1 00:00:00,090 --> 00:00:02,430 The following content is provided under a Creative 2 00:00:02,430 --> 00:00:03,820 Commons license. 3 00:00:03,820 --> 00:00:06,030 Your support will help MIT OpenCourseWare 4 00:00:06,030 --> 00:00:10,120 continue to offer high quality educational resources for free. 5 00:00:10,120 --> 00:00:12,690 To make a donation or to view additional materials 6 00:00:12,690 --> 00:00:16,620 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,620 --> 00:00:17,820 at ocw.mit.edu. 8 00:00:21,318 --> 00:00:24,580 ANDREW LO: First of all, any questions from last lecture? 9 00:00:27,760 --> 00:00:28,976 Yes? 10 00:00:28,976 --> 00:00:38,370 AUDIENCE: [INAUDIBLE] he said he was [INAUDIBLE] possible 11 00:00:38,370 --> 00:00:39,445 [INAUDIBLE]? 12 00:00:44,907 --> 00:00:45,490 ANDREW LO: OK. 13 00:00:45,490 --> 00:00:48,190 So let me repeat the question to make sure everybody heard. 14 00:00:48,190 --> 00:00:52,480 The question about net present value is that, is it possible, 15 00:00:52,480 --> 00:00:56,530 is it possible, that in one currency, 16 00:00:56,530 --> 00:01:00,280 the net present value of a project is positive, 17 00:01:00,280 --> 00:01:04,180 but in a different currency, it is negative? 18 00:01:04,180 --> 00:01:06,240 That's a very interesting question. 19 00:01:06,240 --> 00:01:10,630 And it turns out that the answer is staring us in the face right 20 00:01:10,630 --> 00:01:11,920 here. 21 00:01:11,920 --> 00:01:15,380 Now remember, we're in a world of no uncertainty. 22 00:01:15,380 --> 00:01:18,930 So we know what future cash flows are going to be. 23 00:01:18,930 --> 00:01:22,960 And we know what future discount rates or discount 24 00:01:22,960 --> 00:01:24,430 factors are going to be. 25 00:01:24,430 --> 00:01:26,530 That's my assumption. 26 00:01:26,530 --> 00:01:29,890 And in that world, when I give you 27 00:01:29,890 --> 00:01:33,100 the value of a sequence of cash flows, 28 00:01:33,100 --> 00:01:38,510 this v sub 0, if I wanted denominate it in dollars, 29 00:01:38,510 --> 00:01:43,630 then presumably all the cash flows have to be in dollars. 30 00:01:43,630 --> 00:01:45,560 If I want to denominate it in yen, 31 00:01:45,560 --> 00:01:48,250 then the cash flows have to be in yen. 32 00:01:48,250 --> 00:01:53,680 So strictly speaking, assuming that the exchange 33 00:01:53,680 --> 00:01:57,310 rates don't change over time-- 34 00:01:57,310 --> 00:02:01,340 and that's, again, a big assumption-- 35 00:02:01,340 --> 00:02:05,150 the question is, can I have a different result 36 00:02:05,150 --> 00:02:09,620 in terms of the sign of a net present value by changing 37 00:02:09,620 --> 00:02:11,830 the exchange rate? 38 00:02:11,830 --> 00:02:13,520 Any thoughts on that? 39 00:02:13,520 --> 00:02:14,320 What do you think? 40 00:02:14,320 --> 00:02:15,000 Yeah. 41 00:02:15,000 --> 00:02:16,270 AUDIENCE: No. 42 00:02:16,270 --> 00:02:19,335 ANDREW LO: No, why? 43 00:02:19,335 --> 00:02:22,620 AUDIENCE: Because currency [INAUDIBLE].. 44 00:02:33,730 --> 00:02:34,820 ANDREW LO: OK. 45 00:02:34,820 --> 00:02:37,400 So the answer is no, because currency, 46 00:02:37,400 --> 00:02:39,500 the exchange rates always have to be positive. 47 00:02:39,500 --> 00:02:42,920 And presumably, you're multiplying the cache flows 48 00:02:42,920 --> 00:02:48,360 by the same number, either positive of one number 49 00:02:48,360 --> 00:02:49,990 or positive of another number. 50 00:02:49,990 --> 00:02:54,090 So when you multiply a sequence by a positive number, when 51 00:02:54,090 --> 00:02:57,300 you add that up, it is either still positive 52 00:02:57,300 --> 00:02:58,180 or still negative. 53 00:02:58,180 --> 00:02:59,880 In other words, you can factor it out. 54 00:02:59,880 --> 00:03:01,990 Right? 55 00:03:01,990 --> 00:03:03,250 You sure? 56 00:03:03,250 --> 00:03:04,400 Yeah. 57 00:03:04,400 --> 00:03:05,810 AUDIENCE: I have a question. 58 00:03:05,810 --> 00:03:11,230 When we are doing this in the [INAUDIBLE],, 59 00:03:11,230 --> 00:03:17,680 is it possible to have different [INAUDIBLE]?? 60 00:03:17,680 --> 00:03:19,000 ANDREW LO: Well. 61 00:03:19,000 --> 00:03:21,080 Right now, we're not talking about risk. 62 00:03:21,080 --> 00:03:24,460 So let's hold that off for seven or eight lectures. 63 00:03:24,460 --> 00:03:25,930 I want to ask this question. 64 00:03:25,930 --> 00:03:27,530 Have I got it right? 65 00:03:27,530 --> 00:03:30,224 We agreed that no matter what you multiply it by, 66 00:03:30,224 --> 00:03:31,640 as long as it's a positive number, 67 00:03:31,640 --> 00:03:34,310 it can't change the sign, so the currency doesn't matter. 68 00:03:34,310 --> 00:03:34,810 Yeah. 69 00:03:34,810 --> 00:03:35,581 Ernest? 70 00:03:35,581 --> 00:03:38,467 AUDIENCE: But the exchange rate, so the actuals 71 00:03:38,467 --> 00:03:39,430 are at different times. 72 00:03:39,430 --> 00:03:40,298 ANDREW LO: Yes. 73 00:03:40,298 --> 00:03:42,096 AUDIENCE: So if your exchange rate 74 00:03:42,096 --> 00:03:43,846 is different at different times, then it's 75 00:03:43,846 --> 00:03:47,260 going to stay factored throughout the-- 76 00:03:47,260 --> 00:03:50,312 ANDREW LO: The assumption is that it's fixed. 77 00:03:50,312 --> 00:03:51,270 There's no uncertainty. 78 00:03:51,270 --> 00:03:51,770 But-- 79 00:03:51,770 --> 00:03:53,200 AUDIENCE: [INAUDIBLE]. 80 00:03:53,200 --> 00:03:55,780 ANDREW LO: I didn't say it was the same. 81 00:03:55,780 --> 00:03:57,760 So you said that it was the same. 82 00:03:57,760 --> 00:03:58,660 I didn't. 83 00:03:58,660 --> 00:03:59,450 You're right. 84 00:03:59,450 --> 00:04:01,150 So [? Shlomi, ?] you're right. 85 00:04:01,150 --> 00:04:04,750 If the exchange rate is the same over time, 86 00:04:04,750 --> 00:04:06,610 then when you multiply by one number, 87 00:04:06,610 --> 00:04:08,950 it's the same number for every cash flow. 88 00:04:08,950 --> 00:04:11,840 Then, it factors out. 89 00:04:11,840 --> 00:04:16,540 And then you're multiplying v sub-zero by a positive number. 90 00:04:16,540 --> 00:04:19,180 So if v sub-zero is positive, it stays positive. 91 00:04:19,180 --> 00:04:21,990 If it's negative, it stays negative. 92 00:04:21,990 --> 00:04:28,110 But no uncertainty doesn't mean that it's fixed. 93 00:04:28,110 --> 00:04:29,130 So here's the subtlety. 94 00:04:29,130 --> 00:04:33,230 The subtlety is that if I assume that the exchange rate is fixed 95 00:04:33,230 --> 00:04:40,480 and known, but going up over time, whereas in US dollars, 96 00:04:40,480 --> 00:04:44,510 it stays fixed, that makes a difference. 97 00:04:44,510 --> 00:04:45,230 Right? 98 00:04:45,230 --> 00:04:46,370 So it's possible. 99 00:04:46,370 --> 00:04:49,910 It's possible that if I change currencies and the currency 100 00:04:49,910 --> 00:04:53,420 is rapidly appreciating or rapidly depreciating, 101 00:04:53,420 --> 00:04:56,720 then you can actually change the net present value 102 00:04:56,720 --> 00:04:58,220 of the project. 103 00:04:58,220 --> 00:05:02,480 But it has to be the case that the particular path 104 00:05:02,480 --> 00:05:05,990 of the currency appreciation or depreciation 105 00:05:05,990 --> 00:05:10,770 is exactly opposite what's going on with the NPV. 106 00:05:10,770 --> 00:05:13,820 So the bottom line is, you've got to do the calculation. 107 00:05:13,820 --> 00:05:17,100 And you have to use the currency that you care about. 108 00:05:17,100 --> 00:05:19,550 So if you're in US, you presumably 109 00:05:19,550 --> 00:05:21,509 care about getting paid in US dollars. 110 00:05:21,509 --> 00:05:22,550 You would use US dollars. 111 00:05:22,550 --> 00:05:25,044 If you're in Japan, you get paid in yen. 112 00:05:25,044 --> 00:05:26,210 You'll want to do it in yen. 113 00:05:26,210 --> 00:05:28,620 And you have to do the currency conversion. 114 00:05:28,620 --> 00:05:30,354 Now when we talk about uncertainty, 115 00:05:30,354 --> 00:05:32,270 that's going to make it much more complicated. 116 00:05:32,270 --> 00:05:36,620 It's going to introduce another component of risk 117 00:05:36,620 --> 00:05:39,051 in our calculations that has to be dealt with. 118 00:05:39,051 --> 00:05:40,550 So we're going to come back to that. 119 00:05:40,550 --> 00:05:43,280 But that's a good question. 120 00:05:43,280 --> 00:05:44,360 Anybody else? 121 00:05:44,360 --> 00:05:45,242 Yes? 122 00:05:45,242 --> 00:05:46,991 AUDIENCE: I noticed that you used the term 123 00:05:46,991 --> 00:05:48,404 paper a couple of times. 124 00:05:48,404 --> 00:05:50,288 I just wanted [INAUDIBLE] definition of-- 125 00:05:50,288 --> 00:05:50,759 ANDREW LO: Of what? 126 00:05:50,759 --> 00:05:51,425 AUDIENCE: Paper. 127 00:05:51,425 --> 00:05:52,702 ANDREW LO: Paper. 128 00:05:52,702 --> 00:05:54,160 You mean, this is a piece of paper? 129 00:05:54,160 --> 00:05:55,951 AUDIENCE: Well, I don't think [INAUDIBLE].. 130 00:05:58,450 --> 00:05:59,420 ANDREW LO: Right. 131 00:05:59,420 --> 00:06:03,460 Yeah, so typically by paper, people mean a security. 132 00:06:03,460 --> 00:06:05,800 And commercial paper is a security 133 00:06:05,800 --> 00:06:09,100 that is a debt instrument that is basically an IOU. 134 00:06:09,100 --> 00:06:10,300 It's like a bond. 135 00:06:10,300 --> 00:06:12,010 So we'll come back to that when we talk about fixed income 136 00:06:12,010 --> 00:06:12,670 securities. 137 00:06:12,670 --> 00:06:13,792 But that's what I mean. 138 00:06:13,792 --> 00:06:15,250 By the way, you raise a good point. 139 00:06:15,250 --> 00:06:19,340 When I mention terminology, feel free to ask me. 140 00:06:19,340 --> 00:06:21,494 But in turn, I'm going to feel free to tell you, 141 00:06:21,494 --> 00:06:23,910 you may want to look that up in [? Breeley, ?] [? Myers ?] 142 00:06:23,910 --> 00:06:27,700 and Allan, because I want you to read the book alongside of what 143 00:06:27,700 --> 00:06:30,760 we're doing in class, because you'll need to pick up this 144 00:06:30,760 --> 00:06:34,450 terminology, and we don't have enough time in this 20 lectures 145 00:06:34,450 --> 00:06:37,300 to cover all the terminology that you need to know. 146 00:06:37,300 --> 00:06:42,580 So don't assume that just because I haven't covered it 147 00:06:42,580 --> 00:06:45,010 in class, or that I haven't defined it 148 00:06:45,010 --> 00:06:47,110 that you don't need to know it. 149 00:06:47,110 --> 00:06:50,650 The textbook is there to help you 150 00:06:50,650 --> 00:06:53,890 with the supplementary material that I would like you to cover. 151 00:06:53,890 --> 00:06:55,600 So that's why we assign those chapters. 152 00:06:55,600 --> 00:06:56,230 OK? 153 00:06:56,230 --> 00:06:58,500 Yeah, Justin. 154 00:06:58,500 --> 00:07:01,020 AUDIENCE: [INAUDIBLE]. 155 00:07:01,020 --> 00:07:02,230 ANDREW LO: Yes. 156 00:07:02,230 --> 00:07:05,130 AUDIENCE: Then I read a news article, 157 00:07:05,130 --> 00:07:08,459 and they said the stock market jumps because they're 158 00:07:08,459 --> 00:07:09,250 getting bailed out. 159 00:07:09,250 --> 00:07:09,958 ANDREW LO: Right. 160 00:07:09,958 --> 00:07:13,961 AUDIENCE: So is there a simple reason as to why this is such 161 00:07:13,961 --> 00:07:17,230 a massive increase in stock-- 162 00:07:17,230 --> 00:07:19,500 ANDREW LO: In the stock market, while their stock has 163 00:07:19,500 --> 00:07:20,304 gone down. 164 00:07:20,304 --> 00:07:20,970 AUDIENCE: Right. 165 00:07:20,970 --> 00:07:22,924 So that seems a little counter-intuitive. 166 00:07:22,924 --> 00:07:24,840 I'm going to give you a two minute answer now, 167 00:07:24,840 --> 00:07:27,030 but then I'm going to give you a much deeper answer 168 00:07:27,030 --> 00:07:29,250 in about three or four lectures, when we actually 169 00:07:29,250 --> 00:07:31,200 apply all of the framework we're developing 170 00:07:31,200 --> 00:07:33,210 to pricing common stock. 171 00:07:33,210 --> 00:07:35,280 So as I said with Freddie and Fannie, 172 00:07:35,280 --> 00:07:36,600 there are two components. 173 00:07:36,600 --> 00:07:39,760 There are two sets of issues surrounding those companies. 174 00:07:39,760 --> 00:07:44,840 One is the value of the owner's equity, the folks who owned 175 00:07:44,840 --> 00:07:46,470 a piece of those companies. 176 00:07:46,470 --> 00:07:48,240 What are their investments worth? 177 00:07:48,240 --> 00:07:50,770 And the answer is very little. 178 00:07:50,770 --> 00:07:53,710 The second piece is that Freddie and Fannie 179 00:07:53,710 --> 00:07:57,160 have issued all sorts of IOUs, all sorts of obligations 180 00:07:57,160 --> 00:07:58,570 to counter-parties. 181 00:07:58,570 --> 00:08:02,320 And the question is, what are those securities worth. 182 00:08:02,320 --> 00:08:04,870 The government bailing out Freddie and Fannie 183 00:08:04,870 --> 00:08:09,770 are basically saying, we will stand behind those IOUs. 184 00:08:09,770 --> 00:08:11,960 The shareholders of the company-- 185 00:08:11,960 --> 00:08:14,090 sorry, you guys lost. 186 00:08:14,090 --> 00:08:15,560 The company has not done well. 187 00:08:15,560 --> 00:08:17,082 It suffered a lot of losses. 188 00:08:17,082 --> 00:08:19,040 So the fact that you own a piece of the company 189 00:08:19,040 --> 00:08:21,890 means that what you own is now worthless. 190 00:08:21,890 --> 00:08:25,430 But the pieces of paper that the company has issued, 191 00:08:25,430 --> 00:08:29,510 we will assume that obligation as the US government 192 00:08:29,510 --> 00:08:32,210 and make good on those obligations. 193 00:08:32,210 --> 00:08:35,419 So the fact that those pieces of paper 194 00:08:35,419 --> 00:08:39,500 have much broader impact on the market as a whole, the fact 195 00:08:39,500 --> 00:08:40,970 that the US government is standing 196 00:08:40,970 --> 00:08:45,080 behind those pieces of paper will protect the stock market 197 00:08:45,080 --> 00:08:47,300 as a whole because there's confidence 198 00:08:47,300 --> 00:08:52,740 that business conditions will not be as bad as we thought. 199 00:08:52,740 --> 00:08:54,950 So that's what explains the fact that the stock 200 00:08:54,950 --> 00:08:57,002 market as a whole went up. 201 00:08:57,002 --> 00:08:58,460 It's because the market environment 202 00:08:58,460 --> 00:09:00,350 has been stabilized. 203 00:09:00,350 --> 00:09:02,630 You can imagine what might have happened if Fannie 204 00:09:02,630 --> 00:09:04,640 and Freddie were to go under. 205 00:09:04,640 --> 00:09:07,610 Their pieces of paper, their IOUs, would be worthless. 206 00:09:07,610 --> 00:09:10,710 Which means the folks that own those pieces of paper, 207 00:09:10,710 --> 00:09:13,280 now they have a bunch of worthless paper. 208 00:09:13,280 --> 00:09:17,300 And when that happens, there are repercussion effects 209 00:09:17,300 --> 00:09:19,790 for those businesses, and those businesses 210 00:09:19,790 --> 00:09:21,290 will end up losing money, which will 211 00:09:21,290 --> 00:09:25,520 have repercussions for the entire market as a whole. 212 00:09:25,520 --> 00:09:28,333 AUDIENCE: [INAUDIBLE] the amount that it went up 213 00:09:28,333 --> 00:09:31,219 shows how their paper was distributed 214 00:09:31,219 --> 00:09:32,670 to all these other companies. 215 00:09:32,670 --> 00:09:34,440 ANDREW LO: It's a combination of how 216 00:09:34,440 --> 00:09:35,880 their paper was distributed. 217 00:09:35,880 --> 00:09:36,910 But more than that-- 218 00:09:36,910 --> 00:09:39,190 I mean, there are many companies in the S&P 500, 219 00:09:39,190 --> 00:09:42,660 for example, that don't own any of this paper. 