Next Article in Journal
A Loss-Averse Newsvendor with Cap-and-Trade Carbon Emissions Regulation
Next Article in Special Issue
Financial Risk Measurement and Prediction Modelling for Sustainable Development of Business Entities Using Regression Analysis
Previous Article in Journal
Eco-Efficiency Assessment and Food Security Potential of Home Gardening: A Case Study in Padua, Italy
Previous Article in Special Issue
An Improvement of Gain-Loss Price Bounds on Options Based on Binomial Tree and Market-Implied Risk-Neutral Distribution
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

Market Timing with Moving Averages

1
Faculty of Management, University of Tampere, FI-33014 Tampere, Finland
2
Department of Finance, Asia University, Taichung City 413, Taiwan
3
Discipline of Business Analytics University of Sydney Business School, Sydney NSW 2006, Australia
4
Econometric Institute, Erasmus School of Economics, Erasmus University Rotterdam, 3062 PA Rotterdam, The Netherlands
5
Department of Economic Analysis and ICAE, Complutense University of Madrid, 28040 Madrid, Spain
6
Institute of Advanced Sciences Yokohama National University, Yokohama 240-8501, Japan
*
Author to whom correspondence should be addressed.
Sustainability 2018, 10(7), 2125; https://doi.org/10.3390/su10072125
Submission received: 7 June 2018 / Revised: 18 June 2018 / Accepted: 20 June 2018 / Published: 22 June 2018
(This article belongs to the Special Issue Risk Measures with Applications in Finance and Economics)

Abstract

:
Consider using the simple moving average (MA) rule of Gartley to determine when to buy stocks, and when to sell them and switch to the risk-free rate. In comparison, how might the performance be affected if the frequency is changed to the use of MA calculations? The empirical results show that, on average, the lower is the frequency, the higher are average daily returns, even though the volatility is virtually unchanged when the frequency is lower. The volatility from the highest to the lowest frequency is about 30% lower as compared with the buy-and-hold strategy volatility, but the average returns approach the buy-and-hold returns when frequency is lower. The 30% reduction in volatility appears if we invest randomly half the time in stock markets and half in the risk-free rate.
JEL:
G32; C58; C22; C41; D23

1. Introduction

According to the standard investing separation theorem of Tobin [1], investors allocate investments between risk-free and risky assets. If the risk-free rate is low (high), the investors shift their wealth to (from) the risky assets. Fama [2] divided forecasters into two categories, namely macro forecasters (or market timers) and micro forecasters (or security analysts), who try to forecast individual stock returns relative to the market returns.
Merton [3] defined a market timer to forecast when stocks will outperform (underperform) the risk-free asset, indicating that, when r t m > r t f ( r t m < r t f ) , where r t m is average stock market returns, r t f is the risk-free asset, r t i = r t f + β i ( r t m r t f ) + ε t i , r t i is the return for individual stock i included in the market portfolio m , β i is a positive parameter, and E [ ε t i | r t m ] = E [ ε t i ] . That is, a market timer only forecasts the statistical properties of r t m r t f , indicating that their forecasts contain only the differential performance among individual stocks arising from systematic risk in the markets.
Merton [3] showed theoretically that, when investors have heterogeneous beliefs and imperfect information, the value of a random market timing forecast is zero, and if the forecast variable is distributed independently or the forecast is based on public information, its value is zero, too. In fact, Merton showed that the maximum value of skilled market timing is the value of the protective put against buy-and-hold strategy.
Henriksson and Merton [4] presented an empirical procedure whereby correct forecasts can be analyzed statistically. However, if it is assumed that ε t i follows an approximate normal distribution, this leads to the Capital Asset Pricing Model (CAPM) of Sharpe [5], and Lintner [6].
The purpose of the paper is to detect whether the frequency used in calculating the MA affects the performance of the trading rule. We use a large sample with more than eight million observations for robustness of the empirical results, and a simple MA rule for the timing aspect for individual Dow Jones Industrial Average (DJIA) stocks with different frequencies. We use a simple MA rule for the timing aspect for individual Dow Jones Industrial Average (DJIA) stocks with different frequencies. Zhu and Zhou [7] showed analytically that MA trading rules, as a part of asset allocation rules, can outperform standard allocation rules when stock returns are partly forecastable. The standard rule means investing a fixed proportion of wealth in risky assets and the rest in risk-free assets, with the ratio determined by the risk tolerance of an investor. It is well known that MA is a widely used technical trading rule, which adds value for a risk averse investor if returns are predictable.
This is the well-known reward/risk (or mean-variance) principle in the spirit of Markowitz [8], Tobin [1], and Sharpe [5]. Zhu and Zhou [7] argued that the fixed allocation rule is not optimal if returns are forecastable by using the MA rule. Therefore, assuming that risk tolerance and the forecast performance of stock market returns are constant, the linear combination rule means that, when the MA rule suggests an uptrend (downtrend), the rule suggests that the total weight should be allocated to stock markets (the risk-free rate).
The empirical findings suggest a low volatility anomaly that might be explained by investors’ affection to high volatility, as suggested by Baker et al. [9], and noted in Ang et al. [10]. On the other hand, the reported predictability of risk premia (see, for example, Cochrane [11], and Fama [12]) can explain why, for instance, MA rules forecast better than using random highs and lows in the stock market (as noted in Jagannathan and Korajczyck [13]). The topic is important, as Friesen and Sapp [14], among others, reported that mutual fund investors had negative outcomes, on average, in their timing to invest and withdraw cash from US mutual funds from 1991 to 2004. Munoz and Vicente [15] reported similar results with more recent data in US markets.
The remainder of the paper is organized as follows. Section 2 provides a literature review, and alternative model specifications are presented in Section 3. The empirical analysis is conducted in Section 4, while Section 5 gives some concluding comments.

2. Literature Review

In efficient markets, investors earn above average returns only by taking above average risks (Malkiel [16]). Samuelson [17] conformed with Fama [2] by noting that market efficiency can be divided into micro and macro efficiency. The former concerns the relative pricing of individual stocks, and the latter, for markets as a whole. The CAPM by Sharpe [5], and Lintner [6] argues that beta is a proper definition for systematic risk for stock i, if unexplained changes in risk adjusted returns for the stock follow approximately normal distribution with zero mean.
Black [18] stated that the slope of the security market line (SML) is flatter if there exist restrictions in borrowing, that is, leverage constraints in the model. Starting from Black et al. [19], many studies have reported that the security market line is too flat in US stocks compared with the SML suggested by the CAPM version of Sharpe and Lintner.
Ang et al. [10], Baker et al. [20], and Frazzini and Pedersen [21] found that low-beta stocks outperform high-beta stocks statistically significantly. In fact, Frazzini and Pedersen reported that significant excess profits in US stocks can be achieved by shorting high-beta stocks and buying low-beta stocks with leverage, but that leverage constraints make them disappear. Using Black [18], investors often have leverage constraints, thereby making them place too much weight on risky stocks, which results in lower required return for high-beta stocks than would be justified by the Sharpe–Lintner CAPM.
Markowitz [8] defined portfolio risk simply as the volatility of portfolio returns. Clarke et al. [22] found that the volatility of stock returns contains potentially an additional risk factor with respect to systematic risk that can be defined in the betas of CAPM by Sharpe and Lintner. Moreover, Ang et al. [10] reported that the total volatility of international stock market returns is highly correlated with US stock returns, thereby suggesting a common risk factor for US stocks.
Baker et al. [9] suggested that the low-volatility anomaly is due to investor irrational behavior, mainly because an average fund manager seeks to beat the buy-and hold strategy by overinvesting in high-beta stocks. The explanations include preference for lotteries (Barberis and Huang [23]; Kumar [24]; Bali et al. [25]), overconfidence (Ben-David et al. [26]), and representativeness (Daniel and Titman [27]), which means that people assess the probability of a state of the world based on how typical of that state the evidence seems to be (Kahneman and Tversky [28]).
Baker et al. [9] argued that the anomality is also related to the limits of arbitrage (see also Baker and Wurgler [29]). In fact, the extra costs of shorting prevent taking advantage of overpricing (Hong and Sraer [30]). More importantly, Li et al. [31] reported that the excess returns of low-beta portfolios are due to mispricing in US stocks, indicating that the low-volatility anomaly does not exist because of systematic risk by some rational, stock specific volatility risk factor. They tested the low-volatility anomaly with monthly data from January 1963 to December 2011 in NYSE, NASDAQ, and AMEX stocks.
Market timing is closely related to technical trading rules. Brown and Jennings [32] showed theoretically that using past prices (e.g., the MA rule of Gartley [33]) has value for investors, if equilibrium prices are not fully revealing, and signals from past prices have some forecasting qualities. More importantly, Zhu and Zhou [7] indicated that the MA rules are particularly useful for asset allocation purposes among risk averse investors, when markets are forecastable (quality of signal).
Moskowitz et al. [34] argued that there are significant time series momentum (TSM) effects in financial markets that are not related to the cross-sectional momentum effect (Jegadeesh and Titman [35]). However, TSM is closely related to MA rules, since it gives a buy (sell) signal according to some historical price reference points, whereas MA rules give a buy (sell) signal, when the current price moves above (below) the historical average of the chosen calculated rolling window measure.
Starting from LeRoy [36] and Lucas [37], the literature in financial economics states that financial markets returns in efficient markets are partly forecastable, when investors are risk averse. This leads to the time-varying risk premia of investors, as noted by Fama [12]. For example, Campbell and Cochrane [38] presented a consumption-based model, which indicates that when the markets are in recession (boom), risk averse investors require larger (smaller) risk premium for risky assets. More importantly, Cochrane [11] noted that the forecastability of excess returns may lead to successful market timing rules.
Brock et al. [39] tested different MA lag rules for US stock markets, and found that they gain profits compared with holding cash. On the other hand, Sullivan et al. [40] found that MA rules do not outperform the buy-and-hold strategy, if transaction costs are accounted for. Allen and Karjalainen [41] used a genetic algorithm to develop the best ex-ante technical trading rule model using US data, and found some evidence of outperforming the buy-and-hold strategy. Lo et al. [42] found that risk averse investors benefit from technical trading rules because they reduce volatility of the portfolio without giving up much returns when compared against the buy-and-hold strategy.
More recently, Neely et al. [43] used monthly data from January 1951 to December 2011, and reported that MA rules forecast the risk premia in US stock markets statistically significantly. Marshall et al. [44] found that MA rules give an earlier signal than TSM, suggesting better returns for MA rules, but they both work best with outside of large market value stocks.
Moskowitz et al. [34] used monthly data from January 1965 to December 2009, and reported that TSM provides significant positive excess returns in futures markets. However, Kim et al. [45] reported that these positive excess returns produced by TSM are due to the volatility scaling factor used by Moskowitz et al.

