Position Sizing and the Kelly Criterion

What is Position Sizing?

Position sizing is not about which stock to buy, but rather about deciding what percentage of your total assets to allocate to that stock. Even if two people buy the same stock at the same price, their final returns and psychological states will be entirely different if one allocates 5% of their total wealth and the other allocates 50%. If stock selection answers 'what to buy,' position sizing answers 'how much to buy,' and long-term performance is heavily influenced by the latter.

The core principle is to control the amount you can lose in a single trade upfront. No matter how confident you are in an idea, it can still be wrong. Therefore, setting an upper limit on the allocation to any single stock, to prevent a loss from that stock from devastating your entire account beyond recovery, is the starting point of sizing. This article is for educational purposes and does not recommend specific stocks or allocation percentages.

The Principle of the Kelly Criterion

The Kelly Criterion calculates the optimal betting proportion that maximizes the long-term growth rate of capital in a series of repeated bets. Its basic form is written as f = p - q/b. Here, p is the probability of winning, q is the probability of losing (1-p), and b is the risk/reward ratio (the multiple of the amount won relative to the amount lost) when you win. f represents the proportion of capital to be allocated to the current bet.

For example, let's say the probability of winning p is 0.6, the probability of losing q is 0.4, and the risk/reward ratio b is 1 (meaning you win $1 for every $1 risked). Then f = 0.6 - 0.4/1 = 0.2, so betting 20% of your capital maximizes long-term growth. If the risk/reward ratio b improves to 2, then f = 0.6 - 0.4/2 = 0.4, justifying a larger allocation even with the same winning probability.

If the formula yields a negative number, it means the expected value is negative, signaling that you should not place the bet. In other words, the Kelly Criterion is a tool that tells you not only 'how much to bet' but also 'when not to bet'.

How to Interpret the Results

The f value suggested by the Kelly Criterion is close to a theoretical upper limit. While betting according to this value maximizes long-term growth, it also requires tolerating significant volatility, with your account potentially fluctuating by nearly half. Therefore, in practice, 'Half Kelly' or 'Quarter Kelly' — using half or a quarter of the calculated value — are widely adopted.

Half Kelly is preferred because it maintains approximately 75% of the full Kelly growth rate while reducing volatility by half, offering a better risk-adjusted efficiency. In the previous example, if full Kelly suggested 20%, Half Kelly would be 10%. This conservative application also acts as a safety net, absorbing estimation errors in the input values (winning probability and risk/reward ratio).

Applying to Stock Investing

Unlike a coin toss, the exact winning probability p and risk/reward ratio b are unknown in stock investing. Therefore, one approximates these by estimating the proportion of winning trades and the ratio of average gain to average loss from one's past trading records. For instance, if you set a stop-loss at -10% and a target at +20%, the risk/reward ratio b becomes 2. If your past success rate was 50%, then f = 0.5 - 0.5/2 = 0.25.

However, estimated winning probabilities become less accurate with smaller sample sizes, so it is safer to use a much more conservative allocation in practice. Many retail investors set an upper limit of 5-10% of their assets per stock. This is because even if one stock loses half its value, the total portfolio loss would only be 2.5-5%, remaining within a recoverable range. Combining the Kelly Criterion with a reverse calculation method—first determining a stop-loss price, then limiting the amount lost in a single trade to 1-2% of the account—can make your sizing even more robust.

Pitfalls and Limitations

The biggest risk of the Kelly Criterion is overestimating the input values. If you believe your winning probability is 60% but it's actually 50%, the formula will inflate the appropriate allocation, leading to over-betting. In volatile markets, over-betting can act as the curse of compounding, not the magic of compounding.

Furthermore, the Kelly Criterion is based on the assumption that bets are independent and can be repeated infinitely. However, holding multiple stocks simultaneously exposes them to correlation due to shared industry factors or macroeconomic shocks. Simply summing individual allocations can then underestimate the actual risk. Whether in Korea or the US, when the overall market sharply declines, the correlation coefficient between stocks tends to approach 1.

Moreover, the Kelly Criterion only deals with mathematical expected values and does not account for personal circumstances such as living expenses or psychological limits. Even if the math points to 20%, for someone who cannot withstand the associated volatility and ends up selling at the bottom, 10% might be a better answer.

Checkpoints

In summary, position sizing starts with the understanding that 'how much to allocate' determines performance. The Kelly Criterion, f = p - q/b, provides a theoretical betting proportion based on winning probability and risk/reward ratio. However, due to input value errors and volatility, conservative applications like Half Kelly are more practical.

In practice, simultaneously consider three factors: the upper limit for allocation per stock, the maximum loss per trade, and the correlation between stocks. All figures in this article are hypothetical examples to explain the principles, and actual investment decisions and allocations must be determined by each individual based on their own risk tolerance and financial situation.