Portfolio construction
Growth optimal portfolio
Maximises the average logarithmic return over the actual history, the exact growth rate, without assuming bell-shaped returns.
सरल शब्दों में विचार
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.
गणना कैसे होती है
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_maxइसका उपयोग कैसे करें
- 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.
निष्कर्ष निकालना
- 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.
उदाहरण
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.
उदाहरण के इनपुट केवल समझाने के लिए हैं; वेट और आँकड़े वही हैं जो हमारा ऑप्टिमाइज़र उनके लिए लौटाता है।
इन बातों का ध्यान रखें
- Same overbetting danger as Kelly: a fraction of it is the sensible use.
- Fits the past exactly, so it inherits the past's luck.
थोड़ा इतिहास
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.
केवल शैक्षिक सामग्री। कोई पोर्टफ़ोलियो मॉडल भविष्य नहीं जानता: हर एक अतीत के अनुमानों पर काम करता है, और यहाँ कुछ भी निवेश सलाह नहीं है।