Abstract

This paper will investigate the returns of PSAR and modified PSAR strategies compared to returns of the passive Buy and Hold strategy over several different time periods. Both in- sample and out-of-sample tests are utilized in this process as well as often overlooked market frictions such as transaction costs. Both the active and passive strategies are tested on two different market indexes: S&P 500 and NASDAQ. Results suggest that the passive Buy and Hold strategy outperforms all of the active market timing strategies tested, but also experiences more volatility. Perhaps more interestingly, the newly modified Decelerated PSAR seems to produce higher returns on average than that of its original predecessor, PSAR.

Introduction

The stock market has always garnered attention from the masses. It is an electronic marketplace where anyone can earn above average returns. These grandiose possibilities have led to plenty of people trying their hand at trading and developing trading strategies. However, very few have managed to sustain high profitability consistently over time. This is not because research has failed, but rather it is by design in a free and open market. Any sure strategy (arbitrage opportunity) would quickly be exploited until markets shifted and the strategy became futile. That does not mean a profit cannot be made, just that it is not guaranteed.

The last decade or so has shown a massive influx of capital into Exchange Traded Funds (ETFs) due to the rise in popularity of the passive Buy and Hold method. This has been the tried and true method and for many this is all that is needed/wanted. However, some professional traders find that they have an inability to do nothing. Complex strategies are often picked because of the constant desire of the investor to do something, regardless of whether or not it adds value. All of this is done in the pursuit of Alpha, or the excess return relative to the market. In order to achieve this, many turn to various analysis strategies to give them an edge.

There are two major schools of thought when it comes to approaching the markets, fundamental analysis and technical analysis. Both methods are used to analyze and forecast future movements in stock prices. Fundamental analysis is based on evaluating securities’ intrinsic value. This is done by analyzing companies’ earnings, revenue, assets, liabilities, and even the overall economy. Technical analysts believe that all of these factors are already incorporated in the stock’s price, and as such, they aren’t taken into consideration. Instead, technical analysts use a stock’s price, volume, and charts to identify patterns and trends that can be used to predict future stock movements. These calculations are often referred to as technical indicators.

There are two basic types of technical indicators, overlays and oscillators. Overlays are plotted over the top of stock price charts because they use the same scale and oscillators are plotted below the price chart because they do not use the same scale. Common overlay indicators are Bollinger Bands and Moving Averages (MA). Whereas popular oscillators are Relative Strength Index (RSI), Rate of Change (ROC) and Stochastic Oscillator. Most commonly these indicators are used in tandem with one another to confirm different signals. However, this is on a case by case basis and often can come down to the knowledge of the user and the simplicity/approachability of the indicator.

This paper will look at an overlay indicator, Parabolic Stop and Reverse, herein referred to as PSAR for short, independently from any other indicators. The aim of this paper is to determine the efficacy of two new technical indicators, Support PSAR and Decelerated PSAR, by comparing their cumulative returns to the returns of the aforementioned original PSAR indicator. This paper also will compare the returns of the various PSAR strategies to the returns from the passive Buy and Hold strategy. The Support PSAR and Decelerated PSAR technical indicators are an attempted improvement upon the already derived PSAR technical indicator and will be described further throughout this paper.

Literature Review

Technical analysis represents a set of analysis methods used to examine past prices (and volumes) to predict future price movements in financial markets. It can be defined in several different ways, so for the use of this study the following definition from Ciana will be utilized:

The Dow Theory, created in the early 1900s and named after Charles Henry Dow, was the first introduction of modern technical analysis as we know it today. Although named after Charles Dow, the theory was primarily developed by William Peter Hamilton and then grew in popularity thanks to Robert Rhea. The Dow Theory, predicated upon Dow’s editorials and averages in the Wall Street Journal, attempts to predict prolonged patterns and trends in the stock market. The earliest attempt to test the Dow Theory was by Alfred Cowles. Cowles’ study was based on the trading signals published in the Wall Street Journal in Hamilton’s editorials and attempted to compare the performance of the Dow strategy and the passive strategy. The paper concluded that the passive strategy outperformed the Dow strategy.

Ever since Cowles’ study, the usefulness of Dow Theory (and technical analysis as a whole) has been debated. Academics, in contrast to practitioners, tend to be more critical of technical analysis, with many believing it be closer to art than science. The disparity between a practitioner’s and academic’s adoption of technical analysis can likely be linked to (1) varying acceptance of the efficient market hypothesis, and (2) negative empirical findings in several early and widely cited studies of technical analysis in the stock market, such as Fama and Blume, Van Horne and Parker and Jensen and Benington. However, several recent studies have shown that trading strategies based upon technical analysis would have protected investors throughout highly volatile periods (e.g. the early 2000’s dot com bubble and 2008 global financial crisis) by providing a higher return than the passive buy and hold strategy. These studies were based on several different considerations such as: look-back period, asset class, performance measures and technical indicator utilized. Thus, this literature review will discuss all of the previously stated predictor variables in the following sub-sections.

