5.7.3 Growth & Optimality

Basic Prerequisites: 02 · The Kelly Formula and 03 · Multiplicative Growth.

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1. Intuition & Practical Objective

The past two pages gave the formula. This page answers why full Kelly is the reference standard - the optimality results that justify calling it growth-optimal - and, just as importantly, what it costs. The objective is to internalise four ideas:

1. Kelly is asymptotically dominant. In the long run, the full-Kelly strategy beats every other essentially-different strategy by a factor that diverges - your Kelly-paced bankroll outruns any rival’s almost surely (Breiman 1961; Thorp’s Theorem 1).
2. Kelly is time-optimal. It minimises the expected time to reach any fixed wealth goal - the reason “getting there fastest” and “maximising growth” are the same thing.
3. But full Kelly is aggressive. It maximises growth and maximises the growth-rate volatility; you are paying with drawdown risk for the asymptotic dominance.
4. Growth and the Sharpe ratio are the same object. $g_{\infty }(f^{\ast })=S^2/2+r$: a strategy’s Sharpe ratio is literally the square-root of twice its Kelly growth rate. This is the bridge that makes “growth-optimal” and “mean-variance-optimal” friends, not rivals.

Takeaway. Kelly is the optimal growth strategy for a single-period-per-trade, unchanging-edge world - and its dominance comes with a volatility price, which is precisely why practice de-rates to fractional Kelly (04).

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2. Mathematical Ground Truth & Derivations

2.1 Growth-optimality (Breiman / Thorp Theorem 1)

For an even-money coin with $g(f)>0$ and Kelly strategy ${\Phi }^{\ast }$ betting $f^{\ast }=p-q$ each round:

• if $g(f)>0$ then $X_n\to \infty$ a.s. (Kelly growth is positive);
• dominance: for any essentially different strategy $\Phi$, $\frac{X_n({\Phi }^{\ast })}{X_n(\Phi )}\to \infty$ a.s. - Kelly asymptotically dominates every other strategy;
• time-optimality: Kelly minimises the expected time to reach any fixed goal.

So maximising $\mathbb{E}[\ln X_n]$ is optimal by both growth and goal-reaching. That is the two-part reason Kelly is canonical.

2.2 The fractional-Kelly growth law

Scale the bet to $f=c{f}^{\ast }$. For the continuous case:

$$g_{\infty }(c{f}^{\ast })=r+\frac{(m-r{)}^2}{s^2}c\left(1-\frac{c}{2}\right),\frac{g_{\infty }(c{f}^{\ast })}{g_{\infty }(f^{\ast })}=c(2-c).$$

Half Kelly ($c=1/2$) keeps $c(2-c)=3/4$ of the growth rate with half the volatility. You trade 25% of growth to cut risk 50% - the mathematically precise justification for the industry’s standard half-Kelly default. At $c=1.5$ growth is already identical to $c=0.5$ (the parabola is symmetric about $f^{\ast }$), and $c>2$ pushes growth negative.

2.3 Growth ⟺ Sharpe

From the continuous Kelly optimum,

$$g_{\infty }(f^{\ast })=\frac{(m-r{)}^2}{2s^2}+r=\frac{S^2}{2}+r.$$

A Sharpe ratio of $S$ is worth $S^2/2$ of annualised growth. This is the exact quantitative link to the tangency portfolio of MPT (Tangency & CAPM) and to the multi-asset Kelly of 06 · Advanced Extensions.

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3. Computational Implementation - dominance, the $c(2-c)$ law, and growth↔Sharpe

Stdlib only: evaluate the fractional growth law and simulate full vs. half Kelly to see the volatility-asymmetry trade.

Reading the $g/g^{\ast }$ column: half Kelly compounds at 88% of full Kelly here (with $r>0$ the fall is shallower than the pure $c(2-c)=0.75$; for $r=0$ it is exactly 0.75), while doubling the bet to $c=2$ collapses growth to half its maximum. The growth function is symmetric around $f^{\ast }$ (overbet = underbet in growth) but not in risk: the volatility drag is quadratic in risk, so $c=2$ exposes you to $2\times$ the drawdown for $0.52g^{\ast }$. That asymmetry, not raw growth, is why fractional Kelly is the professional default.

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4. Failure Modes & First-Principles Breakdowns

1. Full Kelly overestimates practical risk appetite. Asymptotic dominance is a long-run, path-averaged guarantee; the median investor on a finite horizon faces drawdowns that full Kelly maximises alongside growth. The dominance is real but does not mean “bet full Kelly.”
2. The optimality is buy-and-hold-in-fraction, not re-estimated-everything. Kelly assumes a fixed, known $p,m,s$. The moment parameters drift or are estimated, “full Kelly from the estimate” is overbetting - optimality for a certainty you do not have.
3. “Growth $=$ Sharpe” tempts a confusion of objectives. Maximising the Sharpe ratio (tangency) maximises growth per unit variance, whereas full Kelly maximises absolute growth - i.e. it levers up. The two optima differ by the capital-scaling decision, which is exactly what fractional Kelly governs.

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5. References

• Breiman, Leo: Optimal Gambling Systems for Favorable Games, Proc. 4th Berkeley Symposium (1961)
• Thorp, Edward O.: The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market (2006)
• Markowitz, Harry: Portfolio Selection, Journal of Finance 7(1) (1952)
• MacLean, Thorp & Ziemba (eds.): The Kelly Capital Growth Investment Criterion (2011)

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6. Connected Graph Bridges

• Back: 02 · The Kelly Formula · Index Hub
• Continue: 04 · Fractional Kelly & Ruin
• Growth foundation: Multiplicative Growth & Volatility Drag
• Sharpe/tangency link: 03 · Tangency & CAPM · MPT Index