# Random Walk Part 4 – Can We Beat a Radically Random Stock Market?

Originally Published October 9, 2017 in Advisor Perspectives

This is the final article of my four-part series into the fallacy of the random-walk paradigm. In Part 1 and Part 2, I showed that asset prices do not follow a tidy bell curve and instead are radically random. In Part 3, I demonstrated that many bad risk management practices are the direct results of equating volatility to risk. In this article, I offer a probability-based framework that captures the true nature of investment reward and risk.

The Efficient Market Hypothesis (EMH) argues that the market is hard to beat because very few people could make better forecasts than the collective market wisdom, which instantly discounts all available information. My new reward-risk framework reveals a little-known secret that market gains and losses have very different distribution profiles. We can beat the S&P 500 not by making better forecasts, but by exploiting the dual personality of Mr. Market.

The random walk theory has been the core of modern finance since Louis Bachelier wrote his 1900 PhD thesis. Economists define reward as the mean return (expected value) and risk as the standard deviation (volatility) of the returns. These mathematical terms may be convenient for academics in formulating their economic theories, but make no sense to the average investor. Reward has a positive overtone but mean could be negative. Risk has a negative undertone but standard deviation weighs gains and losses equally. Investors view reward and risk as two sides of the same coin – reward comes from gains and risk comes from losses. My reward-risk framework quantifies this subtle diametrical symmetry.

The random-walk definitions of investment reward and risk

Modern finance adopted the mean-variance paradigm to frame reward and risk. Appendix A presents the mathematical definitions. Figure 1 illustrates the reward and risk graphically with the annual returns of the S&P 500 from 1928 to 2016 (data sources: MetaStock and Yahoo Finance). The dark blue curve is the random walk probability density function (PDF). Reward (mean or expected value) is computed by integrating the total area under the PDF curve using equation A1 in Appendix A. Risk (the square root of variance), computed with equation A2, is one-half of the width of the light blue central region bounded by ± one standard deviation. The random walk PDF roughly matches the data (the jagged gray area) in the central region except near the peak. Beyond ± one standard deviations, data reside mostly above the PDF curve.

Figure 2 compares the random walk PDFs (blue curves) to the actual S&P 500 returns (gray areas) in a one-, five- and 10-year horizon. The peaks of the PDFs denote the means. The red arrows signify ± one standard deviations. Random walk's notions of mean and volatility bear no resemblance to actual returns and risks in the real world. The longer the return horizons are, the larger the gaps. This is why so many conventional risk management practices derived from the mean-variance paradigm broke down during financial crises. The academics' bell curve paradigm offers investors no protection against financial market risks.

My gain-loss framework for investment reward and riskI offer a new probability-based framework for defining reward and risk. The formulas are presented in Appendix B. Figure 3 illustrates the concept. I define investment reward as the expected gain – the sum of all probability-weighted gains in a return histogram. It is computed by integrating numerically the total green area in Figure 3 using equation B1 in Appendix B. I define risk as the expected loss – the sum of all probability-weighted losses. It is computed by summing the total pink area in Figure 3 using equation B2.

The random-walk paradigm treats both positive and negative dispersions as risks. The expected gain-loss framework only considers losses (red bars) as risks but views the widely disperse green bars (gains) as gainful opportunities. The bell curve does not include all data, especially those at the extremes. The new model accounts for all outlier gains and all tailed risks weighted by their observed probabilities.

The old paradigm versus the new framework

Besides being unrealistic and impractical, the random-walk paradigm has one more subtle fault that is underreported. Mean and variance have different units of measure – mean is in percent but variance is in percent-squared. For unit compatibility, William Sharpe was compelled to use standard deviation – the square root of variance in his Sharpe Ratio. Even so, mean and standard deviation still do not have contextual uniformity. Mean signifies the most probable outcome and standard deviation measures the spread of those outcomes. Comparing reward (mean) to risk (volatility) is like comparing apples and oranges. For instance, Figure 1 shows that the mean-to-volatility ratio of the S&P 500 is 0.48 (dividing a mean of 7.7% by a volatility of 16%). Does this imply that the reward of investing in the S&P 500 is less than half the risk?

By contrast, "expected gain" and "expected loss" are two sides of the same coin – returns with opposite signs. Unlike the Sharpe Ratio that lacks clarity, the expected-gain-to-expected-loss ratio has absolute significance. For instance, Figure 3 shows that from 1928 to 2016, the S&P 500 index has an expected gain of 12.5% versus an expected loss of -4.3%. The expected gain of the S&P 500 is 2.9 times larger than the size of the expected loss.

Challenging the EMH's explanation on why the market is hard to beat

Why is the S&P 500 total-return such a formidable challenge for many active managers and market timers? The Efficient Market Hypothesis (EMH) offers a two-part explanation. First, market prices instantly (efficiently) reflect the collective appraisals of all market participants. Second, one can only beat the market by outsmarting the collective wisdom. On the surface, both points appear logical, but they are not, in fact, as logical as they appear.

The first point may explain why it is difficult for arbitrageurs to make a living because any price gap is instantly exploited. This argument, however, does not apply to the financial markets where prices are not single-valued functions of information. The same news can have multiple meanings and price implications depending on the receivers. Different interpretations of the same news draw buyers and sellers to the table. Price is an equilibrium point where the sellers believe their price is fair but high, while the buyers think is reasonable but low. The market is not a super forecaster, but an efficient auction-clearing house that facilitates buyers and sellers with different appraisals to transact.

The second EMH argument is self-inconsistent. It asserts that few can beat the market because outsmarting the collective forecast is hard to do. No random-walk followers including the EMH faithful should endorse the practice of forecasting because forecasting randomness is a contradiction in terms. Randomness, by definition, is unpredictable.

The real reason why the market is hard to beat

My gain-loss framework offers a painfully obvious explanation of why the market is hard to beat. Figure 4 parses the same data in Figure 2 in terms of gains and losses. It shows that the market offers investors abundant gainful opportunities (green bars), but that they are highly erratic. The probabilities inside the blue rectangle in the middle chart are nearly the same but the gains span from 0% to 80%. The bottom chart shows comparable probabilities for gains ranging from 0% to 200%. To time the market with virtually flat gain distributions is futile. That is why the buy-and-hold approach is unbeatable in the green zone.

The characteristics of market losses (pink bars) are very different. First, the pink zones are much narrower than the green areas. Second, while the green area grows with time (from 12.5% in one year, to 46.6% in five years and 101.7% in 10 years), the pink areas are insensitive to the holding period. In fact, as the holding period expands from five to ten years, the pink area shrinks from -5.2% to -2.9%.

It is a fool's errand to time the market in the green areas, where the probabilities are almost flat and the distributions grow with the holding period. It is prudent to stay in the market and gather those wildly erratic gains. In contrast, the pink areas are confined and insensitive to time. Hence, mitigating losses in the pink areas is much more manageable. My gain-loss framework not only explains why it is hard to beat the market, but also reveals a clue on how to do it logically.

