We previously wrote an article exploring the issues of probability, concentration, and diversification in investing (Probability, Concentration, and Diversification - Investment Reflections (56)). However, we later realized that it was necessary to conduct further detailed research on this issue to better guide our future investments.
casino model
If you have one million dollars and there is an investment opportunity D with a 50% chance of making a 60% profit and a 50% chance of losing 40%, would you invest? (Without considering any restrictions such as bankruptcy or liquidation)
The answer is yes, because the expected rate of return is 50% * 60% + 50% * (-40%) = 10%, which means this is a game with a positive expected return. Regardless of the outcome of the game, making an investment decision is undoubtedly a rational behavior.
One-Round Game
Now, consider another scenario: Suppose there are two identical projects, D1 and D2. You can choose to invest 500,000 in each project, or invest the full amount of 1 million in one of the projects. Which option would you choose?
Let's first calculate the expected return for investing 500,000 in each project: 50%(50%60% + 50%(-40%)) + 50%(50%60% + 50%(-40%)) = 10%. This means that the expected return rate for both investment options is the same. So, what's the difference?
When you invest in only one project, there are only two possible outcomes: a 60% return (with a probability of 50%) or a 40% loss (with a probability of 50%). However, when you invest in both projects, there are three possible outcomes: a 60% return (both projects make a profit, with a probability of 25%), a 40% loss (both projects incur a loss, with a probability of 25%), or a 10% return (one project makes a profit while the other incurs a loss, with a probability of 50%).
Clearly, although the expected return rates of the two strategies are the same, diversifying your investments has a clear advantage: the probability of loss decreases from 50% to 25%, even though the cost is an increased probability of a moderate return.
Two-Round Game
After the first round of the game, your capital has increased to 1.6 million (or decreased to 0.6 million). At this point, you encounter another project D. Would you continue to invest?
The answer is certainly yes, because the expected return is still positive.
Now, let's ask the question in a different way: Suppose you could travel back in time to before the first round of the game, and the game rules were changed to require playing two rounds in a row, and all capital from the previous round must be invested in the next round. Would you still play this game?
Skipping the calculation process, let's take a look at the probability distribution graph:
The probability distribution graph for the results of two consecutive games is shown below. The left graph shows the result of investing in only one project, while the right graph shows the result of investing in both projects at the same time.
When investing in only one project each time, although the compounded return rate for the expected final value is still positive at 10%, the expected compounded return rate is only 4.2%. The difference between the two is that the former calculates the weighted average of the final value after the game is over and then takes the square root to calculate the compounded return rate, while the latter calculates the compounded return rate of each final value first and then calculates the weighted average of these compounded return rates. Due to the impact of compound interest, the corresponding final value is very large when continuously gaining, which greatly increases the average of the final value. Therefore, the compounded return rate for the expected final value overestimates the impact of small probability events and is not suitable as an indicator for evaluating high or low returns. The expected compounded return rate eliminates the impact of compound interest and is more representative. The two are equivalent in a single round of games; however, in multiple rounds of games, the expected compounded return rate is often greater than the compounded return rate for the expected final value.
When investing in both projects at the same time, the capital after each round is divided into two parts and reinvested in the next round of the game. The expected compounded return rate after two rounds is about 7.2% (and the compounded return rate for the expected final value is still positive at 10%). This means that although diversifying investments only increases stability without increasing the expected compounded return rate in single-round investments, both are increased in multiple-round investments.
From the probability distribution graph, it can also be seen that when investing in only one project each time, there is a 75% probability that the compounded return rate will be negative after two rounds, while when investing in both projects at the same time, there is only a 45% probability that the compounded return rate will be negative.
Multiple-Round Game
What about the longer-term results? The compounded return rate distribution graph after ten rounds is shown below:
Among them, when investing in 1, 2, 3, and 5 projects at a time, the expected compounded return rates are -0.8%, 4.5%, 6.4%, and 7.9%, respectively.
If only investing in a single project D can still make a profit after two rounds, in the long run, this investment strategy will definitely lose money. Diversifying investments can greatly increase the expected compounded return rate. From the graph, it can be clearly seen that the peak gradually shifts to the right and the process of turning from negative to positive.
