## Key Concepts

 Numbers A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

A B

Andrew’s Pitchfork
is a trend-channel drawing technique consisting of three lines – parallel upper and lower lines and a median line that bisects the distant between the two. It is named after its developer, Alan Andrews. The chart below shows an example of a Pitchfork drawn from the March 2009 low on a weekly chart of the Dow Jones Industrial Average (DJIA). There are three reference points used to draw the Pitchfork: the initial low (A), a subsequent swing high (B), and a subsequent swing low (C). The process is identical for creating down-sloping Pitchforks from highs. The slope of the Pitchfork is determined by the median line, which connects the initial low with the midpoint of the line connecting points B and C. The Pitchfork is a trend channel intended to provide boundaries to price action, with penetration of the outer lines implying a reversal of the preceding trend. Note that reference highs and lows are chosen subjectively. Different selections will result in different Pitchforks, different trend definitions and different implied reversals.

Average and median:
The mean (or average) of a set of values is the sum of the values divided by the number of values in the set. If a set consists of 10 numbers, add them and divide by 10 to get the mean.
A statistical weakness of the mean is that it can be distorted by exceptionally large or small values. For example, the mean of 1, 2, 3, 4, 5, 6, 7, and 200 is 28.5 (228/8). Take away 200, and the mean of the remaining seven numbers is 4, which is much more representative of the numbers in this set than 28.5.
The median can help gauge how representative a mean really is. The median of a data set is its middle value (when the set has an odd number of elements) or the mean of the middle two elements (when the set has an even number of elements). The median is less susceptible than the mean to distortion from extreme, non-representative values. The median of 1, 2, 3, 4, 5, 6, 7, and 200 is 4.5 ((4+5)/2), which is much more in line with the majority of numbers in the set.

Average true range (ATR)
is simply a moving average of the true range over a certain time period. For example, the five-day ATR would be the average of the true range calculations over the last five days.

Bid-to-cover ratio: The ratio of the number of received bids to the number accepted bids in a Treasury auction. A ratio of 2 (twice as many bids received as accepted) is typically considered a sign of strong demand and a successful auction.

Bollinger Bands: Bollinger Bands are a type of trading “envelope” consisting of lines plotted above and below a moving average, which are designed to capture a market’s typical price fluctuations.
The indicator is similar in concept to the moving average envelope, with an important difference: While moving average envelopes plot lines at a fixed percentage above and below the average (typically 3 percent above and below a 21-day simple moving average), Bollinger Bands use standard deviation to determine how far above and below the moving average the lines are placed. As a result, while the upper and lower lines of a moving average envelope move in tandem, Bollinger Bands expand during periods of rising market volatility and contract during periods of decreasing market volatility.
Bollinger Bands were created by John Bollinger, CFA, CMT, the president and founder of Bollinger Capital Management. By default, the upper and lower Bollinger Bands are placed two standard deviations above and below a 20-period simple moving average.

Upper band = 20-period simple moving average + 2 standard deviations
Middle line = 20-period simple moving average of closing prices
Lower band = 20-period simple moving average - 2 standard deviations

Bollinger Bands highlight when price has become high or low on a relative basis, which is signaled through the touch (or minor penetration) of the upper or lower line. However, Bollinger stresses that price touching the lower or upper band does not constitute an automatic buy or sell signal. For example, a close (or multiple closes) above the upper band or below the lower band reflects stronger upside or downside momentum that is more likely to be a breakout (or trend) signal, rather than a reversal signal. Accordingly, Bollinger suggests using the bands in conjunction with other trading tools that can supply context and signal confirmation.

C D

Carry trades involve buying (or lending) a currency with a high interest rate and selling (or borrowing) a currency with a low interest rate. Traders looking to “earn carry” will buy a high-yielding currency while simultaneously selling a low-yielding currency.

Published weekly by the Commodity Futures Trading Commission (CFTC), the Commitments of Traders (COT) report breaks down the open interest in major futures markets. Clearing member firms, futures commission merchants, and foreign brokers are required to report daily the futures and options positions of their customers that are above specific reporting levels (for each market) set by the CFTC.

