Transcription
Imagine a portfolio with two assets. The first asset doubles, and then in the next year, it halves. This is the first year, this is the second. And the second asset first halves, but then doubles. If at the very beginning both these assets in the portfolio had equal weight, then after the first year, you can calculate it yourself, your portfolio would have grown by 25%. But if you do nothing at this stage, then the first asset will dominate the portfolio, and the second asset will constitute only a small part. Therefore, when the second asset doubles, you return to the same point, and after 2 years, the return on your portfolio will be zero. But if you rebalance the portfolio at this moment, so that the weights of these assets are equal, then your portfolio in the second year will bring another 25% return. Today we will talk with you about investment portfolio management. I will share some of the results of my research from the perspective of a practicing specialist, and not just from a theoretical point of view. Let's look at a brief summary of what I will be talking to you about today. First, I will tell you about what portfolio investing is, starting from the very basics. Then we will talk about what endowment funds are, how we manage endowment funds. Then I will tell you about portfolio theory and illustrate it with a few special and simple examples to make the mathematics easier for you to understand. Then I will switch to discussing the limitations of this theory. After all, this theory was proposed in the 1950s. And, of course, many people have worked in this field, trying to overcome the shortcomings of these theories. I myself have devoted a lot of time to thinking about these problems, so I will tell you about my research work, about how to improve portfolio construction theory, for example, how to improve risk measurement using volatility and the Sharpe ratio, and also about how to model crowd behavior. In addition, you have probably heard about the power law and how this behavior actually arose from crowd behavior. We will also talk about this, and then we will summarize. So, before we start the lecture, I want to conduct one exercise with all of you. I will hand out a sheet of paper to each of you. Yes, please pass it on. Make sure everyone has received a sheet. I want you to write down your own investment portfolio. Just hypothetically. Imagine that you have, say, $10,000 that you can manage and try to write down the investments you would like to have in your portfolio. Also, indicate the percentage share of each investment. And I also want you not to think too long or too hard about it. Just trust your intuition. Imagine that you are now, for example, a fund manager sitting in a pension fund, a charitable foundation, or some hedge fund. Just choose your investments, and when you finish, give me back the sheet, and we will talk about where you started and how the portfolio should actually be managed. Spend no more than a couple of minutes on this. Don't spend too much time, and I will continue my story so that you can write everything down in the process. When you get a blank sheet of paper, you will surely be thinking: "What are the criteria to do this correctly? What needs to be considered? And this is exactly what I want to teach you, how to think about all these issues. First of all, you need to understand what the purpose of this exercise is. But many say: "I want to get a high return on investment." Of course, you have a profit goal, but what is a high profit? 10% or 50, or maybe 100%. What time horizon are we talking about now? And since investments involve a high degree of uncertainty, what is your tolerance for losses? Think about it. If you have $10,000, how much can you afford to lose? These are the questions you should ask yourself. Then, when you choose these investments, you should also ask yourself: are you good at recognizing a winner? Why do you think you will have an advantage in predicting this market or this investment? And then, of course, how many investments should you have in your portfolio? Is it five positions, 50, or 100. How diversified or well-balanced should this investment portfolio be? And finally, the most important question. How to determine the size of each investment? It all comes down to determining the size. I will show you portfolio construction later. When you have a list of investments you like, how will you determine their sizes in proportion to each other and to the entire portfolio? Think about it. And now, please stop writing and return the sheets, because I want you to listen to me, not just waste time thinking and writing down your portfolio. Okay, I'm waiting. You don't have to sign them, with or without a name, it doesn't matter. I won't call you by name, but I will probably use some of your answers as examples to discuss investments. So, while we are collecting these portfolio options, I will also tell you about the decision-making process. When you choose investments, you probably first think about which markets you will choose your investments from and what your instruments will be. Then you think about what data to collect, what signals can be extracted, and what models to build, factors to consider, what forecasts to make, and then combine all this into strategies. Then we talk about the size and allocation of your capital, portfolio construction, and after that, we optimize it. And in the end, we try to understand how to manage risk. So, now you see that ultimately we reduce everything to how to compare different investments to understand the relationship between them, i.e., to understand the return and risk profile of all investments. And what is risk? In this diagram we are talking about, I use volatility, which represents the uncertainty of the outcome. I will come back to this later and tell you that volatility is indeed the best measure of risk. But let's leave that for later. And before we