*3.2. Risk-Adjusted Performances*

The MAP results show that the MA timing strategy outperforms the buy-and-hold strategy. However, we observe that generally, portfolios with higher returns have higher standard deviations, which suggests that their risk (or total risk) levels are di fferent. Therefore, a natural research question is whether the return di fferences are due to exposure to risks. In this sub-section, we concentrate on the risk-adjusted profitability of MAP and TLS using three asset pricing models: the CAPM, FF3F, and LIQ4F models. We use MAP and *TLSMAP*,*t*,*<sup>L</sup>* as dependent variables and run time-series regressions against the risk factors. The results are presented in Table 2.

If the portfolio average return, *MAPj*,*t*,*<sup>L</sup>* is not fully explained by the risk factors, we expect <sup>α</sup>*j*,*<sup>L</sup>* to be significantly di fferent from zero. Otherwise, it suggests that the positive additional return found in the MA strategy is artificial after taking exposure to well-documented risk factors into consideration.


**Table 2.** Time-series regressions with the CAPM and LIQ4F models.

The sample stocks are sorted in ascending order by their BM ratios and assigned into decile portfolios each year. The equally-weighted portfolio daily returns are then obtained. MAP is the difference between the MA(20) strategy and the buy-and-hold strategy. The regression models are *MAPj*,*t*,*<sup>L</sup>* = <sup>α</sup>*j*.*<sup>L</sup>* + β*j*,*L*,*MKTRMKT*,*<sup>t</sup>* + <sup>ε</sup>*j*,*t*,*<sup>L</sup>* and *MAPj*,*t*,*<sup>L</sup>* = <sup>α</sup>*j*.*<sup>L</sup>* + β*j*,*L*,*MKTRMKT*,*<sup>t</sup>* + β*j*,*L*,*SMBRSMB*,*<sup>t</sup>* + β*j*,*L*,*HMLRHML*,*<sup>t</sup>* + β*j*,*L*,*LIQRLIQ*,*<sup>t</sup>* + <sup>ε</sup>*j*,*t*,*L*, where *RMKT*,*<sup>t</sup>* is market excess returns; *RSMB*,*<sup>t</sup>* is the size factor; *RHML*,*<sup>t</sup>* is the book-to-market factor; *RLIQ*,*t*. is the liquidity risk factor proxied by turnover ratio; <sup>ε</sup>*j*,*t*,*<sup>L</sup>* is an error term assumed to have a zero mean and to be uncorrelated with all other explanatory variables; and the factor sensitivities or loadings, β*j*,*L*,*MKT*, β*j*,*L*,*SMB*, β*j*,*L*,*HML*, and β*j*,*L*,*LIQ*, are the slope coefficients for the factors, respectively. <sup>α</sup>*j*.*<sup>L</sup>* is the intercept of the regression. *t*-test statistics are presented in the parentheses.

Table 2 provides the results of the equally-weighted portfolios on the CAPM and LIQ4F models. Because the results of FF3F are similar to those from the LIQ4F model, we do not report them to save space. All of the alphas of the 10 portfolios in the two models are significantly different from zero at the 5% level, which suggests that the profits generated from the timing strategy are not fully captured by these well-known risk factors. The alphas of the CAPM model range from 4.65 to 8.18 points, while the alphas of the LIQ4F model range from 6.57 to 9.08 points.

The betas of the market excess return and size factors are all negative and highly significant, which can partially explain why the alpha becomes larger than the unadjusted return of the MAPs. Half of the coefficients of the HML factor are positive and half are negative. The coefficients of the LIQ factor are all significantly positive.

The alphas of the *TSLMAP* in the CAPM and LIQ4F models are 13.24 and 15.14 points, respectively, which are larger than the unadjusted *TSLMAP* (11.23 points). Interestingly, while the adjusted R<sup>2</sup> for most of the deciles in the two models are larger than 40%, the adjusted R<sup>2</sup> for the *TSLMAP* is extremely low. Overall, the low R<sup>2</sup> and significant alphas indicate that the CAPM and LIQ4F models cannot explain the excess profit from the new BM strategy.

