© 2009 Elsevier Limited. All rights reserved.
Doi:10.1016/B978-0-12-374952-9.00009-9.

2010

Portfolio optimization with

“ Threshold Accepting ” :

a practical guide

Manfred Gilli and Enrico Schumann

9

Executive Summary

Recent years have seen a proliferation of new risk and performance measures in

investment management. These measures take into account stylized facts of finan-

cial time series like fat tails or asymmetric return distributions. In practice, these

measures are mostly used for ex post performance evaluation, only rarely for

explicit portfolio optimization. One reason is that, other than in the case of classical

mean – variance portfolio selection, the optimization under these new risk mea sures

is more difficult since the resulting problems are often not convex and can thus

not be solved with standard methods. We describe a simple but effective optimiza-

tion technique called “ Threshold Accepting (TA), ” which is versatile enough to be

applied to different objective functions and constraints, essentially without restric-

tions on their functional form. This technique is capable of optimizing portfolios

under various recently proposed performance or (downside) risk measures, like

value at risk, drawdown, Expected Shortfall, the Sortino ratio, or Omega, while

not requiring any parametric assumptions for the data, i.e., the technique works

directly on the empirical distribution function of portfolio returns. This chapter

gives an introduction to TA and details how to move from a general description of

the algorithm to a practical implementation for portfolio selection problems.

9.1 Introduction

The aim of portfolio selection is to determine combinations of assets like bonds
or stocks that are “ optimal ” with respect to performance scores based, for
instance, on capital gains, volatility, or drawdowns. For any decision rule, if
it is supposed to be practical, a compromise needs to be found between the
model’s financial aspects, and statistical and (particularly) computational con-
siderations. Thus, the question “ what do I want to optimize? ” has to be traded
off against “ what can I estimate? ” and “ what can I compute? ”
The workhorse model for portfolio selection is mean – variance optimiza-
tion ( Markowitz, 1952 ). To a considerable extent, this specification is owed to