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

3. Part-III: Types of sizing function

4. Part-IV: Strategy characteristics for random walk with a trend

In the last post we analyzed characteristics of a generic momentum strategy in case of an underlying following a random walk with a known trend. In this post we look in to the returns characteristics in case when the underlying is an auto-regressive process, specifically AR(1).

Once again, we assume the sizing function $\Psi$ is linear. Asset return $r_t$ is given by $r_t = \phi.r_{t-1}+\sigma.\epsilon$, where $\epsilon\sim N(0,1)$. Remebering that for linear function the expected return of the strategy is expected value of the signal times return, we get

$$E(R_t)=E(S_tr_t)=E(\sum_{s=t-k}^{t-1}(w_sr_s)r_t)=\sum_{s=t-k}^{t-1} w_s E(r_sr_t)=\sum_{s=t-k}^{t-1} w_s \gamma_s=\Sigma^2\sum_{s=t-k}^{t-1}w_s.\phi^s$$

1. Part-I: Time Series vs Cross-sectional momentum

2. Part-II: Nature of linear time series momentum filters

3. Part-III: Types of sizing function

4. Part-IV: Strategy characteristics for random walk with a trend

In the last post we analyzed characteristics of a generic momentum strategy in case of an underlying following a random walk with a known trend. In this post we look in to the returns characteristics in case when the underlying is an auto-regressive process, specifically AR(1).

Once again, we assume the sizing function $\Psi$ is linear. Asset return $r_t$ is given by $r_t = \phi.r_{t-1}+\sigma.\epsilon$, where $\epsilon\sim N(0,1)$. Remebering that for linear function the expected return of the strategy is expected value of the signal times return, we get

$$E(R_t)=E(S_tr_t)=E(\sum_{s=t-k}^{t-1}(w_sr_s)r_t)=\sum_{s=t-k}^{t-1} w_s E(r_sr_t)=\sum_{s=t-k}^{t-1} w_s \gamma_s=\Sigma^2\sum_{s=t-k}^{t-1}w_s.\phi^s$$

Here $\Sigma^2= \frac{\sigma^2}{1-\phi^2}$ is the unconditional variance of the process. This makes the expected return sensitive to both the weighing scheme, i.e. the signal, and also exponentially sensitive to the auto-correlation coefficient $\phi$. Notice how this differs from the previous case. The higher order moments like variance and skew also vary exponentially with $\phi$ as in the figure below (left-hand one).

Everything (expected return, strategy vol, skew) increases with $\phi$, however increase in return is more than vol hence Sharpe improves as $\phi$ increases. While comparing across different types of positioning function (right hand chart above), the change in the Sharpe ratio is not very significant at reasonable values of $\phi$. The major difference comes in terms of higher order, i.e. skew and excess kurtosis. Again, we see that double-step and sigmoid present a competing choice between Sharpe ratio and positive strategy skew, perhaps with a bias to double-step in this particular case.

The influence of the weighing scheme on the strategy performance of course will depend on the underlying process. Here for the AR(1) process, the shorter the MA lookbacks, better the performance, to the extent that for very large MA (like 50/200) the strategy skew even turns negative (not shown here). Similar to the results above for varying $\phi$, the expected return and the strategy skew is more sensitive to the weighing function than strategy volatility.

The optimization objective here again would be to estimate the process parameters, but perhaps hoping for higher accuracy than the case for a known drift random walk. This is not only because the performance sensitivity is an order higher than before, but we also need to optimize the weighing function (i.e. the signal) which depends on the underlying process parameters. In previous case, we would be happy to have confidence on the sign of the drift term, ignoring the accurate estimation of its value. But in this case, with a given risk/reward budget, we need much more accuracy in the estimated value of $\phi$. Nevertheless, as far as positioning is considered, we again see sigmoid and double-steps are good competing alternatives, with a favor for sigmoid for a implementation with linear instruments and double-steps for non-linear instruments.

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