3. Model Specification

Consider an overlapping generation economy with a continuum of young and old investors [ 0 , 1 ] . A young risk-averse investor j invests their initial wealth, w t j , in infinitely lived risky assets i = 1 , 2 , I , and in risk-free assets that produce the risk-free rate of return, rf. A risky asset i pays dividend D t i , and has x i s outstanding. Assuming exogenous processes throughout, the aggregate dividend is Dt.
A young investor j maximizes their utility from old time consumption through optimal allocation of initial resources w t j , between risky and risk-free assets:
max x t j ( E t ( P t + 1 + D t + 1 ) P t ( 1 + r f ) ) ν j 2 x j 2 σ 2 s . t . x t j P t w t j
where E t is the expectations operator, P t is the price of one share of aggregate stock, ν j is a constant risk-aversion parameter for investor j , σ 2 is the variance of returns for the aggregate stock, and x t j is the demand of risky assets for an investor j. The first-order condition is:
E t ( P t + 1 + D t + 1 ) P t ( 1 + r f ) ν j x t j σ 2 = 0 ,
which results in optimal demand for the risky assets:
x t j = E t ( ( P t + 1 + D t + 1 ) / P t ) ( 1 + r f ) ν j σ 2
Suppose that an investor j is a macro forecaster who allocates their initial wealth, w t j , between risky stocks and risk-free assets according to their forecast about the return of the risky alternative. Then, Equation (1) says that the investor invests in the risky stocks only if the numerator on the right hand side is positive.

4. Empirical Analysis

This section presents the empirical results from seven frequencies for the (MA) trend-chasing rules. The data consist of 29 companies included in the Dow Jones Industrial Average (DJIA) index in January 2018. The trading data (daily closing prices) cover 30 years from 1 January 1988 to 31 December 2017. Choosing the current DJIA companies for the last 30 years creates a “survivor bias” in the buy-and-hold results. However, this should not be an issue, as we intend to compare the performance of the alternative MA frequency rules.
The rolling window is 200 trading days. The first rule is to calculate MA in every trading day; the second frequency takes into account every 5th trading day (thereby providing a proxy for the weekly rule); the third frequency takes into account every 22th trading day (proxy for the monthly rule); the fourth rule is to calculate MA for every 44th trading day (proxy for every other month); the fifth rule takes into account every 66th trading day (proxy for every third month); the sixth rule takes into account every 88th trading day (proxy for every fourth month); and the seventh rule takes into account every 100th trading day (proxy for every fifth month).
For the 29 DJIA companies, 26 of them have daily stock data available from 27 March 1987, thereby giving 4 January 1988 as the first trading day. The data for Cisco are available from 12 February 1990, for Goldman Sachs from 4 May 1999, and for Visa from 19 March 2008. There are 217,569 observations of daily returns from DJIA stocks. Thus, there are 217,569 × 9 = 1,958,121 daily returns for the first three frequencies (rules), 217,569 × 4 = 870,276 daily returns for the fourth rule, 217,569 × 3 = 652,707 daily returns for the fifth rule, 217,569 × 2 = 435,138 daily returns for the sixth rule, and 217,569 daily returns for the seventh rule.
The trading rule for all cases is to use a simple crossover rule. When the trend-chasing MA turns lower (higher) than the current daily closing price, we invest the stock (three-month US Treasury Bills) at the closing price of the next trading day. Thus, the trading rule provides a market timing strategy where we invest all wealth either in stocks (separately, every stock included in DJIA), or to the risk-free asset (three-month U.S. Treasury bill), where the moving average rule advices the timing.
At the first frequency (every trading day), we calculate daily returns for MA200, MA180, MA160, MA140, MA120, MA100, MA80, MA60, and MA40. For example, MA200 is calculated as:
( P t 1 + P t 2 + + P t 200 200 ) = X t 1
At the lowest frequency, where every 100th daily observation is counted, MAC2 is calculated as:
( P t 1 + P t 100 2 ) = X t 1
If X t 1 < P t 1 , we buy the stock at the closing price, P t , thereby giving daily returns as
R t + 1 = ln ( P t + 1 P t )
Table A1, Table A2, Table A3, Table A4, Table A5, Table A6 and Table A7 in Appendix A show that the annualized average log returns of MA200−MA40 are +0.053 after transaction costs (with 0.1% per change of position). Recall that there are 200 closing day prices in the rolling window MA200, whereas MA40 means that there are 44 closing day prices in the window. The respective log returns for MAW40−MAW8 (weekly) are +0.063; for MA10−MA2 (monthly) +0.071; for MAD5−MAD2 (every other month) +0.078; for MAT4−MAT2 (every third month) +0.084; for MAQ3−MAQ2 (every fourth month) +0.094; and for MAC2 (every fifth month) +0.088 after transaction costs.
Table A1, Table A2, Table A3, Table A4, Table A5, Table A6 and Table A7 show that, as the frequency decreases until every fourth month frequency (MAQ3−MAQ2), average returns tend to increase, and decrease thereafter. In comparison, the biased buy-and-hold strategy produces +0.117 with equal weights among all DJIA stocks, and with 0.295 annual volatility. A random investment (half the time in the risk-free rate, and half in the equally weighted portfolio from 4 January 1988) produces ( 0.117 × 0.5 + 0.022 × 0.5 ) = +0.070 annually, on average, with ( 1 0.5 = 0.293 ) = 29.3 % reduction in volatility, indicating 0.209 annual volatility for that portfolio.
The data are dividend excluded, but the average annual dividend yield in DJIA stocks over the last thirty years has been +0.026, so that the biased buy and hold strategy produces +0.143 annually with equal weights among DJIA stocks before taxes. Thus, the random investment strategy produces +0.083 annually, with survivor bias.
Appendix A (namely the second column of Table A1, Table A2, Table A3, Table A4, Table A5, Table A6 and Table A7) also reports the annualized average log returns calculated in the largest sample (full 200 observations) in every category: MA200 +0.065; MAW40 +0.073; MA10 +0.079; MAD5 +0.083; MAT4 +0.089; MAQ3 +0.091; and MAC2 +0.088 after transaction costs and before dividends. Adding +0.013 produces after dividends and before taxes: MA200 +0.078; MAW40 +0.086; MA10 +0.092; MAD5 +0.096; MAT4 +0.102; MAQ3 +0.104; and MAC2 +0.101. These results imply that starting from every fifth trading day frequency, a macro forecaster beats the buy and hold strategy in returns.
Figure 1 illustrates the effects of frequency on the returns to volatility ratio (the second column in Appendix A, Table A1, Table A2, Table A3, Table A4, Table A5, Table A6 and Table A7).
In Figure 1, the straight line illustrates the return to volatility ratio of portfolios, where wealth is randomly invested in combinations of the three-month Treasury Bill (risk-free rate), with stocks included in the DJIA between 4 January 1988 and 31 December 2017. The red crosses represent the average return/volatility points calculated in the 200-day rolling window with the following frequencies: daily, every five days, every 22 days, every 44 days, every 66 days, every 88 days, and every 100 days (with only the most observations in each frequency giving 200, 40, 10, 5, 4, 3, and 2 observations). The red crosses plot a convex curve that deviates increasingly from the straight return to volatility ratio line, thereby symbolizing superior portfolio efficiency.
Table A8, Table A9, Table A10, Table A11, Table A12, Table A13 and Table A14 in Appendix B show that the annualized volatility of daily returns read, on average: MA200−MA40 0.2044; MAW40−MAW8 0.205; MA10−MA2 0.2091; MAD5−MAD2 0.213; MAT4−MAT2 0.219; MAQ3−MAQ2 0.221; and MAC2 0.218. Thus, there is virtually no difference between the MA frequencies, while the biased buy-and-hold strategy produces 0.295.
Figure 1 presents the volatilities calculated in the largest sample (full 200 day rolling window in every category, the second column in Table A8, Table A9, Table A10, Table A11, Table A12, Table A13 and Table A14). They read MA200 0.207; MAW40 0.208; MA10 0.211; MAD5 0.213; MAT4 0.218; MAQ3 0.215; and MAC2 0.218 after transaction costs. Investing randomly half of the time in the risk-free rate and the other half in the equally weighted portfolio, produces 0.209. Thus, the difference between the annual volatilities produced in profitable market timing MA rules (MA10−MAC2) and random market timing (half and half) ranges from 0.009 to 0.002.
In Figure 2, the straight line again presents the return to volatility ratio of portfolios with random investment in the risk-free rate and the stocks in DJIA between 4 January 1988 and 31 December 2017. The red crosses plot the average return to volatility ratios, calculated by using a 200-day rolling window, with the following frequencies: daily, every five days, every 22 days, every 44 days, every 66 days, every 88 days, and every 100 days. The averages of every lag are reported in Table A1, Table A2, Table A3, Table A4, Table A5, Table A6, Table A7, Table A8, Table A9, Table A10, Table A11, Table A12, Table A13 and Table A14, and. Thus, all daily returns from Table A1, Table A2, Table A3, Table A4, Table A5, Table A6, Table A7, Table A8, Table A9, Table A10, Table A11, Table A12, Table A13 and Table A14 are included.
Comparing Figure 1 and Figure 2, it is clear that using the whole 200 daily observation windows in the MA rules produces more efficient results in market timing. That is, comparing the products of shorter and longer MA rule rolling windows, e.g., the last two monthly observations compared with ten monthly observations, average realized returns drop from +0.079 to +0.059 before dividends, while volatility remains approximately unchanged (from 0.211 to 0.207). This suggests that, in both cases, about half and half is invested in the equally-weighted DJIA portfolios and in the risk-free rate, and the MA rules advise the timing. More importantly, Table A8, Table A9, Table A10, Table A11, Table A12, Table A13 and Table A14 in Appendix B show that the range in volatilities with all MA rules varies between 0.202 and 0.227 (with 0.02 difference), whereas Table A1, Table A2, Table A3, Table A4, Table A5, Table A6 and Table A7 in Appendix A show that realized returns vary between 0.096 and 0.033 before dividends (with 0.063 difference).
These results indicate that a macro market timing with 200 days rolling window produces a reduction in volatility from 0.295 (the buy-and hold) to between 0.207 and 0.218, but the average annualized returns (dividends included) tend to rise as the MA frequency falls (+0.078 with all 200 observations to +0.104 with every fourth month observations). Thus, the results indicate that MA market timing finds long term stochastic trends more efficiently than short term stochastic trends.
The Sharpe ratio of random market timing (half and half) with dividends is 0.292; for MA200 0.271; for MAW40 0.308; for MA10 0.332; for the MAD5 0.347; for MAT4 0.370; for MAQ3 0.381; and for MAC2 it is 0.362.
Figure 3 shows that when the volatility changes 1% in the DJIA stocks, then the average returns change is 0.39%. Figure 1 and Figure 2 suggest that the theoretical change should be such that, when the volatility changes 1%, the average returns change is 0.50%, suggesting a flatter SML line in the data. This suggests strongly that DJIA investors have overweight high-beta stocks in the last 30 years.
It is obvious that transaction costs are crucial in MA performance. In the above calculations, the transaction costs are 0.1% per transaction from current wealth. Table A15 and Table A16 in Appendix C report the transaction costs for the MA200−MA40 and MA10−MA2 rules. In the MA200−MA40 rules, the average annualized transaction costs are 0.0133, such that the rules have about 13 changes in positions per year. Meanwhile, for the MA10−MA2 rules, the average annualized transaction costs are 0.0032, suggesting about three changes in positions per year.
Allen and Karjalainen [41] gave reasons for using a cost of 0.2% per transaction in their sample, but since technological progress has reduced transaction costs since the mid-1990s, 0.1% per transaction should be fair, on average. Nevertheless, a trial with 0.2% transaction costs shows that, for example, the average annualized daily returns become 0.0403 for the MA200−MA40 rules, and 0.0674 for the MA10−MA2 rules. Note that the returns grow 67%, on average, for the MA10−MA2 rules (with about the same volatility) compared with costs of 0.1% per transaction.
Note that the model prohibits short selling since we only have long positions in stocks or investing in the risk-free rate. Then, the limits of arbitrage argument of Baker et al. [9] are consistent with our results.

