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Writer's pictureBedRock

Probability, concentration and ergodicity

Updated: Apr 8

Recently, due to the volatile market conditions, especially concerns about future policy changes that have caused a series of fluctuations, we have once again thought about the fundamental methodology of investment strategies:


Regarding diversification and concentration

Buffett is a big proponent of concentrated investments (of course, including Drunkemiller, Greenblatt, David Tepper, etc.). His famous "20-Slot Rule" shows his cautiousness in investing and his support for heavy stock holding. In fact, Buffett is extremely concentrated in his investments. By reviewing his annual letters to shareholders, we can see that his most core holdings often only have five stocks, with the largest position often accounting for 50% of all holdings, such as Apple currently and Coca Cola in the past. In terms of industries, he almost only touches on the consumer and financial industries in his decades-long investment career, showing his strong adherence to his circle of competence.


Extraordinary ability or outstanding luck?

Undeniably, Buffett is not just lucky, but rather someone who has an exceptional ability to see the development of things, which is significantly beyond that of the average investor. Therefore, it is reasonable for him to achieve returns that surpass the average investor's (market index). From another perspective, if someone who has had a more than 60-year investment career solely relies on luck, then they must have been extremely lucky. Moreover, having only excellent cognitive abilities (of course, no one can be a prophet) without a strong investment system as a guarantee is far from sufficient.


Concentrated investment and ergodicity

As we mentioned earlier, Buffett, or value investing, is probably ergodic in the long-term dimension. However, regarding concentrated investments, misunderstanding its meaning may lead to non-ergodic, fragile, or even disastrous results.

Suppose there is a coin-tossing gambling game in which you invest 1 yuan, with a 50% chance of getting 0.6 yuan and a 50% chance of getting 1.5 yuan. According to the expected value calculation, you may lose 40% or gain 50%, with a mathematical expectation of 5%. In popular terms, this is a thing where you can make money with high probability, and you can play this game boldly. However, there are two different ways to play this game:

Method A: You invest 1 yuan every time. Assuming you have an infinite amount of 1 yuan, you can play it continuously. In the long run, you are sure to make money, with an average of 5% of mathematical expectation calculated by a simple math formula. The disadvantage is that it is too slow, and you must have enough time to play.

Method B: You put in the maximum amount of capital you can put in and invest it. This is the so-called "All in" approach. It seems extreme, but many people do it this way.

Let's do a simple calculation:

Your principal is 1 million yuan. You win in the first round, lose in the second round, and win again in the third round, and so on. At first glance, with a principal of 1 million yuan, you can make 500,000 yuan by winning and lose 400,000 yuan by losing, so why not play it? 1 million yuan * (1 + 50%) * (1 - 40%) * (1 + 50%) * (1 - 40%) …

If you keep playing like this, you will find that you will run out of money very quickly.

Isn't this the reality of most ordinary people doing investments?

Therefore, excessive concentration or "all in" is prone to non-ergodic and fragile results. The fact that we see legends in various news media and storybooks is precisely because they have a certain element of chance.


Achieving Ergodicity and Antifragility in Investment

To achieve our mathematical expectation (assuming that our knowledge of things themselves conforms to the mathematical expectation), our investment system must have ergodicity and anti-fragility so that we can eventually achieve our mathematical expectation, especially when our mathematical expectation may be affected by market fluctuations (market consensus expectations) and cause dramatic short-term fluctuations in returns.

The most important means include:

  1. Repeated betting instead of a one-shot deal;

  2. Being able to persist in the long-term dimension, including avoiding excessive leverage (to avoid liquidation), longer-term psychological preparation, longer-term sources of funds, deeper cognitive insights, and better communication with investors;

  3. And appropriate diversification to avoid the impact of errors in the fundamental understanding itself and to reduce the volatility of the system (rather than individual stocks); as the famous economist John Maynard Keynes said: "The stock market can remain irrational longer than you can remain solvent." To achieve the same investment result as the initial mathematical expectation and expectation, in addition to having a strong insight, it is also necessary to ensure that you do not drop out when you have bad luck, so that you can wait until luck returns and eventually achieve the expected return.

Regenerate response


Diversifying investments across opportunities with similar expected returns but different levels of volatility can greatly reduce the overall volatility of the system, improving its ergodicity and antifragility.


Another investment guru, Ray Dalio, also realized that making money through market trading is very difficult, even if your predictions are mostly correct, you can still fail without "ergodicity." His secret is that having 15-20 good, unrelated streams of return can greatly reduce risk. This is also the source of the later "All Weather" Fund.

Another investment guru, Greenblatt, who achieved a 50% compounded return over 10 years, believes that 5 is enough, which is similar to Buffett's approach, but he is unlikely to be more extreme than Buffett.


Buffett, concentrated holdings and ergodicity

Although Buffett's value investment strategy proved to be ergodic, relying solely on a few stocks like Coca Cola and Apple at a single point in time is unrealistic or purely based on luck. In fact, despite the appearance of concentrated holdings, the success of Buffett's strategy still relies on antifragility and ergodicity.


