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0:00 The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare 0:06 continue to offer high quality educational resources for free. To make a donation or view additional materials 0:13 from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. 0:27 PROFESSOR: All right, let’s start. So first of all, I hope you’ve been enjoying the class so far. 0:34 And thank you for filling out the survey. So we got some very useful and interesting feedbacks. 0:43 One of the feedbacks– this is my impression, I haven’t gotten a chance to talk to my co-lecturers 0:50 or colleagues yet, but I read some comments. You were saying that some of the problem sets are quite hard. 0:58 The math part may be a bit more difficult than the lecture. So I’m thinking. 1:05 So this is really the application lecture. And from now, after three more lectures by Choongbum, 1:12 it will be essentially the remainder is all applications. 1:17 The original point of having this class is really to show you how math is applied, 1:23 to show you those cases in different markets, different strategies, and in the real industry. 1:30 So I’m trying to think, how do I give today’s lecture with the right balance? 1:35 This is, after all, a math class. Should I give you more math, or should I– you’ve had enough math. 1:40 I mean, it sounded like from the survey you probably had enough math. So I would probably want to focus a bit more on the application side. 1:47 And from the survey also it seems like most of you enjoyed or wanted to listen to more on the application side. 1:55 So anyway, as you’ve already learned from Peter’s lecture, 2:03 the so-called Modern Portfolio Theory. And it’s actually not that modern 2:08 anymore, but we still call it Modern Portfolio Theory. So you probably wonder, in the real world, 2:14 how actually we use it. Do we follow those steps? Do we do those calculations? 2:20 And so today, I’d like to share with you my experience on that, both in the past, a different area, 2:28 and today probably more focused on the buy side. Oh, come on in. 2:34 Yeah. Actually, these are my colleagues from Harvard Management. 2:39 So– [CHUCKLES] –they will be able to ask me really tough questions. 2:46 So anyway, so how I’m going to start this class. 2:51 You wondered why I handed out to each of you a page. So does everyone have a blank page by now? 2:59 Yeah, actually. Yeah. Could also pass to–? 3:05 Yeah. So I want every one of you to use that blank page 3:10 to construct a portfolio, OK? So you’re saying, well, I haven’t done this before. That’s fine. 3:16 Do it totally from your intuition, from your knowledge base as of now. 3:22 So what I want you to do is to write down, to break down the 100% of what do you 3:28 want to have in your portfolio. OK, you said, give me choices. No, I’m not going to give you choices. 3:34 You think about whatever you like to put down. Wide open, OK? And don’t even ask me the goal or the criteria. 3:41 Base it on what you want to do. And so totally free thinking, but I want 3:48 you to do it in five minutes. So don’t overthink it. And hand it back to me, OK? 3:54 So that’s really the first part. I want you to show intuitively how you 3:59 can construct a portfolio, OK? So what does a portfolio mean? That I have to explain to you. 4:06 Let’s say for undergraduates here, so your parents give you some allowance. You manage to save a $1,000 on the side. 4:12 You decided to put into investments, buying stocks or whatever, or gambling, buy lottery tickets, 4:21 whatever you can do. Just break down your percentage. That could be $1,000, or you could be a portfolio manager 4:27 and have hundreds of billions of dollars, or whatever. Or then and say if they raise some money, start a hedge fund, 4:35 they may have $10,000 just to start with. How do you want to use those money on day one? Just think about it. 4:41 And then so while you’re filling out those pages, please hand it back to me. 4:46 It’s your choice to put your name down or not. And then I will start to assemble those ideas 4:56 and put them on the blackboard. And sometimes I may come back to ask you a question– you know, why did you put this? 5:01 That’s OK. Don’t feel embarrassed. We’re not going to put you on the spot. But the idea is I want to use those examples to show you 5:12 how we actually connect theory with practice. 5:19 I remember when I was a college student I learned a lot of different stuff. But I remember one lecture so well, 5:25 one teacher told me one thing. I still remember vividly well, so I want to pass it on to you. 5:32 So how do we learn something useful, right? You always start with observation. 5:38 5:45 So that’s kind of the physics side. You collect the data. You ask a lot of questions. 5:50 You try to find the patterns. Then what you do, you build models. 5:56 You have a theory. You try to explain what is working, what’s repeatable, 6:03 what’s not repeatable. So that’s where the math comes in. 6:09 You solve the equations. Sometimes in economics, lot of times, unlike physics, the repeatable patterns are not so obvious. 6:18 So what you do after this, so you come back to observations 6:24 again. You confirm your theory, verify your predictions, 6:34 and find your error. Then this feeds back to this rule. 6:39 And a lot of times, the verification process is really about understanding special cases. 6:47 That’s why today I really want to illustrate the portfolio theory using a lot of special cases. 6:54 So can you start to hand back your portfolio construction by now? OK, so just hand back whatever you have. 7:02 If you have one thing on the paper, that’s fine. Or many things on the paper, or you think as a portfolio manager, or you think as a trader, 7:10 or you think simply as a student, as yourself. 7:19 All right, so I’m getting these back. I will start to write on the blackboard. And you can finish what you started. 7:26 8:24 By the way, that’s the only slide I’m going to use today. I’m not concerned– you realize if I show you a lot of slides, 8:30 you probably can’t keep up with me. So I’m going to write down everything, just take my time. And so hopefully you get a chance to think about questions 8:36 as well. 10:16 OK, I think– is anyone finished? Any more? 10:22 OK. All right, OK. 10:33 OK, great. You guys are awesome. 10:40 OK, so let me just have a quick look to see if I missed any, OK? 10:46 Wow, very interesting. So I have to say, some people have high conviction. 10:52 100% of you, one of those. 11:00 I think I’m not going to read your names, so don’t worry, OK? 11:07 OK I’m just going to read the answers that people put down, OK? So small cap equities, bonds, real estate, commodities. 11:16 Those were there. Qualitative strategies, selection strategies, deep value models. 11:23 Food/drug sector models, energy, consumer, S&P index, ETF fund, 11:33 government bonds, top hedge funds. 