WEBVTT 00:00:17.079 --> 00:00:21.079 Today we're going to talk about 00:00:19.239 --> 00:00:22.279 a very important topic 00:00:21.079 --> 00:00:24.839 topic in economics, which is 00:00:22.280 --> 00:00:26.160 expectations. We have barely mentioned 00:00:24.839 --> 00:00:28.879 expectations when we talk about the 00:00:26.160 --> 00:00:30.600 Phillips curve. We talked about 00:00:28.879 --> 00:00:33.719 expectations when we 00:00:30.600 --> 00:00:35.240 when we discussed the UEP and so on. But 00:00:33.719 --> 00:00:37.320 expectations is a much bigger issue in 00:00:35.240 --> 00:00:39.800 economics. In fact, most decisions by 00:00:37.320 --> 00:00:42.759 firms, by consumers, 00:00:39.799 --> 00:00:44.519 governments involve considerations of 00:00:42.759 --> 00:00:46.879 the future. And it plays an even bigger 00:00:44.520 --> 00:00:48.720 role in finance, in which essentially 00:00:46.880 --> 00:00:51.800 everything is about the future. The 00:00:48.719 --> 00:00:53.640 price of an asset today is meaningless 00:00:51.799 --> 00:00:55.079 in itself. You have to compare it with 00:00:53.640 --> 00:00:56.399 what you expect to get out of that asset 00:00:55.079 --> 00:00:58.119 in the future. So, it's all about 00:00:56.399 --> 00:00:59.600 expectations and so on. 00:00:58.119 --> 00:01:01.799 So, that's what we we're going to do 00:00:59.600 --> 00:01:05.439 today. We're going to talk about 00:01:01.799 --> 00:01:07.239 uh expectations, the how to value things 00:01:05.439 --> 00:01:08.599 that that you expect to receive in the 00:01:07.239 --> 00:01:10.679 future, 00:01:08.599 --> 00:01:13.079 uh and how to compare those things with 00:01:10.680 --> 00:01:14.720 things that you have in the present. 00:01:13.079 --> 00:01:16.760 Um 00:01:14.719 --> 00:01:18.400 but before doing that, actually, let's 00:01:16.760 --> 00:01:21.880 talk a little bit about the news. Who 00:01:18.400 --> 00:01:24.160 knows who First Republic Bank is? 00:01:21.879 --> 00:01:27.679 Remember that a few weeks ago I told you 00:01:24.159 --> 00:01:30.519 that um Silicon Valley Bank, 00:01:27.680 --> 00:01:31.440 you read it. I I I I just mentioned it 00:01:30.519 --> 00:01:32.319 that 00:01:31.439 --> 00:01:33.280 uh 00:01:32.319 --> 00:01:35.159 uh 00:01:33.280 --> 00:01:38.840 we'll discuss it that that you know, we 00:01:35.159 --> 00:01:41.159 had the second largest bank by asset in 00:01:38.840 --> 00:01:43.840 in in US history. It was Silicon Valley 00:01:41.159 --> 00:01:46.599 Bank was the second largest 00:01:43.840 --> 00:01:48.560 asset bank in terms of assets 00:01:46.599 --> 00:01:50.599 uh to collapse in the US. The first one 00:01:48.560 --> 00:01:52.760 was uh many years ago. 00:01:50.599 --> 00:01:55.519 Uh and then we had this bank that had 00:01:52.760 --> 00:01:57.520 more than $200 billion in assets that 00:01:55.519 --> 00:02:00.039 essentially collapsed in a few days. It 00:01:57.519 --> 00:02:01.319 was a run on deposits. They had problems 00:02:00.040 --> 00:02:03.359 before, 00:02:01.319 --> 00:02:05.679 but what really did as it always the 00:02:03.359 --> 00:02:07.640 case with banks is they had a run on 00:02:05.680 --> 00:02:08.240 deposits, funding. 00:02:07.640 --> 00:02:11.280 Uh 00:02:08.240 --> 00:02:14.120 well, it's no longer the second largest 00:02:11.280 --> 00:02:15.439 collapse in US bank history. Now we have 00:02:14.120 --> 00:02:17.840 over the weekend 00:02:15.439 --> 00:02:20.520 the the new second largest bank to 00:02:17.840 --> 00:02:22.400 collapse, which is First Republic Bank, 00:02:20.520 --> 00:02:26.000 that was essentially 00:02:22.400 --> 00:02:27.879 liquidated and sold to JP Morgan over 00:02:26.000 --> 00:02:29.719 today morning, very very early in the 00:02:27.879 --> 00:02:31.120 morning. Okay? So, 00:02:29.719 --> 00:02:32.960 you have an account in First Republic 00:02:31.120 --> 00:02:34.759 Bank, you sooner likely to have an 00:02:32.960 --> 00:02:37.719 account in JP Morgan. 00:02:34.759 --> 00:02:39.840 But again, what made it collapse was 00:02:37.719 --> 00:02:41.599 something very similar to what made 00:02:39.840 --> 00:02:44.400 Silicon Valley Bank collapse, which is 00:02:41.599 --> 00:02:46.599 that they had invested on on a series of 00:02:44.400 --> 00:02:48.840 things that were very vulnerable to to 00:02:46.599 --> 00:02:49.719 the fast pace of hikes 00:02:48.840 --> 00:02:51.840 uh 00:02:49.719 --> 00:02:53.080 in interest rates in the US. 00:02:51.840 --> 00:02:55.479 And when they had those losses, 00:02:53.080 --> 00:02:57.040 depositors became became worried about 00:02:55.479 --> 00:02:58.719 it and eventually they decided not to 00:02:57.039 --> 00:03:00.519 wait, just run. 00:02:58.719 --> 00:03:02.439 And see what happened. They 00:03:00.520 --> 00:03:04.719 First Republic Bank lost about $100 00:03:02.439 --> 00:03:07.199 billion in deposits just last week. 00:03:04.719 --> 00:03:10.280 Okay? Um the last few days of last week. 00:03:07.199 --> 00:03:11.759 So, so, so 00:03:10.280 --> 00:03:13.640 so that 00:03:11.759 --> 00:03:15.319 it was obvious that that 00:03:13.639 --> 00:03:16.839 it was not going to survive and that's 00:03:15.319 --> 00:03:18.479 the reason 00:03:16.840 --> 00:03:20.479 something was arranged over the weekend 00:03:18.479 --> 00:03:23.039 to avoid the panics associated with 00:03:20.479 --> 00:03:24.560 collapses of a bank and so on. Okay? But 00:03:23.039 --> 00:03:27.400 anyways, by the way, this is all about 00:03:24.560 --> 00:03:29.280 expectations. This is you know, this is 00:03:27.400 --> 00:03:31.560 if people had expected the deposit to 00:03:29.280 --> 00:03:32.840 remain in the bank, then probably this 00:03:31.560 --> 00:03:34.960 bank would not have collapsed. It's all 00:03:32.840 --> 00:03:38.640 about people anticipating what other 00:03:34.960 --> 00:03:40.120 people will do and so on and so forth. 00:03:38.639 --> 00:03:41.919 Okay, but now let me get into the 00:03:40.120 --> 00:03:43.360 specific of 00:03:41.919 --> 00:03:44.799 of this lecture. 00:03:43.360 --> 00:03:47.600 So, there you have this is the most 00:03:44.800 --> 00:03:50.000 important index of of equity equity 00:03:47.599 --> 00:03:52.879 index in the US, S&P 500. It's a very 00:03:50.000 --> 00:03:55.319 inclusive index that captures all the 00:03:52.879 --> 00:03:56.400 large most of the large companies in the 00:03:55.319 --> 00:03:58.239 US, 00:03:56.400 --> 00:04:00.800 all of the large I think companies in 00:03:58.240 --> 00:04:04.760 the US. And uh that's an index. It's an 00:04:00.800 --> 00:04:07.360 average weighted average by by this 00:04:04.759 --> 00:04:08.799 uh capitalization value of each of the 00:04:07.360 --> 00:04:10.960 shares. It's a weighted average of the 00:04:08.800 --> 00:04:12.480 major the main shares in the US, equity 00:04:10.960 --> 00:04:14.599 shares in the US. 00:04:12.479 --> 00:04:16.959 And one thing you see is that it moves a 00:04:14.599 --> 00:04:18.439 lot around, you know? Here, for example, 00:04:16.959 --> 00:04:20.238 when when 00:04:18.439 --> 00:04:21.839 we became aware that COVID was going to 00:04:20.238 --> 00:04:24.599 be a serious issue, 00:04:21.839 --> 00:04:26.199 the US equity market collapsed by 35% or 00:04:24.600 --> 00:04:28.160 so. That's a very large collapse in a 00:04:26.199 --> 00:04:31.719 very short period of time. 00:04:28.160 --> 00:04:33.720 And then, as a result of lots of policy 00:04:31.720 --> 00:04:37.000 support, actually we had a massive 00:04:33.720 --> 00:04:40.080 rally. Uh so, up to the end of 2021, the 00:04:37.000 --> 00:04:41.399 equity market had rallied by 114%. 00:04:40.079 --> 00:04:44.279 So, a big rally. 00:04:41.399 --> 00:04:45.919 Then we got inflation and the Fed began 00:04:44.279 --> 00:04:47.359 to worry about 00:04:45.920 --> 00:04:48.720 inflation, so they began to hike 00:04:47.360 --> 00:04:50.840 interest rates. And when they hike 00:04:48.720 --> 00:04:52.520 interest rates, that eventually led to a 00:04:50.839 --> 00:04:53.119 very large decline 00:04:52.519 --> 00:04:55.560 uh 00:04:53.120 --> 00:04:58.560 in in asset prices of the order of 30% 00:04:55.560 --> 00:05:00.480 or so, 25% or so, actually, from the 00:04:58.560 --> 00:05:02.519 peak to the bottom. 00:05:00.480 --> 00:05:04.480 And then, since the bottom, which is was 00:05:02.519 --> 00:05:07.359 more or less October of last year, we 00:05:04.480 --> 00:05:09.280 have seen a recovery of about 16% or so 00:05:07.360 --> 00:05:11.280 of the equity market. Okay? And if you 00:05:09.279 --> 00:05:13.639 look at the Nasdaq, which is another one 00:05:11.279 --> 00:05:14.639 index that is very loaded towards 00:05:13.639 --> 00:05:16.319 uh 00:05:14.639 --> 00:05:19.360 technology companies, then you can see 00:05:16.319 --> 00:05:22.639 swings that are even larger than that. 00:05:19.360 --> 00:05:23.800 Now, why do these prices move so much? 00:05:22.639 --> 00:05:27.360 Well, 00:05:23.800 --> 00:05:29.280 a lot of it has to do with expectations. 00:05:27.360 --> 00:05:30.960 You know, are things going to get worse 00:05:29.279 --> 00:05:33.359 in the future? Will the Fed cause a 00:05:30.959 --> 00:05:35.079 recession? Uh 00:05:33.360 --> 00:05:37.480 how much higher will be the interest 00:05:35.079 --> 00:05:38.959 rate? And things like that matter a 00:05:37.480 --> 00:05:41.319 great deal. 00:05:38.959 --> 00:05:42.879 Another thing that matters a great 00:05:41.319 --> 00:05:44.719 deal is 00:05:42.879 --> 00:05:46.000 how much people want to take risk at any 00:05:44.720 --> 00:05:47.320 moment in time. And if you're very 00:05:46.000 --> 00:05:49.759 scared about the environment, you're 00:05:47.319 --> 00:05:51.279 unlikely to want to have something that 00:05:49.759 --> 00:05:52.879 to invest on something that can move so 00:05:51.279 --> 00:05:54.559 much, 00:05:52.879 --> 00:05:56.519 and so risk is well known. So, it's 00:05:54.560 --> 00:05:58.600 called risk off when when people don't 00:05:56.519 --> 00:06:00.919 want to take risk, these asset prices 00:05:58.600 --> 00:06:03.920 tend to collapse. Okay? Of the risky 00:06:00.920 --> 00:06:05.319 asset. Equity is a very risky asset. 00:06:03.920 --> 00:06:06.879 But that's not the only thing that moves 00:06:05.319 --> 00:06:08.480 these assets around. It's not just the 00:06:06.879 --> 00:06:10.360 risk that the companies underlying 00:06:08.480 --> 00:06:11.560 company may go bankrupt or anything like 00:06:10.360 --> 00:06:13.639 that. 00:06:11.560 --> 00:06:14.600 Here you have, for example, the movement 00:06:13.639 --> 00:06:16.599 of a 00:06:14.600 --> 00:06:18.040 for it's an ETF, but it doesn't matter. 