Unlock a little-known market beating secret

How can we differentiate whether the current market is in the green or pink zone? I previously published five models that were designed to do just that. The five models are Holy GrailSuper MacroTR-OscPrimary-ID and Secondary-ID. They detect the green/pink market phases from five orthogonal perspectives – trend, the economy, valuations, major market cycles and minor price movements, respectively.

Figure 5 shows the annual return histograms of the five models against three investment benchmarks – the S&P 500 total-return, the 10-year US Treasury bond, and the 60/40 mix (60% equity and 40% bond rebalanced monthly) (data: Shiller). They were computed from the eight equity curves – the compound growths from January 1900 to September 2017.

Figure 5 shows that my five models share two common features: their green bars are comparable to those of the S&P 500 but with narrower pink areas. In other words, their expected gains are as good as the S&P 500 but their expected losses are much lower. As a result, they offer much higher expected gain-to-loss ratios than that of the S&P 500. A model with a higher expected gain-to-loss ratio than the S&P 500 can surely beat the market return. I will quantify this point a bit later but first, I must challenge yet another modern finance doctrine.

Challenging the Capital Asset Pricing Model

The Capital Asset Pricing Model (CAPM) asserts that first, no return of any asset mix between the S&P 500 and the Treasury bond or Treasury bill can exceed the Capital Market Line (CML); and second, one can only increase return by taking on more risk via leverage. Figure 6 is a plot of expected gains versus expected losses from the data in Figure 5. The dashed red line is the CML connecting the S&P 500 total-return to the 10-year US Treasury bond total-return (bond yields plus bond price changes caused by interest rate changes). Also shown are my five models (light blue dots) and the three benchmarks (red squares). I add a sixth model Cycle-ID (black dot) to show the effects of leverage. All six models are counterexamples to the CAPM. The five unlevered models reside far above the CML. The levered Cycle-ID beats the S&P 500 expected gain by 68% with only 85% of the risk. Hence, both CAPM claims are untrue.

CAGR and maximum drawdown comparisons

It is simple math that investors can beat the S&P 500 total-return if they can achieve close to the S&P 500 expected gains but cut their expected losses sizably relative to the S&P 500. Table 1 lists the compound annual growth rates (CAGRs) of all nine strategies in six sets of bull and bear full market cycles from January 1900 to September 2017. All six models have CAGRs higher than the S&P 500 total-return consistently in different bull and bear full cycles over a century.

My six models not only outperform the total compound returns of the S&P 500 in all cycle sets, they also offer lower risks than the S&P 500 measured by maximum drawdown. Table 2 compares maximum drawdowns of the nine strategies in different market cycles.

A dynamic active-passive investment approachWhich investment approach is better, passive or active? This ongoing debate misses the point. There is a time to be passive and a time to be active. Passive investors underperform active managers in bear markets and active investors are no match to buy-and-holders in bull markets.

Figure 4 reveals that Mr. Market has a dual personality. When he is content, he spreads his random gains over a wide green area. When he is mad, he directs his wrath at a narrow pink zone. Therefore it is feasible to logically beat Mr. Market at his own game – be a passive investor in the green areas to gather the wildly disperse gains but be an active risk manager in the pink areas to trim market losses.

Here is how investors can do that in practice. Do regular checkups on Mr. Market's health. If we detect a mood shift from good to bad, reduce market exposure (actively preserve capital in the pink zones). Otherwise, we stay in the market (passively cumulate wealth in the green areas).

Market health checkups are not market forecasts. Doctors do not forecast our medical conditions at annual exams. They conduct routine diagnoses and look for symptoms. If some tests come back positive, then the doctors actively treat those illnesses. Otherwise, patients would passively count their blessings until the next checkup.

Similarly, in regular market health checkups, we do not forecast market outlook but conduct diagnoses and look for warning signs. For instance, my five models were designed to monitor subtle shifts in trend (Holy Grail), the economy (Super Macro), valuation (TR-Osc), major market cycle (Primary-ID) and minor price movement (Secondary-ID). When a medical test comes back positive, we do not panic but seek second or third opinions. Likewise, investors should not assess market health based on a single indicator, but use the weight-of-evidence from multiple orthogonal models.

Using this dynamic active-passive approach and developing an integrated market monitoring system, investors can achieve the dual objective of capital preservation in bad times and wealth accumulation in good times.

Concluding Remarks

The key findings from all four random-walk series are summarized below:

1. Modern finance assumes that all asset prices follow a random walk. The academics define reward as the mean return at the peak of a bell curve. Data taken from a variety of asset classes (Part 1 and Part 2) with return horizons from one day to 10 years are far too erratic to fit the random walk statistics.

2. The histograms of a variety of asset classes not only reject the bell curve, they do not fit any well-known analytical probability theory. The types of randomness observed are akin to Frank Knight's "radical uncertainties", Donald Rumsfeld's "the unknown unknowns" or Nassim Nicholas Taleb's "Black Swans".

3. In a multimodal histogram with no central tendency, mean and variance are ill defined. The mean-variance paradigm is unfit to depict real-world prices.

4. Modern finance misreads risk as volatility (Part 3). Volatility reflects diversity in market views when the act of buying or selling does not affect the price. Volatility facilitates trades, lubricates liquidity and alleviates financial market risks. In contrast, a market with a single dominant view creates a buyer-seller imbalance. Risks come from uni-directional price movements that freeze liquidity and exacerbate bubbles or panics.

5. Investment risk comes in many forms – market risk, geopolitical events, inflation, currency, interest rate, recession, etc. Regardless the sources, all risks lead to the same outcome – an unacceptable loss in the form of income or capital, or both.

6. High uncertainties and radically random distributional gains are not risks, but represent abundant opportunities and widely scattered investment rewards.

7. I define investment reward as the cumulative probability-weighted gain; and investment risk as the cumulative probability-weighted loss. My new framework accurately captures all observed data and is applicable to probability distributions of any shape and form. More importantly, it has an intuitive appeal to investors.

8. A Random walk is a theory. A theory is supposed to describe and explain empirical observations. It provides analytical formulas that can predict the future. However, theories that hypothesize causation but disregard any aberration that does not fit their paradigms are theoretical landmines for all uninformed followers.

9. My gain-loss framework is not a theory, but a phenomenological model. It truthfully measures observations with statistical tools but offers no causation explanations or analytical formulas. As stated in Part 2, investors' adaptive behavioral dynamics render all analytical models in mathematical finance imprecise at best. An empirical model that objectively captures data with no theoretical bias is more practical for investors.

10. My new framework reveals that Mr. Market has a dual personality. He keeps his losses in time insensitive and confined zones but lets his gains run wild and loose. We can exploit this asymmetry to beat Mr. Market at his own game.

