Multi-Factor Modeling with Technical Indicators
Brynne Fitzgerald
This project analyzes the effectiveness of technical indicators to make trading decisions. The focus is on a specific asset class, Exchange Traded Funds (ETFs) and use historic ETF stock prices to generate three popular indicators: Bollinger Bands, MACD, and RSI. Technical indicator effectiveness is evaluated using the Fama-French 3 Factor model. Results will find that, for the specific asset analyzed, two of the three technical indicators studied provide statistically significant positive correlations to the ETF’s equity risk premium.
Introduction
Fundamental and Technical analysis have historically been divided between the speculative investor and the trend following trader. Technical analysts most often use short term charts and price trends to map over multiple time intervals with the span of hours or even minutes. Alternatively, fundamental analysts use much broader time horizons and macroeconomic factors to adjust for investor sentiment for a given stock price.
The use of technical indicators to predict future price movements is not fully supported by acidemia and is especially criticized by fundamental economists. The fascination in the development in this project is the attempt to bridge the gap between fundamental and technical theories.
The goal of this project is to use technical indicators to test their effectiveness in identifying equity risk premium. This project will focus on Exchange Traded Funds (ETFs), which are similar to mutual funds where they track a diverse selection of securities, typically following an index. ETFs were selected because of their diverse nature and liquidity. Additional motivation for the use of ETFs is my background in ETF operations at JPMorgan.
Technical Indicators
The modern technical analysis used today originated from the Dow Theory, created in the late 1900s by Charles Dow. Dow theory lead to insights to the relationship between stock prices and the economy by identifying market inefficiencies. Dow created a tool which tracked railroad and industrial stocks to find turning points in the business cycle. He was able to identify “bull” and “bear” market trends through this simple idea and later research demonstrated that his idea lead to consistently profitable results over the years (Scott et al., 2017).
Technical analysis involves the use of historical prices and other past data to make investment decisions. Technical analysts believe all fundamental influences are already represented in the stock price; therefore, the importance is weighted heavily on the price trend rather than the underlying fundamentals of the index performance.
Technical analysis only works where there are inefficiencies in the market. This suggests that a security’s price does not accurately reflect all currently available information, and therefore, does not represent the true fair value of the security. This contradicts the Efficient Market Hypothesis (EMH). Although the EMH assumption is believed to be true, it can be affected by unforeseen market events such as news and company announcements. These market movements would affect price on a short-term horizon and the EMH assumption suggests the price would result back to the fair market price. These volatility swings provide traders with the opportunity to speculate and potentially profit from these changes (Neely et al., 2014).
Fama-French 3-Factor Model
The root of fundamental portfolio analysis is the modern portfolio theory (MPT), introduced by Markowitz in 1952, which simplifies modeling asset returns using a multivariate normal distribution. MPT defines the distribution of returns in terms of mean returns, return variances, and return correlations. Following Markowitz, Sharpe introduced the capital asset pricing model (CAPM) in 1964 which models expected return of assets and introduced investors to some influential concepts such as alpha, beta, and systematic risk (DeFusco et.al., Pg. 526).
The CAPM formula is represented below where (ra) is the expected asset’s return as the dependent variable and the expected market return (rm), risk free rate (rf) as independent variables. The (rm-rf) factor is the estimated compensation the investor would receive if they were to invest in the asset while 𝛽𝑎 coefficient is the sensitivity of the asset return in relation to the market.
𝐶𝐴𝑃𝑀: 𝑟𝑎 = 𝑟𝑓 + 𝛽𝑎 ∗ (𝑟𝑚 − 𝑟𝑓)
Since development of Sharpe’s CAPM, the flood gates opened for fundamental theorists to find enhancements to the CAPM since there were clear findings that not all returns were able to be explained by the elegant but simple CAPM structure.
Eugene Fama and Kenneth French expanded on the CAPM model to include two additional company characteristics which they found to accurately predict expected asset returns. They introduced the CAPM-enhanced model, known as the 3-factor Fama-French model by incorporating two more explanatory variables. There are several other factors and CAPM-enhanced models but for the purpose of this project, I will elaborate only on the Fama-French 3-factor model.
The first addition Fama and French incorporated into the CAPM was the SMB factor, or small-minus-big. SMB refers to the small and big capitalization stocks. SMB is calculated by the average return on the fourth quantile of small-cap firms minus the average return on the first quantile of large-cap firms. The incorporation of this size factor to the model provides insights to the sensitivity of the asset returns to the small capitalization firms. For example, if the coefficient for SMB for a given asset were positive, the assets returns are positively correlated to the average small-cap firm returns. SMB is also referred to as being the ‘size effect’ for this reason.
The second addition was the incorporation of HML or high-minus-low. HML refers to the difference between companies with high book-to-market values and companies with low book-to-market values. HML is referred to as the ‘value effect’ since it captures the sensitivity of portfolio returns to high value stocks. For example, if the coefficient for HML is positive, the portfolio returns are suggested to have a positive correlation to the market average high value stock returns.