Therefore, if the investment strategy is to continuously go all in on a single project and only pursue a positive expected return rate (such as project D), the high or low win rate is also an important factor. If there is a project with an expected return rate of 10% and a win rate of 100%, the compounded return rate from continuously going all in on this project will be 10%, which is a good investment opportunity.
However, it is rare to encounter opportunities with a 100% win rate in real life, and there are many projects like project D with a win rate of 50%. At this time, as long as the funds are diversified into independent projects, long-term positive returns can also be achieved. Moreover, the more diversified the investment, the closer the long-term compounded return rate is to the expected return rate of a single investment, which is 10%.
Calculation Formula
So, what combination of returns, losses, and win rates can ensure a positive long-term All in strategy? Or in other words, how do we calculate the expected compounded return rate for a long-term all-in strategy?
Suppose a project has a return rate of a, a loss of b (b<0), and a win rate of p. Then, the expected return rate for a single game is:
When a=60%, b=-40%, and p=50%, E(R)=10%.
And in the long-term all-in strategy (as the number of investments approaches infinity), the expected return rate for a single game is:
When a=60%, b=-40%, and p=50%, E(CR) = -2.02%.
We can observe that when continuously going all in on project D, the expected compounded return rate for two games is 4%, for ten games it is -0.8%, and for 100 games it is -1.91%, which does approach the theoretical value of -2.02%.
Clearly, to increase E(CR), increasing a and p or decreasing b are all feasible options.
When a certain project has m possible outcomes, and the probability of having a return rate (or loss rate) of ak is pk, where k ranges from 1 to m, the expected return rate for a single game in the long-term all-in strategy (as the number of investments approaches infinity) is:
For example, when simultaneously investing in two projects D, where a1=60%, a2=10%, a3=-40%, p1=25%, p2=50%, and p3=25%, the expected return rate for a single game is Ej=3.82%. And the expected compounded return rates for two, ten, and 100 games are 7.2%, 4.5%, and 3.8%, respectively, which gradually approach the theoretical value of 3.82%.
items of different qualities
Assuming there are four different investment projects:
Project A: Return 90%, Loss 10%, Win Rate 50%;
Project B: Return 80%, Loss 20%, Win Rate 50%;
Project C: Return 70%, Loss 30%, Win Rate 50%;
Project D: Return 60%, Loss 40%, Win Rate 50%;
As good investment projects are always scarce, suppose there is only one Project A, two Project B, three Project C, and five Project D available. In this situation, which is more advantageous, a high return or a high degree of diversification?
The probability distribution after one game is shown in the figure below:
The expected value of the compound rate of return is: 40%, 30%, 20%, 10%
he expected compound rates of return are 31.58%, 25.45%, 16.68%, and 7.85%, respectively.
Clearly, while increasing dispersion can bring the expected compound rate of return closer to the expected rate of return for a single investment, it can never surpass it to achieve a higher level. In the long run, even if the win rate is only 50% and dispersion investment is not allowed, project A with high return and low loss is still the best investment target.
Stock Model
The above discussion is more like a game in a casino, which is very different from stocks in reality. The main difference is that in the casino, the two games before and after are completely independent, while in stocks, after a drop, there is often a rebound, as the stock price always tends to return to its value (although the time it takes is highly uncertain) under unchanged fundamentals. Therefore, in the stock model, we consider both the impact of fundamental growth and short-term market sentiment, the former of which is subject to stock selection abilities but is largely predictable, while the latter is completely random.
Simple Model
We should overweight the super bull stocks (35% growth), or several ordinary bull stocks (25%), or many mediocre companies (15%)?
Since the accuracy of fundamental analysis is related to the number of projects examined, the more projects there are, the less accurate the analysis will be. However, diversified investments can bring stability at the overall level.
Assuming there are four stocks:
A, B, C, and D with fundamental growth rates of 35%, 25%, 15%, and 5%, respectively. Six different portfolio allocations are considered: 1 A, 2 A's, 3 B's, 5 B's, 10 C's, and 20 C's.
The accuracy of the fundamental analysis for each stock is 70%, 65%, 60%, 55%, 53%, and 50%, respectively.
