## Friday, November 6, 2015

### Time Series Momentum Strategies | The Spirits Within - Part III

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.