### *3.3. Components of Strategies*

We now establish that the moving average technical analysis can help us obtain better returns than the buy-and-hold strategy. We are curious about how exactly the excess returns are created. To answer this question, we consider the operation of the MA timing strategy. We know that we hold the underlying portfolio when the signal *Pj*,*t*−<sup>1</sup> > *Aj*,*t*−1,*<sup>L</sup>* appears and hold the risk-free asset otherwise. Comparing this with the buy-and-hold strategy, a difference appears only when *Pj*,*t*−<sup>1</sup> < *Aj*,*t*−1,*L*. This difference is between the return generated from the underlying portfolio and the risk-free asset. Therefore, we can express the MAP return in another way:

$$MAP\_{j,t,L} = \begin{cases} 0, & \text{if } P\_{j,t-1} > A\_{j,t-1,L}, \\ R\_{f,t} - R\_{j,t}, & \text{otherwise}, \end{cases} \tag{9}$$

To examine the difference, we separate the sample into two parts: when *Pj*,*t*−<sup>1</sup> > *Aj*,*t*−1,*<sup>L</sup>* (buy signal) and when *Pj*,*t*−<sup>1</sup> ≤ *Aj*,*t*−1,*<sup>L</sup>* (sell signal). We calculate the proportion of days when *Rj* is higher (or lower) than *Rf* for the buying and selling signal states. The results are shown in Table 3.


**Table 3.** Components of strategies.

We separate the MA(20) portfolio sample into two parts. One is when *Pj*,*t*−<sup>1</sup> > *Aj*,*t*−1,*<sup>L</sup>* which indicates a buy signal and the other is when *Pj*,*t*−<sup>1</sup> ≤ *Aj*,*t*−1,*<sup>L</sup>* which suggests a sell signal. We then compare the return of the BM portfolio and the risk-free rate. We calculate the proportion of each situation for buying and selling signal states.

We find that the proportions of the state in which *Rj*,*<sup>t</sup>* > *Rf*,*<sup>t</sup>* are all bigger than 50%, ranging from 56.71% to 60.15%, when *Pj*,*t*−<sup>1</sup> > *Aj*,*t*−1,*L*. However, this does not affect the MAP. In addition, the proportions of the state in which *Rj*,*<sup>t</sup>* < *Rf*,*<sup>t</sup>* are also all bigger than 50% when *Pj*,*t*−<sup>1</sup> ≤ *Aj*,*t*−1,*L*. The highest is 53.97% in decile 6 and the lowest is 50.18% in decile 10. Although the successful rates are not very high (mostly between 50% and 60%), all of them are bigger than 50%, which leads to better performance by MA strategy when a selling signal emerges. These results show that the MA signal is useful in generating positive excess returns and help explain why the moving average can outperform the buy-and-hold strategy.

## *3.4. Alternative Lag Lengths*

In this sub-section, we investigate how the returns of the MAP and TLS change with the lag lengths. We conduct the moving average timing strategy with 5-day, 10-day, 20-day, 50-day, 100-day, and 200-day timing signals. The results are illustrated in Figure 1.

**Figure 1.** (**a**) For each BM portfolio, the moving average timing strategy is imposed with 5-day, 10-day, 20-day, 50-day, 100-day, and 200-day timing signals. The MAP returns for the decile portfolios are then averaged cross-sectionally. The raw returns of the average MAP, risk-adjusted alphas of CAPM and LIQ4F models are plotted; (**b**) The proposed technical analysis enhanced BM strategy is imposed with 5-day, 10-day, 20-day, 50-day, 100-day, and 200-day timing signals. The raw returns of the average TLS strategy, risk-adjusted alphas of CAPM and LIQ4F models are plotted.

Figure 1a plots the average return of MAPs across the decile portfolios with different lag lengths. The 20-day average MAP return is the highest (in fact, although it is not plotted, the 20-day MAP is the highest for each decile portfolio). The 5-day MAP and 10-day MAP are also significantly positive but are slightly smaller than that of the 20-day signals. The returns with signals that are longer than 20 days start to decline and become progressively smaller, and the results of the 100-day lag even become negative. The CAPM and LIQ4F model-adjusted MAP alphas present similar patterns to that of the unadjusted returns.

The returns of the proposed trading strategy, TLS, are plotted in Figure 1b. Again, the 20-day lag strategy generates the highest raw and risk-adjusted profits. The MA(5) and MA(10) strategies also perform well, as their returns are close to that of MA(20). However, when lags longer than 50 days are used, the strategy return drops sharply, almost reaching zero. We conjecture that this is because the Chinese stock market is highly volatile. A shorter moving average signal can capture the fluctuations in information more accurately than a longer moving average signal. The signals with long lag length may miss important information changes, and so the abnormal return produced by these signals becomes flat and small.

The results seem to sugges<sup>t</sup> that a shorter moving average of up to 50 days can produce significant and meaningful positive returns. However, the 20-day lag signal provides the best result by capturing the most information regarding past prices. To sum up, lag length can influence the performance of the moving average timing strategy, and we need to choose the best one for China's stock market.
