### Modeling Cyclical Markets – Part 2

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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.