WEBVTT 00:00:00.090 --> 00:00:02.430 The following content is provided under a Creative 00:00:02.430 --> 00:00:03.820 Commons license. 00:00:03.820 --> 00:00:06.030 Your support will help MIT OpenCourseWare 00:00:06.030 --> 00:00:10.120 continue to offer high quality educational resources for free. 00:00:10.120 --> 00:00:12.690 To make a donation or to view additional materials 00:00:12.690 --> 00:00:16.620 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.620 --> 00:00:17.820 at ocw.mit.edu. 00:00:21.318 --> 00:00:24.580 ANDREW LO: First of all, any questions from last lecture? 00:00:27.760 --> 00:00:28.976 Yes? 00:00:28.976 --> 00:00:38.370 AUDIENCE: [INAUDIBLE] he said he was [INAUDIBLE] possible 00:00:38.370 --> 00:00:39.445 [INAUDIBLE]? 00:00:44.907 --> 00:00:45.490 ANDREW LO: OK. 00:00:45.490 --> 00:00:48.190 So let me repeat the question to make sure everybody heard. 00:00:48.190 --> 00:00:52.480 The question about net present value is that, is it possible, 00:00:52.480 --> 00:00:56.530 is it possible, that in one currency, 00:00:56.530 --> 00:01:00.280 the net present value of a project is positive, 00:01:00.280 --> 00:01:04.180 but in a different currency, it is negative? 00:01:04.180 --> 00:01:06.240 That's a very interesting question. 00:01:06.240 --> 00:01:10.630 And it turns out that the answer is staring us in the face right 00:01:10.630 --> 00:01:11.920 here. 00:01:11.920 --> 00:01:15.380 Now remember, we're in a world of no uncertainty. 00:01:15.380 --> 00:01:18.930 So we know what future cash flows are going to be. 00:01:18.930 --> 00:01:22.960 And we know what future discount rates or discount 00:01:22.960 --> 00:01:24.430 factors are going to be. 00:01:24.430 --> 00:01:26.530 That's my assumption. 00:01:26.530 --> 00:01:29.890 And in that world, when I give you 00:01:29.890 --> 00:01:33.100 the value of a sequence of cash flows, 00:01:33.100 --> 00:01:38.510 this v sub 0, if I wanted denominate it in dollars, 00:01:38.510 --> 00:01:43.630 then presumably all the cash flows have to be in dollars. 00:01:43.630 --> 00:01:45.560 If I want to denominate it in yen, 00:01:45.560 --> 00:01:48.250 then the cash flows have to be in yen. 00:01:48.250 --> 00:01:53.680 So strictly speaking, assuming that the exchange 00:01:53.680 --> 00:01:57.310 rates don't change over time-- 00:01:57.310 --> 00:02:01.340 and that's, again, a big assumption-- 00:02:01.340 --> 00:02:05.150 the question is, can I have a different result 00:02:05.150 --> 00:02:09.620 in terms of the sign of a net present value by changing 00:02:09.620 --> 00:02:11.830 the exchange rate? 00:02:11.830 --> 00:02:13.520 Any thoughts on that? 00:02:13.520 --> 00:02:14.320 What do you think? 00:02:14.320 --> 00:02:15.000 Yeah. 00:02:15.000 --> 00:02:16.270 AUDIENCE: No. 00:02:16.270 --> 00:02:19.335 ANDREW LO: No, why? 00:02:19.335 --> 00:02:22.620 AUDIENCE: Because currency [INAUDIBLE].. 00:02:33.730 --> 00:02:34.820 ANDREW LO: OK. 00:02:34.820 --> 00:02:37.400 So the answer is no, because currency, 00:02:37.400 --> 00:02:39.500 the exchange rates always have to be positive. 00:02:39.500 --> 00:02:42.920 And presumably, you're multiplying the cache flows 00:02:42.920 --> 00:02:48.360 by the same number, either positive of one number 00:02:48.360 --> 00:02:49.990 or positive of another number. 00:02:49.990 --> 00:02:54.090 So when you multiply a sequence by a positive number, when 00:02:54.090 --> 00:02:57.300 you add that up, it is either still positive 00:02:57.300 --> 00:02:58.180 or still negative. 00:02:58.180 --> 00:02:59.880 In other words, you can factor it out. 00:02:59.880 --> 00:03:01.990 Right? 00:03:01.990 --> 00:03:03.250 You sure? 00:03:03.250 --> 00:03:04.400 Yeah. 00:03:04.400 --> 00:03:05.810 AUDIENCE: I have a question. 00:03:05.810 --> 00:03:11.230 When we are doing this in the [INAUDIBLE],, 00:03:11.230 --> 00:03:17.680 is it possible to have different [INAUDIBLE]?? 00:03:17.680 --> 00:03:19.000 ANDREW LO: Well. 00:03:19.000 --> 00:03:21.080 Right now, we're not talking about risk. 00:03:21.080 --> 00:03:24.460 So let's hold that off for seven or eight lectures. 00:03:24.460 --> 00:03:25.930 I want to ask this question. 00:03:25.930 --> 00:03:27.530 Have I got it right? 00:03:27.530 --> 00:03:30.224 We agreed that no matter what you multiply it by, 00:03:30.224 --> 00:03:31.640 as long as it's a positive number, 00:03:31.640 --> 00:03:34.310 it can't change the sign, so the currency doesn't matter. 00:03:34.310 --> 00:03:34.810 Yeah. 00:03:34.810 --> 00:03:35.581 Ernest? 00:03:35.581 --> 00:03:38.467 AUDIENCE: But the exchange rate, so the actuals 00:03:38.467 --> 00:03:39.430 are at different times. 00:03:39.430 --> 00:03:40.298 ANDREW LO: Yes. 00:03:40.298 --> 00:03:42.096 AUDIENCE: So if your exchange rate 00:03:42.096 --> 00:03:43.846 is different at different times, then it's 00:03:43.846 --> 00:03:47.260 going to stay factored throughout the-- 00:03:47.260 --> 00:03:50.312 ANDREW LO: The assumption is that it's fixed. 00:03:50.312 --> 00:03:51.270 There's no uncertainty. 00:03:51.270 --> 00:03:51.770 But-- 00:03:51.770 --> 00:03:53.200 AUDIENCE: [INAUDIBLE]. 00:03:53.200 --> 00:03:55.780 ANDREW LO: I didn't say it was the same. 00:03:55.780 --> 00:03:57.760 So you said that it was the same. 00:03:57.760 --> 00:03:58.660 I didn't. 00:03:58.660 --> 00:03:59.450 You're right. 00:03:59.450 --> 00:04:01.150 So [? Shlomi, ?] you're right. 00:04:01.150 --> 00:04:04.750 If the exchange rate is the same over time, 00:04:04.750 --> 00:04:06.610 then when you multiply by one number, 00:04:06.610 --> 00:04:08.950 it's the same number for every cash flow. 00:04:08.950 --> 00:04:11.840 Then, it factors out. 00:04:11.840 --> 00:04:16.540 And then you're multiplying v sub-zero by a positive number. 00:04:16.540 --> 00:04:19.180 So if v sub-zero is positive, it stays positive. 00:04:19.180 --> 00:04:21.990 If it's negative, it stays negative. 00:04:21.990 --> 00:04:28.110 But no uncertainty doesn't mean that it's fixed. 00:04:28.110 --> 00:04:29.130 So here's the subtlety. 00:04:29.130 --> 00:04:33.230 The subtlety is that if I assume that the exchange rate is fixed 00:04:33.230 --> 00:04:40.480 and known, but going up over time, whereas in US dollars, 00:04:40.480 --> 00:04:44.510 it stays fixed, that makes a difference. 00:04:44.510 --> 00:04:45.230 Right? 00:04:45.230 --> 00:04:46.370 So it's possible. 00:04:46.370 --> 00:04:49.910 It's possible that if I change currencies and the currency 00:04:49.910 --> 00:04:53.420 is rapidly appreciating or rapidly depreciating, 00:04:53.420 --> 00:04:56.720 then you can actually change the net present value 00:04:56.720 --> 00:04:58.220 of the project. 00:04:58.220 --> 00:05:02.480 But it has to be the case that the particular path 00:05:02.480 --> 00:05:05.990 of the currency appreciation or depreciation 00:05:05.990 --> 00:05:10.770 is exactly opposite what's going on with the NPV. 00:05:10.770 --> 00:05:13.820 So the bottom line is, you've got to do the calculation. 00:05:13.820 --> 00:05:17.100 And you have to use the currency that you care about. 00:05:17.100 --> 00:05:19.550 So if you're in US, you presumably 00:05:19.550 --> 00:05:21.509 care about getting paid in US dollars. 00:05:21.509 --> 00:05:22.550 You would use US dollars. 00:05:22.550 --> 00:05:25.044 If you're in Japan, you get paid in yen. 00:05:25.044 --> 00:05:26.210 You'll want to do it in yen. 00:05:26.210 --> 00:05:28.620 And you have to do the currency conversion. 00:05:28.620 --> 00:05:30.354 Now when we talk about uncertainty, 00:05:30.354 --> 00:05:32.270 that's going to make it much more complicated. 00:05:32.270 --> 00:05:36.620 It's going to introduce another component of risk 00:05:36.620 --> 00:05:39.051 in our calculations that has to be dealt with. 00:05:39.051 --> 00:05:40.550 So we're going to come back to that. 00:05:40.550 --> 00:05:43.280 But that's a good question. 00:05:43.280 --> 00:05:44.360 Anybody else? 00:05:44.360 --> 00:05:45.242 Yes? 00:05:45.242 --> 00:05:46.991 AUDIENCE: I noticed that you used the term 00:05:46.991 --> 00:05:48.404 paper a couple of times. 00:05:48.404 --> 00:05:50.288 I just wanted [INAUDIBLE] definition of-- 00:05:50.288 --> 00:05:50.759 ANDREW LO: Of what? 00:05:50.759 --> 00:05:51.425 AUDIENCE: Paper. 00:05:51.425 --> 00:05:52.702 ANDREW LO: Paper. 00:05:52.702 --> 00:05:54.160 You mean, this is a piece of paper? 00:05:54.160 --> 00:05:55.951 AUDIENCE: Well, I don't think [INAUDIBLE].. 00:05:58.450 --> 00:05:59.420 ANDREW LO: Right. 00:05:59.420 --> 00:06:03.460 Yeah, so typically by paper, people mean a security. 00:06:03.460 --> 00:06:05.800 And commercial paper is a security 00:06:05.800 --> 00:06:09.100 that is a debt instrument that is basically an IOU. 00:06:09.100 --> 00:06:10.300 It's like a bond. 00:06:10.300 --> 00:06:12.010 So we'll come back to that when we talk about fixed income 00:06:12.010 --> 00:06:12.670 securities. 00:06:12.670 --> 00:06:13.792 But that's what I mean. 00:06:13.792 --> 00:06:15.250 By the way, you raise a good point. 00:06:15.250 --> 00:06:19.340 When I mention terminology, feel free to ask me. 00:06:19.340 --> 00:06:21.494 But in turn, I'm going to feel free to tell you, 00:06:21.494 --> 00:06:23.910 you may want to look that up in [? Breeley, ?] [? Myers ?] 00:06:23.910 --> 00:06:27.700 and Allan, because I want you to read the book alongside of what 00:06:27.700 --> 00:06:30.760 we're doing in class, because you'll need to pick up this 00:06:30.760 --> 00:06:34.450 terminology, and we don't have enough time in this 20 lectures 00:06:34.450 --> 00:06:37.300 to cover all the terminology that you need to know. 00:06:37.300 --> 00:06:42.580 So don't assume that just because I haven't covered it 00:06:42.580 --> 00:06:45.010 in class, or that I haven't defined it 00:06:45.010 --> 00:06:47.110 that you don't need to know it. 00:06:47.110 --> 00:06:50.650 The textbook is there to help you 00:06:50.650 --> 00:06:53.890 with the supplementary material that I would like you to cover. 00:06:53.890 --> 00:06:55.600 So that's why we assign those chapters. 00:06:55.600 --> 00:06:56.230 OK? 00:06:56.230 --> 00:06:58.500 Yeah, Justin. 00:06:58.500 --> 00:07:01.020 AUDIENCE: [INAUDIBLE]. 00:07:01.020 --> 00:07:02.230 ANDREW LO: Yes. 00:07:02.230 --> 00:07:05.130 AUDIENCE: Then I read a news article, 00:07:05.130 --> 00:07:08.459 and they said the stock market jumps because they're 00:07:08.459 --> 00:07:09.250 getting bailed out. 00:07:09.250 --> 00:07:09.958 ANDREW LO: Right. 00:07:09.958 --> 00:07:13.961 AUDIENCE: So is there a simple reason as to why this is such 00:07:13.961 --> 00:07:17.230 a massive increase in stock-- 00:07:17.230 --> 00:07:19.500 ANDREW LO: In the stock market, while their stock has 00:07:19.500 --> 00:07:20.304 gone down. 00:07:20.304 --> 00:07:20.970 AUDIENCE: Right. 00:07:20.970 --> 00:07:22.924 So that seems a little counter-intuitive. 00:07:22.924 --> 00:07:24.840 I'm going to give you a two minute answer now, 00:07:24.840 --> 00:07:27.030 but then I'm going to give you a much deeper answer 00:07:27.030 --> 00:07:29.250 in about three or four lectures, when we actually 00:07:29.250 --> 00:07:31.200 apply all of the framework we're developing 00:07:31.200 --> 00:07:33.210 to pricing common stock. 00:07:33.210 --> 00:07:35.280 So as I said with Freddie and Fannie, 00:07:35.280 --> 00:07:36.600 there are two components. 00:07:36.600 --> 00:07:39.760 There are two sets of issues surrounding those companies. 00:07:39.760 --> 00:07:44.840 One is the value of the owner's equity, the folks who owned 00:07:44.840 --> 00:07:46.470 a piece of those companies. 00:07:46.470 --> 00:07:48.240 What are their investments worth? 00:07:48.240 --> 00:07:50.770 And the answer is very little. 00:07:50.770 --> 00:07:53.710 The second piece is that Freddie and Fannie 00:07:53.710 --> 00:07:57.160 have issued all sorts of IOUs, all sorts of obligations 00:07:57.160 --> 00:07:58.570 to counter-parties. 00:07:58.570 --> 00:08:02.320 And the question is, what are those securities worth. 00:08:02.320 --> 00:08:04.870 The government bailing out Freddie and Fannie 00:08:04.870 --> 00:08:09.770 are basically saying, we will stand behind those IOUs. 00:08:09.770 --> 00:08:11.960 The shareholders of the company-- 00:08:11.960 --> 00:08:14.090 sorry, you guys lost. 00:08:14.090 --> 00:08:15.560 The company has not done well. 00:08:15.560 --> 00:08:17.082 It suffered a lot of losses. 00:08:17.082 --> 00:08:19.040 So the fact that you own a piece of the company 00:08:19.040 --> 00:08:21.890 means that what you own is now worthless. 00:08:21.890 --> 00:08:25.430 But the pieces of paper that the company has issued, 00:08:25.430 --> 00:08:29.510 we will assume that obligation as the US government 00:08:29.510 --> 00:08:32.210 and make good on those obligations. 00:08:32.210 --> 00:08:35.419 So the fact that those pieces of paper 00:08:35.419 --> 00:08:39.500 have much broader impact on the market as a whole, the fact 00:08:39.500 --> 00:08:40.970 that the US government is standing 00:08:40.970 --> 00:08:45.080 behind those pieces of paper will protect the stock market 00:08:45.080 --> 00:08:47.300 as a whole because there's confidence 00:08:47.300 --> 00:08:52.740 that business conditions will not be as bad as we thought. 00:08:52.740 --> 00:08:54.950 So that's what explains the fact that the stock 00:08:54.950 --> 00:08:57.002 market as a whole went up. 00:08:57.002 --> 00:08:58.460 It's because the market environment 00:08:58.460 --> 00:09:00.350 has been stabilized. 00:09:00.350 --> 00:09:02.630 You can imagine what might have happened if Fannie 00:09:02.630 --> 00:09:04.640 and Freddie were to go under. 00:09:04.640 --> 00:09:07.610 Their pieces of paper, their IOUs, would be worthless. 00:09:07.610 --> 00:09:10.710 Which means the folks that own those pieces of paper, 00:09:10.710 --> 00:09:13.280 now they have a bunch of worthless paper. 00:09:13.280 --> 00:09:17.300 And when that happens, there are repercussion effects 00:09:17.300 --> 00:09:19.790 for those businesses, and those businesses 00:09:19.790 --> 00:09:21.290 will end up losing money, which will 00:09:21.290 --> 00:09:25.520 have repercussions for the entire market as a whole. 00:09:25.520 --> 00:09:28.333 AUDIENCE: [INAUDIBLE] the amount that it went up 00:09:28.333 --> 00:09:31.219 shows how their paper was distributed 00:09:31.219 --> 00:09:32.670 to all these other companies. 00:09:32.670 --> 00:09:34.440 ANDREW LO: It's a combination of how 00:09:34.440 --> 00:09:35.880 their paper was distributed. 