talk about investment decisions, let me look at what you have written down today. So, these are your choices. One portfolio has half in the S&P 500. Oh, this is even somewhat similar to an option strategy. Then in this portfolio, 30% are VI-related ETFs, and 20% are dollar index futures. It seems like someone somewhere interned in option strategies. So, the next portfolio has about eight investments. The largest of them is 20% in cash and 20% in QQQ. This is an ETF on Nasdaq stocks. Let's look further. This consists of 70% bonds and 30% stocks. This is quite interesting because when we usually talk about a standard 60/40 or 70/30 portfolio, the larger portion is usually in stocks. But this portfolio probably has a more risk-neutral, or I would say, less risky structure. We will come back to this. So, and here we have many stocks, i.e., individual companies. And first of all, there is Microsoft. So, and here is 50% S&P and 50% VPL - this is a fund that contains stocks from Japan, Australia, South Korea, and Hong Kong. And this one is 100% in S&P, yes, it reflects the average market very well. So, and this one probably consists of five stocks and 20% cash. So, 50% US Treasury bonds, 25% S&P 500. Okay. So, uh, what can I say? Many portfolios are quite similar. Now, you understand, we usually analyze changes in investment choices from year to year. Just 3 years ago, in 2021, most of the class chose some cryptocurrencies. And by the way, Dogecoin was the most popular choice among all. Today I haven't seen any cryptocurrency investments from you. This suggests that index investment ETFs seem to be the theme of this year. And you know, I don't blame you for that. And I even think it's a pretty good idea. You know, S&P, if you don't have any special advantages, then you feel that it's better for you to invest in an index, because it's a pretty good choice. So, on this chart, I am showing you returns versus volatility. We have already mentioned that many people choose cash. Cash, in essence, can be considered zero volatility with minimal return. You know that the Fed has just lowered the interest rate, which is still above 4%. And I think you know that it was close to 5%. And they may lower it further, but compared to the interest rate that has been in the last 10 years, it's actually not bad for parking capital. Bonds have lower returns, but at the same time, greater certainty regarding returns. Stocks certainly have higher returns, but the index has greater volatility. And in terms of returns, the index may not lag behind the broad stock market. As you know, the index is largely focused on large-cap technology stocks. And recently, these large-cap technology stocks have been performing very well. So, if you hold QQQ or S&P for a long time, then, in essence, you are holding seven stocks from the magnificent seven for a long time. And therefore, choosing ETF investments here is not a bad choice. Or index investments. Further to the right. This is the choice of private equity and venture capital. By the way, I haven't seen anyone mark this in their portfolios. You are probably thinking more about public markets. However, private equity and venture capital are mainly invested in private companies that are not listed on exchanges. These companies are generally not very liquid. You cannot easily withdraw your money. You will have to stay with them for a very long time. But often you will have more influence on the companies' activities. For example, you invest in startups. These companies can grow very quickly, but they can also fail very quickly. Therefore, the range of uncertainty of the outcome there is very large. But overall, over many years, they tend to generate higher returns, which justify all this uncertainty or risk premium you are dealing with. So, now let's return to the goal of return and loss tolerance. We need to think about this when constructing our portfolio. You, of course, try to maximize your return. Does anyone disagree with this? Should you maximize your return? Definitely yes. But at the same time, you must minimize your risk. In this case, minimize your uncertainty. Okay? Thus, you want to maximize the return-to-volatility ratio. And this is one of the goals. But, of course, you may have limitations. There are certain investments in which you may not want to participate. For example, many universities have stated that they do not want to invest in fossil fuels due to environmental concerns. And in the past, as with tobacco companies, some arms companies, people had principled reasons to oppose them. So, there are limitations on what your investment world should be, as well as on what liquidity should be. You know, how much money do you need to use? Those $10,000 you have in your portfolio. Perhaps you don't want to use all of them in certain investments. Or, conversely, you are constantly looking for new and new investments. For example, you want to spend 5% per year. For this, you need to consider your expenses. And on the other hand, a very important question is how much you can afford to lose. If your $10,000 turns into $5,000, you will be very upset, right? You will then say: "I will stop investing altogether." That is, I cannot afford to lose this money. Or maybe this is money that I have set aside for next year's education. I set it aside because I want to get some profit, and of course, I don't want to lose it. In that case, your risk tolerance is very low. This is a completely different matter. For many asset owners, inflation is also an important factor, as wealth represents relative purchasing power. That is, how much wealth do you have? It's not just the dollar amount in your bank account, but how much you can buy. When prices of goods and services rise, your purchasing power decreases. This is inflation. And it is also relative to the next group of people, to how much they can buy. For universities, this means you need a higher-paid professor. For example, you know that professors have competing offers, and you need to understand how much you can offer a professor to come to your university or stay at your university. So, there are certainly absolute returns and certain portfolio requirements. And also now many people talk about bypassing the benchmark, about exceeding the index. You know, many of you have chosen the S&P as an index, but if you can get a return higher than the S&P return, then this is called alpha. Relative return is also often used as a measure of your investment skills. You are not just investing beta or investing with market returns. you can actually get higher returns. Therefore, the ratio of cash flows to liabilities is very important. 