Portfolio construction
Growth optimal portfolio
Maximises the average logarithmic return over the actual history, the exact growth rate, without assuming bell-shaped returns.
用通俗的话解释
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.
计算方法
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_max如何使用
- 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.
得出结论
- 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.
示例
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.
示例的输入仅作说明；权重和数据是我们的优化器针对这些输入给出的结果。
注意事项
- Same overbetting danger as Kelly: a fraction of it is the sensible use.
- Fits the past exactly, so it inherits the past's luck.
一点历史
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.
仅供学习参考。 没有任何投资组合模型知道未来：每一个都基于对过去的估计，本页内容均不构成投资建议。