Kaufman Adaptive Moving Average

Kaufman’s Adaptive Moving Average ( AMA , KAMA , AMkA from Kaufman's Adaptive Moving Average ) is a technical indicator , a kind of adaptive moving average based on an exponentially smoothed moving average and the original method of determining and applying volatility as a dynamically changing smoothing constant ^[1] ^[2] ^[3] ^[4] .

The Adaptive Moving Average indicator was developed by Perry J. Kaufman and first introduced in 1995 in his book “Smart Trading: Improving Performance in a Changing Markets ” ^[1] ^{[ 2]} .

Content

1 Prerequisites for creating an indicator
2 Calculation Method
- 2.1 Basic formula
- 2.2 coefficient of effectiveness
- 2.3 Smoothing constant
- 2.4 Adaptive Moving Average
3 Original parameter values
4 Trading Strategies
5 Filtering
6 Numerical values for the filter
7 Trading strategies using filters
8 Communication with other indicators
9 notes
10 Literature

Prerequisites for creating an indicator

When using classical moving averages as an indicator of technical analysis, traders are faced with the need to choose the optimal window width for their calculations. In the general case, this is a non-trivial task that spawned a whole branch of technical analysis ^[5] . There was a proposal to automate the selection of this parameter. In 1992, Tushar Chande developed an adaptive moving average model ( VIDYA ), in which the width of the window depends on price volatility ^[6] , and in 1995 Perry Kaufman proposed his version of a similar technical indicator ^[2] . The main message of Kaufman was the desire to implement a conservative following in the direction of the trend , while quickly receiving a signal in a dynamic market and timely closing positions when the market becomes non-directional ^[2] .

Calculation Method

Basic Formula

The adaptive Kaufman moving average is a derivative of the classical exponentially smoothed moving average with a variable smoothing coefficient. That is, every time a classic formula is used

${\textit {EMA}}_{t}=\alpha \cdot {\textit {close}}_{t}+(1-\alpha )\cdot {\textit {EMA}}_{t-1},$ ${\ displaystyle {\ textit {EMA}} _ {t} = \ alpha \ cdot {\ textit {close}} _ {t} + (1- \ alpha) \ cdot {\ textit {EMA}} _ {t- one},}$ ${\textit {EMA}}_{t}=\alpha \cdot {\textit {close}}_{t}+(1-\alpha )\cdot {\textit {EMA}}_{t-1},$

in which the smoothing constant $\alpha$ ${\ displaystyle \ alpha}$ $\alpha$ It is calculated dynamically and generally differs for each period .

Performance Ratio

To determine the state of the market, Perry Kaufman introduces the concept of an efficiency coefficient ( ER from the English efficiency ratio ), which is based on the ratio of the total price movement (direction) and the sum of the absolute values of the noise market movements (volatility) for a certain period (n) ^[1] ^[2 ^] ^] :

${\textit {direction}}_{t,n}=|{\textit {close}}_{t}-{\textit {close}}_{t-n-1}|,$ ${\ displaystyle {\ textit {direction}} _ {t, n} = | {\ textit {close}} _ {t} - {\ textit {close}} _ {tn-1} |,}$

${\textit {volatility}}_{t,n}=\sum _{i=0}^{n-1}|{\textit {close}}_{t-i}-{\textit {close}}_{t-i-1}|,$ ${\ displaystyle {\ textit {volatility}} _ {t, n} = \ sum _ {i = 0} ^ {n-1} | {\ textit {close}} _ {ti} - {\ textit {close} } _ {ti-1} |,}$

${\textit {EfficiencyRatio}}_{t,n}={\frac {{\textit {direction}}_{t,n}}{{\textit {volatility}}_{t,n}}}={\frac {|{\textit {close}}_{t}-{\textit {close}}_{t-n-1}|}{\sum _{i=0}^{n-1}|{\textit {close}}_{t-i}-{\textit {close}}_{t-i-1}|}},$ ${\ displaystyle {\ textit {EfficiencyRatio}} _ {t, n} = {\ frac {{\ textit {direction}} _ {t, n}} {{\ textit {volatility}} _ {t, n}} } = {\ frac {| {\ textit {close}} _ {t} - {\ textit {close}} _ {tn-1} |} {\ sum _ {i = 0} ^ {n-1} | {\ textit {close}} _ {ti} - {\ textit {close}} _ {ti-1} |}},}$