For each futures market, report data is traditionally divided into major reporting categories: commercial, non-commercial, and non-reportable positions. The first two groups are those who hold positions above specific reporting levels. The “commercials” are often referred to as “large hedgers.” Commercial hedgers are typically those who actually deal in the cash market (for example, agribusinesses and oil companies, who either produce or consume the underlying commodity) and typically have access to supply and demand information other market players do not.

Non-commercial large traders include large speculators (“large specs”) such as commodity trading advisors (CTAs) and hedge funds. This group consists mostly of institutional and quasi-institutional money managers who do not deal in the underlying cash markets, but speculate in futures on a large-scale basis for their clients. The final COT category is called the non-reportable position category — otherwise known as small traders — i.e., the general public.

In an effort to make the COT data more transparent, in 2009 the CFTC began publishing a more detailed “disaggregated” report that breaks down traders into the following four groups: Producer/Merchant/Processor/User (traditional commercial hedgers); Swap Dealers (traders that deal primarily in commodity swaps and uses the futures markets to manage or hedge their risk); Managed Money (CTAs and other professional funds); and Other Reportables (all other traders). COT data can be accessed at www.cftc.gov.

Confidence levels: Confidence levels offer a more precise estimate of an average value by setting upper and lower limits for a data set’s central tendency. This range can suggest how precise a statistic is (mean, correlation, etc.) according to a certain probability. For example, 98-percent confidence levels suggest that the market is likely to have traded outside of therange just 2 percent of the time.

Assume the S&P 500’s average monthly move is 1.76 percent over the past 12 months, and we want to find the upper and lower confidence levels for this mean at the 95-percent confidence interval. (The higher the confidence interval, the wider this range is.)

Let's say 12 monthly percentage returns are 1.2, 1.5, 1.7, 2.3, 4.5, 3.3, 6.7, -2.4, 1.1, 1.0, 2.1, and -1.9. The standard deviation is 2.46 percent, and for a 95-percent confidence level, you must go 1.96 standard deviations from the mean. The standard error of the mean is 0.71 percent (2.46 percent /(12)). The formula for upper and lower confidence levels are:

Lower limit = Mean - (1.96 standard deviations * 0.71 percent standard error) = 1.76 percent - (1.96)(0.71 percent) = 0.36 percent
Upper limit = Mean + (1.96 standard deviations * 0.71 percent standard error) = 1.76 percent + (1.96)(0.71 percent) = 3.15 percent

The site http://davidmlane.com/hyperstat/index.html offers relatively easy-to-digest definitions of this and other statistical terms.

Correlation coefficient, sometimes referred to simply as correlation, refers to the degree of similarity between two variables. In the markets, correlation is typically used to measure how close the relationship is between two price series (e.g., two distinct stocks or markets), between an individual stock (or trading fund) and an index, and so on. Correlation coefficients range between -1.00 and +1.00, with +1.00 representing perfect positive correlation (i.e., two variables moving precisely in tandem); -1.00 represents perfect negative correlation (i.e., two variables moving exactly opposite to one another). A correlation coefficient of zero means the two variables have no discernible relation. The site http://davidmlane.com/hyperstat/index.html offers relatively easy-to-digest definitions of this and other statistical terms.

Curve-fitting refers to the process of tailoring or optimizing a trading system's rules to produce the best result (typically, the highest profit) on a particular set of price data. However, trading rules that are overly fit to specific data in this fashion almost invariably produce poor results in actual trading because the exact conditions of the test data are never repeated precisely in the future.

