Folio Lab/Documentation

Kelly Optimization

Maximise the expected logarithm of one-period wealth — the growth-optimal portfolio. Kelly weights produce the highest long-run compound growth rate when the joint return distribution is correctly specified.

Overview

The Kelly criterion (Kelly, 1956) chooses bet sizes that maximise the expected logarithm of wealth. In the portfolio context, that translates to selecting weights that maximise the expected log-return of the portfolio over a single period. Asymptotically, the Kelly portfolio dominates any other strategy in long-run growth.

In practice, full-Kelly is famously aggressive. Folio Lab solves the empirical Kelly problem on the historical sample, with long-only and box bounds plus an L2 stabiliser to prevent corner solutions. Users who want fractional Kelly can scale weights post-hoc.

Mathematical Formulation

With historical return matrix of shape , Folio Lab solves the empirical log-utility problem:

subject to:

The default (no per-asset cap), (light L2), (numerical safety against negative wealth). The problem is concave (log is concave) and is solved with cvxpy using exponential-cone solvers (CLARABEL, SCS).

Caveats

Full-Kelly assumes the historical return distribution is exactly correct. In practice the empirical distribution overstates opportunities — a phenomenon often called "Kelly overbet." Industry practice is fractional Kelly (e.g. half-Kelly), which sacrifices some long-run growth for considerably lower drawdowns.

Long-only constraints already truncate Kelly's leverage, but users should still be aware that the resulting portfolio may take considerably more risk than mean-variance.

Advantages & Limitations

Advantages

  • Optimal long-run growth: Asymptotically dominant under correct specification.
  • Convex problem: log-utility maximisation has a unique solution.
  • Nonparametric: Operates directly on the empirical sample.
  • Compatible with constraints: Easy to add box and budget bounds.

Limitations

  • Aggressive: Full-Kelly tolerates very large drawdowns.
  • Distribution-sensitive: Misspecified overestimates growth.
  • Path-dependent realisation: Asymptotic optimality ≠ finite-sample optimality.
  • Compute: Heavier than closed-form mean-variance.

References

  • Kelly, J. L. (1956). "A new interpretation of information rate." Bell System Technical Journal, 35(4), 917-926.
  • Thorp, E. O. (2006). "The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market." In Handbook of Asset and Liability Management, Vol. 1.
  • MacLean, L. C., Thorp, E. O., & Ziemba, W. T. (Eds.). (2011). The Kelly Capital Growth Investment Criterion. World Scientific.