220 00:09:42,660 --> 00:09:45,060 So why would their stock be void? 221 00:09:45,060 --> 00:09:47,400 It's because the business conditions 222 00:09:47,400 --> 00:09:51,060 have been stabilized, and there won't be any knock on effects. 223 00:09:51,060 --> 00:09:53,070 A good example of this is Lehman Brothers. 224 00:09:53,070 --> 00:09:54,930 As many of you know, Lehman Brothers 225 00:09:54,930 --> 00:09:57,390 is a big player in these kinds of securities, 226 00:09:57,390 --> 00:10:00,210 and they are currently under a lot of pressure. 227 00:10:00,210 --> 00:10:02,050 Their stock prices dropped dramatically, 228 00:10:02,050 --> 00:10:06,180 even in the last few days, because they are a big mortgage 229 00:10:06,180 --> 00:10:09,120 lender, and CDO investor, so they're actually 230 00:10:09,120 --> 00:10:10,860 hit pretty hard by all of this. 231 00:10:10,860 --> 00:10:13,500 And while the rescue of Freddie and Fannie 232 00:10:13,500 --> 00:10:17,140 has had some positive effects on Lehman's stock price, 233 00:10:17,140 --> 00:10:19,860 it still is under fire and a lot of people 234 00:10:19,860 --> 00:10:22,390 want to get rid of it. 235 00:10:22,390 --> 00:10:25,830 Imagine if Freddie and Fannie weren't rescued. 236 00:10:25,830 --> 00:10:29,310 It's almost a sure thing that Lehman 237 00:10:29,310 --> 00:10:31,560 would have gone under immediately 238 00:10:31,560 --> 00:10:33,140 as a knock-on effect. 239 00:10:33,140 --> 00:10:34,971 And if Lehman went under, well, I mean, 240 00:10:34,971 --> 00:10:36,720 there are other investment banks out there 241 00:10:36,720 --> 00:10:37,845 that might have gone under. 242 00:10:37,845 --> 00:10:39,660 And now all of a sudden, you have a series 243 00:10:39,660 --> 00:10:41,820 of very large companies that do business 244 00:10:41,820 --> 00:10:44,580 with all of Wall Street that it has gone under. 245 00:10:44,580 --> 00:10:47,010 That's going to have bad repercussions for the stock 246 00:10:47,010 --> 00:10:48,696 market as a whole. 247 00:10:48,696 --> 00:10:50,570 Yeah? 248 00:10:50,570 --> 00:10:57,060 AUDIENCE: [INAUDIBLE] companies, like big companies? 249 00:10:57,060 --> 00:11:00,000 ANDREW LO: Well, the short answer is I don't know. 250 00:11:00,000 --> 00:11:01,260 Nobody knows. 251 00:11:01,260 --> 00:11:06,570 I think that there is a concern that the Fed cannot be viewed 252 00:11:06,570 --> 00:11:09,540 as rescuing every possible financial institution 253 00:11:09,540 --> 00:11:10,950 that's out there. 254 00:11:10,950 --> 00:11:12,327 It's got to stop at some point. 255 00:11:12,327 --> 00:11:14,160 Many people said it should have stopped even 256 00:11:14,160 --> 00:11:17,010 before the Bear Stearns rescue. 257 00:11:17,010 --> 00:11:19,500 So the answer is we don't know. 258 00:11:19,500 --> 00:11:22,470 Wait and see, and we'll find out over the next few days. 259 00:11:22,470 --> 00:11:24,990 As I said last time, these are very 260 00:11:24,990 --> 00:11:27,030 interesting times for financial markets. 261 00:11:27,030 --> 00:11:29,730 Very, very serious issues that are coming to the forefront 262 00:11:29,730 --> 00:11:30,732 literally every day. 263 00:11:30,732 --> 00:11:32,190 So we're going to be watching that, 264 00:11:32,190 --> 00:11:33,680 and we'll be talking about that. 265 00:11:33,680 --> 00:11:34,828 Yeah? 266 00:11:34,828 --> 00:11:36,280 AUDIENCE: [INAUDIBLE]. 267 00:11:40,905 --> 00:11:42,780 ANDREW LO: Where do I think that should stop? 268 00:11:42,780 --> 00:11:47,000 Well, well, there are a couple of issues 269 00:11:47,000 --> 00:11:50,590 that are at the heart of these discussions. 270 00:11:50,590 --> 00:11:52,760 The two issues are, how do you balance 271 00:11:52,760 --> 00:11:57,080 of the cost of bailing out these large organizations 272 00:11:57,080 --> 00:12:00,410 and the implicit moral hazard that it creates, 273 00:12:00,410 --> 00:12:03,230 the kind of potential promises that you're implicitly 274 00:12:03,230 --> 00:12:07,040 making to future equity holders of these organizations 275 00:12:07,040 --> 00:12:12,290 versus letting the market work against the potential disaster 276 00:12:12,290 --> 00:12:16,040 scenario of allowing these kinds of events 277 00:12:16,040 --> 00:12:17,979 to spread like wildfire. 278 00:12:17,979 --> 00:12:19,520 I don't know how many of you actually 279 00:12:19,520 --> 00:12:22,470 know what happens during wildfires, during forest fires. 280 00:12:22,470 --> 00:12:25,160 But when forest fires get started, 281 00:12:25,160 --> 00:12:27,320 they're actually very difficult to stop. 282 00:12:27,320 --> 00:12:31,340 And every once in a while, they try 283 00:12:31,340 --> 00:12:36,210 to stop a forest fire by creating additional fires. 284 00:12:36,210 --> 00:12:36,710 Right? 285 00:12:36,710 --> 00:12:38,330 This may sound counter-intuitive. 286 00:12:38,330 --> 00:12:42,620 But what they will do is around a raging forest fire, 287 00:12:42,620 --> 00:12:47,840 they will burn what's called a firewall. 288 00:12:47,840 --> 00:12:49,794 That term did not come out of IT. 289 00:12:49,794 --> 00:12:51,710 It actually came out of fighting forest fires. 290 00:12:51,710 --> 00:12:55,700 They will burn a ring around that forest fire, 291 00:12:55,700 --> 00:12:59,360 a controlled burn where they target very specific set 292 00:12:59,360 --> 00:13:02,780 of trees, and they would do it in a controlled fashion, 293 00:13:02,780 --> 00:13:06,490 so that when the forest fire gets to that ring, 294 00:13:06,490 --> 00:13:09,560 it burns itself out. 295 00:13:09,560 --> 00:13:13,070 And one could argue that we need a firewall 296 00:13:13,070 --> 00:13:14,960 around these kinds of events. 297 00:13:14,960 --> 00:13:19,580 We need to have certain financial institutions fail 298 00:13:19,580 --> 00:13:24,070 and stop the spread of this kind of problem. 299 00:13:24,070 --> 00:13:28,390 The difficulty with that analogy is that with a forest fire, 300 00:13:28,390 --> 00:13:31,660 all you need is a helicopter to get up there and see 301 00:13:31,660 --> 00:13:33,265 what's going on. 302 00:13:33,265 --> 00:13:34,390 We don't have a helicopter. 303 00:13:34,390 --> 00:13:37,840 There's no helicopter that tells us where the fires are, 304 00:13:37,840 --> 00:13:41,260 and where the fires may be, and where the underground gasoline 305 00:13:41,260 --> 00:13:44,020 tanks are hidden for future explosions. 306 00:13:44,020 --> 00:13:47,860 We don't know because a lot of this stuff is hidden. 307 00:13:47,860 --> 00:13:51,670 So my own opinion is that we are going 308 00:13:51,670 --> 00:13:57,700 to need to have at least one or two additional large failures, 309 00:13:57,700 --> 00:14:02,170 and people will have to lose money before they understand 310 00:14:02,170 --> 00:14:06,250 that this stuff really is risky, and that the price you pay 311 00:14:06,250 --> 00:14:08,830 for the benefits that you've gotten from these very 312 00:14:08,830 --> 00:14:12,370 handsome returns in the years before this kind of an event 313 00:14:12,370 --> 00:14:14,860 is the fact that every once in a while, 314 00:14:14,860 --> 00:14:18,640 in the parlance of Wall Street, you get your face ripped off. 315 00:14:18,640 --> 00:14:21,260 That's the nature of financial markets. 316 00:14:21,260 --> 00:14:24,220 So I think that it's very dangerous 317 00:14:24,220 --> 00:14:26,060 to rescue these companies. 318 00:14:26,060 --> 00:14:28,420 But at the same time, you have to balance 319 00:14:28,420 --> 00:14:30,820 that against the risk of creating a mass panic. 320 00:14:30,820 --> 00:14:33,460 And if we do create that mass panic, 321 00:14:33,460 --> 00:14:36,130 there's virtually no way to stop it, 322 00:14:36,130 --> 00:14:40,030 and then we will run into a very deep recession and depression 323 00:14:40,030 --> 00:14:42,840 of the likes that we haven't seen since 1929. 324 00:14:42,840 --> 00:14:45,594 That's the balance and the danger. 325 00:14:45,594 --> 00:14:46,568 Yeah? 326 00:14:46,568 --> 00:14:48,029 AUDIENCE: [INAUDIBLE]. 327 00:14:58,542 --> 00:15:00,250 ANDREW LO: Well, you know, that might be. 328 00:15:00,250 --> 00:15:03,010 But let me suggest this. 329 00:15:03,010 --> 00:15:05,920 Let me put that off for a discussion point 330 00:15:05,920 --> 00:15:08,004 until we finish fixed income securities. 331 00:15:08,004 --> 00:15:09,670 Because at that point, I'm going to talk 332 00:15:09,670 --> 00:15:12,410 about the subprime problem specifically. 333 00:15:12,410 --> 00:15:14,980 And I'm going to use the tools that we develop== actually, 334 00:15:14,980 --> 00:15:17,890 you guys are going to use the tools that we develop to figure 335 00:15:17,890 --> 00:15:20,740 out exactly what's happened in these markets, 336 00:15:20,740 --> 00:15:23,900 why they're happening, and how maybe we can get around that. 337 00:15:23,900 --> 00:15:26,680 So let me not give you my view now. 338 00:15:26,680 --> 00:15:28,690 I'd rather have you develop your own views 339 00:15:28,690 --> 00:15:30,890 based upon the tools we develop in this course. 340 00:15:30,890 --> 00:15:31,870 OK? 341 00:15:31,870 --> 00:15:33,483 Yeah? 342 00:15:33,483 --> 00:15:37,226 AUDIENCE: Because of all this [INAUDIBLE] 343 00:15:37,226 --> 00:15:41,190 CEOs or executives were fired to get a big handsome buyout 344 00:15:41,190 --> 00:15:43,480 for all their hard work and efforts. 345 00:15:43,480 --> 00:15:44,370 ANDREW LO: Yeah. 346 00:15:44,370 --> 00:15:46,340 AUDIENCE: But now, should the market 347 00:15:46,340 --> 00:15:48,170 be able to self-regulate itself? 348 00:15:48,170 --> 00:15:51,342 Or does there need to be regulation in place? 349 00:15:51,342 --> 00:15:52,426 Or what will become of it? 350 00:15:52,426 --> 00:15:54,050 ANDREW LO: Well, you know, that's again 351 00:15:54,050 --> 00:15:56,380 a very difficult question to answer because we're not 352 00:15:56,380 --> 00:15:59,251 done yet, so we don't know where this is going to end up. 353 00:15:59,251 --> 00:16:01,750 I think that there are some very important issues that we're 354 00:16:01,750 --> 00:16:03,190 going to have to come back to. 355 00:16:03,190 --> 00:16:05,230 Let me put that off for even a bit longer 356 00:16:05,230 --> 00:16:07,960 because when we talk about corporate finance, 357 00:16:07,960 --> 00:16:10,000 we're going to talk about CEO compensation 358 00:16:10,000 --> 00:16:12,520 and ask the question, how do we relate compensation 359 00:16:12,520 --> 00:16:14,260 to performance, and does it make sense? 360 00:16:14,260 --> 00:16:16,900 It turns out that there's some incentive issues, such that 361 00:16:16,900 --> 00:16:19,060 if we don't do that, if we don't allow 362 00:16:19,060 --> 00:16:21,100 them to have these golden parachutes, 363 00:16:21,100 --> 00:16:24,580 then it may end up creating weird incentives 364 00:16:24,580 --> 00:16:26,620 when things are going well. 365 00:16:26,620 --> 00:16:30,550 So every action has some kind of equal and opposite reaction 366 00:16:30,550 --> 00:16:32,200 in some other part of the system. 367 00:16:32,200 --> 00:16:34,015 And unless you know what that system is, 368 00:16:34,015 --> 00:16:36,140 it's hard to figure out the answer to the question. 369 00:16:36,140 --> 00:16:37,600 So by the end of the semester, I'm 370 00:16:37,600 --> 00:16:39,370 hoping that you'll be able to come up with answers 371 00:16:39,370 --> 00:16:40,370 to these questions. 372 00:16:40,370 --> 00:16:42,120 So let me put that off for a little while. 373 00:16:42,120 --> 00:16:42,730 OK. 374 00:16:42,730 --> 00:16:47,158 One more clarifying question maybe, and then we can move on. 375 00:16:47,158 --> 00:16:51,046 AUDIENCE: During the Southeast Asian Crisis in '97, 376 00:16:51,046 --> 00:16:53,962 there was this discussion about the international financial 377 00:16:53,962 --> 00:16:57,364 institutions should risk [INAUDIBLE] countries 378 00:16:57,364 --> 00:17:01,252 and because of the bar [INAUDIBLE].. 379 00:17:01,252 --> 00:17:02,224 ANDREW LO: Right. 380 00:17:02,224 --> 00:17:03,849 AUDIENCE: And they decided they should, 381 00:17:03,849 --> 00:17:06,902 so they rescued them and they survived. 382 00:17:06,902 --> 00:17:09,382 10 years later, Latin America went into a crisis, 383 00:17:09,382 --> 00:17:11,862 and the same discussion started, and 384 00:17:11,862 --> 00:17:14,342 the international financial institutions, 385 00:17:14,342 --> 00:17:19,302 led by the United States decided not to rescue them. 386 00:17:19,302 --> 00:17:21,790 So we went into a crisis. 387 00:17:21,790 --> 00:17:31,060 And so I see now [INAUDIBLE] 388 00:17:31,060 --> 00:17:32,960 ANDREW LO: That's right. 389 00:17:32,960 --> 00:17:35,230 Yeah, that's a very serious issue. 390 00:17:35,230 --> 00:17:38,820 But I would argue that issue actually goes even-- 391 00:17:38,820 --> 00:17:42,850 it goes to an even broader set of issues that have little 392 00:17:42,850 --> 00:17:44,350 to do with economics and finance, 393 00:17:44,350 --> 00:17:48,490 but political and social issues, which I won't comment on 394 00:17:48,490 --> 00:17:51,490 in this class, but which are important for determining 395 00:17:51,490 --> 00:17:53,170 those kinds of policy questions. 396 00:17:53,170 --> 00:17:56,290 That's one of the things that I'd like to get across to you 397 00:17:56,290 --> 00:17:58,240 in terms of thinking about these issues, which 398 00:17:58,240 --> 00:18:01,980 is that there are multiple aspects to every issue. 399 00:18:01,980 --> 00:18:05,770 And rather than trying to come up with a single answer, what 400 00:18:05,770 --> 00:18:07,480 I would propose that you might do 401 00:18:07,480 --> 00:18:10,150 is when you think about a challenge like this, first 402 00:18:10,150 --> 00:18:14,080 of all, you try to identify the different issues 403 00:18:14,080 --> 00:18:18,430 and then come up with an answer for every single perspective 404 00:18:18,430 --> 00:18:19,060 of that issue. 405 00:18:19,060 --> 00:18:21,539 So for example in the case of Latin America, 406 00:18:21,539 --> 00:18:23,830 there is certainly the economic issue and moral hazard. 407 00:18:23,830 --> 00:18:26,390 That's an important one. 408 00:18:26,390 --> 00:18:28,450 But there's also a political and social issue, 409 00:18:28,450 --> 00:18:33,140 which is that if you don't bail out countries that are in need, 410 00:18:33,140 --> 00:18:36,070 that's a recipe for creating social unrest. 411 00:18:36,070 --> 00:18:38,440 And if you don't do it, there is some dictator 412 00:18:38,440 --> 00:18:43,270 waiting with guns and other interesting possibilities 413 00:18:43,270 --> 00:18:46,630 for the people to try to take over. 414 00:18:46,630 --> 00:18:47,290 That's right. 415 00:18:47,290 --> 00:18:49,640 And I mean, it's not rocket science. 416 00:18:49,640 --> 00:18:51,982 I mean, people are looking for solutions. 417 00:18:51,982 --> 00:18:53,440 And if you can't offer one, they'll 418 00:18:53,440 --> 00:18:55,210 go to the next person that has one. 419 00:18:55,210 --> 00:18:57,370 Whether or not it's true or false, 420 00:18:57,370 --> 00:19:02,030 they will try to come up with some kind of leadership. 421 00:19:02,030 --> 00:19:05,380 So how do you balance off the economic considerations 422 00:19:05,380 --> 00:19:06,940 against the political and social? 423 00:19:06,940 --> 00:19:09,160 That's not something that an economist can answer, 424 00:19:09,160 --> 00:19:10,870 so I won't even try to begin. 