Using Z Scores to Trade Foreign Exchange and Other Financial Instruments

PREFACE

Some time ago through many moving average conversations with my honored friend, I decided to investigate further assumptions than what was already known about simple moving averages such as what was the perfect length, what was the perfect distance between averages, what number was the perfect choice in isolation, what was the number that would catch trends, reversals and range markets.

In terms of a perfect choice in isolation, I was bothered by use of the 200 day average because I didn’t understand its purpose and because it didn’t serve a market function in terms of option expiration or six month contracts in any financial instrument especially if trading days are considered against the average. Then came the questions concerning deployment of averages in multiples of 5. Nobody I consulted had an answer except for it was “always done that way”.

Many moving average models exist to trade financial instruments but I didn’t find any useful for many reasons such as wrong choice of numbers, Fibonacci numbers, distance or an added feature of a mathematical term such as k periods. Nor did I find any enthusiasm in many academic papers I read.

My search began with the foundation of all life and numbers, the Bible. So I read “Keys to Scripture Numerics” and “Bible Mathematics” by Ed Vallowe, “Numbers in Scripture” by E.W Bullinger, and “Numbers in the Bible” by Robert D. Johnson. Keys to Scripture Numerics was the book that analyzed and categorized numbers to the last degree and the book I found most fascinating.

I transpired Ed Vallowe”s numbers and applied it to markets to determine the correct choices for moving average lengths. I listed his numbers based on Trends, Continued Trends, Volatile numbers and numbers where Price Rests.

The number 1 for example is a trend number and the place where prices not only begin but it’s the start of the first standard deviation in any number sequence. The number 5 is a trend number due to representation of power and grace.

Continued trends would include 3, 8 and 10 as good numbers. Volatile numbers are 6, 11, 13 and 21. Why 21? 21 represents sin according to Ed Vallowe. Price rest numbers include 7, 12, 15, 16 and 19. 7 is the day God rested, the Ark was built by 15 feet, 7 1/2 feet on each side if cubits are transformed into feet. 16 represents love. In this book was my answer to moving average length choices. I found the number 200 represents insufficiency and an answer to my question. A full list is in chapter 9.

Numbers have symbolic significance in scripture, number equivalents of alphabetical symbols and numerical value of the words formed by the letters. From the Greek word, this is called gematria. Hebrews, Greeks, Romans and other ancient cultures used their alphabets for numbers.

Hebrews used all 22 letters of their alphabet plus 5 finals, the Greeks used a 24 letter alphabet and 3 finals. The Romans used only 6 letters of their alphabet and their combinations to form numbers. l=1, V=5, X=10, L=50, C=100 and D=500. Here was my answer to those 10, 50 and 100 day choices. Further, numbers and letters have a significance to names, Lord=800 (

8 X 100), son = 880 ( 8 x 110) Pastor = 89. Brian= 44, Twomey = 101, all good numbers and combinations.

The common theme among these ancient cultures is to see the transformations of letters into whole numbers but in multiples of 5 such as 10, 50, 100, 500.

Next came use of combinations of numbers and the heart of my answers. For example, 15= 3 x 5 and all good numbers, 7+ 8=15 more good numbers.

So then I sought to understand these numbers in terms of standard deviations but I looked at standard deviations only in multiples of 5 since that was the predominant choice for moving averages.

My polemic with my numbered moving average system after years of use is I wanted different numbers but I sought the most perfect numbers in the most perfect sets. Still not satisfied, I thought it was time to again hit the statistics books.

I bought a 1987 statistics book for 50 cents from my local library’s book sale sponsored by my church. The book was titled Basic Statistical Analysis by Richard C. Sprinthall. The word analysis was my purpose for my purchase.

It was in this book that I began my quest to not only find the perfect number but further review my study of averages. Later I found Basic Statistical Analysis was published in its 9th edition in 2011 so its now a part of my library.

So fascinated with my quest, I bought bunches of statistics books because I found each statistic book had certain twists to them. For example, one book treated the explanation of the standard normal curve in such a more detailed manner than the next book. If one thinks about statistics, its a fairly fixed science so each book had to address fixed numbers and concepts in certain manners than the next book. Over the years MAD was invented, the Mean Average Deviation and Edward Altman’s Z Score but those were small deviations from the foundations of statistics and neither invention serves our purpose in this book.

Through all my work, years of market experience and diligent research, I found and applied the Z Score in all my trades for what seems like forever. My market approach is use of a calculator rather than faulty market chart patterns in all their fallacious forms as well as faulty technical indicators.

What I bring to traders, market professionals, researchers and interested parties is my own study of the Z Score with a step by step approach using my own trades so anyone may replicate my own methods that has served me well for quite some time.

As well, my information is based not only on solid facts but solid statistical foundations. The few articles published on the topic of Z Scores for trading purposes I found faulty in their statistical logic, trades, assessments and overall understanding of the true use of the Z Score.

This book not only looked at the Z Score in all its finest forms but the heart of the Z Score is the standard deviation, one cannot survive without the other.

Further, Z Scores were addressed in terms of Z Score correlation coefficients, Z Score beta coefficients, Z Score

probabilities, Z Score standard vs. normal curves, critical value Z Scores, Z Score slopes, regressions, Z Scores and percent, Z Scores and means, areas of the standard normal curve and Z Scores, Z Scores and medians. I essentially obliterated not only the full spectrum of the Z Score but I offered many calculated formulas and examples with detailed explanations throughout the book so the methods can be replicated by any reader.

The Z score in my examples are used to trade currencies but anyone trading any financial instrument in any market may employ the Z Score methods outlined here since statistics is a universal >

My hope is readers will not only enjoy the read and the information I brought forth but the hope is all would replicate the methods outlined here for use in the financial markets and employ this book as the resource.

ACKNOWLEDGMENTS

My friend Jeanette Schwarz Young, “The Option Queen”, I owe a debt of gratitude for her inspiration for this book. This was a book I never intended to write. Jeanette is the author of the weekly “Option Queen” newsletter, past President of the American Association of Technical Analysts, former option floor trader at NYMEX, assists Certified Market Technician candidates at the Market Technicians Association and played herself in The Pit Movie, a movie with a focus on how commodities are traded on the floor of the exchanges. Always a special thank you to Jeanette for her help and assistance over the years and always an honor to call her my friend.

My friend Peter Wadkins has been involved in currencies and trading since the early 1970’s as an FX professional in corporate banking. Peter is a former bank trader for many years, former longtime member of FX committees and now daily currency market analyst / writer of brilliant, insightful, timely and sometimes amusing commentaries at Thomson Reuters. A special thank you to Peter Wadkins for his time and assistance over the years and as well an honor to call him my friend.

A special thank you to Richard C. Sprinthall for his time. As a long time college professor, author of Basic Statistical Analysis 9th edition and master statistician, it was an honor to have time with Richard.

My friend Troy Beckham, “Mr. T”, is again sincerely appreciated with many thank you’s for his expert computer skills, his time and attention and abilities to assist me in all my work and efforts over the years. An honor to have Troy as my friend.

A special thank you to fxtop.com for use of a resourceful and fabulous website that contains loads of wonderful and highly useful and pertinent data.

