© Elsevier Limited. All rights reserved.
Doi:10.1016/B978-0-12-374952-9.00010-5.

2010

Some properties of averaging

simulated optimization methods

John Knight and Stephen E. Satchell

10

Executive Summary

We consider problems in optimization that are addressed through the use of

Monte Carlo simulated averages. To do this, we revisit the problem of calculat-

ing the exact distribution of optimal investments in a mean – variance world under

multivariate normality. These results help us understand when Monte Carlo – based

averaging methods will work well, and when they will fail. The results derived

allow some exact and numerical analysis, which we use to illustrate the magnitude

of biases that could occur in practical situations. For stock selection optimization,

these can be very large.

10.1 Section 1

The major motivation for this research has been to try and understand the
magnitude of estimation error; i.e., the extent to which the outcome of the
portfolio decision is influenced by parameter uncertainty. The last decade has
seen an overlap in academic and practitioner research in this area. The practi-
tioner research has been exemplified by the work of Michaud. In response to
parameter uncertainty, Michaud (1998) has proposed an averaging simulated
optimization procedure, the outcome of which can only be really understood
by an analysis of the exact properties of optimal portfolios. His innovation was
followed by other commercial products based on similar methods. Michaud’s
procedure purports to solve some of the problems of portfolio optimization
that arise from estimation error. Other authors have criticized Michaud’s
approach, see Scherer (2002) and Harvey, Leichty, Leichty, and Muller (2004).
We investigate the merits of the averaging simulated Mote Carlo approach
using exact distribution theory.
The application of exact distribution theory to mean – variance (MV) analysis
has been undertaken by a number of authors, see Jobson and Korkie (1989) ,
Jobson (1991) , Britten-Jones (1999) , Stein (2002) , Hillier and Satchell (2003) ,
Okhrin and Schmid (2006) , Bodnar and Schmid (2006) , and Frahm (2007). The
usual assumptions are that returns are i.i.d. multivariate normal and that there
may or may not be a riskless asset. However, in all listed cases, the analysis is in