Lecture 13: Portfolio Management | Edited Transcript
A professionally copyedited transcript of Jake Xia’s MIT OpenCourseWare lecture on portfolio construction, endowment management, rebalancing, gain-loss sizing, crowding behavior, and power-law market dynamics.
Chapter Timestamps
00:19 Portfolio construction as sizing, objectives, and lecture roadmap
02:54 Class portfolio exercise: objective, horizon, loss tolerance, edge, diversification, sizing
07:52 From market selection to data, signals, models, strategies, allocation, and risk
09:03 Student portfolios: options, VIX, 70/30 bonds-stocks, ETFs, cash, and the post-crypto shift
11:47 Cash, bonds, stocks, indices, private equity, and venture capital on a return-risk map
14:31 Portfolio constraints: return target, volatility, ethics, liquidity, loss tolerance, inflation, alpha
18:12 Asset-liability matching, time horizon, career risk, and personal versus institutional portfolios
21:21 Endowment math: perpetual horizon, 5% spending, 3% inflation, and the 8% nominal target
23:29 Endowment strategy menu: bonds, credit, hedge funds, CTAs, stat arb, multi-PM, PE, real assets
25:40 Endowment model mechanics: external managers, active management, benchmarks, manager selection
27:18 Classic portfolio construction problem and why managers reduce assets into risk factors
30:43 Two-asset portfolio theory: weights, variance, correlation cases, and the efficient frontier
35:54 Risk-free assets, capital allocation line, Sharpe ratio, alpha, beta, leverage, and risk parity
40:40 Rebalancing example: diversification only pays if you keep the weights aligned with your assumptions
45:19 Limits of Modern Portfolio Theory: fragile assumptions, artificial constraints, volatility as bad risk
48:50 Gain-loss ratio: expected gain, expected loss, Kelly-style sizing, and downside risk budgeting
52:35 Investment game: using daily gains and losses to evaluate portfolio quality
55:43 Capital market assumptions and why finance is harder than physics: adaptive human behavior
57:12 Crowding behavior: flocking, the Millennium Bridge, bubbles, crashes, panic, and greed
1:00:03 Feedback-loop market model: actions, observations, amplification, reactivity, noise, synchronization
1:05:01 Power laws: wealth, venture returns, city size, networks, and rich-get-richer feedback
1:11:40 Final summary: sizing, rebalancing, expected loss, unreliable assumptions, and super agents
1:14:12 Q&A: taxes, expected loss versus worst-case loss, stop-losses, and hedge fund incentive structure
1:19:19 Investment game wrap-up and possible course directions: trading, research process, or fund-building
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Full video:
MIT 18.642 Topics in Mathematics with Applications in Finance, Fall 2024 Instructor: Jake Xia View the complete course: https://ocw.mit.edu/courses/18-642-topics-in-mathematics-with-applications-in-finance-fall-2024 YouTube Playlist: https://www.youtube.com/playlist?list=PLUl4u3cNGP601Q2jo-J_3raNCMMs6Jves
Transcript
00:19-02:10
Jake Xia: Today, we’ll be discussing portfolio management as one of our application lectures. I’ll be sharing some of my research from a practitioner’s point of view, rather than focusing solely on pure theory or mathematics. As we mentioned at the beginning of the term, we’ve been teaching this class since 2012. In 2013, the course was recorded for MIT OpenCourseWare; I’ve included the link to that 2013 lecture here. Some of you may have seen it—it actually turned out to be one of the top five most-viewed courses on OpenCourseWare. However, I promise that today’s lecture will include a lot of new content to make your presence here worthwhile, and we can keep the session interactive.
Jake Xia: Here is the outline for today. First, I’ll explain what portfolio construction is really about, starting from a very basic level. Then, I’ll discuss the endowment model and how we manage an endowment fund. After that, I’ll cover portfolio theory and illustrate it using some very specific, simple cases to help you understand the underlying math. Finally, I’ll switch gears to discuss the limitations of these theories. These models were originally proposed in the 1950s, and while many people have worked in the field since then to refine them...
02:12-04:32
Jake Xia: Many professionals in the field have worked to overcome the shortcomings of these theories. I have spent a significant amount of time thinking about these problems myself, so I will discuss my own research on how to improve portfolio construction theory. For example, we will look at how to improve risk measurement beyond simple volatility and the Sharpe ratio. We will also explore how to model crowding behavior and discuss the power law—specifically, how that behavior actually originates from crowding. Finally, I will provide a summary.
Jake Xia: Before we start the lecture, I want to do an exercise with all of you. I am going to hand out a piece of paper to everyone; please pass them down and make sure everyone gets one. I want you to write down your own investment portfolio. Hypothetically, imagine you have $10,000 to manage. List the investments you would like to have in your portfolio and the percentage allocated to each. Don’t think too hard or take too long on this—just use your intuition. Imagine you are a fund manager at a pension fund, an endowment, or a hedge fund. Just pick a few investments and then hand the paper back to me. Once you are finished, we will discuss how to approach portfolio management from your starting points. Take a couple of minutes, but don’t overthink it. While you are doing that...
Jake Xia: Don’t spend too much time on this. While I’m talking, you can start writing down your choices. When you look at that blank sheet of paper, you’re probably wondering: what are the criteria for doing this correctly? What are the key considerations? This is exactly how I want to lead you into thinking about these problems.
04:32-06:43
Jake Xia: First, you need to consider the objective of this exercise. You might say, “I want to get a great return on my investment.” Sure, but what defines a “great” return? Is it 10%, 50%, or 100%? What is the time horizon we are discussing? Because investing is highly uncertain, you also have to consider your loss tolerance. If you have $10,000, how much of that can you actually afford to lose? These are the essential questions.
Jake Xia: When you select those investments, you should also ask yourself about your own ability to pick a winner. Why do you believe you have an edge in predicting that specific market or asset? Then, of course, you have to decide how many investments should be in the portfolio. Should it be one, five, 50, or 100? How diversified or well-mixed should the portfolio be? Ultimately, the most important question is how you size each investment. It all comes down to sizing.
Jake Xia: Okay, let’s look at portfolio construction. I will show you...
