The simple moving average (SMA) is probably the best-known technical indicator. It is used to smooth price action and highlight trends.

Calculation

The SMA is the average price of an instrument over a specific time period:

n-day moving average = Sum(Pricet, Pricet-1 ...Pricet-N)/n
where,

Price t = today’s price

Price t-n = price n days ago

For example, a 20-day moving average is the average price of the most recent 20 days. The closing price is usually used in the calculation, although the high, low, opening or average price of a price bar can be substituted. To calculate, add the closing prices of the last 20 days and divide by 20.

As a market moves forward in time, the newest price is added to the average and the oldest is dropped from it. It shows a series of daily closing prices and the five day SMA values that result from progressively averaging the five most recent days’ prices.
Moving averages can be calculated for any time frame daily, intraday, weekly, monthly, etc. For a five minute bar chart, for example, a 10-bar moving average would be the average price of the 10 most recent five-minute bars. We will continue the discussion in terms of daily bars for simplicity.

The overnight interest rate at which U.S. banks lend one another funds.

When a country’s government uses its central bank to buy or sell its own currency in the foreign exchange market to push its valuation to a desired level or to prevent it from reaching an undesirable level.

A managed-float currency falls somewhere between a fully floating currency and a fixed-rate currency. The government of a managed-float currency uses capital controls and other measures (including intervention) to keep the value of its currency within a fixed range of levels against other currencies; it changes the range over time according to its own wishes. Therefore, a managed-float currency can vary within the current government set range, and the government can change the range from time to time, but the foreign exchange market cannot push the currency outside the range. The foreign exchange market for a managed-float currency is heavily regulated.

A least squares approach fits a straight line t h rough the series of prices over a lookback period so the difference between each of the individual prices and the straight line is the smallest (least).

This is a “best fit“ line of the data (also re f e r red to as a “re g ression line”).

“Squares“ refers to finding the minimum squared difference between prices and the straight line. The differences are squared because some price points are above the line (positive) and some are below it (negative).

Plotting the final value, or endpoint, of the least squares lines for each bar calculated over a certain lookback period tracks price more closely than a moving average.

The formula for a least squares line is simple. A straight line has a beginning point and rises at a fixed rate. For example, if IBM starts at $100 and moves up for 10 more days, closing at $105, a straight line fit to the data would start at $100 and rise at 50 cents per day, arriving at $105 the final day.

The formula for a straight line is:

a0+a1*t
where

a0 is the initial value of the line;

a1 is the slope of the line;

t is time.

For our example, a0 is $100, a1 is 50 cents, and t is 10 (days). The formula has two terms: coefficients (a0+a1) and variables (t). Mathematicians call this formula a polynomial, which means it has more than one term. Because t is not raised to a power, the polynomial is referred to as a “first-order” polynomial.

Although the least squares line may be an improvement over the moving average, it still changes directions too late for tradable signals. What’s needed is a formula for a line that curves.

A curved line does not change direction at a fixed rate, but at an accelerating or decelerating rate. The mathematical trick to making a line curve is to add an additional term to the formula for a straight line, and have this term be squared (i.e., raised to the second power). The formula for a line that can curve is a0+a1*t+a2*t2. This line takes on the appearance of a parabola. Because it curves, this line fits price data better than a straight line. Because the final term of this polynomial is squared, it is called a “second-order” polynomial. Just as the second-order (parabolic) polynomial calculation displays changes in trend direction sooner than a straight line, higher- order polynomials a p p roximate price better than straight or parabolic lines. The more terms added to a formula, the closer the fit to the price data.

A triangle is a pattern in which price trades in an increasingly narrow range. It represents a period of market congestion or consolidation, and has the same implications as trading ranges (“rectangles”), flags, and pennants.

Triangles and pennants are identical except for their length: Pennants might typically consist of approximately five to 15 bars, while triangles can span dozens of bars.

