© 2009 Elsevier Limited. All rights reserved.

Doi:10.1016/B978-0-12-374952-9.00008-7.

Computing optimal mean/downside

risk frontiers: the role of ellipticity

Tony Hall and Stephen E. Satchell

8

Executive Summary

The purpose of this chapter is to analyze and calculate optimal mean/downside

risk frontiers for financial portfolios. Focusing on the two important cases of

mean/value at risk and mean/semivariance, we compute analytic expressions for

the optimal frontier in the two asset case, where the returns follow an arbitrary

(nonnormal) distribution.

Our analysis highlights the role of the normality/ellipticity assumption in this

area of research. Formulae for mean/variance, mean/expected loss, and mean/semi –

standard deviation frontiers are presented under normality/ellipticity. Computa-

tional issues are discussed and two propositions that facilitate computation are

provided.

Finally , the methodology is extended to nonelliptical distributions where

simulation procedures are introduced. These can be presented jointly with our

analytical approach to give portfolio managers deeper insights into the properties

of optimal portfolios.

2010

(^1) We thank John Knight and Peter Buchen for many helpful comments, particularly on Proposition 4.1.
2 Correspondence to [email protected]

8.1 Introduction

There continues to be considerable interest in the trade-off between portfolio

return and portfolio downside risk. This has arisen because of inadequacies of

the mean – variance framework and regulatory requirements to calculate value

at risk and related measures by banks and other financial institutions.

Much of the literature focuses on the case where asset and portfolio returns

are assumed normally or elliptically distributed, see Alexander and Bapista

(2001) , Campbell, Huisman, and Koedijk (2002) , Kaynar, Ilker Birbil, and

Frenk (2007) , and Kring, Rachev, H ö chst ö tter, Fabozzi, and Bianchi (2009).

In most cases, the problem of a mean/value at risk frontier differs only slightly

from a classic mean – variance analysis. However, the nature of the optimal

mean – risk frontiers under normality is not discussed in any detail and we