Portfolio construction
Growth optimal portfolio
Maximises the average logarithmic return over the actual history, the exact growth rate, without assuming bell-shaped returns.
The idea in plain words
Kelly's formula Σ⁻¹μ is an approximation for normal returns. The growth optimal portfolio maximises the real thing: the average of log(1 + portfolio return) over every past day. Fat tails and skew in the data are taken into account automatically.
How it is calculated
Maximise the mean of log(1 + wᵀrₜ) over the daily returns, weights ≥ 0, total at most the maximum leverage; solved by projected gradient ascent.
maximise (1 ÷ T) Σₜ log(1 + wᵀrₜ)
subject to wᵢ ≥ 0, Σwᵢ ≤ L_maxHow to use it
- As the most faithful growth maximiser when the history is long enough.
- Compare with Kelly: a large difference means the returns are far from normal.
Drawing conclusions
- It usually hits the leverage cap; the cap then decides the size, the model decides the mix.
- A ticker with rare but huge losses is penalised more than in Kelly.
Example
Running example (used in every portfolio guide): US500, GER40, gold (XAUUSD) and EURUSD, two years of daily data. Expected returns 8%, 7%, 7% and 1% a year; volatilities 18%, 20%, 15% and 7%. US500 and GER40 move together (correlation about 0.7); gold hardly moves with the indices (about 0.04). Growth optimal: 100% US500 and 100% gold, at the 2× cap. Return 15.0%, volatility 24%. Close to Kelly's mix: the simulated returns in the example are near-normal.
The example's inputs are illustrative; the weights and figures are what our optimiser returns for them.
Watch out for
- Same overbetting danger as Kelly: a fraction of it is the sensible use.
- Fits the past exactly, so it inherits the past's luck.
A little history
Henry Latané (1959) and Leo Breiman (1961, "Optimal Gambling Systems for Favorable Games") proved that maximising expected log wealth wins in the long run; Nils Hakansson and others developed the growth optimal portfolio in the 1970s.
Educational content only. No portfolio model knows the future: every one of them works on estimates from the past, and nothing here is investment advice.