Optimal solutions for optimization in practice 67
to a target) is part of the prospect theory framework. The theory also includes
that an investor is actually concerned about changes in wealth (and not about
the absolute level of wealth) and that he or she experiences a higher sensitiv-
ity to losses than to gains as described in Cuthbertson and Nitzsche (2004).
The latter is expressed by a factor lambda ( 1) that addresses the loss aversion
within a prospect utility function. For such an investor, who faces asymmetric
distributions and defines his or her utility through a functional combination of
expected gains and losses, we have developed a suitable (Gain/Loss) optimizer.
3.5.2 Omega and GLO
Omega ( Ω ), defined as expected gains divided by expected losses, seems to be a
natural ingredient of a prospect utility function and became very popular since
its introduction in Cascon, Keating, and Shadwick (2002) and Keating and
Shadwick (2002a,b). It also “ takes the whole information set including higher
moments into account ” as in Passow (2005) and, hence, does not base on any
assumptions in this regard (as normality). When it comes to optimization, how-
ever, it appears more sensible to formulate a general utility function as E [ gains ]
(1 l ) E [ losses ], where l loss aversion and “ gains ” are defined as ( ω R )| ω R
and “ losses ” as ( R ω )| ω R , where ω and R denote (final) wealth and tar-
get rate, respectively. The derivation of this formulation compares to the
Sharpe ratio (excess return divided by standard deviation) and its derived utility
function μ
λ
σ
2
(^2).
It is noted that the formulation as “ expected gains minus loss aversion times
expected losses ” contrasts with a ratio such as Ω. In fact, it is currently fash-
ionable to treat Ω as a risk metric, which can be computed against the returns
of the chosen portfolio. However, there is much that is wrong about this proce-
dure. To start with, Ω does not seem to be a globally concave function of port-
folio weights. This means that we will constantly rely on constraints to find
the optimum; indeed generically there should be many optima. Furthermore,
it seems that Ω is a return metric rather than a risk metric. That is to say, if we
could compute our expected utility as a function of Ω , we would expect the
partial derivative of expected utility with respect to Ω to be positive.
A further reference capturing the advantages of expected shortfall (ES) or
CVaR optimization (which is mathematically equivalent to GLO) is Ho, Cadle,
and Theobald (2008) whose conclusions in comparing mean – variance, mean –
VaR, and mean – ES portfolio selections are cited below.
“ We find that the mean – ES framework gives a reasonable and intuitive
solution to the asset allocation problem for the investor. By focusing upon
asset allocations during the recent “ credit crunch ” period, we are able to
develop insights into portfolio selection and downside risk reductions during
high volatility periods.