Terminal Wealth (TWR) ^ | * (Optimal f Peak) | * \ | * \ | * \ <-- The Mathematical Cliff! | * \ | * \ +---------------------------> Risk Fraction (f) Because Optimal
"You can have a system that is right only 20% of the time and make a fortune—if you bet big on the winners and tiny on the losers. The math of ruin does not care about your pride, only your f." — Ralph Vince (paraphrased)
Vince borrowed from Kelly (Bell Labs, 1956) and adapted it for the messy reality of trading—where trades have varying outcomes, not just binary win/loss.
calculates the precise fixed fraction of an account to allocate per contract or share to maximize the geometric growth rate of capital. The Mathematical Formula for Geometric Mean (TWR) To find Optimal Terminal Wealth (TWR) ^ | * (Optimal f
= The fraction being tested (ranging strictly between 0 and 1). TradeiTrade sub i = The profit or loss of the -th trade. Largest LossLargest Loss
The appendices are equally rigorous, including "Using a Negative Mathematical Expectation Market" (a counter-intuitive look at gambling), statistical tables for the Cumulative Normal Distribution, and source code details for "The Portfolio Program," allowing readers to implement the mathematics themselves.
In reality, a trader with $100,000 and a trader with $10,000 face vastly different dynamics. Vince introduced the concept of —the idea that your primary goal is not to maximize average trade return, but to maximize the geometric mean of your account over time. The Mathematical Formula for Geometric Mean (TWR) To
Most professional traders do not trade at full Optimal f. Instead, they trade at a fraction of f (e.g., 0.2f or 0.3f) to smooth the equity curve.
f = (bp - (1 - bp) / r) / r
The year was 1990, and the flickering green phosphorus of trading monitors at the Chicago Board of Trade felt more like a battlefield than a marketplace. While most traders relied on "gut feel" and floor-room adrenaline, a quiet revolution was being printed in the pages of a new book: "Portfolio Management Formulas" Ralph Vince In reality, a trader with $100,000 and a
Portfolio Management Formulas: Mathematical Trading Methods for the Futures, Options, and Stock Markets by Ralph Vince (Nov 1990)
Vince introduced the concept of . This is the fraction of your capital you should risk to maximize the long-term growth of your account.
While Kelly works perfectly for binary outcomes (like coin tosses or blackjack where you either win