Look Back Period

The choice of window that a study will be conducted in has always been of interest and concern to practitioners since the time period and size of the window can lead to different empirical results. Inoue and Rossi looked at assessing the robustness of the predicative capabilities of a model given the choice in window size. Their study shows the significance the chosen window can have on the results of the study and as such they argue that the window should not be arbitrarily selected. Zakamulin found that the active market timing strategy outperforms the passive buy-hold strategy during bear markets and vice versa during bull markets. To account for this finding, Zakamulin proposes that the look-back period includes both bear and bull markets while staying as short as possible to observe both those market conditions. Although Zakamulin argues that the shorter time length that still contains these conditions is better, this paper will use an initial sample length that covers the past twenty years (January 1st, 2000 to January 1st, 2020) because it includes multiple bear and bull markets, some of which were significant; the most recent bull market lasting over a decade. Additional tests will also be conducted with varying smaller sample periods to ensure robustness in the results obtained.

Asset Classes

Technical analysis use is not distributed equally across all asset classes. Menkoff and Taylor found that the majority of foreign exchange (FOREX) traders used technical analysis in some way and Billingsley and Chance found that ~60% of commodity trading advisors used technical trading systems, whereas other markets haven’t adopted its use as eagerly. Although, most studies point to differences in technical analysis use and usefulness across asset classes some studies, such as Moskowitz, Ooi and Pedersen (2012), studied similarities. They found that time series momentum was “remarkably consistent” across varying asset classes. Additionally, Murphy found that there was an enormous overlap between technical analysis of stock markets and futures markets in his argument for the supplemental use of intermarket technical analysis in addition to traditional technical analysis. While there is still a lack of consensus amongst academics, according to a survey of fund managers by Menkoff, technical analysis is the most important form of analysis for forecasting the short-term U.S. equity market. That is why this study will focus on the U.S. equity market.

Performance Measures

According to Cogneau and Hübner, there are over 100 different ways to measure portfolio performance, with the simplest/most common being excess return. It measures the total return less the risk-free rate of return. However, it is not the best performance measure because it disregards risk. A risk-adjusted measure is more appropriate because investors require compensation for risk.

Other popular methods that are commonly utilized in academia and include risk as a factor are the Sharpe Ratio, Jensen’s Alpha, and Fama-French 3-factor model. Each of these methods has varying advantages and disadvantages, but this study will utilize the Sharpe ratio because of the overwhelming precedence set in previous literature. Additionally, drawdown is very commonly used by practitioners to assess the riskiness of any given strategy. Motivated by a desire for practicality, this measure will also be used in this study.

Technical Indicators

This section will assess the former literature on active market timing strategies with especial significance on the profitability of momentum-based trading strategies. Comparable to Modell and Lynngård and Park and Irwin, this paper partitions the reviewed literature into Early Studies: 1960-1987, and Modern Studies: 1988 – 2017. This split point was picked for 3 main reasons: (1) numerous early papers do not conduct a statistical test to validate empirical findings, (2) many studies after the late 1980’s improve upon earlier studies, and (3) the bulk of published studies on technical analysis were submitted following this split point.

Early Studies: 1960 - 1987

Filter rules were developed in the early 1960s by Alexander to forecast market trends. After testing this strategy on the S&P Composite Index and on the DJIA index, Alexander deduced that once a move in stock prices is established, it is inclined to endure. This observation led to an increased interest in the field from academics. After facing criticism from Mandelbrot, Alexander corrected his calculation to include transaction costs, and found that the trading strategy was no longer profitable. Fama and Blume later applied Alexander’s filter rules over a broader time period and also concluded that the filter rules were inferior to the passive strategy when transaction costs were taken into account.

Joseph Granville presented the first money flow indicator in his 1963 book “Granville’s New Key to Stock Market Profits.” Granville believed that volume was one of the most important factors behind market price movements. Thus, he developed the On-Balance Volume (OBV) indicator to attempt to predict major price movements based on coinciding shifts in volume. His belief was that if volume increased sharply without a corresponding significant change in price, then the price would eventually move significantly as well (up or down).

Later, Levy developed an unrelated technique termed “relative strength” to forecast prices. In his study he used relative strength rules on NYSE listed stocks from 1960 to 1965 and found the strategy produced superior results; however, he noted that the lack of statistical tests meant his results were tentative. Jensen and Benington then replicated Levy’s study, but expanded the sample period and adjusted for risk and transaction costs, and concluded that the relative strength trading rules did not produce significant profits greater than the passive buy-and-hold strategy.

Unlike Levy’s “relative strength” which refers to strength in relation to other time series, Welles Wilder Jr. developed the Relative Strength Indicator (RSI) which measures velocity and magnitude of directional price movements. Siu-Man (1987) studied the profitability of RSI on the foreign exchange rate between the Yen and the U.S. Dollar and found that the profits depended on the number of days and RSI limit used, but that a “reasonable” profit could be generated. However, this study did not utilize a risk-adjusted profit measure or transaction costs.

Wilder also developed the Parabolic Stop and Reverse (PSAR) indicator in his same book “New Concepts in Technical Trading Systems” that RSI was first introduced in and stated its purpose was to determine an asset’s momentum direction. The momentum has a higher probability of reversing trends. Although this indicator was developed in 1978, it is only in the modern era that it has gained popularity.

