Figure 1 Histograms of 5-year Returns from Seven Asset Prices

Random Walk Part 2 – Does Any Asset Price Fit the Bell Curve?

Originally Published September 25, 2017 in Advisor Perspectives

This is the second of my four-part empirical research into the fallacy of the random walk paradigm in investments. Part 1focused on the failure of the random walk to depict the Dow Jones Industrial Average. In this article, I expand the study to include six diverse asset classes including large-caps (the S&P500), small-caps (the Russell 2000), emerging markets, gold, the dollar and the 10-year Treasury bond. Asset prices do not walk in tiny steps along an orderly bell curve, but often take giant leaps leaving chaotic turbulence behind.

What are the common features among seven widely diverse asset classes? Why are asset prices so hard to pin down by random walk or other analytical and descriptive models? What are the investment implications if asset returns do not fit the bell curves?

Part 3 will deal with its flaws in defining investment risk. Part 4 offers a new reward and risk framework alternative to the random walk paradigm. The new framework yields new insights on how to logically beat the S&P 500 total return.

Empirical evidence against the random walk model

In Part 1, I presented price return histograms of the Dow Jones Industrial Average from 1900 to 2016. In this paper, I extend the empirical research to six other assets. Detailed results are in the Appendix. Figure 1 summarizes the five-year return histograms (light blue bars) of seven asset classes – the DJIA from Part 1 and six other assets from the Appendix. No return histogram (light blue bar) has any resemblance to its corresponding random walk probability density functions PDF (dark blue curve). All random walk PDFs are unimodal (with a single central mean) with matching wings on both sides. All empirical distributions have multiple peaks with no central symmetry.

Common themes among different asset classes

1. None of the return histograms of the seven asset classes follows the random walk bell curve for periods longer than one day. The longer the return periods, the less bell-shaped they become. Real world prices do not follow a random walk.

2. Even for the one-day returns, all histograms show asymmetric fat tails. Random walk theorists have no explanation for them and treat them as anomalous statistical outliers.

3. All return distributions of time horizons beyond one year do not have a single mean and a definable variance. Random walk's mean-variance paradigm does not reflect reality.

4. Random walk's bell curve underestimates the probabilities of both large losses and gains beyond ± one standard deviation. This has dire consequences in risk management.

5. It is standard practice to scale returns and volatilities in different time horizons by the number of trading days and by the square root of trading days, respectively. Data show that neither return nor the volatility follows such scaling rules.

Why are asset prices so elusive for the theoreticians to capture?

1. Random walk is not the only model that fails to explain price behaviors. Power-lawsgame theoryagent-based modelsbehavioral finance and adaptive market hypothesis are all incomplete theories at best. The rational beliefs equilibria model appears to have the potential for solving the asset-pricing puzzle but it is still too early to tell.

2. Why are prices so hard to pin down? The culprits are the agents involved. Unlike mindless pollens and particles that blindly obey physical laws, humans write their own rules, adapt to their own mistakes and adjust their actions to new encounters with a mix of unscripted rational and emotional responses.

3. Investors appraise prices from the perspective of an imagined future shaped by their past memories and current value judgments. These mental chain reactions transform their decision-making processes into highly complex systems. Our imagined future today can in turn create a new future that could then alter the current reality. The feedback loops continue in real time and exacerbate the already complex systems.

4. To model asset prices properly, economists must track all these nonlinear, multifaceted and interactive dynamics. By comparison, modeling the physical world is a trivial task.

What are the implications for investors?

1. There are many types of randomness described by different analytical probability density functions (PDFs). The bell curve is one of the most well behaved kinds. Figure 1 exhibits a wild form of randomness that does not fit any known PDF. Frank Knight called them "radical uncertainties". Donald Rumsfeld named them "unknown unknowns". Nassim Nicholas Taleb referred to them as "Black Swans".

2. Because asset prices do not follow the bell curve, it comes as no surprise why many random-walk based doctrines stopped working when prices plunged beyond one standard deviation. Markowitz's mean-covariance matrix, Sharpe's beta, Fama's factors and Black-Scholes' volatility neutrality all broke down during market crashes.

3. Because asset prices are radically random, investors should be skeptical of all market forecasts. Figure 2 shows the 10-year return histogram of the S&P 500 from 1928 to 2016. If you think that forecasting returns on a widely disperse bell curve is challenging, then mining for predictive patterns in the erratic randomness in figure 2 is a fool's errand because the probabilities of a +200% gain and a -20% loss are nearly the same.

4. Additionally, statistical flaws are common in many long-term forecasts. For example, analysts use overlapping data to compute the CAPE-based regression lines (here) and researchers gauge secular market cycles with sample sizes of less than a dozen (here). These forecasters not only try to predict something that is statistically unpredictable, they do so with incorrect math.

Concluding remarks

We live our lives every day without forecasting when the next big earthquakes will hit our hometowns. We mitigate quake risks by upgrading building codes and buying insurance.

Likewise, because asset prices exhibit radical randomness, predicting future returns is futile. Investors should focus on managing risk. To manage investment risk properly, however, we must first identify what risk truly is. Unfortunately, the academics view risk as volatility through a distorted random walk lens. In Part 3, I will point out the conceptual flaws and operational pitfalls of viewing volatility as risk. Mistaking risk as volatility has dire financial consequences for investors.

In Part 4, I will present a new framework for defining reward and risk as an alternative to the random walk paradigm. The new reward-risk framework offers investors a probabilistic path for beating the S&P500 total return – a blasphemy in the view of the efficient market orthodoxy. We win not by developing models with superior forecasting ability, but by beating Mr. Market at his own game. Stay tuned for details.