The most important aspect of triangles is that they represent market contraction (i.e., decreasing volatility), a condition typically followed by price thrusts or trends. In technical analysis parlance, there are three types of triangles: symmetrical, ascending, and descending. Symmetrical triangles consist of progressively lower highs and higher lows, so that the upper trendline of the pattern (which represents resistance) slopes downward and the lower trendline of the pattern (which represents support) slopes upward.
An ascending triangle is characterized by a rising support line that reflects progressively higher lows and a horizontal resistance line that reflects equivalent highs. A descending triangle is the opposite: It consists of a falling resistance line that reflects progressively lower highs and a horizontal support line that reflects equivalent price lows.

The linear regression line is a straight line that minimizes the distance between itself and every data point in the series you are working with. 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 pick reasonable starting and ending points. It's still "mathematics" and thus a better way to draw a trendline than using your eye alone, but your choice of starting and ending points is inherently judgmental. Most practitioners chose an obvious lowest low or highest high.

The Fibonacci series is 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 it appearance in natural phenomena (the progression 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, .618 (.62), has a similar significance. Some traders use fairly complex variations of Fibonacci number 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, .382, .50 and .618 –– and then subtracting the results from the high of the price move. In this case, retracement levels of 43.60 [55 - (30*.38)], 40 [55 - (30*.50)] and 36.40 [55 - (30*.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 .382, .50, .618, .786, 1.00, 1.382 and 1.618. Depending on circumstances, other ratios, such as .236 and 2.618, are used.

A candle (either white or black but usually the opposite color of the preceding bar) whose real body is inside the previous candle’s real body.

Reference to a legend of Alexander the Great. There was a prophecy that whoever was able to undo the Gordian Knot, which had supposedly been tied by King Midas, would rule Asia. When in Gordium, Turkey, Alexander supposedly slashed through the knot with the sword and claimed that by doing so he had fulfilled the prophecy.

A tall candle in a downswing with a white real body that engulfs (i.e., overlaps) the previous candle’s real body. The bears were in charge early, as prices opened below the prior candle’s real body, but the bulls took over later, as prices closed above the previous real body.

A price bar with a higher high and lower low than the preceding bar. Because it has a wider range than the previous bar, an outside bar represents a volatility increase.

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 this range 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 the 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/hyprstat/index.html offers relatively easy-to-digest definitions of this and other statistical terms.

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 it appearance in natural phenomena (the progression 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*.38)], 40 [55 - (30*.50)], and 36.40 [55 - (30*.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.

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 a fixed percentage above and below the average (typically three 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 (see Active Trader, April 2003, p. 60).

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.

The simple moving average (SMA) is the standard moving average calculation that gives every price point in the average equal emphasis, or weight. For example, a five-day SMA is the sum of the most recent five closing prices divided by five. Weighted moving averages give extra emphasis to more recent price action. Exponential moving average (EMA) weights prices using 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 the following 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

A measure of price movement 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. The following example uses daily price bars for simplicity.

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.

The lesser of the current low or the previous close.

The greater of the current high or the previous close.

Formulas designed to measure price momentum (the rate at which price changes) and highlight shorter-term overbought and oversold levels — levels at which price is likely to correct or reverse. These indicators typically fluctuate in fixed ranges — for example, 0 to 100 or -100 to +100, although some do not.

Some of the better-known oscillators are the Relative Strength Index, the stochastic oscillator, the Commodity Channel Index (CCI), the True Strength Index (TSI), and the Moving Average Convergence Divergence (MACD) indicator. Related calculations include tools such as the price oscillator and momentum, or rate of change (ROC).

At the heart of every oscillator is momentum, or the rate at which price changes over a given period. Measuring momentum over a given period essentially means comparing the current price to the price n days (or minutes or weeks, etc.) ago.

There are a few basic ways to do this. You can calculate the difference between the price n periods ago from the current price (the momentum indicator), divide the two (rate of change), or calculate the difference or ratio between derivatives of price, such as moving averages. For example, the MACD is the difference between two exponential moving averages, and the stochastic oscillator and the TSI are momentum calculations that are smoothed with moving averages to reduce their volatility.