In this note we present a study of trend following using two centuries of data. We find that trend following is a persistent market anomaly with highly significant performance over this long back-test. We also present a combined portfolio of a trend following investment and a standard basket of equities and bonds.
Large shocks in an equity portfolio are typically driven by correlated (and hence collective) moves of its constituents. This accords correlation matrices a historically central place in numerous studies on portfolio construction and risk management . In this note, we illustrate how certain statistical methods enable us to identify the main market factors (or “modes”) that an equity market neutral portfolio should hedge, in order to extract value from signals, while avoiding exposure to large, collective market moves. These methods rely on the processing of stock returns correlation matrices. However, because time series are finite, measured correlations are subject to the effects of noise: a fact that one must take into account when employing empirical correlation matrices in portfolio construction. Comparing the properties of empirical correlation matrices to those obtained in random cases, and using results from the theory of random matrices, enables us to distinguish genuine characteristics of the dependence structure of a set of stocks from noisy and unreliable features.
Finding a good compromise between the exploitation of known resources and the exploration of unknown, but potentially more profitable choices, is a general problem, which arises in many different scientific disciplines. We propose a stylized model for these exploration-exploitation situations, including population or economic growth, portfolio optimisation, evolutionary dynamics, or the problem of optimal pinning of vortices or dislocations in disordered materials. We find the exact growth rate of this model for tree-like geometries and prove the existence of an optimal migration rate in this case. Numerical simulations in the one-dimensional case confirm the generic existence of an optimum.