How to Invest Like Kelly Without Making Her Look Too Sad
A Cautionary Tale: When Intuition Meets Mathematics in Trading
Have you ever had a day that started off completely ordinary, only to take an unexpected turn? That’s exactly what happened to me on what seemed like a typical Tuesday afternoon.
I was wrapping up my workday, ready to head out, when my boss called me into his office. Little did I know, this impromptu meeting would lead to one of the most interesting encounters of my career.
As a math PhD working in algorithmic trading, my job revolves around evaluating trading strategies from a mathematical perspective. It’s a fascinating field where numbers and market dynamics intersect. So, when my boss introduced me to a middle-aged gentleman and asked me to review his trading ideas, I wasn’t entirely surprised.
What did catch me off guard, however, was my boss’s hasty exit, leaving me alone with the client. Talk about being thrown into the deep end!
The gentleman, despite having no prior trading experience, radiated enthusiasm. His eyes sparkled with determination as he laid out his strategy, clearly passionate about his ideas. It was my job to determine whether his approach held up mathematically.
His strategy? The classic “double or nothing” approach. After each loss, he’d double his bet, convinced that the market’s overall upward trend would eventually turn things in his favor. “It’s simple,” he explained, leaning forward conspiratorially. “The market goes up more than it goes down, right? So if I just keep doubling down, I’m bound to come out ahead!”
I’d heard variations of this before, but something about his unwavering belief made me pause. He was anchoring his hopes on the law of large numbers, convinced that with enough trades, his positive expectancy would shine through.
Intrigued by his conviction, I decided to dive deeper. As I ran the numbers, a sobering reality emerged: his approach had a high probability of completely wiping out his account.
When I shared my findings, he was baffled. “But why?” he asked, genuinely perplexed. “Isn’t the law of large numbers on my side?”
This was my chance to bridge the gap between intuition and mathematical reality.
Unveiling the Pitfalls
I explained that for the law of large numbers to apply, the variance must be finite. His strategy, unfortunately, led to infinite variance, making the law inapplicable. Moreover, he was unknowingly setting himself up as a victim of the Gambler’s Ruin theorem.
“Think of it this way,” I said, “You’re playing against the market, which has essentially infinite resources. Even if the odds are slightly in your favor, with a finite bankroll, you’re almost guaranteed to go broke eventually.” This statement is based on the theory of probability known as the Gambler’s Ruin theorem.
The Gambler’s Ruin theorem is a classical concept in probability theory that describes a scenario in which a gambler with a finite amount of wealth plays a game with fixed odds and stakes. The theorem states that the gambler will eventually go bankrupt (reach zero wealth) if they continue to play indefinitely, provided the game has a negative expected value for the gambler or even if it is fair. This result arises because, in a fair or unfavorable game, random fluctuations will ultimately lead to a situation where the gambler’s losses outweigh their gains, leading to inevitable ruin. The theorem highlights the inherent risk of long-term gambling and underscores the importance of understanding probability and risk management in stochastic processes.
A Light Bulb Moment
I could see the realization dawning on his face. His “foolproof” strategy suddenly seemed a lot less foolproof. But to his credit, instead of walking away defeated, he asked, “So what should I do instead?”
Introducing the Kelly Criterion
Acknowledging the futility of his reckless pursuit, he sought refuge in the realm of wisdom, yearning for a pragmatic yet lucrative alternative. Emphasizing the perils of unwavering commitment and the imperative of calibrated risk-taking, I introduced him to the distinguished beacon of hope in the tumultuous terrain of trading — the venerable Kelly Criterion. Crafted by the visionary John L. Kelly Jr. in 1956, this mathematically grounded formula emerged as a lodestar for determining the optimal bet size in a series of wagers. By weaving together the threads of probability and payoff ratio, the Kelly Criterion crafted a blueprint for traders seeking to maximize the long-term growth of their capital.
Diving Deeper into the Kelly Criterion
After our discussion about the pitfalls of the “double or nothing” strategy, my new acquaintance was eager to learn more about the Kelly Criterion. “You’ve piqued my interest,” he said. “Can you explain how this Kelly Criterion actually works?”
I was more than happy to oblige. After all, the Kelly Criterion is a fascinating tool that combines probability theory and logarithmic utility to optimize betting strategies.
The Basics of Kelly
“At its core,” I explained, “the Kelly Criterion tells you what percentage of your bankroll you should bet to maximize your long-term growth rate.”