5. Concluding Remarks

The analysis suggests that a macro forecaster can obtain higher returns with equal volatility (30% below that of the buy-and-hold strategy) by reducing the frequency used in MA rules. The return to volatility ratio for risk-averse investors with MA market timing significantly outperforms the random benchmark strategy, when the frequency in the MA rules is reduced. This indicates that the forecasts become more accurate as the time frame becomes longer.
The results suggest that a flatter SML in the CAPM can be followed by the irrational preference of investors in high-beta stocks, as suggested by Baker et al. (2011) and Li et al. (2016), since the empirically efficient frontier of portfolios becomes flatter than the theoretically efficient SML (random timing) (see Figure 1). In other words, the empirical results suggests\that market timing with the few past observations (for example, every fourth month) in the past 200 rolling window daily prices, have produced significantly better returns to risk ratio for the portfolio of DJIA equally weighted stocks in the past 30 years than random timing. The finding points to the low-volatility anomaly.
One explanation for the results is that they are due to time-varying risk premiums. This is emphasized by Neely et al. (2014), who claimed that MA rules, in effect, forecast changes in the risk premium. If the results are rational products of time-varying risk premiums, the results suggest that investor sensitivity to risk must be extremely high, and their risk premium is larger (smaller) in downs (ups), as suggested by Campbell and Cochrane (1999). As volatility rises (decreases), usually in downs (ups), the results suggest that, when volatility is high, investors as a group tolerate significantly more risk (that is, volatility) than in calmer periods.
Consider the following numerical example: Assume that the risk premium is 0.08 in volatile downs, and 0.04 in calm ups, and the variance of returns is 0.09 in downs and 0.03 in ups. Then, the risk aversion coefficient must be 0.89 in volatile down periods, and 1.33 in calm up periods. As market timing with MA rules works better in longer periods with few observations, it seems to be more accurate in longer stochastic (up or down) trends.

Author Contributions

This paper is to be attributed in equal parts to the authors.

Funding

This research, for the part of the third author, was funded by the Australian Research Council and the Ministry of Science and Technology (MOST), Taiwan.

Acknowledgments

The authors are most grateful for the helpful comments and suggestions of George Tauchen. The authors also thank the two anonymous referees for their helpful comments and suggestions.

Conflicts of Interest

The authors declare no conflict of interest.