CONTENTS

Preface

Acknowledgments

Chapter 1: Z Score Introduction—Methodologies—Explanations

Chapter 2: Z Score Trades in Action—Methodologies–Explanations

Chapter 3: Standard Deviation of the Range

Chapter 4: Z Score Critical Values–Standard Deviation

Chapter 5: Z Score Correlation Coefficients—Beta Coefficients–Probabilities– Percents

Chapter 6: Z Score Areas of the Curve—P Values–Probabilities Percents and Normal Curve

Chapter 7: Normal vs. Standard Curves—4 Moments of the Curve—Means vs. Medians–Trimmed Means

Chapter 8: Mathematical Stops

Chapter 9: Bible Numbers and Moving Averages

Chapter 10: Additional Paper Findings

Appendix 1:

Appendix 2:

Appendix 3:

Appendix 4:

Appendix 5:

Appendix 6:

Appendix 7:

Appendix 8:

Appendix 9:

Appendix 10:

About the Author:

Chapter 1: Z Score Introduction

I set out to understand why we don’t know first principles about a market price and how to measure that phenomenon in relation to standard deviations and moving averages. After all, what’s in a price and what do we want price to do for us. As technicians, we learned how to isolate price but we didn’t know what to do once that price was isolated. So we employed a correlation strategy which says guilt by association when measured against another financial instrument. Price was now contained within the confines of another financial instrument by measure of the spread but says nothing about price direction. X may move against Y by a certain amount of standard deviations but without a total price understanding in short, intermediate and long term, correlation imparts little for price direction.

A price may experience volatility measured strictly by standard deviations. Empirically, we know returns squared or not are positively correlated to standard deviation but that indicates how far from a mean price relates to standard deviations not price direction. Arch and Garch models measure volatility by the variance to forecast price volatility. The variance, the second moment of the distributional curve, is measured not necessarily price. Unless we know the relationship of price to its average in standard deviation terms, it seems to me we have a difficult time to forecast, to understand present prices in a market context. For these reasons, I’ve devised the Z/DEV as my preferred method because it takes into account a mean price, an average, correlation, probabilities, ranges, X and Y axis and price forecasts in terms of short, intermediate and long.

The Z/DEV is the relationship of a Z-Score, Standard Deviation, a market price in relation to a set of moving averages. It’s a pure price system and if applied correctly, holds extraordinary value. The Z-Score and standard deviation is not new to the markets nor to the understanding of prices but both separately I argue have been incorrectly applied especially the Z-Score.

Data Methodology

The EUR/USD was tested twice daily at 8:00 am / late afternoon under two different exchange rates each day and recorded on 8/21/2012 and 8/22/2012, tested once Friday at 8:00 a.m. on 8/17/2012 and 8:00 a.m. Monday 8/20/2012. A total of four market trading days were tested, chosen randomly. To ensure ample accuracy, USD/CHF and EUR/JPY was tested Monday 8/20/2012. A total of eight averages were tested: 5 Day, 10 Day, 15 Day, 20 Day, 50 Day, 100 Day, 200 Day and 253 Day. Simple Moving Averages is the sine qua non of the experiment due to ability to monitor price changes throughout the day and to determine their overall effectiveness.

Each average’s daily average and Standard Deviations reported are based on a weighted average of European interbank transactions and posted by the ECB at 2:15 p.m. European time, 7:15 am United States. A Z-Score was calculated for each period’s average price and population

standard deviations because a Z-Score calculates a negative or positive value to alert to prices above or below a mean price. A price below a mean signifies a negative Z- Score value, a short, and a positive Z Score signals prices above a mean, along. A standard deviation was then calculated for each average against the present market price to understand present prices and ranges in relation to their average price.

Since a Normal Distribution is not “normal”, a Standard Normal distribution of market prices was transformed into a Z Score so a mean equals 0 and because Z Scores can then measure the spread of prices from the mean in standard deviations on a scale + or minus three standard deviations. Essentially, an X axis with a linear progression is now established that equates to a normal distribution in percentage terms but easier to manage, understand, view and calculate.

For a Y axis, P-Values were recorded for each Z Score’s average to determine not only significant confidence levels at each juncture but I calculated left, right and two tailed P- values due to its instructivnness as an indicator to determine trend and range trading days and as a complement to the Z/DEV to fully understand the Y axis.

Traditional Product Moment correlation coefficients invented by Karl Pearson and popularly known as r has been and will always be an important tool for market purposes due to its powerful determination not only between the correlation of two financial instruments but as a guide to trade two financial instruments together as a measure of the important X and Y axis that two financial instruments share. The measure is between + 1 and minus 1, one standard deviation exists between two instruments whose correlation are completely opposite. I calculated a correlation coefficient of Z-Scores among all pairs. Prior research termed this calculation a Z-Slope, the terms are synonymous when Z Scores are employed in the calculation.

Data Explanation

In order to fully understand a market price, the short, intermediate and long must be seen to alert to all market scenarios. My eight moving averages calculated to a Z-Score allows that view for each average because I know how far prices deviated from their average in standard deviation terms. Each average then provides vital information to the structure of market prices in terms of ranges, trends, and volatilities, short vs. long. More importantly to the view of a spatial relationship and the trade of averages as the first moment of the distributional curve is the early warning signal to a trouble spot within the moving average relationship. A trend or range occurs as moving average alignments dictate. Volatility is nothing more than information priced in the markets. All scenarios are seen from a Z-Score perspective so trades can calculate to the exact point movement. To employ one or even two averages together says little about a full price structure because it completely detracts from the perfect mathematical structure of market prices.

Chapter 2: Z Score Trades in Action

Since the sample size is small in our 8 averages and conducive to a Z score measure because no Z score exceeds 3 in a sample size less than or equal to 10, what is achieved in a Z Score is location, detection and outliers, termed discordancy, that offers trade signals based on the market price in relation to an average. This is accomplished by targeting of the Standard deviation in relation to the market price, the X value.

The Z Score measures the spread of prices from the mean in standard deviations that allows a trade signal to be generated if a market price is above or below the mean. Above and below a mean is an important consideration to answer the question of location.

The general assumption is prices below a mean is a short derived from a negative Z Score and long if prices are above the mean based on a positive Z score. USD/CHF’s 253 day average price on Monday revealed a positive 1.5 Z Score yet a long trade based strictly on the positive Z Score would’ve resulted in a loss, same principle for the EUR/USD and EUR/JPY’s negative Z score.

Location of a Z score is imperative as an evaluative tool in any time series because it reveals if prices are above or below a mean. The principle component of analysis must be the detection of the spread of those Z scores because that is the trade signal. Monday’s EUR/USD 253 day price was below the mean but the spread revealed it was far below the mean with a negative 1.49 Z Score while Monday’s USD/CHF 253 day revealed prices above the average but far above it due to the positive 1.5 Z Score. Positive and negative reveals location of the mean, the spread reveals how far from the mean.

Both Z Scores are considered outliers in relation to a 253 day average so a trade with a target price must be calculated. Monday, I targeted a short with a 1.0 Z Score based on a 1.5 spread for USD/CHF because 1.0 allows prices to remain safely above the mean and at the 1.0 standard deviation, below would’ve revealed a vastly different set of scenarios such as a possible long term short or a range market.

Therefore X= Z X Sigma + mean= target price. 1.00 X 0.0264 + 0.9352=0.9616. A profit of 144 pips, 0.9760 minus 0.9616.

Monday’s long EUR/USD standard deviation target was also 1.0 on a 1.49 spread. So 1.0 X 0.0423 + 1.2941= 1.2518, a profit of 218 pips achieved late Wednesday evening, 1.2518 minus 1.2308.

To answer why not take profit Tuesday evening when the Z Score reached its 1.0 target is answered by the range was secured and the Z Score allows for a benchmark to assess ongoing trades. I evaluated my trade based on my possible X values. Monday’s first benchmark was a minus 1.2 target therefore minus 1.2 X 0.0434 + 1.2941=1.2433. Its an iterative process.

Prices that hover within the mean Z Spread of + or minus 0 are trades either not considered or placed on the “watch list”. The 253 day EUR/JPY was one example.

At a spread of minus 1.10 Monday revealed prices are close enough to their average for trade consideration. Consider a 1.0 target. 1.0 X 4.166 + 102.404 =98.23, a profit of 42 pips, 98.23 minus 97.81 but barely a 1/2 standard deviation trade.

