# Intuitive Explanation for Shannon's Demon?

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.

• This is known as "volatility pumping" in quant finance lingo. You might be interested in this question and the given answers there: quant.stackexchange.com/questions/352/… Feb 25, 2018 at 20:02
• @vonjd Thank you. My initial searches for Shannon's Demon came up empty handed. There is obviously some overlap between that question and this one. Would you recommend any tweaks to this thread to clarify/elaborate on specifically what is being asked? Feb 25, 2018 at 20:09

Whether it's called volatility pumping, rebalancing premium, or Shannon's Demon it would just be a form of replicating a short gamma option strategy (eg. selling straddles). Intuitively, you are systematically selling at higher levels and buying at lower levels.

The payoff for continuously rebalancing an equity/cash portfolio without friction when the equity follows a driftless geometric Brownian motion is shown in the graph. (This was obtained with a 10000 path Monte Carlo simulation). There is a tradeoff between frequent small gains and less frequent large losses, reminiscent of shorting straddles.

Mean reversion would help, trending would hurt.

In the guise of generating positive gains from two losing strategies, this is known as Parrondo's paradox. The examples that demonstrate the effect are typically contrived, exploiting stationarity and a priori knowledge of the parameters.

• I'm assuming the x-axis is annual return on the GBM. Short gamma is very interesting way to look at this phenomenon. Is it fair to say then that rebalancing captures an amount of return which is directly proportional to the variance convexity adjustment between instantaneous and simple returns? If so, it should be possible to replicate the simulation results with a representation of the expectation. Feb 26, 2018 at 2:04
• There's a paper by William Bernstein here that explains 'rebalancing bonus' here. efficientfrontier.com/ef/996/rebal.htm Feb 26, 2018 at 3:03
• @brian: Thank you. I've seen that before. It might be worthwhile to seek a rebalancing premium for special situations (assuming you can overcome transaction costs) if mean reversion dominates. On the other hand, should a large pension fund try to capture this premium by rebalancing frequently to the strategic targets? If the fund has a few asset classes (equity index, bond index, real assets, etc. as with most) it is a bad idea. First of all equity indexes have a tendency to trend. Second you are trying to pick up basis points at the cost of worsening the drawdown if equities crash.
– RRL
Feb 26, 2018 at 3:26
• @vonjd: It's my own simulation results.
– RRL
Feb 26, 2018 at 8:18
• @vonjd Please see the addendum posted to the above question. I think I am pretty close to a closed form approximation of the rebalancing premium in continuous time. I would love your inputs regarding the distribution for the expectation. Feb 26, 2018 at 21:47

You may find the following paper worthwhile. It addresses most of the above points (and many more) in a systematic way:

Dubikovsky, Vladislav and Susinno, Gabriele, Demystifying Rebalancing Premium and Extending Portfolio Theory in the Process (May 20, 2015). Available at SSRN: https://ssrn.com/abstract=2927791 or http://dx.doi.org/10.2139/ssrn.2927791

Abstract
Volatility is usually considered as a synonym for risk. Mainstream financial theory states that higher portfolio volatility is translated into higher expected returns while diversification helps eliminate idiosyncratic risks. This leaves us with an apparent anomaly as low-risk (low-beta) stocks over-perform high-risk (high-beta) stocks over the long term. Is this really an anomaly? What about high conviction investing? Should we dismiss stock-picking as a futile exercise even if such an approach is used by one of the most successful investors of our times? In this paper we answer these questions and propose a framework that encompasses various investment styles and portfolio construction methodologies. Modern Portfolio Theory is a one period approach relating expected returns and volatilities as two independent variables estimated from ensemble averages. Here we focus on a multi-period setting, which is more relevant for the task of maximising investor’s wealth in the long run. Contrary to previous studies based on maximising log returns, we find no contradictions with the results of modern portfolio theory. We show that Markowitz portfolios and Warren Buffett’s investment style are valid special cases of optimal growth portfolios. In addition, we provide insights on rebalancing bonus, showing how and when it is possible to add value from volatility in active portfolio management. As fire can be either dangerous, if uncontrolled, or useful to run a mechanical engine if controlled, in the very same way it should be possible to put volatility to work in a controlled manner in order to produce growth.