11:39 So natural resources, timber land, farmland, checking account, stocks, cash, corporate bonds, 11:59 rare coins, lotteries, collectibles. That’s very unique. 12:07 And Apple’s stock, Google stock, gold, long term saving 12:18 annuities. 12:24 So Yahoo, Morgan Stanley stocks. I like that. 12:29 [LAUGHTER] OK. 12:35 Family trust. OK, I think that pretty much covered it. 12:41 OK, so I would say that list is more or less here. 12:46 So after you’ve done this, when you were doing this, what kind of questions came to your mind? 12:54 Anyone wants to– yeah, please. AUDIENCE: [INAUDIBLE] how do I know what’s the right balance 13:00 to draw in my portfolio? Whether it would be cash, bills, or stuff like that? 13:07 PROFESSOR: How do you do it, really? What’s the criteria? And so before we answer the question how 13:12 you do, how do you group assets or exposures or strategies or even people, traders, together– before we 13:22 ask all those questions, we have to ask ourselves another question. What is the goal? 13:28 What is the objective, right? So we understand what portfolio management is. So here in this class, we’re not talking 13:34 about how to come up with a specific winning strategy in trading or investments. 13:40 But we are talking about how to put them together. So this is what portfolio management is about. So before we answer how, let’s see why. 13:48 Why do we do it? Why do we want to have a portfolio, right? That’s a very, very good point. 13:55 So let’s understand the goals of portfolio management. 14:01 So before we understand goals of portfolio management, let’s understand your situations, 14:09 everyone’s situation. 14:32 So let’s look at this chart. So I’m going to plot your spending 14:38 as a function of your age. So when you are age 0 to age 100, 14:47 so everyone’s spending pattern is different. So I’m not going to tell you this is the spending pattern. So obviously when kids are young, 14:54 they probably don’t have a lot of hobbies or tuition, but they have some basic needs. 15:01 So they spend. And then the spending really goes up. Now your parents have to pay your tuition, 15:07 or you have to borrow– loans, scholarships. And then you have college. 15:13 Now you have– you’re married. You have kids. You need to buy a house, buy a car, pay back student loans. 15:19 You have a lot more spending. Then you go on vacation. You buy investments. 15:25 You just have more spending coming up. So but it goes to a certain point. 15:30 You will taper down, right? So you’re not going to keep going forever. So that’s your spending curve. 15:37 And with the other curve, you think about it. It’s what’s your income, what’s your earnings curve. 15:42 You don’t earn anything where you are just born. I use earning. 15:48 So this is spending. 15:54 So let’s call this 50. Your earning probably typically peaks around age 50, 16:04 but it really depends. Then you probably go down, back up. Right, so that’s your earning. 16:10 And do they always match well? 16:17 They don’t. So how do you make up the difference? You hope to have a fund, an investment on the side, 16:24 which can generate those cash flows to balance your earning 16:30 versus your spending. OK, so that’s only one simple way to put it. 16:35 So you’ve got to ask about your situation. What’s your cash flow look like? 16:41 So my objective is, I’m going to retire at age of 50. Then after the age of 50, I will live free. 16:48 I’ll travel around the world. Now I’ll calculate how much money I need. So that’s one situation. The other situation is, I want to graduate and pay back all 16:55 the student loans in one year. So that’s another. And typically people have to plan these out. 17:02 And if I’m managing a university endowment, so I have to think about what the university’s operating 17:10 budget is like, how much money they need every year drawing from this fund. 17:15 And then by maintaining, protecting the total fund for basically a perpetual purpose, right? 17:22 Ongoing and keep growing it. You ask for more contributions, but at the same time generating 17:28 more return. If you have a pension fund, you have to think about what time frame lot of the people, the workers, 17:37 will retire and will actually draw from the pension. And so every situation is very different. 17:43 Let me even expand it. So you think, oh, this is all about investment. No, no, this is not just about investment. 17:51 So I was a trader for a long time at Morgan Stanley, and later on a trading manager. 17:57 So when I had many traders working for me, the question I was facing is how much money 18:04 I need to allocate to each trader to let them trade. How much risk do they take, right? So they said, oh, I have this winning strategy. 18:11 I can make lots of money. Why don’t you give me more limits? No, you’re not going to have all the limits. 18:17 You’re not going to have all the capital we can give to you. Right, so I’m going to explain. 18:23 You have to diversify. At the same time, you have to compare the strategies 18:28 with parameters– liquidity, volatility, and many other parameters. 18:35 And even if you are not managing people, let’s say– I was going to do this, so Dan, [INAUDIBLE], 18:43 Martin and Andrew. So they start a hedge fund together. So each of them had a great strategy. 18:49 Dan has five, Andrew has four, so they altogether have 30 strategies. 18:55 So they raise an amount of money, or they just pool together their savings. 19:00 But how do you decide which strategy to put more money on day one? 19:05 So those questions are very practical. So that’s all. So you understand your goals, that’s then you’re really clear on how much risk you can take. 19:15 So let’s come back to that. So what is risk? As Peter explained in his lecture, risk is actually not very well defined. 19:24 So in the Modern Portfolio Theory, we typically talk about variance or standard deviation 19:30 of return. So today I’m going to start with that concept, but then try to expand it beyond that. 19:37 So stay with that concept for now. Risk, we use standard deviation for now. 19:45 So what are we trying to do? 20:16 So this, you are familiar with this chart, right? So return versus standard deviation. 20:21 Standard deviation is not going to go negative. So we stop at zero. But the return can go below zero. 20:29 And I’m going to review one formula before I go into it. I think it’s useful to review what previously you learned. 20:37 So you let’s say you have– I will also clarify the notation as well so you don’t get confused. 20:42 So let’s say– so Peter mentioned the Harry Markowitz 20:53 Modern Portfolio Theory which won him the Nobel Prize in 1990, right? 20:59 Along with Sharpe and a few others. So it’s a very elegant piece of work. 21:06 But today, I will try to give you some special cases to help you understand that. So let’s review one of the formulas 21:14 here, which is really the definition. So let’s say you have a portfolio. Let’s call the expected return of the portfolio 21:22 is R of P, equal to the sum, a weighted sum, 21:37 of all the expected returns of each asset. You’ll basically linearly allocate them. 21:44 Then the variance– oh, let’s just look at the variance, 21:58 sigma_P squared. 