# Long Run Returns Predictability and Volatility with Moving Averages

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Model Specification

^{f}= the rate of three-month U.S. Treasury bill. A risky asset i pays dividend ${D}_{t},$ and has ${x}_{i}^{s}$ outstanding.

## 4. Empirical Analysis

**+0.085**with the annualized volatility

**0.167**. Table 1 shows that the annualized average volatility is

**0.115**when MA rules are used, thereby reducing by about 31% compared with the buy and hold returns volatility. Note that $1-\sqrt{0.48}=0.31$, and that the average US three month Treasury bill annualized yield has been

**+0.022**, indicating we invest randomly 48% of time in the DJIA index from 4 January 1988, and 52% of time in the risk-free rate, thereby producing $(0.085\times 0.48+0.022\times 0.52)=$

**+0.053**annually, on average, with

**0.115**volatility.

**+0.026**for the last 30 years. Therefore, the Sharpe ratio of random market timing (48% in stocks and 52% in the risk-free rate), with dividends, is

**0.38**; for MA200, it is

**0.32**; for MAW40, it is

**0.37**; for MA10, it is

**0.51**; for MAD5, it is

**0.47**; for MAT4, it is

**0.55**; for MAQ3, it is

**0.58**; and for MAC2, it is

**0.53**.

**0.255**a year ahead, the average MA trading rule volatility is reduced to

**0.125**, meaning a 51% reduction. When IGARCH is identified, the conditional volatility for 260 trading days ahead is given by:

**0.121**, which means a reduction of 28% compared to the buy and hold strategy returns volatility when MA rules are used. This indicates in random timing strategy that we invest (because $1-\sqrt{0.52}=0.28$) 52% of the time in the DJIA index and 48% in the risk-free rate producing $(0.085\times 0.52+0.022\times 0.48)=$

**+0.055**annually, on average, with

**0.121**volatility. The Sharpe ratio of random market timing (52% in stocks and 48% in the risk-free rate) with dividends is

**0.38**; for MA400

**0.38**; for MAW80

**0.46**; for MA19

**0.58**; for MAD10

**0.63**; for MAT7

**0.58**; for MAQ5

**0.55**; and for MAC4

**0.66**.

**0.129**, which means a reduction of about 23% compared with the buy and hold strategy returns volatility when MA rules are used. This indicates for the random timing strategy that we invest 60% of the time in the index and 40% in the risk-free rate, producing $(0.085\times 0.60+0.022\times 0.40)=$

**+0.060**annually before dividends, on average, with

**0.129**volatility. The Sharpe ratio of random market timing (60% in stocks and 40% in the risk-free rate) with dividends is

**0.42**; for MA600, it is

**0.55**; for MAW121, it is

**0.49**; for MA29, it is

**0.48**; for MAD14, it is

**0.50**; for MAT10, it is

**0.56**; for MAQ7, it is

**0.54**; and for MAC6, it is

**0.67**.

**0.133**, with a reduction of about 20% compared with the buy and hold strategy returns when MA rules are used. This indicates for the random timing strategy that we invest 64% of the time in the index and 36% in the risk-free rate, thereby producing $(0.085\times 0.64+0.022\times 0.36)=$

**+0.062**annually before dividends, on average, with

**0.133**volatility. The Sharpe ratio of random market timing (64% in stocks and 36% in the risk-free rate) with dividends is

**0.43**; for MA800, it is

**0.58**; for MAW161, it is

**0.61**; for MA39, it is

**0.57**; for MAD19, it is

**0.60**; for MAT13, it is

**0.59**; for MAQ10, it is

**0.55**; and for MAC8, it is

**0.58**.

**0.40**for the Sharpe ratio, on average. With the rolling window of 200 trading days, the Sharpe remains the same statistically. However, RW400 produces

**0.55**, RW600 produces

**0.54**, and the rolling window of 800 trading days produces

**0.58**, on average. Table 5, Table 6, Table 7 and Table 8 show that the sample size is 32 for the OLS estimates in Table 9, Table 10, Table 11 and Table 12. According to the small sample adjusted Jarque-Bera test, the residuals are normally distributed, with a p-value of 0.25. In view of the HAC consistent covariance matrix estimators, which are robust against alternative forms of misspecification in heteroscedasticity and autocorrelation, it is not necessary to provide further diagnostic checks.