00:06:16.600 --> 00:06:20.360 It's a portfolio of 00:06:18.040 --> 00:06:22.480 bonds of US Treasury bonds of very long 00:06:20.360 --> 00:06:24.480 duration. Maturities beyond 00:06:22.480 --> 00:06:26.520 uh 20 years and so. 00:06:24.480 --> 00:06:28.000 So, this is incredibly safe bonds, you 00:06:26.519 --> 00:06:29.639 know? Because it's US Treasuries. So, 00:06:28.000 --> 00:06:30.800 there's no risk of default or anything 00:06:29.639 --> 00:06:32.079 like that. 00:06:30.800 --> 00:06:33.680 Still, 00:06:32.079 --> 00:06:35.120 the price swings can be pretty large. I 00:06:33.680 --> 00:06:37.040 mean, in over this period, you know, 00:06:35.120 --> 00:06:39.680 there you have seen an an an increase in 00:06:37.040 --> 00:06:42.920 value of 45%, 00:06:39.680 --> 00:06:45.879 then uh a decline in value of of of 00:06:42.920 --> 00:06:48.439 about 20%. Another increase in 15% here. 00:06:45.879 --> 00:06:50.680 There was a huge decline, 40%, 00:06:48.439 --> 00:06:51.639 since since essentially 00:06:50.680 --> 00:06:53.319 uh uh uh 00:06:51.639 --> 00:06:56.079 What do you think happened here? Why is 00:06:53.319 --> 00:06:57.319 this big decline in in in in bonds? 00:06:56.079 --> 00:06:59.680 You're going to be able to answer that 00:06:57.319 --> 00:07:01.319 very precisely later on, but but I can 00:06:59.680 --> 00:07:02.800 tell you in advance that that was 00:07:01.319 --> 00:07:04.279 essentially the result of monetary 00:07:02.800 --> 00:07:07.120 policy tightening. 00:07:04.279 --> 00:07:09.039 You know, increasing interest rate 00:07:07.120 --> 00:07:10.360 caused the the bonds to decline. So, 00:07:09.040 --> 00:07:12.480 even these instruments that are very 00:07:10.360 --> 00:07:13.840 safe in the sense that you if you hold 00:07:12.480 --> 00:07:15.800 it to maturity, you will get your money 00:07:13.839 --> 00:07:18.159 back and all the promised coupons along 00:07:15.800 --> 00:07:20.360 the path, well, still their price can 00:07:18.160 --> 00:07:22.080 move a lot. And it's obvious that that 00:07:20.360 --> 00:07:23.639 movement in price 00:07:22.079 --> 00:07:25.199 is something you need to explain in 00:07:23.639 --> 00:07:27.800 terms of expectations, what people 00:07:25.199 --> 00:07:29.479 expect things to to happen. In this 00:07:27.800 --> 00:07:31.400 case, it's not whether people expect to 00:07:29.480 --> 00:07:34.160 get paid or not, because you will get 00:07:31.399 --> 00:07:35.799 paid, but it's expect but in this 00:07:34.160 --> 00:07:37.600 particular case, it's about expectations 00:07:35.800 --> 00:07:39.040 about future interest rate. If you think 00:07:37.600 --> 00:07:40.480 the interest rate will be very high, 00:07:39.040 --> 00:07:42.680 then the price of bonds will tend to be 00:07:40.480 --> 00:07:45.879 very low and so on. But it's all about 00:07:42.680 --> 00:07:48.439 the future. Okay? 00:07:45.879 --> 00:07:50.639 So, the a key concept 00:07:48.439 --> 00:07:52.759 uh that we're going to discuss today and 00:07:50.639 --> 00:07:54.959 then we're going going to use it to 00:07:52.759 --> 00:07:57.360 price a specific asset 00:07:54.959 --> 00:07:59.439 uh is the concept of expected present 00:07:57.360 --> 00:08:01.120 discounted value. This this is a loaded 00:07:59.439 --> 00:08:03.480 concept. There's lots of 00:08:01.120 --> 00:08:06.840 terms in there and we need to understand 00:08:03.480 --> 00:08:08.560 what each of these terms means. 00:08:06.839 --> 00:08:10.239 So, the key issue 00:08:08.560 --> 00:08:11.680 that we're going to discuss is how how 00:08:10.240 --> 00:08:13.800 do we decide, for example, if you see 00:08:11.680 --> 00:08:16.720 the price of an asset out there 00:08:13.800 --> 00:08:19.680 that is 100, how do you decide whether 00:08:16.720 --> 00:08:23.160 that price is fair or not, looks cheap 00:08:19.680 --> 00:08:25.199 or not? Okay? Uh and and and and and 00:08:23.160 --> 00:08:26.680 that question means you have to decide 00:08:25.199 --> 00:08:29.680 whether that price that you're paying 00:08:26.680 --> 00:08:31.519 today is consistent with the future cash 00:08:29.680 --> 00:08:32.960 flows that you're going to get from this 00:08:31.519 --> 00:08:34.960 asset. I mean, that's the reason you buy 00:08:32.960 --> 00:08:37.960 an asset is because you'll get something 00:08:34.960 --> 00:08:39.400 in return in the future. Okay? But how 00:08:37.960 --> 00:08:41.639 do we compare that? How do we compare 00:08:39.399 --> 00:08:44.759 the price today with those things that 00:08:41.639 --> 00:08:44.759 will happen in the future? 00:08:46.759 --> 00:08:50.759 So, answering that question, which is 00:08:48.879 --> 00:08:52.879 what we're going to do in this lecture, 00:08:50.759 --> 00:08:55.720 involves the following 00:08:52.879 --> 00:08:57.360 concepts. First, expectations, big 00:08:55.720 --> 00:08:59.279 thing. 00:08:57.360 --> 00:09:01.440 That's a you know, this is expected 00:08:59.279 --> 00:09:04.199 present discounted value. The E part is 00:09:01.440 --> 00:09:05.360 for expectations. That comes there. 00:09:04.200 --> 00:09:07.280 You 00:09:05.360 --> 00:09:08.639 Expectations are really crucial because 00:09:07.279 --> 00:09:11.039 these are things that happen in the 00:09:08.639 --> 00:09:13.159 future. You need to expect. Even if if 00:09:11.039 --> 00:09:16.000 it's a bond that promises you to pay, 00:09:13.159 --> 00:09:17.480 you know, 50 cents per dollar every 6 00:09:16.000 --> 00:09:19.279 month, 00:09:17.480 --> 00:09:20.560 you still may have an expectation that, 00:09:19.279 --> 00:09:23.120 you know, if it is a bond issued by 00:09:20.559 --> 00:09:25.559 First Republic Bank, it may not pay. So, 00:09:23.120 --> 00:09:26.879 so, so you need to have an expectations 00:09:25.559 --> 00:09:28.000 about that. 00:09:26.879 --> 00:09:30.480 Uh 00:09:28.000 --> 00:09:32.840 so, crucial term is expectation. 00:09:30.480 --> 00:09:34.600 Then you need some method 00:09:32.840 --> 00:09:36.840 uh to compare payments received in the 00:09:34.600 --> 00:09:39.040 future with payments made today. I mean, 00:09:36.840 --> 00:09:40.320 if you buy an asset, you pay today, 00:09:39.039 --> 00:09:42.319 but you're going to receive things 00:09:40.320 --> 00:09:45.240 returns on for that asset in the future. 00:09:42.320 --> 00:09:47.000 So, how do I compare that that that 00:09:45.240 --> 00:09:49.720 Suppose I pay one today and I receive 00:09:47.000 --> 00:09:52.600 one 1 year from now. 00:09:49.720 --> 00:09:54.920 Does that seem like a good asset? 00:09:52.600 --> 00:09:56.120 Probably not. I mean, you know, 00:09:54.919 --> 00:09:57.559 probably not. 00:09:56.120 --> 00:09:59.840 Uh 00:09:57.559 --> 00:10:01.559 Um and that's what the word discounted 00:09:59.840 --> 00:10:04.600 really means. You know, when you say 00:10:01.559 --> 00:10:05.719 expected present discounted value 00:10:04.600 --> 00:10:07.680 it says 00:10:05.720 --> 00:10:09.480 somehow that things I receive in the 00:10:07.679 --> 00:10:10.879 future are valued less than things I 00:10:09.480 --> 00:10:12.240 have today. 00:10:10.879 --> 00:10:13.439 Okay? So, if you're going to tell me 00:10:12.240 --> 00:10:15.000 that you're going to pay me a dollar in 00:10:13.440 --> 00:10:17.040 the future and I have to pay you a 00:10:15.000 --> 00:10:18.759 dollar today, most likely I won't take 00:10:17.039 --> 00:10:20.039 that deal. 00:10:18.759 --> 00:10:22.159 So, I need In other words, I'm 00:10:20.039 --> 00:10:23.480 discounting the future. 00:10:22.159 --> 00:10:24.360 How do we discount the future? Well, 00:10:23.480 --> 00:10:26.879 something that we're going to have to 00:10:24.360 --> 00:10:26.879 figure out. 00:10:27.159 --> 00:10:30.759 So, 00:10:28.600 --> 00:10:31.960 let's let me first shut down this part, 00:10:30.759 --> 00:10:33.480 the expectations, and then we'll 00:10:31.960 --> 00:10:34.800 introduce it. So, assume for now that 00:10:33.480 --> 00:10:36.759 you know the future. 00:10:34.799 --> 00:10:39.000 Okay? And I'm going to derive all the 00:10:36.759 --> 00:10:41.080 equations with assuming that you know 00:10:39.000 --> 00:10:42.559 the future. So, there's no issue of 00:10:41.080 --> 00:10:44.360 trying to figure out what the future is. 00:10:42.559 --> 00:10:45.919 You know it. But still you have to 00:10:44.360 --> 00:10:48.440 decide whether 00:10:45.919 --> 00:10:50.919 what is the right value for for an 00:10:48.440 --> 00:10:50.920 asset. 00:10:52.000 --> 00:10:56.320 Okay, so 00:10:54.279 --> 00:10:58.480 let's start with the case where you know 00:10:56.320 --> 00:11:02.440 the future. Sorry. 00:10:58.480 --> 00:11:04.000 And let's do the comparison uh 00:11:02.440 --> 00:11:05.920 Let's try to understand how do we move 00:11:04.000 --> 00:11:07.679 flows, how do we value flows at 00:11:05.919 --> 00:11:10.319 different points in time. 00:11:07.679 --> 00:11:11.759 This is the thing is think first about 00:11:10.320 --> 00:11:13.400 comparing an asset that gives you a 00:11:11.759 --> 00:11:16.120 dollar in the future, 00:11:13.399 --> 00:11:17.519 how much do you think it's worth today? 00:11:16.120 --> 00:11:19.759 Well, the easiest 00:11:17.519 --> 00:11:21.879 way to get to that value is is to think 00:11:19.759 --> 00:11:25.960 on the alternatives. As suppose I have a 00:11:21.879 --> 00:11:25.960 dollar today, what can I do with it? 00:11:26.200 --> 00:11:31.080 Well, in terms of investment. 00:11:28.759 --> 00:11:33.120 Well, suppose that you have available 00:11:31.080 --> 00:11:34.879 one-year bonds, treasury bonds, and that 00:11:33.120 --> 00:11:36.560 the interest rate is I 00:11:34.879 --> 00:11:38.039 t. That's the interest rate on an I 00:11:36.559 --> 00:11:40.919 one-year bond. 00:11:38.039 --> 00:11:42.399 So, if you want if you if you 00:11:40.919 --> 00:11:44.079 have a dollar, 00:11:42.399 --> 00:11:46.199 you have the option to invest it in that 00:11:44.080 --> 00:11:48.960 asset, in that bond, which give will 00:11:46.200 --> 00:11:49.720 give you 1 + I dollars 00:11:48.960 --> 00:11:51.720 uh 00:11:49.720 --> 00:11:52.879 next year. 00:11:51.720 --> 00:11:57.639 Well, 00:11:52.879 --> 00:12:01.799 that means that I can get $1 next year 00:11:57.639 --> 00:12:02.480 by investing 1 over 1 + I dollars today. 00:12:01.799 --> 00:12:05.279 No? 00:12:02.480 --> 00:12:08.000 Because if if I invest 1 + 1 00:12:05.279 --> 00:12:11.720 rather than $1, I invest 1 over 1 + I 00:12:08.000 --> 00:12:14.279 today, then I multiply this by 1 + I 00:12:11.720 --> 00:12:16.200 and I get my dollar in the future. 