2008 taught many asset owners an important lesson. When you don't have the right asset-to-liability ratio, when your portfolio suffers losses, your liabilities begin to dominate your balance sheet. And you then have to cut expenses, stop some projects, and start laying off people. Right. These are all painful lessons, so you need to plan cash flow well. Therefore, the time horizon I mentioned is very important. When I gave you this sheet of paper, I specifically did not specify the time horizon. I want you to think about it. You should ask yourself: for what period is my portfolio designed? For one day, for a year, or for 10 years? In the world of investments, people always compare returns for a year, for 2 years, sometimes even for a quarter. So colleagues try to say: "Oh, I outperformed manager A. Manager B's return is lower for this period, so A is better than B." But in reality, this is often an unhealthy pursuit, because long-term investors should focus on long-term returns. But in the real world, people inevitably look at short-term returns, so you cannot ignore it. But this is pressure from colleagues for many investment managers. This is a career risk, because they are constantly tired of such comparisons. And they think that in the short term, they are not so good. And then people try to replace them, fire them, and so on. Well, let's get back to the question, what to invest for. So, if you look at the left and top of the chart and look at the personal income and expenses of one person depending on age as a function, then your income level will peak somewhere in the middle of your career. Let's call this age around 50 years. Okay? You earn the most, and probably at this time you spend the most. But when you were a student, you probably didn't earn that much, but you didn't spend that much either. The same is true when you, having passed the peak, enter retirement. Therefore, you don't earn that much and you don't spend that much. Therefore, you need to plan your personal portfolio accordingly. This is related to the question of time horizon, as well as the marginal benefit of having great wealth. money starts to decrease over the years. This can be seen on the lower curve. Your benefit starts to sort of stabilize and reaches a plateau. So, do you want to take more risk at this point or not? However, for an organization or a fund, the situation on the right chart is a bit different, because this, you could say, is a perpetual portfolio. This is a very, very long time horizon. And expenses continue to grow as inflation grows. Therefore, you cannot just relax and say: "I have invested the entire portfolio in some conservative investments." After all, you must continue to generate returns. An endowment is created for a university. Usually, they spend about 5% of the portfolio, and in addition to that, inflation of 3% is expected. Thus, the nominal return target is about 8%. As you know, it is not always easy to consistently achieve 8% year after year, especially when interest rates are low and you don't have bonds or safe bonds to invest in. In addition, endowments generally have a long horizon, definitely more than 10 years. You know, universities have hundreds of years of history and are counting on hundreds more years. And in the university budget, in the operating budget, about 40% of the entire budget now comes from its own endowment. So you see that universities are increasingly relying on income from their own endowments, their own funds. And this is because, as you know, research grants from federal agencies and industry are constantly decreasing. Therefore, this must somehow be compensated for by the investment portfolio. So, this is just a list of strategies that endowments typically invest in. And I mentioned cash, government bonds, corporate bonds, credit products. That is, they come from issuers, not from the government. And they also have default risks, so they will pay you higher interest. There are also hedge funds betting on macro markets. Bets on the Fed or the stock market, the general direction. There are also quantitative funds, trend followers. People usually call them CTAs. Commodity trading advisors or those involved in determining the relative value of a pair of stocks or actually many pairs of stocks. This is called statistical arbitrage. That is, it is actually a search for statistical patterns to generate profit. This strategy is largely based on the use of computers. Many large quantitative companies still actively use them. Also here, one can recall fundamental equity hedge funds. They buy stocks and also sell stocks short. And they are also platforms that you have probably heard of. Well, you know, large platforms, they call them multistrat or multiman, short for portfolio manager. They hire hundreds of different teams trading on one platform using very different broad strategies, but achieving a high degree of diversification, and they can significantly enhance your portfolio. So, I mentioned private equity earlier. They also invest in real assets: real estate, natural resources, farmland, timberland, investments of this kind in new assets such as cryptocurrency, intellectual property, legal rights, and so on. In essence, anything you can