The most important aspect in investing is when to commit significant capital.

The essence of substantial losses lies in placing heavy bets on assets where they shouldn't be placed!

In 2022, the U.S. investment firm Archegos Capital Management incurred massive losses of billions of dollars due to misjudged investments. The founder, Bill Hwang, overestimated the valuation of Chinese internet companies, leading to significant losses. Placing too many chips on a single overvalued asset ultimately resulted in a margin call due to China's internet anti-monopoly actions.

In 2008, the U.S. investment bank Lehman Brothers suffered substantial losses of billions of dollars due to misplaced bets. The bank undervalued subprime mortgage-backed securities during its investment, leading to massive losses.

These cases highlight that the essence of substantial losses lies in misjudging investments. Investors should carefully assess the value of investment targets and avoid excessive bets.

The Kelly Criterion, proposed by John L. Kelly Jr. in 1956, initially designed to address signal noise in the telecommunications industry, was later found applicable to gambling and investment, especially in determining bet or investment sizes.

The core of this formula is to maximize long-term growth rates, aiming to balance risk and reward, avoiding bankruptcy from overly large bets while maximizing capital increase.

In essence, the Kelly Criterion calculates the proportion of funds one should wager to maximize the long-term expected growth rate of the investment portfolio. If the result is positive, one should place a bet; if negative, one should avoid betting.

The significance of the Kelly Criterion is that, during each bet, investors should allocate funds to assets with a win rate higher than the odds.

In investing, investors can use the Kelly Criterion to calculate the optimal bet size. However, the Kelly Criterion has a drawback: it assumes investors are entirely rational. In reality, investors are often influenced by emotions, leading to irrational decisions.

In the investment field, the Kelly Criterion helps investors decide how much capital to invest in a single asset or gamble, considering odds and win rates to reduce risk and increase potential returns. However, in practical application, there are several points to note:

1. Accuracy of estimation: The output of the Kelly Criterion highly depends on accurate estimates of win rate (p) and odds (b). If these parameters are inaccurate, the suggested values by the Kelly Criterion may not be optimal.

2. Long-term application: The Kelly Criterion is designed for long-term growth. Short-term fluctuations may involve significant volatility, requiring investors to have corresponding risk tolerance.

3. Fractional Kelly: To reduce volatility and risk, many investors and gamblers opt for a fractional Kelly strategy, investing only a portion of the recommended amount (e.g., half or one-fourth).

While a powerful theoretical tool, the Kelly Criterion is not always the best practical choice, especially when an individual's risk appetite, capital amount, or ability to estimate probabilities does not perfectly match the assumptions of the Kelly Criterion. Therefore, many investors consider the Kelly Criterion as one reference among several, rather than the sole investment decision tool.

Let's interpret the Kelly Criterion through a simplified investment example:

Suppose you have an opportunity to invest in a stock with a 60% probability of rising, providing a 50% return when it rises and a 50% loss when it falls. Now, we need to decide the proportion of your total capital to invest in this stock.

First, define the parameters in the Kelly Criterion:

· b is the net gain rate of the bet, here 50% or 0.5.

· p is the probability of winning, here 60% or 0.6.

· q is the probability of losing, here 40% or 0.4.

Plug these values into the Kelly Criterion:

As the result f∗ is negative, following the Kelly criterion, you shouldn't invest in this stock because it is expected to incur losses in the long term.

Now, let's alter the conditions. If the probability of the stock rising remains at 60%, but if it rises, you can gain a 100% return, and if it falls, you lose 50%.

Now, the parameters for the Kelly Criterion are:

· b is 1 (or 100% return).

· p remains 0.6.

· q remains 0.4.

Plug in the values:

The result f∗=0.2 indicates that to maximize your long-term growth rate, you should invest 20% of your total capital in this stock.

Remember, the Kelly Criterion provides the theoretically optimal solution for maximizing capital growth. However, in practice, factors such as risk tolerance and liquidity must also be considered. Many investors may choose to invest less than 20% of their capital to reduce risk.

