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Expectation and Optimal f: Expected Growth with and without Reinvestment for Discretely-Distributed Outcomes of Finite Length

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What they found

Vince is known to traders for 'optimal f', a position-sizing method that finds the fraction of capital to risk per trade that maximizes geometric growth over a set of historical trade outcomes. This paper generalizes the idea, showing that the growth-optimal fraction depends on the number of trades you expect to make: over a finite horizon, the fraction that maximizes expected wealth is larger than the infinite-horizon Kelly fraction and converges to it as the horizon grows. The paper also treats the case without reinvestment and connects optimal f to Kelly formally.

See it drawn

Original diagrams for the ideas on this page. Illustrative, not real market data.

How a position size is worked outAccount size, risk per trade and stop distance feed into one box giving the number of shares.ACCOUNT SIZE$25,000your capitalRISK PER TRADE1%of the accountSTOP DISTANCE$0.50entry to stopPOSITION SIZE500 sharesrisk budget: $25,000 × 1% = $250position size: $250 ÷ $0.50 = 500 shares
Working out a position size. Three numbers decide how big a trade is: the account, the share of it put at risk, and the distance from entry to stop. One percent of $25,000 is a $250 budget, and a $0.50 stop divides into that 500 times.

What you can use

  • Optimal f is the trader's version of Kelly, computed from your own trade history rather than from an assumed distribution.
  • The growth-optimal fraction depends on how many trades you plan to make; short horizons favor larger bets, but at higher risk.
  • Because optimal f is fitted to past trades, it inherits every flaw of the backtest, including overfitting and the assumption that the worst loss is already in the sample.

Caveats

Working paper by a practitioner; optimal f as commonly used is widely criticized for producing dangerously aggressive sizes because the largest historical loss bounds the calculation. Read with the fractional-Kelly literature.

Tags: risk, position-sizing, optimal-f, kelly

Summaries are our own reading of the paper, not the authors' words. Educational only, not advice. Discuss it in Book Club.