00:12:14.279 --> 00:12:19.360 So, that tells me that say the interest 00:12:16.200 --> 00:12:22.600 rate is 10%, then with $1 today I can 00:12:19.360 --> 00:12:25.600 get 1.1 dollars in the future. 00:12:22.600 --> 00:12:27.240 That means that investing 90% 90 cents 00:12:25.600 --> 00:12:29.519 today, more or less, 00:12:27.240 --> 00:12:31.560 I can get $1 in the future. 00:12:29.519 --> 00:12:34.759 That tells me that a dollar in the 00:12:31.559 --> 00:12:36.959 future is equivalent to 90 cents today. 00:12:34.759 --> 00:12:38.639 That's the assumption. Okay? 00:12:36.960 --> 00:12:40.320 So, that's the reason when I told you 00:12:38.639 --> 00:12:42.799 the deal of me, look, I have an asset 00:12:40.320 --> 00:12:44.400 that cost cost you a dollar, but gives 00:12:42.799 --> 00:12:45.759 you a dollar in the future, well, that's 00:12:44.399 --> 00:12:47.480 not a good deal if the interest rate is 00:12:45.759 --> 00:12:49.559 positive. 00:12:47.480 --> 00:12:51.639 If the interest rate is 10%, then then a 00:12:49.559 --> 00:12:54.039 right a fair comparison is 90 cents with 00:12:51.639 --> 00:12:56.240 $1, not $1 with $1. 00:12:54.039 --> 00:12:58.199 Okay? So, that's the discounting of the 00:12:56.240 --> 00:13:00.159 future. You can The most obvious way of 00:12:58.200 --> 00:13:02.920 discounting the future 00:13:00.159 --> 00:13:04.240 is to discount it by the interest rate. 00:13:02.919 --> 00:13:06.399 Uh 00:13:04.240 --> 00:13:08.279 which interest rate to pick? That's more 00:13:06.399 --> 00:13:09.959 subtle. That depends on risk, depends on 00:13:08.279 --> 00:13:12.480 many other things which we're going to 00:13:09.960 --> 00:13:15.320 discuss to some extent here. But for 00:13:12.480 --> 00:13:16.440 now, let's make it very simple. And in a 00:13:15.320 --> 00:13:17.879 world in which you really know the 00:13:16.440 --> 00:13:19.400 future, really the right interest rate 00:13:17.879 --> 00:13:21.759 to use is the safe interest rate, the 00:13:19.399 --> 00:13:23.159 interest rate of of treasury bonds and 00:13:21.759 --> 00:13:25.799 things like that. 00:13:23.159 --> 00:13:27.799 Okay? So, that's that's that. 00:13:25.799 --> 00:13:29.439 What about a dollar that you receive 00:13:27.799 --> 00:13:31.319 What about if you're thinking about what 00:13:29.440 --> 00:13:32.880 is the value of a dollar two years from 00:13:31.320 --> 00:13:34.080 now? 00:13:32.879 --> 00:13:35.480 Well, 00:13:34.080 --> 00:13:38.000 you know, if I get a dollar to I can do 00:13:35.480 --> 00:13:39.600 the same logic. If I if I 00:13:38.000 --> 00:13:41.159 I can use the same logic. If I get a 00:13:39.600 --> 00:13:45.960 dollar today, 00:13:41.159 --> 00:13:48.759 I can convert that into 1 + I t * 1 + I 00:13:45.960 --> 00:13:52.280 t + 1 dollars. Okay? 00:13:48.759 --> 00:13:55.039 So, say 10% and 10%, I get 1.1 next year 00:13:52.279 --> 00:13:57.000 and then I get 1.1 * 1.1, 1.21 or 00:13:55.039 --> 00:13:58.559 something like that. Okay? 00:13:57.000 --> 00:13:59.639 That's my final 00:13:58.559 --> 00:14:01.479 result. 00:13:59.639 --> 00:14:03.838 So, 00:14:01.480 --> 00:14:05.800 well, then how much is it worth to have 00:14:03.839 --> 00:14:09.120 a dollar, an asset that gives you a 00:14:05.799 --> 00:14:11.240 dollar two years from now? 00:14:09.120 --> 00:14:13.519 Well, it's going to be that dollar 00:14:11.240 --> 00:14:14.600 divided by the product of these interest 00:14:13.519 --> 00:14:16.838 rates. 00:14:14.600 --> 00:14:18.519 Okay? Why is that? Well, because with 00:14:16.839 --> 00:14:20.640 this amount of 00:14:18.519 --> 00:14:22.519 dollars today, 00:14:20.639 --> 00:14:24.639 it's point 80 cents or something like 00:14:22.519 --> 00:14:25.958 that, I can generate a dollar two years 00:14:24.639 --> 00:14:27.679 from now. 00:14:25.958 --> 00:14:29.199 That means a dollar 00:14:27.679 --> 00:14:32.239 two years from now 00:14:29.200 --> 00:14:34.520 is worth about 80 cents today. 00:14:32.240 --> 00:14:34.519 Okay? 00:14:35.958 --> 00:14:38.919 We're going to use a lot this type of 00:14:37.360 --> 00:14:40.959 logic, so 00:14:38.919 --> 00:14:42.439 and and I know that that it may not be 00:14:40.958 --> 00:14:43.359 that intuitive the first time you see 00:14:42.440 --> 00:14:46.079 it, but 00:14:43.360 --> 00:14:46.079 ask questions. 00:14:47.919 --> 00:14:50.838 You want me to repeat it? 00:14:54.440 --> 00:14:57.400 Okay. The 00:14:55.759 --> 00:14:59.159 The final goal is the following. We're 00:14:57.399 --> 00:15:01.439 going to In the what comes next, we're 00:14:59.159 --> 00:15:03.039 going to see if which happens again with 00:15:01.440 --> 00:15:05.320 many decisions in life, but it perhaps 00:15:03.039 --> 00:15:07.159 particularly for financial assets, 00:15:05.320 --> 00:15:08.720 we're going to try to value something 00:15:07.159 --> 00:15:11.360 that 00:15:08.720 --> 00:15:13.680 whose payoff happens at different times 00:15:11.360 --> 00:15:16.000 in the future. And the question is 00:15:13.679 --> 00:15:19.599 how do I value an asset that pays me, 00:15:16.000 --> 00:15:21.919 you know, $5 one year from now, $25 00:15:19.600 --> 00:15:26.240 three years from now, uh 00:15:21.919 --> 00:15:27.519 minus $10 10 years from now, plus $50 00:15:26.240 --> 00:15:29.240 100 years from now? 00:15:27.519 --> 00:15:31.039 What is the value of that? Of having an 00:15:29.240 --> 00:15:33.360 asset like that? 00:15:31.039 --> 00:15:35.639 And so, I needed some method 00:15:33.360 --> 00:15:37.320 to bring it to today's value because 00:15:35.639 --> 00:15:38.600 today I have a meaning of what a dollar 00:15:37.320 --> 00:15:40.079 is, you know? 00:15:38.600 --> 00:15:42.839 And and therefore I can compare it with 00:15:40.078 --> 00:15:44.958 whatever price I mean 00:15:42.839 --> 00:15:47.800 people are asking me for that asset. 00:15:44.958 --> 00:15:49.799 So, what this is doing is is is that is 00:15:47.799 --> 00:15:51.799 doing that. It's telling you how to 00:15:49.799 --> 00:15:54.759 convert a dollar at different parts in 00:15:51.799 --> 00:15:56.759 the future into a dollar today. 00:15:54.759 --> 00:15:59.039 And by that logic, 00:15:56.759 --> 00:16:01.039 the recipe is well, use the interest 00:15:59.039 --> 00:16:02.559 rate because you could always go the 00:16:01.039 --> 00:16:03.879 other way around. You could always with 00:16:02.559 --> 00:16:06.159 a dollar you can ask a question, with a 00:16:03.879 --> 00:16:08.078 dollar today, how many dollars can I get 00:16:06.159 --> 00:16:09.039 two years from now, say? 00:16:08.078 --> 00:16:10.078 That. 00:16:09.039 --> 00:16:13.519 Well, 00:16:10.078 --> 00:16:15.838 say X. Well, then I need 1 over X. Then 00:16:13.519 --> 00:16:18.078 $1 there is worth 1 over X dollars 00:16:15.839 --> 00:16:21.400 today. You know, that's that's the logic 00:16:18.078 --> 00:16:23.838 because 1 over X * X is 1. 00:16:21.399 --> 00:16:23.838 So, 00:16:23.919 --> 00:16:26.919 that's too fast, probably. 00:16:28.639 --> 00:16:34.319 So, 00:16:29.839 --> 00:16:34.320 you know, with $1 today, oops, 00:16:35.480 --> 00:16:41.039 I can generate, say, 00:16:38.039 --> 00:16:41.039 $1.1 00:16:41.720 --> 00:16:44.160 at uh uh 00:16:44.399 --> 00:16:46.879 at t equal to 00:16:45.679 --> 00:16:48.559 Okay? 00:16:46.879 --> 00:16:51.279 Then I'm I'm The question I want to know 00:16:48.559 --> 00:16:53.439 is how much is a dollar worth 00:16:51.279 --> 00:16:56.879 How much is a dollar received at time t 00:16:53.440 --> 00:16:57.720 equal to worth today? 00:16:56.879 --> 00:16:58.838 That's the question I'm trying to 00:16:57.720 --> 00:17:00.399 answer. 00:16:58.839 --> 00:17:01.720 You know, because an asset will be 00:17:00.399 --> 00:17:03.958 something that will pay you in the 00:17:01.720 --> 00:17:08.199 future. So, I want to know how much is 00:17:03.958 --> 00:17:10.240 $1 received in the future worth today. 00:17:08.199 --> 00:17:11.640 And then the answer is 00:17:10.240 --> 00:17:13.799 well, 00:17:11.640 --> 00:17:18.160 then is I know the answer from this 00:17:13.799 --> 00:17:21.678 logic because I know that with one 00:17:18.160 --> 00:17:24.920 if I have 1 over 1.1 dollars today, I 00:17:21.679 --> 00:17:24.920 can convert it 00:17:25.359 --> 00:17:28.519 into one. 00:17:27.078 --> 00:17:31.039 How do I know that? 00:17:28.519 --> 00:17:31.039 Because 00:17:33.200 --> 00:17:36.880 1 over 1.1 00:17:38.679 --> 00:17:43.120 * 1.1 00:17:40.799 --> 00:17:45.319 is equal to 1. 00:17:43.119 --> 00:17:47.678 Okay? This if I invest these dollars 00:17:45.319 --> 00:17:50.039 today, 00:17:47.679 --> 00:17:51.280 I'm going to get this return on that. 00:17:50.039 --> 00:17:54.119 And the product of these two things 00:17:51.279 --> 00:17:54.119 gives me my dollar. 00:17:54.240 --> 00:17:56.599 Okay? 00:17:55.039 --> 00:17:58.359 So, if I tell you, do you prefer to have 00:17:56.599 --> 00:17:59.879 a dollar two days from two years from 00:17:58.359 --> 00:18:02.519 now or today? 00:17:59.880 --> 00:18:05.120 You say, I prefer it obviously prefer it 00:18:02.519 --> 00:18:07.759 today because I can get 1.1 dollars two 00:18:05.119 --> 00:18:09.239 years from now. 00:18:07.759 --> 00:18:10.839 But then then the more relevant question 00:18:09.240 --> 00:18:12.880 is, no, no, but then you do you prefer 00:18:10.839 --> 00:18:15.199 to have 90 cents today 00:18:12.880 --> 00:18:16.960 versus a dollar in the future? And then 00:18:15.200 --> 00:18:18.799 I'm I need to do my multiplication 00:18:16.960 --> 00:18:21.360 because I have to multiply the 90 cents 00:18:18.799 --> 00:18:23.200 by the 1.1 and see whether I get 00:18:21.359 --> 00:18:24.199 something comparable to a dollar or not. 00:18:23.200 --> 00:18:26.840 Okay? 00:18:24.200 --> 00:18:29.880 But that's that's the logic behind that. 00:18:26.839 --> 00:18:33.199 And And that's a So, the interest rate 00:18:29.880 --> 00:18:34.679 is what we discount the future by. 00:18:33.200 --> 00:18:36.120 And it's natural because if the interest 00:18:34.679 --> 00:18:37.880 rate is very high If the interest rate 00:18:36.119 --> 00:18:39.359 is zero, say, 00:18:37.880 --> 00:18:41.159 then a dollar received two years from 00:18:39.359 --> 00:18:42.158 now or a dollar received today is is the 00:18:41.159 --> 00:18:44.040 same 00:18:42.159 --> 00:18:45.360 because I can't If I invest a dollar 00:18:44.039 --> 00:18:46.559 today and the interest rate is zero, I'm 00:18:45.359 --> 00:18:47.879 going to get my dollar two years from 00:18:46.559 --> 00:18:49.119 now. 00:18:47.880 --> 00:18:51.200 If the dollar If the interest rate is 00:18:49.119 --> 00:18:52.918 50%, it makes a big difference receiving 00:18:51.200 --> 00:18:54.360 the dollar today versus receiving it two 00:18:52.919 --> 00:18:56.320 years from now. 00:18:54.359 --> 00:18:58.199 If you're in Argentina, the interest 00:18:56.319 --> 00:19:00.519 rate I don't know what it is. It's 00:18:58.200 --> 00:19:02.880 700%. It makes a huge difference whether 00:19:00.519 --> 00:19:05.720 you receive it, you know, one year from 00:19:02.880 --> 00:19:08.240 now than today. 