Optimal solutions for optimization in practice 73
chooses another than the well-known quadratic utility function) he or she has
to take care of higher moments. Such investors will find our GLO beneficial.
3.6 Combined optimizations
3.6.1 Introduction
Mean – variance (MV) analysis has delivered less interesting portfolios for some
time. Mainly the sensitivity of optimal portfolio weights to estimated inputs
inspired investors to look for “ more robust ” portfolios. Closest to MV, obvi-
ously, as one specific MV-efficient portfolio is the minimum variance portfolio.
Being the only MV portfolio that does not incorporate return expectations, it
delivers the desired robustness. However, it still suffers from the second draw-
back of MV optimization, namely overly concentrated, i.e., not at all diversi-
fied, optimal portfolios. Benartzi and Thaler (2001) and Windcliff and Boyle
(2004) show that a heuristic approach that seems to naturally solve this
diversification weakness is the application of equal weights to the invest-
ment universe, i.e., wn1/ where n is the total number of assets. This also
provides robustness as it does not build on estimated inputs and has been
shown by DeMiguel, Garlappi, and Uppal (2007) to be efficient under cer-
tain circumstances. While achieving diversification with regard to weights,
this approach could come with concentration of risks. Hence, portfolios of
equal risk contributions of underlying assets were a natural consequence as
shown in Maillard, Roncalli, and Teiletche (2009). This, obviously, is also at
best a compromise and not based on optimization processes. A second rea-
son for the consideration of alternatives to MV is the recognition that MV
utility is inadequate in many situations, while alternatives, such as GLO,
seem to come as a remedy. We detail the argument in the discussion below.
These considerations lead us to propose a new utility function that is an
“ amalgam ” (linear combination of MV and GLO). As argued by Brandt (2009),
choosing the appropriate objective function is arguably the most important part
of the entire portfolio optimization problem. Although many different objective
functions have been suggested by practitioners and academics alike, there is little
consensus on the appropriate functional form. Consequently, we can choose a
finite number of “potential” objective functions and assign larger weights to the
objective functions that we believe are most representative of the latent investor
preferences. Indeed, we could even go further and nest this idea within a Bayesian
framework whereby we formally model these weights as prior probabilities using
probability distributions defined upon the unit simplex. Interpreted from this
perspective, our utility function parallels estimation under model uncertainty, i.e.
the portfolio builder is not clear which is the true objective function but has some
intuition as to which objective functions are more reasonable than others.
Alternatively, it may also be possible to derive such a utility function from
aggregating the utility functions of individual investors. These “aggregate” utility