Where ${\textit {direction}}_{t,n},{\textit {volatility}}_{t,n},{\textit {EfficiencyRatio}}_{t,n}$ ${\ displaystyle {\ textit {direction}} _ {t, n}, {\ textit {volatility}} _ {t, n}, {\ textit {EfficiencyRatio}} _ {t, n}}$ - accordingly, the total price movement, the sum of noise movements and the efficiency coefficient at the $t$ ${\ displaystyle t}$ over a period $n$ ${\ displaystyle n}$ ; ${\textit {close}}_{i}$ ${\ displaystyle {\ textit {close}} _ {i}}$ - period closing price $i$ ${\ displaystyle i}$ .

It can be seen from the presented formulas that the efficiency coefficient can vary from 0 to 1. Moreover, its value tends to zero when there is no directed movement in the market, and to unity when the market moves unidirectionally. If the price chart is a straight line, the coefficient of efficiency will be equal to one.

Smoothing Constant

At the next stage, a variable smoothing constant ( SSC from the English scaled smoothing constant ) is calculated, which is built on the assumption that, depending on the efficiency coefficient, it should “remember” data for a different number of previous periods. That is, in the trending market, a fast moving average (calculated on a narrow window) should be used, and on a non-trending market, a slow moving average (calculated on a wide window). Moreover, the specific value of the window width should be obtained automatically based on the value of the efficiency coefficient ^[1] ^[2] :

${\textit {fastest}}={\frac {2}{f+1}}$ ${\ displaystyle {\ textit {fastest}} = {\ frac {2} {f + 1}}}$

${\textit {slowest}}={\frac {2}{s+1}}$ ${\ displaystyle {\ textit {slowest}} = {\ frac {2} {s + 1}}}$

${\textit {smooth}}_{t,n,f,s}={\textit {EfficiencyRatio}}_{t,n}\cdot ({\textit {fastest}}-{\textit {slowest}})+{\textit {slowest}},$ ${\ displaystyle {\ textit {smooth}} _ {t, n, f, s} = {\ textit {EfficiencyRatio}} _ {t, n} \ cdot ({\ textit {fastest}} - {\ textit {slowest }}) + {\ textit {slowest}},}$

Where ${\textit {fastest}},{\textit {slowest}}$ ${\ displaystyle {\ textit {fastest}}, {\ textit {slowest}}}$ Are the classic smoothing coefficients for an exponentially smoothed moving average, and ${smooth}_{t,n,f,s}$ ${\ displaystyle {smooth} _ {t, n, f, s}}$ - the changing smoothing constant calculated for the moment $t$ ${\ displaystyle t}$ using the efficiency coefficient to build ${\textit {EfficiencyRatio}}_{t,n}$ ${\ displaystyle {\ textit {EfficiencyRatio}} _ {t, n}}$ window size $n$ ${\ displaystyle n}$ periods, taking as a quick smoothing factor - ${\textit {fastest}},f$ ${\ displaystyle {\ textit {fastest}}, f}$ periods, and as a slow smoothing coefficient - ${\textit {slowest}},s$ ${\ displaystyle {\ textit {slowest}}, s}$ periods.

For a more effective impact of a changing smoothing constant (SSC) in highly noisy market areas with a weak trend component, Kaufman recommends using the SSC square as a dynamic smoothing coefficient in the formulas of an exponentially smoothed moving average:

$c_{t,n,f,s}={\textit {smooth}}_{t,n,f,s}^{2}=({\textit {EfficiencyRatio}}_{t,n}\cdot ({\textit {fastest}}-{\textit {slowest}})+{\textit {slowest}})^{2}.$ ${\ displaystyle c_ {t, n, f, s} = {\ textit {smooth}} _ {t, n, f, s} ^ {2} = ({\ textit {EfficiencyRatio}} _ {t, n} \ cdot ({\ textit {fastest}} - {\ textit {slowest}}) + {\ textit {slowest}}) ^ {2}.}$

Adaptive Moving Average

The final formula for the adaptive moving average will look as follows ^[1] ^[2] :