If the negative spread widens, a long trade is implemented. If a Z Score lands in positive territory from negative and vice versa and prices firmly cross the mean Z, consideration is assessed for possible directional change but considered in relation to the average. By Wednesday evening, the EUR/USD achieved a Z Score of minus 0.9952, an evident short still remains with limited upside potential until negative Z turns positive.

Z location is important to reveal a troubled average within the matrix due to ability to stop price progression caused by prices approaching its average. The 100 day average signaled an early warning beginning Friday morning and was the explainable troubled average all week due to prices approaching the average. When compared to the EUR/USD 200 day, Friday’s Z spread was .66, .59 Monday, .70 Tuesday morning, .75 Tuesday evening, .72 Wednesday morning and by sheer chance, 0 on Wednesday evening as the average equaled the market price.

To fully compare Standard deviations: square root of sigma squared for each average, divide by the mean and add the result, EUR/USD’s 200 to 100 day comparison for Monday was .053, Tuesday .051 and .050 Wednesday while USD/CHF was .051 and .075 for EUR/JPY. As prices encroached upon and widened against the 200 day and its own 100 day average, EUR/USD prices were fully reflected all week.

The same phenomenon occurred by USD/CHF’s .57 Z spread and EUR/JPY at .40. From a 253 day perspective and short test duration, the EUR/USD and EUR/JPY remains a short on a long term basis due to a negative Z and prices below their averages while USD/CHF remains a long due to a positive Z and prices above its average.

What location signals is position of the averages while Z reinforces those averages to provide a trade signal. Trades were implemented at the 253 day average but a further evaluation of trade signals can be analyzed from short averages vs. a long perspective.

By Tuesday afternoon, EUR/USD short averages from the 50 to 5 day all reached critical levels yet the 253 and 200 day still revealed my initial long was safe but a slow down or correction was needed based on the short averages.

As new daily averages and Standard Deviations are recorded, a new Z score reflects those new averages and standard deviations. A short coming with a Z Score/Standard Deviation is not that it’s a pure price system approach that performs accurately but a daily calculation is a common occurrence. But that’s a matter of preference. The interaction among a new market price, standard deviation in relation to the averages must be recalculated with a new Z Score. The new calculation reinforces ongoing trades, reveals vital information

about future prices from the averages, allows information to determine a further target of a Z Score. The factor to reinforce this new scenario is the Standard Deviation of the range.

Chapter 3: Standard Deviation of the Range

Standard Deviation of the range is calculated from two perspectives: Standard Deviation of the range for each average to know the range of prices in relation to Z Scores and a Standard Deviation of a distribution of Z Scores over each pair’s averages. The formula for the distribution of Z Scores: Square Root of the sum of z squared minus (( sum Z) squared/ average’s mean divided by average’s mean.
Z-(Z)2/average mean/average mean

The results: Monday’s EUR/USD minus 8.917, Tuesday morning minus 9.825, Tuesday afternoon minus 57.935, Wednesday morning minus 5.195. USD/CHF–minus 18.659 and EUR/JPY 2.797.

Standard Deviation of the range of each average allows an accurate range of prices to be known in Standard Deviations and serves as a reinforcement to an ongoing and proposed Z Score trade because its derived directly from calculation of a Z Score as market price minus mean average/ Z Score and because prices are known on a high/low basis above and below the mean.

Both function in conjunction with each other but both share an adverse relationship. As price ranges increase, Z Scores fall and vice versa due to the variability of high/low prices above and below the mean. This says tighter ranges around the averages reduces the Z Spread as Standard deviations increase to account for the high/low basis.

Monday’s Standard Deviation range for the EUR/USD at 2.095 informed to a 1.3003 above and 1.2879 price below the mean against present averages, calculated to a 1.49 Z Score equates to a 490 pip forecast at the minus 1.0 target. The Standard Deviation range at 2.095 must reach slightly above 2.549 in order for my take profit target to be achieved Wednesday at 1.2518. Tuesday evening’s close at 1.2931 was 10 pips off the 1.2941 average from Monday and below Wednesday’s 1.2924 average.

This trade was in its infancy because my 2.549 target actually accounts for the price range above and below the mean rather than a consideration of one side so my take profit in relation to overall price projection and Standard Deviation ranges was taken early. If my target was 1.2879 then I’m short by .361 standard deviations and my Standard deviation range is off by about .65 cumulatively. Therefore the Z Score spread must be equated along with the Standard Deviation range from a one sided mean basis in order to achieve maximum trade potential. My trade derived from a minus Z Score and can only be calculated from the negative side of the mean.

By Tuesday evening, the Standard Deviation range increased from 2.095 to 2.440, .345 for the range compression yet .1071 in my favor or 171 pips. Another view is 1.2518 is the mean of Monday’s 1.2941 average, 1.2508 Tuesday at 1.2931. As a target of the Standard Deviation range, 1.2518 minus 1.0 X 0.0423=1.2095, the range =2.095. The range increased to 0.886 or 886 pips above and below the mean and 0.443 one side. Divide .443 by 2 because a Standard Deviation is an average of an average=.221 minus 210 profit or .210=11. 1.2518 accounts for a moving market price beyond control.

As a target of the average’s standard deviation, 1.2941 minus 1.2518=0.0423.

Monday’s 2.095 Standard Deviation range can reflect directly in present market prices rather than average range prices for a particular mean average. 2.095 divided by 2 is 1.047 or 1.047 pips either side of Monday’s 1.2308 market price and equates to 2.2778 above and 0.1838 below. It’s a daily forecast that changes every day with the market price, Standard deviation and mean for each average.

The range purpose is to understand the width of the distribution and is defined as the difference between the highest and lowest prices therefore the range must be greater than the Standard Deviation. My range was factored from a variation of a Kurtosis distribution by multiplication of the Standard Deviation range by 6 to account for a normal distribution. Calculate the SD X 6 to account for the estimated range. Divide by 2 to equal prices above and below the mean. Add and subtract the answer to and from the mean to establish the range.

The Kurtosis 1/6 rule is a manner to understand the width of a distribution either as a tall and thin price–clustered

Leptokurtic curve or spread apart prices of a Platykurtic curve. The purpose for each curve is to understand compression or expansion of the Standard Deviation range and relationship to the distribution. Fortunately for a Standard Normal distribution of Z Scores is width of distributions remain mesokurtic so allows for strict concentration of the X axis in terms of the Z spread in relation to the Standard Deviation and closing prices of the averages.

The Z Score/ Standard deviation concept can be understood from its mathematical Z minimum and maximum perspective.

Since a Z Score is a Standard Deviation target, a positive/maximum Z Score calculates as ( X subscript n minus mean)/ Standard Deviation subscript n= ( X subscript n minus (X subscript n/ mean)/ Standard Deviation subscript n. (XN-mean) / σN

As Z rises, standard deviation falls.

A Z minimum is the smallest achievable value for the largest negative Z-Score with an accompanying formula: negative ( n minus 1) / square root of n. N for both examples are mean averages and n— 1 accounts for the population Standard Deviation. To simplify, market price minus mean average/ Standard Deviation.

To employ the parameter scale for Monday’s trade based on the 0.0423 Standard Deviation, a minimum negative 3 Z score equates to a EUR/USD price of 1.1672 and a maximum/positive Z score of 3 at the 0.0423 Standard Deviation equates to 1.4210, a bounded range of .2538.

A bounded range should always equate to the average if the skew of the deviation reflects its average. Since Monday’s market price was below the mean and the Z Score reflects the skew by its negative sign then its automatically known the overall direction for price within the negative skew.

Bounded ranges decrease for each average from top to bottom as the plus and minus 3 Standard deviation is factored for each average.