Theodore Wong graduated from MIT with a BSEE and MSEE degree and earned an MBA degree from Temple University. He served as general manager in several Fortune-500 companies that produced infrared sensors for satellite and military applications. After selling the hi-tech company that he started with a private equity firm, he launched TTSW Advisory, a consulting firm offering clients investment research services. For almost four decades, Ted has developed a true passion for studying the financial markets. He applies engineering design principles and statistical tools to achieve absolute investment returns by actively managing risk in both up and down markets. He can be reached at

Appendix: Returns of six Asset Classes – Actual Histograms vs. Random Walk PDFs

Section (A) shows linear and logarithmic histograms of short-horizon returns of three equity assets (the S&P500, the Russell 2000 and emerging markets). Section (B) shows linear and logarithmic histograms of short-horizon returns of three alternative assets (gold, the dollar and the 10-year Treasury bond). Section © shows linear and logarithmic histograms of long-horizon returns of the S&P500, the Russell 2000 and emerging markets. Section (D) shows linear and logarithmic histograms of long-horizon returns of gold, the dollar and the 10-year Treasury bond. Data sources are FREDYahoo Finance and MetaStock.

(A) Short-horizon returns of large-caps, small-caps and the emerging markets

Figure A1 shows linear histograms of the returns in short-term horizons – daily, weekly, monthly and quarterly (top to bottom) across different equity styles – the S&P500 (big-caps), the Russell 2000 (small-caps) and emerging markets (left to right). Big-caps and small-caps (left and center) bear some resemblances to the random walk probability density functions (PDFs). However, for the emerging markets (right), fat tails are visible even in monthly and quarterly returns. Gaps between the data and the PDFs near the peaks are noticeable among all three assets.

Figure A2 displays the same data in logarithmic histograms. All log histograms exhibit fat tails on both sides of the peaks. The random walk PDFs fail to account for the high probabilities at both return extremes across all three assets.

(B) Short-horizon returns of gold, the dollar and the U.S. Treasury bond

Figure A3 shows three different alternatives – gold, the dollar and the 10-year Treasury bond (left to right) in linear histograms of short-horizon returns – daily, weekly, monthly and quarterly (top to bottom). Deviations from the random walk PDFs grow with increasing time horizons. Gold and the dollar have wider spreads than the random walk PDFs while the 10-year Treasury has tall spikes near the peaks.

Figure A4 displays the same data in a log scale. All histograms exhibit higher probabilities at both extremes than those predicted by the PDFs. Gold and bond are asymmetric and skewed to the right relative to the PDFs.

(C ) Long-horizon returns of large-caps, small-caps and the emerging markets

Figure A5 shows linear histograms of long-horizon returns – one year, five years and ten year (top to bottom) across three equity indices – the S&P500, the Russell 2000 and emerging markets (left to right). All charts show multiple peaks and wildly different spreads.

Figure A6 shows the same data in a log scale. Beyond one year, the data distributions show no central tendency and they are skewed to the right. Actual returns offer much better upside opportunities than what random walk predicts.

(D) Long-horizon returns of gold, the dollar and the U.S. Treasury bond

Figure A7 shows linear histograms of different long-horizon returns – one year, five years and ten year (top to bottom) across three alternatives – gold, the dollar and the 10-year Treasury bond (left to right). None of these plots resembles the random-walk bell curve.

Figure A8 shows log histograms of the same data in Figure A7. Most of the fat tails are skewed to the right versus the perfectly symmetric random walk PDFs.


Modeling Cyclical Markets – Part 2

Originally Published November 7, 2016 in Advisor Perspectives

In Part 1 of this series, I presented Primary-ID, a rules- and evidence-based model that identifies major price movements, which are traditionally called cyclical bull and bear markets. This article debuts Secondary-ID, a complementary model that objectively defines minor price movements, which are traditionally called rallies and corrections within bull and bear markets.

The traditional definitions of market cycles

Market analysts define market cycles by the magnitude of price movements. Sequential up and down price movements in excess of 20% are called primary cycles. Price advances more than 20% are called bull markets. Declines worse than -20% are called bear markets. Sequential up and down price movements within 20% are called secondary cycles. Price retracements milder than -20% are called corrections. Advances shy of 20% are called rallies. Talking heads at CNBC frequently use this financial vernacular.

But has anyone bothered to ask how factual these fancy terms and lofty labels really are?

Experts also measure market cycles by their durations. They reported that since 1929, there have been 25 bull markets with gains over 20% with a median duration of 13.1 months, and 25 bear markets with losses worse than 20% with a median duration of 8.3 months.

But is "median duration" the proper statistical yardstick to measure stock market cycle lengths?

Fact-checking the 20% thresholds

Before presenting Secondary-ID, let’s pause to fact-check these two market cycle yardsticks. The ±20% primary cycle rules-of-thumb have little practical use in guiding investment decisions. If we wait for a +20% confirmation before entering the market, we would have missed the bulk of the upside. Conversely, if we wait for an official kick-off of a new cyclical bear market, our portfolios would have shrunk by -20%. The ±20% thresholds may be of interests to stock market historians, but offer no real benefit to investors.

Besides being impractical, the ±20% demarcations are also baseless. This falsehood can be demonstrated by examining the historical evidence. Figures 1A and 1B show the daily closing prices of the S&P 500 from 1928 to 2015. The green bars in Figure 1A are price advances from an interim low to an interim high of over +5%. The red bars in Figure 1B are price retracements from an interim high to an interim low worse than -5%. Price movements less than ±5% are ignored as noise. There were a total of 198 advances and 166 retracements in 88 years. From the figures, it's not obvious why ±20% were picked as the thresholds for bull and bear markets. The distributions of green and red bars show no unique feature near the ±20% markers.


To determine how indistinct the ±20% markers are in the distributions, I plot the same daily data in histograms as shown in Figures 2A and 2B. The probabilities of occurrence are displayed on the vertical axes for each price change in percent on the horizontal axes. For example, Figure 2A shows that a +20% rally has a 3% chance of occurring; and Figure 2B shows that a -20% retreat has near a 3.5% chance. There is no discontinuity either at +20% in Figure 2A that separates bull markets from rallies, nor at -20% in Figure 2B that differentiates bear markets from corrections.