00:09:35.880 --> 00:09:36.910 But more than that-- 00:09:36.910 --> 00:09:39.190 I mean, there are many companies in the S&P 500, 00:09:39.190 --> 00:09:42.660 for example, that don't own any of this paper. 00:09:42.660 --> 00:09:45.060 So why would their stock be void? 00:09:45.060 --> 00:09:47.400 It's because the business conditions 00:09:47.400 --> 00:09:51.060 have been stabilized, and there won't be any knock on effects. 00:09:51.060 --> 00:09:53.070 A good example of this is Lehman Brothers. 00:09:53.070 --> 00:09:54.930 As many of you know, Lehman Brothers 00:09:54.930 --> 00:09:57.390 is a big player in these kinds of securities, 00:09:57.390 --> 00:10:00.210 and they are currently under a lot of pressure. 00:10:00.210 --> 00:10:02.050 Their stock prices dropped dramatically, 00:10:02.050 --> 00:10:06.180 even in the last few days, because they are a big mortgage 00:10:06.180 --> 00:10:09.120 lender, and CDO investor, so they're actually 00:10:09.120 --> 00:10:10.860 hit pretty hard by all of this. 00:10:10.860 --> 00:10:13.500 And while the rescue of Freddie and Fannie 00:10:13.500 --> 00:10:17.140 has had some positive effects on Lehman's stock price, 00:10:17.140 --> 00:10:19.860 it still is under fire and a lot of people 00:10:19.860 --> 00:10:22.390 want to get rid of it. 00:10:22.390 --> 00:10:25.830 Imagine if Freddie and Fannie weren't rescued. 00:10:25.830 --> 00:10:29.310 It's almost a sure thing that Lehman 00:10:29.310 --> 00:10:31.560 would have gone under immediately 00:10:31.560 --> 00:10:33.140 as a knock-on effect. 00:10:33.140 --> 00:10:34.971 And if Lehman went under, well, I mean, 00:10:34.971 --> 00:10:36.720 there are other investment banks out there 00:10:36.720 --> 00:10:37.845 that might have gone under. 00:10:37.845 --> 00:10:39.660 And now all of a sudden, you have a series 00:10:39.660 --> 00:10:41.820 of very large companies that do business 00:10:41.820 --> 00:10:44.580 with all of Wall Street that it has gone under. 00:10:44.580 --> 00:10:47.010 That's going to have bad repercussions for the stock 00:10:47.010 --> 00:10:48.696 market as a whole. 00:10:48.696 --> 00:10:50.570 Yeah? 00:10:50.570 --> 00:10:57.060 AUDIENCE: [INAUDIBLE] companies, like big companies? 00:10:57.060 --> 00:11:00.000 ANDREW LO: Well, the short answer is I don't know. 00:11:00.000 --> 00:11:01.260 Nobody knows. 00:11:01.260 --> 00:11:06.570 I think that there is a concern that the Fed cannot be viewed 00:11:06.570 --> 00:11:09.540 as rescuing every possible financial institution 00:11:09.540 --> 00:11:10.950 that's out there. 00:11:10.950 --> 00:11:12.327 It's got to stop at some point. 00:11:12.327 --> 00:11:14.160 Many people said it should have stopped even 00:11:14.160 --> 00:11:17.010 before the Bear Stearns rescue. 00:11:17.010 --> 00:11:19.500 So the answer is we don't know. 00:11:19.500 --> 00:11:22.470 Wait and see, and we'll find out over the next few days. 00:11:22.470 --> 00:11:24.990 As I said last time, these are very 00:11:24.990 --> 00:11:27.030 interesting times for financial markets. 00:11:27.030 --> 00:11:29.730 Very, very serious issues that are coming to the forefront 00:11:29.730 --> 00:11:30.732 literally every day. 00:11:30.732 --> 00:11:32.190 So we're going to be watching that, 00:11:32.190 --> 00:11:33.680 and we'll be talking about that. 00:11:33.680 --> 00:11:34.828 Yeah? 00:11:34.828 --> 00:11:36.280 AUDIENCE: [INAUDIBLE]. 00:11:40.905 --> 00:11:42.780 ANDREW LO: Where do I think that should stop? 00:11:42.780 --> 00:11:47.000 Well, well, there are a couple of issues 00:11:47.000 --> 00:11:50.590 that are at the heart of these discussions. 00:11:50.590 --> 00:11:52.760 The two issues are, how do you balance 00:11:52.760 --> 00:11:57.080 of the cost of bailing out these large organizations 00:11:57.080 --> 00:12:00.410 and the implicit moral hazard that it creates, 00:12:00.410 --> 00:12:03.230 the kind of potential promises that you're implicitly 00:12:03.230 --> 00:12:07.040 making to future equity holders of these organizations 00:12:07.040 --> 00:12:12.290 versus letting the market work against the potential disaster 00:12:12.290 --> 00:12:16.040 scenario of allowing these kinds of events 00:12:16.040 --> 00:12:17.979 to spread like wildfire. 00:12:17.979 --> 00:12:19.520 I don't know how many of you actually 00:12:19.520 --> 00:12:22.470 know what happens during wildfires, during forest fires. 00:12:22.470 --> 00:12:25.160 But when forest fires get started, 00:12:25.160 --> 00:12:27.320 they're actually very difficult to stop. 00:12:27.320 --> 00:12:31.340 And every once in a while, they try 00:12:31.340 --> 00:12:36.210 to stop a forest fire by creating additional fires. 00:12:36.210 --> 00:12:36.710 Right? 00:12:36.710 --> 00:12:38.330 This may sound counter-intuitive. 00:12:38.330 --> 00:12:42.620 But what they will do is around a raging forest fire, 00:12:42.620 --> 00:12:47.840 they will burn what's called a firewall. 00:12:47.840 --> 00:12:49.794 That term did not come out of IT. 00:12:49.794 --> 00:12:51.710 It actually came out of fighting forest fires. 00:12:51.710 --> 00:12:55.700 They will burn a ring around that forest fire, 00:12:55.700 --> 00:12:59.360 a controlled burn where they target very specific set 00:12:59.360 --> 00:13:02.780 of trees, and they would do it in a controlled fashion, 00:13:02.780 --> 00:13:06.490 so that when the forest fire gets to that ring, 00:13:06.490 --> 00:13:09.560 it burns itself out. 00:13:09.560 --> 00:13:13.070 And one could argue that we need a firewall 00:13:13.070 --> 00:13:14.960 around these kinds of events. 00:13:14.960 --> 00:13:19.580 We need to have certain financial institutions fail 00:13:19.580 --> 00:13:24.070 and stop the spread of this kind of problem. 00:13:24.070 --> 00:13:28.390 The difficulty with that analogy is that with a forest fire, 00:13:28.390 --> 00:13:31.660 all you need is a helicopter to get up there and see 00:13:31.660 --> 00:13:33.265 what's going on. 00:13:33.265 --> 00:13:34.390 We don't have a helicopter. 00:13:34.390 --> 00:13:37.840 There's no helicopter that tells us where the fires are, 00:13:37.840 --> 00:13:41.260 and where the fires may be, and where the underground gasoline 00:13:41.260 --> 00:13:44.020 tanks are hidden for future explosions. 00:13:44.020 --> 00:13:47.860 We don't know because a lot of this stuff is hidden. 00:13:47.860 --> 00:13:51.670 So my own opinion is that we are going 00:13:51.670 --> 00:13:57.700 to need to have at least one or two additional large failures, 00:13:57.700 --> 00:14:02.170 and people will have to lose money before they understand 00:14:02.170 --> 00:14:06.250 that this stuff really is risky, and that the price you pay 00:14:06.250 --> 00:14:08.830 for the benefits that you've gotten from these very 00:14:08.830 --> 00:14:12.370 handsome returns in the years before this kind of an event 00:14:12.370 --> 00:14:14.860 is the fact that every once in a while, 00:14:14.860 --> 00:14:18.640 in the parlance of Wall Street, you get your face ripped off. 00:14:18.640 --> 00:14:21.260 That's the nature of financial markets. 00:14:21.260 --> 00:14:24.220 So I think that it's very dangerous 00:14:24.220 --> 00:14:26.060 to rescue these companies. 00:14:26.060 --> 00:14:28.420 But at the same time, you have to balance 00:14:28.420 --> 00:14:30.820 that against the risk of creating a mass panic. 00:14:30.820 --> 00:14:33.460 And if we do create that mass panic, 00:14:33.460 --> 00:14:36.130 there's virtually no way to stop it, 00:14:36.130 --> 00:14:40.030 and then we will run into a very deep recession and depression 00:14:40.030 --> 00:14:42.840 of the likes that we haven't seen since 1929. 00:14:42.840 --> 00:14:45.594 That's the balance and the danger. 00:14:45.594 --> 00:14:46.568 Yeah? 00:14:46.568 --> 00:14:48.029 AUDIENCE: [INAUDIBLE]. 00:14:58.542 --> 00:15:00.250 ANDREW LO: Well, you know, that might be. 00:15:00.250 --> 00:15:03.010 But let me suggest this. 00:15:03.010 --> 00:15:05.920 Let me put that off for a discussion point 00:15:05.920 --> 00:15:08.004 until we finish fixed income securities. 00:15:08.004 --> 00:15:09.670 Because at that point, I'm going to talk 00:15:09.670 --> 00:15:12.410 about the subprime problem specifically. 00:15:12.410 --> 00:15:14.980 And I'm going to use the tools that we develop== actually, 00:15:14.980 --> 00:15:17.890 you guys are going to use the tools that we develop to figure 00:15:17.890 --> 00:15:20.740 out exactly what's happened in these markets, 00:15:20.740 --> 00:15:23.900 why they're happening, and how maybe we can get around that. 00:15:23.900 --> 00:15:26.680 So let me not give you my view now. 00:15:26.680 --> 00:15:28.690 I'd rather have you develop your own views 00:15:28.690 --> 00:15:30.890 based upon the tools we develop in this course. 00:15:30.890 --> 00:15:31.870 OK? 00:15:31.870 --> 00:15:33.483 Yeah? 00:15:33.483 --> 00:15:37.226 AUDIENCE: Because of all this [INAUDIBLE] 00:15:37.226 --> 00:15:41.190 CEOs or executives were fired to get a big handsome buyout 00:15:41.190 --> 00:15:43.480 for all their hard work and efforts. 00:15:43.480 --> 00:15:44.370 ANDREW LO: Yeah. 00:15:44.370 --> 00:15:46.340 AUDIENCE: But now, should the market 00:15:46.340 --> 00:15:48.170 be able to self-regulate itself? 00:15:48.170 --> 00:15:51.342 Or does there need to be regulation in place? 00:15:51.342 --> 00:15:52.426 Or what will become of it? 00:15:52.426 --> 00:15:54.050 ANDREW LO: Well, you know, that's again 00:15:54.050 --> 00:15:56.380 a very difficult question to answer because we're not 00:15:56.380 --> 00:15:59.251 done yet, so we don't know where this is going to end up. 00:15:59.251 --> 00:16:01.750 I think that there are some very important issues that we're 00:16:01.750 --> 00:16:03.190 going to have to come back to. 00:16:03.190 --> 00:16:05.230 Let me put that off for even a bit longer 00:16:05.230 --> 00:16:07.960 because when we talk about corporate finance, 00:16:07.960 --> 00:16:10.000 we're going to talk about CEO compensation 00:16:10.000 --> 00:16:12.520 and ask the question, how do we relate compensation 00:16:12.520 --> 00:16:14.260 to performance, and does it make sense? 00:16:14.260 --> 00:16:16.900 It turns out that there's some incentive issues, such that 00:16:16.900 --> 00:16:19.060 if we don't do that, if we don't allow 00:16:19.060 --> 00:16:21.100 them to have these golden parachutes, 00:16:21.100 --> 00:16:24.580 then it may end up creating weird incentives 00:16:24.580 --> 00:16:26.620 when things are going well. 00:16:26.620 --> 00:16:30.550 So every action has some kind of equal and opposite reaction 00:16:30.550 --> 00:16:32.200 in some other part of the system. 00:16:32.200 --> 00:16:34.015 And unless you know what that system is, 00:16:34.015 --> 00:16:36.140 it's hard to figure out the answer to the question. 00:16:36.140 --> 00:16:37.600 So by the end of the semester, I'm 00:16:37.600 --> 00:16:39.370 hoping that you'll be able to come up with answers 00:16:39.370 --> 00:16:40.370 to these questions. 00:16:40.370 --> 00:16:42.120 So let me put that off for a little while. 00:16:42.120 --> 00:16:42.730 OK. 00:16:42.730 --> 00:16:47.158 One more clarifying question maybe, and then we can move on. 00:16:47.158 --> 00:16:51.046 AUDIENCE: During the Southeast Asian Crisis in '97, 00:16:51.046 --> 00:16:53.962 there was this discussion about the international financial 00:16:53.962 --> 00:16:57.364 institutions should risk [INAUDIBLE] countries 00:16:57.364 --> 00:17:01.252 and because of the bar [INAUDIBLE].. 00:17:01.252 --> 00:17:02.224 ANDREW LO: Right. 00:17:02.224 --> 00:17:03.849 AUDIENCE: And they decided they should, 00:17:03.849 --> 00:17:06.902 so they rescued them and they survived. 00:17:06.902 --> 00:17:09.382 10 years later, Latin America went into a crisis, 00:17:09.382 --> 00:17:11.862 and the same discussion started, and 00:17:11.862 --> 00:17:14.342 the international financial institutions, 00:17:14.342 --> 00:17:19.302 led by the United States decided not to rescue them. 00:17:19.302 --> 00:17:21.790 So we went into a crisis. 00:17:21.790 --> 00:17:31.060 And so I see now [INAUDIBLE] 00:17:31.060 --> 00:17:32.960 ANDREW LO: That's right. 00:17:32.960 --> 00:17:35.230 Yeah, that's a very serious issue. 00:17:35.230 --> 00:17:38.820 But I would argue that issue actually goes even-- 00:17:38.820 --> 00:17:42.850 it goes to an even broader set of issues that have little 00:17:42.850 --> 00:17:44.350 to do with economics and finance, 00:17:44.350 --> 00:17:48.490 but political and social issues, which I won't comment on 00:17:48.490 --> 00:17:51.490 in this class, but which are important for determining 00:17:51.490 --> 00:17:53.170 those kinds of policy questions. 00:17:53.170 --> 00:17:56.290 That's one of the things that I'd like to get across to you 00:17:56.290 --> 00:17:58.240 in terms of thinking about these issues, which 00:17:58.240 --> 00:18:01.980 is that there are multiple aspects to every issue. 00:18:01.980 --> 00:18:05.770 And rather than trying to come up with a single answer, what 00:18:05.770 --> 00:18:07.480 I would propose that you might do 00:18:07.480 --> 00:18:10.150 is when you think about a challenge like this, first 00:18:10.150 --> 00:18:14.080 of all, you try to identify the different issues 00:18:14.080 --> 00:18:18.430 and then come up with an answer for every single perspective 00:18:18.430 --> 00:18:19.060 of that issue. 00:18:19.060 --> 00:18:21.539 So for example in the case of Latin America, 00:18:21.539 --> 00:18:23.830 there is certainly the economic issue and moral hazard. 00:18:23.830 --> 00:18:26.390 That's an important one. 00:18:26.390 --> 00:18:28.450 But there's also a political and social issue, 00:18:28.450 --> 00:18:33.140 which is that if you don't bail out countries that are in need, 00:18:33.140 --> 00:18:36.070 that's a recipe for creating social unrest. 00:18:36.070 --> 00:18:38.440 And if you don't do it, there is some dictator 00:18:38.440 --> 00:18:43.270 waiting with guns and other interesting possibilities 00:18:43.270 --> 00:18:46.630 for the people to try to take over. 00:18:46.630 --> 00:18:47.290 That's right. 00:18:47.290 --> 00:18:49.640 And I mean, it's not rocket science. 00:18:49.640 --> 00:18:51.982 I mean, people are looking for solutions. 00:18:51.982 --> 00:18:53.440 And if you can't offer one, they'll 00:18:53.440 --> 00:18:55.210 go to the next person that has one. 00:18:55.210 --> 00:18:57.370 Whether or not it's true or false, 00:18:57.370 --> 00:19:02.030 they will try to come up with some kind of leadership. 