If the market price exceeded its average above or below during a day’s trading, a range price won’t reflect its average until the next day’s calculation of the range due to the factor of the closing price.

The last view to be used as a guide only is for every average in multiples of 5 adds 1 Standard Deviation to the range beginning at the 5 multiple. A 5 day average equals a 50 point range, 10 day equals 100, 15 day equals 150, 20 equals 200, 50 equals 500 and 100 equals 1000.

Chapter 4: Z Score Critical Values

Critical values is a characterization of not only the height of confidence intervals based on Y axis probability distributional areas under the curve but certain moving averages customarily reach critical values and its assumed by prior writings critical values refers to absolute buy and sell points because its assumed the outlier effect in any Z Score reached its maximum distance from a particular average.

This deviation is a distortion since it fails to consider the full symmetrical distribution of a curve within the context of normal curve properties 68, 95 and 97.5. The purpose to transform any Z Score is interest in location by scale parameters of the full curves’ data and averages measured by standard deviations. Means and standard deviations are the measure, Z score the trade signal and the curve is the operational context to pictorially view the distribution. +-1.96 is most popular because it represents 1.96 Standard Deviations from the mean, a 95% confidence level.

As the name implies, moving averages are constantly in motion so average paths change regularly for each period. Sometimes long averages push up shorter averages, sometimes long and short averages work against each other, other times an intermediate average may be misaligned and wholly responsible for market prices. Notice the Euro in the appendices and notice Short averages may reach a +-1.96 Z-Score but absolute buy and sell points is a misnomer whether employed against a Bollinger Band, another indicator or one or two averages in conjunction with each other. Risk must be considered.

A short EUR/USD trade taken Tuesday morning at the 5, 10, 15 or 20 day average at a 1.96 or complementary Critical Value Z Score would’ve resulted in a 61 point loss by Tuesday afternoon since the path of prices continued a forward motion and greatly exceeded 1.96. Statistical analysis terms this outlier phenomenon swamping and masking.

Swamping is that one outlier that swamps another so a possible cluster forms while masking is the one outlier that

masks another where less extreme outliers go undetected because of the most extreme. I call this the price trap if not taken within the full context of all averages.

Tuesday’s price and purpose for shorter averages to reach critical levels was explainable Tuesday morning by the EUR/USD 253 day average’s minus 1.2302 Z-Score and the 200 day Z Score of minus 1.0690. More importantly to Tuesday’s Z values was Monday’s 253 day Z score alert of minus 1.4965, 200 day at minus 1.3723 and shorter averages that barely exceeded or alerted to expectations of prices passing the Z Mean level from negative to positive territory. This alerted to the widely expected large point movement because prices had wide latitude to reach intended destination. Longer averages in this instance actually pushed up prices and explained the levels of critical values.

The complementary EUR/USD trade Monday was USD/CHF whose level of 1.54 and 1.37 Z Scores at 253 and 200 was far more favorable due to the smoothness of Z Scores across most of the pertinent averages.

EUR/JPY price levels at Z scores of 1.58 at 20 day, 1.35 at the 15 day and 1.36 at 10 day were approaching 1.96 critical levels but again the minus 1.1012 Z Score at the 253 day average exclusively explained the near 100 point EUR/JPY rise for the day and due further to the 200 day Z Score of minus 0.9340.

The EUR/USD Friday morning witnessed a minus 1.35 Z score, 84 % confidence level while the 5 day average saw a + 1.35 Z Score and 84 % confidence level. The day’s range market was clearly witnessed yet a long or short taken at 1.35 for extended periods would’ve resulted in a loss.

The actual and theoretical foundation of a Z score Standard Normal distribution is a Z Score for each average’s X value has a range of +- 3 standard deviation units centered with a Z Score mean of 0 defined by the Central Limit Theorem by U mean                                                    (subscript) =u  (Um=U )

(Appendix) X values convert to Z where u=0 so u minus sigma = minus 1 and u+                          sigma = + 1.U-σ=-1 and U+σ=+1

Since U mean (subscript) =u, sigma mean (subscript) = sigma / square root of N.                            Since Um=U, σm  σ/√N

So Z = mean minus u mean (subscript)/ sigma mean (subscript)

Z=M-UM/σM

+-1.96 as my example is just one number within the context of a larger set of average numbers that represents possible ranges of price. A price within any average can then range from a + 3 Z score to minus 3. Along this spectrum, price must cross the mean Z in order to travel to its intended destination. But the intended price destination is unknowable unless the perspective of one average price is viewed from the larger sets of averages in the distribution.

I’m confident with a short or long trade at a Z score of 1.00, 68.26 confidence interval, or even 1.35 at a 200 or 253 day average than a short trade at lower averages at 1.96 unless I know the full spectrum of averages. A host of Critical value Z Scores exist from 1.28, 80% confidence interval, to plus or minus 3.00, but should serve as alerts to future prices and evaluated closely in relation to other average’s relationships and positions before an actual trade. The direction of a trade is found in the sign of the Z score Value.

Standard Deviation of the Average

Standard deviation of an average is the average of an average price and measures the spread of prices from the mean in standard deviations. As prices deviate further from the mean, the standard deviation of the particular moving average falls and it rises as prices return closer to the mean to employ the 253 day example.

Standard Deviations in shorter term averages experience shorter spreads due to calculation of the mean for shorter time periods. A perfect market scenario exists when Standard Deviations of the averages have smoothness between all averages, a comfortable distance. Any average not uniform within the matrix is an alert to a troubled average that won’t perform to market expectations of price. As will be further seen, the 100 day average has been one such average that literally dictated prices within the four trading days.

If a price meets its average such as Wednesday evening’s EUR/USD 100 day average, the Standard Deviation will be at its lowest point.

A crossover occurs when a market price crosses its average; more typical for short time periods but the Standard Deviation matrix remains constant since a spread is still the overall measure.

A method to transform the standard deviation from each average is to factor an actual price so the average of an average is understood in relation to how far and how much prices deviated. A comparison of price to price is known.

For example, the Euro’s Monday morning 253 day reported a .0423 Standard Deviation in relation to the average but by Wednesday afternoon the average dropped to .0414 with a rise in prices closer to their averages. Monday’s .0423 transforms to 42.3 pips and Wednesday’s .0414 calculates to 41.4, a difference of 00.9 between averages.

A comparison of Standard Deviations and means between averages is found by addition of the Square Root of sigma 1 Squared/ N + Sigma 2 squared N.

The objective is to determine if an outlier exists to warn to future market scenarios with an average.

Chapter 5: Z Score Correlation Coefficients

The slope of the regression line of Y on X in Z-Score form answers how many Standard Deviations Y changes when X changes 1 SD, a positive relationship equates to an increase in X is an increase in Y, a negative relationship is a decrease in X and decrease in Y and a negative correlation between X and Y is an X increase, Y decrease. The Z correlation formula is Sum of Zx X Zy / N and Z Scores on X and Y to chart results is Zx=price minus mean divided by SDx and Zy=y minus mean Y / divided by SDy.

This slope formula alone accomplishes first Pearson r correlations which doesn’t change in the manner its calculated since the correlation results will remain the same.

By taking each Zx and Zy Score for each of 8 moving averages and dividing by 8, a standard deviation for each Zx and Zy moving average was correlated against each other since the Z Score was transformed into standard deviations. Essentially a correlation now exists for the entire distribution.

Secondly, the slope of the regression line is formally termed the beta coefficient or standardized coefficient due to the transformation of Z Scores. The slope formula is b= r X SDy / SDx. The beta indicates the amount of change of Y for a change in X but the slope and point intercept of the line has also been established.

