Optimizing Optimization: The Next Generation of Optimization Applications and Theory (Quantitative Finance)

Modeling, estimation, and optimization of equity portfolios with heavy-tailed distributions 131

among n assets in the reward – risk plane consists of minimizing a given risk
measure ρ provided that the expected reward ν is constrained by some minimal
value m ; i.e.,

min

..

;;

x b

bi

i

n

i

xr r

st

vxr r m x x

ρ()

() ,



   



01

1

∑

(5.8)

where r b denotes the return of a given benchmark, and xr xr
i

n

∑ 1 ii stands for

the returns of a portfolio with composition x  ( x 1 , ... , x n ). The portfolio that
provides the maximum expected reward ν per unit of risk ρ is called the market
portfolio and is obtained from problem (5.8) for one value m among all admis-
sible portfolios. In particular, when the reward and risk are both positive
measures, the market portfolio is obtained as the solution of the optimization
problem:

max

x

b

b

i i

n

i

vxr r

xr r

st

xx

()

()

..

,









ρ

01

1

∑

(5.9)

Clearly , there exist many possible performance ratios G ( X )  v ( X )/ ρ ( X ).
A first classification with respect to the different characteristics of reward
and risk measures is given in Rachev, Ortobelli, Stoyanov, Fabozzi, and
Biglova (2008). The most important characteristic is the isotony (consistency)
with an order of preference; i.e., if X is preferable to Y , then G ( X )  G ( Y )
( G ( X )  G ( Y )) Although the financial literature on investor behavior agrees
that investors are nonsatiable, there is not a common vision about the inves-
tors ’ aversion to risk. Thus, investors ’ choices should be isotonic with nonsa-
tiable investors ’ preferences (i.e., if X  Y , then G ( X )  G ( Y )).
Several behavioral finance studies suggest that most investors are neither
risk averse nor risk loving. 19 Thus , according to Bauerle and M ü ller (2006), if
risk and reward measures are invariant in law (i.e., if X and Y have the same
distribution, then ρ ( X )  ρ ( Y ) and v ( X )  v ( Y )), and the risk measure is posi-
tive and convex (concave) and the reward is positive and concave (convex),
then the performance ratio is isotone with risk-averse (lover) preferences.

19 See Friedman and Savage (1948) , Markowitz (1952) , Tversky and Kahneman (1992) , Levy and
Levy (2002) , and Ortobelli et al. (2009).