If the model is able to completely explain the asset’s performance, the 𝑎𝑖 (abnormal rate of return) is going to be near zero and statistically insignificant. The next section provides the modification made to the Fama-French 3-factor model to incorporate technical indicators.
The Models and Hypothesis
The model used in this research will use the Fama-French 3-Factor model along with an additional pseudo-variable for each of the three technical indicators. The additional variable is referred to as signal with the specific indicator name preceding it. Please note, ‘signal’ and ‘indicator’ are used inter-changeably but refer to the same pseudo-variable which I elaborate on shortly.
There are 3 indicators which will be tested separately for a total of 3 models where only the signal term will be changed to represent the technical indicator being tested. The following sections will provide the calculation and strategies used for each indicator. The models are as follows:
\((R_i - RF_t )= a_i + b_i (Mkt-RF) + s_iSMB + h_iHML + g_i MACDsignal + e_i\) \((R_i - RF_t )= a_i + b_i (Mkt-RF) + s_iSMB + h_iHML + g_i RSIsignal + e_i\) \((R_i - RF_t )= a_i + b_i (Mkt-RF) + s_iSMB + h_iHML + g_i BBsignal + e_i\)
Ho: The asset returns do not reflect positive sensitivity to the indicator
Ha: The asset returns are positively sensitive to the indicator
The Method
For each of three models, the indicators use a bivariate pseudo-variable. The assignment (0 or 1) is defined by the assumption that the model is not holding the asset on day 1 (indicator = 0 to start) and will flip to 1 on the day which the long/buy strategy is satisfied. If the preceding days do not create a short/sell signal as per the strategy mentioned, the indicator remains as 1 to indicate holding. Alternatively, if the model shows t-1 as holding (or at 1) and a sell signal is received on t, the indicator will flip to 0.
The purpose of using the bivariate (zero and one) pseudo-variable is to represent excess returns of an investor who takes a long or hold position with the pseudo-variable 1. Alternatively, a short position is indicated by the pseudo-variable 0.
The Indicators
It’s worth noting that a single indicator is rarely used by analysts for investment decisions. Most often, indicators are paired since each indicator has their own respective objective. There many types of indicators but for the purpose of this project’s objective, I’ve selected one indicator from three popular groups based on their objectives; trend-following, momentum, and volatility.
Moving Average Convergence – Divergence (MACD)
Trend-following indicators help to identify trends that a specific asset price is following. They are a lagging type of indicator because they are based on past price data to provide a signal when the trend has already been established. One of the most common trend-following indicators, and the one used in this paper is the moving average convergence-divergence (MACD).
Moving average convergence-divergence (MACD) was created by Gerald Appel in the 1970s. Appel used the exponential moving average (EMA) technique.The indicator is created by the difference between two moving averages: a fast-moving average to reflect short term trends, and a long moving average to reflect long term trends. The long-term EMA is usually calculated using 26 periods while the short-term is typically 12 periods.
Next, requires a signal line which is the EMA of the MACD line. The signal line is typically 9 periods. These period lengths are not hard rules but for simplicity, this project uses the standard period lengths (26,12,9) proposed by Appel.
With the two lines, MACD and the signal line, we can then generate signals (or indicators) which are when the two lines intersect. Appel suggests that these signals are more robust when the long crossover takes place below 0 and short crossover takes place above 0 (Appel 2008). The trading indicator method for MACD is:
Long/Buy at period t: MACDt-1 < Signalt-1 and MACDt > Signalt Short/Sell at period t: MACDt-1 >Signalt-1 and MACDt < Signalt
Relative Strength Index (RSI)
Momentum indicators are often called oscillators because the line derived by the calculation oscillates above and below a pre-defined level. The objective of this type of indicator is to measure the relative strength of the recent price moves and to indicate when an asset is oversold or overbought. The momentum indicator selected for this project is the Relative Strength Index (RSI).
The Relative Strength Index or RSI was developed by the mechanical engineer turned technical analyst, J. Welles Wilder in 1978. RSI measures both speed and rate of change in price movements. The RSI will fluctuate between 0 and 100. The RSI indicates that an asset is oversold when below 30 and overbought when the RSI is over 70. Therefore, the RSI trading method for this project has been set to the following:
Long/Buy at period t: RSIt-1 < 30 and RSIt > 30 Short/Sell at period t: RSIt-1 >70 and RSIt < 70
Bollinger Bands (BB)
Volatility indicators measure the rate of price changes usually in reference to the exponential moving average or EMA. Volatility indicators rely on the assumption that the price is mean reverting. The indicator selected from this group to be used in this project is Bollinger Bands (BB).