Jake Xia: I will show you more about portfolio construction later. It is essentially about taking a list of investments you like and determining how to size them relative to each other and to the portfolio as a whole.
Jake Xia: Please hand those back now. I want you to focus on the lecture rather than spending your time writing down your portfolio. It’s your choice whether or not to include your name; I won’t call on you individually, but I will likely use some of your answers as examples when we discuss investments.
06:43-09:02
Jake Xia: While we are collecting these portfolio choices, I want to discuss your decision-making process. When you choose investments, you are likely thinking about which markets you should select from and which specific instruments to use.
Jake Xia: Are we mostly finished? For those of you just joining, we are doing an exercise where we ask you to write down your investment portfolio and the percentages allocated to each. If you need more time, that’s fine. I’m just going to start using some of these as examples. As I mentioned, the decision-making process involves deciding which markets to pick for your investments.
Jake Xia: Once you have decided which market to pick for your investment, you need to think about what data to collect, what kind of signals you can derive, and what models you need to build. You must consider which factors to include and what predictions you need to make. Then, you assemble those into strategies. From there, we move on to position sizing, capital allocation, and portfolio construction, followed by optimization. Finally, the goal is to effectively manage the risk.
Jake Xia: So, what this really comes down to is how to compare different investments. To understand the relationship between various investments, you have to understand their respective return and risk profiles. But what is risk? In the chart we are looking at here, I am using volatility for the time being, which represents the uncertainty of the outcome. Later on, I will circle back and explain why volatility is not actually the best measure for risk, but let’s save that for later.
09:03-11:13
Jake Xia: Before we discuss investment choices further, let me look at some of the portfolios you have written down. One portfolio here has 50% in an S&P straddle—which is an options strategy—then 30% in VIX-related ETFs, and 20% in Dollar Index futures. It sounds like someone here has done an internship somewhere and gained some experience.
Jake Xia: Some of you may have done internships involving option strategies. Looking at this portfolio, it contains about eight investments. The largest holding is cash at 20%, followed by another 20% in QQQ, which is the NASDAQ ETF.
Jake Xia: Let’s look at this next one: it’s 70% bonds and 30% stocks. This is quite interesting because when we typically discuss a 60/40 or 70/30 portfolio, the larger allocation is usually the equity piece. This specific construction likely follows a risk-neutral or risk-parity approach. We’ll dive into that later.
Jake Xia: This portfolio contains many single-name stocks, with Microsoft at the top. Another one here is split 50% in the S&P 500 and 50% in a Vanguard fund—at least, that’s what’s written here. Then we have one that is 100% S&P 500, which very much represents the broader market.
Jake Xia: This one has five stocks and 20% cash. Another consists of 50% US Treasury bonds and 25% S&P 500. So, they are all fairly similar. You get the idea.
11:13-13:39
Jake Xia: Peter and I used to review the year-to-year changes in investment selections. Only three years ago, in 2021, most of the class chose cryptocurrencies, with Dogecoin being the top pick for almost everyone. I haven’t seen any crypto investments today, which tells you that index investing and ETFs seem to be the dominant theme this year.
Jake Xia: That seems to be the theme of this year. I don’t blame you; I think it’s a pretty good idea. If you don’t feel you have a particular edge in investing, choosing an index is a solid choice. On this graph, I’m showing return versus volatility. We already mentioned that a lot of people chose cash. You can view cash as having zero volatility and a minimum return. Even though the Fed just cut interest rates, they are still over 4%—close to 5%. They may cut more, but compared to where interest rates have been over the last ten years, this is actually not a bad place to park your capital.
Jake Xia: Bonds offer a slightly higher return but come with a bit more uncertainty. Stocks definitely offer more, but an index has less volatility than individual stocks. In terms of returns, the index isn’t necessarily lagging behind the broader stock market. As you know, the index is currently heavily weighted toward large-cap tech stocks, and lately, those have been performing exceptionally well. If you are long on the QQQ or the S&P 500, you are essentially long on the “Magnificent Seven” stocks.
Jake Xia: So, choosing an ETF or index investment is a good move. Further up and to the right on the graph, we have private equity and venture capital. I haven’t seen anyone here list those as a choice yet. You might be focusing more on the public markets, but private equity and venture capital are essentially...
13:41-16:05
Jake Xia: Venture capital involves investing in private companies that are not publicly listed on stock exchanges. Consequently, these investments tend to be illiquid; you cannot easily withdraw your money and must commit to them for a very long time. However, you often have more influence over the company’s operations. For example, when you invest in startups, these companies can grow very rapidly, but they can also fail just as quickly. The range of uncertainty regarding the outcome is quite large, but historically, they tend to produce higher returns to justify the uncertainty and the risk premium you are taking on.
Jake Xia: Returning to the topics of return objectives and loss tolerance: when you are constructing your portfolio, you naturally want to maximize your return. I doubt anyone here would disagree with that. At the same time, you want to minimize your risk—or in this case, minimize your uncertainty. Essentially, one of your primary objectives is to maximize the ratio of return relative to volatility.
Jake Xia: However, you may also face certain constraints. There might be specific investments you do not want to participate in. For example, many universities have declared that they will not invest in fossil fuels due to environmental concerns. In the past, similar restrictions were often applied to the tobacco industry.
Jake Xia: There are also ethical concerns. In the past, people have had principled reasons to avoid investing in tobacco or weapons companies. These values create constraints on what your investment universe should be.
Jake Xia: Then there is liquidity. You need to consider how much of that money you actually need to use. For example, if you have $10,000 in a portfolio, you might not want to keep it all invested at all times; you might need to spend 5% of it per year. You must take your spending needs into account.
16:05-18:12
Jake Xia: On the downside, a very important question is how much you can afford to lose. If your $10,000 becomes $5,000, will you feel so bad that you decide to stop investing altogether? If you are saving for next year and just want to earn a small return without losing your principal, your risk tolerance is very low. Everyone’s situation is different.
Jake Xia: For many asset owners, inflation is also a major concern because wealth represents relative purchasing power. The value of your wealth isn’t just the dollar amount in your bank account; it’s about what you can actually buy. When the price of goods and services goes up, your purchasing power goes down. Inflation also affects your wealth relative to other people or groups.