In the early 1980’s, Marc Chaikin developed the Chaikin Money Flow (CMF) indicator that looked at buying and selling pressures of a stock over time. The indicator was designed to show signs of institutional buying or selling even though it only technically looks at price and volume activity. Chaikin’s CMF was a further development of his other indicator: Accumulation and Distribution (A/D).

Around the same time as Wilder and Chaikin developed these indicators, influential papers such as Fama were rising in popularity causing the prevalence of the Efficient Market Hypothesis in finance to skyrocket. This led to fewer papers on technical analysis being published in the 1970’s and 1980’s and consequently hardly any papers (at the time) analyzing the performance of these indicators.

Modern Studies: 1988-2017

Unlike earlier studies, Lukac, Brorsen, and Irwin implemented an out-of-sample test, a statistical significance test, and adjusted for transaction costs. It is perhaps the first study on technical analysis to do so. Lukac et al. utilized Jensen’s alpha to estimate significant risk adjusted profits. They were able to conclude that 4 out of the 12 trading systems they studied produced statistically significant profits. Lukac and Brorsen extended their original study to include more trading strategies and a longer time horizon and found more positive results with 7 out of the now 23 strategies they studied producing statistically significant positive returns.

Brock et al. is regarded as one of the most influential modern studies on technical analysis. The paper analyzed 26 trading rules on the DJIA from 1897 to 1986 and found that all of the trading rules they analyzed generated significant profits and has predictive power. However, like several previous studies examined in this literature review, this study did not account for transaction costs and only simulated returns in-sample. Only simulating returns in-sample can lead to overfitting and an inferior market timing strategy out of sample with likely negative expected returns. Park and Irwin found that net returns on the Brock et al. study was no longer significant after examining break even transaction costs from Bessembinder and Chan study that used the same trading rules. However, one year later Sullivan et al. once again tested the same data as Brock et al. to analyze returns, but they extended the sample period to include an out-of-sample test and assessed the level of “data mining” by utilizing White’s Reality Check methodology. Sullivan et. al found statistically significant profits (similar to Brock et al.) and, arguably just as important, their results showed no evidence of “data mining”. “Data mining” is when the optimal combination of parameters (that yield the best performance) are chosen when searching for outperformance in an in-sample-test. These results are significant because “data mining” although not a new concept is often not adjusted for despite it being a well-known problem. This can lead to overestimation of performance and lower returns when the strategy is tested out- of-sample.

As opposed to trying to prove market timing strategies superiority like the majority of technical analysis studies, Zakamulin attempted to disprove it. An out-of-sample test was utilized to examine the practical performance of two trading strategies over time. Zakamulin found that none of the active market timing strategies tested produced statistically significant results that outperformed the buy-and-hold strategy to which they were compared.

There is very little consensus in regards to the profitability of trading strategies based on volume and momentum rules. Multiple studies do show significant positive returns; however, certain deficiencies seem to endure. Specifically, many of the reviewed studies do not take into consideration important market frictions such as transaction costs. Furthermore, many of the reviewed studies do not utilize an out-of-sample test to correct for data mining bias.

As an alternative, some researchers have conducted studies to prove that market timing strategies do produce superior returns to the passive buy-and-hold strategy by attempting to improve upon popular technical indicators despite varying findings regarding their profitability in their current state. This is discussed in the following section.

Modified Technical Indicators

Plenty of studies have attempted to modify popular technical indicators, but two of the more successful attempts on momentum-based indicators are as follows:

In 1989, Gene Quong and Avrum Soudack developed the Money Flow Index (MFI) which is a modification of RSI and also commonly referred to as volume-weighted RSI. They state that MFI measures the strength of money entering and leaving the market (for each asset). Marek and Marková (2020) found that MFI trading strategy can yield higher returns than the passive buy-and-hold strategy. Additionally, they compared their results to those of Marek and Šedivá (2017) and concluded that MFI outperformed the RSI trading strategy. Lastly, they found that there was room for improvement in the recommended parameters of MFI and suggested several optimized parameters.

Twiggs developed the Twiggs Money Flow which is an adaptation of Chaikin Money Flow indicator, and is designed to overcome some of the CMF’s shortcomings, such as failing to identify price gaps. It does this by using true range instead of daily high minus low and EMA instead of SMA. Srichawla and Romprasert found that as a standalone indicator Twiggs Money Flow performed the best out of the indicators tested and statistically significantly outperformed the buy-and-hold benchmark strategies. Their study also included transaction costs, so these results are more realistic.

The main goal of this paper is to extend the existing literature on momentum-based trading strategies by modifying Parabolic SAR (PSAR) and also taking into account transaction costs and data mining bias when testing the performance of the new strategies. This should provide an unbiased comparison of cumulative returns generated by the new trading strategies in real-life conditions against the passive buy-and-hold strategy as well as the previously derived PSAR strategy.

