困在"赌徒心态"中的人 你是不是经常有下面的想法： 1、连抛三次硬币都是正面，下一次背面的概率更高 4、不同时间出门，却在这个路口连续三天遇到红灯，今天应该不会再遇到红灯了吧； 5、最近三次打板都吃大面了，换了一个账户，下一次转运！ 你危险了！那些把身家性命都输光的赌徒也是这么想的。 其实高中数学就学过，抛硬币、掷骰子、彩票号码、股票打板，等等，每一次都是独立事件，过去的结果对未来没有影响。 但学归学，很多人在运气连续不好时，就会觉得运气"欠"自己一次，将随机性拟人化，感觉硬币、骰子或股票记住了它之前的小概率"行为"，会通过反向 操作来平衡之前的失衡。 这个想法好像也有点科学道理，抛硬币的正反面不是应该各占50%吗？出现了连续几次正面，后面不是应该多几个反面吗？不是有一个规律叫"均值回 归"吗？ 所以赌徒输得越多，越是相信只要继续下注，运气就能帮自己翻本，正是这种心态，让他们越输越要赌，直到输光。 要理解这个现象，我们首先应该理解"独立事件概率不变"与"均值回归"这两个规律各自适用的范围： A连开四次"小" 如果赌徒有"无穷多"的钱，那当然一定可以翻本，可如果他有"无穷多"的钱，那还来赌什么呢？ 很多人道理都 ...

Group 1 - The core idea of the article is to compare investing in the stock market to gambling in a casino, emphasizing the importance of probability and strategy in achieving long-term success [12][19][20]. - The article highlights the concept of "mathematical advantage" in casinos, where the odds are structured to favor the house, and this principle can be applied to investing by focusing on long-term strategies [8][11][22]. - It discusses the "law of large numbers," which states that with enough participants and time, outcomes will converge to expected probabilities, underscoring the need for investors to stay in the market long enough for their strategies to work [10][29][30]. Group 2 - The article suggests that investors should align themselves with high-probability outcomes, such as investing in fundamentally strong companies or low-cost index funds, rather than speculating on short-term price movements [21][24][27]. - It emphasizes the necessity of maintaining a long-term perspective in investing, as the probability of making a profit increases significantly over extended holding periods [22][32]. - The importance of position sizing and diversification is stressed, advising against concentrated bets on single stocks and advocating for spreading investments across multiple assets to mitigate risks [18][28][36].

Core Viewpoint - The article emphasizes the importance of understanding the underlying logic of mathematics as a fundamental tool for comprehending the essence of business and economic phenomena [2][4][30]. Group 1: Mathematical Concepts in Business - Mathematics serves as a universal language to describe the essence of various fields, including economics and business [4][30]. - The article introduces six mathematical concepts that are highly relevant to business: four basic operations, Cartesian coordinates, exponentiation and powers, variance and standard deviation, probability and statistics, and game theory [13][30]. Group 2: Application of Mathematical Concepts - The four basic operations in business can be understood through the lens of competition and cooperation, where addition represents cooperation and multiplication signifies collaborative efforts across different dimensions [33][39]. - The Cartesian coordinate system is used to analyze complex decisions, such as hiring employees based on multiple dimensions like attitude and ability, rather than a one-dimensional perspective [56][62]. - Exponentiation and powers illustrate the differences in market dynamics between industries, such as the contrast between the restaurant industry and the internet sector, highlighting the challenges of achieving significant market share in labor-intensive sectors [71][88]. - Variance and standard deviation are crucial for assessing quality and consistency in business operations, as they provide a quantitative measure of variability within a dataset [90][102]. - Probability and statistics are essential for understanding risks and making informed decisions in business, with concepts like the law of large numbers helping to predict outcomes over time [104][115]. Group 3: Game Theory in Business - Game theory is presented as a mathematical framework for analyzing strategic interactions between multiple decision-makers, emphasizing the importance of understanding the payoff matrix in competitive scenarios [122][124]. - The article discusses how concepts from game theory, such as dominant strategies and Nash equilibrium, can aid in making strategic business decisions [129][131].

