M.9.6 Advanced Extensions

Basic Prerequisites: 04 · Kelly Criterion and 05 · Ruin & Drawdown.

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1. Intuition & Practical Objective

Pages 04–05 sized a single bet. Real investing is a vector decision across correlated assets with estimated parameters, in a world that does not hold still. This page is the launchpad to the three extensions that turn Kelly from a casino rule into a portfolio discipline:

1. Multi-asset (vector) Kelly. The optimal fractions solve ${\mathbf{f}}^{\ast }={\Sigma }^{-1}(\mathbf{\mu }-r1)$ - the growth-optimal analogue of mean-variance, with the same matrix inverse but a log objective. It naturally produces leverage and short positions.
2. Estimation error → fractional Kelly. Because $f^{\ast }$ is computed from noisy estimates and the growth function is asymmetric (overbetting is punished harder than underbetting), the practical optimum is systematically below the plug-in Kelly fraction. This is the single most important robustness lesson in the folder.
3. Ergodicity economics. The deeper theoretical extension: abandon expected-utility maximisation entirely and instead maximise the time-average growth rate of the actual dynamic - which, for multiplicative wealth, derives log utility rather than assuming it.

Takeaway. Three out of four real-world blow-ups in leveraged growth strategies come from estimation error, not from the model being wrong. Multi-asset Kelly is elegant, but the fraction you should actually trade is smaller than the formula, by a factor you choose for survival.

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2. Mathematical Ground Truth & Derivations

2.1 Multi-asset (vector) Kelly

Let $\mathbf{f}$ be the vector of wealth fractions in $n$ risky assets, $\mathbf{\mu }$ the drift vector, $\Sigma$ the covariance matrix, $r$ the riskless rate. The continuous growth rate generalises $g_{\infty }(f)=r+f(m-r)-\frac{1}{2}{s}^2{f}^2$ to

$$g_{\infty }(\mathbf{f})=r+{\mathbf{f}}^{\top }(\mathbf{\mu }-r1)-\frac{1}{2}{\mathbf{f}}^{\top }\Sigma \mathbf{f}.$$

Maximising (the function is concave in $\mathbf{f}$) gives the first-order condition $\Sigma \mathbf{f}=\mathbf{\mu }-r1$, hence

$$\boxed{{\mathbf{f}}^{\ast }={\Sigma }^{-1}(\mathbf{\mu }-r1)}{g}_{\infty }({\mathbf{f}}^{\ast })=r+\frac{1}{2}(\mathbf{\mu }-r1{)}^{\top }{\Sigma }^{-1}(\mathbf{\mu }-r1).$$

This is the log-utility cousin of the Markowitz tangency portfolio: same ${\Sigma }^{-1}$, but the objective is growth, not mean-variance utility. Note the fractions can exceed $1$ (leverage) and go negative (shorts) - the constraints $1^{\top }\mathbf{f}=c$ and $\mathbf{f}\ge 0$ must be added for real portfolios (Thorp §7).

2.2 Estimation error and the case for fractional Kelly

In practice $(\mathbf{\mu },\Sigma )$ are estimated, so $f^{\ast }$ is itself random. Two structural problems make the plug-in Kelly fraction an overbet:

• Mean reversion / data mining: estimated expected returns are biased upward ($\mathbb{E}[\hat{m}]>m$), so $f^{\ast }=(m-r)/s^2$ is too large.
• Asymmetry of $g$: the growth function is flat at its maximum and steep at the edges, so overbetting by $\delta$ costs more growth than underbetting by $\delta$ gains. The optimal response to uncertainty is to shrink the fraction.

Thorp’s recommendation (§7.3): choose $f$ so that even if the true edge is, say, $m_t=\frac{1}{2}\hat{m}$, you stay in the growth region - i.e. cap $f$ well below ${\hat{f}}^{\ast }$; practice uses $\mathbf{f}=0.25{\hat{\mathbf{f}}}^{\ast }$ to $0.5{\hat{\mathbf{f}}}^{\ast }$.