Methodology

Market Timing Rules and Parabolic SAR

Technical analysis practitioners typically believe that prices move in patterns and that these patterns can not only be determined in advance, but also traded upon. Thus, by doing so, traders can attempt to profit from the price movements. In its most rudimentary form this looks like buying (selling) when the asset is trending upward (downward). However, as many practitioners can probably say from experience, this is easier said than done. That is why traders turn to momentum indicators to help them determine the trend and good entry and exit points. One specific indicator that practitioners use for this purpose is the Parabolic Stop and Reverse (PSAR) indicator. PSAR is almost exclusively calculated using computers because of the difficulty and time involved in calculating it. The formula is best shown as follows:

Where, EP (Extreme Price) is the highest high for the uptrend and the lowest low for the downtrend and AF (Acceleration Factor) is set at a default of 0.02 and increases by 0.02 each time a new EP is reached. Each time a new EP is reached it is updated in the formula, which causes the AF to increase by the step of 0.02. This happens until a maximum value of 0.20 is reached. This calculation creates a dot above or below the price line. If the dots are below then it indicates an upwards trend and if the dots are above it indicates a downwards trend. This can be seen in Figure 1 below, for historic values of the S&P 500 in 2019. The daily prices are shown as the blue line and the PSAR values are shown as the orange hollow diamonds.

Chart 1: Original PSAR indicator over a 1-year period of S&P 500 daily prices in 2019.

PSAR Modifications

While PSAR is a commonly used technical indicator, it does have some limitations. In particular, it doesn’t work as well in sideways markets (when price is range bound with limited movement up or down). This is because PSAR moves regardless of price and the acceleration factor can cause PSAR to “catch up” to the price after enough time has passed. This can cause a reversal signal to be produced even if the price is not actually reversing. Additionally, PSAR can tend to overproduce trade signals in sideways markets, causing lots of smaller less profitable trades. This is an issue when transaction costs are taken into consideration. This paper proposes two modifications to attempt to remedy or improve upon these limitations: Support PSAR and Decelerated PSAR.

Support PSAR

Support PSAR attempts to improve upon the over eagerness of PSAR to identify trends in a sideways market by stipulating that two bars must cross the price line (with the second bar being fully across) before the trade signal is produced; the original bar and then one supporting the trend prediction. This second bar is meant to confirm the trend that is predicted by the first, thus supporting the original signal and providing a stronger basis on which to trade. This is graphically shown below:

Figure 2: Original PSAR indicator compared to Support PSAR indicator over a 4-month period of S&P 500 daily prices in 2019.

The above graphs show the exact same time period with two different indicators overlaid; the Original PSAR (left) and the modified Support PSAR (right). To highlight the differentiating factor of the modified indicator (stricter parameter before a trade signal is produced) a specific set of candles on the graphs are called to attention. On the left, for the Original PSAR, the first candle breaks the PSAR trend line and thus a trade signal is produced and the PSAR trend line is reversed.

However, on the right, for the Support PSAR, although the first candle breaks the same PSAR trend line as in the case of the original, the next candle does not fully break the line thus no trade signal is produced and the trend line continues.

Decelerated PSAR

Decelerated PSAR also attempts to improve upon the false signals produced by PSAR in sideways markets. It works by decreasing the acceleration factor (used in the original formula) whenever there has not been a large enough price change over a specific time frame. Although these parameters can be flexed, this strategy was tested with a 2% price change cushion within a 3-day period. If the price does not change by this amount (Δprice <2%) then the acceleration factor would be decreased by 0.05. This decrease is classified as a backwards step and in effect nullifies the constant forward step of 0.02 of the acceleration factor (AF). The choice to use a 2% price change in a 3-day period was based on the prevalence of 3-day periods experiencing price changes over this threshold in times of trend reversal.

Figure 3: Price change of 3-day sample periods compared to macro trends in S&P 500

Figure 3 shows the price change of all of the consecutive 3-day periods from January 1st, 2000 to January 1st, 2020. Overlaid on top of that are two red line thresholds showing ± 2% and a blue line showing the price of the S&P 500 over the same period of time. Thus, it is clear to see that whenever the price change was larger than 2% (up or down) it correlated with periods of higher volatility and trend reversals. On the contrary whenever price changes were range bound between the +2% and -2% thresholds, the price trend of the S&P 500 was fairly consistent. These periods of consistent uptrends are when trading transactions should be minimized to optimize profitability. This finding is what motivated the creation of the Deceleration PSAR.

The purpose of the implementation of a backwards step in the Deceleration PSAR is to slow down the process of PSAR catching up to the price movement in sideways markets and producing a false signal. This would in turn produce fewer trade signals in consistently trending markets. This is executed by using various loops and if-statements in the code of the modified strategy (attached in Appendix A).

Trading Signals

This trading signal set will also only be utilized for long trades. This paper will not include short selling. Fewer trade signals are expected to be seen for the two modifications than for the original PSAR indicator, as the two modified versions have more stringent parameters in place prior to the creation of a trade signal.

Testing The Profitability

To evaluate the profitability of the above stated market timing strategies, this paper utilizes the Back Test (In Sample Test) and the Forward Test (Out of Sample Test). These tests were chosen because of their extensive use in reviewed literature as well as the robustness they provide when deployed in tandem.

