Portfolio construction
Growth optimal portfolio
Maximises the average logarithmic return over the actual history, the exact growth rate, without assuming bell-shaped returns.
La idea en pocas palabras
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.
Cómo se calcula
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_maxCómo usarlo
- 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.
Sacar conclusiones
- 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.
Ejemplo
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.
Los datos de entrada del ejemplo son ilustrativos; los pesos y cifras son los que devuelve nuestro optimizador para ellos.
Ten cuidado con
- Same overbetting danger as Kelly: a fraction of it is the sensible use.
- Fits the past exactly, so it inherits the past's luck.
Un poco de historia
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.
Solo contenido educativo. Ningún modelo de cartera conoce el futuro: todos trabajan con estimaciones del pasado, y nada de lo aquí expuesto es asesoramiento de inversión.