> 00:38:48.820 align:middle line:84% they should give you more or less the same return. 00:38:48.820 --> 00:38:52.870 align:middle line:84% That's what you-- so let me do this arbitrage thing. 00:38:52.870 --> 00:38:55.950 align:middle line:84% Suppose you're considering investing $1 for one year. 00:38:55.950 --> 00:38:56.950 align:middle line:90% So that's your decision. 00:38:56.950 --> 00:38:57.992 align:middle line:90% I'm going to invest one-- 00:38:57.992 --> 00:39:02.060 align:middle line:84% I have a dollar, which I want to invest for one year. 00:39:02.060 --> 00:39:05.090 align:middle line:84% But I have two options to do that. 00:39:05.090 --> 00:39:08.840 align:middle line:84% I can invest $1 in a 1-year bond. 00:39:08.840 --> 00:39:11.710 align:middle line:84% I know exactly what I'm going to get in that bond. 00:39:11.710 --> 00:39:17.440 align:middle line:84% Or I can invest in a 2-year bond and sell it at the end 00:39:17.440 --> 00:39:18.640 align:middle line:90% of the first year. 00:39:18.640 --> 00:39:23.465 align:middle line:84% Those are two ways of investing for one year. 00:39:23.465 --> 00:39:25.840 align:middle line:84% Arbitrage has to be compared over the same period of time 00:39:25.840 --> 00:39:26.260 align:middle line:90% and everything. 00:39:26.260 --> 00:39:27.843 align:middle line:84% It's not the return of a bond that you 00:39:27.843 --> 00:39:30.760 align:middle line:84% hold for 10 years versus one that you hold for one year. 00:39:30.760 --> 00:39:32.780 align:middle line:84% It has to be something a similar investment. 00:39:32.780 --> 00:39:34.870 align:middle line:84% Suppose I need to invest for one year. 00:39:34.870 --> 00:39:38.640 align:middle line:90% 00:39:38.640 --> 00:39:39.140 align:middle line:90% OK. 00:39:39.140 --> 00:39:41.750 align:middle line:84% Then if I have these two bonds, the option 00:39:41.750 --> 00:39:44.822 align:middle line:84% is not buy one or the other and then 00:39:44.822 --> 00:39:46.280 align:middle line:84% hold to maturity because that would 00:39:46.280 --> 00:39:48.655 align:middle line:84% be comparing an investment of one year with an investment 00:39:48.655 --> 00:39:49.550 align:middle line:90% of two years. 00:39:49.550 --> 00:39:52.310 align:middle line:84% I need to compare the strategies of getting 00:39:52.310 --> 00:39:54.240 align:middle line:90% my return in one year. 00:39:54.240 --> 00:39:57.320 align:middle line:84% In the 1-year bond that's trivial because I get my return 00:39:57.320 --> 00:39:59.120 align:middle line:84% at the end at the maturity of the bond. 00:39:59.120 --> 00:40:02.210 align:middle line:84% In the 2-year bond, it means I need to sell it in between after 00:40:02.210 --> 00:40:03.290 align:middle line:90% one year. 00:40:03.290 --> 00:40:05.870 align:middle line:84% So those are the two strategies I want to compare. 00:40:05.870 --> 00:40:08.630 align:middle line:90% And since I'm not-- 00:40:08.630 --> 00:40:11.598 align:middle line:84% considering risk here as a central element, 00:40:11.598 --> 00:40:13.640 align:middle line:84% those two strategies are going to have to give me 00:40:13.640 --> 00:40:16.120 align:middle line:90% the same expected return. 00:40:16.120 --> 00:40:17.050 align:middle line:90% That's arbitrage. 00:40:17.050 --> 00:40:19.430 align:middle line:90% That's what we call arbitrage. 00:40:19.430 --> 00:40:25.910 align:middle line:84% The two strategies have to give me the same expected return. 00:40:25.910 --> 00:40:30.350 align:middle line:84% So what do we get from this strategies? 