The analysis is only done in the first year, and if the analysis is incorrect, the stock is downgraded by one level (A becomes B, B becomes C), while if it is correct, the stock is not downgraded. The impact of market sentiment is modeled with PE ratios of 10, 15, 20, 30, and 40 with probabilities of 10%, 20%, 40%, 20%, and 10%, respectively. The initial PE ratio for all stocks is 20. When the target compound return falls below 15%, the current stock is sold, and a new stock with a PE ratio of 20 is bought. If the net value is less than 0.7 in any year, it is marked as a liquidation and the liquidation rate is calculated.
The probability distribution after one year is：
The probability distribution after five years is：
From the results, although diversification can affect the accuracy of project evaluation and increase the probability of project downgrading, leading to lower returns, the biggest advantage of diversification is to avoid extreme risks, such as liquidation. Combining two completely independent projects can reduce the liquidation rate from about 10% to less than 2%. If too many low-quality stocks are chosen in pursuit of a zero liquidation rate, the overall rate of return will decrease significantly due to the quality decline of the stocks. In addition, the difference in quality between stocks is crucial, and the difference in returns between high-growth and low-growth stocks is significant, which cannot be compensated by diversification. This is similar to the gambling model, where diversification can only increase the stability of returns, but cannot increase the absolute value of returns.
Complex Model
The above model has many flaws, such as the market's valuation of different quality stocks cannot be the same 20 times, so we introduce the concept of individual stock PE multiples, multiplying different coefficients on the basis of the market PE. In addition, low-growth projects may not be downgraded even if they are judged incorrectly because achieving such growth is not difficult, so we introduce the concept of project retention probability. The project will only be downgraded when a judgment error and failure to retain occur at the same time. Other improvements include: judging the project for downgrading once a year; when investing in multiple projects, the market PE in the same year is the same, which increases the correlation of projects when diversifying.
The new model：
Assuming there are five projects S, A, B, C, D:
Their respective basic growth rates are: 50%, 35%, 25%, 15%, 5%.
The higher the growth rate, the shorter the duration it can be sustained. The different growth rate stages can be sustained for 2, 3, 4, 5 years, and 5% growth can be perpetual, but they will eventually be downgraded.
The individual PE multiples are +250%, +175%, +125%, 100%, 75% compared to the market PE.
The market PE (assumed to be random) has probabilities of 5%, 10%, 15%, 40%, 15%, 10%, 5% for the values of 10, 12, 16, 20, 25, 33, and 40.
The actual PE is calculated as market PE multiplied by the individual PE multiple.
Buy at the median market PE (i.e., 20 times), which means the initial PEs of the five projects are 50, 35, 25, 20, and 15, respectively. Sell the current stock and buy a new project with the same level of initial PE when the target compound rate of return is less than 15% (calculated over the next 10 years).
Consider six investment portfolios: 1 S, 2 S, 1 A, 2 A, 3 B, and 5 B.
The accuracy of fundamental analysis judgment is 75%, 70%, 75%, 70%, 65%, and 60% per year (assuming research can improve the accuracy of selection, but the accuracy of judgment decreases as the number of selected stocks increases due to limited resources).
The probability of self-maintenance for each S, A, B, C, and D project is 10%, 30%, 50%, 80%, and 100%, respectively (D project will not be downgraded).
If the judgment is wrong and the project is not self-maintaining, the project level will be downgraded (A becomes B, B becomes C, etc.). For example, the probability of downgrading for 1 S project is (1-10%) * (1-75%) = 22.5%. In other cases, the project level remains unchanged.
When the net value is less than 0.7, clear the position, and the market value at the time of clearing is the final value；
After one year the result is：
After five years the result is：
retracement is：
Consideration of projects B and C:
Project types: S, S, A, A, B, C
Number of projects: 1, 2, 1, 2, 10, 20
Accuracy of judgment: 75%, 70%, 75%, 70%, 60%, 55%
Accuracy of judgment: 75%, 70%, 75%, 70%, 60%, 55% one year：
five years：
retracement is：
It should be noted that when diversifying investments in many stocks, due to their correlation with the overall stock market, it seems that it cannot significantly reduce drawdowns. Moreover, as the underlying assets themselves have weaker growth prospects, the median, average, and maximum drawdowns do not appear to be lower than those of fewer but better-performing concentrated investments, as these superior assets can greatly reduce volatility through their excellent growth prospects.