Back Testing

A back-test is tantamount to a simulation, in which a specific historical time frame is utilized to analyze how a specific trading strategy would have performed. Back testing is often employed by traders and researchers to attempt to optimize various trading strategies. This is done by using the full range of historic data available. However, this process often leads to “data-mining” bias. Meaning that the proposed “best” trading strategy from the plethora of tested strategies will be biased upward because it was chosen specifically as the optimal for the examined time frame. This upward bias is most commonly referred to as “data mining bias” and is known for increasing the probability of Type 1 error. Type 1 error is best explained as a false positive. This bias can best be understood by examining the two components of the observed performance of the trading rule; true performance and randomness. When a dataset is overused in an attempt to extract the best trading strategy that maximizes return over that time period, then the resulting strategy likely benefits the most from this random component as opposed to the efficacy of the trading strategy in general. To mitigate “data mining bias” the Sharpe ratio of the portfolio (asset) utilizing the proposed trading strategy should be reduced by 50%.

Forward Testing

Given the overestimation of the profitability of trading strategies produced from back-testing, an out-of-sample test (forward test) is also utilized. This is done by analyzing the performance of the proposed trading strategy in an alternative time frame than that which the previous dataset contained. This allows for the performance of the trading strategy to either be validated or negated. It works by dividing the historical dataset into two distinct subsections. One subsection is used for the in-sample test and the other subsection is used for the out-of- sample test. For the purposes of this paper we will denote the two subsections as in-sample [1, s] and out-of-sample [s + 1, T], where s is the split point and T is the last point of the dataset (for daily data this would be the last day).

This annotation is similar to that used in Modell and Lynngård 2017. As discussed earlier in this paper, the window chosen to evaluate the performance of the proposed strategy makes a difference to the produced results. Likewise, the split point is equally important. Sullivan et al. uses a split point towards the end of the historical sample. Given how widely cited this paper was, we have chosen to go with a similar set up. The first split point, January 2020, means 20 years for the in-sample and 8 months for the out-of-sample. Although this seems like a short time frame for the out of sample, it does consist of a very clear bear and bull market, meaning it should encapsulate multiple market conditions to ensure a fair assessment. However, motivated by the understanding that the split point can have a measurable effect on the performance we have decided to employ a second split point as well. The second split point is January 2017, which means 17 years for the in-sample test and not quite 4 years for the out-of-sample. This second split point gives a longer out-of-sample period without adversely affecting the in-sample test since it does not remove all of the historic bull run (2010-2019) from the in-sample period.

Transaction costs

Real-world investors have to consider transaction costs; thus, this paper will include them in our analysis of profitability for the various strategies considered. Not only will this make this analysis more realistic, but additionally it is one of the differentiating factors from previous papers in the cited literature. Transaction costs are commonly broken up into two subsets; direct and indirect costs.

Direct costs are things such as commissions, fees and taxes. Indirect costs are the factors such as market impact costs (related to liquidity) and opportunity costs. Frazzini et al. found that the key driver behind market impact cost heterogeneity is liquidity in the form of the size of the trade as a proportion of the daily volume traded. Given that all of these factors vary significantly given the specifics of any individual trade, overall generalizations had to be made for this study. This paper used a transaction cost of 0.25% (per side) throughout. This was chosen based off of its use in other papers reviewed.

Performance Measures

As previously discussed, there are a number of different ways to measure performance; the simplest being excess return. The excess return is the return less the risk-free rate of return (typically US Treasury Bills are used here). However, there are better performance measures that include risk in their analysis. Risk is important to include because without it, investors could over leverage their portfolio as much as possible (borrowing limits being the main constraint) to achieve greater returns. In the real world we know that this is not possible and that investors not only encounter risk, but actually require compensation for taking it on. Thus, instead of simply being able to assess which strategy makes the most money, this dissertation implements the Sharpe Ratio, a widely utilized performance measure in academic research. However, drawdowns seem to be far more important to practitioners so that will also be calculated for each of the strategies tested.

The Sharpe Ratio

The Sharpe Ratio, created by William Sharpe in 1966 (Sharpe, 1966) and later generalized in 1994. It is widely used by academics in the prevailing literature and is formally written as:

Drawdown

As opposed to the Sharpe Ratio that uses standard deviation which handicaps upside probable gains at the same rate as prospective downside losses, drawdown only analyzes the downside risk. Maximum drawdown measures the percentage decrease of a portfolio from its highest point (peak) to its subsequent lowest point (trough). This is formulaically shown as:

Where Ppeak is the price of the portfolio at its highest point, and Ptrough is the price of the portfolio at its subsequent lowest point. Maximum drawdown was chosen as the performance measure of choice to transparently show what extreme losses investors would have to be able to stomach in order to stay invested in each corresponding strategy. This is not only important because of its measure of financial risk, but additionally in regards to behavioral finance and at what point investors choose to sell to try and cut their losses.

Statistical Analysis for Outperformance

To determine if any potential outperformance of active market-timing strategies is statistically significant and not just pure coincidence, this paper implements box and whisker graphs that plot the mean, median, and quartiles of a dataset. Figure 4 below shows how to read these graphs.

Figure 4: Depiction of a generic Box and Whisker Graph (sometimes referred to as a Box Plot) with all of the individual parts labeled. IQR stands for Interquartile Range. Lower Quartile is sometimes called Q1 or 1st quartile and Upper Quartile is sometimes called Q3 or 3rd quartile.