E F

Elliott Wave is a descriptive form of technical analysis based on the concept that price action unfolds in identifiable, structured waves that define both trend and countertrend moves.
Ralph Nelson Elliott (1871-1948) introduced his ideas through a series of letters to Charles J. Collins, who help Elliott publish The Wave Principle in 1938. Also with Collins’ aid, Elliott published a series of articles in Financial World magazine in 1939. Today, Elliott Wave theory is probably best known through the work of Robert R. Prechter Jr., who in 1978 coauthored with A.J. Frost Elliott Wave Principle: Key to Stock Market Profits (John Wiley & Sons, 10th edition, 2001).
Elliott Wave theory contains elements of a concept known as a fractal, which is an object or shape that has self-similarity on different scales. Fractals are found in a variety of phenomena. For example, if you look at a mountain from a distance, you may see a peak with relatively smooth sides. As you move closer, you begin to see how the sides of the mountain are actually made up of smaller sub-peaks and sides, which consist of even smaller peaks and sides, all sharing a similar basic structure or pattern.
Similarly, part of wave theory is the idea that any wave cycle is part of a larger wave cycle that adheres to the same rules, and is also composed of smaller wave cycles with the same structure. Many devotees of Elliott Wave consider price action to be a natural phenomena driven by human emotion, which makes the fractal aspect of wave patterns a valid way to understand and describe the price movement.

Exponential moving average (EMA): While the simple moving average (SMA) calculation gives every price point in the average equal emphasis — for example, a five-day SMA is the sum of the most recent five closing prices divided by five — weighted moving averages emphasize more recent price action. An exponential moving average is a type of weighted moving average that uses the following formula:

EMA = SC * price + (1 - SC) * EMA(yesterday)
where: SC is a “smoothing constant” between 0 and 1, and EMA(yesterday) is the previous day’s EMA value.

You can approximate a particular SMA length for an EMA by using this formula to calculate the equivalent smoothing constant:

SC = 2/(n + 1)
where: n = the number of days in a simple moving average of approximately equivalent length.

For example, a smoothing constant of 0.095 creates an exponential moving average equivalent to a 20-day SMA (2/(20 + 1) = 0.095). The larger n is, the smaller the constant, and the smaller the constant, the less impact the most recent price action will have on the EMA. In practice, most software programs allow you to simply choose how many days you want in your moving average and select either simple, weighted, or exponential calculations.

Fibonacci series: A number progression in which each successive number is the sum of the two immediately preceding it: 1, 2, 3, 5, 8, 13, 21, 34, 55, and so on. As the series progresses, the ratio of a number in the series divided by the immediately preceding number approaches 1.618, a number that is attributed significance by many traders because of its appearance in natural phenomena (the progression of a shell’s spiral, for example), as well as in art and architecture (including the dimensions of the Parthenon and the Great Pyramid). The inverse, 0.618 (0.62), has a similar significance.
Some traders use fairly complex variations of Fibonacci numbers to generate price forecasts, but a basic approach is to use ratios derived from the series to calculate likely price targets. For example, if a stock broke out of a trading range and rallied from 25 to 55, potential retracement levels could be calculated by multiplying the distance of the move (30 points) by Fibonacci ratios –– say,  0.382, 0.50, and 0.618 –– and then subtracting the results from the high of the price move. In this case, retracement levels of 43.60 [55 - (30 * 0.38)], 40 [55 - (30 * 0.50)], and 36.40 [55 - (30 * 0.62)] would result.
Similarly, after a trading range breakout and an up move of 10 points, a Fibonacci follower might project the size of the next leg up in terms of a Fibonacci ratio –– e.g., 1.382 times the first move, or 13.82 points in this case.
The most commonly used ratios are 0.382, 0.50, 0.618, 0.786, 1.00, 1.382, and 1.618. Depending on circumstances, other ratios, such as 0.236 and 2.618, are used.

While Fibonacci retracements are used to calculate the possible partial correction levels of a previous price move (i.e., a reversal of up to 100 percent of a previous price swing), Fibonacci extension levels are used to extrapolate moves in the same direction as a previous price swing — for example, projecting a target for a new upswing that represents a 161.8-percent gain from a certain price level based on the size of the previous upswing.