11. The active-passive investment approach capitalizes on this gain-loss asymmetry and tilts the betting odds in our favor. We actively mitigate losses in the pink zones via regular market health checkups. Otherwise, we stay as passive investors and pick up the radically random profits Mr. Market leaves behind.

12. How can we detect Mr. Market's mood changes? My six rules-based market monitors demonstrate that early warning detection is possible.

13. Warren Buffett consistently beats the market. He could be a practitioner of the dynamic active-passive approach because his favorite holding period is "forever" (a passive investor) subject to his first and second rules of "don't lose money" (an active manager).

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 mailto:ted@ttswadvisory.com.

Modern finance defines reward as the mean return (also known as the expected value). Mean return is the cumulative probability weighted return defined as follows:

where r is return (a continuous random variable) and Prob(r) is the Gaussian probability density function (PDF) of r. The integration limits are - infinitive to + infinitive. The cumulative sum of Prob(r) is normalized to 100%.

Modern finance defines risk as volatility (also known as standard deviation). It is the cumulative probability weighted root-mean-squared (RMS) of the deviation of each r from Mean. The mathematical formula for risk is:

where Mean is given by Eqn (A1). The integration limits are - infinitive to + infinitive.

The reward-to-risk ratio is therefore the mean divided by the standard deviation. Replacing r in Eqn(A1) and Eqn(A2) by the quantity r minus the risk-free interest rate (this quantity is also known as the equity risk premium), the reward-to-risk ratio becomes the Sharpe Ratio.

Appendix B: The Gain-Loss Framework

Investors view reward and risk from a gain-loss perspective. The best way to capture this intuitive view is with a pair of complementary formulas called "Expected Gain" and "Expected Loss". Investment reward is the expected gain, which is the sum of all probability weighted positive returns. The formula is:

where r* is return (a discrete random variable) and Prob*(r*) is the observed probability of return r*. The cumulative sum of Prob*(r*) is normalized to 100%. The limits of integration are from zero to + infinitive, so only positive returns (gains) are summed. In Eqn (A1), r and Prob(r) are Gaussian function variables. In Eqn (B1), r* and Prob*(r*) are measured data.

Correspondingly, investment risk is defined as the expected loss, which is the sum of all cumulative probability weighted negative returns. The formula for expected loss is:

The integration limits are - infinitive and zero, namely, only negative returns (losses) are summed.

The reward-to-risk ratio is the expected gain divided by the expected loss, both of which are in percent.

# Random Walk Part 3 – What’s Wrong with Depicting Risk as Volatility?

Originally Published October 2, 2017 in Advisor Perspectives

This is the third of my four-part empirical research into the fallacy of the random walk view of investment reward and risk. Part 1 and Part 2 discussed the random walk's failure in portraying asset returns. A random walk depicts risk as volatility.

This article explains why this view is problematic. Risk and volatility are different conceptually. The bell-curve understates actual risks by huge amounts. For asset returns that are not bell-shaped, volatility has no meaning. Finally, volatility is akin to noise, which alleviates instead of elevating risk.

In part 4, I will define investment reward and risk mathematically. I will demonstrate how this probability-based framework enables investors to beat the S&P 500 total-return with less risk.

Risk and volatility are conceptually different

Volatility is a measure of uncertainty in a bell-shaped distribution. The academics define risk as uncertainty primarily for mathematical convenience. Glyn Holton argued that if risk were akin to uncertainty, then a man jumping out of an airplane without a parachute would face no risk because his death was 100% certain. Figure 1 illustrates Holton's argument in an investment context. The green dune at the bottom right is an investment with an expected value of 100% and a volatility of 10%. The pink spike on the left is another investment with an expected value of -99% and a volatility of 1%.

From a risk perspective, modern finance favors the pink investment that offers a 10 times lower volatility than the green one. Investors, however, would intuitively avoid the pink investment because they see the 100% odds of a total loss with no chance of any gain. The academics view the green investment as more risky because it is 10-times more volatile than the pink one. Investors would jump on the green one because they see near 100% odds of doubling their money with zero chance of any loss.

Figure 1 illustrates the conceptual flaw in the random walk notion of risk. Outcome uncertainty is not necessarily risk. Risk is an unacceptable loss.

Random walk grossly underestimates risk

Benoit Mandelbrot points out in The Misbehavior of Markets that Gaussian statistics (random walks) that are behind modern finance grossly underestimate the probabilities of many stock market crashes. To find out how way-off the random walk predictions are, I computed the probability density function (PDF) of the daily returns of the Dow Jones Industrial Average (DJIA) using a measured mean of 0.03% and a standard deviation of 1.24% (from Figure 7 in Part 1). The dark blue curve in figure 2 is the random walk PDF showing the model's predictions of the DJIA's one-day price changes in 116 years (data sources: MetaStock and Yahoo Finance).

The inset table in figure 2 lists 10 of the worst one-day drops in 116 years. On October 19, 1987, for instance, the DJIA dropped -22.61%, a decline that the random walk PDF predicted to have a probability of only 4.2E-075 (the fifth column in the table). To grasp the magnitude of this miss, I use the age of our universe (roughly 14 billion years or 1.4E+010) as the unit of measure. A 4.2E-075 probability means that the Black Monday crash should have occurred only once in a billion billion billion billion billion billion billion times the age of our universe – one followed by 64 zeros. The probability is practically zero.

In reality, the actual frequency of occurrence of each of the seven worst events including Black Monday is 3.2E-005 or 0.0032% (one event out of 31,793 trading days from 1/2/1900 to 12/30/2016). A probability of 0.0032% may be small, but the impact of a one-day crash of -23% was enormous. To make things worse, those bad days tended to cluster – five times from 1928 to 1933, twice in 1987 and twice in 2008.

Volatility is a meaningless metric except for bell-shaped distributions

Even if we disregard the conceptual flaw and ignore the huge errors of viewing risk as volatility, we still face with an operational issue. Volatility only works as a statistical yardstick in an ideal bell curve. In the real world, most asset histograms are not bell-shaped. Figure 3 (taken from Part 1 and Part 2) shows five-year return histograms of four widely diverse asset classes (light blue bars). None of them resembles the corresponding random walk PDFs (dark blue curves). The notions of mean and variance are not workable in these wild histograms.

Modern finance doctrines such as Markowitz's portfolio selection, Sharpe's beta and Black-Scholes' risk neutrality view the world through a random walk lens describing everything in terms of means and variances. Figure 3 shows that in the real world, there is no central mean or recognizable variance. It comes as no surprise that many investment strategies based on these doctrines failed to protect investors against risk during financial market meltdowns.