00:19:02.030 --> 00:19:05.380 So how do you balance off the economic considerations 00:19:05.380 --> 00:19:06.940 against the political and social? 00:19:06.940 --> 00:19:09.160 That's not something that an economist can answer, 00:19:09.160 --> 00:19:10.870 so I won't even try to begin. 00:19:10.870 --> 00:19:14.160 And by the way, my opinion is no better or worse than anybody 00:19:14.160 --> 00:19:14.660 else's. 00:19:14.660 --> 00:19:16.960 So I won't waste your time with that. 00:19:16.960 --> 00:19:20.210 But what I would suggest is from looking at these issues, 00:19:20.210 --> 00:19:22.930 first of all, try to think clearly 00:19:22.930 --> 00:19:25.870 about what the economic issues are, and then 00:19:25.870 --> 00:19:28.600 what the social and political issues are, and separate them 00:19:28.600 --> 00:19:29.530 out. 00:19:29.530 --> 00:19:33.100 And then you can answer each of those questions in isolation 00:19:33.100 --> 00:19:36.220 and, at the end, decide on how you want to balance 00:19:36.220 --> 00:19:37.930 these kind of considerations. 00:19:37.930 --> 00:19:41.230 But don't use economics to try to answer a political question, 00:19:41.230 --> 00:19:44.295 and don't use politics to try to answer an economic question. 00:19:44.295 --> 00:19:45.670 You should use the tools that you 00:19:45.670 --> 00:19:49.180 have to answer the questions that those tools are designed 00:19:49.180 --> 00:19:49.810 for. 00:19:49.810 --> 00:19:51.184 And in the case of Latin America, 00:19:51.184 --> 00:19:53.710 I would argue that's a very complex set of issues that 00:19:53.710 --> 00:19:56.910 economics alone cannot answer. 00:19:56.910 --> 00:19:59.040 The economic answer, never bail out 00:19:59.040 --> 00:20:01.500 countries that are failing, because you'll 00:20:01.500 --> 00:20:04.950 create moral hazard and increase the cost of borrowing 00:20:04.950 --> 00:20:07.680 for future generations in other countries. 00:20:07.680 --> 00:20:10.410 That sounds good until you see what 00:20:10.410 --> 00:20:12.300 happens when you don't, and you get 00:20:12.300 --> 00:20:14.400 these socialist dictatorships that 00:20:14.400 --> 00:20:18.920 end up creating all sorts of dislocation 00:20:18.920 --> 00:20:21.780 for the people in the country. 00:20:21.780 --> 00:20:24.695 I mean, that there's a very big cost to that as well. 00:20:24.695 --> 00:20:26.820 And I'm going to have to beg the question about how 00:20:26.820 --> 00:20:29.610 you balance those costs against the benefits. 00:20:29.610 --> 00:20:33.450 Again, that's something for politicians and for voters 00:20:33.450 --> 00:20:34.790 to hopefully to decide. 00:20:38.470 --> 00:20:38.970 Yeah? 00:20:38.970 --> 00:20:39.922 Which? 00:20:39.922 --> 00:20:43.730 AUDIENCE: [INAUDIBLE]. 00:20:43.730 --> 00:20:44.960 ANDREW LO: No. 00:20:44.960 --> 00:20:46.418 Sorry. 00:20:46.418 --> 00:20:52.286 AUDIENCE: [INAUDIBLE] has renounced the United States 00:20:52.286 --> 00:20:53.270 treasury-- 00:20:53.270 --> 00:20:53.951 ANDREW LO: Yeah. 00:20:53.951 --> 00:20:57.567 AUDIENCE: [INAUDIBLE]. 00:20:57.567 --> 00:20:59.900 ANDREW LO: That sounds good, but that wasn't my handout. 00:20:59.900 --> 00:21:03.320 So that might be my handout in about three weeks. 00:21:03.320 --> 00:21:05.270 But we have work to do now. 00:21:05.270 --> 00:21:07.050 So let me let me stick to that. 00:21:07.050 --> 00:21:09.050 And we'll come back to these interesting issues. 00:21:09.050 --> 00:21:11.540 But I want to give you the framework and the tools 00:21:11.540 --> 00:21:12.960 to be able to think about them. 00:21:12.960 --> 00:21:13.460 OK. 00:21:16.320 --> 00:21:19.350 So let me continue on. 00:21:19.350 --> 00:21:21.120 This is Lecture Three. 00:21:21.120 --> 00:21:22.890 And we're going to continue looking 00:21:22.890 --> 00:21:27.330 at present value relationships and the time value of money. 00:21:27.330 --> 00:21:29.910 Last time, we were left with the expression 00:21:29.910 --> 00:21:34.410 for the value of an asset as simply being equal to the cash 00:21:34.410 --> 00:21:38.250 flows discounted with the appropriate discount factors, 00:21:38.250 --> 00:21:42.750 where I've assumed for simplicity that the discount 00:21:42.750 --> 00:21:47.250 rate between one year and the next is constant 00:21:47.250 --> 00:21:52.020 and given by the interest rate, or discount factor, or cost 00:21:52.020 --> 00:21:57.200 of capital, or user cost, or opportunity cost, r. 00:21:57.200 --> 00:21:59.880 Fancy terms for the simple concept 00:21:59.880 --> 00:22:04.560 of the number that you use to construct these exchange 00:22:04.560 --> 00:22:09.840 rates between cash at different points in time. 00:22:09.840 --> 00:22:16.220 Now the solution of how you make management decisions given 00:22:16.220 --> 00:22:19.910 this simple framework becomes trivial. 00:22:19.910 --> 00:22:22.490 Take projects that have positive NPV. 00:22:22.490 --> 00:22:23.900 That's it. 00:22:23.900 --> 00:22:26.180 When you figure out what the value of a project 00:22:26.180 --> 00:22:30.350 is as a function of all of these exchange rates, 00:22:30.350 --> 00:22:33.410 you calculate what the present value is. 00:22:33.410 --> 00:22:35.060 And if the cost of the investment 00:22:35.060 --> 00:22:38.720 is included as a cash flow, possibly a negative cash flow, 00:22:38.720 --> 00:22:40.400 you've got the net present value. 00:22:40.400 --> 00:22:44.120 And for things that are positive NPV, you want them, 00:22:44.120 --> 00:22:45.350 you want to take them. 00:22:45.350 --> 00:22:47.760 For things that are negative NPV, you don't want them, 00:22:47.760 --> 00:22:51.050 you don't take them, or if you can, you sell them. 00:22:51.050 --> 00:22:52.880 All right? 00:22:52.880 --> 00:22:54.710 Now, there are many different assumptions 00:22:54.710 --> 00:22:56.430 that got us to this point. 00:22:56.430 --> 00:22:57.620 We understand that. 00:22:57.620 --> 00:23:00.110 We're going to make those assumptions more 00:23:00.110 --> 00:23:01.410 and more realistic over time. 00:23:01.410 --> 00:23:03.920 That's, in fact, what the rest of the course 00:23:03.920 --> 00:23:05.310 is going to be doing. 00:23:05.310 --> 00:23:08.862 We're going to be focusing on picking this expression 00:23:08.862 --> 00:23:10.070 and making it more realistic. 00:23:10.070 --> 00:23:13.010 And it's going to take us 12 more weeks to do that. 00:23:13.010 --> 00:23:16.490 So it's non-trivial, but that's exactly the objective. 00:23:16.490 --> 00:23:17.138 Yes? 00:23:17.138 --> 00:23:22.635 AUDIENCE: Last week, you said [INAUDIBLE] summation 00:23:22.635 --> 00:23:23.840 of cash flow. 00:23:23.840 --> 00:23:24.460 ANDREW LO: No. 00:23:24.460 --> 00:23:28.460 I said the asset was a sequence. 00:23:28.460 --> 00:23:30.180 What is an asset? 00:23:30.180 --> 00:23:33.390 An asset is a sequence of cash flows. 00:23:33.390 --> 00:23:36.890 That's the definition of an asset, not the value. 00:23:36.890 --> 00:23:38.630 The value of the asset, remember, 00:23:38.630 --> 00:23:42.590 is that function that you stick in a cash flow sequence, 00:23:42.590 --> 00:23:45.320 and out pops a number. 00:23:45.320 --> 00:23:48.770 So the value of an asset is not the same thing as the asset 00:23:48.770 --> 00:23:51.460 itself, right? 00:23:51.460 --> 00:23:54.520 You can have a rocket ship that can go to the moon. 00:23:54.520 --> 00:23:56.450 That is an asset. 00:23:56.450 --> 00:23:59.110 The value of a rocket ship that goes to the moon, that's 00:23:59.110 --> 00:24:01.030 a different thing, right? 00:24:01.030 --> 00:24:03.850 You need to have this v function in order to figure out 00:24:03.850 --> 00:24:05.420 the value of an asset. 00:24:05.420 --> 00:24:07.600 But I can't really talk about the value of an asset 00:24:07.600 --> 00:24:11.480 unless I've defined the asset to begin with. 00:24:11.480 --> 00:24:15.590 So v sub-zero is the value of the asset. 00:24:15.590 --> 00:24:16.690 It's not the asset itself. 00:24:16.690 --> 00:24:17.990 It's the value of the asset. 00:24:17.990 --> 00:24:23.150 The asset itself is the sequence of cash flows. 00:24:23.150 --> 00:24:27.060 Now, here's a simple example about how 00:24:27.060 --> 00:24:28.560 these discount factors work. 00:24:28.560 --> 00:24:30.460 This is just an interest rate example. 00:24:30.460 --> 00:24:33.900 If you let little r equal 5%, then you 00:24:33.900 --> 00:24:37.470 can figure out what the value of a dollar is in the future, 00:24:37.470 --> 00:24:41.171 or you can figure out what the value today of a future dollar 00:24:41.171 --> 00:24:41.670 is. 00:24:41.670 --> 00:24:44.310 It's just using simple arithmetic 00:24:44.310 --> 00:24:45.880 to be able to do that. 00:24:45.880 --> 00:24:49.020 So this is just a simple concrete illustration. 00:24:49.020 --> 00:24:54.060 And if you graph the present value of a dollar, over time, 00:24:54.060 --> 00:24:57.450 you'll notice that as time goes out farther, 00:24:57.450 --> 00:24:59.880 the present value of a dollar declines. 00:24:59.880 --> 00:25:02.640 Not surprisingly, $1 today is worth more 00:25:02.640 --> 00:25:03.930 than the dollar tomorrow. 00:25:03.930 --> 00:25:07.380 But $1 tomorrow is worth more than $1 two years from now. 00:25:07.380 --> 00:25:09.000 And $1 two years from now is worth 00:25:09.000 --> 00:25:12.771 much more than $1 an infinite number of years from now. 00:25:12.771 --> 00:25:13.270 Right? 00:25:18.270 --> 00:25:23.280 Now here's an example of how you use this valuation approach. 00:25:23.280 --> 00:25:26.250 And the problems that we handed out last time 00:25:26.250 --> 00:25:29.190 will give you practice in how to think about present value. 00:25:29.190 --> 00:25:31.260 So I urge you to do those problems 00:25:31.260 --> 00:25:34.220 to make sure you really understand these concepts. 00:25:34.220 --> 00:25:36.690 Here's an example where a firm spends 00:25:36.690 --> 00:25:39.680 $800,000 every single year for electricity 00:25:39.680 --> 00:25:41.210 at its headquarters. 00:25:41.210 --> 00:25:44.810 And by installing some kind of specialized computer lighting 00:25:44.810 --> 00:25:48.110 system, it turns out that you can reduce your electricity 00:25:48.110 --> 00:25:53.000 bills by $90,000 in each of the next three years. 00:25:53.000 --> 00:25:55.250 Now, of course, it costs money to install that system. 00:25:55.250 --> 00:25:59.220 It costs $230,000 to install that system. 00:25:59.220 --> 00:26:02.120 So the question is, is this a good deal? 00:26:02.120 --> 00:26:03.960 Should you do it? 00:26:03.960 --> 00:26:07.010 That's a management decision. 00:26:07.010 --> 00:26:10.910 And the management decision relies on valuation first. 00:26:10.910 --> 00:26:14.590 Once you value it, then you can make a decision. 00:26:14.590 --> 00:26:19.570 So you've got 90,000, 90,000, 90,000 in the three years 00:26:19.570 --> 00:26:22.000 as your cost savings, but it's going 00:26:22.000 --> 00:26:25.880 to cost you $230,000 upfront. 00:26:25.880 --> 00:26:30.340 Now if it turns out that the interest rate is 4%, 00:26:30.340 --> 00:26:33.430 you can figure out what the answer is. 00:26:33.430 --> 00:26:40.510 At 4%, it turns out that the NPV of this project 00:26:40.510 --> 00:26:42.252 is about $20,000. 00:26:44.935 --> 00:26:47.720 So it's a good deal. 00:26:47.720 --> 00:26:52.320 On the other hand, if you change the assumptions, 00:26:52.320 --> 00:26:54.390 and you make the interest rate something else, 00:26:54.390 --> 00:26:59.090 well, it might not be a good deal. 00:26:59.090 --> 00:27:01.490 How would you have to change the interest rate 00:27:01.490 --> 00:27:03.260 to make this a terrible deal? 00:27:06.000 --> 00:27:08.190 Increase or decrease it? 00:27:08.190 --> 00:27:09.090 Increase it. 00:27:09.090 --> 00:27:09.810 Why? 00:27:09.810 --> 00:27:13.070 Why does that make sense? 00:27:13.070 --> 00:27:13.582 Yeah? 00:27:13.582 --> 00:27:15.159 AUDIENCE: [INAUDIBLE]. 00:27:15.159 --> 00:27:15.950 ANDREW LO: Exactly. 00:27:15.950 --> 00:27:18.010 With a higher interest rate, money now 00:27:18.010 --> 00:27:22.852 is more valuable than the cost savings to your electricity. 00:27:22.852 --> 00:27:24.310 How do you know it's more valuable? 00:27:24.310 --> 00:27:26.749 AUDIENCE: [INAUDIBLE] 00:27:26.749 --> 00:27:27.540 ANDREW LO: Exactly. 00:27:27.540 --> 00:27:30.999 The opportunity cost is 10% as opposed to 4%. 00:27:30.999 --> 00:27:32.040 It's a lot more valuable. 00:27:32.040 --> 00:27:34.690 If you stick it in the bank, you get 10%. 00:27:34.690 --> 00:27:42.640 So the cost savings depends on the interest rate at hand. 00:27:42.640 --> 00:27:47.350 Once you have the interest rate, you can make a decision. 00:27:47.350 --> 00:27:48.875 Where does interest rate come from? 00:27:48.875 --> 00:27:49.929 AUDIENCE: [INAUDIBLE] 00:27:49.929 --> 00:27:50.720 ANDREW LO: Exactly. 00:27:50.720 --> 00:27:52.460 The market. 00:27:52.460 --> 00:27:54.460 You don't pick the interest rate out of the air. 00:27:54.460 --> 00:27:58.060 You don't say, I sort of feel like it's a 2% kind of day. 00:27:58.060 --> 00:28:00.910 The interest rate is what you can get on the open market. 00:28:00.910 --> 00:28:03.210 See, that's why the market matters. 00:28:03.210 --> 00:28:05.380 It's because if that's a market interest 00:28:05.380 --> 00:28:07.720 rate, by saying it's a market interest rate, 00:28:07.720 --> 00:28:13.140 it means you can actually get that rate from the market. 00:28:13.140 --> 00:28:16.890 And therefore, it's a real number that can be actionable. 00:28:16.890 --> 00:28:19.260 It's not a fictitious theoretical construct 00:28:19.260 --> 00:28:21.610 that may or may not have any practical bearing. 00:28:21.610 --> 00:28:25.660 It's a number that actually you can achieve. 00:28:25.660 --> 00:28:27.630 And as a manager, if you're trying 00:28:27.630 --> 00:28:30.192 to increase the value of shareholder wealth, 00:28:30.192 --> 00:28:31.650 if that's the objective, is to make 00:28:31.650 --> 00:28:35.516 more money for the shareholders, this is the way to do it. 00:28:35.516 --> 00:28:37.140 So this is what I meant when I told you 00:28:37.140 --> 00:28:38.490 at the very beginning of this course 00:28:38.490 --> 00:28:40.500 that finance is the most important subject you'll ever 00:28:40.500 --> 00:28:41.145 study. 