Bollinger Bands (BB) was invented by John Bollinger, CFA, CMT in the 1980s. The key to these bands is that the price movements over time will fall within the upper and lower bands. The area between the upper and lower bands are often referred to as the “envelope”.The standard deviation is used to express the difference of the price movement from its mean value. The term ‘breakout’ is used when the price changes above the upper band or below the lower band. The breakout is used to signal a long or short position. The trading method is as follows:
Long/Buy at period t: Pricet-1 < Upper Bandt-1 and Pricet > Upper Bandt Short/Sell at period t: Pricet-1 > Lower Bandt-1 and Pricet < Lower Bandt
Data Description
Heteroskedasticity
Heteroskedasticity occurs when the variance of the errors differs across observations. To test my model for conditional or unconditional heteroskedasticity, I’ve used the Breush-Pagon and White tests. The Breush-Pagon test for conditional heteroskedasticity regresses the squared residuals from the estimated regression equation on the independent variables in the regression. Although the Breush-Pagon is reliable and a robust test, I sed the White test as well for reference to see the differences among each test result.
Both the Breush-Pagon and the Whites tests identified statistically significant conditional heteroskedasticity.
Autocorrelation
The second test on my original model was for autocorrelation. Autocorrelation or serial correlation occurs when regression errors are correlated across observations. I chose to use a visual test for autocorrelation in my project by plotting the errors and using the python stats models module. This is an alternative to the Durbin-Watson test since this is working with a large data set over a time-series. The Durbin-Watson results in the original regression are misleading since the result was close to 2 and could have easily been dismissed as non-serially correlated. This approach provides a visual of the autocorrelation bias.
As you can see from the plot, there are some autocorrelations which are statistically significantly different from zero. The light-blue bands provide the standard deviation from 0 and there are residuals which fall outside of that band.
Results
The following OLS regression results for MACD, RSI, and BB signals were corrected for autocorrelation and heteroskedasticity by incorporating the stats model robust covariance, HAC.
Conclusion
MACD
The MACD pseudo-variable for the indicator results in a high R2 (0.928) which suggest that about 93% of the risk-free returns of the fund are explained by this model. The F-statistic is large and statistically significant. The model as a whole appears stable and ready for analysis of the results.
The intercept term, previously explained as the alpha excess returns, is statistically significant which suggests that not all excess returns are explained by these factors and indicator. The market beta coefficient (Mkt_RF) is 3.2118 this is in-line with the fund’s prospectus which is to provide fund returns 3x the market.
The SMB factor is sensitivity is negative and statistically significant This suggests the fund is more sensitive to large-cap stock price changes. This again, makes sense based on the prospectus since the fund exposure is of the 100 top performers of the Nasdaq. The HML factor is also statistically significant with a negative coefficient, which suggest the fund is negatively correlated to value stocks. The portfolio of TQQQ primarily holds growth stocks which is in-line with these results.
The MACD indicator is statistically significant with a t-statistic well above the rejection level of 2. As previously mentioned, the signal (or indicator) is a constructed pseudo-variable which is 1 for a long/buy, 1 for hold, and 0 for short/sell. The coefficient is positive in this result, therefore if the indicator were to be used and traded when a signal occurs, it will add up to a small amount of the total ETF return. The null hypothesis for the MACD indicator can be rejected and the alternative that the asset returns are positively sensitive to the indicator is accepted at the 99% confidence level.
RSI
The OLS regression model for using RSI is also stable with a slightly lower R-squared, high F-statistic with p-value of 0. The alpha or intercept term is no longer significant in this model. The risk-free market beta, SMB, and HML are all generally the same conclusions as the ones stated for the MACD results.
This result fails to reject the null hypothesis since the RSI indicator is statistically insignificant based on the low t-test results and would otherwise be removed from this model.
It is worth noting that the RSI performed poorly in almost all ETFs I’ve regressed. Based on the results from this test, RSI is not a statistically significant indicator, but I think it is worth further testing to conclude if the insignificant result is due to the application of the signal or its performance (See general comments section 1 for additional comments on this).
Bollinger Bands
The Bollinger Bands indicator performed the best in this model with the highest coefficient compared to RSI and MACD. The coefficient is positive with the value of 0.0048 with a t-value well above the rejection level of 2. I can reject the null hypothesis at the 99% confidence level and conclude that Bollinger Bands are the best of the 3 indicators for identifying excess returns for trading the ETF ticker TQQQ.
The results of the OLS regression are mixed based on the indicator used. I was able to reject the null hypothesis for MACD and Bollinger Band indicators. However, the RSI indicator does not reflect a statistically significant sensitivity in relation to this fund’s returns. Although only two of the three indicators tested in this model were statistically significant, these findings support motivation for further study.
References
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Bollinger, J. (2018). John bollinger’s Official Bollinger band website. Retrieved April 18, 2021, from https://www.bollingerbands.com/
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