Appendix A

Table A1. Annualized daily returns of MA40–MA200, average annualized returns.
Table A1. Annualized daily returns of MA40–MA200, average annualized returns.
Buy and HoldMA200MA180MA160MA140MA120MA100MA80MA60MA40
3M0.0900.0420.0340.0170.0150.0190.0140.006−0.0096 × 10−4
American Express0.0940.0350.0370.0390.0550.0390.0420.0430.0410.008
Apple0.1570.1470.1450.1470.1420.1560.1490.1500.1460.164
Boeing0.1190.0880.0890.0600.0550.0610.0610.0580.0460.048
Caterpillar0.1000.0750.0790.0580.0580.0490.0340.0280.0390.025
Chevron0.0840.0050.0130.002−0.000−0.0000.003−0.01−0.025−0.05
Coca-Cola0.0990.0580.0550.0300.0350.0390.0270.0230.0090.003
Walt Disney0.1030.0720.0780.0790.0740.0770.0740.0760.0560.048
Exxon0.072−0.011−0.010−0.020−0.030−0.020−0.025−0.01−0.044−0.05
GE0.0520.0720.0710.0580.0390.0390.0330.0180.0139 × 10−4
Home Depot0.1900.1250.1160.1020.0920.0870.0760.0670.0680.058
IBM0.0550.0160.0290.0330.0280.0160.0210.0310.0290.048
Intel0.1340.0830.0820.0830.0730.0910.0820.0800.0770.078
Johnson & Johnson0.1130.0620.0580.0530.0420.0320.0440.0280.008−0.00
JP Morgan0.0900.0130.0140.0030.0100.0170.0130.0310.0380.025
McDonalds0.1140.0470.0480.0400.0440.0400.0350.0430.0300.018
Merck0.0630.0500.0480.0440.0320.0330.0290.0220.016−0.02
Microsoft0.1800.1170.1280.1050.1020.1040.0950.0900.0700.062
Nike0.1770.0870.0930.0850.1020.1080.1070.1190.1330.112
Pfizer0.0970.0590.0560.0430.0420.0520.0440.0400.0240.009
Procter & Gamble0.0950.0370.0450.0370.0360.0370.0290.0230.0040.017
Travellers0.0820.0360.0350.0380.0290.008−0.004−9 × 10−4−0.0010.006
United Technologies0.1130.0510.0570.0460.0590.0570.0490.0490.0410.017
United Health Group0.2520.1810.1820.1570.1470.1360.1300.1180.1250.076
Verizon0.043−0.017−0.020−0.010−0.000−0.020−0.020−0.02−0.029−0.02
Wal-Mart0.1130.0190.0160.0100.0120.0120.0160.0120.0200.024
Cisco0.2100.1980.1940.2100.2080.1980.2050.1520.0960.085
Goldman Sachs0.0610.0380.0290.0330.0380.0500.0570.0780.0760.063
Visa0.2360.1120.1180.1290.1410.1280.1320.1200.0940.085
Average0.1170.0650.0660.0590.0580.0570.0530.050.0410.0330.054
Table A2. Annualized daily (every fifth trading day) returns of MAW8–MAW40 (W = number of weeks), average annualized returns.
Table A2. Annualized daily (every fifth trading day) returns of MAW8–MAW40 (W = number of weeks), average annualized returns.
Buy and HoldMAW40MAW36MAW32MAW28MAW24MAW20MAW16MAW12MAW8
3M0.0900.0350.0330.0200.0210.0190.0120.0190.0320.026
American Express0.0940.0580.0530.0620.0630.0470.0460.0350.0340.015
Apple0.1570.1300.1370.1430.1310.1340.1310.1880.1740.144
Boeing0.1190.0890.0790.0750.0740.0800.0820.0660.0740.076
Caterpillar0.1000.0570.0620.0580.0580.0610.0540.0490.0430.023
Chevron0.0840.0050.0153 × 10−40.0040.0080.0090.0040.004−0.03
Coca-Cola0.0990.0550.0540.0540.0410.0540.0470.0470.0290.011
Walt Disney0.1030.0710.0730.0620.0800.0760.0800.0780.0650.051
Exxon0.0720.0180.0160.0070.0080.0100.0130.0200.0110.005
GE0.0520.0610.0460.0470.0470.0450.0230.0180.0310.023
Home Depot0.1900.1350.1330.1240.1120.1100.0880.0760.0960.077
IBM0.0550.0200.0370.0440.0400.0510.0270.0280.0080.016
Intel0.1340.0880.0910.0750.0610.0750.0730.0700.0760.085
Johnson & Johnson0.1130.0740.0790.0710.0590.0500.0500.0480.0420.027
JP Morgan0.0900.0400.0360.0270.0330.0330.0480.0510.0420.020
McDonalds0.1140.0860.0680.0590.0580.0520.0520.0590.0580.044
Merck0.0630.0510.0390.0290.0340.0340.0300.0330.0240.029
Microsoft0.1800.1280.1250.1160.1160.1160.1050.0990.0620.078
Nike0.1770.0870.0910.0980.0930.0870.0940.1020.1190.091
Pfizer0.0970.0700.0610.0570.0530.0630.0490.0500.0440.050
Procter & Gamble0.0950.0500.0440.0500.0510.0400.0430.0420.0310.033
Travellers0.0820.0200.0060.0100.0140.0060.0050.0080.0170.015
United Technologies0.1130.0710.0770.0620.0720.0710.0560.0610.0510.053
United Health Group0.2520.1710.1330.1300.1510.1240.1340.1230.1130.087
Verizon0.043−0.00−0.010.0020.006−0.01−0.01−0.01−0.009−0.00
Wal-Mart0.1130.0500.0490.0450.0380.0280.0330.0260.0380.029
Cisco0.2100.2090.2110.2190.2220.2190.2040.1640.1200.094
Goldman Sachs0.0610.0500.0300.0310.0400.0360.0710.0890.0780.077
Visa0.2360.1430.1420.1310.1710.1670.1590.1130.1190.080
Average0.1170.0730.0690.0660.0670.0650.0620.0610.0560.0460.063
Table A3. Annualized daily (every 22th trading day) returns of MA2–MA10, average annualized returns.
Table A3. Annualized daily (every 22th trading day) returns of MA2–MA10, average annualized returns.
Buy and HoldMA10MA9MA8MA7MA6MA5MA4MA3MA2
3M0.0900.0330.0350.0230.0230.0240.0230.0380.0210.012
American Express0.0940.0860.0870.0910.1070.0880.0620.0620.0360.038
Apple0.1570.0570.0690.0560.0760.0760.0940.0690.0990.071
Boeing0.1190.1220.1220.1020.0990.1150.1100.1000.0910.077
Caterpillar0.1000.0650.0620.0710.0830.0810.0630.0570.0090.051
Chevron0.0840.0220.0210.0250.0260.0190.0320.0320.0130.005
Coca-Cola0.0990.0830.0720.0870.0710.0730.0720.0690.0460.026
Walt Disney0.1030.0610.0660.0730.0770.0710.0790.0810.0730.057
Exxon0.0720.0400.0380.0280.0280.0340.0200.0270.0250.026
GE0.0520.0790.0780.0800.0720.0700.0630.0180.0380.037
Home Depot0.1900.1260.1330.1340.1360.1200.140.1190.1180.110
IBM0.0550.0290.0330.0320.0380.0360.0260.0330.0260.03
Intel0.1340.0790.0800.0960.0950.0850.0630.0820.1100.116
Johnson & Johnson0.1130.0780.0760.0710.0590.0570.0580.0500.0520.031
JP Morgan0.0900.0570.0510.0510.0630.0460.0700.0790.0670.067
McDonalds0.1140.0770.0770.0570.0550.0450.0560.0420.0450.033
Merck0.0630.0690.0690.0540.0590.050.0450.0270.0113 × 10−4
Microsoft0.1800.1220.1270.1230.0990.1120.0930.0950.0900.108
Nike0.1770.1280.1360.1300.1270.1150.1110.1090.0820.089