00:40:30.350 --> 00:40:32.400 align:middle line:84% Well, if I go through the 1-year bond, 00:40:32.400 --> 00:40:36.080 align:middle line:84% I know I'm going to get my dollar times 1 plus i1t. 00:40:36.080 --> 00:40:39.950 align:middle line:84% That's what I get for one year-- out of investing $1 in a 1-year 00:40:39.950 --> 00:40:40.580 align:middle line:90% bond. 00:40:40.580 --> 00:40:43.220 align:middle line:84% If I go through the 2-year bond strategy, 00:40:43.220 --> 00:40:45.080 align:middle line:84% buy it and sell it at the end of the year, 00:40:45.080 --> 00:40:46.520 align:middle line:90% then I'm going to get-- 00:40:46.520 --> 00:40:52.670 align:middle line:84% I invest $1 today, I'm going to pay P2t-- 00:40:52.670 --> 00:40:55.700 align:middle line:84% that's what I pay today for a 2-year bond, 00:40:55.700 --> 00:40:57.530 align:middle line:84% that's what I pay here for a 2-year bond. 00:40:57.530 --> 00:41:02.660 align:middle line:84% And I expect to get the price of a 1-year bond one year from now. 00:41:02.660 --> 00:41:06.530 align:middle line:84% I mean, the 2-year bond will be a 1-year bond after a year has 00:41:06.530 --> 00:41:08.120 align:middle line:90% passed. 00:41:08.120 --> 00:41:10.910 align:middle line:84% It's a 2-year bond today, but after one year, 00:41:10.910 --> 00:41:13.470 align:middle line:84% it's going to have only one year to mature. 00:41:13.470 --> 00:41:17.810 align:middle line:84% So that's the reason the price I need to forecast is the price 00:41:17.810 --> 00:41:20.210 align:middle line:84% of a 1-year bond one year from now. 00:41:20.210 --> 00:41:22.430 align:middle line:90% That's what this is here. 00:41:22.430 --> 00:41:24.020 align:middle line:84% And that's my return on this strategy 00:41:24.020 --> 00:41:27.900 align:middle line:84% because I'm going to pay this today, these dollars, 00:41:27.900 --> 00:41:31.020 align:middle line:84% and I expect to get that one year from now. 00:41:31.020 --> 00:41:35.835 align:middle line:84% OK so arbitrage means I need to set these to equal. 00:41:35.835 --> 00:41:42.280 align:middle line:90% 00:41:42.280 --> 00:41:47.643 align:middle line:84% So that means I have to get the same return with two strategies. 00:41:47.643 --> 00:41:49.060 align:middle line:84% That means I'm investing the same. 00:41:49.060 --> 00:41:52.980 align:middle line:84% So you only need to compare the returns. 00:41:52.980 --> 00:41:57.310 align:middle line:90% This needs to be equal to that. 00:41:57.310 --> 00:42:00.100 align:middle line:90% That's what I have here. 00:42:00.100 --> 00:42:04.530 align:middle line:84% Which tells you that you are solving from here that the price 00:42:04.530 --> 00:42:08.610 align:middle line:84% of a 2-year bond at time t is equal to the expected price 00:42:08.610 --> 00:42:13.960 align:middle line:84% of a 1-year bond at t plus 1 discounted by 1 plus the 1-year 00:42:13.960 --> 00:42:16.770 align:middle line:90% interest rate. 00:42:16.770 --> 00:42:18.870 align:middle line:90% This was like my cash flow-- 00:42:18.870 --> 00:42:20.830 align:middle line:84% my cash flow now is not the cash flow-- 00:42:20.830 --> 00:42:23.340 align:middle line:84% it's just the price I'm going to get a price for that asset. 00:42:23.340 --> 00:42:26.880 align:middle line:84% That's like the z's I had in my formula. 00:42:26.880 --> 00:42:30.660 align:middle line:84% And for a 1-year strategy, I only needed to worry about the z 00:42:30.660 --> 00:42:31.690 align:middle line:90% plus 1. 00:42:31.690 --> 00:42:34.780 align:middle line:90% There was no dividend at day 0. 00:42:34.780 --> 00:42:37.110 align:middle line:90% And that's exactly that formula. 00:42:37.110 --> 00:42:43.730 align:middle line:84% But notice that at t plus 1, that will hold. 00:42:43.730 --> 00:42:44.968 align:middle line:90% So at t plus 1-- 00:42:44.968 --> 00:42:46.760 align:middle line:84% I'm at t plus 1, I don't need expectations. 00:42:46.760 --> 00:42:51.470 align:middle line:84% I know that P1t plus 1 will be equal to 100 divided by 1 plus 00:42:51.470 --> 00:42:54.340 align:middle line:84% i1, the 1-year rate, at t plus 1. 00:42:54.340 --> 00:42:57.830 align:middle line:84% Therefore, the expected is something 00:42:57.830 --> 00:42:59.090 align:middle line:90% like this, approximately. 00:42:59.090 --> 00:43:02.490 align:middle line:84% The expected price is something like that. 00:43:02.490 --> 00:43:05.890 align:middle line:84% I expect-- I mean, this will be-- without the e, 00:43:05.890 --> 00:43:09.400 align:middle line:84% will be the price of this 1-year bond at t plus 1. 