22:13 So these are vectors. This is a matrix. The sigma in the middle is a covariance matrix. 22:19 OK that’s all you need to know about math at this point. So I want us to go through an exercise on that piece of paper 22:27 I just collected back to put your choice of the investment on this chart. 22:33 OK, so let’s start with one. So what is cash? 22:39 Cash has no standard deviation. You hold cash– so it’s going to be on this axis. 22:45 It’s a positive return. So that’s here. So let’s call this cash. 22:53 Where is– and let’s me just think about another example. Where’s lottery? Say you buy Powerball, right? 23:02 So where’s lottery falling? Let’s assume you put everything in lottery. 23:11 So you’re going to lose. So your expected value is very close to lose 100%. 23:18 And your standard deviation is probably very close to 0. So you will be here. 23:24 So some of you say, oh, no, no. It’s not exactly zero. So OK, fine. So maybe it’s somewhere here, OK? 23:30 So not 100%, but you still have a pretty small deviation from losing all the money. 23:37 What is coin flipping? So let’s say you decide to put all your money to gamble 23:43 on a fair coin flip, fair coin. So expected return is zero. 23:50 What is the standard deviation of that? AUDIENCE: 100%? PROFESSOR: Good. 23:56 So 100%. 24:06 So we got the three extreme cases covered. OK, so where is US government bond? 24:17 So let’s just call it five-year note or ten-year bond. So the return is better than cash with some volatility. 24:26 Let’s call it here. 24:32 What is investing in a start up venture capital fund like? 24:39 Pretty up there, right? So you’ll probably get a very high return, by you can lose all your money. 24:45 So probably somewhere here, you see. 24:52 Buying stocks, let’s call it somewhere here. 24:58 Our last application lecture, you heard about investing in commodities, right? Trading gold, oil. 25:04 So that has higher volatility, so sometimes high returns. 25:10 So let’s call this commodity. And the ETF is typically lower than single stock volatility, 25:19 because it’s just like index funds. So here. Are there any other choices you’d like to put on this map? 25:26 OK. 25:31 So let me just look at what you came up with. Real estate, OK. 25:38 Real estate, I would say probably somewhere around here. 25:45 Private equity probably somewhere here. 25:54 Or investing in hedge funds somewhere. So I think that’s enough examples to cover. 26:01 So now let me turn the table around and ask you a question. 26:06 Given this map, how would you like to pick your investments? 26:13 So you learned about the portfolio theory. As a so-called rational investor, 26:19 you try to maximize your return. At the same time, minimize your standard deviation, right? 26:28 I hesitate to use the term “risk,” OK? Because as I said, we need to better define it. 26:33 But let’s just say you try to minimize this but maximize this, the vertical axis. 26:40 OK, so let’s just say you try to find the highest possible return for that portfolio 26:46 with the lowest possible standard deviation. So would you pick this one? 26:53 Would you pick this one? OK, so eliminate those two. 26:58 But for this, that’s actually all possible, right? So then that’s where we learn about the efficient frontier? 27:06 27:12 So what is the efficient frontier? It’s really the possible combinations 27:18 of those investments you can push out to the boundary that you can no longer find another combination– given 27:27 the same standard deviation, you can no longer find a higher return. So you reached the boundary. 27:33 And the same is true that for the same return, you can no longer minimize your standard deviation 27:40 by finding another combination. OK, so that’s called efficient frontier. 27:55 How do you find the efficient frontier? That’s what essentially those work were done 28:01 and it got them the Nobel Prize, obviously. It’s more than that, but you get the flavor 28:07 from the previous lectures. So what I’m going to do today is really reduce 28:13 all of these to the special case of two assets. Now we can really derive a lot of intuition from that. 28:21 28:53 So we have sigma, R. We’re going to ignore what’s below this now, right? We don’t want to be there. 28:59 And we want to stay on the up-right. So let’s consider one special case. 29:05 So again for that, let’s write out for the two assets. So what is R of P? 29:11 It’s w_1 R_1 plus 1 minus w_1 R_2, right? 29:19 Very simple math. And what is sigma_P? 29:24 So the standard deviation of the portfolio– or the variance of that, which is a square– we know that’s 29:46 for the two asset class special case. So let me give you a further restriction– which, let’s 29:55 consider if R_1 equal to R_2. 30:06 Again, here meaning expected return. I’m simplifying some of the notations. 30:12 And sigma_1 equal to 0, and sigma_2 30:21 not equal to 0, so what is rho? What is the correlation? 30:26 Zero, right? 30:32 Because you have no volatility on it. OK, so what is– what’s that? 30:41 AUDIENCE: It’s really undefined. PROFESSOR: It’s really undefined, yes. Yeah. AUDIENCE: [INAUDIBLE] no covariance. 30:47 PROFESSOR: There’s no– yeah, that’s right. OK, so let’s look at this. So you have sigma_2 here. 30:53 Sigma_1 is 0. And you have R_1 equal to R_2. 31:03 What is all R of P? It’s R, right? 31:08 Because the weighting doesn’t matter. So you know it’s going to fall along this line. 31:18 So here is when weight one equal to 0. So you weight everything on the second asset. 31:25 Here you weight the first asset 100%. 31:31 So you have a possible combination along this line, along this flat line. Very simple, right? 31:37 I like to start with a really a simple case. So what if sigma_1 also is not 0, but sigma_1 equal 31:53 to sigma_2. And further, I impose– impose– the correlation to be 0, OK? 32:00 What is this line look like? So I have sigma_2 equal to sigma_1. 32:09 And R_1 is still equal to R_2, so R_P is still equal to R_1 or R_2, right? 32:15 What does this line look like? 32:27 So volatility is the same. Return is those are the same of each of the asset class. 32:32 You have two strategies or two instruments. They are zero-ly correlated. 32:39 How would you combine them? So you take the derivative of this variance 32:47 with regarding to the weight, right? And then you minimize that. 32:52 So what you find is that this point is R_1 equal to 0, 33:01 or– I’m sorry, w_1, or w_1 equal to 1. You’re at this point, right? 