${\textit {AMA}}_{t,n,f,s}=c_{t,n,f,s}\cdot {\textit {close}}_{t}+(1-c_{t,n,f,s})\cdot {\textit {AMA}}_{t-1},$ ${\ displaystyle {\ textit {AMA}} _ {t, n, f, s} = c_ {t, n, f, s} \ cdot {\ textit {close}} _ {t} + (1-c_ { t, n, f, s}) \ cdot {\ textit {AMA}} _ {t-1},}$

Where ${\textit {AMA}}_{t,n,f,s},{\textit {AMA}}_{t-1,n,f,s}$ ${\ displaystyle {\ textit {AMA}} _ {t, n, f, s}, {\ textit {AMA}} _ {t-1, n, f, s}}$ - values of adaptive moving average at time $t$ ${\ displaystyle t}$ and $t-1$ ${\ displaystyle t-1}$ (current and previous values), $c_{t,n,f,s}$ ${\ displaystyle c_ {t, n, f, s}}$ - the second degree of the changing smoothing constant, ${\textit {close}}_{t}$ ${\ displaystyle {\ textit {close}} _ {t}}$ - closing price of the current period $t$ ${\ displaystyle t}$ .

Original parameter values

Kaufman used ^[1] as the original parameters:

$n=10$ ${\ displaystyle n = 10}$ (for the window for calculating the efficiency coefficient),
$f=2$ ${\ displaystyle f = 2}$ (for fast moving average)
$s=30$ ${\ displaystyle s = 30}$ (for a slow moving average).

When substituting the specified parameters in the formulas, we obtain (with the original rounding):

${\textit {direction}}_{t,10}=|{\textit {close}}_{t}-{\textit {close}}_{t-9}|$ ${\ displaystyle {\ textit {direction}} _ {t, 10} = | {\ textit {close}} _ {t} - {\ textit {close}} _ {t-9} |}$

${\textit {volatility}}_{t,10}=\sum _{i=0}^{9}|{\textit {close}}_{t-i}-{\textit {close}}_{t-i-1}|$ ${\ displaystyle {\ textit {volatility}} _ {t, 10} = \ sum _ {i = 0} ^ {9} | {\ textit {close}} _ {ti} - {\ textit {close}} _ {ti-1} |}$

${\textit {EfficiencyRatio}}_{t,10}={\frac {{\textit {direction}}_{t,10}}{{\textit {volatility}}_{t,10}}}={\frac {{\textit {close}}_{t}-{\textit {close}}_{t-9}}{\sum _{i=0}^{9}|{\textit {close}}_{t-i}-{\textit {close}}_{t-i-1}|}}$ ${\ displaystyle {\ textit {EfficiencyRatio}} _ {t, 10} = {\ frac {{\ textit {direction}} _ {t, 10}} {{\ textit {volatility}} _ {t, 10}} } = {\ frac {{\ textit {close}} _ {t} - {\ textit {close}} _ {t-9}} {\ sum _ {i = 0} ^ {9} | {\ textit { close}} _ {ti} - {\ textit {close}} _ {ti-1} |}}}$

${\textit {fastest}}={\frac {2}{2+1}}={\frac {2}{3}}=0,6667$ ${\ displaystyle {\ textit {fastest}} = {\ frac {2} {2 + 1}} = {\ frac {2} {3}} = 0.6667}$

${\textit {slowest}}={\frac {2}{30+1}}={\frac {2}{31}}=0,06452$ ${\ displaystyle {\ textit {slowest}} = {\ frac {2} {30 + 1}} = {\ frac {2} {31}} = 0.06452}$

${\textit {smooth}}_{t,10,2,30}={\textit {EfficiencyRatio}}_{t,10}\cdot ({\textit {fastest}}-{\textit {slowest}})+{\textit {slowest}}={\textit {EfficiencyRatio}}_{t,10}\cdot 0,6021+0,0645$ ${\ displaystyle {\ textit {smooth}} _ {t, 10,2,30} = {\ textit {EfficiencyRatio}} _ {t, 10} \ cdot ({\ textit {fastest}} - {\ textit {slowest }}) + {\ textit {slowest}} = {\ textit {EfficiencyRatio}} _ {t, 10} \ cdot 0.6021 + 0.0645}$

