A quantitative investor allocates capital across many assets and over time, from forecasts that are weak, noisy and numerous. Risk is imperfectly known, trading costs money, and the market reacts to what is traded. The course separates this problem into the scientific questions it contains and treats each with the theory available for it and with real market data.
Three questions structure the course. The first is statistical. It asks what can be inferred from returns that are dependent in time and correlated across assets, when the number of assets and of candidate predictors is comparable to or larger than the number of observations: inference under dependence, regression in high dimension, estimation of large covariance matrices, and the selection of a few signals among many. The second concerns decision under constraints: how forecasts, risk, costs and limits enter a portfolio programme, and what its optimality conditions say about the value of each constraint. The third concerns optimal control. Holdings evolve, moving them has a cost, and the course examines the conditions under which a policy that trades towards a target over time improves on a sequence of single-period decisions, which requires a horizon, assumed dynamics, and a programme that can be solved at scale. A final session takes up the combination of forecasts and the evidence itself: how a strategy selected among many is validated, and how its results are defended.
