16-Port Selection Mean Var Page 490 Wednesday, February 4, 2004 1:09 PM
490 The Mathematics of Financial Modeling and Investment Management+∞U= EUx[ ()]= ∫px()Ux()dx
- ∞
From this definition, it is clear why concavity represents risk aver-
sion. To see this point, it is useful to imagine a discrete world where
only a discrete set of states is possible. In a discrete setting, utility is
defined as a discrete, finite or infinite sum:U= EUx[ ()]= ∑px()iUx()i
To each state corresponds a discrete finite probability. A risk-neutral
investor does not require any compensation for risk-taking: the investor is
indifferent to choices where the increment in the variable is inversely pro-
portional to the decrease in probability. For instance, a risk-neutral investor
will be indifferent to choices where the halving of probability is compen-
sated with the doubling of consumption. However, a risk-averse investor
will require more than a simple proportionality: a halving of probability
must be compensated with more than a doubling of consumption.Optimizers
An optimizer is a software program that searches the maximum of a
(multivariate) function. If we know both the analytical expression of the
function to be optimized and the constraints to be applied, the method
of Lagrange multiplier yields closed-form solutions. However, if no ana-
lytical expression is available or if the function is too complex, numeri-
cal optimization techniques must be used. Numerical optimizers work
by searching a space of likely maxima or minima.
Mathematical optimization is a well-established technology and,
outside of finance, is also used in many areas of science and technology.
Different optimization technologies are employed, depending on the
functions to be optimized and the constraints to be imposed. Statistical
optimization technologies such as simulated annealing and genetic algo-
rithms have been employed to allow the optimization of generic func-
tions with multiple local minima and/or maxima. Chapter 7 provides a
brief introduction to optimization technology.A Global Probabilistic Framework for Portfolio Selection
We are now ready to state the global principles of portfolio selection.
Portfolio selection works by finding those portfolio weights that maxi-
mize expected portfolio utility. Formally, we will have a joint probabil-