00:19:05.720 --> 00:19:09.440 And and and uh 00:19:08.240 --> 00:19:10.440 So, that's that's the role of the 00:19:09.440 --> 00:19:12.000 interest rate. The higher is the 00:19:10.440 --> 00:19:13.360 interest rate, 00:19:12.000 --> 00:19:15.599 the less 00:19:13.359 --> 00:19:17.639 is a dollar received in the future worth 00:19:15.599 --> 00:19:19.199 relative to a dollar received today. 00:19:17.640 --> 00:19:21.320 Because you can get a much higher return 00:19:19.200 --> 00:19:22.559 from the dollar you have today 00:19:21.319 --> 00:19:24.039 if the interest rate is high. If the 00:19:22.559 --> 00:19:26.079 interest rate is low, 00:19:24.039 --> 00:19:28.319 you don't get that much. Okay? 00:19:26.079 --> 00:19:30.480 Much difference. Okay, good. 00:19:28.319 --> 00:19:31.759 So, this is a big principle. And and I I 00:19:30.480 --> 00:19:33.679 mean 00:19:31.759 --> 00:19:36.240 everything I'll say next builds on this 00:19:33.679 --> 00:19:36.240 logic. 00:19:38.640 --> 00:19:43.320 So, let me give you a general formula. 00:19:40.720 --> 00:19:44.799 So, let's ask what is the value 00:19:43.319 --> 00:19:46.119 of an asset 00:19:44.799 --> 00:19:49.319 that gives 00:19:46.119 --> 00:19:51.918 payouts of Z 00:19:49.319 --> 00:19:55.399 t dollars this year, 00:19:51.919 --> 00:19:57.160 Z t + 1 one year from now, ZT plus two, 00:19:55.400 --> 00:20:01.960 two years from now, and so on and so 00:19:57.160 --> 00:20:03.279 forth for N periods more. Okay? 00:20:01.960 --> 00:20:04.880 Well, 00:20:03.279 --> 00:20:06.879 I just need to do several of these 00:20:04.880 --> 00:20:09.120 operations. I know that the dollar 00:20:06.880 --> 00:20:10.760 received this year is is worth a dollar. 00:20:09.119 --> 00:20:13.519 Okay? That's ZT. 00:20:10.759 --> 00:20:14.720 A dollar received one year from now 00:20:13.519 --> 00:20:16.720 is not 00:20:14.720 --> 00:20:18.920 is not the same as a dollar received 00:20:16.720 --> 00:20:22.079 today. It's the same as one over one 00:20:18.920 --> 00:20:23.920 plus IT dollars received today. 00:20:22.079 --> 00:20:26.519 So, that cash flow I'm going to receive 00:20:23.920 --> 00:20:28.120 from this asset is worth this amount. 00:20:26.519 --> 00:20:30.839 For a two something that I receive two 00:20:28.119 --> 00:20:32.759 years from now, then it's not 00:20:30.839 --> 00:20:34.679 it's not certain, it's much less than 00:20:32.759 --> 00:20:38.200 receiving a dollar today. It's going to 00:20:34.680 --> 00:20:39.120 be one over one plus IT one plus IT plus 00:20:38.200 --> 00:20:40.559 one. 00:20:39.119 --> 00:20:42.159 And that I have to multiply by the 00:20:40.559 --> 00:20:46.000 number of dollars I will receive two 00:20:42.160 --> 00:20:48.320 years from now. Okay? And I keep going. 00:20:46.000 --> 00:20:51.200 So, that's that's the 00:20:48.319 --> 00:20:53.559 the present value. Present discounted 00:20:51.200 --> 00:20:56.559 value. Present because I'm bringing all 00:20:53.559 --> 00:20:57.799 these future cash flows to the present. 00:20:56.559 --> 00:21:00.440 That's what each of these terms is 00:20:57.799 --> 00:21:01.680 doing. The one over that is bringing it 00:21:00.440 --> 00:21:03.640 to the present. 00:21:01.680 --> 00:21:05.160 Discounted because the interest rate is 00:21:03.640 --> 00:21:06.400 discounting things. It's making them a 00:21:05.160 --> 00:21:08.600 smaller. 00:21:06.400 --> 00:21:12.560 And value because I'm trying to reduce 00:21:08.599 --> 00:21:14.199 them to the current value. Okay? 00:21:12.559 --> 00:21:15.919 That's the general formula. So, it's a 00:21:14.200 --> 00:21:18.279 formula you need to understand. 00:21:15.920 --> 00:21:21.720 It's just So, that that was an asset 00:21:18.279 --> 00:21:23.920 that gives you Z dollars today, 00:21:21.720 --> 00:21:26.920 ZT plus one, one year from now, so you 00:21:23.920 --> 00:21:29.160 use this formula. ZT plus two, two years 00:21:26.920 --> 00:21:32.360 from now, so you use this formula, and 00:21:29.160 --> 00:21:32.360 then you keep going. Okay? 00:21:33.200 --> 00:21:37.480 What if we don't know the future? 00:21:35.640 --> 00:21:39.080 You know, I have to remove the expected 00:21:37.480 --> 00:21:40.039 part. 00:21:39.079 --> 00:21:41.679 Well, 00:21:40.039 --> 00:21:43.960 if we don't know the future, then the 00:21:41.680 --> 00:21:45.080 best we can do, in fact, we do fancier 00:21:43.960 --> 00:21:47.120 things, but that's what we're going to 00:21:45.079 --> 00:21:50.199 all that we'll do in this course. 00:21:47.119 --> 00:21:52.759 Uh all that you can do is just replace 00:21:50.200 --> 00:21:54.400 the known quantities we have here 00:21:52.759 --> 00:21:56.759 for the expectations. 00:21:54.400 --> 00:21:58.800 Okay? So, that's the closest. So, you 00:21:56.759 --> 00:22:00.079 know, I know ZT, that's the cash flow I 00:21:58.799 --> 00:22:02.079 get now, 00:22:00.079 --> 00:22:04.119 but I don't know ZT plus one. So, I can 00:22:02.079 --> 00:22:05.519 replace it by expectation. 00:22:04.119 --> 00:22:08.239 I do know the interest rate on a 00:22:05.519 --> 00:22:09.319 one-year bond from today to one year. 00:22:08.240 --> 00:22:10.880 So, that's the reason I don't need an 00:22:09.319 --> 00:22:12.759 expectation here. 00:22:10.880 --> 00:22:14.520 But I don't know what the one-year rate 00:22:12.759 --> 00:22:17.160 will be one year from now. So, that's 00:22:14.519 --> 00:22:18.400 the reason I need an expectation there. 00:22:17.160 --> 00:22:20.160 And so on. 00:22:18.400 --> 00:22:21.360 And I don't know what the cash flow will 00:22:20.160 --> 00:22:22.920 be two years from now. I have an 00:22:21.359 --> 00:22:24.639 expectation about what the cash flow 00:22:22.920 --> 00:22:26.960 will be, but I don't know it. 00:22:24.640 --> 00:22:29.960 So, I have an expectation there. Okay? 00:22:26.960 --> 00:22:31.000 So, so all that I've done here is say, 00:22:29.960 --> 00:22:32.279 "Okay, 00:22:31.000 --> 00:22:33.759 I acknowledge that this guy knew a 00:22:32.279 --> 00:22:35.399 little bit too much. You know, he knew 00:22:33.759 --> 00:22:37.279 exactly what the cash flows were going 00:22:35.400 --> 00:22:38.880 to be in the future, and he knew what 00:22:37.279 --> 00:22:40.559 the one-year rates were going to be in 00:22:38.880 --> 00:22:42.440 the future." 00:22:40.559 --> 00:22:44.039 This guy here knows less. He knows the 00:22:42.440 --> 00:22:46.360 cash flow today. He knows the interest 00:22:44.039 --> 00:22:48.279 rate today, but he doesn't know the cash 00:22:46.359 --> 00:22:50.519 flows. Really, he has a hunch, but he 00:22:48.279 --> 00:22:51.920 doesn't know the cash flows one year, 00:22:50.519 --> 00:22:53.519 two years, three years, and so on for 00:22:51.920 --> 00:22:56.560 the future, and he doesn't know the 00:22:53.519 --> 00:22:58.400 one-year interest rate in the future. 00:22:56.559 --> 00:23:00.159 So, all these expectations, here's 00:22:58.400 --> 00:23:02.640 important the concept of time. This is 00:23:00.160 --> 00:23:04.160 an expectation as of time T. At time T, 00:23:02.640 --> 00:23:07.000 you have some information and you make 00:23:04.160 --> 00:23:08.640 forecast about the future. Okay? 00:23:07.000 --> 00:23:10.160 Use whatever you want, machine learning, 00:23:08.640 --> 00:23:11.400 whatever, but you have information at 00:23:10.160 --> 00:23:13.080 time T, 00:23:11.400 --> 00:23:14.800 and then you have a forecast for the 00:23:13.079 --> 00:23:16.079 future. At T plus one, you have you'll 00:23:14.799 --> 00:23:17.399 have more information, so you make 00:23:16.079 --> 00:23:18.199 another forecast, and so on and so 00:23:17.400 --> 00:23:20.360 forth. 00:23:18.200 --> 00:23:23.160 But in this we're valuing an asset at 00:23:20.359 --> 00:23:26.240 time T, then all these expectations are 00:23:23.160 --> 00:23:28.679 taken as of time T. That means given the 00:23:26.240 --> 00:23:30.120 information you have available at time 00:23:28.679 --> 00:23:31.280 T. 00:23:30.119 --> 00:23:33.000 That's the reason these guys don't have 00:23:31.279 --> 00:23:36.559 expectations in front of them because 00:23:33.000 --> 00:23:36.559 you know this at time T. 00:23:36.599 --> 00:23:40.079 Had we taken the value at T minus one, 00:23:38.679 --> 00:23:41.800 we would have not known that, and then 00:23:40.079 --> 00:23:43.799 we would have had to expectation because 00:23:41.799 --> 00:23:46.480 it would have been expectation as of T 00:23:43.799 --> 00:23:46.480 minus one. 00:23:46.640 --> 00:23:49.720 Okay, so that's your big formula there. 00:23:48.960 --> 00:23:51.799 So, 00:23:49.720 --> 00:23:53.839 there are some examples that are sort of 00:23:51.799 --> 00:23:55.440 well known and and and 00:23:53.839 --> 00:23:56.599 and and 00:23:55.440 --> 00:23:58.640 and and 00:23:56.599 --> 00:24:00.959 So, let me let me show you. They have 00:23:58.640 --> 00:24:02.400 nicer expressions. So, that's that's an 00:24:00.960 --> 00:24:04.720 example 00:24:02.400 --> 00:24:05.960 of the valuation of of this the same 00:24:04.720 --> 00:24:08.400 asset, 00:24:05.960 --> 00:24:09.200 but when the interest rate is constant, 00:24:08.400 --> 00:24:10.880 then 00:24:09.200 --> 00:24:13.559 then obviously I don't need all these 00:24:10.880 --> 00:24:16.120 products in the denominator. 00:24:13.559 --> 00:24:18.599 I have a constant interest rate, then I 00:24:16.119 --> 00:24:20.119 just get powers of that interest rate. 00:24:18.599 --> 00:24:22.678 That's one in which you have constant 00:24:20.119 --> 00:24:23.839 payments. So, the interest rate may be 00:24:22.679 --> 00:24:26.440 different, 00:24:23.839 --> 00:24:27.399 but the payments are the same over time. 00:24:26.440 --> 00:24:29.000 Okay? 00:24:27.400 --> 00:24:30.519 So, that's that. 00:24:29.000 --> 00:24:32.559 So, those are two 00:24:30.519 --> 00:24:33.480 easy formulas. That's one in which you 00:24:32.559 --> 00:24:36.079 have 00:24:33.480 --> 00:24:37.720 both constant, the interest rate 00:24:36.079 --> 00:24:39.439 and the payment. 