Modern finance confuses risk with noise

The left chart in figure 4 is a typical noise output of an electronic amplifier measured with an oscilloscope. The right chart is the corresponding histogram. Noise in an amplifier shares the same mathematical root as volatility in asset prices.

I constructed a simulated price series using the random walk pricing model outlined in the Appendix. Figure 5 is a simulated DJIA index computed with Equation (A1) along with its daily returns computed with equation (A2). The band labeled "Random-Walk Daily % Change" in Figure 5 looks very similar to the amplifier noise band in Figure 4 because both are Gaussian. What the academics define as investment risk is what electrical engineers call white noise.

Then what is risk?

Figure 6 shows the actual DJIA and its daily percent changes from 1929 to 1934. The white noise band labeled "Random-Walk Daily % Change" in Figure 5 is uniform and well behaved within ± 1.5%. The band labeled "Actual DJIA Daily % Change" in Figure 6 is erratic and thorny with sharp spikes extended beyond ± 10%. The uniform band in Figure 5 is noise. The negative spikes in Figure 6 are risks. Noise is akin to volatility – uncertainty in the outcomes. Risk is associated with the harmful impact from a negative outcome.

Finally, I use two NASA incidences to illustrate the key difference between noise and risk. The Hubble Space Telescope had a defective mirror that caused spherical aberration that distorted its signals. As a result, Hubble's signals were noisy. The Space Shuttle Challenger broke apart shortly after launch due to improper O-rings. As a result, seven astronauts lost their lives. The Hubble fiasco was a noise issue leading to fuzzy images. The Challenger disaster was a risk matter resulting in deaths. Noise and risk are materially different.

Concluding remarks

The academics envision investors strolling down Wall Street with random movements, but at a smooth and orderly pace. Such tidy bell-shaped randomness is volatility. In reality, investors do not walk in small steps; they jog, run, jump and even take giant leaps, and deep dives, creating turbulence and chaos in the financial markets. Such violent footprints are risks.

Volatility measures fluctuations and risk signifies dangers. They are not synonymous as modern finance claims. Fischer Black in his 1986 article entitled Noise argued that volatility was a risk-reducing agent. When there is diversity (noise) in market views, prices will fluctuate, giving rise to volatility. Volatility is the mechanism through which buyers and sellers carry out normal price discoveries. Noise facilitates transactions and volatility alleviates risk in the financial markets. In contrast, a market that has only one prevailing belief is noise-free. Prices no longer fluctuate but charge in the direction of the dominant view. When the bears coalesce and become the majority, liquidity will dry up, panic selling will set in and risk will surge.

By defining risk as volatility, the random walk theorists create a paradox. When price fluctuations are less than ± one standard deviation from the mean, the bell curve has some resemblance to the real world. In the central region of the bell curve, however, volatility is not risk but a risk stabilizer. At the tail ends of the bell curve, where real risks reside, volatility is a meaningless metric because Gaussian statistics no longer work.

In the next and final article, I will define both investment reward and risk using a framework different from the random walk paradigm. Like the random walk, my new framework has firm probability underpinnings. Unlike the random walk, my new formulas are applicable to all probability distributions regardless of their shapes and forms. More importantly, the new model yields new insights on how to increase the odds of beating the S&P500 total return with less risk – a direct challenge to Efficient Market Hypothesis.

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 mailto:ted@ttswadvisory.com.

Appendix: The Random-walk Pricing Model

The random walk pricing model relates the price at time T + ΔT (Price T + ΔT) to the price at time T (Price T) with the following formula:

Price T + ΔT = Price T x [1 + Mean ± Volatility]. (A1)

Dividing both sides of Equation (A1) by Price T and rearranging terms, one obtains:

(Price T + ΔT / Price T) - 1 = Mean ± Volatility. (A2)

Equation (A1) is a simulated random walk price series. Equation (A2) is the return (rate of change) on that price series within a time increment of ΔT.

In the above two equations,

Mean = Expected Value (Cumulative Probability Weighted Return); and (A3)

Volatility = N x Standard Deviation, (A4)

The sign and the amplitude of N can be simulated using a Gaussian noise generator with a bell-shaped probability distribution.

To create a daily (ΔT = 1 day) DJIA time series, I use a mean of 0.03% and a standard deviation of 1.24%, the same parameters obtained from Figure 7 in Part 1, which were used to compute the DJIA daily return PDF in figure 2.

# 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 mailto:ted@ttswadvisory.com.

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 1

Originally Published October 24, 2016 in Advisor Perspectives

The proverbial wisdom is that there are two types of stock market cycles – secular and cyclical. I argued previously that secular cycles not only lacked statistical basis to be credible, but their durations of 12 to 14 years are also impractical for most investors. We live in an internet age with a time scale measured in nanoseconds. Wealth managers often turn over their portfolios after only a few years. Simply put, secular cycles can last longer than financial advisors can retain their clients.

The second type of cycle is called a “cyclical market” and is believed to comprise both primary and secondary waves. Economic cycles are thought to drive primary waves. According to the National Bureau of Economic Research (NBER), the average economic cycle length is 4.7 years, which would be more suitable for the typical holding periods of most investors.

To succeed in accumulating wealth in bull markets and preserving capital in bear markets, we must first define and detect primary and secondary markets. In this article, I present a modeling approach to spot primary cycles. Modeling secondary market cycles will be the topic of Part 2.

Common flaws in modeling financial markets

Before presenting my model on primary markets, I must digress to discuss two common mistakes in modeling financial markets. For example, when modeling secular market cycles and market valuations, analysts use indicators such as the Crestmont P/E, the Alexander P/R and the Shiller CAPE (cyclically adjusted price-earnings ratio). By themselves, these indicators are fundamentally sound. It's the modeling approach using these indicators that is flawed.

Models on valuations and secular cycles cited above share two assumptions. First, they assume that the amplitude (scalar) of the indicators can be relied on to indicate market valuations and secular outlook. Extremely high readings are interpreted as overvaluations or cycle crests, and extremely low readings, undervaluation or cycle troughs. Second, it's assumed that mean reversion will always drive the extreme readings in the models back into line.

Figure 1A shows the S&P 500 from 1881 to mid-2016 in logarithmic scale. Figure 1B is the Shiller CAPE overlay. The solid purple horizontal line is the mean from 1881 to 1994 and has a value of 14.8. The upper and lower dashed purple lines represent one standard deviation above and below the mean of 14.8, respectively. The solid brown line to the right is the mean from 1995 to mid-2016 and has a value of 26.9. The upper and lower dashed brown lines are one standard deviation above and below the post-1995 mean of 26.9, respectively. One standard deviation above the pre-1995 mean is 19.4 and one standard deviation below the post-1995 mean is 20.4. The data regimes in the two adjoining timeframes do not overlap. The statistically distinct nature of the two regimes invalidates the claim by many secular cycle advocates and CAPE-based valuations practitioners that the elevated CAPE readings after 1995 are just transitory statistical outliers and will fall back down in due course.

