Notional Value and the Kelly Criterion: A Controversial Marriage-rmm2
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In the world of quantitative finance, the Kelly Criterion is a legendary and controversial formula. It is a mathematical formula for bet sizing that is designed to maximize the long-term growth rate of a portfolio. While it has its roots in information theory and has been successfully applied in the world of professional gambling, its application to the world of futures trading is a topic of much debate. This article will provide a detailed analysis of the Kelly Criterion and explore the pros and cons of its use in a professional trading context.
The Kelly Criterion: A Formula for Optimal Growth
The Kelly Criterion was developed by John Kelly, a scientist at Bell Labs, in the 1950s. It is a formula for determining the optimal size of a series of bets in order to maximize the long-term growth rate of a bankroll. The formula is as follows:
Kelly % = W - [(1 - W) / R]
Kelly % = W - [(1 - W) / R]
Where:
- W: The probability of a winning trade.
- R: The average gain on a winning trade divided by the average loss on a losing trade (the win/loss ratio).
The Kelly percentage is the fraction of the trading account that should be risked on a single trade. For example, a trader with a 60% win rate and a win/loss ratio of 2 would have a Kelly percentage of:
0.60 - [(1 - 0.60) / 2] = 0.40
This means that the trader should risk 40% of their account on each trade in order to maximize the long-term growth rate of their account. This is a shockingly high number, and it highlights the primary criticism of the Kelly Criterion: it is far too aggressive for most traders.
The Dangers of the Full Kelly
A trader who attempts to use the full Kelly Criterion is likely to experience a wild and uncomfortable ride. The formula is designed to maximize long-term growth, but it does so at the cost of extreme short-term volatility. A trader using the full Kelly can expect to experience gut-wrenching drawdowns, and they may even be forced to liquidate their account if they experience a string of losing trades.
For this reason, many proponents of the Kelly Criterion advocate for the use of a fractional Kelly. A fractional Kelly is simply a fraction of the full Kelly percentage. For example, a trader might use a half Kelly (20% in our example) or a quarter Kelly (10%). A fractional Kelly will result in a lower long-term growth rate, but it will also result in a much smoother equity curve and a lower risk of ruin.
The Challenge of Applying Kelly to Futures Trading
The application of the Kelly Criterion to futures trading is further complicated by the fact that the inputs to the formula (win rate and win/loss ratio) are not known in advance. They must be estimated from historical data, and these estimates can be highly uncertain. A small error in the estimation of the win rate or the win/loss ratio can lead to a large error in the Kelly percentage.
Furthermore, the Kelly Criterion assumes that the returns of the trading strategy are independent and identically distributed. This is rarely the case in the real world. Financial markets are subject to sudden shifts in volatility and correlation, and a trading strategy that performs well in one market regime may perform poorly in another.
A Notional Value-Based Kelly
Despite these challenges, the Kelly Criterion can be a valuable tool for the professional futures trader, provided that it is used with caution and in conjunction with a sound risk management framework. One way to do this is to use a notional value-based Kelly. In this approach, the Kelly percentage is applied to the notional value of the position, rather than to the account equity. This can help to control the leverage of the position and to prevent the trader from taking on excessive risk.
The following table provides an example of how a notional value-based Kelly might be used in practice:
| Parameter | Value |
|---|---|
| Account Equity | $100,000 |
| Win Rate | 55% |
| Win/Loss Ratio | 1.5 |
| Kelly % | 25% |
| Fractional Kelly | 10% (Quarter Kelly) |
| Max. Notional Exposure | $1,000,000 (10x Account Equity) |
| Notional Value per Trade | $100,000 (10% of Max. Notional Exposure) |
In this example, the trader is using a quarter Kelly to limit their risk. They are also capping their total notional exposure at 10 times their account equity. This is a prudent approach that combines the growth-optimizing properties of the Kelly Criterion with a disciplined, notional value-based risk management framework.
Conclusion
The Kelly Criterion is a effective and controversial tool. It is not a magic formula for success, and it should not be used by the undisciplined or the faint of heart. However, for the professional futures trader who is willing to do the work and to use the formula with caution, it can be a valuable component of a comprehensive and profitable trading strategy.
References
[2] Thorp, E. O. (1966). Beat the Dealer: A Winning Strategy for the Game of Twenty-One. Vintage.