These graphs will be implemented to compare the relative performance of each strategy to each other. The plots will allow for quick assessment of whether or not one strategy outperformed the other and by how much. This will be assessed by looking at the median and IQR position compared to each as well as assessing whether certain returns are outliers. Lastly, these plots will be useful in determining the skewness and dispersion of each strategy’s set of returns.

Results

This section illustrates the results of the empirical analysis. The methods described in Section 3 were used to present results for all four strategies described for both in-sample and out-of- sample tests conducted, and for both the Standard and Poor’s 500 (S&P 500) and the National Association of Securities Dealers Automated Quotations (NASDAQ) indices. All results were found and presented using a combination of MATLAB and Excel and all historical data used was daily frequency data.

Figure 5: Price changes in S&P 500 and the NASDAQ indices over the past twenty years from January 1st, 2000 to August 21st, 2020, the same timeframe used to test our strategies.

The above graph is important in relation to the relative performance of strategies tested on both indices. It is apparent from the graph that both the S&P 500 and NASDAQ experienced periods of both bear and bull markets. However, the NASDAQ clearly has had a larger bull market in recent years.

S&P 500

When testing the four different strategies (Buy and Hold, PSAR, Support PSAR, and Decelerated PSAR) on the S&P 500 it became apparent that the passive Buy and Hold strategy outperformed the three active market-timing strategies (PSAR and the two modifications) for all of the sample periods tested, except for the forward test of the 1st split point, as seen in Table 1 below.

Table 1: Cumulative Returns, Sharpe Ratios, and Max Drawdowns for the four tested strategies with two different split points for in-sample and out-of-sample simulations for the S&P 500.

The Decelerated PSAR strategy seemed to perform the best out of the three PSAR strategies tested. The one exception being, once again, the forward test of the 1st split point. However, it should be noted that the degree to which the original PSAR strategy outperformed the Decelerated PSAR strategy in that sample is not as large as the degree to which the Decelerated PSAR outperformed it in the three other sample periods. Additionally, the Decelerated PSAR had the highest Sharpe Ratio out of all of the strategies tested, over all of the time frames tested. The Max Drawdowns showed that all of the tested strategies experienced significant losses at some point in the tested period. The overall performance of the four tested strategies can better be seen in Figure 6 below.

Figure 6: Box and Whisker graph of cumulative returns of the four trading strategies tested on the S&P 500. The “X” in the boxes represents the mean of each respective strategy. The horizontal line in the middle of the boxes represents the median of each strategy.

Figure 6 above clearly shows that the first modification, Support PSAR, was not as successful as the second modification, Decelerated PSAR. Both the mean and median cumulative return for the PSAR and Support PSAR are negative, whereas the mean and median for Decelerated PSAR are positive. The comparative downside risk seems to have been negated for the Decelerated PSAR in comparison to the other two PSAR-based strategies.

NASDAQ

Similar to the testing on the S&P 500, the buy and hold strategy outperformed the PSAR, Support PSAR, and Decelerated PSAR strategies across the board, except for in the forward test of the 1st split point. In this sample period both the Support PSAR and Decelerated PSAR outperformed it, as shown in Table 2 below.

Table 2: Cumulative Returns, Sharpe Ratios, and Max Drawdowns for the four tested strategies with two different split points for in-sample and out-of-sample simulations for the NASDAQ index.

Decelerated PSAR once again had the highest Sharpe Ratio out of the tested strategies. The Max Drawdowns showed that once again the tested strategies had similar downside risk, but they were significantly higher for the NASDAQ than for the S&P 500. Also similar to the tests run on the S&P 500, the Decelerated PSAR performed the best out of the active market- timing strategies and Support PSAR was an improvement upon the Original PSAR, albeit not as big of an improvement as Decelerated PSAR, as shown in Figure 7 below. Figure 7 also shows how much closer the cumulative returns of Decelerated PSAR were to those of the Buy and Hold strategy for the NASDAQ tests than in Figure 6 for the S&P 500.

Arguably the most important result for the NASDAQ tests compared to the S&P 500 is that the Decelerated PSAR strategy didn’t produce a negative return during any of the time periods or split points tested above. This was not the case for the tests on the S&P 500 when it produced a negative return for the 2nd Split back-test (Table 1).

Figure 7: Box and Whisker graph of cumulative returns of the four trading strategies tested on the NASDAQ. The “X” in the boxes represents the mean of each respective strategy. The horizontal line in the middle of the boxes represents the median of each strategy.

Additional Tests

Motivated by the outperformance of the Decelerated PSAR strategy compared to original PSAR strategy and the improved performance of the market timing strategies in the in the shorter time frames of the out-of-sample tests compared to the longer time frames of the back-tests, we chose to run a few more tests of the Decelerated PSAR and PSAR strategy in shorter time frames randomly selected throughout our initial time frame. For this exercise in data robustness, a six-month, one- year, two-year, and five-year time length was chosen and each was to be tested ten times on randomly chosen time frames within the already sampled data.

Figure 8: These graphs show the price changes of the S&P 500 and the NASDAQ for the ten different 6-month periods selected for comparison.

Figure 9: These graphs show the price changes of the S&P 500 and the NASDAQ for the ten different 1-year periods selected for comparison.

