One reason why we prefer models with a high Sharpe ratio and short maximum drawdown duration is that this almost automatically ensures that the model will pass the cross-validation test: the only subsets where the model will fail the test are those rare drawdown periods.

The mathematical description of a mean-reverting price series is that the change of the price series in the next period is proportional to the difference between the mean price and the current price. This gives rise to the ADF test, which tests whether we can reject the null hypothesis that the proportionality constant is zero.

A clear mathematical exposition of the ADF and Variance Ratio tests can be found in Walter Beckert's course notes (Beckert, 2011). Here, we are interested only in their applications to practical trading strategies.

If we are trading the cross-rate AUD.ZAR, then AUD is called the base currency, and ZAR is the quote currency. (My personal mnemonic for this: B is ahead of Q alphabetically, so the order is B.Q.) If AUD.ZAR is quoted at 9.58, it takes 9.58 South African rand to buy 1 Australian dollar. Buying 100,000 AUD.ZAR means buying 100,000 Australian dollars, while selling the equivalent amount (100,000 × 9.58 = 958,000 at the preceding quote) of South African rand.

Researchers Coval and Stafford (2007) found that mutual funds experiencing large redemptions are likely to reduce or eliminate their existing stock positions. This is no surprise since mutual funds are typically close to fully invested, with very little cash reserves. More interestingly, funds experiencing large capital inflows tend to increase their existing stock positions rather than using the additional capital to invest in other stocks, perhaps because new investment ideas do not come by easily. Stocks disproportionately held by poorly performing mutual funds facing redemptions therefore experience negative returns. Furthermore, this asset “fire sale” by poorly performing mutual funds is contagious. Since the fire sale depresses the stock prices, they suppress the performance of other funds holding those stocks, too, causing further redemptions at those funds. The same situation occurs in reverse for stocks disproportionately held by superbly performing mutual funds with large capital inflows. Hence, momentum in both directions for the commonly held stocks can be ignited.

Even though we do not know the true distribution of R, we can use the so-called Pearson system (see www.mathworks.com/help/toolbox/stats/br5k833-1.html or mathworld.wolfram.com/PearsonSystem.html) to model it. The Pearson system takes as input the mean, standard deviation, skewness, and kurtosis of the empirical distribution of R, and models it as one of seven probability distributions expressible analytically encompassing Gaussian, beta, gamma, Student's t, and so on. Of course, these are not the most general distributions possible. The empirical distribution might have nonzero higher moments that are not captured by the Pearson system and might, in fact, have infinite higher moments, as in the case of the Pareto Levy distribution. But to capture all the higher moments invites data-snooping bias due to the limited amount of empirical data usually available. So, for all practical purposes, we use the Pearson system for our Monte Carlo sampling.

As the oft-quoted Daniel Kahneman wrote, experts are uniformly inferior to algorithms in every domain that has a significant degree of uncertainty or unpredictability, ranging from deciding winners of football games to predicting longevity of cancer patients. One can hope that the financial market is no exception to this rule.