425 00:19:10,870 --> 00:19:14,160 And by the way, my opinion is no better or worse than anybody 426 00:19:14,160 --> 00:19:14,660 else's. 427 00:19:14,660 --> 00:19:16,960 So I won't waste your time with that. 428 00:19:16,960 --> 00:19:20,210 But what I would suggest is from looking at these issues, 429 00:19:20,210 --> 00:19:22,930 first of all, try to think clearly 430 00:19:22,930 --> 00:19:25,870 about what the economic issues are, and then 431 00:19:25,870 --> 00:19:28,600 what the social and political issues are, and separate them 432 00:19:28,600 --> 00:19:29,530 out. 433 00:19:29,530 --> 00:19:33,100 And then you can answer each of those questions in isolation 434 00:19:33,100 --> 00:19:36,220 and, at the end, decide on how you want to balance 435 00:19:36,220 --> 00:19:37,930 these kind of considerations. 436 00:19:37,930 --> 00:19:41,230 But don't use economics to try to answer a political question, 437 00:19:41,230 --> 00:19:44,295 and don't use politics to try to answer an economic question. 438 00:19:44,295 --> 00:19:45,670 You should use the tools that you 439 00:19:45,670 --> 00:19:49,180 have to answer the questions that those tools are designed 440 00:19:49,180 --> 00:19:49,810 for. 441 00:19:49,810 --> 00:19:51,184 And in the case of Latin America, 442 00:19:51,184 --> 00:19:53,710 I would argue that's a very complex set of issues that 443 00:19:53,710 --> 00:19:56,910 economics alone cannot answer. 444 00:19:56,910 --> 00:19:59,040 The economic answer, never bail out 445 00:19:59,040 --> 00:20:01,500 countries that are failing, because you'll 446 00:20:01,500 --> 00:20:04,950 create moral hazard and increase the cost of borrowing 447 00:20:04,950 --> 00:20:07,680 for future generations in other countries. 448 00:20:07,680 --> 00:20:10,410 That sounds good until you see what 449 00:20:10,410 --> 00:20:12,300 happens when you don't, and you get 450 00:20:12,300 --> 00:20:14,400 these socialist dictatorships that 451 00:20:14,400 --> 00:20:18,920 end up creating all sorts of dislocation 452 00:20:18,920 --> 00:20:21,780 for the people in the country. 453 00:20:21,780 --> 00:20:24,695 I mean, that there's a very big cost to that as well. 454 00:20:24,695 --> 00:20:26,820 And I'm going to have to beg the question about how 455 00:20:26,820 --> 00:20:29,610 you balance those costs against the benefits. 456 00:20:29,610 --> 00:20:33,450 Again, that's something for politicians and for voters 457 00:20:33,450 --> 00:20:34,790 to hopefully to decide. 458 00:20:38,470 --> 00:20:38,970 Yeah? 459 00:20:38,970 --> 00:20:39,922 Which? 460 00:20:39,922 --> 00:20:43,730 AUDIENCE: [INAUDIBLE]. 461 00:20:43,730 --> 00:20:44,960 ANDREW LO: No. 462 00:20:44,960 --> 00:20:46,418 Sorry. 463 00:20:46,418 --> 00:20:52,286 AUDIENCE: [INAUDIBLE] has renounced the United States 464 00:20:52,286 --> 00:20:53,270 treasury-- 465 00:20:53,270 --> 00:20:53,951 ANDREW LO: Yeah. 466 00:20:53,951 --> 00:20:57,567 AUDIENCE: [INAUDIBLE]. 467 00:20:57,567 --> 00:20:59,900 ANDREW LO: That sounds good, but that wasn't my handout. 468 00:20:59,900 --> 00:21:03,320 So that might be my handout in about three weeks. 469 00:21:03,320 --> 00:21:05,270 But we have work to do now. 470 00:21:05,270 --> 00:21:07,050 So let me let me stick to that. 471 00:21:07,050 --> 00:21:09,050 And we'll come back to these interesting issues. 472 00:21:09,050 --> 00:21:11,540 But I want to give you the framework and the tools 473 00:21:11,540 --> 00:21:12,960 to be able to think about them. 474 00:21:12,960 --> 00:21:13,460 OK. 475 00:21:16,320 --> 00:21:19,350 So let me continue on. 476 00:21:19,350 --> 00:21:21,120 This is Lecture Three. 477 00:21:21,120 --> 00:21:22,890 And we're going to continue looking 478 00:21:22,890 --> 00:21:27,330 at present value relationships and the time value of money. 479 00:21:27,330 --> 00:21:29,910 Last time, we were left with the expression 480 00:21:29,910 --> 00:21:34,410 for the value of an asset as simply being equal to the cash 481 00:21:34,410 --> 00:21:38,250 flows discounted with the appropriate discount factors, 482 00:21:38,250 --> 00:21:42,750 where I've assumed for simplicity that the discount 483 00:21:42,750 --> 00:21:47,250 rate between one year and the next is constant 484 00:21:47,250 --> 00:21:52,020 and given by the interest rate, or discount factor, or cost 485 00:21:52,020 --> 00:21:57,200 of capital, or user cost, or opportunity cost, r. 486 00:21:57,200 --> 00:21:59,880 Fancy terms for the simple concept 487 00:21:59,880 --> 00:22:04,560 of the number that you use to construct these exchange 488 00:22:04,560 --> 00:22:09,840 rates between cash at different points in time. 489 00:22:09,840 --> 00:22:16,220 Now the solution of how you make management decisions given 490 00:22:16,220 --> 00:22:19,910 this simple framework becomes trivial. 491 00:22:19,910 --> 00:22:22,490 Take projects that have positive NPV. 492 00:22:22,490 --> 00:22:23,900 That's it. 493 00:22:23,900 --> 00:22:26,180 When you figure out what the value of a project 494 00:22:26,180 --> 00:22:30,350 is as a function of all of these exchange rates, 495 00:22:30,350 --> 00:22:33,410 you calculate what the present value is. 496 00:22:33,410 --> 00:22:35,060 And if the cost of the investment 497 00:22:35,060 --> 00:22:38,720 is included as a cash flow, possibly a negative cash flow, 498 00:22:38,720 --> 00:22:40,400 you've got the net present value. 499 00:22:40,400 --> 00:22:44,120 And for things that are positive NPV, you want them, 500 00:22:44,120 --> 00:22:45,350 you want to take them. 501 00:22:45,350 --> 00:22:47,760 For things that are negative NPV, you don't want them, 502 00:22:47,760 --> 00:22:51,050 you don't take them, or if you can, you sell them. 503 00:22:51,050 --> 00:22:52,880 All right? 504 00:22:52,880 --> 00:22:54,710 Now, there are many different assumptions 505 00:22:54,710 --> 00:22:56,430 that got us to this point. 506 00:22:56,430 --> 00:22:57,620 We understand that. 507 00:22:57,620 --> 00:23:00,110 We're going to make those assumptions more 508 00:23:00,110 --> 00:23:01,410 and more realistic over time. 509 00:23:01,410 --> 00:23:03,920 That's, in fact, what the rest of the course 510 00:23:03,920 --> 00:23:05,310 is going to be doing. 511 00:23:05,310 --> 00:23:08,862 We're going to be focusing on picking this expression 512 00:23:08,862 --> 00:23:10,070 and making it more realistic. 513 00:23:10,070 --> 00:23:13,010 And it's going to take us 12 more weeks to do that. 514 00:23:13,010 --> 00:23:16,490 So it's non-trivial, but that's exactly the objective. 515 00:23:16,490 --> 00:23:17,138 Yes? 516 00:23:17,138 --> 00:23:22,635 AUDIENCE: Last week, you said [INAUDIBLE] summation 517 00:23:22,635 --> 00:23:23,840 of cash flow. 518 00:23:23,840 --> 00:23:24,460 ANDREW LO: No. 519 00:23:24,460 --> 00:23:28,460 I said the asset was a sequence. 520 00:23:28,460 --> 00:23:30,180 What is an asset? 521 00:23:30,180 --> 00:23:33,390 An asset is a sequence of cash flows. 522 00:23:33,390 --> 00:23:36,890 That's the definition of an asset, not the value. 523 00:23:36,890 --> 00:23:38,630 The value of the asset, remember, 524 00:23:38,630 --> 00:23:42,590 is that function that you stick in a cash flow sequence, 525 00:23:42,590 --> 00:23:45,320 and out pops a number. 526 00:23:45,320 --> 00:23:48,770 So the value of an asset is not the same thing as the asset 527 00:23:48,770 --> 00:23:51,460 itself, right? 528 00:23:51,460 --> 00:23:54,520 You can have a rocket ship that can go to the moon. 529 00:23:54,520 --> 00:23:56,450 That is an asset. 530 00:23:56,450 --> 00:23:59,110 The value of a rocket ship that goes to the moon, that's 531 00:23:59,110 --> 00:24:01,030 a different thing, right? 532 00:24:01,030 --> 00:24:03,850 You need to have this v function in order to figure out 533 00:24:03,850 --> 00:24:05,420 the value of an asset. 534 00:24:05,420 --> 00:24:07,600 But I can't really talk about the value of an asset 535 00:24:07,600 --> 00:24:11,480 unless I've defined the asset to begin with. 536 00:24:11,480 --> 00:24:15,590 So v sub-zero is the value of the asset. 537 00:24:15,590 --> 00:24:16,690 It's not the asset itself. 538 00:24:16,690 --> 00:24:17,990 It's the value of the asset. 539 00:24:17,990 --> 00:24:23,150 The asset itself is the sequence of cash flows. 540 00:24:23,150 --> 00:24:27,060 Now, here's a simple example about how 541 00:24:27,060 --> 00:24:28,560 these discount factors work. 542 00:24:28,560 --> 00:24:30,460 This is just an interest rate example. 543 00:24:30,460 --> 00:24:33,900 If you let little r equal 5%, then you 544 00:24:33,900 --> 00:24:37,470 can figure out what the value of a dollar is in the future, 545 00:24:37,470 --> 00:24:41,171 or you can figure out what the value today of a future dollar 546 00:24:41,171 --> 00:24:41,670 is. 547 00:24:41,670 --> 00:24:44,310 It's just using simple arithmetic 548 00:24:44,310 --> 00:24:45,880 to be able to do that. 549 00:24:45,880 --> 00:24:49,020 So this is just a simple concrete illustration. 550 00:24:49,020 --> 00:24:54,060 And if you graph the present value of a dollar, over time, 551 00:24:54,060 --> 00:24:57,450 you'll notice that as time goes out farther, 552 00:24:57,450 --> 00:24:59,880 the present value of a dollar declines. 553 00:24:59,880 --> 00:25:02,640 Not surprisingly, $1 today is worth more 554 00:25:02,640 --> 00:25:03,930 than the dollar tomorrow. 555 00:25:03,930 --> 00:25:07,380 But $1 tomorrow is worth more than $1 two years from now. 556 00:25:07,380 --> 00:25:09,000 And $1 two years from now is worth 557 00:25:09,000 --> 00:25:12,771 much more than $1 an infinite number of years from now. 558 00:25:12,771 --> 00:25:13,270 Right? 559 00:25:18,270 --> 00:25:23,280 Now here's an example of how you use this valuation approach. 560 00:25:23,280 --> 00:25:26,250 And the problems that we handed out last time 561 00:25:26,250 --> 00:25:29,190 will give you practice in how to think about present value. 562 00:25:29,190 --> 00:25:31,260 So I urge you to do those problems 563 00:25:31,260 --> 00:25:34,220 to make sure you really understand these concepts. 564 00:25:34,220 --> 00:25:36,690 Here's an example where a firm spends 565 00:25:36,690 --> 00:25:39,680 $800,000 every single year for electricity 566 00:25:39,680 --> 00:25:41,210 at its headquarters. 567 00:25:41,210 --> 00:25:44,810 And by installing some kind of specialized computer lighting 568 00:25:44,810 --> 00:25:48,110 system, it turns out that you can reduce your electricity 569 00:25:48,110 --> 00:25:53,000 bills by $90,000 in each of the next three years. 570 00:25:53,000 --> 00:25:55,250 Now, of course, it costs money to install that system. 571 00:25:55,250 --> 00:25:59,220 It costs $230,000 to install that system. 572 00:25:59,220 --> 00:26:02,120 So the question is, is this a good deal? 573 00:26:02,120 --> 00:26:03,960 Should you do it? 574 00:26:03,960 --> 00:26:07,010 That's a management decision. 575 00:26:07,010 --> 00:26:10,910 And the management decision relies on valuation first. 576 00:26:10,910 --> 00:26:14,590 Once you value it, then you can make a decision. 577 00:26:14,590 --> 00:26:19,570 So you've got 90,000, 90,000, 90,000 in the three years 578 00:26:19,570 --> 00:26:22,000 as your cost savings, but it's going 579 00:26:22,000 --> 00:26:25,880 to cost you $230,000 upfront. 580 00:26:25,880 --> 00:26:30,340 Now if it turns out that the interest rate is 4%, 581 00:26:30,340 --> 00:26:33,430 you can figure out what the answer is. 582 00:26:33,430 --> 00:26:40,510 At 4%, it turns out that the NPV of this project 583 00:26:40,510 --> 00:26:42,252 is about $20,000. 584 00:26:44,935 --> 00:26:47,720 So it's a good deal. 585 00:26:47,720 --> 00:26:52,320 On the other hand, if you change the assumptions, 586 00:26:52,320 --> 00:26:54,390 and you make the interest rate something else, 587 00:26:54,390 --> 00:26:59,090 well, it might not be a good deal. 588 00:26:59,090 --> 00:27:01,490 How would you have to change the interest rate 589 00:27:01,490 --> 00:27:03,260 to make this a terrible deal? 590 00:27:06,000 --> 00:27:08,190 Increase or decrease it? 591 00:27:08,190 --> 00:27:09,090 Increase it. 592 00:27:09,090 --> 00:27:09,810 Why? 593 00:27:09,810 --> 00:27:13,070 Why does that make sense? 594 00:27:13,070 --> 00:27:13,582 Yeah? 595 00:27:13,582 --> 00:27:15,159 AUDIENCE: [INAUDIBLE]. 596 00:27:15,159 --> 00:27:15,950 ANDREW LO: Exactly. 597 00:27:15,950 --> 00:27:18,010 With a higher interest rate, money now 598 00:27:18,010 --> 00:27:22,852 is more valuable than the cost savings to your electricity. 599 00:27:22,852 --> 00:27:24,310 How do you know it's more valuable? 600 00:27:24,310 --> 00:27:26,749 AUDIENCE: [INAUDIBLE] 601 00:27:26,749 --> 00:27:27,540 ANDREW LO: Exactly. 602 00:27:27,540 --> 00:27:30,999 The opportunity cost is 10% as opposed to 4%. 603 00:27:30,999 --> 00:27:32,040 It's a lot more valuable. 604 00:27:32,040 --> 00:27:34,690 If you stick it in the bank, you get 10%. 605 00:27:34,690 --> 00:27:42,640 So the cost savings depends on the interest rate at hand. 606 00:27:42,640 --> 00:27:47,350 Once you have the interest rate, you can make a decision. 607 00:27:47,350 --> 00:27:48,875 Where does interest rate come from? 608 00:27:48,875 --> 00:27:49,929 AUDIENCE: [INAUDIBLE] 609 00:27:49,929 --> 00:27:50,720 ANDREW LO: Exactly. 610 00:27:50,720 --> 00:27:52,460 The market. 611 00:27:52,460 --> 00:27:54,460 You don't pick the interest rate out of the air. 612 00:27:54,460 --> 00:27:58,060 You don't say, I sort of feel like it's a 2% kind of day. 613 00:27:58,060 --> 00:28:00,910 The interest rate is what you can get on the open market. 614 00:28:00,910 --> 00:28:03,210 See, that's why the market matters. 615 00:28:03,210 --> 00:28:05,380 It's because if that's a market interest 616 00:28:05,380 --> 00:28:07,720 rate, by saying it's a market interest rate, 617 00:28:07,720 --> 00:28:13,140 it means you can actually get that rate from the market. 618 00:28:13,140 --> 00:28:16,890 And therefore, it's a real number that can be actionable. 619 00:28:16,890 --> 00:28:19,260 It's not a fictitious theoretical construct 620 00:28:19,260 --> 00:28:21,610 that may or may not have any practical bearing. 621 00:28:21,610 --> 00:28:25,660 It's a number that actually you can achieve. 622 00:28:25,660 --> 00:28:27,630 And as a manager, if you're trying 623 00:28:27,630 --> 00:28:30,192 to increase the value of shareholder wealth, 624 00:28:30,192 --> 00:28:31,650 if that's the objective, is to make 625 00:28:31,650 --> 00:28:35,516 more money for the shareholders, this is the way to do it. 626 00:28:35,516 --> 00:28:37,140 So this is what I meant when I told you 627 00:28:37,140 --> 00:28:38,490 at the very beginning of this course 628 00:28:38,490 --> 00:28:40,500 that finance is the most important subject you'll ever 629 00:28:40,500 --> 00:28:41,145 study. 630 00:28:41,145 --> 00:28:43,680 It's because with proper valuation, 631 00:28:43,680 --> 00:28:45,750 management decisions are easy. 632 00:28:45,750 --> 00:28:47,970 Now, it's not always easy to get to the point 633 00:28:47,970 --> 00:28:50,400 where the numbers tell you so much. 634 00:28:50,400 --> 00:28:53,280 And so, management is trying to understand 635 00:28:53,280 --> 00:28:56,164 all of the various different factors and balancing them out. 636 00:28:56,164 --> 00:28:58,080 Like, the kind of questions you were asking me 637 00:28:58,080 --> 00:29:00,930 at the very beginning of class, I can't answer many of them 638 00:29:00,930 --> 00:29:01,740 in the abstract. 