# Strategic Portfolio Optimization Using Simulated, Digital, and Quantum Annealing
## 3 Quantitative Investment Strategies  
Quantitative trading can be defined as the systematic implementation of trading strategies that humans create through rigorous research. In this context, systematic is defined as a disciplined, methodical, and automated approach [22] (p. 15). The quantitative investment process starts by testing a possible strategy. Therefore, one has to start by gathering the necessary historical data, which might contain several flaws. For example, the data might be not adjusted to stock splits or dividends, there might be missing values, reported values might not be correct, and so on. If this is the case, one has to include a data cleaning step before one proceeds [22] (pp. 135-136). After establishing the data set for the potential strategy, the strategy is fitted to the historical data. There are additional pitfalls that need to be considered. For example, a historical database of stock prices might not include stocks that have \"disappeared\" due to bankruptcies, delistings, mergers, or acquisitions, and thus suffer from a so-called survivorship bias, because only \"survivors\" of those often unpleasant events remain in the database [23] (p. 33). Another example of a possible bias is the so-called data-snooping bias. This bias can be thought of as a kind of overfitting of the parameters to given historical data. It occurs when one tries to optimize the parameters of a strategy such that it performs exceptionally well on historical data. It is very likely that such an optimization can be performed [23] (p. 35). The look-ahead bias is about situations where information is used that was only available after the investment has been made. This information was not available when an asset was bought or sold [23] (p. 58). After the strategy shows acceptable performance on historical data, the strategy is tested on an out-of-sample data set, usually taken from the near past. Finally, the out-of-sample test (so-called backtest) is evaluated, and the strategy will either be used for investments or discarded [23] (p. 29). In our work, the price series of assets are the foundation of a trend strategy and a mean reversion strategy. Trend strategies (also called momentum strategies) are based on the theory that sometimes markets move long enough in one direction such that the trend can be identified and exploited. Mean reversion strategies assume, however, that there exists a center of a price series (e.g., the mean of a price) and that the price including its fluctuation always returns to its center [22] (pp. 39-45). The price series must be stationary for a mean reversion strategy. Stationarity can mathematically be tested by the augmented Dickey-Fuller (ADF) test [24] (p. 49). Price changes in a price series are described as  
\[\Delta y(t)=\lambda y(t-1)+\mu+\beta t+\alpha_{1}\Delta y(t-1)+\ldots+\alpha_{ k}\Delta y(t-k)+\epsilon_{t} \tag{4}\]  
The ADF test finds out if \(\lambda=0\). If the hypothesis \(\lambda=0\) can be rejected, then the next price change \(\Delta y(t)\) depends on \(y(t-1)\). Therefore, the series is not a random walk. The test statistics is the regression coefficient \(\lambda\) divided by the standard error of the regression \(\lambda/SE(\lambda)\). Since mean reversion is expected, \(\lambda/SE(\lambda)\) must be negative and lower than the critical \(z\)-_score_ of the hypothesis. Variable \(k\) denotes price lags of the price series, \(\beta t\) is the constant drift of the price series, and \(\alpha\) is the auto-regressive parameter. Error term \(\epsilon_{t}\) is independent and identically distributed with a mean of 0. Intuitively formulated, being a stationary price series means that the price moves away from its initial value slower than a geometric random walk.