There are, however, two distinct distribution patterns in both up and down markets. Figure 2A shows an exponential drop in the probability distribution with increasing rally sizes from +10% to +40%. Above +45%, the histogram is flat. Figure 2B shows a similar exponential decline in the probability distribution with increasing retracements from -5% to -40%. Beyond -45%, the histogram is again flat. The reasons behind the exponential declines in the distributions and the two-tier histogram pattern are beyond the scope of this paper. It's clear, however, that there is no distinct inflection point at ±20% from Figures 2A and 2B. In fact, it would be more statistically correct to use the ±45% as the thresholds for bull and bear markets. But such large thresholds for primary cycles would be worthless for investors.

Figures 2A and 2B also expose one other fallacy. It's often believed that price support and resistance levels are set by the Fibonacci ratios. One doesn't have to read scientific dissertations using advanced mathematical proofs to dispel the Fibonacci myth. A quick glance at Figure 2A or 2B would turn any Fibonacci faithful into a skeptic. If price tops and bottoms are set by the Fibonacci ratios, we would have found such footprints at ±23.6%, ±38.2%, ±50.0%, ±61.8%, or ±100%. No Fibonacci pivot points can be found in 88 years of daily S&P 500 data.

Fact-checking the market duration yardstick

I now turn to the second cyclical market yardstick-cycle duration. It's been reported that since 1929, the median duration for bull markets is 13.1 months and the median duration for bear markets is 8.1 months. The same report also notes that the spread in bull market durations spans from 0.8 to 149.8 months; and the dispersion among bear market durations extents from 1.9 to 21 months. When the data is so widely scattered, the notion of median is meaningless. Let me explain why with the following charts.

Figures 3A and 3B show duration histograms for all rallies and retreats, respectively. The vertical axes are the probabilities of occurrence for each duration shown on the horizontal axes. The notions of median and average are only useful when the distributions have a central tendency. When the frequency distributions are skewed to the extent seen in Figure 3A or both are skewed and dispersed like in Figure 3B, median durations cited in those reports are meaningless.


Figures 3A and 3B also expose one other myth. We often hear market gurus warning us that the bull (or bear) market is about to end because it's getting old. Chair Yellen was right when she said that economic expansions don't die of old age. Cyclical markets don't follow an actuarial table. They can live on indefinitely until they get hit by external shocks. Positive shocks (pleasant surprises) end bear markets and negative shocks (abrupt panics) kill bull markets. These black swans follow Taleb distributions in which average and median are not mathematically defined. In my concluding remarks I further expand on the origin of cyclical markets.

Many Wall Street beliefs and practices are just glorified folklores decorated with Greek symbols and pseudo-scientific notations to puff up their legitimacy. Many widely followed technical and market-timing indicators are nothing but glamorized traditions and legends. Their theoretical underpinnings must be carefully examined and their claims must be empirically verified. It's unwise to put ones' hard earned money at risk by blindly following any strategy without fact-checking it first, no matter how well accepted and widely followed it may be.

Envisioning cyclical markets through a calculus lens

Now that I have shown how absurd these two yardsticks are in gauging market cycles, I would like to return to the subject at hand – modeling cyclical markets. The methodology is as follows: First, start with a metric that is fundamentally sound. The Super Bowl indicator, for example, is an indicator with no fundamental basis. Next, transform the metric into a quasi range-bound indicator. Then devise a set of rational rules using the indicator to formulate a hypothesis. High correlations without causations are not enough. Causations must be grounded in logical principles such as economics, behavioral finance, fractal geometry, chaos theory, game theory, number theory, etc. Finally, a hypothesis must be empirically validated with adequate samples to be qualified as a model.

Let me illustrate my modeling approach with Primary-ID. The Shiller CAPE (cyclically adjusted price-earnings ratio) is a fundamentally sound metric. But when the CAPE is used in its original scalar form, it is prone to calibration error because it's not range-bound. To transform the scalar CAPE into a range-bound indicator, I compute the year-over-year rate-of-change of the CAPE (e.g. YoY-ROC % CAPE). A set of logically sound buy-and-sell rules is devised to activate the indicator into actionable signals. After the hypothesis is validated empirically over a time period with multiple bull and bear cycles, Primary-ID is finally qualified as a model.

This modeling approach can be elucidated with a calculus analogy. The scalar Shiller CAPE is analogous to "distance." The vector indicator YoY-ROC % CAPE is analogous to "velocity." When "velocity" is measured in the infinitesimal limit, it's equivalent to the "first derivative" in calculus. In other words, Primary-ID is similar to taking the first derivative of the CAPE. There are, however, some differences between the YoY-ROC % CAPE indicator and calculus. First, a derivative is an instantaneous rate-of-change of a continuous function. The YoY-ROC % CAPE indicator is not instantaneous, but with a finite time interval of one year. Also, the YoY-ROC % CAPE indicator is not a continuous function, but is based on a discrete monthly time series – the CAPE. Finally, a common inflection point of a derivative is the zero crossing, but the signal crossing of Primary-ID is at -13%.

Secondary-ID – a model for minor market movements

I now present a model called Secondary-ID. If Primary-ID is akin to "velocity" or the first derivative of the CAPE and is designed to detect major price movements in the stock market, then Secondary-ID is analogous to "acceleration/deceleration" or the second derivative of the CAPE and is designed to sense minor price movements. Secondary-ID is a second-order vector because it derives its signals from the month-over-month rate-of-change (MoM-ROC %) of the year-over-year rate-of-change (YoY-ROC %) in the Shiller CAPE metric.

Figures 4A to 4D show the S&P 500, the Shiller CAPE, Primary-ID signals and Secondary-ID signals, respectively. The indicator of Primary-ID (Figure 4C) is identical to that of Secondary-ID (Figure 4D), namely, the YoY-ROC % CAPE. But their signals differ. The signals in Figures 4C and 4D are color-coded – bullish signals are green and bearish signals are red. The details of the buy and sell rules for Primary-ID were described in Part 1. The bullish and bearish rules for Secondary-ID are presented below.