639 00:29:01,740 --> 00:29:03,554 It depends on the situation. 640 00:29:03,554 --> 00:29:05,470 And I'm hoping that by the end of this course, 641 00:29:05,470 --> 00:29:07,440 you will know enough about the basic framework 642 00:29:07,440 --> 00:29:10,320 to make those trade-offs yourself. 643 00:29:10,320 --> 00:29:13,350 And then, the art of management works together 644 00:29:13,350 --> 00:29:17,150 with the science of management to come up with good decisions. 645 00:29:17,150 --> 00:29:17,940 OK. 646 00:29:17,940 --> 00:29:19,440 So this is simple. 647 00:29:19,440 --> 00:29:21,990 And in the next few slides, I'm going 648 00:29:21,990 --> 00:29:24,330 to ask you to take a look at examples on your own. 649 00:29:24,330 --> 00:29:27,300 Here's an example, a real live example, 650 00:29:27,300 --> 00:29:29,820 where CNOOC, the Chinese oil company, 651 00:29:29,820 --> 00:29:33,000 made an offer to acquire Unocal about a year, year 652 00:29:33,000 --> 00:29:34,110 and a half ago. 653 00:29:34,110 --> 00:29:36,570 And I would suggest you take a look at this example 654 00:29:36,570 --> 00:29:38,850 and just do the back of the envelope calculation 655 00:29:38,850 --> 00:29:41,280 to see whether or not they provided 656 00:29:41,280 --> 00:29:45,480 a good deal or a bad deal. 657 00:29:45,480 --> 00:29:48,390 But I want to turn now to the main subject of today's 658 00:29:48,390 --> 00:29:53,220 lecture, which is one of the most beautiful formulas 659 00:29:53,220 --> 00:29:54,870 in this entire course. 660 00:29:54,870 --> 00:29:57,210 Now it might seem strange for me to talk about a formula 661 00:29:57,210 --> 00:30:00,270 as being beautiful. 662 00:30:00,270 --> 00:30:04,770 You know, a while ago, Paul Samuelson, the great economist 663 00:30:04,770 --> 00:30:08,700 here at MIT, once said that, you know, 664 00:30:08,700 --> 00:30:12,240 either you think that probability theory is beautiful 665 00:30:12,240 --> 00:30:13,000 or not. 666 00:30:13,000 --> 00:30:14,583 And if you don't think it's beautiful, 667 00:30:14,583 --> 00:30:15,940 then I feel sorry for you. 668 00:30:15,940 --> 00:30:19,020 And I suppose the same can be said for this formula. 669 00:30:19,020 --> 00:30:22,350 It's hard to believe that a formula can be beautiful, 670 00:30:22,350 --> 00:30:24,730 but trust me, it is. 671 00:30:24,730 --> 00:30:27,242 And if you don't think so, I feel sorry for you. 672 00:30:27,242 --> 00:30:28,950 By the end of the semester, hopefully you 673 00:30:28,950 --> 00:30:31,050 will think it's beautiful. 674 00:30:31,050 --> 00:30:34,410 Let me explain what we're about to do. 675 00:30:34,410 --> 00:30:40,710 I want to come up with the value of a very specific asset, 676 00:30:40,710 --> 00:30:44,181 an asset with a very, very simple and interesting cash 677 00:30:44,181 --> 00:30:44,680 flow. 678 00:30:44,680 --> 00:30:47,230 So this is one of the two special cash flows 679 00:30:47,230 --> 00:30:49,830 that we're going to analyze in this class. 680 00:30:49,830 --> 00:30:53,140 And this cash flow is known as a perpetuity. 681 00:30:53,140 --> 00:30:56,340 A perpetuity is exactly what it sounds like. 682 00:30:56,340 --> 00:31:01,470 It pays cash forever. 683 00:31:01,470 --> 00:31:04,970 Now we can debate whether or not forever really exists. 684 00:31:04,970 --> 00:31:09,680 I won't try to argue with you that we will live forever. 685 00:31:09,680 --> 00:31:12,330 But it's a hypothetical construct. 686 00:31:12,330 --> 00:31:12,830 OK? 687 00:31:12,830 --> 00:31:16,040 So this is a figment of our imaginations. 688 00:31:16,040 --> 00:31:20,150 There exists in my imagination a piece of paper 689 00:31:20,150 --> 00:31:22,970 that has a claim, such that whoever 690 00:31:22,970 --> 00:31:26,150 holds the piece of paper will be entitled 691 00:31:26,150 --> 00:31:30,570 to a cash payment of C dollars every year 692 00:31:30,570 --> 00:31:35,174 forever, out to infinity. 693 00:31:35,174 --> 00:31:35,927 OK? 694 00:31:35,927 --> 00:31:37,760 And the question is, how much is this worth? 695 00:31:37,760 --> 00:31:39,980 How much is this piece of paper worth? 696 00:31:39,980 --> 00:31:42,830 It's an asset, because it's a sequence of cash flows. 697 00:31:42,830 --> 00:31:45,830 It just turns out that this cash flow is an infinite sequence. 698 00:31:45,830 --> 00:31:47,420 It never ends. 699 00:31:47,420 --> 00:31:51,090 It's the gift that keeps on giving. 700 00:31:51,090 --> 00:31:55,020 So you would think that it should be worth 701 00:31:55,020 --> 00:31:58,380 an infinite amount, because it pays an infinite amount 702 00:31:58,380 --> 00:32:00,830 of cash, right? 703 00:32:00,830 --> 00:32:02,270 No, that's not right. 704 00:32:02,270 --> 00:32:06,500 And the reason it's not right is because $1 today 705 00:32:06,500 --> 00:32:11,120 is worth more than $1 tomorrow, which is worth more than $1 706 00:32:11,120 --> 00:32:13,445 a year from now, which is worth more than $1 707 00:32:13,445 --> 00:32:14,570 two years from now. 708 00:32:14,570 --> 00:32:17,570 And so the value of a dollar paid out 709 00:32:17,570 --> 00:32:20,510 into the far future declines. 710 00:32:20,510 --> 00:32:23,820 And it turns out that it declines 711 00:32:23,820 --> 00:32:26,940 at a rate for which you can actually figure out 712 00:32:26,940 --> 00:32:29,050 what the value is today. 713 00:32:29,050 --> 00:32:30,630 So here's we're going to do. 714 00:32:30,630 --> 00:32:35,820 Using the same principle of discounting 715 00:32:35,820 --> 00:32:40,590 that we did for the previous set of cash flows, 716 00:32:40,590 --> 00:32:45,390 we're going to take a sequence and discount it. 717 00:32:45,390 --> 00:32:47,220 I'm assuming with the perpetuity, 718 00:32:47,220 --> 00:32:49,452 that it starts paying next year. 719 00:32:49,452 --> 00:32:50,910 So that's the very first cash flow. 720 00:32:50,910 --> 00:32:52,830 We're sitting here at date zero, and it 721 00:32:52,830 --> 00:32:56,070 pays C dollars next year, and then another C dollars the year 722 00:32:56,070 --> 00:32:58,530 after, and then another C dollars the year after that, 723 00:32:58,530 --> 00:32:59,310 and so on. 724 00:32:59,310 --> 00:33:03,840 So we're going to dis count them by 1 plus r, 1 plus r squared, 725 00:33:03,840 --> 00:33:06,630 dot, dot, dot, forever. 726 00:33:06,630 --> 00:33:09,870 And so this is an infinite sequence. 727 00:33:09,870 --> 00:33:13,590 And those of you who were on your high school math team, 728 00:33:13,590 --> 00:33:16,770 you'll know that a quick and dirty way of some summing 729 00:33:16,770 --> 00:33:19,470 that infinite sequence is basically 730 00:33:19,470 --> 00:33:22,290 to multiply both sides by 1 plus r. 731 00:33:22,290 --> 00:33:24,600 And you'll notice that when you do that, 732 00:33:24,600 --> 00:33:28,410 you get the series back again, but with an extra C. 733 00:33:28,410 --> 00:33:31,510 And when you do the subtraction and division, 734 00:33:31,510 --> 00:33:35,040 you end up with this incredibly simple formula 735 00:33:35,040 --> 00:33:38,670 that says that the present value of this claim that 736 00:33:38,670 --> 00:33:42,630 pays C dollars forever is not infinite. 737 00:33:42,630 --> 00:33:44,010 In fact, it's quite finite. 738 00:33:44,010 --> 00:33:46,350 It's C divided by r. 739 00:33:46,350 --> 00:33:49,950 What a simple formula. 740 00:33:49,950 --> 00:33:53,960 If I have a piece of paper that pays $100 a year forever, 741 00:33:53,960 --> 00:33:56,300 and the interest rate is 10%, what 742 00:33:56,300 --> 00:33:59,670 is this piece of paper worth? 743 00:33:59,670 --> 00:34:00,170 Yes? 744 00:34:00,170 --> 00:34:00,960 AUDIENCE: $1,000. 745 00:34:00,960 --> 00:34:01,751 ANDREW LO: Exactly. 746 00:34:01,751 --> 00:34:02,550 $1,000. 747 00:34:02,550 --> 00:34:05,160 Isn't that amazing, that we could actually value something 748 00:34:05,160 --> 00:34:07,410 like that? 749 00:34:07,410 --> 00:34:11,210 If the interest rate is 5%, what is it worth then? 750 00:34:11,210 --> 00:34:12,090 Yeah. 751 00:34:12,090 --> 00:34:12,761 $2,000. 752 00:34:12,761 --> 00:34:13,260 Right. 753 00:34:13,260 --> 00:34:14,350 Simple. 754 00:34:14,350 --> 00:34:19,110 We have complete analytical solution for a cash flow 755 00:34:19,110 --> 00:34:21,449 that, on the surface of it, seems like it should be 756 00:34:21,449 --> 00:34:23,820 worth a huge amount of money. 757 00:34:23,820 --> 00:34:26,272 It's not that huge. 758 00:34:26,272 --> 00:34:26,772 Yeah? 759 00:34:26,772 --> 00:34:29,085 AUDIENCE: [INAUDIBLE] 760 00:34:29,085 --> 00:34:29,960 ANDREW LO: Well, yes. 761 00:34:29,960 --> 00:34:33,199 We're assuming-- assume that interest rates are constant. 762 00:34:33,199 --> 00:34:34,310 Absolutely. 763 00:34:34,310 --> 00:34:35,900 So if interest rates vary. 764 00:34:35,900 --> 00:34:37,040 This formula is not right. 765 00:34:37,040 --> 00:34:38,780 We're going to come to the case where 766 00:34:38,780 --> 00:34:40,909 interest rates vary over time. 767 00:34:40,909 --> 00:34:41,659 So, absolutely. 768 00:34:41,659 --> 00:34:43,699 This is still under the simplistic assumption 769 00:34:43,699 --> 00:34:45,560 that interest rates are the same. 770 00:34:45,560 --> 00:34:47,989 But under that case, I think it's still pretty cool 771 00:34:47,989 --> 00:34:50,530 that we're able to come up with the formula for value, right? 772 00:34:50,530 --> 00:34:51,030 Yeah. 773 00:34:51,030 --> 00:34:56,409 AUDIENCE: [INAUDIBLE] 774 00:34:56,409 --> 00:34:58,430 ANDREW LO: Well, that's a good question. 775 00:34:58,430 --> 00:35:00,550 I was afraid you were going to ask that. 776 00:35:00,550 --> 00:35:01,600 But I am prepared. 777 00:35:01,600 --> 00:35:03,780 I am prepared to answer that. 778 00:35:03,780 --> 00:35:06,880 In the United Kingdom, there is a bond 779 00:35:06,880 --> 00:35:10,720 issued by the government called a console. 780 00:35:10,720 --> 00:35:13,120 And this bond is a perpetuity. 781 00:35:13,120 --> 00:35:18,060 That is, it pays to the holder a fixed amount every year 782 00:35:18,060 --> 00:35:19,430 forever. 783 00:35:19,430 --> 00:35:21,100 Now in that case, forever means as long 784 00:35:21,100 --> 00:35:24,950 as the British government is still in existence. 785 00:35:24,950 --> 00:35:27,660 You know, it's still around. 786 00:35:27,660 --> 00:35:29,052 But that's an example. 787 00:35:29,052 --> 00:35:29,969 AUDIENCE: [INAUDIBLE]. 788 00:35:29,969 --> 00:35:31,093 ANDREW LO: Yes, absolutely. 789 00:35:31,093 --> 00:35:31,720 It trades. 790 00:35:31,720 --> 00:35:34,400 You can buy it, sell it, observe the price. 791 00:35:34,400 --> 00:35:34,970 Absolutely. 792 00:35:34,970 --> 00:35:37,380 Yeah. 793 00:35:37,380 --> 00:35:37,880 Yes? 794 00:35:37,880 --> 00:35:49,760 AUDIENCE: [INAUDIBLE] 795 00:35:49,760 --> 00:35:50,970 ANDREW LO: Right. 796 00:35:50,970 --> 00:35:52,320 Good question. 797 00:35:52,320 --> 00:35:54,560 Where do we get the interest rate? 798 00:35:54,560 --> 00:35:55,250 The market. 799 00:35:55,250 --> 00:35:56,660 Exactly. 800 00:35:56,660 --> 00:36:00,180 So you can either get it from the marketplace-- 801 00:36:00,180 --> 00:36:01,810 so I have a piece of paper. 802 00:36:01,810 --> 00:36:04,460 It pays $1 a year forever. 803 00:36:04,460 --> 00:36:07,490 Who will pay me $5 for this piece of paper. 804 00:36:07,490 --> 00:36:08,090 $6? 805 00:36:08,090 --> 00:36:08,830 $7? 806 00:36:08,830 --> 00:36:11,340 I'll auction it off to the highest bidder, 807 00:36:11,340 --> 00:36:16,850 and that price will translate into an interest rate 808 00:36:16,850 --> 00:36:18,494 determined by the marketplace. 809 00:36:18,494 --> 00:36:19,910 So the short answer is the market. 810 00:36:19,910 --> 00:36:23,030 Now you're asking me probably a deeper question, which 811 00:36:23,030 --> 00:36:24,556 is where does that come from? 812 00:36:24,556 --> 00:36:26,180 Because there are all sorts of factors, 813 00:36:26,180 --> 00:36:29,870 like future, famine, and plagues, and wars, 814 00:36:29,870 --> 00:36:32,280 and all these other issues. 815 00:36:32,280 --> 00:36:35,510 And the answer is, it's an approximation 816 00:36:35,510 --> 00:36:39,110 that market participants make, and they're 817 00:36:39,110 --> 00:36:40,330 willing to live with. 818 00:36:40,330 --> 00:36:41,030 Right? 819 00:36:41,030 --> 00:36:42,470 I'll give you an example. 820 00:36:42,470 --> 00:36:47,000 A few years ago, Walt Disney, the entertainment company, 821 00:36:47,000 --> 00:36:49,840 issued bonds, corporate bonds. 822 00:36:49,840 --> 00:36:52,830 They were 100 year bonds. 823 00:36:52,830 --> 00:36:56,760 They were going to mature in 100 years. 824 00:36:56,760 --> 00:36:58,590 Now, I don't know how many of you 825 00:36:58,590 --> 00:37:00,810 are high school math team jocks, but if you 826 00:37:00,810 --> 00:37:05,110 are, one test is to ask the question, 827 00:37:05,110 --> 00:37:09,430 with this infinite series, if you take it out to 100 terms, 828 00:37:09,430 --> 00:37:11,470 instead of all the way out to infinity, 829 00:37:11,470 --> 00:37:15,580 what percentage of the total market value will you capture? 830 00:37:15,580 --> 00:37:19,270 It turns out that 100 terms is pretty darn close 831 00:37:19,270 --> 00:37:22,120 to infinite in this grand scheme of things with interest rates 832 00:37:22,120 --> 00:37:23,350 that we use. 833 00:37:23,350 --> 00:37:26,440 So that's an example, where when they issued that bond, 834 00:37:26,440 --> 00:37:29,410 and they auctioned if off to the market participants, whoever 835 00:37:29,410 --> 00:37:33,220 bought those bonds, whoever the highest bidders were, 836 00:37:33,220 --> 00:37:34,660 they set the price. 837 00:37:34,660 --> 00:37:38,320 Once you have the price, you can back out and calculate the r. 838 00:37:38,320 --> 00:37:39,550 In fact, let me ask you this. 839 00:37:39,550 --> 00:37:43,200 If I tell you what C is, C is $100, 840 00:37:43,200 --> 00:37:48,130 and I tell you the market price, say it's $500, 841 00:37:48,130 --> 00:37:49,630 what's the interest rate? 842 00:37:49,630 --> 00:37:51,360 Can you figure that out? 843 00:37:51,360 --> 00:37:51,859 Yeah? 844 00:37:51,859 --> 00:37:53,190 AUDIENCE: [INAUDIBLE]. 845 00:37:53,190 --> 00:37:53,940 ANDREW LO: Right. 846 00:37:53,940 --> 00:37:57,310 Exactly it's basically determined. 847 00:37:57,310 --> 00:38:00,930 So the market price for an instrument like this 848 00:38:00,930 --> 00:38:03,930 will give you the market's assessment 849 00:38:03,930 --> 00:38:06,182 of what that interest rate is. 850 00:38:06,182 --> 00:38:21,690 AUDIENCE: [INAUDIBLE] 851 00:38:21,690 --> 00:38:24,060 ANDREW LO: Let me repeat the question in case people 852 00:38:24,060 --> 00:38:25,020 didn't hear. 853 00:38:25,020 --> 00:38:29,640 The question is, am I telling you that with all the PhDs 854 00:38:29,640 --> 00:38:32,910 out there, there is nothing more sophisticated in terms 855 00:38:32,910 --> 00:38:35,550 of pricing these instruments than simply auctioning them 856 00:38:35,550 --> 00:38:38,760 off, as we did to a bunch of MBAs? 857 00:38:38,760 --> 00:38:44,460 Well, first of all, I wouldn't denigrate MBAs that way. 858 00:38:44,460 --> 00:38:49,170 I would argue that the PhDs who are doing research 859 00:38:49,170 --> 00:38:53,280 are ultimately advising the MBAs as to what to bid, 860 00:38:53,280 --> 00:38:55,380 and then the MBAs take into account the business 861 00:38:55,380 --> 00:38:58,410 considerations, as well as the analytics. 