Kelly Optimization
Long-only growth-optimal portfolio that maximises the expected logarithm of terminal wealth. Kelly (1956) showed that this rule is asymptotically optimal: over long horizons it dominates every other betting strategy that does not go bankrupt with probability one.
Overview
The Kelly criterion was introduced by Kelly (1956) in the context of information-theoretic gambling, generalised by Thorp (1969) to favourable games, and extended to portfolio choice in a long line of work culminating in MacLean, Thorp and Ziemba (2010). The idea is simple: instead of maximising one-period expected return or one-period Sharpe, maximise the expected log-return per period. By the strong law of large numbers, the log-optimal portfolio dominates all other constant-rebalancing strategies in terminal wealth almost surely as the horizon grows.
The penalty for deviating from log-optimality is asymmetric: under-betting leaves growth on the table linearly; over-betting (above the so-called full-Kelly weight) reduces growth super-linearly and increases the probability of large drawdowns. This is why practitioners commonly use "fractional Kelly" (e.g. half-Kelly), trading some long-run growth for a tighter tail.
FolioLab implements long-only Kelly via a convex programme that maximises the empirical-mean log-return over the historical sample, with a small ridge regulariser to keep the solution well-defined when the sample covariance is near-singular.
Mathematical Formulation
Notation
- — vector of asset returns at time
- — portfolio weights, summing to 1, with
- — number of historical periods used as scenarios
- — small ridge regulariser
Empirical-mean log-wealth objective
is concave in because the logarithm is concave and the inner argument is affine, so the objective is concave and a global maximum is reached by any convex solver. The ridge term stabilises the solution when the sample covariance is near-singular; FolioLab's default sets .
Constraints
The simplex constraint and an optional per-asset cap are standard. The third constraint forces the gross portfolio return on every historical scenario to stay above a small floor , which keeps the logarithm bounded and the problem strictly feasible.
Quadratic approximation
In the small-return regime the second-order Taylor expansion gives the familiar approximation: maximising expected log-return is approximately equivalent to mean-variance with risk-aversion coefficient one. The exact Kelly is preferred over the approximation when the universe contains instruments with material tail exposure (commodities, options proxies) where the higher-order terms matter.
Fractional Kelly
Cover (1991) and the survey in MacLean, Thorp and Ziemba (2010) argue that full-Kelly is the optimal stationary strategy under log-utility but that most real-world investors should bet less. A fractional-Kelly portfolio invests a fraction in the Kelly weight and holds the remainder in cash. The half-Kelly portfolio () captures three-quarters of the long-run growth at half the variance, which is a much friendlier risk-return profile in practice.
FolioLab's default optimisation reports the full-Kelly weights; users wanting a fractional implementation typically combine the Kelly weights with a cash overlay at the rebalance step, which preserves the asset-mix recommendation while controlling the leverage actually deployed.
Advantages & Limitations
Advantages
- Asymptotically optimal: Maximises long-run growth almost surely.
- Convex: Single global optimum reached by standard solvers.
- Distribution-free: Uses the empirical distribution directly, no Gaussianity assumption.
- Scale-aware: Naturally captures the cost of compounding losses.
Limitations
- Aggressive in good markets: Full-Kelly often calls for high concentration.
- Sensitive to tail estimation: Mis-specifying the empirical distribution at the tail can over-bet.
- Slow convergence: "Long run" can be longer than any investor's horizon.
- Asymmetric penalty: Over-betting punishes more than under-betting.
References
- Kelly, J. L. (1956). "A New Interpretation of Information Rate." Bell System Technical Journal, 35(4), 917-926.
- Thorp, E. O. (1969). "Optimal Gambling Systems for Favorable Games." Review of the International Statistical Institute, 37(3), 273-293.
- MacLean, L. C., Thorp, E. O., & Ziemba, W. T. (Eds.) (2010). The Kelly Capital Growth Investment Criterion: Theory and Practice. Singapore: World Scientific.
- Cover, T. M. (1991). "Universal Portfolios." Mathematical Finance, 1(1), 1-29.
- Rotando, L. M., & Thorp, E. O. (1992). "The Kelly Criterion and the Stock Market." The American Mathematical Monthly, 99(10), 922-931.