33:08 Agreed? So you choose either, you will be ending up– the portfolio exposure in terms of return and variance will be right here. 33:16 But what if you choose– so when you try to find the minimum variance, you actually end up– 33:22 I’m not going to do the math. You can do it afterwards. You check by yourself, OK? You will find at this point, that’s 33:32 when they are equally weighted, half and half. 33:42 So you get square root of that. 33:49 So you actually have a significant reduction of the variance of the portfolio by choosing half and half, 33:57 zero-ly correlated portfolio. So what’s that called? What’s that benefit? Diversification, right? 34:03 When you have less than perfectly correlated, positively correlated assets, you 34:09 can actually achieve the same return but having a lower standard deviation. 34:16 I’ll say, OK, that’s fairly straightforward. So let’s look at a few more special cases. 34:21 I want really to have you establish this intuition. 34:28 So let’s think about what if in the same example, 34:34 what if rho equals to 1, perfectly correlated? 34:43 Then you can’t, right? So you end up at just this one point. 34:49 You agree? OK. 34:55 What if it’s totally negatively correlated? Perfectly negatively correlated. 35:00 What’s this line look like? 35:05 35:17 Right? 35:24 So you if you weight everything to one side, you’re going to still get this point. 35:29 But if you weight half and half, you’re going to achieve basically zero variance. 35:37 I think we showed that last time, you learned that last time. OK, so let’s look beyond those cases. 35:46 So what now? Let’s look at– so R_1 does not equal to R_2 anymore. 35:58 36:09 Sigma_1 equal to 0. There’s no volatility of the first asset. 36:14 So that’s cash, OK? So that’s a riskless asset in the first one. 36:19 So let’s even call that R_1 is less than R_2. So that’s the– right? 36:24 You have the cash asset, and then you have a non-cash asset. Rho equal to 0, zero correlation. 36:30 So let’s look at what this line looks like. 36:54 So R_1, R_2, sigma_2 here. When you weight asset two 100%, you’re 37:06 going to get this point, right? When you weight asset one 100%, you’re 37:15 going to get this point, right? So what’s in the middle of your return 37:24 as a function of variance? Can someone guess? 37:30 AUDIENCE: A parabola? Should it be a parabola? PROFESSOR: Try again. 37:35 AUDIENCE: A parabola. PROFESSOR: Yeah, I know, I know. Thank you. Are there any other answers? 37:45 OK, this is actually I– let me just derive very quickly for you. Sigma_1 equal to 0, rho equal to 0. 37:52 What’s sigma_P? 38:03 Right? And sigma_P is essentially proportional to sigma_2 38:16 with the weighting. OK, and what’s R? 38:21 R is a linear combination of R_1 and R_2. 38:30 So it’s still– so it’s linear. 38:37 OK, so because in these cases, you actually– you 38:47 essentially– your return is a linear function. 38:52 And the slope, what is the slope of this? 39:00 Oh, let’s wait on the slope. So we can come back to this. This actually relates back to the so-called capital market 39:09 line or capital allocation line, OK? Because last time we talked about the efficient frontier. 39:14 That’s when we have no riskless assets in the portfolio, right? 39:25 When you add on cash, then you actually can select. 39:31 You can combine the cash into the portfolio by having a higher boundary, higher Efficient Frontier, 39:42 and essentially a higher return with the same exposure. 39:50 So let’s look at a couple more cases, then I will tell you– so I think let’s look at– so R_1 40:02 is less than R_2. And volatilities are not 0. 40:08 Also, sigma_1 is less than sigma_2, but it has a negative correlation of 1. 40:28 So you’ll have asset one, asset two. And as we know, where you pick half and half, this goes to 0. 40:38 So this is a quadratic function. You can verify and prove it later. 40:43 And what if when rho is equal to 0– 40:54 and actually, I want to– so sigma_1 should be here, OK? 41:02 So when rho is equal to zero, this no longer goes to– the variance can no longer be minimized to 0. 41:10 So this is your efficient frontier, this part. 41:16 41:23 I think that’s enough examples of two assets for the efficient frontier. 41:28 So you get the idea. So then what if we have three assets? So let me just touch upon that very quickly. 41:34 If you have one more asset here, essentially you can solve the same equations. 41:41 And when the– special case: you can verify afterwards, 41:48 if all the volatilities are equal, and zero correlation among the assets. 41:54 You’re going to be able to minimize sigma_P equal to 1 42:00 over the square root of three of sigma_1. OK. 42:07 So it seems pretty neat, right? The math is not hard and straightforward. 42:13 But it gives you the idea how to answer your question, how to select them when you start with two. 42:20 So why are two assets so important? What’s the implication in practice? 42:26 It’s actually a very popular combination. Lot of the asset managers, they simply 42:34 benchmark to bonds versus equity. And then one famous combination is really 60/40. 42:41 They call it a 60/40 combination. 60% in equity, 40% in bonds. And even nowadays, any fund manager, you have that. 42:49 People will still ask you to compare your performance with that combination. So the two-asset examples seem to be quite easy and simple, 42:59 but actually it’s a very important one to compare. 43:05 And that will lead me to get into the risk parity discussion. But before I get to risk parity discussion, 43:13 I want to review the concept of beta and the Sharpe ratio. 44:21 So your portfolio return, this is your benchmark return, 44:28 R of m, expected return. R_f is the risk-free return, so essentially a cash return. 44:36 And alpha is what you can generate additionally. So let’s even not to worry about these small other terms– 44:45 or not necessarily small, but for the simplicity, I’ll just reveal that. So that’s your beta. 44:52 Now what is your Sharpe ratio? 45:14 OK. And you can– so sometimes Sharpe 45:21 ratio is also called risk-weighted return, 45:26 or risk-adjusted return. And how many of you have heard of Kelly’s formula? 45:36 So Kelly’s formula basically gives you that when you have– let’s say in the gambling example, 45:44 you know your winning probability is p. 45:55 So this basically tells you how much to size up, how much you want to bet on. 46:01 So it’s a very simple formula. 46:07 So you have a winning probability of 50/50, 46:13 how much you bet on? Nothing. So if you have p equal to 100%, you bet 100% of your position. 46:23 If you have a winning probability of negative 100%, so what does it mean? 46:29 That means you have a 100% probability of losing it. What do you do? You bet the other way around, right? 46:36 You bet the other side, so that when p is equal to negative– I’m sorry, actually what I should 46:42 say is when p equal to 0, your losing probability becomes 100%, right? 46:47 So you bet 100% the other way, OK? So that I leave to you to think about. 46:54 That’s when you have discrete outcome case. But when you construct a portfolio, 47:00 this leads to the next question. It’s in addition to the efficient frontier discussion, 47:08 is that really all about asset allocation? Is that how we calculate our weights of each asset 47:14 or strategy to choose from? The answer is no, right? So let’s look at a 60/40 portfolio example. 