862 00:38:58,410 --> 00:39:02,670 And so it's actually a highly complex and sophisticated 863 00:39:02,670 --> 00:39:04,480 process by which the bidding occurs. 864 00:39:04,480 --> 00:39:06,840 In other words, you're not getting amateurs doing it. 865 00:39:06,840 --> 00:39:10,020 You're getting professionals who know how to price these things. 866 00:39:10,020 --> 00:39:12,570 That said, are they going to make mistakes? 867 00:39:12,570 --> 00:39:13,650 Absolutely. 868 00:39:13,650 --> 00:39:17,700 So market pricing is an imperfect mechanism. 869 00:39:17,700 --> 00:39:21,190 But the imperfect mechanism actually works pretty well. 870 00:39:21,190 --> 00:39:23,190 And so far, nobody else has figured out anything 871 00:39:23,190 --> 00:39:25,140 that works any better. 872 00:39:25,140 --> 00:39:27,034 So, yeah? 873 00:39:27,034 --> 00:39:35,228 AUDIENCE: [INAUDIBLE] price [INAUDIBLE] $1 [INAUDIBLE].. 874 00:39:38,120 --> 00:39:40,292 Obviously, they're not just issuing one 875 00:39:40,292 --> 00:39:42,270 to the highest bidder. 876 00:39:42,270 --> 00:39:44,340 ANDREW LO: So the question is, isn't there 877 00:39:44,340 --> 00:39:47,430 a problem in terms of the auction if what we're doing 878 00:39:47,430 --> 00:39:50,131 is determining the price based upon the highest bidder. 879 00:39:50,131 --> 00:39:52,380 Because the highest bidder is typically the individual 880 00:39:52,380 --> 00:39:54,720 that's the most confident. 881 00:39:54,720 --> 00:39:59,180 Or it's possible that that particular bidder knows 882 00:39:59,180 --> 00:40:01,800 something that the rest of the market doesn't. 883 00:40:01,800 --> 00:40:04,230 So I don't know which of those two possibilities 884 00:40:04,230 --> 00:40:06,370 might be the case. 885 00:40:06,370 --> 00:40:08,166 It depends on the market circumstances. 886 00:40:08,166 --> 00:40:09,540 One of the things about auctions, 887 00:40:09,540 --> 00:40:13,080 though, is that the design of the auction 888 00:40:13,080 --> 00:40:15,690 can actually have a big impact on how informative 889 00:40:15,690 --> 00:40:16,770 the price is. 890 00:40:16,770 --> 00:40:21,930 So the standard auction is actually very, very complicated 891 00:40:21,930 --> 00:40:23,910 in terms of the various incentive effects. 892 00:40:23,910 --> 00:40:26,490 But there are more intelligent auctions 893 00:40:26,490 --> 00:40:31,280 that are designed to elicit true responses based upon not just 894 00:40:31,280 --> 00:40:35,770 kind of anxiousness to win, but on what the economic valuation 895 00:40:35,770 --> 00:40:36,270 is. 896 00:40:36,270 --> 00:40:39,510 In fact, there's an example of an auction 897 00:40:39,510 --> 00:40:42,390 that works something like this. 898 00:40:42,390 --> 00:40:46,680 You have bidders bidding for a particular commodity. 899 00:40:46,680 --> 00:40:54,880 And it turns out that the highest bidder wins. 900 00:40:54,880 --> 00:40:59,110 But the highest bidder will pay a price 901 00:40:59,110 --> 00:41:01,895 that is the second highest bidder's price. 902 00:41:04,470 --> 00:41:09,240 So that actually has a very interesting incentive 903 00:41:09,240 --> 00:41:14,100 in the sense that it ends up forcing you to actually reveal 904 00:41:14,100 --> 00:41:16,080 your true preferences. 905 00:41:16,080 --> 00:41:19,350 And we'll come back to that as we talk later on about market 906 00:41:19,350 --> 00:41:21,137 mechanisms and pricing. 907 00:41:21,137 --> 00:41:21,636 Yeah? 908 00:41:21,636 --> 00:41:24,011 AUDIENCE: [INAUDIBLE] mechanisms in auction, for example, 909 00:41:24,011 --> 00:41:25,194 for public contracts-- 910 00:41:25,194 --> 00:41:25,860 ANDREW LO: Yeah. 911 00:41:25,860 --> 00:41:28,170 AUDIENCE: In which they do the average 912 00:41:28,170 --> 00:41:30,660 and they rule out people who have more than 15% 913 00:41:30,660 --> 00:41:31,910 deviation from that. 914 00:41:31,910 --> 00:41:34,456 So it could really go for a very low price. 915 00:41:34,456 --> 00:41:35,290 ANDREW LO: Yeah. 916 00:41:35,290 --> 00:41:37,210 AUDIENCE: It's interpretative that you're like, [INAUDIBLE].. 917 00:41:37,210 --> 00:41:38,460 So you're kicked off the deal. 918 00:41:38,460 --> 00:41:39,460 ANDREW LO: That's right. 919 00:41:39,460 --> 00:41:42,150 So there are mechanisms to try to make the auctions smarter. 920 00:41:42,150 --> 00:41:43,450 And that's one example. 921 00:41:43,450 --> 00:41:45,210 Another example of that. 922 00:41:45,210 --> 00:41:48,420 But we're going to assume for now that the auction mechanism 923 00:41:48,420 --> 00:41:50,520 produces a good price. 924 00:41:50,520 --> 00:41:52,599 Later on, after we figure out how markets work, 925 00:41:52,599 --> 00:41:54,390 we're going to come back and question that. 926 00:41:54,390 --> 00:41:57,590 And the very end of this course, I'm 927 00:41:57,590 --> 00:42:00,590 going to question all of this and confront you 928 00:42:00,590 --> 00:42:05,180 with empirical evidence that describes psychological biases 929 00:42:05,180 --> 00:42:06,830 that all of us have that are hardwired 930 00:42:06,830 --> 00:42:09,710 into us that would make you think that markets 931 00:42:09,710 --> 00:42:11,210 don't work well at all. 932 00:42:11,210 --> 00:42:13,880 And we'll give you a framework for thinking about those two 933 00:42:13,880 --> 00:42:15,155 kinds of phenomenon. 934 00:42:15,155 --> 00:42:16,520 Yeah? 935 00:42:16,520 --> 00:42:20,508 AUDIENCE: I'm just curious to see-- 936 00:42:20,508 --> 00:42:24,780 [INAUDIBLE] would you have bought this [INAUDIBLE] 937 00:42:24,780 --> 00:42:26,202 at market price. 938 00:42:26,202 --> 00:42:31,890 [INAUDIBLE] 939 00:42:31,890 --> 00:42:32,650 ANDREW LO: OK. 940 00:42:32,650 --> 00:42:34,670 The question is, do people's bids actually 941 00:42:34,670 --> 00:42:36,410 reflect interest rates over time? 942 00:42:36,410 --> 00:42:39,600 Well, remember that market conditions are changing. 943 00:42:39,600 --> 00:42:43,790 So the question is, do they reflect people's information 944 00:42:43,790 --> 00:42:45,480 as of when. 945 00:42:45,480 --> 00:42:48,570 I mean, you never step into the same river twice. 946 00:42:48,570 --> 00:42:51,730 So what you bought last year at last year's price 947 00:42:51,730 --> 00:42:54,510 may have no bearing on what you're willing to buy 948 00:42:54,510 --> 00:42:56,250 at this year's price, right? 949 00:42:56,250 --> 00:42:57,730 Things change. 950 00:42:57,730 --> 00:43:00,960 So I'm not sure that that question is well-posed. 951 00:43:00,960 --> 00:43:03,060 At every point in time, if an individual 952 00:43:03,060 --> 00:43:08,560 will pay this C divided by r for a security that pays C forever, 953 00:43:08,560 --> 00:43:09,820 that's the fair market price. 954 00:43:09,820 --> 00:43:13,390 Now in the future, if interest rates change, 955 00:43:13,390 --> 00:43:15,300 the price will change. 956 00:43:15,300 --> 00:43:17,910 But what this does say is a very interesting point 957 00:43:17,910 --> 00:43:19,530 that I think you're getting to, which 958 00:43:19,530 --> 00:43:25,260 is that suppose that C never changes by contract. 959 00:43:25,260 --> 00:43:31,440 If interest rates never change, then this security 960 00:43:31,440 --> 00:43:34,860 will never change in price. 961 00:43:34,860 --> 00:43:40,530 It will have absolutely no price growth. 962 00:43:40,530 --> 00:43:43,560 So here's an example where you buy a piece of paper-- 963 00:43:43,560 --> 00:43:47,640 let's say the interest rate is 10% and C is $100. 964 00:43:47,640 --> 00:43:50,980 You pay $1,000 today. 965 00:43:50,980 --> 00:43:55,310 If next year the interest rate is 10%, this piece of paper's 966 00:43:55,310 --> 00:43:57,110 still worth $1,000. 967 00:43:57,110 --> 00:44:00,530 And then five years from now, if he interest rate is 10%, 968 00:44:00,530 --> 00:44:04,190 the piece of paper's still worth $1,000. 969 00:44:04,190 --> 00:44:05,820 Does that makes sense? 970 00:44:05,820 --> 00:44:08,120 Does that seem to suggest that you got stiffed 971 00:44:08,120 --> 00:44:13,260 because you bought a security and it didn't grow in price? 972 00:44:13,260 --> 00:44:17,670 In fact, the rate of return on that security is 0. 973 00:44:17,670 --> 00:44:18,170 Right? 974 00:44:21,600 --> 00:44:23,560 AUDIENCE: [INAUDIBLE]. 975 00:44:23,560 --> 00:44:25,630 ANDREW LO: Or I mean, a $100 payment. 976 00:44:25,630 --> 00:44:27,570 AUDIENCE: You have one coupon payment plus-- 977 00:44:27,570 --> 00:44:28,680 ANDREW LO: Every year. 978 00:44:28,680 --> 00:44:30,090 Right, exactly . 979 00:44:30,090 --> 00:44:33,890 So it's wrong that the return is zero. 980 00:44:33,890 --> 00:44:35,600 The price return is zero. 981 00:44:35,600 --> 00:44:37,700 There's no price growth. 982 00:44:37,700 --> 00:44:41,790 But meanwhile every year, you've been getting checks for $100. 983 00:44:41,790 --> 00:44:45,440 And if the piece of paper was $1,000 984 00:44:45,440 --> 00:44:49,430 and you've been getting checks for $100 every year, 985 00:44:49,430 --> 00:44:50,924 what's your annual return? 986 00:44:53,610 --> 00:44:54,670 10%. 987 00:44:54,670 --> 00:44:56,520 What's the interest rate? 988 00:44:56,520 --> 00:44:58,530 Oh, funny how that works, huh? 989 00:44:58,530 --> 00:44:59,610 That's great. 990 00:44:59,610 --> 00:45:01,680 You get a 10% return. 991 00:45:01,680 --> 00:45:02,760 Why? 992 00:45:02,760 --> 00:45:04,890 Because you're holding this piece of paper 993 00:45:04,890 --> 00:45:09,510 that generates coupons, and the coupons 994 00:45:09,510 --> 00:45:12,130 end up giving you a 10% rate of return, 995 00:45:12,130 --> 00:45:14,250 because the price of the security 996 00:45:14,250 --> 00:45:19,630 is those coupons discounted at 10%. 997 00:45:19,630 --> 00:45:20,680 Nothing magic about it. 998 00:45:20,680 --> 00:45:21,310 It all adds up. 999 00:45:21,310 --> 00:45:22,870 It all works together. 1000 00:45:22,870 --> 00:45:25,032 OK? 1001 00:45:25,032 --> 00:45:25,978 Yes? 1002 00:45:25,978 --> 00:45:31,227 AUDIENCE: [INAUDIBLE] for example-- 1003 00:45:31,227 --> 00:45:32,810 ANDREW LO: We're going to get to that. 1004 00:45:32,810 --> 00:45:33,710 Yes, we're going to get to that. 1005 00:45:33,710 --> 00:45:35,100 That's my next example. 1006 00:45:35,100 --> 00:45:36,320 Let me hold off on that. 1007 00:45:36,320 --> 00:45:37,360 I want to make sure everybody understands 1008 00:45:37,360 --> 00:45:38,690 the perpetuity though. 1009 00:45:38,690 --> 00:45:42,590 And then we're going to get to the example where C changes. 1010 00:45:42,590 --> 00:45:46,410 Now to your example, what happens if C changes. 1011 00:45:46,410 --> 00:45:52,100 In fact, let's be optimistic and let's say that C grows. 1012 00:45:52,100 --> 00:45:55,010 So not only am I going to pay you something 1013 00:45:55,010 --> 00:45:58,160 forever, but that something, I'm going 1014 00:45:58,160 --> 00:46:02,310 to let that grow by a rate of growth of g. 1015 00:46:02,310 --> 00:46:07,150 So next year, I pay you C. But the year after, I'm 1016 00:46:07,150 --> 00:46:10,900 going to pay you C, multiplied by 1 plus g. 1017 00:46:10,900 --> 00:46:14,670 So let's say g is 5%. 1018 00:46:14,670 --> 00:46:17,090 Then next year, I pay you $100. 1019 00:46:17,090 --> 00:46:20,000 The year after, I pay you $105. 1020 00:46:20,000 --> 00:46:22,130 And the year after that, I'll pay you 1021 00:46:22,130 --> 00:46:29,150 whatever 1.05 squared is times 100, and so on. 1022 00:46:29,150 --> 00:46:33,110 Now, what is this piece of paper worth? 1023 00:46:33,110 --> 00:46:36,160 And if you do the same kind of high school math team 1024 00:46:36,160 --> 00:46:43,000 trick and solve for the present value, you get an answer, 1025 00:46:43,000 --> 00:46:49,040 PV is equal to C divided by r minus g. 1026 00:46:49,040 --> 00:46:51,100 r minus g. 1027 00:46:51,100 --> 00:46:53,000 So you subtract this growth rate. 1028 00:46:53,000 --> 00:46:57,250 Now when you subtract the growth rate, 1029 00:46:57,250 --> 00:46:59,990 that makes the denominator smaller, 1030 00:46:59,990 --> 00:47:02,590 which makes the whole thing bigger, 1031 00:47:02,590 --> 00:47:05,380 which is the right direction because you're 1032 00:47:05,380 --> 00:47:09,010 getting a cash flow that is not steady over time, 1033 00:47:09,010 --> 00:47:10,300 but it's growing over time. 1034 00:47:10,300 --> 00:47:12,670 So it should be worth more. 1035 00:47:12,670 --> 00:47:16,480 And it's worth r minus g more. 1036 00:47:16,480 --> 00:47:19,490 All right? 1037 00:47:19,490 --> 00:47:22,130 Now you notice, I have a little condition at the end of that. 1038 00:47:22,130 --> 00:47:25,220 r has to be greater than g. 1039 00:47:25,220 --> 00:47:27,669 Why do I have that condition? 1040 00:47:27,669 --> 00:47:29,028 Yeah? 1041 00:47:29,028 --> 00:47:33,370 AUDIENCE: [INAUDIBLE] infinite [INAUDIBLE] 1042 00:47:33,370 --> 00:47:36,900 the infinite [INAUDIBLE]. 1043 00:47:36,900 --> 00:47:37,900 ANDREW LO: That's right. 1044 00:47:37,900 --> 00:47:39,550 So let's suppose that r equals g. 1045 00:47:39,550 --> 00:47:40,870 Let's see what happens. 1046 00:47:40,870 --> 00:47:47,440 If r equals g, then the infinite series on top, c divided by 1 1047 00:47:47,440 --> 00:47:53,880 plus r plus C times 1 plus g over 1 plus r squared, 1048 00:47:53,880 --> 00:47:56,670 that's just C over 1 plus r, because I'm assuming g and r 1049 00:47:56,670 --> 00:47:58,890 are the same. 1050 00:47:58,890 --> 00:48:03,210 Plus C over 1 plus r, plus C over 1 plus r, plus C over 1 1051 00:48:03,210 --> 00:48:04,870 plus r. 1052 00:48:04,870 --> 00:48:07,780 I have an infinite number of C over 1 plus r. 1053 00:48:07,780 --> 00:48:10,520 And C over 1 plus r is a finite constant. 1054 00:48:13,640 --> 00:48:17,620 The sum is infinite. 1055 00:48:17,620 --> 00:48:19,900 So at some point, that's going to exceed 1056 00:48:19,900 --> 00:48:23,660 total world GDP, and then beyond it, and then 1057 00:48:23,660 --> 00:48:28,020 the other planets of the solar system, and so on. 1058 00:48:28,020 --> 00:48:29,460 What's going on here? 1059 00:48:29,460 --> 00:48:30,725 Why is it happening? 1060 00:48:33,340 --> 00:48:35,889 Anybody give me the intuition for what's happening? 1061 00:48:35,889 --> 00:48:41,268 AUDIENCE: Because the numbers are going smaller and smaller 1062 00:48:41,268 --> 00:48:45,670 [INAUDIBLE] 1063 00:48:45,670 --> 00:48:46,404 ANDREW LO: Right. 1064 00:48:46,404 --> 00:48:49,652 AUDIENCE: But compared just to zero, the amount of 1065 00:48:49,652 --> 00:48:51,510 [INAUDIBLE]. 1066 00:48:51,510 --> 00:48:52,320 ANDREW LO: Right. 1067 00:48:52,320 --> 00:48:52,820 Yes. 1068 00:48:52,820 --> 00:48:54,060 AUDIENCE: [INAUDIBLE]. 1069 00:48:54,060 --> 00:48:55,140 ANDREW LO: Yeah. 1070 00:48:55,140 --> 00:48:55,740 That's right. 1071 00:48:55,740 --> 00:48:56,940 It's growing. 1072 00:48:56,940 --> 00:49:00,210 But what's the intuition for why that can't persist? 1073 00:49:00,210 --> 00:49:02,118 AUDIENCE: Sounds like you're [INAUDIBLE] 1074 00:49:02,118 --> 00:49:03,550 the 10,000 [? quantity. ?] 1075 00:49:03,550 --> 00:49:05,699 ANDREW LO: Right. 1076 00:49:05,699 --> 00:49:06,240 That's right. 1077 00:49:06,240 --> 00:49:09,240 It's basically working against the time value of money 1078 00:49:09,240 --> 00:49:12,510 because the numerator is growing as fast as the denominator is 1079 00:49:12,510 --> 00:49:13,840 growing. 