Buffett's early diversified investments

For some reason, many of Buffett's letters to investors were written during his time at Berkshire, probably because that period was long enough and the content was rich enough, and many investors thought that this was his entire investment career. However, before Berkshire, Buffett also operated a fund (limited partnership) between 1957-1962, similar to most fund operating models today, by raising funds for investment. Unlike later periods, Buffett's investment model during this period was not as concentrated and diversified in investment types, with about 1/3 in each category, although it varies depending on the opportunities. The first category is the so-called "deep value," or the so-called "cigarette butt" investment, with a shorter investment period than the later period, often selling when the value returns. The second category is arbitrage trading, such as obtaining returns through company mergers, liquidation, and restructuring. The third category is controlling investments, mainly obtaining returns through the company's annual dividends. The second and third categories of investments are basically unrelated to short-term market volatility, and only the pricing of the first category fluctuates with short-term market prices. The difference in the path of value discovery and value realization among these three categories is significant. In 1969, Buffett closed his limited partnership company, believing that the stock market had already deviated from its fundamental values, and he could not find good investment opportunities. He introduced his investors to Bill Ruance, the founder of Sequoia Fund, and wrote a lot of praise for him in his letter to investors, but at the same time, he pointed out that although his investment performance was quite amazing, with an average annual return of 40% for the investment portfolio in 1956-1961 and 1964-1968, he also suffered a 50% loss in 1962. The biggest difference between early Buffett and Bill is that Bill's strategy is relatively single, only investing in deep value assets, and there is no arbitrage or controlling investments, while Buffett is more diversified in risk-reward sources.



Concentrated investments during the Berkshire period

Undeniably, Buffett's investment style has changed to become more extreme and concentrated during the Berkshire period. As mentioned earlier, he often only holds five major stocks, with the first major holding often in 30-50% positions. But in addition to Buffett's profound understanding of business models and targets, his investment strategy itself also makes it possible to achieve ergodicity.



Buffett, cash flow, and ergodicity

Buffett's emphasis on cash flow is not only because companies with strong cash flow have better investment value, but also because it enhances the traversability and anti-fragility of his investment system.

Buffett's company, with only 25 employees, manages many companies worth billions of dollars. He delegates management and only requires one thing: companies continuously turn over their earnings. Buffett's controlling investments actually separate management's management and capital allocation capabilities, with management focusing on creating cash flow and Buffett focusing on capital allocation.

One of Buffett's standards for selecting good companies is that they have low capital expenditures and large free cash flow, in addition to the float of insurance companies. Buffett has a spring-like ammunition of funds. Coupled with the moat of value investing and the discipline of not using leverage, in the stock market game of "traversability," Buffett can keep betting.

From this perspective, for funds that do not have the support of insurance float and a large amount of controlling company cash flow, the issuance and sales department of the fund has great advantages in improving the systematic traversability of fund investments. For example, well-known funds such as Hillhouse Capital and Sequoia Capital have adopted similar practices, issuing new funds to achieve new investment opportunities. If the fund can continuously increase in size and obtain opportunities for repeated betting, it is obviously beneficial to increase the traversability. Of course, the stability of fund issuance is subject to market conditions and other factors, making it less controllable. If there is no inflow from fund sales, cash can be obtained by reducing previous holdings to achieve repeated betting. However, during a market downturn, it may result in insufficient liquidity and an increase in the cost of panic selling, making it difficult to achieve.


Buffett's advice and methods for achieving traversability:


  1. Strong cognitive ability and a steadfast circle of competence;

  2. Firm and solid fundamentals with a margin of safety in valuation;

  3. Never use leverage and maintain stable emotions;

  4. Almost perpetual insurance funds, constantly inflowing float, and cash flow from invested companies. Without these foundations to achieve traversability, it is very difficult to achieve concentrated holdings and long-term investment according to Buffett's approach, especially when most funds and individuals do not have such strong insight and fluctuating emotions, and funding sources themselves are volatile, etc., which may make it difficult to achieve traversability even when the expected win rate is positive.


Building systematic traversability:

For the evaluation of specific single investment opportunities, investors can conduct research and judgment based on potential returns and probabilities. However, the biggest problem we have encountered in investing over the past period is in the construction of the systematic return rate and system, which has caused some problems in realizing the traversability of the mathematical expectations of single investment opportunities through systematic investment. When the expected return and win rate are satisfactory, how to achieve the regression of mathematical expectations in the unlucky times mainly depends on "endurance" to satisfy the traversability conditions. This means that stability must also be maintained in investment strategy and investment teams. Imagine if the investment strategy itself is not stable, it is difficult to ensure that it will not withdraw during difficult and unlucky times, and thus cannot achieve traversability. If the team does not highly recognize the core strategy, it cannot guarantee the realization of the traversability of the investment strategy at the bottom and information support.

AQR's ergodic enhancement composition

AQR has an interesting research report called "Building a Better Core Equity Portfolio", which categorizes investors into three types based on their investment attributes: value, momentum, and profitability. The report argues that these three types of investors have significant differences in biases, which can result in differences in volatility and enhance the risk-return profile of the overall portfolio.


Due to differences in style preferences, the three types rarely fluctuate synchronously, so portfolio management can effectively reduce volatility and enhance the risk-return of the overall portfolio.


AQR's strategy is to maintain stability in the three investment styles, even in situations where certain styles may perform poorly, rather than simply buying low and selling high or heavily rotating styles. This is what enables the achievement of investment return traversal. However, while the theory is sound, the challenge for investment teams is still significant, as they must simultaneously master different investment style types and avoid being influenced by short-term performance drift.


Anti-fragility in investing in China

When investing in China (and many other developing countries), it is essential to recognize the possibility that the rapid pace of development and significant state intervention may lead to situations where the actual probability of achieving the expected return is lower than anticipated. It is also possible that there may be unstable expectations of the expected return in such situations.

In such cases, selecting traversal opportunities may also result in a situation where it becomes impossible to achieve regression (due to the involvement of unknown external forces). Therefore, blindly concentrating investments on a single option may pose significant risks in such situations.

We believe that in the current situation, moderate diversification, especially in terms of style characteristics, as well as increasing allocation to different countries, may be important ways to achieve anti-fragility and realize the traversal of value investments.

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