How to apply the Kelly criterion when expected return may be negative?

Question

My concern is how to handle a negative value for the Kelly formula. Even when you have a system that has positive expectancy, you can (and usually will) sustain a number of losses, sometimes consecutively. This is usually most relevant at the beginning of trading system signals, each loss has a large impact on the formula. How would one handle trade signals when the Kelly formula turns negative? This is assuming a long only system. It seems to me if you ignore the trade signal you would never recover back to a positive probability since your choice to not take the trade would negate any chance to recover.

Some clarifications: I am writing software for a mechanical trading system. I can run backtest simulations to get a sense of historical "edge" and "odds". My confusion is how to apply the Kelly formula once the system goes live and I am making trades based on the system signals. I want to use actual trade data to calculate the Kelly %. Should I use backtest data for the previous x trades and then walk that data forward as I make new trades? Should I just use backtest odds and edge until I have enough live data to use?

Answer 1

You are trying to apply the Kelly Criterion, supposedly to maximize how aggressively to bet, and you are having trouble when the Kelly Value turns negative. The naive answer to your question is that when your kelly value turns negative, then $f=\frac{bp-q}{b}$ turning negative means the instantaneous expected return is negative, which means you should not bet any of your wealth at this moment if you don't have the ability to take the opposite bet (you said you're long only) or alter your instantaneous strategy or timing to change your estimated p, b and q.

The more accurate answer is that if your estimated probabilities are changing sufficiently for the calculated kelly ratio to turn negative and positive at various times, then that means you are violating one of the key assumptions of the strategy. The strategy only works if you are able to sustain both the edge and the odds (p and b) for the long term, and therefore maximize your long term wealth. You should aim to estimate probabilities for a longer time horizon for your strategy to make sense. In the case where that is negative too, then you should abstain from trading for that period of time, or find a way to reverse your strategy or go short.

Finally, note that using such trade signals is likely to lead to large error rates in the odds (b) if you're able to eventually fit your system's behavior to produce accurate "edge" estimates (p).

If you insist that your system has positive expectancy, then the only for you to have positive expectancy and negative kelly ratio is if your calculating those two on different maturity horizons. That is, with a positive expectancy over the longer horizon, if you want to keep trading, and want to know how aggressively to bet in the short term, then you need to estimate the kelly ratio for the probabilities and odds with appropriate maturity horizon, and the kelly ratio will NOT be negative, by definition. You can work through the math to convince yourself by calculating expected payoff as a function of probability of the probability distribution of gains and losses.