1080 00:49:13,840 --> 00:49:16,740 So what it says is that the cash that you're presumably 1081 00:49:16,740 --> 00:49:19,260 going to be paying to somebody is actually 1082 00:49:19,260 --> 00:49:21,930 increasing at the exact same rate 1083 00:49:21,930 --> 00:49:25,650 that the discount rate is growing. 1084 00:49:25,650 --> 00:49:28,750 So there's no way to sustain that forever. 1085 00:49:28,750 --> 00:49:31,320 You can't do that forever. 1086 00:49:31,320 --> 00:49:35,520 So it has to be the case that the amount that the cash is 1087 00:49:35,520 --> 00:49:41,470 growing can never exceed the discount rate. 1088 00:49:41,470 --> 00:49:43,590 Now remember, these are all theoretical concepts 1089 00:49:43,590 --> 00:49:46,590 where I'm assuming that growth rate stays the same forever, 1090 00:49:46,590 --> 00:49:49,430 and the interest rate stays the same forever. 1091 00:49:49,430 --> 00:49:54,050 This doesn't rule out for short periods of time growth rates 1092 00:49:54,050 --> 00:49:55,970 exceeding interest rates. 1093 00:49:55,970 --> 00:49:58,370 You just can't do it forever. 1094 00:49:58,370 --> 00:50:02,030 For the last 15 years, China has been growing 1095 00:50:02,030 --> 00:50:04,130 at a rate of approximately 10%. 1096 00:50:04,130 --> 00:50:08,330 Their entire economy has been growing at 10% a year 1097 00:50:08,330 --> 00:50:13,520 for every single year over the past 15 years. 1098 00:50:13,520 --> 00:50:15,690 That can't persist. 1099 00:50:15,690 --> 00:50:19,800 If it did, not only would we all be speaking Chinese, 1100 00:50:19,800 --> 00:50:23,130 but all of the planets in this entire galaxy 1101 00:50:23,130 --> 00:50:25,350 would end up speaking Chinese. 1102 00:50:25,350 --> 00:50:28,650 I mean, growth rates cannot persist forever. 1103 00:50:28,650 --> 00:50:31,860 But here, we're assuming, we're assuming, that this growth rate 1104 00:50:31,860 --> 00:50:34,030 is an infinite growth rate. 1105 00:50:34,030 --> 00:50:35,920 It applies forever. 1106 00:50:35,920 --> 00:50:40,610 So in that sense, it has to be smaller than the discount rate. 1107 00:50:40,610 --> 00:50:41,870 Question? 1108 00:50:41,870 --> 00:50:44,046 OK. 1109 00:50:44,046 --> 00:50:58,380 AUDIENCE: [INAUDIBLE] rest of the world. 1110 00:50:58,380 --> 00:51:01,960 ANDREW LO: Well, there are a couple of problems with that. 1111 00:51:01,960 --> 00:51:05,970 So the question is, what happens when r is actually less than g? 1112 00:51:05,970 --> 00:51:06,691 Right? 1113 00:51:06,691 --> 00:51:08,940 You would think that that gives you a negative number. 1114 00:51:08,940 --> 00:51:11,640 In fact, it doesn't, because there's 1115 00:51:11,640 --> 00:51:13,800 a discontinuity at zero. 1116 00:51:13,800 --> 00:51:18,090 And so this formula is-- that doesn't even apply. 1117 00:51:18,090 --> 00:51:23,020 What happens, if you do the infinite sum, when 1118 00:51:23,020 --> 00:51:28,670 g approaches r, this infinite sum already goes to infinity. 1119 00:51:28,670 --> 00:51:33,890 When g gets above r, it gets to be even more infinite, 1120 00:51:33,890 --> 00:51:36,660 whatever that means. 1121 00:51:36,660 --> 00:51:37,160 Right? 1122 00:51:37,160 --> 00:51:40,010 Because the numerator is then not growing at the same rate, 1123 00:51:40,010 --> 00:51:42,900 but it's growing at a faster rate than the denominator. 1124 00:51:42,900 --> 00:51:44,450 So the formula, you wouldn't even 1125 00:51:44,450 --> 00:51:47,720 get the formula, because now you're dealing with infinities. 1126 00:51:47,720 --> 00:51:48,866 OK? 1127 00:51:48,866 --> 00:51:50,740 AUDIENCE: [INAUDIBLE]. 1128 00:51:50,740 --> 00:51:52,670 ANDREW LO: Right. 1129 00:51:52,670 --> 00:51:53,370 Right. 1130 00:51:53,370 --> 00:51:55,164 AUDIENCE: [INAUDIBLE]. 1131 00:51:55,164 --> 00:51:57,080 ANDREW LO: It would just be an infinite value, 1132 00:51:57,080 --> 00:52:00,330 but an even bigger infinity, whatever that means. 1133 00:52:00,330 --> 00:52:04,340 And so, this formula really only holds under this condition. 1134 00:52:04,340 --> 00:52:08,130 If it were equal to or negative, this formula 1135 00:52:08,130 --> 00:52:09,380 just would not be appropriate. 1136 00:52:09,380 --> 00:52:11,120 You'd have to go back to that formula. 1137 00:52:11,120 --> 00:52:12,620 And what that formula would show you 1138 00:52:12,620 --> 00:52:14,831 is that you're getting an infinity. 1139 00:52:17,130 --> 00:52:17,630 OK? 1140 00:52:17,630 --> 00:52:18,520 So that's a perpetuity. 1141 00:52:18,520 --> 00:52:20,186 And we're going to use this, by the way. 1142 00:52:20,186 --> 00:52:21,722 This may seem kind of theoretical. 1143 00:52:21,722 --> 00:52:23,180 But trust me, it's going to come in 1144 00:52:23,180 --> 00:52:26,520 very handy when we start pricing bonds and stocks. 1145 00:52:26,520 --> 00:52:30,520 So we're going to use this quite a bit. 1146 00:52:30,520 --> 00:52:34,060 Now I want to tell you about a formula that 1147 00:52:34,060 --> 00:52:37,420 is my second favorite formula in this entire course. 1148 00:52:37,420 --> 00:52:39,970 And in a way, this is much more practical, 1149 00:52:39,970 --> 00:52:41,980 and it's very closely related to the perpetuity. 1150 00:52:41,980 --> 00:52:46,360 This formula is a formula for an annuity. 1151 00:52:46,360 --> 00:52:51,190 An annuity is a security that pays a fixed amount every year 1152 00:52:51,190 --> 00:52:54,190 for a finite number of years, and then it stops paying. 1153 00:52:54,190 --> 00:52:56,950 So an example of an annuity is a bond. 1154 00:52:56,950 --> 00:52:58,660 Another example is an auto loan. 1155 00:52:58,660 --> 00:53:00,320 Another example is a mortgage. 1156 00:53:00,320 --> 00:53:04,600 And I think I told you that this mortgage valuation formula is 1157 00:53:04,600 --> 00:53:06,100 one that you're going to use when 1158 00:53:06,100 --> 00:53:08,540 you start thinking about making a home purchase decision. 1159 00:53:08,540 --> 00:53:10,090 And it will actually be this formula 1160 00:53:10,090 --> 00:53:12,280 exactly that you're going to need to use. 1161 00:53:12,280 --> 00:53:13,048 Question? 1162 00:53:13,048 --> 00:53:15,340 AUDIENCE: [INAUDIBLE]. 1163 00:53:15,340 --> 00:53:16,150 ANDREW LO: Yes. 1164 00:53:16,150 --> 00:53:17,620 AUDIENCE: Just one question. 1165 00:53:17,620 --> 00:53:20,944 The principle is returned within these payments, or at the end? 1166 00:53:20,944 --> 00:53:22,610 ANDREW LO: Let me talk about that later. 1167 00:53:22,610 --> 00:53:24,250 Right now, we don't know what principle is. 1168 00:53:24,250 --> 00:53:26,500 So when I talk about bonds, I'm going to come back to that. 1169 00:53:26,500 --> 00:53:27,110 OK? 1170 00:53:27,110 --> 00:53:28,610 So let's not get ahead of ourselves. 1171 00:53:28,610 --> 00:53:30,190 I want to make sure we understand the formula 1172 00:53:30,190 --> 00:53:30,970 and then I'll come to that. 1173 00:53:30,970 --> 00:53:32,590 That's an important point that we're 1174 00:53:32,590 --> 00:53:35,008 going to get to in about a lecture and a half. 1175 00:53:35,008 --> 00:53:36,910 OK? 1176 00:53:36,910 --> 00:53:37,410 OK. 1177 00:53:37,410 --> 00:53:41,460 So let me explain what a perpetuity and an annuity 1178 00:53:41,460 --> 00:53:43,410 are in relationship to each other. 1179 00:53:43,410 --> 00:53:47,260 A perpetuity pays a fixed amount forever. 1180 00:53:47,260 --> 00:53:53,560 An annuity pays a fixed amount for a finite period of time. 1181 00:53:53,560 --> 00:53:58,550 So there's a relationship between the two. 1182 00:53:58,550 --> 00:54:03,080 And in particular, you can think about the value 1183 00:54:03,080 --> 00:54:10,740 of an annuity as the value of a perpetuity 1184 00:54:10,740 --> 00:54:15,270 where you only get to have it for a finite period of time. 1185 00:54:15,270 --> 00:54:16,290 Right? 1186 00:54:16,290 --> 00:54:18,270 Let me explain. 1187 00:54:18,270 --> 00:54:21,360 An annuity, the value of that, is 1188 00:54:21,360 --> 00:54:23,400 given by the expression on the top line. 1189 00:54:23,400 --> 00:54:24,090 Right? 1190 00:54:24,090 --> 00:54:28,230 C, C, C, C for T periods, discounted 1191 00:54:28,230 --> 00:54:30,670 at the appropriate discount rate. 1192 00:54:33,610 --> 00:54:36,180 Now, it turns out that you can come up 1193 00:54:36,180 --> 00:54:39,570 with an expression for what that present value is, again, 1194 00:54:39,570 --> 00:54:43,140 using the high school math team kind of an approach. 1195 00:54:43,140 --> 00:54:46,170 You simply multiply both sides by 1 plus r, 1196 00:54:46,170 --> 00:54:50,310 and then you solve for the present value, 1197 00:54:50,310 --> 00:54:55,640 and you get an expression that looks like this. 1198 00:54:55,640 --> 00:54:58,820 Well, this looks an awful lot like it's related 1199 00:54:58,820 --> 00:55:01,280 to the perpetuity formula. 1200 00:55:01,280 --> 00:55:03,290 You've got to C over r here, but then there's 1201 00:55:03,290 --> 00:55:05,690 some annoying other terms over here. 1202 00:55:09,320 --> 00:55:12,430 So let me give you a thought experiment that will show you 1203 00:55:12,430 --> 00:55:16,300 how to derive this formula in less than one minute 1204 00:55:16,300 --> 00:55:19,320 without any kind of high school math team tricks. 1205 00:55:19,320 --> 00:55:22,500 And the experiment goes like this. 1206 00:55:25,390 --> 00:55:29,590 Suppose that you want to create an annuity, 1207 00:55:29,590 --> 00:55:33,910 but you don't have an annuity at hand. 1208 00:55:33,910 --> 00:55:42,380 Well, one way you can do it is to buy a perpetuity, 1209 00:55:42,380 --> 00:55:47,200 hold it for T periods, and then get rid of it and sell it. 1210 00:55:51,800 --> 00:55:54,500 Now look at the cash flows that you get. 1211 00:55:54,500 --> 00:55:58,800 If you were to take a perpetuity, 1212 00:55:58,800 --> 00:56:02,820 which is the top cash flow, and you 1213 00:56:02,820 --> 00:56:07,830 would subtract from it a perpetuity as of date 1214 00:56:07,830 --> 00:56:10,170 T plus 1-- so you've gotten rid of the perpetuity 1215 00:56:10,170 --> 00:56:12,040 at this point. 1216 00:56:12,040 --> 00:56:15,100 When you take the top cash flow sequence 1217 00:56:15,100 --> 00:56:18,910 and you subtract from it the next cash flow sequence, 1218 00:56:18,910 --> 00:56:21,760 you get the bottom cash flow sequence, 1219 00:56:21,760 --> 00:56:25,370 which is just an annuity. 1220 00:56:25,370 --> 00:56:26,300 Right? 1221 00:56:26,300 --> 00:56:32,340 So an annuity is a perpetuity on borrowed time. 1222 00:56:32,340 --> 00:56:33,630 So what is it worth? 1223 00:56:33,630 --> 00:56:41,080 Well, it's worth whatever it is to buy a perpetuity, 1224 00:56:41,080 --> 00:56:45,250 hold it for T periods, and as soon as it pays off 1225 00:56:45,250 --> 00:56:50,000 in that Tth date, you sell it. 1226 00:56:50,000 --> 00:56:51,950 OK? 1227 00:56:51,950 --> 00:56:54,740 So what's it going to cost? 1228 00:56:54,740 --> 00:56:57,080 What's the value of that? 1229 00:56:57,080 --> 00:57:03,600 The value of that is this is what it costs 1230 00:57:03,600 --> 00:57:07,000 to purchase the annuity today-- 1231 00:57:07,000 --> 00:57:09,131 the perpetuity, sorry. 1232 00:57:09,131 --> 00:57:09,630 Right? 1233 00:57:09,630 --> 00:57:10,620 C over r. 1234 00:57:10,620 --> 00:57:14,640 That's what it costs to purchase the perpetuity today. 1235 00:57:14,640 --> 00:57:18,530 And you're going to hold on to that perpetuity for T days or T 1236 00:57:18,530 --> 00:57:20,180 periods. 1237 00:57:20,180 --> 00:57:26,870 And at date T plus 1, you're going to sell the perpetuity. 1238 00:57:26,870 --> 00:57:29,230 What are you going to get when you sell it? 1239 00:57:29,230 --> 00:57:33,780 What would you get as the payment? 1240 00:57:33,780 --> 00:57:34,519 C over r. 1241 00:57:34,519 --> 00:57:36,810 That's right, because that's the price of a perpetuity. 1242 00:57:36,810 --> 00:57:38,260 The price never changes. 1243 00:57:38,260 --> 00:57:40,080 It's always C over r. 1244 00:57:40,080 --> 00:57:43,950 When do you get paid that price? 1245 00:57:43,950 --> 00:57:45,820 At T or T plus 1? 1246 00:57:45,820 --> 00:57:46,800 AUDIENCE: T plus 1. 1247 00:57:46,800 --> 00:57:49,720 ANDREW LO: Because I want to have T periods a cash flow. 1248 00:57:49,720 --> 00:57:52,080 So I've got to hold onto that perpetuity at least 1249 00:57:52,080 --> 00:57:53,430 until T periods. 1250 00:57:53,430 --> 00:57:55,860 After the Tth date, I sell it, which 1251 00:57:55,860 --> 00:57:59,680 means I sell at the next date, which is T plus 1. 1252 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. 1253 00:58:06,880 --> 00:58:08,582 What is that cash flow worth today? 1254 00:58:12,772 --> 00:58:14,730 Remember, it's at two different points in time. 1255 00:58:14,730 --> 00:58:16,604 I need to use the exchange rate to convert it 1256 00:58:16,604 --> 00:58:17,910 to the same currency. 1257 00:58:17,910 --> 00:58:23,050 What's the exchange rate between date 0 and date t plus 1? 1258 00:58:23,050 --> 00:58:24,803 Yeah? [? Scholmi? ?] 1259 00:58:24,803 --> 00:58:29,720 AUDIENCE: [INAUDIBLE] 1260 00:58:29,720 --> 00:58:31,826 ANDREW LO: By t, or by t plus 1? 1261 00:58:31,826 --> 00:58:33,110 AUDIENCE: By t. 1262 00:58:33,110 --> 00:58:34,310 ANDREW LO: No. 1263 00:58:34,310 --> 00:58:37,000 Close, but no cigar. 1264 00:58:37,000 --> 00:58:37,880 AUDIENCE: t plus 1. 1265 00:58:37,880 --> 00:58:38,360 ANDREW LO: Why t plus 1? 1266 00:58:38,360 --> 00:58:39,830 AUDIENCE: That's the period where you're getting paid. 1267 00:58:39,830 --> 00:58:41,400 ANDREW LO: That's the period where you're getting paid. 1268 00:58:41,400 --> 00:58:42,109 So let's go back. 1269 00:58:42,109 --> 00:58:43,650 And remember, the first thing you do? 1270 00:58:43,650 --> 00:58:44,360 Draw a time line. 1271 00:58:44,360 --> 00:58:45,230 Right? 1272 00:58:45,230 --> 00:58:46,495 So here's the timeline. 1273 00:58:46,495 --> 00:58:48,290 And you see why it's confusing. 1274 00:58:48,290 --> 00:58:51,050 You know, I don't blame you for thinking it's t, 1275 00:58:51,050 --> 00:58:52,754 because I said two periods and you're 1276 00:58:52,754 --> 00:58:54,170 going to sell it after two periods 1277 00:58:54,170 --> 00:58:56,780 but when I say sell it after two periods 1278 00:58:56,780 --> 00:59:00,310 if it's after two periods it's plus 1. 1279 00:59:00,310 --> 00:59:02,060 So take a look at this diagram, and you've 1280 00:59:02,060 --> 00:59:03,200 got to draw the diagram. 1281 00:59:03,200 --> 00:59:05,930 You've got to draw the diagram to really get this. 1282 00:59:05,930 --> 00:59:07,160 OK? 1283 00:59:07,160 --> 00:59:10,620 The top part is a perpetuity. 1284 00:59:10,620 --> 00:59:15,650 The middle part is that same perpetuity at day T plus 1. 1285 00:59:15,650 --> 00:59:21,860 So if you own the top piece, and at the same time 1286 00:59:21,860 --> 00:59:26,330 you sell the middle piece, that means at time T plus 1, 1287 00:59:26,330 --> 00:59:29,020 you're going to give up all of your future cash flows 1288 00:59:29,020 --> 00:59:31,160 because you're going to sell the perpetuity. 1289 00:59:31,160 --> 00:59:39,150 Then you're left with the actual annuity cash flow that we want. 1290 00:59:39,150 --> 00:59:42,740 So the question is, what does this transaction cost? 1291 00:59:42,740 --> 00:59:45,500 I buy that it's going to cost me c over r. 1292 00:59:45,500 --> 00:59:47,720 I sell this. 1293 00:59:47,720 --> 00:59:50,470 This is a sequence of cash flows. 