^{2}value is

**0.34**, indicating that the size of the rolling window explains about one-third of the variations in the Sharpe ratios. The empirical results show that even the stochastic trend information from three years ago seems to improve the performance of the trading strategies. Moreover, the random timing (efficient market hypothesis) performance is beaten by MA trading strategies, using the long run rolling window. This indicates that stock returns are more predictable in the long run.

**0.40**, on average. However, using monthly frequencies, the Sharpe ratio increases to

**0.54**, every other month produces

**0.55**, every third month frequency produces

**0.57**, every fourth month produces

**0.56**, and every fifth month produces a Sharpe ratio of

**0.61**, on average. According to the small sample adjusted Jarque-Bera test, the residuals are normally distributed, with a p-value of 0.78.

**51%**increase in the Sharpe ratio, on average. The results suggest that using daily and weekly frequencies are practically useless, except when the widest rolling window is used. The adjusted R

^{2}value is

**0.38**, which indicates that the frequency explains 38% of the variations in the Sharpe ratios.

**0.26**for the $CS{R}_{i}$, on average. However, when the rolling window of 400 trading days is used, the $CS{R}_{i}$ increases to

**0.68**, RW600 produces

**0.50**, and the rolling window of 800 trading days produces

**0.53**, on average. The small sample adjusted Jarque-Bera test shows that the residuals are normally distributed, with a p-value of 0.48.

^{2}value is

**0.31**, indicating that the size of the rolling window explains about one-third of the variations in the $CS{R}_{i}$. However, the empirical findings suggest that even the stochastic trend information from three years ago seems to improve the statistical performance of the trading strategies. Moreover, the random timing (efficient market hypothesis) performance is beaten by MA trading strategies, with the long run rolling window increasing by 100%. This outcome indicates that the stock returns are indeed predictable in the long run.

**0.26**, on average. However, with daily, weekly, and monthly frequencies, the $CS{R}_{i}$ does not increase significantly (at the 5% level of significance). On the other hand, every other month produces

**0.51**, every third month frequency produces

**0.62**, every fourth month produces

**0.72**, and every fifth month produces a Sharpe ratio of

**0.64**, on average. The small sample adjusted Jarque-Bera test shows that the residuals are normally distributed, with a p-value of 0.72. The realized volatilities for the 800 trading days rolling window are given in Figure 4.

**173%**in the $CS{R}_{i}$, on average. The adjusted R

^{2}value is

**0.38**, indicating that the frequency explains 38% of the variations in the Sharpe ratios when the conditional volatilities are accommodated.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Realized volatilities in 200 trading days rolling window are used. The straight line is the 2% realized volatility bound.

**Figure 2.**Realized volatilities in 400 trading days rolling window are used. The straight line is the 2% realized volatility bound.

**Figure 3.**Realized volatilities in 600 trading days rolling window are used. The straight line is the 2% realized volatility bound.

**Figure 4.**Realized volatilities in 800 trading days rolling window are used. The straight line is the 2% realized volatility bound.

**Table 1.**The 200 trading-day rolling window, average annualized returns, volatilities, and conditional volatilities in Dow Jones Industrial Average (DJIA) index, 4 January 1988 to 31 December 2017.