The Kelly Criterion and Bayes' Theorem are both important decision-making tools in investing, although they address different types of problems. Let's explain their applications in investment decisions through examples.

Investment Application Example of the Kelly Criterion:

Suppose you are considering investing in a startup technology company. This company will either succeed, bringing substantial returns to investors, or fail, resulting in a total loss. After analyzing the company's business model, market potential, and team background, you estimate a 30% probability of the company's success. If successful, the return on investment will be three times the initial investment; if it fails, you will lose the entire investment.

Here:

Investment Application Example of Bayes' Theorem:

Bayes' Theorem is a method for updating prior probabilities based on new evidence or information. Suppose you are investing in large-cap stocks in an industry typically closely related to macroeconomic indicators. Initially, you might have a prior belief (probability) based on historical data and analysis that, in the current economic environment, the probability of this stock performing well in the next quarter is 60%. However, the latest employment data unexpectedly improves, a strong indicator of the stock's positive performance.

Using Bayes' Theorem, you can update the probability of the stock performing well. If historical data shows that, when similar employment data is announced, the probability of the stock performing well is 80%.

Therefore, considering the new employment data, with the Bayesian update, you now believe the probability of the stock performing well this quarter is 96%.

In investment decisions, the Kelly Criterion helps you determine the optimal investment ratio based on your win rate and odds, while Bayes' Theorem helps you update your probability assessment of an event based on new evidence. Both are powerful tools in investment decision-making, aiding investors in making more informed decisions amid uncertainty.

From a probability theory perspective, combining the Kelly Criterion and Bayes' Theorem, how to increase the expected return of the portfolio and when to place significant bets?

The Kelly Criterion and Bayes' Theorem are crucial tools in decision theory, particularly in investment decisions. Combining these two tools can allow for more sophisticated portfolio management, potentially increasing the expected return of the portfolio.

Combined Use:

1. Prior Probability Assessment: Use Bayes' Theorem to assess prior returns and risks for each investment. Collect market data, historical performance, economic indicators, and other relevant information as evidence to update your beliefs about asset returns.

2. Posterior Probability Updates: Continuously update posterior probabilities of asset returns using Bayes' Theorem as market conditions change and new information emerges. This helps you quickly adapt to market fluctuations.

3. Capital Allocation: Use the Kelly Criterion to determine how much capital to invest in each asset based on updated probabilities. This optimization allows you to refine your investment portfolio according to the latest market information and your beliefs about future market performance.

4. Risk Management: Utilize the Kelly Criterion to avoid overinvesting, maintaining risk control even in high-odds situations. This is because the Kelly Criterion considers the probability of failure as a crucial parameter in capital allocation.

Combining the Kelly Criterion and Bayes' Theorem can help investors increase the expected return of the portfolio while maintaining risk control. However, caution is needed in practical application, as accurate estimation of returns and probabilities is required, and the ability to adapt to new information poses a significant challenge to investors' judgment and execution.

Investment opportunities validated through Bayes' Theorem and the Kelly Criterion are often the most worthwhile opportunities to place significant bets.

If the odds are high but the win rate is low, it may not be worth risking too much, as in the long term, you may lose more times than you win. Conversely, if the win rate is high but the odds are low, it may not be worth investing too much, as losses could be significant when you do lose.

Only when both the odds and win rate are relatively high will the Kelly Criterion recommend investing a larger amount of capital – this is when it's worth placing significant bets.

With such complex formulas, personal calculations are impractical. Hence, the AI application tool named "FinTech&AI Turbo" from the AI Wealth Club, powered by super AI, accomplishes this task excellently.

Whether in the stock market or the cryptocurrency market, especially in cryptocurrency contract trading, maximizing profits can be achieved. Place bets wisely for returns!

Daniel Will's Approach to Long-Term Growth