So, after applying a basic logarithmic transformation and differentiating it, we find that the maximum growth rate occurs at:
Where:
- f∗ is the fraction of the current bankroll to bet
- p is the probability of winning
- b is the net odds received on the bet (that is, odds are bbb to 1)
A Practical Example
To illustrate, I used a simple coin flip scenario. “Let’s say you’re offered a game where you win $2 for every $1 bet if a coin comes up heads, and you lose your $1 bet if it comes up tails. The coin is slightly biased, giving you a 51% chance of winning.”
I then walked him through the calculation:
- p=0.51 (probability of winning)
- b=1 (net odds: you win $2 for a $1 bet, so net odds are 1 to 1)
Plugging these into our formula:
“This means,” I concluded, “that to maximize your long-term growth rate, you should bet 2% of your bankroll on each flip.”
The Power of Kelly
My new friend looked intrigued but slightly skeptical. “That seems like a very small bet,” he mused.
I nodded in agreement. “It does seem small, especially compared to your original double-or-nothing strategy. But here’s the beautiful thing about Kelly: it balances the competing goals of growing your bankroll and protecting against ruin.”
I went on to explain that betting more than the Kelly Criterion suggests increases your risk of ruin while betting less reduces your expected growth rate. It’s the sweet spot for long-term success.
Kelly in Trading
“But how does this apply to actual trading?” he asked. “The market isn’t a simple coin flip.”
“You’re absolutely right,” I replied. “In trading, estimating your edge (your ‘p’ in the formula) is much more challenging. It requires careful analysis of your strategy’s historical performance and current market conditions.”
I explained that many traders use a fraction of the full Kelly bet (often called “fractional Kelly”) to account for uncertainties in their probability estimates.
“For instance,” I said, “if the full Kelly suggests betting 10% of your bankroll, you might choose to bet only 5% as a more conservative approach.”
The Limitations
As we wrapped up our impromptu Kelly Criterion seminar, I made sure to highlight its limitations.
“Remember,” I cautioned, “Kelly assumes you know the true probabilities and payoffs of your bets. In real trading, these are often estimates. It also assumes you can tolerate high volatility, as full Kelly betting can lead to significant short-term drawdowns.”
To connect this with real-world application, I brought up the example of Warren Buffett, often regarded as one of the most successful investors of all time. Buffett’s approach to investment management, emphasizing value investing, aligns well with the principles behind the Kelly Criterion.
Buffett’s strategy involves identifying undervalued companies with strong fundamentals and holding onto these investments for the long term. He believes in thorough research and understanding of the companies he invests in, focusing on their intrinsic value rather than market fluctuations. This meticulous approach mirrors the careful analysis required to estimate your edge in trading as dictated by the Kelly Criterion.
Buffett’s investment philosophy is built on patience, discipline, and a keen eye for quality businesses with sustainable competitive advantages. His management of Berkshire Hathaway, a multinational conglomerate holding company, exemplifies his ability to achieve consistent returns by investing in a diverse portfolio of well-managed, financially sound companies. Through his prudent and insightful investment decisions, Buffett has consistently demonstrated the power of long-term value investing.
A New Perspective
By the end of our conversation, I could see a shift in my new friend’s approach. The allure of doubling down had been replaced by a more nuanced understanding of risk and reward.
“This is a lot to take in,” he admitted, “but it makes so much sense. I’ve got some serious rethinking to do about my trading strategy.”
As he left, I felt a sense of satisfaction. Another trader had been introduced to the power of mathematical thinking in finance. It’s moments like these that remind me why I love working at the intersection of math and trading.
An Unexpected Compliment
The following morning, my boss called me to his office.
“I have no idea how you did it, but you are the first one who managed to talk some sense into the gentleman last night. Investors like him are often convinced they know better than anyone else. We need to ensure they don’t lose all their money on one hand and invest wisely on the other. You did it in an impressive manner.”
My boss’s praise highlighted the importance of not only doing my job well but also communicating effectively. This is a lesson I continue to practice to this day.
In Summary
While intuitive trading strategies may exude a veneer of appeal, they often crumble when subjected to the piercing gaze of mathematical scrutiny. The fusion of the Gambler’s Ruin theorem and the Kelly Criterion serves as a bulwark of reliability, offering traders a dependable framework for risk management and a pathway to sustained success. Traders are thus implored to delve deep into the mathematical bedrock underpinning their trading tactics, equipping themselves with the cognitive ammunition to orchestrate judicious, well-informed decisions in the ever-evolving landscape of trading.