00:28:41.145 --> 00:28:43.680 It's because with proper valuation, 00:28:43.680 --> 00:28:45.750 management decisions are easy. 00:28:45.750 --> 00:28:47.970 Now, it's not always easy to get to the point 00:28:47.970 --> 00:28:50.400 where the numbers tell you so much. 00:28:50.400 --> 00:28:53.280 And so, management is trying to understand 00:28:53.280 --> 00:28:56.164 all of the various different factors and balancing them out. 00:28:56.164 --> 00:28:58.080 Like, the kind of questions you were asking me 00:28:58.080 --> 00:29:00.930 at the very beginning of class, I can't answer many of them 00:29:00.930 --> 00:29:01.740 in the abstract. 00:29:01.740 --> 00:29:03.554 It depends on the situation. 00:29:03.554 --> 00:29:05.470 And I'm hoping that by the end of this course, 00:29:05.470 --> 00:29:07.440 you will know enough about the basic framework 00:29:07.440 --> 00:29:10.320 to make those trade-offs yourself. 00:29:10.320 --> 00:29:13.350 And then, the art of management works together 00:29:13.350 --> 00:29:17.150 with the science of management to come up with good decisions. 00:29:17.150 --> 00:29:17.940 OK. 00:29:17.940 --> 00:29:19.440 So this is simple. 00:29:19.440 --> 00:29:21.990 And in the next few slides, I'm going 00:29:21.990 --> 00:29:24.330 to ask you to take a look at examples on your own. 00:29:24.330 --> 00:29:27.300 Here's an example, a real live example, 00:29:27.300 --> 00:29:29.820 where CNOOC, the Chinese oil company, 00:29:29.820 --> 00:29:33.000 made an offer to acquire Unocal about a year, year 00:29:33.000 --> 00:29:34.110 and a half ago. 00:29:34.110 --> 00:29:36.570 And I would suggest you take a look at this example 00:29:36.570 --> 00:29:38.850 and just do the back of the envelope calculation 00:29:38.850 --> 00:29:41.280 to see whether or not they provided 00:29:41.280 --> 00:29:45.480 a good deal or a bad deal. 00:29:45.480 --> 00:29:48.390 But I want to turn now to the main subject of today's 00:29:48.390 --> 00:29:53.220 lecture, which is one of the most beautiful formulas 00:29:53.220 --> 00:29:54.870 in this entire course. 00:29:54.870 --> 00:29:57.210 Now it might seem strange for me to talk about a formula 00:29:57.210 --> 00:30:00.270 as being beautiful. 00:30:00.270 --> 00:30:04.770 You know, a while ago, Paul Samuelson, the great economist 00:30:04.770 --> 00:30:08.700 here at MIT, once said that, you know, 00:30:08.700 --> 00:30:12.240 either you think that probability theory is beautiful 00:30:12.240 --> 00:30:13.000 or not. 00:30:13.000 --> 00:30:14.583 And if you don't think it's beautiful, 00:30:14.583 --> 00:30:15.940 then I feel sorry for you. 00:30:15.940 --> 00:30:19.020 And I suppose the same can be said for this formula. 00:30:19.020 --> 00:30:22.350 It's hard to believe that a formula can be beautiful, 00:30:22.350 --> 00:30:24.730 but trust me, it is. 00:30:24.730 --> 00:30:27.242 And if you don't think so, I feel sorry for you. 00:30:27.242 --> 00:30:28.950 By the end of the semester, hopefully you 00:30:28.950 --> 00:30:31.050 will think it's beautiful. 00:30:31.050 --> 00:30:34.410 Let me explain what we're about to do. 00:30:34.410 --> 00:30:40.710 I want to come up with the value of a very specific asset, 00:30:40.710 --> 00:30:44.181 an asset with a very, very simple and interesting cash 00:30:44.181 --> 00:30:44.680 flow. 00:30:44.680 --> 00:30:47.230 So this is one of the two special cash flows 00:30:47.230 --> 00:30:49.830 that we're going to analyze in this class. 00:30:49.830 --> 00:30:53.140 And this cash flow is known as a perpetuity. 00:30:53.140 --> 00:30:56.340 A perpetuity is exactly what it sounds like. 00:30:56.340 --> 00:31:01.470 It pays cash forever. 00:31:01.470 --> 00:31:04.970 Now we can debate whether or not forever really exists. 00:31:04.970 --> 00:31:09.680 I won't try to argue with you that we will live forever. 00:31:09.680 --> 00:31:12.330 But it's a hypothetical construct. 00:31:12.330 --> 00:31:12.830 OK? 00:31:12.830 --> 00:31:16.040 So this is a figment of our imaginations. 00:31:16.040 --> 00:31:20.150 There exists in my imagination a piece of paper 00:31:20.150 --> 00:31:22.970 that has a claim, such that whoever 00:31:22.970 --> 00:31:26.150 holds the piece of paper will be entitled 00:31:26.150 --> 00:31:30.570 to a cash payment of C dollars every year 00:31:30.570 --> 00:31:35.174 forever, out to infinity. 00:31:35.174 --> 00:31:35.927 OK? 00:31:35.927 --> 00:31:37.760 And the question is, how much is this worth? 00:31:37.760 --> 00:31:39.980 How much is this piece of paper worth? 00:31:39.980 --> 00:31:42.830 It's an asset, because it's a sequence of cash flows. 00:31:42.830 --> 00:31:45.830 It just turns out that this cash flow is an infinite sequence. 00:31:45.830 --> 00:31:47.420 It never ends. 00:31:47.420 --> 00:31:51.090 It's the gift that keeps on giving. 00:31:51.090 --> 00:31:55.020 So you would think that it should be worth 00:31:55.020 --> 00:31:58.380 an infinite amount, because it pays an infinite amount 00:31:58.380 --> 00:32:00.830 of cash, right? 00:32:00.830 --> 00:32:02.270 No, that's not right. 00:32:02.270 --> 00:32:06.500 And the reason it's not right is because $1 today 00:32:06.500 --> 00:32:11.120 is worth more than $1 tomorrow, which is worth more than $1 00:32:11.120 --> 00:32:13.445 a year from now, which is worth more than $1 00:32:13.445 --> 00:32:14.570 two years from now. 00:32:14.570 --> 00:32:17.570 And so the value of a dollar paid out 00:32:17.570 --> 00:32:20.510 into the far future declines. 00:32:20.510 --> 00:32:23.820 And it turns out that it declines 00:32:23.820 --> 00:32:26.940 at a rate for which you can actually figure out 00:32:26.940 --> 00:32:29.050 what the value is today. 00:32:29.050 --> 00:32:30.630 So here's we're going to do. 00:32:30.630 --> 00:32:35.820 Using the same principle of discounting 00:32:35.820 --> 00:32:40.590 that we did for the previous set of cash flows, 00:32:40.590 --> 00:32:45.390 we're going to take a sequence and discount it. 00:32:45.390 --> 00:32:47.220 I'm assuming with the perpetuity, 00:32:47.220 --> 00:32:49.452 that it starts paying next year. 00:32:49.452 --> 00:32:50.910 So that's the very first cash flow. 00:32:50.910 --> 00:32:52.830 We're sitting here at date zero, and it 00:32:52.830 --> 00:32:56.070 pays C dollars next year, and then another C dollars the year 00:32:56.070 --> 00:32:58.530 after, and then another C dollars the year after that, 00:32:58.530 --> 00:32:59.310 and so on. 00:32:59.310 --> 00:33:03.840 So we're going to dis count them by 1 plus r, 1 plus r squared, 00:33:03.840 --> 00:33:06.630 dot, dot, dot, forever. 00:33:06.630 --> 00:33:09.870 And so this is an infinite sequence. 00:33:09.870 --> 00:33:13.590 And those of you who were on your high school math team, 00:33:13.590 --> 00:33:16.770 you'll know that a quick and dirty way of some summing 00:33:16.770 --> 00:33:19.470 that infinite sequence is basically 00:33:19.470 --> 00:33:22.290 to multiply both sides by 1 plus r. 00:33:22.290 --> 00:33:24.600 And you'll notice that when you do that, 00:33:24.600 --> 00:33:28.410 you get the series back again, but with an extra C. 00:33:28.410 --> 00:33:31.510 And when you do the subtraction and division, 00:33:31.510 --> 00:33:35.040 you end up with this incredibly simple formula 00:33:35.040 --> 00:33:38.670 that says that the present value of this claim that 00:33:38.670 --> 00:33:42.630 pays C dollars forever is not infinite. 00:33:42.630 --> 00:33:44.010 In fact, it's quite finite. 00:33:44.010 --> 00:33:46.350 It's C divided by r. 00:33:46.350 --> 00:33:49.950 What a simple formula. 00:33:49.950 --> 00:33:53.960 If I have a piece of paper that pays $100 a year forever, 00:33:53.960 --> 00:33:56.300 and the interest rate is 10%, what 00:33:56.300 --> 00:33:59.670 is this piece of paper worth? 00:33:59.670 --> 00:34:00.170 Yes? 00:34:00.170 --> 00:34:00.960 AUDIENCE: $1,000. 00:34:00.960 --> 00:34:01.751 ANDREW LO: Exactly. 00:34:01.751 --> 00:34:02.550 $1,000. 00:34:02.550 --> 00:34:05.160 Isn't that amazing, that we could actually value something 00:34:05.160 --> 00:34:07.410 like that? 00:34:07.410 --> 00:34:11.210 If the interest rate is 5%, what is it worth then? 00:34:11.210 --> 00:34:12.090 Yeah. 00:34:12.090 --> 00:34:12.761 $2,000. 00:34:12.761 --> 00:34:13.260 Right. 00:34:13.260 --> 00:34:14.350 Simple. 00:34:14.350 --> 00:34:19.110 We have complete analytical solution for a cash flow 00:34:19.110 --> 00:34:21.449 that, on the surface of it, seems like it should be 00:34:21.449 --> 00:34:23.820 worth a huge amount of money. 00:34:23.820 --> 00:34:26.272 It's not that huge. 00:34:26.272 --> 00:34:26.772 Yeah? 00:34:26.772 --> 00:34:29.085 AUDIENCE: [INAUDIBLE] 00:34:29.085 --> 00:34:29.960 ANDREW LO: Well, yes. 00:34:29.960 --> 00:34:33.199 We're assuming-- assume that interest rates are constant. 00:34:33.199 --> 00:34:34.310 Absolutely. 00:34:34.310 --> 00:34:35.900 So if interest rates vary. 00:34:35.900 --> 00:34:37.040 This formula is not right. 00:34:37.040 --> 00:34:38.780 We're going to come to the case where 00:34:38.780 --> 00:34:40.909 interest rates vary over time. 00:34:40.909 --> 00:34:41.659 So, absolutely. 00:34:41.659 --> 00:34:43.699 This is still under the simplistic assumption 00:34:43.699 --> 00:34:45.560 that interest rates are the same. 00:34:45.560 --> 00:34:47.989 But under that case, I think it's still pretty cool 00:34:47.989 --> 00:34:50.530 that we're able to come up with the formula for value, right? 00:34:50.530 --> 00:34:51.030 Yeah. 00:34:51.030 --> 00:34:56.409 AUDIENCE: [INAUDIBLE] 00:34:56.409 --> 00:34:58.430 ANDREW LO: Well, that's a good question. 00:34:58.430 --> 00:35:00.550 I was afraid you were going to ask that. 00:35:00.550 --> 00:35:01.600 But I am prepared. 00:35:01.600 --> 00:35:03.780 I am prepared to answer that. 00:35:03.780 --> 00:35:06.880 In the United Kingdom, there is a bond 00:35:06.880 --> 00:35:10.720 issued by the government called a console. 00:35:10.720 --> 00:35:13.120 And this bond is a perpetuity. 00:35:13.120 --> 00:35:18.060 That is, it pays to the holder a fixed amount every year 00:35:18.060 --> 00:35:19.430 forever. 00:35:19.430 --> 00:35:21.100 Now in that case, forever means as long 00:35:21.100 --> 00:35:24.950 as the British government is still in existence. 00:35:24.950 --> 00:35:27.660 You know, it's still around. 00:35:27.660 --> 00:35:29.052 But that's an example. 00:35:29.052 --> 00:35:29.969 AUDIENCE: [INAUDIBLE]. 00:35:29.969 --> 00:35:31.093 ANDREW LO: Yes, absolutely. 00:35:31.093 --> 00:35:31.720 It trades. 00:35:31.720 --> 00:35:34.400 You can buy it, sell it, observe the price. 00:35:34.400 --> 00:35:34.970 Absolutely. 00:35:34.970 --> 00:35:37.380 Yeah. 00:35:37.380 --> 00:35:37.880 Yes? 00:35:37.880 --> 00:35:49.760 AUDIENCE: [INAUDIBLE] 00:35:49.760 --> 00:35:50.970 ANDREW LO: Right. 00:35:50.970 --> 00:35:52.320 Good question. 00:35:52.320 --> 00:35:54.560 Where do we get the interest rate? 00:35:54.560 --> 00:35:55.250 The market. 00:35:55.250 --> 00:35:56.660 Exactly. 00:35:56.660 --> 00:36:00.180 So you can either get it from the marketplace-- 00:36:00.180 --> 00:36:01.810 so I have a piece of paper. 00:36:01.810 --> 00:36:04.460 It pays $1 a year forever. 00:36:04.460 --> 00:36:07.490 Who will pay me $5 for this piece of paper. 00:36:07.490 --> 00:36:08.090 $6? 00:36:08.090 --> 00:36:08.830 $7? 00:36:08.830 --> 00:36:11.340 I'll auction it off to the highest bidder, 00:36:11.340 --> 00:36:16.850 and that price will translate into an interest rate 00:36:16.850 --> 00:36:18.494 determined by the marketplace. 00:36:18.494 --> 00:36:19.910 So the short answer is the market. 00:36:19.910 --> 00:36:23.030 Now you're asking me probably a deeper question, which 00:36:23.030 --> 00:36:24.556 is where does that come from? 00:36:24.556 --> 00:36:26.180 Because there are all sorts of factors, 00:36:26.180 --> 00:36:29.870 like future, famine, and plagues, and wars, 00:36:29.870 --> 00:36:32.280 and all these other issues. 00:36:32.280 --> 00:36:35.510 And the answer is, it's an approximation 00:36:35.510 --> 00:36:39.110 that market participants make, and they're 00:36:39.110 --> 00:36:40.330 willing to live with. 00:36:40.330 --> 00:36:41.030 Right? 00:36:41.030 --> 00:36:42.470 I'll give you an example. 00:36:42.470 --> 00:36:47.000 A few years ago, Walt Disney, the entertainment company, 00:36:47.000 --> 00:36:49.840 issued bonds, corporate bonds. 00:36:49.840 --> 00:36:52.830 They were 100 year bonds. 00:36:52.830 --> 00:36:56.760 They were going to mature in 100 years. 00:36:56.760 --> 00:36:58.590 Now, I don't know how many of you 00:36:58.590 --> 00:37:00.810 are high school math team jocks, but if you 00:37:00.810 --> 00:37:05.110 are, one test is to ask the question, 00:37:05.110 --> 00:37:09.430 with this infinite series, if you take it out to 100 terms, 00:37:09.430 --> 00:37:11.470 instead of all the way out to infinity, 00:37:11.470 --> 00:37:15.580 what percentage of the total market value will you capture? 00:37:15.580 --> 00:37:19.270 It turns out that 100 terms is pretty darn close 00:37:19.270 --> 00:37:22.120 to infinite in this grand scheme of things with interest rates 00:37:22.120 --> 00:37:23.350 that we use. 00:37:23.350 --> 00:37:26.440 So that's an example, where when they issued that bond, 00:37:26.440 --> 00:37:29.410 and they auctioned if off to the market participants, whoever 00:37:29.410 --> 00:37:33.220 bought those bonds, whoever the highest bidders were, 00:37:33.220 --> 00:37:34.660 they set the price. 00:37:34.660 --> 00:37:38.320 Once you have the price, you can back out and calculate the r. 00:37:38.320 --> 00:37:39.550 In fact, let me ask you this. 00:37:39.550 --> 00:37:43.200 If I tell you what C is, C is $100, 00:37:43.200 --> 00:37:48.130 and I tell you the market price, say it's $500, 00:37:48.130 --> 00:37:49.630 what's the interest rate? 00:37:49.630 --> 00:37:51.360 Can you figure that out? 00:37:51.360 --> 00:37:51.859 Yeah? 00:37:51.859 --> 00:37:53.190 AUDIENCE: [INAUDIBLE]. 00:37:53.190 --> 00:37:53.940 ANDREW LO: Right. 00:37:53.940 --> 00:37:57.310 Exactly it's basically determined. 00:37:57.310 --> 00:38:00.930 So the market price for an instrument like this 00:38:00.930 --> 00:38:03.930 will give you the market's assessment 00:38:03.930 --> 00:38:06.182 of what that interest rate is. 00:38:06.182 --> 00:38:21.690 AUDIENCE: [INAUDIBLE] 00:38:21.690 --> 00:38:24.060 ANDREW LO: Let me repeat the question in case people 00:38:24.060 --> 00:38:25.020 didn't hear. 00:38:25.020 --> 00:38:29.640 The question is, am I telling you that with all the PhDs 00:38:29.640 --> 00:38:32.910 out there, there is nothing more sophisticated in terms 00:38:32.910 --> 00:38:35.550 of pricing these instruments than simply auctioning them 00:38:35.550 --> 00:38:38.760 off, as we did to a bunch of MBAs? 00:38:38.760 --> 00:38:44.460 Well, first of all, I wouldn't denigrate MBAs that way. 00:38:44.460 --> 00:38:49.170 I would argue that the PhDs who are doing research 00:38:49.170 --> 00:38:53.280 are ultimately advising the MBAs as to what to bid, 00:38:53.280 --> 00:38:55.380 and then the MBAs take into account the business 00:38:55.380 --> 00:38:58.410 considerations, as well as the analytics. 