Jake Xia: The next factor is purchasing power. For a university, this translates to hiring faculty. If there are competing offers, you need to know how much more you can offer a professor to convince them to join your institution.
Jake Xia: There is definitely an absolute return requirement for a portfolio, but many people today also focus on beating a benchmark or an index. Many of you have chosen the S&P 500 as your index. If you can generate returns above the S&P, that is called “alpha.” Relative return is often used as a measurement of investment skill, showing that you aren’t just relying on “beta” or broad market exposure, but are actually extracting additional returns.
18:12-20:16
Jake Xia: Asset-liability matching and cash flow management are also critical. The 2008 financial crisis taught many asset owners a painful lesson. When you don’t have the right asset-liability match and your portfolio suffers a drawdown, the liability side begins to dominate your balance sheet. You are forced to cut your spending budget, halt building projects, and lay off staff. To avoid these outcomes, you must have well-planned cash flows. Time horizon is another vital factor. When I gave you that assignment, I intentionally didn’t specify a time horizon because I wanted you to think about it yourselves.
Jake Xia: I’m mentioning the time horizon on purpose. You should always ask: how long is this portfolio for? Is it for one day, one year, or ten years?
Jake Xia: In the investment world, people are constantly comparing returns year-over-year, or even quarter-over-quarter. Peers will say, “A outperformed B, therefore A is better than B.”
Jake Xia: This is actually a very unhealthy exercise most of the time. Long-term investors should really focus on long-term returns, but in the real world, people inevitably look at short-term performance. You can’t ignore it because that peer pressure creates career risk for investment managers. A board of directors might look at short-term results, decide a manager isn’t performing well, and try to replace them.
Jake Xia: Let’s get back to the core of why we invest. If you look at the chart on the left, it shows personal income and spending as a function of a person’s age.
20:16-22:35
Jake Xia: Your income level typically peaks somewhere in the middle of your career—let’s say around age 50. That is when you are earning the most, and you are likely spending the most as well. As students, you probably don’t earn much, but you don’t spend much either. The same trend holds true once you pass that peak and enter retirement; your earnings decrease, and your spending typically goes down as well.
Jake Xia: When you don’t earn much, you don’t spend much, so you need to plan your personal portfolio accordingly. This relates to your time horizon and the fact that the marginal benefit of accumulating more wealth starts to diminish once you have reached a certain minimum amount. As shown in the bottom curve, your benefit starts to plateau. At that point, you have to decide whether or not you want to take on more risk.
Jake Xia: However, for an organization or an endowment, the situation is a bit different. These are perpetual portfolios with very long time horizons. Spending requirements keep increasing as inflation rises, so you cannot afford to be complacent and simply put the entire portfolio into conservative investments. You have to consistently generate higher returns.
Jake Xia: A university endowment, for example, typically spends about 5% of the portfolio annually. On top of that, there is an expected inflation rate of 3%. This means the nominal return target is 8%. As you know, it is not always easy to consistently make 8% year after year, especially when interest rates are low and you don’t have safe bonds to rely on. Endowments also tend to have very long time horizons—definitely more than ten years. That is what people always say.
22:39-24:41
Jake Xia: People often point out that universities have hundreds of years of history and look forward to many more centuries ahead. Currently, roughly 40% of a university’s operating budget comes from endowment spending. As you can see, universities are becoming increasingly reliant on endowment returns.
Jake Xia: This shift is occurring because research grants from federal agencies and private industries are declining. Consequently, the shortfall must be made up by the investment portfolio. Here is a list of the strategies that endowments typically invest in.
Jake Xia: I mentioned cash, government bonds, corporate bonds, and credit products. With credit products, unlike the U.S. government, the issuer carries a default risk and therefore pays a higher interest rate.
Jake Xia: There are also hedge funds that bet on macro markets, such as the direction of the Federal Reserve or the general stock market. Additionally, there are quantitative funds that engage in trend following, which people usually refer to as CTAs (Commodity Trading Advisors).
Jake Xia: Then there is statistical arbitrage, or “stat arb,” which captures the relative value between pairs of stocks—actually, many pairs of stocks. This strategy focuses on finding statistical patterns to generate returns. These are heavily computer-driven strategies. Many big...
Jake Xia: Many large quantitative shops still rely heavily on computer-driven strategies. There are also fundamental equity hedge funds that engage in both buying stocks and selling them short.
24:41-26:45
Jake Xia: You have likely heard of the major platforms known as multi-strategy or multi-PM (portfolio manager) platforms. These firms hire hundreds of different teams to trade on a single platform. They employ a wide range of diverse strategies, which allows them to achieve a high degree of diversification and significantly leverage the portfolio.
Jake Xia: As I mentioned earlier, there is private equity, as well as real asset investments like real estate, natural resources, farmland, and timberland. We also see newer asset classes such as crypto, intellectual property, and legal claims. Essentially, anything you can think of that has the potential to generate a return can be considered an investment.
Jake Xia: In our first lecture, I highlighted some of the key features of the endowment model. Nowadays, the endowment model tends to focus on hiring external managers and investing in both public and private markets. We hire both generalists, who look at the broad market, and specialists in specific domains such as biotech or AI. We focus on both absolute return and...
Jake Xia: We focus on both absolute return and relative return compared to the benchmark. We are primarily active managers. While we do have some passive beta index investments, they represent a minority of our portfolio and are not always permanent. Whenever we can find an active strategy or a manager capable of beating a benchmark or index, we will always opt for the active investment.
26:45-29:03
Jake Xia: As you can see, this investment process is centered around manager selection—specifically, identifying and selecting high-performing fund managers. In the past, Harvard used to employ many internal fund managers, but that model changed about seven years ago to focus more heavily on external managers.
Jake Xia: Now, I’m going to shift gears a bit and discuss the portfolio construction problem, moving more toward the mathematical side of the discussion. The classic portfolio construction problem is stated as follows: assume you know your return objectives and your loss tolerance, and you can predict each investment’s return volatility (which is the standard deviation) as well as the correlation matrix of these investments.