00:24:37.720 --> 00:24:40.839 Then you get a nice expression. That's 00:24:39.440 --> 00:24:44.200 just uh 00:24:40.839 --> 00:24:46.839 that. Okay? You'll recognize that. 00:24:44.200 --> 00:24:47.600 If if you have a constant 00:24:46.839 --> 00:24:50.119 uh 00:24:47.599 --> 00:24:52.159 constant interest rate here, you see 00:24:50.119 --> 00:24:54.559 that the value 00:24:52.160 --> 00:24:56.440 is is declining is a is a geometric 00:24:54.559 --> 00:24:58.240 series. You know? The value of a two 00:24:56.440 --> 00:25:00.720 years from now is a square 00:24:58.240 --> 00:25:02.519 of one over one plus some 00:25:00.720 --> 00:25:03.559 it's a square of a a number less than 00:25:02.519 --> 00:25:05.400 one. 00:25:03.559 --> 00:25:07.519 You know? One over one plus I is some 00:25:05.400 --> 00:25:09.120 number less than one. This is a square 00:25:07.519 --> 00:25:10.519 of that, then the cube, and so on. So, 00:25:09.119 --> 00:25:12.319 it's a geometric series that is 00:25:10.519 --> 00:25:14.480 declining at the rate one plus I, one 00:25:12.319 --> 00:25:16.039 over one plus I. Okay? Or declining at 00:25:14.480 --> 00:25:18.200 the rate one plus I. 00:25:16.039 --> 00:25:19.440 So, that's your geometric series. 00:25:18.200 --> 00:25:22.759 Okay? 00:25:19.440 --> 00:25:25.920 That's the value of that. 00:25:22.759 --> 00:25:27.960 Constant rate and payment forever. 00:25:25.920 --> 00:25:28.840 Suppose you have an asset that 00:25:27.960 --> 00:25:31.120 it 00:25:28.839 --> 00:25:32.559 that lives forever. 00:25:31.119 --> 00:25:36.799 There are some bonds like that called 00:25:32.559 --> 00:25:38.919 perpetuities. Uh uh 00:25:36.799 --> 00:25:40.159 The US hasn't issued one, but the UK 00:25:38.920 --> 00:25:40.840 has, and so on. 00:25:40.160 --> 00:25:42.440 I 00:25:40.839 --> 00:25:44.919 So, that's an asset, for example, that 00:25:42.440 --> 00:25:46.880 pays you a fixed amount 00:25:44.920 --> 00:25:48.840 forever. And if the interest rate is 00:25:46.880 --> 00:25:50.760 constant, that's the trickier thing, 00:25:48.839 --> 00:25:53.399 then the value of that asset you can see 00:25:50.759 --> 00:25:55.480 that this this is going to zero. 00:25:53.400 --> 00:25:57.040 So, the value of that asset 00:25:55.480 --> 00:25:59.440 is that. 00:25:57.039 --> 00:26:02.079 And actually a formula that you may see 00:25:59.440 --> 00:26:04.840 that is very oftenly used as as a first 00:26:02.079 --> 00:26:07.559 approximation is this one. This is 00:26:04.839 --> 00:26:09.879 is is the same asset, but it's called 00:26:07.559 --> 00:26:12.000 ex-dividend or ex-coupon. It's it's 00:26:09.880 --> 00:26:13.440 after the coupon of this year has been 00:26:12.000 --> 00:26:15.240 paid. 00:26:13.440 --> 00:26:17.480 Okay? So, it's an asset that starts 00:26:15.240 --> 00:26:19.120 paying at T plus one. It's ZT plus one, 00:26:17.480 --> 00:26:20.599 ZT plus two, and so on. 00:26:19.119 --> 00:26:22.199 Well, that 00:26:20.599 --> 00:26:25.799 is the same as this minus the first 00:26:22.200 --> 00:26:28.120 coupon, so is equal to that. 00:26:25.799 --> 00:26:28.119 Okay? 00:26:29.119 --> 00:26:32.159 That's an interesting thing, huh? Look, 00:26:31.039 --> 00:26:35.200 what happened to this asset as the 00:26:32.160 --> 00:26:35.200 interest rate goes to zero? 00:26:36.000 --> 00:26:39.799 So, this is an asset that lasts for a 00:26:37.440 --> 00:26:41.759 very long time. 00:26:39.799 --> 00:26:43.678 And and and look, we got to a valuation 00:26:41.759 --> 00:26:45.079 formula. 00:26:43.679 --> 00:26:46.960 What hap- what is happening as the 00:26:45.079 --> 00:26:49.159 interest rate goes to zero? 00:26:46.960 --> 00:26:51.319 To the value. 00:26:49.160 --> 00:26:53.400 Very large. It goes to infinity. 00:26:51.319 --> 00:26:55.279 And a lot of what has happened in in 00:26:53.400 --> 00:26:57.759 global financial markets 00:26:55.279 --> 00:26:58.759 in the last few years has to do with 00:26:57.759 --> 00:27:01.279 that. 00:26:58.759 --> 00:27:02.720 Interest rates were very very very low. 00:27:01.279 --> 00:27:05.799 And so, most assets that had long 00:27:02.720 --> 00:27:07.240 duration had very high values. 00:27:05.799 --> 00:27:09.039 Okay? 00:27:07.240 --> 00:27:10.880 And it has a lot to that. Monetary 00:27:09.039 --> 00:27:12.599 policy had a lot to do 00:27:10.880 --> 00:27:13.720 whether it was the right monetary policy 00:27:12.599 --> 00:27:15.519 or not, 00:27:13.720 --> 00:27:16.720 that's something to be discussed. I 00:27:15.519 --> 00:27:17.920 think on average it was the right 00:27:16.720 --> 00:27:20.440 monetary policy, but one of the things 00:27:17.920 --> 00:27:22.160 it did, it increased the value of many 00:27:20.440 --> 00:27:24.200 assets. In fact, that's one of the 00:27:22.160 --> 00:27:25.720 mechanisms through which monetary policy 00:27:24.200 --> 00:27:27.200 works in practice. It's not something we 00:27:25.720 --> 00:27:29.319 have discussed, but you can begin to see 00:27:27.200 --> 00:27:31.200 here. Because if the value of all assets 00:27:29.319 --> 00:27:32.720 go up a lot, people feel wealthier, and 00:27:31.200 --> 00:27:34.480 that they will tend to consume more, and 00:27:32.720 --> 00:27:36.480 so on. Well, this is one of the channels 00:27:34.480 --> 00:27:38.519 monetary policy does. By the way, this 00:27:36.480 --> 00:27:40.960 effect happens also to this asset that 00:27:38.519 --> 00:27:41.720 has finite N. It's just that this goes 00:27:40.960 --> 00:27:43.559 is 00:27:41.720 --> 00:27:44.920 it's maximized when this asset lasts 00:27:43.559 --> 00:27:46.519 forever. You know? 00:27:44.920 --> 00:27:47.840 This this asset literally goes to 00:27:46.519 --> 00:27:51.759 infinity 00:27:47.839 --> 00:27:54.480 if the interest rate goes to zero. 00:27:51.759 --> 00:27:56.640 Well, if an asset lasts for N periods, 00:27:54.480 --> 00:27:58.360 it doesn't go to infinity. It goes to N 00:27:56.640 --> 00:27:59.960 times Z. 00:27:58.359 --> 00:28:01.479 You know? It's the sum. 00:27:59.960 --> 00:28:03.200 If the interest rate is zero, you just 00:28:01.480 --> 00:28:04.640 sum things. 00:28:03.200 --> 00:28:07.679 See that? 00:28:04.640 --> 00:28:09.880 If I if if an asset lasts for N periods, 00:28:07.679 --> 00:28:11.720 and it gives me a payment of Z in every 00:28:09.880 --> 00:28:13.120 single period, 00:28:11.720 --> 00:28:15.759 then when the interest is zero, that 00:28:13.119 --> 00:28:18.319 asset is worth N times Z. 00:28:15.759 --> 00:28:19.720 Because I will receive Z coupons. 00:28:18.319 --> 00:28:21.639 And I don't discount the future because 00:28:19.720 --> 00:28:23.319 the interest rate is zero. 00:28:21.640 --> 00:28:25.600 What happens is when the asset lasts 00:28:23.319 --> 00:28:27.519 forever, then N times Z is a really 00:28:25.599 --> 00:28:29.039 large number, you know? And that's 00:28:27.519 --> 00:28:31.279 that's what this expression captures 00:28:29.039 --> 00:28:31.279 here. 00:28:31.799 --> 00:28:36.079 Okay. 00:28:34.319 --> 00:28:39.200 So, let's talk about bonds now. We're 00:28:36.079 --> 00:28:42.960 going to start pricing bonds. 00:28:39.200 --> 00:28:45.000 Well, so bonds differ uh uh uh 00:28:42.960 --> 00:28:46.840 along many dimensions, but one of them 00:28:45.000 --> 00:28:49.799 is is very important for bonds is 00:28:46.839 --> 00:28:52.480 maturity, the N that I had there 00:28:49.799 --> 00:28:55.720 in the previous expression. Okay? 00:28:52.480 --> 00:28:57.599 Uh so, so maturity means essentially how 00:28:55.720 --> 00:28:59.880 long the bond lasts. Okay? When when 00:28:57.599 --> 00:29:02.000 does it pay you back the principal? The 00:28:59.880 --> 00:29:03.760 bonds typically pay coupons, and then 00:29:02.000 --> 00:29:05.279 there's a final payment, which we call 00:29:03.759 --> 00:29:08.759 face value of the bond or something like 00:29:05.279 --> 00:29:10.399 that. And and when that final payment 00:29:08.759 --> 00:29:12.400 takes place, that's the maturity of a 00:29:10.400 --> 00:29:14.120 bond. Okay? 00:29:12.400 --> 00:29:15.960 So, a bond that promises to make a 00:29:14.119 --> 00:29:17.000 thousand-dollar final payment in six 00:29:15.960 --> 00:29:20.400 months 00:29:17.000 --> 00:29:20.400 has a maturity of six months. 00:29:20.919 --> 00:29:24.159 A bond that promised to pay a hundred 00:29:22.679 --> 00:29:26.560 dollars for twenty years and then one 00:29:24.159 --> 00:29:28.960 thousand dollars final payment in twenty 00:29:26.559 --> 00:29:30.399 years has a maturity of twenty years. 00:29:28.960 --> 00:29:31.640 Maturity is different from duration. I 00:29:30.400 --> 00:29:33.759 don't think I'm going to talk about 00:29:31.640 --> 00:29:36.320 duration here, but but that's maturity. 00:29:33.759 --> 00:29:38.200 Just when the when is the final payment 00:29:36.319 --> 00:29:39.519 of of a 00:29:38.200 --> 00:29:42.400 of a loan. 00:29:39.519 --> 00:29:44.359 Of a of a bond. Okay? 00:29:42.400 --> 00:29:45.720 Bonds of different maturities each have 00:29:44.359 --> 00:29:47.039 a price 00:29:45.720 --> 00:29:48.799 and an associated interest rate. We're 00:29:47.039 --> 00:29:51.119 going to look at those things. 00:29:48.799 --> 00:29:53.159 And the associated interest rate is 00:29:51.119 --> 00:29:55.759 called the yield to maturity, or simply 00:29:53.160 --> 00:29:57.279 the yield of a bond. 00:29:55.759 --> 00:30:00.319 This is terminology, but we're going to 00:29:57.279 --> 00:30:02.319 calculate these things later on. 00:30:00.319 --> 00:30:03.879 The The relationship between maturity 00:30:02.319 --> 00:30:05.759 and yield 00:30:03.880 --> 00:30:07.560 is called the yield curve. Very 00:30:05.759 --> 00:30:09.519 important concept. Big fuss about the 00:30:07.559 --> 00:30:11.799 yield curve these days. 00:30:09.519 --> 00:30:13.240 Talk a little bit more about that. 00:30:11.799 --> 00:30:15.240 Or sometimes it's called the term 00:30:13.240 --> 00:30:17.759 structure of interest rate. 00:30:15.240 --> 00:30:19.680 Term, in the language of bonds, is 00:30:17.759 --> 00:30:21.640 really maturity. 00:30:19.680 --> 00:30:23.880 So, term structure of interest rate 00:30:21.640 --> 00:30:26.520 really tells you what is the yield in a 00:30:23.880 --> 00:30:29.360 1-year bond, 2-year bond, 3-year bond, 4 00:30:26.519 --> 00:30:30.519 5 6 so on. You plot them, and that gives 00:30:29.359 --> 00:30:31.639 you a curve. 00:30:30.519 --> 00:30:33.039 Okay. 00:30:31.640 --> 00:30:34.880 So, 00:30:33.039 --> 00:30:36.279 uh for example, 00:30:34.880 --> 00:30:38.640 look at the those These are two 00:30:36.279 --> 00:30:40.039 different yield curves. This is November 00:30:38.640 --> 00:30:42.560 2000, 00:30:40.039 --> 00:30:43.879 and this is June 20 00:30:42.559 --> 00:30:47.079 2001. 