*5.3. Performance Measures*

The following measures widely used in finance to evaluate portfolio strategies [43] are chosen. The portfolio return at time *t* is defined as *Rt* = ∑*n j*=1 *rj*,*<sup>t</sup>wj*,*t*−<sup>1</sup> where *rj*,*<sup>t</sup>* is the monthly return of *j* asset at time *t*, *wj*,*t*−<sup>1</sup> is the weight of *j* asset in the portfolio at time *t* − 1, and *n* is the total number of assets.

The annualized return (AR), annualized risk as the standard deviation of return (RISK), and risk-adjusted return (R/R) are defined as follows:

$$\text{AR} = \frac{12}{T} \times \sum\_{t=1}^{T} R\_{t\prime} \tag{26}$$

$$\text{RISK} = \sqrt{\frac{12}{T - 1} \times \sum\_{t=1}^{T} (R\_t - \hat{\mu})^2}, \quad \hat{\mu} = 1/T \times \sum\_{t=1}^{T} R\_{t\prime} \tag{27}$$

$$\text{R/R} = \text{AR/RISK.}\tag{28}$$

Among them, R/R is the most important measure for a portfolio strategy. We also evaluate the maximum draw-down (MaxDD), which is another widely used risk measure [44] for the portfolio strategy. In particular, MaxDD is the largest drop from a peak defined as

$$\text{MaxDD} = \min\_{t \in [1, T]} \left( 0, \frac{\mathcal{W}\_t}{\max\_{\tau \in [1, t]} \mathcal{W}\_\tau} - 1 \right), \tag{29}$$

where *Wk* is the cumulative return of the portfolio until time *k*; that is, *Wt* = ∏*tt*=<sup>1</sup>(<sup>1</sup> + *Rt*).

The turnover (TO) indicates the volumes of rebalancing [18]. Since a high TO inevitably generates high explicit and implicit trading costs, the portfolio return tends to be reduced. The TO is a proxy for the transaction costs of the portfolio. The one-way annualized turnover is calculated as an average absolute value of the rebalancing trades over all the trading periods:

$$\text{TO} = \frac{12}{2(T - 1)} \sum\_{t=1}^{T-1} ||w\_t - w\_t^-||\_1 \tag{30}$$

where *T* − 1 indicates the total number of the rebalancing periods and *<sup>w</sup>*<sup>−</sup>*t* = *wt*−1<sup>⊗</sup>(<sup>1</sup>+*rt*) <sup>1</sup>+*wt*−1*rt* is the re-normalized portfolio weight vector before rebalance. Here, *rt* is the return vector of the assets at time *t*, *wt*−1 is the weight vector at time *t* − 1, and the operator ⊗ denotes the Hadamard (element-wise) product.
