QQ 102, 2 min read

Position sizing and leverage

Two traders can take the same signals and end in different places because one risked 1% per trade and the other risked 20%.

Fixed fractional sizing

The common rule is to risk a fixed fraction of the account on each trade.

amount at risk = account x risk fraction
position size  = amount at risk / (entry price - stop price)

With a 10,000 dollar account risking 1%, the amount at risk is 100 dollars. If the stop is 2 dollars below the entry, the position is 50 units.

Because the fraction is of the current account, positions shrink after losses and grow after gains.

Why the fraction has to be small

A run of losses is certain to happen eventually. Ten losses in a row at 2% each leave 82% of the account. Ten in a row at 10% each leave 35%.

0.98 to the power 10 = 0.817
0.90 to the power 10 = 0.349

The Kelly fraction

For a bet with win probability p, loss probability q = 1 - p, and a win that pays b times the amount risked:

Kelly fraction = p - q / b

A bet that wins 55% of the time and pays 1 to 1 gives 0.55 - 0.45 = 0.10. Kelly says risk 10%.

Kelly maximises long run growth if p and b are known exactly. They never are. Estimated from a backtest they are usually too optimistic, and betting above the true Kelly fraction reduces growth and raises the chance of ruin. People who use it usually bet a half or a quarter of what the formula says.

Leverage

Leverage lets a position be larger than the account. It multiplies gains, losses and drawdown by the same factor, so it cannot make a strategy better, and a drawdown that was survivable at 1x can close the account at 5x.

Further reading

• Kelly, J. L. (1956). "A New Interpretation of Information Rate." Bell System Technical Journal 35(4).
• Thorp, E. O. (2006). "The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market." In Handbook of Asset and Liability Management, volume 1. Elsevier.