$c_{t,10,2,30}={\textit {smooth}}_{t,10,2,30}^{2}=({\textit {EfficiencyRatio}}_{t,10}\cdot 0,6021+0,0645)^{2}$ ${\ displaystyle c_ {t, 10,2,30} = {\ textit {smooth}} _ {t, 10,2,30} ^ {2} = ({\ textit {EfficiencyRatio}} _ {t, 10} \ cdot 0.6021 + 0.0645) ^ {2}}$

${\textit {AMA}}_{t,10,2,30}=c_{t,10,2,30}\cdot {\textit {close}}_{t}+(1-c_{t,10,2,30})\cdot {\textit {AMA}}_{t-1,10,2,30}.$ ${\ displaystyle {\ textit {AMA}} _ {t, 10,2,30} = c_ {t, 10,2,30} \ cdot {\ textit {close}} _ {t} + (1-c_ { t, 10,2,30}) \ cdot {\ textit {AMA}} _ {t-1,10,2,30}.}$

Trading Strategies

Trading strategies based on Kaufman’s adaptive moving average are common to all trend-following indicators ^[1] :

Open a long position (close short) when the price chart crosses the AMA chart from the bottom up.
Close a long position (open short) when the price chart crosses the AMA chart from top to bottom.

It is important to note that the AMA changes its direction of movement exactly at the intersection of its chart with the price chart, that is, for trading it is enough to compare the current and previous value of the indicator ^[2] :

Open a long position (close a short one) when the current AMA value is greater than its previous value.
Close a long position (open short) when the current AMA value is less than its previous value.

Filtering

Despite the dynamic adaptability of the adaptive moving average to market volatility, Kaufman believed that his indicator gives too many false signals ^[1] . Therefore, he proposed an additional filtering technique based on the estimation of the standard deviation of the difference of the adaptive moving average for neighboring periods ^[1] ^[2] .

For this, the change in AMA between periods is taken as a randomly studied variable:

$\Delta _{i}={\textit {AMA}}_{i}-{\textit {AMA}}_{i-1}.$ ${\ displaystyle \ Delta _ {i} = {\ textit {AMA}} _ {i} - {\ textit {AMA}} _ {i-1}.}$

Then, the standard deviation of this change is calculated:

$\sigma _{t}={\sqrt {{\frac {1}{d}}\sum _{i=0}^{d-1}\left(\Delta _{t-i}-{\bar {\Delta _{t}}}\right)^{2}}},$ ${\ displaystyle \ sigma _ {t} = {\ sqrt {{\ frac {1} {d}} \ sum _ {i = 0} ^ {d-1} \ left (\ Delta _ {ti} - {\ bar {\ Delta _ {t}}} \ right) ^ {2}}},}$

Where $\sigma _{t}$ ${\ displaystyle \ sigma _ {t}}$ - standard deviation of the change $A M A$ ${\ displaystyle AMA}$ in neighboring periods - $\Delta _{i}$ ${\ displaystyle \ Delta _ {i}}$ , ${\bar {\Delta }}_{t}={\frac {1}{d}}\sum _{i=0}^{d-1}\Delta _{t-i}$ ${\ displaystyle {\ bar {\ Delta}} _ {t} = {\ frac {1} {d}} \ sum _ {i = 0} ^ {d-1} \ Delta _ {ti}}$ - expectation $\Delta _{i}$ ${\ displaystyle \ Delta _ {i}}$ behind $d$ ${\ displaystyle d}$ periods.

As a filter, the share of the obtained standard deviation is used:

${\textit {filter}}_{t}=K\cdot \sigma _{t},$ ${\ displaystyle {\ textit {filter}} _ {t} = K \ cdot \ sigma _ {t},}$

Where ${\textit {filter}}$ ${\ displaystyle {\ textit {filter}}}$ - filter value based on the standard deviation of the indicator movements, $K$ ${\ displaystyle K}$ - percentage ratio.

