May 16, 2011
On Optimal Arbitrage
Some might be interested in reading: “On Optimal Arbitrage” by Daniel Fernholz and Ioannis Karatzas. It can be downloaded from HERE. (update: seems to not be available). But you can download a copy from HERE. It is a recent preprint and well done.
In the above paper, the authors make the case that the most probable outcome on portfolio return is to achieve an overall rate that is close to the long-term average market return (meaning close to the secular trend or, in plain text, close to a 10% average over the long haul). Also, a mathematical demonstration is made that it is the best you can expect and, therefore, should consider a combination of an index fund and a money market fund. Their presentation leads to a relatively constant Sharpe ratio as price leaders are partially sold to increase the holdings of laggers which in turn will have a tendency of maintaining a relatively stable risk ratio. The optimal trading strategy is presented as a mix of risky and riskless assets leading to the search for the optimal portfolio residing on the efficient frontier.
In short, if you needed arguments to torpedo my own research paper, the above-mentioned paper (and many others like it) could possibly provide all the ammo you need.
The paper I presented has far-reaching implications and may require reformulating some basic equations of modern portfolio theory. It is not only the exponentially adjusted Sharpe ratio that is concerned; even the Capital Market Line (CML) will need to be re-adjusted to reflect the exponential Sharpe since, technically, the Sharpe ratio represents the risk premium over volatility - which in turn is the slope of the CML. This would imply that the CML can rise exponentially (up to a limit; again, see my paper: Alpha Power ), which goes against accepted notions of portfolio management theory. And yet, my paper makes that claim. It also states that trading methods can greatly improve performance when the literature on this subject has a hard time extracting an edge in any mechanical trading methodology as, more often than not, it is shown that the stock market game is a zero-sum game (to which I agree). And yet, my paper shows that trading methods can be found and implemented that produce better than average returns even in the worst possible trading environment where all price series are randomly generated, where no selection bias, curve fitting, or over-optimization is possible.
When designing a trading strategy, one must maintain: 1) Feasibility, 2) Marketability, 3) Sustainability, 4) and remain realistic in a real-world trading environment. There is no trading of a million shares of a penny stock that should be considered realistic. I’ve covered these points before. There has to be someone on the other side to take your trade, whatever your intended volume in whichever direction. Selection survivability has also to be addressed in order to contain market risk. Putting 100% equity on a downer is not a realistic way to generate portfolio profits, as once in a blue moon, this downer (a black swan) has the potential to blow up your account, which in turn puts you out of the game with a score of zero. One has to manage risk at all times, whatever may happen.