47:22 47:42 So again, two asset stock. Stock is, let’s say, 60% percent, 40% bonds. 47:56 48:06 So on this– so typically your stock volatility 48:31 is higher than the bonds, and the return, expected return, is also higher. 48:36 So your 60/40 combinations likely fall on the higher 48:41 return and the higher standard deviation part of the efficient frontier. 48:48 So the question was– so that’s typically what people do before 2000. A real asset manager, the easiest way or the passive way 48:56 is just to allocate 60/40. But after 2000, what happened was when after the equity 49:05 market peaked and the bond had a huge rally as first Greenspan 49:12 cut interest rates before the Y2K in the year 2000. 49:20 You think it’s kind of funny, but at that time everybody worried about the year 2000. 49:25 All the computers are going to stop working because old software were not prepared for crossing 49:32 this millennium event. So they had to cut interest rates for this event. But actually nothing happened, so everything was OK. 49:41 But that left the market with plenty of cash, and also after the tech bubble burst. 49:46 So that was a good portfolio, but then obviously in 2008 when the equity market crashed, 49:54 the bond market, the government bond hybrid market, had a huge rally. And so that made people question that. 50:03 Is this 60/40 allocation of asset simply by the market 50:09 value the optimal way of doing it, even though you are falling on the Efficient Frontier? 50:15 But how do you compare different points? Is that simple choice of your objectives, your situation, 50:22 or there’s actually other ways to optimize it. 50:28 So that’s where the risk parity concept was really– the concept has been around, but the term 50:34 was really coined in 2005, so quite recently, by a guy named Edward Qian. 50:41 He basically said, OK, instead of allocating 60/40 based on market value, why shouldn’t we 50:49 consider allocating risk? Instead of targeting a return, targeting asset amount– 50:55 let’s think about a case where we can have equal weighting of risk between the two assets. 51:02 So risk parity really means equal risk weighting rather than equal market exposure. 51:09 And then the further step he took was he said, OK. 51:17 So this actually, OK, is equal risk. So you have lower return and a lower risk, a lower 51:24 standard deviation. But sometimes you will really want a higher return, right? How do you satisfy both? 51:31 Higher return and lower risk. Is there a free lunch? 51:37 So he was thinking, right? There is, actually. It’s not quite free, but it’s the closest thing. 51:45 You’ve probably heard this phrase many times. The closest thing in investment to a free lunch 51:50 is diversification. OK, and so he’s using a leverage here as well. 52:00 let me talk about it a bit more, about diversification, give you a couple more examples, OK? 52:07 That phrase about the free lunch and diversification was actually from– was that from Markowitz? 52:14 Or people gave him that term. OK, but anyway. So let me give you another simple example, OK? 52:24 So let’s consider two assets, A and B. In year one, 52:34 A goes up to– it basically doubles. 52:39 And in year two, it goes down 50%. So where does it end up? 52:46 52:58 So it started with 100%. It goes up to 200%. 53:04 Then it goes down 50% on the new base, 53:10 so it returns nothing, right? It comes back. So asset B in year one loses 50%, then doubles, up 100% 53:21 in year two. So asset B basically goes down to 50% 53:30 and it goes back up to 100%. So that’s when you look at them independently. 53:39 But what if you had a 50/50 weight of the two assets? So if someone who is quick on math can tell me, 53:46 what does it change? So A goes up like that, B goes down like that. 53:53 Now you have a 50/50 A and B. So let’s look at magic. 53:59 So in year one, A, you have only 50%. 54:05 So it goes up 100%. 54:12 So that’s up 50% on the total basis. B, you’ll also weight 50%, but it goes down 50%. 54:25 So you have lost 25%. So at the end of year one, you’re 54:31 actually– so this is a combined 50/50 portfolio, year one 54:38 and year two. So you started with 100. You’re up to 1.25 at this point, OK? 54:48 So at the end of year one, you rebalance, right? So you have to come back to 50/50. 54:55 So what do you do? So this becomes 75, right? 55:01 So you no longer have the 50/50 weight equal. So you have to sell A to come back to 50 55:08 and use the money to buy B. So you have a new 50/50 percent weight asset. 55:17 Again, you can figure out the math. But what happens in the following year 55:23 when you have this move, this comes back 50%, this goes up 100%. 55:29 You return another 25% positively without volatility. 55:36 So you have a straight line. You can keep– so this two year is 55:43 a– so that’s so-called diversification benefit. And in the 60/40 bond market, that’s really the idea 55:53 people think about how to combine them. And so let me talk a little bit about risk parity 56:00 and how you actually achieve them. 56:06 I’ll try to leave plenty of time for questions. 56:12 So that’s the return, and so let’s forget about these. 56:17 56:24 So let’s leave cash here, OK? So the previous example I gave you, when you have two assets, 56:35 one is cash, R_1, the other is not. 56:43 The other has a volatility of sigma_2. You have this point, right? 56:49 So and I said, what’s in between? It’s a straight line. That’s your asset allocation, different combination. 56:57 Did it occur to you, why can’t we go beyond this point? 57:04 So this point is when we weight w_2 equal to 1, w_1 equal to 0. 57:16 That’s when you weight everything into the asset two. What if you go beyond that? 57:22 What does that mean? OK. So let’s say, can we have w_1 equal to minus 1, w_2 equal 57:31 to plus 2? So they still add up to 100%. 57:37 But what’s negative 1 mean? Borrow, right? 57:42 So you went short cash 100%, you borrow money. 57:48 You borrow 100% of cash, then put into to buy equity or whatever, risky assets, here. 57:56 So you have plus 2 minus 1. What does the return looks like when you do this? 58:02 So R_P equal to w_1 R_1 plus w_2 R_2. 58:10 So minus R_1 plus 2R_2. 58:19 That’s your return. It’s this point here. What’s your variance look like, or standard deviation 58:28 look like? As we did before, right? 58:34 So sigma_P simply equal to w_2 sigma_2. 58:39 So in this case, it’s 2sigma_2. So you’re two times more risky, two times as risky 58:47 as the asset two. 58:53 So this introduces the concept of leverage. Whenever you go short, you introduce leverage. 58:59 You actually– on your balance sheet, you have two times of asset two. 59:05 You’re also short one of the other instrument, right? OK so that’s your liability. 