Bullish signals are triggered by a rising YoY-ROC % CAPE indicator or when the indicator is above 0%. For bearish signals, the indicator must be both falling and below 0%. "Rising" is defined as a positive month-over-month rate-of-change (MoM-ROC %) in the ROC % CAPE indicator; and "falling", a negative MoM-ROC %. Because it is a second-order vector, Secondary-ID issues more signals than Primary-ID. It's noteworthy that the buy and sell signals of Secondary-ID often lead those of Primary-ID. The ability to detect acceleration and deceleration makes Secondary-ID more sensitive to changes than Primary-ID that detects only velocity.

For ease of visual examination, Figures 5A shows the S&P 500 color-coded with Secondary-ID signals. Figure 5B is the same as Figure 4D describing how those signals are triggered by the buy and sell rules. Since 1880, Secondary-ID has called 26 of the 28 recessions (a 93% success rate). The two misses were in 1926 and 1945, both were mild recessions. Secondary-ID turned bearish in 1917, 1941, 1962, 1966 and 1977 but no recessions followed. However, these bearish calls were followed by major and/or minor price retracements. If Mr. Market makes a wrong recession call and the S&P 500 plummets, it's pointless to argue with him and watching our portfolios tank. Secondary-ID is designed to detect accelerations and decelerations in market appraisal by the mass. Changes in appraisal often precede changes in market prices, regardless of whether those appraisals lead to actual economic expansions or recessions.


Secondary-ID not only meets my five criteria for robust model design (simplicity, sound rationale, rule-based clarity, sufficient sample size, and relevant data), it has one more exceptional merit – no overfitting. In the development of Secondary-ID, there is no in-sample training involved and no optimization done on any adjustable parameter. Secondary-ID has only two possible parameters to adjust. The first one is the time-interval for the second-order rising and falling vector. Instead of looking for an optimum time interval, I choose the smallest time increment in a monthly data series – one month. One month in a monthly time series is the closest parallel to the infinitesimal limit on a continuous function. The second possible adjustable parameter is the signal crossing. I select zero crossing as the signal trigger because zero is the natural center of an oscillator. The values selected for these two parameters are the most logical choices and therefore no optimization is warranted. Because no parameters are adjusted, there's no need for in-sample training. Hence Secondary-ID is not liable to overfitting.

Performance comparison: Secondary-ID, Primary-ID and the S&P 500

The buy and sell rules of Secondary-ID presented above are translated into executable trading instructions as follows: When the YoY-ROC CAPE is rising (i.e. a positive MoM-ROC %), buy the S&P 500 (e.g. SPY) at the close in the following month. When the YoY-ROC CAPE is below 0% and falling (i.e. a negative MoM-ROC %), sell the S&P 500 at the close in the following month and use the proceeds to buy U.S. Treasury bond (e.g. TLT). The return while holding the S&P 500 is the total return with dividends reinvested. The return while holding the bond is the sum of both bond coupon and bond price changes caused by interest rate movements.

Figures 6A shows the S&P 500 total return index and the total return of the U.S. Treasury bond. In 116 years, return on stocks is nearly twice that of bonds. But in the last three decades, bond prices have risen dramatically thanks to a steady decline in inflation since 1980 and the protracted easy monetary policies since 1995. Figures 6B shows the cumulative total returns of Primary-ID, Secondary-ID and the S&P 500 on a $1 investment made in January 1900. The S&P 500 has a total return of 9.7% with a maximum drawdown of -83%. By comparison, Primary-ID has a hypothetical compound annual growth rate (CAGR) of 10.4% with a maximum drawdown of -50% and trades once every five years on average. The performance stats on Primary-ID are slightly different from that shown in Figure 5B in Part 1 because Figure 6B is updated from July to August 2016.

Secondary-ID delivers a hypothetical CAGR of 12.8% with a -36% maximum drawdown and trades once every two years on average. Note that Primary-ID and Secondary-ID are working in parallel to avoid most if not all bear markets. Secondary-ID offers an extra performance edge by minimizing the exposure to bull market corrections and by participating in selected bear market rallies.


I now apply the same buy and sell rules in the recent 16 years to see how the model would have performed in a shorter but more recent sub-period. This is not an out-of-sample test since there's no in-sample training. Rather, it's a performance consistency check with a much shorter and more recent period. Figures 7A shows the total return of the S&P 500 and the U.S. Treasury bond price index from 2000 to August 2016. The return on bonds in this period is higher than that of the S&P 500. Record easy monetary policies since 2003 and large-scale asset purchases by global central banks since 2010 pumped up bond prices. Two severe back-to-back recessions dragged down the stock market. Figures 7B shows the cumulative total returns of Primary-ID, Secondary-ID and the S&P 500 on a $1 investment made in January 2000.


Since 2000, the total return index of the S&P 500 has returned 4.3% compounded with a maximum drawdown of -51%. By comparison, Primary-ID has a CAGR of 7.7% with a maximum drawdown of -23% and trades once every five years on average. Again, the performance stats on Primary-ID shown in Figure 7B are slightly different from that shown in Figure 5B in Part 1 because Figure 7B is updated to August 2016. Secondary-ID delivers a hypothetical CAGR of 10.5% with a maximum drawdown of only -16% and trades once every 1.4 years on average. The performance edge in return and risk of Secondary-ID over both Primary-ID and the S&P 500 total return index is remarkable. The consistency in performance gaps in both the entire 116-year period and in the most recent 16-year sub-period lends credence to Secondary-ID.

Theoretical support for both cyclical market models

The traditional concepts of "primary cycles" and "secondary cycles" rely on amplitude and periodicity yardsticks to track market cycles in the past and to predict market cycles in the future. Primary-ID and Secondary-ID do not deal with primary or secondary market cycles. Their focus is on cyclical markets – major and minor price movements. All market movements are driven by changes in investors' collective market appraisals. The Shiller CAPE is selected as the core metric because it is a value assessment gauge-based fundamental indicator – appraising the inflation adjusted S&P 500 price relative to its rolling 10-year average earnings. Although the scalar-CAPE is prone to overshoot and valuations misinterpretation, the first- and second-order vectors of the CAPE are not. Primary-ID and Secondary-ID sense both major changes and minor shifts in investors' collective market appraisal that often precede market price action.

