Portfolio Optimization via Risk-Sensitive Control

This research concerns investment: how should an investor construct and manage a portfolio or fund of risky assets so as to maximize returns? Mathematical theories of portfolio optimization have been the object of intensive study, both in industry and academia, for at least the last 50 years. The initial breakthrough was made by Harry Markowitz with the introduction of 'mean-variance analysis', quantifying precisely the relationship between risk and return. Markowitz' ideas pervade the finance industry and are familiar to everyone concerned with portfolio management. Nonetheless, the mean-variance theory suffers from certain well-known drawbacks. Firstly, the optimization requires as inputs the mean vector and covariance matrix of asset returns. While it may be possible to estimate the latter, it is impossible to estimate the former from historical data with any usable degree of accuracy. Thus a further modelling step is required to relate the mean returns to other market or economic factors. Secondly, this is a one-period theory in which, conceptually, investors buy a portfolio, hold it for some period and then cash out. What is needed is a dynamic theory in which the investors' optimal response to the movement of market prices over time is determined.Systematic study of a dynamic theory of portfolio optimization was initiated in 1969 in ground-breaking work by Robert Merton who formulated it as a stochastic control problem where the objective was to maximize the lifetime utility of consumption. This led to wide-ranging developments in the mathematics of utility maximization and stochastic control, but little of this has found its way into the practical world of asset management, mainly perhaps because it is too dependent on a stylized mathematical model. A different but related approach is to maximize the long-run growth rate (this is sometimes known as the 'Kelly criterion'). The theory for this is less model-dependent, but it leads to strategies that can be very risky over realistic investment horizons.In this research we develop an approach which, in our view, combines the virtues of Markowitz, Merton and Kelly and addresses the deficiencies of each.The approach based on the theory of risk-sensitive control (RSC). This subject was introduced in the control systems literature by Jacobson in the 1970s. Its application to asset management was pioneered by Bielecki and Pliska in the 1990s. In traditional stochastic control one seeks to maximize the expected value of some performance index. In RSC this is replaced by maximizing the exponential of a performance index. In the asset management application this index is the growth rate, and it turns out that the objective is equivalent to maximizing the expected growth rate with a penalty for variance (i.e. risk). This research is a joint project with New Bond Street Asset Management (NBSAM), whose business is managing portfolios involving credit risk, so it is essential that our mathematical models include credit events, i.e. jumps in asset prices. At present RSC theory only does this in limited ways, so new mathematics is required.There are four components to the work, (a) developing the theory of RSC, (b) producing demonstrably robust algorithms for solving the equations of RSC, (c) studying the mathematical modelling of the various classes of credit-risky securities traded by NBSAM and (d) undertaking an econometric analysis to establish the dependence of prices on economic factors. Part (a) involves extending the theory of so-called 'viscosity solutions' of nonlinear partial differential equations to cover integro-differential operators and solutions in unbounded domains with quite weak growth conditions, while part (b) will cover finite-difference schemes which are related to control problems for Markov chains.When completed, this work will give portfolio managers a fully dynamic asset allocation model that represents a huge improvement over current techniques.