On Long Term Investment Optimality

摘要

We study the problem of optimal long term investment with a view to beat a benchmark for a diffusion model of asset prices. Two kinds of objectives are considered. One criterion concerns the probability of outperforming the benchmark and seeks either to minimise the decay rate of the probability that a portfolio exceeds the benchmark or to maximise the decay rate that the portfolio falls short. The other criterion concerns the growth rate of an expected risk-sensitised utility of wealth which has to be either minimised, for a risk-averse investor, or maximised, for a risk-seeking investor. It is assumed that the mean returns and volatilities of the securities are affected by an economic factor, possibly, in a nonlinear fashion. The economic factor and the benchmark are modelled with general Ito differential equations. The results identify asymptotically optimal portfolios and produce the decay, or growth, rates. The proportions of wealth invested in the individual securities are time-homogeneous functions of the economic factor. Furthermore, a uniform treatment is given to the out-and under-performance probability optimisation as well as to the risk-averse and risk-seeking portfolio optimisation. It is shown that there exists a portfolio that optimises the decay rates of both the outperformance probability and the underperformance probability. While earlier research on the subject has relied, for the most part, on the techniques of stochastic optimal control and dynamic programming, in this contribution the quantities of interest are studied directly by employing the methods of the large deviation theory. The key to the analysis is to recognise the setup in question as a case of coupled diffusions with time scale separation, with the economic factor representing the fast motion.