K L

Keltner channels: Price bands formed by placing lines above and below a moving average of the “typical price” (the average of a bar’s high, low, and closing prices) by a distance equal to the moving average of the daily ranges over an n-day period. The following definition uses a 10-day simple moving average (SMA) for both calculations.
1. Upper Keltner channel: Today’s 10-day moving average of the typical price + 10-day SMA of the daily ranges (high-low).
2. Lower Keltner channel: Today’s 10-day moving average of the typical price - 10-day SMA of the daily ranges (high-low).

Key reversal: A basic key reversal high is a bar that makes a significantly higher high than the preceding high but reverses intrabar to close near the bottom of the bar’s range. (Different definitions require the close to be below the close of the preceding bar, etc.) A key reversal is essentially a “spike” high that closes weakly. The traditional implication is that price has expended upside momentum and turned lower, foreshadowing further selling. (A key reversal low is the opposite pattern.) However, although such bars can often be observed – in retrospect – at market turning points, traders are apt to overlook them when they are followed by more price movement in the same direction. As with all patterns, objectifying such concepts as a “significantly higher high” and “near the bottom” are necessary to be able to consistently identify and test key reversal bars in different markets.

Linear regression (“best-fit”) line:
A way to calculate a straight line that best fits a set of data (such as closing prices over a certain period) — that is, a line that most accurately reflects the slope, or trend, of the data.
A regression line is calculated using the “least squares” method, which refers to finding the minimum squared (x*x, or x2) differences between price points and a straight line. For example, if two closing prices are 2 and 3 points away (the distance being calculated vertically) from a straight line, the squared differences between the points and the line are 4 and 9, respectively.
The squared differences are used (instead of just the differences) because some differences are negative (for points below the line) and others are positive (for points above the line). Squaring all the differences creates all-positive values and allows you to calculate a formula for the straight line.
The “best-fit” line is the line for which the sum of the squared differences between each price and the straight line are minimized.
The formula for a straight line (y) is:

y = a + b*t
where:
t = time
a = the initial value of the line when “t” is equal to zero (sometimes called the “intercept” value — i.e., the point at which the line intercepts the vertical y-axis) or the point at which a specific line begins
b = the slope of the line, which is the rate at which the line rises or falls (e.g., 0.75 points per day).

M N

Moving average convergence-divergence (MACD): Although it is often grouped with oscillators, the MACD is more of an intermediate-term trend indicator (although it can reflect overbought and oversold conditions). The default MACD line (which can also be plotted as a histogram) is created by subtracting a 26-period exponential moving average (EMA) of closing prices from a 12-period EMA of closing prices; a nine-period EMA is then applied to the MACD line to create a “signal line.”

MACD = EMA(C,12)-EMA(C,26)
Signal line = EMA(MACD,9)

Moving average crossover: A trend-following approach that signals a trend change when a shorter-term moving average crosses above or below a longer-term moving average.

O P

Open interest: The number of outstanding (unclosed) positions in a given futures or options market.

Optimization (and walk-forward optimization, walk-forward testing, in-sample, and out-of-sample: The process of testing a range of system parameters on historical price data to find the "best" values for use in trading. For example, a trading system that enters long when price makes an n-bar high might be optimized to find the value of n, from 10 to 100, that produces the most favorable results. There is a great deal of debate over the benefits of optimization, because the process is often simply used to find system parameter values that produce the highest net profit on a certain set of historical data, with the expectation these values will produce similar results in the future. However, this is rarely the case. Highly optimized parameters typically do nothing more than reflect the specific circumstances of the data set from which they were derived; they will typically underperform in the future because price action never unfolds exactly as it has in the past.
Some traders believe optimization of any kind is potentially damaging. However, there is an argument that optimization can be used to find less-idealized but more reliable parameter settings by identifying ranges of values with comparable, positive results. In the case of the aforementioned n-bar breakout system, it might turn out that a 23-bar breakout produced the best results over the initial test period, but the results for 20-, 21-, 22-, 24-, 25-, and 26-bar breakouts were all negative, and even n values up to 30 produced sub-par results. This suggests the 23-bar results were an anomaly, and using this parameter would likely result in poor performance in the future. By contrast, if all n values from 35 to 47 produced positive, comparable results — even if they were much less profitable than the 23-bar breakout — a representative value from this stable range (i.e., the median value, 41) is much more likely to produce success in actual trading.
Typically, parameter values are optimized on an initial data set (the "test," "sample" or "in-sample" data), and then tested on a new, different data set (the "out-of-sample" data) to simulate the process of trading the system. This can be repeated several times — the process of "walking the system forward" on new data is the reason this approach is referred to as "walk-forward" optimization. If the optimized values do not perform consistently on new, out-of-sample data, they are not suited for trading. Some system designers choose to re-optimize system parameters on relatively short data periods and use these parameters on the next set of data, believing the price behavior from the more recent past has a greater chance of persisting into the near future. Others believe this simply results in a highly optimized system that will always be one step behind the market.
Finally, among the many problems of optimization is the challenge of determining what constitutes a "best-performing" value. Traders are apt to look exclusively at net profit, without incorporating measures of risk (e.g., drawdown depth and duration) and consistency (e.g., winning percentage, average trade) that are likely to make future profitability more likely.

