This is first part of a small series on some observations on general time series momentum strategies. In this I lay threadbare generic time series momentum strategies with the objective of establishing general theoretical underpinning on the strategy performance and optimization approach.

The class of "time series momentum" strategies are very common and popular among investors. At its simplest, it means buy high and sell low. Or more precisely, buy high (an asset that has recently appreciated) to sell at a even higher price. Similarly sell low (an asset that has recently sold off) to buy back at a even lower price. From this point of view, this is fundamentally different than the value investing paradigm. In other words, any returns generated from these class of strategies should be independent and hence should represent an independent risk factor to the investors.

Most of the theoretical interest in momentum investing started with the classic paper by Jegadeesh and Titman in 1993, where they found strong evidence of momentum profit. Further research followed these interesting observation since then. However, what is described as momentum in this case is usually termed as cross-sectional momentum (XSMOM). This is fundamentally different than what is usually understood to be time series momentum (TSMOM).

An XSMOM strategy looks at the relative performance of a basket of assets, and invests in the winners and shorts the losers. The buy/sell signal is generated from relative performance of different assets (cross-section) within a given time interval. On the other hand, a TSMOM strategy looks at past performance of an asset, and buys if has been a winner, or sell otherwise. A TSMOM strategy may or may not involve a basket of assets, it does not depend on a basket crucially for the strategy implementation (unlike the XSMOM which is meaningless without the context of a basket). When a basket is used for a TSMOM it is for diversification and risk management.

This fundamental difference in construction shows how the performances of these two strategies can be similar and different. For sake of comparison, let's assume in both cases we have a basket of two assets. As it is evident if we indeed have strong correlated changes in asset prices (past performance predicts future), then both strategies will perform, as we will be buying and selling the right assets by construction. However, even if we have a change in the momentum, if there is an increase in dispersion of the asset performance (e.g. winner becomes losers, but losers become even more so - not a complete reversal, as in winners become losers and losers become winners) then the XSMOM will still perform. Similarly, even if we do not have strong auto-correlation, but persistent trends the TSMOM will perform better.

It seemed to me the first major theoretical insights in to TSMOM strategy was presented by Moskowitz, Ooi and Pedersen in 2012. The paper also captures the essence of the above paragraph, by breaking down the sources of profits in XSMOM and TSMOM strategies, as follows (in terms of one period expected returns):

$$E[r_{t,t+1}^{XSMOM}] = \frac{N-1}{N^2}tr(\Omega) - \frac{1}{N^2}[l^T\Omega l - tr(\Omega)] + 12\sigma_{\mu}^{2}$$

$$E[r_{t,t+1}^{TSMOM}] = \frac{tr(\Omega)}{N} + 12\frac{\mu^T\mu}{N}$$

$$E[r_{t,t+1}^{TSMOM}] = \frac{tr(\Omega)}{N} + 12\frac{\mu^T\mu}{N}$$

Here $\Omega$ is the covariance matrix, $\mu$ is the mean returns vector, $\sigma_{\mu}^{2}$ is the cross-sectional variance of the means, $N$ is the number of assets in the basket and $tr()$ is sum of the diagonals. The above expression clearly shows the points made in the previous paragraphs. The TSMOM returns is driven by auto-covariance and strength of the drift terms. Where as XSMOM, in addition to auto-covariance, also depends on cross-variance (dispersion) and cross-variance of the mean returns (dispersion, again), but not particularly on the strength of the mean returns. These results are valid for linearly weighted basket (linear in returns), but in general give good guidance.

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