Let's examine the investment impacts from these two assumptions. The first assumption is that extreme amplitudes can be used to track cycle turning points. Figure 1B shows that both high and low extremes are arbitrary and relative. As such, they cannot be used as absolute valuation markers. For example, after 1995, the entire amplitude range has shifted upward. Secular cyclists and value investors would have sold stocks in 1995 when the CAPE first pierced above 22, exceeding major secular bull market crests in 1901, 1937 and 1964. They would have missed the 180% gain in the S&P 500 from 1995 to its peak in 2000. More recently, the CAPE dipped down to 13 at the bottom of the sub-prime crash. Secular cycle advocates and value investors would consider a CAPE of 13 not cheap enough relative to previous secular troughs in 1920, 1933, 1942, 1949, 1975 and 1982. They would have asked clients to switch from stocks to cash only to miss out the 200% gain in the S&P 500 since 2010. These are examples of huge upside misses caused by the first flawed assumption used in these scalar-based models.

The second assumption is that mean reversion always brings the out-of-bound extremes back into line. This assumption falters on three counts. First, mean reversion is not mean regression. The former is a hypothesis and the latter, a law in certain statistics like Gaussian distributions (the bell curve). Second, mean regression is guaranteed only for distributions that resemble a bell curve. If the distributions follow the power-law or the Erlang statistics, even mean regression is not guaranteed. Finally, neither mean regression nor mean reversion is a certainty if the overshoots are not by chance, but are the results of causation. Elevated CAPE will last as long as the causes (Philosophical Economics, Jeremy Siegel and James Montier) remain in place. The second assumption creates a false sense of security that could be very harmful to your portfolios.

The confusion caused by both of these false assumptions is illustrated in Figure 1B. For the 26.9 mean, reversion has already taken place in 2002 and 2009. But for the 14.8 mean, reversion has a long way to go. All scalar models that rely on arbitrary amplitudes for calibration and assume a certainty of mean reversion are doomed to fail.

A vector-based modeling approach

The issues cited above are the direct pitfalls of using scalar-based indicators. One can think of scalar as an AM (amplitude modulation) radio in a car. The signals can be easily distorted when the car goes under an overpass. Vector, on the other hand is analogous to FM (frequency modulation) signals, which are encoded not in amplitude but in frequency. Solid objects can attenuate amplitude-coded signals but cannot corrupt frequency-coded ones. Likewise, vector-based indicators are immune to amplitude distortions caused by external interferences such as Fed policies, demographics, or accounting rule changes that might cause the overshoot in the scalar CAPE. Models using vector-based indicators are inherently more reliable.

Instead of creating a new vector-based indicator from scratch, one can transform any indicator from a scalar to a vector with the help of a filter. Two common signal-processing filters used by electronic engineers to condition signals are low-pass filters and high-pass filters. Low-pass filters improve lower frequency signals by blocking unwanted higher frequency chatter. An example of a low-pass filter is the moving average, which transforms jittery data series into smoother ones. High-pass filters improve higher frequency signals by detrending irrelevant low frequency noise commonly present in the physical world. The rate-of-change (ROC) operator is a simple high-pass filter. ROC is defined as the ratio of the change in a variable over a specific time interval. Common time intervals used in financial markets are year-over-year (YoY) or month-over-month (MoM). By differentiating (taking the rate-of-change of) a time series, one transforms it from scalar to vector. A scalar only shows amplitude, but a vector contains both amplitude and direction contents. Let me illustrate how such a transformation works.

Figures 2A is identical to Figure 1B, the scalar version of the Shiller CAPE. Figure 2B is a vector transformation, the YoY-ROC of the scalar Shiller CAPE time series. There are clear differences between Figure 2A and Figure 2B. First, the post-1995 overshoot aberration in Figure 2A is no longer present in Figure 2B. Second, the time series in Figure 2B has a single mean, i.e. the mean from 1881 to 1994 and the mean from 1995 to present are virtually the same. Third, Figure 2B shows that the plus and minus one standard deviations from the two time periods completely overlap. This proves statistically that the vector-based indicator is range-bound across its entire 135-year history. Finally, the cycles in Figure 2B are much shorter than that in Figure 2A. Shorter cycles are more conducive to mean reversion.

It's clear that the YoY-ROC filter mitigates many calibration issues associated with the scalar-based CAPE. The vector-based CAPE is range-bound, has a single and stable mean and has shorter cycle lengths. These are key precursors for mean reversion. In addition, there are theoretical reasons from behavioral economics that vectors are preferred to scalars in gauging investors' sentiment. I will discuss the theoretical support a bit later.

The vector-based CAPE periods versus economic cycles

Primary market cycles are believed to be driven by economic cycles. Therefore, to detect cyclical markets, the indicator should track economic cycles. Figure 3A shows the S&P500 from 1950 to mid-2016. The YoY-ROC GDP (Gross Domestic Product) is shown in Figure 3B and the YoY-ROC CAPE in Figure 3C. The Bureau of Economic Analysis (BEA) published U.S. GDP quarterly data only after 1947.

The waveform of the YoY-ROC GDP is noticeably similar to that of the YoY-ROC CAPE. In fact, the YoY-ROC CAPE has a tendency to roll over before the YoY-ROC GDP dips into recessions, often by as much as one to two quarters. The YoY-ROC GDP and the YoY-ROC CAPE are plotted as if the two curves were updated at the same time. In reality, the YoY-ROC CAPE is nearly real-time (the S&P 500 and earnings are at month-ends and the Consumer Price Index has a 15-day lag). GDP data, on the other hand, is not available until a quarter has passed and is revised three times. The YoY-ROC CAPE indicator is updated ahead of the final GDP data by as much as three months. Hence, the YoY-ROC CAPE is a true leading economic indicator.

Although the waveforms in Figures 3B and 3C look alike, they are not identical. How closely did the YoY-ROC CAPE track the YoY-ROC GDP in the past 66 years? The answer can be found with the help of regression analysis. Figure 4 shows an R-Squared of 29.2%, the interconnection between GDP growth rate and the YoY-ROC CAPE. A single indicator that can explain close to one-third of the movements of the annual growth rate of GDP is truly amazing considering the simplicity of the YoY-ROC CAPE and the complexity of GDP and its components.

Primary-ID – a model for primary market cycles

Finding an indicator that tracks economic cycles is only a first step. To turn that indicator into an investment model, we have to come up with a set of buy and sell rules based on that indicator. Primary-ID is a model I designed years ago to monitor major price movements in the stock market. In the next article, I will present Secondary-ID, a complementary model that tracks minor stock market movements. I now illustrate my modeling approach with Primary-ID.