Pfizer0.0970.0700.0690.0670.0680.0660.0680.0560.0400.034
Procter & Gamble0.0950.0570.0600.0550.0420.0430.0210.0240.0380.039
Travellers0.0820.0450.0490.0470.0410.0340.0160.0090.0020.017
United Technologies0.1130.0640.0620.0740.0780.0630.0460.0370.0500.050
United Health Group0.2520.1580.1620.1670.1540.1680.1760.1740.1800.158
Verizon0.0430.0029 × 10−40.0110.0170.025−0.000.01−0.00−0.02
Wal-Mart0.1130.0460.0460.0400.0440.0320.0410.0370.0230.038
Cisco0.2100.2280.2270.2220.2210.1910.1860.1840.1600.134
Goldman Sachs0.0610.0290.0300.0200.0520.0670.0650.0700.0410.068
Visa0.2360.1710.1610.1620.1490.1220.1130.1150.1420.097
Average0.1170.0790.0790.0780.0780.0730.0690.0660.0590.0550.071
Table A4. Annualized daily (every other month) returns of MAD2–MAD5 (D = every other month, and 5, 4, 3, 2 are the numbers of observations in the rolling window), average annualized returns.
Table A4. Annualized daily (every other month) returns of MAD2–MAD5 (D = every other month, and 5, 4, 3, 2 are the numbers of observations in the rolling window), average annualized returns.
Buy and HoldMAD5MAD4MAD3MAD2
3M0.0900.0620.0630.0420.049
American Express0.0940.0890.0980.0520.041
Apple0.1570.0400.0420.0300.085
Boeing0.1190.1120.1100.1020.110
Caterpillar0.1000.0790.090.0890.084
Chevron0.0840.0330.0360.0260.028
Coca-Cola0.0990.0930.1020.0800.078
Walt Disney0.1030.0680.0740.0800.084
Exxon0.0720.0220.0180.0100.009
GE0.0520.0670.0660.0410.033
Home Depot0.1900.1740.1750.1560.160
IBM0.0550.0160.0230.0170.021
Intel0.1340.0930.0980.0890.112
Johnson & Johnson0.1130.0830.0860.0480.071
JP Morgan0.0900.0530.0520.0480.054
McDonalds0.1140.0940.0980.0710.070
Merck0.0630.0840.0670.0360.031
Microsoft0.1800.1380.1360.1060.088
Nike0.1770.1400.1440.1330.122
Pfizer0.0970.0620.0510.0610.059
Procter & Gamble0.0950.0480.0540.0480.034
Travellers0.0820.0180.0150.0182 × 10−4
United Technologies0.1130.0660.0730.0960.060
United Health Group0.2520.1810.1790.1910.207
Verizon0.043−0.018−0.01−0.02−0.02
Wal-Mart0.1130.0670.0650.0500.061
Cisco0.2100.2170.2260.2070.196
Goldman Sachs0.0610.0410.0590.0600.039
Visa0.2360.1740.1730.1510.120
Average0.1170.0830.0850.0730.0720.078
Table A5. Annualized daily (every third month) returns of MAT2–MAT4 (T = every third month, and 4, 3, 2 are the numbers of observations in the rolling window), average annualized returns.
Table A5. Annualized daily (every third month) returns of MAT2–MAT4 (T = every third month, and 4, 3, 2 are the numbers of observations in the rolling window), average annualized returns.
Buy and HoldMAT4MAT3MAT2
3M0.0900.0610.0550.039
American Express0.0940.1130.0910.066
Apple0.1570.0890.0730.096
Boeing0.1190.1270.1310.114
Caterpillar0.1000.0700.0690.078
Chevron0.0840.0470.0530.037
Coca-Cola0.0990.0770.0780.072
Walt Disney0.1030.0430.0420.068
Exxon0.0720.0550.0490.037
GE0.0520.0840.0800.047
Home Depot0.1900.1610.1630.128
IBM0.0550.0540.0480.028
Intel0.1340.1070.1150.072
Johnson & Johnson0.1130.0940.0940.074
JP Morgan0.0900.0580.0760.007
McDonalds0.1140.0800.0820.069
Merck0.0630.0620.0620.049
Microsoft0.1800.1270.1280.080
Nike0.1770.1460.1510.099
Pfizer0.0970.0780.0700.056
Procter & Gamble0.0950.0680.0720.076
Travellers0.0820.0410.0430.025
United Technologies0.1130.0770.0890.079
United Health Group0.2520.1470.1610.178
Verizon0.043−0.00−0.00−0.02
Wal-Mart0.1130.0810.0810.083
Cisco0.2100.2110.2170.213
Goldman Sachs0.0610.0440.0260.030
Visa0.2360.1830.1990.177
Average0.1170.0890.0890.0750.084
Table A6. Annualized daily (every fourth month) returns of MAQ2–MAQ3 (Q = every fourth month, and 3 and 2 are the numbers of observations in the rolling window), average annualized returns.
Table A6. Annualized daily (every fourth month) returns of MAQ2–MAQ3 (Q = every fourth month, and 3 and 2 are the numbers of observations in the rolling window), average annualized returns.
Buy and HoldMAQ3MAQ2
3M0.0900.0560.058
American Express0.0940.0890.094
Apple0.1570.0940.094
Boeing0.1190.1220.128
Caterpillar0.1000.0640.084
Chevron0.0840.0600.054
Coca-Cola0.0990.0830.093
Walt Disney0.1030.0610.062
Exxon0.0720.0560.064
GE0.0520.0690.081
Home Depot0.1900.1520.157
IBM0.0550.0480.031
Intel0.1340.0640.070
Johnson & Johnson0.1130.0800.079
JP Morgan0.0900.0850.091
McDonalds0.1140.0960.112
Merck0.0630.0560.061
Microsoft0.1800.1430.145
Nike0.1770.1810.199
Pfizer0.0970.0590.045
Procter & Gamble0.0950.0730.077
Travellers0.0820.0510.051
United Technologies0.1130.0800.077
United Health Group0.2520.1850.218
Verizon0.0430.0270.023
Wal-Mart0.1130.0870.076
Cisco0.2100.1950.180
Goldman Sachs0.0610.0420.056
Visa0.2360.1950.228
Average0.1170.0910.0960.094
Table A7. Annualized daily (every fifth month) returns of MAC2 (C = every fifth month, and 2 = observations accounting in the rolling window), average annualized returns.
Table A7. Annualized daily (every fifth month) returns of MAC2 (C = every fifth month, and 2 = observations accounting in the rolling window), average annualized returns.
Buy and HoldMAC2
3M0.0900.076
American Express0.0940.088
Apple0.1570.132
Boeing0.1190.080
Caterpillar0.1000.094
Chevron0.0840.047
Coca-Cola0.0990.094
Walt Disney0.1030.044
Exxon0.0720.049
GE0.0520.048
Home Depot0.1900.143
IBM0.0550.032
Intel0.1330.057
Johnson & Johnson0.1130.081
JP Morgan0.0900.045
McDonalds0.1140.079
Merck0.0630.080
Microsoft0.1800.094
Nike0.1770.141
Pfizer0.0970.099
Procter & Gamble0.0950.039
Travellers0.0820.068
United Technologies0.1130.056
United Health Group0.2520.152
Verizon0.0430.048
Wal-Mart0.1130.093
Cisco0.2100.225
Goldman Sachs0.0610.053
Visa0.2360.217
Average0.1170.088