00:43:09.400 --> 00:43:12.020 align:middle line:84% I don't know exactly what the interest rate will be next year, 00:43:12.020 --> 00:43:15.310 align:middle line:84% so I have-- the best I can do is have an expectation-- that's 00:43:15.310 --> 00:43:18.400 align:middle line:90% my expectation, approximately. 00:43:18.400 --> 00:43:23.120 align:middle line:84% But now I can stick this expression in here. 00:43:23.120 --> 00:43:27.770 align:middle line:84% I have this I'm going out and I can stick that in there 00:43:27.770 --> 00:43:29.940 align:middle line:90% and I get this expression. 00:43:29.940 --> 00:43:32.810 align:middle line:84% So that's the price for the 2-year bond. 00:43:32.810 --> 00:43:33.920 align:middle line:90% Do you recognize this? 00:43:33.920 --> 00:43:37.805 align:middle line:90% 00:43:37.805 --> 00:43:38.555 align:middle line:90% You saw it before. 00:43:38.555 --> 00:43:46.970 align:middle line:90% 00:43:46.970 --> 00:43:48.740 align:middle line:84% That's the same expression that we 00:43:48.740 --> 00:43:51.260 align:middle line:84% got when we used the expected present discounted value 00:43:51.260 --> 00:43:53.430 align:middle line:90% formula. 00:43:53.430 --> 00:43:55.980 align:middle line:84% We said we're going to go-- say $100 years-- 00:43:55.980 --> 00:43:58.290 align:middle line:84% 100 years-- two years from now, I 00:43:58.290 --> 00:44:00.450 align:middle line:84% know that discount factor for that is 1 over 1 00:44:00.450 --> 00:44:05.460 align:middle line:84% plus i1t times 1 plus i1t plus 1 expected. 00:44:05.460 --> 00:44:08.040 align:middle line:90% Well, that's what I got. 00:44:08.040 --> 00:44:09.460 align:middle line:90% Get some arbitrage. 00:44:09.460 --> 00:44:10.760 align:middle line:90% OK. 00:44:10.760 --> 00:44:11.990 align:middle line:90% From an arbitrage logic. 00:44:11.990 --> 00:44:15.830 align:middle line:90% This is used a lot in finance. 00:44:15.830 --> 00:44:17.600 align:middle line:84% I'm going to say something complicated, 00:44:17.600 --> 00:44:23.290 align:middle line:90% but just ignore it if it's-- 00:44:23.290 --> 00:44:26.310 align:middle line:90% 00:44:26.310 --> 00:44:29.470 align:middle line:84% it's not relevant for the quiz or anything. 00:44:29.470 --> 00:44:33.440 align:middle line:84% But there is a big debate in the US today about-- 00:44:33.440 --> 00:44:42.000 align:middle line:84% not big debate, a big concern about the US Treasury debt 00:44:42.000 --> 00:44:43.950 align:middle line:84% because there is a debt ceiling, meaning 00:44:43.950 --> 00:44:46.620 align:middle line:84% there is a maximum amount that the government can-- 00:44:46.620 --> 00:44:52.090 align:middle line:84% of debt they can issue, and that ceiling 00:44:52.090 --> 00:44:54.640 align:middle line:84% has been moved over time, but every time we 00:44:54.640 --> 00:44:57.700 align:middle line:84% get close to a deadline, when this needs to be agreed again, 00:44:57.700 --> 00:45:00.974 align:middle line:84% there is a concern and there's negotiations and so on, and-- 00:45:00.974 --> 00:45:03.700 align:middle line:90% 00:45:03.700 --> 00:45:06.310 align:middle line:84% well, I mean, everyone at this moment at least 00:45:06.310 --> 00:45:10.390 align:middle line:84% thinks that, as in every instance in the past, 00:45:10.390 --> 00:45:13.000 align:middle line:84% they're going to reach some sort of agreement the day 00:45:13.000 --> 00:45:15.920 align:middle line:90% before of the deadline or not. 00:45:15.920 --> 00:45:18.550 align:middle line:84% But if they don't, and there is a mess, 00:45:18.550 --> 00:45:20.320 align:middle line:90% this is huge for finance. 00:45:20.320 --> 00:45:23.410 align:middle line:84% It's huge for finance because US Treasury bonds, especially 00:45:23.410 --> 00:45:26.590 align:middle line:84% short-term bonds, are used for pricing everything 00:45:26.590 --> 00:45:28.430 align:middle line:90% through arbitrage and so on. 00:45:28.430 --> 00:45:30.070 align:middle line:90% So you get a mess there. 