# John Kelly, Jr And His Formula

July 25, 2021December 9th, 2021Uncategorized

This work is known as turnpike theory for reasons that never really made sense to me. While time series momentum is a well-studied phenomenon in finance, common strategies require the explicit definition of both a trend estimator and a position sizing rule. In this paper, we introduce Deep Momentum Networks — a hybrid approach which injects deep learning based trading rules into the volatility scaling framework of time series momentum. The model also simultaneously learns both trend estimation and position sizing in a data-driven manner, with networks directly trained by optimising the Sharpe ratio of the signal.

Traditional Kelly betting is about limiting your exposure to a risky bet. The wager in question is usually a “bet”, in that when you lose, you lose everything on the table. After all Kelly measures position size, Sharpes measure relative asset return out- view it performance standardized by return volatility. Excess returns, volatility and position sizing should not be looked at separately. There is logical intuition that drives the relationship between the Sharpe ratio and the Kelly leverage; you want to increase your allocation to a strategy that you believe to have better risk-adjusted returns .

## The Expected Payoff

I have three questions for you about how much of your bankroll you would wager in three different circumstances. If you accepted this bet, you would have a value bet because you would be getting odds of 5/4 on an event which should be priced at evens. The real benefits of the Kelly formula can be seen when used over the long term as opposed to just over the course of a few weeks. If you are looking for a Kelly criterion example, it is simply a case of substituting the B, P and Q from above and replacing them with the odds from a bet that you have in mind.

Let’s say they have 100 actual bets wherein they win, say, 58 and lose 42. Remember, according to the Kelly criterion if the deck goes ‘negative’ and you do not have a positive expectation don’t bet anything. Just flip the next card and the next until you do have a positive expectation. Size your Kelly bets exactly as you do against sports, and to make the exercise more realistic, as when actually betting against sports, flip several cards at once. After all, NFL, NBA, MLB and NHL games often go off several at a time and cannot be bet sequentially. Try flipping 3 or 4 or more cards at once – maybe even a dozen or so – just like when you’re betting on sports.

1,000 bets with 1%, 2% and 4% of net worthIt’s not surprising that as the bet size increases, so does the profit and the variability of our gambler’s net worth. Imagine a betting opportunity which offers positive expected value with known payouts and probabilities. For example, a card counter playing blackjack who knows that the current Running count and True count imply a win/loss probability for the next hand of 52% vs 48%. Under-betting less than 20%, on the other hand, would lead to a smaller profit, which means that adhering to the Kelly criterion will maximize the rate of capital growth for the long-term. The formula is therefore suggesting that 20% of the portfolio be stake 20% of your bankroll.

## Why Does Kelly Maximise \$e

An important aspect of the Kelly Criterion is that it can also tell you when a bet offers value. Broadly speaking, you have found a value bet when the probability of its being successful is higher than the implied probability of the odds for that bet. The coin however is slightly biased, so it has a 52% chance of actually landing on tails.

## Fractional Kelly Probabilities

You don’t want to follow the Kelly formula by the book. Use it as an indicator of how good the odds are and apply 25% to 50% of the recommended sizing. This is a clear example of where there are difference with investing. The Kelly Criterion clearly sends the message that this is a stock you should bet a lot on.

To give a common example, assume that someone decides to make you a generous offer in a coin-tossing game. The coin is fair, and if it comes up heads, you will win \$2 for every \$1 you bet; if it comes up tails, you will lose each dollar you bet. Moreover, you are permitted to keep playing the game under these terms for as long as you’d like, as long as you don’t run out of money. You, rightly, conclude that this is a game worth playing and gather all your money together to play.