Mathematical Finance Seminar
October 5, 2000 , 5:30 PM to 7:00 PM
Michael Stutzer, University of Iowa
A Large Deviations
Approach to Portfolio Analysis
Fund managers may sensibly be averse to realizing a time averaged
portfolio return that is less than the average return of some
trustee-designated benchmark portfolio or return target.
When a portfolio is expected to earn a higher average return than the
benchmark, the probability that this won't happen approaches zero
asymptotically, at a decay rate computable via the
Cramer (in many IID return processes) and the Gartner-Ellis (in many
stationary ergodic return processes) Large Deviation Theorems. The size
of the decay rate is thus proposed as a
new index of portfolio performance.
When returns are arithmetically averaged, it is shown that the decay
rate maximizing portfolio is the one that maximizes the expected
exponential utility of the portfolio's excess return over
the benchmark's return. When log returns are arithmetically averaged
(i.e. the analysis is done on growth rates), it is shown that the decay
rate maximizing portfolio is the one that
maximizes the expected power utility of the portfolio's excess return.
In both cases, the expected utility maximization must be conducted over
the space of portfolios AND the utility
function's coefficient of risk aversion. The latter effect is absent in
standard portfolio theory, and the resulting endogeneity of the
coefficient of risk aversion invalidates the usual econometric
estimates of it.
I compute several parametric examples that show the usefulness of the
theory, and demonstrate that it is easily nonparametrically implemented
on a computer spreadsheet, using time
series of past asset returns.