00:45:30.070 --> 00:45:32.650 align:middle line:84% That's a mess in every single financial market. 00:45:32.650 --> 00:45:36.640 align:middle line:84% You wouldn't know how to price many financial assets, actually. 00:45:36.640 --> 00:45:38.020 align:middle line:90% So it would be a disaster. 00:45:38.020 --> 00:45:42.340 align:middle line:84% But the reason I mention this here 00:45:42.340 --> 00:45:44.770 align:middle line:84% is because, again, lots of prices 00:45:44.770 --> 00:45:47.140 align:middle line:84% are priced in reference-- in finance 00:45:47.140 --> 00:45:49.610 align:middle line:84% are priced in reference, especially derivatives, options, 00:45:49.610 --> 00:45:52.880 align:middle line:84% and stuff like that, you price them relative to something 00:45:52.880 --> 00:45:54.660 align:middle line:90% using this type of logic. 00:45:54.660 --> 00:45:56.390 align:middle line:84% So if the thing you use as a base, 00:45:56.390 --> 00:45:58.640 align:middle line:84% as a reference becomes highly unstable 00:45:58.640 --> 00:46:01.790 align:middle line:84% and uncertain and risky, then obviously everything 00:46:01.790 --> 00:46:04.420 align:middle line:84% becomes very complicated, very risky, 00:46:04.420 --> 00:46:09.790 align:middle line:84% and financial markets do not like risk, that's for sure. 00:46:09.790 --> 00:46:12.970 align:middle line:84% Anyways, ignore that, that's irrelevant for your quiz, 00:46:12.970 --> 00:46:15.220 align:middle line:84% but that's the reason this-- the whole discussion 00:46:15.220 --> 00:46:17.800 align:middle line:84% can over the summer can get to be very, very 00:46:17.800 --> 00:46:20.850 align:middle line:90% tricky for finance. 00:46:20.850 --> 00:46:23.340 align:middle line:84% So the yield to maturity-- remember I mentioned this 00:46:23.340 --> 00:46:27.750 align:middle line:84% concept before-- of an n-year bond is also-- 00:46:27.750 --> 00:46:31.780 align:middle line:84% whenever you hear the 3-year rate, it's that. 00:46:31.780 --> 00:46:34.390 align:middle line:90% It's the yield to maturity. 00:46:34.390 --> 00:46:35.700 align:middle line:90% A which is different from-- 00:46:35.700 --> 00:46:37.350 align:middle line:90% OK, let me tell you-- 00:46:37.350 --> 00:46:40.050 align:middle line:84% show you a formula that's easy to explain, then. 00:46:40.050 --> 00:46:42.330 align:middle line:84% And it's defined-- it's important-- 00:46:42.330 --> 00:46:46.350 align:middle line:84% as the constant annual interest rate that makes the bond price 00:46:46.350 --> 00:46:50.010 align:middle line:84% today equal to the present discounted value or expected 00:46:50.010 --> 00:46:51.250 align:middle line:90% discounted value. 00:46:51.250 --> 00:46:53.910 align:middle line:84% So notice the highlight there, it's 00:46:53.910 --> 00:46:56.730 align:middle line:84% defined as the constant annual interest 00:46:56.730 --> 00:47:00.240 align:middle line:84% rate that makes the bond price today equal to the present 00:47:00.240 --> 00:47:04.370 align:middle line:84% discounted value of future payments of the bond. 00:47:04.370 --> 00:47:10.510 align:middle line:84% So, for example, in our 2-year bond, that's the price. 00:47:10.510 --> 00:47:12.450 align:middle line:90% This is the price of the asset. 00:47:12.450 --> 00:47:14.400 align:middle line:90% That we know the price. 00:47:14.400 --> 00:47:18.010 align:middle line:84% We already got the price from the previous slides of the bond, 00:47:18.010 --> 00:47:20.520 align:middle line:84% which was based on the short-term interest rate, 00:47:20.520 --> 00:47:23.790 align:middle line:84% 1-year interest rate, and our forecast of the short-term-- 00:47:23.790 --> 00:47:26.790 align:middle line:84% the 1-year interest rate one year from now. 00:47:26.790 --> 00:47:29.080 align:middle line:84% I know that price, take that as a number. 