Like Primary-ID, Secondary-ID also finds support from many of the behavioral economics principles. First, prospect theory shows that a -10% loss hurts investors twice as much as the pleasure a +10% gain brings. Such reward-risk disparities are recognized by the asymmetrical buy and sell rules in both models. Second, both models use vector-based indicators. This is supported by the findings of Daniel Kahneman and Amos Tversky that investors are sensitive to the relative changes (vectors) in their wealth much more so than to the absolute levels of their wealth (scalars). Finally, the second-order vector in Secondary-ID is equivalent to the second derivative of the concave and convex value function described by the two distinguished behavioral economists in 1979.

Concluding remarks – cyclical markets vs. market cycles

I developed rules- and evidence-based models to assess cyclical markets and not market cycles. The traditional notion of market cycles is defined with a prescribed set of pseudo-scientific attributes such as amplitude and periodicity that are neither substantiated by historical evidence nor grounded in statistics. Cyclical markets, on the other hand, are the outcomes of random external shocks imposing big tidal waves and small ripples on a steadily rising economic growth trend. Cyclical markets cannot be explained or predicted using the traditional cycle concepts because past cyclical patterns are the outcomes of non-Gaussian randomness. Let me illustrate with a simple but instructive narrative.

Cyclical markets can be visualized with a simple exercise. Draw an ascending line on a graph paper with the y-axis in a logarithmic scale and the x-axis in a linear time scale. The slope of the line is the long-term compound growth rate of the U.S. economy. Next, disrupt this steadily rising trendline with sharp ruptures of various amplitudes at erratic time intervals. These abrupt ruptures represent man-made crises (e.g., recessions) or natural calamities (e.g., earthquakes). Amplify these shocks with overshoots in both up and down directions to emulate the cascade-feedback loops driven by the herding mentality of human psychology. You now have a proxy of the S&P 500 historical price chart.

This descriptive model of cyclical markets explains why conventional market cycle yardsticks – the ±20% thresholds and median durations will never work. Unpredictable shocks will not adhere to a prescribed set of amplitude or duration. Non-Gaussian randomness cannot be captured by the mathematical formulae defining average and median. The conceptual framework of market cycles is flawed and that's why it fails to explain cyclical markets.

Looking from the perspective of Primary-ID and Secondary-ID, cyclical bull markets can last as long as the CAPE velocity is positive and/or accelerating. Cyclical bear markets can last as long as the CAPE velocity is negative and/or decelerating. Stock market movements are not constrained by the ±20% thresholds or cycle life-expectancy stats. Primary-ID detects the velocity of the stock market valuation assessment by all stock market participants that drives bull or bear markets. Secondary-ID senses subtle accelerations and decelerations in the same collective market valuation assessment. These second-order waves manifest themselves in stock market rallies and corrections. It doesn't matter whether the market is down less than -20%, labeled by experts as a correction, or plunges by worse than -20%, which is called a cyclical bear market, Primary-ID and Secondary-ID capture the price movements just the same.

Does synergy exist between Primary-ID and Secondary-ID? Would the sum of the two offer performance greater than those of the parts? A composite index of the two models enables the use of leverage and short strategies that pave the way for more advanced portfolio engineering and risk management tactics. Do these more complex strategies add value? For answers, please stay tuned for Part 3.

Theodore Wong graduated from MIT with a BSEE and MSEE degree and earned an MBA degree from Temple University. He served as general manager in several Fortune-500 companies that produced infrared sensors for satellite and military applications. After selling the hi-tech company that he started with a private equity firm, he launched TTSW Advisory, a consulting firm offering clients investment research services. For almost four decades, Ted has developed a true passion for studying the financial markets. He applies engineering design principles and statistical tools to achieve absolute investment returns by actively managing risk in both up and down markets. He can be reached at



Secular Market Cycles – Fact or Illusion?

Originally Published October 3, 2016 in Advisor Perspectives

Conventional wisdom dictates that equity markets adhere to long-term secular cycles and that investors should adjust their allocations based on whether valuation metrics, such as the Shiller CAPE, are relatively high or low. But what if the notion of secular market cycles is misguided because, for example, the sample size of past cycles is insufficient to attach any statistical significance?

I have been a student of the stock market for over 40 years. I rarely heard the term "secular market cycles" before the 1970s. In 1991, Angus Maddison used the term "long waves" to describe economic activities in 16 advanced capitalist countries since 1820. The term "secular cycles" has gained popularity since 2000 when Robert Shiller published the first edition of Irrational Exuberance. Figure 1 is taken from page 8 of his book, which shows the now famous Shiller CAPE (cyclically adjusted price-earnings ratio). Shiller's chart featured four major tops from 1881 to 2000. The last peak was spot-on in nailing the dot-com bubble.

Many believed that Shiller had deciphered the incoherent S&P 500 chart into a comprehensible rhythmic waveform with CAPE.


Since then, the notion of "secular market cycles" has been increasingly accepted as an undisputable fact in both academic research and investment circles. Experts are busy giving meaning to such cyclical patterns. They rationalize causal connections between secular market cycles and socioeconomic shifts and attribute those cycles to structural factors such as technological advances, demographic waves, inflation trends, political reforms or wars.

The problem arises when analysts advise clients to deploy different investment strategies depending on whether the current phase is a secular bull or bear market. In order to know which strategy is appropriate, investors must first identify where they are in a secular market cycle. Unfortunately, the same experts who can explain past cycles are in total disarray regarding the current cycle. Since 2010, analysts have been debating if the secular bear market that started in 2000 is still in place or if a new secular bull market has already begun. During their six-year debate, the S&P 500 has melted up over 200%.

A deep dive into the secular market debate

Are we currently in a secular bull or secular bear market? You can find experts with a wide dispersion of opinions –– the bearish camp, the bullish camp and those on the agnostic fence.