00:30:43.880 --> 00:30:49.240 So, this tells you what the yield is 00:30:47.079 --> 00:30:51.399 in on a 3-month bond, so a bond that 00:30:49.240 --> 00:30:53.240 matures in three in 3 months, on a 00:30:51.400 --> 00:30:56.000 6-month bonds and so forth, up to 00:30:53.240 --> 00:30:57.519 30-year bonds. Okay. 00:30:56.000 --> 00:30:58.680 What is the big difference between these 00:30:57.519 --> 00:31:00.879 What do you think happened here in 00:30:58.680 --> 00:31:02.799 between? Notice that these two curves 00:31:00.880 --> 00:31:04.360 are more or less the same long-term 00:31:02.799 --> 00:31:06.039 interest rate. 00:31:04.359 --> 00:31:08.000 But they have very different This curve 00:31:06.039 --> 00:31:12.039 This is a very steep curve, and this is 00:31:08.000 --> 00:31:12.039 a very flat or even inverted curve. 00:31:12.839 --> 00:31:16.359 What do you think may have happened 00:31:14.319 --> 00:31:20.119 there? 00:31:16.359 --> 00:31:22.399 Between November 2000 and June 2001. 00:31:20.119 --> 00:31:25.319 People changed their expectations then. 00:31:22.400 --> 00:31:28.600 Yeah, it's That's true. That's for sure 00:31:25.319 --> 00:31:30.319 true about that. But look also that But 00:31:28.599 --> 00:31:32.159 that that that this 3-month There is 00:31:30.319 --> 00:31:34.359 very little uncertainty about 3 months. 00:31:32.160 --> 00:31:35.800 It was a lot lower than that. 00:31:34.359 --> 00:31:37.240 So, yes, people changed their 00:31:35.799 --> 00:31:40.240 expectation, but why do you think they 00:31:37.240 --> 00:31:40.240 changed their expectation? 00:31:42.319 --> 00:31:46.159 Well, it's rising inflation. We have a 00:31:44.799 --> 00:31:47.559 lot 00:31:46.160 --> 00:31:48.880 Rising inflation from here to here. 00:31:47.559 --> 00:31:50.519 These are These are nominal interest 00:31:48.880 --> 00:31:51.800 rates. 00:31:50.519 --> 00:31:54.759 Up to now I've been talking about 00:31:51.799 --> 00:31:54.759 nominal interest rate. 00:31:56.679 --> 00:32:00.519 What happens here 00:31:58.519 --> 00:32:03.319 is there was a mini recession. 00:32:00.519 --> 00:32:05.319 So, the Fed cut interest rate. 00:32:03.319 --> 00:32:08.000 When you're in recessions, the curve 00:32:05.319 --> 00:32:09.039 tend to look like this. 00:32:08.000 --> 00:32:10.519 Because 00:32:09.039 --> 00:32:12.480 the central bank is cutting interest 00:32:10.519 --> 00:32:14.160 rates in the in the short run to deal 00:32:12.480 --> 00:32:15.640 with the current recession. 00:32:14.160 --> 00:32:16.880 What happens 30 years from now has 00:32:15.640 --> 00:32:18.400 nothing to do with the business cycle 00:32:16.880 --> 00:32:20.400 today, so that interest rate doesn't 00:32:18.400 --> 00:32:22.440 need to move a lot. But the Fed is 00:32:20.400 --> 00:32:24.640 bringing interest rate down a lot in the 00:32:22.440 --> 00:32:27.080 front end. Okay. So, that's the typical 00:32:24.640 --> 00:32:29.840 shape of a curve in a recession. 00:32:27.079 --> 00:32:31.159 That's the typical shape of a of a curve 00:32:29.839 --> 00:32:32.959 in the opposite situation where the 00:32:31.160 --> 00:32:34.679 inflation is too high and so on. Because 00:32:32.960 --> 00:32:36.480 what happens? The Fed is trying to The 00:32:34.679 --> 00:32:37.840 Fed really controls the very front end 00:32:36.480 --> 00:32:39.160 of the curve. 00:32:37.839 --> 00:32:40.720 That's what the Fed really control. The 00:32:39.160 --> 00:32:42.040 central bank in general, but the Fed. 00:32:40.720 --> 00:32:43.200 They control the very front end of the 00:32:42.039 --> 00:32:44.960 curve because they're setting the very 00:32:43.200 --> 00:32:46.200 short-term interest rate. 00:32:44.960 --> 00:32:47.880 So, this is a situation where they're 00:32:46.200 --> 00:32:48.840 tightening the monetary policy very 00:32:47.880 --> 00:32:51.120 tight. 00:32:48.839 --> 00:32:53.039 Because they are a situation of uh 00:32:51.119 --> 00:32:54.319 overheating in the economy. And in fact, 00:32:53.039 --> 00:32:55.759 they got too carried away. That's the 00:32:54.319 --> 00:32:57.000 reason they we ended up in a recession 00:32:55.759 --> 00:32:59.440 here. 00:32:57.000 --> 00:32:59.440 Okay. 00:33:00.200 --> 00:33:03.440 How do you think it looks today? 00:33:06.200 --> 00:33:09.679 That Do you think it looks more like 00:33:07.400 --> 00:33:13.360 this or more like that? 00:33:09.679 --> 00:33:13.360 Is inflation low or high today? 00:33:14.119 --> 00:33:17.199 High. I mean, that's a problem, you 00:33:15.720 --> 00:33:19.759 know? The Fed is trying to hike interest 00:33:17.200 --> 00:33:21.720 rate. Now, recently, because of the the 00:33:19.759 --> 00:33:23.119 mess in the banking sector, then the 00:33:21.720 --> 00:33:25.559 expectations of interest rate began to 00:33:23.119 --> 00:33:27.319 decline a little, but but but the 00:33:25.559 --> 00:33:28.399 situation was was very important. Here 00:33:27.319 --> 00:33:30.678 you are. 00:33:28.400 --> 00:33:34.040 That's The green line is today. 00:33:30.679 --> 00:33:35.519 Okay. So, it's very inverted. 00:33:34.039 --> 00:33:38.240 Okay. 00:33:35.519 --> 00:33:39.879 A year ago, it looked like that. 00:33:38.240 --> 00:33:41.799 So, you see the the long end hasn't 00:33:39.880 --> 00:33:43.640 changed much, but a year ago, there was 00:33:41.799 --> 00:33:46.440 no sense that the inflation was getting 00:33:43.640 --> 00:33:47.960 so much out of line. 00:33:46.440 --> 00:33:49.320 It happened a little later than that. 00:33:47.960 --> 00:33:51.360 There was some concern that interest 00:33:49.319 --> 00:33:53.599 rate would would rise, 00:33:51.359 --> 00:33:55.240 but but but now it's very clear the 00:33:53.599 --> 00:33:57.159 economy is overheating. And this I 00:33:55.240 --> 00:33:58.359 should have plotted you something for 00:33:57.160 --> 00:34:00.120 for 00:33:58.359 --> 00:34:03.319 a month ago. It would have been even 00:34:00.119 --> 00:34:05.199 steeper. Okay. 00:34:03.319 --> 00:34:06.879 Anyways, but that's because the Fed is 00:34:05.200 --> 00:34:08.559 trying to slow down the economy. It's 00:34:06.880 --> 00:34:12.320 hiking interest rates. That's the reason 00:34:08.559 --> 00:34:12.320 the curve is very very inverted today. 00:34:12.960 --> 00:34:16.599 So, let me let me calculate these rates. 00:34:14.760 --> 00:34:17.520 How do we go about it? So, the first 00:34:16.599 --> 00:34:19.480 thing we're going to do is we're going 00:34:17.519 --> 00:34:22.719 to use the expected present discounted 00:34:19.480 --> 00:34:24.240 value formula to calculate the price 00:34:22.719 --> 00:34:25.639 of a bond. 00:34:24.239 --> 00:34:27.559 And then we want to start 00:34:25.639 --> 00:34:29.239 doing it for different bonds, 00:34:27.559 --> 00:34:30.759 and we're going to construct uh the 00:34:29.239 --> 00:34:33.759 yield curve. 00:34:30.760 --> 00:34:35.760 So, suppose you have a bond that pays 00:34:33.760 --> 00:34:37.800 $100, 00:34:35.760 --> 00:34:39.679 nothing in between, $100 1 year from 00:34:37.800 --> 00:34:41.800 now. So, this is a bond with maturity 00:34:39.679 --> 00:34:43.720 1-year maturity. 00:34:41.800 --> 00:34:45.159 I'm going to call that bond with 1-year 00:34:43.719 --> 00:34:47.158 maturity 00:34:45.159 --> 00:34:50.200 P1 the price of a bond with a 1-year 00:34:47.159 --> 00:34:51.480 maturity at time T, P1T. 00:34:50.199 --> 00:34:53.480 Well, that's easy to calculate. It's 00:34:51.480 --> 00:34:55.240 expected present discounted value for If 00:34:53.480 --> 00:34:57.519 you have the interest rate, whatever you 00:34:55.239 --> 00:34:59.279 say, 1-year interest rate, then I know 00:34:57.519 --> 00:35:01.519 that the price of the bond is 100 00:34:59.280 --> 00:35:04.400 divided by 1 plus the interest rate, the 00:35:01.519 --> 00:35:06.039 1-year interest rate today. 00:35:04.400 --> 00:35:07.720 Okay. That's the price. That's expected 00:35:06.039 --> 00:35:08.759 discounted value. So, I tell you what 00:35:07.719 --> 00:35:11.399 I'm showing you is the relationship 00:35:08.760 --> 00:35:13.960 between interest rates and prices. 00:35:11.400 --> 00:35:16.480 Okay. Our price of a bond. The price of 00:35:13.960 --> 00:35:19.119 that bond is just 100 00:35:16.480 --> 00:35:23.079 uh divided by 1 plus the 1 plus the 00:35:19.119 --> 00:35:23.079 1-year interest rate today. Okay. 00:35:23.719 --> 00:35:28.239 So, important observation is that the 00:35:25.880 --> 00:35:29.559 price of a 1-year bond varies inversely 00:35:28.239 --> 00:35:31.839 with the current 00:35:29.559 --> 00:35:34.519 1-year nominal interest rate. This is 00:35:31.840 --> 00:35:34.519 all nominal, huh? 00:35:34.599 --> 00:35:38.519 Why is it an inverse relationship? 00:35:36.719 --> 00:35:40.439 Why is it the price of a 1-year bond is 00:35:38.519 --> 00:35:42.679 inversely related to the 1-year interest 00:35:40.440 --> 00:35:42.679 rate? 00:35:44.719 --> 00:35:48.000 In other words, I'm asking 00:35:46.760 --> 00:35:49.680 what do you think happens to the price 00:35:48.000 --> 00:35:50.840 as a nominal as a nominal interest rate 00:35:49.679 --> 00:35:52.279 rises? 00:35:50.840 --> 00:35:54.880 And why do you think that's what happens 00:35:52.280 --> 00:35:54.880 to the price? 00:35:55.039 --> 00:35:58.358 Well, the first question doesn't have a 00:35:57.199 --> 00:35:59.839 I mean, it's very easy, you know, the 00:35:58.358 --> 00:36:01.519 answer to the first question. What 00:35:59.840 --> 00:36:03.559 happens if I goes up? Well, it's obvious 00:36:01.519 --> 00:36:06.119 that this price comes down. 00:36:03.559 --> 00:36:06.119 But why? 00:36:08.599 --> 00:36:12.279 And and and I'm And you use the concept 00:36:11.039 --> 00:36:13.440 we have developed here. Remember we 00:36:12.280 --> 00:36:15.680 spent 00:36:13.440 --> 00:36:19.679 like 20 minutes in one slide. Well, you 00:36:15.679 --> 00:36:19.679 start the slide for that answer. 00:36:22.840 --> 00:36:26.280 Hint. 00:36:24.119 --> 00:36:28.719 This $100 you're not receiving today, 00:36:26.280 --> 00:36:30.040 you're receiving a year from now. 00:36:28.719 --> 00:36:31.719 What happens with a dollar received a 00:36:30.039 --> 00:36:33.519 year from now? 00:36:31.719 --> 00:36:35.480 What is the value of a year 00:36:33.519 --> 00:36:38.440 dollar received 1 year from now when the 00:36:35.480 --> 00:36:38.440 interest rate is high? 00:36:39.039 --> 00:36:42.559 Slow, because, you know, 00:36:41.119 --> 00:36:44.239 you'd much rather have the dollar today, 00:36:42.559 --> 00:36:46.320 invest it, and get this big return on 00:36:44.239 --> 00:36:48.358 the on on the dollar. 