59:11 So your net is still one. So what this risk parity says is, OK, 59:21 so we can target on the equal risk weighting, which will give you somewhere around– let’s called it 25. 59:31 25% bonds, 75%– 25% equity, 75% of fixed income. 59:39 Or in other words, 25% of stocks, 75% of bonds. So you have lower return. 59:45 But if you leverage it up, you actually 59:51 have higher return, higher expected return, 59:58 given the same amount of standard deviation. You achieved by leveraging up. 1:00:05 Obviously, you leverage up, right? That’s the other implication of that. 1:00:10 We haven’t talked about the liquidity risk, but that’s a different topic. 1:00:16 So what’s your Sharpe ratio look like for risk parity portfolio? 1:00:23 1:01:08 So you essentially maximized the Sharpe ratio, or risk-adjusted return, by achieving the risk parity 1:01:17 portfolio. So 60/40 is here. You actually maximize that, and this is– does leverage matter? 1:01:28 When you leverage up, does Sharpe ratio change, or not? 1:01:35 AUDIENCE: It splits in half. So you’ve got twice the [? variance ?] [INAUDIBLE]. 1:01:41 PROFESSOR: So let’s look at that straight line, this example, OK? 1:01:51 So we said Sharpe ratio equal to– right? 1:02:02 So R_P, what is sigma_P? 1:02:17 It’s 2sigma_2, right, when you leverage up. 1:02:24 So this equals to R_2 minus R_1, divide by sigma_2. 1:02:34 So that’s the same as at this point. 1:02:41 So that’s essentially the slope of the whole line. It doesn’t change. 1:02:47 OK, so now you can see the connection between the slope of this curve and the Sharpe ratio 1:02:55 and how that links back to beta. So let me ask you another question. 1:03:00 When the portfolio has higher standard derivation of sigma_P, 1:03:08 will beta to a specific asset increase or decrease? 1:03:16 So what’s the relationship intuitively between beta– so let’s take a look at the 60/40 example. 1:03:23 Your portfolio, you have stocks, you have bonds in it. So I’m asking you, what is really the beta of this 60/40 1:03:32 portfolio to the equity market? When equity market, it becomes– when the portfolio 1:03:38 becomes more volatile. Is your beta increasing or decreasing? 1:03:43 So you can derive that. 1:03:49 I’m going to tell you the result, but I’m not going to do the math here. So beta equals to– [INAUDIBLE] in this special case, 1:04:06 is sigma_P over sigma_2. OK. 1:04:13 All right, so so much for all these. I mean, it sounds like everything is nicely solved. 1:04:19 And so coming back to the real world, and let me bring you back, OK? 1:04:24 So are we all set for portfolio management? We can program, make a robot to do this. 1:04:30 Why do we need all these guys working on portfolio management? 1:04:37 Or why do we need anybody to manage a hedge fund? 1:04:44 You can just give money, right? So why do you need somebody, anybody, to put it together? So before I answer this question, 1:04:50 let me show you a video. 1:05:01 [VIDEO PLAYBACK] 1:05:51 [HORN BLARING] 1:06:02 [END VIDEO PLAYBACK] OK. 1:06:08 Anyone heard about the London Millennium Bridge? 1:06:13 So it was a bridge built around that time and thought it had the latest technology. 1:06:21 And it would really perfectly absorb– you heard about soldiers just marching across a bridge, 1:06:28 and they’ll crush the bridge. When everybody’s walking in sync, your force gets synchronized. 1:06:35 Then the bridge was not designed to take that synchronized force, so the bridge collapsed in the past. 1:06:42 So when they designed this, they took all that into account. But what they hadn’t taken into account 1:06:49 was the support of that is actually– so they allow the horizontal move to take that tension away. 1:06:59 But the problem is when everybody’s sees more people walking in sync, then the whole bridge 1:07:07 starts to swell, right? Then the only way to keep a balance for you standing on the bridge is 1:07:13 to walk in sync with other people. So that’s a survival instinct. 1:07:20 And so I got this– I mean, that’s actually my friend at Fidelity, Ren Cheng. Dr. Ren Cheng brought this up to me. 1:07:26 He said, oh, you’re doing– how do you think about the portfolio risk, right? 1:07:32 This is what happened in the financial market in 2008. When you think you got everything figured out, 1:07:40 you have the optimal strategy. When everybody starts to implement the same optimal strategy for your own as individual, 1:07:50 the whole system is actually not optimized. It’s actually in danger. Let me show you another one. 1:07:56 [VIDEO PLAYBACK] [CLACKING] OK. These are metronomes, right? 1:08:02 So can start anywhere you like. 1:08:09 Are they in sync? Not yet. 1:08:18 What is he doing? 1:08:39 You only have to listen to it. You don’t have to see it. 1:08:44 So what’s going on here? This is not– metronomes don’t have brains, right? 1:08:49 They don’t really follow the herd. Why are they synchronizing? 1:09:11 OK, if you’re expecting they are getting out of sync, it’s not going to happen. OK, so I’m going to stop right here. 1:09:17 OK. [END VIDEO PLAYBACK] You can try as many– how do I get out of this? 1:09:25 1:09:30 OK, so you can try it. You can look at– there’s actually a book written on this 1:09:36 as well, so. But the phenomena here is nothing new. 1:09:41 But what when he did this, what’s that mean? 1:09:46 When he actually raised that thing on the plate and put it on the Coke cans? 1:09:53 What happened? Why is that is so significant? 1:09:59 AUDIENCE: Because now they’re connected. PROFESSOR: They’re connected. Right. So they are interconnected. Before, they were individuals. 1:10:05 Now they’re connected. And why did I show you the London Bridge and this 1:10:12 at the same time? What’s this to do with portfolio management? What’s this to do with portfolio management? 1:10:19 AUDIENCE: [INAUDIBLE] people who are trading, if they have the same strategy, [INAUDIBLE] affect each other, 1:10:25 they become connected in that way– PROFESSOR: Right. AUDIENCE: If as an individual, you are doing a different strategy, if everybody 1:10:31 has been doing something different, you can maximize [? in the space. ?] PROFESSOR: Very well said. 1:10:37 So if you’re looking for this stationary best way of optimizing your portfolio, 1:10:44 chances are everybody else is going to figure out the same thing. And eventually, you end up in the situation 1:10:50 and you actually get killed. OK, so that’s the thing. 1:10:58 What you learned today, what you walk away was this. OK, today is not what I want you to know that all 1:11:06 the problems are solved. 1:11:12 Right? So you say, oh, the problem’s solved. The Nobel Prize was given. So let’s just program them. 1:11:18 No, you actually– it’s a dynamic situation. You have to. So that makes the problem interesting, right? 1:11:25 As a younger generation, you’re coming to the field. The excitement is there are still a lot of interesting problems out there unsolved. 1:11:32 You can beat the others already in the field. And so that’s one takeaway. 1:11:39 And what are the takeaways you think by listening to all these? 