Core Insights - Anzocapital emphasizes that probability theory is crucial in forex trading, rather than relying solely on luck [1][3] - The article discusses the importance of understanding the law of large numbers and the central limit theorem in making informed trading decisions [3] Group 1: Law of Large Numbers - The law of large numbers states that as the number of trials increases, the frequency of an event approaches its probability [3] - In forex trading, a trader's winning probability will converge to their actual profitability as the number of trades increases [3] - Evaluating trading performance should focus on long-term results rather than individual trades [3] Group 2: Central Limit Theorem - The central limit theorem indicates that the sum of a large number of independent random variables will tend to follow a normal distribution [3] - To achieve positive average returns, traders must ensure that their average profits exceed their average losses, effectively managing risk [3] - The conditions for the central limit theorem require traders to have sufficient capital for numerous trades and a mindset that is not affected by past outcomes [3] Group 3: Trading Strategy - Successful traders should adopt a disciplined approach, akin to skilled marksmen, by waiting for the right opportunities and treating each trade with care [3] - Many traders fail to survive in futures trading due to over-leveraging, which undermines their ability to conduct a large number of trades [3] - Anzocapital believes that understanding and applying these two principles is essential for investors in forex trading [3]

Core Viewpoint - Goldman Sachs hedge fund manager Tony Pasquariello emphasizes that the current AI-driven U.S. stock market remains a bull market, but investors should avoid blindly chasing gains at these high levels. The market is expected to consolidate in the short term due to record high valuations and a decrease in short-term capital inflows [5]. Macroeconomic and Corporate Earnings - Goldman Sachs predicts that U.S. GDP growth will slow to 1.3% by 2025, significantly lower than recent levels, particularly as the labor market is in a "stalling state." However, growth is expected to rebound to 1.8% in 2026 and 2.1% in 2027 [6]. - The report highlights that a loose financial environment, strong fiscal support, deregulation, and a surge in capital expenditure in the AI sector provide significant upside potential for economic growth [6]. - Despite uncertainties like tariffs, Goldman Sachs forecasts a steady 7% growth in S&P 500 earnings per share (EPS) for the next two years, reaching $262 and $280 respectively [6]. - The strong performance of corporate earnings contrasts sharply with the pessimistic macro narrative, with S&P 493 (excluding the seven tech giants) showing a 7% year-on-year earnings growth in the first half of 2025, while the tech giants' earnings surged by 28% [6]. Valuation and Capital Flows - The report warns of short-term alerts regarding U.S. stocks based on valuation and capital flows. The S&P 500 is currently trading at a price-to-earnings ratio of 22, which is in the 96th percentile since 1980, indicating a "harsh" valuation [8]. - High valuations are seen more as a "roadmap" for future returns rather than a signal to short the market, as sustained high valuations have not prevented significant market gains in the past three years [8]. - The technical buying momentum that supported the market over the summer is weakening, with systematic trading funds reaching "saturated" positions and stock buybacks expected to be limited in the coming months [8]. Key Variables: Federal Reserve, AI, and the Law of Large Numbers - The report identifies three significant variables that could impact the market: the Federal Reserve's interest rate cuts, the influence of AI, and challenges posed by the law of large numbers [9]. - Goldman Sachs anticipates about five interest rate cuts by the Federal Reserve from now until mid-2026, which historically has been favorable for the S&P 500, suggesting that investors should not go against the Fed, especially without an economic recession [11]. - The ongoing debate about AI's impact on the market continues, with some viewing it as a new phase while others see it as a significant capital misallocation since the tech bubble [11]. - The report raises concerns about whether the most explosive growth days for major tech stocks are over, citing Nvidia's stock performance as an example of the challenges in maintaining high growth rates at large scales [12].

Core Viewpoint - The report emphasizes that the current AI-driven U.S. stock market is supported by two main pillars: technology giants and loose monetary and fiscal policies, but warns against blindly chasing high valuations as the market may need to consolidate in the short term [2][4]. Group 1: Macroeconomic and Corporate Earnings - Goldman Sachs predicts that U.S. GDP growth will slow to 1.3% by 2025, significantly lower than recent years, particularly as the labor market is in a "stalling state" [3]. - However, the economy is expected to return to trend growth levels of 1.8% and 2.1% in 2026 and 2027, respectively, supported by a loose financial environment, strong fiscal support, deregulation, and a surge in capital expenditure in the AI sector [4]. - Despite uncertainties such as tariffs, Goldman Sachs forecasts a steady 7% growth in S&P 500 earnings per share (EPS) for the next two years, reaching $262 and $280, respectively [4][5]. Group 2: Valuation and Capital Flows - The report highlights two warning signals for the short-term outlook of U.S. stocks: market valuation and capital flows [6]. - The S&P 500 index currently has a price-to-earnings ratio of 22 times based on expected earnings for the next 12 months, placing it in the 96th percentile since 1980, indicating a "harsh" valuation [7][8]. - The report notes that high valuations serve more as a "roadmap" for future returns rather than a short-term sell signal, as sustained high valuations have not prevented significant market gains in the past [9]. - Systematic trading funds are reported to be "quite saturated," and stock buybacks are expected to be limited in the coming months, suggesting that capital will not be the primary market driver in the short term [10]. Group 3: Key Variables - The report identifies three significant variables that could impact the market: the Federal Reserve, AI, and the law of large numbers [12]. - Goldman Sachs anticipates approximately five interest rate cuts by the Federal Reserve from now until mid-2026, which historically has been favorable for the S&P 500 index, advising not to go against the Fed, especially without an economic recession [14][15]. - AI is highlighted as a major swing factor, with ongoing debates about whether the market is in the early stages of a new era or experiencing significant capital misallocation since the tech bubble [16]. - The report acknowledges the exceptional performance of U.S. tech stocks but raises concerns about sustaining high growth rates at such large scales, using Nvidia as an example of the challenges faced [17][18].