Jake Xia: The question then becomes: what percentage of your capital should you allocate to each investment? That is the core of the portfolio construction problem. As we mentioned earlier, the objective is to maximize the return for the entire portfolio while simultaneously minimizing the portfolio risk.
Jake Xia: In this classic problem, minimizing portfolio risk means minimizing the uncertainty or volatility of the entire portfolio.
Jake Xia: As you can see, the portfolio construction problem is essentially a sizing problem for investments at various levels. You can relate this to the asset allocation problem, where you group investments into different asset classes. As I showed you earlier with the list of strategies, you can also think of those as a list of different asset classes. From a top-down perspective, the discussion needs to focus on how much capital you want to allocate to each category or asset class.
29:03-30:43
Jake Xia: Sometimes people argue that there are still too many asset classes. If you have 10 to 15 asset classes, it makes the optimization problem very difficult. Consequently, people tend to reduce them to major risk factors. Usually, this involves three to five risk factors—typically equities and bonds, plus others such as inflation, currency, or credit. However, factor analysis can also expand into much more detailed levels. You could have 50 or even hundreds of factors, but these are simply different ways of grouping investments to make the optimization problem more manageable.
Jake Xia: This makes the optimization problem a little easier. So, what exactly is the risk management problem for a portfolio? In many ways, risk management is the same as portfolio construction—it’s a sizing problem. However, you also need to consider how to avoid concentration, which is itself a sizing issue. Additionally, you must account for illiquidity, unacceptable losses, and unwanted risks to properly constrain the portfolio.
30:43-32:58
Jake Xia: I’m going to spend a few minutes on portfolio theory, illustrated using two assets. Asset 1 has a return $R_1$, volatility $\sigma_1$, and a portfolio weight of $W_1$. We have the same parameters for Asset 2. The sum of the two weights, $W_1 + W_2$, must equal one. The portfolio return, denoted here as $R_p$, is the weighted return of $R_1$ and $R_2$. The variance, or $\sigma_p^2$, can be expressed in terms of the weights, the volatility of each investment, and the correlation $\rho$ between them. This is a very common equation that you have likely seen many times in any probability or statistics class. Let’s take a look at the...
Jake Xia: Let’s look at some special cases. First, consider when the correlation, rho, is positive one. This means the two assets are perfectly correlated: when one moves up, the other moves up at the same time; when one goes down, the other goes down as well. That is what we call perfect correlation.
Jake Xia: If rho equals one, we can simplify the formula for sigma P, which is the portfolio volatility. It becomes the weighted sum of the two individual volatilities: $W_1 \sigma_1 + W_2 \sigma_2$. You can also express $W_1$ as $(1 - W_2)$. By doing this, you can solve for $W_2$ in terms of sigma P, sigma 1, and sigma 2.
32:58-35:01
Jake Xia: From there, you can derive the overall portfolio return, $R_p$, in terms of $W_2$. Since $W_2$ is a linear function of sigma P, the portfolio return $R_p$ is also a linear function of sigma P. You can see this on the chart as a straight line representing the special case where rho equals one.
Jake Xia: Now, consider the case where rho equals negative one. This means the assets are perfectly negatively correlated: when one goes up, the other goes down. In this scenario, you can easily derive that sigma P behaves differently.
Jake Xia: You can actually easily derive that $\sigma_p$ has two branches of solutions. It depends on the value of $w_1 \sigma_1$ compared to $w_2 \sigma_2$. Those are the two separate lines you see on the left; that is the other solution. In the middle, we have the other values for the correlation, ranging between negative one and plus one, including a zero correlation where $\rho$ equals zero.
Jake Xia: Now, what if the first asset has zero volatility, meaning $\sigma_1 = 0$? From our earlier equation, we can derive that the portfolio volatility, $\sigma_p$, is simply $w_2 \times \sigma_2$. This is because $\sigma_1$ is zero. Therefore, $w_2$—the weighting of the second asset—is just $\sigma_p$ divided by $\sigma_2$. That is a very straightforward result.
35:01-37:22
Jake Xia: The portfolio return, $R_p$, can now be written in terms of $R_1$, which is the return on the first asset with zero volatility. You can think of this as cash. As we mentioned earlier, cash has zero volatility, so this cash or riskless asset has a return of $R_1$. In this scenario, $R_p$ is still a linear function of $\sigma_p$, but it appears as a straight line. You can see that the slope of that line is determined by the difference between $R_2$ and $R_1$, divided by $\sigma_2$.
Jake Xia: By the way, this is what is known as the Capital Allocation Line in portfolio theory. So, let’s look at how we construct this.
Jake Xia: Let’s look at how we construct a portfolio. Suppose you already have a portfolio with a specific volatility, $\sigma_p$, and a return, $R_p$, which consists of many different assets. By adding a riskless asset or cash—which returns $R_f$, the risk-free rate—you can draw a tangent line from that $R_f$ point to a point on the efficient frontier.
I should explain that the efficient frontier represents the boundary on a return-versus-volatility chart. Because you can choose different weightings for your assets—let’s look at this curve here—different values for $W_1$ and $W_2$ will result in different portfolio returns and volatilities. The upper part of this curve is called the efficient frontier. Your objective is to improve the portfolio by lowering risk and increasing return. Therefore, any point on this line represents a potential optimal solution for portfolio construction.
37:22-39:16
Jake Xia: Now, why do we add a risk-free asset? By starting at the $R_f$ point on the y-axis, we can combine that risk-free asset with an existing risky portfolio to further improve the efficient frontier, moving it up to that straight tangent line. The slope of that line—calculated as $(R_p - R_f) / \sigma_p$—is what we call the Sharpe Ratio. In simple terms, think of it as your excess return per unit of risk.
Jake Xia: In simple terms, think about your return relative to volatility. That is essentially the definition of the Sharpe ratio. You will also frequently hear the terms “alpha” and “beta,” which are used when comparing performance against a benchmark. In this context, you can express the portfolio return minus the risk-free rate as alpha plus beta, multiplied by the benchmark return minus the risk-free return. Here, beta is a function of the correlation between the portfolio and the benchmark, as well as the ratio of their volatilities.
