In the world of finance and betting, understanding how to size your bets can make or break your success. One powerful tool to guide this decision is the Kelly Criterion. Originally developed for gambling, it’s now widely applied in portfolio management to maximize long-term capital growth while managing risk.

What is the Kelly Criterion?

The Kelly formula helps determine what percentage of your capital to invest in a given opportunity when you have an edge. The basic version is:

Where:

• : probability of winning
• : net odds (reward-to-risk ratio)

If you estimate your probability of success and your upside vs downside correctly, Kelly tells you how much to invest. But real investing rarely gives you those clear odds.

Enter the Sharpe Ratio

In finance, we often use the Sharpe Ratio as a proxy to adapt Kelly to investment portfolios. The modified formula becomes:

Where:

• : expected return
• : risk-free rate
• : volatility

This gives you the optimal leverage you should use to grow capital most efficiently. For example, if your expected return is 12%, volatility is 20%, and the risk-free rate is 3%, Kelly suggests investing 225% of your capital—a signal to consider fractional Kelly for real-life applications.

Simulating the Future: Monte Carlo Method

To account for uncertainty, we use Monte Carlo simulations—a method that generates thousands of possible future scenarios based on random returns. You define assumptions like expected return and volatility, then simulate portfolio growth over decades.

This shows the range of outcomes, highlights potential downside risk, and helps answer critical questions like:

• Will I have enough to retire?
• What are the chances of losing money?

Monte Carlo visuals often reveal that even a solid plan can have a wide range of possible futures, which is a powerful reminder that risk management matters as much as return.

在投资和赌博的世界里，如何正确“下注”决定了最终的命运。凯利公式（Kelly Criterion） 是一个经典工具，帮你在有优势时决定投入多少资金，从而 最大化长期收益，并控制风险。

什么是凯利公式？

凯利公式如下：

其中：

• ：成功的概率
• ：赔率（收益与风险的比值）

如果你能准确判断成功的概率和盈亏比例，凯利公式会告诉你最合理的投资比例。但在实际投资中，这种精确的数据往往难以获得。

引入夏普比率（Sharpe Ratio）

在金融中，凯利公式常通过夏普比率进行扩展。新版凯利公式变为：

其中：

• ：预期收益率
• ：无风险利率
• ：波动率

这个公式告诉你应投入多少资金（甚至杠杆多少倍），以实现最优长期增长。例如：若预期年收益率为 12%、波动率为 20%、无风险利率为 3%，则凯利公式建议投资 225% 的本金。现实中，多数投资者会选择 半凯利 或 四分凯利，来降低风险。

蒙特卡洛模拟：预见未来的工具

为了评估不确定性，我们用 蒙特卡洛模拟 来生成成千上万个投资“平行宇宙”。通过设置预期收益和波动率，模拟几十年后的投资结果。

它展示了投资可能的全部路径，让我们看到：

• 最糟糕和最好的结果
• 有多少概率亏损或暴富

这些结果提醒我们：回报很重要，但控制风险同样关键。