2.3 Ergodicity economics (the theoretical extension)

Peters & Gell-Mann propose replacing “maximise expected utility” with “maximise the time-average growth rate of the dynamic”. For additive dynamics $dx=\mu dt+\sigma dW$, the time-average growth equals the ensemble drift $\mu$ - reproduce the standard answer. For multiplicative dynamics $dx=\mu xdt+\sigma xdW$, the time-average growth rate is $\mu -\frac{1}{2}{\sigma }^2$ (page 03) - so maximising it derives logarithmic utility rather than assuming it. Ergodicity economics thus gives the axiomatic foundation under Kelly: log utility is not an arbitrary risk-aversion choice, it is the utility that makes the ensemble average equal the time average.

2.4 Non-stationarity and path dependence

All the closed forms assume i.i.d. returns. Real markets are non-stationary: $m$ and $s$ drift, correlations change, and regimes switch. The practical consequences are (i) $f^{\ast }$ should be re-estimated and revised over time (Thorp §7.2), and (ii) drawdowns path-depend - leverage that looked safe is re-levered automatically as equity falls, exactly the mechanism of 1987 and 2008.

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3. Computational Implementation - vector Kelly and the estimation-error penalty

Stdlib only. Block A solves the two-asset Kelly system and verifies the closed-form growth. Block B simulates a trader who estimates the edge with noise and evaluates full Kelly versus fractional Kelly.

The punchline is in block B: with a modest estimation error, full Kelly has almost a 20% chance of a negative twenty-year growth rate and a lower median outcome ($+0.600$) than half Kelly ($+0.965$). Fractional Kelly is not merely safer - it is better once you admit you do not know $\mu$. A quarter Kelly nearly eliminates the ruin probability at a small growth cost. This is the empirical case for $\mathbf{f}\approx 0.5{\hat{\mathbf{f}}}^{\ast }$ in production.

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4. Failure Modes & First-Principles Breakdowns

1. Plugging in ${\hat{\mathbf{f}}}^{\ast }$ from estimated moments. The single most common way to lose money with Kelly. Overbetting is asymmetric; the estimate is biased high; shrink the fraction (block B).
2. Ignoring the $1^{\top }\mathbf{f}=c$ and $\mathbf{f}\ge 0$ constraints. Unconstrained vector Kelly can demand huge leverage or violent shorts; the constrained problem needs KKT (page 04 of calculus) and its solution is not a simple rescaling.
3. Treating $\Sigma$ as known. A poor covariance estimate (noisy, ill-conditioned) inverts into explosive weights; shrinkage/denoising is mandatory (Covariance Estimation).
4. Assuming stationarity. $f^{\ast }$ moves as $\mu ,s$ move; a static full-Kelly lever is a bet on regime stability as much as on the edge.
5. Confusing vector Kelly with mean-variance optimality. They share ${\Sigma }^{-1}$ but maximise different objectives; a mean-variance “optimal” portfolio is generally not growth-optimal (it ignores the $\frac{1}{2}{\mathbf{f}}^{\top }\Sigma \mathbf{f}$ curvature of log wealth).

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5. References

• Thorp, Edward O.: The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market (2006)
• MacLean, Thorp & Ziemba (eds.): The Kelly Capital Growth Investment Criterion (World Scientific, 2011)
• Peters, Ole: The Ergodicity Problem in Economics, Nature Physics 15 (2019), and Peters & Gell-Mann, Evaluating Gambles Using Dynamics, Chaos 26 (2016)
• Glasserman, Paul: Monte Carlo Methods in Financial Engineering
• Tsay, Ruey S.: Analysis of Financial Time Series

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6. Connected Graph Bridges

• Back: 05 · Ruin & Drawdown
• Hub: Index Hub
• Base: Constrained Optimization (constrained vector Kelly) · Probability & Measure Theory
• Applied destinations: Portfolio Optimization · Covariance Estimation, Shrinkage & RMT · Extreme Value Theory & Fat Tails · Quantitative Research (overfitting & estimation error)
• Forward topic-folder (in Pillar 5): kelly-criterion-and-bet-sizing - the applied treatment of this material.