I am reading Fortune's Formula by William Poundstone, and I am puzzled by a phenomenon called "Shannon's Demon", which Claude Shannon allegedly proposed in a series of lectures, and preserved only by mimeographed lecture notes. Basically, Shannon's Demon proposes a way of harvesting volatility even when the underlying is driftless and/or one is agnostic regarding the underlying's drift.
The simplest example starts with a portfolio's allocation is split between two assets which can be represented as (uncorrelated) martingales or semi-martingales. Assume that trading is costless and frictionless. The allocation changes, however, when prices change. After each period, the allocations are rebalanced back to their original “target” allocation. As a result of the costless and frictionless rebalancing, the portfolio's expected logarithmic growth rate is greater than the arithmetic mean of the two assets, while its variance is less than (about ~2/3 of) the mean of the variances.
One can easily simulate such a scenario using GBM (or some simpler martingale property, as this example shows) to demonstrate its veracity. I ran Monte Carlo simulations for two risky assets under ABM and GBM. My results were consistent with Shannon’s.
Mathematically, this phenomenon is due to the assumption that portfolio variance is a quadratic function of asset weights. This is not intuitive to me, however.
I partly wonder whether Shannon’s demon, being an apparent free lunch, is valid only for simulated processes... meaning it will not hold up empirically (because either price processes are not actually martingales or the phenomenon is subsumed by transaction costs and trading frictions).
Anyhow, I am looking for intuitive explanations for the phenomenon and/or evidence which shows that frequent rebalancing outperforms using real world assets.
Good references are also appreciated.
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