00:38:58.410 --> 00:39:02.670 And so it's actually a highly complex and sophisticated 00:39:02.670 --> 00:39:04.480 process by which the bidding occurs. 00:39:04.480 --> 00:39:06.840 In other words, you're not getting amateurs doing it. 00:39:06.840 --> 00:39:10.020 You're getting professionals who know how to price these things. 00:39:10.020 --> 00:39:12.570 That said, are they going to make mistakes? 00:39:12.570 --> 00:39:13.650 Absolutely. 00:39:13.650 --> 00:39:17.700 So market pricing is an imperfect mechanism. 00:39:17.700 --> 00:39:21.190 But the imperfect mechanism actually works pretty well. 00:39:21.190 --> 00:39:23.190 And so far, nobody else has figured out anything 00:39:23.190 --> 00:39:25.140 that works any better. 00:39:25.140 --> 00:39:27.034 So, yeah? 00:39:27.034 --> 00:39:35.228 AUDIENCE: [INAUDIBLE] price [INAUDIBLE] $1 [INAUDIBLE].. 00:39:38.120 --> 00:39:40.292 Obviously, they're not just issuing one 00:39:40.292 --> 00:39:42.270 to the highest bidder. 00:39:42.270 --> 00:39:44.340 ANDREW LO: So the question is, isn't there 00:39:44.340 --> 00:39:47.430 a problem in terms of the auction if what we're doing 00:39:47.430 --> 00:39:50.131 is determining the price based upon the highest bidder. 00:39:50.131 --> 00:39:52.380 Because the highest bidder is typically the individual 00:39:52.380 --> 00:39:54.720 that's the most confident. 00:39:54.720 --> 00:39:59.180 Or it's possible that that particular bidder knows 00:39:59.180 --> 00:40:01.800 something that the rest of the market doesn't. 00:40:01.800 --> 00:40:04.230 So I don't know which of those two possibilities 00:40:04.230 --> 00:40:06.370 might be the case. 00:40:06.370 --> 00:40:08.166 It depends on the market circumstances. 00:40:08.166 --> 00:40:09.540 One of the things about auctions, 00:40:09.540 --> 00:40:13.080 though, is that the design of the auction 00:40:13.080 --> 00:40:15.690 can actually have a big impact on how informative 00:40:15.690 --> 00:40:16.770 the price is. 00:40:16.770 --> 00:40:21.930 So the standard auction is actually very, very complicated 00:40:21.930 --> 00:40:23.910 in terms of the various incentive effects. 00:40:23.910 --> 00:40:26.490 But there are more intelligent auctions 00:40:26.490 --> 00:40:31.280 that are designed to elicit true responses based upon not just 00:40:31.280 --> 00:40:35.770 kind of anxiousness to win, but on what the economic valuation 00:40:35.770 --> 00:40:36.270 is. 00:40:36.270 --> 00:40:39.510 In fact, there's an example of an auction 00:40:39.510 --> 00:40:42.390 that works something like this. 00:40:42.390 --> 00:40:46.680 You have bidders bidding for a particular commodity. 00:40:46.680 --> 00:40:54.880 And it turns out that the highest bidder wins. 00:40:54.880 --> 00:40:59.110 But the highest bidder will pay a price 00:40:59.110 --> 00:41:01.895 that is the second highest bidder's price. 00:41:04.470 --> 00:41:09.240 So that actually has a very interesting incentive 00:41:09.240 --> 00:41:14.100 in the sense that it ends up forcing you to actually reveal 00:41:14.100 --> 00:41:16.080 your true preferences. 00:41:16.080 --> 00:41:19.350 And we'll come back to that as we talk later on about market 00:41:19.350 --> 00:41:21.137 mechanisms and pricing. 00:41:21.137 --> 00:41:21.636 Yeah? 00:41:21.636 --> 00:41:24.011 AUDIENCE: [INAUDIBLE] mechanisms in auction, for example, 00:41:24.011 --> 00:41:25.194 for public contracts-- 00:41:25.194 --> 00:41:25.860 ANDREW LO: Yeah. 00:41:25.860 --> 00:41:28.170 AUDIENCE: In which they do the average 00:41:28.170 --> 00:41:30.660 and they rule out people who have more than 15% 00:41:30.660 --> 00:41:31.910 deviation from that. 00:41:31.910 --> 00:41:34.456 So it could really go for a very low price. 00:41:34.456 --> 00:41:35.290 ANDREW LO: Yeah. 00:41:35.290 --> 00:41:37.210 AUDIENCE: It's interpretative that you're like, [INAUDIBLE].. 00:41:37.210 --> 00:41:38.460 So you're kicked off the deal. 00:41:38.460 --> 00:41:39.460 ANDREW LO: That's right. 00:41:39.460 --> 00:41:42.150 So there are mechanisms to try to make the auctions smarter. 00:41:42.150 --> 00:41:43.450 And that's one example. 00:41:43.450 --> 00:41:45.210 Another example of that. 00:41:45.210 --> 00:41:48.420 But we're going to assume for now that the auction mechanism 00:41:48.420 --> 00:41:50.520 produces a good price. 00:41:50.520 --> 00:41:52.599 Later on, after we figure out how markets work, 00:41:52.599 --> 00:41:54.390 we're going to come back and question that. 00:41:54.390 --> 00:41:57.590 And the very end of this course, I'm 00:41:57.590 --> 00:42:00.590 going to question all of this and confront you 00:42:00.590 --> 00:42:05.180 with empirical evidence that describes psychological biases 00:42:05.180 --> 00:42:06.830 that all of us have that are hardwired 00:42:06.830 --> 00:42:09.710 into us that would make you think that markets 00:42:09.710 --> 00:42:11.210 don't work well at all. 00:42:11.210 --> 00:42:13.880 And we'll give you a framework for thinking about those two 00:42:13.880 --> 00:42:15.155 kinds of phenomenon. 00:42:15.155 --> 00:42:16.520 Yeah? 00:42:16.520 --> 00:42:20.508 AUDIENCE: I'm just curious to see-- 00:42:20.508 --> 00:42:24.780 [INAUDIBLE] would you have bought this [INAUDIBLE] 00:42:24.780 --> 00:42:26.202 at market price. 00:42:26.202 --> 00:42:31.890 [INAUDIBLE] 00:42:31.890 --> 00:42:32.650 ANDREW LO: OK. 00:42:32.650 --> 00:42:34.670 The question is, do people's bids actually 00:42:34.670 --> 00:42:36.410 reflect interest rates over time? 00:42:36.410 --> 00:42:39.600 Well, remember that market conditions are changing. 00:42:39.600 --> 00:42:43.790 So the question is, do they reflect people's information 00:42:43.790 --> 00:42:45.480 as of when. 00:42:45.480 --> 00:42:48.570 I mean, you never step into the same river twice. 00:42:48.570 --> 00:42:51.730 So what you bought last year at last year's price 00:42:51.730 --> 00:42:54.510 may have no bearing on what you're willing to buy 00:42:54.510 --> 00:42:56.250 at this year's price, right? 00:42:56.250 --> 00:42:57.730 Things change. 00:42:57.730 --> 00:43:00.960 So I'm not sure that that question is well-posed. 00:43:00.960 --> 00:43:03.060 At every point in time, if an individual 00:43:03.060 --> 00:43:08.560 will pay this C divided by r for a security that pays C forever, 00:43:08.560 --> 00:43:09.820 that's the fair market price. 00:43:09.820 --> 00:43:13.390 Now in the future, if interest rates change, 00:43:13.390 --> 00:43:15.300 the price will change. 00:43:15.300 --> 00:43:17.910 But what this does say is a very interesting point 00:43:17.910 --> 00:43:19.530 that I think you're getting to, which 00:43:19.530 --> 00:43:25.260 is that suppose that C never changes by contract. 00:43:25.260 --> 00:43:31.440 If interest rates never change, then this security 00:43:31.440 --> 00:43:34.860 will never change in price. 00:43:34.860 --> 00:43:40.530 It will have absolutely no price growth. 00:43:40.530 --> 00:43:43.560 So here's an example where you buy a piece of paper-- 00:43:43.560 --> 00:43:47.640 let's say the interest rate is 10% and C is $100. 00:43:47.640 --> 00:43:50.980 You pay $1,000 today. 00:43:50.980 --> 00:43:55.310 If next year the interest rate is 10%, this piece of paper's 00:43:55.310 --> 00:43:57.110 still worth $1,000. 00:43:57.110 --> 00:44:00.530 And then five years from now, if he interest rate is 10%, 00:44:00.530 --> 00:44:04.190 the piece of paper's still worth $1,000. 00:44:04.190 --> 00:44:05.820 Does that makes sense? 00:44:05.820 --> 00:44:08.120 Does that seem to suggest that you got stiffed 00:44:08.120 --> 00:44:13.260 because you bought a security and it didn't grow in price? 00:44:13.260 --> 00:44:17.670 In fact, the rate of return on that security is 0. 00:44:17.670 --> 00:44:18.170 Right? 00:44:21.600 --> 00:44:23.560 AUDIENCE: [INAUDIBLE]. 00:44:23.560 --> 00:44:25.630 ANDREW LO: Or I mean, a $100 payment. 00:44:25.630 --> 00:44:27.570 AUDIENCE: You have one coupon payment plus-- 00:44:27.570 --> 00:44:28.680 ANDREW LO: Every year. 00:44:28.680 --> 00:44:30.090 Right, exactly . 00:44:30.090 --> 00:44:33.890 So it's wrong that the return is zero. 00:44:33.890 --> 00:44:35.600 The price return is zero. 00:44:35.600 --> 00:44:37.700 There's no price growth. 00:44:37.700 --> 00:44:41.790 But meanwhile every year, you've been getting checks for $100. 00:44:41.790 --> 00:44:45.440 And if the piece of paper was $1,000 00:44:45.440 --> 00:44:49.430 and you've been getting checks for $100 every year, 00:44:49.430 --> 00:44:50.924 what's your annual return? 00:44:53.610 --> 00:44:54.670 10%. 00:44:54.670 --> 00:44:56.520 What's the interest rate? 00:44:56.520 --> 00:44:58.530 Oh, funny how that works, huh? 00:44:58.530 --> 00:44:59.610 That's great. 00:44:59.610 --> 00:45:01.680 You get a 10% return. 00:45:01.680 --> 00:45:02.760 Why? 00:45:02.760 --> 00:45:04.890 Because you're holding this piece of paper 00:45:04.890 --> 00:45:09.510 that generates coupons, and the coupons 00:45:09.510 --> 00:45:12.130 end up giving you a 10% rate of return, 00:45:12.130 --> 00:45:14.250 because the price of the security 00:45:14.250 --> 00:45:19.630 is those coupons discounted at 10%. 00:45:19.630 --> 00:45:20.680 Nothing magic about it. 00:45:20.680 --> 00:45:21.310 It all adds up. 00:45:21.310 --> 00:45:22.870 It all works together. 00:45:22.870 --> 00:45:25.032 OK? 00:45:25.032 --> 00:45:25.978 Yes? 00:45:25.978 --> 00:45:31.227 AUDIENCE: [INAUDIBLE] for example-- 00:45:31.227 --> 00:45:32.810 ANDREW LO: We're going to get to that. 00:45:32.810 --> 00:45:33.710 Yes, we're going to get to that. 00:45:33.710 --> 00:45:35.100 That's my next example. 00:45:35.100 --> 00:45:36.320 Let me hold off on that. 00:45:36.320 --> 00:45:37.360 I want to make sure everybody understands 00:45:37.360 --> 00:45:38.690 the perpetuity though. 00:45:38.690 --> 00:45:42.590 And then we're going to get to the example where C changes. 00:45:42.590 --> 00:45:46.410 Now to your example, what happens if C changes. 00:45:46.410 --> 00:45:52.100 In fact, let's be optimistic and let's say that C grows. 00:45:52.100 --> 00:45:55.010 So not only am I going to pay you something 00:45:55.010 --> 00:45:58.160 forever, but that something, I'm going 00:45:58.160 --> 00:46:02.310 to let that grow by a rate of growth of g. 00:46:02.310 --> 00:46:07.150 So next year, I pay you C. But the year after, I'm 00:46:07.150 --> 00:46:10.900 going to pay you C, multiplied by 1 plus g. 00:46:10.900 --> 00:46:14.670 So let's say g is 5%. 00:46:14.670 --> 00:46:17.090 Then next year, I pay you $100. 00:46:17.090 --> 00:46:20.000 The year after, I pay you $105. 00:46:20.000 --> 00:46:22.130 And the year after that, I'll pay you 00:46:22.130 --> 00:46:29.150 whatever 1.05 squared is times 100, and so on. 00:46:29.150 --> 00:46:33.110 Now, what is this piece of paper worth? 00:46:33.110 --> 00:46:36.160 And if you do the same kind of high school math team 00:46:36.160 --> 00:46:43.000 trick and solve for the present value, you get an answer, 00:46:43.000 --> 00:46:49.040 PV is equal to C divided by r minus g. 00:46:49.040 --> 00:46:51.100 r minus g. 00:46:51.100 --> 00:46:53.000 So you subtract this growth rate. 00:46:53.000 --> 00:46:57.250 Now when you subtract the growth rate, 00:46:57.250 --> 00:46:59.990 that makes the denominator smaller, 00:46:59.990 --> 00:47:02.590 which makes the whole thing bigger, 00:47:02.590 --> 00:47:05.380 which is the right direction because you're 00:47:05.380 --> 00:47:09.010 getting a cash flow that is not steady over time, 00:47:09.010 --> 00:47:10.300 but it's growing over time. 00:47:10.300 --> 00:47:12.670 So it should be worth more. 00:47:12.670 --> 00:47:16.480 And it's worth r minus g more. 00:47:16.480 --> 00:47:19.490 All right? 00:47:19.490 --> 00:47:22.130 Now you notice, I have a little condition at the end of that. 00:47:22.130 --> 00:47:25.220 r has to be greater than g. 00:47:25.220 --> 00:47:27.669 Why do I have that condition? 00:47:27.669 --> 00:47:29.028 Yeah? 00:47:29.028 --> 00:47:33.370 AUDIENCE: [INAUDIBLE] infinite [INAUDIBLE] 00:47:33.370 --> 00:47:36.900 the infinite [INAUDIBLE]. 00:47:36.900 --> 00:47:37.900 ANDREW LO: That's right. 00:47:37.900 --> 00:47:39.550 So let's suppose that r equals g. 00:47:39.550 --> 00:47:40.870 Let's see what happens. 00:47:40.870 --> 00:47:47.440 If r equals g, then the infinite series on top, c divided by 1 00:47:47.440 --> 00:47:53.880 plus r plus C times 1 plus g over 1 plus r squared, 00:47:53.880 --> 00:47:56.670 that's just C over 1 plus r, because I'm assuming g and r 00:47:56.670 --> 00:47:58.890 are the same. 00:47:58.890 --> 00:48:03.210 Plus C over 1 plus r, plus C over 1 plus r, plus C over 1 00:48:03.210 --> 00:48:04.870 plus r. 00:48:04.870 --> 00:48:07.780 I have an infinite number of C over 1 plus r. 00:48:07.780 --> 00:48:10.520 And C over 1 plus r is a finite constant. 00:48:13.640 --> 00:48:17.620 The sum is infinite. 00:48:17.620 --> 00:48:19.900 So at some point, that's going to exceed 00:48:19.900 --> 00:48:23.660 total world GDP, and then beyond it, and then 00:48:23.660 --> 00:48:28.020 the other planets of the solar system, and so on. 00:48:28.020 --> 00:48:29.460 What's going on here? 00:48:29.460 --> 00:48:30.725 Why is it happening? 00:48:33.340 --> 00:48:35.889 Anybody give me the intuition for what's happening? 00:48:35.889 --> 00:48:41.268 AUDIENCE: Because the numbers are going smaller and smaller 00:48:41.268 --> 00:48:45.670 [INAUDIBLE] 00:48:45.670 --> 00:48:46.404 ANDREW LO: Right. 00:48:46.404 --> 00:48:49.652 AUDIENCE: But compared just to zero, the amount of 00:48:49.652 --> 00:48:51.510 [INAUDIBLE]. 00:48:51.510 --> 00:48:52.320 ANDREW LO: Right. 00:48:52.320 --> 00:48:52.820 Yes. 00:48:52.820 --> 00:48:54.060 AUDIENCE: [INAUDIBLE]. 00:48:54.060 --> 00:48:55.140 ANDREW LO: Yeah. 00:48:55.140 --> 00:48:55.740 That's right. 00:48:55.740 --> 00:48:56.940 It's growing. 00:48:56.940 --> 00:49:00.210 But what's the intuition for why that can't persist? 