Appendix B

Table A8. Annualized daily volatility of MA40–MA200, average annualized volatility.
Table A8. Annualized daily volatility of MA40–MA200, average annualized volatility.
Buy and HoldMA200MA180MA160MA140MA120MA100MA80MA60MA40
3M0.2250.1640.1650.1610.1610.1590.1590.1580.1580.157
American Express0.3450.2270.2280.2210.2250.2240.2250.2240.2280.229
Apple0.4510.3170.3210.3150.3150.3130.3150.3150.3100.305
Boeing0.2940.2010.2030.1990.2010.1990.1980.1980.2010.204
Caterpillar0.3110.2160.2180.2160.2160.2140.2150.2140.2130.215
Chevron0.2440.1670.1680.1660.1660.1660.1650.1640.1670.168
Coca-Cola0.2250.1640.1660.1610.1600.1590.1580.1580.1560.155
Walt Disney0.2910.1960.2010.1990.2000.1990.1980.2030.2040.203
Exxon0.2300.1620.1630.1590.1590.1590.1570.1560.1550.157
GE0.2750.1740.1750.1720.1730.1730.1710.1680.1680.168
Home Depot0.3140.2260.2280.2230.2210.2210.2190.2170.2170.214
IBM0.2710.1870.1890.1850.1840.1810.1790.1770.1760.174
Intel0.3820.2730.2750.2670.2650.2630.2600.2570.2560.254
Johnson & Johnson0.2150.1630.1640.1610.1590.1570.1550.1530.1520.149
JP Morgan0.3750.2230.2260.2230.2240.2270.2370.2420.2450.248
McDonalds0.2400.1830.1840.180.1780.1770.1760.1760.1750.174
Merck0.2690.1770.1790.1730.1730.1740.1720.1740.1740.177
Microsoft0.3230.2480.2490.2430.2410.2370.2360.2330.2320.231
Nike0.3270.2430.2450.2380.2360.2350.2350.2320.2320.233
Pfizer0.2660.1880.190.1870.1860.1870.1860.1870.1870.187
Procter & Gamble0.2250.1690.1690.1640.1630.1610.1580.1570.1560.156
Travellers0.2680.1740.1750.1740.1750.1780.1800.1840.1820.185
United Technologies0.2610.1790.1810.1790.1780.1770.1770.1770.1760.173
United Health Group0.3860.2900.2930.2900.2900.2830.2820.2820.2800.273
Verizon0.2460.1630.1650.1640.1630.1630.1630.1610.1610.163
Wal-Mart0.2630.2030.2040.2000.1980.1950.1910.190.1890.191
Cisco0.4150.3000.3020.2970.2950.2910.2900.2850.2820.275
Goldman Sachs0.3730.2220.2260.220.2220.2230.2280.2300.2270.229
Visa0.2600.2090.2120.2090.2080.2120.2080.2060.2050.197
Average0.2950.2070.2090.2050.2050.2040.2030.2030.2020.2020.204
Table A9. Annualized daily (every fifth trading day) volatility of MAW8–MAW40 (W = number of weeks), average annualized volatility.
Table A9. Annualized daily (every fifth trading day) volatility of MAW8–MAW40 (W = number of weeks), average annualized volatility.
Buy and HoldMAW40MAW36MAW32MAW28MAW24MAW20MAW16MAW12MAW8
3M0.2250.1650.1650.1630.1630.160.1590.1570.1570.159
American Express0.3450.2270.2240.2240.2270.2250.2230.2280.2320.234
Apple0.4510.3160.3160.3130.3180.3160.3430.3170.3120.309
Boeing0.2940.2040.2030.2040.2040.2030.2030.2010.2010.206
Caterpillar0.3110.2160.2150.2150.2170.2140.2150.2150.2130.214
Chevron0.2440.1690.1680.1690.1680.1680.1670.1660.1680.172
Coca-Cola0.2250.1650.1650.1640.1620.1600.1590.1590.1570.155
Walt Disney0.2910.1950.1980.1970.1970.1990.2000.2020.2030.204
Exxon0.2300.1630.1610.1600.1610.1600.1570.1560.1530.158
GE0.2750.1740.1740.1740.1750.1740.1700.1690.1710.166
Home Depot0.3140.2280.2280.2260.2250.2220.2240.2190.2190.214
IBM0.2710.1900.1880.1850.1840.1830.1780.1770.1780.177
Intel0.3820.2670.2670.2680.2640.2630.2590.2560.2590.259
Johnson & Johnson0.2150.1640.1630.1620.1600.1580.1560.1560.1520.15
JP Morgan0.3750.2220.2250.2240.2300.2360.2390.2430.2410.252
McDonalds0.2400.1850.1820.1810.1790.1770.1770.1760.1740.171
Merck0.2690.1790.1750.1740.1730.1730.1720.1750.1760.175
Microsoft0.3230.2500.2470.2450.2440.240.2360.2360.2300.232
Nike0.3270.2440.2410.2390.2400.2410.2380.2350.2320.232
Pfizer0.2660.1890.1870.1860.1870.1880.1900.1890.1890.184
Procter & Gamble0.2250.1700.1680.1670.1650.1640.1610.1580.1600.156
Travellers0.2680.1750.1750.1750.1780.1770.1770.1840.1840.185
United Technologies0.2610.1810.1790.1780.1770.1770.1770.1770.1760.172
United Health Group0.3860.2920.2910.2920.2910.2900.2890.2870.2820.278
Verizon0.2460.1630.1620.1620.1620.1640.1620.1610.1600.159
Wal-Mart0.2630.2050.2020.2010.1980.1940.1910.1910.1900.192
Cisco0.4150.3070.3050.3000.2960.2920.2930.2880.2850.281
Goldman Sachs0.3730.2250.2230.2210.2210.2200.2300.2330.2410.241
Visa0.2600.2030.2100.2090.2080.2100.2080.2060.2030.195
Average0.2950.2080.2070.2060.2060.2050.2050.2040.2030.2030.205
Table A10. Annualized daily (rule in every 22th trading day) volatility of MA2–MA10, average annualized volatility.
Table A10. Annualized daily (rule in every 22th trading day) volatility of MA2–MA10, average annualized volatility.
Buy and HoldMA10MA9MA8MA7MA6MA5MA4MA3MA2
3M0.2250.1670.1690.1620.1630.1610.1610.1570.1560.156
American Express0.3450.2320.2350.2220.2180.220.2190.220.2430.235
Apple0.4510.3430.3470.3420.3390.3390.3380.3420.3350.331
Boeing0.2940.2070.2100.2020.2020.1990.2000.1970.2070.205
Caterpillar0.3110.2160.2200.2170.2150.2140.2170.2180.2210.224
Chevron0.2440.1690.1710.1720.170.1690.1690.1670.1810.171
Coca-Cola0.2250.1680.1710.1690.1680.1660.1610.1610.1610.156
Walt Disney0.2910.2030.2070.2020.2030.2030.2100.2120.2150.211
Exxon0.2300.1660.1670.1650.1640.1630.1620.1570.1610.160
GE0.2750.1770.1770.1750.1750.1750.1720.1690.1720.180
Home Depot0.3140.2340.2350.2280.2210.2300.2280.2330.2250.219
IBM0.2710.1940.1960.1960.1960.1960.190.1940.1950.190
Intel0.3820.2730.2770.2720.2720.2680.2660.2660.2640.259
Johnson & Johnson0.2150.1680.1690.1670.1670.1620.1580.1580.1540.150
JP Morgan0.3750.2220.2230.2170.2200.2300.2330.2340.2440.234
McDonalds0.2400.1890.1890.1860.1850.1850.1790.1700.1710.180
Merck0.2690.1770.1780.1730.1730.1740.1730.1810.1820.192
Microsoft0.3230.2500.2510.2470.2390.2330.2350.2370.2330.234
Nike0.3270.2470.2480.2440.2410.2400.2350.2360.2380.248
Pfizer0.2660.1880.1900.1860.1860.1860.1870.1870.1910.189
Procter & Gamble0.2250.1730.1740.1710.1670.1650.1630.1640.1580.155
Travellers0.2680.1710.1720.170.1690.1710.1910.1860.1920.198
United Technologies0.2610.1780.1790.1780.1770.1770.1750.1780.1760.173
United Health Group0.3860.3000.3020.2990.2980.2940.2890.2800.2830.275
Verizon0.2460.1670.1670.1640.1620.1600.1640.1570.1600.163
Wal-Mart0.2630.2080.2100.2050.1990.1960.1970.1980.1980.189
Cisco0.4150.3040.3070.3010.2980.3000.2920.2900.2810.278
Goldman Sachs0.3730.2300.2320.2250.2320.2450.2390.2530.2680.256
Visa0.2600.2040.2030.2120.2250.2210.2190.2170.2170.196
Average0.2950.2110.2130.2090.2080.2080.2080.2080.2100.2070.209
Table A11. Annualized daily (every other month) volatility of MAD2–MAD5 (D = every other month, and 5, 4, 3, 2 are the numbers of observations in the rolling window), average annualized volatility.
Table A11. Annualized daily (every other month) volatility of MAD2–MAD5 (D = every other month, and 5, 4, 3, 2 are the numbers of observations in the rolling window), average annualized volatility.
Buy and HoldMAD5MAD4MAD3MAD2
3M0.2250.1680.1690.1620.159
American Express0.3440.2220.2260.2160.211
Apple0.4500.3510.3630.3570.338
Boeing0.2940.2100.2160.2110.208
Caterpillar0.3110.2180.2290.2150.211
Chevron0.2440.1680.1750.1660.165
Coca-Cola0.2250.1680.1730.1650.158
Walt Disney0.2910.1970.2000.1980.203
Exxon0.2300.1720.1740.1590.156
GE0.2740.1750.1810.1760.182
Home Depot0.3140.2290.2300.2210.237
IBM0.2710.1960.1990.2000.200
Intel0.3820.2740.2860.2670.265
Johnson & Johnson0.2150.1730.1750.1650.154
JP Morgan0.3750.2360.2410.2460.237
McDonalds0.2400.1820.1860.1780.169
Merck0.2690.1850.1960.1880.199
Microsoft0.3230.2450.2490.2380.250
Nike0.3270.2520.2580.2530.253
Pfizer0.2660.1990.2030.1910.189
Procter & Gamble0.2250.1730.1770.1690.166
Travellers0.2680.1760.1780.1830.191
United Technologies0.2610.1820.1870.1780.177
United Health Group0.3860.3130.3130.2990.305
Verizon0.2460.1630.1710.1650.153
Wal-Mart0.2630.1970.1990.1940.193
Cisco0.4150.3120.3170.3150.285
Goldman Sachs0.3730.2290.2450.2390.265
Visa0.2600.2150.2150.2250.222
Average0.2950.2130.2180.2120.2100.213
Table A12. Annualized daily (every third month) volatility of MAT2–MAT4 (T = every third month, and 4, 3, 2 are the numbers of observations in the rolling window), average annualized volatility.
Table A12. Annualized daily (every third month) volatility of MAT2–MAT4 (T = every third month, and 4, 3, 2 are the numbers of observations in the rolling window), average annualized volatility.
Buy and HoldMAT4MAT3MAT2
3M0.2250.1720.1740.171
American Express0.3440.2300.2370.206
Apple0.4500.3450.3570.349
Boeing0.2940.2060.2190.200
Caterpillar0.3110.2190.2230.214
Chevron0.2440.1760.1820.170
Coca-Cola0.2250.1770.1790.181
Walt Disney0.2910.2200.2280.205
Exxon0.2300.1680.1760.158
GE0.2740.1780.1850.177
Home Depot0.3140.2360.2510.241
IBM0.2710.2050.2090.193
Intel0.3820.2850.2960.274
Johnson & Johnson0.2150.1850.1880.165
JP Morgan0.3750.2420.2480.240
McDonalds0.2400.1980.2040.192
Merck0.2690.1910.1910.180
Microsoft0.3230.2570.2670.258
Nike0.3270.2640.2650.258
Pfizer0.2660.1950.2060.208
Procter & Gamble0.2250.1770.1810.168
Travellers0.2680.1870.1880.198
United Technologies0.2610.1920.1990.187
United Health Group0.3860.3000.3080.315
Verizon0.2460.1760.1760.160
Wal-Mart0.2630.2020.2080.208
Cisco0.4150.3100.3110.303
Goldman Sachs0.3730.2260.2320.235
Visa0.2600.2040.2150.208
Average0.2950.2180.2240.2140.219
Table A13. Annualized daily (every fourth month) volatility of MAQ2–MAQ3 (Q = every fourth month, 3 and 2 are the number of observations in the rolling window), average annualized volatility.
Table A13. Annualized daily (every fourth month) volatility of MAQ2–MAQ3 (Q = every fourth month, 3 and 2 are the number of observations in the rolling window), average annualized volatility.
Buy and HoldMAQ3MAQ3
3M0.2250.1680.176
American Express0.3440.2200.226
Apple0.4500.3600.373
Boeing0.2940.2130.224
Caterpillar0.3110.2220.239
Chevron0.2440.1670.177
Coca-Cola0.2250.1730.182
Walt Disney0.2910.2060.218
Exxon0.2300.1600.176
GE0.2740.1800.195
Home Depot0.3140.2370.242
IBM0.2710.1940.218
Intel0.3820.2740.293
Johnson & Johnson0.2150.1810.186
JP Morgan0.3750.2180.227
McDonalds0.2400.1770.193
Merck0.2690.2040.212
Microsoft0.3230.2480.260
Nike0.3270.2580.265
Pfizer0.2660.1980.207
Procter & Gamble0.2250.1730.174
Travellers0.2680.1820.192
United Technologies0.2610.1810.188
United Health Group0.3860.2990.314
Verizon0.2460.1670.177
Wal-Mart0.2630.1940.207
Cisco0.4150.3410.349
Goldman Sachs0.3730.2400.260
Visa0.2600.2120.225
Average0.2950.2150.2270.221
Table A14. Annualized daily (every fifth month) volatility of MAC2 (C = every fifth month, 2 = observations in rolling window), average annualized volatility.
Table A14. Annualized daily (every fifth month) volatility of MAC2 (C = every fifth month, 2 = observations in rolling window), average annualized volatility.
Buy and HoldMAC2
3M0.2250.176
American Express0.3440.226
Apple0.4500.323
Boeing0.2940.218
Caterpillar0.3110.227
Chevron0.2440.165
Coca-Cola0.2250.168
Walt Disney0.2910.206
Exxon0.2300.166
GE0.2740.187
Home Depot0.3140.242
IBM0.2710.202
Intel0.3820.296
Johnson & Johnson0.2150.187
JP Morgan0.3750.244
McDonalds0.2400.182
Merck0.2690.194
Microsoft0.3230.250
Nike0.3270.249
Pfizer0.2660.191
Procter & Gamble0.2250.187
Travellers0.2680.183
United Technologies0.2610.204
United Health Group0.3860.298
Verizon0.2460.170
Wal-Mart0.2630.223
Cisco0.4150.333
Goldman Sachs0.3730.218
Visa0.2600.220
Average0.2950.218