Figure 10: These graphs show the price changes of the S&P 500 and the NASDAQ for the ten different 2-year periods selected for comparison.

Figure 11: These graphs show the price changes of the S&P 500 and the NASDAQ for the ten different 5-year periods selected for comparison.

Figures 8, 9, 10, and 11 show all of the different sample periods selected for the additional tests of the Decelerated PSAR and PSAR strategies, as well as the subsequent market conditions (bear, bull, sideways) that were happening during those smaller periods being tested. It is clear from these graphs that some of the shorter sample lengths did not cover the entire span of the initial sample period, which is to be expected. Additionally, from these graphs you can see which tests spanned over what different market conditions (i.e. bear or bull market).

No parameters of the tested strategies were altered before running these tests. This is an important clarification, especially in the case of the newly modified Decelerated PSAR strategy, to ensure that data mining bias should not be a consideration despite the same historical dataset being used. This is true because the dataset was not used to further optimize the strategy, rather just to obtain further results to base any observations off of. These additional tests are particularly helpful in determining any differences in performance due to the length of the sample period tested, and specific market conditions faced compared to the broad-brush strokes of the initial tests conducted. The results of these additional tests can be seen in Table 3 below:

Table 3: Cumulative returns from the Buy and Hold, PSAR, and Decelerated PSAR strategies for both the S&P 500 and NASDAQ indices over the 40 different additional selected sample periods (broken up into 6-month, 1-year, 2-year, 5-year categories). The asterisks (*) denote periods where market crashes occur (i.e. Dot Com Bubble or Financial Crisis of ’08).

The above table shows all of the cumulative returns results from the additional tests which were run. From this table one can ascertain that the Buy and Hold strategy outperformed both the Decelerated PSAR and Original PSAR strategies on average with a few exceptions. Most of those exceptions came during periods that incurred some sort of market crash (denoted with an asterisk in Table 3), however this was not true across the board. While the Buy and Hold strategy did outperform both of the other two tested strategies, the Decelerated PSAR strategy still performed better than the Original PSAR strategy that it was modified from. This can be better visualized in Figure 12 below:

Figure 12: Box and Whisker graph of the cumulative returns of the Buy and Hold, PSAR, and Decelerated PSAR strategies for the 40 additional sample periods on the S&P 500 and NASDAQ. Each dot represents an actual return from one of the additional tests. The ‘x’ in the box represents the mean of the returns and the horizontal line within the box represents the median.

Once again, the highs of the Decelerated PSAR were closer to that of the highs of the Buy and Hold when tested on the NASDAQ index versus the S&P 500. However, both indices show that the Decelerated PSAR outperformed the Original PSAR strategy on average over the 40 additional tests performed. To analyze this outperformance further, Figure 13 breaks up these returns by the time length of the sample periods tested. This visual allows us to compare returns over different time horizons.

Figure 13: Box and Whisker graphs comparing the cumulative returns of the Buy and Hold, PSAR, and Decelerated PSAR strategies over the differing time lengths of the sample periods tested. Cumulative returns for the different strategies on the S&P 500 and NASDAQ were combined for each time length analyzed.

The above figure shows a few interesting trends. First, it shows that as the length of the sample period increased, so did the median of the Buy and Hold and Decelerated PSAR strategies. Conversely, as the length of the sample period increased, the median of the Original PSAR strategy decreased. This demonstrates that investors with longer time horizons would see better performance from the Decelerated PSAR strategy or buy and hold strategy than from the original PSAR strategy. Additionally, Figure 12 shows a trend of increasing dispersion of the cumulative returns as the sample length increased. This is to be expected as higher returns and larger losses do correlate with longer investment periods. Perhaps most interestingly, the original PSAR strategy never had a positive mean return for the 4 different time lengths tested.

Statistics

The statistical box and whisker graphs above show that on average the Buy and Hold strategy is the best performer, with Decelerated PSAR second best and the Original PSAR strategy in last place.

The claims of outperformance by the Decelerated PSAR strategy against the Original PSAR strategy as well as the other modification, Support PSAR, when tested on the NASDAQ and S&P 500 indices is justified by the median and mean being higher for it than the others for all of the different tests, with the closest being the averaged 10 tests in the 6-month sample periods.

One other additional statistical observation made from the box and whisker plots above is that most of the samples show a normal distribution. The only exceptions are when the data is broken up by sample period length (Figure 13) as opposed to the indices tested.

Investors also often look at standard deviation to help them determine the volatility of returns from assets, portfolios, and/or trading strategies. To better gauge the performance of the tested trading strategies (excluding Support PSAR) the standard deviation was calculated. The tables below (Table 4 and 5) show these standard deviations as well as the respective mean for each of the relevant test samples performed.

Table 4: Standard Deviation and Mean values for Buy and Hold, PSAR, and Decelerated PSAR strategies on the S&P 500 index for differing time lengths.

Table 5: Standard Deviation and Mean values for Buy and Hold, PSAR, and Decelerated PSAR strategies on the NASDAQ index for differing time lengths.