Leading the bearish camp are many renowned analysts who believe that the secular bear market started in 2000 continues today. Members include Ed Easterling, Michael Alexander, John Hussman,Jeremy Grantham, John Mauldin, Russell Napier, Joseph Calhoun, Van Tharp and Martin Pring (who might have turned bullish recently). Many of them justify their bearish stances with only one or two secular cycles of data supported by anecdotal evidence. Easterling and Alexander extended the database to over a century but could only increase the number of cycles to four or eight.

Such sample sizes are too small for any meaningful statistical analysis.

In the bullish camp, there is a contingent of prominent experts who believe that a new secular bull market began sometime after 2009. Doug Short, Jill Mislinski, Guggenheim Partners and others presented calendar tables that depicted the periods of their secular cycles in the last century. Others include Chris Puplava, Liz-Ann Sounders, Craig Johnson, Jeffrey Saut, Barry Ritholtz, Ralph Acamporaand Tim Hayes. Institutional members include Fidelity, INVESCO and Bank of America Merrill Lynch. Most of the analysts in this camp turned bullish in the period from 2012 to 2014, after the March 2009 price had been firmly established as the bottom of the preceding secular bear market.

Members sitting on the agnostic fence are harder to find. It takes honesty, humility and, above all, guts to admit publically that you don't have the answer. Doug Ramsey turned from bearish to neutral in 2014.Alex Planes hedged his mildly bearish stance by acknowledging that no one could be certain about the exact cycle phase except in hindsight.

Easterling and Alexander are the only two researchers in all three camps who applied rule-based models on more than a century of data to define secular market cycles. The transparency of their methodologies allows peer reviews. Their work and findings are summarized below.

Ed Easterling's secular market cycle model

Ed Easterling of Crestmont Research is a recognized authority on the subject of secular market cycles and has written extensively on the subject. According to Easterling, secular bull markets start at the troughs of below-average price-to-earnings ratios (P/Es) or the Crestmont P/Es and secular bear markets start at the peaks of above-average P/Es. Based on these "rules," Easterling tabulated a secular cycle calendar from 1901 to 2015 with four secular bull and five secular bear markets. The performances of secular bull versus bear markets is tabulated in Table 1.


In Figure 2A, the S&P 500 is in green to depict Easterling's secular bull markets and in red for his bear markets. Figure 2B is an overlay of the Shiller CAPE and is the same as Figure 1 above but is extended to 2015. According to Easterling, the current secular bear market that began in 2000 shows no sign of ending soon. His basic premise is that secular bull markets in the past didn't begin until either the Shiller CAPE or the Crestmont P/E bottomed at below-average levels.


The mean of the Shiller CAPE from 1881 to mid-2016 is 16.7. The CAPE dipped down to 13.3 in March 2009, but that was not "below-average" enough for Easterling. He noted that in all four previous bear market bottoms in the 1920s, 1930s, 1940s and the 1980s, the CAPE dropped to at least 10 and, most of the time, close to 5.

Michael Alexander's secular market cycle model

Michael Alexander wrote a ground-breaking book in 2000 entitled: Stock Cycles: Why Stocks Won't Beat Money Markets Over the Next Twenty Years. Alexander developed a database of over 200 years, much longer than other researchers. As a result, he was able to show more supporting evidence that linked his secular cycles to economic fundamentals. Alexander argued that there were two alternate types of secular cycles – monetary cycles followed by real cycles. In monetary cycles, falling inflation produced secular bull markets and rising inflation, secular bear markets. In real cycles, strong or consistent earnings growth fueled secular bull markets, and weak or inconsistent earnings drove secular bear markets. The secular bear market that began in 2000 and continues to the present day is a weak and inconsistent earnings phase of a real cycle.

Alexander developed a new metric called the P/R ratio (price-to-resource ratio) to detect secular market turning points. His metric is grounded on sound fundamentals and the derivation of P/R was detailed in the Appendix in his book. His P/R ratio resembles Easterling's P/E and he uses a similar rule narrative – secular bull markets start after P/R ratios have bottomed, and secular bear markets start after P/R ratios have peaked.

Table 2 summarizes Alexander's original findings, which ended in 2000. I updated his table through 2015, which is consistent with his bearish market stance posted in a recent blog.


The common thesis Easterling and Alexander share

Both Easterling and Alexander applied quantitative metrics to define secular cycles. From their statements, we can find a common thesis in their bearish arguments.

In April 2013, Easterling affirmed that "the current secular bear will continue at least for another five to ten years until the CAPE reaches 10 or lower."

In July 2015, Easterling reaffirmed that "Crestmont Research identifies – without hesitation or doubt – the current cycle as the continuation of a secular bear market...we have a strong conviction that the prospect of a secular bull is far away...this secular bear, however, started at dramatically higher levels due to the late 1990s bubble... the reality is that the level of stock market valuation (i.e., P/E) is not low enough to provide the lift to returns that drives secular bull markets....the current P/E is at or above the typical starting level for a secular bear market."

In August 2013, Alexander wrote, "I sold my last position last month when the S&P 500 was in the low 1600's. The P/R graph shows that the market has reached roughly the same position relative to past secular bear markets as it had in 2007...The bet I am making is that there will be another downturn as there was in the past and this downturn will send the S&P 500 down to 1250."

In June 2015, Alexander published a blog entitled "10,000 point decline in the Dow in the cards over the next three years." Based on the declines from P/R peaks to P/R troughs in previous secular bear markets, he projected a secular bear bottom for the Dow Jones Industrial Average to be around 8000 and the S&P 500 around 900 by 2018.

The self-assurance expressed in the statements by Easterling and Alexander is admirable. But their doomsday forecasts are misplaced. When we become too personally or professionally invested in a supposition, we fall into an overconfidence trap.

Philip Tetlock in his book Superforecasting: The Art and Science of Prediction identifies key traits that separate good forecasters from bad. Hedgehogs are lousy forecasters because they are overconfident on their immutable grand theories and stubbornly cling to their confirmation biases despite contradictory evidence. Foxes, on the other hand are much better forecasters primarily because they are skeptical about grand theories, diffident in their beliefs and ready to adjust their convictions based on actual events. Foxes are true Bayesians.