A robust model must meet five criteria: simplicity, sound rationale, rule-based clarity, sufficient sample size, and relevant data. Primary-ID meets all five criteria. First, Primary-ID is elegantly simple – only one adjustable parameter for "in-sample training." Second, the vector-based CAPE is fundamentally sound. Third, buy and sell rules are clearly defined. Forth, the Shiller CAPE is statistically relevant because it covers over two dozen samples of business cycles. Fifth, the Shiller database is quite sufficient because it provides over a century of monthly data.

Figure 5A shows both the S&P 500 and the YoY-ROC CAPE from 1900 to 1999. This is the training period to be discussed next. The curves are in green when the model is bullish and in red when bearish. Bullish signals are generated when the YoY-ROC CAPE crosses above the horizontal orange signal line at -13%. Bearish signals are issued when the YoY-ROC CAPE crosses below the same signal line. The signal line is the single adjustable parameter in the in-sample training.

Figure 5B compares the cumulative return of Primary-ID to the total return of the S&P 500, a benchmark for comparison. A \$1 invested in Primary-ID in January 1900 hypothetically reached \$30,596 at the end of 1999, a compound annual growth rate (CAGR) of 10.9%. The S&P 500 over the same period earned \$23,345, a CAGR of 10.3%. The 60 bps CAGR gap may seem small, but it doubles the cumulative wealth after 100 years. The other significant benefit of Primary-ID is that its maximum drawdown is less than two third of that of the S&P 500. It trades on average once every five years, very close to the average business cycle of 4.7 years published by NBER.

The in-sample training process

Figures 5A and 5B show a period from 1900 to 1999, which is the back-test period used to find the optimum signal line for Primary-ID. The buy and sell rules are as follows: When the YoY-ROC CAPE crosses above the signal line, buy the S&P 500 (e.g. SPY) at next month's close. When the YoY-ROC CAPE crosses below the signal line, sell the S&P 500 at next month's close and park the proceeds in US Treasury bond. The return while holding the S&P 500 is the total return with dividends reinvested. The return while holding bond is the sum of both bond yields and bond price percentage changes caused by interest rate changes.

Figure 6 shows the back test results in two tradeoff spaces. The plot on the left is a map of CAGR versus maximum drawdown for various signal lines. The one on the right is CAGR as a function of the position of the signal line. For comparison, the S&P 500 has a total return of 10.6% and a maximum drawdown of -83% in the same period. Most of the blue dots in Figure 6 beat the total return and all have maximum drawdowns much less than that of the S&P 500.

Figures 6A and 6B only show a range of signal lines that offers relatively high CAGR. What is not shown is that all signal lines above -10% underperform the S&P 500. The two blue dots marked by blue arrows in both charts are not the highest returns nor the lowest drawdowns. They are located in the middle range of the CAGR sweet spot. I judiciously select a signal line at -13% that does not have the maximum CAGR. An off-peak parameter gives the model a better chance to outperform the optimized performance in the backtest. Picking the optimized adjustable parameter would create an unrealistic bias for the out-of-sample test results. Furthermore, an over-optimized model even if it passes the out-of-sample test is prone to underperform in real time. A parameter that is peaked during back-tests will likely lead to inferior out-of-sample results as well as actual forecasts.

Why do all signal lines above -10% give lower CAGR's than those within -10% and -19%? There is a theoretical reason for such an asymmetry to be discussed a bit later.

The out-of-sample validation

The out-of-sample test is a guard against the potential risk of over-fitting during in-sample optimization. It's like a dry-run before applying the model live with real money. Passing the out-of-sample test, however, does not necessarily guarantee a robust model but failing the out-of-sample test is certainly ground for model rejection.

Here’s how out-of-sample testing works. The signal line selected in the training exercise is applied to a new set of data from January 2000 to July 2016 with the same buy and sell rules. Figure 7A shows both the S&P 500 and the YoY-ROC CAPE.

Figure 7B compares the cumulative return of Primary-ID to the total return of the S&P 500. A \$1 invested in Primary-ID in January 2000 would hypothetically make \$3.50 in mid-2016, a CAGR of 7.8%. Investing \$1 in the S&P 500 over the same period would have earned \$2.02, a CAGR of 4.3%. An added perk Primary-ID offers is the maximum drawdown of -23%, half of that of the S&P 500’s -51%. It trades on average once every five years, similar to that in the in-sample test, and therefore profits are taxed at long-term capital gains rates.

Primary-ID sidestepped two infamous bear markets: the dot-com crash and the sub-prime meltdown. It also fully invested in equities during the two mega bull markets in the last 16 years. The value of the YoY-ROC CAPE as a leading economic indicator and the efficacy of Primary-ID as a cyclical market model are validated.

Theoretical support for Primary-ID

The theoretical support for Primary-ID can be found in prospect theory proposed by Daniel Kahneman and Amos Tversky in 1979. Prospect theory offers three original axioms that lend support to Primary-ID. The first axiom shows that there is a two-to-one asymmetry between the pain of losses versus the joy of gains – losses hurt twice as much as gains bring joy. Recall from Figure 2B that the sweet spot for CAGR comes from signal lines located between -10% and -19%, more than one standard deviation below the mean near 0%. Why is the sweet spot located that far off center? The reason could be the result of the asymmetry in investors' attitude toward reward versus risk. Prospect theory explains an old Wall Street adage – let profits, run but cut losses short. Primary-ID adds a new meaning to this old motto – buy swiftly, but sell late. In other words, buy quickly once YoY-ROC CAPE crosses above -13% but don't sell until YoY-ROC CAPE crosses below -13%.

The second prospect theory axiom deals with scalar and vector. The authors wrote, "Our perceptual apparatus is attuned to the evaluation of changes or differences rather than to the evaluation of absolute magnitudes." In other words, it's not the level of wealth that matters; it's the change in the level of wealth that affects investors' behavior. This explains why the vector-based CAPE works better than the original scalar-based CAPE. The former captures human behaviors better than the latter.

The third prospect theory axiom proposed by Kahneman and Tversky is that "the value function is generally concave for gains and commonly convex for losses." Richard Thaler explains this statement in layman's terms in his 2015 book entitled "Misbehaving." The value function represents investors' attitudes toward reward and risk. The terms concave and convex refer to the curve shown in Figure 3 in the 1979 paper. A concave (or convex) value function simply means that investors' sensitivity to joy (or pain) diminishes as the level of gain (or loss) increases. The diminishing sensitivity is observed only on the change in investors' attitude (vector) and not on the investors' attitude itself (scalar). Investors' diminishing sensitivity toward both gains and losses is the reason that the YoY-ROC CAPE indicator is range-bound and why mean reversion occurs more regularly. The original Shiller CAPE is a scalar time series and does not benefit from the third axiom. Therefore the apparent characteristics of range-bound and mean reversion of the scalar Shiller CAPE in the past are the exceptions, not the norms.

