Mandelbrot–Kelly: from theory to position sizing

1. The problem with textbook Kelly

The Kelly criterion gives the bet fraction $f^{\ast }$ that maximises the expected logarithm of wealth over many independent trials. For a binary outcome with probability $p$ of winning at odds $b:1$, the classical result is

$$f^{\ast }=\frac{pb-q}{b},q=1-p.$$

Generalised to a continuous return $R$ with mean $\mu$ and variance ${\sigma }^2$, a common Gaussian approximation gives

$$f^{\ast }\approx \frac{\mu }{{\sigma }^2},$$

a result that practitioners often apply directly to position sizing in financial markets. It is wrong in a way that matters.

The derivation assumes the distribution of returns is well-described by its first two moments. For most liquid futures the assumption fails. Returns are heavy-tailed, asymmetric, and not even guaranteed to have a finite second moment. Plugging an empirical sample variance into the Gaussian formula produces a $f^{\ast }$ that is systematically too large and that, under any reasonable drawdown constraint, blows up the account during the next regime that does not look like the calibration window.

2. Mandelbrot, fractal markets, and L-stable returns

Mandelbrot's 1963 study of cotton prices was the first systematic argument that financial returns are not Gaussian but follow a Lévy stable distribution — a family parameterised by

$$S_{\alpha }(\beta ,\gamma ,\delta ),\alpha \in (0,2],\beta \in [-1,1].$$

Here $\alpha$ is the characteristic exponent (tail heaviness), $\beta$ is the skewness, $\gamma$ the scale, and $\delta$ the location. The Gaussian distribution is the special case $\alpha =2$. For $\alpha <2$, the variance is infinite and the standard Kelly formula has no defensible derivation.

Empirically, fitted $\alpha$ for daily futures returns sits in the range $1.6$ to $1.9$ across most of the liquid universe. That is far enough from $2$ to invalidate Gaussian sizing in practice, even though the deviation is small enough that it is easy to ignore.

The point of fitting $\alpha <2$ is not to produce more accurate point forecasts. It is to produce more accurate bet-size constraints.

3. A fractional Kelly correction for stable returns

Because variance is undefined for $\alpha <2$, the Gaussian Kelly formula cannot be applied directly. A practical workaround is to replace the population variance with a truncated scale parameter — the variance of the distribution conditioned on $|R|<c$ for some censoring level $c$. This recovers a finite quantity but introduces a sensitivity to $c$ that the practitioner must own.

A more honest path is to keep the stable parameters and fractionalise more aggressively. We use

$$f=\lambda (\alpha )\cdot f_{\text{Gauss}}^{\ast },$$

where $\lambda (\alpha )$ is a shrinkage factor calibrated empirically from out-of-sample drawdown behaviour. For $\alpha =2$ we take $\lambda =0.25$ (the conventional quarter-Kelly safety margin). For $\alpha <2$ we shrink further:

$$\lambda (\alpha )=0.25\cdot \exp (-k(2-\alpha )),$$

with $k\approx 1.5$ chosen so that the realised maximum drawdown over rolling out-of-sample windows is bounded by the design limit. The exponential form is not derived from first principles — it is a tractable family that fits the empirical relationship between tail thickness and drawdown amplification.

The result is that an instrument with $\alpha =1.7$ runs at roughly 40 percent of the Gaussian quarter-Kelly size. An instrument with $\alpha =1.9$ runs at roughly 70 percent. Models calibrated on Gaussian Kelly without this correction systematically allocate too much capital to the assets that hurt them most.

4. Implementation

In practice we do not compute $f$ each day from scratch. The model registry stores, per instrument and per strategy:

• A rolling estimate of $\alpha$ over a long window
• A short-window scale parameter $\gamma$ that responds to current volatility
• A precomputed $\lambda (\alpha )$ table

Position sizes are then computed as

$$\text{size}=\lambda (\alpha )\cdot \frac{\hat{\mu }}{{\hat{\gamma }}^2}\cdot {\text{capital}}_{\text{available}},$$

subject to drawdown, concentration, and contract caps that may bind before the Kelly result does. In most regimes, the secondary constraints bind first. Kelly sizing only becomes the active constraint for trades that the model considers exceptionally favourable.

5. What this does not solve

Three honest caveats:

1. Estimation error. Stable parameter estimation is harder than variance estimation. Confidence intervals on $\alpha$ are wide on small samples, and the $\lambda (\alpha )$ shrinkage assumes the estimate is roughly correct.
2. Regime changes. If the underlying $\alpha$ shifts over time, a rolling window will lag. We address this with a fast-window $\gamma$ that responds to volatility regime changes even when $\alpha$ is held constant.
3. Joint behaviour. The framework above is single-instrument. Cross-asset tail dependence — the empirical fact that fat-tail events cluster across markets — requires a separate copula-style treatment that we will address in a future note.

6. Why this matters for sizing

The headline implication is simple. Standard Gaussian Kelly, applied naively to financial returns, produces a position size that is too large by a factor that depends on $\alpha$. The correction is not exotic; it is a one-parameter shrinkage informed by empirical tail behaviour. It is also the difference between a system that survives the next fat-tail event and one that does not.

Mandelbrot's original argument was that markets are wild. Kelly's original argument was that bet sizing should maximise long-run growth. Combining the two gives a sizing rule that is both growth-optimal and survivable. We treat survival as the binding objective.

References

Kelly, J. L. (1956). A New Interpretation of Information Rate. Bell System Technical Journal, 35, 917–926.

Mandelbrot, B. (1963). The Variation of Certain Speculative Prices. Journal of Business, 36(4), 394–419.

Mandelbrot, B. & Hudson, R. L. (2004). The (Mis)Behavior of Markets: A Fractal View of Financial Turbulence. Basic Books.

Thorp, E. O. (1966). Beat the Market: A Scientific Stock Market System. Random House.

MacLean, L. C., Thorp, E. O., & Ziemba, W. T. (Eds.). (2010). The Kelly Capital Growth Investment Criterion: Theory and Practice. World Scientific.

Samuelson, P. A. (1979). Why we should not make mean log of wealth big though years to act are long. Journal of Banking & Finance, 3(4), 305–307.