Numerical values for the filter

Often, for the filter period d, the same number of periods is taken as for constructing the efficiency coefficient ^[1] ^[2] :

$d=n=10.$ ${\ displaystyle d = n = 10.}$

As for the percentage for the filter - $K$ ${\ displaystyle K}$ , Kaufman recommended using different values, for example, for futures and in the forex market, using values of about 10% ( $K=0,1$ ${\ displaystyle K = 0,1}$ ), and on the stock market - up to 100% ( $K=1$ ${\ displaystyle K = 1}$ )

Trading Strategies Using Filters

When using an adaptive moving average with a filter, analysts recommend adhering to the following strategy ^[1] ^[2] :

Open long position (close short) when ${\textit {AMA}}_{t}-{\textit {min}}({\textit {AMA}})>{\textit {filter}}_{t}$ ${\ displaystyle {\ textit {AMA}} _ {t} - {\ textit {min}} ({\ textit {AMA}})> {\ textit {filter}} _ {t}}$ .
Close long position (open short) when ${\textit {max}}({\textit {AMA}})-{\textit {AMA}}_{t}>{\textit {filter}}_{t}$ ${\ displaystyle {\ textit {max}} ({\ textit {AMA}}) - {\ textit {AMA}} _ {t}> {\ textit {filter}} _ {t}}$ .

In these formulas ${\textit {min}}({\textit {AMA}})$ ${\ displaystyle {\ textit {min}} ({\ textit {AMA}})}$ - the minimum value of the AMA at the pivot from the bottom up, ${\textit {max}}({\textit {AMA}})$ ${\ displaystyle {\ textit {max}} ({\ textit {AMA}})}$ - the maximum value of the AMA at the pivot from top to bottom, ${\textit {filter}}$ ${\ displaystyle {\ textit {filter}}}$ - filter value based on the standard deviation of indicator movements.

Link to other indicators

In addition, the Kaufman Adaptive Moving Average is a type of moving average indicators using the method of exponentially smoothed moving average. It is worth noting that the rate of change indicator (for the period for direction and the sum of single-period for volatility ) is used to calculate the efficiency coefficient .

You can also pay attention to the fact that it was Kaufman who was the first to use estimates based on standard deviations (here to construct the filter), which were subsequently used in one form or another by many analysts, in particular, Bollinger Bands ^[2] .

Notes

↑ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹ ¹⁰ ¹¹ Perry J. Kaufman Smarter Trading: Improving Performance in Changing Markets - McGraw-Hill, Inc., 1995, 257 p. - ISBN 0-07-034002-1
↑ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹ ¹⁰ ¹¹ ¹² Dynamic moving averages. Part 2. // Konstantin Kopyrkin, “Modern Trading”, No. 5–6, 2001. P. 8–12.
↑ Article Adaptive Moving Average Archived on June 4, 2012. on the KROUFR website.
↑ Do Adaptive Moving Averages Lead To Better Results? (Eng.) // Investopedia.com , November 26, 2008.
↑ Erlich A. A. Technical analysis of commodity and financial markets: Applied manual. - 2nd ed. - M .: INFRA-M, 1996 .-- 176 p. ISBN 5-86225-346-7
↑ Jeffrey Owen Katz, Donna L. McCormick. Encyclopedia of Trading Strategies - M. Alpina Publisher, 2002.400 p. - ISBN 5-94599-028-0

Literature

Perry J. Kaufman Smarter Trading: Improving Performance in Changing Markets - McGraw-Hill, Inc. - 1995—257 p. - ISBN 0-07-034002-1 .

[Smarter.Trading-1] ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹ ¹⁰ ¹¹ Perry J. Kaufman Smarter Trading: Improving Performance in Changing Markets - McGraw-Hill, Inc., 1995, 257 p. - ISBN 0-07-034002-1

[konkop.narod.ru.7_74_79-2] ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹ ¹⁰ ¹¹ ¹² Dynamic moving averages. Part 2. // Konstantin Kopyrkin, “Modern Trading”, No. 5–6, 2001. P. 8–12.

[kroufr.ru.1177.124-3] Article Adaptive Moving Average Archived on June 4, 2012. on the KROUFR website.

[investopedia.com.adaptive-moving-averages-4] Do Adaptive Moving Averages Lead To Better Results? (Eng.) // Investopedia.com , November 26, 2008.

[erlih.tech-5] Erlich A. A. Technical analysis of commodity and financial markets: Applied manual. - 2nd ed. - M .: INFRA-M, 1996 .-- 176 p. ISBN 5-86225-346-7

[kats.encyclopedia-6] Jeffrey Owen Katz, Donna L. McCormick. Encyclopedia of Trading Strategies - M. Alpina Publisher, 2002.400 p. - ISBN 5-94599-028-0