imagine that can generate income, and this can also be discussed with you. Currently, the endowment model generally focuses on hiring external managers and making both public and private investments. We hire both generalists who look at everything, and specialists in specific areas such as biotechnology, artificial intelligence, and so on. We focus on both absolute returns and relative returns to the benchmark. Therefore, we are mainly active managers. We have some passive index investments, but they are the minority. In addition, this is not always a constant practice. Therefore, when we find an active strategy or manager who can outperform the benchmark or index, we always choose active investments. As you can see, this investment process focuses on selecting managers, selecting good investment funds. Harvard used to have many internal fund managers, but this model changed about 7 years ago. Now we focus more on external managers. So, and now I will move on to discussing the problem of portfolio construction. That is, I will try to shift the focus a bit towards the mass part of our discussion. The classic portfolio construction problem is formulated as follows. Assume that you know your return goals. You have loss tolerance, and you can predict the volatility, the return of each investment, which represents some standard deviation, as well as the correlation matrix of these investments. The question is, what percentage should you invest in each of your investments? So, this is the portfolio construction problem. And the goal of this, as we mentioned earlier, is to maximize the return of the entire portfolio while minimizing the risk of the portfolio in this classic problem, to minimize the uncertainty or volatility of the entire portfolio. As you can see, the portfolio construction problem is actually about determining the size of investments at all levels. You can relate this to the asset allocation problem. So, what is the asset allocation problem? You group these investments by different asset classes. As I showed you earlier, the list of these strategies can also be considered as a list of different asset classes. I already showed this slide. So, from top to bottom, it is necessary to discuss how much you want to invest in each category or asset class. And sometimes people will say: "Well, this is still too many asset classes." Right? But if you have 10 to 15 asset classes, it greatly complicates the optimization task. Therefore, people tend to reduce them to major risk factors. Sometimes, let's say, three to five risk factors. Usually, these are stocks and bonds. And plus one of some other factors, perhaps currency, credit, or some other aspects, but factor analysis can also be extended to many detailed levels. You can have 50 factors or even hundreds of factors, but these are just different ways of grouping investments to simplify the optimization task a bit. Okay. So, what is the problem of portfolio risk management? In many respects, the problem of risk management is actually the same as the problem of portfolio construction, i.e., the problem of sizing. But in addition, one needs to think about how to avoid excessive concentration. Concentration, of course, is also a sizing problem, as well as illiquidity and what is unacceptable. and acceptable losses and undesirable risks to limit the portfolio. I will spend a few minutes on portfolio theory, which is illustrated by two assets. Asset one has return R1 and volatility Sigma1, and it will have a weight in the portfolio W1. The same for asset 2. Thus, the sum of the two weights must be equal to one W1 + W2. And the portfolio return is denoted here as RP, and it is the sum of returns R1 and R2. Right? >> And the variance here is denoted as sigma squared pi. The square of volatility can be expressed as the weighted volatility of each of the two investments, as well as the correlation RO between the two investments. This is a very common equation. You have probably seen it in one of the MIT lectures on the Finance channel, where we discussed portfolio theory in practice and talked about probability and statistics. Let's consider some special cases. When, for example, the correlation RO is positive, it means that the two assets are perfectly correlated. When one rises, the other rises simultaneously. When one asset falls, the other falls with it. This is called perfect correlation. If rho equals one, then we can simplify sigma pi, which is the portfolio volatility, and it becomes a weighted sum of the two volatilities W1 x si1 + W2 on si2. Or you can also express W1 as 1 - W2. So, and in the next graph, you can see a straight line when ρ equals one. And there is also the case when ρ equals minus one. This means that the two assets are perfectly negatively correlated. That is, when one quantity rises, the other falls. In this case, you can easily derive that for sigmapi there are two possible solutions. It depends on the value of sigma1 compared to W2 and sigma 2. So, these are two separate lines on the left. This is another solution. In the middle are other correlation values between -1 and plus1, including zero correlation, when ρ equals zero. Well, and if the first asset has zero volatility, sigma1 = 0, then from the previous equation we can derive sigmapi. The portfolio volatility is simply w2 already on si2, because sigma1 equals 0. And therefore, W2, the weight of asset 2, equals sigmapima 2. This is simple. And the return RP can now be written as R1, which is the first asset with zero volatility, which can be considered as cash. We mentioned earlier that cash has zero volatility. Thus, this cash or risk-free asset has a return R1. And RP is still a linear function of sigmapi, but it is a straight line. In this case, you can see that the slope of this line is determined by the difference R2 - R1/2. By the way, this is the so-called capital allocation line in portfolio theory. So, let's look, say, at portfolio construction. You