00:49:00.210 --> 00:49:02.118 AUDIENCE: Sounds like you're [INAUDIBLE] 00:49:02.118 --> 00:49:03.550 the 10,000 [? quantity. ?] 00:49:03.550 --> 00:49:05.699 ANDREW LO: Right. 00:49:05.699 --> 00:49:06.240 That's right. 00:49:06.240 --> 00:49:09.240 It's basically working against the time value of money 00:49:09.240 --> 00:49:12.510 because the numerator is growing as fast as the denominator is 00:49:12.510 --> 00:49:13.840 growing. 00:49:13.840 --> 00:49:16.740 So what it says is that the cash that you're presumably 00:49:16.740 --> 00:49:19.260 going to be paying to somebody is actually 00:49:19.260 --> 00:49:21.930 increasing at the exact same rate 00:49:21.930 --> 00:49:25.650 that the discount rate is growing. 00:49:25.650 --> 00:49:28.750 So there's no way to sustain that forever. 00:49:28.750 --> 00:49:31.320 You can't do that forever. 00:49:31.320 --> 00:49:35.520 So it has to be the case that the amount that the cash is 00:49:35.520 --> 00:49:41.470 growing can never exceed the discount rate. 00:49:41.470 --> 00:49:43.590 Now remember, these are all theoretical concepts 00:49:43.590 --> 00:49:46.590 where I'm assuming that growth rate stays the same forever, 00:49:46.590 --> 00:49:49.430 and the interest rate stays the same forever. 00:49:49.430 --> 00:49:54.050 This doesn't rule out for short periods of time growth rates 00:49:54.050 --> 00:49:55.970 exceeding interest rates. 00:49:55.970 --> 00:49:58.370 You just can't do it forever. 00:49:58.370 --> 00:50:02.030 For the last 15 years, China has been growing 00:50:02.030 --> 00:50:04.130 at a rate of approximately 10%. 00:50:04.130 --> 00:50:08.330 Their entire economy has been growing at 10% a year 00:50:08.330 --> 00:50:13.520 for every single year over the past 15 years. 00:50:13.520 --> 00:50:15.690 That can't persist. 00:50:15.690 --> 00:50:19.800 If it did, not only would we all be speaking Chinese, 00:50:19.800 --> 00:50:23.130 but all of the planets in this entire galaxy 00:50:23.130 --> 00:50:25.350 would end up speaking Chinese. 00:50:25.350 --> 00:50:28.650 I mean, growth rates cannot persist forever. 00:50:28.650 --> 00:50:31.860 But here, we're assuming, we're assuming, that this growth rate 00:50:31.860 --> 00:50:34.030 is an infinite growth rate. 00:50:34.030 --> 00:50:35.920 It applies forever. 00:50:35.920 --> 00:50:40.610 So in that sense, it has to be smaller than the discount rate. 00:50:40.610 --> 00:50:41.870 Question? 00:50:41.870 --> 00:50:44.046 OK. 00:50:44.046 --> 00:50:58.380 AUDIENCE: [INAUDIBLE] rest of the world. 00:50:58.380 --> 00:51:01.960 ANDREW LO: Well, there are a couple of problems with that. 00:51:01.960 --> 00:51:05.970 So the question is, what happens when r is actually less than g? 00:51:05.970 --> 00:51:06.691 Right? 00:51:06.691 --> 00:51:08.940 You would think that that gives you a negative number. 00:51:08.940 --> 00:51:11.640 In fact, it doesn't, because there's 00:51:11.640 --> 00:51:13.800 a discontinuity at zero. 00:51:13.800 --> 00:51:18.090 And so this formula is-- that doesn't even apply. 00:51:18.090 --> 00:51:23.020 What happens, if you do the infinite sum, when 00:51:23.020 --> 00:51:28.670 g approaches r, this infinite sum already goes to infinity. 00:51:28.670 --> 00:51:33.890 When g gets above r, it gets to be even more infinite, 00:51:33.890 --> 00:51:36.660 whatever that means. 00:51:36.660 --> 00:51:37.160 Right? 00:51:37.160 --> 00:51:40.010 Because the numerator is then not growing at the same rate, 00:51:40.010 --> 00:51:42.900 but it's growing at a faster rate than the denominator. 00:51:42.900 --> 00:51:44.450 So the formula, you wouldn't even 00:51:44.450 --> 00:51:47.720 get the formula, because now you're dealing with infinities. 00:51:47.720 --> 00:51:48.866 OK? 00:51:48.866 --> 00:51:50.740 AUDIENCE: [INAUDIBLE]. 00:51:50.740 --> 00:51:52.670 ANDREW LO: Right. 00:51:52.670 --> 00:51:53.370 Right. 00:51:53.370 --> 00:51:55.164 AUDIENCE: [INAUDIBLE]. 00:51:55.164 --> 00:51:57.080 ANDREW LO: It would just be an infinite value, 00:51:57.080 --> 00:52:00.330 but an even bigger infinity, whatever that means. 00:52:00.330 --> 00:52:04.340 And so, this formula really only holds under this condition. 00:52:04.340 --> 00:52:08.130 If it were equal to or negative, this formula 00:52:08.130 --> 00:52:09.380 just would not be appropriate. 00:52:09.380 --> 00:52:11.120 You'd have to go back to that formula. 00:52:11.120 --> 00:52:12.620 And what that formula would show you 00:52:12.620 --> 00:52:14.831 is that you're getting an infinity. 00:52:17.130 --> 00:52:17.630 OK? 00:52:17.630 --> 00:52:18.520 So that's a perpetuity. 00:52:18.520 --> 00:52:20.186 And we're going to use this, by the way. 00:52:20.186 --> 00:52:21.722 This may seem kind of theoretical. 00:52:21.722 --> 00:52:23.180 But trust me, it's going to come in 00:52:23.180 --> 00:52:26.520 very handy when we start pricing bonds and stocks. 00:52:26.520 --> 00:52:30.520 So we're going to use this quite a bit. 00:52:30.520 --> 00:52:34.060 Now I want to tell you about a formula that 00:52:34.060 --> 00:52:37.420 is my second favorite formula in this entire course. 00:52:37.420 --> 00:52:39.970 And in a way, this is much more practical, 00:52:39.970 --> 00:52:41.980 and it's very closely related to the perpetuity. 00:52:41.980 --> 00:52:46.360 This formula is a formula for an annuity. 00:52:46.360 --> 00:52:51.190 An annuity is a security that pays a fixed amount every year 00:52:51.190 --> 00:52:54.190 for a finite number of years, and then it stops paying. 00:52:54.190 --> 00:52:56.950 So an example of an annuity is a bond. 00:52:56.950 --> 00:52:58.660 Another example is an auto loan. 00:52:58.660 --> 00:53:00.320 Another example is a mortgage. 00:53:00.320 --> 00:53:04.600 And I think I told you that this mortgage valuation formula is 00:53:04.600 --> 00:53:06.100 one that you're going to use when 00:53:06.100 --> 00:53:08.540 you start thinking about making a home purchase decision. 00:53:08.540 --> 00:53:10.090 And it will actually be this formula 00:53:10.090 --> 00:53:12.280 exactly that you're going to need to use. 00:53:12.280 --> 00:53:13.048 Question? 00:53:13.048 --> 00:53:15.340 AUDIENCE: [INAUDIBLE]. 00:53:15.340 --> 00:53:16.150 ANDREW LO: Yes. 00:53:16.150 --> 00:53:17.620 AUDIENCE: Just one question. 00:53:17.620 --> 00:53:20.944 The principle is returned within these payments, or at the end? 00:53:20.944 --> 00:53:22.610 ANDREW LO: Let me talk about that later. 00:53:22.610 --> 00:53:24.250 Right now, we don't know what principle is. 00:53:24.250 --> 00:53:26.500 So when I talk about bonds, I'm going to come back to that. 00:53:26.500 --> 00:53:27.110 OK? 00:53:27.110 --> 00:53:28.610 So let's not get ahead of ourselves. 00:53:28.610 --> 00:53:30.190 I want to make sure we understand the formula 00:53:30.190 --> 00:53:30.970 and then I'll come to that. 00:53:30.970 --> 00:53:32.590 That's an important point that we're 00:53:32.590 --> 00:53:35.008 going to get to in about a lecture and a half. 00:53:35.008 --> 00:53:36.910 OK? 00:53:36.910 --> 00:53:37.410 OK. 00:53:37.410 --> 00:53:41.460 So let me explain what a perpetuity and an annuity 00:53:41.460 --> 00:53:43.410 are in relationship to each other. 00:53:43.410 --> 00:53:47.260 A perpetuity pays a fixed amount forever. 00:53:47.260 --> 00:53:53.560 An annuity pays a fixed amount for a finite period of time. 00:53:53.560 --> 00:53:58.550 So there's a relationship between the two. 00:53:58.550 --> 00:54:03.080 And in particular, you can think about the value 00:54:03.080 --> 00:54:10.740 of an annuity as the value of a perpetuity 00:54:10.740 --> 00:54:15.270 where you only get to have it for a finite period of time. 00:54:15.270 --> 00:54:16.290 Right? 00:54:16.290 --> 00:54:18.270 Let me explain. 00:54:18.270 --> 00:54:21.360 An annuity, the value of that, is 00:54:21.360 --> 00:54:23.400 given by the expression on the top line. 00:54:23.400 --> 00:54:24.090 Right? 00:54:24.090 --> 00:54:28.230 C, C, C, C for T periods, discounted 00:54:28.230 --> 00:54:30.670 at the appropriate discount rate. 00:54:33.610 --> 00:54:36.180 Now, it turns out that you can come up 00:54:36.180 --> 00:54:39.570 with an expression for what that present value is, again, 00:54:39.570 --> 00:54:43.140 using the high school math team kind of an approach. 00:54:43.140 --> 00:54:46.170 You simply multiply both sides by 1 plus r, 00:54:46.170 --> 00:54:50.310 and then you solve for the present value, 00:54:50.310 --> 00:54:55.640 and you get an expression that looks like this. 00:54:55.640 --> 00:54:58.820 Well, this looks an awful lot like it's related 00:54:58.820 --> 00:55:01.280 to the perpetuity formula. 00:55:01.280 --> 00:55:03.290 You've got to C over r here, but then there's 00:55:03.290 --> 00:55:05.690 some annoying other terms over here. 00:55:09.320 --> 00:55:12.430 So let me give you a thought experiment that will show you 00:55:12.430 --> 00:55:16.300 how to derive this formula in less than one minute 00:55:16.300 --> 00:55:19.320 without any kind of high school math team tricks. 00:55:19.320 --> 00:55:22.500 And the experiment goes like this. 00:55:25.390 --> 00:55:29.590 Suppose that you want to create an annuity, 00:55:29.590 --> 00:55:33.910 but you don't have an annuity at hand. 00:55:33.910 --> 00:55:42.380 Well, one way you can do it is to buy a perpetuity, 00:55:42.380 --> 00:55:47.200 hold it for T periods, and then get rid of it and sell it. 00:55:51.800 --> 00:55:54.500 Now look at the cash flows that you get. 00:55:54.500 --> 00:55:58.800 If you were to take a perpetuity, 00:55:58.800 --> 00:56:02.820 which is the top cash flow, and you 00:56:02.820 --> 00:56:07.830 would subtract from it a perpetuity as of date 00:56:07.830 --> 00:56:10.170 T plus 1-- so you've gotten rid of the perpetuity 00:56:10.170 --> 00:56:12.040 at this point. 00:56:12.040 --> 00:56:15.100 When you take the top cash flow sequence 00:56:15.100 --> 00:56:18.910 and you subtract from it the next cash flow sequence, 00:56:18.910 --> 00:56:21.760 you get the bottom cash flow sequence, 00:56:21.760 --> 00:56:25.370 which is just an annuity. 00:56:25.370 --> 00:56:26.300 Right? 00:56:26.300 --> 00:56:32.340 So an annuity is a perpetuity on borrowed time. 00:56:32.340 --> 00:56:33.630 So what is it worth? 00:56:33.630 --> 00:56:41.080 Well, it's worth whatever it is to buy a perpetuity, 00:56:41.080 --> 00:56:45.250 hold it for T periods, and as soon as it pays off 00:56:45.250 --> 00:56:50.000 in that Tth date, you sell it. 00:56:50.000 --> 00:56:51.950 OK? 00:56:51.950 --> 00:56:54.740 So what's it going to cost? 00:56:54.740 --> 00:56:57.080 What's the value of that? 00:56:57.080 --> 00:57:03.600 The value of that is this is what it costs 00:57:03.600 --> 00:57:07.000 to purchase the annuity today-- 00:57:07.000 --> 00:57:09.131 the perpetuity, sorry. 00:57:09.131 --> 00:57:09.630 Right? 00:57:09.630 --> 00:57:10.620 C over r. 00:57:10.620 --> 00:57:14.640 That's what it costs to purchase the perpetuity today. 00:57:14.640 --> 00:57:18.530 And you're going to hold on to that perpetuity for T days or T 00:57:18.530 --> 00:57:20.180 periods. 00:57:20.180 --> 00:57:26.870 And at date T plus 1, you're going to sell the perpetuity. 00:57:26.870 --> 00:57:29.230 What are you going to get when you sell it? 00:57:29.230 --> 00:57:33.780 What would you get as the payment? 00:57:33.780 --> 00:57:34.519 C over r. 00:57:34.519 --> 00:57:36.810 That's right, because that's the price of a perpetuity. 00:57:36.810 --> 00:57:38.260 The price never changes. 00:57:38.260 --> 00:57:40.080 It's always C over r. 00:57:40.080 --> 00:57:43.950 When do you get paid that price? 00:57:43.950 --> 00:57:45.820 At T or T plus 1? 00:57:45.820 --> 00:57:46.800 AUDIENCE: T plus 1. 00:57:46.800 --> 00:57:49.720 ANDREW LO: Because I want to have T periods a cash flow. 00:57:49.720 --> 00:57:52.080 So I've got to hold onto that perpetuity at least 00:57:52.080 --> 00:57:53.430 until T periods. 00:57:53.430 --> 00:57:55.860 After the Tth date, I sell it, which 00:57:55.860 --> 00:57:59.680 means I sell at the next date, which is T plus 1. 00:57:59.680 --> 00:58:06.880 And so I get paid a cash flow of c over r at day T plus 1. 00:58:06.880 --> 00:58:08.582 What is that cash flow worth today? 00:58:12.772 --> 00:58:14.730 Remember, it's at two different points in time. 00:58:14.730 --> 00:58:16.604 I need to use the exchange rate to convert it 00:58:16.604 --> 00:58:17.910 to the same currency. 00:58:17.910 --> 00:58:23.050 What's the exchange rate between date 0 and date t plus 1? 00:58:23.050 --> 00:58:24.803 Yeah? [? Scholmi? ?] 00:58:24.803 --> 00:58:29.720 AUDIENCE: [INAUDIBLE] 00:58:29.720 --> 00:58:31.826 ANDREW LO: By t, or by t plus 1? 00:58:31.826 --> 00:58:33.110 AUDIENCE: By t. 00:58:33.110 --> 00:58:34.310 ANDREW LO: No. 00:58:34.310 --> 00:58:37.000 Close, but no cigar. 00:58:37.000 --> 00:58:37.880 AUDIENCE: t plus 1. 00:58:37.880 --> 00:58:38.360 ANDREW LO: Why t plus 1? 00:58:38.360 --> 00:58:39.830 AUDIENCE: That's the period where you're getting paid. 00:58:39.830 --> 00:58:41.400 ANDREW LO: That's the period where you're getting paid. 00:58:41.400 --> 00:58:42.109 So let's go back. 00:58:42.109 --> 00:58:43.650 And remember, the first thing you do? 00:58:43.650 --> 00:58:44.360 Draw a time line. 00:58:44.360 --> 00:58:45.230 Right? 00:58:45.230 --> 00:58:46.495 So here's the timeline. 00:58:46.495 --> 00:58:48.290 And you see why it's confusing. 00:58:48.290 --> 00:58:51.050 You know, I don't blame you for thinking it's t, 00:58:51.050 --> 00:58:52.754 because I said two periods and you're 00:58:52.754 --> 00:58:54.170 going to sell it after two periods 00:58:54.170 --> 00:58:56.780 but when I say sell it after two periods 00:58:56.780 --> 00:59:00.310 if it's after two periods it's plus 1. 00:59:00.310 --> 00:59:02.060 So take a look at this diagram, and you've 00:59:02.060 --> 00:59:03.200 got to draw the diagram. 00:59:03.200 --> 00:59:05.930 You've got to draw the diagram to really get this. 00:59:05.930 --> 00:59:07.160 OK? 00:59:07.160 --> 00:59:10.620 The top part is a perpetuity. 00:59:10.620 --> 00:59:15.650 The middle part is that same perpetuity at day T plus 1. 00:59:15.650 --> 00:59:21.860 So if you own the top piece, and at the same time 00:59:21.860 --> 00:59:26.330 you sell the middle piece, that means at time T plus 1, 00:59:26.330 --> 00:59:29.020 you're going to give up all of your future cash flows 00:59:29.020 --> 00:59:31.160 because you're going to sell the perpetuity. 