Point-and-figure charts display price independently of time. Price advances and declines are represented by columns of Xs and Os, respectively. (Note: Charting programs often replace Os with squares.) Each time price advances by a certain amount, called the “box size,” an X is added to the ascending column of Xs. For example, when analyzing a stock, a box size of 1.00 point would mean an X would be added every time price gained 1.00 point. If the box size was .50, an X would be added every time price rallied 0.50 points, and so on.

To begin plotting a column of Os, price first has to decline a certain amount, called the reversal size. A reversal size of “2” would mean price would have to drop by two box sizes before you would end a column of ascending Xs and begin plotting a column of descending Os. Similarly, to begin a new column of Xs, price would have to turn back up at least 2.00 points.

The larger the box and reversal sizes, the more price fluctuations a point-and-figure chart will filter out. For more detail, you simply decrease those values.

Purchasing power parity: The idea that an exchange rate should reflect the level that results in the same price (in the two currencies) for a product purchased in two countries. For example, if a certain automobile costs 50,000 British pounds in Great Britain, it should cost 25,000 U.S. dollars in the United States if the current British pound/U.S. dollar rate (GBP/USD) is 2.0000.

Q R

Quantitative easing is a tool a central bank uses to attempt to stimulate the economy when cutting interest rates is not feasible — such as when rates are already at or near zero. Through quantitative easing, the central bank purchases assets (e.g., treasuries, mortgages, securities) from financial institutions to pump money into the financial system. Quantitative easing is often referred to as “printing money.” Critics contend the practice runs a high risk of creating high inflation, among other drawbacks.

R2 (r-squared): A measure of correlation strength. It is the square of the correlation coefficient, which is the measure of association of two variables ranging from +1 (perfect positive correlation) to -1 (perfect negative correlation), where zero represents no correlation. R-squared is always a positive number, ranging from zero to +1.

S T

Simple moving average (SMA). The simple moving average (SMA) is the mean price of a stock, futures contract, currency, or other instrument over a specific time period:

N-day moving average = Sum (Pricet, Pricet-1 … Pricet-N)/N

where,

Pricet  = today’s price
Pricet-N = price N days ago

For example, a 20-day SMA is the sum of the prices of the most recent 20 days, divided by 20.
The closing price is usually used in moving average calculations, although the high, low, opening price, or average price of a price bar is sometimes substituted. As the market moves forward in time, the newest price is added to the average and the oldest is dropped from it.
Moving averages can be calculated for any time increment – daily, intraday, weekly, monthly. For a five-minute bar chart, for example, a 10-bar SMA would be the average price of the 10 most recent five-minute bars. Moving averages are typically used to highlight trend direction and filter out smaller fluctuations (or “noise”) from a data series. The degree to which they do this depends on the length of the average: A longer moving average reflects a longer-term trend and filters out shorter-term fluctuations, while a shorter moving average reflects the more immediate trend and filters out less noise. However, this filtering comes at a cost: The longer the average, the more it lags behind the movements of the data series.