00:47:29.080 --> 00:47:32.250 align:middle line:84% So then the yield the yield to maturity 00:47:32.250 --> 00:47:35.790 align:middle line:84% is calculated as that constant interest rate-- 00:47:35.790 --> 00:47:37.432 align:middle line:84% constant-- how do I see this constant? 00:47:37.432 --> 00:47:39.390 align:middle line:84% Because, well, I'm using the same interest rate 00:47:39.390 --> 00:47:41.490 align:middle line:84% for the first period and the second period, 00:47:41.490 --> 00:47:42.990 align:middle line:90% and now I'm calling it i2t. 00:47:42.990 --> 00:47:45.510 align:middle line:84% It's a 2-year interest rate, but it's constant. 00:47:45.510 --> 00:47:48.730 align:middle line:84% Constant doesn't mean that it doesn't move over time. 00:47:48.730 --> 00:47:50.560 align:middle line:84% It means I'm discounting all the cash flows 00:47:50.560 --> 00:47:51.880 align:middle line:90% as a constant interest rate. 00:47:51.880 --> 00:47:57.770 align:middle line:84% It means I'm using this equation. 00:47:57.770 --> 00:47:58.310 align:middle line:90% OK. 00:47:58.310 --> 00:48:02.930 align:middle line:84% So the yield to maturity is find an interest rate that allows 00:48:02.930 --> 00:48:04.640 align:middle line:90% me to use this constant thing-- 00:48:04.640 --> 00:48:07.820 align:middle line:84% constant-- assume-- use this formula 00:48:07.820 --> 00:48:10.820 align:middle line:84% and get back the same price as I got 00:48:10.820 --> 00:48:15.560 align:middle line:84% by using the expected discounted value or the arbitrage 00:48:15.560 --> 00:48:16.970 align:middle line:90% or something like that. 00:48:16.970 --> 00:48:19.280 align:middle line:90% So that's the definition. 00:48:19.280 --> 00:48:20.940 align:middle line:90% You this price. 00:48:20.940 --> 00:48:23.360 align:middle line:84% Now you look for that interest rate that 00:48:23.360 --> 00:48:24.890 align:middle line:90% allows you to match that price. 00:48:24.890 --> 00:48:27.510 align:middle line:90% 00:48:27.510 --> 00:48:28.740 align:middle line:90% And that's called the yield. 00:48:28.740 --> 00:48:33.780 align:middle line:84% That's the thing-- remember, I plotted some curves. 00:48:33.780 --> 00:48:36.090 align:middle line:84% Well, those interest rates and those curves 00:48:36.090 --> 00:48:39.340 align:middle line:90% were computed this way. 00:48:39.340 --> 00:48:42.850 align:middle line:84% Now notice that we know what this price is. 00:48:42.850 --> 00:48:45.400 align:middle line:84% This price is, by the expected discounted value 00:48:45.400 --> 00:48:49.760 align:middle line:84% or the arbitrage approach, is equal to 100 divided by this. 00:48:49.760 --> 00:48:51.670 align:middle line:84% So I know that these two things are-- 00:48:51.670 --> 00:48:56.820 align:middle line:84% this is equal to that, which means that this denominator is 00:48:56.820 --> 00:48:58.660 align:middle line:90% equal to that. 00:48:58.660 --> 00:49:01.420 align:middle line:84% And that implies, for a small interest rate, 00:49:01.420 --> 00:49:04.900 align:middle line:84% that these two-year interest rate is approximately equal 00:49:04.900 --> 00:49:07.510 align:middle line:84% to the average of the expected interest rate-- 00:49:07.510 --> 00:49:08.320 align:middle line:90% 1-year rates. 00:49:08.320 --> 00:49:11.310 align:middle line:90% 00:49:11.310 --> 00:49:14.130 align:middle line:84% So this is called actually the expectation or hypothesis, 00:49:14.130 --> 00:49:14.890 align:middle line:90% by the way. 00:49:14.890 --> 00:49:18.420 align:middle line:84% Is that the 2-year rate is approximately equal 00:49:18.420 --> 00:49:23.160 align:middle line:84% to the average of the 1-year rate this year plus the expected 00:49:23.160 --> 00:49:25.500 align:middle line:90% 1-year rate one year from now. 00:49:25.500 --> 00:49:30.130 align:middle line:90% 00:49:30.130 --> 00:49:31.910 align:middle line:90% So that's an important concept. 00:49:31.910 --> 00:49:36.030 align:middle line:84% I'm going to start from here again in the next lecture. 00:49:36.030 --> 00:49:49.000 align:middle line:90%