A Comparison of Kelly and Shannon Strategies vs Buy and Hold
To dig deeper into the previous post, we will take a look at how the “demon” strategy actually performed. We will also introduce the concept of a Kelly methodology introduced in the paper by Ed Thorp. The math is a little hairy for most, but for the brave and diligent the link is here http://www.bjmath.com/bjmath/thorp/paper.htm This post was developed by my research associate– Henry Bee, who besides being a whiz with numbers is also a good trader/writer. You can check out his blog on Seeking Alpha for some fundamental insight and trading ideas.
So first we have the demon portfolio, where we used 1%/-1% as thresholds for rebalancing. These levels were chosen to make this realistic rather than a thought experiment—daily rebalancing obviously cannot survive trading costs. We made assumptions concerning transaction costs that were consistent with institutional execution. Each day, the demon rebalances back to equal weight if the portfolio is greater than or equal to 1% in absolute value terms in either direction. The Stupid Kelly methodology is derived using expected returns and variance assumptions from the previous 3 years. Much like the typical armchair retail investor–the Stupid Kelly assumes that what happened in the last 3 years will repeat in the future–thus if we had a poor 3-year period its time to go into the bomb shelter, and if the last 3 years everyone has been making money, its time to back up the truck and get into the market. The bet size is dictated using a formula mentioned in the paper. This bet size is constrained at 2x leverage– a fairly typical adjustment designed to prevent overestimation bias—– in the markets we do not have any certainty concerning the future distribution of returns (unlike games of chance). The Smart Kelly methodology is derived using forward looking data such as the beta, treasury bond yields and other inputs to derive an equity risk premium. The risk assumptions incorporated implied volatility or variance (historical was used at earlier dates). Lets look at the results:
The demon portfolio outpeformed buy and hold on a risk-adjusted basis, and is beginning to improve versus buy and hold in recent years. Impressively it still performed reasonably well from 1965-2000 in the “pre mean-reversion days.” This is because it capitalizes on brownian motion versus a pure reversion assumption. Though not visible in this data, the rolling sharpe ratio for the demon portfolio versus buy and hold was actually underperforming until the last decade. The recent acceleration in performance can be attributed in my opinion mainly to greater randomness in the markets, and only partially to increased mean-reversion. The “stupid” kelly portfolio was the worst method, and only made significant progress during the giant bull market from 1982-1999. Unsurprisingly that was one of the few periods where retail investors actually made money. In contrast, the “smart” kelly portfolio which used forward looking data and was long-term contrarian performed the best of all three on an absolute and risk-adjusted basis. Combining the “smart” kelly portfolio and the shannon’s demon portfolio obviously would be a nice way to capitalize on the “academic” notions that 1) markets are efficient and exhibit brownian motion patterns (randomness) 2) there is a “risk premium” in equity markets–investors must be compensated for bearing risk. So negative longer-term historical returns, high expected variance, and high long-term bond yields mean that investors require a higher premium (reward) to invest. So with all the technical analysis and fundamental analysis methods we use to try to beat the market sometimes its nice to have back up in case those crazy professors are right all along!