The European currency “snake:” A short-lived agreement among European member states in the early 1970s to limit currency fluctuations within a proscribed percentage band.

Standard error channel (linear regression channel): The linear regression line is a straight line that minimizes the distance between itself and every data point in a series. A standard error measures the variance from the linear regression. Subtracting the standard error from the linear regression line yields the bottom of the standard error channel, and adding it to the linear regression value gives you the top of the channel.
The standard error channel is a parallel concept to Bollinger Bands, which use the standard deviation calculation to set boundaries above and below a moving average to capture variance away from the average. Because the moving average is a wavy line, the Bollinger bands are wavy, too, and also widen or narrow as variability rises or falls. The standard error does the same thing, only with straight lines. The critical difference is that you don't need to choose a starting and ending point for Bollinger Bands, because they track a moving average that constantly discards old data and refreshes itself with new data. To construct a useful linear regression channel, however, you have to subjectively select starting and ending points.

Support and resistance: Support is a price level that acts as a “floor,” preventing prices from dropping below that level. Resistance is the opposite: a price level that acts as a “ceiling;” a barrier that prevents prices from rising higher.
Support and resistance levels are a natural outgrowth of the interaction of supply and demand in any market. For example, increased demand for a stock will cause its price to rise, creating an uptrend. But when price has risen to a certain level, traders and investors will take profits and short sellers will come into the market, creating “resistance” to further price increases. Price may retreat from and advance to this resistance level many times, sometimes eventually breaking through it and continuing the previous trend, other times reversing completely.
Support and resistance should be thought of more as general price levels rather than precise prices. For example, if a stock makes a low of 52.15, rallies slightly, then declines again to 52.15, then rallies again, a subsequent move down to 52 does not violate the “support level” of 52.15. In this case, the fact that the stock retraced once to the exact price level it had established before is more of a coincidence than anything else.

True range (TR): A measure of price movement or volatility that accounts for the gaps that occur between price bars. This calculation provides a more accurate reflection of the size of a price move over a given period than the standard range calculation, which is simply the high of a price bar minus the low of a price bar. The true range calculation was developed by Welles Wilder and discussed in his book New Concepts in Technical Trading Systems (Trend Research, 1978).True range can be calculated on any time frame or price bar — five-minute, hourly, daily, weekly, etc. Using daily price bars as an example, true range is the greatest (absolute) distance of the following:
1. Today’s high and today’s low.
2. Today’s high and yesterday’s close.
3. Today’s low and yesterday’s close.

Average true range (ATR) is simply a moving average of the true range over a certain time period. For example, the five-day ATR would be the average of the true range calculations over the last five days.

U V

Ulcer Index: A measure of the depth and duration of a drawdown from an equity peak.

Variance and standard deviation: Variance measures how spread out a group of values are — in other words, how much they vary. Mathematically, variance is the average squared “deviation” (or difference) of each number in the group from the group’s mean value, divided by the number of elements in the group. For example, for the numbers 8, 9, and 10, the mean is 9 and the variance is:

{(8-9)2 + (9-9)2 + (10-9)2}/3 = (1 + 0 + 1)/3 =   0.667

Now look at the variance of a more widely distributed set of numbers: 2, 9, and 16:

{(2-9)2 + (9-9)2 + (16-9)2}/3 = (49 + 0 + 49)/3 =  32.67

The more varied the prices, the higher their variance — the more widely distributed they will be. The more varied a market’s price changes from day to day (or week to week, etc.), the more volatile that market is.
A common application of variance in trading is standard deviation, which is the square root of variance. The standard deviation of 8, 9, and 10 is: sq. rt. 0.667 = .82; the standard deviation of 2, 9, and 16 is: sq. rt. 32.67 = 5.72.

Volatility: The level of price movement in a market. Historical (“statistical”) volatility measures the price fluctuations (usually calculated as the standard deviation of closing prices) over a certain time period — e.g., the past 20 days. Implied volatility is the current market estimate of future volatility as reflected in the level of option premiums. The higher the implied volatility, the higher the option premium.