Group 1: Market Outlook - Goldman Sachs hedge fund chief Tony Pasquariello emphasizes that the current AI-driven tech giants and loose monetary and fiscal policies are the two main pillars supporting the bull market in U.S. stocks [1] - Despite the strong support, record high valuations and a decrease in short-term capital inflows suggest that the market needs to "consolidate" in the short term [1] - Investors are advised to remain patient and not to chase prices at current high levels, while considering low-cost options for hedging in preparation for a potential market rally in Q4 [1] Group 2: Economic Growth and Corporate Earnings - Goldman Sachs predicts that U.S. GDP growth will slow to 1.3% by 2025, significantly lower than recent levels, particularly as the labor market is in a "stalling state" [2] - However, the economy is expected to return to trend growth levels of 1.8% and 2.1% in 2026 and 2027, respectively, supported by a loose financial environment, strong fiscal support, deregulation, and a surge in capital expenditures in the AI sector [3] - The report highlights that despite uncertainties like tariffs, S&P 500 earnings per share (EPS) are projected to grow by 7% in both 2024 and 2025, reaching $262 and $280, respectively [3] Group 3: Corporate Profitability - The strong performance of corporate earnings contrasts sharply with the pessimistic macro narrative, as the "S&P 493" (excluding the seven tech giants) saw a 7% year-over-year profit growth in H1 2025, while the tech giants experienced an impressive 28% growth [4] - This indicates that corporate profitability is a solid foundation supporting the market [4] Group 4: Valuation and Capital Flows - The current price-to-earnings (P/E) ratio of the S&P 500 is at 22 times expected earnings for the next 12 months, placing it in the 96th percentile since 1980, which is described as a "harsh" valuation [6] - The report notes that high valuations serve more as a "roadmap" for future returns rather than a short-term sell signal, as sustained high valuations have not prevented significant market gains in the past three years [6] - There is a warning regarding the weakening technical buying momentum that supported the market during the summer, with systematic trading funds now "fairly saturated" and stock buybacks expected to be limited in the coming months [7] Group 5: Key Variables Impacting the Market - The report identifies three key variables that could significantly impact the market: the Federal Reserve, AI, and the law of large numbers [9] - The Federal Reserve is expected to implement about five rate cuts from now until mid-2026, which historically has been favorable for the S&P 500, suggesting that investors should not go against the Fed, especially without an economic recession [11] - AI continues to be a major swing factor, with ongoing debates about whether the market is in the early stages of a new era or experiencing significant capital misallocation [12] - The report acknowledges the challenges of maintaining high growth rates for large tech stocks, as evidenced by Nvidia's stock performance relative to its earnings expectations [13][14]

Group 1 - The core viewpoint of the article emphasizes the harsh realities of the futures market, indicating that a significant majority of participants are unlikely to achieve long-term success, with estimates suggesting that only 0.1% will be profitable over three years and 0.01% over ten years [3][5][8] - In 2024, the domestic futures trading volume reached 619 trillion, with an estimated total fee of around 80 billion, leading to the disappearance of approximately 140,000 medium-sized accounts annually [6][7] - The article discusses the impact of the "Big and Beautiful" Act on commodity investments, particularly how it may reduce demand for industrial metals like silver and copper while benefiting traditional energy sources like crude oil [9][10] Group 2 - The article outlines the current market dynamics for various commodities, indicating that the futures market is predominantly long for financial indices and certain metals, while short positions dominate in others like paper pulp and pure alkali [12] - It highlights the potential for a rebound in the glass market driven by policy and sentiment, suggesting specific trading strategies for both long positions and hedging [20][21][22] - The discussion on copper emphasizes the tactical implications of tariffs and the need for a realistic understanding of market conditions, suggesting that the current situation is more about short-term volatility rather than long-term direction [15][16]