1:11:46 AUDIENCE: Diversification is a free lunch. 1:11:52 [CHUCKLES] PROFESSOR: Diversification is a free lunch, yes. Not so free, right, in the end. It’s free to a certain extent. 1:11:58 But it’s something– you know, it’s better than not diversified, right? It depends on how you do it. 1:12:04 But there is a way you can optimize. And so it’s– I want to leave with you, 1:12:10 I actually want to finish a few minutes earlier so that you can ask me questions. You can ask. It’s probably better to have this open discussion. 1:12:19 And so I want you to walk away, to really keep in mind is in the field of finance, 1:12:27 and particularly in the quantitative finance, 1:12:32 it’s not mechanical. It’s not like solving physics problems. It’s not like you can get everything figured so it 1:12:39 becomes predictable, right? So the level of predictability is actually very much linked 1:12:47 to a lot of other things. Physics, you solve Newton’s equations. You have a controlled environment 1:12:53 and you know what you’re getting in the outcome. But here, when you participate in the market, 1:12:59 you are changing the market. You are adding on other factors into it. So think more from a broader scope kind of view 1:13:09 rather than just solve the mathematics. That’s why I come back to the original– 1:13:14 if you walk away from this lecture, you’ll remember what I said at the very beginning. Solving problems is about observe, 1:13:22 collecting data, building models, then verify and observe again. 1:13:28 OK, so I’ll end right here, so questions. 1:13:35 AUDIENCE: Yeah, just [INAUDIBLE] question. Does this have anything to do with– it kind of sounds 1:13:40 like game theory, but I’m not exactly too sure. Because you have a huge population and no stable equilibrium. 1:13:48 Does it have anything to do with game theory, by any chance? PROFESSOR: It has a lot to do with game theory, but not only to game theory. 1:13:55 So game theory, you have a pretty well-defined set of rules. 1:14:01 Two people play chess against each other. That’s where a computer actually can become smarter, right? 1:14:07 So in this market situation, you have so many people participating without clearly defined rules. 1:14:16 There are some rules, but not always clearly defined. And so it’s much more complex than game theory. 1:14:26 But it’s part of it, yeah. Dan, yeah? AUDIENCE: Can you talk a little bit about why some of the risk 1:14:33 parity portfolios that did so poorly in May and June when rates started to rise and what about their portfolio 1:14:38 allowed them do that? PROFESSOR: Good question, right. So as you can see here, what the risk parity approach does 1:14:47 is essentially to weight more on the lower volatility asset. 1:14:53 In this case, the question is, how do you know which asset has low volatility? 1:14:58 So you look at historical data, which you conclude bonds have the lower volatility. 1:15:05 So you overweight bonds. That’s the essence of them, right? So then when bonds to start to sell off 1:15:11 after Bernanke, Fed chairman Bernanke, said he’s going to taper quantitative easing. 1:15:17 So bonds from a very low high yield, a very low yield level, 1:15:22 the yield went much higher, the interest rate went higher. Bonds got sold off. So this portfolio did poorly. 1:15:30 So now the question is, does that prove the risk parity approach wrong, or does it prove right? 1:15:39 Does the financial crisis of 2008 prove the risk parity approach a superior approach, 1:15:46 or does the June/May experience prove this as the less-favored approach? 1:15:53 What does it tell us? Think about it. So it really is inconclusive. 1:16:00 So you observe, you extrapolate from your historical data. But what you are really doing is you’re 1:16:10 trying to forecast volatility, forecast return, forecast 1:16:15 correlation, all based on historical data. It’s like– a lot of people use that example. 1:16:22 It’s like driving by looking at the rear view mirror. 1:16:27 That’s the only thing you look at, right? You don’t know what’s going on, happening in front of you. 1:16:33 You have another question? AUDIENCE: Given all this new information, do you find that people are still 1:16:38 playing similar [INAUDIBLE] strategy with portfolio management? 1:16:44 PROFESSOR: Very much true. Why? Right, so you said, people should be smarter than that. 1:16:51 It’s very difficult to discover new asset classes. It’s also very difficult to invent 1:16:58 new strategies in which you have a better winning probability. The other risk, the other very interesting phenomenon, 1:17:05 is most of the traders and the portfolio managers, the investors, they are career investors– 1:17:12 meaning just like if I’m a baseball coach, I’m hired to coach a baseball team. 1:17:21 My performance is really measured against the other teams when I win or lose, right? 1:17:27 A portfolio manager or investor is also measured against their peers. 1:17:32 So the safest way for them to do is to benchmark to an index, 1:17:37 to the herd. So there’s very little incentive for them to get out 1:17:43 of the crowd, because if they are wrong, they get killed first. They lose their jobs. 1:17:49 So the tendency is to stay with the crowd. It’s for survival instinct. 1:17:55 It’s, again, the other example. It’s actually the optimal strategy for individual portfolio manager is really to do the same thing 1:18:04 as other people are doing because you stay with the force. 1:18:10 AUDIENCE: So you said given that we have all these groups, 1:18:16 in the end, it’s not just that we could leave it to the computers. We need managers. So what different are the managers 1:18:23 doing, other than [INAUDIBLE]? PROFESSOR: Can you try to answer that question yourself? What’s the difference between a human and a computer? 1:18:31 That’s really– what can human add value to what a computer can do? 1:18:37 AUDIENCE: Consider the factors, the market factors and news and what’s going on. 1:18:43 PROFESSOR: So taking more information, processing information, make a judgment on a more holistic approach. 1:18:51 So it’s an interesting question. I have to say that computers are beating 1:18:56 humans in many different ways. Can a computer ever get to the point actually beating 1:19:04 a human in investment? I can’t confidently tell you that it’s not going to happen. 1:19:10 It may happen. So I don’t know. 1:19:18 Any other questions? Yeah? AUDIENCE: Just to add to that. I think there is some more to management than just investing. 1:19:25 I think managers also have key roles in their HR, key roles in 1:19:33 just like managing people and ensuring that they’re maximizing their talents, not just like, oh, how much money did you make? 1:19:39 But I mean, are you moving forward in your career while you’re there? So I think management has a role to play in that as well, 1:19:46 not just investment. PROFESSOR: Yeah, I think that’s a good point. 