Each Z Score has been correlated by a standard deviation so each Zx, Zy equals a beta weight and each can factor to a regression line at any point of Standard Deviations. This point is mentioned for those that trade other markets, other financial instruments. But by virtue of the correlation between the EUR/USD x and USD/CHF y for example, the entire distribution of correlations is known therefore price movements are known. What remains is to plot the points on a chart. The point intercepts are those points that establish price points on a chart. For example, a correlation of a slope of r 1 says prices move on a one for one basis between each other’s standard deviations. A slope of 2 says a regression line is steep and not steep if the correlation was say .50.

Monday’s correlations: EUR/USD vs. EUR/JPY r=0.810, EUR/USD vs. USD/CHF r = minus 0.91 and EUR/JPY vs. USD/CHF r=0.20.

The X, Y relationship should be viewed pair to pair in the context of each nation’s interest rates since interest rates trade in increments of 0.025 or 25 pips.

Probabilities and Percents

Probabilities and percents factors to an x axis Z Score to reveal percent areas under the curve, probability/percent confidence intervals and tail areas for either one or both sides of the mean. Probabilistic boundary lines are calculated above and below the mean as another trade reinforcement since both sides, 50%, reveals different yet vitally important information and because the general lines of the overall bell curve doesn’t move due to its fixed mesokurtic/standard normal shape, a trade off in exchange for the Z-Score so a Z Score is again

transformed to describe areas and confidence areas of the curve.

The X axis/ Z score is dominant since it reveals a moving market standard deviation price while probabilities and percents is an explanation of that moving price in a closely related form and is marked on the X axis line to understand the boundary of possibilities for a Z Score since the X and probability / percent Y share a direct relationship not only in its calculation but each informs the other.

Since probabilities and percents are each explained by 0.00 to 1, a curve is explained by both as 68.26% equates to + or minus 1 standard deviations or 34.13% either side of the mean and .3413 probabilities either side, +- or minus 47.7 explains 95.44% and + or minus 2 standard deviations or .1359 probabilities and +- or minus 49.9% or 99.8% + or minus 3 standard deviations equates to .0215 either side. The remaining 1% or .01 explains the tails.

Chapter 6: Normal Vs Standard Curves

Normal Curve and Probabilities

Since a normal curve is Y ==    or  1 / sigma square root 2 pi (3.14159) e (2.71828) raised to minus (x – u) squared / 2 sigma squared, the probability height is measured by the constants pi as a ratio of a circumference of a circle to its diameter and Napier’s e. X – u is the difference between the height of X and Y and the height of X and Y at the mean, x = 0 for the pair at the mean so e minus 0 / 2 sigma squared = 1.

The maximum height at the mean = Yu= 1 / sigma square root of 2 pi = 1. Height then equals probabilities charted on the left associated with each deviation against the right mean deviations as Yu, Y1 sigma, Y2 sigma and Y3 sigma.

Heights vary in probabilities due to expansion of the base meaning a larger sigma or it contracts to signify a smaller sigma. Heights must increase or decrease to allow areas to remain constant. So Y equals the match of X under the umbrella of the curve.

Z Score and Normal Curve

A Z-Score calculates to a Standard Normal curve as Y=
 

1 / Square root of 2 pi e power of minus ( Z squared / 2).

If Z = 1 from our original targeted example, Y= 1 / square root of (2) X (3.14), 2.72 power of minus (1.00 squared / 2) = 1 / square root of 6.28, 2.72 power of minus (.50) = (1/2.51) X (.61) = (.40) X (.61) = .24.

The height of Y when Z equals 1 = .24. At a .50 Z Score which is the ongoing trade, the calculation is as follows.

Y = 1 / square root of (2) X (3.14) 2.72 power of minus (.50 squared / 2) = 1 / square root of 6.28 2.72 power of minus (.13) = (1 / square root of 2.51) X (.88) = (.40) X (.88) = .35. The height equals .35. No matter how far the curve deviates from the mean, it never touches the x axis.

Moments of the Curve

Trading a Z score is the trade of averages, the X, and defined as addition of closing prices divided by number of days. A mean is the first moment of the curve where average deviations equal 0 so a deviation equals price minus mean as X = X minus mean and calculated as sum X divided by N=0.

The second moment defines the variance so the mean now = sum X squared / N. The square root of the variance= the Standard Deviation so SD = square root of sum X squared divided by N.

The third moment defines skewness, sk, calculated as Sum X cubed / N and as a standard value Sk= mean cubed / SD cubed so a standard value greater than + minus 1 signifies significant skewness. By cubing deviations, Sk can take negative values.

The fourth moment is kurtosis calculated as Sum X 4th power / N. To determine deviations from Mesokurtosis Ku = Mean subscript 4 / SD 4th power minus 3 because the formula defines 3.00 for mesokurtosis as a standard value.

Kurtosis is a measure of skewness, how peaked or flat of a distribution either as a positive Leptokurtic curve with small standard deviations or negative Platykurtic curves with large standard deviations. The higher peakedness the leptokurtic curve, the smaller the standard deviations because prices cluster around the mean. The more spread of prices, the larger the standard deviation and flatter curve because prices deviate further from the mean.

Skewness is measured by deriving signals from kurtosis calculations as a +3 or minus 3 to derive indications of price moves based on the skew. Skewed right will eventually skew left and vice versa. Therefore flat Platykurtic curves will eventually form higher leptokurtic peaks as prices revert back to the mean. High peaked leptokurtic curves will eventually form flat platykurtic curves as prices move away from the mean. A kurtosis of 0 equals a Mesokurtic curve. The + – 3 indicates when a peaked or a flattened curve reached its zenith or its bottom and is used as buy and sell signals.

All define a mean as a measure of central tendency in relation to the four moments yet all moments build on each other through understanding and further evaluation of the mean. Means egress and regress along with the standard deviations as heights of curves rise and fall. The only option beside a mean is the median.

Median

The median is the midpoint of a distribution that separates upper and lower bounds. Its more accurate as a measure of central tendency in skewed distributions than the mean. But averages skew means and is more volatile due to an extreme high or low price point for example while the median corrects for those extreme prices due to its focus on middle bounds. Therefore the X= Z X Sigma + mean performs far better than employment of the median because it catches the extreme values by a skew of the averages. If averages skew so therefore must Standard deviations. The question must be what does one wish for a price to do, to move in a particular direction, to remain stasis or remain in a middle bound.

Trimmed Means

Central banks employ trimmed means frequently to measure interest rates such as Libor and as a measure to capture a distribution of prices more accurately by eliminating the tails of the curves.

Trimmed means eliminates prices in both tails of a distribution either as a 2% or 5% basis to reflect the tails. A 2% trimmed mean eliminates the highest and lowest 2% of prices in all prices while a 5% trimmed mean eliminates the highest and lowest 5% of prices in all prices. A trimmed mean factors the same as any mean calculation; it just eliminates certain prices in the calculation.

Chapter 7 Z Score Areas of the Curve

Friday morning’s EUR/USD Z Score recorded a + 1.35 for the 253 day average and a negative 1.35 for the 5 day average, essentially a range bound market determined by areas of the curve in percentage terms and quite an interesting circumstance.

The + 1.35 Z Score factors to 41.15% or 41.15% of prices between +1.35 and the positive mean, the right side.

If 50% falls below the mean as is the case here, add 50% to 41.15= 91.15% or the 91st percentile. For positive Z’s, add 50% and subtract 50% for negative Z’s to account for percentage areas of the curve.

The negative 1.35 also factors to 41.15% so if 41.15% accounts for percent areas above the mean in the 91st percentile, 91.15% minus 41.15%=50% to account for areas left of the mean.

So 50% of prices fell below the mean while 41.15% fell above the mean, a classic example of a range bound market where highs are sold and lows are bought for the day.

