Lessons › Advanced track › Fixed fraction versus optimal-growth sizing

World 10 · Advanced track · lesson 10 · level 5

Fixed fraction versus optimal-growth sizing

A formula can tell you the bet size that grows money fastest, but real traders bet much less because their numbers might be wrong.

In one line

A formula says to risk 25% per trade for the fastest growth. Why would a sensible trader use a small slice of that?

Explained simply

The optimal-growth formula is like a recipe for the fastest-growing plant: lots of fertilizer, but a little too much burns the roots. Because you never know exactly what your plant needs, gardeners use much less than the maximum. Traders do the same with bet size.

The lesson

The optimal-growth formula from betting maths gives the fraction of capital to risk as f = p - (1 - p) / b, where p is the win rate and b is the average win divided by the average loss. For a 50% win rate at 2R it suggests risking 25% of capital per trade, which brings enormous swings, and if you overestimate your edge, those full-size bets can shrink the account. Many traders use a half or a quarter of the result, or simply a fixed 0.5% to 2% risk per trade. Smaller fractions give up some growth in exchange for much smaller drawdowns and room for mistakes in your estimates.

A worked example

Illustrative example: a strategy wins 50% of the time and its average win is twice its average loss, so p = 0.5 and b = 2. The formula gives f = 0.5 - 0.5 / 2 = 0.25, or 25% per trade; half of that is 0.25 / 2 = 0.125 and a quarter is 0.25 / 4 = 0.0625. But suppose the true win rate is only 0.4. Then the real optimum is 0.4 - 0.6 / 2 = 0.1, so betting 0.25 is 0.25 / 0.1 = 2.5 times too big. A fixed 1% rule, 0.01, is 0.01 / 0.1 = 0.1 of even the true optimum, a tenth.

The same idea at four levels

1. Beginner. How much you bet on each trade matters as much as how often you win.
2. Foundation. The optimal-growth formula is f = p - (1 - p) / b, where p is the win rate and b is the size of wins compared with losses.
3. Intermediate. For example, p = 0.5 and b = 2 gives f = 0.5 - 0.5 / 2 = 0.25, which means risking 25% per trade.
4. Advanced. Full-size bets bring huge drawdowns, and if your true edge is smaller than you think the 'optimal' size can shrink your account, so traders use a half or a quarter of it, or less.
5. Expert. In this example, betting twice the optimal fraction, 50%, gives zero long-run growth even though the edge is real, and because estimates from small samples are shaky, a fixed 0.5% to 2% risk is a sensible cap for most people.

Mistakes to avoid

• Plugging in a win rate from a small sample and betting the full result.
• Thinking the fastest-growth bet suits a real person, who has limits on how much pain they can take.
• Forgetting that betting above the optimal fraction lowers growth and raises risk at the same time.

Goal of this lesson: Compare fixed-fraction sizing with the optimal-growth formula and see why traders use a fraction of it or less.