*5.1. Incorporating Anomaly Scores into Portfolio Management*

We first demonstrated how anomaly scores can be incorporated into portfolio optimization to form portfolios with lower volatility. Even when forming a diversified portfolio among cryptocurrencies, its volatility as measured by standard deviation is too high compared to traditional assets, because each cryptocurrency is volatile on its own and the correlation among cryptocurrencies are relatively high, as already discussed in Figures 5 and 6. However, anomaly scores can help reduce portfolio volatility. Anomaly scores reflect abnormal market movements, so avoiding these periods reduces portfolio volatility even when forming a portfolio that only invests in cryptocurrencies.

For this backtest, rolling optimization was performed with weekly re-optimization and a lookback period of either 52 or 104 weeks. In order to focus on portfolio models with low risk, global minimum-variance (GMV) and risk-parity (equal risk contribution) models were used for optimizing portfolio weights. These are two popular models for forming an investment portfolio based on investment risk rather than expected return. The GMV portfolio model finds the optimal weights *<sup>ω</sup>* <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* with the smallest risk in the mean-variance optimization framework [40,45] and is written as:

$$\min\_{\boldsymbol{\omega}} \frac{1}{2} \boldsymbol{\omega}^T \boldsymbol{\Sigma} \boldsymbol{\omega}$$

where <sup>Σ</sup> <sup>∈</sup> <sup>R</sup>*n*×*<sup>n</sup>* is the covariance matrix of returns for *<sup>n</sup>* assets. The risk-parity formulation can be written as:

$$\min\_{\omega} \sum\_{i=1}^{n} \sum\_{j=1}^{n} \left( R\mathbb{C}(\omega\_{i}) - R\mathbb{C}(\omega\_{j}) \right)^{2} \text{ where } R\mathbb{C}(\omega\_{i}) = \omega\_{i} \frac{\partial \sigma(\omega)}{\partial \omega\_{i}}.$$

that minimizes discrepancies among risk contributions (*RC*) of each asset, where *RC* is measured with respect to the standard deviation *σ* of a portfolio [46]. The feasible portfolios were restricted to non-negative weights that sum to one, which is the most basic setting in portfolio construction [47].

On each re-optimization date, the portfolio strategy decided not to invest in cryptocurrencies (i.e., sell all positions) if the anomaly score was above a certain pre-determined limit (e.g., 1 or 2), and ex ante anomaly scores with shrinkage were computed each time from either previous 52-week or 104-week returns. A 52-week lookback period results in portfolio performance from January 2019 to February 2022, and a 104-week lookback provides performance from January 2020 to February 2022. Portfolios were constructed with no-shorting constraints, and USDT was excluded in the backtest because it had negative expected returns during this period.

Table 1 presents weekly standard deviations, annualized standard deviations, and the number of weeks over limit for several anomaly limits. The third column shows results for an equally weighted portfolio of the top 14 cryptocurrencies. The annualized volatility was above 90% without incorporating anomaly scores, but decreased to below 50% with an anomaly limit of 0.5. GMV, and risk-parity portfolios had lower standard deviation compared to the equally weighted portfolio. In particular, GMV had the lowest risk and the annualized volatility was near 40% when an anomaly limit of 0.5 was imposed. Therefore, portfolios with annualized volatility above 80% are unreasonably risky for all rational investors, which is the case without any anomaly limit, but reducing volatility to 40% may provide a viable investment option for investors with minimal risk aversion.


**Table 1.** Risk performance of portfolios based on various anomaly limits.

\* Standard deviation is annualized by multiplying <sup>√</sup>52.