Interestingly, the above tables show that the Original PSAR strategy actually had the smallest Standard Deviations on average, followed by Decelerated PSAR and then the Buy and Hold strategy had the highest. This means that the Decelerated PSAR strategy is actually less volatile than the passive Buy and Hold strategy, but more volatile than the Original PSAR. This is fairly regular given that the returns of these three strategies were inverse to this list with the highest average cumulative returns being made by the Buy and Hold and then the Decelerated PSAR, followed last by the Original PSAR and investors require higher returns for taking on more risk.

Discussion

This dissertation looked at comparing modified PSAR trading strategies not only to the original PSAR, but also to the passive buy-and-hold strategy. From the initial tests, it appeared that the Buy and Hold strategy was the superior strategy as far as cumulative returns were concerned. This is likely because the active market-timing strategies lagged the “perfect” entry and exit points when executing trades and thus missed out on potential profits. However, the Decelerated PSAR strategy was shown to outperform the Buy and Hold strategy on occasion. These occasional outperformances occurred in the forward test of the 1st split point for the initial tests conducted. This is significant because of the differing length of that sample period compared to the others used; it was considerably shorter than the other periods tested.

Motivated by this outperformance in the shorter time horizon, as well as the known importance of the sample period tested, several other shorter sample periods were picked to run additional tests on. This allowed for further analysis to test the robustness of the initial observations. After conducting these additional tests, it was found that the Buy and Hold strategy still achieved superior returns to those of either the Original PSAR strategy or the Decelerated PSAR strategy. However, the additional tests also provided much more insight into the improvement of the Decelerated PSAR strategy on the Original PSAR strategy. These additional tests suggested that, on average, the Decelerated PSAR strategy had higher returns than the original PSAR strategy.

Important to note is the degree of outperformance by the Buy and Hold strategy in the initial tests (shown in figures 7 and 6) versus that of the additional tests (shown in Figure 12). When compared, it is easy to see that the Buy and Hold strategy outperformed by much more in the initial tests (especially on the S&P 500) than in the additional tests. This coincides with the findings in Zakamulin that the passive strategy performs better in bull markets, which dominated the sample period for the initial test. Zakamulin wrote that because of this expected outperformance in these conditions, the sample periods should be minimized and include both bear and bull market conditions to get a fairer comparison. That is likely why the returns in the shorter length additional tests are more comparable, with the Decelerated PSAR even beating the Buy and Hold strategy on occasion.

One potential reason for the higher returns from the Decelerated PSAR strategy over the Original PSAR strategy is that it consistently had fewer trades made, likening it more to the buy and hold and also subjecting it to fewer transactions’ costs overall. The specific number of trades per strategy per sample period is broken up below in Table 6 by the index the strategies were tested on:

Table 6: Number of trades made for each market timing strategy in the initial tests performed on both the S&P 500 and the NASDAQ.

Also, of note, the volatility as shown through the standard deviations in Tables 4 and 5, shows that the Decelerated PSAR strategy is on average less volatile than the Buy and Hold strategy. Faber describes this observation as equity-like returns with bond-like volatility. While this paper’s results do not suggest the same bond-like standard deviations and drawdowns as expressed in Faber’s results, it does follow the same pattern of lower volatility while maintaining similar returns. The most likely reason for this observation is that the active strategy will have time spent in a cash position whereas the passive strategy will not. This time spent in a cash position will lower the volatility of the overall strategy because cash does not have the same downside risk (outside of inflation which rarely has noticeable affects in the short term). The breakdown of the strategy’s time spent in a trade versus time spent in cash can be seen in Table 7 below:

Table 7: Percentage time spent in cash for each market-timing strategy in the initial tests conducted on both the S&P 500 and the NASDAQ.

This time spent in cash has another side effect though, the return for periods held in cash are dependent on interest rates. For the purposes of this study, no return was allocated for the periods of time held in cash, because interest rates are so low. This reduces the overall return the strategy could produce. However, interest rates have not always been low and will not always be low, so this could cause these active strategies, especially Decelerated PSAR, to catch up to the overall returns of the passive strategy if interest rates rise. Additionally, this paper did not include short selling in its scope, this almost certainly adversely affected the calculated returns of the market timing strategies. As you can see from the above graph, these strategies spent a significant amount of time in cash, often for lengthy periods on end. Some of this time could have been spent shorting the market (when the sell signal was given) to provide for further returns on the long positions already taken. Taking this into consideration, the Decelerated PSAR strategy potentially could have outperformed the Buy and Hold strategy on a consistent basis.

Conclusion

This study has been built on substantial previous literature debating the efficacy of technical analysis. It has focused in on the Parabolic Stop and Reverse (PSAR) indicator and attempted to modify this indicator two different ways. The resulting modifications, Support PSAR and Decelerated PSAR, were then compared not only to the original indicator PSAR but also to the passive Buy and Hold strategy. These tests suggested that the Buy and Hold strategy is the superior strategy as far as cumulative returns are concerned, when only long positions are considered. However, the results also suggested that the newly modified Decelerated PSAR strategy has been a semi-successful improvement upon the original PSAR strategy. Further studies should be conducted to confirm these results by expanding upon the sample periods and split points tested. Further studies should also be conducted to determine the efficacy of these strategies on individual stocks rather than just indices. Lastly, further studies should attempt to flex the return on cash positions due to changing interest rates as well as the inclusion of short positions to the scope of the active market timing strategies.

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