already have Sigma P, which has many assets, right? as well as RP, to which one risk-free asset or cash is added, which brings a return S. I already have that, which is basically like a risk-free return. So, the upper part of this line is called the efficient frontier, because by reducing risk and increasing return, you achieve your goal, i.e., you improve the portfolio. Thus, any point on this line is a potential solution for portfolio construction. So, why, when we add a risk-free asset, can we essentially combine this risk-free asset with an existing portfolio to further improve the risk and improve the efficiency of our portfolio? And you just need to move the boundary to go up to this straight line. In general, to put it simply, think about your return in relation to volatility. This, in fact, is the definition of the Sharpe ratio. Sometimes you may also hear the terms alpha and beta, which are compared to a benchmark. In this case, you can write the portfolio return minus the risk-free asset return in terms of alpha p beta multiplied by the benchmark minus the risk-free return. Beta is a function of the portfolio's correlation with the benchmark and the volatility coefficient. What if you want to leverage the portfolio? and borrow money, i.e., make the weight of the first risk-free asset equal to minus 100%, and the other asset 200%. So that the sum still equals one.
This means that you borrow money in the form of cash and invest the proceeds in a risky asset. This way, you increase volatility to this level by investing more in the second asset. This is also possible. This is a typical exercise. People do this in high-risk portfolios. By the way, if you don't understand something yet, it's okay. I want to talk about this to give you an idea of what portfolio theory is, because later I will tell you that in practice, a lot of this can be improved. So, an example of a two-asset portfolio we are talking about is 60/40 stocks and bonds. Some of you wrote 70% bonds, 30% stocks. This is very similar to a risk parity portfolio, where they try to equalize the risk contribution from bonds and stocks by borrowing. They invest more in bonds because bonds typically have lower volatility than stocks. Therefore, you need to have more to equalize the risk contribution. So, this is again one of the ideas. But today I will not go into the mathematics, because I want to talk to you about other things. And before we move on to the next question, I want to talk about the importance of rebalancing. Imagine you have two assets, and you hold them for 2 years. In the first year, the first one doubles, and in the second year, it halves. I think it would be easier to show this on the board, but the other asset behaves the opposite. So, let me draw this out now. Both of them started here. The first asset doubles, and then in the next year, it halves. This is the first year, this is the second. And the second asset first halves, but then doubles. So, these were R1 and R2. If at the very beginning both of these assets in the portfolio had equal weight, then after the first year, you can calculate it yourself, your portfolio would have grown by 25%. This is a rather simple mathematical calculation. You know that the first asset has a weight of 50%. Asset 2 also has a weight of 50%. One doubles, one halves. In total, you earned 25%. Okay. But if you do nothing at this stage, then the first asset will dominate the portfolio, and the second asset will be only a small part. Therefore, when the second asset doubles, you return to the same point, and after 2 years, your portfolio's return will be zero. Do you get it? Even though you had a diversified portfolio, both assets were negatively correlated. But if you rebalance the portfolio at this point so that the weights of these assets are equal, then your portfolio in the second year will bring another 25% return. Thus, you will get a return of 25% for 2 years in a row. Some of you may notice that this is simply a bet on mean reversion. And if you believe this is a trend, then you should not do this. You should ask yourself: on what is your determination of the size or weight of these two assets based? So, you started with equal weighting, which means you believe that the two assets have the same probability of return and the same level of risk. Therefore, you value them equally. And, of course, you also know that they are negatively correlated. So, at the end of the first year, do you maintain this view or not? If you believe that the forecast of risk and return remains the same, then you should maintain an equal weighting. You should not change the weighting unless your forecast has completely changed during the first year. Do you believe that asset one will continue to outperform asset two? If yes, then in that case, you can move to unequal weighting. This is the point. If you believe they have the same probability distribution, then there is no reason not to rebalance. People say it's a free lunch. The only free lunch is diversification. But to eat it, you just need to rebalance.