Friday’s price for the EUR/USD was 1.2357 while Monday morning’s opening price was 1.2308, Friday’s shorts were favored 50% to 41.15% yet the difference of the lower 1.2308 price was the perfect set up for Monday’s trade since the lower price saw a Z Score that drifted further away from the mean in standard deviations at minus 1.49.

Since negative 1.49 equates to 43.19% of prices between negative 1.49 and the mean, add 50% to 43.19% = 93.19% to signify the percent of prices above a Z Score of negative 1.49 while the Z Score of 1.00 = 34.13% of prices between the Z

Score and the mean therefore add 50% to 34.13 = 84.13% of prices above a Z Score of 1.00.

If 93.19% of prices fell above a negative 1.49 Z Score, this signifies 93.19% at least that a bottom of prices has arrived and a trade back to 84.13% still allows room for prices to travel since at 93.19 says not much room is left of the area of the curve for further prices to travel.

Monday’s EUR/JPY Z Score at negative 1.10 at the 253 day average reveals 34.38% of prices fall between the mean and the Z Score therefore 50 minus 34.38%= 15.62% of prices fall below. On Monday, EUR/JPY prices were contained.

USD/CHF 253 day Z Score of +1.54 reveals 43.83% of prices fall between the mean and the +1.54 Z Score therefore 50 + 43.83 = 93.83%. Compare USD/CHF at 93.83% to the EUR/USD at 93.19% and correlations hopefully begin to make sense to the reader.

Percentage Prices between Z Scores on Opposite Sides

For the EUR/USD at 43.19% and USD/CHF at 43.83%, 43.83% + 43.19% = 87.02% of prices between both Z Scores. For EUR/JPY and USD/CHF, 15.62% + 43.83%= 59.45%. Add percentages on opposite sides of the mean.

Percentage Prices On Same Side of the Mean

For the EUR/USD and EUR/JPY, 43.19% minus 34.38% = 8.81% of prices between Z Scores. For the EUR/USD at 43.19% and the Z Score 1.00 target at 34.13%, 43.19% minus 34.13%= 9.06% of prices between Z Scores. Subtract percentages on the same side of the mean.

Percent Prices between Exchange Rates and the Mean

Suppose 1.2618 was the target, 1.2618 minus 1.2941 (average) / 0.0423 (SD) = .7635 = 27.64% between the Z Score and 1.2618. Our 1.00 target = 34.13% between the Z Score and the mean. Z Score formula= X minus Mean / SD or price minus mean average / SD of average.

Percent of Prices Above / Below an Exchange Rate

Since the Z Score is negative, 50 minus 27.64% = 22.36% of prices below the mean. Suppose the Z Score was positive, 50 + 27.64%= 77.64 % of prices below the mean. Had we wanted to know prices above the mean for example at 1.5 Z score then 50 minus 43.32% = 6.68% above the mean.

Z Score Probabilities

Probability that the EUR/USD exchange rate at 1.2308 falls between the USD/CHF exchange rate at 0.9760. Negative 1.49 = 43.19% and 1.54% = 43.83%. Add 43.83% to 43.19% =87.02% so 87.02 / 100 = .87 probability.

Probability that the EUR/USD exchange rate at 1.2308 falls between 1.2357. 43.19% minus 41.15% = 2.04%. 2.04 / 100= 0.02 probability.

Critical Values / P Values

If 1.96 equals a 95% confidence interval, a p value formula is p ( minus 1.96 is less than or = to Z is less than or equal to 1.96) = 0.95.

A Z Score is then 1/2 the confidence interval. A confidence interval says an exchange rate in probabilities will fall between a probability Z Score minus 1.96 and + 1.96. The number defines the area under the curve within the confines of the population standard deviation and not in the extremes as a general rule.

In critical value terms, 1.96 = 95% or 0.095 and a .4750 Z Score, 0.095 divided by 2 = .4750. 2.32 critical value = 98%, .098 and a .4900 Z Score or .098 divided by 2 = .4900.

2.58 = 99%, .099 and a .4950 Z Score, .099 divided by 2= .4950. The Z Score defines the area between 0 and the Z Score and the majority of the area between 2 Z Scores.

Probabilities are estimations of the Z Score where for example Z (0, .05) asks what is the area between 0 and .05. So a 1.645 for example is the Z Score that has an area .05 to the right and .4500 between the areas of 0 and the Z Score. The .05 represents the tail in this particular distribution, .05 divided by 2 = 0.025 to equal each tail areas in the distribution. An area is found by area = 2 X the probability table value of the Z Score. Probability tables range from 0– .5000.

Critical value 2.58 has a p value of .4950 and is not listed in the table, .4949 and .4951 is listed so 2.58 equates to .4951.

P values found in both sides of the tail for example in the 2.58 Z Score in the 99% confidence level is a Z Score between +2.58 and negative 2.58, the .01 or .099 are extremes in relation to prices that fall outside the + or minus 2.58 Z Scores.

The p value in the tails is the probability of prices hitting the tails or falling outside the bounds of the Z Score at a certain confidence level. Prices within the bounds of the Z Score is normal or random in statistical terms yet a trade signal if market prices is the focus. If a market price hits a confidence level, the reason must be an extreme occurrence or market phenomenon that would force prices into those extremes. Those trade signals are perfect scenarios.

A 1.96 Z Score = a left tailed p value of 0.9750 or a Z Score is less than 1. A right tailed p value = 0.0250 or a Z Score that is greater than 1. A 2 tailed p value= 0.0500 or the null hypothesis of the Z Score is greater than 1, normal occurrence in simple terms. A 2 tailed confidence level = 0.9500 or the null hypotheses of the Z Score is less than 1.

To calculate, transform a Z Score into a p value. A 1 tail calculation =1.00 minus p. the probability. A 2 tailed probability = 2 X p.

The purpose for the measurement of the p value is to factor the heights of a distribution on the Y axis in relation to Z Scores, standard deviations and prices.

A price may hit a + or minus 2.58 Z Score at the 99% confidence level and trail back to say the 1.96 Z Score at the 95% confidence level then travel further back to say the 1.28 or 85% level.

Each time prices travel, Z Scores and standard deviations expand and contract along with the various heights of the confidence levels. It’s a continuous motion as prices

trade and it’s a monitor or reinforcement to any trade because both X and Y are measured in relation to each other. The height of Y can both measure on the height level at the top of the distribution as well as in the tails so the trade confidence is paramount. It’s a solid relationship and one that cannot change based on two centuries of statistics when mathematical genius Carl Fredrich Gauss invented the normal curve in 1809.

Chapter 8: Mathematical Stops

Since the Z Score is a standard deviation target of a market price in relation to a particular average, a stop has not only a mathematical foundation but the mathematical stop as I call it is a benchmark and should be moved in relation to an ongoing trade and set at the start of a proposed trade.

Since Monday’s entry price was 1.2308 based on the Z Score of negative 1.49, 0.0423 Standard Deviation and 1.2941 mean, an initial stop must be calculated before actual entry. Suppose 1.2298 was a decided stop placement. So negative 1.52 X 0.0423 + 1.2941 = 1.2298.

Now that stop must be benchmarked and moved in relation to the overall target. Let’s target negative 1.39 from our negative 1.49 entry, negative 1.39 X 0.0423 + 1.2941= 1.2353. Assume the market price is negative 1.30, negative 1.30 X 0.0423 + 1.2941= 1.2391 or a 38 pip stop. How about negative 1.20, negative 1.20 X 0.0423 + 1.2941= 1.2433 or negative 1.10 X 0.0423 + 1.2941= 1.2475. Remember our EUR/USD target is negative 1.0 or 1.2518 so all examples here represent perfectly calculated mathematical stops.