00:59:31.160 --> 00:59:39.150 Then you're left with the actual annuity cash flow that we want. 00:59:39.150 --> 00:59:42.740 So the question is, what does this transaction cost? 00:59:42.740 --> 00:59:45.500 I buy that it's going to cost me c over r. 00:59:45.500 --> 00:59:47.720 I sell this. 00:59:47.720 --> 00:59:50.470 This is a sequence of cash flows. 00:59:50.470 --> 00:59:52.320 So if I'm selling a sequence of cash flows, 00:59:52.320 --> 00:59:54.030 I'm selling that value. 00:59:54.030 --> 00:59:56.920 I'm going to receive that value as payment. 00:59:56.920 --> 01:00:01.200 So it's going to reduce my cost, and so like any other sequence 01:00:01.200 --> 01:00:04.290 of cash flows, when I sell this, I have to value it, 01:00:04.290 --> 01:00:07.620 and it turns out that this is equal to the value 01:00:07.620 --> 01:00:11.760 at this date, the value of the perpetuity at this date. 01:00:11.760 --> 01:00:14.850 And what is that value? 01:00:14.850 --> 01:00:16.680 C over r. 01:00:16.680 --> 01:00:20.150 And if it's C over r at this date, what 01:00:20.150 --> 01:00:22.460 is the value at this date? 01:00:22.460 --> 01:00:27.680 I've got to discount it by 1 plus r to the T plus 1, 01:00:27.680 --> 01:00:34.400 because it's T plus 1 periods going back. 01:00:34.400 --> 01:00:37.352 OK? 01:00:37.352 --> 01:00:39.110 Well, actually, sorry. 01:00:39.110 --> 01:00:44.560 T periods, the convention is a little confusing. 01:00:44.560 --> 01:00:48.290 It's T periods because you're at t plus 1, 01:00:48.290 --> 01:00:51.340 and you want to figure out what the value is. 01:00:51.340 --> 01:00:55.600 And the value of the perpetuity at T 01:00:55.600 --> 01:00:59.440 is a perpetuity that starts paying off at T plus 1. 01:00:59.440 --> 01:01:01.170 So you're right. 01:01:01.170 --> 01:01:03.580 It's T, but you're discounting it 01:01:03.580 --> 01:01:07.260 as of the payment as of T plus 1. 01:01:07.260 --> 01:01:07.760 OK? 01:01:10.580 --> 01:01:12.850 How many people are confused? 01:01:12.850 --> 01:01:14.420 OK. 01:01:14.420 --> 01:01:15.574 Yes. 01:01:15.574 --> 01:01:16.490 AUDIENCE: [INAUDIBLE]. 01:01:16.490 --> 01:01:17.660 ANDREW LO: Let me-- 01:01:17.660 --> 01:01:19.110 let me do this on the board. 01:01:19.110 --> 01:01:19.610 Right. 01:01:19.610 --> 01:01:20.109 Exactly. 01:01:20.109 --> 01:01:23.060 Let me do this on the board, because the notation is 01:01:23.060 --> 01:01:23.670 confusing. 01:01:23.670 --> 01:01:25.481 Let me just switch on the light. 01:01:25.481 --> 01:01:25.981 Whoops. 01:01:29.377 --> 01:01:29.877 OK. 01:01:35.730 --> 01:01:38.270 So we're going to start by assuming that we've 01:01:38.270 --> 01:01:40.580 got a perpetuity at date 0. 01:01:40.580 --> 01:01:42.020 So this is date 0. 01:01:42.020 --> 01:01:43.970 And remember, the definition of perpetuity 01:01:43.970 --> 01:01:46.340 is that it starts paying the next period. 01:01:48.890 --> 01:01:55.870 And so it pays C,C until this date. 01:01:55.870 --> 01:01:57.875 And sorry. 01:01:57.875 --> 01:02:06.220 T Plus 1 pays T plus 2, and so on. 01:02:06.220 --> 01:02:08.380 The annuity that we want to value 01:02:08.380 --> 01:02:14.830 is an annuity that is just consisting of the first T cash 01:02:14.830 --> 01:02:16.070 flows. 01:02:16.070 --> 01:02:18.790 Right? 01:02:18.790 --> 01:02:22.960 So what I claim is that if you engage 01:02:22.960 --> 01:02:33.110 in the following transaction, at date 0, you buy a perpetuity, 01:02:33.110 --> 01:02:40.100 and you also agree to sell that perpetuity at date 01:02:40.100 --> 01:02:48.540 after date T. So you sell after T. What that means 01:02:48.540 --> 01:02:50.940 is that you will hold onto the perpetuity 01:02:50.940 --> 01:02:55.080 until it pays you C dollars. 01:02:55.080 --> 01:02:58.780 And as soon as it does that, after it does that, 01:02:58.780 --> 01:03:00.980 you sell it. 01:03:00.980 --> 01:03:03.170 Now when you sell it, what do you get for it? 01:03:03.170 --> 01:03:04.670 You get C over r. 01:03:04.670 --> 01:03:08.150 But the question is, when do you get that C over r? 01:03:08.150 --> 01:03:11.690 If you have a sequence of cash flows 01:03:11.690 --> 01:03:19.970 that starts in year T plus 1, then the value of it at day T 01:03:19.970 --> 01:03:21.620 is C over r right? 01:03:21.620 --> 01:03:25.190 Because a perpetuity by assumption is a piece of paper 01:03:25.190 --> 01:03:30.560 that starts paying off not today, but next year. 01:03:30.560 --> 01:03:34.390 So if it starts paying off next year, 01:03:34.390 --> 01:03:36.340 for every single year thereafter, 01:03:36.340 --> 01:03:42.400 the value at that point is going to be equal to C over r. 01:03:42.400 --> 01:03:45.020 Any questions about that? 01:03:45.020 --> 01:03:45.610 OK . 01:03:45.610 --> 01:03:48.580 So we've now established that when 01:03:48.580 --> 01:03:53.440 you sell these cash flows going out into the infinite future, 01:03:53.440 --> 01:04:01.260 the value at date T is C over r. 01:04:05.090 --> 01:04:07.280 And therefore, if the value at date 01:04:07.280 --> 01:04:12.560 T is C over r, what is the value of date 0? 01:04:12.560 --> 01:04:15.320 You have to bring it back to date 0. 01:04:15.320 --> 01:04:19.080 You're discounting it by T periods. 01:04:19.080 --> 01:04:25.160 So it's C over r times 1 over 1 plus r to the t. 01:04:25.160 --> 01:04:29.060 That's what you get when you sell this perpetuity. 01:04:29.060 --> 01:04:32.360 And what you paid for it is C over r. 01:04:32.360 --> 01:04:36.560 So the value of this particular set of actions 01:04:36.560 --> 01:04:42.260 that you've engaged in is C over r minus C over r times 1 over 1 01:04:42.260 --> 01:04:47.350 plus r to the T. 01:04:47.350 --> 01:04:51.430 That's the annuity discount formula in a nutshell. 01:04:51.430 --> 01:04:55.930 And this formula is the basis of how you figure out 01:04:55.930 --> 01:04:57.610 your mortgage payments. 01:04:57.610 --> 01:05:03.130 Because a mortgage is where you have an obligation every month 01:05:03.130 --> 01:05:06.820 to pay something to the bank in exchange for a pile of money, 01:05:06.820 --> 01:05:09.080 the money that you used to buy your house. 01:05:09.080 --> 01:05:11.530 And the first time I was buying my, house I 01:05:11.530 --> 01:05:13.626 actually went through this transaction. 01:05:13.626 --> 01:05:16.000 I decided that I was going to just calculate this myself, 01:05:16.000 --> 01:05:17.920 because the interest rate was not 01:05:17.920 --> 01:05:23.374 exactly given by what was in the particular banker's table. 01:05:23.374 --> 01:05:24.790 So I went to the mortgage company. 01:05:24.790 --> 01:05:26.600 It was a bank. 01:05:26.600 --> 01:05:29.080 And I think the interest rate that day 01:05:29.080 --> 01:05:32.410 was something like, I don't know, 8%, 8 and 1/2%, 01:05:32.410 --> 01:05:34.750 or 8 and 3/4%. 01:05:34.750 --> 01:05:37.750 And it turns out that the table, this book, 01:05:37.750 --> 01:05:39.950 that had all of these calculations, all 01:05:39.950 --> 01:05:43.090 of these numbers, didn't have that interest rate. 01:05:43.090 --> 01:05:44.320 It didn't have 8 and 3/4. 01:05:44.320 --> 01:05:48.400 It had 8 and 1/2 or and 9, but it didn't have 8 and 3/4. 01:05:48.400 --> 01:05:51.820 And so I just used this formula, punched in a few numbers, 01:05:51.820 --> 01:05:53.620 and I got my monthly payment. 01:05:53.620 --> 01:05:55.810 And you know, I told the banker, well, you know, 01:05:55.810 --> 01:05:57.940 this is what I'll pay every month. 01:05:57.940 --> 01:06:01.349 And he said, well, you can't just do that. 01:06:01.349 --> 01:06:02.640 I said, well, what do you mean? 01:06:02.640 --> 01:06:04.270 And he says, well, you know, I don't know 01:06:04.270 --> 01:06:05.478 that that's the right number. 01:06:05.478 --> 01:06:08.200 We have to wait for the senior vice president 01:06:08.200 --> 01:06:10.089 to tell me what the right number is. 01:06:10.089 --> 01:06:11.380 Because we don't have the book. 01:06:11.380 --> 01:06:12.960 And he contacted the senior vice president. 01:06:12.960 --> 01:06:14.709 It turned out he did have the book either. 01:06:14.709 --> 01:06:16.900 So they had to call the main branch, 01:06:16.900 --> 01:06:19.900 and somebody had to look it up in this book. 01:06:19.900 --> 01:06:23.350 And sure enough, when they came back with the number, 01:06:23.350 --> 01:06:26.930 it was my number down to the fourth decimal place. 01:06:26.930 --> 01:06:29.890 And so he was amazed like, wow, how did you do that? 01:06:29.890 --> 01:06:31.760 You know, this is amazing. 01:06:31.760 --> 01:06:33.620 You're incredible. 01:06:33.620 --> 01:06:37.960 It's incredible if you don't know this very basic secret. 01:06:37.960 --> 01:06:39.430 So you're going to do this. 01:06:39.430 --> 01:06:40.400 You're going to do this in the problems. 01:06:40.400 --> 01:06:42.649 You're going to calculate mortgage payments, auto loan 01:06:42.649 --> 01:06:44.170 payments, consumer finance payments. 01:06:44.170 --> 01:06:46.930 All of it is based upon this simple formula. 01:06:46.930 --> 01:06:52.990 And you can construct tables, as people have done, 01:06:52.990 --> 01:06:56.480 of what are called annuity discount factors. 01:06:56.480 --> 01:06:58.420 So the annuity discount factor is simply 01:06:58.420 --> 01:07:02.180 separating the interest rate from the cash flow. 01:07:02.180 --> 01:07:05.950 And so when you're going out for a mortgage, the amount 01:07:05.950 --> 01:07:07.630 that you're borrowing, you borrow 01:07:07.630 --> 01:07:12.010 $200,000 for your house, that's the left-hand side. 01:07:12.010 --> 01:07:15.970 Your monthly payment, C, that's the right-hand side. 01:07:15.970 --> 01:07:18.610 And the prevailing interest rate, that's the r. 01:07:18.610 --> 01:07:22.450 So if you know the annuity discount factor, which 01:07:22.450 --> 01:07:24.850 is based just on the interest rate, 01:07:24.850 --> 01:07:27.340 and you know the amount of the loan that you want, 01:07:27.340 --> 01:07:29.500 PV, you can divide and figure out 01:07:29.500 --> 01:07:32.090 what your monthly payments are, or vice versa. 01:07:32.090 --> 01:07:35.314 If you have a particular set of cash flows every month, 01:07:35.314 --> 01:07:36.730 and you have an interest rate, you 01:07:36.730 --> 01:07:39.760 can figure out what your total value of that loan 01:07:39.760 --> 01:07:42.730 is going to be in terms of market terms. 01:07:42.730 --> 01:07:45.650 And so once you have this expression, 01:07:45.650 --> 01:07:49.090 you can use a simple table of numbers 01:07:49.090 --> 01:07:51.626 to calculate these annuity discount factors. 01:07:51.626 --> 01:07:53.750 And then you can compute mortgage payment yourself. 01:07:53.750 --> 01:07:56.200 So this is the kind of number I was talking about. 01:07:56.200 --> 01:07:58.420 Given various different interest rates, 01:07:58.420 --> 01:08:01.300 you can come up with these particular annuity discount 01:08:01.300 --> 01:08:03.060 factors. 01:08:03.060 --> 01:08:06.540 And once you do, you can calculate monthly payments 01:08:06.540 --> 01:08:07.960 very easily. 01:08:07.960 --> 01:08:10.490 So you only need one set of tables. 01:08:10.490 --> 01:08:13.680 And for any kind of mortgage, for any kind of consumer loan, 01:08:13.680 --> 01:08:15.790 you can compute the monthly payments. 01:08:15.790 --> 01:08:16.290 Right? 01:08:16.290 --> 01:08:18.806 Whether it's an auto loan, or a mortgage, it doesn't matter. 01:08:18.806 --> 01:08:20.430 What you need is this table right here. 01:08:20.430 --> 01:08:21.840 Nowadays, we can do it in Excel. 01:08:21.840 --> 01:08:23.220 It's not a big deal. 01:08:23.220 --> 01:08:26.069 But still, you should know what the underlying basis 01:08:26.069 --> 01:08:30.160 is for those calculations. 01:08:30.160 --> 01:08:31.189 OK. 01:08:31.189 --> 01:08:35.120 Now before you finish this, I would 01:08:35.120 --> 01:08:36.979 like you to take a look at a few examples. 01:08:36.979 --> 01:08:39.859 I've given you some here, numerical examples. 01:08:39.859 --> 01:08:43.939 And I want to close this class with a discussion 01:08:43.939 --> 01:08:46.279 about compounding, and then next time, 01:08:46.279 --> 01:08:49.160 finish up with a discussion of inflation. 01:08:49.160 --> 01:08:52.279 Because I don't think we'll have time to do both. 01:08:52.279 --> 01:08:57.080 Compounding is a matter of convention. 01:08:57.080 --> 01:08:59.960 And I want to explain what that convention is and try 01:08:59.960 --> 01:09:02.520 to give you a little bit of motivation for the logic of it, 01:09:02.520 --> 01:09:04.436 so that at least it doesn't look like I'm just 01:09:04.436 --> 01:09:07.060 making it up out of the blue. 01:09:07.060 --> 01:09:10.640 The idea behind convention is to take into account calendar 01:09:10.640 --> 01:09:14.260 effects, and in particular, the possibility 01:09:14.260 --> 01:09:15.420 of early withdrawal. 01:09:15.420 --> 01:09:16.550 Let me explain. 01:09:16.550 --> 01:09:20.680 When I tell you that an interest rate is 10%, 01:09:20.680 --> 01:09:25.029 typically, that quote is in terms of an annual interest 01:09:25.029 --> 01:09:25.899 rate. 01:09:25.899 --> 01:09:28.660 Almost all interest rates in the world 01:09:28.660 --> 01:09:31.930 are quoted on an annualized basis, meaning 01:09:31.930 --> 01:09:33.939 if you were to keep an investment 01:09:33.939 --> 01:09:37.270 for a 12-month period, that's what the rate of return 01:09:37.270 --> 01:09:40.649 for that investment would be. 01:09:40.649 --> 01:09:44.069 The problem with that quote, 10%, 01:09:44.069 --> 01:09:47.340 is that what do you do if you want to withdraw 01:09:47.340 --> 01:09:50.870 your money after six months? 01:09:50.870 --> 01:09:53.890 What should you get paid then? 01:09:53.890 --> 01:09:58.440 Well, it would seem fair, if you agree to a 10% interest 01:09:58.440 --> 01:10:01.920 rate per year, to say, all right, if I take it 01:10:01.920 --> 01:10:04.890 out in six months instead of a year, maybe 01:10:04.890 --> 01:10:07.501 you should only pay me half the interest rate. 01:10:07.501 --> 01:10:08.000 Right? 01:10:08.000 --> 01:10:09.450 That seems like a fair deal. 01:10:09.450 --> 01:10:09.950 Right? 01:10:09.950 --> 01:10:14.100 Instead of 10%, pay me 5%. 01:10:14.100 --> 01:10:19.110 And maybe if I keep it in for only a month, 01:10:19.110 --> 01:10:21.870 it would be fair not to pay me 10% for that month, 01:10:21.870 --> 01:10:28.390 but to pay me 10% divided by 12 for the month. 