The key to successful forecasts is to keep an open mind. In his book Sapiens: A Brief History of Humankind, Yuval Noah Harari argues that a new mindset in the 16th century based on the Latin word ignoramus – the willingness to admit ignorance -- was the catalyst that set in motion the Scientific Revolution that continues today. As Mark Twain said, “It ain’t what you don’t know that gets you into trouble. It’s what you know for sure that just ain’t so.”

A common thesis behind the bearish stances of both Easterling and Alexander is that the current levels of their metrics – Easterling's P/E and Alexander's P/R – are still too high relative to the starts of all previous secular bull markets in the past century. This logic compels them to reject any possibility of a new secular bull market. I will challenge their logic a bit later. But I would like to clarify two common misconceptions first.

Misconceptions about the term "secular cycles"

Cycle advocates claim that the existence of secular cycles is self-evident as proven by the large performance gap between secular bull and bear markets shown in Tables 1 and 2. Large differences in the returns exist, but they don't necessarily prove the existence of secular cycles. Secular bull markets are defined as the periods from troughs (either in price, P/E or P/R ratio) to peaks, and secular bear markets, from peaks to troughs. By such definitions, returns in bull markets must be higher than those in bear markets. The self-evidence argument is a circular logic and can be illustrated with a simple analogy. The temperatures from June to August are relatively high not because of the summer season. Rather, the summer season is defined from June to August because the temperatures in those months are relatively high. Claiming that secular bull markets create wealth and secular bear markets destroy wealth is as trivial as saying that June, July and August are hot because of summer and the winter months are cold because of winter. Sequential high and low returns do not prove the existence of secular cycles because those patterns are used to define secular cycles in the first place.

The term "cycles" in engineering and sciences refers to events with regular periodicity or at a uniform frequency. The term "cycle" used by stock market researchers refers to contiguous pairs of up and down markets. Investment analysts claimed that the stock market exhibits cycles at an interval of 17 to 18 years. One analyst even calculated an average cycle as precisely 17.6 years.

The notion of "average" is only meaningful when the sample distribution has a central tendency, i.e., not flat, multi-modal or skewed. When the spreads on the "half cycle" are so widespread (from 3 to 25 years shown in columns 3 and 7 in Tables 1 and 2, respectively) and the sample sizes are so small (4 and 8 "full cycles" shown in Tables 1 and 2), the term "average" may not even be mathematically definable. The stock market does exhibit pseudo sine-wave oscillatory patterns because investors’ sentiment fluctuates between greed and fear emotional extremes. Such extremes are captured by my TR-Osc and several other models to be presented in my future articles. But there's no evidence of any periodicity. The term "cycles" is highly misleading and grossly misused and claims such as 17.6 years cycle length are absurd.

Why are secular cycles so illusive?

Let's return to the question – why is there no consensus among analysts on the current secular cycle phase? Is that because, when standing in the middle of a cycle, one cannot see the future direction of the market? It's understandable that if price turning points are used to define cycles, a cycle in progress cannot be identified until a higher high or a lower low has been clearly established.

But the lack of consensus is not limited to the cycle currently in progress. Experts couldn't even agree with the benefit of hindsight on past secular cycles. For example, none of the secular chronology published by Easterling, Alexander, Short, Guggenheim, Ramsey, Hussman, Maddison and Fidelitylooks exactly alike. For those analysts who used anecdotal evidence, descriptive arguments and only a few decades of supporting data to define their cycles, different hindsights should not be a total surprise. But one would expect the two cycle calendars from Easterling and Alexander to be similar because both researchers apply similar quantitative metrics and objective rules on over a century of market data to determine their secular cycles. How different are their secular calendars?

Compare Easterling's secular calendar shown in Tables 1 to Alexander's in Table 2. From 1900 to 2015, Easterling counted five bear markets and four bull markets, while Alexander identified only four bear markets and three bull markets. That's a whopping 30% discrepancy. An average investor has only 30 to 40 years to build his or her retirement nest egg, missing or adding one full secular cycle with an "average cycle of 17.6 years" could mean a world of difference.

Flawed assumptions common to both valuations and secular cycle models

Secular cycle metrics used by Easterling and Alexander share many common attributes with the traditional valuations gauges such as the Shiller CAPE, the Tobin-Q, the Buffet market-cap-to-GNP ratio, price-to-earnings, price-to-dividend and price-to-book ratio. I previously argued that their uniformly high readings in the past 20 years indicated two common flaws. Many experts have begun to question whether the two-decade long elevated CAPE readings really reflect high market valuations or if they are signs of possible calibration malfunction. Many "fixes" have been proposed to adjust the high levels back down (see Philosophical Economics, Jeremy Siegel and James Montier). When a gauge needs fixing, it means that users have lost confidence in its accuracy. The same critiques I made to challenge the validity of many of the valuations models also apply to the bearish secular market thesis of Easterling and Alexander.

Easterling's P/E, Alexander's P/R and all of the valuations gauges cited above share two key operating assumptions. First, they rely on the absolute levels of the readings in their metrics to appraise future market outlook. Second, they assume that mean reversion will always bring the outliers back to the normal range. The first assumption – high absolute levels (relative to the historical means) translate to low future returns – will only hold when the time series has a stable mean (a single mean that is constant in time). The means of all those valuations gauges cited above have shifted upward significantly in the last two decades. With multiple means, the out-of-bound data wouldn't know which mean to revert to. The elevation anomaly observed in the CAPE also appeared in both the Crestmont P/E and the Alexander P/R, which led both researchers to hold their secular bear market stances for over a decade.

