Kelly Optimization
Long-only growth-optimal portfolio that maximises the expected logarithm of terminal wealth. Breiman (1961) 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.
Near the so-called full-Kelly weight, the growth penalty is quadratic on both sides: a small under-bet and an equally small over-bet cost about the same growth. The two sides differ in risk. Over-betting increases the volatility and the probability of large drawdowns for no gain in growth, and in the standard continuous-time approximation twice the full-Kelly weight earns no growth above the risk-free rate. 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 unique when the return scenarios are nearly collinear. The scenarios are the daily returns of the run window, and the method needs at least 30 of them.
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 return scenarios are nearly collinear; FolioLab's default sets . The logarithm makes this an exponential-cone programme, solved with CLARABEL first, then SCS, then ECOS.
Constraints
The simplex constraint and a per-asset cap are standard. The method's own cap is 1.0 on every run, so it does not bind; a per-name bound submitted with the run sits beside it, and the tighter of the two applies. Asset exclusions and group sum limits enter the same programme as extra rows. 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
The contributions collected 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 about three-quarters of the long-run growth at a quarter of the variance (half the volatility), in the standard continuous-time approximation.
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.
- Breiman, L. (1961). "Optimal Gambling Systems for Favorable Games." Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, 1, 65-78.
- 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.
Not investment advice. Past performance is not indicative of future results.