Concluding remarks

The stock market is influenced by different driving forces including economic cycles, credit cycles, Fed policies, seasonal/calendar factors, equity premium anomaly, risk aversion shifts the equity premium puzzle and bubble/crash sentiment. At any point in time, the stock market is simply the superposition of the displacements of all these individual waves. Economic cycle is likely the dominant wave that drives cyclical markets, but it is not the only one. That's why the R-squared is only at 29.2% and not all bear markets were accompanied by recessions (such as 1962, 1966 and 1987).

The credibility of the Primary-ID model in gauging primary cyclical markets is grounded on several factors. First, it is based on a fundamentally sound metric – the Shiller CAPE. Second, its indicator (YoY-ROC CAPE) is a vector that is more robust than a scalar. Third, the model tracks the cycle dynamics between the market and the economy relatively well. Forth, the excellent agreement between the five-year average signal length of Primary-ID (0.2 trades per year shown in Figures 5B and 7B) and the average business cycle of 4.7 years reported by NBER adds credence to the model. Finally, the Primary-ID model has firm theoretical underpinnings in behavioral economics.

It's a widely held view that the stock market exhibits both primary and secondary waves. If primary waves are predominantly driven by economic cycles, what drives secondary waves? Can we model secondary market cycles with a vector-based approach similar to that in Primary-ID? Can such a model complement or even augment Primary-ID? Stay tuned for Part 2 where I debut a model called Secondary-ID that will address all these questions.

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 atted@ttswadvisory.com.

# A Market Valuation Gauge That Works

Originally Published March 15, 2016 in Advisor Perspectives

In my previous article, I examined many popular metrics that all show that U.S. equities have been overvalued for over 20 years. The conventional explanation is that the overvaluation and its unusually long duration is a statistical outlier. But those aberrations were observed in only 15% of the data population (20 out of 134 years) and are unlikely to be statistical outliers. The root cause is not yet known. Until the anomaly is better understood, naively equating the lack of mean reversion with overvaluations will lead to misguided valuations and ill-advised investment strategies.

A decade ago, I began searching for a valuation indicator that is immune to possible mean-reversion malfunction. The challenge proved to be much more difficult than anticipated. I ultimately had to abandon my search and developed my own valuations gauge, the total return oscillator (TR-Osc) and present it here.

Oscillatory gauge

Mean reversion is the underpinning of all valuations metrics. The basic concept of valuations relies on the notion that value oscillates between an upper bound (overvalued) and an lower bound (undervalued) around a median (fair-valued). How do you calibrate a gauge that has an unbounded output or with a drifing median that confuses mean reversion? A functioning valuations gauge should resemble a pseudo sine-wave oscillator with quasi-periodicity.

Although the cyclically adjusted price-to-earnings ratio (CAPE) oscillated around a stable geometric mean of 14 from 1880 to 1994, its mean has risen to 26.2 since 1995 (Figure 1A) – a telltale sign of mean reversion malfunction. By contrast, my TR-Osc has been bounded by well-defined upper and lower demarcations for over a century. The mean of TR-Osc measured from 1875 to 1994 is almost identical to the value computed over the last 20 years (Figure 1B). After reaching either extreme, TR-Osc always reverts toward its long-term historical mean.

From 1880 to 1950, TR-Osc and CAPE were almost in sync. After 1955, the two indicators began to diverge. Although both the CAPE and TR-Osc detected the dot-com bubble in 2000 (red squares), only the TR-Osc warned us about the 1987 Black Monday crash (red circle). After the 2000 peak, CAPE stayed elevated and came down only once in mid-2009 to touch its historical mean at 14. The TR-Osc, however, dropped to its lower bound in January 2003 (green arrow) getting ready for the six-year bull market from 2003 to 2008. The TR-Osc did it again after the housing bubble when it dipped below the lower bound of 0% in February 2009, just in time to reenter the market at the start of a seven-year bull market from 2009 to present.

In 2008, the TR-Osc reached a minor summit (red triangle) while CAPE exhibited no peak at all. Both TR-Osc and CAPE indicate that the meltdown in global financial markets did not stem from an overvalued equity market. I will expand on this later when I discuss the real estate sector.

Common deficiencies in all contrarian indicators

There are two common deficiencies shared by all contrarian indicators including all traditional valuations models. First, their signals are often premature because the market can stay overvalued or undervalued for years. Greenspan's 1996 irrational exuberance speech alluded to an overvalued market but it was four years too early. From 1996 to the dot-com peak in 2000, the S&P500 surged 87% and the NASDAQ 288%.

The second deficiency of all contrarian indicators is that the market can reverse direction without hitting either extreme at all. The CAPE, for example, was not undervalued in 2002 or 2009. Value investors would have missed out on huge gains of 90% and 180% from the two spectacular bull markets in the 2000s.

The dual gauges of the TR-Osc: scalar and vector

Before I explain how the TR-Osc overcomes these two deficiencies, let me first describe the TR-Osc. The TR-Osc captures what investors in the aggregate earn by investing in the S&P 500. That is the sum of two components – the first from price changes and the second from dividend yields. Price return is the trailing five-year compound annual growth rate (CAGR). Dividend yield is the annual return from the dividends investors received. The look-back period doesn't necessarily have to be five years. All rolling periods from 2 to 20 years can do the job. In addition, both real (inflation-adjusted) and nominal TR-Osc's work equally well because inflation usually does not change much over a five-year period.

The TR-Osc overcomes the two deficiencies by having two orthogonal triggers, a scalar marker and a vector sensor. The oscillatory and mean-reverting attributes of the TR-Osc allow overvaluation and undervaluation markers to be clearly defined (Figure 2). When the TR-Osc was near the upper bound (the 20% overvalued marker), the S&P 500 often peaked. When the TR-Osc was near the lower bound (the 0% undervalued marker), the market soon bottomed. But in 2008, the TR-Osc only reached 12% and the market was not overvalued. Investors had no warning from the valuation marker to avert the impending subprime meltdown. Valuation markers (scalar) alone are not enough. The TR-Osc needs a second trigger, a motion sensor (vector) that tracks the up or down direction of valuations.

Let me illustrate how the scalar and vector triggers work in concert and how buy/sell signals are executed. When the TR-Osc is rising (an up-vector) or drops below the lower bound at 0% (an undervalued marker), a bullish market stance is issued. When the TR-Osc is falling (a down vector) but stays above 0% (not undervalued), or when it exceeds the upper bound at 20% (an overvalued marker), a bearish alarm is sounded. The asymmetry in the buy/sell rules stems from prospect theory, which contends that losses have more emotional impact to people than an equivalent amount of gains.

When a bullish signal is issued, buy the S&P 500 (e.g. SPY). When a bearish alarm is sounded, sell the S&P 500. After exiting the stock market, park the proceeds in 10-year Treasury bonds. The return while holding the S&P 500 is the total return with dividends reinvested. The return while holding U.S. Treasury bonds is the geometric sum of both bond yields and bond price percentage changes caused by interest rate changes.