Jake Xia: What if you want to lever up the portfolio by borrowing more? For example, you could set the weight of the risk-free asset to negative 100% and allocate 200% to the risky asset. The total weight still sums to one, but it means you are borrowing cash and putting those proceeds into the risky asset. By investing more heavily in that second asset, you are effectively moving the volatility up to a higher point on the curve.
39:19-41:49
Jake Xia: This is a common practice, particularly in risk parity portfolios. If you aren’t following all of this yet, don’t worry. The reason I’m introducing these concepts is to give you a flavor of what portfolio theory entails, because I’m going to show you how many of these models need to be improved in practice. Returning to our example of a two-asset portfolio, we previously discussed the 60/40 equity-bond split. Some of you also suggested a 70/30 split.
Jake Xia: Some of you suggested a portfolio of 70% bonds and 30% stocks. That is very similar to a risk parity portfolio, where the goal is to ensure that bonds and equities contribute equally to the overall risk. Since bonds typically have lower volatility than equities, you have to borrow more to invest in bonds to equalize that risk contribution. That is the general idea. I won’t dive into the mathematics of it today; in previous years, I spent more time on the formulas, but today I want to focus on other topics.
Jake Xia: Before we move on, I want to discuss the importance of rebalancing. This actually relates to a 2013 OCW video we looked at two weeks ago. Someone recently made a clip of that class, sped it up to 1.5x, and created a three-minute video about diversification being the “only free lunch” in finance. This example illustrates that perfectly, so I’m going to walk you through it quickly.
Jake Xia: Imagine you have two assets over a two-year period. In the first year, the first asset doubles in value, and in the second year, it loses half its value. It might be easier to show this on the blackboard, but essentially, the second asset does the exact opposite. They both start at the same point. The first asset doubles...
41:51-43:40
Jake Xia: Let’s look at a scenario where the first asset doubles in year one and then halves in year two. The second asset does the opposite: it halves in year one and then doubles in year two. These are our returns, $R_1$ and $R_2$. If you start with an equal weight in both assets, you can calculate that after the first year, the portfolio is up 25%.
Jake Xia: This is just simple math. Asset one has a 50% weight and asset two has a 50% weight. When one doubles and the other halves, your total gain is 25%. However, if you don’t do anything at that point, asset one will dominate the portfolio while asset two will only represent a small portion.
Jake Xia: Because asset one then halves and asset two doubles, you end up right back where you started. After two years, your portfolio’s total return is exactly 0%. This happens even though you had a diversified portfolio with perfectly negatively correlated assets.
Jake Xia: But if you rebalance to equal weights at the end of year one, your portfolio will make another 25% return in year two. By doing this, you achieve a compounded return of 25% for two years in a row.
Jake Xia: Now, some of you might point out that this looks like betting on mean reversion. You might argue that if you believe the market is trending, you shouldn’t do this. That is a fair point. The real question you should ask is...
43:40-45:18
Jake Xia: The question you should ask yourself is: what is the basis for your sizing or weighting of these two assets? You started with equal weighting, which implies you believe both assets have the same likelihood of performing and carry the same amount of risk. That is why you sized them equally, and of course, you also know they are negatively correlated.
At the end of year one, do you still maintain that view? If you believe the risk and return projections remain the same, then you should maintain the equal weighting. You shouldn’t change the weights unless your projection has totally changed by the end of the year. If you think asset one will continue to outperform asset two, then yes, you could switch to a non-equal weighting.
Jake Xia: That is the key point. If you believe they share the same probability distribution, there is no reason not to rebalance. People often say that diversification is the only “free lunch” in finance, but you have to rebalance to actually eat it.
Now, I’m going to shift gears and talk about the limitations of what is known as Modern Portfolio Theory, or MPT for short. As I mentioned, this was mostly developed in the 1950s, with Harry Markowitz pioneering the field.
45:19-47:46
Jake Xia: Harry Markowitz pioneered this area and actually received a Nobel Prize in the early 90s for his work. Everything I have described to you so far regarding the mathematical framework is largely based on his contributions. However, as you can probably tell, much of this depends on your confidence in your Capital Market Assumptions—specifically your projections for returns, volatility, and the correlation matrix of these investments. The mean-variance optimization problem often yields an unlimited number of solutions, forcing you to apply artificial constraints to bound them. Furthermore, these solutions are extremely sensitive to even small changes in your Capital Market Assumptions. Consequently, in practice, this model is very difficult to use, and honestly, not many people rely on it directly.
Jake Xia: Given those challenges, I have always been curious about this field and have conducted various research projects on the subject. Today, I am going to highlight a few key areas. The first is related to the question of position sizing using a different approach, which I call the gain-loss ratio. In the second part, I will discuss the effects of crowding behavior and how to model it. Finally, I will talk about the power law distributions that emerge from these crowded market dynamics.
Jake Xia: Let’s discuss why these crowd interactions are relevant to portfolio construction. The first issue with Modern Portfolio Theory (MPT) is that volatility is a poor measure of risk.
Jake Xia: Why do I say that? Volatility is simply the standard deviation or the range of uncertainty. Consider a scenario where you are long on an out-of-the-money call option. Stephen, you’ve worked with options, so you’ll likely find this intuitive. If you are long on an out-of-the-money option, do you want the volatility to go up or down?
47:58-50:18
Jake Xia: Do I want my portfolio to have high volatility or low volatility?
Stephen: Higher volatility.
Jake Xia: Exactly. The higher the volatility, the better. On the other hand, if I am short an out-of-the-money option, I would prefer lower volatility so that the option doesn’t get exercised.
Jake Xia: It really depends on your specific payout structure and your potential for gains or losses. Since the Sharpe ratio is derived from volatility, it suffers from the same fundamental issue. Over the years, methods like the Sortino ratio have been developed to differentiate between upside and downside volatility. I believe that is a much better measurement, but it has its own complexities.
Jake Xia: While it is a good measurement, it doesn’t provide a direct link to sizing. Ultimately, sizing is the key question we need to answer. To compare different investments, I’ve developed a method using expected gain and expected loss, which I refer to as G and L. I use positive numbers for both, so when I refer to “loss,” I am talking about the absolute value.