Appendix C

Table A15. Transaction costs per year of MA40–MA200, with one transaction costing 0.1% of total wealth, average annualized transaction costs.
Table A15. Transaction costs per year of MA40–MA200, with one transaction costing 0.1% of total wealth, average annualized transaction costs.
MA200MA180MA160MA140MA120MA100MA80MA60MA40
3M0.0100.0110.0100.0110.0120.0130.0160.0190.022
American Express0.0110.0110.0110.0120.0120.0130.0160.0170.023
Apple0.0070.0080.0080.0090.0100.0120.0140.0150.020
Boeing0.0080.0090.0100.0110.0110.0120.0140.0150.020
Caterpillar0.0080.0090.0100.0110.0120.0130.0150.0150.019
Chevron0.0110.0120.0120.0130.0140.0160.0180.0200.024
Coca-Cola0.0090.0100.0110.0110.0110.0120.0150.0180.022
Walt Disney0.0070.0080.0090.0110.0120.0120.0130.0170.021
Exxon0.0110.0130.0160.0170.0170.0180.0190.0230.028
GE0.0070.0080.0090.0100.0110.0120.0140.0170.023
Home Depot0.0080.0090.0100.0110.0130.0140.0160.0180.021
IBM0.0090.0100.0100.0100.0120.0120.0130.0140.019
Intel0.0070.0090.0100.0100.0120.0140.0140.0160.019
Johnson & Johnson0.0090.0080.0090.0100.0120.0140.0160.0200.024
JP Morgan0.0100.0100.0110.0120.0120.0140.0150.0160.020
McDonalds0.0100.0110.0110.0130.0120.0140.0160.0180.023
Merck0.0080.0090.0090.0110.0110.0130.0150.0170.022
Microsoft0.0080.0090.0100.0100.0110.0130.0150.0150.020
Nike0.0090.0090.0100.0100.0110.0120.0130.0140.019
Pfizer0.0080.0100.0100.0110.0110.0120.0140.0170.021
Procter & Gamble0.0100.0100.0100.0110.0120.0140.0160.0190.022
Travellers0.0100.0110.0120.0120.0130.0150.0160.0180.024
United Technologies0.0090.0100.0110.0110.0120.0140.0150.0180.021
United Health Group0.0080.0080.0100.0100.0110.0120.0140.0170.021
Verizon0.0110.0110.0110.0110.0130.0140.0170.0180.023
Wal-Mart0.0100.0100.0120.0130.0130.0140.0150.0190.022
Cisco0.0060.0060.0080.0100.0090.0100.0140.0170.023
Goldman Sachs0.0080.0100.0120.0120.0140.0150.0220.0260.035
Visa0.0080.0080.0090.0090.0080.0100.0110.0140.022
Average0.0090.00100.0100.0110.0120.0130.0150.0180.0220.013
Table A16. Transaction costs per year of MA2–MA10, average annualized transaction costs.
Table A16. Transaction costs per year of MA2–MA10, average annualized transaction costs.
MA10MA9MA8MA7MA6MA5MA4MA3MA2
3M0.0030.0030.0030.0030.0030.0040.0040.0050.006
American Express0.0020.0020.0020.0020.0020.0030.0030.0040.006
Apple0.0020.0020.0020.0020.0030.0030.0040.0050.006
Boeing0.0020.0020.0020.0020.0020.0030.0040.0040.006
Caterpillar0.0020.0020.0020.0020.0030.0030.0040.0050.006
Chevron0.0020.0030.0030.0030.0030.0030.0040.0050.007
Coca-Cola0.0020.0020.0020.0020.0020.0030.0030.0040.006
Walt Disney0.0020.0020.0020.0020.0030.0030.0030.0040.006
Exxon0.0020.0020.0030.0030.0030.0040.0040.0050.006
GE0.0020.0020.0020.0020.0030.0030.0040.0040.006
Home Depot0.0020.0020.0020.0020.0030.0030.0030.0040.006
IBM0.0030.0020.0030.0020.0030.0030.0040.0040.006
Intel0.0020.0030.0030.0030.0030.0030.0040.0040.006
Johnson & Johnson0.0020.0020.0020.0020.0030.0030.0040.0050.006
JP Morgan0.0020.0030.0030.0030.0030.0030.0030.0040.006
McDonalds0.0020.0020.0030.0030.0030.0030.0040.0050.006
Merck0.0020.0020.0020.0030.0030.0030.0040.0050.006
Microsoft0.0020.0020.0020.0030.0030.0030.0040.0040.006
Nike0.0020.0020.0020.0020.0030.0030.0040.0040.006
Pfizer0.0020.0020.0020.0030.0030.0030.0040.0040.006
Procter & Gamble0.0020.0020.0030.0030.0030.0040.0040.0050.006
Travellers0.0030.0020.0030.0030.0030.0040.0040.0050.007
United Technologies0.0020.0020.0020.0020.0030.0030.0040.0040.006
United Health Group0.0020.0020.0020.0030.0030.0030.0030.0040.006
Verizon0.0030.0030.0030.0030.0030.0040.0040.0050.006
Wal-Mart0.0030.0030.0030.0030.0030.0040.0040.0050.006
Cisco0.0020.0020.0020.0020.0030.0030.0030.0050.006
Goldman Sachs0.0020.0020.0020.0030.0030.0030.0030.0040.005
Visa0.0020.0010.0020.0020.0020.0030.0030.0030.005
Average0.0020.0020.0020.0030.0030.0030.0040.0040.0060.003