1294 00:59:50,470 --> 00:59:52,320 So if I'm selling a sequence of cash flows, 1295 00:59:52,320 --> 00:59:54,030 I'm selling that value. 1296 00:59:54,030 --> 00:59:56,920 I'm going to receive that value as payment. 1297 00:59:56,920 --> 01:00:01,200 So it's going to reduce my cost, and so like any other sequence 1298 01:00:01,200 --> 01:00:04,290 of cash flows, when I sell this, I have to value it, 1299 01:00:04,290 --> 01:00:07,620 and it turns out that this is equal to the value 1300 01:00:07,620 --> 01:00:11,760 at this date, the value of the perpetuity at this date. 1301 01:00:11,760 --> 01:00:14,850 And what is that value? 1302 01:00:14,850 --> 01:00:16,680 C over r. 1303 01:00:16,680 --> 01:00:20,150 And if it's C over r at this date, what 1304 01:00:20,150 --> 01:00:22,460 is the value at this date? 1305 01:00:22,460 --> 01:00:27,680 I've got to discount it by 1 plus r to the T plus 1, 1306 01:00:27,680 --> 01:00:34,400 because it's T plus 1 periods going back. 1307 01:00:34,400 --> 01:00:37,352 OK? 1308 01:00:37,352 --> 01:00:39,110 Well, actually, sorry. 1309 01:00:39,110 --> 01:00:44,560 T periods, the convention is a little confusing. 1310 01:00:44,560 --> 01:00:48,290 It's T periods because you're at t plus 1, 1311 01:00:48,290 --> 01:00:51,340 and you want to figure out what the value is. 1312 01:00:51,340 --> 01:00:55,600 And the value of the perpetuity at T 1313 01:00:55,600 --> 01:00:59,440 is a perpetuity that starts paying off at T plus 1. 1314 01:00:59,440 --> 01:01:01,170 So you're right. 1315 01:01:01,170 --> 01:01:03,580 It's T, but you're discounting it 1316 01:01:03,580 --> 01:01:07,260 as of the payment as of T plus 1. 1317 01:01:07,260 --> 01:01:07,760 OK? 1318 01:01:10,580 --> 01:01:12,850 How many people are confused? 1319 01:01:12,850 --> 01:01:14,420 OK. 1320 01:01:14,420 --> 01:01:15,574 Yes. 1321 01:01:15,574 --> 01:01:16,490 AUDIENCE: [INAUDIBLE]. 1322 01:01:16,490 --> 01:01:17,660 ANDREW LO: Let me-- 1323 01:01:17,660 --> 01:01:19,110 let me do this on the board. 1324 01:01:19,110 --> 01:01:19,610 Right. 1325 01:01:19,610 --> 01:01:20,109 Exactly. 1326 01:01:20,109 --> 01:01:23,060 Let me do this on the board, because the notation is 1327 01:01:23,060 --> 01:01:23,670 confusing. 1328 01:01:23,670 --> 01:01:25,481 Let me just switch on the light. 1329 01:01:25,481 --> 01:01:25,981 Whoops. 1330 01:01:29,377 --> 01:01:29,877 OK. 1331 01:01:35,730 --> 01:01:38,270 So we're going to start by assuming that we've 1332 01:01:38,270 --> 01:01:40,580 got a perpetuity at date 0. 1333 01:01:40,580 --> 01:01:42,020 So this is date 0. 1334 01:01:42,020 --> 01:01:43,970 And remember, the definition of perpetuity 1335 01:01:43,970 --> 01:01:46,340 is that it starts paying the next period. 1336 01:01:48,890 --> 01:01:55,870 And so it pays C,C until this date. 1337 01:01:55,870 --> 01:01:57,875 And sorry. 1338 01:01:57,875 --> 01:02:06,220 T Plus 1 pays T plus 2, and so on. 1339 01:02:06,220 --> 01:02:08,380 The annuity that we want to value 1340 01:02:08,380 --> 01:02:14,830 is an annuity that is just consisting of the first T cash 1341 01:02:14,830 --> 01:02:16,070 flows. 1342 01:02:16,070 --> 01:02:18,790 Right? 1343 01:02:18,790 --> 01:02:22,960 So what I claim is that if you engage 1344 01:02:22,960 --> 01:02:33,110 in the following transaction, at date 0, you buy a perpetuity, 1345 01:02:33,110 --> 01:02:40,100 and you also agree to sell that perpetuity at date 1346 01:02:40,100 --> 01:02:48,540 after date T. So you sell after T. What that means 1347 01:02:48,540 --> 01:02:50,940 is that you will hold onto the perpetuity 1348 01:02:50,940 --> 01:02:55,080 until it pays you C dollars. 1349 01:02:55,080 --> 01:02:58,780 And as soon as it does that, after it does that, 1350 01:02:58,780 --> 01:03:00,980 you sell it. 1351 01:03:00,980 --> 01:03:03,170 Now when you sell it, what do you get for it? 1352 01:03:03,170 --> 01:03:04,670 You get C over r. 1353 01:03:04,670 --> 01:03:08,150 But the question is, when do you get that C over r? 1354 01:03:08,150 --> 01:03:11,690 If you have a sequence of cash flows 1355 01:03:11,690 --> 01:03:19,970 that starts in year T plus 1, then the value of it at day T 1356 01:03:19,970 --> 01:03:21,620 is C over r right? 1357 01:03:21,620 --> 01:03:25,190 Because a perpetuity by assumption is a piece of paper 1358 01:03:25,190 --> 01:03:30,560 that starts paying off not today, but next year. 1359 01:03:30,560 --> 01:03:34,390 So if it starts paying off next year, 1360 01:03:34,390 --> 01:03:36,340 for every single year thereafter, 1361 01:03:36,340 --> 01:03:42,400 the value at that point is going to be equal to C over r. 1362 01:03:42,400 --> 01:03:45,020 Any questions about that? 1363 01:03:45,020 --> 01:03:45,610 OK . 1364 01:03:45,610 --> 01:03:48,580 So we've now established that when 1365 01:03:48,580 --> 01:03:53,440 you sell these cash flows going out into the infinite future, 1366 01:03:53,440 --> 01:04:01,260 the value at date T is C over r. 1367 01:04:05,090 --> 01:04:07,280 And therefore, if the value at date 1368 01:04:07,280 --> 01:04:12,560 T is C over r, what is the value of date 0? 1369 01:04:12,560 --> 01:04:15,320 You have to bring it back to date 0. 1370 01:04:15,320 --> 01:04:19,080 You're discounting it by T periods. 1371 01:04:19,080 --> 01:04:25,160 So it's C over r times 1 over 1 plus r to the t. 1372 01:04:25,160 --> 01:04:29,060 That's what you get when you sell this perpetuity. 1373 01:04:29,060 --> 01:04:32,360 And what you paid for it is C over r. 1374 01:04:32,360 --> 01:04:36,560 So the value of this particular set of actions 1375 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 1376 01:04:42,260 --> 01:04:47,350 plus r to the T. 1377 01:04:47,350 --> 01:04:51,430 That's the annuity discount formula in a nutshell. 1378 01:04:51,430 --> 01:04:55,930 And this formula is the basis of how you figure out 1379 01:04:55,930 --> 01:04:57,610 your mortgage payments. 1380 01:04:57,610 --> 01:05:03,130 Because a mortgage is where you have an obligation every month 1381 01:05:03,130 --> 01:05:06,820 to pay something to the bank in exchange for a pile of money, 1382 01:05:06,820 --> 01:05:09,080 the money that you used to buy your house. 1383 01:05:09,080 --> 01:05:11,530 And the first time I was buying my, house I 1384 01:05:11,530 --> 01:05:13,626 actually went through this transaction. 1385 01:05:13,626 --> 01:05:16,000 I decided that I was going to just calculate this myself, 1386 01:05:16,000 --> 01:05:17,920 because the interest rate was not 1387 01:05:17,920 --> 01:05:23,374 exactly given by what was in the particular banker's table. 1388 01:05:23,374 --> 01:05:24,790 So I went to the mortgage company. 1389 01:05:24,790 --> 01:05:26,600 It was a bank. 1390 01:05:26,600 --> 01:05:29,080 And I think the interest rate that day 1391 01:05:29,080 --> 01:05:32,410 was something like, I don't know, 8%, 8 and 1/2%, 1392 01:05:32,410 --> 01:05:34,750 or 8 and 3/4%. 1393 01:05:34,750 --> 01:05:37,750 And it turns out that the table, this book, 1394 01:05:37,750 --> 01:05:39,950 that had all of these calculations, all 1395 01:05:39,950 --> 01:05:43,090 of these numbers, didn't have that interest rate. 1396 01:05:43,090 --> 01:05:44,320 It didn't have 8 and 3/4. 1397 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. 1398 01:05:48,400 --> 01:05:51,820 And so I just used this formula, punched in a few numbers, 1399 01:05:51,820 --> 01:05:53,620 and I got my monthly payment. 1400 01:05:53,620 --> 01:05:55,810 And you know, I told the banker, well, you know, 1401 01:05:55,810 --> 01:05:57,940 this is what I'll pay every month. 1402 01:05:57,940 --> 01:06:01,349 And he said, well, you can't just do that. 1403 01:06:01,349 --> 01:06:02,640 I said, well, what do you mean? 1404 01:06:02,640 --> 01:06:04,270 And he says, well, you know, I don't know 1405 01:06:04,270 --> 01:06:05,478 that that's the right number. 1406 01:06:05,478 --> 01:06:08,200 We have to wait for the senior vice president 1407 01:06:08,200 --> 01:06:10,089 to tell me what the right number is. 1408 01:06:10,089 --> 01:06:11,380 Because we don't have the book. 1409 01:06:11,380 --> 01:06:12,960 And he contacted the senior vice president. 1410 01:06:12,960 --> 01:06:14,709 It turned out he did have the book either. 1411 01:06:14,709 --> 01:06:16,900 So they had to call the main branch, 1412 01:06:16,900 --> 01:06:19,900 and somebody had to look it up in this book. 1413 01:06:19,900 --> 01:06:23,350 And sure enough, when they came back with the number, 1414 01:06:23,350 --> 01:06:26,930 it was my number down to the fourth decimal place. 1415 01:06:26,930 --> 01:06:29,890 And so he was amazed like, wow, how did you do that? 1416 01:06:29,890 --> 01:06:31,760 You know, this is amazing. 1417 01:06:31,760 --> 01:06:33,620 You're incredible. 1418 01:06:33,620 --> 01:06:37,960 It's incredible if you don't know this very basic secret. 1419 01:06:37,960 --> 01:06:39,430 So you're going to do this. 1420 01:06:39,430 --> 01:06:40,400 You're going to do this in the problems. 1421 01:06:40,400 --> 01:06:42,649 You're going to calculate mortgage payments, auto loan 1422 01:06:42,649 --> 01:06:44,170 payments, consumer finance payments. 1423 01:06:44,170 --> 01:06:46,930 All of it is based upon this simple formula. 1424 01:06:46,930 --> 01:06:52,990 And you can construct tables, as people have done, 1425 01:06:52,990 --> 01:06:56,480 of what are called annuity discount factors. 1426 01:06:56,480 --> 01:06:58,420 So the annuity discount factor is simply 1427 01:06:58,420 --> 01:07:02,180 separating the interest rate from the cash flow. 1428 01:07:02,180 --> 01:07:05,950 And so when you're going out for a mortgage, the amount 1429 01:07:05,950 --> 01:07:07,630 that you're borrowing, you borrow 1430 01:07:07,630 --> 01:07:12,010 $200,000 for your house, that's the left-hand side. 1431 01:07:12,010 --> 01:07:15,970 Your monthly payment, C, that's the right-hand side. 1432 01:07:15,970 --> 01:07:18,610 And the prevailing interest rate, that's the r. 1433 01:07:18,610 --> 01:07:22,450 So if you know the annuity discount factor, which 1434 01:07:22,450 --> 01:07:24,850 is based just on the interest rate, 1435 01:07:24,850 --> 01:07:27,340 and you know the amount of the loan that you want, 1436 01:07:27,340 --> 01:07:29,500 PV, you can divide and figure out 1437 01:07:29,500 --> 01:07:32,090 what your monthly payments are, or vice versa. 1438 01:07:32,090 --> 01:07:35,314 If you have a particular set of cash flows every month, 1439 01:07:35,314 --> 01:07:36,730 and you have an interest rate, you 1440 01:07:36,730 --> 01:07:39,760 can figure out what your total value of that loan 1441 01:07:39,760 --> 01:07:42,730 is going to be in terms of market terms. 1442 01:07:42,730 --> 01:07:45,650 And so once you have this expression, 1443 01:07:45,650 --> 01:07:49,090 you can use a simple table of numbers 1444 01:07:49,090 --> 01:07:51,626 to calculate these annuity discount factors. 1445 01:07:51,626 --> 01:07:53,750 And then you can compute mortgage payment yourself. 1446 01:07:53,750 --> 01:07:56,200 So this is the kind of number I was talking about. 1447 01:07:56,200 --> 01:07:58,420 Given various different interest rates, 1448 01:07:58,420 --> 01:08:01,300 you can come up with these particular annuity discount 1449 01:08:01,300 --> 01:08:03,060 factors. 1450 01:08:03,060 --> 01:08:06,540 And once you do, you can calculate monthly payments 1451 01:08:06,540 --> 01:08:07,960 very easily. 1452 01:08:07,960 --> 01:08:10,490 So you only need one set of tables. 1453 01:08:10,490 --> 01:08:13,680 And for any kind of mortgage, for any kind of consumer loan, 1454 01:08:13,680 --> 01:08:15,790 you can compute the monthly payments. 1455 01:08:15,790 --> 01:08:16,290 Right? 1456 01:08:16,290 --> 01:08:18,806 Whether it's an auto loan, or a mortgage, it doesn't matter. 1457 01:08:18,806 --> 01:08:20,430 What you need is this table right here. 1458 01:08:20,430 --> 01:08:21,840 Nowadays, we can do it in Excel. 1459 01:08:21,840 --> 01:08:23,220 It's not a big deal. 1460 01:08:23,220 --> 01:08:26,069 But still, you should know what the underlying basis 1461 01:08:26,069 --> 01:08:30,160 is for those calculations. 1462 01:08:30,160 --> 01:08:31,189 OK. 1463 01:08:31,189 --> 01:08:35,120 Now before you finish this, I would 1464 01:08:35,120 --> 01:08:36,979 like you to take a look at a few examples. 1465 01:08:36,979 --> 01:08:39,859 I've given you some here, numerical examples. 1466 01:08:39,859 --> 01:08:43,939 And I want to close this class with a discussion 1467 01:08:43,939 --> 01:08:46,279 about compounding, and then next time, 1468 01:08:46,279 --> 01:08:49,160 finish up with a discussion of inflation. 1469 01:08:49,160 --> 01:08:52,279 Because I don't think we'll have time to do both. 1470 01:08:52,279 --> 01:08:57,080 Compounding is a matter of convention. 1471 01:08:57,080 --> 01:08:59,960 And I want to explain what that convention is and try 1472 01:08:59,960 --> 01:09:02,520 to give you a little bit of motivation for the logic of it, 1473 01:09:02,520 --> 01:09:04,436 so that at least it doesn't look like I'm just 1474 01:09:04,436 --> 01:09:07,060 making it up out of the blue. 1475 01:09:07,060 --> 01:09:10,640 The idea behind convention is to take into account calendar 1476 01:09:10,640 --> 01:09:14,260 effects, and in particular, the possibility 1477 01:09:14,260 --> 01:09:15,420 of early withdrawal. 1478 01:09:15,420 --> 01:09:16,550 Let me explain. 1479 01:09:16,550 --> 01:09:20,680 When I tell you that an interest rate is 10%, 1480 01:09:20,680 --> 01:09:25,029 typically, that quote is in terms of an annual interest 1481 01:09:25,029 --> 01:09:25,899 rate. 1482 01:09:25,899 --> 01:09:28,660 Almost all interest rates in the world 1483 01:09:28,660 --> 01:09:31,930 are quoted on an annualized basis, meaning 1484 01:09:31,930 --> 01:09:33,939 if you were to keep an investment 1485 01:09:33,939 --> 01:09:37,270 for a 12-month period, that's what the rate of return 1486 01:09:37,270 --> 01:09:40,649 for that investment would be. 1487 01:09:40,649 --> 01:09:44,069 The problem with that quote, 10%, 1488 01:09:44,069 --> 01:09:47,340 is that what do you do if you want to withdraw 1489 01:09:47,340 --> 01:09:50,870 your money after six months? 1490 01:09:50,870 --> 01:09:53,890 What should you get paid then? 1491 01:09:53,890 --> 01:09:58,440 Well, it would seem fair, if you agree to a 10% interest 1492 01:09:58,440 --> 01:10:01,920 rate per year, to say, all right, if I take it 1493 01:10:01,920 --> 01:10:04,890 out in six months instead of a year, maybe 1494 01:10:04,890 --> 01:10:07,501 you should only pay me half the interest rate. 1495 01:10:07,501 --> 01:10:08,000 Right? 1496 01:10:08,000 --> 01:10:09,450 That seems like a fair deal. 1497 01:10:09,450 --> 01:10:09,950 Right? 1498 01:10:09,950 --> 01:10:14,100 Instead of 10%, pay me 5%. 1499 01:10:14,100 --> 01:10:19,110 And maybe if I keep it in for only a month, 1500 01:10:19,110 --> 01:10:21,870 it would be fair not to pay me 10% for that month, 1501 01:10:21,870 --> 01:10:28,390 but to pay me 10% divided by 12 for the month. 