Jake Xia: We are looking for the best “skill,” where the expected gain is much larger than the expected loss. This can be written as (G - L) / (G + L), where G - L represents the return. If you rearrange the formula slightly, it becomes 1 - 2L / (G + L).
Jake Xia: This is exactly the sizing percentage derived from the Kelly Criterion for binary betting. This value is bounded between -1 and +1, so it can be directly translated into sizing. Let me give you a few examples to help you understand this.
50:18-52:31
Jake Xia: Consider flipping a coin. You have a certain probability for heads and a certain probability for tails. If it lands on heads, there is a specific payout you either receive or lose, and the same applies to tails. The expected gain is simply the probability of heads multiplied by the payout, assuming you are betting on heads.
Jake Xia: If you bet on heads, your expected loss is the probability of tails multiplied by the payout on that side. In a discrete example, this is fairly easy to understand. However, in a continuous example, you have to look at the terminal distribution. Let’s say you have a probability density function describing all possible outcomes. You would take that density function, multiply it by the payout, and integrate over the losing portion and the gaining portion separately to find the expected loss and the expected gain.
It is important to note that these don’t always have to be separated by zero. If you have a specific target return, you would integrate from negative infinity up to that target return. That represents your expected loss, which is what needs to be included in the variable $L$.
When you approach it this way, your optimization problem is no longer a trade-off between return and volatility. Instead, the vertical axis represents gain ($G$) and the horizontal axis represents loss ($L$). Your goal is to maximize $G$ while minimizing $L$. This is highly relevant because, as I explained earlier, minimizing volatility isn’t always to your benefit. For instance, when you are long an option, you actually want higher volatility. By specifying expected loss instead, you can more accurately budget and control your risk.
52:35-54:42
Jake Xia: You need to budget and control your investments. Regarding the investment game we started at the beginning of the term, I hope you have all submitted your choices from September, as well as any rebalancing you did in October. I want you to track your daily gains and losses in dollar amounts. At the end of the game in mid-November, you will summarize all daily gains and losses to derive your total gain (G) and total loss (L).
Jake Xia: Some of you might think that to win this game, you should just pick the stock with the highest volatility to maximize your potential return. While that might produce a high return, a high-volatility investment will suffer in the denominator of our ratio (G + L). This ratio indicates the quality of your investment. Even if your net return (G - L) is good, the denominator will normalize it, making it comparable to other investments. This leads us to the optimization problem. Yes, go ahead?
Student: In practice, what do you use to estimate the expected gain? It seems difficult to estimate the actual price distribution.
Jake Xia: Yes, it is. However, it is no more difficult than estimating the expected return and the standard deviation.
Jake Xia: It requires the same amount of work. You are simply separating the numbers and calculating half the distribution instead of the whole distribution. When you approach the optimization problem, as I’m about to explain, these lines are key. For example, the 45-degree line indicates where the gain minus the loss equals zero—the zero-return line.
54:43-57:12
Jake Xia: Naturally, you don’t want to pick any investments below that 45-degree line. You want lines with a higher slope, indicating a higher G/L ratio. However, when you combine two assets into one portfolio, calculating the combined G and L becomes path-dependent.
Jake Xia: This path dependency is expected. When you have two investments, the final outcome of the expected gain and loss will indeed depend on the path taken. This means you will likely need to run many Monte Carlo simulations to derive those values. However, it isn’t any more work than a traditional distribution calculation.
Jake Xia: Okay, let’s move on to the second issue with Modern Portfolio Theory: How do we predict the future based on the past? How confident can we truly be in our capital market assumptions regarding returns?
Jake Xia: When we look at capital market assumptions—specifically return, volatility, and correlation—we have to consider the human thinking process. Our minds are naturally wired to extract patterns from data. Typically, this process involves making observations and gathering measurable data, which we then try to quantify. From there, we attempt to extract useful information or recognize specific patterns.
Next, we build models with defined inputs, outputs, and conditions, using them to predict outcomes. We then repeat our observations and iterate to calibrate our model parameters. Once you become proficient at this, you can actually control the conditions and inputs to achieve a desired outcome. This approach applies to any subject, whether it’s physics, engineering, or finance. However, finance involves human behavior, which makes it significantly more complicated.
57:12-59:09
Jake Xia: In finance, historical patterns may not repeat themselves. Factors like crowding behavior and the adaptive nature of human participants make the entire process much more complex. To illustrate this, I want to show you a video. Let’s see if this works. What you are watching here is a video of birds—or perhaps bats.
Jake Xia: Think of how birds or bats move in a totally self-organized way to form group behavior. What they do is watch their neighbors and adjust their actions accordingly. As a result, the group as a whole displays certain patterns.
Jake Xia: Financial markets are very similar to this, except there is a central observation point you can actually see: the stock price. While you can still look at what your neighbors are doing, nowadays it is much easier to focus on that central observation.
59:09-1:01:31
Jake Xia: In the past, I’ve shown a video of the London Millennium Bridge. How many of you have heard of it? It opened around the year 2000, and on the very first day, as people walked across, the bridge started to sway. To keep their balance, everyone instinctively synchronized their steps, which made the swaying even more intense. The market behaves the same way. When panic feeds into panic or greed feeds into greed, the market forms bubbles and crashes. These types of behaviors are not easily predictable, but they are a fundamental part of crowding behavior. So, how do we understand and model this?
Jake Xia: How can we understand and model this type of behavior? We can look at it as a feedback loop. In this model, $S$ represents the agent’s action, and $O$ is the observation. $A$ acts as an amplifier, representing how the agent’s action feeds through into the observation. As you can see, the change in the observation is the sum of the agent’s actions multiplied by their power, $A$. The change in the agent’s action, $dS$, is influenced by external forces as well as the feedback loop from the observation, plus some random noise.
Jake Xia: That is the basic framework for crowd modeling. When agents behave under normal conditions, they are more rational and less reactive to observations. In these cases, the feedback loop parameter, $B$, takes on a lower value. However, when observations become more volatile, agents tend to react more strongly and enter a reactive state. As more agents in the crowd enter this reactive state, the crowd itself becomes more volatile.