The performance data presented in this article assume that all buy and sell signals issued at the end of the month were executed at the close in the following month. When the TR-Osc signals were executed closer to the issuance dates, both return and risk performances were slightly better.

TR-Osc performance stats

Figure 3 shows two hypothetical cumulative returns from 1880 to 2015 – the TR-Osc with the buy/sell rules stated above and the S&P 500 total return. Over 135 years, the TR-Osc has a 190 basis point CAGR edge over the buy-and-hold benchmark with less than half of the drawdown risk.

The TR-Osc traded infrequently – less than one round trip a year on average. The TR-Osc is an insurance policy that protects investors against catastrophic market losses while preserving their long-term capital gain tax benefits.

Let's take a closer look at the TR-Osc signals in two more recent time windows. Since 1950, there have been 10 recessions. Figure 4A shows that the TR-Osc kept investors out of the market in all 10 of them. Figure 4B shows that the latest TR-Osc bearish call was issued in September 2015. The TR-Osc sidestepped the recent stock market turmoil and has kept investors' money safe in Treasury bonds.

Table 1 shows performance stats for various sets of bull and bear market cycles. TR-Osc beats the S&P500 total return in CAGR, maximum drawdown, and volatility. The consistency in outperforming the S&P500 in returns and in risk over different sets of full bull/bear cycles demonstrates the robustness of TR-Osc.

TR-Osc has universal applicability

Like the CAPE, the TR-Osc’s efficacy is not limited to the S&P 500. It can also measure valuations in overseas markets (developed and emerging), hard assets and currencies. For example, Figure 5 shows three alternative spaces – raw materials (Figure 5A), oil and gas (Figure 5B) and real estate (Figure 5C) (data source: Professor Kenneth French). This universal applicability of the TR-Osc also enables intermarket synergies. Recall in Figure 2 that the stock market was not overvalued in 2008 according to both the CAPE and the S&P 500 TR-Osc. Note that the real estate TR-Osc correctly detected the housing bubble (red square in Figure 5C). When the systemic risk spread to the stock market, the S&P 500 TR-Osc vector sensed the danger and turned bearish.

Figures 6A to 6C shows that the TR-Osc improves both the return and drawdown in two distinctively different spaces – precious metals (data source: Professor Kenneth French), the Canadian dollar and the Australian dollar (data source: FRED). Prices in precious metals fluctuate widely at rapid speeds while foreign currencies crawl in narrow ranges at a snaillike pace. It's remarkable that the TR-Osc works equally well across drastically different investment classes. How does the TR-Osc help a diverse group of characters with different personalities perform better?

The analytics of TR-Osc

You may say that TR-Osc is just a five-year rolling total return. But what breathes new life into an otherwise ordinary formula is the analytics behind the TR-Osc. The adaptability of buy and sell rules is the reason behind the TR-Osc's universal applicability. As indicated previously, the TR-Osc has two triggers: valuation markers (scalar) and valuation directional sensor (vector). How did I pick the values for these triggers? The vector is obvious – up is bullish and down is bearish – but how do I select the valuation markers?

In Figures 2, 5 and 6, the middle blue line is the mean. The upper blue lines are the overvaluation markers and the lower blue lines, undervaluation markers. The upper blue lines are M standard deviations above the mean and the lower blue lines, N standard deviations below the mean. Each time series has a unique personality. For example, the means of most currencies are near 0% while the mean of the S&P 500 is near 9%. More volatile investments like precious metals, oil and gas would have larger standard deviations than the serene currency space. The values of M and N are selected to match the personality of each underlying investment. The general range for both M and N is between 1 and 2.

A common flaw in the design of engineering or investments systems is over-fitting. I have developedfive criteria to minimize this bad practice. The five criteria are simplicity, sound rationale, rule-based clarity, sufficient sample size, and economic cycle stability. The TR-Osc not only meets all of these criteria but offers one additional merit – universal applicability. It works not only on the S&P 500, but on overseas markets and across a diverse set of alternative investments.

Theoretical support for TR-Osc

Traditional valuation metrics rely on fundamentals, which often experience paradigm shifts across secular cycles. Fundamental factors can be influenced by generational changes – technological advances, demographic waves, socioeconomic evolutions, structural shifts, political reforms or wars. Therefore the means in many of the traditional valuation metrics can drift when the prevailing fundamentals change.

The TR-Osc downplays the importance of the external fundamental factors and focuses primarily on the internal instinct of the investors. Investors' value perception has two behavioral anchors. The first anchor drives investors toward the greed/fear emotional extremes. For example, when the S&P 500 delivers a five-year compound annual return in excess of the 20%, euphoria tends to reach a steady state and investors become increasingly risk adverse. When their returns get stuck at 0% five years in a row, investors are in total despair and the market soon capitulates. Both greed and fear extremes can be quantified by the TR-Osc's over- and undervaluation markers.

The second behavioral anchor is the tendency of herding with the crowd. When neither greed nor fear is at extreme levels, investors have a behavioral bias toward crowd-herding. Once a trend is established in either up or down direction, more investors will jump onboard the momentum train and price momentum will solidify into sustainable trends. The collective movement of the masses is tracked by the TR-Osc's vector sensor.

Concluding remarks

Unlike fundamental factors which can be altered by paradigm shifts over long arcs of time, human behaviors which are hardwired into our brains have not changed for thousands of years. The efficient market hypothesis assumes that markets are made up of a large number of rational investors efficiently digesting all relevant information to maximize their wealth. Behavioral finance theory suggests that investors are often driven by the inherent cognitive psychology of people whose decisions are often irrational and their actions exhibit behavioral biases. Perhaps the aberration (the malfunctioned mean reversion) observed in many of the traditional valuations ratios suggests that investors are not 100% homo economicus beings after all. More often than not, investors behave irrationally when they are besieged by emotions.

The TR-Osc captures the essence of both traditional finance and behavioral economics by reading investors' value perception from both the rational and the emotional wirings of their brains. It elucidates many valuable but abstract concepts from both schools into quantitative, objective and actionable investment strategies. As long as humans continue to use their dual-process brains (see also Dr. Daniel Kahneman) in decision making, TR-Osc will likely endure as a calibrated valuation gauge until humans evolve into the next stage.

The TR-Osc asserts that the current stock market is not overvalued. Instead, since mid-2015, its vector has been reverting towards its stable historical mean.

Theodore Wong graduated from MIT with a BSEE and MSEE degree. 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 over three decades, Ted has developed a true passion in the financial markets. He applies engineering statistical tools to achieve absolute investment returns by actively managing risk in both up and down markets. He can be reached at ted@ttswadvisory.com.