00:36:46.320 --> 00:36:50.120 That means, naturally, a bond that is 00:36:48.358 --> 00:36:51.519 paying you $100 tomorrow is going to be 00:36:50.119 --> 00:36:53.079 worth less 00:36:51.519 --> 00:36:54.480 when the interest rate is very high. 00:36:53.079 --> 00:36:55.440 It's going to be worth less today when 00:36:54.480 --> 00:36:57.119 the interest rate is very high. You'd 00:36:55.440 --> 00:36:58.880 rather have the money today, invest it 00:36:57.119 --> 00:37:00.400 in the in in the interest rate, and get 00:36:58.880 --> 00:37:02.400 the interest rate. 00:37:00.400 --> 00:37:05.039 And and uh 00:37:02.400 --> 00:37:07.280 No, I need to invest 1 over 1 plus I1T 00:37:05.039 --> 00:37:09.719 dollars to get $100. That's another way 00:37:07.280 --> 00:37:09.720 of saying it. 00:37:10.480 --> 00:37:15.199 What about with the bond that pays $100 00:37:13.358 --> 00:37:17.759 in 2 years? 00:37:15.199 --> 00:37:19.559 Well, I need to discount that by this, 00:37:17.760 --> 00:37:21.080 which is a You know, it's a product of 00:37:19.559 --> 00:37:23.079 the two interest rate. And since I don't 00:37:21.079 --> 00:37:25.119 know what the 1-year rate 00:37:23.079 --> 00:37:26.799 will be 1 year from now, I have to use 00:37:25.119 --> 00:37:28.599 expectation here rather than the actual 00:37:26.800 --> 00:37:30.080 rate. But look at the notation. I'm 00:37:28.599 --> 00:37:31.639 calling 00:37:30.079 --> 00:37:34.840 P2T, 00:37:31.639 --> 00:37:36.960 dollar P2T, the price of a 2-year bond, 00:37:34.840 --> 00:37:39.039 a bond with maturity of 2 years, 00:37:36.960 --> 00:37:40.280 as of time T. 00:37:39.039 --> 00:37:41.960 Okay. 00:37:40.280 --> 00:37:44.720 And this is a bond that has no coupons. 00:37:41.960 --> 00:37:47.840 So, yes, pays you $100 00:37:44.719 --> 00:37:50.719 at the end of the 2 years. 00:37:47.840 --> 00:37:53.840 Now, note Note that this price 00:37:50.719 --> 00:37:56.599 is inversely related to both 00:37:53.840 --> 00:37:59.000 the 1-year rate today 00:37:56.599 --> 00:38:01.519 and the expectation of the 1-year rate 1 00:37:59.000 --> 00:38:04.079 year from now. 00:38:01.519 --> 00:38:05.719 If either one of these goes up, 00:38:04.079 --> 00:38:08.319 the bond is worth less today. You 00:38:05.719 --> 00:38:09.399 discount more a dollar received 00:38:08.320 --> 00:38:10.680 uh 00:38:09.400 --> 00:38:12.559 um 00:38:10.679 --> 00:38:13.319 2 years from now. I don't care which 00:38:12.559 --> 00:38:14.159 one. 00:38:13.320 --> 00:38:16.240 You know, 00:38:14.159 --> 00:38:19.079 either of them that goes up is is bad 00:38:16.239 --> 00:38:21.279 news for the for the price of a bond. 00:38:19.079 --> 00:38:21.279 Okay. 00:38:24.239 --> 00:38:26.839 Is this clear? 00:38:28.358 --> 00:38:31.119 So, 00:38:29.440 --> 00:38:33.559 there's an alternative So, this is the 00:38:31.119 --> 00:38:35.880 way you price a bond bonds using just 00:38:33.559 --> 00:38:38.320 expected discounted value 00:38:35.880 --> 00:38:40.240 uh approach. Now, it turns out that in 00:38:38.320 --> 00:38:42.440 practice, a lot of the asset pricing is 00:38:40.239 --> 00:38:45.479 done by arbitrage. Meaning, you you 00:38:42.440 --> 00:38:47.039 compare different assets, and that that 00:38:45.480 --> 00:38:48.960 have similar risk, they should give you 00:38:47.039 --> 00:38:51.599 more or less the same return. That's 00:38:48.960 --> 00:38:53.559 what you do. So, let me let me do this 00:38:51.599 --> 00:38:56.079 arbitrage thing. Suppose you're 00:38:53.559 --> 00:38:57.239 considering investing $1 for 1 year. So, 00:38:56.079 --> 00:38:59.199 that's your decision. I'm going to 00:38:57.239 --> 00:39:02.159 invest one I need I have a dollar, which 00:38:59.199 --> 00:39:04.439 I want to invest for 1 year. 00:39:02.159 --> 00:39:07.480 But I But I I have two options to do 00:39:04.440 --> 00:39:08.720 that. I can invest a dollar in a 1-year 00:39:07.480 --> 00:39:10.440 bond. 00:39:08.719 --> 00:39:11.879 I know exactly what I'm going to get, 00:39:10.440 --> 00:39:13.400 you know, in that bond. 00:39:11.880 --> 00:39:16.079 Or 00:39:13.400 --> 00:39:17.800 I can invest in a 2-year bond 00:39:16.079 --> 00:39:18.799 and sell it at the end of the first 00:39:17.800 --> 00:39:20.600 year. 00:39:18.800 --> 00:39:23.480 That's Those are two ways of, you know, 00:39:20.599 --> 00:39:24.880 investing for 1 year. 00:39:23.480 --> 00:39:26.320 Arbitrage has to be compared over the 00:39:24.880 --> 00:39:27.960 same period of time and everything. It's 00:39:26.320 --> 00:39:30.039 not the return of a bond that you hold 00:39:27.960 --> 00:39:31.559 for 10 years versus one that you hold 00:39:30.039 --> 00:39:33.759 for a 1 year. It has to be something a 00:39:31.559 --> 00:39:36.799 similar investment. Suppose I need to 00:39:33.760 --> 00:39:39.760 invest for 1 year. 00:39:36.800 --> 00:39:43.360 Or you know, then then Okay, then if I 00:39:39.760 --> 00:39:45.160 have these two bonds, the option is not 00:39:43.360 --> 00:39:46.760 buy one or the other and then hold to 00:39:45.159 --> 00:39:48.039 maturity because that would be comparing 00:39:46.760 --> 00:39:49.720 an investment of 1 year with an 00:39:48.039 --> 00:39:52.000 investment of 2 years. 00:39:49.719 --> 00:39:54.279 I need to compare the strategies of 00:39:52.000 --> 00:39:55.800 getting my return in 1 year. 00:39:54.280 --> 00:39:57.960 In the 1-year bond, that's trivial 00:39:55.800 --> 00:39:59.920 because I get my return at the end of at 00:39:57.960 --> 00:40:01.240 the maturity of the bond. In the 2-year 00:39:59.920 --> 00:40:03.639 bond, it means I need to sell it in 00:40:01.239 --> 00:40:06.159 between after 1 year. Okay? So, those 00:40:03.639 --> 00:40:08.839 are the two strategies I want to compare 00:40:06.159 --> 00:40:10.759 and since I'm not take 00:40:08.840 --> 00:40:11.960 considering riskier as a central 00:40:10.760 --> 00:40:13.200 element, 00:40:11.960 --> 00:40:15.880 those two strategies are going to have 00:40:13.199 --> 00:40:17.519 to give me the same expected return. 00:40:15.880 --> 00:40:18.800 Okay? That's arbitrage. That's what we 00:40:17.519 --> 00:40:19.639 call arbitrage. 00:40:18.800 --> 00:40:20.680 Okay? 00:40:19.639 --> 00:40:23.039 Two 00:40:20.679 --> 00:40:26.000 the two strategies have to give me the 00:40:23.039 --> 00:40:28.480 same expected return. 00:40:26.000 --> 00:40:28.480 So, 00:40:28.519 --> 00:40:32.360 what do we get from this strategies? 00:40:30.320 --> 00:40:34.760 Well, if I go through the 1-year bond, I 00:40:32.360 --> 00:40:37.519 know I'm going to get my dollar times 1 00:40:34.760 --> 00:40:39.880 plus I1T. That's what I get off a 1 year 00:40:37.519 --> 00:40:40.880 out of investing a dollar in a 1-year 00:40:39.880 --> 00:40:42.559 bond. 00:40:40.880 --> 00:40:44.280 If I go through the 2-year bond 00:40:42.559 --> 00:40:47.199 strategy, buy it and sell it at the end 00:40:44.280 --> 00:40:48.080 of the year, then I'm going to get I I I 00:40:47.199 --> 00:40:49.919 I 00:40:48.079 --> 00:40:52.880 invest a dollar today, 00:40:49.920 --> 00:40:54.800 no? I'm going to pay P2T. 00:40:52.880 --> 00:40:56.640 That's what I paid today for a 2-year 00:40:54.800 --> 00:40:59.440 bond. That's what I pay here for a 00:40:56.639 --> 00:41:02.759 2-year bond and I expect to get the 00:40:59.440 --> 00:41:04.519 price of a 1-year bond 1 year from now. 00:41:02.760 --> 00:41:05.560 I mean, the 2-year bond will be a 1-year 00:41:04.519 --> 00:41:07.360 bond 00:41:05.559 --> 00:41:08.320 after a year has passed. 00:41:07.360 --> 00:41:10.280 No? 00:41:08.320 --> 00:41:11.960 It's a 2-year bond today, but 00:41:10.280 --> 00:41:13.640 after 1 year, it's going to have only 1 00:41:11.960 --> 00:41:15.639 year to mature. 00:41:13.639 --> 00:41:18.000 So, that's the reason the price I need 00:41:15.639 --> 00:41:20.480 to forecast is the is the price of a 00:41:18.000 --> 00:41:21.840 1-year bond 1 year from now. That's what 00:41:20.480 --> 00:41:23.480 this is here. 00:41:21.840 --> 00:41:25.039 Okay? And that's my return on this 00:41:23.480 --> 00:41:26.840 strategy because I'm going to pay this 00:41:25.039 --> 00:41:28.079 today, 00:41:26.840 --> 00:41:30.000 these dollars, 00:41:28.079 --> 00:41:30.880 and I expect to get that 1 year from 00:41:30.000 --> 00:41:32.280 now. 00:41:30.880 --> 00:41:35.360 Okay? 00:41:32.280 --> 00:41:37.880 So, arbitrage means I need to set these 00:41:35.360 --> 00:41:37.880 two equal. 00:41:38.559 --> 00:41:40.679 Okay? 00:41:42.280 --> 00:41:46.720 So, 00:41:44.679 --> 00:41:48.199 that means I have to get the same return 00:41:46.719 --> 00:41:49.599 with the two strategies. That means I'm 00:41:48.199 --> 00:41:52.079 investing the same, so I only need to 00:41:49.599 --> 00:41:54.880 compare the the the the 00:41:52.079 --> 00:41:57.239 the returns. This needs to be equal to 00:41:54.880 --> 00:41:57.240 that. 00:41:57.280 --> 00:42:00.040 That's what I have here. 00:42:00.119 --> 00:42:03.960 Which tells you 00:42:01.639 --> 00:42:06.960 that you're solving from here that the 00:42:03.960 --> 00:42:08.760 price of a 2-year bond at time T 00:42:06.960 --> 00:42:11.559 is equal to the expected price of a 00:42:08.760 --> 00:42:14.240 1-year bond at T plus 1 00:42:11.559 --> 00:42:15.480 discounted by 1 plus the 1-year interest 00:42:14.239 --> 00:42:16.799 rate. 00:42:15.480 --> 00:42:18.920 No? 00:42:16.800 --> 00:42:20.640 This was like my cash flow. 00:42:18.920 --> 00:42:22.280 My cash flow now is not the cash flow. 00:42:20.639 --> 00:42:23.920 It's It's just a price. I'm going to get 00:42:22.280 --> 00:42:27.120 a price for that asset. That's like the 00:42:23.920 --> 00:42:29.599 Zs I had in my formula. Okay? 00:42:27.119 --> 00:42:31.679 And for a 1-year strategy, I only need 00:42:29.599 --> 00:42:34.319 to worry about the ZT plus 1. 00:42:31.679 --> 00:42:37.279 There was no dividend at day zero. 00:42:34.320 --> 00:42:41.320 Okay? And that's exactly that formula. 00:42:37.280 --> 00:42:43.240 But notice that at T plus 1, 00:42:41.320 --> 00:42:45.800 that will hold. 00:42:43.239 --> 00:42:47.719 No? So, at T plus 1, I'm at T plus 1, I 00:42:45.800 --> 00:42:50.600 don't need expectations. I know that P1T 00:42:47.719 --> 00:42:55.639 plus 1 will be equal to 100 divided by 1 00:42:50.599 --> 00:42:57.759 plus I1, the 1-year rate at T plus 1. 00:42:55.639 --> 00:42:59.639 Therefore, the expected is something 00:42:57.760 --> 00:43:01.920 like this, approximately. The expected 00:42:59.639 --> 00:43:04.199 price is something like that. 00:43:01.920 --> 00:43:06.039 Okay? I expect 00:43:04.199 --> 00:43:07.399 I mean, this will be without the E will 00:43:06.039 --> 00:43:09.679 be the price 00:43:07.400 --> 00:43:11.320 of this 1-year bond at T plus 1. I don't 00:43:09.679 --> 00:43:13.000 know exactly what the interest rate will 00:43:11.320 --> 00:43:15.360 be next year, so I have the best I can 00:43:13.000 --> 00:43:17.840 do is have an expectation. That's my 00:43:15.360 --> 00:43:19.680 expectation, approximately. 