In this talk I discuss the problem of managing a portfolio of financial assets in time, using techniques from convex optimization and model predictive control. First, I describe a framework for single- and multi-period optimization, where the trades are found by solving a convex optimization problem trading off expected return, risk, transaction cost and holding cost such as the borrowing cost for shorting assets. The single-period model, which traces back to Markowitz, only considers the next step in time. The multi-period model considers many steps in the future, using predictions of uncertain quantities, and optimizes over a sequence of trades, with only the first one executed. This traces back to model predictive control. Second, I consider the classic Kelly gambling problem, as an alternative model for portfolio management. I show how to solve the problem, with general distribution of outcomes, using convex optimization. I then introduce a risk constraint that limits the probability of a drawdown of wealth to a given undesirable level. I develop a bound on the drawdown probability, which yields a convex optimization problem that guarantees the drawdown risk constraint holds. In numerical experiments I find that the bound is close to the actual drawdown risk. Finally, I show that a natural quadratic approximation of this convex problem is closely connected to the Markowitz problem. [1][2]
[1] Multi-Period Trading via Convex Optimization, S. Boyd, E. Busseti, S. Diamond, R. Kahn, K. Koh, P. Nystrup, J. Speth, Foundations & Trends in Optimization, 2017
[2] Risk-Constrained Kelly Gambling, E. Busseti, E. Ryu, S. Boyd, Journal of Investing, 2016