1:19:52 Yeah. All right, so– oh, sure. Jesse? AUDIENCE: What is your portfolio breakdown? 1:19:59 PROFESSOR: My personal portfolio? Well, I am actually very conservative at this point, 1:20:04 because if you look at my curve of those spending and earning 1:20:10 curve, I’m basically trying to protect principals rather 1:20:16 than try to maximize return at this point. So I would be sliding down more towards this part 1:20:25 rather than try to go to this corner, yeah. So I haven’t really talked much about risk. 1:20:32 What is risk, right? So I talk about volatility or standard deviation. 1:20:37 But as we all know that, as Peter mentioned last time as well, there are many other ways to look at risk– value at risk 1:20:46 or half distribution or truncated distribution, or simply maximum loss you can afford to take, right? 1:20:55 But looking at standard deviation or volatility 1:21:01 is an elegant way. You can see. I can really show you in very simple math about how 1:21:09 the concept actually plays out. But in the end, actually volatility 1:21:14 is really not the best measure, in my view, of risk. Why? Let me give you another simple example before we leave. 1:21:24 So let’s say this is over time. 1:21:31 This is your cumulative return or you dollar amount. 1:21:43 So you start from here. If you go flat, then– does anyone 1:21:53 like to have this kind of a performance? Right? 1:21:59 Of course, right? This is very nice. You keep going up. You never go down. But what’s the volatility of that? 1:22:08 The volatility is probably not low, right? And then on the other hand, you could 1:22:16 have– what I’m trying to say, when 1:22:26 you look at expected return matching expected return and the volatility, you can still really not 1:22:38 selecting the best combination. Because what you really should care about 1:22:43 is not just your volatility. And again, bear in mind all the discussion about the Modern 1:22:51 Portfolio Theory is based on one key assumption here. It’s about Gaussian distribution, OK? 1:22:59 Normal distribution. The two parameters, mean and standard deviation, 1:23:04 categorize the distribution. But in reality, you have many other sets of distributions. 1:23:11 And so it’s a concept still up for a lot 1:23:17 of discussion and debate. But I want to leave that with you as well. 1:23:24 Yeah? AUDIENCE: Just going back to the same question about what these guys were asking about management 1:23:30 and how do they add value, I think the people who added value– there were some people who added a tremendous amount of value in the financial crisis. 1:23:38 And they were doing the same mathematics. But a difference was in their expected return of various assets was different from the entire– 1:23:44 the broad market. So if you can just know what expected return is that, 1:23:50 probably that is the only answer to the whole portfolio management debate. PROFESSOR: Yes. If you can forecast expected return, then that’s– yeah, 1:24:00 now you know the game. You solved it. You solved the big part of the puzzle. Yeah? AUDIENCE: What management does is 1:24:07 how good it can do [INAUDIBLE] expected return, full stop. Nothing more. 1:24:14 PROFESSOR: I disagree on that. That’s the only thing. Because given two managers, they have the same expected return, 1:24:21 but you can still further differentiate them, right? So that’s– yeah. And that’s what all this discussion is about. 1:24:29 But yes, expected return will drive lot of these decisions. If you know one manager’s good expected return, three years 1:24:37 later, he’s going to make 150%. You don’t really care what’s in between, right? 1:24:43 You’re just going to ride it through. But the problem is you don’t know for sure. 1:24:48 You will never be sure. AUDIENCE: I’d like to comment on that. 1:24:53 PROFESSOR: Sure. AUDIENCE: What [INAUDIBLE] looked at in simplified settings, estimating returns and volatilities. 1:25:00 And the problem, the conclusion for the problem, was basically cannot estimate returns very well, 1:25:09 even with more data, over a historical period. But you can estimate volatility much better with more data. 1:25:15 So there’s really an issue of perhaps luck in getting the return estimates right with different managers, 1:25:23 which are hard to prove that there was really an expertise behind that. 1:25:28 Although with volatility, you can have improved estimates. 1:25:33 And I think possibly with a risk parity portfolio, those portfolios are focusing not on return expectations, 1:25:40 but saying if we’re going to consider different choices based on just how much risk they have 1:25:46 and equalize that risk, then the expected return should be comparable across those, perhaps. 1:25:54 PROFESSOR: Yeah. So that highlights the difficulty of forecasting return, forecasting volatility, 1:25:59 forecasting correlation. So risk parity appears to be another elegant way of proposing the optimal strategy 1:26:07 but it has the same problems. Yeah? AUDIENCE: Actually, I also wanted to highlight. You mentioned the Kelly criterion, 1:26:14 which we haven’t covered the theory for that previously. But I encourage people to look into that. 1:26:21 It deals with issues of multi-period investments as opposed to single-period investments. 1:26:26 And most– all this classical theory we’ve been discussing, or that I discuss, covers just a single period analysis, 1:26:34 which is an oversimplification of an investment. And when you are investing over multiple periods, 1:26:41 the Kelly criterion tells you how to optimally basically bet with your bank roll. 1:26:47 And actually there’s an excellent book, at least 1:26:53 I like it, called Fortune’s Formula that talks about– [? we already ?] 1:26:58 said the origins of options theory in finance. But it does get into the Kelly criterion. And there was a rather major discussion between Shannon, 1:27:08 a mathematician at MIT, who advocated applying the Kelly criterion, and Paul Samuelson, one of the major economists. 1:27:17 PROFESSOR: Also from MIT. AUDIENCE: Also from MIT. And there was a great dispute about how you should 1:27:23 do portfolio optimization. PROFESSOR: That’s a great book. And a lot of characters in that book 1:27:30 actually are from MIT– and Ed Thorp, for example. And it’s really about people trying to find the Holy Grail 1:27:41 magic formula– not really to that extent, but finding something other people haven’t figured out. 1:27:47 But it’s very interesting history. Big names like Shannon, very successful in other fields. 1:27:56 In his later part of his career and life really devoted most of his time to studying this problem. 1:28:06 You know Shannon, right? Claude Shannon? He’s the father of information theory 1:28:15 and has a lot to do with the later information age invention of computers and very successful, yeah. 1:28:23 So anyway, so we’ll end the class right here. No homework for today, OK? So you just need to– yeah, OK. 1:28:31 All right, thank you.