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Statisticians say large changes in stock prices occur more often than would occur in random behavior, giving a distribution chart "long tails" or "fat tails." If this is true stock prices would move in exaggerated trends, not in random walks.

To determine average trend lengths in a random walk, 100,000,000 random numbers were processed with RANDWALK.BAS. The figures for Avg are unmodified, not expressed as boxes:

Generally, trend lengths are about twice Min. For low Min's, trend lengths are twice Min plus a bias of about 0.3 because of the finite pace size. In a true random walk each pace would be infinitesimal. Avg grows increasingly more erratic with increasing Min because of the diminishing number of trends.

The largest Min that would have found the current trend is equal to the length of the trend. It can be expected that the current trend will extend until it doubles. Consequently, strong trends are more persistent than weak trends.

All available daily prices through 7/1/2016 for stocks and funds with 2018 options in the CBOE Symbol Directory were analyzed with TRENDS.BAS. All bond, commodity, currency, and precious metal funds that were not leveraged or inverse were included:

Generally, trend lengths are significantly more than twice Min. This is most evident for 5 Box Min, whose results are less distorted by the upward bias of the market than the other results.

Stocks from the Zacks screener were analyzed to see if there is any correlation between price and performance:

To reduce risk, do not to invest in stocks under $10. This would have eliminated the weakest stocks in 2008, 2011, and 2015. Even though this would have eliminated the strongest stocks in other years, there would still have been good investing candidates.

The same stocks from the Zacks screener were analyzed to determine behavior during January Barometer defensive periods:

The January Barometer was right in 2008, 2010, 2014, and 2015. Most of the loss in 2008 and 2015 occurred during the defensive periods. The defensive periods in 2010 and 2014 were weak while the entire year was strong.

The January Barometer was wrong in 2009. Most of the gain in 2009 occurred during the defensive period.

To reduce risk, do not hedge with stocks under $10. This would have reduced the damage in 2009. Even though this would have eliminated the weakest stocks in other years, there would still have been good hedging candidates.

Investing in options is risky. An option loses percentage value faster than the underlying security. Risk can be reduced by limiting the amount of options in an account or by offsetting call options with put options.

Investopedia says: "Six primary factors influence option pricing: the underlying price, strike price, time until expiration, volatility, interest rates and dividends." No mention is made of trend.

Volatility is based on investors' expectations for only the next 30 days. Trend can persist substantially longer than 30 days and dominate volatility. Premiums on long-term options should not only incorporate trend, they should weight trend more heavily than volatility.

Premium ratio is the premium of an option at the money relative to the price of the underlying security:

R = 100 * [(M*(U-S)+V*(S-L)] / (U-L)*S

R: premium ratio (percentage)

S: price of underlying security

L: lower strike price, just below S

M: lower strike ask price

U: upper strike price, just above S

V: upper strike ask price

The last price is ignored because some options are so inactive the price of the underlying security might have changed significantly since the last option transaction. Bid prices are ignored because many are unrealistically low and some are even zero.

Long-term call options on stocks with strong uptrends should have higher premium ratios than long-term call options on stocks with weak uptrends, but they have lower premium ratios:

Merit, which is twenty times the absolute value of the trend divided by the premium ratio, assesses option premiums. High merit requires a strong trend or a low premium ratio. Very high merit requires both.

The Real Estate (#realty) Fund Strategy uses VNQ and IYR. The #Easy Strategy uses SPY. The Treasury bond Strategy (#choices) uses IEF.

Long-term options on these funds are underpriced. VNQ, IYR, and SPY were in strong uptrends. The premium ratios were lower than the average for the selection list (#lists) of stocks with very high merit, and much lower than the average for the other preselected stocks:

I assumed stocks with low PEG ratios would outperform stocks with high PEG ratios. PEG ratio is the P/E (price to earnings) ratio divided by the expected percentage earnings growth rate for the next three to five years.

Tests on PEG ratios from the FinViz and Zacks screeners failed to validate this assumption. This reflects badly on the ability of professional analysts to predict earnings growth.

I tested advisory service ratings and Motley Fool CAPS ratings. The advisory service performances were abysmal:

The Motley Fool CAPS ratings showed promise. There were five ratings levels with about 1000 securities in each level. Securities with the top rating declined an average of 2.86 boxes, lower-rated securities declined progressively more, and bottom-rated securities declined an average of 6.08 boxes.

However, CAPS rating underperformed trends. The 1,000 securities with the strongest uptrends, using a threshold of 30 boxes, declined an average of 0.73 boxes.

Incidentally, the CAPS test provides further evidence that price trends persist. The 1,000 securities with the strongest downtrends declined an average of 12.36 boxes.

I again tested advisory service ratings and Motley Fool CAPS ratings. Securities in the higher CAPS ranks outperformed securities in the lower ranks, but all ranks underperformed broad market indices.

Most of the advisory services underperformed the indices. McLean Capital did about as well as the indices. Ned Davis was the only one that clearly outperformed the indices, but it even more clearly underperformed the stocks with the strongest uptrends.