74 Optimizing Optimization
functions were first explored by Harsanyi (1955) who argued that if both social
welfare and individual utility functions take the von Neumann – Morgenstern
form, then any social welfare function that satisfies the Pareto principle must be
a weighted sum of individual utility functions. What is more the notion of an
‘individual’ can be made quite general depending on the context; for instance,
we can endow a pension fund with one utility function reflecting short run
investment and another reflecting long run investment.
We discuss the literature issues in the next section, followed by a discussion
of the model, and concluded by a discussion of the incorporation of alpha and
risk information.
3.6.2 Discussion
Von Neumann and Morgenstern (1944, hereafter vNM) define utility as a
function U ( W ) over an investor’s wealth W. To make use of this decision theo-
retic foundation in portfolio construction, Markowitz approximated the vNM
expected utility theory by his well-known MV utility function from Levy and
Markowitz (1979). This is justified for normal return distributions or quadratic
utility functions. However, as demonstrated by Mandelbrot (1963) , the assump-
tion of normal return distributions does not hold for many assets. Also, many
investors would not describe their perception of risk through variance. They
relate risk rather to “ bad outcomes ” than to a symmetrical dispersion around
a mean. Sharpe (1964) shows that even Markowitz suggested a preference of
semivariance over variance due to computational constraints at the time.
One attempt, Jorion (2006) , to describe risk more suitably was the use of
VaR. While focusing on the downside of a return distribution and considering
nonnormal distributions, it comes with other shortcomings. VaR pays no atten-
tion to the loss magnitude beyond its value, implying indifference of an inves-
tor with regard to losing the VaR or a possibly considerably bigger loss, and it
is complicated to optimize VaR. Using VaR for nonnormal distributions can
cause the combination of two assets to have a higher VaR value than the sum
of both assets ’ VaR, i.e., VaR lacks subadditivity. Subadditivity is one of four
properties for measures of risk that Artzner, Delbaen, Eber, and Heath (1999)
classify as desirable. Risk measures satisfying all four are then called coherent.
Contrary to VaR, lower partial moments are coherent. Fishburn (1977) intro-
duced these and used them as the risk component in his utility function. Here, an
investor would formulate his or her utility relative to target wealth, calling for sign
dependence. Final wealth above the target wealth has a linearly increasing impact
on utility, while outcomes below the target wealth decrease utility exponentially.
Closely related is the expected utility function Kahneman and Tversky (1979)
proposed under another descriptive model of choice under uncertainty, as an
alternative to vNM. They provide strong experimental evidence for the phenom-
enon that an investor’s utility is rather driven by the impact of expected gains
and losses than by expected return and variance. Considering expected gains and
losses (i.e., expected returns relative to a target) is part of their prospect theory