Now, I will probably switch to the limitations of so-called modern portfolio theory. I mentioned that it was largely developed in the 1950s. And Harry Markowitz was undoubtedly a pioneer in this field. He received the Nobel Prize, I think, in the early nineties somewhere for his work. I have just described to you what he did. The mathematical part is largely related to his work. But, as you have already guessed, a lot depends on your confidence in the assumptions about the capital market, i.e., your forecasts of volatility, return, and the correlation matrix of all your investments. And the mean-variance optimization problem often has an infinite number of solutions, and you have to set artificial constraints for some solutions. They are very sensitive to small changes. to small changes in assumptions about the capital market, so they are very difficult to use in practice, and probably few people use them. So, problem number one for modern portfolio theory. Volatility is a bad measure of risk. Why do I say that? Because volatility is just the standard deviation or the range of uncertainty. So, if I have a long position in a cash option, think about it. You have one investment, Stephen, I think you studied options and even worked with them. I think you will understand this easily. So, if I have a cash option, do I want volatility to increase or decrease? Do I want my portfolio to have high volatility or low volatility? And yes, I want my portfolio to have high volatility. The higher the volatility, the better. On the other hand, if I suddenly shorted a cash option, then I probably want volatility to be as low as possible, because I don't want these options to be exercised. So, it all depends on your situation, your payout, or your potential loss. The Sharpe ratio is derived from volatility, so essentially it suffers from the same problem. Over the years, all methods, such as the Sortino ratio, have tried to differentiate the upside range from the downside range. I think this is a good indicator, but it does not give you a direct link to size. You know, the final question we need to answer is still about size. A new way of comparing different investments that I mainly use or that I have come up with is to use expected gain and expected loss. I call them G for gain, and L for loss. I use positive numbers. When I say loss, I mean absolute numbers. We are looking for the best shift, i.e., a gain that is much larger than the expected gain, which is much larger than the expected loss, or return in different forms, i.e., G - L, which is the return on deals to G + L. Okay? Let me give you a few examples to help you understand this. If you flip a coin, then, as you know, heads has a certain probability and tails. And when heads comes up, you get a win. You either win or lose. And when you still have an expected win, the gain is the probability of heads multiplied by the win, if you bet on heads. And the expected loss is the probability of tails multiplied by the win in this case. If you have a target return, then you, say, integrate from minus infinity to the target return. This will be your expected loss. This is what needs to be included in your calculation. When you do this, your optimization task is no longer about the ratio of return to volatility. But on the vertical axis is return, on the horizontal axis L is loss. Therefore, you basically need to maximize G while minimizing L. All of this is relevant because, as I just explained, minimizing volatility is not always beneficial to you. Right? When you own options, you need higher volatility. But in this case, when you specify the expected loss, this is what you need to budget for and control your investments. Let's move on to problem number two of modern portfolio theory. How can we predict the future based on the past? How much can we trust the capital market? Assumptions about volatility and return and correlation. After all, we are humans, we have constant thought processes. We always think, always find patterns in some data. We have an observation, and I am just describing a typical thought process. You have some observations, you have measurable data. Try to quantify them. Then try to extract useful information or recognize patterns. Then try to build models with input data, output data, and conditions. Then predict the output based on the input and conditions, and then repeat your observations, and after that, repeat the process again. Try to calibrate the parameters of your model. When you become very good at this, you will be able to actually control the conditions and input data to get the desired result. And this is true for any field: physics, engineering, finance. But finance is closely related to human behavior, and that is much more complex, so historical patterns may not repeat themselves. And also, crowd behavior and adaptive behavior of people and players will make the process much more complex. And I want to show you a video. So, what do you see? This is a video about birds or bats that are completely self-organizing. This is a way of forming group behavior. They observe their neighbors, their actions, and the group as a whole exhibits certain patterns. In fact, on financial markets, everything happens in a very similar way, except that you have a central object of observation that you can see, and that is the stock price.
You can look at your neighbors, but currently it is much easier to look at the central object of observation. In the previous lecture, I also showed you a video about the London Bridge. You probably remember it. Or perhaps you have heard of the Millennium Bridge in London. The Millennium Bridge opened around 2000. And on the first day, when people walked on it, it started to sway. Then people tried to maintain balance. They all synchronized, started to act synchronously. Their steps made the oscillations even stronger. And the same happens in the markets. When panic feeds panic, and greed feeds greed, the market can form bubbles, crashes, and similar phenomena that are very difficult to predict. But all of this is partly crowd behavior. So, how can we understand and model such behavior so that we can consider it as a feedback loop? Here we can distinguish S, the action of the agent. O is observation. A is a kind of amplifier or the action of the agent, and it is transmitted to the observation. As you can see, the change in observation is the sum of the agent's actions or the force they possess. Changes in agent action are subject to external forces, as well as feedback loops, observation, and some random noise. This is the basic structure of crowd modeling. When agents behave under normal conditions, they are more rational, so they react less to observation. Or one can say that the feedback loop parameter B actually takes a lower value. But a change in observation is