Random Timing | MA200 Daily | MAW40 Weekly | MA10 Monthly | MAD5 Every Other Month | MAT4 Every Third Month | MAQ3 Every Fourth Month | MAC2 Every Fifth Month | |
---|---|---|---|---|---|---|---|---|

Returns | 0.053 | 0.045 | 0.051 | 0.066 | 0.064 | 0.075 | 0.078 | 0.072 |

Volatility | 0.115 | 0.110 | 0.111 | 0.111 | 0.116 | 0.120 | 0.118 | 0.119 |

Conditional volatility | 0.177 | 0.168 | 0.150 | 0.143 | 0.128 | 0.100 | 0.078 | 0.110 |

**Table 2.**The 400 trading-day rolling window, average annualized returns, volatilities, and conditional volatilities in DJIA index, 4 January 1988 to 31 December 2017.

Random Market Timing | MA400 Daily | MAW80 Weekly | MA19 Monthly | MAD10 Every Other Month | MAT7 Every Third Month | MAQ5 Every Fourth Month | MAC4 Every Fifth Month | |
---|---|---|---|---|---|---|---|---|

Returns | 0.055 | 0.053 | 0.063 | 0.077 | 0.085 | 0.076 | 0.080 | 0.087 |

Volatility | 0.121 | 0.118 | 0.118 | 0.119 | 0.122 | 0.125 | 0.125 | 0.123 |

Conditional volatility | 0.184 | 0.100 | 0.098 | 0.097 | 0.101 | 0.098 | 0.090 | 0.099 |

**Table 3.**The 600 trading-day rolling window, average annualized returns, volatilities, and conditional volatilities in DJIA index, 4 January 1988 to 31 December 2017.

Random Market Timing | MA600 Daily | MAW121 Weekly | MA29 Monthly | MAD14 Every Other Month | MAT10 Every Third Month | MAQ7 Every Fourth Month | MAC6 Every Fifth Month | |
---|---|---|---|---|---|---|---|---|

Returns | 0.060 | 0.078 | 0.068 | 0.068 | 0.072 | 0.079 | 0.077 | 0.093 |

Volatility | 0.129 | 0.130 | 0.127 | 0.129 | 0.130 | 0.129 | 0.131 | 0.130 |

Conditional volatility | 0.198 | 0.135 | 0.189 | 0.221 | 0.229 | 0.103 | 0.101 | 0.129 |

**Table 4.**The 800 trading-day rolling window, average annualized returns, volatilities, and conditional volatilities in DJIA index, 4 January 1988 to 31 December 2017.

Random Timing | MA800 Daily | MAW161 Weekly | MA39 Monthly | MAD19 Every Other Month | MAT13 Every Third Month | MAQ10 Every Fourth Month | MAC8 Every Fifth Month | |
---|---|---|---|---|---|---|---|---|

Returns | 0.062 | 0.081 | 0.085 | 0.081 | 0.085 | 0.083 | 0.080 | 0.084 |

Volatility | 0.133 | 0.131 | 0.131 | 0.132 | 0.133 | 0.132 | 0.136 | 0.135 |

Conditional volatility | 0.204 | 0.138 | 0.136 | 0.144 | 0.140 | 0.175 | 0.152 | 0.148 |

**Table 5.**Rolling window of 200 trading days, Sharpe ratios, data frequencies, and Sharpe ratios with conditional volatilities (bold denotes significance).

Sharpe Ratio | Frequency | Sharpe Ratio with Conditional Volatility |
---|---|---|

0.38 | random timing | 0.25 |

0.32 | 200 | 0.21 |

0.37 | 40 | 0.28 |

0.51 | 10 | 0.39 |

0.47 | 5 | 0.43 |

0.55 | 4 | 0.65 |

0.58 | 3 | 0.88 |

0.53 | 2 | 0.57 |

**Table 6.**Rolling window of 400 trading days, Sharpe ratios, data frequencies, and Sharpe ratios with conditional volatilities (bold denotes significance).

Sharpe Ratio | Frequency | Sharpe Ratio with Conditional Volatility |
---|---|---|

0.38 | random timing | 0.25 |

0.38 | 400 | 0.45 |

0.46 | 80 | 0.56 |

0.58 | 19 | 0.71 |

0.63 | 10 | 0.76 |

0.58 | 7 | 0.69 |

0.55 | 5 | 0.79 |

0.66 | 4 | 0.79 |

**Table 7.**Rolling window of 600 trading days, Sharpe ratios, data frequencies, and Sharpe ratios with conditional volatilities (bold denotes significance).

