3.5 Estimation risk and many bets

Motivation

Kelly's formula takes the edge as known. A trader knows only an estimate, and M02 · 2.4 showed that mean returns are estimated very imprecisely. Separately, traders hold several positions at once, and correlated positions are one large bet.

Shrinking for estimation error#

Setup

Let the true excess return be $\mu$ (write $\mu$ for $\mu - r$ in this subsection) and the volatility $\sigma$ known. The trader estimates $\hat\mu \sim N(\mu, \text{se}^2)$ , independent of future returns, and bets $f = k\hat\mu/\sigma^2$ for a chosen multiplier $k$ . Let $t = \mu/\text{se}$ be the true signal-to-noise ratio of the estimate.

Proposition 3.9shrinking the estimated Kelly fraction

The expected excess growth is

E\big[g(k\hat\mu/\sigma^2)\big] - r = \frac{k\mu^2}{\sigma^2} - \frac{k^2(\mu^2 + \text{se}^2)}{2\sigma^2}, \tag{3.13}

maximised at

k^* = \frac{\mu^2}{\mu^2 + \text{se}^2} = \frac{t^2}{1 + t^2}. \tag{3.14}

At $k = 1$ (plug-in Kelly) the expected excess growth is $(\mu^2 - \text{se}^2)/(2\sigma^2)$ , which is negative if $t < 1$ .

Proof

From (3.9), $g(f) - r = f\mu - \frac12 f^2\sigma^2$ . With $f = k\hat\mu/\sigma^2$ : $E[f] = k\mu/\sigma^2$ and $E[f^2] = k^2(\mu^2 + \text{se}^2)/\sigma^4$ . Hence $E[g(f)] - r = k\mu^2/\sigma^2 - k^2(\mu^2 + \text{se}^2)/(2\sigma^2)$ . This is a concave quadratic in $k$ with maximum at $k = \mu^2/(\mu^2 + \text{se}^2)$ .∎

The optimal multiplier depends on the unknown $t$ , but its message is robust. With $T$ years of data, $\text{se} = \sigma/\sqrt T$ and $t = S\sqrt T$ (M02 Proposition 2.9), so $k^* = S^2T/(1 + S^2T)$ . For the Nifty example ( $S = 0.225$ , $T = 19$ years), $S^2T = 0.96$ and $k^* = 0.49$ : the data justify half the estimated Kelly leverage. For a 3-year backtest with $S = 1$ , $k^* = 0.75$ ; with $S = 0.5$ , $k^* = 0.43$ . A Bayesian treatment with prior $\mu \sim N(0, \tau^2)$ gives the same form: in continuous time the growth rate is linear in $\mu$ , so the Bayes-optimal fraction is $E[\mu \mid \text{data}]/\sigma^2$ , and the posterior mean shrinks $\hat\mu$ by $\tau^2/(\tau^2 + \text{se}^2)$ .

Estimation error in $\sigma$ also matters. If the true volatility is $\lambda$ times the estimate, a bettor aiming at $c$ times Kelly is actually at $c\lambda^2$ . A regime shift that doubles volatility (Nifty daily volatility went from 0.77% in Jan–Feb 2026 to 1.48% in March–April 2026, M02 · 2.10.4) turns half Kelly into double Kelly: zero growth.

Multi-asset Kelly#

Proposition 3.10multi-asset Kelly

With $n$ risky assets with drift vector $\mu$ , covariance matrix $\Sigma$ (positive definite) and portfolio weights $f \in \mathbb R^n$ ,

\begin{aligned} g(f) &= r + f^\top(\mu - r\mathbf 1) - \tfrac12 f^\top\Sigma f, \qquad f^* = \Sigma^{-1}(\mu - r\mathbf 1), \\ g(f^*) &= r + \tfrac12(\mu - r\mathbf 1)^\top\Sigma^{-1}(\mu - r\mathbf 1). \end{aligned} \tag{3.15}

Proof

$dW/W = r\,dt + f^\top(dS/S - r\mathbf 1\,dt)$ has drift $r + f^\top(\mu - r\mathbf 1)$ and instantaneous variance $f^\top\Sigma f$ . By Itô's lemma on $\ln W$ (M08), the log drift is $r + f^\top(\mu - r\mathbf 1) - \frac12 f^\top\Sigma f$ . The gradient is $(\mu - r\mathbf 1) - \Sigma f$ , zero at $f^*$ ; the Hessian $-\Sigma$ is negative definite, so this is the unique maximum. Substitute to get $g(f^*)$ .∎

The quantity $(\mu - r\mathbf 1)^\top\Sigma^{-1}(\mu - r\mathbf 1)$ is the squared Sharpe ratio of the tangency (maximum-Sharpe) portfolio of mean–variance theory (M14). The Kelly portfolio is the tangency portfolio, levered to the point where its own Kelly fraction is 1.

Correlated positions as one bet#

Correlation and simultaneous bets For two assets with the same excess return $m$ , volatility $\sigma$ and correlation $\rho$ , $f^*_1 = f^*_2 = m/(\sigma^2(1+\rho))$ , total exposure $2m/(\sigma^2(1 + \rho))$ . At $\rho = 0$ the total is twice one asset's Kelly fraction; at $\rho = 1$ it equals one asset's. A trader who sizes each of $n$ positions as if it were alone, when they are perfectly correlated, is at $c = n$ times Kelly. With $n = 3$ , growth is negative.

Worked example 3.5.1

Two assets with excess returns 6% and 4% ( $\mu = (12\%, 10\%)$ , $r = 6\%$ ), volatilities 20% and 15%, correlation 0.4. $\Sigma = \begin{pmatrix}0.04 & 0.012\\ 0.012 & 0.0225\end{pmatrix}$ , $\det\Sigma = 0.000756$ , $\Sigma^{-1}(0.06, 0.04)^\top = \frac{1}{0.000756}(0.0225 \times 0.06 - 0.012 \times 0.04,\ -0.012 \times 0.06 + 0.04 \times 0.04) = (1.151, 1.164)$ , and $g^* = 0.06 + \frac12(0.06 \times 1.151 + 0.04 \times 1.164) = 11.78\%$ . At $\rho = 0.9$ the solution becomes $(1.58, -0.12)$ : with high correlation, Kelly shorts the weaker asset to hedge the stronger one. Small changes in an estimated correlation produce large changes in the weights, which is the instability of mean–variance optimisation that M14 treats with shrinkage and risk budgets.

Practitioner notes

Use $k^*$ as a ceiling: estimation error is only one reason to shrink. Fat tails (the continuous model has none) and correlation in stress are others.
Count correlated positions as one bet. Five long positions in Indian private banks are one bet on Indian private banks.
Many professional allocators run strategies at a volatility target rather than a Kelly multiple. A volatility target fixes $\sigma_{\text{target}} = f\sigma$ , so $c = f/f^* = f\sigma^2/(\mu - r) = f\sigma/S = \sigma_{\text{target}}/S$ : a 10% volatility target on a Sharpe-0.5 strategy is $c = 0.2$ .