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We consider different formulas for Wilder's Relative Strength Index (RSI) and the relation of the RSI to the Chande Momentum Oscillator (CMO) and to the Kaufman Efficiency Ratio (KER). The implications in writing PCFs for RSI are discussed.
Define:
Up = MA((C0>C1)*(C1-C0))
and
Dn = MA((C0<C1)*(C0-C1))
where MA is any linear moving average. Notice that by linearity:
Up-Dn = MA((C0-C1))
= MA(C0) - MA(C1)
and
Up+Dn = MA(ABS(C0-C1)) Consider the following three equivalent formulas for RSI:
1:
RSI = 100*{1 - 1/(1+Up/Dn)}
2:
RSI = 100*{Up/(Up+Dn)}
3:
RSI = (100/2)*{(Up-Dn)/(Up+Dn) + 1}
Formula 1 is called the Wilder Relative Strength Index (RSI).
Formula 2 is also called the Chande Momentum Oscillator (CMO).
In Formula 3, the portion 100*(Up-Dn)/(Up+Dn) is called the Kaufman Efficiency Ratio (KER), particularly when MA is a Simple Moving Average (SMA).
When writing a PCF for RSI:
Formula 1: Requires two series expansions, one for Up and one for Dn.
Formula 2: Requires two series expansions, one for Up and one for Up+Dn.
Formula 3: Requires two series expansions, one for Up-Dn and one for Up+Dn. In special cases, no series expansion is required for Up-Dn. For example, this is true for an Exponential Moving Average (EMA) or an SMA.
Consider the KER when the MA is an SMA of Period P. Then:
KER = 100*{(SMA(C0,P)-SMA(C1,P))/SMA(ABS(C0-C1),P)}
= 100*{(C0-CP)/(P*SMA(ABS(C0-C1),P))}
The (C0-CP) is considered the net distance traveled to get from CP to C0. The P*SMA(ABS(C0-C1),P) is the total distance traveled, up and down, in going from CP to C0 via of the daily closes. Thus, the name "Efficiency Ratio".
It seems to me that the easiest way to write the PCF and to describe RSI is in terms of the Efficiency Ratio.
Any comments will be appreciated.
Thanks,
Jim Murphy
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