The second assumption is mean reversion, which is misunderstood to imply that any data that is temporarily out-of-bound will always self-correct and migrate towards the mean. They have mistakenmean regression for mean reversion. Mean regression is a law in probability that states that random outliers in a normal probability distribution have a tendency to move towards the mean driven by random statistical processes. Mean reversion, on the other hand, is the result of causation, not randomness. Mean reversion is a causal hypothesis (not a law) postulated to explain certain observed tendency towards the mean. Jeremy Siegel, Philosophical Economics, James Montier and others have proposed various causes to explain the elevation in the Shiller CAPE. If causations are involved, the elevations in the CAPE and other metrics are not random, and therefore mean regression has no jurisdiction. Past mean reversion episodes in the Crestmont P/E, the Alexander P/R and the Shiller CAPE are no guarantees for their future reappearances. Since there is no mathematical law to mandate mean reversion, these metrics could stay elevated or suppressed indefinitely. The means could also step up or down to different plateaus if a new cause merges and shifts the baselines of their previous means.

Concluding remarks

The Shiller CAPE, the Crestmont P/E and the Alexander P/R are all good metrics built on solid economic fundamentals. The problem arises when these metrics are wrongly applied to gauge market valuations or to define long-term market cycles. Over the years, these widely held but misconceived models have become the sacred cows in the theology of investments. Any challenge to the cardinal truth would be denounced by the high priests as a heresy. Observations that cannot be explained by the traditional doctrines are conveniently casted as one-off anomalies. The fact is that secular cycles and the other related valuations models are not infallible axioms based on first principles but are merely hypotheses yet to be validated. Perhaps it would take a heretic from outside the investment circles with no career risk to point out the obvious flaws in this “cardinal truth.” I argue that the elevated readings in various secular cycle and valuations metrics since 1995 are not anomalous aberrations but are empirical evidence against the orthodoxy.

In fact, the dispersion in opinions among all secular cycle advocates could be viewed as a nullification of the secular cycle hypothesis. Analysts used price data from 1800 to 2000 for the "in-sample training" of their models. These models are "trained" to interpret the past. It is therefore no surprise that they can depict past cycles. Market behaviors from 2009 to present, however, could be looked at as the "out-of-sample" test results of these models. The confusion among analysts on their post-2009 market stances could be considered as a form of inconsistency between the out-of-sample test outcome and their in-sample data mining. When the out-of-sample reality stirs up a controversy that lasts for six years, it raises the presumption of doubt whether the secular cycle notion is a good approximation of realty.

There are two mathematical explanations for why these models give contradicting out-of-sample market stances even though they were trained with the same in-sample data. Any model that is constructed with fewer than a dozen input samples is deemed to be unreliable. First, the smaller the sample size, the more susceptible the model is to curve-fitting. Second, small in-sample sizes mathematically guarantee out-of-sample predictions to have low confidence intervals, high margins of error or both. The 2008 sub-prime meltdown was a horrific example of insufficient and irrelevant in-sample data. All credit rating agencies used U.S. housing market data from the 1970s to the 1990s to model the default risk of mortgage-backed securities. During this training period, mortgage default rates were very low and the U.S. real estate market was booming. If these credit agencies were to incorporate U.S. housing data from 1890 to 1950 (both bull and bear housing markets) or housing data from Japan since the 1970s (bear housing market) in their Gaussian copula credit risk models, we may not have had the sub-prime crash.

Daniel Kahneman in Thinking, Fast and Slow described two distinct human mental faculties – a spontaneous pattern recognition ability followed by a reflective aptitude to rationalize. Our ancestors survived in the savannah jungles mainly with their first mental faculty – extracting camouflaged signals from noise swiftly to outwit both stronger predators and faster prey. Having survived the jungles, humans had more time to indulge in contemplation. It's our propensity for ex-post rationalizations that gave birth to culture, religion, philosophy and sciences. Today, however, living in an internet maze packed with terabytes of data, our innate pattern perceptive intuition and our natural rationalization tendency are often fooled by randomness.

These two human traits manifest in the behaviors of secular cycle advocates. They first visualize a handful of apparent cyclical patterns like those in the Shiller CAPE. They then draw causal connections that link these observations to fundamental causes without bothering to check for statistical significance. According to Angus Maddison (see p. 16 in the reference), long-wave patterns are not caused by any periodic structural forces proposed by secular theorists, but rather, the results of accidental systematic shocks and subsequent attempts to stabilize the aftermaths by monetary and fiscal policies. Secular patterns are the reflection of these random wave-like disturbances on an otherwise continuously rising economic growth curve.

If those perceived cyclical patterns are purely accidental and caused by unpredictable random shocks, it's entirely plausible that the imaginative secular cycles could be mirages misconceived by the overzealous cycle advocates. Their faithful followers could be searching for something nonexistent. The Crestmont P/E, the Alexander P/R and the Shiller CAPE might stay elevated indefinitely and never revert to or undershoot their historical means. Long-lasting elevations could place the misguided forecasters in a special class of perpetual permabears.

Market watchers love the secular market controversy because a protracted debate keeps them relevant. Unfortunately, investors only have 30-plus years to accumulate wealth. Should we entrust our hard earned money to a hypothesis that might take an average secular period of 17.6 years to pan out? If by then the hypothesis is proven wrong, investors would have wasted half of their investing life-cycle.

It's a common belief that the stock market exhibits both secular and cyclical waves. If the concept of secular markets is dubious and the nature of the in-progress secular phase is always unclear, we should shift our attention to the shorter version called cyclical markets. Does the notion of cyclical markets share the same flawed premises as their secular cousin? What drives cyclical markets? Can they be defined, identified and modeled objectively? Modeling cyclical markets and the efficacy of such models will be the topics of my next articles.

Theodore Wong graduated from MIT with a BSEE and MSEE degree and earned an MBA degree from Temple University. He served as general manager in several Fortune-500 companies that produced infrared sensors for satellite and military applications. After selling the hi-tech company that he started with a private equity firm, he launched TTSW Advisory, a consulting firm offering clients investment research services. For almost four decades, Ted has developed a true passion for studying the financial markets. He applies engineering design principles and statistical tools to achieve absolute investment returns by actively managing risk in both up and down markets. He can be reached at