You derived Kelly's intuition in M6.03 §4.10; here you operationalize it, and to do that you need to see clearly where it comes from, because the derivation is what tells you why the formula behaves as it does.
What does the Kelly criterion maximise?
What Kelly maximizes, and why that is the right objective. The Kelly criterion asks: what fraction f of my capital should I stake on a repeated favorable bet to maximize the long-run growth rate of my wealth? The key word is growth rate. Because wealth compounds multiplicatively (each period multiplies the last, it does not add to it) the quantity that governs where you end up after many periods is not the arithmetic average of your returns but the geometric average, equivalently the average of the logarithm of your wealth multiplier. (This is the ergodicity point from M6.03: the arithmetic mean describes the average across many parallel lives; the geometric mean describes the single life you actually live through, in sequence.) So Kelly maximizes the expected logarithm of the wealth multiplier, and log-wealth maximization turns out to have a beautiful built-in safety property: because ln(0) is negative infinity, an objective that maximizes expected log-wealth will never choose a bet that has any chance of taking you to zero. Kelly refuses ruin as a mathematical reflex, not as an add-on.
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How is the Kelly formula derived?
The derivation, once, so the formula is not a mystery. Take the investment form directly (the betting form is a special case of it). You put fraction f of wealth into a position; the rest sits in cash. With probability p the position gains fraction u (so wealth becomes 1 + f·u); with probability q = 1 − p it loses fraction d (wealth becomes 1 − f·d). The expected growth per period is:
g(f) = p · ln(1 + f·u) + q · ln(1 − f·d)
To find the growth-maximizing f, differentiate and set to zero:
g'(f) = p·u/(1 + f·u) − q·d/(1 − f·d) = 0
Cross-multiplying: p·u·(1 − f·d) = q·d·(1 + f·u), which expands to p·u − p·u·f·d = q·d + q·d·f·u. Collect the f terms: p·u − q·d = f·u·d·(p + q) = f·u·d (since p + q = 1). Therefore:
f* = (p·u − q·d) / (u·d) = p/d − q/u
That is the investing Kelly fraction, the same f* = p/d − q/u you saw in M6.03, now derived. Set u = b and d = 1 (you lose the whole stake on a loss) and it collapses to the gambling form f* = (b·p − q)/b = p − q/b. Read the formula's message: the size rises with p (conviction), rises as u grows (upside), and, the point that matters most for investors, rises sharply as d shrinks (a capped downside). This is the mathematics beneath Pabrai's Dhandho asymmetry: it is not the upside that earns a big Kelly size, it is the small downside. "Heads I win, tails I don't lose much" is the same statement as "small d, therefore large f*."
The continuous approximation you will actually reach for. Real investments do not have two outcomes; they have a distribution. For a position with expected excess return μ (over cash) and return variance σ², a Taylor expansion of the log-growth objective gives the workhorse approximation:
f* ≈ μ / σ² and the growth it earns is g_max ≈ μ² / (2σ²)
This is the form most practitioners keep in their heads: size in proportion to your edge and in inverse proportion to the variance of the outcome. Double your edge, double your size; double the uncertainty (variance), halve it. It is Kelly's DNA in one line, and it is the skeleton of the conviction × downside heuristic.
What is the overbetting cliff?
The growth curve and the overbetting cliff: the single most important picture in sizing. Substitute a fractional size f = c · f* (bet c times the full-Kelly fraction) into the approximation and the growth rate becomes:
g(c·f*) = g_max · (2c − c²)
Trace this curve and burn it into memory:
Fraction of Kelly, c | Growth as % of max, 2c − c² | What it means |
|---|---|---|
| 0.25 (quarter-Kelly) | 0.44 | 44% of the growth, a quarter of the volatility |
| 0.50 (half-Kelly) | 0.75 | 75% of the growth for half the volatility, the sweet spot |
| 1.00 (full Kelly) | 1.00 | Maximum growth, brutal drawdowns |
| 1.50 | 0.75 | Same growth as half-Kelly, three times its volatility, strictly worse |
| 2.00 (double Kelly) | 0.00 | Zero growth: the entire edge is consumed by variance drag |
| >2.00 | negative | You now lose money, on a positive-edge bet, through pure oversizing |
Two facts leap out. First, the curve is a hump: it rises to full Kelly and falls symmetrically after, so betting 1.5× Kelly gives you exactly the growth of 0.5× Kelly while carrying three times the risk. Every point past the peak is strictly dominated. Second, and decisively, the penalty for overbetting is catastrophically asymmetric with the reward for getting it right. Underbet to half-Kelly and you keep 75% of the growth; overbet to double-Kelly and you keep none of it. Since your inputs (p, u, d, μ, σ) are estimates and you will sometimes overestimate your edge, prudence is not a matter of taste; it is forced by the shape of this curve. You want to be on the left of the peak, where errors are cheap, never the right, where they are fatal. That is the entire case for fractional Kelly, which fractional Kelly makes explicit. (In worked example 1 you will see that with real, large, skewed outcomes the overbetting penalty is even worse than this quadratic approximation says: double-Kelly can produce sharply negative growth, not merely zero.)