Diversifying investments in a single market does not seem to effectively reduce drawdowns, and there are two solutions to this: either find suitable hedging strategies or look for investment targets that are independent of the stock market.
The PE distribution we have assumed above may not match the actual valuation system in different markets, as we have observed significant historical differences between markets (although this does not necessarily mean that it will repeat in the future, it is for reference only).
Market PE of CSI 300
According to the statistical results from 2006 to 2021, the probability of the market PE of the CSI 300 being 9, 10, 11, 13, 15, 30, and 40 are 5.88%, 11.76%, 11.76%, 29.41%, 23.53%, 11.76%, and 5.88% respectively. The highest market PE was in December 2007, at 43.41 times, while the lowest was in December 2013, at 8.92 times.
Market PE of the main board of Hong Kong stocks
According to the statistical results from 2006 to 2021, the probability of the market PE for Hong Kong's main board being 8, 10, 12, 14, 16, 19, and 23 are 8.5%, 34.5%, 16.5%, 15%, 15%, 8.5%, and 2%, respectively. The highest market PE occurred in October 2007, at 25 times, while the lowest was in February 2009, at 6.5 times.
Market PE of the S&P 500
According to the statistical data from 1999 to 2021, the probability of market PE being 15, 17, 18, 20, 23, 28, and 35 are 13.04%, 17.39%, 13.04%, 17.39%, 21.74%, 13.04%, and 4.35%, respectively. The lowest market PE was 14.8 times in 2011, while the highest was 34.9 times in 2021, followed by 28.9 times in 1999.
The market PE of Nasdaq:
According to the statistical data from 2004 to 2021, the probability of market PE being 24, 26, 31, 34, 36, 40, and 51 is 11.11%, 11.11%, 16.67%, 16.67%, 16.67%, 11.11%, 16.67% respectively. The highest was in 2004, which was 53.38 times, and the lowest was in 2012, which was 23.32 times. Currently, it is 50.67 times.
Impact of Different Markets
The comparison of the same project in different markets is as follows：
Obviously, market environment has a significant impact on investment returns. With the same stock selection and investment strategy, the compound return on A-shares can exceed 60%, while in Hong Kong and US stocks it is less than 40%. This is due to the fact that A-share investors are relatively more optimistic (perhaps due to the absence of short selling mechanism and higher proportion of individual investors), which leads to higher incentives for dreams and smaller punishments for bad companies, making it easier to get lucky with the same stock selection logic, and less severe punishments when unlucky. Of course, this is only a regression analysis of the past valuation distribution, and is greatly influenced by extreme valuations, and does not mean that such valuations and reward-punishment distributions will continue in the future, and it also requires observation of investor structure and overall market governance to adjust.
With the same stock selection criteria, diversified investment can greatly reduce the clearing rate and avoid the worst results, even though the accuracy of stock selection may decrease slightly, leading to a slightly smaller expected return.
Conclusion:
In the gambling game:
For the same type of project, diversification cannot increase the absolute value of returns, but it can improve the stability of returns and the expected compound returns under the continuous all-in strategy;
In a single round game, expected returns can evaluate the results well, but in multi-round games, expected compound returns are more appropriate;
In multi-round games, due to the effect of compound interest, expected returns often overestimate the performance of the strategy.
In stock investment：
The difference in returns between different quality investment targets is crucial, and high-quality targets should be invested in as much as possible;
When all-in on a high-quality target, although the expected return is very high, the risk is high, and the clearing rate is high. However, only two completely independent projects can greatly reduce the clearing rate, even if it is two incomplete. Independent projects also have a significant effect;
The impact of the market on returns is significant and depends mainly on the distribution of market PE. The return on A-shares is significantly higher than that of other markets (due to higher incentives for dreams and smaller punishment for poor-performing companies);
Diversification of many stocks in stock investments may not significantly reduce drawdowns due to the correlation of the stock market as a whole. There are two solutions to this: either find a suitable hedging solution or find investment targets that are mutually independent of the stock market.
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