As a general rule under my own assumptions, a mathematical stop at a 1 / 2 standard deviation or 0.025 may serve well and protect the trade if an outlier bid or offer may hit the market or a central bank or head of state may say something outlandish to cause a slight deviation in normal functional market prices. Suppose the original target then was benchmarked at negative 1.33 on our negative 1.30 market price of 1.2391, negative 1.33 X 0.0423 + 1.2941= 1.2365, a 26 pip stop at 1.2391 minus 1.2365.

Monday’s Z Score was based on a particular standard deviation and average and since trade entry was also based on these figures, mathematical stops must be targeted based on those same averages and standard deviations using the Z Score as the guide.

Tuesday’s Z Score hit 1.0 but that figure was based on a new Standard Deviation and average. That situation presents a new set of trade entries and targets and can’t be benchmarked against Monday’s average and standard deviation. Had averages and Standard deviations remained the same, the Z Score would’ve remained the same so Monday’s scenario would’ve been the exact scenario for Tuesday.

So the question does a Z Score have to be calculated daily, the answer is it depends if the calculation is based on an ongoing trade and benchmark a stop or to search for the perfect trade for an entry. The answer remains to the market trader and possible trade or benchmark purpose.

Chapter 9: Bible Numbers and Moving Averages

Following first are numbers and meanings based on biblical scripture whose work and interpretations was published by Ed Vallowe originally in 1972 with updated editions in 1988 and 1997. The book is called Keys to Numerics of Bible Scripture. The follow up to this version was published in 1997 in Biblical Mathematics. The purpose of this exercise is to strictly highlight numbers and their significance as a choice for moving average lengths employed in the traded markets. Therefore I transformed Vallowe’s numbers into classifications of Volatile, Price Rest, Trend and Trend Continuation to highlight what I thought were acceptable number choices.

1. Unity
2. Union, division
3. Resurrection
4. World, creation
5. Grace, power
6. man weakness, sin
7. Complete, perfection
8. Beginning
9. Divine Completeness
10. Law, Judgment
11. Disorder, Judgment
12. Government perfection
13. Rebellion
14. Salvation
15. Rest
16. Love
17. Victory
18. Bondage
19. Faith
20. Redemption
21. Sin
22. Light
23. Death
24. Priesthood
25. Forgive sins
26. Gospel of Christ
27. Preach Gospel
28. Eternal life
29. Departure
30. Blood of Christ
31. Offspring
32. Covenant
33. 34 Son

35.Hope

36.Enemy

37.Word of God

38.Slavery

39.Disease

40.Trials, Testing

42. Lord Coming
43. Preservation

50.Holy spirit

60.Pride

66 Idol Worship

70.Restoration

100. God Elecetion of Grace
101. Resurrection Day
102. Divine probation period
103. Guided life
104. Bear fruit

200.Insufficieny

600. Warfare
601. Saints Resurrection

Volatile Numbers

6.Sin

11.Disorder

13.Rebellion

21.Sin

36. Enemy

38.Slavery

39.Disease

40.Trials, testing

66.Idol worship

600.Warfare

Price Rest

1.Unity

5.Grace,power

7.Complete, perfection

9.Divine completeness

14.Salvation

15.Rest

16.Love

19.Faith

20.Redemption

30.Blood of Christ

45.Preservation

Trend-Price reverts to

2.Union, division

10.Law, judgement

22.Light

24.Priesthood

25.Forgive sins

26.Gospel of Christ

30.Blood of Christ

31.Offspring

32.Covenant

33. Promise

34.Son

35. Hope
36. 37. Word of God

    50.Holy Spirit

    70.Restoration

    119. Resurrection

     

    Trend Continutaion-Price departs from

    2.Union, Division

    3.Resurrection

    4.World, creation

    8.Beginning

    9.Divine completeness

    20.Redemption

    23.Death

    28.Eternal life

    29.Departure

    30. Blood of Christ

    33.Promise

    42.Lord coming

    60.Pride

    100.God grace

    120.Probation

    144.Guided Life

    153. Bear fruit
    154. Saints resurrection

     

     

     

    Chapter 10: Additional Paper Findings

     

    Prices in four days were tracked both by the Z/Dev methods and Simple Moving averages. Only the weekly chart revealed SMA averages in proper alignment with prices and lines while the daily and all time frames below it revealed improper alignments. The monthly chart was okay but just okay and never a means for me to track prices and SMA averages ever again.

    The 253 and 5 day averages were by far the most essential especially as they pertain to forecast of range markets as well as topside vs. shorter average trades and forecasts. The 15 day average underperformed and wasn’t a crucial element within the four days because the average length was too close to the 10 and 20 day average to reveal a trade signal and because the range of prices within the 10 and 20 day encroached upon the 15 day. As a tool to track averages and Z Scores, it then may be considered.

    Since the 3 SD price distributions are fixed, the next sized average as a complement to another must be that average length that doesn’t encroach upon the prior distribution of prices. An encroachment is the definition of a crossover and may be the intended purpose for two sized average closeness and employed as a price monitor. Yet only one of the two averages will result in a Z/Dev trade signal. So the appropriate sized average length is not a question for higher or intermediate sized averages since their distributions are wide with sufficient distance but a question for shorter averages with smaller distributions. I found the 5, 10 and 20 day averages sufficient especially as the 20 relates to the intermediate 50. But I’ve always questioned that assumptionAn interest to include another number between 20 and 50 may be acceptable as further protection. As a number, 34 is popular but notice its not that its a factor of a Fibonacci number, 34 represents 34% of the area between 0 and 1 standard deviation on one side of the mean. The number 35 may be acceptable due to its multiple of 5.

    The Z/Dev must work in conjunction with each other. It informs strictly for prices in ranges from a short, intermediate and long perspective, it informs for longs vs. shorts, forward and backward pricing are all immediately determined by simple calculations.

    Time was not addressed however a time factor calculates as Z X SD X time. Normally time is square rooted. This specialization encroaches upon questions of risk and possible losses over time. Parametric VAR is one such example that employs a Z-Score and Standard Deviation and calculates as Mean X Time + ( Z-Score X SD X square root of time.). A percentage based risk assessment is rendered.

    If an intended recipient of these methods doesn’t have a grasp of at least standard deviation concepts than this may not be the way for those persons.

    Questions of weighted or exponential averages was not tested nor addressed in this paper however since exponential averages have front loaded means, I would not hold much hope that Z/Dev methods would be accurate in their signals if one wished to obliterate the Z/Dev concept. The Z/Dev is a pure mathematical trade of averages based on closing prices and nothing more.

    The question of Bollinger Bands used in conjunction with Z/ Dev methods I find so far a highly questionable concept on its face because what allows Bollinger Bands to be its wonderfully accurate trade signal is not based on volatility but based on Bandwidths. Bandwidths are frequencies and an important concept in mathematics and normal/standard normal curves. A frequency asks how often is a particular occurrence of prices. So unless one understands the concepts of frequencies and bandwidths, trading Z/Dev methods are questionable.

37.  
    

