Why has this formula been hidden from investors?

Gamblers Ruin

In the world of professional betting, which encompasses gambling and investing, the cornerstone is having a statistical edge — a measurable advantage that predicts a positive return. Without this edge, participating in betting activities is generally unwise. However, possessing a statistical edge is merely one piece of a larger puzzle. A critical, yet often neglected component, is the art of bet sizing or "money management."

The significance of adept money management cannot be emphasised enough. Even the most skilled handicapper can find themselves on the brink of financial disaster or severely underutilising their abilities if they fail to manage their funds wisely. It's a stark reality that many skilled gamblers encounter financial troubles due to overlooking the importance of sound money management.

This brings us to the concept known as 'gambler's ruin’. Consider a hypothetical scenario of 1,000 bets. Surprisingly, the journey towards bankruptcy can be alarmingly rapid. It's essential to remember that going broke just once means being out of the game for good. And this holds true even for those with a slight statistical advantage — if the betting strategy is flawed, financial ruin is still a significant risk.

Take, for example, the martingale strategy — doubling your bet following each loss until you win. Though it may appear to be a practical approach, it actually hastens the onset of financial collapse.

The solution to avoiding gambler’s ruin lies in proportional betting — adjusting your bet size according to the fluctuations in your bankroll. But even this strategy has its pitfalls. If the bet proportions are too aggressive compared to your statistical advantage, the threat of going to zero remains present.

Investors face a constant dilemma in effectively allocating their capital, especially in bullish markets brimming with opportunities. This leads to a crucial inquiry:

What is the ideal bet size that not only protects against a gambler's ruin but also optimises long-term wealth growth?

The answer to this question is complex and demands a careful equilibrium between risk and potential gains. This balance is essential for successful betting and investment strategies, a balance that the Kelly Criterion Formula aims to achieve.

Kelly Criterion Formula Explained

Enter the Kelly Criterion Formula, a method that has remained relatively under the radar. Developed in 1956 by John Kelly, a scientist at Bell Labs, this formula is designed for sizing bets or investments with expected positive returns.

At its core, the Kelly Criterion is a simple formula that calculates the optimal proportion of bankroll in order to maximise the geometric growth rate of your wealth. It encompasses two primary elements: the probability of winning (the likelihood that a given trade will yield a positive return) and the win/loss ratio (the total amount of positive trades divided by the total negative trades).

^{Please note: the ‘W’ and ‘R’ are decided by the investors independently, which leads the formula’s outcome quality to be down to user discretion.}

These components feed into Kelly's equation, producing an output known as the "Kelly percentage," a useful tool for various real-world applications.

The Kelly Percentage represents the amount of your portfolio to devote to each investment.

While it initially gained traction among gamblers for optimising betting sizes in horse racing, its usage extends beyond.

Why is the formula hidden?

Despite its practicality and adoption by renowned investors like Warren Buffet, Charlie Munger, and Bill Gross, the Kelly Criterion is seldom a topic of discussion in business academia. This can be attributed to two factors:

1. Information Theory vs. Economic Theory: the formula's roots are in information theory, a field often at odds with conventional economic theory, contributing to its marginalisation by economists.

2. Business School Curriculum: there is a predominant focus on returns accounting for market volatility and on the principle of diversification in business education. Even Warren Buffet touches on the issue with this approach.

In essence, unlike traditional investment strategies, the Kelly Criterion prioritises the rate of wealth growth as opposed to maximising returns adjusted for market volatility, which goes against traditional investment principles.

The problem behind the Kelly Criterion Formula

The formula's real-world application also faces empirical challenges. For instance, it doesn't consider the magnitude of profits and losses, only their ratio, leading to significant mathematical limitations. The below example illustrates this flaw: regardless of the change in the volatility of an investment, the formula may yield the same allocation percentage, which is not optimal in practice.

If we insert the numbers into the formula for the table below we get an answer of 20%. In other words, a 20% allocation to the investment maximises the portfolio’s potential long-term growth.

If we change the loss and profit to 100% we still receive an answer of 20% as the ratio between wins and losses remains the same, which is inherently wrong as the investment is 5 times more volatile (we won’t bore you with the math). Below is a simulation ran with different amounts of capital invested highlighting this flaw as it is clear that 20% investment does not yield the optimal results for investors.

As such, the Kelly Criterion is often criticised, with investors ignoring it. However, some thought about how to apply the formula to the more volatile world of investing can yield tremendous results for investors. 

Applying the Kelly Criterion to Long Term Investing

When applying the Kelly Criterion formula to investing we must first ensure to not overlook a key caveat - the more bets one can place, the more accurate the formula will be due to the Law of Large Numbers.

In a casino, you could either double or lose your bet every few seconds. On the other hand, investing isn’t a casino game and you won’t have many sequential bets, in fact in investing you don’t know when the next opportunity arises.

As such a slight derivation of the formula presents a much more sophisticated solution to investors. 

Unfortunately, the restructured formula presents investors with a higher Kelly percentage than the original. As such, investors often choose to avoid it and opt for the simpler flawed version which does not account for downside risk.

Thus, the above formula still requires some further thought to ensure it is viable for investors. Some key points to consider:

1. Conceptual Understanding: We must recognise that the Kelly Criterion is a thought experiment rather than a concrete, infallible solution. Experience and market understanding often eclipse mathematical models in investment decision-making.

2. Managing Overconfidence: Investors must temper their confidence in the probabilities input into the formula, remaining mindful of inherent biases that can render the formula ineffective.

3. Margin of Safety: Implementing a margin of safety is crucial. As investors near the optimal Kelly Percentage, the risk-to-return trade-off intensifies. Using a margin can help balance this by trading off a larger amount of risk for a lower reduction in potential returns. Additionally, this is a solution to the altered formulas’ higher percentage.

Conclusion

The Kelly Criterion Formula, while a notable tool in the arsenal of investors and gamblers alike, is not a panacea for investment strategy. Its roots in information theory rather than conventional economics, and its focus on maximising wealth growth rather than accounting for market volatility, have contributed to its relative obscurity in mainstream finance.

Investors should view the Kelly Criterion as a starting point for strategic thinking, not as a definitive guide. Real-world investing is complex and requires more than mere mathematical formulas as evidenced by the fact calculating the winning probability and potential win/loss size demands experience, intuition, and a deep understanding of market dynamics.