1502 01:10:28,390 --> 01:10:31,210 The reason that that discussion matters 1503 01:10:31,210 --> 01:10:35,050 is that if you agree that that's the fair thing to do, 1504 01:10:35,050 --> 01:10:40,140 well then 10% is not what you're going to get. 1505 01:10:40,140 --> 01:10:45,180 Because if you get paid 5% interest over the first six 1506 01:10:45,180 --> 01:10:50,450 months, you take your money out of the bank, 1507 01:10:50,450 --> 01:10:53,840 and they give you that 5%, and then 1508 01:10:53,840 --> 01:10:57,560 you put the money back in the bank literally the next minute 1509 01:10:57,560 --> 01:10:59,850 and keep it in for the next six months, 1510 01:10:59,850 --> 01:11:05,670 you're going to earn another 5% on your original amount, 1511 01:11:05,670 --> 01:11:10,290 plus you're going to earn 5% on the first six month's 5%. 1512 01:11:10,290 --> 01:11:13,890 You're going to earn interest on the interest. 1513 01:11:13,890 --> 01:11:15,240 And the banks know that. 1514 01:11:15,240 --> 01:11:19,360 And after a while, they were OK with that. 1515 01:11:19,360 --> 01:11:23,580 That's a convention there's no reason it has to be that way. 1516 01:11:23,580 --> 01:11:27,130 The bank could say, I'm going to give you 10% interest. 1517 01:11:27,130 --> 01:11:29,610 But if you want to withdraw your money in six months, 1518 01:11:29,610 --> 01:11:32,369 I'm going to give you the amount of interest such 1519 01:11:32,369 --> 01:11:34,410 that if you were to take the money out and put it 1520 01:11:34,410 --> 01:11:38,840 right back in and hold it, you would get 10%. 1521 01:11:38,840 --> 01:11:41,570 Does anybody know what that interest rate would be? 1522 01:11:41,570 --> 01:11:42,841 How you figure that out? 1523 01:11:42,841 --> 01:11:43,590 Yeah, [INAUDIBLE]? 1524 01:11:43,590 --> 01:11:45,081 AUDIENCE: [INAUDIBLE]. 1525 01:11:47,752 --> 01:11:48,460 ANDREW LO: Roots. 1526 01:11:48,460 --> 01:11:49,930 That sounds painful . 1527 01:11:49,930 --> 01:11:51,940 Is that like a root canal? 1528 01:11:51,940 --> 01:11:53,329 What root do you mean? 1529 01:11:53,329 --> 01:11:53,870 You're right. 1530 01:11:53,870 --> 01:11:54,411 You're right. 1531 01:11:54,411 --> 01:11:55,078 What is it? 1532 01:11:55,078 --> 01:11:58,360 AUDIENCE: [INAUDIBLE]. 1533 01:11:58,360 --> 01:11:59,692 ANDREW LO: Yes. 1534 01:11:59,692 --> 01:12:03,100 AUDIENCE: [INAUDIBLE]. 1535 01:12:03,100 --> 01:12:07,790 ANDREW LO: Yes The square root. 1536 01:12:07,790 --> 01:12:08,290 Right. 1537 01:12:08,290 --> 01:12:11,461 AUDIENCE: [INAUDIBLE] you get [INAUDIBLE].. 1538 01:12:11,461 --> 01:12:12,460 ANDREW LO: That's right. 1539 01:12:12,460 --> 01:12:13,376 AUDIENCE: [INAUDIBLE]. 1540 01:12:13,376 --> 01:12:14,050 Exactly. 1541 01:12:14,050 --> 01:12:17,230 So what you would do in order to figure out what the six month 1542 01:12:17,230 --> 01:12:20,980 interest rate would be so that when you held 1543 01:12:20,980 --> 01:12:25,240 the interest on the interest over through the whole year, 1544 01:12:25,240 --> 01:12:28,150 it would add up to exactly 10%. 1545 01:12:28,150 --> 01:12:34,630 The way you would do it is 1.10, take the square root of that. 1546 01:12:34,630 --> 01:12:38,240 Minus 1, that's the interest rate 1547 01:12:38,240 --> 01:12:40,747 that you would have for the first six months 1548 01:12:40,747 --> 01:12:41,830 and the second six months. 1549 01:12:41,830 --> 01:12:45,500 A little less than 5%, such that that number, 1550 01:12:45,500 --> 01:12:51,380 when you add 1 and multiply it by itself, you'll get 1.10. 1551 01:12:51,380 --> 01:12:54,410 They don't do that, mainly because nobody likes dealing 1552 01:12:54,410 --> 01:12:56,510 with roots, except dentists. 1553 01:12:56,510 --> 01:12:58,130 OK? 1554 01:12:58,130 --> 01:13:01,417 So what they do is they say, OK, as a matter of convention, 1555 01:13:01,417 --> 01:13:03,000 here's what we're going to do for you. 1556 01:13:03,000 --> 01:13:04,708 This is the deal we're going to give you. 1557 01:13:04,708 --> 01:13:08,510 When we say 10% on an annualized basis, what we mean 1558 01:13:08,510 --> 01:13:11,750 is that it's going to be compounded, 1559 01:13:11,750 --> 01:13:16,700 typically on a monthly basis, and nowadays on a daily basis. 1560 01:13:16,700 --> 01:13:18,470 What that means is that the interest 1561 01:13:18,470 --> 01:13:20,750 rate that you're actually going to get 1562 01:13:20,750 --> 01:13:25,470 is the stated equivalent. 1563 01:13:25,470 --> 01:13:30,340 It's the stated annual rate divided by the compounding 1564 01:13:30,340 --> 01:13:33,140 interval. 1565 01:13:33,140 --> 01:13:37,480 Now that's a good deal when you're a depositor. 1566 01:13:37,480 --> 01:13:41,490 That's not a good deal if you're a borrower. 1567 01:13:41,490 --> 01:13:44,580 Because when they tell you, you want to borrow money from me, 1568 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%. 1569 01:13:48,400 --> 01:13:50,650 But when you actually look at how much interest you're 1570 01:13:50,650 --> 01:13:53,710 paying, you're going to find out that, actually, it's 1571 01:13:53,710 --> 01:13:55,810 more than 10%. 1572 01:13:55,810 --> 01:14:01,550 So that's where the term APR and ERA came from. 1573 01:14:01,550 --> 01:14:02,720 What does APR stand for? 1574 01:14:02,720 --> 01:14:04,430 Anybody know? 1575 01:14:04,430 --> 01:14:07,580 When you see this ad on TV for auto loans, you know, 1576 01:14:07,580 --> 01:14:09,970 [? buyer ?] loans, buy a car, no money down, 1577 01:14:09,970 --> 01:14:15,322 we'll give you a loan, the APR is x%. 1578 01:14:15,322 --> 01:14:16,280 Annual percentage rate. 1579 01:14:16,280 --> 01:14:17,870 That is the stated rate. 1580 01:14:17,870 --> 01:14:22,910 That's not the rate including the effects of compounding. 1581 01:14:22,910 --> 01:14:27,350 So as a depositor, when you're lending money to the bank-- 1582 01:14:27,350 --> 01:14:29,760 that's what it means to deposit money in the bank-- 1583 01:14:29,760 --> 01:14:30,710 that's a good thing. 1584 01:14:30,710 --> 01:14:33,681 Because the annual percentage rate of 10% 1585 01:14:33,681 --> 01:14:35,180 is actually not what you're getting. 1586 01:14:35,180 --> 01:14:36,440 You're getting more than that, because it's 1587 01:14:36,440 --> 01:14:38,780 going to be compounded on a monthly, or in some cases, 1588 01:14:38,780 --> 01:14:40,880 on a daily basis. 1589 01:14:40,880 --> 01:14:41,540 OK? 1590 01:14:41,540 --> 01:14:45,050 In other words, the compounding means you get interest 1591 01:14:45,050 --> 01:14:48,380 on your interest on your interest's interest going 1592 01:14:48,380 --> 01:14:49,200 forward. 1593 01:14:49,200 --> 01:14:51,530 Right? 1594 01:14:51,530 --> 01:14:55,100 So you've got to keep in mind that when you see these 1595 01:14:55,100 --> 01:14:59,870 discount rates being quoted, ask whether or not they are APR, 1596 01:14:59,870 --> 01:15:03,560 annual percentage rate-- that's like the 10% stated rate-- 1597 01:15:03,560 --> 01:15:04,986 or EAR. 1598 01:15:04,986 --> 01:15:08,670 EAR is the equivalent annual rate. 1599 01:15:08,670 --> 01:15:10,760 That's what you're really going to get. 1600 01:15:10,760 --> 01:15:14,060 That's what you would actually get in terms of literal dollars 1601 01:15:14,060 --> 01:15:17,030 at the end of the year if you did nothing but left the money 1602 01:15:17,030 --> 01:15:18,886 in there for that entire year. 1603 01:15:18,886 --> 01:15:21,260 It would include the interest on the interest on interest 1604 01:15:21,260 --> 01:15:22,550 on the interest and so on. 1605 01:15:22,550 --> 01:15:23,153 Yeah? 1606 01:15:23,153 --> 01:15:25,542 AUDIENCE: [INAUDIBLE]. 1607 01:15:25,542 --> 01:15:27,000 ANDREW LO: Annual percentage yield. 1608 01:15:27,000 --> 01:15:28,850 Yeah, that's right. 1609 01:15:28,850 --> 01:15:31,540 Now, this is a very clear example of it. 1610 01:15:31,540 --> 01:15:32,600 OK? 1611 01:15:32,600 --> 01:15:37,460 If you've got $1,000, and there's no compounding effects 1612 01:15:37,460 --> 01:15:41,390 and the interest rate is 10%, you're going to get $1,100. 1613 01:15:41,390 --> 01:15:45,110 If you compound twice a year, which is what the old banks 1614 01:15:45,110 --> 01:15:47,870 used to do because they didn't have calculators in those 1615 01:15:47,870 --> 01:15:51,000 days-- it was kind of hard to compute these numbers-- 1616 01:15:51,000 --> 01:15:53,370 they would compound it twice a year. 1617 01:15:53,370 --> 01:15:56,030 And so you would get credit for the interest, 1618 01:15:56,030 --> 01:15:58,460 and then you would get interest on that interest 1619 01:15:58,460 --> 01:16:02,720 as well as on the original deposit or principle. 1620 01:16:02,720 --> 01:16:05,870 Then that turns into $1,103. 1621 01:16:05,870 --> 01:16:09,260 So being able to compound more frequently 1622 01:16:09,260 --> 01:16:11,630 gives you an additional bonus, right? 1623 01:16:11,630 --> 01:16:13,580 Not much. $3. 1624 01:16:13,580 --> 01:16:16,670 But if you think about this as billions of dollars, 1625 01:16:16,670 --> 01:16:19,430 this starts adding up to be real money. 1626 01:16:19,430 --> 01:16:23,270 Now, if you compound on a quarterly basis, it's $4. 1627 01:16:23,270 --> 01:16:25,770 If you compound on a monthly basis, it's $5. 1628 01:16:25,770 --> 01:16:28,460 That's actually starting to add up to something important. 1629 01:16:28,460 --> 01:16:29,310 Right? 1630 01:16:29,310 --> 01:16:30,492 Yeah? 1631 01:16:30,492 --> 01:16:31,965 AUDIENCE: [INAUDIBLE]. 1632 01:16:43,270 --> 01:16:45,700 ANDREW LO: Well, I mean, I think it's six of one, 1633 01:16:45,700 --> 01:16:47,500 or half a dozen of the other, as they say. 1634 01:16:47,500 --> 01:16:49,060 Banks will compete with each other 1635 01:16:49,060 --> 01:16:52,150 to offer ultimately what the market rate is. 1636 01:16:52,150 --> 01:16:55,060 So they won't play any tricks with this kind of stuff some. 1637 01:16:55,060 --> 01:16:57,610 Banks did play tricks with this early 1638 01:16:57,610 --> 01:16:59,410 on in the early days of banking. 1639 01:16:59,410 --> 01:17:02,230 That's why banking is such a highly regulated industry, 1640 01:17:02,230 --> 01:17:04,300 to make sure that no funny business goes on. 1641 01:17:04,300 --> 01:17:07,660 And frankly, that's why banks are forced now 1642 01:17:07,660 --> 01:17:11,800 to tell you what whether it's an APR or an EAR. 1643 01:17:11,800 --> 01:17:14,740 It's a truth in lending kind of a commitment 1644 01:17:14,740 --> 01:17:17,300 that they are now being forced to make. 1645 01:17:17,300 --> 01:17:21,000 So nowadays, when you get an auto loan or a mortgage, 1646 01:17:21,000 --> 01:17:23,380 they have to tell you, yeah, this is NPR. 1647 01:17:23,380 --> 01:17:25,240 This is the annual percentage rate. 1648 01:17:25,240 --> 01:17:29,770 But your actual rate earned may vary, 1649 01:17:29,770 --> 01:17:32,980 and it may vary because of compounding effects. 1650 01:17:32,980 --> 01:17:35,230 And if you ask them what the effective annual rate is, 1651 01:17:35,230 --> 01:17:37,224 they are obligated to tell you. 1652 01:17:37,224 --> 01:17:39,660 AUDIENCE: [INAUDIBLE] all the information. 1653 01:17:39,660 --> 01:17:43,544 Because with APR, you also need to know the compound-- 1654 01:17:43,544 --> 01:17:44,710 ANDREW LO: Compounding rate. 1655 01:17:44,710 --> 01:17:48,280 But it's now taken for granted that compounding 1656 01:17:48,280 --> 01:17:50,150 happens on a daily basis. 1657 01:17:50,150 --> 01:17:51,890 So that's a given. 1658 01:17:51,890 --> 01:17:52,390 OK? 1659 01:17:52,390 --> 01:17:54,010 Any questions about that? 1660 01:17:54,010 --> 01:17:55,450 AUDIENCE: [INAUDIBLE]. 1661 01:17:55,450 --> 01:17:57,162 ANDREW LO: Compounding does, yes. 1662 01:17:57,162 --> 01:17:58,495 AUDIENCE: For checking accounts? 1663 01:17:58,495 --> 01:18:01,360 ANDREW LO: For checking accounts, for savings accounts. 1664 01:18:01,360 --> 01:18:02,140 Yes, it's daily. 1665 01:18:02,140 --> 01:18:03,096 And you know why? 1666 01:18:03,096 --> 01:18:04,720 It's because they allow you to take out 1667 01:18:04,720 --> 01:18:06,400 money on a daily basis. 1668 01:18:06,400 --> 01:18:08,440 So if they didn't do it on a daily basis, 1669 01:18:08,440 --> 01:18:10,507 they'd have to figure out on a one-off, 1670 01:18:10,507 --> 01:18:13,090 if you were to take your money out in the middle of the month, 1671 01:18:13,090 --> 01:18:15,640 and I was to take my money out after the first three days, 1672 01:18:15,640 --> 01:18:18,070 and you were to take your money out after five days, 1673 01:18:18,070 --> 01:18:20,620 they'd have to do all these custom calculations for each 1674 01:18:20,620 --> 01:18:21,769 of those circumstances. 1675 01:18:21,769 --> 01:18:22,810 So now they do it simply. 1676 01:18:22,810 --> 01:18:23,980 They say, fine, we're going to give you your interest 1677 01:18:23,980 --> 01:18:24,891 rate every day. 1678 01:18:24,891 --> 01:18:26,890 Every day, we're going to compute your interest. 1679 01:18:26,890 --> 01:18:29,650 So whether you come or go, you will figure out 1680 01:18:29,650 --> 01:18:30,940 when you get the interest. 1681 01:18:30,940 --> 01:18:34,060 For certain market applications, people 1682 01:18:34,060 --> 01:18:36,190 compute interest intraday. 1683 01:18:36,190 --> 01:18:38,492 Like the number of hours you borrow money, 1684 01:18:38,492 --> 01:18:39,700 they will calculate interest. 1685 01:18:39,700 --> 01:18:42,310 There are cases where you need to borrow money, only 1686 01:18:42,310 --> 01:18:45,232 for four hours or three hours. 1687 01:18:45,232 --> 01:18:46,690 I know this sounds like drug money. 1688 01:18:46,690 --> 01:18:49,180 But that's not-- that's not what I'm talking about. 1689 01:18:49,180 --> 01:18:50,980 There are cases where you need very, very 1690 01:18:50,980 --> 01:18:54,520 short-term financing, and you need to borrow the money. 1691 01:18:54,520 --> 01:18:57,160 And in those cases, they compute it on a minute to minute. 1692 01:18:57,160 --> 01:18:59,390 And in some cases, on a continuous basis. 1693 01:18:59,390 --> 01:19:03,100 So I'm going to leave you with a little puzzler, which 1694 01:19:03,100 --> 01:19:08,530 is if this tells you what the effective annual rate is, where 1695 01:19:08,530 --> 01:19:12,040 you're compounding at intervals of n-- 1696 01:19:12,040 --> 01:19:16,330 so if r is an APR, an annual percentage rate, 1697 01:19:16,330 --> 01:19:18,910 and n is denominated in months-- 1698 01:19:18,910 --> 01:19:20,725 so monthly would be 12-- 1699 01:19:23,260 --> 01:19:28,850 what would happen, what would your effective annual rate be, 1700 01:19:28,850 --> 01:19:34,460 if you compounded not every day, not every hour, not 1701 01:19:34,460 --> 01:19:37,610 every minute, not every femtosecond, 1702 01:19:37,610 --> 01:19:40,940 but literally every possible time slice, 1703 01:19:40,940 --> 01:19:42,920 the narrowest time slice you can think of. 1704 01:19:42,920 --> 01:19:49,090 If you did it continuously, if n were to go to infinity, 1705 01:19:49,090 --> 01:19:51,760 what would you get? 1706 01:19:51,760 --> 01:19:52,480 Think about that. 1707 01:19:52,480 --> 01:19:53,650 That's a little puzzle. 1708 01:19:53,650 --> 01:19:57,760 It turns out that's called continuous compounding. 1709 01:19:57,760 --> 01:20:00,990 So you're compounding continuously. 1710 01:20:00,990 --> 01:20:03,780 It turns out that you actually get a number. 1711 01:20:03,780 --> 01:20:08,530 And what that number is is really bizarre. 1712 01:20:08,530 --> 01:20:10,750 So I want you to think about that, 1713 01:20:10,750 --> 01:20:12,510 and we'll take that up next time. 1714 01:20:12,510 --> 01:20:13,970 Thank you.