more volatile. In a more volatile manner, agents or people tend to react more strongly. They enter some reactive state. Thus, the more agents in the crowd enter a reactive state, then the crowd actually becomes more synchronized. So, if we look at the order parameter, which I define as the ratio of the sum of all actions to the sum of the absolute number of actions, if they are all synchronized, then this number should be equal to one. Right? If they are not synchronized, then this number can be very close to zero. The first graph on the left shows the order parameter as a function of the number of agents in the crowd who are in a reactive state. Thus, when more agents react to observation, the order parameter can become very high. The crowd synchronizes, and on the right you can see that the order parameter is also a function of random noise. The higher the noise, the less likely it is that agents, i.e., the crowd, will be synchronized. So, by the way, this type of behavior is not something new in physics. In some of these, well, you know, engineering chain systems, when you introduce noise, you can also reduce the order parameter in a synchronized manner. But in the social sciences, it is actually similar behavior. Therefore, I conducted further modeling of the bubble process when the external force, shown in the upper left corner, increases, then decreases, and then oscillates. The number of agents in a reactive state begins to grow, and then also decreases as the external shock begins to decrease. Observation as a result also gradually decreases, but if the external shock on the other side remains at a higher level forever, eventually all members of the crowd, i.e., all agents, enter a reactive state, so the entire crowd becomes synchronized. Observation essentially boils down to it skyrocketing, so this system becomes very unstable. It is at this moment that the bubble forms, and when the system becomes unstable, this bubble bursts. This is part of crowd behavior. And I am showing you this so that when we look at portfolio management, this type of bubble formation and bursting process is a very important aspect of our understanding, going beyond fundamental analysis. But all of this is about behavioral finance, and it has nothing to do with fundamental economic indicators or fundamental company indicators. But you have probably also heard of power-law distribution. In short, power-law distribution can be explained as: the winner takes all. You can also take the 80/20 rule or effect. Many different terms describing essentially social phenomena where a small number or a small percentage of agents occupy a large share of something in a particular group. Let's think about wealth distribution. 20% of people probably own 80% of the wealth. And this is also true if we look at the top 20%. Within this 20%, the top 20% will also own 80% of the wealth. And if we look even deeper inside, it will also be true inside. This is so-called scale-invariance or scale invariance. This is a unique property of power-law distribution. So, if we speak in mathematical terms, then the probability pbx is a random variable p, which is a scaling factor equal to some function g of b, multiplied by the probability pi of x. So, you can mathematically prove this only if pi x is a power distribution. This property of scale invariance is preserved. An example of power-law distribution that I mentioned is wealth distribution. The same can be seen in venture fund returns. Top-tier funds generally receive the largest share of profits from startup companies. 2 years ago, a book was published called "The Power Law" or "The Law of Degree." Its author is Sebastian Manabe. You should read it, it is very interesting. It is not a mathematical book, but it talks about how top-tier venture companies take everything. And the winner-takes-all phenomenon. The size of cities, by the way, tells us the same thing. Large cities become even larger because all nodes want to connect to a more popular node to attract more traffic. So, in network space, for example, internet connections, web pages, or social networks and posts, this type of power-law distribution is also very common. But in nature, we observe many normal distributions, such as human height. So, many people are probably concentrated around a height, you know, from 5 to 6 feet, maybe a little less than 6 feet. This is the majority of the population. It's not that most people are very short, but there are a few giants. Indeed, they are just much taller than other people. And here the power law appears. The power law exists even in the distribution of height. So, now we see that many social phenomena obey power-law distribution. But why? This is always a question. I am also very interested. So, what is the reason for such an uneven distribution of wealth? In the end, I realized that it is actually largely related to human interaction. Just like in the mechanism we considered earlier, agents have a feedback loop. This is the reason for the cost of this type of distribution law in nature. You don't have such a feedback mechanism. As soon as you are born, you grow to a certain height. Your height does not change. But in social phenomena, it constantly changes. Agents have different forces influencing the market. In the financial market, the rich can become even richer. Therefore, we need to model each agent differently. We call them heterogeneous agents. If we look at the feedback mechanism, we can conclude that agents with more power, in this case, which is represented as changing over time, can gain more power when they bet in the right direction, when observations show that bets are made in the right direction. This is how the rich get richer. This is how the concentration of power is formed. In this case, super-agents appear, and the system may or may not become stable. When does this happen? I mean, this can actually be extended to a discussion of entropy, you know, about