1:01:33-1:04:00
Jake Xia: The crowd actually becomes more synchronized. If we look at the order parameter, which I defined as the ratio of the sum of all actions divided by the sum of the absolute number of those actions, we can see this effect. If everyone is perfectly synchronized, that number should be one. If they are not synchronized at all, that number can be very close to zero.
Jake Xia: The first chart on the left shows the order parameter as a function of the number of agents in the crowd who are in a reactive state. As more agents become reactive to the observation, the order parameter can become very high, meaning the crowd becomes synchronized.
Jake Xia: On the right, you can see that the order parameter is also a function of the random noise term we discussed on the previous page. The higher the noise, the less likely the crowd is to be synchronized. By the way, this type of behavior is not new in physics. In certain engineering circuit systems, for example, injecting noise can also reduce synchronization.
Jake Xia: By injecting noise, you can reduce the synchronization of the order parameter. In social science, we actually observe very similar behavior.
Jake Xia: I further simulated the bubble process. As you can see in the top left, when an external force is applied, it ramps up, comes back down, and then begins to oscillate.
Jake Xia: The number of agents in a reactive state starts to peak and then drops once the external shock diminishes. Consequently, the observed results follow a similar pattern before declining.
Jake Xia: However, if the external shock remains at a high level indefinitely, every member of the crowd—all the agents—eventually enters a reactive state.
1:04:00-1:06:30
Jake Xia: At that point, the entire crowd becomes synchronized. The observation essentially shoots through the roof, and the system becomes highly unstable. That is exactly how a bubble forms; once the system becomes unstable, the bubble will eventually burst.
Jake Xia: That covers the section on crowding behavior. The purpose of demonstrating this is to show that when we look at portfolio management, these types of bubble formations and—
Jake Xia: The process of bubbles forming and bursting is a crucial aspect of our understanding that goes beyond fundamental analysis. This is really the domain of behavioral finance; it has nothing to do with the underlying fundamentals of the economy or a specific company.
Jake Xia: You have probably also heard of the power law distribution. This refers to “winner-take-all” scenarios, the 80/20 rule, or the Matthew effect. There are many different terms used to describe this social phenomenon.
Jake Xia: Essentially, it describes a situation where a small number or percentage of agents hold the majority share of something within a group. For example, let’s consider wealth distribution.
Jake Xia: Typically, 20% of the people hold 80% of the wealth. This pattern is recursive: if you look at that top 20%, the top 20% within that group also holds 80% of that group’s wealth. This continues to hold true as you scale down, which is why it is referred to as being “scale-free.”
Jake Xia: This is what we call scale-free or scale invariance, which is a unique property of the power law distribution. In mathematical terms, if $X$ is a random variable and $b$ is a scaling factor, the probability of $bX$ is equal to some function $g(b)$ times the probability of $X$. You can prove mathematically that this scale-free property holds if and only if $P(X)$ follows a power law distribution.
1:06:30-1:08:51
Jake Xia: I previously mentioned wealth distribution as an example of a power law, but venture capital fund returns are another great example. The top-tier funds tend to capture the vast majority of profits from startup companies. There is a very interesting book published two years ago called *The Power Law* by Sebastian Mallaby that you should check out. It isn’t a mathematical text, but it discusses the “winner-take-all” phenomenon among top-tier VC firms. City sizes follow the same pattern—big cities just keep getting bigger. In network theory, this happens because all nodes want to connect to the most popular nodes to attract more connections.
Jake Xia: In network spaces—such as internet connections, web pages, or social media posts—it is common for a popular node to attract more traffic, leading to a power law distribution. However, in nature, we frequently observe normal distributions. Take human height, for example. Most people are centered around a specific range, perhaps between five and six feet. The vast majority of the population falls within this bracket. It isn’t a situation where most people are extremely short while a few “giants” are exponentially taller. That would be a power law, but the power law does not exist in the distribution of human height.
Jake Xia: Now, let’s consider social phenomena. Many social phenomena obey the power law distribution. But why? This is a question that has always made me very curious. What is the underlying cause of such a skewed wealth distribution? Eventually, I realized that it has a lot to do with crowd interaction. It stems from the same mechanism we looked at earlier: the agents operate within a feedback loop. That feedback loop is the primary cause of this type of power law.
1:08:52-1:11:03
Jake Xia: In nature, you don’t typically see this kind of feedback mechanism. Once you are born and grow to a certain height, your height remains relatively constant. However, in social phenomena, things are constantly changing. In financial markets, different agents have varying levels of power to influence the market, and the rich can get richer. Because of this, we need to model each agent differently, which is why we refer to them as heterogeneous agents.
Jake Xia: We are all familiar with the normal distribution for a random variable $X$, which follows the form of $e$ raised to the power of negative $(x - \mu)^2 / 2\sigma^2$. However, when it comes to power laws, people often get confused by the terminology. Pareto’s Law refers to the cumulative distribution function of the probability. In contrast, Zipf’s Law describes the rank of those outcomes—essentially looking at the random variable $X$ versus its rank, as displayed in this chart. While they focus on different aspects, they are all related to how we typically view power laws within a probability density function. The behavior looks like this, as opposed to...
Jake Xia: This behaves quite differently compared to a normal distribution. Of course, a uniform distribution would be represented by a delta function. The parameters for these three different ways of representing the power law are all linked by a specific formula. The probability density function for a power law is $C/x^a$. Here, $a$ is related to the power $k$ in the cumulative distribution function, as well as the parameter $b$ in Zipf’s law.
1:11:03-1:13:33
Jake Xia: If we examine the feedback mechanism, we can derive how agents with more power—represented here by a variable that changes over time—can gain even more power. When they bet in the right direction and observations confirm their strategy, they accumulate more influence. This is essentially the “rich get richer” phenomenon, which explains how a concentration of power is formed. In this scenario, “super agents” are born, and the system may ultimately become unstable.