    Z	0	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08
    0	0	0.4	0.8	1.2	1.6	1.99	2.39	2.79	3.19
    0.1	3.98	4.38	4.78	5.17	5.57	5.96	6.36	6.75	7.14
    0.2	7.93	8.32	8.71	9.1	9.48	9.87	10.26	10.64	11.03
    0.3	11.79	12.17	12.55	12.93	13.31	13.68	14.06	14.43	14.8
    0.4	15.54	15.91	16.28	16.64	17	17.36	17.72	18.08	18.44
    0.5	19.15	19.5	19.85	20.19	20.54	20.88	21.23	21.57	21.9
    0.6	22.57	22.91	23.24	23.57	23.89	24.22	24.54	24.86	25.17
    0.7	25.8	26.11	26.42	26.73	27.04	27.34	27.64	27.94	28.23
    0.8	28.81	29.1	29.39	29.67	29.95	30.23	30.57	30.78	31.06
    0.9	31.59	31.86	32.12	32.38	35.08	32.9	33.15	33.4	33.65
    1	34.13	34.38	34.61	34.85	37.29	35.31	35.54	35.77	35.99
    1.1	36.43	36.65	36.86	37.08	39.25	37.49	37.7	37.9	38.1
    1.2	38.49	38.69	38.88	39.07	40.99	39.44	39.62	39.8	39.97
    1.3	40.32	40.49	40.66	40.82	42.51	41.15	41.31	41.47	41.62
    1.4	41.92	42.07	42.22	42.36	43.83	42.65	42.79	42.92	43.06
    1.5	43.32	43.45	43.57	43.7	44.95	43.94	44.06	44.18	44.29
    1.6	44.52	44.63	44.74	44.84	45.91	45.05	45.15	45.25	45.35
    1.7	45.54	45.64	45.73	45.82	46.71	45.99	46.08	46.16	46.25
    1.8	46.41	46.49	46.56	46.64	47.38	46.78	46.86	46.93	46.99
    1.9	47.13	47.19	47.26	47.32	47.93	47.44	47.5	47.56	47.61
    2	47.72	47.78	47.83	47.88	48.38	47.98	48.03	48.08	48.12
    2.1	48.21	48.26	48.3	48.34	48.75	48.42	48.46	48.5	48.54
    2.2	48.61	48.64	48.68	48.71	49.04	48.78	48.81	48.84	48.87
    2.3	48.93	48.96	48.98	49.01	49.27	49.06	49.09	49.11	49.13
    2.4	49.18	49.2	49.22	49.25	49.45	49.29	49.31	49.32	49.34
    2.5	49.38	49.4	49.41	49.43	49.59	49.46	49.48	49.49	49.51
    2.6	49.53	49.55	49.56	49.57	49.69	49.6	49.61	49.62	49.63
    2.7	49.65	49.66	49.67	49.68	49.77	49.7	49.71	49.72	49.73
    2.8	49.74	49.75	49.76	49.77	49.84	49.78	49.79	49.79	49.8
    2.9	49.81	49.82	49.82	49.83	49.84	49.84	49.85	49.85	49.86
    3	49.87	49.81	49.87	49.88	49.88	49.89	49.89	49.89	49.9

    Above % Area Under the Curve between mean and 0

     

    8-17-2012 EUR/USD	Markat Price	Price Average	Standard Deviation Average	Standard Deviation Range	Z Score	Left  Tail	Right Tail	2 Tail	Confidence Interval
    253	1.2357	1.2854	0.0367	2.185	-1.354	0.9121	0.0879	0.1757	0.8243
    200	1.2357	1.2822	0.0397	2.330	-1.1713	0.8793	0.1207	0.2415	0.7585
    100	1.2357	1.2468	0.0215	3.650	-0.5163	0.6972	0.3028	0.6056	0.3944
    50	1.2357	1.2311	0.0124	2.082	0.3710	0.6447	0.3553	0.7106	0.2899
    20	1.2357	1.2312	0.0052	0.186	0.8654	0.8066	0.1934	0.3868	0.6132
    15	1.2357	1.2322	0.0054	0.665	0.6481	0.7415	0.2585	0.5169	0.4831
    10	1.2357	1.2322	0.0052	0.594	0.6731	0.7496	0.2504	0.5009	0.4991
    5	1.2357	1.2311	0.0034	0.325	1.3529	0.9120	0.0880	0.1761	0.8239
    8-20-2012 EUR/USD
    253	1.2308	1.2941	0.0423	2.095	-1.4965	0.9327	0.0673	0.1345	0.8655
    200	1.2308	1.2824	0.0377	2.165	-1.3723	0.9150	0.0850	0.1700	0.8300
    100	1.2308	1.2546	0.0303	2.827	-0.7855	0.7839	0.2161	0.4322	0.5678
    50	1.2308	1.2323	0.0124	11.415	-0.1210	0.5482	0.4518	0.9037	0.0963
    20	1.2308	1.2313	0.0049	13.302	-0.1020	0.5406	0.4594	0.9188	0.0812
    15	1.2308	1.2320	0.0051	6.466	-0.2353	0.5930	0.4070	0.8140	0.1860
    10	1.2308	1.2309	0.0030	38.194	-0.0333	0.5133	0.4867	0.9734	0.0266
    5	1.2308	1.2298	0.0024	1.720	0.4167	0.6616	0.3384	0.6789	0.3231
    8-20-2012 USD/CHF
    253	0.9760	0.9352	0.0264	0.3708	1.5455	0.9389	0.0611	0.1222	0.8778
    200	0.9760	0.9396	0.0265	0.2919	1.3736	0.9152	0.0898	0.1696	0.8304
    100	0.9760	0.9579	0.0225	0.2148	0.8044	0.7894	0.2106	0.4212	0.5788
    50	0.9760	0.9747	0.0096	6.2226	0.1354	0.5539	0.4461	0.8923	0.1077
    20	0.9760	0.9755	0.0038	6.4366	0.1316	0.5523	0.4477	0.8953	0.1047
    15	0.9760	0.9749	0.0040	2.5690	0.2750	0.6083	0.3917	0.7833	0.2167
    10	0.9760	0.9757	0.0024	12.0333	0.0750	0.5299	0.4901	0.9402	0.0598
    5	0.9760	0.9766	0.0019	4.0684	-0.3158	0.6239	0.3761	0.7522	0.2478
    8-20-2012 EUR/UPY
    253	97.816	102:404	4.1664	190.8090	-1.1012	0.8646	0.1354	0.2708	0.7292
    200	97.816	101.991	4.4702	207.0140	-0.9340	0.8248	0.1752	0.3503	0.6497
    100	97.816	99.522	3.1997	284.4664	-0.5332	0.7031	0.2969	0.5939	0.4061
    50	97.816	97.286	1.5293	182.8706	0.3466	0.6356	0.3644	0.7289	0.2711
    20	97.816	96.768	0.6628	36.6169	1.5812	0.9431	0.0569	0.1138	0.8862
    15	97.816	96.970	0.6242	26.2672	1.3553	0.9123	0.0877	0.1753	0.8247
    10	97.816	97.025	0.5794	26.7458	1.3652	0.9139	0.0861	0.1722	0.8278
    5	97.816	97.537	0.3880	37.8216	0.7191	0.7640	0.2360	0.4721	0.5279
    8-21-2012 EUR/USD A.M
    253	1.2418	1.2931	0.0417	2.292	-1.2302	0.8907	0.1093	0.2186	0.7814
    200	1.2418	1.2821	0.0377	2.441	-1.0690	0.8575	0.1425	0.2857	0.7149
    100	1.2418	1.2521	0.0280	4.645	-0.3679	0.6435	0.3565	0.7129	0.2871
    50	1.2418	1.2326	0.0124	0.419	0.7419	0.7709	0.2291	0.4581	0.5419
    20	1.2418	1.2320	0.0055	0.550	1.7818	0.9626	0.0374	0.0748	0.9252
    15	1.2418	1.2335	0.0054	0.439	1.5370	0.9379	0.0621	0.1243	0.8757
    10	1.2418	1.2321	0.0050	0.606	1.9400	0.9738	0.0262	0.0524	0.9476
    5	1.2418	1.2236	0.0057	0.384	1.4384	0.9249	0.0751	0.1503	0.8497

Using the Z Score to trade Foreign Exchange and other Financial Instruments: The Step by Step

By Brian Twomey

Copyright c 2012  by Brian Twomey

Probability Table

A missing Tables and Chapter Pictures and hopefully added later

Brian Twomey