01:10:28.390 --> 01:10:31.210 The reason that that discussion matters 01:10:31.210 --> 01:10:35.050 is that if you agree that that's the fair thing to do, 01:10:35.050 --> 01:10:40.140 well then 10% is not what you're going to get. 01:10:40.140 --> 01:10:45.180 Because if you get paid 5% interest over the first six 01:10:45.180 --> 01:10:50.450 months, you take your money out of the bank, 01:10:50.450 --> 01:10:53.840 and they give you that 5%, and then 01:10:53.840 --> 01:10:57.560 you put the money back in the bank literally the next minute 01:10:57.560 --> 01:10:59.850 and keep it in for the next six months, 01:10:59.850 --> 01:11:05.670 you're going to earn another 5% on your original amount, 01:11:05.670 --> 01:11:10.290 plus you're going to earn 5% on the first six month's 5%. 01:11:10.290 --> 01:11:13.890 You're going to earn interest on the interest. 01:11:13.890 --> 01:11:15.240 And the banks know that. 01:11:15.240 --> 01:11:19.360 And after a while, they were OK with that. 01:11:19.360 --> 01:11:23.580 That's a convention there's no reason it has to be that way. 01:11:23.580 --> 01:11:27.130 The bank could say, I'm going to give you 10% interest. 01:11:27.130 --> 01:11:29.610 But if you want to withdraw your money in six months, 01:11:29.610 --> 01:11:32.369 I'm going to give you the amount of interest such 01:11:32.369 --> 01:11:34.410 that if you were to take the money out and put it 01:11:34.410 --> 01:11:38.840 right back in and hold it, you would get 10%. 01:11:38.840 --> 01:11:41.570 Does anybody know what that interest rate would be? 01:11:41.570 --> 01:11:42.841 How you figure that out? 01:11:42.841 --> 01:11:43.590 Yeah, [INAUDIBLE]? 01:11:43.590 --> 01:11:45.081 AUDIENCE: [INAUDIBLE]. 01:11:47.752 --> 01:11:48.460 ANDREW LO: Roots. 01:11:48.460 --> 01:11:49.930 That sounds painful . 01:11:49.930 --> 01:11:51.940 Is that like a root canal? 01:11:51.940 --> 01:11:53.329 What root do you mean? 01:11:53.329 --> 01:11:53.870 You're right. 01:11:53.870 --> 01:11:54.411 You're right. 01:11:54.411 --> 01:11:55.078 What is it? 01:11:55.078 --> 01:11:58.360 AUDIENCE: [INAUDIBLE]. 01:11:58.360 --> 01:11:59.692 ANDREW LO: Yes. 01:11:59.692 --> 01:12:03.100 AUDIENCE: [INAUDIBLE]. 01:12:03.100 --> 01:12:07.790 ANDREW LO: Yes The square root. 01:12:07.790 --> 01:12:08.290 Right. 01:12:08.290 --> 01:12:11.461 AUDIENCE: [INAUDIBLE] you get [INAUDIBLE].. 01:12:11.461 --> 01:12:12.460 ANDREW LO: That's right. 01:12:12.460 --> 01:12:13.376 AUDIENCE: [INAUDIBLE]. 01:12:13.376 --> 01:12:14.050 Exactly. 01:12:14.050 --> 01:12:17.230 So what you would do in order to figure out what the six month 01:12:17.230 --> 01:12:20.980 interest rate would be so that when you held 01:12:20.980 --> 01:12:25.240 the interest on the interest over through the whole year, 01:12:25.240 --> 01:12:28.150 it would add up to exactly 10%. 01:12:28.150 --> 01:12:34.630 The way you would do it is 1.10, take the square root of that. 01:12:34.630 --> 01:12:38.240 Minus 1, that's the interest rate 01:12:38.240 --> 01:12:40.747 that you would have for the first six months 01:12:40.747 --> 01:12:41.830 and the second six months. 01:12:41.830 --> 01:12:45.500 A little less than 5%, such that that number, 01:12:45.500 --> 01:12:51.380 when you add 1 and multiply it by itself, you'll get 1.10. 01:12:51.380 --> 01:12:54.410 They don't do that, mainly because nobody likes dealing 01:12:54.410 --> 01:12:56.510 with roots, except dentists. 01:12:56.510 --> 01:12:58.130 OK? 01:12:58.130 --> 01:13:01.417 So what they do is they say, OK, as a matter of convention, 01:13:01.417 --> 01:13:03.000 here's what we're going to do for you. 01:13:03.000 --> 01:13:04.708 This is the deal we're going to give you. 01:13:04.708 --> 01:13:08.510 When we say 10% on an annualized basis, what we mean 01:13:08.510 --> 01:13:11.750 is that it's going to be compounded, 01:13:11.750 --> 01:13:16.700 typically on a monthly basis, and nowadays on a daily basis. 01:13:16.700 --> 01:13:18.470 What that means is that the interest 01:13:18.470 --> 01:13:20.750 rate that you're actually going to get 01:13:20.750 --> 01:13:25.470 is the stated equivalent. 01:13:25.470 --> 01:13:30.340 It's the stated annual rate divided by the compounding 01:13:30.340 --> 01:13:33.140 interval. 01:13:33.140 --> 01:13:37.480 Now that's a good deal when you're a depositor. 01:13:37.480 --> 01:13:41.490 That's not a good deal if you're a borrower. 01:13:41.490 --> 01:13:44.580 Because when they tell you, you want to borrow money from me, 01:13:44.580 --> 01:13:48.400 I'll give it to you at a great rate, it's going to be at 10%. 01:13:48.400 --> 01:13:50.650 But when you actually look at how much interest you're 01:13:50.650 --> 01:13:53.710 paying, you're going to find out that, actually, it's 01:13:53.710 --> 01:13:55.810 more than 10%. 01:13:55.810 --> 01:14:01.550 So that's where the term APR and ERA came from. 01:14:01.550 --> 01:14:02.720 What does APR stand for? 01:14:02.720 --> 01:14:04.430 Anybody know? 01:14:04.430 --> 01:14:07.580 When you see this ad on TV for auto loans, you know, 01:14:07.580 --> 01:14:09.970 [? buyer ?] loans, buy a car, no money down, 01:14:09.970 --> 01:14:15.322 we'll give you a loan, the APR is x%. 01:14:15.322 --> 01:14:16.280 Annual percentage rate. 01:14:16.280 --> 01:14:17.870 That is the stated rate. 01:14:17.870 --> 01:14:22.910 That's not the rate including the effects of compounding. 01:14:22.910 --> 01:14:27.350 So as a depositor, when you're lending money to the bank-- 01:14:27.350 --> 01:14:29.760 that's what it means to deposit money in the bank-- 01:14:29.760 --> 01:14:30.710 that's a good thing. 01:14:30.710 --> 01:14:33.681 Because the annual percentage rate of 10% 01:14:33.681 --> 01:14:35.180 is actually not what you're getting. 01:14:35.180 --> 01:14:36.440 You're getting more than that, because it's 01:14:36.440 --> 01:14:38.780 going to be compounded on a monthly, or in some cases, 01:14:38.780 --> 01:14:40.880 on a daily basis. 01:14:40.880 --> 01:14:41.540 OK? 01:14:41.540 --> 01:14:45.050 In other words, the compounding means you get interest 01:14:45.050 --> 01:14:48.380 on your interest on your interest's interest going 01:14:48.380 --> 01:14:49.200 forward. 01:14:49.200 --> 01:14:51.530 Right? 01:14:51.530 --> 01:14:55.100 So you've got to keep in mind that when you see these 01:14:55.100 --> 01:14:59.870 discount rates being quoted, ask whether or not they are APR, 01:14:59.870 --> 01:15:03.560 annual percentage rate-- that's like the 10% stated rate-- 01:15:03.560 --> 01:15:04.986 or EAR. 01:15:04.986 --> 01:15:08.670 EAR is the equivalent annual rate. 01:15:08.670 --> 01:15:10.760 That's what you're really going to get. 01:15:10.760 --> 01:15:14.060 That's what you would actually get in terms of literal dollars 01:15:14.060 --> 01:15:17.030 at the end of the year if you did nothing but left the money 01:15:17.030 --> 01:15:18.886 in there for that entire year. 01:15:18.886 --> 01:15:21.260 It would include the interest on the interest on interest 01:15:21.260 --> 01:15:22.550 on the interest and so on. 01:15:22.550 --> 01:15:23.153 Yeah? 01:15:23.153 --> 01:15:25.542 AUDIENCE: [INAUDIBLE]. 01:15:25.542 --> 01:15:27.000 ANDREW LO: Annual percentage yield. 01:15:27.000 --> 01:15:28.850 Yeah, that's right. 01:15:28.850 --> 01:15:31.540 Now, this is a very clear example of it. 01:15:31.540 --> 01:15:32.600 OK? 01:15:32.600 --> 01:15:37.460 If you've got $1,000, and there's no compounding effects 01:15:37.460 --> 01:15:41.390 and the interest rate is 10%, you're going to get $1,100. 01:15:41.390 --> 01:15:45.110 If you compound twice a year, which is what the old banks 01:15:45.110 --> 01:15:47.870 used to do because they didn't have calculators in those 01:15:47.870 --> 01:15:51.000 days-- it was kind of hard to compute these numbers-- 01:15:51.000 --> 01:15:53.370 they would compound it twice a year. 01:15:53.370 --> 01:15:56.030 And so you would get credit for the interest, 01:15:56.030 --> 01:15:58.460 and then you would get interest on that interest 01:15:58.460 --> 01:16:02.720 as well as on the original deposit or principle. 01:16:02.720 --> 01:16:05.870 Then that turns into $1,103. 01:16:05.870 --> 01:16:09.260 So being able to compound more frequently 01:16:09.260 --> 01:16:11.630 gives you an additional bonus, right? 01:16:11.630 --> 01:16:13.580 Not much. $3. 01:16:13.580 --> 01:16:16.670 But if you think about this as billions of dollars, 01:16:16.670 --> 01:16:19.430 this starts adding up to be real money. 01:16:19.430 --> 01:16:23.270 Now, if you compound on a quarterly basis, it's $4. 01:16:23.270 --> 01:16:25.770 If you compound on a monthly basis, it's $5. 01:16:25.770 --> 01:16:28.460 That's actually starting to add up to something important. 01:16:28.460 --> 01:16:29.310 Right? 01:16:29.310 --> 01:16:30.492 Yeah? 01:16:30.492 --> 01:16:31.965 AUDIENCE: [INAUDIBLE]. 01:16:43.270 --> 01:16:45.700 ANDREW LO: Well, I mean, I think it's six of one, 01:16:45.700 --> 01:16:47.500 or half a dozen of the other, as they say. 01:16:47.500 --> 01:16:49.060 Banks will compete with each other 01:16:49.060 --> 01:16:52.150 to offer ultimately what the market rate is. 01:16:52.150 --> 01:16:55.060 So they won't play any tricks with this kind of stuff some. 01:16:55.060 --> 01:16:57.610 Banks did play tricks with this early 01:16:57.610 --> 01:16:59.410 on in the early days of banking. 01:16:59.410 --> 01:17:02.230 That's why banking is such a highly regulated industry, 01:17:02.230 --> 01:17:04.300 to make sure that no funny business goes on. 01:17:04.300 --> 01:17:07.660 And frankly, that's why banks are forced now 01:17:07.660 --> 01:17:11.800 to tell you what whether it's an APR or an EAR. 01:17:11.800 --> 01:17:14.740 It's a truth in lending kind of a commitment 01:17:14.740 --> 01:17:17.300 that they are now being forced to make. 01:17:17.300 --> 01:17:21.000 So nowadays, when you get an auto loan or a mortgage, 01:17:21.000 --> 01:17:23.380 they have to tell you, yeah, this is NPR. 01:17:23.380 --> 01:17:25.240 This is the annual percentage rate. 01:17:25.240 --> 01:17:29.770 But your actual rate earned may vary, 01:17:29.770 --> 01:17:32.980 and it may vary because of compounding effects. 01:17:32.980 --> 01:17:35.230 And if you ask them what the effective annual rate is, 01:17:35.230 --> 01:17:37.224 they are obligated to tell you. 01:17:37.224 --> 01:17:39.660 AUDIENCE: [INAUDIBLE] all the information. 01:17:39.660 --> 01:17:43.544 Because with APR, you also need to know the compound-- 01:17:43.544 --> 01:17:44.710 ANDREW LO: Compounding rate. 01:17:44.710 --> 01:17:48.280 But it's now taken for granted that compounding 01:17:48.280 --> 01:17:50.150 happens on a daily basis. 01:17:50.150 --> 01:17:51.890 So that's a given. 01:17:51.890 --> 01:17:52.390 OK? 01:17:52.390 --> 01:17:54.010 Any questions about that? 01:17:54.010 --> 01:17:55.450 AUDIENCE: [INAUDIBLE]. 01:17:55.450 --> 01:17:57.162 ANDREW LO: Compounding does, yes. 01:17:57.162 --> 01:17:58.495 AUDIENCE: For checking accounts? 01:17:58.495 --> 01:18:01.360 ANDREW LO: For checking accounts, for savings accounts. 01:18:01.360 --> 01:18:02.140 Yes, it's daily. 01:18:02.140 --> 01:18:03.096 And you know why? 01:18:03.096 --> 01:18:04.720 It's because they allow you to take out 01:18:04.720 --> 01:18:06.400 money on a daily basis. 01:18:06.400 --> 01:18:08.440 So if they didn't do it on a daily basis, 01:18:08.440 --> 01:18:10.507 they'd have to figure out on a one-off, 01:18:10.507 --> 01:18:13.090 if you were to take your money out in the middle of the month, 01:18:13.090 --> 01:18:15.640 and I was to take my money out after the first three days, 01:18:15.640 --> 01:18:18.070 and you were to take your money out after five days, 01:18:18.070 --> 01:18:20.620 they'd have to do all these custom calculations for each 01:18:20.620 --> 01:18:21.769 of those circumstances. 01:18:21.769 --> 01:18:22.810 So now they do it simply. 01:18:22.810 --> 01:18:23.980 They say, fine, we're going to give you your interest 01:18:23.980 --> 01:18:24.891 rate every day. 01:18:24.891 --> 01:18:26.890 Every day, we're going to compute your interest. 01:18:26.890 --> 01:18:29.650 So whether you come or go, you will figure out 01:18:29.650 --> 01:18:30.940 when you get the interest. 01:18:30.940 --> 01:18:34.060 For certain market applications, people 01:18:34.060 --> 01:18:36.190 compute interest intraday. 01:18:36.190 --> 01:18:38.492 Like the number of hours you borrow money, 01:18:38.492 --> 01:18:39.700 they will calculate interest. 01:18:39.700 --> 01:18:42.310 There are cases where you need to borrow money, only 01:18:42.310 --> 01:18:45.232 for four hours or three hours. 01:18:45.232 --> 01:18:46.690 I know this sounds like drug money. 01:18:46.690 --> 01:18:49.180 But that's not-- that's not what I'm talking about. 01:18:49.180 --> 01:18:50.980 There are cases where you need very, very 01:18:50.980 --> 01:18:54.520 short-term financing, and you need to borrow the money. 01:18:54.520 --> 01:18:57.160 And in those cases, they compute it on a minute to minute. 01:18:57.160 --> 01:18:59.390 And in some cases, on a continuous basis. 01:18:59.390 --> 01:19:03.100 So I'm going to leave you with a little puzzler, which 01:19:03.100 --> 01:19:08.530 is if this tells you what the effective annual rate is, where 01:19:08.530 --> 01:19:12.040 you're compounding at intervals of n-- 01:19:12.040 --> 01:19:16.330 so if r is an APR, an annual percentage rate, 01:19:16.330 --> 01:19:18.910 and n is denominated in months-- 01:19:18.910 --> 01:19:20.725 so monthly would be 12-- 01:19:23.260 --> 01:19:28.850 what would happen, what would your effective annual rate be, 01:19:28.850 --> 01:19:34.460 if you compounded not every day, not every hour, not 01:19:34.460 --> 01:19:37.610 every minute, not every femtosecond, 01:19:37.610 --> 01:19:40.940 but literally every possible time slice, 01:19:40.940 --> 01:19:42.920 the narrowest time slice you can think of. 01:19:42.920 --> 01:19:49.090 If you did it continuously, if n were to go to infinity, 01:19:49.090 --> 01:19:51.760 what would you get? 01:19:51.760 --> 01:19:52.480 Think about that. 01:19:52.480 --> 01:19:53.650 That's a little puzzle. 01:19:53.650 --> 01:19:57.760 It turns out that's called continuous compounding. 01:19:57.760 --> 01:20:00.990 So you're compounding continuously. 01:20:00.990 --> 01:20:03.780 It turns out that you actually get a number. 01:20:03.780 --> 01:20:08.530 And what that number is is really bizarre. 01:20:08.530 --> 01:20:10.750 So I want you to think about that, 01:20:10.750 --> 01:20:12.510 and we'll take that up next time. 01:20:12.510 --> 01:20:13.970 Thank you.