This is part of a series on time series momentum. Previous post on this are:
1. Part-I: Time Series vs Cross-sectional momentum
2. Part-II: Nature of linear time series momentum filters
The second phase of designing a momentum strategy is designing the positioning function \Psi. This is the function that converts the signal in to a positions. The common choices are:
1. Sign/ Binary function (i.e. maximum long position allowed if positive signal, maximum short otherwise): The simplest of the lot. Sharp change in positioning near 0 level of signal (ambiguous zone) which can lead to increasing turnover and related costs. \Psi = Sign(S)
2. Linear (including constant): Simple, but no limit on maximum position. \Psi =c S, c is a constant scaling factor.
3. Step function: Sudden change in direction (although no longer around the ambiguous zone). \Psi=+1|S>\epsilon, -1|S<-\epsilon. Here \epsilon is the threshold.
4. Sigmoid function (error function) or tangent hyperbolic filtering: Smooth combination of linear and binary, moving gradually from one to another depending on parameters. \Psi=erf(S), or \Psi=\frac{-e^{-S} + e^S}{e^{-S} + e^S}
5. Reverse sigmoid function: Sigmoid with peak sizing in long or short zone. \Psi=e^{1/2}S.e^{-S^2/2}
Given this set up, now we are ready to look in to the performance of such a strategy. By definition, the one-period return of the strategy is R_t=\Psi(S_t).r_{t}. The expression for one-period mean is as below.
\mu = E\left(\sum{\Psi(S_t).r_t}\right)
The k-th moment is given by
M(k) = E\left(\sum{(\Psi(S_t).r_t)^k}\right)
As we have seen already, S can be (in case of linear filters) expressed as S_t=\sum(w.r_t). Also for linear \Psi we can take the coefficient outside the summation notation.
1. Part-I: Time Series vs Cross-sectional momentum
2. Part-II: Nature of linear time series momentum filters
The second phase of designing a momentum strategy is designing the positioning function \Psi. This is the function that converts the signal in to a positions. The common choices are:
1. Sign/ Binary function (i.e. maximum long position allowed if positive signal, maximum short otherwise): The simplest of the lot. Sharp change in positioning near 0 level of signal (ambiguous zone) which can lead to increasing turnover and related costs. \Psi = Sign(S)
2. Linear (including constant): Simple, but no limit on maximum position. \Psi =c S, c is a constant scaling factor.
3. Step function: Sudden change in direction (although no longer around the ambiguous zone). \Psi=+1|S>\epsilon, -1|S<-\epsilon. Here \epsilon is the threshold.
4. Sigmoid function (error function) or tangent hyperbolic filtering: Smooth combination of linear and binary, moving gradually from one to another depending on parameters. \Psi=erf(S), or \Psi=\frac{-e^{-S} + e^S}{e^{-S} + e^S}
5. Reverse sigmoid function: Sigmoid with peak sizing in long or short zone. \Psi=e^{1/2}S.e^{-S^2/2}
Given this set up, now we are ready to look in to the performance of such a strategy. By definition, the one-period return of the strategy is R_t=\Psi(S_t).r_{t}. The expression for one-period mean is as below.
\mu = E\left(\sum{\Psi(S_t).r_t}\right)
The k-th moment is given by
M(k) = E\left(\sum{(\Psi(S_t).r_t)^k}\right)
As we have seen already, S can be (in case of linear filters) expressed as S_t=\sum(w.r_t). Also for linear \Psi we can take the coefficient outside the summation notation.
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