Sharpe Ratio | Frequency | Sharpe Ratio with Conditional Volatility |
---|---|---|

0.42 | random timing | 0.27 |

0.55 | 600 | 0.53 |

0.49 | 121 | 0.33 |

0.48 | 29 | 0.27 |

0.50 | 14 | 0.29 |

0.56 | 10 | 0.70 |

0.54 | 7 | 0.70 |

0.67 | 6 | 0.67 |

**Table 8.**Rolling window of 800 trading days, Sharpe ratios, data frequencies, and Sharpe ratios with conditional volatilities (bold denotes significance).

Sharpe Ratio | Frequency | Sharpe Ratio with Conditional Volatility |
---|---|---|

0.43 | random timing | 0.28 |

0.58 | 800 | 0.55 |

0.61 | 161 | 0.59 |

0.57 | 39 | 0.53 |

0.60 | 19 | 0.57 |

0.59 | 13 | 0.44 |

0.55 | 10 | 0.49 |

0.58 | 8 | 0.53 |

Coefficients | Heteroskedasticity and AutoCorrelation (HAC) | t-Values | |
---|---|---|---|

Constant | 0.403 | 0.012 | 32.5 |

RW200 | 0.073 | 0.048 | 1.52 |

RW400 | 0.146 | 0.038 | 3.84 |

RW600 | 0.139 | 0.027 | 5.16 |

RW800 | 0.180 | 0.013 | 13.9 |

Coefficients | HAC | t-Values | |
---|---|---|---|

Constant | 0.403 | 0.013 | 30.6 |

Daily | 0.055 | 0.055 | 1.01 |

Weekly | 0.080 | 0.046 | 1.76 |

Monthly | 0.133 | 0.028 | 4.79 |

Every other month | 0.148 | 0.041 | 3.62 |

Every third month | 0.168 | 0.016 | 10.8 |

Every fourth month | 0.153 | 0.018 | 8.31 |

Every fifth month | 0.208 | 0.027 | 7.58 |

Coefficients | HAC | t-Values | |
---|---|---|---|

Constant | 0.262 | 0.008 | 34.9 |

RW200 | 0.225 | 0.115 | 1.95 |

RW400 | 0.416 | 0.060 | 6.96 |

RW600 | 0.237 | 0.087 | 2.70 |

RW800 | 0.266 | 0.018 | 14.8 |

Coefficients | HAC | t-Values | |
---|---|---|---|

Constant | 0.262 | 0.008 | 32.9 |

Daily | 0.172 | 0.072 | 2.37 |

Weekly | 0.174 | 0.078 | 2.24 |

Monthly | 0.213 | 0.094 | 2.26 |

Every other month | 0.248 | 0.101 | 2.45 |

Every third month | 0.361 | 0.061 | 5.95 |

Every fourth month | 0.454 | 0.084 | 5.39 |

Every fifth month | 0.379 | 0.056 | 6.74 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Chang, C.-L.; Ilomäki, J.; Laurila, H.; McAleer, M.
Long Run Returns Predictability and Volatility with Moving Averages. *Risks* **2018**, *6*, 105.
https://doi.org/10.3390/risks6040105

**AMA Style**

Chang C-L, Ilomäki J, Laurila H, McAleer M.
Long Run Returns Predictability and Volatility with Moving Averages. *Risks*. 2018; 6(4):105.
https://doi.org/10.3390/risks6040105

**Chicago/Turabian Style**

Chang, Chia-Lin, Jukka Ilomäki, Hannu Laurila, and Michael McAleer.
2018. "Long Run Returns Predictability and Volatility with Moving Averages" *Risks* 6, no. 4: 105.
https://doi.org/10.3390/risks6040105