what stability is, but that is beyond the scope of our lecture. So, let's get back to why all of this is relevant to portfolio management. I want to summarize on this page what we have learned today. Portfolio management is about determining the size of each investment. You need to clarify your goals. You need to know your loss tolerance and be sure to subscribe to Financegram. Diversification helps, but you need to rebalance in time. Volatility is not risk. The Sharpe ratio does not always reflect size and does not provide a size assessment. We should use the new ratio that I presented to you. Capital market assumptions are not reliable, and crowd behavior can drive markets to extremes. You should pay attention to influential super-agents who are the most influential. You can think about it. Does anyone have an idea about the market? Who could it be? Who has enough power to move the market? What do you think? Yes, of course, it's the Fed. Yes, so when you invest in the market, you should be careful because they have such great power to change the direction of markets. And, you know, some hedge funds have even more power, but I would say that currently it is definitely the government. I want to end my lecture here. We have covered a lot of material today, but I tried to give you general concepts. Remember that portfolio construction is about determining size, investing correctly, and determining size is about comparison, it's all about investment. But how to compare them? That's what you really need to understand. So, let me try to answer a few questions, and then we will finish. The question about investment taxation. After all, these are also significant sums. Do you need to plan your taxable base in some way or choose investments that will allow you, so to speak, to avoid paying capital gains tax? Well, that's a great question. By the way, donations and charity are not taxed, so keep that in mind. But if you manage a family office, you need to plan your capital gains income very carefully. Family offices or private investors tend to invest more in long-term private equity because they don't need to realize these gains. They invest less in public stocks for tax reasons. So, this is a very important parameter. Are there any other questions? And what is the best measure of risk? How do you measure risk? I still believe that we should try to understand potential losses or expected losses. I talked about this earlier, because people tend to think that gains and losses are symmetrical, but usually they are not, because you can build your positions with very asymmetric payouts. Therefore, you need to understand that when you buy an option, then, of course, your maximum loss is the option premium. Your payout is very clear and distinct, but when you buy stocks, it may not be so obvious, right? Therefore, you need to understand how much stocks can fall and what is the most likely range to which they can fall. Therefore, I think that risk assessment should be shifted or focused more on downside protection and understanding your downside, i.e., what you can actually lose from. Oh, thank you. But usually when I buy something, I don't think about losses, I only think about future growth. Or do I always need to keep the worst-case scenario in mind, is that it? Well, these are different things. Roughly speaking, one can think about the worst loss. Possibly, possibly about the limits of two standard deviations. I mean, the expected loss is really the weighted average of the outcome multiplied by the payout. Therefore, you cannot build your portfolio based solely on the worst-case scenario. Then you will do nothing. Therefore, expected loss is probably the best way to measure and size your investments. But you should be aware of your worst loss in case something suddenly goes wrong. And can't the worst-case scenarios be limited by a stop-loss, or not? Well, I would say that this is also a very good question. You have raised an interesting topic of stop-loss. A stop-loss is generally not set at the level of the worst loss, but rather at a level where you should close. possibly even closer than the expected loss. If you trade on hedge fund platforms, they usually give you a very tight stop-loss because they try to artificially create an asymmetric payout for portfolio managers or traders. When you lose a little, they halve your position. Then you lose a few more percentage points, they halve it again, and then they take you out of the game. Therefore, when you make money, they try to give you an opportunity to take advantage of the situation to create such an asymmetric optionality. But this forces you to choose investments with such a probability distribution. So, if you think that your investment has a worst-case loss that is, say, very bad, but has a very low probability, then you should consider whether your company is taking these risks for you. But if you believe that the potential gain is very large, then you should probably still take this risk. Right? Do you understand, yes, how it works?
This is, you know, one of the many problems in hedge funds. They get a share of the profits, but are not responsible for losses. This is usually how it works. So, it's more like free options. That's why many of you want to become hedge fund managers. Many people tell me that they just want to learn more about how to become a manager, how to raise funds, how to build a team, how to negotiate terms, how to create systems and back-office, how to handle legal and tax issues. Therefore, I think that we, as an organization, should simply help young, promising people to become the best fund managers. If any of you want to ask questions after the lecture or discuss an idea, you can always talk to me, email me, or just come up to me after class. Maybe, if something goes beyond what we are discussing in this class, it might be useful for any of you. Well, that's all. Thank you very much. Goodbye, everyone.