00:43:17.840 --> 00:43:22.559 Okay? But now I can stick this 00:43:19.679 --> 00:43:24.519 expression in here. 00:43:22.559 --> 00:43:26.759 No? I have this. 00:43:24.519 --> 00:43:28.000 I'm going to go out and I can stick that 00:43:26.760 --> 00:43:30.120 in there 00:43:28.000 --> 00:43:32.760 and I get this expression. So, that's 00:43:30.119 --> 00:43:35.679 the price for the 2-year bond. 00:43:32.760 --> 00:43:35.680 Do you recognize this? 00:43:37.840 --> 00:43:40.640 You saw it before. 00:43:46.119 --> 00:43:49.039 You know? 00:43:47.079 --> 00:43:50.440 That's the same expression that we got 00:43:49.039 --> 00:43:52.239 when we used the expected present 00:43:50.440 --> 00:43:53.559 discounted value formula. 00:43:52.239 --> 00:43:54.439 Right? 00:43:53.559 --> 00:43:57.599 We said, "Well, I'm going to get the 00:43:54.440 --> 00:43:59.159 $100 100 years a 100 years a 2 years 00:43:57.599 --> 00:44:02.559 from now. I know that discount factor 00:43:59.159 --> 00:44:05.440 for that is 1 over 1 plus I1T times 1 00:44:02.559 --> 00:44:08.000 plus I1T plus 1 expected." 00:44:05.440 --> 00:44:09.440 Well, that's what I got. 00:44:08.000 --> 00:44:10.719 That's from arbitrage. 00:44:09.440 --> 00:44:12.639 Okay? 00:44:10.719 --> 00:44:15.559 From an arbitrage logic. This is used a 00:44:12.639 --> 00:44:15.559 lot in finance. 00:44:15.760 --> 00:44:20.640 I I I'm going to say something 00:44:16.960 --> 00:44:20.639 complicated, but but um 00:44:21.519 --> 00:44:25.039 just ignore it if it's 00:44:26.119 --> 00:44:28.599 uh 00:44:26.719 --> 00:44:30.679 not really up for the for the for the 00:44:28.599 --> 00:44:32.199 quiz or anything, but 00:44:30.679 --> 00:44:34.519 you know, there's a big debate in the US 00:44:32.199 --> 00:44:36.199 today about uh not big debate, a big 00:44:34.519 --> 00:44:37.039 concern about 00:44:36.199 --> 00:44:40.039 uh 00:44:37.039 --> 00:44:40.039 the the 00:44:40.358 --> 00:44:45.000 the US Treasury debt because there is a 00:44:42.840 --> 00:44:46.760 debt ceiling, meaning there's a maximum 00:44:45.000 --> 00:44:48.960 amount that the government can 00:44:46.760 --> 00:44:52.680 of debt they can issue. 00:44:48.960 --> 00:44:54.760 And and the and that ceiling has been 00:44:52.679 --> 00:44:56.399 moved over time, but every time we get 00:44:54.760 --> 00:44:58.800 close to a deadline when this needs to 00:44:56.400 --> 00:45:00.800 be agreed again, there's a concern and 00:44:58.800 --> 00:45:03.160 there's negotiations and so on. 00:45:00.800 --> 00:45:05.600 And the and the 00:45:03.159 --> 00:45:07.519 and the well, I mean, everyone at this 00:45:05.599 --> 00:45:09.960 moment at least thinks that 00:45:07.519 --> 00:45:11.440 as every as in every instance in the 00:45:09.960 --> 00:45:14.280 past, they're going to reach some sort 00:45:11.440 --> 00:45:16.079 of agreement the day before 00:45:14.280 --> 00:45:18.640 of the deadline or not. 00:45:16.079 --> 00:45:20.880 But if they don't and there is a mess, 00:45:18.639 --> 00:45:22.960 this is huge for finance. It's huge for 00:45:20.880 --> 00:45:25.000 finance because US Treasury bonds, 00:45:22.960 --> 00:45:26.760 especially short-term bonds, are used 00:45:25.000 --> 00:45:28.599 for pricing everything 00:45:26.760 --> 00:45:30.320 through arbitrage and so on. 00:45:28.599 --> 00:45:32.239 So, you get a mess there, 00:45:30.320 --> 00:45:34.120 that's a mess in every single financial 00:45:32.239 --> 00:45:36.839 market. You wouldn't know how to price 00:45:34.119 --> 00:45:39.960 many financial assets, actually. 00:45:36.840 --> 00:45:42.079 So, it would be a disaster. But uh 00:45:39.960 --> 00:45:43.440 but the reason I describe I mention this 00:45:42.079 --> 00:45:45.279 here is because 00:45:43.440 --> 00:45:47.639 again, lots of prices are priced in 00:45:45.280 --> 00:45:49.000 reference in as in finance are priced in 00:45:47.639 --> 00:45:50.799 reference, especially derivatives, 00:45:49.000 --> 00:45:52.760 options, and stuff like that. 00:45:50.800 --> 00:45:54.960 Uh you price them relative to something 00:45:52.760 --> 00:45:57.320 using this type of logic. So, if the 00:45:54.960 --> 00:45:59.159 thing you use as a base as a reference 00:45:57.320 --> 00:46:01.600 becomes highly unstable and uncertain 00:45:59.159 --> 00:46:03.440 and risky, then obviously everything 00:46:01.599 --> 00:46:05.440 becomes very complicated, 00:46:03.440 --> 00:46:09.280 very risky, and and financial markets do 00:46:05.440 --> 00:46:09.280 not like risk. That's for sure. 00:46:09.760 --> 00:46:13.560 Anyway, ignore that. That's 00:46:11.920 --> 00:46:15.680 irrelevant for your quiz, but that's the 00:46:13.559 --> 00:46:17.719 reason this the whole discussion then 00:46:15.679 --> 00:46:20.679 over the summer can get to be very very 00:46:17.719 --> 00:46:20.679 tricky for finance. 00:46:21.000 --> 00:46:24.358 So, the yield to maturity, remember I 00:46:22.719 --> 00:46:25.879 mentioned this concept before, of an 00:46:24.358 --> 00:46:27.799 N-year bond, 00:46:25.880 --> 00:46:31.119 but it's also what we When you see 00:46:27.800 --> 00:46:34.160 Whenever you hear the 3-year rate, 00:46:31.119 --> 00:46:36.319 is that. It's the yield to maturity. 00:46:34.159 --> 00:46:37.399 Uh which is different from Okay, let me 00:46:36.320 --> 00:46:38.640 tell you 00:46:37.400 --> 00:46:40.240 show you a formula that's easy to 00:46:38.639 --> 00:46:42.679 explain then. 00:46:40.239 --> 00:46:44.199 And it's defined, it's important, as the 00:46:42.679 --> 00:46:45.960 constant 00:46:44.199 --> 00:46:48.639 annual interest rate that makes the bond 00:46:45.960 --> 00:46:50.400 price today equal to the present 00:46:48.639 --> 00:46:51.440 discounted value or expected discounted 00:46:50.400 --> 00:46:53.680 value. 00:46:51.440 --> 00:46:56.240 So, notice notice the highlighted 00:46:53.679 --> 00:46:57.719 is defined as the constant annual 00:46:56.239 --> 00:46:59.719 interest rate 00:46:57.719 --> 00:47:01.519 that makes the bond price today equal to 00:46:59.719 --> 00:47:03.079 the present discounted value of future 00:47:01.519 --> 00:47:04.480 payments of the bond. 00:47:03.079 --> 00:47:06.239 Okay? 00:47:04.480 --> 00:47:09.039 So, 00:47:06.239 --> 00:47:10.839 for example, in our 2-year bond, 00:47:09.039 --> 00:47:12.519 that's the price. Right? This is the 00:47:10.840 --> 00:47:14.720 price of the asset. 00:47:12.519 --> 00:47:16.800 We that we know the price. We already 00:47:14.719 --> 00:47:18.000 got the price from the previous slides 00:47:16.800 --> 00:47:19.880 of the bond, 00:47:18.000 --> 00:47:21.320 which was based on the short-term 00:47:19.880 --> 00:47:23.079 interest rate, 1-year interest rate, and 00:47:21.320 --> 00:47:25.160 our forecast of the 00:47:23.079 --> 00:47:26.840 short-term the 1-year interest rate 1 00:47:25.159 --> 00:47:28.159 year from now. 00:47:26.840 --> 00:47:29.240 I know that price. Take that as a 00:47:28.159 --> 00:47:31.559 number. 00:47:29.239 --> 00:47:34.000 So, then I the yield the yield to 00:47:31.559 --> 00:47:35.799 maturity is calculated as that constant 00:47:34.000 --> 00:47:37.400 interest rate 00:47:35.800 --> 00:47:38.960 constant How do I see this constant? 00:47:37.400 --> 00:47:40.160 Because, well, I'm using the same 00:47:38.960 --> 00:47:42.119 interest rate for the first period and 00:47:40.159 --> 00:47:44.639 the second period. I And now I'm calling 00:47:42.119 --> 00:47:46.440 it I2T. It's a 2-year interest rate, but 00:47:44.639 --> 00:47:48.679 it's constant. Constant doesn't mean 00:47:46.440 --> 00:47:50.240 that it doesn't move over time. 00:47:48.679 --> 00:47:51.799 It means I'm discounting all the cash 00:47:50.239 --> 00:47:55.000 flows as a constant interest rate. This 00:47:51.800 --> 00:47:55.000 It means I'm using 00:47:55.280 --> 00:47:58.560 I'm using this equation. 00:47:57.719 --> 00:48:00.439 Okay? 00:47:58.559 --> 00:48:02.358 So, the yield to maturity is 00:48:00.440 --> 00:48:03.920 find an interest rate 00:48:02.358 --> 00:48:08.039 that allows me to use this constant 00:48:03.920 --> 00:48:10.039 thing constant assume use this formula 00:48:08.039 --> 00:48:13.039 and get back the same price 00:48:10.039 --> 00:48:14.800 as I got by using the the 00:48:13.039 --> 00:48:17.159 the expected discounted value or the 00:48:14.800 --> 00:48:19.519 arbitrage or something like that. Okay? 00:48:17.159 --> 00:48:21.000 So, that's that's the definition. Okay? 00:48:19.519 --> 00:48:23.119 You have this price. 00:48:21.000 --> 00:48:26.719 Now you you look for that interest rate 00:48:23.119 --> 00:48:27.599 that allows you to match that price. 00:48:26.719 --> 00:48:29.319 Okay? 00:48:27.599 --> 00:48:32.199 And that's called the yield. That's the 00:48:29.320 --> 00:48:33.440 thing I Remember I plotted this the some 00:48:32.199 --> 00:48:35.399 curves? 00:48:33.440 --> 00:48:39.200 Well, those those interest rates in 00:48:35.400 --> 00:48:39.200 those curves were computed this way. 00:48:39.480 --> 00:48:43.039 Now, 00:48:40.599 --> 00:48:44.960 notice that we know what this price is. 00:48:43.039 --> 00:48:46.960 This price is by the expected discounted 00:48:44.960 --> 00:48:49.679 value or the arbitrage approach is equal 00:48:46.960 --> 00:48:51.840 to 100 divided by this. 00:48:49.679 --> 00:48:54.239 So, I know that these two things are 00:48:51.840 --> 00:48:56.760 this is equal to that, 00:48:54.239 --> 00:48:58.719 which means that this denominator is 00:48:56.760 --> 00:49:00.760 equal to that, 00:48:58.719 --> 00:49:03.919 and that implies for a small interest 00:49:00.760 --> 00:49:06.320 rate that this 2-year interest rate is 00:49:03.920 --> 00:49:09.519 approximately equal to the average of 00:49:06.320 --> 00:49:11.440 the expected interest rate 1-year rates. 00:49:09.519 --> 00:49:12.719 Okay? 00:49:11.440 --> 00:49:15.039 So, this is called actually the 00:49:12.719 --> 00:49:17.480 expectation hypothesis, by the way. Is 00:49:15.039 --> 00:49:19.039 that the the 2-year rate 00:49:17.480 --> 00:49:21.358 is approximately equal to the average of 00:49:19.039 --> 00:49:24.759 the 1-year rate this year 00:49:21.358 --> 00:49:26.358 plus the expected 1-year rate 1 year 00:49:24.760 --> 00:49:28.480 from now. 00:49:26.358 --> 00:49:28.480 Okay? 00:49:30.320 --> 00:49:34.080 So, that's an important concept. And I'm 00:49:31.960 --> 00:49:36.720 going to start from here again 00:49:34.079 --> 00:49:36.719 in the next lecture. 00:49:48.360 --> 00:49:50.360 Mhm.