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
In this post we look in to the returns characteristics of a generic time series momentum (TSMOM) strategy. We have the expressions for the returns and moments for the previous post. We here consider two cases of the behaviors of the underlying asset - one where the asset behave like a Gaussian random walk, and in the second where the asset returns are autogressive (of the order 1).
Gaussian Random Walk: Let's assume our sizing function $\Psi$ is linear and the underlying asset is a random walk. That is asset return $r_t$ is given by $r_t=\mu + \sigma.\epsilon$, where $\epsilon\sim N(0,1)$ is Gaussian noise. In this case we can find the expected return from a TSMOM strategy as below
$$E(R_t)=E(S_tr_t)=E(\sum_{s=t-k}^{t-1}(w_sr_s)r_t)=\sum_{s=t-k}^{t-1} w_s E(r_sr_t)=\sum_{s=t-k}^{t-1} w_s (\gamma_s+\mu^2)=\mu^2\sum_{s=t-k}^{t-1}w_s=\mu^2$$
Here $\gamma_s$ is the autocovariance of underlying returns at a lag $s$. We obtain the results using the facts that $\gamma_s=0$ for $s\neq0$ in our particular case, and also that $\sum_{s=t-k}^{t-1}w_s=1$ by design. This is a case of strict TSMOM strategy in the sense all $w$ are positive. The result is intuitive. The position size is proportional to the expected return $\mu$ and so is the return on this size, hence the square of $\mu$ term. Note, this result does not depend on the exact type of signals, as long as the weights are positive and adds up to one. Similarly we can show the volatility (square root of variance) of this strategy is proportional to $\mu\sigma$. Figure below shows simulated results for different parameters
As we can observe, the expected returns and strategy volatility is as discussed above. The skew of the strategy is positive and increases with decreasing $\mu$ (till a certain threshold) and increasing $\sigma$. Excess kurtosis increases with decreasing $\mu$. The signal function $S$ is a 10 vs 50 period simple moving average cross-over signal. Since for this special case of random walk, all the individual terms under the summation evaluates to the same expression for all terms (this is true for all moments), the underlying signal function parameters (i.e. simple vs. exponential or 5/10 period vs 50/250 period) do not influence the performance.
However, the positioning function $\Psi$ will influence the performance. The above results are for a linear function $\Psi=S$. Below is the performance comparison for different types of $\Psi$.
As we can see, there are variations in statistical characteristics across different choices of $\Psi$. The sigmoid function behaves similar to the linear function we have already seen. This is expected, for example, sigmoid can be made to resemble a linear function (with position cut-off) with appropriate choice of parameters. In general for the random walk case, binary function will show similar expected returns and variance as the underlying itself and little skew or excess kurtosis, Compared to both, linear will have higher skew due to higher potential position on the extreme. Sigmoid usually will show a reduced expected return (but maintaining the Sharpe Ratio more or less).
However, the double-step shows markedly lower Sharpe and higher skew (in spite of the position limit). It has a lower vol but an even lower expected return makes the Sharpe lower overall (compared to the benchmark linear case). The higher skew comes from the sharp increase in position at a relatively lower threshold of signal (compared to, again, a linear function). Also higher the threshold $\epsilon$, higher is the skew.
So in the case of random walk with deterministic drift, the optimization problem is rather trivial. The underlying signal function does not affect the strategy performance much. That includes the type of the signal and the parameter space of the signal function. The choice then reduces to finding appropriate positioning function $\Psi$. Usually the linear is NOT preferred because of potentially very large exposure. Sigmoid is a good choice for position limiting with a higher Sharpe. On the other hand double-step is a good choice for a high skew strategy. Depending on the trading style (confidence in underlying process estimates, along with risk management), instruments (linear or convex) and trading horizon (we will come back to trading horizon later in details), a combination of sigmoid and double-step can deliver the desired mix of Sharpe and positive skew.
$$E(R_t)=E(S_tr_t)=E(\sum_{s=t-k}^{t-1}(w_sr_s)r_t)=\sum_{s=t-k}^{t-1} w_s E(r_sr_t)=\sum_{s=t-k}^{t-1} w_s (\gamma_s+\mu^2)=\mu^2\sum_{s=t-k}^{t-1}w_s=\mu^2$$
Here $\gamma_s$ is the autocovariance of underlying returns at a lag $s$. We obtain the results using the facts that $\gamma_s=0$ for $s\neq0$ in our particular case, and also that $\sum_{s=t-k}^{t-1}w_s=1$ by design. This is a case of strict TSMOM strategy in the sense all $w$ are positive. The result is intuitive. The position size is proportional to the expected return $\mu$ and so is the return on this size, hence the square of $\mu$ term. Note, this result does not depend on the exact type of signals, as long as the weights are positive and adds up to one. Similarly we can show the volatility (square root of variance) of this strategy is proportional to $\mu\sigma$. Figure below shows simulated results for different parameters
As we can observe, the expected returns and strategy volatility is as discussed above. The skew of the strategy is positive and increases with decreasing $\mu$ (till a certain threshold) and increasing $\sigma$. Excess kurtosis increases with decreasing $\mu$. The signal function $S$ is a 10 vs 50 period simple moving average cross-over signal. Since for this special case of random walk, all the individual terms under the summation evaluates to the same expression for all terms (this is true for all moments), the underlying signal function parameters (i.e. simple vs. exponential or 5/10 period vs 50/250 period) do not influence the performance.
However, the positioning function $\Psi$ will influence the performance. The above results are for a linear function $\Psi=S$. Below is the performance comparison for different types of $\Psi$.
As we can see, there are variations in statistical characteristics across different choices of $\Psi$. The sigmoid function behaves similar to the linear function we have already seen. This is expected, for example, sigmoid can be made to resemble a linear function (with position cut-off) with appropriate choice of parameters. In general for the random walk case, binary function will show similar expected returns and variance as the underlying itself and little skew or excess kurtosis, Compared to both, linear will have higher skew due to higher potential position on the extreme. Sigmoid usually will show a reduced expected return (but maintaining the Sharpe Ratio more or less).
However, the double-step shows markedly lower Sharpe and higher skew (in spite of the position limit). It has a lower vol but an even lower expected return makes the Sharpe lower overall (compared to the benchmark linear case). The higher skew comes from the sharp increase in position at a relatively lower threshold of signal (compared to, again, a linear function). Also higher the threshold $\epsilon$, higher is the skew.
So in the case of random walk with deterministic drift, the optimization problem is rather trivial. The underlying signal function does not affect the strategy performance much. That includes the type of the signal and the parameter space of the signal function. The choice then reduces to finding appropriate positioning function $\Psi$. Usually the linear is NOT preferred because of potentially very large exposure. Sigmoid is a good choice for position limiting with a higher Sharpe. On the other hand double-step is a good choice for a high skew strategy. Depending on the trading style (confidence in underlying process estimates, along with risk management), instruments (linear or convex) and trading horizon (we will come back to trading horizon later in details), a combination of sigmoid and double-step can deliver the desired mix of Sharpe and positive skew.
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