Jake Xia: In some cases, a system may not remain stable. This could actually be expanded into a broader discussion on entropy—defining what constitutes order versus stability—but that is beyond the scope of this lecture. To bring this back to why it is relevant to portfolio management, I want to summarize what we have learned today on this final page.
Jake Xia: Portfolio management is ultimately about how to size each investment. To do it effectively, you must clarify your objectives and understand your loss tolerance. While diversification helps, you need to rebalance your positions. It is also important to remember that volatility is not the same as risk. The Sharpe ratio doesn’t capture skill, nor does it provide guidance on position sizing; instead, you should use the new ratio I presented to you today. Furthermore, capital market assumptions are often unreliable, and crowding behavior can push markets to extremes.
Jake Xia: You should also pay close attention to powerful “super agents.” Who are the most powerful super agents you can think of in the market? Does anyone have an idea?
Student: The Fed.
Jake Xia: Yes, exactly—the Fed. When you invest in the market, you have to pay close attention because they have such immense power to change the direction of the markets. In the old days, certain hedge funds held more sway, but nowadays, the government is definitely the primary driver. You need to understand the key factors driving your portfolio.
Jake Xia: I’m going to stop the lecture here. I’d like to take some questions, and then I’ll save a few minutes at the end to talk about the investment game. But before we get to that, let’s open the floor for questions.
1:13:33-1:15:21
Jake Xia: I know I packed a lot of content into today’s session. The goal isn’t necessarily for you to follow every single detail right now, as you’ll likely need to derive some of the math yourself to really get a feel for it. However, I wanted to give you the core concepts of what portfolio construction is all about. Ultimately, it’s about sizing your investments, and sizing is about comparing different investment opportunities. How you compare them is what it really comes down to. So, with that in mind, are there any questions?
Student: I have a question regarding other factors that influence investments, specifically taxes. Capital gains taxes can vary based on how long you hold an asset, and certain types of investments are more tax-sensitive than others. Is tax efficiency something that is typically considered when making investment decisions?
Jake Xia: That’s a great question. Endowments are tax-exempt, so for them, it isn’t a consideration. However, if you are managing a family office, you have to be very careful about planning for capital gains. Because of this, family offices and private investors tend to favor long-term private equity, as they don’t have to realize those gains immediately. They often invest less in public equities for that very reason. So yes, taxes are a very important parameter. Are there any other questions?
Student: What is the best way to measure risk? How should you actually quantify it?
Jake Xia: Well, I still believe we should try...
1:15:22-1:17:31
Jake Xia: I still believe we should strive to understand potential or expected loss. This refers to the “L” I discussed earlier. People often assume that gains and losses are symmetrical, but that is usually not the case. You can construct positions with highly asymmetrical payouts. For example, when you buy an option, your maximum loss is clearly the premium you paid. However, when you buy a stock, it may not be as clear. You need to understand how much that stock can lose, how far it can move down, and what the likely range of that downward movement is. Risk measurement should be shifted to focus more on downside protection and understanding your exposure, which is essentially your expected loss.
Student: That’s a good point. Regarding the question of expected loss versus worst-case loss, how do you—
Jake Xia: Yes, those are two very different things.
Jake Xia: Yeah, those are two different things. Roughly speaking, you can think of the worst-case loss as being a few standard deviations away. Expected loss, on the other hand, is the weighted average of outcomes multiplied by their respective payouts. You can’t construct your entire portfolio based solely on the worst-case scenario, or you’d never do anything. That’s why expected loss is generally a better metric for measuring and sizing your investments, though you must remain aware of your worst-case loss in case something catastrophic happens.
Student: Building on that idea of worst-case loss, when would you typically implement stop-losses? In your experience, how effective have they been? I’d love to hear a bit more about that.
Jake Xia: That’s a very good question from both of you. Stop-loss is an interesting concept. Typically, a stop-loss isn’t set at the absolute worst-case loss level; it’s usually set...
1:17:33-1:19:52
Jake Xia: Stop-losses are usually set quite close to your entry point, often even closer than the expected loss. If you are trading on a hedge fund platform, they typically give you a very tight stop-loss. The reason for this is that they are trying to artificially construct an asymmetric payout for the portfolio managers or traders. When you lose a small amount, they cut your position in half; if you lose another few percentage points, they cut it in half again and eventually get you out. Conversely, when you are making money, they try to let you ride your wins to create that kind of asymmetric optionality.
Jake Xia: However, that structure forces you to select investments that fit a specific probability distribution. For example, if you are looking at an investment with a very bad worst-case loss but a very low probability of that happening, you have to consider how that risk is managed. In a sense, the firm takes that risk for you because they are responsible for cutting your position. If you believe the upside is significant enough, you should take that bet.
Student: You should take that bet, right? I mean, worst-case scenario, you get fired, but for them...
Jake Xia: Yeah, exactly. Right.
Jake Xia: That highlights a major issue with hedge funds. Managers get a cut of the profits, but they don’t bear responsibility for the downside. Essentially, they are long a free call option. That’s exactly why so many people want to be hedge fund managers.
Jake Xia: I’ll briefly touch upon the investment game we’ve already discussed. Once we pass November 15th, you should keep track of your P&L numbers and let us know how you performed; I will try to verify the results.
Jake Xia: For the last slide, I want to return to a question many people ask: how do you actually become a hedge fund manager? I’d like to poll the audience here for a moment. Do you think there is real value to be found in that kind of trading, or...
1:19:54-1:21:11
Jake Xia: I’m curious to hear your thoughts on the direction of this course. Would you prefer to focus on trading and investment simulations, such as paper trading games? Or would you rather have more lectures on the research process—specifically data collection, signal generation, backtesting, and the portfolio optimization techniques we discussed today?
Alternatively, some of you might want to learn the business side of the industry: how to become a manager, how to handle fundraising, how to build a team, and how to negotiate terms. We could also cover how to assemble systems for the back office, legal requirements, and taxes.
I’ve been thinking that perhaps our organization, or others like it, should do more to help younger, potential portfolio managers by providing an incubator to help them become better fund managers. If any of you think that is a valid idea, please feel free to email me or come speak with me after class. Perhaps there is something beyond the scope of our current curriculum that could be useful to you. With that, thank you very much.
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