References

  1. Tobin, J. Liquidity preference as behavior towards risk. Rev. Econ. Stud. 1958, 67, 65–86. [Google Scholar] [CrossRef]
  2. Fama, E. Components on investment performance. J. Financ. 1972, 27, 551–568. [Google Scholar]
  3. Merton, R. On market timing and investment performance. I. An equilibrium theory of value for market forecast. J. Bus. 1981, 54, 363–406. [Google Scholar] [CrossRef]
  4. Henriksson, R.; Merton, R. On market timing and investment performance II: Statistical procedures for evaluating forecasting skills. J. Bus. 1981, 54, 513–533. [Google Scholar] [CrossRef]
  5. Sharpe, W. Capital asset prices: A theory of market equilibrium under conditions of risk. J. Financ. 1964, 19, 425–442. [Google Scholar]
  6. Lintner, J. The valuation of risky assets and the selection of risky investments in stock portfolios and capital budgets. Rev. Econ. Stat. 1965, 47, 13–37. [Google Scholar] [CrossRef]
  7. Zhu, Y.; Zhou, G. Technical analysis: An asset allocation perspective on the use of moving averages. J. Financ. Econ. 2009, 9, 519–544. [Google Scholar] [CrossRef]
  8. Markowitz, H. Portfolio selection. J. Financ. 1952, 7, 77–91. [Google Scholar]
  9. Baker, M.; Bradley, B.; Wurgler, J. Benchmark as limits to arbitrage: Understanding the low-volatility anomaly. Financ. Anal. J. 2011, 67, 1–15. [Google Scholar] [CrossRef]
  10. Ang, A.; Hodrick, R.; Xing, Y.; Zhang, X. High idiosyncratic volatility and low returns: International and further U.S. evidence. J. Financ. Econ. 2009, 61, 1–23. [Google Scholar] [CrossRef]
  11. Cochrane, J. The dog that did not bark: A defense of return predictability. Rev. Financ. Stud. 2008, 21, 1533–1573. [Google Scholar] [CrossRef]
  12. Fama, E. Two pillars of asset pricing. Am. Econ. Rev. 2014, 104, 1467–1485. [Google Scholar] [CrossRef]
  13. Jagannathan, R.; Korajczyk, R. Market timing. In Portfolio Construction, Measurement and Efficiency; Guerard, J., Ed.; Springer International Publishing: Cham, Switzerland, 2017; pp. 49–71. ISBN 978-3-319-33974-0. [Google Scholar]
  14. Friesen, G.; Sapp, T. Mutual fund flows and investor returns: An empirical examination of fund investor timing ability. J. Bank. Financ. 2007, 31, 2796–2816. [Google Scholar] [CrossRef] [Green Version]
  15. Munoz, F.; Vicente, R. Hindsight effect: What are actual cash flow timing skills of mutual fund investors? J. Empir. Financ. 2018, 45, 181–193. [Google Scholar] [CrossRef]
  16. Malkiel, B. The efficient market hypothesis and its critics. J. Econ. Perspect. 2003, 17, 59–82. [Google Scholar] [CrossRef]
  17. Samuelson, P. Summing up on business cycles: Opening address. In Beyond Shocks; What Causes Business Cycles? Fuhrer, J., Schuh, S., Eds.; Federal Reserve Bank of Boston: Boston, MA, USA, 1998. [Google Scholar]
  18. Black, F. Capital market equilibrium with restricted borrowing. J. Bus. 1972, 45, 444–455. [Google Scholar] [CrossRef]
  19. Black, F.; Jensen, M.; Scholes, M. The capital asset market model: Some empirical tests. In Studies in the Theory of Capital Markets; Jensen, M., Ed.; Praeger: New York, NY, USA, 1972; pp. 79–121. [Google Scholar]
  20. Baker, M.; Bradley, B.; Taliaferro, R. The Low-risk anomaly: A decomposition into micro and macro effects. Financ. Anal. J. 2014, 70, 45–58. [Google Scholar] [CrossRef]
  21. Frazzini, A.; Pedersen, L. Betting against betas. J. Financ. Econ. 2014, 111, 1–25. [Google Scholar] [CrossRef]
  22. Clarke, R.; de Silva, H.; Thorley, S. Know your VMS exposure. J. Portf. Manag. 2010, 36, 52–59. [Google Scholar] [CrossRef]
  23. Barberis, N.; Huang, M. Stocks as lotteries: The implications of probability weighting for security prices. Am. Econ. Rev. 2008, 98, 2066–2100. [Google Scholar] [CrossRef]
  24. Kumar, A. Who gambles in the stock markets? J. Financ. 2009, 64, 1889–1933. [Google Scholar] [CrossRef]
  25. Bali, T.; Cakici, N.; Whitelaw, R. Maxing out: Stock as lotteries and the cross-section of expected returns. J. Financ. Econ. 2011, 99, 427–446. [Google Scholar] [CrossRef]
  26. Ben-David, I.; Graham, J.; Harvey, C. Managerial miscalibration. Quart. J. Econ. 2013, 128, 1547–1584. [Google Scholar] [CrossRef]
  27. Daniel, K.; Titman, S. Market reactions to tangible and intangible information. J. Financ. 2006, 61, 1605–1643. [Google Scholar] [CrossRef]
  28. Kahneman, D.; Tversky, A. Judgement under uncertainty: Heuristics and biases. Science 1974, 185, 1124–1131. [Google Scholar]
  29. Baker, M.; Wurgler, J. Do strict requirements raise the cost of capital? Bank regulation, capital structure, and the low-risk anomaly. Am. Econ. Rev. Pap. Proc. 2015, 105, 315–320. [Google Scholar] [CrossRef]
  30. Hong, H.; Sraer, D. Speculative betas. J. Financ. 2016, 71, 2095–2144. [Google Scholar] [CrossRef]
  31. Li, X.; Sullivan, R.; Garcia-Feijoo, L. The low-volatility anomaly: Market evidence on systematic risk vs. mispricing. Financ. Anal. J. 2016, 72, 36–47. [Google Scholar] [CrossRef]
  32. Brown, D.; Jennings, R. On technical analysis. Rev. Financ. Stud. 1989, 2, 527–551. [Google Scholar] [CrossRef]
  33. Gartley, H. Profits in the Stock Markets; Lambert-Gann Publishing: Washington, DC, USA, 1935. [Google Scholar]
  34. Moskowitz, T.; Ooi, Y.; Pedersen, L. Time series momentum. J. Financ. Econ. 2012, 104, 228–250. [Google Scholar] [CrossRef]
  35. Jegadeesh, N.; Titman, S. Returns to buying winners selling losers: Implications for stock market efficiency. J. Financ. 1993, 48, 65–91. [Google Scholar] [CrossRef]
  36. LeRoy, S. Risk aversion and the martingale property of stock prices. Int. Econ. Rev. 1973, 14, 436–446. [Google Scholar] [CrossRef]
  37. Lucas, R. Asset prices in an exchange economy. Econometrica 1978, 46, 1429–1445. [Google Scholar] [CrossRef]
  38. Campbell, J.; Cochrane, J. By force of habit: Consumption-based explanation of aggregate stock market behavior. J. Political Econ. 1999, 107, 205–251. [Google Scholar] [CrossRef]
  39. Brock, W.; Lakonishok, J.; LeBaron, B. Simple technical trading rules and the stochastic properties of stock returns. J. Financ. 1992, 47, 1731–1764. [Google Scholar] [CrossRef]
  40. Sullivan, R.; Timmermann, A.; White, H. Data-snooping, technical trading rule performance and the bootstrap. J. Financ. 1999, 53, 1647–1691. [Google Scholar] [CrossRef]
  41. Allen, F.; Karjalainen, R. Using genetic algorithms to find technical trading rules. J. Financ. Econ. 1999, 51, 245–271. [Google Scholar] [CrossRef]
  42. Lo, A.; Mamaysky, H.; Wang, J. Foundations of technical analysis: Computational algorithms, statistical inference, and empirical implementation. J. Financ. 2000, 54, 1705–1770. [Google Scholar] [CrossRef]
  43. Neely, C.; Rapach, D.; Tu, J.; Zhou, G. Forecasting equity risk premium: The role of technical indicators. Manag. Sci. 2014, 66, 1772–1791. [Google Scholar] [CrossRef]
  44. Marshall, B.; Nguyen, N.; Visaltanachoti, N. Time series momentum and moving average trading rules. Quant. Financ. 2017, 17, 405–421. [Google Scholar] [CrossRef]
  45. Kim, A.; Tse, Y.; Wald, J. Time series momentum and volatility scaling. J. Financ. Mark. 2016, 30, 103–124. [Google Scholar] [CrossRef]
Figure 1. Returns to volatility ratio in MA200, MAW40, MA10, MAD5, MAT4, MAQ3, MAC2, and the theoretical random timing efficient SML.
Figure 1. Returns to volatility ratio in MA200, MAW40, MA10, MAD5, MAT4, MAQ3, MAC2, and the theoretical random timing efficient SML.
Sustainability 10 02125 g001
Figure 2. Returns to volatility ratio in MA200 − MA40, MAW40 − MAW8, MA10 − MA2, MAD5 − MAD2, MAT4 − MAT2, MAQ3 − MAQ2, MAC2, and the theoretical random timing efficient SML.
Figure 2. Returns to volatility ratio in MA200 − MA40, MAW40 − MAW8, MA10 − MA2, MAD5 − MAD2, MAT4 − MAT2, MAQ3 − MAQ2, MAC2, and the theoretical random timing efficient SML.
Sustainability 10 02125 g002
Figure 3. Returns to volatility ratio in current DJIA stocks, annual averages from 4 January 1988 to 31 December 2017.
Figure 3. Returns to volatility ratio in current DJIA stocks, annual averages from 4 January 1988 to 31 December 2017.
Sustainability 10 02125 g003

Share and Cite

MDPI and ACS Style

Ilomäki, J.; Laurila, H.; McAleer, M. Market Timing with Moving Averages. Sustainability 2018, 10, 2125. https://doi.org/10.3390/su10072125

AMA Style

Ilomäki J, Laurila H, McAleer M. Market Timing with Moving Averages. Sustainability. 2018; 10(7):2125. https://doi.org/10.3390/su10072125

Chicago/Turabian Style

Ilomäki, Jukka, Hannu Laurila, and Michael McAleer. 2018. "Market Timing with Moving Averages" Sustainability 10, no. 7: 2125. https://doi.org/10.3390/su10072125

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop