#### Abstract

A dynamically adjusted strategy between mean-reversion and momentum (2008, 2009). Stocks are grouped together using PCA. The idiosyncratic returns is calculated by comparing the returns of the stock to the returns of the entire group. This residual return often oscillates around a long-term mean. This strategy is dollar neutral and have high turnover. The medium-term momentum strategy trade the 9 sector ETFs, based on technical trading rules. Dynamic allocation was done between the 11 strategies, with rebalancing at the end of each month. Out of sample IR of 2.27, with beta 35%#### Equity mean reversion model

The decomposition of the stock returns is given by $$r_t = \alpha + \sum_{j=1}^n \beta_j F_t + \epsilon_t.$$ PCA of the normalized returns (after data centering and normalization in 252 day moving window) is used and the first 12 factors are retained. The Eigenportfolio returns $F_{jt}$ are given by $\sum_i \frac{v^{(j)}_i}{\sigma_i}R_{it}$. We, further, neglect the drift in returns. The model we implement is $dX_t=k(m-X_t)dt+\sigma dW_t$. The mean reversion time is $\tau = 1/k$. Use stock with mean reversion within 20 days, and for the s-score $s=\frac{X_t-m}{\sigma_{eq}}$ at +1.25 go short and get out at +0.75 (similarly for long). Trading cost of 10 bps. The model is two-times levered per side or four-times levered gross (industry standard).#### Momentum strategy

S&P500 industry sector ETFs, S&P500 ETF and SPY. 60 and 5 day exponentially moving average is used. Signal long if 5d EMA is above 60d EMA for the previous 4 or more trading days. In all other scenarios the signal is short. There is no rebalancing the trade and 10 bps cost assumed.#### In-sample analysis

2005-2007 in sample show mean-reversion strategy being much better than momentum with an IR of 1.28. The equally weighted strategy has an IR of 0.49.#### Optimization and out-of-sample results

There are returns to be made by dynamically optimizing the weights of different strategies. We can use Quadratic programming with the objective function and constraints as $$\min_x \frac{1}{2}x^THx+f^Tx \quad Ax \le b, \quad A_{eq}x=b_{eq}, \quad lb \le x \le ub.$$An important input into the process is lower and upper bounds for each variable. Using expected returns and allocation targets, we can customize the optimization process to best suit our portfolio specifications. The goal of this optimization is to maximize the Sharpe ratio of the diversified portfolio with a penalty for marginal risk contribution. The portfolio was optimized at the end of each month using the returns from the previous 252 trading days. There was no transaction cost used, except flat 10 bps per trade. The diversified strategy IR is 2.27 vs static allocation IR of 1.56, out-of-sample. The mean reversion strategy has a beta exposure. Optimization can be used to control beta, volatility and leverage as well to control drawdowns.

#### Conclusion

- Potential benefit of including both mean-reversion and momentum in portfolio.
- Did not hedge the beta risk using SPY, but can be done.
- Momentum signal using PCA eigen-portfolios is not apparent at individual stock level.
- Potentially greater alpha at finer time scales.
- Varying time-scales with signal decay for both momentum and mean reversion can be useful.

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